Fully developed anelastic convection with no-slip boundaries

TL;DR

This paper develops scaling laws for heat transport and flow in high Rayleigh number anelastic convection with no-slip boundaries, highlighting the role of boundary layer structures and stratification effects.

Contribution

It introduces explicit scaling laws for anelastic convection considering boundary layer differences and stratification, supported by numerical simulations up to Ra = 10^7.

Findings

Boundary layers differ significantly at top and bottom in anelastic convection.

Scaling laws depend on whether dissipation occurs mainly in boundary layers or the bulk.

Numerical results agree with theoretical predictions up to Ra = 10^7.

Abstract

Anelastic convection at high Rayleigh number in a plane parallel layer with no slip boundaries is considered. Energy and entropy balance equations are derived, and they are used to develop scaling laws for the heat transport and the Reynolds number. The appearance of an entropy structure consisting of a well-mixed uniform interior, bounded by thin layers with entropy jumps across them, makes it possible to derive explicit forms for these scaling laws. These are given in terms of the Rayleigh number, the Prandtl number, and the bottom to top temperature ratio, which measures how stratified the layer is. The top and bottom boundary layers are examined and they are found to be very different, unlike in the Boussinesq case. Elucidating the structure of these boundary layers plays a crucial part in determining the scaling laws. Physical arguments governing these boundary layers are presented, concentrating on the case in which the boundary layers are thin even when the stratification is large, the incompressible boundary layer case. Different scaling laws are found, depending on whether the viscous dissipation is primarily in the boundary layers or in the bulk. The cases of both high and low Prandtl number are considered. Numerical simulations of no-slip anelastic convection up to a Rayleigh number of $10^7$ have been performed and our theoretical predictions are compared with the numerical results.