Note on the No-stress Boundary Condition at the Edge of the Ice Pack

Résumé

The theoretical modelling of the large-scale motion of the arctic ice pack is receiving increasing attention as the economic importance of the region increases. One of the most widely used types of model is the so-called "viscous fluid" model. ... The boundary condition at the edge of the ice pack is an important feature of most such models. In some cases a no-slip condition seems appropriate, but in others, when the ice near the boundary has a low compactness (fraction of ice coverage) or the boundary occurs away from a coast, some other condition may be more appropriate. One that is often suggested is a no-stress condition, which is often assumed to imply that there is no velocity gradient perpendicular to the boundary. When the edge occurs away from a coast, the latter assumption is wrong. It suffices for present purposes to assume that we are dealing with an incompressible two-dimensional fluid. In this case the viscous force per unit of area (corresponding to volume in three dimensions) is del·(A del v), where A is an isotropic but possibly variable coefficient of eddy viscosity, and v, the large-scale averaged horizontal ice velocity, has components u and v in the x and y directions respectively. The notation del v ... is equivalent to the tensor [partial derivative of vi with respect to xj], where i and j vary independently over all coordinate directions, and (del·del v)i = Sum over the index j of (partial derivative with respect to xj of the partial derivative of vi with respect to xj). ... Since the viscous force is the divergence of the stress, the quantity A del v is often thought of as the eddy stress (or "internal ice stress"). That this is not true is easily seen by noting that the tensor A del v, to be referred to here as the "pseudo-stress" tensor, is not symmetrical. The non-diagonal elements of the stress tensor, which must be equal, are ½A(partial derivative of v with respect to x + partial derivative of u with respect to y). The distinction made here is irrelevant in determining the viscous forces, since the stress tensor and the pseudo-stress tensor differ by a tensor of zero divergence .... In large-scale ocean models which employ eddy viscosity, the stress itself is often required in connection with boundary conditions, particularly at the sea surface, or naviface .... Here, however, those who use the pseudo-stress are saved both by scale considerations and by the fact that w=0 (where w is the vertical or z-component of velocity), hence the (partial derivative of w with respect to x)=0 and the (partial derivative of w with respect to y)=0, at the naviface, so that the stress components there reduce to A(partial derivative of u with respect to z) and A(partial derivative of v with respect to z). In the "viscous liquid" model of an ice pack bounded by open water, we at last have a case in which the distinction between real stress and pseudo-stress assumes geophysical importance. ... Assuming (without loss of generality) that the edge is oriented with its outward normal in the first quadrant at an angle of theta to the x-axis, we have for the direction cosines: n1 = +cos(theta), n2 = +sin(theta), t1 = +sine(theta), t2= -cos(theta). The appropriate expression of the condition that there be no tangential stress at the boundary becomes: sin(2theta)(partial derivative of u with respect to x - partial derivative of v with respect to y) - cos(2theta)(partial derivative of v with respect to x - partial derivative of u with respect to y) = 0. ... If the boundary is oriented along a coordinate axis this reduces to (partial derivative of v with respect to x) + (partial derivative of u with respect to y) = 0, which qualitatively means that shears at the boundary are permitted, provided that they are part of a locally uniform rotation and do not produce deformation of the ice field. If one also wishes to assume zero normal stress at the boundary, there is an additional condition given by: (partial derivative of u with respect to x)cos² (theta) + (partial derivative of v with respect to y)sin²(theta) + ½(partial derivative of v with respect to x + partial derivative of u with respect to y)sin(2theta) = 0. These are purely mathematical deductions; the appropriateness of the physical conditions is a more difficult question which can only be answered experimentally. The physical condition of zero tangential stress qualitatively means that no deformation of the ice field can take place at the boundary. Techniques for measuring the deformation of the ice fields are now under development. It is suggested that it would be interesting to measure the deformation of ice fields near the boundary, even though a measurement of non-zero deformation (which the author suspects would be found, since external driving forces will in general tend to produce deformation) would not distinguish critically between the correctness of the boundary condition and the basic validity of the "viscous liquid" type of model.