The effect of stratification and compressibility on anelastic convection in a rotating plane layer

Abstract

We study the effect of stratification and compressibility on the threshold of convection and the heat transfer by developed convection in the nonlinear regime in the presence of strong background rotation. We consider fluids both with constant thermal conductivity and constant thermal diffusivity. The fluid is confined between two horizontal planes with both boundaries being impermeable and stress-free. An asymptotic analysis is performed in the limits of weak compressibility of the medium and rapid rotation ( À1/12 ( jj ( 1, where is the Taylor number and is the dimensionless temperature jump across the fluid layer). We find that the properties of compressible convection differ significantly in the two cases considered. Analytically, the correction to the characteristic Rayleigh number resulting from small compressibility of the medium is positive in the case of constant thermal conductivity of the fluid and negative for constant thermal diffusivity. These results are compared with numerical solutions for arbitrary stratification. Furthermore, by generalizing the nonlinear theory of Julien and Knobloch [Fully nonlinear three-dimensional convection in a rapidly rotating layer. Phys. Fluids 1999Fluids , 11, 1469Fluids -1483 to include the effects of compressibility, we study the Nusselt number in both cases. In the weakly nonlinear regime we report an increase of efficiency of the heat transfer with the compressibility for fluids with constant thermal diffusivity, whereas if the conductivity is constant, the heat transfer by a compressible medium is more efficient than in the Boussinesq case only if the specific heat ratio is larger than two.

Onset of rotating and non-rotating convection in compressible and anelastic ideal gases

“…We have also made a comparison of the full anelastic equations with the entropy diffusion anelastic model that has been employed frequently (e.g., Jones et al 2009, Mizerski andTobias 2011). The linear stability analysis showed that the anelastic equations and entropy diffusion model possess qualitatively similar stability behavior as a function of stratification, though the two models give quantitatively different results.…”

“…To provide a close connection with previous work that has utilized the entropy diffusion model (e.g., Jones et al 2009, Mizerski andTobias 2011), the above equations are supplemented with isentropic boundary conditions given by…”

“…Here we present a simplified form of the anelastic equations that is commonly employed in numerical simulations and analysis (Jones et al 2009, Mizerski and Tobias 2011. To distinguish this model from the anelastic equations presented previously we refer to it as the entropy diffusion model.…”

“…For most geophysical and astrophysical systems the large length scales involved result in strong variations in fluid density across the convecting region due to changes in the fluid pressure; in such cases the Boussinesq approximation is no longer valid and compressibility must be considered (Spiegel and Veronis 1960, Mihaljan 1962, Gray and Giorgini 1976. Previous work has shown that one effect of fluid compressibility on convection dynamics is the occurrence of asymmetric flows with respect to the midplane of the fluid layer in the case of plane layer geometries (Spiegel 1965, Heard 1973, Gough et al 1976, Mizerski and Tobias 2011; analagous effects occur in spherical geometries (e.g., Gilman 1981a, Jones et al 2009). Another important consequence of compressibility is the excitation of convective Rossby waves in rotating convection when the gravity and rotation vectors are not antiparallel (Glatzmaier and Gilman 1981b); the resulting vortex stretching mechanism may be an important process for generating mean flows in gaseous atmospheres (Glatzmaier et al 2009, Verhoeven andStellmach 2014).…”

Mean flow generation in rotating anelastic two-dimensional convection

“…Typically, in Boussinesq formulations the vertical axis increases upwards, whereas in compressible studies it is taken downwards. In order to ease the comparison with Boussinesq models, we take the vertical axis to increase upwards in line with Mizerski and Tobias 32 . The existing studies of anelastic convection, linear and nonlinear, each consider a different formalism (discussed below) and therefore care has to be taken when comparing across the different models.…”