Loss of Hyperbolicity and Ill-posedness of the Viscous–Plastic Sea Ice Rheology in Uniaxial Divergent Flow

Abstract

Local contact interactions between sea ice floes can be modeled on the large scale by treating the pack as a two-dimensional continuum with granular properties. One such model, which has gained prominence, is the viscous plastic constitutive rheology, using an elliptical yield curve and normal flow law. It has been used extensively in ice and coupled ice?ocean studies over the past two decades. It is shown that in uniaxial flow this model reduces to a system of three quasi-linear first-order partial differential equations, which are hyperbolic in convergent flow and have mixed elliptic/hyperbolic behavior in divergence with two imaginary wave speeds. A linear stability analysis shows that the change in type causes the equations to be unstable and ill posed in uniaxial divergence. The root cause is a positive feedback mechanism that becomes stronger and stronger with smaller wavelengths. Numerical computations are used to demonstrate that fingers form and break the ice into discrete blocks. The frequency and growth rate of the fingers increase as the numerical resolution is increased, which implies that the model does not converge to a solution as the grid is refined. Two new models are proposed that are well posed. The first retains the positive feedback mechanism and introduces higher-order derivatives to suppress the unbounded growth rate of the instability. The second eliminates the positive feedback mechanism, and the instability, by repositioning the elliptical yield curve in principal stress space. Numerical simulations show that this model diverges without becoming unstable.