Tests of the Ice Thickness Distribution Theory

Tests of the Ice Thickness Distribution Theory A. S. THORNDIKE INTRODUCTION Early investigators of sea ice distinguished many types of ice, the main categories being multiyear ice, first-year ice, various forms of very thin ice, and open water (which can be considered to be ice of zero thickness). A large region (100 x 100 km, say) contains many ice types. It is natural to describe the ice in such a region in terms of the proportion of the different ice types. Since to some extent the ice types can be labeled by their thickness, the term thickness distribution was used by Wittmann and Schule (1966) to refer to the composition of the ice pack in a given region. Many local properties of the ice pack depend on its thickness for instance, its surface temperature, its albedo, its strength. Large-scale averages of these quantities therefore depend on the thickness distribution. This fact was an incentive for developing a theory to show how the thickness distribution depends on the thermal and mechanical history of a region. During AIDJEX a theory was developed and incorporated into a more general sea ice model (see Coon et al., 1974; Thorndike et al., 1975). Unfortunately no satisfactory technique exists for measuring the ice thickness distribution, and therefore the theory itself has not been verified, despite attempts by several investigators to test parts of the theory. This note outlines the techniques and results of these first attempts in the hope that such an outline may be useful to future investigators. THEORY The theory for the ice thickness distribution is described in Thorndike et al. (1975). A function g(h) is defined for a fixed, Eulerian, regioni? such that g(h)dh is the fraction of the area in R for which the thickness satisfies h < <h + dh. Equivalently, G(h) is the fraction of the area of R for which < h. 144

Tests of the Ice Thickness Distribution Theory 145 Changes in thickness occur in response to heat fluxes from the atmosphere and ocean in a prescribed way: dt Changes in distribution occur in response to deformation of the ice. In divergence, for example, new regions of open water are created and some existing ice is exported through the boundaries of/?. In convergence, on the other hand, some ice within R is "redistributed" to form pressure ridges and to make room for ice imported into/? through its boundaries. The creation of leads and pressure ridges is parameterized in terms of a redistribution function é, depending on the large-scale average strain rate è and on G and h. é = é(è,g,h) The fluxes through the boundaries of /? depend only on the divergence of the velocity field, div u. With these definitions, changes in thickness distribution are given by theory as = -f hé - G div u - u grad G at ah Putting aside for the moment the possibility of measurement errors, we ask how well this equation represents the physical situation. The redistribution term é is assumed to depend only on the mean strain rate tensor e. It has been learned from AIDJEX deformation measurements (Thorndike and Colony, 1977; Hibler et al., 1975) that the mean strain rate cannot fully describe the deformation activity within a region of 100 km dimensions. It accounts only for the linear variation of velocity with position, which is typically 1 cm sec^1 per 100 km. Superimposed on this linear variation are fluctuations of order 0.5 cm sec -1 which cannot be represented bye. Thus, even at times when e is very small, there may be significant redistribution due to unresolved motions. In the summer, when the area of open water is, say, 10%, this will not have much effect on G since both convergence and divergence change only the area of open water. Over a large region, the average change in open water will be related to the average deformation. But at other times of the year convergence affects the regions of thin ice and divergence creates open water. Thus, local variations in deformation will produce somewhat more open water and more redistribution of thin ice than would the mean deformation alone. A second assumption imbedded in é is that it depends on the principal values of e but not on the principal directions. The model is isotropic, in other words. It cannot distinguish uniaxial compression in a direction parallel to an existing system of leads from uniaxial compression in the perpendicular direction. The thermodynamic term is also based on limiting assumptions. The growth

146 A. S. THORNDIKE rate is not in reality simply a function of time and thickness. In addition it depends on snow cover and on the geometry of the ice surface. Ablation rates for jagged pressure ridges of some thickness differ from rates for level ice of the same thickness. The fraction of area covered by thin ice or open water also affects growth rates. Large amounts of heat can be absorbed in the upper ocean, causing on the average an increased flux of heat to the lower ice surface. To model this effect the growth rates would need to depend on G itself. Some of this heat can be used to melt ice at the edges of open leads, increasing the proportion of open water while decreasing the proportions of the other ice types. This is a thermodynamic process not modeled at all by/ = dç/dt. TESTING THE THEORY The tests described below neither test these assumptions one by one nor evaluate the size of the errors introduced by each approximation. They address the question: how well do results from the model as now conceived agree with actual measurements? I. Test with Drifting Station Data Climatological growth rates and strain data collected from May 1962 to May 1964 at drifting stations T-3, NP-10, and ARLIS II were used to calculate the thickness distribution history for this period (Thorndike et al., 1975). The results were compared with distributions estimated from submarine profiles at various times and places. Only in a weak sense is this a test: the calculated behavior is completely unverified by simultaneous measurement. The distribution as calculated remains similar to observed distributions after two years, and it responds to the annual cycle of thermal forcing in/. This results in 15% open water by summer's end, in rough agreement with other observations. However, there is a steady decline in the proportion of ice thicker than 6 m. No very thick ice was formed by ridging; ridging was assumed in those calculations to convert ice of thickness h into thickness 5h. A somewhat larger multiplier 15, say appears to be appropriate based on Parmerter and Coon's (1972) ridging study. This might be enough to maintain the thick ice balance. 2. Test with Landsai Data Remote sensing data from Landsat were used by Rothrock and Hall (1975) to measure the strain and the change in areas of thin ice. The procedure has the advantage of testing the terms xjj G div u indpendently of the growth rates/ (see Nye, 1975, for details). By theory a particular deformation should cause a certain increase dg 0 in thin ice by the creation and immediate freezing of leads and a certain decrease dg r by the formation of ridges. Each of these quantities was measured from the Landsat images and then used to define values of parameters in the assumed redistribution functions. From their study Rothrock and Hall (1975) concluded that (a) the changes in

Tests of the Ice Thickness Distribution Theory 147 areas of thin ice can be modeled in terms of the large-scale mean deformation and (b) reasonable forms of the redistribution functions fit the observations within experimental error. In fact, the errors were just small enough to discriminate between different trial redistributors (see their Figures 8, 9, and 11). This procedure is a test of how much open water-thin ice is created or destroyed by deformation. This typically involves only 10% or less of the study area. The procedure does not test the assumed ridging model. Despite the few (four) examples considered, the study is valuable for its confirmation that the amounts of opening and ridging depend on proportions of shear and divergence in the large-scale deformation. More examples should be studied. 3. Testing with AIDJEX Data, April-August 1975 A thickness distribution calculation was performed by Roger Colony (personal communication) using strain data from AIDJEX, April-August 1975. At the end of the run, comparison data for areas of open water were available from Landsat images. The results were not favorable: the thickness distribution model predicted 23% open water, as opposed to the measured value of 11%. According to Colony, very little open water was formed as a result of thin ice melting away, because there was very little thin ice available at the beginning of the melt season. Instead, the open water was created by divergence. In the theory, once the fraction of open water has reached a prescribed value (G* = 5%, say), the redistribution function assumes the trivial form i» = div u if G(0) > G* This means that all deformations are accomplished by changing the proportion of open water, with no need for ridging. In this calculation, then, the large proportion of open water results from large divergence and would not be affected by modification to the redistribution function or to the assumed growth rates. The erroneous values are probably due to poor strain estimates. These estimates were based on the motion of the four manned ice stations, and other work has shown that those strain estimates are vulnerable to large sampling errors (Thorndike and Colony, 1977). 4. Test with AIDJEX Winter Data Another test was possible using winter data from AIDJEX. A sequence of Landsat images from March 1976 showed changes in the area of thin ice. Ground truth measurements had been made so that the thickness of the ice corresponding to the grey scales on Landsat could be defined. Strain data were assembled by fitting third-order polynomials to the measured displacements of all available buoy and manned camp data and then examining the coefficients. During the last 9 days of the 21-day test, the ice in the manned camp region was motionless. Both strain rate invariants were fixed at zero then.

148 A. S. THORNDIKE The measured and modeled values for several categories of thin ice are presented in Table 1. The calculated results follow the measurements closely, resolving an increase in the area of very thin ice on 7 and 8 March and, by the end of the run, an increase in the area of ice of roughly 50 cm in thickness. A direct borehole measurement was made of ice thickness in one of the grey areas included in the Landsat measurements. That ice was 55 cm thick on 26 March. In a second calculation, to evaluate the importance of the shear term in the redistribution, the shear invariant was set to zero, but the divergence was left unchanged. The tabulated results do not agree as well with the measurements. This run is evidence that large-scale mean strains provide acceptable inputs, that the shear invariant makes an important contribution to the redistribution, and that the assumed growth rates for thin ice are reasonable. 5. Test with Ice Model The thickness distribution theory has been used in a full dynamic ice model, where strains are determined internally to the model. The resulting changes in thickness distribution are used to adjust the average thickness of the ice and to calculate the ice strength. These quantities feed back into the dynamic model. The ice strength, following arguments by Rothrock (1975), is a function of the lowest 5th percentile of the thickness distribution. Work during compression is done against gravity in building ridges out of the thin ice. Ice strengths predicted by these arguments are typically less than the value required by Pritchard (1976) to balance forces on the ice. We suspect this is a shortcoming of the argument relating strength to thickness distribution, not a shortcoming of the thickness distribution theory. In one such full model run, some Landsat comparisons were possible (Coon et al., 1977). The measured area of ice thinner than an estimated 1O-20 cm was bracketed by the calculated values for G (10 cm) and G (30 cm), implying that the measurements and the calculations were consistent. TABLE 1 MEASURED AND MODELED VALUES (%) OF THIN ICE Thickness Category Landsat Measure - l nents (cm) Thickness Distribution Theory (cm) Theory with No Shear (cm) Date 0-10 10-20 20-50 0-10 10-20 20-75 0-10 10-20 20-75 5 March 76 6 March 76 7 March 76 8 March 76 1.3 1.2 0.4 0.5 1.3 0.5 0.7 1.7 0.8 0 0.6 0 1.1 0 0.2 0.5 0.5 26 March 76 0 0 2.6 0 0 2.6 0 0 3.4 NOTE: All Landsat measurements were made at 2100 GMT.

Tests of the Ice Thickness Distribution Theory 149 DISCUSSION What are the prospects for better tests of this theory? An ideal test would be over a region of roughly 100 km square for several days. Strain data, growth rate data, and measured thickness distributions would be needed. None of these is easy to obtain, but in particular the thickness distribution itself is beyond our capability. The only operational technique for measuring the full range of thickness is by upward-looking sonar mounted on a nuclear submarine. To sample fully a 100 x 100 km region would require a greater commitment of submarine time than we are likely to get. Of course, a good test would need to be repeated for several regions with different distributions and for different deformations. Until some appropriate satellite sensor comes along, we will have to settle for weak tests of the thin ice-open water part of this model. Improvements in the treatment of thick ice must come after a more complete understanding of the "life cycle" of pressure ridges. How are they formed? What is the statistical relationship between the thickness of ice going into the ridge and the thickness of the ridge itself? How important is the presence of large cavities of sea water in a ridge or of unconsolidated ice blocks in the keel? Do ridge keels change their shape primarily through erosion or ablation? Some of these processes depend on other properties than thickness on age or slope, for instance and so they cannot be treated within the present framework. An essential ingredient of the model is that thick ice decreases in thickness over the course of a year. Yet there is very little observational evidence to confirm this. The ablation rates for thick ice that are now being used (see table in Thorndike et al., 1975) need to be tested against field measurements. No tests have been made yet in regions of low ice concentrations. At times, the periphery of the ice pack will be defined by a region of low concentration. There the model takes a simpler form with ip = div u. To compensate for the simple redistribution, we can expect more difficult thermodynamics, including positive feedback between the area of open water and the process of lateral melting. The strength of the present model rests in its clean mathematical statement of the interaction of thermal and mechanical processes which control the ice pack. When tests are done using measurements, the results are in rough agreement. The following tentative conclusions are offered. They have been suggested, but certainly not proven, by the tests outlined here. 1. Prescribed growth rates for thin ice work well in the Beaufort Sea in winter. 2. The present treatment depending on two invariants of strain rate (related to divergence and to the rate of shearing) seems adequate. No need is seen to include the direction of maximum shear. 3. Significant redistribution of ice is accomplished through large-scale shearing. 4. The model does not produce enough very thick ice. The level of detail in the model seems roughly appropriate to our level of understanding of the physical processes and to our present ability to confirm by measurement.

150 A. S. THORNDIKE ACKNOWLEDGMENT This work was supported by the National Science Foundation Grant OPP71-04031, formerly GV 28807, to the University of Washington for the Arctic Sea Ice Study. REFERENCES Coon, M. D., G, A. Maykut, R. S. Pritchard, D. A. Rothrock, and A. S. Thorndike. 1974. Modeling the pack ice as an elastic-plastic material. AIDJEX Bulletin, 24, 1-105. Coon, M. D., R. T. Hall, and R. S. Pritchard. 1977. Predictions of arctic ice conditions for operations. In Proceedings of the Ninth Annual Offshore Technology Conference, vol. 4, pp. 307-14, Offshore Technology Conference, Dallas, Texas. Hibler, W. D., W. F. Weeks, A. Kovacs, and S. F. Ackley. 1974. Differential sea ice drift. I. Spatial and temporal variations in sea ice deformation. Journal ofglaciology, 13(69), 437-55. Nye, J. F. 1975. A test of the ice thickness redistribution equations by measurements of ERTS pictures. AIDJEX Bulletin, 28, 141-49. Parmerter, R. R., and M. D. Coon. 1972. A model of pressure ridge formation in sea ice. Journal of Geophysical Research, 77(33), 6565-75. Pritchard, R. S. 1976. An estimate of the strength of arctic pack ice. AIDJEX Bulletin, 34, 94-115. Rothrock, D. A. 1975. The energetics of the plastic deformation of pack ice by ridging. Journal of Geophysical Research, 80(33), 4514-19. Rothrock, D. A., and R. T. Hall. 1975. Testing the redistribution of sea ice from ERTS photographs. AIDJEX Bulletin, 29, 1-19. Thorndike, A. S., and R. Colony. 1977. Estimating the deformation of sea ice. AIDJEX Bulletin, 37, 25-36. Thorndike, A. S., D. A. Rothrock, G. A. Maykut, and R. Colony. 1975. The thickness distribution of sea ice. Journal of Geophysical Research, 80(33), 4501-13. Wittmann, W., and J. J. Schule. 1966. Comments on the mass budget of arctic pack ice. In Proceedings of a Symposium on Arctic Heat Budget and Atmospheric Circulation (ed. J.,0. Fletcher), pp. 215-46, RM-5233-NSF, Rand Corp., Santa Monica, Calif.