Dynamics of density patches in infinite Prandtl number convection

Abstract

This work examines the dynamics of density patches in the 2D zero-diffusivity Boussinesq system modified such that momentum is in a large Prandtl number balance. We establish the global well-posedness of this system for compactly supported and bounded initial densities, and then examine the regularity of the evolving boundary of patch solutions. For $k \in \{0,1,2\}$, we prove the global in time persistence of $C^{k+\mu}$-regularity, where $\mu \in (0,1)$, for the density patch boundary via estimates of singular integrals. We conclude with a simulation of an initially circular density patch via a level-set method. The simulated patch boundary forms corner-like structures with growing curvature, and yet our analysis shows the curvature will be bounded for all finite times.