Abstract

This talk is about the analysis of an asymptotically reduced system for rotationally constrained convection. The presence of a dominant balance in equations for fluid flow can be exploited to derive a simpler set of governing equations that permits analytical explorations. For rotation dominated flows, the geostrophic balance occurs: the pressure gradient force is balanced by the Coriolis effect. The Taylor-Proudman constraint suggests that the dominant Coriolis force leads to flows that are organized into vertical plumes or columns whose horizontal scale is small compared to the layer height. Applying the asymptotic theory for small Rossby number and tall columnar structures, Julien and Knobloch derived a closed set of reduced equations from the three-dimensional Boussinesq equations. Also, Sprague et al. numerically simulated this reduced model to study the equal populations of cyclonic and anticyclonic structures in rapid rotating convection. This reduced system is interesting yet challenging for analytical study. On the one hand, the nonlinear convection term has a reduced complexity since it contains only the horizontal gradient. On the other hand, the physical domain remains three dimensional, while the regularizing viscosity acts in the horizontal direction only, creating a major difficulty for establishing the global existence theory. Another difficulty arises due to a linear term involving the vertical derivative of the stream function, reflecting the balance of the Coriolis force by the pressure. I will present some of our results motivated by the global regularity problem. We show that the model is globally well-posed if regularized by a very weak dissipation. I will also discuss the case of infinite Prandtl number convection, and the situation when both of the Prandtl and Rayleigh numbers approach infinity. This is a joint project with Cao and Titi.