Analysis of a sea-ice thickness distribution equation: modeling and dynamics

We develop a theoretical analysis of the sea-ice thickness distribution equation as a nonlinear transport problem in the extended phase space of horizontal position, time, and ice thickness. The thickness distribution is first formulated rigorously as a Borel measure, obtained as the normalized pushforward of planar area measure under the ice-thickness field. We then study the sea-ice thickness distribution equation with horizontal advection, thermodynamic growth or melting, and a nonlocal mechanical redistribution function. Due to the presence of thermodynamic the governing equation is represented by a conservation law in the extended phase space. Using extended characteristics, we derive a Green-kernel/Duhamel representation including initial and inflow boundary contributions, and establish local well-posedness criteria based on transport regularity, measurability, Lipschitz continuity, and compatibility in the space of absolute integrable functions. We examine classical ridging-type redistribution models and show how regularization of the open-water Dirac term and of the velocity-dependent ridging coefficient is required for the hypotheses of the well-posedness theorem. A Fourier expansion in the thin-ice, divergence-free regime reveals the infinite-dimensional coupled structure generated by the nonlocal dependence on the cumulative distribution. We further derive reduced criteria for the evolution, displacement, and possible splitting of peaks in the thickness profile, including conditions for the transition from unimodal to bimodal distributions under thermodynamic and mechanical forcing. Finally, we formulate the coupled sea-ice thickness, momentum, and elastic-viscoplastic rheology as a quasi-linear evolution problem and discuss local strong solutions, energy bounds, possible global continuation, and mechanisms of blow-up. The analysis identifies the mathematical assumptions needed for predictive sea-ice thickness models and clarifies the limitations imposed by singular open-water states, non-Lipschitz ridging closures, and derivative-loss mechanisms.