Linear well posedness of regularized equations of sea-ice dynamics

Abstract

The viscous–plastic equations (VPE) of Hibler [J. Geophys. Res. 82(27), 3932–3938 (1977)] are widely adopted and used in Earth system models to represent sea-ice drift due to surface winds, ocean currents, and internal stresses. However, it has been reported by various investigators, at least in one space dimension, that both Hibler’s original equations and their variant using a pressure replacement are ill posed in divergent flow regimes. Especially, Guba et al. [J. Phys. Oceanogr. 43(10), 2185–2199 (2013)] shows that both variants are ill-posed when the flow divergence exceeds a minimum threshold and their results seem to extend to two dimensions when a tensile cut-off is used. In particular, Hibler uses a Heaviside function cut-off for the viscosity coefficients of the VPE’s to avoid a singularity at infinity. Lemieux et al. [J. Comput. Phys. 231(17), 5926–5944 (2012)] regularized the Heaviside function by a hyperbolic tangent for numerical efficiency. Here, we show that, for periodic data, the linearized one-dimensional regularized VPE’s, in which the Heaviside function is replaced with a hyperbolic tangent, is well posed in the case of Hibler’s original equations. Moreover, we prove that the linearization procedure, for the regularized equations, is consistent, in the sense that the residual converges to zero that the perturbation of the solutions goes to zero, in suitable norms.