MODELLING SEA ICE FLOE FIELDS by Iain Linklater Bratchie A dissertation submitted for the degree of Doctor of Philosophy in the University of Cambridge UNIVERSITY LIBRARY CAMBRIDGE Churchill College, Cambridge . January 1984 - i - - ii - PREFACE This dissertation is an account of my work carried out while a research student at the Scott Polar Research Institute and at the Meteorological Office during the period October 1980 to January 1984. During this time my supervisor at the Meteorological Office was Dr Howard Cattle. At the Scott Polar Research Institute during the period October 1980 to October 1981 my supervisor was Dr Vernon Squire, and from October 1981 to January 1984 my supervisor was Dr Peter Wadhams. The dissertation does not exceed the regulations in length, and has not been submitted fo·r a degree at any other university. It is the result of my own work and includes nothing which is the outcome of work done in collaboration. Iain Linklater Bratchie January 1984 - iii - ABSTRACT This thesis is concerned with the modelling of sea ice, particularly in regions where it is composed of individual floes interacting through collisions. This has been done by modifying and extending existing models that have demonstrated their ability to simulate sea ice in various Arctic and Antarctic regions. The purpose of this study is the introduction of the representation of floes, in terms of their size and number, into a sea ice model, thus adding a feedback mechanism and a further output to the output fields normally produced by sea ice models, the ice velocity and the ice thickness distribution and the ice conc en tration. Many of the physical processes concerning floes that are relevant to a sea ice model have not yet been investigated quantitatively. These aspects of floe behaviour used as model input are calculated from idealized mechanical models of a floe field . These include determinations of floe collision rates, side melting of floes and the cracking of floes in high winds. The strength of the pack ice is investigated, and in particular the effect of open water on the strength is considered. The shape of a plastic yield curve used in the model to determine the ice interaction forces is derived theoretically. The model used includes both thermodynamics and dynamics. The ice thickness characteristics and floe sizes change due to growing and melting, advection, floe cracking, floe collisions, and redistribution processes such as ridging and rafting. Daily wind and temperature data together with long term ocean currents are used as input to drive the model. The results of a six month simulat ion of the sea ice development in the (Eas t) Greenland region are presented and discussed together with a compari son with the observations. - i v - ' ACKNOWLEDGEMENTS I would like to thank Dr Peter Wadhams for all his advice, encouragement and interest throughout my stay at the Scott Polar. Thanks also to Dr Vernon Squire who started me on the road to sea ice modelling. I am grateful to Dr Howard Cattle of the Me teorological Office for useful discussions and for his co-operation in all aspe~ts of my work. I would like also to mention Dr Margaret Bottomley who tracked down some material, and who provided me with a place to live while in Bracknell. Pat Roberts deserves special mention for the hours of work she put into preparing the magnetic tapes containing the data. Th e photographic work was done by Rob Massom of the Sco tt Polar. I received grants from the Natural Environment Research Council and the Meteorological Office in Bracknell as part of a CASE (Co-operative Awards in Sciences of the Environment) award, and since Octob~r 1983 from the Arctic section of the Office of Naval Research, Washington. I am grateful for all their support. CONTENTS ' PREFACE ABSTRACT ACKNOWLEDGEMENTS 1. INTRODUCTION TO SEA ICE AND ITS MODELLING l·l Introduction 1·2 History of sea ice modelling 1·3 Sea ice description 1·4 Marginal ice zones 1·5 Some general considerations regarding modelling 2. THE BASICS OF SEA ICE MODELLING 2·1 Introduction 2•2 An ice model description 2•2•1 Basic equations 2•2•2 Internal ice stress 2•2•3 Ice distribution 2•3 The heat budget 2·3·1 Introduction 2•3•2 Determining growth rates in general 2•3•3 Short-wave radiation 2·3·4 Long-wave radiation 2•3•5 Bulk latent and sensible heat 2•3•6 Conductive flux 2•3•7 Testing the heat budget 2•4 Ice thickness distribution theory 3. ICE FLOES 3•1 Spatial distribution of floes 3•2 Floe collisions 3·3 Floe size change 3·3·1 Irrtroduc tion 3·3·2 Floe break up 3·3·3 Ther modynamic change in f l oe number densi ty 3·3·4 Lateral melting 4. PACK ICE DYNAMICS 4·1 Introduction 4·2 Ridging and r af ting 4•2•1 A s impl e ridge building model 4•2•2 Rafting 4·3 Ice strength 4·3·1 Evaluating ice strength 4·3·2 Effect of open water on ice strength 4·4 Ice interaction 4·4·1 Introduction ii iii iv 1 4 9 10 11 13 13 13 15 17 19 19 20 21 23 24 26 26 29 38 43 49 49 50 65 66 74 74 74 88 89 89 100 106 106 - vi - 4·4 ·2 A yield curve 108 5 . INCORPORATING THEORY INTO THE NUMERICAL CODE 5·1 Introduction 116 5·2 Ice thickness distribution 11 6 5•2•1 Representation of the ice thickness distribution 117 5·2·2 Ice redistribution 121 5•2•3 Analytic form of redistribution matrix 124 5·2·4 Numerical evaluation of the redistribution coefficients 126 5·2·5 A finer grid 126 5·3 Strength in the code 127 5•4 Floe number densities in the code 130 5~·1 Floe size distribution 130 5•4•2 Floe size change 131 5~·3 Continuous ice in a floe model 133 5 ·5 Thermodynamics in the code 134 5·5·1 Vertical changes 134 5·5·2 Lateral melting 137 6. THE MODEL SIMULATIONS 6•1 Model inputs 142 6•1•1 The model grid 142 6•1 •2 Initial thickness distribution 144 6•1•3 Oceanic heat flux and initial mixed layer temperature 146 6•1•4 Input winds and temperatures 149 6•1 •5 Other thermodynamic inputs 149 . 6·1·6 Ocean currents 6•2 Application of the model to a standard simulation . 6·2 ·1 Introduction 6·2·2 Variability of the model results 6•2•3 General features of the output fields 6•2•4 Distributions within single grid squares 6·2·5 Transects 6•3 Variations to the model 6•3•1 Introduction 6·3·2 Drift trajectories 6•3•3 Wind-induced currents 7. CONCLUSIONS AND FURTHER WORK 7· 1 Summary of the new features in this model 7•2 Future development of sea ice modelling APPENDIX BIBLIOGRAPHY PLATES 150 152 152 152 154 173 181 185 185 190 192 195 197 199 227 236 1. INTRODUCTION TO SEA ICE AND ITS MODELLING 1.1 Introduction The problem of concern in this thesis is the modelling of sea ice, particularly in regions where it is composed of individual floes interacting through collisions. This has been done by modifying and extending existing models that have demonstrated their ability to simulate sea ice in various Arcti~ and Antarctic regions. The purpose of this study is the introduction of the representation of floes, in terms of their size and number, into a sea ice model, thus adding a feedback mechanism and a · further output to the , output fields normally produced by sea ice mo dels, the ice velocity, the ice thickness distribution and the ice concentration. Many of the physical processes concerning floes that are relevant to a sea ice model have not yet been investigated quantitatively. These aspects of floe behaviour used as model input have been calculated from idealized mechanical models of a floe field. Both thermodynamic and dynamic processes are important in sea ice variability. The model used for the present study was based upon a dynamic thermodynamic sea ice model that included a two-layer ice thickness parameterization (Hibler 1979). The computer code for this model has been published (Hibler 1980b) and was adapted to produce a 6-layer model , in which the ice is specif ied by a probability. density function for the ice thickness. The floe number density was included as an add itional parameter . The ice thickness categories evolve in time due to growth and melt, through horizontal adv ection t o and from neighbouring grid squares , and by redi s tribution of ice during ridging and rafting. The floe si zes are parameterized by specifying the average radius of the fl oes within each thickness category. The floe size and the floe number density are related to the thickness distribution by assuming that the floes are circular. In a similar way to I E I y hickness distribution, the floe LIBRARY CAMBrllDGE number density distribution evolves because of melting and growing of the ice. If floes grow in thickness, the floe number density in a thick ice category increases at the expense of the floe number density in a thinner category. In addition, lateral melting (included in the model) allows floes to decrease in radius while their number density remains unchanged. The floe number densities also change due to advection, from collisions, and from the cracking of floes due to the action of high winds. The model has been applied to the (East) Greenland Sea where many ice types exist, from strong thick ice near the coast to thin ice near the ice edge which is observed to consist of floes of various sizes. Chapter 1 includes an outline of the historical development of ice models as various levels of complexity were introduced to explain the observed ice drift patterns in the Arctic. In chapter 2, Hibler's (1979) sea ice model is described. This model, although considerably modified for this study, includes many of the aspects now regarded as necessary for inclusion in any reasonable sea ice model. Because of the large spatial variation in the thermodynamic parameters off East Greenland, the ice growth rates are calculated at each grid point. Chapter 2 includes also a description of the various thermodynamic factors and the calculation of ice growth rates. The method described utilizes formulae obtained from various thermodynamic models. In chapter 3, the idea of a floe number density distribution is introduced and a number of analytic results obtained concerning floe fields. By considering the motion of a set of randomly scattered discs within a two-dimensional velocity field, an expression is obtained for the number of floe collisions occurring where · there is an arbitrary strain rate expressed in terms of a shear and divergence. Using the theory of flexible beams on elastic foundations, together with data available regarding floe tilting in wind, the cracking length of floes is calculated as a function of the applied wind speed and the floe thickness. The results indicate that there is a strong dependence of the cracking length upon the floe thickness, necessitating the introduction of a distribution of floe sizes in the model parameterization, rather than using a single representative value. Lateral melting of floes in summer is an important form of ice a rea loss. The relationship between th~ vertical and the lateral melt rate for floes is not known, sri a simple theoretical treatment of floe melting was constructed to investigate the best suitable lateral melt rate for a model in which the floes are assumed to remain cylindrical. In chapter 4 we consider the dynamics of pack ice deformation. In particular, a force model of the ice ridging process is presented that is simple enough to give an analytic expression for the ridge height in terms of the ice sheet thickness from which it was formed, and yet gives good quantitative and qualitative agreement with observed ridge heights. The ridging process is important in determining the ice strength needed in the momentum equation that gives the ice velocity. The determination of the ice strength in the context of ice thickness distribution theory is described and an analytic expression for the strength of an ice sheet of a single thickness is obtained that uses the ridge height formula calculated previously. The strength of a single ice thickness together with a fraction of open water is also obtained. By calculating the strength in this case by a method that takes into account the floe sizes present, a partial solution to one of the main problems of ice thickness distribution theory is obtained, that of the determination of the distribution of the ice involved in ridging as a function of the original ice thickness distribution. The form of the collision rate obtained in chapter 3 is used to derive the shape of a yield · curve which determines the relative amounts of internal shear and compressive stress that. occur when the ice velocity field has non-zero shear or diverging strain rates. The choice of yield curve for sea ice models has in the past been a matter of intuition and this represents the first attempt to derive a yield curve shape from a physical model. I n chapter 5 , the ways in which i d e as and r esult s der ived in previous c h ap t ers may b e included in pract i ce into a numerical model are discussed. Chapter 6 contains the details of the inputs to the model runs for the (East) Greenland Sea area and the setting up of the initial conditions . Results are presented and compared with the observed data. The conclusions are outlined in chapter 7. The appendix contains the complete listing of the code as used for a typical run together with the output. 1. 2 History of sea ice modelling The early phase of sea ice modelling was characterized by initial observations leading to the formulation of theories to explain them. A classic example of this is the observation of Nansen that free ice did not drift in the direction of the wind, but consistently at an angle to the right of it. This led Ekman to formulate his boundary layer theories that explained the angular deviation of the ice drift as a consequence of the balance between the air and water stresses and the Coriolis force. The situation regarding the relationship between observations and theory has changed with experimental programmes designed to seek .evidence to test already formulated theories. There are however many phenomena still without satisfactory physical explanations. After Nansen's initial studies, further work on ice drift was carried out by Sverdrup ( 1928) who found evidence that ice-land and ice-ice interaction forces were causing significant departures from the Nansen theories of ice drift near the North Siberian coast. He assumed this resistance to be given by a coefficient of friction multiplied by the ice velocity. Using observed drift data from the . Arctic, Zubov (1943) derived an empirical formula for ice drift which states that the ice moves in a direction parallel to the atmospheric isobars, and with a speed given by V = 13000 lip (1) wher e V is the drift speed in kilometres per month and lip is the atmospheric pressure gradient in millibars per kilometre. Because of the fairli high degree of accuracy of this formula and its simplicity, it was used for many years for long term Ar~tic drift t rajectory calculat i ons by Soviet researchers (Gordienko 1958). The formula does not work so we ll 1n regions where the ocean current is a dominant forcing . Reed and Campbell (1962) demonstrated from observations of the drift of i ce station Al pha that internal ice resistance as well as the effect of gradient currents was important to the drift calculations. Campbell (1964) introduced terms for these two forces (the ice interaction term being more sophisticated than that assumed by Sverdrup) together with the three forces originally considered by Nansen, and obtained solutions to the resulting momentum equation. He also reviews the previous attempts to solve the momentum equation by leaving out various combinations of the forcing terms. Sea ice models today generally include a solution to the ice momentum equation, at various positions on a two-dimensional grid, in an attempt to derive the ice velocity field over some geographic region. The development of models in this sense can be traced back to before computers became generally used. For example Zubov (1943) considers the deformation of a Lagrangian grid moving with ice that is deflected by the passage of a ,_,.,: ..,,r s torm. Another element of sea ice modelling that required consideration was the prediction of ice thicknesses or its distribution. Models that --included these factors emerged in the 1950's. Some of these models are mentioned below. The ability to reproduce observed features of the ice c i rculation and distribution then became possible. Drogaicev (1956) used a simple compactness ice model together with geostrophic wind fields and assuming ice dr i ft 10° to t he r i ght of the i sobars to give zones of convergence and divergence in the Arctic . Campbe l l ( 1964), using an internal ice stress term of the form ,ce velocit':1 (2 ) simu l a te d t he a nt i cyc l onic gy r e observed 1n t he Beaufort Sea, a lthough Fe lzenba um (19 58) had obt a ined this purely from long term wind input, the average pressure field having a high centred in the Beaufort. However, the inclusion of the ice stress term resulted in a shift of the position of the gyre that more closely fitted the observations . Models which include ice interaction terms proportional to V2u a im to parameterize the effects of fioe collisions by assuming that the assoc i ated energy losses give rise to viscous behaviour on the larger scale. This wou l d give rise to a smoothing out of the resulting velocity field in that departures from locally averaged values would be resisted. This assumption may be valid for pack ice where there is a fair amount of open water but not for a compact ice cover. Russian models with viscous terms include that of Ovsiyenko (1976) who investigated the wind drift of pack ice with a free boundary, using a constitutive law of the form a .. lJ = -(p + µtkk)o .. + µt .. lJ lJ where the pressure term p is a function of the ice compactness evaluated from a simple random collision model of ice floes. Ovsiyenko concluded that the terms µ and y were negligible except for maximum compactness, thus leaving only a pressure term. Such an approach together with the more sophisticated collision calculations dealt with later in this study may prove to be a good approach to modelling ice very near the ice edge. Models for forecasting need to be economical and simple to use so that care has to be taken with the choice of the physical parameters to include. Neralla and· Liu (1979) have developed a sea ice model for use in predicting the ice compactness for short term local forecasting . They include the ice acceleration term, air and water stress, the Coriolis force and an internal ice resistance of the form. f\h\l·(K\]u) where K, the horizon t a l kinema tic eddy viscosity, has a linear dependence upon the ice concentration. The mode l can be run on a microcomputer produc ing reasonable results. Lepparanta (1981) describes a forecasting model for use in the Baltic Sea. The output giv es ice type (level ice, r idged ice, open water) and i s thus o f direct use for shipping. * er~ o.nd €.!j are t.he <:.\:,es.s o.od st,o.'.i~ r-a..\::e. 1::-ensors o-.ocl o.re deo..lt w.:..1:h in.. sec.bon 1·1 · 2. . 6 inclusion of the ice stress term resulted in a shift of the position of the gyre that more closely fitted the observations. Models which include ice interaction terms proportional to V2 u aim to parameterize the effects of f~oe collisions by assuming that the associated energy losses give rise to viscous behaviour on the larger scale. This would give rise to a smoothing out of the resulting velocity field in that departures from locally averaged values would be resisted. This assumption may be valid for pack ice where there is a fair amount of open water but not for a compact ice cover . Russian models with viscous terms include that of Ovsiyenko (1976) who investigated the wind drift of pack ice with a free boundary, using a constitutive law of the form a .. lJ = -(p + µtkk)o .. + µt .. lJ lJ (3) If where the pressure term p is a function of the ice compactness evaluated from a simple random collision model of ice floes. Ovsiyenko concluded that the terms JJ and y were negligible except for maximum compactness, thus leaving only a pressure term. Such an approach together with the more sophisticated collision calculations dealt with later in this study may prove to be a good approach to modelling ice very near the ice edge, Models for forecasting need to be economical and simple to use so that care has to be taken with the choice of the physical parameters to include. Neralla and Liu (1979) have developed a sea ice model for use 10 predicting the ice compactness for short term local forecasting . They include the ice acceleration term, air and water stress, the Coriolis force and an internal ice resistance of the form f\h\J·(K\Ju) where K, the horizontal kinematic eddy vi scosi ty, has a linear dependence upon the ice concen t rat ion. The model can be run on a microcomputer produc ing reasonable results. Lepparanta (1981) describes a forecasting mode l f or use in the Baltic Sea. The output gives ice type (level ice, ridged ice, open water) and is thus of direct use for shipping. -it O"~ o.oc\ E.~ are \:.he s\:,ess o.nd sl:.,O.:,n. (""a..te. t-ensors o-ool a.re deo...lt wt-1:h in. sec.t;on 1-1-2.. Another approach to sea ice modelling is the purely thermodynamic one in which the amount of ice in a region is determined by the amount of growing and melting which enable the ice to vary over a seasonal cycle. The most comprehensive study of this type was made by Maykut and Untersteiner (1971) in a one-dimensional simulation. Economical versions were produced by Semtner (1976). Such one-dimensional models have been made into three dimensional models (Washington et al 1976, Parkinson and Washington 1979, Parkinson and Herman 1980, Parkinson and Good 1982) to study the seasonal changes in the Arctic and Antarctic, but have limited ice dynamics. These sea ice models are useful in climate studies in that they may be ea sily incorporated into global circulation models. The effects on the ice sheets of changes in the atmospheric co2 concentration may be investigated in this way (Manabe and Wetherald 1980). Predictions of ice edge positions where ice dynamics plays an important part are not valid though from such models. However, thermodynamic models have been combined with full dynamic sea ice models (Doronio 1970, Hibler 1980a). The ice model described in this study includes a heat budget calculation to determine time varying growth rates. The sea ice model developed as part of the AIDJEX field programme (see for example Colony 1976, Pritchard 1977, Pritchard et al. 1977) included a thickness distribution, and a momentum equation with the five forces already discussed, in which the internal ice stress is determined using an elastic plastic stress strain constitutive law. Hibler (1979, 1980a) developed the AIDJEX model further to produce a large scale dynamic thermodynamic sea ice model with non-linear advection terms that allows long term simulations. He used the model initially to simulate the seasonal cycle in the Arctic Basin although further studies have been carried out with the model on other geographical regions (Hibler and Ackley 1982) . Hibler used a viscous plastic rheology for the ice interaction. Hibler's (1979) model is described in more detail later as it forms the basis of the dynamics used in the model developed here. The more complicated plastic rheologies were introduced with the aim of being ab le to model the zones of intense shearing observed near coasts (Hibler et al 1974). For these plastic rheologies the ice yields only for ice of sufficient weakness (for a given forcing) and so should be able to 7 cope with areas of large shear, where viscous models would find difficulty. The first application of a large scale numerical sea ice model to the East Greenland region was by Tucker (1982) using Hibler's two-layer mode l . He later repeated his study (Tucker 1983) using three different input datasets concluding that some available data fields contained incorrectly derived pressure fields which can substantially effect the resulting sea ice velocities. Gaskill et al (1980) compared the free drift characteristics of a number of models. They concentrated on the wind and water stress formulations, concluding that those models incorporating stress terms proportional to the vertical air or water velocity gradients at the surface (Doronio (1970), Odin and Ullerstig (1976)) performed better than those employing stresses proportional to the square of relative air or water velocities. (Reed and Campbell 1962, Neralla et al 1980, Brown 1973, McPhee 1975, Hibler 1979). However, the former type requires more knowledge of the vertical structure of the surface boundary layers. The quadratic drag law formulations are simpler to apply. The only models to take account of the finite size of floes, rather than simply using the ice area, are those of Timokhov (1967a, 1967b) and Solomon (1973). Timokhov derives equations based on the idea of collisions between floes caused by stochastic variations in their velocities. Solomon obtains ice interaction terms that depend on floe size, and concentration. However, both Timokhov and Solomon include only a one-dimensional treatment. Recently model studies have attempted to simulate the processes occurring near the ice edge where there is much physical activity. These studies include that of R6ed and O'Brien (1981, 1983) which predicts oceanic upwelling near an ice edge. This is a one dimens i onal model and has not yet been applied t o a two-dimensional geographical region. Muench et al (1983) introduce a wave radiation term into the momentum balance to model the ac tivity in the extreme ice edge zone. As far as performance is concerned, the model best able to simulate observed ice edge position particularly in winter is the recently developed ice model of Hibler and Bryan (1983) that incorporates an 8 interacting ocean model to provide oceanic heat flux and time varying currents that are missing from previous models. 1.3 Sea ice description The area of ice off East Greenland has been chosen for particular study in this thesis. This region is useful for testing a sea ice model because of the large variation in ice type present. Also a marginal ice zone and ice edge form part of the ice distribution there introducing an extra challenge to the ice modeller. Substantial areas of the ice consist of floes, a feature of the model developed here being the prediction of floe size ~istribution. We first describe the kinds of ice observed off East Greenland giving guidance to the kind of results that the model must be able to produce. The East Greenland sea ice was described by Koch (1945), and a review including more recent observations was given by Wadhams (1981). Apart from a very narrow area of ice frozen to the coast, which is called the ice foot, there are two main types of ice off East Greenland. They are 1) the fast ice, which is essentially fixed compared to 2) the pack ice, which comprises the majority of the ice area. Most of the fast ice is less than two metres in thickness although very thick (about 10m) ice can form after growing for many years in fjords and is known as sikussak (Walker and Wadhams 1979). In the Fram Strait region the average thickness of the drifting ice may reach 6m as indicated by submarine sonar data (Wadhams 1983b). Ice movement in the fast ice zone is prevented by grounded keels, giving rise to an apparent ice strength greater than that expected from ice thickness considerations alone. The characteristic feature of the ice distribution and movement of ice in the Greenland Sea is a strong southwards transport by the East Greenland Current that advects ice into regions where the thermodynamic conditions are such that ice would not grow there or survive unless it was continuously replaced. The pack ice can move with the currents and is susceptib le to t he effec t s of the winds . I t cons i sts of multiyear ice, gener a lly abo u t 3 y ea r s old, tha t ori g inates in the Arctic an d move s 9 across the pole in the Trans-Polar Drift Stream and passes together with slightly younger ice from north of Spitsbergen, through Fram Strait before moving south along the Greenland Current where the various ice t ypes become mixed up. Ice divergence takes place in the vicinity of Fram Strait, where the East Greenland Current appears to accelerate. Here the opening of leads and polynyas allow new ice to grow, particularly 10 winter when the average air temperature can reach -28°c (Crutcher and Merserve 1970). Some of the new ice occurs in the form of frazil ice because of the general level of turbulence. Thus new ice types will be expected within the areas of mul tiyear ice. The physical properties of the ice 10 the Greenland Sea have only recently been studied (Overgaard et al 1983). 1.4 Marginal ice zones Man's interaction with the ice covered portions of the globe occur mostly near the ice edge, and in those regions which are ice-covered only for some part of the year. Such regions, known as seasonal sea ice zones, or marginal ice zones thus merit particular study. Knowledge of such regions is important for such activities as shipping and fishing as well as to the oil industry with its need to exploit all areas of the world. In Iceland for example the harbours become ice-bound 10 some winters but not others. The ability to forecast such situations then becomes of economic importance. Marginal ice zones include the Greenland and Bering Seas , as well as parts of the Barents a nd the Labrador Seas, A large proportion of the area sur r ounding the Antarctic in winter is also a margi nal ice zone. Marginal ice zones may be characterized by their appearance. The ice in these regions is generally composed of floes of various shapes , sizes , and thicknesses. Using the Greenland Sea as an ex ample , floes as large as 60 km in diameter have been observed in Fram Strait (Vinje 1977) . Further south, the ice lies in a zone (in winter) parallel to the East Greenland coast. The area towards the edge is characterized by a gradual decrease in the average floe size until, very near the ice edge, the action of ocean waves becomes significant to very small broken up floes. Floes larger than a few metres in diameter do not survive near the ice edge because of the level of activity there. This applies even to the thickes t sea ice floes (Wadhams 197 8). This scheme is complicated by the motion of the floes whereby areas of small floes may be advected back away from the ice edge into the interior giving rise to large variations in floe size over comparatively short distances. Another complicating feature is that of bands of ice that break away from the ice edge. They remain relatively coherent as they drift away from the edge until they begin to melt in areas of warm water. Various mechanisms have been suggested for the formation and persistence of bands (Muench and Charnell (1977), McPhee (1982), Wadhams (1983a)). Along the ice edge, eddies have been observed, the effect of which is to disrupt any smoothness in the horizontal variation in ice properties, such as floe size (Wadhams et al 1979, Wadhams and Squire 1983). The physics of marginal ice zones and ice edges include a number of additional complexities, concerned with the effects of the ice upon the factors that force it. In fact the ice-ocean-atmosphere system is totally interacting. Experimental programmes are now in progress (MIZEX-84) to study these interactions. To model all these interactions simultaneously would necessitate coupling ice, ocean and atmosphere models. The difficulties involved suggest that such a coupled model will not be achieved in the near future. 1.5 Some general considerations regarding modelling With numerical models, it is possible to investigate the effects on sea ice of physical factors in a way not possible experimentally. This can be done by adjusting the physical parameters in the model and comparing the result s with a standard set of results . There are, however, many factors preventing numerical sea ice models from ever truly representing reality. For example, stochastic variations in physical parameters such as the tensile strength of ice mean that their representation by single values 11 necessarily introduces errors. There are very few mechanical properties of sea ice for which there is little variation. The degree of complexity of a model is another factor that should be considered. There is no point including highly sophisticated physics in a model if it has very little effect upon the results. Of course it may not be known to what extent some physical process 1.s important, and here modelling can play a part. Where a physical process is expected to be significant but its precise mathematical treatment is too difficult or beyond the scope of the model, it may be better to include some kind of approximate parameterization rathe~ than nothing at all. Many of the early ice models included this sort of parameterization. 12 2. THE BASICS OF SEA ICE MODELLING 2.1 Introduction In this chapter we introduce some of the details of fairly well established theory with regard to 1.ce modelling. Firstly, we describe Hibler's (1979) two-layer model which serves as the basis of further development. A description of a simple heat budget calculation is given. Here there are a few minor modifications' to existing models. Finally, an outline of 1.ce thickness distribution theory of which extensive use 1.s made later, 1.s given. 2.2 An 1.ce model descript{on 2.2.1 Basic equations In this section the major features of Hibler's sea 1.ce model are outlined. Some of the components of the model have subsequently been altered to suit this particular study but the basic momentum equation and its method of numerical solution remain. Sea 1.ce 1.s modelled as a two-dimensional continuum with spatially varying velocity u(x), a~d the ice thickness characteristics specified by two quantities h, the mean ice thi ckness over the grid square, and the compactness A which is the fraction of open water covered by ice. Thus , what is essentially a two-layer ice thickness distribution is set up, thin ice or open water and thicker ice. The ice motion 1.s determined from a momentum balance expressed by the following equation. 13 where Du m--Dt -mfkAu + T + T - mgVH + F -a --w m Ice mass per unit area f - The Coriolis parameter k - Unit vector normal to the plane of the ice mo tion T - Force on ice due to air stress _ a T Force on ice due to water stress _ w g Acceleration of gravity H - Sea surface dynamic height F - Force due to internal ice resistance (1) We consider the forcing terms on the right hand side of (1) in turn. The first term -mfk-u is the Coriolis force and is a significant contribution to the momentum balance. The terms ~a and ~w are computed as follows (2) (3) where ~g - Geostrophic wind ~w - Geostrophic ocean currents Ca - Air drag coefficient Cw Water drag coeff i cient ra - Air density rw - Sea water density 1 - Air turning angle 8 - Water turning angle The geostrophic wind can be obtained from pressure data, and similarly, the geostrophic currents are obtained from maps of the sea surface dynamic height. Formulae (2) and (3) are thus quite convenient methods of estimating the water and wind stress for a long term climate model. The term -mgVH represents the component of the gravitational force on the ice in the direction parallel to the sea surface which is tilted in response to the ocean geostrophic currents. The sea surface dynamic height H is related to _Ew by the equation (4) so that (5) and the tilt effects can be combined with the Coriolis forces in the single term mf_!:A (~w - u) (6) The final term_!, the internal ice stress, depends on a number of other components of the model, specifically the rheology, the constitutive law and the ice strength. The evaluation of Fis now considered in more detail. 2.2.2 Internal ice stress The choice of rheology determines how F, the ice interaction term depends upon the motion of the ice. The frictional forces set up in the ice depend upon the relative velocities at various places, expressed in terms of a two-dimensional strain rate tensor £ ..• Also the magnitude of the lJ forces set up in the ice due to its motion will depend upon the strength of the ice, denoted p*. For normal strain rates Hibler's model employs a viscous plastic rheology in wh(ch the internal ice stress a- has a value. that lies on a particular curve (the yield curve) in a suitably defined coordinate system. The value of the stress is independent of the magnitude of the strain rate , and it is this property that characterizes plasticity. For very sma l l strain rates , Hibler uses a linear viscous rheology so that the stress drops linearly t o zero from its plastic value as the magnitude o f t h e s tr a in r a t e t ends to z e r o. An e lliptical yield curve is used (in the 15 (a-,, 0--11) plane, where o--, and 0--11 are the principle components of stress) . In a later chapter, the meaning of the shape of the yield curve 1s discussed. Hibler expresses the stress tens or er· · 1 n terms o f the s t r a in r a t e lJ tensor t .. according to a non-linear constitutive law of the form lJ a .. lJ 2nf .. + {i;; - nHkko .. - ~p*o .. lJ lJ lJ ( 7) where n and s are the shear and bulk viscosities respectively. These viscosities depend upon the strength as well as the the strain rate. The forms that n and r;; take as functions of E .. and p* are determined by the lJ choice of yield curve. For the ellip~ical yield curve with eccentricity e, as used by Hibler, the viscosities are z;; and n = z;;/e 2 p* -2·2 e £" (8) ( 9) The terms€:, and {. 11 are functions of the components of the strain rate tensor and are defined by E .. = (10) Once the stress O-·· has been obtained from the given strain rate and lJ the ice strength, the components of the ice interaction term 1n the momentum equation (1) are given by 16 F. l. 2.2.3 Ice distribution (11) The momentum equation when solved (numerically) gives the ice velocity field u(x). Some of the terms in the equation, in particular the ice interaction term, depend upon the amount and thickness of the ice. These quantities evolve in time by advection which is determined by the velocity field. In addition, thermodynamic effects alter the ice thickness characteristics and so affect the ice strength. Thus the momentum equation giving u and depending upon p*, and the advection and thermodynamic equations giving hand A and depending upon u, give a coupled system of equations. The evolution of hand A· are given by (using a one-dimensional version of the equations for illustration) and clh clt clA clt a (uh) dX a (uA) dX + + where the thermodynamic terms are given by = and f(h/A)A + (1-A)f(O) UNIVERSITY LIBRARY CAMBRIDGE (12) (13) (14) 17 I (f(O)/h0 )(1-A) 0 0 + (A/2h)Sh if f(O) > 0 if f(O) < 0 (15) Here, f(h) is the growth rate of ice of thickness h, and h0 is the demarkation thickness between thick and thin ice. Also, the compactness A is forced to remain less than or equal to one. Equations (12) to (15) represent a simple form of the ice thickness distribution equation which is examined in detail later so that here it suffices to mention that the equations are designed to include such concepts as the increase in thickness during ridging when the compactness becomes unity and the ice field converges. Also the term SA allows rapid freezing of open water to occur by letting its fraction (1-A) decay exponentially, and, during melting conditions, the amount of thin ice resulting from the melting of the thick ice will occur relatively slowly. The values for the growth rate function f(h) used in Hibler's (1979) Arctic study were those calculated by Thorndike et al (1975) using Maykut and Untersteiner's (1971) numerical thermodynamic model. The values however may also be obtained by performing a complete heat budget calculation at each time step. Semtner (1976) and Hibler (1980a) show how this may be done efficiently. A similar heat budget calculation is dealt with later for use in the model developed here and is described in more detail then. Finally, the ice strength is determined by the formula p* Ph exp[-K( 1-A)] (16) where P and K are fixed empirical constants. This equation was not derived from a s tudy of the mechanics of deforming ice but rather, chosen just to give a sharp drop in strength as the amount of open water increases from zero, and also to incorporate an increase in strength as the ice thickness 18 increases. A more detailed treatment of the 1.ce strength 1.s given 1.n section (4·3 ). 2.3 The heat budget 2.3.1 Introduction As we have seen, the thermodynamics, 1.n Hibler's models and 1.n the thickness distribution evolution equations, enter 1.n the form of a growth rate function f(h,t). This growth rate (dh/dt) is a function of the ice thickness h, but not the ice type. Different growth rates might be expected for new 1.ce compared to multiyear 1.ce due to their relative surface albedos. Such modifications will not be made in this treatment and average values of albedo and othe~ thermodynamically important quantities will be used. rhal: The general problem of determining the growth rates f(h) isAof solving the time dependent heat equation within the ice with appropriate boundary conditions at the top and bottom surfaces, as well as including heat source terms within the interior of the ice itself. A direct attack on the one-dimensional problem was attempted by Maykut and Untersteiner (1971). This model included the effects of a snow cover and salinity inputs. The surface boundary conditions were computed by means of evaluating the heat fluxes, which govern the growth rates. Semtner (1976) constructed versions of the Maykut and Untersteiner model which included just a few ice layers with linear temperature profiles. These yielded results reasonably close to those of Maykut and Untersteiner but considerably more economically, making it possible to evaluate the growth rates over a horizontal grid, as was done by Parkinson and Washington ( 1979 ). In their model, useful parameterizations of the long-wave and short-wave radiation terms as functions of obtainable climatic variables were given. 2.3.2 Determining growth rates 1n general As part of a numerical climate mode~, growth rates are needed not only at every time step and at each grid point, but also for a variety of ice thicknesses. Thus, an efficient ·method is required to generate these quantities. Semtner (1976) considers a simple method of obtaining growth rates from a one layer (slab) model which assumes a temperature profile linear with depth. Hibler (1980a) also assumes a linear temperature profile with a surface heat budget calculation i~ order to calculate the ice growth rates. In this study, the heat budget is calculated in a way similar to that given by Hibler, except that the latent heat transfer is given in terms of the relative humidity, and an albedo dependence on ice thickness is also included. Also stability-dependent heat transfer coefficients will be used (including these can affect the ice edge position in large scale numerical models; personal communication, H. Cattle). The details of the heat budget calculations are now discussed. Using the convention that fluxes towards the ice surface are taken positive, the surface heat balance equation may be written (io) (io) (To) (1-cx)Fs + FL + Fsen /\+ Flat " - Fd " + (K/h)(Tmix-To) = 0 (17) where cx is the ice surface albedo, F and FL represent the incoming short-s . wave and long-wave radiation, F (T) and F1 t(T) are the sensible and o.nd ore fu.ncho("IS of bhesTie. s0u.c-fo.c.e,. tt_.~pe.r~b.A.re io. latent heat flux terms. The final term included here is the heat " conductivity to the surface through the ice from the bottom surface. After solving (17) ~umerically for T0 , the heat budget growth rate fb(h) may be evaluated from = (18) where each of the terms within the bracket is a heat input or output t o the Q:r i.,; the volu.r,,etric. heel+ oJ fusio,n. of ice ('302 l"IS r0-l). ice slab as a whole. hThe add itional term F0 represents the oceanic heat flux transmitted through the mixed layer to the ice and is included at this stage in Hibler's (1980a) thermodynamic model although in the model developed here, the term enters into the heat budget by raising the 'I temperature of an oceanic mixed layer which subsequently produces bottom and side melting of the floes. The ideal situation as far as a numerical sea ice model incorporating thermodynamics is concerned, would be to have the growth rat e s themselves as observed input. Failing this, di'rect observational data for the terms in (17) and (18) would be useful . However , even this situatio n 1s somewhat unrealistic in terms of the amount and scale of the data needed to run a climate simulation. Fortunately, most of the quantities used 1n the thermodynamics are expressible using theoretical and empirical relationships 1n terms of more basic thermodynamic quantities that are available in the form of long term datasets for substantial geographic regions. These relationships we now outline. 2.3.3 Short-wave radiation The short-wave radiation, denoted F8 in equation (18) originates from the sun, and is a maJor component of the heat balance. Although Fs 1s a measurable quantity, it is more convenient in a large scale numerical model to use the empirical formula of Zillman (1972), l·085cosZ + (2·7+cosZ)ealo-5 + O·l (19) where S0 is the solar constant taken to be 1353Wm- 2. Z 1s the solar zenith angle and can be calculated as a function of the hour angle HA, the latitude~ and the declination o by the formula cosZ = sin~sino + cos~cosocosHA (20) ea 1s the atmospheric vapour pressure 1n Pascals. The declination o can be calculated as a function of the time of year approximately as follows . 8 = 23·44°cos[(l72-DAY)TI/180] (21) '.21 The hour angle HA can be determined by the formula HA = (12-Solar time 1.n hours)TI/12 (22) It should be noted that (19) applies only if cosZ ~ 0, for otherwise the sun is below the horizon and Qs = O. In (19), Qs is the incoming short-wave radiation for clear skies. For cloud covered skies characterized by the quantity C, the fraction of the celestial dome with cloud, the short-wave radiation Qs is modified by multiplying by the factor l -0•6c3, thus = (23) Various empirical formulae giving the change in short-wave radiation with cloudiness have been suggested (see Maykut 1983 for a review), the form used in equation (23) being due to Laevastu (1960). Monthly averaged and hourly data give different results when used to obtain cloudiness- radiation relationships. The reasons for this are not yet known. Monthly or seasonally averaged values are suitable for climate models. Factors such as increased cloudiness as is observed near an ice edge may have to be taken into account when modelling ice edge phenomena. 1:.hose Forhclimate models in which the equations are integrated using a time step of the order of a day it is better to average Fs over a 24 hour period. In practice, since Fs is symmetric about solar noon, the average over a suitably chosen 12 hour period may be used. The term Fs is multiplied by the factor 1-~ to account for the short-wave radiation reflected at the surface. For T 0 ( 273•16°K an ice thickness dependent albedo is used of the form (Maykut 1983) ~(h) 0•44h0 • 28 + 0 •08 (24) 22 derived from the albedo measurements of refreezing ice in l e ads (Weller 1972). This formula indicates the rapid change of albedo as the ice thickness varies near zero. The albedo for the case when the surface temperature is 273·16°K (or above) is set slightly lower than given by (24). To reduce the albedo by the . same proportion as does Hibler (1980a), we multiply the albedo from (24) by the factor 0·8213 when T0 = 273•16°K. The purpose of this is to reduce the albedo when, particularly in summer, melt pools appear on the ice surface. 2.3.4 Long-wave radiation As with the case of short-wave radiation, empirical or theoretically derived formulae for long-wave radiation are most useful in determining the heat balance in a climate model. The incoming long-wave energy at the ice surface, consisting of the black body radiation from the atmosphere, almost balances the upward radiation emitted by the ice itself. Usually there is a net loss of radiation at the ice surface. The long-wave radiation considered has wavelengths in the range 5 - 50pm corresponding to the black body spectrum for a temperature of about 250 K, whereas the short-wave radiation considered in the previous section has wavelengths of the order O•l - 3pm due to a black body temperature of that of the sun's surface of 6000 K (Fleagle and Businger 1963). The downward long-wave radiation depends on the temperature and humidity profiles through the atmosphere, from which FL may be calculated although in polar regions there is sparse data concerning the vertical structure of the atmosphere. A convenient derived formula for FL was obtained by Idso and Jackson (1969) which depends on air temperature Ta, of the form (25) where er is Stefan's constant. The term in the curly brackets is the emissivity E* 0 of the atmosphere for clear skies. For cloudy skies an effective emissivity E can be used where 0 a and so E 0 E* (1 +nC) 0 E o-T4 o a (26) ( 27) C is the cloudiness, as before, and n is an empirical parameter that varies slightly depending upon the time of year. The upward long-wave radiation F1 t depends upon the surface temperature according to E* o-T4 S 0 where E* 1s the surface emissivity. s 2.3.5 Bulk latent and sensible heat (28) The sensible heat transfer F sen occurs because of the temperature difference between the ice surface T0 and that of the atmosphere, Ta• It is parameterized by the relation, (29) where n1 1s the bulk sensible heat transfer coefficient, and lugl is the geos tr oph i c wind spee d. The coe ffici ent n1 can be expressed 1n the form (30) where Pa i s the de ns ity of air , cp 1s the specif i c he a t of dr y a i r , a nd CH is the tran s fer c oefficien t o f sens i ble heat , or the St an t on number. The · values of use d 1n the AI DJEX models, by Parkinson and Washington (1979), and Hibler (1980a) are 1004 J kg - l K- l 1·7510- 3 (31) From measurements by Joffre (1982), the Stanton number can vary depending on whether the surface boundary layer is stable (Tair > T0 ) or unstable (Tair < T0 ). The value given in (31) corresponds to unstable situations, whereas for stable conditions the Stanton number CH is rather less, giving rise to less heat transfer. We thus drop CH to 10-3 if Tair > T0 , this value corresponding to the lower values of CH as measured by Joffre in near neutral conditions. The latent heat Flat is associated with the heat released during phase changes, and has a bulk parameterization of the form (Andreas and Ackley 1982) (32) where qs(T) is the saturation specific humidity at temperature T, and f is the relative humidity. The terms qs(Ta) and qs(T 0 ) may be calculated using Murry's (1967) formula, (33) p - (1-E:)es where E: is the ratio of the molecular weight of water vapour to that of dry air , taken to be 0·622, and with es the saturation vapour pressure (Pascals), given by 611 x lOa[(T-273.16)/(T-b)] (34) where the empirical parameters a and b are given by (a, b) = (9·5, 7·66) when there is an ice cover and (a,b)= (7•3,35·86) for open water . In equation (32) the bulk latent heat transfer coefficient n2 takes values that depe nd on whe ther there is ice or not, and so whether sublimation or evaporation is taking place. Over water n2 = 5·69 x 10 3 Jm- 3 and ove r i c e n2 = 6 ·45 x 10 3 Jm- 3 (Hibler 1980a). 2.3.6 Conductive flux In the simple slab model considered here, a temperature profile, linear with depth is assumed, so that there is a constant conductive heat flux (upward) in the ice of magnitude (35) where K is the conductivity and, Tmix is the upper water temperature, or the temperature of the ice at its lower boundary. More sophisticated models include a number of points through the ice for which the temperature is solved, a linear temperature profile being assumed to exist between these points. Semtner (1976) compares the results obtained from the various models. The single slab models perform favourably compared to the sophisticated models, despite their simplicity. 2.3.7 Testing the heat budget The heat budget calculation as described here was . tested with various input parameters to determine their relative importance. The results are best described pictorially. The growth rates fb(h) were obtained for values of ice thickness ranging from Oto Sm. The thermodynamic conditions were calculated with short-wave radiation corresponding to different times 'of the year (Julian days 50, 100 and 150), together with various humidities, cloudiness factors , and air temperaiures. Figures (2.1), (2.2) and (2.3) show the growth rates pl otted against the ice thickness obtained by varying the values of each of the thermodynamic inputs in turn from a standard se t. The growth rates for zero ice thickness (open water) in the diagrams correspond to water at freezing point. If the water temperature is above freezing the growth rate (heat absorption) would be different. The standard se t plotted with a solid 1 ine were 20 Figure(2•1) 2.5 --->.. Day 50 {'j '"O 2.0 "' s C) '--"' 1.5 Q.) ,+J {'j 1.0 h ,..q ,+J ~ 0.5 0 h 0 -- -- =~:-:-~--------- -- -- -- -- -- -... ___ _ 0.0 0 1 2 3 4 5 Ice thickness (m) Figure(2•2) 1.0-,-------------------------, (l) ,+J 0.5 0.0 ~ -0.5 ,..q ,+J ~ -1.0 0 h c., Day 100 ' I 'l ------ '----··-=- ,' 0 , , 1 2 3 4 Ice thickness (m) Testing the sensitivity of the calculations to variations in parameters . growth rate the input See the next page for the key to the diagrams. 5 27 ,..__ 2 >.. f co I ""CJ I "' 0 sI I () I ....__ I (l) -2 -+,J I co H I ..q -+,J -4 I ~ I 0 H CJ -6 Figure(2·3) Day 150 - - - - -------r- / / ---- I ---r I , , 0 1 2 3 4 Ice thickness (m) Testing the sensitivity of the growth rate calculations to variations 1n the input parameters. The 1 ines tyles indicate the growth rates as calculated by changing each of the following input parameters. ----- - Standard Cloudiness Humidity Wind speed Temperature Pressure 28 5 calculated for an atmospheric pressure of 970mbar, an air temperature of 270°K, a wind speed of lms- 1, a relative humidity of 50% and a cloudiness factor of 50%. Tests were done increasing the pressure to 1030mbar, increasing the air temperature to 275°K, increasing the wind speed to 5ms-l and changing the humidity and cloudiness to 70%. The wind speed and temperature changes were found to have the greatest effect on the growth rates, especially for thin ice. The humidity and cloudiness changes which represent the range of observed monthly average values had only a small effect on the growth rates. The pressure change had almost no effect at all. Thus in calculating the thermodynamic heat budget for the growth rates, the wind speed and air temperatures should be specified at each grid point and time step, whereas for the cloudiness, humidities and pressures, single · monthly average values would be sufficient. 2.4 Ice thickness distribution theory An area of sea ice has, in a local region, many thicknesses of ice as well as areas of open water. It would clearly be inadequate to model such an area by considering it to be composed of one thickness, say the mean thickness. For example, an area consisting of 50% open water and 50% 2m ice would not be able to resist deformation, whereas an area consisting of 100% lm ice would be comparatively strong. The first remedy would be to parameterize the area by giving the fraction of open water together with the mean thickness of the ice present. Although a considerable improvement; - there is still difficulty in choosing the appropriate thermodynamic growth rates for such a representation. In growing conditions, thin ice grows an order of magnitude faster than thick ice so that it is not possible to obtain the change to the mean thickness caused by such growth . Calculations by Thorndike et al (1975) using Maykut and Untersteiner's (1971) ice growth rate model gave a growth of 1 · 95 cm/day for 0 · 5m ice but for 3m ice this growth rate is down to 0•27cm/day (the conditions use d i n t he calculation were for January l in the Central Arctic ) . 29 30 A description of sea ice in terms of the thickness distribution stems from observations that ice is not uniform and a desire to indicate the relative amounts of the various kinds of ice present in a region. Wittmann and Schule (1966) produced cumulative ice thickness diagrams for various regions in the Arctic. They concluded from the observations that between 13 and 18% of the ice . area consisted of ridges of thickness considerably greater than the average ice thickness. Using submarine data Williams et a l ( 1975 ) we re able to estimate percentages of various categories of ice cover along transects in the Arctic. Submarine data has been obtained on subsequent cruises (Wadhams 1980a) to provide detailed thickness distribution graphs. These kinds of data prompted the development of a theory to explain the time evolution of the thickness categories due to thermodynamic and dynamic causes. Thorndike et al (1975) describe such a theory. Here we describe the development of the theory in a way that leads to its use in a numerical sea ice model. The thickness distribution g(h) is defined such that the quantity f h2 g(h)dh h1 (36) is the fraction of the area of ice of thickness in the range (h 1 ,h2 ). From this definition g(h) acts as a probability density function for thickness. In particular integrating over the entire range of thicknesses gives a probability of one, thus 00 f g(h) dh = 1 0 (37) The thickness distribution may vary from place to place and may evolve in time so t hat g g(h ,x,t ) (38) and is defined at a point x by measuring g(h) within some region R that includes x and taking the limit as R - O. Thorndike et al (1975) derive the following equation to represent the evolution of g(h). -V• (~g) cl(fg) + ijJ clh (39) The three terms on the right represent, respectively, the flux divergence of g, the changes in g due to thermodynamic processes and the changes in g due to mechanical redistribution. Hibler (1980a) introduced a fourth term on the right hand side of (39) to account for lateral melting of floes. This term F1 is such that (40) because lateral melting conserves area in that open water is created to compensate for the loss of ice. It is assumed that the amount of ice in all the thickness categories are reduced by the same percentage with the amount depending on the quantity of heat available. This assumes that the lateral melt rate is independent of ice thickness, so that the volume loss is linearly dependent on the ice thickness. This assumption is consistent with the lateral melting analysis outlined in the next chapter. The quantity u in the first term -V· (~g) is the two-dimensional ice velocity so that -g'v·~ ~·Vg (41) represents influx of ice due to convergence and changes due to adve c tion. The s econd term 31. a (fg) dh (42) , includes f which is a function of hand is the ice growth rate dh/dt. This term is thus analogous to the flux divergence term with f corresponding to the velocity u, except that it represents the transfer of ice between vertical thickness levels rather than horizontal ones. 32 The final term~ accounts for the changes in g that do not change the ~ ice volume within a region but merely alter the relative amounts in each category. Thus, this term parameterizes redistributive processes such as ridging and rafting.~ depends on g and u in a rather complicated way and so is discussed in more detail.' is a function of h because it describes the change in g(h) for each thickness category (h,h+dh). We can write down two general properties of the redistribution function. Firstly, by integrating equation (39) and noting (37) we obtain (43) The volume of ice is expressed as (44) so that to assume that no ice volume is created during ridging gives (45) Note that here the symbols~ and~ are equivalent. Now to derive a more €Xplicit form for~ showing its dependence on the ice velocity we consider first the simplest cases of pure divergence and pure convergence. Horizontal variations 1n u are expressed by the (symmetric) strain rate tensor £ .. 1] (aui au.) ~-+__J ax. ax. J 1 (46) It is often found more convenient to express the strain rate 1n terms of the sum and difference of the eigenvalues of E ..• Thus we define the two 1] strain rate invariants£, and £ 11 by £ .. (47) r., 1s the divergence of the velocity field, V·u, and E 11 1s a measure of the rate of shear of the field. Sometimes another set . of strain rate invariants proves useful. These are denoted by I EI and 8. v·2 EI ·2 + E,, (48) gives the amount of deformation and 8 = tan- 1 U:.,,/t,) depends on the relative amounts of shear and divergence. If the strain rate is specified as a point in the (t , ,t,,) plane, then \r.\ and 8 are its polar coordinates. In terms of the velocity gradients, £ .. = au av + ax ay + /au + av)2 \ay ax (49) 33 During pure divergence (jEj = V·u), open water 1s cr e ated so that the thickness distribution changes by increasing the value of g(h) at h = O only. This means that in these circumstances~ is proportional to a delta function o(h) and from (43) we see that o(h) !El (50) During pure convergence ( jEj = -V·u), ice redistribution will have to take place due to the flux of ice into the region considered. Then the redistribution function~ is written (51) where w (h,g), called the ridging mode, describes the way each thickness r -level 1s changed during ice redistribution. For ice of a particular thickness, area will be lost as it ridges to thicker ice, but there will also be a gain 1n area due to the thinner ice ridging to that thickness. The function wr(h,g) gives the net rate of change of area of thickness h per unit strain rate V·u. The way the thickness categories change described by wr(h,g) depends on the thickness distribution present so that the redistribution function 1s a functional dependent on g(h). The dependence of wr upon g(h) is still an open question but here we describe the form assumed by Thorndike et al (1975). Firstly, we note that with (51) , conditions (43) and (45 ) become (52) and Then wr(h,g) may be split into two components -a(h) + z(h) 00 f {a(h)-z(h)}dh 0 (53) (54) where a(h) is the distribution of ice ridged, and z(h) is the distribution thus formed. The denominator in (54) ensures that wr(h,g) is normalized to -1 as in (52). a(h) is a probability density function and so is normalized to 1, thus (55) Thus it must be remembered that a(h) is normalized with respect only to the ice involved in ridging so that a(h)dh does not give the actual amount of area lost in ridging, that quantity being given by a(h) la ~a (h )-z (h)) dh (56) Thorndike~ al (1975) hypothesized that a(h) could be related to g(h) by a relation of the form a(h) = b(h)g(h) (57) ______________ ......... _ • where b(h) is some weighting factor. Noting that thin ice tends to . deform more easily than thick ice, Thorndike et al (1975) chose b(h) 2 G(h) max { 0 , - ( l - --) } G* G* (58) This assumes that only a fraction G* of the thinnest ice is ridged with a bias toward the thinnest ice within that range. A reasonable choice for G* is 0·15. G(h) is the cumulative thickness distribution so that G(h) (59) The factor 2/G* in (58) means that (60) as required by (55). The function z(h) can be deduced from a(h) from the way ridges are assumed to be formed. If ridges are produced as described in section (4•2), then analytic expressions for z(h) can be obtained for some distributions a(h). In section (4•3) regarding ice strength, z(h) is calculated when a(h) is a delta function. Having established, with equations (50) and (51), the redistribution function for pure divergence and pure convergence, the next step is to postulate that for an arbitrary strain rate £, which is composed of a divergence £, and a shear £.,, the redistribution function is a linear combination of the two forms. Thus lfi = !tlfo (8)cS(h) + a (8)w} o r r (61) Thus the coefficients a (8) and a (8) of the ridging modes jEjcS(h) and o r !E!wr, depend only on 8. Substituting (61) into (43) and noting (52) we obtain a (8) - a (8) = cos8 o r (62) so that' is known to within one function. For pure divergence 8 = 0, so from (61) we immediately see that a (0) = 1 0 a (0) = O r In the case of pure convergence, 8 = TI so similarly, a (n) = 0 0 a (n) = 1 r (63) (64) Rothrock ( 197 5) shows how a knowledge of the amount of ridging, specified by the coefficient a (8), in an arbitrary deformation can be r related to the yield function and yield curve of plasticity theory. He equates the rate of working OijEij in deforming the material to rates of production of potential energy and the loss of energy in frictional processes. The energy equation he derives expressed in terms of the stress and strain rate invariants is ~.E, + O.,E., jt!a (6)p* r (65) where p* is the strength of the material in pure compression. In a later section an explicit expression for a (8) is derived from physical r arguments so that' may be determined completely. In subsequent chapters the ideas introduced with thickness distribution theory are used to calculate p* for an area of pack ice containing many different thicknesses of ice . 37 I I I I I I I I I I I l 3. ICE FLOES 3.1 Spatial distribution of floes Some areas of sea ice, such as marginal ice zones, contain distributions of floes of various shapes and sizes. This situation is modelled here by considering an idealized floe field in which variations in floe shape are neglected - all the floes being considered circular. A floe field, within a region R, is described by the quantities A, n and r. The compactness, A, is the fraction of the sea surface covered by ice, and n is the number of floes per unit area. These may be related to a third quantity r, the average floe radius in R, according to A = nnr 2 (1) Later we split n into a distribution n(h), of floe number densities for each ice thickness category, but for the moment only the average floe number density need be considered. Within R, the discs or floes can be considered randomly distributed in the sense that there is no preferred position for any given floe, but with the constraint that no two floe centres can be closer than 2r apart. The information provided by a knowledge of the spatial distribution of a set of floes together with their velocities at some moment is sufficient to determine the instantaneous collision rate. In fact it is necessary only to consider the floes that at some time are just about to touch. We can regard the floe field as homogeneous so that we need consider only one typical floe which we subsequently refer to as the reference floe. We take the centre of the reference floe as the origin of the coordinate system. In particular, polar coordinates Cr,W) may be used so that the reference floe edge satisfies r = r. Also, it is found convenient to express floe 38 • - velocities relative to this or1g1n. The floe collision proj:>lem is thus reduced to determining the distribution of floes that at some mome nt are almost touching the reference floe. The concept of closeness of floes can be made more precise by defining a floe to be close to the reference flo e if its centre lies at a distance in the range 2r to 2(r+c5p) from the or1g1n, where cSp << r . For a region sparsely populated with floes, the expected number of floe centres in the annulus (? = [ 2r, 2( r+op )] is the product of its area, oS, and the floe number density and is thus ncSS = 8nnrop. This value is accurate only in the limit A -- 0 since we have not restricted the possibility of floes overlapping. If non-overlapping discs are scattered randomly then each one has about 6 'neighbours', and of course, for close packing each disc has exactly 6 neighbours. From this we can find an expression for the expected number of floe centres 1n the annulus oS that is accurate not just for low floe densities. The problem becomes tractable if one assumes that near the reference floe there are 6 floe centres uniformly distributed in the annulus defined by the range 2r 2 [ 2r , - . ( 1- ~A ) ] , c,,. ,M I A= nnr2 (2) ~and~ are as yet undetermined scaling constants . The constant~ scales the upper limi t simply because as the floe number density decreases, the floes get further apart . The constant~ accounts for the finite size of the floes further out restricting the distance that the neighbouring floe can drift . The e x pe cted number of floes with centres · 1n the annulus p = [2r,2 (r+op)] is 39 1. I· I I 6(8nrop) (3) From the low density approximation, this expression must tend to 8Tinr6p as A -- O which implies that cx.2 = 2/3. For close packing we let the upper limit in (2) tend to 2r as A -- 1 which gives p = 1-oc. The expression (3) becomes LnoS (4) where 1 L(A) = (5) and oS is the area of the annulus p [2r,2(r+op)]. This means that close to a given floe the local floe number density is L(A) times as great as the overall floe number density n. The function L(A) increases rapidly near A= 1 as is indicated in figure (3.1). The previous calculation for discs may be done analogously for line segments in one dimension. In the one-dimensional case however, the situation is sufficiently simple for an exact analytical solution for the expected number of 'close' floes to be obtained as a function of the compactness . lt would be useful to compare the exact solution with that obtained by a method for one dimension that is analogous to the approximate method employed in the two-dimensional case. Suppose that along an infinite line, there are distributed n line segments (floes) per unit length, and that each floe is of length 2r. The compactness of the floes is then A= 2nr and the expected gap between the floes is 40 ,,---.... ~ '--"' ~ H 0 _, CJ cd '+-l o.O ~ ...... ~ CJ cd ~ 1000 Figure(3•1) 100 10 1 -L-;==~~=====~~~~~~~~-,-J 0.0 0.2 0.4 0.6 0.8 Ice concentration The rapid increa?e of L(A) as the compactness reaches unity indicates the increased number of collisions likely to occur between floes. 1.0 ,, I 1 - 2r 1-A (6) n n The probability density function p(x) for the width of the gap between floes is thus the Poisson distribution for points on a line with density n/ (1-A) per unit length. Hence p(x) = l:A expG::J (7) The probability that a gap is less than 2op, or that two floe centres are closer than 2(r+op) apart is i2op p(x) dx 0 2nop 1-A ' 1 - A op « (8) 2n Now to obtain the value of p(x) at x = 2r by the method employed for the case of discs, assume that p(x) for the centre of a floe neighbouring the reference floe is uniformly distributed in the range (9) As before, ex and p are scaling constants to be determined by considering limiting compactnesses. Thus given the range in which the nearest floe centre can lie, the expected number of floe centres in the range [2r,2(r+op] is 4op ar(l-SA) - 2r A (lo) 4.2 ex and p can then be determined by considering the limiting cases for which A - 0 and A -- 1. For A - 0 the expected number of floe centres within [2r,2(r+op)] is simply 2nop For (10) to give this limit, ex = 4. In the case A --1 the upper limit (cxr/A)(l-pA) should tend to 2r, hence p = 1/2. Hence the expected number of floes within a distance 2op from the reference floe is 2nop/(l-A). We see that in one dimension, the exact and approximate solutions are in fact identical. This suggests that the approximate solution to the two- dimensional problem may be usable. The equality of the implied collision rates for the two methods is thus assured. 3.2 Floe collisions We are not concerned here with the 'random' collisions caused by the small scale variation in the floe velocity field but rather the collisions due to the differential mean drift of neighbouring floes. The energy losses associated with 'random' bumping of floes may be calculated in a way similar to the analysis given below for the 'strain rate' collisions. This requires additional information regarding the magnitude of the random components of the floe velocity field. The problem then is to determine for a floe in a velocity field with strain rate specified by E, and E11 , the resulting rate at which collisions occur with other floes. We have to determine the area within which a floe centre has to be for it to involved in a collision in some small time ot. Suppose that a two-dimensional ice velocity field u(x) varies on a scale much larger than the floe radius r. If the reference floe with centre at X 0 has velocity u(x0 ) then the velocity field close by at x is u(x) (11) neglecting terms small compared to (x-x 0 )·'vu. With the centre of the reference floe as the origin, x 0 to the origin is thus given by 0, the relative velocity of a floe close u(x) x·Vu(O) (12) Thus, close to the reference floe we consider only linear variations in the velocity field. Expressing the relative velocity of a floe in polar coordinates Cr,,),and using equation (12), the radial velocity ue is given by lJ. z up = l~I } [(~~x + :~+ + (;:x + ;;Y)YJ since X = rcos, and y = rsir:,- Hence if we define and £ 11 cos~ au ax av ay au av = - + ay ax then we have u = ~p{f, + £ 11 sin(~+2W)} p The transverse velocity u~ can be similarly found to be uw ~p{w + f.,cos(~+2w)} wher e * (14) (15) (16) (17) Note that here the symbol s Wand~ are equivalent. I , I I I 11 I w = clv tlu clx cly (18) is the magnitude of the vorticity of the floe velocity field . However onl y the radial velocity component is of interest for the purpose of calculating the floe collision rate . For another floe to collide with the reference floe at an angle in the range c,,, + o,), the radial velocity ue has to be negative for that value of,, and its centre has to be within the shaded parallelogram shown in figure (3.2). The total area within which a floe centre has to be to collide in a time ot is oA = - s~:n{O,u (i/J,p=2r)}2r di/Jot 0 p 0 \TI 4 S (cos8 + sin8sin2i/J') di/J' i/Jo f2TI J~cos8 + sin8sin2i/J')di/J' 0 < 8 < \TI where' has been replaced by,,=~+ t/2 and the angle ' 0 is such that -co t e Then we have where ,o. ( 8 ) (19) (20) (21) (22) Figure(3·2) Reference Floe The shaded parallelogram contains the centres of floes about to collide with the reference floe in a time ot. 0 0 < e < l.,in - ~ cas-l (cate) case + ~. /sin2e - cas2e 7f 1T ~ J...n < e < \n (23) - case %n < e < 7T The inverse cosine function takes its principal value. The function a(e) increases monotonically from Oto 1 as 8 varies from Oto TI and is shown in figure (3.3). The number of coll is ions in time ot is the product of oA = 4nr 2 li:la(e)ot and the local floe number d~nsity , L(A)n, thus the collision rate is 4nr2nL Ii: j a( e) (24) For a unit area containing n floes, the collision rate is thus 2nr 2n 2Lli:la(e) 2LA2 li:la(e) = nr2 = 2LAIE:la(e)n (25) where a factor of 2 in (24) is lost because each collision involves two floes . Results (24) and (25) are, in principle, experimentally verifiable. Also they are a convenient starting point for describing several aspects of floe behaviour, and will be used in this and the next chapter to derive further theoretical results. The MIZEX-84 experimental programme includes the deployment of 3-axis accelerometer arrays on floes to measure collisions . The results will be supplemented by 'other measurements such as the floe drift rate and floe size. 'I 111 I I 1 I! I I I! I 1 II Figure(3 · 3) 1 a(9) 9 0 TT The function a(8) giving the floe collision rate for an arbitrary deformation type relative to that for pure convergence. 48 3.3 Floe size change 3.3.1 Introduction The average floe radius r may change thermodynamically or dynamically. The floe radius decreases during periods of lateral melting. Floes also change size due to collisions. A number of outcomes to collisions have been observed. The floes may bounce apart elastically, coalesce, or break up. However, when considering floe sizes in the context of a climate model, the collisions considered are those due to the differential mean drift of the floes. These collisions would be expected to give rise to the coalescing of floes with a ridge produced along the region of contact. Collisions are thus considered to increase the average floe size. For thin floes such as found in the Bering Sea, floe collisions often result in rafting, the overriding of one floe by another. This has the effect of producing a single larger floe. With these assumptions, the floe number density n would satisfy a two- dimensional continuity equation of the form Dn + n\] • u = Sn Dt (26) where Sn is a source term for floe number density, due to collisions. From the mean velocity field within which the floes move, the collision rate per unit area ·may be expressed, from the previous section as, 2LAnllla(e) (27) The assumption that the floes stay together when they collide implies that n, the floe number density , decreases by 1 for each collision. Hence -2LAIE!a(8)n (28) 49 The sink term Sn giving the rate of decrease of floes is thus proportional to the number of floes. So, neglecting other methods of floe size change, an exponential decay in the number of floes is expected with the decay constant depending on the ice concentration and the strain rate of the velocity field. This is not strictly true because changes in compactness would be expected to accompany changes in floe number density. Equation (26) may then be written cln clt + V• (un) = -2LAjtja(8)n (29) For a full sea ice model in which the compactness A undergoes time evolution, the average floe size may be calculated from A=nnr 2 using the current value of A. 3.3.2 Floe break up Often, in a particular region, there is a characteristic floe size, suggesting that there is some mechanism, determined by the local physical conditions, responsible for their formation. However, there is a very large range of observed floe sizes (from tens of kilometres down to just a few metres) so that it is unlikely that a single mechanism is at work. We must therefore look for several possible mechanisms that could cause a uniform and infinite ice sheet to crack into floes or indeed for a large floe to crack into smaller units. We must consider each mechanism in turn and determine the physical conditions required to cause floe cracking together with the size of the floe thus produced. The characteristic floe size will be - the least value of the floe size predicted by the various mechanisms for which local physical conditions suggest floe cracking to occur. For the very large floes observed, there is some doubt about whether they are actually a single entity. Observations by · Ackley and Hibler (1974) taken from two surface stations on what appeared, from aerial photographs, to be continuous ice, experienced a relative drift. 50 There is some difficulty in explaining the size of the largest floes produced. It is possible that their scale is determined by variations in the forcing, of which a major component is the wind. The situation is complicated by the fact that no ice field exists that is formed of perfectly uniform ice of constant strength. Suppose some mechanism suggests the formation of a floe ~f the order of a hundred kilometres in diameter. Such a result must be treated with caution since over these distances the physical properties are likely to vary considerably. Consider the following simple example. A semi-infinite ice sheet of thickness h is subjected to an off-ice wind giving rise to a surface stress T. The internal stress at x = L would then be TL. Equating this with o-maxh where o-max is the tensile strength of sea ice gives a breaking length of L o-maxh (30) ' T Nm-2 For h = lm, o-max = 10 51'\and T = O·SNm- 1, this gives L = 200Km. If o-max and h were not constant along the floe then depending on the amount of variation, it is possible that the tension Tx/h(x) exceeds o-max(x) for some x much less than L giving rise to floes of the order of a few kilometres. Because of the large variation in the physical properties of ice over large distances it is not feasible to predict the sizes of extremely large ice features other than, as we have done here, to estimate the orders of magnitude involved. A more complete analysis would have to take into account the stochastic nature of the ice properties. This would require more comprehensive datasets than exist at present concerning the variation in sea ice properties. Sodhi (1977) investigates the arched fracture lines that are visible from sate 11 i te imagery of the ice which is forced through the Bering Strait and in the Amundsen Gulf. He explains these using the theory of granular media forced through a narrowing passage. In this way large floes of the order of 20 - 60 km may be produced. 51 We now turn our attention to mechanical processes that would pe expected to produce floes less than one kilometre in length. Evans and Untersteiner (1971) considered the formation of thermal cracks in an ice sheet due to a sufficiently large difference in air and water temperatures. For 3m ice, a temperature difference of 20°K was able to give rise to cracks 200m apart. · ThP. analysis was performed using the coefficient of thermal expansion for brine-free ice. Evans (1971) extended the theory to include a salinity dependent coefficient of thermal expansion. The results were similar to the brine free case suggesting that this mechanism could explain cracking in sea ice. Schwaegler (1974) suggests that cracking due to isostatic imbalances caused by the variation in ice sheet thickness is possible. However, results presented by Ackley et al (i976) using observed ice profiles show that the internal ice stresses would not be enough to cause fracture. They also point out that stresses caused by isostatic imbalances would have sufficient time to subside due to plastic creep. We next consider the wind-induced flexure of ice sheets. This flexure may be directly caused by the wind or indirectly in that the wind-induced motion of the floe causes interaction with the ocean in such a way as to produce vertical forces. Also, winds increase the sea surface roughness -and swell which may similarly cause flexure and break up of floes. There are little data and no satisfactory theories to explain wind- induced flexure and fracturing of ice floes. Browne and Crary (1958) observed the tilting of Fletcher's ice island, T-3, and found a correlation between the time series for this tilt and the atmospheric pressure. Weber and Erdelyi (1976), as part of the AIDJEX pilot study, placed tilt measuring devices on an ice floe and found a good correlation between the tilt and the wind speed. They put forward the suggestion that the floe tilt was caused by the couple exerted by the wind on the top surface and the water traction on the floe bottom. This, however, predicted tilts far smaller than those actually observed. In fact, subsequently it became known (personal .communication from Weber to Wadhams) that the sense of the tilt was opposite to that predicted by their calculations and that the floe was behaving as a planing dinghy in that the down-wind end of the 52 floe tilted up. Coon and Evans (1977) criticised the assumption made by Weber and Erdelyi that the floe behaved rigidly. They suggested that consideration should be made of the elastic character of ice floes. They thus utilized the theory of non-rigid beams on elastic foundations to represent a floating ice floe. They concluded that for normal winds the bending stresses were insufficient to cause cracking but that the angle of deflection of the end of the floe was of the order of magnitude of the Weber and Erdelyi observations, and hence that wind-induced ti 1 ting of floes could not cause cracking. The slope at the end of the floe is largest and the magnitude quickly decays away from the edge. For a wind stress of 0·5Nm- 2 and for 3m ice they calculated a tilt at the end of the floe of 3•16 10-6 radians. This should according to their equation (13) be 0·316 10-6 radians. Thus they cannot satisfactorily explain the tilts of 30 10-6 radians actually observed by Weber and Erdelyi ( 1976), who presumably did not place their tilt measuring device at the edge of the floe. One can cone lude from their analysis, however, that the wind and water surface tractions alone are not sufficient to explain the observed tilts even when consideration is taken of the flexure of ice floes. This means that one must accept that floes do tilt in a way that is closely dependent on the wind speed but that the tilt is caused indirectly by the wind. It is possible that features on the underside of the floe cause deflection of water as the floe moves giving rise to vertical forces on the floe . In this .section, we attempt to use the observations of Weber and Erdelyi (1976), bearing in mind that the floe might have been flexing, to find a maximum floe size that can exist for a given wind field. Beams on elastic foundations behave rigidly or flexibly depending on their length. A length scale A~l, called the characteristic plate length, occurs in beam theory and only those beams whose length is greater than about x- 1 behave flexibly. X i s defined by 53 -k 4D - (31) where k is the foundation modulus, which is the upthrust per unit length of the beam required to cause a unit vertical deflection. In the case of a . . § i!. l::he o.c.ce\eto-\-ion of '3'°'c:a.V'i ~~-body floating in water, k is fw~·AD is the flexural rigidity, which for a plate is D == (32) Here, Eis Young's modulus and Vis Poisson's ratio for sea ice. If we take N m-2. E/ (l-v 2 ) = 1010 /\then for 3m ice, the characteristic plate length A-1 is about 55m. Weber and Erdelyi ( 1976) found the tilt of a floe by measuring the variations in the levels of two points on the floe at a distance 120m apart . Thus we see that the flexure of the floe must be taken into account when interpreting their data. Suppose a semi-infinite beam has an applied load Pai x = 0. The angle of deflection of the beam and the bending moment, for x > 0, are given by Hetenyi ( 1946), with a sign change because we consider an upthrus t, as 8(x) 2PA 2 = ex p(-)vc) [cos\x + sin\x] k M(x) p = exp(-\x)sin\x >- (33) The maximum bending moment occurs at x = (1r/4))_-l and has the value (p/)_.J!)expT-1r/4]. In terms of the deflection at x = 0, 8 · , 0 ~ax k8 0 2 ./1exp [ TI /4] A3 (34) I I Now the bending stress a- is related to the bending moment by a- = (35) so that from (34) we obtain (36) Thus, for an observed tilt e0 , we expect a maximum bending stress given by (36). A floe tilting sufficiently to give a bending stress exceeding a-max' the tensile strength of sea ice, will break. If we substitute A-l = 55m and e 0 = 30 10-6 radians, which is the extreme range of tilt measured by Weber and Erdelyi (1976), equation (36) gives er ;::; 5 103 Nm- 2 which is about 20 times smaller than the breaking stress. Thus the floe was in no danger of breaking up due to wind tilting. Now to proceed with the analysis, an estimate of the relation between the tilt, o, measured by Weber and Erdelyi's device and the tilt at the floe end, e0 , must be made. For the moment we assume they are equal (o = e0 ) and if necessary make adjustments once the results have been obtained. Equation (36) should thus be replaced with the following expression for the maximum bending moment in the floe, 3f>w~< A-1 )3 ./2.exp [n/4 ]h2 0 where o is the deflection measured in Weber an·d Erdelyi's (1976) data. (37) Clearly, the floe for which Weber and Erdelyi (1976) made observations was long with regard to its rigidity in that its length exceeded the characteristic plate length, A-l, for sea ice of its thickness. Thus the analysis for a semi-infinite beam as has been carried out so far is applicable. 55 To deal with the situation for a short floe, a different theory is needed. The presence of the second free end to the floe has, as we shall see below, a substantial effect on the relation between the applied force P and the angle of tilt. The analysis proceeds as for the case of the semi-infinite beam except that it has a free end at x = L. From Hetenyi (1946), the tilt e(x) and bending moment M(x) are 1 e (x) = + cos}vcSinh)vc') + sinAI,(Sinh)vccos)vc' + Cosh}vcsin)vc')] M(x) P SinhAI,sin)vcSinh}vc' - sinAI,Sinh}vcsin)vc' A Si~h2 AL sin2 AL (38) where x' = 1-x. Now if we assume that the beam is short in that AL<< 1, then the trigonometric and hyperbolic trigonometric terms may be replaced by the first two terms of their series expansions. Keeping only the highest order terms, this yields the following expressions for the tilt and bending moment for the beam. 6P e(x) = kL2 M(x) = PLX(l-X) 2 (39) where X = x/1. The bending moment has a maximum at X = 1 . h J wit a value 4 ~ax = - PL (40) 27 Hence, noting (35), the maximum bending stress is given 1n terms of the angle of tilt by er == (41) I I 11 11 It is worth noting at this point that Weber and Erdelyi (1916) gave a similar formula for the maximum bending moment in a tilting 'rigid' beam. However, they obtained = f>igL3 e 6f3h2 (42) Coon and Evans (1977) gave a maximum bending stress which is half that of Weber and Erdelyi (1976 ). In fact there is no unique formula for the maximum bending moment as a function only of the tilt; it depends on the method by which the beam is tilted. For example, in the case of a beam tilted by applying a moment about its centre, the relationship between o- and(:) is (43) which is similar to the Coon and Evans (1977) value except that they have the factor f>i instead of f>w· Since floe tilting seems more likely to be caused by an upthrust at the leading end of the floe. rather than an applied couple (either by way of a concentrated couple or from an upthrust in combination with a downthrust at another part of the floe), formula . . (41) would seem more applicable. The only tilt measurements that are related to the wind speed are those of Weber and Erdelyi (1976) so we again have to use these with the floe bending models to predict the tilt of a small floe. Weber and Erdelyi (1976) made observations on only one floe. To obtain an empirical law for the tilt of a floe of arbitrary length and thickness, based on these observations, it is necessary to assume a mechanism to explain the tilt and then determine theoretically the dependence on factors such as the floe thickness hand the wind speed u10• The procedure is to use (33) to determine the upthrust P needed to give rise t o a til t at the floe end of 8 = o, where o is t he tilt meas ured on 0 57 Weber and Erdelyi's (1976) floe for a given wind condition. We obtain (44) We now assume that a short floe ·Of length L is subjected to the same loading Pat one end. Equation (39) then gives the tilt of this short floe as e = (45) from which the maximum bending stress may be found from (41) to be (46) Summarizing, given a floe in a wind field, we first determine the tilt expected from the Weber and Erdelyi (1976) floe if it had the same thickness as the given floe. This tilt is denoted o. We next calculate A-l for the floe and so determine if it is to be classified as a 'long' or 'short' floe. If the floe is long then er in equation (37) is calculated and if it is greater than crmax the floe will crack producing floes of length (1r/4)A-l. The short floe theory with L = (1r/4)A-l is then used to determine if subsequent cracking of the pieces occurs. If the given floe is already short then formula (46) is used immediately without first using the semi-infinite beam theory. The method is to solve (46) for L with cr = crmax to determine the largest floe that can exist in the wind field without cracking. There is still one implicit assumption here that needs to be checked. That is, that a floe of length ( 1r/4) A-l produced according to the semi- infinite floe theory will not, if subjected to the same forcing that produced it, continue to break according to the short floe theory. This is checked as follows. Suppose a semi-infinite beam cracks due t o a load Pat 58 its end. From (33), l:;n~mc,...~i""u.M \oe..i"lchl'\';i ,rno"'e.n.\:; ino. \on~ floe. M:.,_v ·1s. ~i-Je.n.. lo~ ~ax (47) A short beam of length L moment of ( n/4 )A-1 will from (40) have a maximum bending L ~ax 4 = - P( n / 4) A- l 27 so that since M~ax/~ax is TI ./2exp [ 7T /4] 27 ::: 0 · 361 (48) (49) which is less than unity, the bending moment in the fragments of the large floe does not exceed the maximum bending moment. We now turn our attention to the mechanisms that might cause floe bending. We have assumed that an upthrust is produced at the leading edge of the floe . The tilt produced will again depend on whether we have a long or short floe. Considering first the long floe, the tilts will depend on the thickness hand the wind speed u10• Suppose the upthrust at the end of the floe is caused by the downward deflect ion of water by a number of keels at the down-wind end of the floe. In this case the upthrust would be expected to be proportional to the square of the wind speed. As fa r as the thickness is concerned, we see from (31), (32) and (33) that the tilt is proportional to h- 3/ 2• Thus an empirical law relating the wind speed with the tilt shou ld be of the form, 0 (50) Weber and Erdelyi (1976) suggest that the tilt angle o is linearly related to u10• The corresponding data curves agree quite well. However, by 59 I I 11 I 1.0 +J 0.8 ,.....-; . ,-; +J 'D 0.6 ·~ co 'D 0.4 ~ . ,-; ~ 0.2 0.0 Figure (3 ·4) The square of the wind velocity plotted against the magnitude of the tilt, the sealing having been adjusted to give the best eye fit between the two curves. (After Weber and Erdelyi 1976) I 11 11 I J I I I I I I --- - - Square of wind speed Magnitude of tilt I \ \ 0 20 40 60 80 100 Day number Go 11 2 rescaling the graphs a good fit between the curves for tilt and u10 can be obtained (see figure (3.4)). From the data, it is possible to derive the approximate law o = 0 · 66 10-6 uf0 ( 51) where o is measured in radians. So, taking the thickness of the Weber and Erdelyi (1976) floe to be 3m, the express ion for tilt, ( 50), becomes 0 (52) This may be substituted into (37) and (46) to give the maximum bending stresses for long and short floes as c:, = (53) and c:, = (54) where C = 3•43 10- 6• Thus a long floe will break into floes of length (,r/4)).-l if er in (53) excee ds o-max· Solving (54) for L with o- = o-max will give the maximum size that the floes can be in the wind field u10. If L/2 is taken to be the radius of the floe, then (54) may be rearranged to give the maximum size of short floes in a wind u10 as = (55) The line of re ason ing regarding the breaking of long and short floes is 61 I 11 Figure(3·5) Is floe long or short? Long r > ,--1 Will it break? r unchanged Short WIii It break? r unchanged The line of reasoning involved in determining whether a floe will break up or not. ' 62 I I j 1 Figure(3·6) 30 ----- 25 ~ (/) ~ --1 Q) N • r-1 s 20 ....___., 'D Q) 15 Q) P.. (/) 'D 10 ~ • rl ~ 5 0 10000 tl) 1 OOO Q) 0 . .--. t+-t 100 0.0 0.5 1.0 1.5 2.0 2.5 3 .0 Ice thickness (n1.) The wind speed needed to break a floe acc~rding to the long floe equations . Figure( 3· 7) ---·-------- ---- ------- - ~ 0 10 20 30 Wind speed (m/s) The radius of a floe three metres thick able to survive a given wind according to sh~rt floe theory. I I 40 I 11 traced in figure (3.5) Figures (3.6) and (3.7) show the relationship between wind speed and the cracking of floes. Figure (3.6) shows the wind speed needed to crack a long floe. From this we see that only rarely will floes of more than 1 m in thickness crack when in the interior of the pack. A wind speed of about 16ms- 1 is needed for this. As is mentioned below, wave action will be responsible for breaking the thicker floes. Another complicating factor is the variation of the tensile strength of ice crmax which may be important during summer when melting may cause weakness of the ice structure. In figure (3.7) we see the maximum size rmax that floes can be in a given wind field, according to the short floe formulae. The plotted curve is applicable to a floe of thickness 3m. The part of the curve for which large values of rmax are given is not relevant since only short floes will be tested for breaking using the formula. The comments regarding the variation in ice tensile strength apply to short floe breaking as well as to long floe breaking. The values that these curves indicate for breaking of floes seem reasonable, so without further evidence, we can take our assumption leading to equation (37), that the tilt as measured by Weber and Erdelyi is equal to the tilt at the floe end. When considering a floe field consisting of many thickness categories, the dependence of rmax upon h in (55) means that the floe radius r averaged over a region depends on the ice thickness distribution g(h ). Suppose an area of pack ice initially consisting of short floes with a distribution g(h) is subjected to a wind of speed u10• Any floes with radius larger than rmax given by equation (55) would then crack. The fraction of floes with thickness in the range (h,h+dh) is g(h)dh, and if initially the floes have radius r, then subsequently they will have radius min{r, rmax(h) }. The average floe size resulting from the action of the wind is thus co r =~min{r,rmax(h))g(h) dh 0 Note that if min{r,rmax(h)} = r for all g(h) =f= 0, then r = r. (56) The preceding analysis, which stems from a single set of observations relating wind speed and floe tilt, concludes that if a thick floe tilts with the wind, then a sufficiently thin floe would break in the same wind. However, no mechanism has been suggested which fully explains the tilting. One mechanism which 'involves flexure and floe tilting (see page 52) , but which can explain the break-up of floes is that due to the transfer to and subsequent propagation of ocean waves in the pack ice (Robin 1963). Measurements in the Fram Strait region (Wadhams 1978) suggest that the wave energy decays exponentially with distance into the pack, the waves of shorter wavelength decaying most rapidly. Wadhams found the decay rate such as to imply floe breaking to a distance of a few tens of kilometers into the pack. Kozo and Tucker (1974) found that further south in Denmark Strait, it was necessary to travel 165km into the pack before its character was similar to that of the Central Arctic. We should perhaps note that the resulting sizes of floes produced according to wave-breaking theory are similar to that given in this section since both rely on the elastic property of ice floes which depend upon the characteristic plate length A-l. The full theory of wave-induced break-up could be incorporated into a sea ice model by applying observed ocean wave amplitude fields which decay within the pack. In the model developed for this study, no explicit consideration of wave-induced break- up is included. However, to simulate the extreme break-up of floes observed near the ice edge, an ad hoe method is used in which the equations derived here for floe break-up are used with the ice tensile strength reduced in proportion to the ice concentration. Al though this method of parameterizing wave-induced break-up has the advantage of being simple and requires knowledge only of the local wind speeds, it is not based upon any direct physical arguments. Because of this, we must regard the equations for floe break-up only as empirically derived, to be modified as and when observations of floe break-up in connection with wind speed become available. Such observations should note the break-up of any thin ice present in a region, even if it contributes only a small fraction of the total area. As part of an empirical approach we may regard the wind speed as a measure of the level of physical activity, the wind acting indirectly to cause floe break-up. For example, wind causes local ocean surface waves which, as we have mentioned , can cau s e floe break- up. Alternatively, the wind can produce waves in the ice directly (Hunkins 1962). Also, general turbulence can cause random floe collisions resulting in impulsive forces on floes. There are more complicated mechanisms · · f f th mechanisms already mentioned, such involving a combination o some o e · veloc~ty associated with changes in the as the variation in wave group L · · th · fl es (Mollo-Christensen 1983). horizontal compressive stress wi in o h d 1 ~ tudes in regions where the group These can lead to en ance wave amp L · h "bl fractur~ng of the floe. Before we can velocity is zero wit possi e L determine quantitatively the relationship between floe tilt and wind speed, we cannot do better than to use empirical relationships and will continue to use the floe breaking formulae derived here, but bearing in mind the limitations involved. 3.3.3 Thermodynamic changes 10 floe number density When dealing with collisions, only the average floe number density, n, has been considered. However, in order to more fully model the spatial distribution and changes in floe size it is necessary to consider a distribution of floe sizes at each point. This is motivated from observations of floe fields which show large variations iµ the floe sizes present at any location. The analysis of the previous section suggests that floe break up and the subsequent size of the residual pieces depend strongly upon the thickness of the · floe involved. Thus a suitable way of considering a floe distribution is to consider the floe number density for each thickness category (h, h+dh). Thus we obtain a floe number density distribution n(h) such that n (57) It is possible for the floe size distribution to change without altering the total number of floes. Such redistributions occur, for instance, during vertical melting and growing in exactly the same way as when the relative areas of each thickness level change during thermodynamic redistribution. Thus dur ing growing, t h e numbers of floes 10 t h e t hic ker ic e c a te gorie s increase at the ex pense of a thinner category. There are some d i ff e r enc es in behaviour between the thickness distribution g(h) and the floe numb e r d e nsity distribution n(h) . The distribution n(h) is not normal i z e d as is g ( h) : there is no limit to the numb e r of pieces a floe may break up into. When the thinnest floes melt, open water is created which changes g(h), but for n(h), the floes simply disappear. By adding a thermodynamic term to the floe number density equation (29) we obtain an equation for the evolution of n(h) that is analagous to the ice thickness distribution equation (2.39), thus = . ljJ + F n n (58) where the third term on the left indicates the thermodynamic changes ton. The term w represents the changes ton occurring during collisions and n fracture. The final term only loosely similar to the lateral melting term F1 represents the changes to n occurring when new ice forms from open water or melts completely. The functional form of w 1s complicated and b'l'le. c-e.cl isttib-....\-ic:,n fu.nc.hon. of €C(u.o.+iol'I l'Z·JI)? totally different from that of W,~Handling Wn in a model · is best _done by converting the floe number densities into the equivalent floe sizes for each thickness category and make the necessary changes to the floe radius as outlined 10 this chapter before converting back to floe number densities . 3.3.4 Lateral melting Consider the situation 10 which heat is absorbed i nto the ocean mix ed layer, through leads . The heat is assumed to mix underneath the ice floes and melt t hem from below as well as from the sides . This melting from the sides, known as lateral melting, is an important componen t of the change in ice geometry through thermody namic effects , espec i all y when t here 1s a hi gh flo e number de nsity and there ex ists a la r ge length of floe e dge per unit area. A specific problem assoc i a t ed with l ateral melting 1s t o determine the r a te at which me l ting o c c ur s l a teral ly compared to that at 66 which it occurs vertically. The more general problem is to determine how and at what rate an arbitrarily shaped ice floe will melt when immersed or floating in water which is at a temperature above freezing , such that melting will occur. This problem cannor be solved analytically even in simple cases because of the difficulties introduced by the boundary condition, namely the shape of the boundary of the ice, being one of the unknowns. However, we make some assumptions here which simplify the problem and enable some kind of a solution to be obtained for the case of a disc or a large floe with thickness small compared to its horizontal extent. The method consists of assuming that the upper surface of the ice floe is maintained at a constant low temperature and the surfaces below the water line are at a temperature slightly higher than that necessary to initiate melting. Laplace's equation v2T = 0, is solved for the body with these Dirichlet boundary conditions. The next step is to look at the isotherm for which melting will occur. The temperature of the water surrounding the ice is assumed to be just enough to make the surface layer melt, so that the melting-isotherm will be very close to the original surface. We can assume that the body will melt in a short time so that the new surface coincides with this isotherm. The solution to Laplace's equation for the original body may be modified to give the solution of the newly melted body by specifying the temperature on the old melt ing-isotherm as zero. The new isotherms will have the same shape as before but with slightly different values of T. We thus reach the conclusion that a body will melt in such a way as to assume the shape of its isotherms in the solution to Laplace's equation. We assume also that the body melts slowly compared to the time required for the ice to reach thermal equilibrium internally, so that a quasi-static situation is obtained and the heat equation may be solved fo r the body without including tQe time dependent term. We can obtain the solution to Laplace 's equation for a disc with boundary conditions as shown in figure (3.8). To simplify matters we t ake the temper a ture at the top surface to be 1 and at the bottom and curved surfaces to be zero. The height of the disc is hand its radius is 1. The solution may be obtained by separation of variables and is expressed as a series of Bessel functions thus 67 h --------- T=O I t 0 r = l T=O Figure(3•8) The boundary conditions used to determine the temperature distribution within a disc. 68 00 ~ 2 J 0 (\r) Sinh \z LJ Ap J 1(\) Sinh\ p=l (59) However this solution for a given isotherm, T = constant, does not give a very useful indication of the relation between the variables rand z . We thus solve the problem in a rectangular region shown in figure (3.9) . For simplicity we take the height as n with boundary conditions T = 0 for x = 0 and y = n and T = 1 for y = 0 1n the region x > 0. The solution of Laplace's equation 1.n this region may be solved by summing two known solutions. Firstly, the solution in which the two longer sides have zero temperature and the side x = 0 has a Dirichlet boundary condition T(O,y) = (y/rr) -1. The second solution 1s simply T(x,y) = 1-(y/n). The first solution can be obtained by Fourier superposition which together with the second solution gives T(x,y) = 1 (60) The isotherms for this solution are shown in figure (3.9) . Thus we assume that for a rectangular floe left undisturbed within well mixed water of a temperature sufficient to melt the ice, the floe will melt so that its profile at subsequent times follows the isotherms as they appear in figure (3.9). The figure shows the 'bathtub' shape that is observed in laboratory experiments with melting ice blocks (Russell-Head 1980, Gebhart et al 1983) It i s convenient in the following analysis to r e-express (60) in the form T( x,y) 1 -1 [ siny Sinhx] tan rr 1-cos yCosh x (61 ) where ( - n / 2 ) < tan - 1 < n / 2 in ( 6 0 ) and O < tan - 1 < n rn ( 61 ) • Thus the tan- 1 function in (60) is the principle value of the function, whereas the tan-l function in (61) is a modified tan-I function. A typical isotherm 69 y T = 0 Figure(3·9) ~------~ (l) Ol -0 (l) (l) 0 LL 0 y The isotherms or the lines of melting at the edge of an infinite rectangular floe when the temperature distribution is assumed to obey Laplace's equation in two dimensions. Y=n / / / C Top of the floe 1 [2Jj_ _______________________ --'-------3>-X Q Xo Figure(3•10) An approximation to the mode by which floes melt laterally. The arrows give the relative speeds with which various parts of the floe melt. 70 71 T = constant starts at the origin at an angle Tn/2. I t th e n mee ts th e asymptote y=nT at x = oo. In a simple model in which vertical melting of floes as well as lateral me lting (but retaining vertical edges) is allowed, the problem is to determine the relation between the vertical and lateral melt rate that best describes the melting patterns shown in figure (3.9). One way to approach this problem is to assume that the floe melts very slightly so that it decreases in thickness by o << 1. If we further assume that the floe remains rectangular, then in principle a calculable amount of lateral melt is needed to give the correct total volume melted. We consider the isotherm T = o(n and expressing (61) in the form y = f(x) we need to find x 0 such that lx 0 0 f(x) dx = 1~~-0-f{x)] dx XO (62) x 0 (o) would then give, as a function of the vertical melt o, the amount of lateral melt needed to retain a rectangular cross section and give correct total volume melt . Solving (62) in practice is difficult because there is no simple way to re-express (61) in the form y = f(x) . However, since o << 1 we may assume tano :: o and write (61) in the form o(l - cosy Coshx) = siny Sinhx (63) Equa t ion (63 ) may be used to. give the o/n i sotherm for the following cases . 1) x smal l ( The e dge of t he floe) y = 2 t a n - l ( x / o ) 2) X small and y:: 7T (The corner of the floe) y = 7T - ( 20/x) 3) x not small and y :::: TI (The bottom of the floe) y TI - 6 Coth(x/2) By examining these three cases, we see that it is possible to enclose the slightly melted floe by a polygonal region as shown in figure (3.10) . Case (1) gives a tilting of the floe edge away from the vertical. This is shown by line A in figure (3.10) which hai a gradient 2/6. From case (2) which is the equation of a rectangular hyperbola, we obtain line B of slope 1, touching the curve at (x,y)=(./26,n- ../26). Case (3) tells us that line C, y = n-6, encloses the curve, n-6 being the value that the isotherm asymptotes to as x -oo. As the floe melts, line A tilts, its angle changing at a rate with order of magnitude o. Similarly line C drops at a rate proportional to 6. However line B proceeds away from the corner (O,n) at a significantly faster rate (proportional to Jo)· Thus we would expect to see any corners of the floes melt most rapidly. Now we redistribute the ice of the floe so that it is again vertical (along the line x = x 0 ), and determine x 0 such that the two shaded regions in figure (3.10) are equal. x0 can be determined fr~m simple geometry to be [i + ;Jc + (Higher order terms in 6) (64) The higher order terms in 6 in (64) may be neglected if we consider the limit as c-o. We then obtain the relation (x0 /c) = (n/4)+(4/n) :::: 2·05 which gives us the rate at which lateral melting occurs compared to vertical melting. Thus in a model of floes consisting of squat cylinders with vertical sides, including a lateral melt rate equal to 2·05 times the vertical melt rate will give the best approx imation to the melting suggested in figure (3.9). It should be mentioned that the factors that we have ignored here if included would tend to increase the lateral melt rate compared to the vertical melt . For instance, if the heat was not effectively transported beneath the ice in the mixed layer, then more melting would be expected near th e leads. Another factor would be the breaking off o f small pieces of 72 the floe at the edge which would melt more quickly in the water. Also a significant amount of melting occurs due to wave action (Wadhams et al 1979). This acts at the water line producing a 'wave-cut' which destroys the profile suggested by the calculations here. The result of the long term wave action and melting is to produce an underwater sill with a flat top (Alekseev and Buzuev 1973) . Again an additional contribution to the loss of ice from the side of the floe would result from including wave- induced melting in the calculations. Such melting could occur even when there is net heat loss from the leads. Thus if we decide to use the factor (1r/4)+(4/7r) to give the lateral melt rate, it should be borne in mind that it is really a lower limit. 73 4. PACK ICE DYNAMICS 4.1 Introduction A full understanding of the large scale deformation of pack ice, which is important in climate studies, can be achieved only when a study has been made of the small scale processes that accompany such deformation. These include ridging and rafting. The average thickness of the pack ice in a region where there is convergence, will tend to increase (by ice conservation). In general an area of pack ice will consist of a variety of thicknesses, and it is the thinnest ice that will be crushed in a converging flow field. Determining which thicknesses are crushed and by how much was discussed in section (2·3) concerning redistribution theory. The size and shape of ridges produced during pack ice deformation determines the strength of the ice, an important factor in determining the magnitude of the internal ice resistance, or ice interaction term. In this chapter we aim to derive the ice interaction terms using, where possible, mechanistic models of the underlying physical processes. 4.2 Ridging and rafting 4.2.1 A simple ridge building model Although relatively thick ice is able to ridge, it will do so substantially only after any near by thin ice has ridged. The thin ice, often newly grown ice in leads, breaks up into rubble and piles into a ridge- like structure. The thin i ce con tin ues to break up and feed rubble into the ridge until some limiting ridge height i s r eached, a heigh t that 74 depends on the strength of the original ice sheet. In this section the thickness of the original, or parent, ice sheet is h 1 and the ridge height which is the distance from the tip of the sail to the bottom of the keel, is h 2• For the situation in which thin ice in a lead is ridging, h1 will be the thicknes~ of the ice in the lead. The frequent occurrence of ridges in pack ice suggests that ridging is an important method of ice redistribution. Tucker and Govoni (1981) conclude from observations of 30 ridges in 5 locations off Prudhoe Bay, Alaska, that there is a dependence of the ridge heights on the maximum block thickness from which the ridge is composed. In other words, that the ridge height is a function of the parent ice thickness. In early studies of ice redistribution theory (Thorndike et al, 1975; Rothrock, 1975), a linear ridging law was used, so that h2 the ridge height is related to the parent ice thickness h 1 by (1) where k is a constant. This rule was chosen not as an accurate formula for ridge heights but to simplify the interpretation of the results obtained using redistribution theory. The Tucker and Govoni (1981) observations suggest that (1) is far from adequate and that a square root dependence such as (2) used in Hibler's (1980a) variable ice thickness model, is a better formula for an appropriate constant H, than is equation (1). Equation (2) has the additional advantage that it can be derived (Hibler 1980a) from a geometrical argument by assuming that ridges have triangular cross section and are formed from leads of constant width. 75 Results from a numerical mod~l of the ridge building process (Parmerter and Coon, 1972) tend to confirm that kin equation (1), rather than being constant, should decrease with increasing h 1• Rothrock (1975) calculates the potential energy and frictional losses occurring when an ice sheet of thickness h slides into a rubble field of -thickness kh. In this way he derived an estimate of the ice strength. Here mechanical arguments are used similar to those of Rothrock (1975) but for a different purpose: Firstly some idealized shape is chosen to represent the cross-section of a typical ridge. An ice strength dependent on its thickness is assumed, and the size of the resulting ridge calculated. A ridge shape is chosen that is conveniently handled mathematically and is also reasonably in accord with observations. Figure (4 . 2) shows a possible idealization of a 'typical' ridge cross section. A ridge of this shape is however not in isostatic equilibrium and, if the ice were to deform, an adjustment due to the buoyancy of the keel would take place resulting in a shape something like that shown in figure (4.3). An altogether more convenient shape to deal with mathematically is shown in figure (4 . 1). The ridge is assumed to be in isostatic equilibrium at each point. This model represents the keel well, but not the small steep-sided sail shown in figure (4.2). This is perhaps not such a serious deficiency because the keel contains a far greater proportion of ice than the sail. Also since the shapes in figures (4 . 3) and (4. 1) are in isostatic equilibrium, the volumes of ice in the keels and sails are correctly represented. It is important in a redistribution model to ensure that the correct volume of thin ice is used up in forming a ridge. Figure (4. 1) is thus chosen to represent a 'typical' ridge, and the theoretical results obtained should be compared with keel data -rather than sail data . The angle of inclination of the keel ~bis fixed, and is chosen to agree with observed values. Other dimensions are calculated by assuming that throughout the ridging process, the ice remains in isostatic balance. By definition h~+h9 = h -J J J J = 1,2 (3) The superscripts a and b r efer to quantiti e s above and below sea-level respective l y. 76 Figure (4.l) ___:i' __ ......-; _____ ft ___ -µa h1 )J, -,-~--~~--------µb ,!, Sea level~------------ ' - --------~ - - t t h~ hb 2 The assumed cross section of a ridge used . in calculating the height of ridges produced h2 as a function of the original ice thickness h1• 77 Figure (4 . 2) ---- ----- - -- - - - -- ------------ - ---Sea level Idealization of a ridge cross section Figure (4 . 3) t ~ Sea level Isostatic adjustment The cross section of a ridge after undergoing isostatic adjustment . 78 During ridging, the ice will not have a continuous structure, but will contain many air gaps between the blocks. Below the sea surface, the gaps will contain water . The fractional volume of the ridge structure consisting of the air and water gaps is defined as the porosity n 0 • The simplifying assumption is made that the porosity is constant throughout the ridge and in particular above and below the the water line. In fact the porosity varies considerably within the pack ice, but here a single 'average' value is chosen. With this assumption isostasy implies, a £).h. ,- i J b (o -D·)h· ,w ,- i J Combining (3) and (4) gives, h~ = crw-t>i) h· J rw J hl? J ri h. rw J From the ~eometry of the ridge and from (5) and (6), J = 1,2 J = 1,2 J 1, 2 Here (>i and t>w are the densities of ice and sea water respectively. (4) (5) (6) (7) The ridging strength is regarded as the horizontal stress that an ice sheet can exert before it yields. Intuitively it is expected that this strength should increase with increasing ice thickness. The ideas behind ice strength are dealt with in more detail in the next section. The force required to increase the gravitational potential energy of the ridge Fpot together with the frictional forces Ffric involved in the ridge building are calculated. It is assumed that whe n (8) 79 the ridge has reached its maximum height h2• Consider the left half of t he ridge. The gravitational potential energy of the sail is ha ( l-n0 )E\~ cot eaf 2y(h27y ) dy hi = ! c1-n0 )ei~cotea(h2+2h1)Ch2-h1)2 l arew-eiJ 3 ( 2 = 6 Cl-n0 )ei~cote l ew h2+2h 1Hh 2-h 1 ) wher-e. g ic;. the o..c.c.elero-tion. of ~r-o.,til:~. The potential energy of the keel can be similarly calculated, to give ( 9) (lo) The potential energy of the keel due to buoyancy is similarly found to be Adding (9), (10) and (11) and using (7) to eliminate ea in favour of eb gives the total isostatic potential energy as It is not necessary to consider the potential energy of the ice of thickness hl since only changes in potential energy are of concern. Since (13) 80 where 1 is the length of pack ice fed into the ridge, it is necessary to relate 1 to changes in h 2• From (14) which represents volume conservation during ridging, one gets using (5), (6) and (7) From which So d(P.E.) dl d(P.E.)/dh 2 dl/dh2 (15) (16) (17) The frictional forces are calculated by assuming that the incoming ice (generally ice in leads) slides along in contact with the forming ridge (shown in figure 4. 1 ) as it breaks up and that the coefficient of friction at the surface above sea-level is pa and that ~elow is pb. Cohesion between the surfaces is not assumed. The friction on the upper surface is 8 1 ( 18) and on the lower surface is (19) The total friction is thus, (20) Substituting (17) and (20) into (8) gives (21) where, = 2 1 r· b -(1ta+l1b)(l-n )~(D -D.)2cot8 2 r , 0 5 ,-W ,- 1 r; (22) and (23) Equation (21) is a simple quadratic equation in k=h2 /h 1, the physically realistic solution of which is 82 k ' 1 - cb [ c~ 2 cb r < h 1 l J l (24) = - + -2--+~ 2 2Cf 4Cf Cf Cfhl for h1 < (" p* /2Cb, (24) predicts either h2 < h1 or that h2 is complex, both of which are not physically meaningful, and indicates that no ridging can take place. Indeed if such a ridge were artificially constructed from rubble blocks, it would immediatel y collapse if unsupported. Hence if (30) Figure (4.4) shows h2 plotted against h1 for the following values of the physical constants. pa = Pb = 0·5 eb = 26° ri = 922 kg m-3 rw = 1025 kg rn - 3 no = 0 · 3 g = 9•82 m-2 p* = 9000 Nrn- 2 The+ signs in figure (4.4) represent the adjusted observations of Tucker and Govoni (19£1). Figure (4 .5) shows the ratio k = h 2 /h 1 plot te d against h 1 and indicates the drop ink as h1 increases . If h2 is plotted for values of h 1 outside the range of the Tucker and r;ovoni ( 1981) observations (figure 4 .6) then a maximum ridge height of about 9m is predicted. This is not consis t en t with observations that have been made of much larger ridges hl).O o.. l O·l7m. Parameterization of rafting may be incorporated into a sea ice model that includes 1.ce thickness redis t ribution by considering a redistribution of the form for h 1 < O·l7m (31) Note that the redistribution suggested by (31) does not describe the pr o duc ti on of tri ang u la r ri dge s and s o must b e tr e at e d in a diffe ren t wa y to equation (24) when considering the ic e thickness distribution changes during deforma t ion. 4.3 Ic~ strength 4.3.1 Evaluating ice strength It is vital to incorporate the idea of ice strength into any dynamic sea ice model. Expressed simply, thin ice is weak and thick is strong. The strength of ice in a model is a measure of its resistance to deformation. The dependence of the resistance on the type of deformation is dealt with in the next section. In this section, we are concerned with the magnitude of, say, the compressive resistance of ice and its dependence on the ice thickness. In fact we consider the ice strength in terms of the more general idea of an ice thickness distribution. Qualitatively, for a region where the greatest percentage of the area consists of thin ice categories and open water, the ice strength would be small, and ice would thus deform easily. Strong ice would include that near coastal regions which is thick and relatively difficult to deform. The variation in ice strength from strong to weak as one moves seaward, accounts for the shear zone observed in such regions. Hibler's (1979) two layer sea ice model represents the ice in a region by two quantities; the average ice thickness h, and the compactness A. As mentioned before, strength is included by the term p* = Ph exp{-K(l-A)} (32) The quantity (1-A) is the amount of open water and P is an empirically determined constant. The constant K is taken to be 20. This equation, which has been adopted by subsequent ice modellers · (R!6ed and O'Brien, 1983) expresses, in mathematical terms, the idea that the strength decreases rapidly when even a small amount of open water appears. The notion that the strength increases for thicker ice is included in equation (32) by a linear dependence of the ice strength on the thickness h. We shall investigate the dependence of the ice strength upon the amount of open water in more detail later. In a general area of pack ice , many different 89 thic~nesses of ice are present, so that during a process of deformation, different thickness categories of ice will be involved. A full description of the ice strength should thus be dependent on the ice thickness distribution g(h). The gravitational potential energy of an area of sea ice containing many thicknesses is 00 PE[g(h)] = cbf h 2g(h)dh 0 (33) Following Rothrock (1975) we evaluate the rate of production of potential energy by distribution by differentiating (33) with respect to time and eliminating the time rate of change of g(h) by using the ice thickness distribution equation (2.39). This gives, (34) The redistribution function W can be expressed as tjJ = IE!{a (8)o(h) + a (8)w (h,g)} o r r (35) where the terms are defined in chapter 2 . By substituting (35) into (34) one obtains (36) where r1p is defined as 90 I I 11 111 (37) This quantity p'p is the strength of the ice with respect to potential energy considerations. Rothrock (19j5) thus regards the strength as the potential energy produced per unit area per unit strain 10 pure convergence. By expressing the rate of loss of energy by frictional work, during ice re dis tr ibu t ion, in the form = /E/a (8)P1 r (38) Rothrock (1975) obtains an expression for ifi, the strength due to frictional considerations. Firstly, he expresses Rfric in the form (39) where Efric is the frictional energy loss in ridging per unit area of pack ice fed into the ridge, and Rarea is the rate of loss of this area per unit area per unit width of the ridge. Rothrock (1975) calculates Efric by considering a model of ridging, similar to that used in section (4.2), but in which he assumes that ice of thickness h breaks up during deformation, producing a ruhble field of thickness kh, where k is taken as a constant. The corresponding strength derived from frictional energy loss considerations, Rothrock (1975) derived as 00 1 h 2 a(h) Cf 1-(1/k) . 0 dh (40) ~here 91 11 Ill (41) The total ice strength is given by ~+P1". Later we derive expressions for~ and P1" as in (37) and (40) bu t based upon t he ridge model desc ribed previousl y. In his multi-layer model Hibler (1980a) calculated the strength numerically, but essentially used the formula given by equation (37) . His method consists of artificially the thickness distribution . . increasing term g(h) to g'(h) so that the distribution is no longer normalized to unity, thus If g'(h) CX) f g'(h)dh > l 0 (42) (l+~)g(h), then by redistributing the thin ice categories of g ' (h), as described in the section on redistribution theory, so that a new nor malized distribution g"(h) is obtained, the strength may be obtained from the ex pression (43) From the de finition of the redist r ibutor wr(h) of redistribu t ion theory, i t may be seen that in this case it is given by (l/1){g" (h) - g ' (h)} (44) so .t ha t ( 43) i s simpl y an application of equation (33) fo r th i s part i c u la r redistributor. Equat i on (43 ) may a l so be seen as t he d i ff ere nce in potential e nergies given by (3 3 ) o f the di s tri butions g" (h) and g'(h) , divided by the increment factor 1· 92 Our aim now is to calculate the strength of a single thickness of ice with the method used by Rothrock (1975), but assuming that ice deforms to produce ridges as described in the previous section. In the process of this we derive expressions for determining the strength p* of more general distributions. Equation (37) may be used directly to give the strength pi once wr(h) has been calculated. However, equation (40) must be modified since its form depends on the particular method of ridge production assumed by Rothrock. Instead of assuming that ice of thickness h 1 ridges to a single valued thickness h 2 , we assume that the ice involved in the ridging process is uniformly distributed between the values h 1 and h 2 (h 1). If h 2 is the ridge height function introduced previously, (equation 24) then our redistribution assumption is consistent with that of having triangular ridges. An area of ice of one thickness h 1, has a thickness distribution given by (45) Consider a unit area of ice of thickness h 1 totally converted into ridges. If the resulting distribution is g(h) = 0 C 0 then from volume conservation, 00 f ho(h-h 1 )dh 0 We thus obtain Jh2 = eh dh h . 1 (46) (4 7) 93 C = (48) By conserving volume in this way, and noting that the redistributor wr(h) is normalized to -1 in that O'.) .( wr(h)dh = -1 0 (49) it is possible to obtain the redistributor for the distribution given by (45) as h2+h1 o(h-h1 ) h2-h1 h + l [H(h-h 1 )+H(h 2-h)] Chrh1 )2 His the Heaviside step function defined by H(h) -- { 01 h > 0 h < 0 (50) (51) We recapitulate here on the meaning of the redistributor wr(h). For any ice distribution g(h), the redistributor wr[g(h);h] tells us the relative changes that the various categories of ice that make up g(h) undergo when the ice deforms in pure compression. By specifying the type of ridges to be formed in deforming an initially uniform ice sheet, we essentially specify the redistributor for the distribution (45). Equation (50) can be compared with the general expression for the mechanical redistributor wr(h) given by 91-1 -a(h)+z(h) (52) -~'f-a(h)+z(h)}dh so that a(h) (53) 1.s the distribution of 1.ce that 1.s ridged, and z(h) (54) 1.s the distribution of ice formed when a(h) is redistributed. Sine~ wr(h) 1.s normalized, the first and second terms in equation (50) give the actual areas of ice involved in ridging to form the distribution (46). We can immediately obtain the strength Vi, by substituting (50) in (37) to obtain (55) This gives the strength i:i'p of the a-function thickness distribution. For a general distribution g(h), and an assumed distribution of the ice involved in ridging, it is possible to calculate the new ice distribution z(h) and proceed as before to obtain Vi> from equation (37). However, a more direct method is to n9te from equations (45) to (48) that a unit area of 1.ce of thickness h 1 which ridges, undergoes a potential energy change given by 2h1 dh h2-h2 2 1 (56) where the potential energies for the distributions before and after ridging are given by equation (33). The expression (56) simplifies to 95 ( 5 7) Redistribution of ice occurs when g(h) becomes non-normalized by the introduction of areas of various · thickness categories by advection. The function a(h) enables us to deduce from g(h) the distribution of ice thicknesses that are lost when ridging occurs. The actual areas of ice involved in the redistribution process depend upon the amount by which g(h) becomes non-normalized. The function a(h) gives us only the relative area losses from each of the thickness levels. Suppose that it is known by how much each thickness category is reduced through redistribution and that these areas are described by the distribution a'(h). Thus the area of ice lost from the (h,h+dh) thickness level is given by a'(h) dh. Since not all the ice in a distribution would be expected to ridge, the distribution a'(h) is not normalized to 1, and in fact (58) If a'(h) is taken to be the areal distribution that is lost when the distribution (l+t)g(h) is renormalized, then by using Hibler's method for determining ice strength from equation (43) we have P1> (59) To determine the frictional strength, JYl of a general distribution g(h) in terms of a'(h), we first note, as does Rothrock (1975), that the frictional energy loss per unit area los s , Efric• is given by the frictional force per unit ridge width. Hence, from equation (20) E = Cf(h2-h1)2 fric (60) Now, we note that during deformation described by IEI and 8, the rate of area loss of ice of thickness (h 1,h 1+dh 1 ) is (61) so that Rfric becomes, from equation (39), (60), (61) and (38), (62) which immediately gives J>1. For the a-function thickness distribution, a'(h) is, from the first term in equation (50), given by (63) By substituting (63) into (62) and comparing the result with equation (38), the frictional strength is found to be (64) The total strength of an area of pack of a single thickness h 1 1.s simply p* P1> + J>1 l 3 Cbh1(2h2+h1) (65) Figures (4. 7) and (4.8) show the two components of the predicted 1.ce strength for 1.ce of uniform thickness. Figure (4. 9) shows the total strength , 97 I I I a Figure(4·7) Z 60000 -.----------------------, '-" ...-4 a:S •r-i 50000 +> ~ $ 40000 0 P-t 30000 ,Cl +> ~ 20000 Cl) ,,b 10000 U) 0 ,.-._ 0 2 4 . 6 8 Ice Thickness (m) The component of the strength of a single thickness of ice obtained from a consideration of the potential energy losses during ice redistribution. 10 -f 3 35000 -----------------------, ...-4 ~ 30000 0 ~ 25000 C.) •r-i ~ 20000 ... .Cl 15000 +> 0.0 ~ 10000 Cl) ~ en 5000 Figure(4•8) 0-+--,--,--~..--,--,--.---.---,-~-.-~.,--..---...--,,---r---r--r--,--"r--,--,--'r--i 0 2 4 6 8 Ice Thickness (m) As above but only the part of the strength obtained by considering the frictional energy losses during red i str i bution. 10 98 ---~ 60000 z "-' ,....,c 50000 (0 .+-) 0 40000 E-t . .. .c: 30000 .+-) ~ ~ Q) 20000 f...t .+-) Cl) 10000 0 Figure(4·9) 0 2 4 . 6 8 Ice Thickness (m) The strength of a single thickness of ice according to ice redistribution theory when assuming ridges produced by deformation are of a height predicted by force model. 99 10 4.3.2 The effect of open water on ice strength We have employed Rothrock's (1975) method to obtain the strength of ice of a single thickness. If a certain amount of open water is present, then a considerably lower strength would be expected. Take as the initial distribution g(h) ao(h) + (l-a)o(h-h1) (66) This distribution represents a fraction~ of the area as open water, and the rest, ice of thickness h 1• If the distribution is subjected to pure compression then some open water will be lost and some of the ice will ridge. There is a choice in determining the relative amounts of open water loss and ice ridging. This choice can be expressed by a free parameter pin the annihilator a(h) of the distribution (66). Thus a(h) is expressed a(h) = Bo(h) + (l-B)o(h-h1) (67) We consider two methods of evaluating p later. Proceeding _ as before and noting that if the distribution (67) were to ridge totally, a distribution 0 z(h) = 2(1-p)hl h2-h2 2 1 0 would be produced. Since (X) -.({-a(h)+z(h)}dh 0 we c an wr it e t he r edistributor as h < h1 hl < h < h2 h > h2 h2+C2p-l)h1 , h2+h1 (68) (69) 100 h2+h1 h2+h1 (2p-1) [-po(h)-( 1-p)o(h-h1) + 0-p)hl h2-h2 {H(h-h1)+H(h2-h)}] 2 1 Equation (37) then gives the potential energy derived strength as Pp 1 Cb(1-p)(2h2+h1)Ch2-h1)h1 h2-h1+2ph1 (70) (71) The frictional strength JJ1 is found from the redistributor to be (using analysis similar to that used in deriving (64)) Cf(h2-h1) 2h1Ch2+h1)C1-p) h 2+h 1C2p-1) (72) Writing the total strength of the distribution containing a fraction ex of open water as p*(cx), we see from (65), (71) and (72) that p* (ex) = (73) When p is known as a function of ex, (73) demonstrates the effect on the ice strength of the presence of open water. Thorndike et al (1975) suggest that the distribution of ice that becomes ridged may be obtained by weighting the original distribution with the factor 2 G(h) G* max { 1-~, 0} (74) where G(h) is the cumulative thickness distribution defined by 101 G(h) - ~:(h')dh' (75) and c* is a constant usually taken to be 0·15. This means that 15% of the ice undergoes ridging, with the thinner ice ridging in preference to the thicker. For the initial distribution (66), the distribution ridged is thus (a G , then only open water is removed and no ice is ridged, so that a(h) = o(h) implying that p = 1 (ex > c*) With p given by (77) and (79), the strength p*(ex) given by (73) becomes p* (a} and (G* - a.)2 G* 2 + 2a(2G* - a) k - 1 (a. < G*) (78) (79) (80) 102 I I p*(ex) = 0 (ex > c*) (81) where k = h2 (h 1)/h 1• Thus with the Thorndike et al (1975) assumptions, the strength drops from the value p*[ o(h-h 1 )] as the amount of open water increases until it reaches zero when ex= 1-A = c*. Figure (4.10) shows this drop in strength for three different ice thicknesses, with c* taken to be 15%. The picture of a floe field introduced in section (3.1) may be used to give a method of calculating p to give another estimate of the ice strength when open water is present. The factor~ depends on how much ice area is lost during each floe collision. Thus we must extend our ideas of collisions in terms of the circular floe model, in which a collision between two floes is assumed to produce one floe consisting of two touching discs with any ice area lost accounted for independently in the ice redistribution equation. We must modify this assumption in order to calculate p. Suppose that two floes of radius rand thickness h 1 collide and that a triangular ridge of length l(r) and height h2 (h 1 ) is produced. For our original picture of a floe collision l(r) would be zero no matter how large the floes. It is more reasonable to suppose that l(r) depends on the floe size. A linear dependence would be expected on the basis that fields of floes of differing radii appear similar. We thus obtain 1 = C r (82) where c is a constant. For floes touching at a single point c = 0. Only for elongated floes colliding side-on could a ridge larger than 2r be produced so that in normal conditions 0 < C < 2 (83) Until measurements are made relating floe size to ridge length, the value c must be guessed. c = 1 or perhaps a little less would seem reasonable . 103 ..q +> QI) p Q.) H +> rn '"O Q.) N •r--i ~ ro s H 0 z ..q +> QI) p Q.) H +> if] '"O Q) N • r--i r---l ro s H 0 z Figure(4• 10) 1.0 0.8 0.6 0.4 0.2 0.0 0.00 0.05 0.10 0.15 1.0 0.8 0.6 0.4 0.2 0.0 Open water fraction The drop in strength with increased open water amount according to the Thorndike et al (197 5) ice distribution theory . Figure(4·11) 0.00 0 .05 0.1 0 · 0.15 0.20 Open water fraction The drop in strength as the amount_ of open water is increased when floe size is taken into account. The volume of ice i 'n a length 1 of r i dge of the type describe d 1. n section (4 . 2) would be If the area lost 1.s Ar, then by equating volumes Now the rate of collisions per unit a r ea found 1.n section (3·3) 1.s So that the rate of loss of area per unit area 1.s which can be equated with (see equation (67)) to give S (a) 7Tr p w a ( 8) r b cot 8 . c (84) (85) (86) (87) (88) (89 ) The strength of the ice predicted by this method is thus ob t a ined by substituting this expression for µ(ex) into equation (73). Figure (4.11) 10 5 1111 i ri 1 shows the resulting strengths for floes of radius 300 m and for three thicknesses of ice, in which also, a(S) = a (8) is assumed. This assumption r is discussed in the next section. Comparing figure (4 . 10) with figure (4.11), we see a rather sharper drop in ice strength as the open water amount 1ncreases, al though the strength never drops to zero due to collisions occurring, albeit infrequently, even in low concentrations. A maximum strength is predicted until open water exceeds about 3% in this case because the number of collisions predicted for the high 1ce concentrations, becomes too large for a full sized ridge to form for each collision. 4.4 Ice interaction 4.4.1 Introduction In this section, we are concerned with the determination of the 1ce interaction term, E, 1n the momentum equation. This term depends on the velocity gradients that occur during 1ce deformation. Energy lo s ses accompany such deformation so that the 1ce behaves as if it has a viscosity. No energy losses of this kind occur during pure translation, only when there is shear or divergence. Stresses 1n a material can be calculated by considering the energy losses associated with the physical processes that accompany deformation. If the energy loss during some process depends on the amount of deformation b~t not on the rate at which the deformat i on occurs, then the resulting stresses are independent of the magnitude of the strain rate. A material with this property is described as plastic. Ice ridging is a small scale physical process that occurs during large scale deformation of pack ice. Assume, such as in Parmerter and Goon's (1972) ridge building model that the energy required to form a ridge depends on its potential energy and the work done against frictiona l forces during its production. The amount of energy thus needed 1s independent of the rate at which the ridge 1s built, supporting the plastic hypothesis for pack ic e . 106 I I In a two-dimensional ice field, the stress state can be represented by the st re ss tensor with components O-·· which have dime nsions of force per iJ unit length. Stress invariants o-, and 0-11 can be defined in a way similar to the strain rate invariants EI and Err . Thus a, = ~(01 1 + 022) a,. ~~ {(022 - 011) 2 + 4012021} (90) (91) wher e -o-, is the pressure component of the ice stress and 0-11 is a measure of the shear stress . The stress o-ij can be related to the strain rate Eij by a constitutive equation of the form o .. lJ (92) where n and s are the shear and bulk viscosities and p* is a pressure or ice strength term. The stress o-ij gives E according to the rule dO . . F . = ~ (93) l dX. J The stress depends on the strain rate in a more complicated way than at first equation (92) suggests. This is because the viscosity terms n and~ themselves are functions of t .. lJ · The way n and s depend on t .. lJ is determ i ned by "the yield func tion chosen to describe the way the ice behaves plastically. For a two-dimensional plastic medium, a yield function F(o-,,0-11) can be defined such that no strain occur s for values of.:, with F < 0 and that the medium yi elds when F = O. The curve in the (0-,,0-11) plane defined by F(a-) = 0 is known as a yield curve . It is plotted in the ( 0-,,0-11 ) plane which covers all possible stress states in two dimensions. The curve defines all the possible stress states that can exist in the medium when it is deforming. The normal flow rule states that the stress at a point on 1 O 7 p the yield curve occurs when the material is deforming with a strain rate which has a direction normal to the yield curve at that point. The normal flow rule can be deduced from the assumption that non-negative work is done on the material when the point representing its stress state passes along a closed curve (Drucker 1950). Another general property of yield curves implied by this is that they are convex. This assumption implies a further property that the directions of the (t::.,t:: 11 ) axes and the (o-,,o-,,) axes coincide. Because the sign of the shear rate is arbitrary, so the yield curve would be expected to be symmetric about the o-, axis . Having mentioned some of the general properties of yield curves, we go on to determine a particular example from physical considerations, in particular the idea that the ice is composed of finite sized floes that interact by colliding. The viscosities that this yield curve implies are then calculated so that they may be used in equation (92) to give o-ij from which E may be found from (93). 4.4. 2 A yield curve In this section we aim to derive the viscosities that could be used in a viscous plastic sea ice model, based on the notion that the ice field is composed of finite floes. The collision rate given by (86) is a product of kinematic variables ltl and a(e) and variables associated with the physical structure of the floe field, A, n and r. We assume that ice ridges are produced as a result of collisions and that the rate of production of ridges, and hence the rate of loss of ener_gy, is proportional to itla(e), the kinematic term in (86). Thus a,t, + 0 11 £ 11 a: jtja(8) (94) The ridging coefficient a. (8) used by Rothrock (1975) in (2.65) is r normalized so that a. (n) = 1. The quantity a(e) is likewise normalized so r by comparing (94) with (2.65), we can write 108 a,£, + a,,£,, (95) Equating ·a(6) and a (8) is the simplest possible assumption we can make r about these functions. There are many points that should be included when making a fuller study of ice inter.action. Firstly, the model used in this calculation does not incorporate an ice thickness distribution except that the amount of open water is specified. Thus we cannot properly include ridging. The occurrence of collisions gives information regarding the initiation of a ridge but not its subsequent development. In particular, in this model, less ridging occurs as a result of glancing collisions because of the smaller area in which a floe has to be for it to collide, but no account is taken of smaller ridges being produced as a result, requiring less energy. The derivation of a(6) is based on a model of the pack that is most applicable in the case where there is enough open water present to significantly reduce the ice strength, i.e., its resistance to pure compression. Thus in fact we are dealing with the case in which there is very little ice ridge formation. The importance of a(6) is that it gives a indication of relative amounts of ridging for the various types of flow, even though the amount of ridging is small. It is a(e) that will be used to give the shape of a plastic yield curve whereas its size, which depends on the ice strength, is not determined by this model. The function a (8) in the case where there is no open water and the r ice strength is high, cannot be deduced from a collision model. Despite all these difficulties, further examination of the derivation of a( 6) in section (3.2) reveals an unexpected bonus. Suppose that a continuous ice cover is deforming with uniform spatial gradients. Suppose also that the ice cannot support tension so that open water is produced by divergence with no loss of energy. Then the loss of area of ice due to ridging from a circular region 2r in diameter is given by (3.22). Thus the rate of loss of area due to ridging per is 1Ela(6) . Hence a (8) = a(e). Thus we r unit area (which defines /i::/a (e)) r see that al though a( 6) refers to the ridging amount for weak ice in which there is a significant amount of open water, it may also refer to that for a continuous ice cover. The intermediate cases, such as when the ice cover consists of highly compacted floes , require further study. 109 An attempt to find a function a (8) from satellite observations has r be e n made (Pritchard and Coon 1981) and the following function has bee n suggested as a fairly good representation of the situation. ,a ( e) = r 0 1 - 38/2n - cose -cos e o < e < n/ 3 n/3 < e i 2 n/ 3 2n/3 < 8 i TI (96) The comparison between a (8) given by (96) and a(S) as evaluated in r (3 .23) is shown in figure (4 . 12). The functions differ slightly and in particular, equation (96) pr edicts no ice interaction for strain r ates with e in the range (n/4) < 8 < (n/3) whereas the derived function a(S) does indicate some interaction in this range. By considering general properties of plastic materials, Drucker (1950) derived the normal flow rule, which specifies that for a stress on the yield curve, e is equal to the angle between the normal to the yield curve at that point and the er, axis, and can be written in the form l, A 3F I ao , F =O • A ~r dO II F=O I (97) The normal flow rule states nothing abou t the magnitude of E, only its direction with respect to the t, and E11 axes. Hence ). is just a constant that remains undetermined. Th e e ne r gy equa ti on (95) can be writ t en (Rothrock 1975) = -cotea-, + s i ne p*a ( e) (98) wh e r e a(S ) i s give n by (3.23) and the norma l flow rule is expressed in the f orm 1 10 a(9) 0 0 Figure(4·12) a(8) suggested by observations Calculated a(8) I I I I I I .!.rr 4 I I I I I ln 2 I 9 I I I I I I I I I I I I I 1n 4 TT The function a(8) giving the relative amounts of area loss through ridging as a function of the deformation type. 111 ' d 0-11 do-, = -cote where 0-11 has been regarded as a function of a-,. (99) Equation (98) defines a family of straight lines in the (0-1,0"""11) plane with the parameter e. The envelope of this family of lines satisfies (99) and is thus the yield curve. For O < 0 < ± lT the lines all pass through the 3 ( * ) 1 origin and for 4 lT < e < lT they all pass through the point -p , 0 • For 4 TI< e < Z lT, e can be eliminated from (99) and (98) to give o,. do,. 0 _ p* )cos -1(-do,.)do,, + do I I 1T i . do I do I (100) Since the yield curve is the envelope of the straight line solutions of (98) we require the singular solution of the differential equation (100) which is 0 II = (101) The full yield curve is symmetric about the o-, axis and is thus as shown in figure (4.13). The pointed ends of the yield curve indicate that a range of values of e rather than a single value give rise to a particular stress . To calculate the bulk and shear viscosities that this yield curve implies we first note that the express ion for the stress tensor in equation (92) gives o, z:e::, - 1:!p* (102) and 112 113 Figure (4 ·13) 0 11 - P* The derived sine wave lens y ield curve. Figure(4 •14) !P* - ...... 0 11 - ...... - ' - ' -- -- ...... ...... 9 0 ...... 1n !n ~n n 4, 4 ' '\ \ \ \ - !P* \ o, \ \ \ \ '\ ' ' ' -p* -- - - The shear stress and negative pressure as a function of the type of deformation as implied by the sine wave lens yield curve. The dashed line shows the same for an elliptical yield curve of eccentricity 2. a,, T]E II (103) For± 1T < e < Z lT, we use the flow rule in the form of equation (97) to obtain (104) and l 11 = >.n (105) Thus a, /_p*) -l(E ) -\n cos "i.: (106) The minus sign occurs so that the inverse cosine function can take its principal value. From (101) and (106) we obtain p* \fr.~ - €:.~ (107) all ( E II >Q) 7f E II Comparing (106) and (107) with (102) and (103) we obtain the required viscosities , valid and I'; \)t~ ·2 p* - £.11 Tl = ·2 1T E11 r; p* p* . -l(r. ) = 2€:. , - ---.-- cos .:;-i.. TIE I E II 1 3 1 for £,, > O and 4 1T < e < 4 1T. For e < 4 1T the = p * / 2 €:.,and for Z 1T < e < 1T we haven= O and th e vi s cosities at the end points of the yield c urve. (108) (109 ) viscos i t i es are Tl = 0 r;=-p* / 2 €:. ,, which a re 114. r Figure (4.14) shows the stresses o-, and 0-11 as functions of 8. The dashed curves are the results obtained from Hibler's (1979) elliptical yield curve with e = 2. The similarities in the stress curves between those derived from the sine wave yield curve and Hibler's elliptical yield curve, reflect the similarities in the shapes of the yield curves themselves. Both satisfy the deductions from Drucker's (1950) hypothesis regarding plastic materials, in that they are closed convex curves. In addition, they both pass through the origin and the point (-p*, 0). For the sine wave yield curve zero stress occurs for 8 < ± 1T for which £, > E,,. In this state no collisions occur and leads of all orientations within the pack ice open up. One conclusion that can be drawn from the shape of the sine wave lens yield curve is that as long as the convergence is larger than the shear, then the actual value of the shear has no effect on the ice rheology, and similarly, if the divergence is larger than the shear, then the pack ice is essentially drifting freely and the ice interaction term is zero. If we assume that energy is lost through ridging in proportion to the area loss in the regions where neighbouring points approach, and that where neighbouring points move apart, there is no energy loss, then a sine wave lens yield curve is implied. Thus, compared to other yield curves, the sine wave lens yield curve makes the least number of assumptions about the nature of the physical processes involved in ice deformation, so that it is a canonical yield curve. If, for instance, account is taken of the differing types of ridges produced in pure compression and in shear, then a modified yiel~ curve would be expected. 11 5 5. INCORPORATING THEORY INTO NUMERICAL CODE 5.1 Introduction The theory that has been developed in the previous chapters must be modified before it can be included into a numerical model. There are a number of reasons for this, of which perhaps the most important is the necessity to construct discrete versions of continuous functions, such as the ice thickness distribution. Integrals become finite sums, necessarily incurring certain amounts of inaccuracies. During the development of the code, techniques can be utilized that minimize such errors. In this chapter we also mention some of the factors that have to be included in the code that are not connected with any real physics but are necessary only to avoid non-physical results. During a redistribution of ice thicknesses various amounts of ice are lost from each category. The amounts lost determine the ·amounts of ice added to higher categories as well as the potential energy changes and frictional energy losses. Thus by obtaining coefficients giving the amounts of ice and energy produced by destroying unit areas of each category, the necessary changes to each of these quantities may be evaluated for arbitrary deformations. The first part of this chapter deals with the evaluation of these coefficients. 5. 2 Ice thickness distribution 116 5 . 2.1 Representation of the ice thickness d i stribu t i o n Solv ing the thickness distribution equation analytically is not possible because of its highly complex nature. A numerical solu t ion within the contex t of a climate model demands the c onstruction of a finite set of thickness levels . Hibler (1980a) uses 10 thi c kn ess l evels for his variable t hickness Arctic ice model in which the ca t egory widths are small for thin ice and i ncrease for the th i cke r ice levels . This is because the growth rate function f (h) dh dt (1) varies mostly for small values of h. Rothrock (1983) points out some other difficulties associated with the numerical integration of the thickness distribution equation (2.39). In particular, the formation of thin ice from the freezing of open water must be handled carefully. Some of these problems may be overcome by constructing more thickness levels so that the vert i cal grid is finer. In some. cases however, this tends to smear out some of the structure of the distribution as the integration proceeds. Also, in climate models where the sea ice is specified on a two-dimensional grid, the addition of a large number of levels in the third dimension increases considerably the computational burden, particularly with respect to the amount of store needed. Better results can be obtained by letting the thickness levels float so that they follow the characteristics of equation ( 1 ) . This allows the correct thermodynamic changes to be made to the thickness levels . However, in this case, grid squares in the horizontal direction will -in general have thic kne ss distribution levels t ha t do no t mesh together so that difficulties are int r oduced i n modelling horizon t al redistribution processes such as advection. With all these shor t comings we have to compromise with the choice of the form of the vert i cal grid used . Th e mo de l used here i s general with respect to the number of levels , as l ong a s there a re three or mo r e . There is no l imi t to the maximum numb e r of l eve l s except for the constraints introduced by stora ge requirements. It i s possible to have useful results with a s few as four leve ls. 117 For a grid consisting of four levels, the thickness distribution is expressed by the four quantities (2) The first of these, G1, is taken to be the fraction of open water present. The quantities G2, c3, and G4 are the fractional areas of ice within thickness categories defined below, whereas G1 is the area of a single thickness. For this reason G1 is by nature different from G2, G3, and G4 and so is treated differently. We consider the thickness levels to be specified by their limits, so that there are four thickness values (3) and that Hi-Hi-l is the width of the i'th thickness level. Since the lowest level represents open water, H1 = 0. The values in (3) are considered fixed except for the top level H4 , which is allowed to vary. H4 is thus taken to be the value of the maximum thickness of ice present in the region that g(h) describes. The variable top level is useful in that a knowledge of G4 and H4 together, gives an indication of the character and amount of ridges present in the distribution g(h). A disadvantage of a totally fixed grid is that any ice thicker than the maximum thickness level will have to be redistributed within the lower thickness levels. To reduce the amount of error introduced in this way, the top level will have to be made large compared with the maximum ice thickness expected. In that case, unless a large number of thickness levels are added to represent the thicker ice, there ~ill be little information given regarding g(h) fo r large h . By choosing approp r iate values for Hi , the thickness levels can be chosen to represent physically different categories of ice. For example , if the following values are chosen J18 Hl = 0 H2 0 • Sm H3 = 2 ·Orn H4 = > 2 ·Orn (4) then G1 would represent the fraction of open water as always, G2 would represent young ice, G3 would be the fraction of older first-year ice and finally, G4 would give the fraction of multiyear or ridged ice. Of course these can only be approximate categorizations but they should be enough to give a reasonably full description of the ice field. A problem remaining is to adequately represent the formation of thin ice, G2, by the quick freezing of open water G1• Suppose that during one time step, which can be as little . as a few hours, all the open water freezes over producing a layer of ice a few centimetres thick. This means that G1 becomes reduced to zero, and that the area of thin ice increases by G1• However, from (4), the thin ice has thickness O·Sm, so that regardless of how thin the newly formed ice really is, the model takes it to be on average 0•25m thick. In other words, too much ice volume would be created This can be partially overcome if some modification is made to our physical picture of the freezing process. We suppose that newly formed thin ice, rather than remaining as a coherent sheet of nilas, is blown by the wind and buffeted by the waves until it piles up against other more substantial pieces of ice. We thus postulate a minimum thickness that ice in the form of a solid structure can be. A fifth thickness level can then be introduced at this value. Then when new ice forms at a certain rate, we calculate the increase to the lowest ice level by forcing volume conservation and replacing the requisite amount of open water. This concept is consistent with the formation of 'grease' ice which does not have a solid consistency until a reasonable thickness is built up, often as a result of piling up against solid floes (Bauer and Martin 1983). If there are five or more vertical thickness levels, then included in the code is a routine that calculates the values H1 , H2 , ... , Hn in such a way that the level widths increase smoothly as h increases. Hibler (1980a) used such a grid in which the level widths Hi-Hi-l increases according to a Gaussian formula. If there are only four thickness levels , then choosing 119 (m-1) Open Figure(5·1) 4 Top level varies water 3 t 2 1 5 I 6 H1 H2 Ice thickness (m) Schematic representation of the vertical thickness distribution grid used in the code. Figure(5•2) The construction of a distribution of ridged ice from an arbitrary initial distribution by summing the resulting distribution from elemental strips each consisting of the ice distribution within a set of triangular ridges. 120 a value for the thinnest ice level and the highest fixed ice l evel, completely determines t .he vertical grid. Figure (5.1) shows the spacings for a 6 level grid. 5.2.2 Ice redistribution At each grid point, the information about the ice distribution is given in the form of four or five numbers representing the thickness distribution, and similarly for the floe number density. At each time step during the model run it is necessary to update each of these quantities. We concentrate now on the thickness distribution. Although representing a continuous distribution with just a few values is a fairly crude approximation, we would like to retain as much accuracy as possible when calculating the various redistributions at each time step. The method employed is to consider the actual continuous thickness distribution to be constructed by assuming that the ice is uniformly distributed within the thickness levels. This means that the distribution would resemble a histogram. It is then possible, analytically or numerically, to any desired accuracy to determine the resulting continuous thickness distribution. In order to proceed with the next time step, the new thickness distribution is interpolated back to the finite grid for subsequent use by the model. In order to keep to a minimum the calculations performed at each time step, as much as possible concerning the redistribution is calculated at the start of the model run. We describe here the way the mechanical redistribution is performed for a finite grid and is essentially the same as that used by Hibler's (1980a) multi-layer model. For an N-layer model, the areas of ice in each category are denoted by (5) where G1 is the area of open water. Suppose t hat during the ~ourse of the time step , areas have been added and subtracted until the distribut i on i s no longer normalized to 1, so that 121 a value for the thinnest ice level and the highest fixed ice level, completely determines the vertical grid, Figure (5.1) shows the spacings for a 6 level grid, 5.2.2 Ice redistribution At each grid point, .the information about the ice distribution is given in the form of four or five numbers representing the thickness distribution, and similarly for the floe number density. At each time step during the model run it is necessary to update each of these quantities. We concentrate now on the thickness distribution. Although representing a continuous distribution with just a few values is a fairly crude approximation, we would like to retain as much accuracy as possible when calculating the various redistributions at each time step. The method employed is to consider the actual continuous thickness distribution to be constructed by assuming that the ice is uniformly distributed within the thickness levels. This means that the distribution would resemble a histogram. It is then possible, analytically or numerically, to any desired accuracy to determine the resulting continuous thickness distribution. In order to proceed with the next time step, the new thickness distribution is interpolated back to the finite grid for subsequent use _by the model. In order to keep to a minimum the calculations performed at each time step, as much as possible concerning the redistribution is calculated at the start of the model run. We describe here the way the mechanical redistribution is performed for a finite grid and is essentially the same as that used by Hibler's (1980a) multi-layer model. For an N-layer model, the areas of ic_e in each category are denoted by (5) where G1 is the area of open water. Suppose that during . the course of the time step, areas have been added and subtracted until the distribution is no longer normalized to 1, so that 121 (6) If ( 7) then the area of open water is increased until the distribution is normalized. Thus G" 1 = G' + ( 1 - 2G ! ) 1 i n = 1 - ~ c! 2 i and the other areas remain unaltered thus G'.' = G! i i i = 2, ••• ,N If however (8) (9) (10) then mechanical redistribution takes place. An NxN matrix Y is constructed such that Yij is the increase in area in level j caused by the loss through redistribution of a unit area from level i. A set of category reductions G~ i are calculated from the values Gi by applying the annihilator function a(h) of redistribution theory. These category reductions Gi give only the relative losses from each level, i, so to determine the actual reductions, they are multiplied by a constant such that the resulting distribution is normalized. Writing the relative loss from level i as Ai for which the net gain to level J. is Y· ·"- we write the net loss to level J0 as lJ l' G* = ). . j J I y .. ). . i l] 1 (11) 122 Thus G~ given by J l.S G'.' J c! - J because then the reiulting distribution G~ 1.s normalized to 1, J L c'.' = 1 J (12) (13) It is quick computationally to calculate G'.' at each time step from (12) J once the values Yij are known. These values can be calculated at the start of the run and remain constant. We now describe how the values of Y may be calculated analytically. From the definition of Y, the coefficients Yij may be obtained by considering a unit area of 1.ce uniformly distributed 1.n the range (Hi-I• H1 ), and allowing all the 1.ce to become ridged. Al though, from a real distribution, only a fraction of the 1.ce will ridge, we consider a hypothetical situation in which all the ice ridges. This 1.s because the coefficients annihilated . y .. lJ refer to areas of 1.ce The distribution of the 1.ce annihilated 1.s 1 a I (h) = formed per ·unit Hi-1 < h < o t herwise H· l. area 1.ce (14) The 1.ce described by a'(h) is ridged to produc~ a new distribution n' (h) . The coe fficients of Y are then given by y .. lJ = (15) 123 5.2.3 Analytic form of redistribution matrix. We firstly describe a method of obtaining an analytic expression for n'(h) 1n terms of a general distribution a'(h) assuming that ice of thickness h 1 forms triangular ridges of height h 2 (h 1). Although this 1s interesting in its own right, even for a simple distribution a'(h) as 1n (14), the function n'(h) obtained is too complicated for the integral 1n (15) to be evaluated to give Yij• Thus, we also describe a numerical method of calculating Yij that is completely general with respect to the manner in which ice redistribution is assumed to take place. The concept of considering an entire area of ice of thickness h 1 forming ridges as described in section (4•2) was introduced in section (4•3) to derive the strength of a general thickness distribution. This was done by straightforwardly integrating the potential energy changes that occurred during the loss of ice from each thickness element (h,h+dh). This idea can be extended to calculate the actual distribution a'(h) of ridges. The difficulty here arises because the ice from the element (h1 ,h1+dh 1 ) redistributes through a range of thicknesses (h 1 ,h2 ). The new distribution can be obtained by summing up the elemental strips produced by the redistribution from each thickness element (h,h+dh). This is shown in. figure (5.2). From the element (h 1 ,h1+dh 1) an area a'(h 1 )dh 1 1s redistributed uniformly between h 1 and h 2• Volume conservation gives the value of the new distribution between h 1 and h 2 as (16) We imagine that the elemental strips are stacked on top of one another starting with the strips near h = 0. The left and right hand ends of the stacked strips trace out two curves ( in the limit as the elemen t s dh 1 ~ 0) A(h) and B(h) . Constructed in this way curve A(h) would be give n b y 121± A(h) (17) since the height of the stack at his just the sum of the heights of each of the elements whose left hand end starts before h. The curve B(h) would then be given by B(h) (18) where h21(h) is the function inverse to h2 in that if h2 = h2 (h 1 ) then h 1 = h21(h2 ). The resulting distribution would then be given by n'(h) = max{A(h),B(h)}-min{A(h),B(h)} (19) Taking a ' (h) to be the distribution in (14), A(h) may be evaluated analytically for certain simple functions h2 (h1). By substituting the ridge height formula (4.24) into (16), with a'(h) given by (14) the function A(h) can be evaluated. The result f dx ../x.Jb - x + ax (20) is needed for -this, where a and b are constants. B(h) i s then calculated from (18) using the inverse function h1 = h21(h) given by (21) The resulting expression is too cumbersome to deal with f urther so that it is at this point that the limit of the analytic approach i s reached. Since the evaluation of the integral obtained by substituting (21) into the 125 I I expression obtained for A(h) is not possible, we can only evaluat e (15) numerically. If however a numerical method is needed a more dir ect me thod applicable to arbi t rary functions h2 (h 1 ) may be used and is described below. 5.2.4 Numer ical evaluation of redistribution coefficients The coefficients Yij are evaluated numerically by dividing (Hi-l•Hi) into a large number of elements. The ice originally contained within the element (h, h+dh) is uniformly redistributed in the range h+~oh to The area of ice then created within the range (Hj-l•Hj) is noted, and such areas summed for each thickness element in the range (Hi-l•Hi) . The total area obtained is then Yij• This method gives the coefficients Yij for any function h2 (h). Also, the method may easily be extended to include modes of redistribution other than by the formation of triangular ridges. 5.2.5 A finer grid A particular kind of error is likely to occur when evaluating the changes to the thickness distribution through ridging, especially when there are few thickness levels . This occurs when H1 -H1 _ 1 is large enough for there to be a significant difference between h2 (H1 _ 1) and h2 (H1 ) . Suppose that during an ice redistribution process, that ice up to a maximum h* is involved in ridging. If H1 _ 1 < h* < H1 then only the ice in t he re gion H1~1 to h* should ridge. However , because the thickness levels a r e ridged as a whole , so the whole of the level (H1 _ 1 , H1 ) will in th i s case be ridged. Thus , although the correct los~ of ice from the L'th level is evaluated, incorrect information regarding the increase in ice t o the upper l evels i s calculated. For th is r ea s on, be fo r e ri dg ing t ake s p lace, the thi cknes s grid is interpo l ated t o~ fine th ickness gr id in which each o f the coarse grid thickness levels i s uniformly div i ded into a number of finer levels . The change s to the fine grid thickness levels are ca lcu l ated due to ridg i ng 126 and the new thickness distribution interpolated back to the coarse grid again. The number of fine thickness levels within each coarse thickness level is a matter of choice. The larger the number chosen, the greater is the amount of computer time required. Later 1n this chapter, we show results of some experiments in which the number of thickness levels within the coarse and the fine grids are changed, giving an indication of how f ew levels may be included without incurring significant errors. 5 . 3 Strength 1n the code In section (4•3) the theory concerning the determination of the strength of the ice, p* as a functional of the ice thickness distribution g(h) was dealt with. We saw that for an arbitrary distribution g(h) the strength may be found by renormalizing, by ridging, the distribution (l+~)g(h) where~ is small. If a'(h) represents the ice destroyed through ridging then formulae (4.59) and (4.62) may be used to determine p* by p* = (22) This procedure may be adapted to the determination of the strength of an. ice thickness distribution described as in the code by the areal fractions G1 , G2 , ... , GN' The method is to increase each of the ~ by the increment ~G1 as for the continuous case and allow the distribution to ridge. If the actual fractional areas of ice from each thickness category are recorded d~ring the ridging process, and if they are denoted A1 , then the strength may be determined from p* = ClN) Ir A L L L (23) where r1 are constant strength coefficients that are calculated at the start of the run. They are given by 127 C (2h2 (h) +h)(h 2 (h)-h)h]dh b h2(h)+h (24) and are evaluated numerically. The coefficient rN does have to be calculated at each evaluation of the strength because of the variable value of RN" Equation (24) still applies though once RN is known. Because the strength determination relies on ridging being carried out, which as we have seen may be inaccurate when there are only a few thickness levels, the coefficients r L are calculated for the fine grid used when evaluating the changes to the thickness distribution from ridging. To investigate the kind of errors introduced by taking few thickness levels, a number of runs of the model were performed on a small (3 x 3 velocity field) test grid in which the number of refinements to each layer for the ridging procedure, NFINE were varied. The strength of the ice in one of the grid cells after 40 half-day time steps was evaluated in each run. All the grid cells initially contained a 100% cover within the thinnest ice level (G2 = 1). This is so that the same initial conditions could be applied for each of the runs, the thinnest ice level always having the same width, in this case 0•25m. Ice growth conditions were applied . The October growth rates for the Arctic as calculated by Thorndike et al (1975) were used. This test would thus pick up errors introduced during the ridging process, the thermodynamic growing of the ice, and the strength evaluation, by changing the number of thickness levels used. Figure (5.3) shows the variation of the model results for NL from 5 to 12, and for NFINE = 1,2,3,5 and 8. The curves show that whe n there are more than 6 levels reasonable accuracy is a chieved when NFINE is greater than one. For 5 or 6 levels accurate results may be obtained by taking NFINE to be 3 or more. For the standard simulations described in the next chapter, 6 levels were used (NL = 6) with NFINE, the number of refinements to each level during ridging, equal to four . In this case, we are essentially using an 18 level model to evaluate ice strength and to perform ridging. J 28 1.0 0.0 Figure(5·3) NF=1 NF=2 5 6 7 ·8 9 10 11 Number of levels, NL The value of the strength as measured in one grid cell after 40 time steps of growth. Values have been plotted for a range of thickness levels NL and for various numbers of refinements to the vertical grid used when evaluating strength or performing ridging. 129 12 5.4 Floe number densities in the code 5.4.l Floe size distribution The average floe number density in a region, or a grid square, is denoted n. A real floe field consists of many different floe sizes, usually from the very small up to some maximum cut-off size. (Goodman et al 1980). Thus, there may be some difficulties introduced by trying to model the floe field by considering only the average floe number density, n. In section (3·3) we saw that many of the factors that determined the size of newly formed floes, depended upon the ice thickness h. Wind-induced fracture of floes would give small floes for thinner ice and conversely, large floes for thicker ice (if the thick ice can be made to break at all). Having divided up the thickness into a number of levels, or thickness categories, in order to increase the amount of information provided by the representation of the ice thickness, it would seem that an analogous procedure could be carried out for floe sizes. One method would be to give the fractional area covered by each floe size category. Another would be to give the number of floes per unit area of each floe size. There would be a difficulty involved here in delimiting the range of floe sizes to be used for each size range. In addition there would be a los~ of information involved in taking a finite number of possible floe sizes, particularly because. of the large number of pass ible floe sizes that exist and would have to be coped with in a model. Instead of choosing to represent the range of floe sizes present at a single place by one of the two methods mentioned so far, we take advantage of the fact, already pointed out, that the floe sizes depend very much upon the ice thicknesses involved and consider floes of various thicknesses rather than of various lateral extents. Thus we specify the average floe number density n1 for each thickness category. This preserves more information regarding the actual floe sizes making up the pack ice. The disadvantage of this method is that it implies that each floe is of more or less uniform thickness, and that the thicker ice, which generally consists of ridged ice in contact with thinner ice, is considered as separate thick floes. It would not be easy to overcome this difficulty because, given a thickness distribution g(h) we 130 .:. I I I I I I I I I I ..._ cannot a priori decide whether or not the upper levels (h large) consist of ridged ice or genuinely thick floes of uniform ice thickness. Formally, we consider floe number densities n2 ,n3 , ... ,nN such that n1 1.s the number of floes per unit area with thickness lying 1.n the range N I: nL = n (25) 2 An average floe size r 1 for each thickness level may be defined by the relation = (26) for which r 1 would be the floe radius for circular floes. Note that n1 1.s not defined. In the code only the values of n1 and Gr, are held in store and a conversion to r 1 is made when it is more convenient to deal with floe sizes (such as for floe breaking) and then convert back to n1 . Processes such as advection are best handled by varying n1 directly. · 5.4.2 Floe size change Equation (3.58) 1.s split into two stages in the code. Firstly, the advection is performed according to the equation 3n ____k + V • (un ) = 0 3t - L (27) for each of the thickness levels L = 2, ... ,N. This is done 1.n the same way as the advection of the thickness level areas g(h). The change 1.n floe number density due to the coalescing of floes within a large scale straining velocity field was obtained in chapter 3 and is given by 13 1 I I II 11 I I I I I I I I dn dt -2LAj£ja(e)n This equation may be solved simply to give n(t) = n(O)exp{-2LAj£1a(8)t} (28) (29) where n(O) is the floe number density at time t = 0, and it has been implicitly assumed that (30) is constant in time. This is clearly not true for long periods of time where any of the factors L, A, 1£1 or a(e) in (30) may vary. However, in practice (29) is used to determine n(t+lit) in terms of n(t), where lit is the length of the time step, and because of the nature of the staggered time marching used in this model, the values of L, A, J £ I and a( 8) are those evaluated at time t+;lit, giving second order accuracy in time. Thus, when the floes are not being broken up by the action of the wind or the waves, we assume an exponential decay in their number within each time step. Whether or not the number of floes in a region is able to decrease, with a corresponding increase in floe size, according to this mechanism, depends upon whether the wind is strong enough to break the floes up again. For this reason we allow the floe number to decrease according to (29) before subjecting the floes to the effects of the wind and waves. When A= l-G1 is close to unity, the function L(A) may become very large , thus giving rise to a very -sudden drop inn. This represents the situation that occurs when the pack ice stops behaving as floes and becomes continuous. Thus, in a floe model we must include the possibility of a continuous pack being formed . The transfer from a floe model to a continuous model occur s when equation (29) predicts a sudden drop inn. This idea is discussed in more detail below. 132 I ii I I :: I I' I ! - r 5.4.3 Continuous ice 10 a floe model Consider a floe field described by an array of grid squares, within each of which is specified the relative areas of open water and the various ice thicknesses together with the average floe sizes for each thickness category. There needs to be special care taken if the implied floe sizes become of the same order of magnitude or larger than the grid size. For, in this case we are dealing with a continuous ice cover, not a floe field. If the predicted floe size, obtained from the relation G1 = nn1rl, is larger than some specified value, related to the grid size, then the code abandons the floe modelling for that particular grid square and thickness level Land sets G 1T ( L'ix) 2 (31) where L'ix is the grid length. Thus where we do not expect to find floes the model does not try to predict sizes. Instead of using the grid length L'ix here, any large value may be chosen. The value only serves to indicate the floe size beyond which they are no longer considered as floes. A newly formed ice cover is considered to be continuous in that its initial value for the floe number density is given by (31). Conversely, if all the floes of some thickness range melt then the floe number density, n1 , is set to zero. We have considered one circumstance in which a floe field would have to be thought of a_s continuous . That is when the predicted floe size becomes large. A floe field would also have to be thought of as continuous if its compactness became unity. In this model, this would occur in a region experiencing a converging ice velocity field. If the fraction of open water drops below 10-12 then the ice cover is taken to be continuous and the floe number density dropped so that the floe radius is equal to · the grid length. The value 10-12 is arbitrary - any low value will do . Even a value of 10-9 would represent only 10 square metres of open water in a 100 km grid square. 133 I The following figure shows schematically the range of floe siz es best dealt with in a model with grid size of the order 10km to 100km. 0 .1 m lm 10m 100m 1 km 10km 100 km ? <-----------Floes-----------> <-----Continuous---- Although the floe model is able to produce floes of less than about lm, such values should be regarded as meaning only that the ice consists of very broken up fragments. 5.5 Thermodynamics in the code 5.5.1 Vertical changes We now consider ·the numerical treatment of the thermodynamic term in the thickness distribution equation. The thermodynamic part of the thickness distribution equation may be written clg cl (fg) ~+ah = 0 (32) where f(h) is the growth rate (dh/dt) of ice of thickness h. The second term in (32) behaves as an advection term for g with f analogous to velocity and h representing distance. Thus we expect many of the problems associated with the numerical representation of advection or conservation laws in general. We will discuss the way in which growth and melt are handled with respect to the variable thickness grid described above, and in particular, mention ways in which some of the inherent problems associated with numerical representation of (32) are dealt with to suit the particular problems we are concerned with. If we consider a typical thickness level , number L, then growth and rnel t may be achieved very simply . If the growth rate f(h) for that level is positive, so that we have growth, then the area of ice 1.n that level G1 will decrease and be added to that of the L+ l 'th level. Similarly, if melting occurs then f(h) < 0 and a decrease in G1 is accompanied by an increase in the L- l'th level. The special cases where the L'th level is the top or bottom level are treated separately. So far, we have talked about the growth or melt of level L without reference to which value of h we are considering when calculating f(h). For level L which extends from H1 _ 1to H1 , there may be a range of values of f(h), especially if the thickness level is large. In particular, f(h) may even change sign within the range (H1 _ 1 ,H1 ). So, do we take f(H1 _ 1), f(H1 ) or perhaps f{!(H1 _ 1 +H1 )} as the representative growth rate of level L? 1 Hibler (1980a) chose f{ 2(H1 _ 1 +H1 )} as the representative value. This would be reasonable when a large number of thickness levels are used (in this context, about 10 levels, as used by Hibler, would be enough). If fewer levels are desired, then problems will arise, problems best illustrated by an example. Suppose we have four levels in all, the first representing open water. Suppose further that ice 1.n lev~l 2 is growing (f > 0) and that ice in level 3 is melting. Clearly we would expect an equilibrium thickness lying somewhere within levels 2 or 3. If the equilibrium thickness lay in level 2 then it would be reasonable that during a time integration of the thermodynamic equation (32), all the ice from level 3 woul-d be transferred to level 2. However, this would not occur within the regime so far mentioned. Instead, a balance between the rate at which the ice was being transferred from level 2 to level 3 and that in the opposite direction would be set up. It wastes levels to represent a particular thickness of ice by their relative values, especially when two levels may span the entire thickness range from zero to the maximum fixed value. This situation 1.s overcome by concentrating not on the growth or melt of particular levels but rather, on the growth or melt at the thickness values between the levels, i.e. at the values H1 • Thus the value of f 1.s evaluated at H1 and the growth performed by reducing the area of 1.ce 1.n level L by the appropriate amount and increasing that of level L+l by the same. Melt would occur with a reduction from level L+l and an increase to level L. 135 As an example, suppose we have growth at H1. So f(H1 ) > 0, then in time bt,, the value of G1 would drop by the amount G1 f(H1 )Lit HL - HL-1 (33) and the value GL+l would increase by the same amount. If f(H1 ) < 0, then G1 would increase by -GL+lf(HL)Lit (34) hL+l - h1 and GL+l would decrease by this amount. The cases of the thermodynamic changes occurring to the upper and lower levels need now to be considered. Melting of the thinnest ice layer is straightforward except that open water is formed rather than a thinner layer of ice. The increase in open water would be given by (34) with L = 1. In addition to melting of the thinnest ice, the value f(H 1) = f(O) determines the amount of heat absorbed by open water used to melt the ice from below and laterally. This is described in more detail in the section on lateral melting. The situation in which new ice is formed from open water has been discuss~d previously in connection with the choice of the type of thickness grid to be used. The increase in the ice of level 2 due to freezing of open water in time lit is given by (35) The situation that occurs for the top two thickness levels is complica t ed by t he fact that the top thickness level is variable and so must vary in response to growth and melt . If there are N thickness levels , then HN-l is the value of the maximum fixed thi ckness lev el, and HN will vary. If f( HN_ 1) > 0 then GN- l will decrease and GN will increase . Also, fN 136 may need to be increased if f(HN-l) is sufficiently large. This is the one case in which f(h)~t may exceed the level width. This is because we allow the possibility of the top level reducing in width to zero if no ice exists there. If GN-l > 0, then HN is readjusted to the value (36) If f(HN) > 0, then no change occurs to GN but providing ~ > 0, then hN is increased by f(HN)~t. In the case of melting of the top levels (f(HN_ 1) < 0, f(HN) < 0), we must allow for the possibility of the entire top level melting and the top level HN dropping down to the value HN-l· Thus we have considered all the possibilities of growth and melt of all the layers, including the possible total melt of the top variable layer or its increase from the growth of the N- l 'th layer. 5.5.2 Lateral melting The growth rate as calculated by the heat budget is given by the function fb(h). The growth rate for ice of zero thickness is thus fb(O). If fb(O) > 0 then ice is grown at that rate. If however, fb(O) < 0 then no ice can be melted. Instead, it is assumed that heat is absorbed through the leads (areas of open water) into the upper mixed layer of the ocean. This heat is then able to melt ice from the bottom of the floes as well as from their vertical sides. If all the ice melts or there is none to begin with, then the heat absorbed but not used to melt ice is recorded as a mixed layer temperature increase. In the case where ice is melted, the volume lost from the floe edges depends upon the perimeter of the floes per unit area, which depends on the floe sizes and number densities. The total floe perimeter per unit area may be written 137 s (37) where h1 is the average thickness of the L'th ice thickness category. Now suppose the rate of bottom melting of the floes is fb. The sides of the floes will then melt at a rate [(n/4) + (4/n)]fb so that the total rate of loss of volume through ice melting can be expressed (38) This may be equated with the rate of loss of ice volume as calculated from the heat absorbed through leads (39) By equating the two volume melt rates (38) and (39), the vertical bottom melt rate fb may be evaluated. The code is written so that if any artificial adjustment of the thickness distribution occurs, then an adjustment is made to the thermodynamic budget so that total energy is conserved. The type of thickness distribution changes envisaged here are those concerned with the adjustments necessary to correct the effects of numerical errors, such as the production of areas containing small negative values of the thickness distribution (for example, as produced by the second order accurate advection routines). In this case the thickness distribution value would be set to zero and the ext ra ice needed to be melted to conserve energy would be added to the term in ' equation (39) . If an ex tra volume rate of melt of v were needed to balance the energy budget, then from (38) and (39) the final value for fb would be given by (40) p . [n 4] ~ L4 + -; s + ( 1 - G 1 ) w 138 I I I The oceanic heat flux F 0 which is assumed to enter the mixed layer at a depth of 30m, gives another source of both lateral and bottom me lting. Numerically this term can be handled by including it with the~ term. Having now established the rate fb at which the bot t om of the floe wi l l melt , the lateral melt is handled indirectly. During the model run, fb gives the melting rate at the start of a time step. As the floe melts, fb would in fact change as the floe became th i nner. This is the error caused when modelling a continuously changing process with forward time steps at discrete intervals. Thus instead of calculating the increase in open water according to the lateral melt at the start of the time step which can be deduced from fb, rather we use an i ndirect approach taken whereby the difference between the volume melted due to bottom melting fb and the volume that should be melted according to the heat budget, is calculated. The difference between the two volumes is converted into a mixed layer temp e rature rise. Ice is removed from each thickness category equally and at the same time the amount of open water is increased to compensate, until either the temperature has reduced to freezing or all the ice has melted, in which case the water temperature remains above freezing. Using the lateral melting term F1 introduced by Hibler (1980a) into the ice thickness distribution equation, the changes to the mixed layer temperature may be described by the following equations aT . mix = at + (1 where 0) QI { f F1 (g, h, Tmix )h dh Cwdmix 0 - G1)fb - G1min(O,fb(O)) +F0 } -C(T . )g(h) mix C(T . )(l-G1 )o(h) mix (41) h > 0 (42) h = 0 (43) 139 I I II ,II 111 : " II II JI! ii 111 I• ·,,, 111 1 I . Ill and his the mean ice thickness 00 Fi = f g(h)hdh 0 (44) Here, dmix is the mixed layer depth (taken to be 30m), QI is the volumetric heat of fusion of ice (302 MJm-3) and Cw is the volumetric heat capacity of water (4•19 MJm-3), and the freezing temperature of sea water has been taken as 271 •2 K. The term F O is the upward oceanic heat flux into the mixed layer. The factor Cwdmix/QI when multiplied by a temperature difference gives an ice volume difference. Equation (41) is a modification of Hibler ' s (1980a) equation (6). The last two terms of (41) are dealt with simply by evaluating the volumes of ice before and after melting by amount fb6t and comparing the result with the volume that should be melted according to the heat budget (as described above) rather than attempting a direct numerical integration. The changes to the thickness distribution due to the melting of all the thickness categories by fb6t cannot be done by using the methods described in the previous section. This is because the melting rate here may be very much larger than the heat budget growth and melt rates, especially when the ice concentration is low so that large amounts of heat can enter the ocean. Thus the amount of melting required during a single time step may be considerably larger than the width of the thickness levels involved. Indeed i t may be the case that an entire thickness distribution is melted . b If a distribution, initially described by g(h), is melted by an amount f 6t f r om all levels , then the subsequent distribution is g ' (h) fb6t o(h) f g(h) dh + 0 b g( h+ f 6t) (45 ) Thi s r epre sen t s a shif t ing t o t he lef t of the func tion g(h) and is a chieved in th e code by in t erpola t ing the va lue s GL from the old thickness grid onto a new grid with values s pecified a t the node s 140 (46) The new amount of open water is obtained by adding the value (4 7) to the old value, and in the code is evaluated in the obvious manner, by summing up the values of g1 for which h1 < r\'it together with the appropriate fractional part of the thickness distribution level within h . b 1 · w ich h = f ~t ies. 141 6. THE MODEL SIMULATIONS 6.1 Model inputs 6.1.1 The model grid The model described thus far is applicable to any geog r aphic region. In this section we describe the input fields, boundaries and other special features associated with modelling the sea ice off East Greenland . Figure (6.1) shows the grid employed for this purpose. The grid i s rectangular if plotted on a map with a Lambert equal area projection. There are 21 x36 velocity points or 22 x 37 cells in which the _ physical properties are defined. The grid length is 35 km. The northern boundary represents the Greenland-Spitsbergen passage (Fram Strait). The western boundary includes part of the east coast of Greenland extending from Nordostrundingen 1n the north to roughly Scoresby Sund at the south. The east and south sides of the grid 1 ie in the open sea. This· grid was chosen so that the sea ice off Greenland would cover as much of the grid as possible, such that both the coast of Greenland and, for as much of the year as possible, the ice edge are included. The orientation of the grid is chosen so that t he initial ice edge position lies roughly vertically from the top to bottom of the grid. The northern , eastern and southern boundari e s consist of ou t flow cells 1n which the ice viscosities are set to zero and natural ou t flow or inflow 1s allowed. At the northern boundary, the ice - thickness distribution is specified as a function of the time of the year. This is because the ice thicknesses there cannot be predicted by a local model since they depend upon t he pr ocesses occurring in the Central Arctic . The ice flux through Fram St r ait is not specified as it depends upon the ice velocities ther e , and the s e a re ca lc u l a t ed i n t he model . 14 2 10W 0 Figure(6·1) The grid representing Greenland Sea modelled simulations. the part of the in the standard 6.1.2 Initial thickness distribution For a short term simulation, the initial variables used for input are of greater importance than in a long term climate study where an equilibrium solution is obtained from arbitrary initial data. In this study, the initial conditions were those of November 30, 1978. The initial ice thickness distribution was obtained from charts produced by Vinje (1979). These charts however, gave only the the position of the 3/8 ice concentration line. Thus, the ice concentration was assumed to be total from the coast out to a certain distance at which point it drops as a cosine function until zero half as far away again from the coast. The average ice thickness was taken to drop linearly from a maximum at the coast to zero where the ice concentration is zero. The linear change in thickness across Fram Strait is consistent with the observations of Wadhams (1983b). This data, which was obtained from a submarine, suggests a maximum mean thickness in Fram Strait of 6 m near the western coast. The average ice thickness of the pack ice at its western limit was taken to vary as the function a+ by3 (a and b constants) where y is the distance along the vertical axis of the grid (with increasing y towards north). This is so that the initial ice thickness increases from south to north, with enhanced thickness in the Fram Strait region. The initial ice thickness distribution at each point was then determined by assuming that the probability density decreases linearly from h = 0 to h = hmax• the maximum thickness. For such a linear distribution, hmax is three times the mean ice thickness. In addition, where the compactness was set to unity, the thin ice in level 2 was redistributed to level 3, and very close to the coast it was fu!ther redistributed to level 4. Figure (6.2) shows contours of the mean ice thickness, and the compactness used to initialize the model run. The thickness distribution at the northern boundary is specified not only initially, but throughout the integration. Th.e procedure for determining this is the same as that used in obtaining the initial distribution except that now the position of the 3/8 compactness line is specified as a function of time . Monthly values of the position of this 144. Figure(6 · 2) Ice thickness (m) Compactness - - - - - - - - - - - I 0•1 0•5 [] Contours of the initial ice thickness and ~ompactness used in the standard model simulation. 'The c.onlour in~ervo.1 for l:he c.ompCAdness clio..,~ro..rn is 0 · 1'25. D 14-5 - - ' 1' I 1 • line were obtained from Vinje (1979, 1980), and interpolated to weekly values. Initially, the floe sizes are set to the largest value allowed in the model, the winds acting immediatly to break up floes that are too large. In addition, the floes entering the grid through Fram Strait are specified to be equal to the maximum allowable floe size. 6.1.3 Oceanic heat flux and initial mixed layer temperature The mixed layer temperature in the model is defined where an ice cover exists, to be 271 •2°K, the freezing point of sea water. Where the initial ice distribution consisted entirely of open water, the initial mixed layer temperature was increased 1 inearly away from the ice edge, the heat contained in the mixed layer at a given distance away from the ice edge being just enough to melt all the ice at an equivalent distance into the pack. This is the result of extrapolating the linear ice thickness profile away from the ice edge and converting the resulting negative ice thicknesses into a mixed layer temperature increase. A rise of one degree Kelvin for a 30m mixed layer would be enough to melt a layer of ice of about 42cm in thickness, before the mixed layer temperature returned to freezing point. Little is known of the magnitude and distribution of the oceanic heat flux, the heat entering the surface mixed layer from the deep sea levels. Thermodynamic sea ice models have been found to produce excessive ice growth if the oceanic heat flux is set to zero. Maykut and Untersteiner ( 1971) used a constant value of 2 Wm- 2 (equivalent to 0•057 cm/day of melting) in their standard simulation although the model can be made to give reasonable ice thicknesses in the Central Arctic without oceanic heat flux if the other forcing parameters are suitably tuned (personal communication, K. Shine, University of Liverpool). Tucker (1982), in modelling the East Greenland area uses an oceanic heat flux equivalent to a melt of lOcm/day in the regions east of a band roughly corresponding to the expected position of the ice edge. By adjusting the oceanic heat flux in this way, one can control the ice edge position, and for the model here , Tucker's (1982) value of the heat flux is high enough to do this . It was H6 felt undesirable to force the results of the model in this way so the simulations here were run with lower heat flux values. The heat flux dis tr ibution used in the model was such as to have a 3 ·5cm day-l melt a t the grid point (20,30) with a ~(1-cose) decrease away from this maximum, reaching zero on the southern edge of the grid and at the East Greenland coast. This oceanic heat flux distribution was found to stabilize the ice distribution changes for early December and is also similar to the oceanic heat flux derived from the Hibl er and Bryan (1983) ocean-ic e mod e l . To give an idea of how much the oceanic heat flux can influence the position of the ice edge when melting and advection are taking place, a simplified version of the ice thickness distribution equation may be considered. = f(x) (1) where his the average ice thickness (the volume of ice per unit area). If u is constant, then to maintain an ice edge at a constant position, take = = 0 (2) sci that f (3) Fo r the initial conditions near the northern part of the grid, :: -8 xlo-6 (4) for which a velocity of about 5 cms- 1 away from the ice edge would be H7 Month January February March April May June July August September October November December Table 1 Values of humidity and cloudiness (percentage) used as input to the thermodynamic calculations. Humidity Month Jan Mayen Myggbukta Average January 82 76 79 February 81 74 78 March 81 76 78 April 83 81 ·82 ·May 86 78 82 June 90 82 86 July 87 83 85 August 84 75 80 September 81 70 76 November 82 74 78 December 82 78 80 Cloudiness N.E. Greenland Svalbard Jan Mayen Myggbukta 53 66 83 58 45 73 82 48 48 64 78 49 50 59 81 44 51 79 83 83 58 83 83 64 58 86 88 61 71 87 86 66 72 78 80 57 67 78 82 57 60 79 82 62 51 69 81 51 148 Average 65 62 60 58 61 72 73 78 72 61 71 63 r I I I I I I I I I I ! I I -- sufficient to balance a melt of 3·5 cm/day. 6 . 1.4 Input winds and temperatures Both the temperature and the wind speed are important in determining the thermodynamic growth rates, the wind field also being important in the dynamics. Spatially varying temperature fields are needed in this study because of the large variation that can exist across the Greenland Sea. This is particularly the case in the winter months when the mean monthly temperatures can vary from -28°c in the north-western corner of the grid to about o0 c in the south-east (Crutcher and Merserve 1970). Both the spatial and temporal variations in the wind field are important. Wind and temperature data was available from the ECMWF FGGE IIIb dataset, which had a spatial resolution of 1·875° (ECMWF: The Global Weather Experiment daily global analysis). The data was given for 12 hour intervals, zero and 12 hours GMT. Surface temperature and lOOOmbar winds were used. Hibler's (1980b) model code was written for geostrophic wind input and so the input winds used here were converted to geostrophic winds using a turning angle and modifying the magnitude (McPhee 1980). The input fields were interpolated to the model grid using simple bilinear interpolation. 6.1 .5 Other thermodynamic inputs To complete the list of inputs, we consider those for which average values over the grid are sufficient, and a representative value is more important to the results than are the variations from that value. Monthly averaged cloud cover values are used and obtained from Huschke (1969) and Hovm~ller (1945) . The values obta i ned are those of stations in North Greenland (Huschke 1969) and Jan Mayen and Myggbukta (Hovm~ller 1945) which correspond roughly to the four corners of the grid used. The mean of the four sets of data are used and t he monthl y values a r e given in t abl e 1. The humidi t ies are those given by Hovm~ller averaged from the t wo stations at Myggbuk ta and Jan Mayen. ill jl I I I I ,II J The atmospheric pressure values were obtained from the FGGE dataset and averaged over the East Greenland grid. All the quantities described here were either interpolated from monthly values or averaged from half-day intervals to weekly values. 6.1.6 Ocean currents The geostrophic current with its associated sea surface tilt are of sufficient magnitude to affect the ice distribution in the Arctic (Hibler 1979). In one of Tucker's (1982) sensitivity studies however, he found that for the East Greenland region the inclusion of geostrophic currents had negligible effect upon the ice distribution simulation. The geostrophic current field as used in Tucker's (1982) simulations was not of a sufficiently high resolution to include the high speed (-30cms-l) parts of the East Greenland Current. The highest speeds occur just to the south of Fram Strait in the region near the shelf break (Wadhams 1981 ). In another sensitivity study Tucker used long term wind driven ice velocities obtained from a previous run, as an ocean current input for a test run. Although there is no physical justification for such a procedure, in this particular case the current velocities used more closely resembled the observed long term currents than did the geostrophic currents that he used. This test indicated that by not including the high speed East Greenland Current, the results could be adversely affected. In view of these points, it was decided to use modified observed long term currents as the current input. The observed currents used are those of Kiilerich (1945), adjusted to include some measurements given by Coachman and Aagaard (1974) . For the model run, the values were reduced by a constant factor to parameterize the effect of meandering or snaking of the path of the current and to attempt to counter the effect by which parameterizing any time varying forcing by using constant time averaged values overestimates the true response. The resulting field is shown in f igure (6.3). Time -varying wind-induced currents were also parameterized a s described in section 6.3.3. The sensitivity of the motion of sea ice to ocean currents , together with the fact that that long term currents are specified, mean that errors in the oceanic forcing will build u p. As an e xamp le , e r r o rs i n the 150 East Greenland Current Greenland Ocean current input :7~ 'j 10 cm/sec \~1{ { L___J I. ~\ \ \ q ~~::: l ~\: ~ " l I\~~\' \ \ \ \ \ \ . \~~~"-,.' - - - , , I ' \ ~~ :--.----- --, D ' r l ~~:;:--:-:-:::=:.:: : , \ ~'--..,.~----.--/ / I ! I \~ ~------------..---..--_,,.,, / ./' / / I \ "-,_ ~----..~'-....,.~--.....--_,,.,, / / / / .............. ~~'--.....,.~'-.....,.",,--..--+/" ./' ./' / ., I \ '\. ~ ~ ~ ",. '.. ,-.__.__. __. _... - J \ \ ............ "-.,."" '"' '" '-,. ........ - - - - - -I- j- \- -.....- ".- ·-.. ".- ~ "'- '". ........ " --... ..... - I I I - Figure(6·3) The long term ocean current input for the model interpolated from a map produced by Kiilerich (1945) supplemented with data · from 1 Coachman and Aagaard (1974). 151 152 specified currents can upset the delicate balance between advection and melting at the ice edge. Also, near coasts, specifying too large a current could result in excessive build up rif ice there. In the standard run we have opted to have currents on the low side. In this way, errors are not compounded, and also it allows the time varying wind fields to be the dominant forcing, the effects of a long term current field being more intuitively obvious. The lack of accurate time varying data for the oceanic forcing terms (including the heat flux) is perhaps the major source of error in the model. Coupled ocean-sea ice models which are now being developed should be the answer to these difficulties. 6.2 Application of the model to a standard simulation 6.2.1 Introduction The model was run through a standard simulation with initial conditions for 30 November 1978. The time step used was six hours and with a grid length of 35km this meant that for expected ice velocities, the Courant- Friedrichs-Lewy stability requirement for the advection equations would be satisfied. This states that the following inequality involving the ice velocity should hold, (5) where ~x is the grid length and ~t is the time step. 6.2.2 Variability of the model results There is considerable day to day variabil~ty in some of the output fields . Figure (6.4) shows ice velocity fields 84 hours apart. The applied wind field changes from day to day and it is this that is responsible for the rapid changes in the resulting ice velocity fiel~; The average floe s ize varies in some places fairly rapidly. This would be expected to occur where the compactness was high and converging conditions occurred abruptly giving rise to large numbers of floes coming together. The ice 1 Veloci t y fi e l d ·. \ \. ·,. :. ' '.. ' · ... ·· .. ' •, - - -- - .- ,. I I I I ', ·. \ \ '\ .\ , ~)I > .. '. \ .. - -- - ~ ~ ::::(~} :~ ::~ .: ~ ~ :~ = :~ = \ i ! ' ' -..... ..... · .... · ... ·· ~. \ \'.'::i. ·, • .. - - - -- -- I / i r - ·- ..._ ' , \ \ \ \ ·., '" ·, ·- .. • . • -. '. I / /. I 1 i ! ' ·, \ \ \ \ , \ ·,, .... - - - - ' \ J J l : ,,.\\\\,,- --..,11/f, , ·, \ \ \ \ ·, ' · - ' ' \ I i ~ j. , \ \ \ \ \ ·., .'- ' ' \ I J j / . /, I \\\\\\'- '- '\ I J.//f., \ \ \ \ \ \ ' \ I j / / / /. I I \ '1 ·1 \ \ \ \ I I J. /. / f i. 1 I \ \ \ \ I I J / / / / / 1 I\\ \ \IJ///// / , \ I \ I i J. / / //// 1 Iii II J}I///,!/, \\ i J//,///,1/1 I i I j I /. / IJ.///, ' : : i : ii HH?f:tft , , \ \ i , .. ,· I • ,.. n / / , ' I , , . ,. ·•./.'< .· LJ •·•. ~ ~ : f L,·:: ~•: ~:: /-:~_,> .. . _:;_·:·:<: : I \ .. , .. "- "i 1 · ., : . · . •. , i..;. _,i }. :::·.:~ :- .. ,_·~· ... , ~· ~, I , · • .- 1' 1 ... •/ : l -.:· / .;: · " . • ' , -· • 1 ••. _..· 1 I ' . i./ ./, ·-1 / .,~· ·/ ;. - · _ .'_ •' .. • __ .,,.._/_ . '_t' · __ . t' • . • · / / /,/// ..... .....-......-.......--..-/ //.,,..,,, ................. .....-..... D__.___ , / I////._/"...----...--_..- __.___, I I////._/'.....-- .._.__ , I / ////._/".,..-.:. F .,__ ..,__.__ I 'I //////...----~..__, , 'I!/////...----~'-.'-. ' l / / //:.(/ <" '("'/-..-..........,_, '. I / ./// /_,-....-__.__...._..._ ' ; I// ,W / .,-.....----.-.. ' ' : I _ - - - - - - - - - - - - - - - . 2 Ve loci Ly fi el d / / / / / / / ... " ..... - ' ' ' '' '"'' ' // ///////.--,,,,,,,,, //////////-,,, ,,,,,, //// ///, //_,- , ,,"'''"'' / / / / li'.i'.I,//-'' ''\_ '\\._'-0 I ~~~~~~~%~~= =:~~~: /~;;~~{~~~==~~~~~~: ': ~?~1~%~;:::~~* : 1/ ]j,!Jj,//--~,'...._~~I '. ~?1,~%~;::=~: . l/j,!JI,/// /- -, ,"-;_\, . 111,%Z/I///--,,~, Figure(6•4) I JJ,1,/,/II/I/-'-'- \, 1 J Z Z J j 1 I I ( I ~ - ' \ , ,,zj J j i/Jjj , ·\, 0.2 m/s · 1 I 7 I 1 1 1 1 \ ) \ - , , ' I I l J j j j l ~ .\ ·' ·, ' - I I I I I J j j J i ,I .\ ·' \ , ·, - I / I I J J I j I j I I \ \ ..... ·,- I ' I I i J j j J j i \ \ " ·---.·-.....---. I / j j j j j i I \ ' ' ,·- I I 'Iii ii l i i ID,-, ' I I j j i j i i \ I - - I I I j I I i I - - I ,, , ll i\\\,-.-- 1 I I I J J I I I \ \ ' ....... - - 1 I I I j I I i \ \ ' ....... - - ' ////111 \\,, -- , •,1111 ;, 1,,---, - "' -' 7 7 7 7 7 7 1 l,, -:_-_-:_· 4 Velocity field Instantaneous ice velocity vector plots from the model for periods 84 hours apar t showing the rapid changes possible. The period covered by the figures is from 13 to 23 April 1979. 153 thickness distribution does not change so rapidly, although where the ice 1s thin, the higher velocities mean that features can advect tens of kilometres in a few days. The floe size fields show a greater spatial variation than the ice thickness fields, as might be expected (see figures 6.11, 6.12 and 6.13 for some examples). First of all, there is a large range of possible values that floe size can take, and secondly it 1s quite possible for neighbouring grid squares to have floe size values that span this entire range (the wind field may break large floes in one grid square but not in the neighbouring square). Despite this, some coherent structures appear to be visible, particularly in regions of large floes. For the regions 1n which very large floes are indicated, the interpretation should be that the region consists essentially of a continuous ice cover. At the other end of the floe size spectrum, predicted floe sizes of less than 1 or 2m should be regarded as meaning simply that the ice cover is highly broken up. This occurs for thin ice very near the ice edge where the effects of ocean waves have been parameterized by artificially weakening the ice there. 6.2.3 General features of the output fields We must be careful when interpreting features of the output that occur near the start of the simulation because they may be due to the form of the initial distribution specified. We thus regard the first two or three months of the simulation as the (settling down) period during which the thickness distributions develop their own characteristic profiles. During the first week or so of the simulation, which would be expected to be a period of ice growth (in thickness and extent), the position of the 3/8 ice concentration line advanced at an expected rate. However the fairly high initial sea surface temperatures 1n some regions dropped despite the imposed oceanic heat flux term, down to freezing when a thin ice layer of high compactness (greater than 90%) formed over most of the formerly open water region. The layer remained thin for a considerable amount of time and a sharp increase in the ice thickness within the pack persisted. Figure (6 . 5) shows the ice characteristics along a transect 154 10000 m 1000 100 10 1 0.1 0 1.0 ] 0 q · '-' j 0 .6 j 0.4 0 .2 ~ 00 0 nm ! j 0 Observed Floe radius ice edge 1/ position 5 10 15 Grid lengths Compactness J 10 15 Grid lengths Ice thickness 5 10 15 Grid lengths Figure(6·5) Ice characteristics along a transect during a . period when the model output included a large area of thin ice ( in this example the date is April 12 1979). The observed ice edge at that time is also indicated. 155 20 20 20 across the Greenland Sea toward the end of this period. It is this thin layer of ice that can be removed by specifying an oceanic heat flux as high as that specified by Tucker (1982) in his Greenland Sea simulation. An oceanic heat flux that effectively defines the position of the ice edge might also undesirably affect the type of ice edge profiles developed. Since the ice edge profile is of interest in this study, the lower values of heat flux used here allowed more natural ice edges to appear during the spring melting. The compactness field for the end of December (after one month's simulation) is shown in figure (6.6a) which also includes the observed 3/8 concentration line (Vinje 1979). It shows that by then, the observed and simulated ice edge positions had already started to diverge. The initial oceanic heat flux and sea surface temperature distributions could of course be adjusted (by trial and error) to give the correct ice edge position after a month's simulation, but this might be regarded as excessive 'tuning'. The inability of the model to predict the correct lateral extent of the thin ice during winter meant that observed features of the ice edge for the year 1979 were not modelled. These include a large tongue of sea ice extending northward from the ice edge north of Jan Mayen (Vinje 1980). The same feature occurs in other years (it previously appeared in 1971) in the same location and has acquired the name "Odden" (Vinje 1980). In 1979, Odden appeared in February and became detached by April, melting completely by May. The compactness values which in winter were found difficult to simulate show much better agreement at the start of June. This can be seen clearly if we compare the model results with Vinje's (1980) data (figure 6.6b). An energy source, not yet considered, that is available in the winter for ice melting is that due to wave action, the energy ultimately deriving from the wind. This would have the desirable effect, as far as this model is concerned, of melting ice near areas of open water, but leaving unaffected the thick ice away from the ice edge. Wadhams et al ( 1979) calculate that for typical conditions near the ice edge in the Greenland Sea, wave-induced melting can destroy a 3 metre floe in a matter of days (if the wave has a Sm amplitude, the floes will melt in about 1-2 days; for a 1 m amplitude wave, the melting would take about one week). The melting is caused by a turbulent heat exchange between the lower surface of the ice 156 I I 1 1 I a 0 Compactness b Compactness \ \ ~ J\ 8 \'I/ - - I Figure(6•6) .._., I Observed Simulated \)8 ~> Observed 1 Comparison of observed and simulated 3/8 concentration line. Diagram (a) refers to the end of December after one months simulation. Diagram (b) is for the end of May after the spring melting of the thin ice. The compactness contours are measured in eighths. [] - - I I-' Vl ---J a 0 Compactness - - - - - - - - - -. b Con1pactness d'\ 8 \ \ ~ '~ - - - - - - - - - Figure(6•6) < 11 Observed Simulated \)e ~' [] Observed I I / lOo I Comparison of observed and simulated 3/8 concentration line. Diagram (a) refers to the end of December after one months simulation. Diagram (b) is for the end of May after the spring melting of the thin ice. The compactness contours are measured in eighths. - , I-' \JI --..J and the ocean. This requires the water near the ice surface to be above the melting point of 1.ce. For the model simulations, no such temperature difference 1.s possible so that more elaborate oceanic boundary lay er formulations would be required to model wave and shear current-induced melting. Some features of the ice thickness distribution near the East Greenland coast are interesting. When the ice velocity is towards the coast, the 1.ce thickness there increases. However, when wind changes result 1.n an 1.ce velocity away from the coast, the rheology in the model is such that the formation of open water occurs comparatively easily. This, as was shown in chapter 4, considerably reduces the resistance of the ice to deformation, which results in a large shear and further opening. The effect, which can be seen in figure (6.7) is to peel off a layer of thick ice which becomes caught in the East Greenland drift stream and is advected south. This situation stops, usually in a matter of days, when the wind again turns. The open water formed near _the coast soon becomes frozen in winter (figure (6.7) starts in March). The ice still has a low compressive strength so that the thick ice can return to the coast. Figures ( 6.8a-f) show the changes to the mean 1.ce thickness distribution, the compactness and the mean floe size distribution at times throughout the model integration. For comparison, observed ice limits from charts produced by the United States Naval Polar Oceanography Centre are shown which include more detail than Vinje's data. We see from figure (6.8a) that one month after the start of the simulation, a large amount of new ice has formed near the centre-right of the grid. In the model simulations, a large area of new ice forms, but in a region too far south. By early February ( figure ( 6.8b)) , the initial errors have compounded and there are few features, other than very general ones, in the simulated output that can be equated with the observed Sea ice distribution. Four weeks later (figure 6.8c) things have started to improve. The region of open sea 1.n the map of the observed conditions seems to correspond to the areas of reduced compactness and very small floes suggested from the model output. Figure (6.8c) shows a much reduced area of large floes compared to that in figure (6.8b). In addition, the area of the observed ice of high concentration (7/8 to total) also undergoes a considerable reduction. In 158 1 lee lh1ck n ess if l(i~ ;;--·· --: \' '{ . ' / . ' 'I ' \i, \\ ' - \\1 I I l ·1 ~ ·r: : . f .))1)\ : i r;' d/J : \ : I I ,;/(; ,__/', , 1 1 '' 1 ' - o' · 1 l\\ '---: I ' ')0-J \ ' 1 // \ I (~ --- -·- ___ : 5 Ice lh1ckness - - - - - - - - - - - - - . ' ' ' ' ' ' ' ' ' ' ' ' lJ) : 9 . Ice thickness 1 3 , lee thickness , ' ' ' 2 Ice lh1ckness .6 lee thickness 1 0 lee thickness ~: 1 4 ' Ice thickness ~. ' ' I 7 ' Ice thickness ' cmi ' ~: 1 5 . lee thickness ' I Figure(6•7) 4 Ice thickness --- -- -------- , ' ' I ' ' ', : ' Cffi' : ~ / : 1 6 lee thickness I I The contours are (from left to right) Sm, 4m, 3m, 2m, lm and O•Sm. Continued on the next page. 159 Figure (6.7) Continued from the previous page. The development of features near the coast of East Greenland. Notice the changes occurring in the top left area of each pie ture. An area of thick ice becomes detatched (Picture 3), gets advected south until it hits a larger area of thick ice (Picture 6). Eventually (Picture 16) the ice thickness pattern again resembles the original (Picture 1). Other smaller areas of thin ice near the coasts can be seen further south. (Pictures 1-9). Picture 1 refers to 12 March 1979 and picture 16 refers to 4 May 1979, each picture being 84 hours apart. Figure(6•8) The next 6 figures show the development of ice characteristics throughout the model run. For comparison, we show the observed ice conditions as presented by the United States Naval Polar Oceanography Centre. Sui tland, Md. 160 Ice thickness ' ) ' \ ; • ' \ I • 0\:\lt ~ ) ~~1/ i ~ (/// ~ '/ / '~, I ·. I . ; I I j I ( \ I I " \ ) ,-, , , , D Contours of compactness from left to right: 0.98 0.925 0.875 0.75 0.625 0.5 0.375 0.25 0.125 Figure (6.8a ) 26 December 1978 CompaC'tness . I I Key to symbols S Sea F Land and fast ice OW Open water (Within ice liIDi t) 7- 8 Fractional ice area (Eighths) Floe size t Contours of floe radius from left to right 1 km 500 ID 200 ID 100 ID 50 ID 20 ID 10 ID 5 ID 1 m 161 11 ke thickness I \1 \ I I \ \ / \ < ----~ff) \ ' Figure ( 6. Sb ) 7 February 1979 Compactness 5-7 7-8 F Key to symbols s Sea F Land and fast ice ow Open water (Within ice limit) 7-8 Fractional ice area (Eighths) Floe size :1 t Contours of floe radius from left to right 1 km 500 ID 200 m 100 m 50 m 20 m 10 m 5 m 1 m 162 111 II II 11 I Figure (6.8c) Ice thickness 6 March 1979 Compactness ' . ' ' ' 0-98 0·925 I /, I ; I I : ' ' I ' \ I . ,q \~ \, ... _'""·. f D I \ ·-.. Key to symbols S Sea F Land and fast ice OW Open water (Within ice limit) 7-8 Fractiona l ice area (Eighths) Floe size I Contours of floe radius from left to right 1 km 500 m 200 m 100 m 50 m 20 m 10 m 5 m 1 m Ice thickness / // //! / / ' I, 4 \3~2, : 'I ... , ' I! \ \·\ i \ I I \ \ \ 'i : I//: I//: /I ! ; ,/ ) : II,' I I/ 10·5 . s r.1.,. \ I" '/ /1 . !, ; "I \1 1 \I\\ \ \ '\ \ \, \ '11 \ ) ' I, ' F Figure (6.8d) 3 April 1979 Compactness 1-3 5-7 Belts Strips 5-7 Key to symbols s Sea F Land and fast ice OW Open water (Within ice limit) 7-8 Fractional ice area (Eighths) Floe. size I Contours of floe radius from left to right 1 km 500 . m 200 m 100 m 50 m 20 m 10 m 5 m 164 I I II Figure (6.8e) 1 May 1979 Ice thickness Compadness f J- ----------- \ ~ :0-1 I ' ' \ .. · · ... Contours of compactness from left to right 0.98 0.925 0.875 0.75 0.625 0.5 0.375 0.25 0.125 0/ :1 ... Key to symbols S Sea F Land and fast ice OW Open water (Within ice limit) 7-8 Fractional ice area (Eighths) Flo e size I Contours of floe radius from left to right 1 kin 500 m 200 m 100 m 50 m 20 m 10 m 5 m 1 m I I 11 II I I Ii ke thickness Contours of compactness from left to right 0.98 0.925 0.875 0.75 0.625 0.5 0.375 0.25 0.125 Figure (6.8f) 29 May 1979 Compac-lness / s F -s Key to symbols S Sea F Land and fast ice OW Open water (Within ice limit) 7-8 Fractional ice area (Eighths) Floe size . : ~ : 11 : :, : ., ' i Contours of floe radius from left to right 1 km 500 ID 200 . m 100 ID 50 ID 20 ID 10 ID 5 ID 1 ID 166 I I I I I I l•O 0•9 0•8 ro 0•7 Cl) I-; ro "Cl 0•6 ....... I-; on '- 0•5 0 c= .9 0•4 ...... I-; 0 o. 0•3 0 I-; 0... 0•2 O•l O•O Figure(6•9) L Observed • L Simulated • 0 50 100 Day of the year The observed extent of ice in the region covered by the model grid (given as a proportion of the area of sea in the grid) is compared with the extent predicted by the model. Also shown are the extents of ice of thicknesses greater than certain values (10cm, 20cm, 30cm and 40cm). The proportions can be multiplied by 0·69 106 to give actual areas (km2) . --~ ~ ---- ----=--- -~ 150 p °' --.J figure (6.8c), the area of large floes at the top left of the grid area seems to correspond quite closely with the observed area of high concentration. However, the large area of ice of high concentration observed near the south-west part of the grid is not simulated. After a further 4 weeks (figure (6.8d)), the area covered by compact ice increases slightly. Also, the simulated area of large floes shows a similar increase in extent, the resulting area corresponding very closely to the observed high compactness region. This correspondence remained close for the rest of the simulation. The area of compact ice at the eastern side of the grid started to melt at this time (early April 1979) so that by the start of May (figure (6.8e)) a general agreement between the observed and simulated compactness fields occurs, although the details of the observed picture are not simulated. The best agreement between the observed and simulated compactness fields occurs in the second half of May by which time most of the thin ice areas have melted (both in reality and in the model) (figure (6.8f)). By May the ice edge is essentially determined by the limit of the extent of thick ice. The representation of the thick ice which is naturally less responsive to forcing is also less susceptible to errors in the forcing and this could explain the improvement in the results compared to those of winter. A sharp change in ice concentration and floe size at the ice edge is then predicted, in contrast with the situation for most of the winter months. We have seen that in the winter months, the model produces too large an extent of ice and that most of the excess area consists of very thin ice. Figure (6.9) shows the variation throughout the model run of the proportion of the sea area within the grid that is ice covered. The observed areas are those obtained from figures (6.8a-f). Figure (6.9) also shows values for total area obtained by not including ice less than a certain thickness. We see that if ice less than about 30cm is neglected in the area evaluation, then the model would predict more or less correct values for total ice extent during most of the run. Figure (6.10) is a satellite photograph of part of the area covered by the model grid. The model floe size distribution for the same date (9 May 1979) is shown in figure (6.11). The large area of ice classified by the model as consisting of very large floes, and is marked A in the diagram 168 I I p A Q Floe size , ,~- ' \ //'- I / ., • I //~,:-;.-=! ; t ,, ' , I ,2,,: .. I •7-0 , •' I ,.. ( I 1i: ; V /; :s ·. ~ :1 \ "10 1~, I \• ' "' . , . , : '\ ,, -- ) Figure(6•11) Comparison of features visible satellite picture and a floe contour diagram for 9 May 1979. 1.n a size Floe size I :1 -:o·:.\, · .. ,): . '' , ... ~~~ ..... J ', · .• ,; Figure(6•12) The average floe size of thickness categorie s (levels 4 the model grid) for 9 May 1979. the wh ite ar e a s t o the right in the photogra ph are cl ouds rather than sea ice. Figure(6•10) - , selected and 5 of r-' °' \.0 170 corresponds well with what appears to be the main pack in the satellite image. A narrow area slightly lighter than the sea touching the main pack in the photograph suggests an area of loose pack. A larger area of loose pack ( smal 1 floes) is indicated from the model results. This may be because there was ice there that cannot be detected in the image or that there was no ice there in reality. The area marked Bin figure (6.11) also indicates the presence of very large floes, but again there is no sign in the image of any features there. The ice in Bis however very much thinner (0·3m) than that in A (about 4m) and would thus be expected to appear (if at all) differently than thick ice in the image. Another point to note from the satellite photograph is the detailed structure of the sea ice distribution that is visible, and in particular, the ice edge. We would not expect to simulate these details with the grid resolution used in the model. The wind fields used for forcing the model were not sufficiently detailed to include, for instance, the small scale wind anomalies that occur near the ice edge due to variations in the atmospheric drag coefficients and boundary layer stability (Overland et al 1983). Although figure (6.11) shows a large area of continuous ice, this does not mean that there are no small floes there but that large floes dominate the field. This is illustrated in figure (6.12) which sho~lS the average floe size for thickness levels 4 and 5 revealing that there are floes even near the coast. The diagram must be interpreted with care because it gives the floe size without any indication of the proportion of area covered by those floes. In this example, the actual number of floes near the coast would be very small. During the period April 29 to May 2, a storm passed through the region covered by the model and at this time the ice edge had returned to the model grid. The model response to the storm is shown in figure (6.13) which gives the floe size distributions during a period of a week. The floe size maps show the effects of the storm more graphically than the thickness distribution or compactness maps. The dramatic effect of the storm can be seen by the considerable reduction in the area of large floes. The surviving large floes are confined to a narrow band near the coast . Floe Flo e size ~ b--' C /.,··· 011' ' 0 . , 0 : : '' '. \ \:''1-..,o ~0. 20 t, I 50 i1 \ ~ '11 ;'~...;,--~ . .. : •,<> ,.-~ . t \:, ... 1\~\ r \ , . \ I \..,.. J , r-, : , -,,; •· ·., , \ /!\..'{~ ' \ ~ ~~ . Before storm - , The lett e r s a-i show whe r e the f l oe si ze d i s tribut i ons are g i ven i n figur e 6. 18 Floe s1ze - I d 200 500 e , I : , ~[li '\ ' ' l - - - - Dur i n g s torm Figur e( 6• 13) The effect of a storm on the Sea. the d i stribution in the Gr eenland d i agrams correspond roughly to be fore, during and after the storm. floe The s iz e three situa t ion g [ _ Fl o e s1ze ·.- -. _._·_·::-.. ,. . . I r· · · '., \ ~' • J \ 10_,· ' 1 -~-~ ...... , ~~:_'· 1>:t' •. ., If : Af t er stor m - I 1-'- -J ....... Figure(6•14) A large floe, well within the ice pack in the Greenland Sea, starting to break up during a storm. Photogr aph from Ketchum and Wittmann 1972. 172 n 1 1 I I breaking like this, w~ll away from the ice edge, h~s been observed from aircraft by Ke·tchum and Wittmann (1972) who show photographs of fragments of a large floe that appears to have broken up in a way consistent with the long floe break up mechanism suggested in chapter 3 (see figure (6.14)). Ketchum and Wittmann were fortunate enough to have made their observations around the time of a severe storm similar to that occurring during the time of the simulation. An area of reduced ice concentration and ice thickness appears at the beginning of May and persists until the ice edge approaches when it develops an eddy-like structure (figure (6.15)). This does not seem to correspond to any real feature in that location, although polynyas and eddies are observed in the East Greenland pack ice (Wadhams 1981, Wadhams and Squire 1983). The feature 1.n the model seems to derive from a divergence of the applied ocean current field. The map from which the current field was digitized has the diverging region, but whether or not this is a permanent feature is not known. 6.2.4 Distributions within single grid squares The full information regarding floe sizes produced by the model can be seen only by looking at the floe size distribution at each grid point. Figures (6.16a-c and 6.17) show floe size distributions · obtained from digitized photographs along a 37km transect of a region near the ice edge (data provided by A. Cowan, Scott Polar Research Institute). The area from which the data was collected (72°N) falls near the southern edge of the model grid. The areas of a total of 42 56 floes were sampled. Figures (6.16a-c) are the floe size distributions of three 12km sections of the transect . Figur~ (6.17) shows the floe size distribution for the transect as a whole. We see from these diagrams that variations in the floe size distribution would be expected even within one model grid square. We also see that although there is quite a variation in the range of floe sizes, there does seem to be a dominating floe size in thi s region. The data for figures (6.16a-c and 6.17) were obtained from photographs taken from aircraft. Much larger floes, further north, have been measured from satellite imagery (Vinje 1977). 173 Ice thickness (m) ------, _,,.---- -- , I ' ' : ~ I \\: /3 , \' \ ...... 2 1l' ;0-1 ', l- : 1 ' ' I I ' "- \ . : ......... ~.._ ;', \ ·, : '-----\,1\ I 4 1'11.: ! ': I • 1t; 1t: It: 1'r.: !}l'w-0·5 11 Ii : /1 ,\.·· I) ,( 1/ ~:.\ I ~ \, . \v )1·' ''-(.) I/. ,.'_: ) / 1: .. ' I I/;' '/·/ '/ I : ' ///:I \ \. \'.·~-. ~ ' ' ·. \ \ \ "-; ': ) \ ' : : \ I: : / II: .' , I I I • I .' .' I / ' ' D Figure(6•15) Compactness D Anomalies 1.n the long term oceanic current input as used by the model produce features at the ice edge. Contours of compactness fr om l eft to right 0.98 0.925 0.875 0. 75 0.625 0.5 0.375 0.25 0.125 174 - I re (l) i..., re -0 C 0 - .--< -+-' i..., 0 p.. 0 i..., 0.. re (l) ..... re -0 C 0 - .-< -+-' i..., 0 p.. 0 ..... 0.. Figure (6.16a) 0.25 · 0 .20 Bin size 2 rn 0 15 0 . 10 lJ 0 .0~ 0 .00 0 10 20 30 40 50 GO Floe radius (m) Observed floe size distribution in a region 0- 12 km from the ice edge in the Greenland Sea. Figure (6.16b) 0 .14 - 0 .12 0 10 J 0 "P .U,; J 0 .06 · Bin size 2m 0 .04 0 .02 o ob nn 0 10 20 30 40 50 GO Floe radius (m) Observed floe size distribution in a region 12-24 km from the ice edge in the Greenland Sea. 175 re (l) l,.., ~ -0 ~ 0 . ,-t ..., i.... 0 p, 0 l,.., 0... ~ (l) l,.., ~ -0 ~ 0 . ,-< ..., ..... 0 ~ 0 l,.., 0... Figure (6.16c) 0 10 J 0 08 J 0 .0G · Bin size 2 m 0 .04 0 02 lr~ 0 00 0 10 20 30 40 50 60 Floe radius (m) Observed floe size distribution in a region 24-36km from the ice edge in the Greenland Sea. Figure(6•17) 0 10 J 2m 0 08 J Bin size O.OG · 1~ 0 04 0 02 ~1 n 0 .00 , I 0 10 20 30 40 50 60 Floe radius (m) Observed floe size distribution in a region 0- 37km from the ice edge in the Greenland Sea. 176 ro a Cl) H 1.0 ·, ro o) 'H OCJ 0 ,:I 0 0.4 .,., +' H 02 0 ;:i.. 0 C 0 - 1 H p... ro d Cl) H o.c; ' ro 'H O.'i 0 · 0 4 .:: 0 0.3 .,., +' 0.2 H 0 0.1 ;:i.. 0 O.C H p... -1 ro g Cl) H 0,; ro 'H 04 0 .0 3 .:: 0 0 .2 .,., +' H 0 l 0 ;:i.. G.C 0 H - 1 p... Before storm Flo e size distribution b Floe size distribution C Flo<" S! Z l' distribution C 2 3 4 3 Lo:;(r) 1.0 , OJ ' j ''i 04 02 0.G - 1 C 2 Log(r) 3 During storm 0 14 j 0 12 0 10 j 0 ,. . .., 0 ;~J 0.C4] G C:? J 0 :: +-, 4 ) - I C 2 3 4 Log(r) Floe size distribution e Floe size distribution f Floe SIZl' distribution o., · 0 35 J 'U 0 30 0.4 025] 03 0 20 i 0 !5 j 0 .2 0.10 0 1 i 0 C) -i o.: ' 0 c: , C 2 3 4 =; -1 C 2 3 4 ) -1 C 2 3 4 Log(r) Log(r) Lo ,:;(r) After storm Floe size distribution h Floe stze distribution i Floe siZ E' distribution C o,; ''l _J 0.4 JULJl 0 .3 0.3 02j 0 .2 0 .1 0 l l 0 .G G.C · 2 3 4 5 - 1 C 2 3 4 5 - 1 C Log(r) Log (r) Figure(6·18) These diagrams show the shift of ·the floe size distribution toward the lower end of the spectrum due to a storm, and a slight redjustment back to larger floes after the storm. The letters a-i refer to positions i n the gr i d as indicated in figure 6.13 2 3 4 Log(r) 177 ) =; 3 To illustrate the relative areal proportions of floes of various radii, the floe distribution within each thickness cat e gory is needed (the model gives only the mean floe size within each thickness l eve l). Here, we have assumed a rectangular distribution for each of the thickness levels, although this 1s not the only possibility. The total distribution is obta ined by summing the separate distributions for each of the thickness levels. Figures (6.18) showing the simulated floe size distributions indicate the same kind of variation as 1n the observed data. There are large variations 1n the floe size 1n a single region with perhaps some particular floe sizes dominating. There can, however, be no real comparison between this sort of data and the model results unless far more than 6 ice thickness levels are used. A one grid cell study with perhaps 100 ice thickness levels would be more useful in trying to generate results to compare with figure (6.17), however a three-dimensional model with this resolution is not yet fe?sible. The model ice thickness distributions show the characteristic shape seen in observed profiles of a tailing off towards larger ice thicknesses. The values of the thickness distribution for thin ice will depend on its recent history. Divergence with freezing will cause large amounts of thin ice. Convergence will cause ridging of the thin ice reducing the thin ice amounts and increasing the thick ice. Figure (6.19) shows some typical profiles obtained by the model and for comparison we show some profiles obtained from upward-looking sonar data from submarine (figure (6.20)) . These profiles were observed in the shear zone north of Fram Strait (shown 1n segments 1-4 in figure (6.20)) and so generally have more thick ice than 1n most of the .Greenland Sea. During summer the melting of the thin ice in regions where the average ice thickness was high gives rise to flatter distributions . This could be an artifact of the model due to the lack of th ickness distribution structure defined within the top level which could be overcome by increasing the thickness of the maximum fixed thickness level and increasing the number of levels. However, this would involve using much larger amounts of computer store. 178 0 ./'. l 0 .6 · m-1 0.4 02 O. 'i Thicknf'ss distrihution a 0 'j 04 0 3 0 .2 0 l O.C 0 2 3 kE Thicknf'ss m Thickness distrihution d 0; 179 Thickness distrihution Thickness d1stnhut10n b C 04 C 2 3 4 '> r, 0 .3 I . 0 2 j I I : ~1'1: ,, 0 2 3 4 '> G ke Thickness kt': Thickness m m Thickness distribution Thicknf'ss distribution e oc+.~..-4-i~..___,.,.__--,,-_.'"'T'",..,...-, 0. () -t, ~-~ ......... r'-----.--........ -.---, -0 '> 0 .C O."i I O J.5 2 .0 -0.'i CC O:'\ 10 I 5 2.0 0 2 3 lcE Thickness m ke Thi ckness m ke Thickness m Figure(6•19) Figures (a), (b) and (c) show the thickness distribution at point 2 of the grid for the following dates: 17 March, 12 April and 28 April 1979. Figures (d) and (e) are for point 3 which is nearer the ice edge and for 17 March and 12 April 1979. Figure (f) is the thickness distribution further south (point 1) and later (26 May) when melting has started. The positions of the points 1, 2 and 3 within the grid are ~hown in figure ( 6.15 ). 4 \-;;-----"s,,_5• _ __ -iss• ,e c 90°W lfl 17 <1 0 ~! lo• '\ 2 0. I,~ P(h) 0 2 4 24 2 • • • ,. 0. I ,, P(h) I\ A 3 ( 4 0. . I\ 1 ·,,.\ ii I '"' I \ 1- I ~'"- "J ... ~ ... lo V\ .J/ \A, , "v-v'\..,... "---· " 0 2 • • B ,. " h m h m Figure(6•20) Observed ice thickness distributions along four 100km sections of the track of H.M.S. Sovereign. The cruise was made during October 1976 (Wadhams 1980a), The lables 1-4 refer to the numbered 100km sections marked in the map starting at point A. 180 Wittmann and Schule (1966), in their early study on the proportions of ice types , in the Arctic gave seasonal values of the percentages of three categories of ice in the East Greenland region. The categories were new ice, thick winter ice and a third category consisting of second-year and multiyear ice. Their values were made from a number of visual observations from aircraft so at best their data should be regarded only as estimates. If we assume that the three categories correspond to level 2, levels 3 to 5 and level 6 of the model thickness distribution, then a comparison can be made between the observations and the model results. The amounts were summed over the ice covered portions of the grid and the results plotted for every two weeks. The result is shown in figure ( 6.21 ). Definite seasonal responses of the model are shown, particularly the rapid decrease in the thin ice proportion at the onset of summer melting. The amount of thin ice drops to well below the Wittmann and Schule results during spring; Wittmann and Schule's data for East Greenland included more of the area north of Fram Strait than in the model grid and this is where the extra amounts of thin ice would be expected in spring. 6.2.5 Transects The change of some ice properties along a transect in Denmark Strait was investigated by Kozo and Tucker (1974). The observations were carried out from a submarine and amongst the properties they measured was floe size. Figure (6.22) shows a representation of their data, in which the boxes denote the sizes and distances from the ice edge of various regimes of ice types observed. They collected data to a distance of 229km from the ice edge which would be between 6 and 7 of the grid lengths used in the model. Although the grid employed for this study did not include Denmark Strait, a comparison between the Tucker and Kozo data and model results for transects further north would still be useful. The data suggests an order of magnitude increase in floe size for each 150km or so into the pack. At the same rate, the floes would become large at a distance of about 12 grid squares from the edge. Figure (6.23) shows some profiles from the model for times when the ice edge appeared on the grid. In fact, these are transects for the period during the storm discussed above. There are differences due to latitude changes but the general trend shows some agreement . A steeper profile is suggested for the spring simulations 18 1 ·7 ·6 ·5 C .Q ..... .... ·4 0 0. 0 .... a.. ·3 ·2 ·1 N Figure(6•21) The seasonal changes of the proportion of various ice thickness categories. Observations of Wittmann and Schule (1966) are compared with the model output. D J F M A M Month J J New ice 2 Thick winter ice 3 Multiyear ice • Observed • Simulated A s 0 """" 00 l'v 400 300 - 200 E -j 100 ~ I I I "O Q} 50 0 u: 20 Figure(6•22) Observed floe size variation away from the ice edge in Denmark Strait (From data given by Koza and Tucker (1974)), >td1~. I I ~ I .... ............ , ' ' ' 10 ', 200 150 100 50 Distance from the ice edge (km) ' ' (1) c:n "O (1) (1) (.) 0 1--' . C1J v-1 10000 m 1000 100 10 1 0.1 0 10000 m 1000 100 10 1 0.1 0 10000 m 1000 100 10 1 0.1 0 Floe size Before storm 5 10 15 Grid lengths Floe> size> During storm 5 10 15 Grid lengths Floe size After storm 5 10 15 Grid lengths Figure(6•23) Changes in floe size distribution as a result of a storm. The · straight line in the third transect shows the gradient of the floe size line if there is an order of magnitude increase in floe size for each 150km into the pack (as is the case for the Kozo and Tucker (1974) data) . 184 .:zo 20 20 however, Figure (6.24) shows observed ice velocity vectors south of Fram Strait during a period of fairly strong northerly winds of up to 15 ms- 1. As we have seen, the model ice velocities can vary over over a few days but figure (6.25) shows simulated ice speed contours at a time when the general wind pattern was similar to that occurring when the observed drift rates were obtained. The distribution of ice drift speeds in the observed and simulated cases is similar although in the model results, a greater area of the region is contoured because the model included a larger area of ice than observed at that time. Wadhams (1983b) estimated the ice volume flux across Fram Strait using ice thickness data derived from measurements that were made from the submarine Sovereign during the period April-May 1979. The ice velocity profile across Fram Strait was obtained from Vinje (1982) which together with the ice thickness gives the ice volume flux. Figures (6.26, 6.27 and 6.28) show the results obtained by Wadhams (1983b) (dashed lines) together with the results obtained from the model (solid lines). The agreement is quite good both qualitatively and quantitatively, although the model does not produce a zone of fast ice in the grid cell closest to the coast of Greenland. Also the position at which the ice flux curve reaches a maximum is slightly displaced. The total ice volume flux across Fram Strait according to the model is 0.25 Sverdrup ( 1 Sverdrup is equivalent to 106m3s-l), This is a little lower than Wadhams' (1983b) value of 0.29 Sv. 6 . 3 Variations to the model 6.3.1 Introduction Here we present some more model results obtained by varying some of the input parameters. Ideally the model should be run over a number of seasonal cycles for an equilibrium solution to be reached. In addition, the entire Arctic should also be included in the model simulation to provide an ice volume transport through Fram Strait. The effect of modifying a model parameter should be tested by running the modified model through an 185 . 20° . . GREE_NL_AND 10° 0 0 Fast ice (2J Ice edge O 20 40 10° OCEAN c-1) SPEED(cm-se 100 · re(6•24) · 1 F1.gu Apr1. t imagery, . s from Landsa d Wadhams "ft observation SCOR 1979 an Dn 1976 [after ' . 21 to May 7' 1981] • 186 Figure(6·25) Contours of ice drift speed (cms- 1 ) for 30 April 1979 (produced by the model). The square shows the area covered by the map in the previous figure . 187 "I Figure( 6•26) 6 - ....--.. ---s 5 ..._ ..__, -- --- [/J -- '- [/J '-- (l) 4 ....... 12 ....... C) ....... •rl 3 '-,..q ....... +> " ' (l) 2 " C) ' · rl Observed ----- ' ~ " Simulated " co 1 (l) ::.:: 0 0 100 200 300 400 Distance from the coast (Inn) Comparison of the observed (Wadhams 1983b) mean ice thickness across Fram Strait with the model results. The ice thickness along the northern edge of the grid is specified in the model run , however the thickne sses given here correspond to the position of th e submarine track which was a little further south. ....--.. [/J ............ s ..__, >-, +> •rl C) 0 ,...., (l) :> +> 'H ·rl H "d (l) C) H Figure( 6·27) 0 .4 0 .3 / / / / / / Obs e rve d -----/ / Simul a ted I I I I I 0 0 I 0 100 200 300 . 4 00 Di stance f r om t h e c oast (Inn) Comparison of the observe d (Wadhams 1983b) ice v elocity prof i le across Fram Stra i t with the model resul t s . +> (.) Q.) r/J ~ ro H +> '+-I 0 ] H Q.) Pi ,......_ ~ I r/J "' s '--' ~ ~ .-i '+-I Q.) (.) H Figure(6·28) I 14 00 j 1200 j --- --.,..- ' 1000 j / ' / ' I / " 8 00 " GOO j I I ' I ' I " 400 j I " " I Observed ----- " 200 j I I Simulated I 0 0 100 200 300 400 Di s tance f r om t he coas t (km) Comparison of the ice flux profile across Fram Strait der ived from observations (Wadhams 1983b) and the model results. 189 I 1 I I, 1 ! 11 entire equilibrium run, but at present this is prohibitively expensive in terms of computer resources. In this section we restrict our attention to particular parts of the model run and compare the standard results with those from a modified program. This is found to produce results of a quality sufficient to suggest improvements to the model. 6.3.2 Drift trajectories During the time of the model simulation, a satellite-tracked buoy (number 1924 of the Arctic Ocean Buoy Program, Thorndike and Colony 1980) drifted southwards near the East Greenland coast. It passed through the region of fairly thick ice where the ice interaction term is important. In the standard simulation the observed ocean currents were reduced so that the resulting ice velocities along the East Greenland Current reach O·Jms-1. However the mean velocities that resulted from this were too low to give the correct mean drift rates. Thus, for the drift tests no such adjustments to the ocean currents were made. There is some difficulty in interpreting the results of drift track calculations in regions where the thickness distribution has not been verified. Studies of this sort are more useful where a concurrent set of drift track measurements and wind and current velocities are made. Such studies made during the AIDJEX programme resulted in the determination of the air and water turning angles used in the model described here. There, calculated and observed drift tracks were made to coincide by adjusting these turning angles (McPhee 1980). Figure (6. 29) shows the simulated drift trajectories and the observed trajectory of the buoy for a 9 day period. The simulated drift rates are too smal 1, something that may be due to a number of causes. The most probable cause is from errors in the imposed ocean current field. This could also explain the slight difference in the direction of the real and simulated drift. Also, because the simulated buoy moves too quickly towards the coast where the ice is thicker it slows down too much, compounding the error. 190 I , I Figure( 6•29) 21 April 1979 A Observed • 35 km 29 April 1979 Drift trajectories: The track of a buoy is compared to that predicted by the model. The dots along the paths show the positions at times one day apart. Point A shows the pos i tion reached by the buoy for a run in which the shear viscosity is reduced by a factor of four . The arrow shows the simulated initial direction of the buoy if the ice strength is set t o zero. 191 ., ...... 192 Changing the viscosities used can affect the drift. Point A in figure (6.29) shows the position reached by the buoy in a model run in which the shear viscosity is reduced by a factor of four. This gives a result significantly closer to the observed buoy position. Neglecting the shear viscosity altogether would in this case improve the result further. The bulk viscosity term cannot be neglected however and this is shown by the arrow at the start of the drift tracks which shows the direction of a buoy obtained by setting the ice strength to zero (which gives rise to small viscosity terms). The effect of the bulk viscosity term is seen to prevent excessive drift towards the coast. It should be noted that the buoy in question was within only one or two grid lengths of the coast suggesting the possibility of problems associated with the resolution of the grid. 6.3.3 Wind-induced currents The momentum equation in Hibler's (1979) model inciudes forcing due to long term geostrophic currents. No attempt had been made to account for the effect of the long term ice circulation upon the current, and neither had the effect of short term wind conditions upon the surface current been included. The surface current is in fact a combination of the long term geostrophic, wind-induced (directly or indirectly via the ice) and tidal currents. The Greenland Sea is fairly open and not shallow enough for the tidal current to be significant there, although in other areas, the tides can affect the sea ice velocities (for example, in the Laptev Sea (Zubov (1943)). The influence of the ice motion on the ocean currents would repay study as part of a fully interacting ocean ice model. As far as the short term wind-induced current is concerned, an attempt could be made to simulate this by adding an Ekman surface current with a magnitude modified by the amount of ice present. This was done for the standard simulations, and here we compare results obtained by running the model without a wind- induced current parameterization. Ekman's solution for the water velocity due to a steady wind stress acting on an infinitely deep ocean with no boundaries and an eddy viscosity constant with depth , gave a surface current acting at 45° to the II' I ! II I H I Ill I I I II 11 I I 11 . II I right of the surface wind (in the northern Hemisphere) (see for example Pond and Pickard 1978). Ekman found experimentally that, the magnitude of the surface current is related to the surface wind speed lul by the relation 0·0127 = ~sin!~! where ~ 1s the latitude. Thus in complex notation u e = 0·0127 U -in/4 ~sinl~I ge (6) (7) The simplest way to include the modified Ekman current into the the current stress formulation of the model is to turn the Ekman current by the geostrophic current turning angle (but with the opposite sense) and add it to the geostrophic current. The quadratic water stress formula for the geostrophic current will then turn the geostrophic current, as it should do, and also turn the Ekman component back again to its surface direction before evaluating the ice-water stress. In figure (6.30), the effect on the ice velocity field due to the introduction of a wind-induced current parameterization 1.s shown. The figure shows the difference between the velocity solutions after running the model through 18 time steps both with and without a modified current. As can be seen, the effect is to enhance the velocity toward the ice pack near .the 1.ce edge. The change is large enough near to the ice edge to significantly affect the solution there. The velocity within the pack is almost unchanged. The effect of wind-induced currents is thus of importance when modelling the ice pack very near the ice edge. 193 Difference velocity field (5 May 1979) - - ' J • I Figure(6•30) 4 cm/s 194 The diagram shows the difference between the resulting ice velocity v ector fields after running the model for 28 t i mesteps wi t h and without i ncluding a parameterization of the effects of wind-induced currents . 7. CONCLUSIONS AND FURTHER WORK 7. 1 A summary of the new features in this model We have shown that it is possible to construct a sea ice model that predicts floe size, as well as ice thickness and velocity, giving a comprehensive description of the ice conditions in a region . The floe size is a particularly useful parameter to include as it is easily measurable from aerial photographs and this model is the first to incorporate specific representation of floes. The Greenland Sea region was chosen for the simulations, for which reasonable agreement with observations has been obtained. The floe size distribution maps seem to give a very good indication of the sea ice conditions with respect to what . . is seen in satellite imagery. Where the results are not so good, the path towards more complete coupled models is indicated. The introduction of floes into an already fairly complex sea ice model allows for the possibility of many feedback mechanisms. Included in the model here is a lateral melting feedback in which the amount of lateral melting occurring in the model depends upon the average floe size. The model developed here represented the ice with a multi-level formulation, this type of model not having previously been applied to the East Greenlan? region. This enabled estimates to be made of the various ice types present in the Greenland Sea, and their variation with time, giving a general agreement with observations. The model represents an advance over those large scale sea ice models that do not include a full rheology as part of the momentum equation. This means that better ice thicknesse s are predicted (particularl y where any interaction with land is likely) than models such as that of Parkinson and Washington (1979) which uses ad hoe methods when dealing with the ice interaction. Parkinson and Washington solve the momentum equation without I 'JG the ice interaction terms, and subsequently adjust th e velocity solution so that the resulting ice concentrations do not exceed a given max imum value. Compared to those models that do have sophisticated rheologies (Hibler 1979, Tucker 1982), our model is of more direct use in that easily verifiable sea ice properties are dealt with. Also, because many of the processes included are based on physics at a fundamental level, there is less tuning needed to run the model. For instance, the ice strength is determined completely from the ice thickness distribution, whereas Hibler's model has an adjustable parameter in the strength determination. Many theoretical results concerning the behaviour of randomly distributed circular floes have been obtained. Previous studies of the mathematics of finite sized floes have been very limited and are not easily extended. In particular, they were restricted to one dimension and thus avoided the difficulties involved in the more realistic two- dimensional problem. The treatment here is two-dimensional and is such that many aspects of pack ice behaviour not derivable from a consideration of ice as a continuum can be made. In particular, the idea of pack ice as consisting of floes was used to derive relationships for the ice strength for various conditions. A number of results concerning the physics of ice floes were obtained that are of interest not just from the modelling point of view, but also in their own right. In particular, very little work on the wind-induced break up of floes has been done before. A full treatment of the ice dynamics applicable to ice modelling was made, resulting in the theoretical derivation of a plastic yield curve for the determination of the ice interaction terms. The results from the drifting buoy calculations suggest that factors neglected in the yield curve calculation may yet be important; however the fact that a yield curve has been derived at all is significant here. Previous studies have assumed the shape of a yield curve as the starting point. The derivation here can serve as a basis for designing new yield curves . [I 7.2 Future development of sea ice modelling There are essentially two directions in which the futur e course of sea ice modelling can go . One direction is towards complexity and t he inclusion of more and more factors to try to represent the many ph ysical processes occurring in nature . To model fully the entire range of ice types found in the (East) Green land Se a, from the fast ice at the coast to the active region very near the ice edge with its complicated physics, more comprehensive mod e ls will be needed than exist at present. Inst e ad of making models more complicated, we could aim for simplicity, to mod e l sea ice using only the most important factors . The advantages would be ec onomy and speed of operation. Such models , if they can be mad e to provide good results, would be most useful for coupling with ocean and atmosphere models. The most difficult aspect of the simplification would be to retain ice interaction terms which, as we have seen, are vital. The most important part of the rheology appears from the results here to be the bulk viscosity. A simplified rheology involving only the bulk viscosity and neglecting the shear viscosity may be more economical and produce reasonable results in the East Greenland region, although it may not be valid in the Central Arctic. There is no reason why the model cannot be applied to regions other than the Greenland area. The model could be used in the Arctic Basin although the parts of the model concerned with floes would be wasted since large floes would be predicted everywhere. A more useful exercise would be to model the marginal ice zone in the Bering Sea where floes are observed and have been measured. The ice there is generally thinner near the ice edge than in t he Greenland Sea , and so different floe size prof iles would be expected. The Antarctic with its large seasonal sea ice zone and unique climate could also be usefully modelled. The mqdel is sufficiently general to cope with a wide range of conditions. The representation of floes in t he model can be used to provide a feedback mechanism for the dynamics, in that different rheologies may be use d in regions of smal 1 floes and where the ice is continuous. The research has not yet been done for determining the various rheologies to use in various regions, but the model here provides a framework in which the results of such studies may be incorporated. Without going as far as coupling ice models with ocean and atmosphere models, the most promising improvement to the model for studying the East Greenland region would be to have a more dynamic upper ocean layer forced by the factors already used for the ice forcing. This has been partly investigated here with the introduction of a parameterization of the wind- induced currents. Further improveme nts could involve the lateral transport of the oceanic mixed layer together with its heat content. This could, for instance, advect warmer water with the West Spitsbergen Current to increase the ice-free area within Fram Strait. Without an ocean model however, no forcing of the mixed layer from below can be achieved. Although the problems of modelling sea ice are great, particularly in marginal ice zones, there are many avenues open for further research. Improvement to the models and interesting results will surely come. 1, APPENDIX APPENDIX Listing of the code In this appendix, we give an outline of the contents of the model code with some details of the numerical methods used, not mentioned in the text. A description of the main subroutines are given, although the code contains sufficient comment lines for the logic to be followed. The figure here shows the subroutine structure of the program and is up to four levels deep. BNDRY ~ LEVELS~ MAIN ~ RELAX~::~::: ~ADVECT >DIFFUS REDIST GVECT > DIFFG UVT FINE fauNFINE VOLICE FORM DIST~AREA PLAST~LEN ~ JPH~:::::T--=~::~::: Subroutines ADVECT, BNDRY, DIFFUS, FORM, PLAST and VOLICE are essentially the same as used in Hibler's (1980b) code. However PLAST is retained only for comparison with SINLEN which has replaced it. Subroutine RELAX and its associated subroutines FDIFFl and FDIFF2 have had only minor modifications, as outlined below. GVECT and DIFFG are simply multi-level versions of ADVECT and DIFFUS and are trivially obtained. The rest of the subroutines are new and the other subroutines , in Hibler's code are no longer needed. The basic numerical methods for performing the time integration to second order accuracy are retained as are the second order accurate spatial finite differences. The reader 1.s referred to Hibler (1979, 1980b) for a full description of the numerical methods, but here we will mention some of the salient features. 199 The advection (ADVEC1) is performed by a conservative 2-2 (second order accuracy in space and time) explicit scheme (modified Euler step) together with small harmonic and biharmonic diffusion (DIFFUS) terms that prevent numerical instabilities. The advection equations are staggered in time with respect to the momentum equations which gives an efficient scheme for coupled equations of this sort. The momentum equations are also 2-step so that two relaxation solutions are needed at each time step (RELAX), the first giving an approximate velocity solution at half the time step to use in the non-linear velocity terms during the second relaxation solution to give the solution after the full time step. For each grid cell, the velocities are specified at the corners, and physical properties such as the thickness distribution and floe number density are specified at the centr~s of the grid cells . Near coasts, the velocity points are forced to be zero in the solution to the momentum equation. The cells are of four types. 1) ordinary ice or open water, 2) land cells 1.n which no 1.ce 1.s allowed to advect or diffuse, 3) open boundary cells which allow free inflow and outflow of ice by setting the viscosities and strengths to zero, and resetting the ice properties within the cell to be a weighted average of the properties 1.n the non-open boundary cells neighbouring it, and finally 4) cells in which the 1.ce amounts are specified as input boundary conditions . Subroutines BNDRY and NORTH deal with type 4) cells. Of the other subroutines , the most important 1.s DIST which deals with the redistribution of the ice thickness levels due to both dynamic and thermodynamic causes. The corresponding changes to the number of floes within each thickness level are also handled here. In addition, any temperature changes to the mixed layer are made at this stage. Subroutine DIST calls subroutine RIDGE which redistributes ice between the thickness levels during ridging to produce a normalized thickness distribution. Subroutines SINLEN and ALPHA take the 1.ce velocity field as input and give the viscosities for use in solving the momentum equation for which subroutines FORM and RELAX are used. Subroutines BUDGET and BALANCE evaluate the ice growth rates as functions of the ice thickness , the time of the year, and thermodynamic inpu t parame t ers such as the wind speed, the 200 cloud amount, the reiative humidity, and the air temperature. Subroutine NORTH provides the boundary conditions for the northern part of the grid corresponding to Fram Strait. A subroutine of this type would not be needed in a study of the central Arctic where all the ice would be generated within the system. Subroutine GRNLND reads in data concerning the ocean currents and heat flux specific to the Greenland grid used in this study, Finally, subroutine UVT reads in the winds and surface air temperatures when they are needed. 1. Solution of the momentum equation: in RELAX The solution of the momentum equation by over-relaxation involves considerable computation, especially when the grid is large. Thus any time saving techniques for this part of the model would be useful. At those points in the model grid at which there is no ice, one method is to set the ice mass and strength to zero and solve for the ice velocity in the usual way. The solutions obtained at the no-ice ~oints are then disregarded or set to zero. If however, solutions at such points can be specified a priori then the computations need be performed only at the points where there is ice. To do this though, one must be careful not to specify an unreasonable ice velocity that would occur if a small amount of ice were allowed to drift there. This is because an obviously incorrect velocity woul9 influence the solution in the region where there is ice. For example, specifying the velocity to be zero at no-ice points would make the ice edge behave as a coastline. Thus although the value of the velocity solution at these points is not of interest, their influence on other solution points is . A possible method is to specify the solution at the no-ice points to be the drift velocity suggested by Zubov (1943) where the ice velocity is g iven as 2% of the wind speed and in the direction of the geostrophic wind, A modern study of the data from buoys in the Arctic ocean (Thorndike and 201 Colony 1982) sugg~st that for summer the ice velocity 1s related to the geostrophic wind according to u (1) although the constant of proportionality and the turning angle (18° to the right 1n this case) vary seasonally. The situation 1n Greenland more closely resembles the situation in the Arctic during summer than for other times so (1) will be used here. It should be noted that near the ice edge, the ice interaction term is negligible so that empirical formulae of the form of (1) are more applicable there. When modelling real geographical . regions, the time saving involved by passing the solution calculation at land points and zero-ice points may be significant. Also, because fewer unknowns are calculated the number of iterations needed for convergence may well be less than otherwise. Another way 1n which slightly faster evaluation of the velocity solution can be obtained is to use values of the relaxation parameter w that varies over the grid. Hibler's model uses w = 1·5 for all the points. The relaxation parameter is used as follows. Suppose f(u) . = u requires a numerical solution for u. If un is the solution after n iterations, then un+l the solution at the next iteration is given by (2) If w = 1, un+l _= un and there 1s no relaxation. For w > 1 and w < 1, we have over-relaxation and under-relaxation respectively. The optimal relaxation parameter to use depends on the particular equation involved. The best r esults obtained in practice with this model were obtained by initializing the relaxation parameter for each equation of the system tow .. k= 1•48. The suff i ces i and j refe r t o the g r id poin t lJ and the suffix k= 1 or 2 , for the u or the v equation. At each i te r a tio n a simple test is done to compare t he changes to t h e veloc ity solution. If t he differences appear to be . . 1ncreas 1ng , lO 1s decreas ed by 0•0 2. If the 202 differences are, decreasing, indicating stability, w is allowe d to increase by 0·01. However, w is kept within the limits (l •0,1 · 5). Formally , 1 •48 n+l w = wn + 0·01 if f(un)-un < f(un-1)-un-l wn - 0 • 02 otherwise (3) Over a number of time steps, this scheme showed a 4 % decrease in the number of iterations needed for convergence, compared with taking w = 1·5 everywhere. 2 0 3 l 2 3 4 5 6 7 8 9 10 ll 12 l 3 14 15 16 l 7 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 The listing C Mai n drivin g program for viscous plastic sea ice mod e l C i nco rporating floe size number densities · IMPLICIT REAL*8 (A-H,0-Z) DIMENSION U(2l ,36, 3), V(21,36 , 3) ,ETA(22 ,37) ,PRESS(22, 37) ,ZETA(22, 37 1), DRAGS( 21, 36), DRAGA( 21, 36) ,GAIRX( 21, 36), GAIRY(2.l, 36) ,GWATX(21, 36) 2,GWATY(21,36),G(6,22,37,3),GAHMA(l8,l8),AMASS(2l,36),FORCEX(2l,36) 3 , FORCEY(21,36),TMIX(22,37),Gl(6),G2(l8),G3(18),UC(21,36),VC(21,36) 4 , FND(5,22,37,3),GAMMAF(l8),EKMANX(21,36),EKMANY(21,36),PHI(2l,36), 5TAIR(21,36) , C(52) ,P (5 2) ,RHUM(52),EDGE(52) ,HTSEA(22,37) ,AVV(22,37 ,3 6) ,UAV(21,36), VAV (21,36) , COMMON/EE/ECCEN COMMON/GRIDI/NX, NY, NXJ, NY l, NXMl, NYM l, NL, NLMl, NFINE, NS, NSMl COMMON/GRIDR/HI(22,37,3),HMS(l8),HWS(l8),HS(l7),HM(6),HW(6),H(6),T lOP COMMON/STEP/DT,DX,DY COMMON/ARRAY /GMASK(22 , 37),UVM(2 1, 36) COMMON/OUTFLO /OUT(22,37) COMMON /SIGMA/ALPHR(22 , 37),ALPH0(22,37),EI ,EII2, COTT COMMON /PHYS/CB ,CF,STREN,RHOICE,RHOWAT,GRAV,COT,UA,UB,AH,BH,RHOAIR, lQI,CW,DMIX,SIDER,TI,SIGMAX,PI,ARMAX,CPLF,CONl,CON2,STFN,CON,ERROR2 2 , Dl, D3, TING, CH, S1N20, COS20, SIN25, COS25 C Various modes of printout C Set IPl•l for output of info rmation at start of run I Plml · C Set IP2•1 for basic informat i on every NFULL'th time step I P2• l C Set IPJ•l for more information every NFULL'th time step IP3 .. J C Set INIT•l if reading in input from previous run INI T•O C Set IWIND•l if using new wind file IWIND= l C Set IOUT•l if final variable s are to be output IOUT=l C Set ITMIX•l if using artific i al TMIX field as input ITMIX•l C Decide on .basic parameters NUM1Ts 6 NFULL• S NX=21 NY=36 NL=6 NX [cNX+l NYl=NY+l NXM l=NX-1 NYMl=NY-1 NLMl•NL-·l NFINE• 4 NSm((NL-2)*NFINE)+2 - NSMl•NS-l 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 NG=5 TOP= l. OD-06 STREN=9.0D+03 DELTT=2.l6D+04 DT=2.l6D+04 DX=3.5D+04 DY=3.5D+04 ERROR= I • OD- 02 ERROR2=0.0]D0 TINC=ERROR2*1.0D-02 DIFF1=4.0D-03*DX ECCEN=2.0D0 RHOAIR=l.3DO RHOICE=0.922U+03 RHOWAT=0.1025D+04 GRAV=9,82868DO COT=l .428DO SIN20•0.342DO COS20u0,9397D0 SIN25=0 .4226DO COS25=0.9063DO UA=0,5DO UB=0.5DO C8=((RHOWAT-RH0ICE)/RHOWAT)*RHOICE*GRAV*D.5D0 CF~CB*COT*(UA+UB)*RHOICE/RHOWAT AH=RHOWAT*0.5DO/(COT*(UA+UB )*RHOICE) BH=2,0DO*STREN/CB CHzBH/ (4. ODO-AH) QinJ.02D+08 CW=4. l 9D+06 DMIX=J.OD+Ol TI=QI/(CW*DMIX) SIGMAX=l.OD+05 PI=J.14159265358979300 STFN=5.67D-08 CON=2. 165600 . Dl=2.28DO D3=5.5D-08 ' CON1=4.5DO*S1GMAX/(RHOWAT*GRAV*3.43D-06) CPLF=(l.OD+l0 /(3.0DO*RHOWAT*GRAV))**(0.2500) CON2=3.0D0*RHOWAT*GRAV*3.43D-06/(DSQRT(2.0DO)*DEXP(PI/4.0D0)) SIDER=(PI/4.0D0)+(4 . 0DO/PI) RRMJ\X:l.0D+04 ARMAX=PI*RRMAX*RRMAX LAD=2 IUVT"'l C Initialise counter DAY=334.75DO ICOUNT•O TOUT• O.ODO I\) 0 .i::- I lOl i 102 1 103 104 105 106 107 108 109 110 111 112 113 114 115 116 11.7 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 1'33 134 135 136 13 7 138 139 140 141 142 143 144 145 146 147 148 149 l ~O C Define boundaries , CALL BNDRY C Set up vertical thickness levels CALL LEVELS C Calculate coefficients used in redistribution CALL REDIST(GAMMA,GAMMAF) C Input test data 2 3 4 5 DO 2 J=l ,NYl DO 2 I=l ,NXl TMIX(I,J)=2. 712D+02 DO 2 K=l, 3 DO 1 L•2,NL G(L,I,J,K)=O.ODO CONTINUE G(2,I,J,K) a0.25DO*OUT(I,J) G(3,I,J,K)=0.25DO*OUT(I , J) G(~ 1 I,J,K)•0.25D0*0UT(I,J) G(5,I,J,K)•0.25DO*OUT(I , J) G(l,I , J,K)cO.ODO*OUT(I,J) CONTINUE DO 3 J•l , NY DO 3 I•l ,NX PHI(I , J) • 75.0DO*PI/180 . 0DO TAIR(I,J)•274 . 0DO CONTINUE DO 4 NWEEK= 1, 52 C(NWEEK) 0 0.75DO RHUM(NWEEK)c0 . 7DO P(NWEEK) =l01400 . 0DO CONTINUE DO 5 J•l,NY DO 5 I•l ,NX GAIRX(I,J) • S,000 GAIRY(I,J)=O,ODO GWATX(l,J)=O.ODO GWATY(l,J)=O.ODO CONTINUE C Call data specific to Greenland CALL GRNLND(GWATX,GWATY,PHI,C,RHUM,P,EDGE,HTSEA) C Read initial thickness distribution and mixed layer temperature IF(INIT.EQ.O)READ(l3)G , Hl,TMIX C If observed surface temperature field is available, read it in IF(ITMIX.EQ.O)READ(12)TMIX 6 DO 6 K=l,3 DO 6 J= 1, NY 1 DO 6 I=l ,NXl IF(OUT(I,J).EQ.0.0DO)HI(I , J,K)=H(NLMl) +TOP DO 6 L=l, NL G(L,I,J,K)=G(L,I,J,K)*OUT ( I,J) CONT IN UI': 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 DO 7 K•l,3 DO 7 J•l,NYl DO 7 I,. 1, NX 1 . DO 7 L=l,NLMl FND(L,I,J,K)=G(L+l,I,J,K)/ARMAX 7 CONTINUE C Zero out velocity 8 DO 8 J=l,NY DO 8 I=l ,NX GWATX(I,J)=GWATX(I,J)+l.OD-08 U(I,J,3)=0.0DO . U(I,J,2)=0.0DO U(I,J,1)=0.0DO V(I,J,3)=0.0DO V(I,J,2)=0.0DO V(I,J,l)=O.ODO UC(I,J)=O.ODO VC(l,J)aQ.ODO UAV(I,J)=O.ODO VAV(I,J)=O.ODO CONTINUE C Zero out viscosities DO 9 J=l ,NYl 9 DO 9 I• l, NXl ETA(I,J)=O,ODO ZETA(I,J)=O,ODO CONTINUE C Get first value of U and V THETA• 1. ODO C Evaluate strength DO 11 J • 1 , NY l DO 11 1=1,NXl DO 10 L= 1, NL Gl(L)=G(L,I,J,l)*(l,ODO+l.OD-04) 10 CONTINUE C Refine the grid CALL FINE(Gl,G2) C Call RIDGE to determine the initial ice strength CALL RIDGE(I,J,G2,G3,GAMMA,GAMMAF,PRESS(I,J)) 1.1 CONTINUE C Read in winds and air temperature CALL UVT(GAIRX,GAIRY,TAIR,ICOUNT,INIT,IUVT) CALL FORM(U,V,ETA,ZETA,DRAGS,DRAGA,GAIRX,GAIRY,GWATX,GWATY , FORCEX , lFORCEY,G,AMASS,PRESS,P HI,EKMANX,E KMANY) DO 13 J= l , NY l DO 13 I=l,NXl HM(NL)=(HI(I,J,l)+H(NLM1))*0.5DO C Define initial viscosity ZETA(I,J)=O.ODO DO 12 L"'2,NL t\J 0 Vl 201 ZETA(I,J)•ZETA(I,J) +(G(L,I , J .,l)* l,OD+ll*HM(L)) . 202 . 12 203 CO NTINUE . ETA(I,J) • ZETA(I,J)/(ECCEN*ECCEN) CONTINUE 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 13 C Write out t hickness leve ls 14 15 IF(IPl.EQ.l) WRITE (6,14) H,HI(2,2 ,l) FORJ.1AT( ////,1X, ' Thickness levels 1 , 9X,6(2X , F7,4)) IF(IPl.EQ.l) WR ITE(6,15)HM FORJ.1AT(lX, 'M i dpoint s of l eve l s' , 7X,6(2X , F7 , 4)) IF(IPl.EQ.l) WRITE(6,16)HW 16 FORJ.1AT(lX, 'l eve l wid ths' ,14X,6 (2X,'F7 .4 ),//) C IF(IPl.EQ.l) WRITE(6,ll) . Cl l . FORJ.1AT(/// ,31X, 'Redistribution coe fficients, GAMMA(Ll,12) 1 /) C IF(IPl.EQ.l) WRITE(6,12)((GAMMA(Ll,L2),Ll•l,NS),L2=1,NS) Cl2 FORMAT(lX, 18( 1X,F6.4)) . C Ca l culate ice velocities if this is th~· f irst run I FIRST= l IF(INIT.EQ,0) CALL RELAX(U ,V ,ETA,ZETA,DRAGS,DRAGA,AMASS,FORCEX,FOR lCE.Y, ERROR, THETA, UC, VC, IFIRST ,GAIRX, GAIRY, EKMANX , EKMANY) IFIRSTaO ERRORm 1. 0.D-05 DO 17 J~l, NY DO 17 I~l,NX U(I,J,2) • U( I , J,l) U(I,J,3)=U(I,J,2) V(I,J,2) aV(I ,J, l) 228 17 V( I ,J,3)=V (I ,J,2 ) CONTINUE 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 C Re ad in output from previous run if available. lF(lNIT,EQ.l) READ(lO)G,HI,FND,U,V,TMIX,TOUT,DAY,ICOUNT,POSX,POSY P0SXz5, l5 DO P0SYal 6 ,2DO IF ( IWIND.EQ;l ) ICOUNT• O CALL VOLICE(G,VOL) DO 18 J•l ,NY DO 18 I=l, NX UC ( l,J )• U( I, J,l) VC( I ,J)=V(I ,J ,l) 18 CONTINUE C Predictor corrector procedure starts here 19 CONTINUE C Update timeste p and calculate day and week of the year ICOUNT=ICOUNT+l . NUMIT=NUM IT-1 DAY=DAY+DELTT/8 .64D+04 DAY=DMOD(DAY,365.0DO) NW=IDINT(DAY*52.0D0/365.0D0)+ 1 WRITE(6,58)ICOUNT,DAY C Re ad in wi nd s and temperature CALL UVT(GAIRX,GAIRY,TAIR,ICOUNT,INIT,IUVT) 251 252 253 254 255 256 257 258 259 260 261 262 263 264 26 5 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 IUVT~O C Set reasonable thickness and floe number dist ribution at outflow cells DO 22 L=l,NL DO 20 J=2,NY DO 20 1=2,NX . UC(I,J)=(G(L,I-l,J-1,l)*OUT(I-l , J-l)+G(L~I+l , J-l,l)*OUT(l+l , J-l)+G l(L,I-l,J+l, l )*OUT(l-l , J+l)+G(L,l+l,J+l,l)*OUT(l+l , J+l)+4, 0DO* (G(L, 21,J-l,l)*OUT(I,J-l)+G(L,I-l,J , l)*OUT(I-l,J)+G(L,I+l,J , l) *OUT (l+l ,J 3)+G(L,I,J+l,l)*OUT(I,J+l)))/(OUT(I-t,J-l)+OUT(l+l , J-l)+OUT(I- l,J+l 4)+0UT(I+l,J+l)+4.0D0*(00T(I,J-l)+OUT(l-l,J)+OUT(l+l,J)+OUT(l,J+l)) 5+1.0D-15) 20 CONTINUE DO 21 J=2, NY DO 21 1=2,NX G(L,I,J,l)•G(L,I , J,l)+(GMASK(I,J) - OUT(I,J))*UC(l , J) 21 CONTINUE 22 CO NTINUE DO 25 L=l,NLMl DO 23 J =2,NY DO 23 1=2,NX UC(I,J)z(FND(L,I-l,J-l,l)*OUT(I-l , J- l )+FND(L,l+l,J-l,l) *OUT(I+l,J - ll)+FND(L,I-l,J+l,l)*OUT(I-l,J+l)+FND(L,I+l,J+l , l)*OUT(I+l ,J+ l)+4 , 0 2DO*(FND(L,l , J-l,l)*OUT(I,J-l)+FND( L,l- l ,J, l) *OUT(I-l,J) +FND (L ,I+ l, 3J,l)*OUT(I+l,J)+FND(L,I,J+l,l)*OUT(l , J+l)))/(OUT(l-l,J-l)+OUT(I+l, 4J-l)+OUT(I-l,J+l)+OUT(I+l , J+l)+4 . 0DO*( OUT(l,J-l)+OUT(I-l , J)+OUT(I+ 51,J)+OdT(I,J+l))+l,OD-15) 23 CONTINUE DO 24 J=2,NY DO 24 1=2,NX FND(L,I,J,l)•FND(L,I,J,l)+(GMASK( l, J )-OUT(I ,J ))*UC(l ,J ) 24 CO NTIN UE 25 CONTINUE DO 26 J • 2,NY DO 26 1=2,NX UC(l,J)=(HI(l-l , J-1,l)*OUT(l-l , J-l)+HI(I+l , J-l,l)*OUT(I+l,J-l )+HI( lI-l,J+l,l)*OUT(l-l,J+l)+Hl(l+l,J+l,l) *OUT(l+l ,J+l) +4,0DO*(HI(l, J-l 2,l)*OUT(I,J-l)+Hl(I-1,J,l)*OUT(I-l,J)+HI(I+l,J,l)*OUT(l+l,J)+HI(i, 3J+l, l)*OUT(I,J+l) ))/(OUT(I-l ,J-1 ) +OUT(I +l ,J-1 )+OUT(I-1,J+l )+OUT(I+ 41,J+l)+4 , 0DO*(OUT(l,J- l)+OUT(I-l,J)+OUT(I+l , J)+OUT(I ,J+l) )+l ,OD-15 5) 2'6 CONTINUE DO 27 J=2,NY DO 27 1=2 ,NX Hl(I,J,l)=HI(I,J,l)+(GMASK(l,J)-OUT(I,J))*UC(l,J) 27 CONTINUE C Read in imposed nor t hern boundary condiations (Greenland) CALL NORTH(G,EDGE(NW),FND,TMIX) CALL VOLICE(G,VOLl) VOLl=VOLl-VOL C First do predictor t\J 0 °' 301 302 303 304 305 306 307 308 309 310 3 l l 312 313 31 4 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 34 1 342 343 344 345 346 347 348 349 350 DO 28 J • l ,NY DO 28 I•l ,NX U(I,J,3)=U(I,J,l) V(I,J,3)=V(I,J,l) UC(I , J)=U(I,J,l) VC(I,J)=V(I,J,l) 28 CONTINUE THETA=l . ODO DT=DELTT/2.0DO DO 30 J=l ,NYl DO 30 I =l,NXl DO 29 L=l ,NL Gl(L)=C(L , I,J,l)*(l.ODO+l.OD-04) 29 CONTINUE CALL FINE(Gl,G2) C Det ermin e ice strength CALL RIDGE(I,J,G2,G3,GAMMA,GAMMAF,PRESS(I,J)) 30 CONTINUE Ca lcu l ate · forcing CALL FORM(U,V , ETA,ZETA,DRAGS,DRAGA,GAIRX,GAIRY,GWATX,GWATY,FORCEX, lFORCEY,G,AMASS,PRESS,P HI,EKMANX,EKMANY ) CALL RELAX(U,V,ETA,ZETA,DRAGS,DRAGA,AMASS,FORCEX,FORCEY,ERROR,THET l A,UC , VC,!FIRST,GAIRX,GAIRY,EKMANX,EKMANY ) C Do r egular time step C Do backwards time step I THETA= l. ODO DT=DELTT CALL FORM(U,V,ETA,ZETA,DRAGS,DRAGA,GAIRX,GAIRY,GWATX,GWATY,FORCEX, lFORCEY,G,AMASS,PRES S,PHI,E KMANX,EKMANY ) C Set U(l) ~U(2) a nd same for V DO 31 J:l,NY DO 31 Izl ,NX U(l,J,3)aU(I,J,l) V(I,J,3)=V(I,J,l) UC(I , J):U(I,J,l) VC(l,J)=V(I,J,l) U(l,J,l)=U(l,J,2} V(l,J,l):V(I,J,2) 31 CONTINUE CALL RELAX(U,V,ETA,ZETA,DRAGS,DRAGA,AMASS,FORCEX,FORCEY,ERROR,THET lA,UC,VC,IFIRST,GAIRX,GAIRY,EKMANX,EKMANY) C Advect each thickness level NUM=NL CALL GVECT(U,V,G,DIFFl,LAD,NUM) C Calculate new maximum thickness by advecting total volume, C and usin g conservation. DO 35 J= l ,NYl DO 35 l= l ,NXl DO 32 K=l ,3 AVV(l,J,K)=O.ODO 32 33 34 35 36 37 38 39 40 CONTINUE DO 34 K:1,3 DO 33 L:2,NLMl AVV(1,J,K)=AVV(I,J,K)+G(L,I,J,K)*HM(L) CONTINUE AVV(I,J,K)=AVV(l,J,K)+G(NL,I,J,K)*0.5DO*(H(NLMl)+Hl(l,J,K)) CONTINUE CONTINUE CALL ADVECT(U,V,AVV,DIFFl,LAD) DO 36 J=I ,NYl DO 3 6 I= 1 , NX 1 HI(l,J,2)=HI(I,J ,l) CONTINUE DO 40 J=l,NYl DO 40 I=l ,NXl IF(G(NL,I,J,l).LE.O.ODO) GOTO 38 SLiM=O.ODO DO 37 L=2,NLM1 SUM• SUM+G(L ,l, J ,l )*HM(L) CONTINUE HI(I,J,1)=(2.0DO*(AVV(I ,J,l )-SUM)/G(NL,I,J , 1))-H(NLMl) GOTO 39 · HI(I,J,l)•H(NLMl)+TOP CONTINUE CONTINUE 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 C 389 390 391 C Advect floe number densities NUM=NLMl 392 393 394 395 396 397 398 399. 400 41 CALL GVECT(U,V ,FND,DIFFl ,LAD,NUM) DO 41 J • l,NYl DO 41 I=l ,NXl IF(GMASK(I,J).EQ.O.ODO)HI(I,J,l)•H(NLMl)+TOP HI(I,J,l)•DMAXl(HI(I,J,l),H(NLMl)+TOP) HI(l,J,l)=DMINl(HI(I,J,l),25.0DO) CONTINUE C Carry out thickness and floe size redistributions 42 43 44 CALL DIST(U , V,G,GAMMA,GAMMAF,PRESS ,TMIX,FND, GAIRX,GAIRY,DAY ,NW,PHI l,TAIR,C,P,RHUM,HTSEA) Correct outflow points and get outflow i ce CALL VOLICE(G,VOL) DO 44 J=l ,NYl DO 44 I=l,NXl HI(I,J,l)=DMAXl(Hl(l,J,l),H(NLMl)+TOP) DO 42 LF=l,NLMl FND(LF,I,J,l)=FND(LF,l,J,l)*OUT(I,J) CONTINUE DO 43 LG=l ,NL G(LG,l,J,l)=G(LG,l,J,l)*OUT(I,J) CONTINUE IF(G(NL,I,J , l).EQ.O.ODO)Hl(l,J,l) • H(NLMl)+TOP CONTINUE l'v 0 -..J 401 402 403 404 405 406 407 408 409 410 411 412 4 13 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 43 1 432 433 434 43'5 436 43 7 43 8 439 440 441 442 443 444 45 46 CALL VOLICE(G,VOL2) TOUTl • VOL-VOL2-VOLl VOL•VOL2 l'OUT 2 TOUT+TOUTl II • POSX JJ=POSY SY=POSY-DFLOAT(JJ) SX=POSX-DFLOAT(II) Ul•SY*U(II+l,JJ+2,l)+(l,OD0-SY)*U (II+l,JJ+l,l) U2 • SY*U(II+2,JJ+2,l)+(l.ODD-SY)*U(II+2,JJ+l,l) UJ=SX*U2+(1.0DO-SX)*Ul Vl=SY*V(II+l,JJ+2,l)+(l.ODO~SY)*V(Il+l, JJ+l,l) V2s SY*V(II+2,JJ+2,l)+(l,ODO-SY)*V(II+2 ,JJ+l,l ) ·v3=SX*V2+(1 ; 0DO-SX)*Vl POSX=POSX+DT*UJ/DX POSY=POSY+DT*VJ/DY WRITE(6,45)DAY,POSX,POSY FORMAT(l X, 'POSITION' ,3G20 .6 ) DO 4·6 J _. l, NY DO 46 I•l ,NX UAV(I,J)=UAV(I,J)+U(I,J,1) VAV(I,J) • VAV(I,J)+V(I,J,l) CONTINUE WRITE (6, 66)VOL WRITE(6,67)TOUTl 1 WRITE(6,68)TOUT C Print out every K'th point KTH=MOD(ICOUNT,NFULL) IF(KTH.EQ,O) GO TO 47 GO TO 54 47 CONTINUE C Outp ut information a t ea ch time ste p 48 IF(IP2,EQ.l) WRITE~6,~9) IF(IP2,EQ.l ) WRITE(6,60) IF(IP2.EQ,l) WRITE(6,56)((U(I,J,l),I•l,NX,NG),J•l,NY,NG) IF(IP2.EQ.1) WRITE(6,61) IF( I P2,EQ.l) WRITE(6,56)((V(I,J,l),I•l,NX ,NG),J•l,NY,NG) IF(IP2,EQ.l) WRITE(6,48) FORMAT(/ , lX, 'FLOE NUMJIER DENSITY',/) IF(IP2,EQ.l) WRITE(6,49)((FND(l,I,J,l),I• l,NX1,NG) , J=l,NY1,NG) IF( I P2.EQ.l)WRITE(6 ,57) IF(IP2,EQ.l) WRITE(6,49)((FND(2,I,J,l ) ,I=l,NX1,NG), J=l,NY1 ,NG) IF( I P2 ,EQ. l )WRITE(6,57) 445 49 446 IF(IP2.EQ.l) WRITE(6,49)((FND(J,I,J,l),I=l,NX1,NG),J=i,NY1,NG) FORMAT(lX,5Gl7 . 10) IF(IP2,EQ,l) WRITE(6,62) 447 448 449 450 so IF(IP2.EQ . l) WRITE(6,50)(G(L,10,10,l),G (L ,20,1 0,l),L=l,NL) IF(IP2.EQ.l) WRITE(6,50)(G(L,10,20,l),G(L,20,20,l),L•l,NL) FORMAT(//,(1X,G20 . 12,5X,G20,12)) IF(IP3.EQ . l) WRITE(6,63) 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 49 4 495 496 497 498 499 500 51 52 53 IF(IP3,EQ , l) WRITE(6,49)((ZETA(l,J),l•l,NX1,NG),J•l,NY1,NG) IF(IP3.EQ,l) WRITE(6,51) FORMAT(/,lX, 'SHEAR VISCOSITY, ETA',/) IF(IP3.EQ.1) WRITE(6,49)((ETA( I,J),I• l,NX1,NG) , J • l,NY1,NG) IF(IP3,EQ,l) WRITE(6,52) FORMAT(/,lX, 'RIDGING AMOUNT, ALPHR' ,/) IF(IP3.EQ .l) WRITE(6,49)((ALPHR(I,J),I•l,NX1,NG), J • l,NY1 ,NG) IF(IP3 ,EQ.l ) WRITE(6,53) FORMAT(/,lX, 'OPEN WATER OPENING, ALPHO' , /) IF(IP3 .EQ.l) WRITE(6,49)((ALPHO(I,J ) ,I•l,NX1,NG),J• l, NY1,NG) IF(IP3.EQ.l) WRITE(6,64) IF(IP3.EQ.l) WRITE(6,49)({PRESS(I,J),I=l,NX1 , NG),J=l,NY1,NG) IF(IP3.EQ.l) WRITE(6,65) IF(IP3.EQ.l) WRITE(6,49)((TMIX(I,J),I=l,NXl, NG),J • l,NYl,NG) C Check count er a nd decide if new winds needed C Decide if done 54 CONTINUE 55 IF(NUMIT.EQ,O) GO TO 55 GO TO 19 CO NTINUE C Output information for nect run 56 57 58 59 60 61 62 63 64 65 66 67 68 C C C C C IF(IOUT.EQ.l) WRITE(l4)UAV,VAV IF(IOUT.EQ,l) WRITE(ll)G,HI,FND,U,V,TMIX, TOUT,DAY,ICOUNT,POSX,POSY STOP FORMAT(lX,5Gl7,10) FORMAT(/) . FORMAT(lX,'**** TIME STEP ',16,' DAY ',F7 ,2 ) FORMAT(6X, 'FULL DATA PRINTED FOR THIS TIME STEP ' ) FORMAT(/,IX,'X-COMPONE NT OF ICE VELOCITY') FORMAT(/,lX, 'Y-COMPONENT OF ICE VELOCITY') FORMAT(/,lX, 'ICE THICKNE SS DISTRIBUTION AT SELECTED POINTS') FORMAT(/,lX, 'BULK VISCOSITY, ZETA') FORMAT(/,lX, 'STRENGTH') FORMAT(/, IX, 1 MIXED LAYER TEMPERATURE 1 ) FO~~AT(6X, 'TOTAL VOLUME ',G20. 12) FORMAT(6X,'OUTFLOW 1 ,Gl4.7) FORMAT(6X, 'NET: ',4X,Gl4,7) END SUBROUTINE ADVECT(U ,V,AD,DIFFl,LAD) C Advection IMPLICIT REAL*B (A-H,O- Z) DIMENSION AD(22,37,3) , U(2l,36,3),V(2I , 36,3) COMMON/ GRIDI/NX,NY,NXl , NYl,NXMl ,NYM l,NL,NLMl ,NFI NE,NS,NSMl COMMON /GRIDR/HI(22,37,3) , HMS(l8),HWS(l8),HS(l7) ,HM (6),HW(6) , H(6) , T !OP . I\) 0 (J) . S01 S02 S03 S04 505 S06 S07 508 509 510 511 51 2 S13 51 4 515 51 6 517 518 51 9 520 S2 1 522 523 524 525 526 527 52 8 529 530 531 S32 53 3 534 535 536 537 538 539 ·540 541 54 2 543 544 545 546 547 548 549 550 / COMMON/STEP/DT,DX,DY COMMON/ARRAY/GMASK(22,37),UVM(21,36) C Decide if backward euler or leapfrog LL• LAD IF(L~ . EQ,l) GO TO 1 C Ba ckward euler DELTT=DT K3a2 GO TO 2 C Leapfrog l DELTT=DT*2,0DO K3=3 2 CO NTINUE C Rearrange DO 3 J=l ,NYl DO 3 I =l , NXl AD(I,J,3)=AD(I,J,2) AD(I,J,2)mAD(I,J,l) 3 CONTINUE C Go 4 5 through conse r vative advection DELTXm DELTT/(4,0DO*DX) DELTY=DELTT/(4 . 0DO*DY) CONTINUE DO 5 J•2 , NY DO S· I • 2,NX AD(l,J , l)•AD(I,J,K3)-DELTX*((AD(I,J,2)+AD(I+l,J,2))*(U(I,J,l)+U(I 1 1J-l , l))-(AD(I,J,2)+AD(I-l,J,2))*(U(I-l,J,l)+U(I-l,J-l,l)))-DELTY*( 2(AD(I,J,2)+AD(I,J+l,2))*(V(I-l,J,l)+V(l,J , l))-(AD(l,J,2)+AD(I,J-l, 32))* (V(l-l , J - l,l)+V(I , J-l,l))) CONTINUE C Decide if done 6 7 8 C Do 9 GO TO (10,8,6),LL CONTINUE DO 7 J • l ,NYl DO 7 1ml ,NXl AD( I ,J,2) • AD(I,J,3) CONTINUE GO TO 10 CONTINUE backward euler correction DO 9 J=l,NYl DO 9 I=l,NXl AD(l,J,3)=AD(I,J,2) AD(I,J,2) =0.SDO*(AD(I,J , l)+AD(I,J,2)) CONTINUE LL=3 K3=3 GO TO 4 10 CONTINUE · DO 14 KD=l ,2 551 SS2 553 554 SSS S56 SS7 558 SS9 S60 S61 S62 563 564 565 566 S67 568 569 570 S71 572 S73 574 575 S7 6 577 578 579 580 581 582 S83 584 50·s · S86 587 S88 589 590 591 592 593 S94 S95 596 597 598 599 600 IF(KD.EQ , 2) GO TO 11 CALL DIFFUS(AD,DIFFl,DELTT) GOTO 12 11 'DIFF2=-(DX*DX)/DELTT CALL DIFFUS(AD,DIFF2,DELTT) 12 CONTINUE DO 13 J=l ,NYl DO 13 I=l,NXl AD(I,J,l)=(AD(I,J,l )+AD(I,J,3))*GMASK( I, J) 13 CONTINUE 14 CONTINUE C C C RETURN END SUBROUTINE DIFFUS(AD,DIFFl,DELTT) IMPLICIT REAL*8 (A-H,0-Z) DIMENSION AD(22,37,3),AD1(22,37) COMMON/GRIDI/NX,NY,NXl,NYl,NXMl , NYMl,NL,NLMl,NFINE,NS,NSMl COMMON/GRIDR/HI(22,37,3),HMS(l8),HWS(l8),HS(l7),HM(6),HW(6),H(6),T lOP COMMON/STEP/DT,DX,DY COMMON/ARRAY/GMASK(22,37),UVM(21,36) C Subroutine diffuses AD,multiplies by DELT,and puts result in AD C Zero out ADl DO l J=l,NYl DO l I=l, NX l ADl(I , J)•O.ODO CONTINUE C Do diffusion DELTXX•DELTT*DIFFl/(DX*DX) DELTYY=DELTT*DIFFl/(DY*DY) DO 2 J=2,NY DO 2 1=2,NX ADl(I,J)=DELTXX*((AD(I+l,J,3)-AD(I,J,3))*GMASK(I+l,J)-(AD(I,J,3)-A lD( 1-1, J, 3) )*GMASK( 1-1 ,J) )+DELTYY>~((AD( 1, J + 1, 3 )-AD(l, J, 3) )*GMAS K(I, 2J+l)-(AD(I,J , 3)-AD(l,J-l,3))*GMASK(l,J-l)) 2 CONTINUE DO 3 J=l , NY l DO 3 I=l,NXl AD(I,J,3)=ADl(I,J) 3 CONTINUE C C RETURN END SUBROUTINE GVECT(U,V,G,DIFFl,LAD,NUM) IMPLICIT REAL*8 (A-H,0-Z) DIMENSION G(NUM,22 , J7,3),U(21,36,3),V(21 , 36 , 3) tv 0 I..O 601 602 60 3 604 605 606 607 608 609 610 611 612 613 614 6l5 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 COMMON/GRIDI/NX,NY,NXl,NYl,NXMl,NYMl,NL,NLMl,NFINE,NS,NSMl COMMON/GRIDR/HI(22,37,3),HMS(18),HWS(l8),HS(l7),HM(6),HW(6),H(6),T lPP COMMON/STEP/DT,DX,DY COMMON/ARRAY/GMASK(2 2,37) ,UVM (21,36) C Decide if bac kward euler or l eapfrog LL=LAD IF(LL.EQ.l) GO TO 1 C Backwa rd euler DELTT=DT K3=2 GO TO 2 C Leapfrog 1 DELTT=DT*2.0DO K3=3 2 CONTINUE C Rearrange DO 3 J=l, NYl DO 3 Ial,NXl DO 3 L•l ,NUM G(L,I, J,3) =G (L , I,J,2) G(L,I,J,2)•G(L,I,J,l) 3 CONTINUE C Go through con·s erva t i ve advec t i on ' DELTX•DELTT/(4,0DO*DX) DELTY•DELTT/(4,0DO*DY) 4 CONTINUE DO 5 J • 2 ,NY DO 5 I •2,NX DO 5 L• l, NUM G(L,I,J,l)•G(L,I,J,K3)-DELTX*((G(L,I,J,2)+G(L,I+l,J,2))*(U(I,J,l)+ lU(I,J-l,l))-(G(L,I,J,2)+G(L,I-l,J,2))*(U(I-1,J,l)+U(I-l,J-l,l)))-D 2ELTY*((G(L,I,J , 2) ~G(L , I,J+l, 2))*(V(I-1,J,l)+V(I , J,l))-(G(L,I,J,2)+ 3G(L,I,J-1,2))*(V(I-l,J-l,l)+V(I,J-l,l))) 5 CONTINUE C Decide if done GO TO (10 , 8,6),LL 6 CONTINUE DO 7 J:l,NYl DO 7. I = 1 , NX 1 DO 7 L=l, NUM G(L , I ,J,2 )=G(L,I , J , 3) 7 CONTINUE GO TO 10 8 CONTINUE C Do backwa rd euler correction D09J=l,NY1 DO 9 I•l, NXl DO 9 L=l , NUM G(L , I,J,3)=G(L,I,J,2) 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 G(L,I,J,2)•0.SDO*(G(L,I,J , l)+G(L , I , J , 2)) 9 CONTINUE LL=3 K3=3. GO TO 4 10 CONTINUE DO 15 L=l ,NUM DO 14 KD=l,2 IF(KD.EQ.2) GO TO 11 CALL DIFFG(G,DIFFl,DELTT , L,NUM) GOTO 12 11 D1FF2=-(DX*DX)/DELTT CALL DIFFG(G,DIFF2,DELTT,L,NUM) 12 CONTINUE DO 13 J=l ,NYl DO 13 I=l,NXl G(L,I,J,l)=(G(L,I ,J,l)+G(L,I,J,3))*GMASK(I , J) 13 CONTINUE 14 CONTINUE 15 CONTINUE C C C RETURN END SUBROUTINE DIFFG(G,DIFFl,DELTT,L ,NUM) IMPLICIT REAL*B (A-H,0-Z) DIMENSION G(NUM,22,37,3),Gl(22,37) COMMON/GRIDI/NX,NY,NXl,NYl,NXMl,NYMl,NL ,NLMl,NFINE,NS,NSMl COMMON/GRIDR/HI(22, 37 ,3) ,HMS( 18) ,HWS( 18) ,HS( 17), HM(6) ,HW (6) ,H(6), T !OP COMMON/STEP/DT,DX,DY COMHON/ARRAY/GMASK(22,37),UVM(21,36) C Subroutine diffuses ad,rnultiplies by delt,and puts result in ad C Zero out ad! DO 1 J=l ,NYl DO 1 I=l,NXl Gl(I,J)=O.ODO CONTI NU E C Do diffusion DELTXX=DELTT*DIFFl/(DX*DX) DELTYY=DELTT*DIFFl/(DY*DY) DO 2 J=2,NY DO 2 I=2,NX Gl(I,J)=DELTXX*((G(L,I+l,J,3)-G(L,I,J,3))*GMASK(I+l,J) -(G (L,I ,J,3 ) l-G(L,I-l,J,3))*GMASK(I-l,J))+DELTYY*((G(L , I,J+l,3)-G(L ,I, J , 3)) *GMA 2SK(I,J+l)-(G(L , I,J,3)-G(L , I,J-l , 3))*GMASK(I ,J- l)) 2 CONTINUE . DO 3 J=l , NY l DO 3 I=l ,NXl I\J I-' 0 ~ - . ~ r - --- -, 701 · 702 703 704 705 706 707 708 709 710 711 71 2 713 714 71 5 71 6 717 718 719 720 721 722 723 72 4 725 72 6 72 7 728 729 730 73 1 73 2 73 3 73 4 73 5 736 73 7 73 8 739 740 741 742 743 744 745 746 74 7 748 749 750 -= G(L , I,J,3) • Gl(I,J) . 3 CONTINUE C R~TURN END SUBROUTINE BNDRY C C Subrout i ne sets up boundary mask IMPLICIT REAL*B (A-H,0-Z) COMMON/GRIDI/NX, NY, NXl, NY 1, NXMl, NYMi, NL, NLMl, NFINE, NS-, NSMl COMMON/GRIDR/HI( 22, 37, 3), HMS (18), HWS(l8), HS (17), RM( 6), HW( 6), H( 6), T lOP . COMMON/ARRAY/GMASK(22,37),UVM(21,36) COMJ-lON/OUTFLO/OUT( 22, 37) READ(5 , l)((UVM(I , J),I=l,NX),J=l,NY) FORMAT(21Gl.0) READ(5,2)((GMASK(I,J),I=l,NXl),J= l ,NYl) REAq(5,2)((0UT(I,J) , Ial,NXl),J=1,NY1) 2 FORMAT(22G1.0) RETURN END C C SUBROUTINE LEVELS C IMPLICIT REAL*8 (A-H,0-Z) _ COMMON/GRIDI/NX,NY,NXl,NYl,NXMl,NYMl,NL,NLMl,NFINE,NS,NSMl COMMON/GRIDR/HI(22,37,3),HMS(18),HWS(18),HS(l7),HM(6),HW(6),H(6),T lOP C De te rmin e fixed vertical thickness spacings ERR• l,ODO C Cl is width of thinnest ice l ayer C1•0,25DO . C C2 is a constant to be determined iteratively C2•0,0DO C C3 is a scaling constant C3=6.6D+Ol C HNLMl is max of the fixed thickness levels HNLMl.=2.0DO HM(2)=Cl*0.5DO C Convenient definitions of hm(l) etc HM(l)=O , ODO HW(l) ,. l.ODO H(l)=O.ODO H(2) =Cl DO 2 La3,NLM1 HM(L)=HM(L- 1) +C1+C2*(1,0DO- DEXP(-(DFLOAT(L)-2.0D0)*(DFLOAT(L)-2 . 0D 10)/C3)) H(L) =(2 . 0DO*HM(L))-H(L-1) 2 CONTINUE 751 752 753 754 755 756 757 758 4 759 3 5 C Set 6 IF(DABS(H(NLMl)-HNLMl),LT.l,OD-15) IF(H(NLM1).LT,HNLM1) GO TO 3 C2=C2-ERR ERR=ERR/2,5DO GO TO 1 . C2=C2+ERR GO TO l DO 5 L=2,NLM1 HW(L)=H(L)-H(L-1) CONTINUE max thickness just above hnlml D06K=l,3 DO 6 J= 1, NY 1 DO 6 I=l ,NX1 HI(I,J,K)=HNLMl+TOP CONTINUE C Set up fine grid NN=l HS(l )=H(l) DO 7 L=2,NLM1 DO 7 N=l,NFINE NN,.NN+ l GO TO 4 HS(NN)=H(L-l)+(DFLOAT(N)*HW(L)/DFLOAT(NFINE)) 7 8 C C CONTINUE DO s· L=2,NSM1 HMS(L)•0,5DO*(HS(L-1)+HS(L)) HWS(L)=HS(L)-HS(L-1) CONTINUE HMS(l )"HM(l) HWS( l )=HW( 1) RETURN END SUBROUTINE FINE(GL,GS) C C Refines the grid from 6 to 18 layers IMPLICIT REAL*B (A-H,0-Z) 760 , 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 BOO DIMENSION GL(6),GS(l8) COMMON/GRIDI/NX,NY,NXl,NYl,NXMl,NYMl,NL,NLMl,NFINE,NS,NSMl COMMON/GRIDR/HI(22,37,3),HMS(l8),HWS(l8),HS(l7),HM(6),HW(6) , H(6),T 10!' GS(! )=GL( l) NN=l DO l L=2,NLM1 DO l N=l,NFINE NN=NN+l GS(NN)=GL(L)/DFLOAT(NFINE) CONTINUE GS(NS)=GL(NL) - ~= ~ -- --- - Kl f.-" f.-" 801 802 803 804 80 5 806 807 808 809 810 8 11 812 813 814 815 816 8 17 818 81 9 820 821 822 823 824 825 826 827 8 28 829 830 831 83 2 833 834 835 836 837 838 839 840 841 842 843 8411 84 5 846 847 848 849 850 C C RETURN END SUBROUTINE UNFINE(GS,GL) C C In te r polates th e thickness distribution from fine to normal grid IMPLICIT REAL*8 (A-H,O-Z) DIMENS ION GL(6),GS(l8) COMMON/GRIDI/NX,NY,NXl,NYl,NXMl,NYMl,NL,NLMl,NFINE,NS,NSMl COMMON/GRIDR/HI(22,37,3),HMS(l8),HWS(l8),HS(l7),HM(6),HW(6),H(6),T lOP GL( 1 ) aGS(l) NNml DO 1 L• 2 , NLM1 GL(L) =O. ODO DO l Ne[ ,NFINE NN=NN+l GL(L)=GL(L)+GS(NN) l CONTINUE C GL(NL) 0 GS(NS) RETURN END SUBROUTINE REDIST(GAMMA,GAMMAF) c, C Calculate s the coee ficients used in redistributing C i ce between th e layers when ridging , Also the coeeficients C us ed in evaluating the ice strength are determined IMPLICIT REAL,~8 (A-H,O-Z) DIMENSION GAMMA(l8,18),GAMMAF(l8),0VRLP(18) COMMON/GRIDI/NX,NY,NXl,NYl,NXMl,NYMl,NL,NLMl,NFINE,NS,NSMl COMMON/GRIDR/H I (22,37,3),HMS(l8),HWS(l8),HS(l7),HM(6),HW(6),H(6),T lOP COMMON/PHYS/CB , CF , STREN,RHOICE,RHOWAT,GRAV,COT,UA,UB,AH,BH,RHOAIR, lQI,CW , DMIX,SIDER,TI,SIGMAX,PI,ARMAX,CPLF,CONl,CON2,STFN,CON,ERROR2 2 , Dl,D3 ,TINC,CH , SIN20,COS20,SIN25,COS25 DO 1 Llcl -,NS GA MMAF(Ll)cO,ODO DO 1 L2 a l ,NS GAMMA(Ll,L2)a0 , 0DO CONTINUE NDH=25 DO 5 Ll=2,NSM1 DO 4 M=l ,NDH Hl =HS(Ll-l)+(((DFLOAT(M)-0 , SDO)/DFLOAT(NDH))*HWS(Ll)) CALL HEIGHT(Hl , H2) GAMMAF(Ll)cGAMMAF(Ll)+(CF*(H2-Hl)*(H2-Hl)/DFLOAT(NDH)) GAMMAF (Lt) aGAMMAF(Ll)+(CB*Hl*( 2. 0DO*H2+Hl)*(H2-Hl)/(3 . 0DO*(H2+Hl)* lDFLOAT(NDH))) 2 851 852 853 854 855 856 857 858 859 3 860 4 5 C C DO 2 L2mL1, NSMl OVRLP(L2)=DMINl(H2,HS(L2))-DMINl(H2 , HS(L2-l)) CONTiNUE OVRLP(NS)=DMA.Xl(H2,HS(NSM1))-HS(NSM1) OVRLP(Ll)=HS(Ll)-Hl DO 3 L2=Ll,NS . GAMXA(Ll,L2)=GAMMA(Ll,L2)+(2.0DO*Hl*OVRLP(L2)/(DFLOAT(NDH)*((H2*H2 1)-(Hl*Hl)))) CONTINUE CONTINUE CONTINUE RETURN END 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 SUBROUTINE HEIGHT(Hl,H2) C C Ridge height H2 in terms of the parent ice thickness Hl C C IMPLICIT REAL*8 (A-H,0-Z) COMMON/PHYS/CB,CF,STREN,RHOICE,RHOWAT,GRAV , COT,UA,UB,AH,BH,RHOAIR, lQI,CW,DMIX,SIDER,TI,SIGMAX,PI,ARMAX,CPLF,CONl,CON2,STFN , CON , ERROR2 2,Dl,D3,TINC,CH,SIN20,COS20,SIN25,COS25 IF(Hl.GE.CH) GOTO 1 H2°Hl*(l , ODO-AH)+DSQRT(((A~-4.0DO) *Hl+BH)*AH*Hl) H2=DMA.Xl(Hl,H2) RETURN H2nHl RETURN END SUBROUTINE RIDGE(I,J,Gl,G2,GAMMA,GAMMAF,PR) C C Initial distribution Gl(L) is renormalized to G2(L) IMPLICIT REAL*8 (A-H ,0-Z) C Up DIMENSION GAMMA(l8,18),GAMMAF(l8),Gl(l8),G 2(18),GD(l8),TOTAL(l8) COMMON/GRIDI/NX,NY,NXl,NYl,NXMl,NYMl , NL , NLMl, NFINE,NS,NSMl COMMON/GRIDR/HI(22,37,3),HMS(l8),HWS(l8) , HS(l7),HM(6),HW(6) , H(6),T lOP COMMON/PHYS/CB,CF,STREN,RHOICE,RHOWAT,GRAV,COT,UA,UB,AH , BH,RHOAIR, lQI,CW,DMIX,SIDER,Tl,SIGMAX,PI,ARMAX,CPLF,CONl,CON2,STFN , CO N, ERROR2 2,Dl,D3,TINC,CH,SIN20 , COS20,SIN25,COS25 INEG=O to a fraction ARIDGE of the ice ar ea is involved in ridging. ARIDGE=0 . 15DO HI(I,J,3)=HI(I,J,l) PR=O,ODO HMAX=O.ODO GTOT=Gl(l) HHS(NS) • (HI(I,J,t)+HS(NSHl))*O,SDO I\J f.->. I\J 2 3 HWS(NS) • HI(I,J,1)-HS(NSMl) DO 2 1 2 1,NS GD(L) •O .ODO G2(L)=Gl (L) CONTINUE . TOT=O.ODO DO 3 L=l, NS TOT=TOT+G l(L) TOTAL(L)•TOT CONTINUE C Find cumulative · thickness totals IF(TOT,EQ,l,ODO,OR,TOT,LT,0,lDO)RETURN I F((TOT .GT,l , ODO,AND , Gl(l),LT ,ARIDGE),OR,TOT.GT,(l,ODO+ARIDGE))GOT 901 9o'2 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 4 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 94 7 948 949 950 10 4 . C Add ope n water if initial distribution is undernormalized. 5 G2(1)•1.0DO-TOT+G2(1) RE;TURN DO 5 1=2 ,NS G2( L)=O,ODO CONTINUE C Determine relative amounts of each thickness category to be ridged DO 8 La2 , NS IF(GTOT , GE,ARIDGE.OR,(GTOT+Gl(L)),LT.ARIDGE) GOTO 7 , HST•HS(L-l)+((ARIDGE-GTOT)*HWS(L)/Gl(L)) C Find maximum height of new ridges f ormed 6 7 8 CALL HEIGHT(HST,HMAX) GD(L)•((l,ODO-(GTOT/ARIDGE))**2 ) IF(L,EQ,2) GOTO 7 DO 6 11 • 3,L GD ( Ll - 1)•(2,0DO-((TOTAL(Ll-2)+TOTAL(Ll-l))/ARIDGE))*Gl(Ll-l)/ARIDG lE CONT INUE GTOT• GTOT+G 1 ( L) CO NTINUE C Tentative open water loss G2( l) 2 -(2 , 0DO-(Gl(l)/ARIDGE))*(Gl(l)/ARIDGE) C Eva l uate GAMMA(NS,NS) HMID=H(NLMl)+((DMAXl(HST,H(NLMl))-H(NLMl))/3.0DO) CALL HEIGHT(HMID , H2) C Redistribution coefficient for the top layer GAMMA(NS,NS)=2 . 0DO*HMID/(HMID+H2) C Ca lc ul a te net changes to each layer, G2(L) DO 10 L2E2,NS 9 10 DO 9 Ll=2,L2 G2 (L2)=G2(L2)+GD (Ll)*GAMMA(Ll,L2) CO NTI NUE G2(L2)=G2(L2)-GD(L2) CO NTINUE DEN=O.O DO GN UM=GTOT-1 . 0DO 951 952 953 954 955 956 957 958 959 960 961 962 .963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 DO 11 L•l,NS DEN=DEN+G2(L) 11 CONTINUE C Update maximum ice thickness if HMAX exceeds . current value HI(I,J,3)=DMAXl(HMAX,HI(I,J,l)) DO 12 L=l,NS C Modify the changes G2(L) so resulting distribution is normali zed G2(L)=(-(GNUM/DEN)*G2(L))+Gl(L) 12 CONTINUE DO 13 L=l,NS IF(G2(L),LT.O.ODO) GOTO 14 13 CONTINUE GOTO 19 C If more that ARIDGE needs to be ridged allow this and start again 14 ARIDGE=ARIDGE+0.05DO IF(ARIDGE . LE,3.0DO)GOTO C Debuggin ·g aid WRITE(6,I5) 15 FORMAT(IX, 'NO RIDGING POSSIBLE, INVESTIGATE',/,lX,'LAST ICE THICKN lESS DISTRIBUTION WAS ••• ') WRIT-E(6,16)Gl,HI(I ,J,l) 16 FORMAT(lX,Gl7,10) WRITE ( 6 , l 7) I , J 17 FORMAT( lX, 1 GRID POINT' ·, 2I4) ARIDGE•0, 15DO !NEG= INEG+ l Gl (I )=O, ODO Gl (NS)•l ,ODO DO 18 L=2,NSMl Gl(L)=O.ODO 18 CONTINUE IF(INEG.EQ,4)STOP GOTO l C Calulate strength 19 GAMMAF(NS)=(CF*(H2-HMID)*(H2-HMID)) GAMMAF(NS)=GAMMAF(NS)+(CB*HMID*(2,0DO*H2+HMID)*(H2-HMID)/(3,0DO*(H 12+HMID))) DO 20 L=2,NS PR=PR-(GAMMAF(L)*GD(L)*GNUM/(DEN*l.OD-04)) 20 CONTINUE C C RETURN END SUBROUTINE DIST(U,V,G,GAMMA,GAMMAF,PRESS,TMIX,FND,GAIRX,GAIRY,DAY, lNW,PHI,TAIR,C,P,RHUM,HTSEA) C C Thermodynamic and Dynamic redistribution C Floe size and number density are updated IMPLICIT REAL*8 (A-H,0-Z) N 1-'- \..,l 1001 1002 1003 1004 1005 1006 1007 1008 1009 101 0 1011 10 12 1013 1014 1015 1016 101 7 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 103 5 1036 1037 1038 1039 1040 1041 104 2 1043 1044 1045 1046 104 7 1048 1049 1050 DIMENSION G(6 ,2 2,37,3),GAMMA(l8,18),GAMMAF(l8),PRESS(22,37),FGROW( 121),F(6),TMIX(22,37),FND(5,22,37,3),GAIRX(2l,36),GAIRY(21,36),U(21 2,36,3 ),V(21,36,3) ,Gl(6),G2(18),G3(1 8), RR(22,37),PHI(21,36),TAIR(21 3, 36) , C(52),P(52),RHUM(52),HTSEA(22,37) COMMON/GRIDI/NX,NY , NXl ,NYl ,NXMl ,NYM l ,NL,NLMl ,NFINE,NS,NSMl COMMON/GRIDR/HI(22,37,3),HMS(l8),HWS(l8),HS(l7),HM(6),HW(6),H(6),T !OP COMMO N/STEP/DT,DX,DY COMMON/ARRAY /GMASK(22,37),UVM(2 1,36) COMMON/S IGMA/ALPHR(22,37),ALPH0(22 , 37) ,E I,EII2,COTT COMMON /PHYS/C B, CF,STREN,RHOICE ,RHOWAT , GRAV,COT,UA,UB,AH,BH,RHOAIR, lQI , CW,DMIX,SIDER,TI,SIGMAX,PI,ARMAX,CPLF,CONl,CON2,STFN,CON,ERROR2 2, Dl,D3,TINC,CH,SIN20,COS20,SIN25,COS25 DO 36 J=2,NY DO 36 1=2,NX C Ca l culat e actual area of open water produced C when ice area is lost through ridging CALL ALPHA(I,J,U,V) , IF(G(NL, I ,J,l) . EQ.O.ODO . OR .GMASK(I,J) . EQ . O. ODO)HI(I,J,l) • H(NLMl)+T l OP HM(NL) • (HI(I ,J ,2)+H(NLM1)) *0 . 5DO H(NL)•HI(I , J ,2 ) C Eva l uute wind s peed UG• ( DSQRT(GAIRX(I-1,J-l )*GAIRX( I-l , J-1 )+GAIRY( I-1,J-l )*GAIRY( I-1, J 1-1 ) )+DSQRT( GAI RX (I, J-1 )*GAI RX ( I, J-1 )+GAIRY( I, J-1 ),,~F(L) /HW(L+MELT+Ll) G(L+ l , I ,J,3)=G( L+l,I,J , 3)+DG 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 10 79 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 10 98 1099 1100 G(L,I,J,3)•G(L,I,J,3) - DG Ll=O 3 CONTINUE GOTO 5 C Melt of possibly thin top level 4 DG=DMINl(G(NL,I,J , 2),-DT*G(NL,I,J,2)*F(NL)/HW(NL)) G(NL,I,J , 3)•G(NL,I,J,3) - DG G(NLMl,I,J,3)=G(NLMl,I;J,3)+DG C Adjust HI due to possible growth from NLMl level 5 IF(G(NLMl,I,J , 2).GT . 0.0DO . AND.G(NL ,I, J,2). LE. 0 . 0DO . AND.F(NLMl).GT. -10.0DO)HI(I,J,l)=H(NLMl)+DMAXl(TOP ,DT*F(NLMl)) IF(F(NL),LT.O.ODO)GOTO 6 C Growth of top laye r IF(G(NL,I,J,2).GT . 0.0DO)HI(I,J,l)•HI(I , J,l)+(DT*F(NL)) GOTO 7 C Melt of top laye r 6 HI(I,J,l)•DMAXl(HI(I,J,l)+(DT*F(NL)),H(NLMl)+TOP) 7 CONTINUE C Thermodynamic changes to the floe number density DO 8 L=l,NLMl FND(L,I , J,3)•0.0DO 8 CONTINUE DO 9 L"' 2,NLM1 MELT•-! IF(F(L).LT . O.ODO)MELT•O IF(L , EQ.NLMl,AND.MELT,EQ.O)GOTO 10 DFND=DT*FND(L+MELT,I,J,2)*F(L)/HW(L+MELT+l) FND(L , I,J,3)=FND(L , I,J,3)+DFND FND(L-l,I,J,3) • FND(L-l,I , J,3)-DFND 9 CONTINUE GOTO 11 10 DFN.D• DMIN 1 ( FND( NLM l, I, J, 2) ,-DT*FND( NLM 11 I, J , 2 ) *F (NL) /HW(NL)) FND(NLM1,I,J,3)=FND(NLM1,I,J , 3)-DFND FND(NLM1-l,I,J , 3)=FND(NLM1-l,I,J,3)+DFND 11 CONTI NUE C Number density increas es in level 2 only if level 3 floes melt FND(l,I,J,3)=FND(l , I,J,3)+DT*DMIN1(F(l) , 0.0DO)*FND(l,I,J ,2 )/ HW (2) DO 12 L=l,NLMl FND(L,I,J , l)=FND(L,I,J ,l )+FND(L,I,J,3) 12 CONTINUE DO 13 L=l,NL G(L,I,J,l)=G(L , I,J , l)+G(L,I,J,3) 13 CONTINUE C calculate mixed layer warming C Eliminate negative ice areas (Resulting from Numerical C errors as sociated with advection). Store amount a s a C heat input. GNEG=O.ODO DO 14 L=l,NL IF(G(L,I,J,l).LE . l.OD-18.AND.G(L,I,J , l).GT.0 . 0DO)G(L , I,J,1) • 0. 0DO Kl f...>. ~ 1101 ll 02 1[03 1104 1 ios 1[06 1107 1108 1109 1110 1111 1112 111 3 11 [4 1115 1116 111 7 1118 1119 1120 1121 1122 1123 11 24 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 GNORM• DMAX l(G(L , I,J , -1) , 0.0DO ) GNEGcGNEG+((GNORM-G(L,I,J,l))*HM(L)) G(L,I,J,l )• GNO RM Gl(L)•G(L,I,J,l) 14 CONTINUE C Perform r i dging CALL FINE (Gl,G 2) CALL RI DGE(I,J , G2,G3,GAMMA,GAMMAF,PR) CALL UNFINE(G3,Gl) C Update G due to r idging DO 15 Lal,NL G(L,I ,J, ·l) • Gl(L)*GMASK(I,J) 15 CONTINUE C Impose maximum pos s ible floe si ze DO 16 L=2 ,NL FND(L-l,I,J,l)=DMAXl(G(L,I,J,l)/ARMAX,FND(L-1,I,J,l))*GMASK(I,J) I F(G(L , I,J,l). LE. O.ODO)FND(L-l , I,J , l)cO . ODO 16 CONTI NUE C Must use 'total flo e number density here C Fl oes increase in s i ze due to coa lescing TOTFNDcO.ODO DO 17 L•2 ,NL TOTFND• TOTF ND+FN D(L-1,I, J,l) 17 CONTIN UE C If compactness is high, tr eat cover as continuous 0 I F(G(l,I , J,l).LT . l . OD- 12)GOTO 18 A• l.ODO- G(l , I,J , l) CL• -((2.0DO*A*ALPHR(I,J)*DT)/((l.ODO-(l.ODO-DSQRT(2,0D0/3.0DO))*A* 1A)*(l . OD0-(l.OD0-DSQRT(2 . 0D0/3. 0DO))*A*A)-(2 . 0DO*A/3.0DO)+l.OD-l2) 2 ) . C If -CL is large, cover becomes cont i nuous IF (CL.LT.-5.0D+Ol) GOTO 18 C Integra tion of flo e number density FNDl•TOTFND*DEXP(CL) GOTO 19 C Treat ice cover as cont i nuous 18 FNDl=(l . ODO- G(l ,I ,J , 1))/ARMAX 19 FNDl=DMAXl(FNDl , (l.ODO- G(l , I,J ,1 ))/ARMAX) DO 20 L=2,NL FND( L-l , I,J,l) c(FND(L-l , I , J,l)/(TOTFND+l.OD- 18))*FNDl 20 CONTINUE HI (I,J ,l )•HI (I, J, 3) AVG=O.ODO HM (NL )=(H I(I,J,l)+H( NLM1))*0 . 5DO H(NL)=HI(I,J,l) C Adjust thickness di s tribution due t o lateral me lting HW (NL)=HI(I , J,1)-H(NLMl ) VOLUM=O.ODO DO 21 L=2,NL VOLUM=VOLUM+G (L , I,J,l)*HM(L) CONTINUE SIDES=O. ODO DO 22 L=2, NL 1151 . 21 1152 1153 1154 1155 11 56 1157 C Calculate total floe perimeter SIDES=SIDES+2.0DO*HM(L)*SIDER*DSQRT(PI*FND(L-l, I,J,l)*G(L , I,J , l))* lRHOICE/RHOWAT 22 CONTINUE , 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 117 2 1173 1174 1175 1176 1177 1178 1179 1180 1181 11 82 1183 1184 1185 1186 1187 1188 11 89 1190 1191 1192 1193 1194 119 5 119 6 1197 1198 1199 12 00 C VMELT is volume to be melted ' VMELT=GNEG-DMINl(F(l) , 0 . 0DO)*G(l , I ,J,l) *DT-HTSEA(I,J)*DT IF(VMELT.LE.0.0DO)GOTO 30 FBDTO=VMELT/(l.ODO-G(l , I,J,l)+l . OD- 18) 23 NUM=NL CALL AREA(I,J,G,O.ODO,FBDTO,Gl(l),NUM) DO 23 L=l,NLMl Xl=FBDTO+H(L) X2=FBDTO+H(L+l) CALL AREA(I,J,G,Xl,X2,Gl(L+l),NUM) CONTINUE C VDIFFO is vertical volume melt if SIDES• O Hl=DMAXl (HI(I,J,1)-FBDTO,H(NLMl)+TOP) VDIFFO=O. ODO DO 24 Lc2,NLM1 VDIFFO=VDIFFO+(G(L,I,J,1)-Gl(L))*HM(L) 24 CONTI NUE VDIFFO=VDIFFO+G(NL,I,J,l)*0.5DO*(H(NL)+H(NLM1))-Gl(NL)*0.5DO*(H(NL !Ml )+Hl) C FBDT is vertical melt FBDT=VMELT/(SIDES+l.ODO-G(l,I,J,l) +l. OD-18) C Melt all ice by amount FBDT NUM=NL CALL AREA(I,J,G,0.0DO,FBDT , G(l,I,J,3) ,NUM ) DO 25 L•l,NLMl Xl=FBDT+H(L) X2=FBDT+H(L+l) CALL AREA(I,J,G,Xl,X2,G(L+l,I,J,3),NUM) 25 CONTI NUE C VDIFF is vertical melt with SIDES VDIFF=O.ODO DO 26 L=2,NLM1 VDIFF=VDIFF+(G(L,I , J,l)-G(L,I,J,3)) *HM( L) 26 CONTINUE VDIFF=VDIFF+G(NL,I,J,l)*0.5DO*(H(NL)+H(NLM1))-G(NL,I,J,3)*0.5DO*(H l(NLMl)+Hl(I,J , l)) NUM=NLMl DO 27 L=2,NL Xl=FBDT+H(L-1) X2=FBDT+H(L) FND(L-1,I,J,3)=0.0DO IF(G (L, I,J,3) . GT . O.ODO)CALL AREA(I,J , FND,Xl,X2,FND(L-l,I,J,3), NUM) 27 CONTI NUE K) I-< V1 I 1 2 0 1 H I ( I , J , l ) • D M A X l ( H I ( I , J , 1 ) - F B D T , H ( N L M l ) + T O P ) C . V M E L T i s v o l u m e t o b e m e l t e d b y h e a t a b s o r b e d b y V D I F F i s A c t u a l v o l u m e m e l t e d v e r t i c a l l y 2 8 2 9 U p d a t e G ( l ) D O 2 8 L a l , N L G ( L , I , J , l ) = G ( L , I , J , 3 ) C O N T I N U E D O 2 9 L ml , N L M l F N D ( L , I , J , l ) c F N D ( L , I , J , 3 ) C O N T I N U E C R a i s e t e m p e r a t u r e b y a m o u n t n o t H ( N L ) = H I ( I , J , l ) H M ( N L ) = 0 . 5 D O * ( H ( N L M l ) + H ( N L ) ) HW( N L ) = H ( N L ) - H ( N L M l ) C O NT I N U E 3 0 3 1 D O 3 1 L = 2 , N L A V G = A V G + H M ( L ) * G ( L , I , J , l ) C O NT I N U E u s e d t o m e l t i c e l e a d s C T U P i s r e s i d u a l t e m p e r a t u r e i n c r e a s e t o m i x e d l a y e r . a f t e r C p e r f o r m i n g v e r t i c a l m e l t . I f a l l t h e i c e i s l o s t , a n e t 1 .2 0 2 1 2 0 3 C 1 2 0 4 C 1 2 0 5 1 2 0 6 1 2 0 7 1 2 0 8 1 2 0 9 1 2 1 0 1 2 1 1 1 2 1 2 1 2 1 3 1 2 1 4 1 2 1 5 1 2 1 6 1 2 1 7 1 2 1 8 1 2 1 9 1 2 2 0 1 2 2 1 1 2 2 2 1 2 2 3 1 2 2 4 1 2 2 5 1 2 2 6 1 2 2 7 1 2 2 8 1 2 2 9 1 2 3 0 1 2 3 1 1 2 3 2 1 2 3 3 1 2 3 4 1 2 3 5 1 2 3 6 1 2 3 7 1 2 3 8 1 2 3 9 1 2 4 0 1 2 4 1 1 2 4 2 1 2 4 3 1 2 4 4 1 2 4 5 1 2 4 6 1 2 4 7 1 2 4 8 1 2 4 9 1 2 5 0 C t e m p e r a t u r e i n c r e a s e t o t h e m i x e d l a y e r r e s u l t s . I F ( A V G . G T . O . O D O ) T UP n T I * ( V D I F F O - V D I F F ) I F ( A V G , L E , O . O D O ) T U P a T I * ( V M E L T - V O L U M ) T M I X ( I , J ) • T M I X ( l , J ) + T U P C M I X • D M I N l ( ( T M I X ( I , J ) - 2 , 7 1 2 D + 0 2 ) / ( ( A V G * T l ) + l , O D - 1 8 ) , l , O D O ) 1 C C a l c u l a t e n e w a m o u n t o f o p e n w a t e r f r o m l a t e r a l m e l t G ( l , I , J , l ) a G ( l , I , J , l ) + ( C M I X * ( l , O D O - G ( l , I , J , l ) ) ) C D o l a t e r a l m e l t D O 3 2 L • 2 , N L G ( L , I , J , 3 ) ~ G ( L , I , J , l ) * ( l , O D 0 - C M I X ) C A d j u s t m i x e d l a y e r t e m p e r a t u r e f r o m l a t e r a l m e l t i n g T M I X ( I , J ) • T M I X ( l , J ) + ( ( G ( L , I , J , 3 ) - G ( L , I , J , l ) ) * H M ( L ) * T I ) C U p d a t e G ( l ) 3 2 C C G ( L , I , J , l ) a G ( L , I , J , 3 ) C O N T I N U E I n c l u d e n e x t l i n e s i n c e r o u n d i n g e r r o r s m a y p u t T M I X b e l o w f r e e z i n g T M I X ( I , J ) = D M A X 1 ( 2 , 7 1 2 D + 0 2 , T M I X ( I , J ) ) C C r a c k i n g o f f l o e s i n a w i n d U l O U l O = U G / l . 5 D O H M ( N L ) = O . S D O * ( H ( N L M l ) + H I ( I , J , l ) ) D O 3 5 L = 2 , N L R H = D S Q R T ( G ( L , I , J , l ) / ( P I * F N D ( L - l , I , J , l ) + l , O D - 1 8 ) ) C C h a r a c t e r i s t i c p l a t e l e n g t h C P L C P L = C P L F * ( H M ( L ) * * 0 , 7 5 D O ) C I s f l o e l o n g o r s h o r t I F ( R H . L E . C P L ) G O T O 3 3 C F i n d t h e m a x i m u m b e n d i n g m o m e n t i n l o n g f l o e A = l . O D 0 - G ( l , I , J , l ) S I G = C O N 2 * C P L * C P L * C P L * U l O * U l O / ( ( H M ( L ) * * 3 . S D O ) * A + l . O D - 1 8 ) 1 2 5 1 1 2 5 2 1 2 5 3 1 2 5 4 1 2 5 5 1 2 5 6 1 2 5 7 1 2 5 8 1 2 5 9 1 2 6 0 1 2 6 1 1 2 6 2 1 2 6 3 1 2 6 4 1 2 6 5 1 2 6 6 1 2 6 7 1 2 6 8 1 2 6 9 1 2 7 0 1 2 7 1 1 2 7 2 1 2 7 3 1 2 7 4 1 2 7 5 1 2 7 6 1 2 7 7 1 2 7 8 1 2 7 9 1 2 8 0 1 2 8 1 1 2 8 2 1 2 8 3 1 2 8 4 1 2 8 5 1 2 8 6 1 2 8 7 1 2 8 8 1 2 8 9 1 2 9 0 1 2 9 1 1 2 9 2 1 2 9 3 1 2 9 4 1 2 9 5 1 2 9 6 1 2 9 7 1 2 9 8 1 2 9 9 1 3 0 0 C W i l l t h e f l o e b r e a k I F ( S I G , L T . S I G M A X ) G O T O 3 4 C I f s o t h e n p i e c e s h a v e r a d i u s R H ! l i ! = C P L * P I / 8 . 0 D O C S h o r t f l o e f o r m u l a e 3 3 R M A X = C O N l * ( H M ( L ) * * 3 , 5 D O ) * A / ( C P L * C P L * U l O * U l 0 + 1 , 0 D - 1 8 ) R H = D M I N l ( R H , R . ~ A X ) 3 4 F N D ( L - l , 1 , J , l ) = G ( L , I , J , l ) / ( P I * R H * R H + l , 0 D - 1 8 ) 3 5 C O N T I N U E 3 6 C O N T I N U E C O u t p u t i n f o r m a t i o n a b o u t f l o e s i z e s C W R 1 T E ( 6 , 9 9 7 ) C 9 9 7 F O R M A T ( l X , ' F L O E R A D I I ' ) C D O 7 4 L = 2 , N L C W R I T E ( 6 , 9 9 8 ) L C 9 9 8 F O R M A T ( l X , ' L E V E L N U M B E R 1 , 1 4 ) C D O 7 3 J = 2 , N Y C D O 7 3 I = 2 , N X C R R ( I , J ) a D S Q R T ( G ( L , I , J , l ) / ( P I * F N D ( L - l , I , J , l ) + l , O D - 1 8 ) ) C 7 3 C O N T I N U E C W R I T E ( 6 , 9 9 6 ) ( ( R R ( I , J ) , I • l , N X , 5 ) , J • l , N Y , 5 ) C 9 9 6 F O R M A T ( l X , 5 G l 7 , 1 0 ) C 7 4 C O N T I N U E C C C R E T U R N E N D S U B R O U T I N E R E L A X ( U , V , E T A , Z E T A , D R A G S , D R A G A , A M A S S , F O R C E X , F O R C E Y , E R R O l R , T H E T A , U C , V C , I F I R S T , G A I R X , G A I R Y , E K M A N X , E K M A N Y ) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) D I M E N S I O N U ( 2 l , 3 6 , 3 ) , V ( 2 1 , 3 6 , 3 ) , E T A ( 2 2 , 3 7 ) , Z E T A ( 2 2 , 3 7 ) , D R A G S ( 2 1 , 3 6 l ) , D R A G A ( 2 1 , 3 6 ) , G A I R X ( 2 1 , 3 6 ) , G A I R Y ( 2 1 , 3 6 ) , F O R C E X ( 2 1 , 3 6 ) , F O R C E Y ( 2 1 , 3 2 6 ) , F X E T A ( 4 ) , F X Z E T A ( 4 ) , F Y E T A ( 4 ) , F Y Z E T A ( 4 ) , U E R R ( 2 1 , 3 6 ) , V E R R ( 2 1 , 3 6 ) , C 3 0 E F ( 2 1 , 3 6 ) , E K M A N X ( 2 1 , 3 6 ) , E K M A N Y ( 2 1 , 3 6 ) , A M A S S ( 2 1 , 3 6 ) , F X M ( 2 1 , 3 6 ) , F Y M 4 ( 2 1 , 3 6 ) , U C ( 2 I , 3 6 ) , V C ( 2 1 , 3 6 ) , F X E ( 4 , 2 1 , 3 6 ) , F Y E ( 4 , 2 1 , 3 6 ) , F X Z ( 4 , 2 1 , 3 6 ) 5 , F Y Z ( 4 , 2 l , 3 6 ) , W F A ( 2 I , 3 6 , 2 ) C O M M O N / G R I D I / N X , N Y , NX l , N Y 1 , NXM l , N Y M l , N L , N L M l , N F I N E , N S , N S M l C O M M O N / D I N V / D E L I N 2 C O M M O N / S T E P / D T , D X , D Y C O M M O N / A R R A Y / G M A S K ( 2 2 , 3 7 ) , U V M ( 2 1 , 3 6 ) I C O U N T = O C I n i t i a l i z e r e l a x a t i o n p a r a m e t e r D O l K X Y = l , 2 D O l J = l , N Y D O l 1 = 1 , N X W F A ( I , J , K X Y ) = l . 4 8 D O I F ( A M A S S ( 1 , J ) , L T , 1 . O D - 0 8 ) W F A ( l , J , K X Y ) = O . O D O C O N T I N U E D E L I N = l , O D O / D X f ' v I - ' ° ' 130 1 1302 1303 1304 1305 1306 1307 1308 1309 13 10 1311 1312 1313 1314 13 1 S 1316 1317 1316 1319 1320 1321 1322 13 23 1324 1325 13 26 1327 1328 13 29 13 30 1331 1332 1333 13 34 1335 1336 1337 1336 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 DELIN2=0.5DO/(DX*DX) C First se t U(2)=U(I) DO 2 J=l,NY DO 2 I=l ,NX C Make s ur e bdry pts are equa l to zero U(I,J,2j=U(I,J,l) V(I, J,2)=V(I,J,l) U(I,J, 1 )=U(I,J ,3),~UVM(I,J) V(I,J,l)=V(I,J,3)*UVM(I,J) C Use Zubov's l aw for velocity solution where there is no ice IF(AMASS(I,J).LT.I.OD-08)U(I,J,l)=O.Ol 1DO* (GAIRX(I,J)*0.95106D0+GA 1IRY(I,J)*0.30902DO)*UVM(I ,J ) IF(AMASS(I,J).LT.l. OD-08)V(I,J,l)a0 . 0l1DO* (GAIRY(I,J)*0,95106DO-GA 1I RX (I,J)*0.30902DO)*UVM(I ,J ) UERR(I,J)=l , OD+l2 VERR(I,J)=l.0D+ l 2 IF(tFIRST.EQ,l)AMASS(I, J) =O,ODO C Se t up coef fa of di agonal comp on ~nt ! COEF(I,J)=AMASS(I,J)/DT+2 . 0DO*THETA*(0.5D0*DRAGS(l,J)+2,0D0*((ETA( II,J)+ETA(I+l,J)+ETA(I,J+l)+ETA(I+l, J+l))+ . SDO*(ZETA(I,J)+ZETA(I+I, 2J)+ZETA(I,J+l)+ZETA(I+l,J+l)))/(4,0DO*(DX*DX)))+l, 0D-08 2 CO NTINUE C Ca lculate all functions of prev ious U and V values TTHETA=2.0DO*(l,ODO-THETA) DO 3 J •2, NYM1 DO 3 1=2,NXMl IF(WFA(I,J~l).EQ.O.ODO)GOTO 3 CALL FDIFFl(U,V , ETA,FXETA,I, J ) CALL FDIFFl(U ,V ,ZETA,FXZETA,I,J) CALL FDIFFl( V,U ,ETA,FYETA , I,J) CALL FDIFFl(V , U,ZETA,FYZETA,I,J) FXO= O.SDO*TTHETA*(FXETA(l)+FXZETA(l)+FXETA(2)+FXETA(3)+FXZETA(4)-F 1XETA(4)) FX l=(AMASS(I ,J )/DT-TTHETA*O.SDO*DRAGS(I,J))*U(I,J,2) FX2=TTHETA*O. 5DO*DRAGA(I, J )*V ( I ,J, 2) FY0=0.5D0*TTHETA*(FYETA(l)+FYETA(2)+FYZETA(2)+FYZETA(3)-FYETA(3)+F 1YETA(4)) FYl=(AMASS(I,J)/DT-TTHETA*O,SDO*DRAGS(I,J))*V(I, J,2) FY2=-TTHETA*O.SDO*DRAGA( I ,J)*U(I,J,2) FXC=AMASS(I,J)*0.5DO*TTHETA*(UC(I,J)*(U(l+l,J,2)~U(I-l,J,2))+VC(I , · lJ)*(U(I,J+l, 2)-U(I,J-1,1)))/(2.0DO*DX) . FXM(I,J)=FXO+FXl+FX2+FORCEX(I,J)+FXC FYC =AMASS(I,J)*O.SDO*TTHETA*(UC(l,J)*(V(I+l,J,2)-V(I-l,J , 2))+VC(I, lJ)* (V(I,J+l ,2 ) - V(I,J-1,2)))/(2.0DO*DX) FYM(I,J)=FYO+FYl+FY2+FORCEY(I , J)+FYC 3 CONTI NUE C Se t u(3) a u(l) 4 CONTI NUE DO 5 J • l, NY DO 5 I•l,NX U (I, J , 3) zU ( 1, J , I ) V(I,J,3)zV(l,J,l) CONTINUE C Begin sweep 1351 1352 1353 '5 1354 1355 1356 1357 1358 1359 1360 13 61 1362 1363 1364 1365 1366 136 7 1368 1369 1370 1371 137 2 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 13 86 1387 1388 1389 1390 1391 1]92 1393 1394 1395 1]96 1397 1398 1399 1400 CALL FD1FF2(U,V,ETA,FXE,WFA) CALL FDIFF2(U,V,ZETA,FXZ,WFA) CALL FDIFF2(V,U,ETA,FYE,WFA) CALL FDIFF2(V,U,ZETA , FYZ,WFA) DO 6 J=2,NYMI DO 6 I=2,NXMl IF(WFA(I,J,l).EQ . 0.0DO)GOTO 6 FXETA(l)=FXE(l,I,J)+DELIN2*(U(I-l,J,l)*(ETA(I , J +l) +ETA(I,J))) FXETA(2)=FXE(2,I,J)+DEL1N2*(U(I,J-l,l)*(ETA(I,J)+ETA(l+l,J))) FXETA(3)•FXE(3,l,J)+0.5DO*DELIN2*(V(I-l,J-l,l )*ETA(I,J)+V(l ,J-l,l) l*(-ETA(I,J)+ETA(I+l,J)) -V(I+l,J-1,l)*ETA ( I+l,J)+V(I-l,J,l)*(-ETA(I 2,J+l)+ETA(I,J))-V(I-1,J+l,l)*ETA(I , J+l)) FXETA(4)=FXE(4,I ,J)+DEL IN 2*0.5DO*(V(l-l,J-l,l )*ETA(I,J)+V( I,J-l,I) l*(-ETA(I+l,J)+ ETA(l,J))-V(l+l,J-l,l) *ETA(I+l, J )+V(l-l,J,l)*(ETA(I, 2J+l)-ETA(I,J))-V(I-l,J+l,l)*ETA(I,J+I)) FYETA(l)•FYE(l,I,J)+DELIN2*(V(I-l,J,l) * (ETA(I , J+l)+ETA(I,J))) FYETA(2)sfYE(2, I,J)+DELIN 2*(V(I ,J-l ,l)*(ETA(I,J)+ETA(l+l,J))) FYETA(3)zFYE(3,I,J)+0.5DO*DELlN2*(U(I-l,J-l,l ) *ETA(I , J)+U(I,J - l ,I) l*(-ETA(I,J)+ETA(I+l,J))-U(I+l,J-l,l)*ETA(I+l,J)+U(I-l ,J,l)* (-ETA(I 2,J+l)+ETA(I,J))-U(I-1,J+l,l)*ETA(l,J+I)) FYETA(4)zFYE(4, I ,J)+DELIN2*0.5DO*(U(l-l,J-l , l )*ETA(I,J)+U(I,J-l,l) l *(-ETA(I+l ,J)+ETA(I ,J) )-U(l+l ,J-l, l )*ETA (l+ l ,J)+U(I-1,J, !)*(ETA( I, 2J+l) -ETA(l,J)) -U (I-l,J+l ,l)*ETA(l,J+l)) FXZETA(l)=FXZ(l,I,J)+DELIN2*(U(l-l,J,l)*(ZETA(I,J+l)+ZETA(I,J))) FXZETA(4)•FXZ(4,I,J)+DELIN2*0.5DO*(V(l-l ,J-I, l)*ZETA(l,J)+V(I,J-1, 11 )*(-ZETA(I+ l , J) +ZETA(I, J) )-V(I+ l , J-1 , I )*Z ETA(I+ l , J )+V(I-L ,J, l )*( Z 2ETA(l,J+l)-ZETA(l ,J) )-V(I-l ,J+l,l)*ZETA(I,J+l)) FYZETA(2)=FYZ(2,I,J)+OELIN2*(V(I,J-l,l)*(ZETA(I,J)+ZETA(I+l,J))) FYZETA(3)=FYZ(3,l , J)+0.5DO*DELIN2* (U(I-l,J-l,l)*Z ETA(I,J)+U(I,J-l, ll)*(-ZETA(I,J)+ZETA(I+l,J))-U(I+l,J-l,l) *ZETA ( I+l , J)+U(I-l , J,l)*(-2ZETA(I ,J+I )+ZETA(I, J) )-U( I-1 ,J+ l, l )*ZETA(I ,J+ l)) FX3zTHETA*(FXETA(l)+FXZETA(l)+FXETA(2)+FXETA(3)+FXZETA(4)-FXETA(4) I) FXCP=AMASS(l ,J)*TH ETA*( UC(l,J)*(U(I+l ,J, l)-U(I-l,J , l))+VC(l,J)*(U( ll,J+l,l)-U(I , J-l,1)))*0 , 5DO*DELIN FX3=FX3-FXCP FY3=THETA*(FYETA(l)+FYETA(2)+FYZETA(2)+FYZETA(3)-FYETA(3)+FYETA(4) I) FYCP=AMASS(I ,J)*TH ETA*(UC(I,J) *( V(l+l ,J,l)-V(I- 1,J, l)) +VC(l,J)*(V( ll ,J+l,l)-V(I,J-l,l) ))*0.5DO*DELIN FY3=FY3-FYCP FLll=THETA*DRAGA(I , J) /COEF(I,J) Fll=(FXM(l,J)+FX3) /COEF( I,J ) F22z(FYM(I,J)+FY3)/COEF(I,J) FLllS=l.ODO+(FLll *FLll) FLllSI=I.ODO/FLll S f..,;) f-' --.J 1:.01 1 !..02 1403 1404 140 5 140 6 140 7 1408 1409 14 10 141 I 14 I 2 141] 14 14 I:. I 5 14 16 14 I 7 I.'; 18 I.'; 19 ' 14 20 142 1 142 2 142) ' 1424 1425 14 26 1427 142 8 1429 14 30 14) I 1432 14 )) 14 34 14 35 1436 14 37 l 4 38 14 39 11,40 144 I 144 2 144 3 1444 141, 5 14 46 144 7 I 1,48 1449 11150 lJ l CO R= ( ( f' I I + F LI I* F2 2) * f LI I SI ) *UVM ( I , J ) V 1 COR= ( ( F2 2- F LI I* FI I ) * I' LI I S [ ) * UVM ( [ , J ) U ( l , J , I ) = U ( [ , J , I ) +W FA ( [ , J , I ) * ( U [CO R - U ( l , .J , I ) ) V( [ ,J, I )=V( l ,J, I )+WFA( I , J, 2 )* (VTCOR-V( I , .J , I)) 6 CONTINUE LCOUNT=[COUNT + I lf(LCOUNT.GT.200) GO TO 9 Sl=0.0 00 llO 7 J= I , NY llO 7 l=l,NX I f'(WFA ( l ,J, I). EQ.O. ODO)GUTO U K=U ( l, J , I ) - U ( [, J, 3) VK=V( [ ,J, I )-V ( [ , J , 3) DW=0.0100 lf(DABS(UK).GT.DABS(UERR(l,J)))DW=-0.02DO WFA (I, J, I)= OM l N l ( I . 500, DMAX I ( WFA ( I , J , I ) + DW , I . ODLJ ) ) OW =O. 0 IDO lf(DAHS ( VR).GT.DABS(VERR(l,J)))DW=-U. 0200 WFA( l ,J, 2)=DMINI ( I. 5DO, l>MAX I (WFA ( l , J, 2)+DW , I . Ull*O. 28DO )+0 . 08DO C Red uce albedo due to me lt ponds IF(T . EQ.273.1 6DO) ALB=ALB*0 . 8213DO IF(H . EQ . 0 . 05DO)ALB•0.08DO C Cons ider stability of atmospheric boundary layer. C C STB=l . ODO IF(T , LE.TAIR)STB=0 . 571428DO BALA•(l . OD0-ALB)*FS+FL+UG*STB*(Dl*(TAIR-T)+D2*(QA-QS))-D3*(T**4) BAL•BALA+(CON/H)*(TMIX-T) RETURN END SUBROUTINE NORTH(G , X38,FND,TMIX) C C Cal culat e input parameters for Fram Strait IMPLICIT REAL*8 (A- H,0-Z) 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 2 1887 3 1888 1889 1890 4 1891 1892 1893 1894 1895 1896 1897 5 1898 1899 1900 DIMENSION G(6,22,37 ,3) ,FND(S,22,37 ,3) ,TMIX(22,37) COMMON/PHYS/CB, CF, STREN, RflOICE, RHOWAT, GRAV, COT, UA, UB ,AH, BH, RHOAIR, lQI,CW,DMIX,SIDER,TI,SIGMAX,PI,AR}IAX,CPLF,CONl,CON2,STFN,CON,ERROR2 2,Dl,D3,TINC,CH,SIN20,COS20,SIN25,COS25 COMMON/GRIDI/NX,NY,NXl,NYl,NXMl,NYMl,NL,NLMl,NFINE,NS,NSMl COMMON/GRIDR/HI(22,37,3),HMS(l8),HWS(l8),HS(l7),HM(6),HW(6),H(6),T !OP J=36 ALP=O.SDO BF.TA=0.7552854129DO Xlc(l.ODO+ALP-ALP*BETA)*X38 X2=(1 . 0DO-ALP*BETA)*X38 Y=J6.0D0-0.5DO DO 7 1=2,NX HI(I,J, l)=H(NLMl)+TOP H(NL)=HI(I,J,l) HW(NL)=H(NL)-H(NLMl) HM(NL)=0.5DO*(H(NL)+H(NLM1)) X=DFLOAT(I)-0.5DO A•DCOS((X-X2)*l.570796327DO/(Xl-X2)) IF(X.LE.X2)A=l.ODO IF(X.GF..Xl)A=O.ODO G(l,I,J,l)al.ODO-A HXO=l.OD-04*Y*Y*Y+0.9DO HMAX=o·. ODO IF(X.LT.Xl)HMAX=((Xl-X)/Xl)*HX0*3.0DO IF(A.EQ.O.ODO)GOTO 4 NLMAX=NLMl DO 1 L=l ,NLMl G(L+l,I,J,l)•O.ODO IF(H(L).LE . HMAX.AND.HMAX.LT.H(L+l))NLMAXmL CONTINUE IF(NLMAX.EQ.l)GOTO 3 DO 2 L=2, NLMAX G(L,I,J,1)=2.0DO*A*HW(L)*(l.ODO-(HM(L)/HMAX))/HMAX CONTINUE G (NLMAX+ l, I ,J, l )=2 . ODO*A*(HMAX-H(NLMAX) )*( l. ODO-((HMAX+H(NLMAX)) / ( 12. ODO*IIMAX))) /HMAX TMIX(I,J)=271.2DO HI(I,J, 1 f=DMAXl(HI(I,J, 1) ,HMAX) IF(G(l,I,J,l).NE.0.0DO) GOTO 5 G(3,I,J,l)=G(3,I,J,l)+G(2,I,J,l) G(2,I,J,1)=0.0DO IF(I.GT.5) GOTO 5 G(4,I,J,l)=G(4,I,J,l)+G(3,I,J,l) G(3,I , J,l)=O.ODO CONTINUE DO 6 L=2,NL FND(L-1,I,J,l) • G(L,I,J,l)/ARMAX IF(G(L,I,J,l) . LE.O.ODO)FND(L-1,I,J,l) • O.ODO K) l'Q K) 1901 1902 1903 1904 1905 1906 1907 1908 1909 19 10 1911 1912 19 13 19 14 19 15 1916 1917 19 18 1919 1920 192 1 1922 19 23 1924 1925 1926 1927 19 28 19 29 1930 19 31 1932 19 33 1934 19 35 1936 19 37 1938 1939 1940 194 1 1942 194 3 1944 1945 1946 1947 1948 1949 1950 6 CONTINUE 7 CONTINUE C C C RETURN END SUBROUTINE GRNLND(GWATX,GWATY,PHI,C,RHUM,P,EDGE,HTSEA) IMPLICIT REAL*8 (A- H,O-Z) COMMON/GRIDI/NX,NY,NXl,NYl,NXMl , NYMl,NL,NLMl,NFINE,NS,NSMl COMMON/PHYS/CB,CF,STREN,RHOICE, RHOWAT,GRAV,COT,UA,UB,AH,BH,RHOAIR, lQI , CW , DMIX,SIDER,TI,SIGMAX,PI,ARMAX,CPLF,CONl,CON2,STFN,CON,ERROR2 2 , Dl , D3 ,TINC , CH,SIN20,COS20,SIN25 , COS25 DIMENSION GWATX( 21, 36), GWATY( 21', 36), PHI ( 21, 36), C( 52), RHUM( 52), P ( 52 l ), EDGE(52),HTSEA(22,37) RE AD ( 1 , 1 )( C ( I) , RHUM (I) , P (I) , I • l , 5 2) FORMAT(3G8.0) READ(l , 2)(EDGE(I), Ial ,52) 2 .FORMAT(G6 . 0) C Read i n long term ocean currents RE AD(3)GWATX,GWATY DO 3 J • l ,NY DO 3 I 2 l ,NX GWATX(I,J) • GWATX(I,J) GWATY(I,J) • GWATY(I,J) GX•GWATX(I,J)*COS25+GWATY(I,J)*SIN25 GY=GWATY(I,J)*COS25-GWATX(I,J)*SIN25 GWATX(I ,J) • GX GWATY(I,J) • GY 3 CONTINUE C Read i n la t itud e values a t grid poin t s READ(4)PHI C Se t up oce anic heat flux field Xl= l .ODO X2•20,0DO SEAa-3 . 5D0/8 , 64D+06 DO 4 J xl, NY l SA=SEA*0.5DO*(l . OD0-DC0S(PI*DFLOAT(J)/30.0DO)) DO 4 I=l , NXl HTSEA(I,J)=0,5DO*SA*(l . ODO-DC0S(PI*(DFLOAT(I)-Xl)/(X2-Xl))) I F(I.GE . 20)HTSEA(I,J)=SA I F(I , LE , l)HTSEA(I , J)=O . ODO 4 CONTINUE C C RETURN END SUBROUTI NE UVT(GAIRX , GAIRY,TAIR,ICOUNT,INIT , IUVT) C REAL*4 SGAIRX(21 1 36) , SGAIRY(21,36),STAIR(21 , 36) I I 19~1 1952 1953 1954 C 1955 1956 1957 1958 19 59 1960 · 1961 1962 1 1963 2 1964 1965 1966 1967 1968 1969 3 1970 1971 REAL*8 GAIRX(21,36),GAIRY(21,36),TAIR(21,36) COMMON/GRIDI/NX,NY,NXl,NYl,NXMl,NYMl,NL,NLMl,NFINE,NS,NSMl IF(ICOUNT , EQ.l.AND.INIT,EQ.O)RETURN Read in winds and temperature NUMFLD=O .IF((INIT.EQ.O.AND.ICOUNT . EQ . O).OR.MOD(ICOUNT,2).EQ.l)NUMFLD• l IF(INIT.EQ.l.AND.IUVT.EQ , l)NUMFLDa(ICOUNT/2)+1 IF((INIT.EQ . l.AND.ICOUNT.EQ.O) . OR.NUMFLD . EQ . O)RETURN DO 2 N=l,NUMFLD READ(2)SGAIRX,SGAIRY,STAIR WRITE(6 , l) FORMAT(lX , 'WIND AND TEMPERATURE FILE READ ONCE') CONTINUE DO 3 J=l,NY DO 3 I=l ,NX GAIRX(I,J)=l,5*(SGAIRX(I,J)*0.9063+SGAIRY(I , J)*0.4226) GAIRY(I,J)=l.5*(SGAIRY(I,J)*0.9063-SGAIRX(I,J)*0.4226) TAIR(I,J) • STAIR(I,J) CONTINUE RETURN END tv tv vl 224 The output WIND AND TEMPERATURE FILE READ ONCE Thickness levels o.o 0 .2500 0.6055 1.1626 2. 0000 -0 . 0000 2.0000 Midpoint s of levels o.o 0.1250 0.4278 0 . 8841 1. 5813 2.0000 level widths 1.0000 0.2500 0.3555 0.5571 0.8374 -0 . 0000 RELAX CALLED: NO. OF ITERATIONS 32 MAX ERROR .97085D-02 **** TIME STEP 1 DAY 335.00 RELAX CALLED: NO . OF ITERATIONS 31 MAX ERROR .8 1424D-05 RELAX CALLED: NO. OF ITERATIONS 31 MAX ERROR .98272D- 05 POSITION 335.000 5.27889 16.1800 TOTAL VOLUME 431.356046103 OUTFLOW - 6.181303 NET: -6.181303 **** TIME .STEP 2 DAY 335.25 . RELAX CALLED: NO. OF ITERATIONS 25 MAX ERROR .91648D-05 RELAX CALLED: NO. OF ITERATIONS 24 MAX ERROR .62637D-05 POSITION 335.250 5.43457 16.1576 .TOTAL VOLUME 431.000494675 OUTFLOW -6.914162 NET: -13 .09546 **** TIME STEP 3 DAY 335.50 WIND AND TEMPERATURE FILE READ ONCE RELAX CALLED: NO. OF ITERATIONS 33 MAX ERROR . 84667D-05 RELAX CALLED: NO. OF ITERATIONS 35 HAX ERROR .73470D-05 POSITION 335.500 5.46213 16.0752 TOTAL VOLUME 435.370187203 OUTFLOW -8 . 766854 NET: -21. 86232 **** TIME STEP 4 DAY 335.75 RELAX CALLED: NO. OF ITERATIONS 20 MAX ERROR .79765D- 05 RELAX CALLED: NO. OF ITERATIONS 29 MAX ERROR .86755D-05 POSITION 335.750 5.46323 16.0044 TOTAL VOLUME 440.555927308 OUTFLOW -9.040284 NET: -30.90260 **** TIME STEP 5 DAY 336.00 WIND AND TEMPERATURE FILE READ ONCE RELAX CALLED: NO. OF ITERATIONS 24 MAX ERROR .57374D-05 RELAX CALLED: NO. OF ITERATIONS 32 MAX ERROR .86341D-05 POSITION 336.000 5.49240 16.0044 TOTAL VOLUME 438 . 7485 79 794 OUTFLOW -.9579669 NET: -31. 86057 FULL DATA PRINTED FOR THIS TIME STEP X-COMPONENT OF ICE VELOCITY .o .0 .o .0 .o .o -.3105038395D-Ol -. 7090668542D-02 -.5200650416D-02 .0 .o .9293394750D-Ol .1109645415 .6619184922D-Ol .o .o • 48718435 70D- 0 l .1944705420 .2506536305 . 0 .o .5005692442D-01 .3338169365 .1296760889 .0 .o .1132368337 . 4068779663 . 1319461303 .0 .o .1288521988 • 2881 886907 .1274435170 .o .o .o .o .o .o Y-COMPONENT OF ICE VELOCITY .o .o .o .o .o . ci . 1023818950D-01 . 1107732253 .1139191078 .o .o .1366843052D-01 • 8795392231D-Ol .1726953539 .o .0 • 7595177703D-03 . 79 31579803D-Ol . 3206200 784 .o .o .5634955024D-01 • 2638644639 .o .7451490554D-Ol . 3484362157 .o -.2674565943D-01 .2022760537 .o .o . o FLOE NUMBER DENSITY .o .o .o .o .2750180735D-03 .1096185814D-01 . 0 .9981429823D-04 .2332027623D-03 .o .1014151682D-03 .4871 l 9 l 795D-03 .0 . 1803283251D-03 .6810533466D-01 . 0 • l 78 l l 63404D-03 .1272327791 .o .4848284624D-03 .1575645400D-01 .0 .o .o .0 .0 .o .0 .2022082297D-02 • l 9CJ.5831061D-02 .o .3168191094D-02 . 152i679427D-02 · .o .1557761745D-02 .1946563369D-02 . o .1799266279D-02 .4876829533D-02 .o .1209163148D-02 . 1211953314D-01 . o .5329698482D-04 . 2196141150D-02 .o .0 .0 .o .0 .o .0 .1310137030D-04 .5731718733D-05 .o .1518286356D-02 .2548198015D-03 .0 .6851716353D-03 .3936485051D-03 .0 .1730699351D-04 .6378890565D-03 .0 • 9283334952D-ll .8675465480D-03 .o .2423516246D-05 .5486861522D-03 .o .o . o ICE THICKNESS DISTRIBUTION AT SELECTED POINTS .713857542288D-Ol .124933091651D-01 .503079942980 l.00000000000 .o • 298952916710 .113920381882 .167695033636D-03 .o .o .0 .o .169063889 104 .115517607501 • 265092112425 .242459828699 • 207866562271 .o 1.00000000000 .o . o . o .o .o BULK VISCOSITY, ZETA . 1000000000D+l5 .1000000000D+l5 .1000000000D+l5 400000000.0 .1000000000D+l5 400000000.0 ·.1ooooooooon+1s 400000000.0 .1000000000D+l5 400000000 . 0 .1000000000D+l5 446184565.9 .lOOOOOOOOOD+lS 475716525;0 .lOOOOOOOOOD+l5 . O SHEAR VISCOSITY, ETA .lOOOOOOOOOD+l5 400000000.0 1261337622. 400000000.0 400000000.0 400000000.0 400000000.0 .o .6829193563D-Ol .4387192074D-01 - . 1701246158D-01 .o .0 . o .6611309131D-Ol .1996824763 .0 .o .o .0 .0 .o . 2051673547D-02 . 5030122683D-02 .o .o . o .o . o .o • 334 789889 lD-04 .9643199641D-04 .0 .o . o . o .lOOOOOOOOOD+lS 400000000.0 400000000.0 400000000.0 400000000.0 400000000.0 400000000.0 .o .o . o .o .o .6 .o .o .o .o .o .o .o .0 .0 .o .o .o . o . o .0 .o .0 .o .0 .o .o .o .o .1000000000D+l5 .0 .o .0 .o .o . 0 .o 225 .1000000000D+l5 .1000000000D+l5 • lOOOOOOOOOD+ 15 .1000000000D+l5 95998407. 77 87251015.07 .JOOOOOOOOOD+l5 63655008.29 28942)850.9 . 1000000000D+l5 .0 86897564.25 .1000000000D+l5 92416697.77 .o .JOOOOOOOOOD+l5 80444759.70 .0 .1000000000D+l5 104407294. l .o .JOOOOOOOOOD+ l5 .o . 0 RIDGING AMOUNT, ALPHR ·-.2154116179D-26 -.2154116179D-26 -.2154116179D-26 - .2154116179D-26 .2327202660D-06 · .5165083634D-06 -.2154116179D-26 .3999514791D-07 -.2154116179D-26 .3352887927D-06 - .2 154116179D-26 .4497742 760D-06 - .2 l54ll6l79D-26 .6717434115D-06 -.2154116179D-26 • 594104 l 654D-06 -.2154116179D-26 .3499299728D-06 OPEN WATER OPENiNG, ALPHO -.2l5.4ll6l 79D-26 -.2154116179D-26 -.2154116179D-26 -. 2154116179D-26 -. 2154116179D-26 -.2154116179D-26 -.2154116179D-26 -.2154116179D-26 STRENGTH .o .o .o .o .o .o .o .o -.2154116179D-26 .2973621442D-06 .6054287125D-06 • 206165 7913D-06 .3963688412D-06 .4324915321D-07 .376848584 2D-06 .2744526737D-05 .o 710.9509204 300.6551840 658.3601275 3288.410915 3676.058916 2355 .159303 .o MIXED LAYER TEMPERATURE 271.2000000 271.2000000 271.2000000 271.2000000 271.2000000 271 . 2000000 271.2000000 271.2000000 271.2000000 271.2000000 271.2000000 271~2000000 271 . 2000000 271.2000000 271.2000000 271.2000000 • 39 3971 l 896D-06 .2420644460D-06 .3599316456D-07 • 83I 8341972D-06 .1226072867D-05 .6827316801D-06 -.2154116179D-26 .3543722124D-06 .3126197932D-06 .l l53104473D-05 .5052280725D-05 .3363626913D-05 .2076650835D-05 .4558018235D-05 .o 33.76574347 1574.998346 839.5225336 .o .0 .0 .0 272.4780163 271. 2000000 2 71. 2000000 2 71. 2000000 2 71 . 2000000 271. 2000000 2 71. 2000000 2 71. 2000000 **** TIME STEP 6 DAY 336.25 .lOOOOOOOOOD+l5 .o .0 .o .o .0 .o .0 - .2154116179D-26 .1970238047D-06 . 5996104804D-07 .1377650517D-05 .2827201339D-06 .3668635776D-06 .454890512 2D-06 .468 16l 1Jl8D-05 -.2154116179D-26 .3883281488D-07 .1180541237D-05 .o . 8963723064D-09 .1206300990D-07 ·. ll 17424776D-06 .1782523318D-05 .o .0 .o .o .o .o .o .o 275.9183538 271. 8526248 271.2000000 271. 2000000 2 72. 3314632 272.7513367 273.5532844 273.4011479 RELAX CALLED: NO. OF ITERATIONS RELAX CALLED: NO. OF ITERATIONS 23 31 5.53519 MAX ERROR MAX ERROR .79660D-05 .85239D-0,5 16.0145 POSITION 336.250 TOTAL VOLUME 437.306506862 OUTFLOW - . 6669196 NET: -32.52749 226 . 1000000000D+l5 .o .0 .0 .o .o .o .o -.2154116179D-26 .7855160803D-06 : 1077869592D-05 .I765303628D-05 .23762J5088D-05 .3286920362D-05 .3674979915D-05 • l 147183998D- 05 -.2154116179D-26 .I234058187D-05 .4745627325D-06 • 6351697152D-07 .o .o .22 70364608D-07 .1098585468D- 05 .o .o .o .o .o .0 .o .o 279.3586912 273.1358492 2 71. 8650621 272.3528315 274.5447666 275.6227700 277 .4405386 278.3582854 BIBLIOGRAPHY 227 BIBLIOGRAPHY Ackley, S. F. and Hibler, W. D. III., 1974: Measurement of Arctic Ocean ice deformation and fracture patterns from satellite imagery. AIDJEX Bull. 26, 33-4 7. Ackley, S. F., Hibler, W. D. III, and Kugzrug, F. K., 1976: Misgivings on isostatic imbalance as a mechanism for sea ice cracking. AIDJEX Bull. 33, 85-94. Alekseev, G. V. and Buzuev A. 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Wadhams, P., 1980b: -Ice characteristics in the seasonal sea ice zo n e . Col d Regions Sci. Tech., 2, 37-88 . Wadhams, P., 1981: The ice cover in the Greenla~d and Norwegian Seas, Reviews of Geophysic s and Space Phys i cs, 19, No. 3, 345-393 . Wadhams, P., 1983a: A mechanism f or the formation of ice e dge bands. Journal of Geophysical Resea rch , 88, No. CS, 2813-2818. Wadhams, P. , 1983b: Sea ice thickness distribution in Fram Strait. Nature, 305, No . 5930 , 108-111. I I I I 235 Wadhams, P., Gill, A. E., and Linden, P. F., 1979: Transects by submarine of the East Greenland Polar front. Deep-Sea Research, 26A, 1311-1327. Wadhams, P., and Squire, V. A., 1983; An ice-water vortex at the edge of the East Greenland Current. Journal of Geophysical Research. 88, CS, 2770-2780. Walker, E. R. , and Wadhams, P., 1979: On thick sea-ice floes, Arctic, 32(2), 140-147. Washington, W. M., Semtner, A. J. Jr., Parkinson, C., and Morrison, L., 1976: On the development of a Seasonal Change Sea-Ice model. Journal of Physical Oceanography, 6, No. 5, 679-685. Weber, J. R., and Erdelyi, M., 1976: Ice and ocean tilt measurements in the Beaufort Sea. Journal of Glaciology, 17, 61-71. Weller, G., 1972: Radiation flux investigations. AIDJEX Bull. 14, 28-30. (Available as PB 220/859, National Technical Information Service, Springfield, Virginia) Wittmann, W. I., and Schule, J. J. J r., 1966: Comments on the mass budget of arctic pack ice. Proc Syrop. on the Arctic Heat Budget and Atmospheric Circulation, RM-5233-NSF, ed. J. O. Fletcher, Rand Corp., Santa Monica, Calif., 215-246. Zillman, J. W. 1972: A study of some aspects of the radiation and heat budgets .of the southern hemisphere oceans, Meteorol. Stud • . 26, 562pp., Bureau of Meteorology, Dept . of the interior, Canberra, Australia. Zubov, N. N., 1943 : Arctic Ice (in Russian), Izdatel ' stvo Glavsermorputi, Moscow. (English translation, U. S. Naval Oceanographic Office, Washington, D. C., 195.) LIST OF ACRONYMS AIDJEX Arctic Ice Dynamics Joint Experiment ECMWF European Centre for Medium-range Weather Forecasting FGGE First GARP [Global Atmospheric Research Program] Global Experiment MIZEX Margin~l Ice Zone Experiment SCOR Scientific Committee on Oceanic Research I' I; 1 I :, 11 Ii 236 ADDENDUM Hunkins, K., 1962: Waves on the Arctic Ocean. Journal of Geophysics, 67 , 24 77. Mollo-Christensen E., 1983: Interactions between waves and mean drift 1n an ice pack. Journal of Geophysical Research, 88 , 2971-2972. Robin G. de Q., 1963: Wave propagation through fields of pack ice. Philosophical Transactions of the Royal Society of London, A255, 313-339. Williams, E., Swithinbank, C.W.M., and Robin G. de Q. 1975: A submarine study of the Arctic pack ice. Journal of Glaciology, 15 , 349-362. PLATES A selection of views of floes from the air in a region near the ice edge in Fram Strait in summer. Photographs supplied by Vernon Squire, All the photographs were taken on 27 June 1983. . 1 11 I I I 1 I, I I I I 100m i . I i: I I I I I I , I I : I 'i 100m 100m IOOm 100 m E;E~SI ~7'!\ LIBRARY MB;:,/ r"--··-·· _,, ...