A two-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 responsive water drag on the base of the floes. Integration of the mass and momentum balances through the layer thickness result in a two-dimensional theory for the interface between ocean and atmosphere. The theory is presented for a plane horizontal interface, but the construction is readily extended to a non-planar interface. An interacting continua framework is adopted to describe the layer mixture of ice and water, which introduces the layer thickness h and ice area fraction A as smoothly varying functions of the plane coordinate and time, on a pack length scale and weather system timescale. It is shown how an evolution equation for A which ignores ridging can lead to the area fraction exceeding unity in maintained converging flow, which is physically invalid. This is a feature and weakness of current models, and is eliminated by artificial cut-off in numerical treatments. Here we formulate a description of the ridging process which redistributes smoothly the excess horizontal ice flux into increasing thickness of a ridging zone of area fraction Ar, and a simple postulate for the vertical ridging flux yields an evolution equation for A which shows how A can approach unity 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 horizontal momentum “balance involves the gradients of the extra stress integrated through the layer thickness, extra to the integrated water pressure over the depth of a local floe edge below sea level. These extra stresses 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 provides an interpretation of the local extra stress in the pack. The interpretation introduces the further model function f (A) which defines the fraction of ice-ice contact length over the boundary of a floe, describing an increase of the contact fraction as A increases. Model interaction 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 motion of a failure criterion on the stresses in a ridging zone where the convergence greatly exceeds a threshold value. We have therefore defined precisely the two-dimensional ice pack stress arising in the momentum balance, and determined its relation to the contact forces between adjacent floes. The foregoing analyses hinge on the introduction of dimensionless variables and coordinate scalings which reflect the orders of magnitude of the many physical variables and their gradients in both individual floe and ice pack motions. A variety of small dimensionless parameters arise, which allows the derivation of leading-order equations defining a reduced model which describes the major balances in the motion. The distinct equations for diverging and converging flow regions indicates the existence of moving boundaries (in the two-dimensional pack domain) in the flow, satisfying appropriate matching conditions to be determined as part of the complete evolution. This feature appears to have been ignored in previous treatments. Here we illustrate the evolution of a moving boundary by constructing an exact solution 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 demonstrates the influence of various parameters, but, importantly, will provide a test solution for numerical algorithms which must be constructed to determine more complex one and two-dimensional motions.