A 2-DIMENSIONAL MODEL FOR THE DYNAMICS OF SEA-ICE

This paper develops a systematic analysis of a sea ice pack viewed as a thin layer of coherent ice floes and open water regions at the ocean surface. The pack is driven by wind stress and Coriolis force, with r esponsive water drag on the base of the floes. Integration of the mass and momentum balances through the layer thickness result in a two-dim ensional theory for the interface between ocean and atmosphere. The th eory is presented for a plane horizontal interface, but the constructi on is readily extended to a non-planar interface. An interacting conti nua framework is adopted to describe the layer mixture of ice and wate r, which introduces the layer thickness h and ice area fraction A as s moothly varying functions of the plane coordinate and time, on a pack length scale and weather system timescale. It is shown how an evolutio n equation for A which ignores ridging can lead to the area fraction e xceeding unity in maintained converging flow, which is physically inva lid. This is a feature and weakness of current models, and is eliminat ed by artificial cut-off in numerical treatments. Here we formulate a description of the ridging process which redistributes smoothly the ex cess horizontal ice flux into increasing thickness of a ridging zone o f area fraction Ar, and a simple postulate for the vertical ridging fl ux yields an evolution equation for A which shows how A can approach u nity asymptotically, but not exceed unity, in a maintained converging flow. This is a significant feature of the new model, and eliminates a serious physical and numerical flaw in existing models. The horizonta l momentum balance involves the gradients of the extra stress integrat ed through the layer thickness, extra to the integrated water pressure over the depth of a local floe edge below sea level. These extra stre sses are zero in diverging flow and arise as a result of interactions between floes during converging flow. It is shown precisely how a mean stress in a floe is determined by such edge tractions, and in turn pr ovides an interpretation of the local extra stress in the pack. The in terpretation introduces the further model function f(A) which defines the fraction of ice-ice contact length over the boundary of a floe, de scribing an increase of the contact fraction as A increases. Model int eraction mechanisms then suggest a qualitative law for the pack stress in terms of relative motions of the floes which define the pack-scale strain rates. A simple viscous law is presented for illustration, but it is shown that even this simple model can reflect a conventional mo tion of a failure criterion on the stresses in a ridging zone where th e convergence greatly exceeds a threshold value. We have therefore def ined precisely the two-dimensional ice pack stress arising in the mome ntum balance, and determined its relation to the contact forces betwee n adjacent floes. The foregoing analyses hinge on the introduction of dimensionless variables and coordinate scalings which reflect the orde rs of magnitude of the many physical variables and their gradients in both individual floe and ice pack motions. A variety of small dimensio nless parameters arise, which allows the derivation of leading-order e quations defining a reduced model which describes the major balances i n the motion. The distinct equations for diverging and converging flow regions indicates the existence of moving boundaries (in the two-dime nsional pack domain) in the flow, satisfying appropriate matching cond itions to be determined as part of the complete evolution. This featur e appears to have been ignored in previous treatments. Here we illustr ate the evolution of a moving boundary by constructing an exact soluti on to a one-dimensional pack motion which describes onshore drift due to increasing, then decreasing, wind stress. During the second phase a region of diverging flow expands from the free edge. The solution dem onstrates the influence of various parameters, but, importantly, will provide a test solution for numerical algorithms which must be constru cted to determine more complex one and two-dimensional motions.