A model for convection in the evolution of under-ice melt ponds

A model for convection in the evolution of under-ice melt ponds is presented. The system exhibits two competing effects namely, a temperature gradient which is destabilising and a salt gradient which is stabilising. Density is assumed to have a dependence quadratic in temperature and linear in concentration. A linear instability analysis and a nonlinear stability analysis are performed. The standard energy method does not yield unconditional stability so a weighted energy analysis is employed to achieve global results. The global stability bound is found to be independent of the salt field and a presentation of the region of possible subcritical instabilities is given.