On the Profile of Singularity Formation for the Incompressible Hydrostatic Boussinesq system

The primitive equations (PEs) model planetary large-scale oceanic and atmospheric dynamics. While it has been shown that there are smooth solutions to the inviscid PEs (also called the hydrostatic Euler equations) with constant temperature (isothermal) that develop stable singularities in finite time, the effect of non-constant temperature on the singularity formation has not been established yet. This paper studies the stability of singularity formation for non-constant temperature in two scenarios: when there is no diffusion in the temperature, or when a vertical diffusivity is added to the temperature dynamics. For both scenarios, our results indicate that the variation of temperature affects neither the formation of singularity, nor its stability, in the velocity field, respectively.