The Cooling Box Problem: Convection with a quadratic equation of state

Abstract

We investigate the convective upper-boundary cooling of a fluid with a nonlinear equation of state. We performed three-dimensional direct numerical simulations of the flow with a fixed top-boundary temperature, less than the initial fluid temperature, and determine the conditions under which the system is linearly stable. When unstable, the resultant convection leads to a well-mixed lower-layer, beneath a strongly stratified boundary layer. The convective system quickly reaches a quasi-steady state and is controlled by three dimensionless parameters: the Rayleigh number (Ra), the Prandtl number (Pr), and the dimensionless bottom water temperature (TB). We develop scaling laws for the thickness of the top boundary-layer and the kinetic energy of the system and demonstrate that they agree well with the numerical simulations.