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 also measures how much the density varies across the layer. 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 so thin that temperature and density vary little across them, even though there may be substantial temperature and density variations across the whole layer. 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.
Highlights
The problem of the influence of density stratification on developed convection is of great importance from the astrophysical point of view
These scaling laws are formulated in terms of the Rayleigh number, the Prandtl number and the temperature ratio Γ which measures the strength of the stratification
The existence of the well-mixed entropy state, with the entropy changes being mainly confined to thin boundary layers, makes it possible to estimate the terms in the entropy balance equation, so allowing Nusselt number and Reynolds number relationships to be established
Summary
The problem of the influence of density stratification on developed convection is of great importance from the astrophysical point of view. Developed anelastic convection with no-slip boundaries phase diagram, figure 2 of Grossmann & Lohse (2000), correspond to Rayleigh numbers less than about 1012 in the Boussinesq case It must be noted, that contrary to the Boussinesq case, which is well established by numerous experimental and numerical investigations, there are to date no experiments on fully turbulent stratified convection, due to the difficulties of achieving significant density stratification in laboratory settings. There have been some numerical investigations of anelastic convection in a plane layer, mostly focused on elucidating how well the anelastic approximation performs compared with fully compressible convection, Verhoeven et al (2015) and Curbelo et al (2019) This latter paper notes that the top and bottom boundary layer structures that occur in the case of a high Prandtl number are different.
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