Jmnt. Gray et Lw. Morland, A 2-DIMENSIONAL MODEL FOR THE DYNAMICS OF SEA-ICE, Philosophical transactions-Royal Society of London. Physical sciences and engineering, 347(1682), 1994, pp. 219-290
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.