Loss of hyperbolicity and ill-posedness of the viscous-plastic sea ice rheology in uniaxial divergent flow

Local contact interactions between sea ice flees can be modeled on the larg e scale by treating the pack as a two-dimensional continuum with granular p roperties. One such model, which has gained prominence, is the viscous plas tic 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 decodes. It is shown that in uniaxial flow this model reduces to a system of three quasi-linear first-order partial differential equatio ns, which are hyperbolic in convergent flow and have mixed elliptic/hyperbo lic behavior in divergence with two imaginary wave speeds. A linear stabili ty analysis shows that the change in type causes the equations to be unstab le and ill posed in uniaxial divergence. The root cause is a positive feedb ack 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 m odel 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 curv e in principal stress space. Numerical simulations show that this model div erges without becoming unstable.