Anelastic Versus Fully Compressible Turbulent Rayleigh-Bénard Convection

Abstract

Numerical simulations of turbulent Rayleigh-Bénard convection in an ideal gas, using either the anelastic approximation or the fully compressible equations, are compared. Theoretically, the anelastic approximation is expected to hold in weakly superadiabatic systems with $\epsilon =\Delta T/T_r\ll 1$, where $\Delta T$ denotes the superadiabatic temperature drop over the convective layer and $T_r$ the bottom temperature. Using direct numerical simulations, a systematic comparison of anelastic and fully compressible convection is carried out. With decreasing superadiabaticity $\epsilon$, the fully compressible results are found to converge linearly to the anelastic solution with larger density contrasts generally improving the match. We conclude that in many solar and planetary applications, where the superadiabaticity is expected to be vanishingly small, results obtained with the anelastic approximation are in fact more accurate than fully compressible computations, which typically fail to reach small $\epsilon$ for numerical reasons. On the other hand, if the astrophysical system studied contains $\epsilon \sim O(1)$ regions, such as the solar photosphere, fully compressible simulations have the advantage of capturing the full physics. Interestingly, even in weakly superadiabatic regions, like the bulk of the solar convection zone, the errors introduced by using artificially large values for $\epsilon$ for efficiency reasons remain moderate. If quantitative errors of the order of $10\%$ are acceptable in such low $\epsilon$ regions, our work suggests that fully compressible simulations can indeed be computationally more efficient than their anelastic counterparts.

arXiv comment24 pages, 9 figures

References in corpus (14)

• Astrophysical magnetic fields and nonlinear dynamo theory
• Scaling in thermal convection: A unifying theory
• Global-Scale Turbulent Convection and Magnetic Dynamo Action in the Solar Envelope
• Structure and Evolution of Giant Cells in Global Models of Solar Convection
• Approaching the Asymptotic Regime of Rapidly Rotating Convection: Boundary Layers vs Interior Dynamics
• Low Mach Number Modeling of Type Ia Supernovae
• Active Carbon and Oxygen Shell Burning Hydrodynamics
• Convective dynamos in spherical wedge geometry
• Energy Conservation and Gravity Waves in Sound-proof Treatments of Stellar Interiors: Part I Anelastic Approximations
• Anelastic and Compressible Simulations of Stellar Oxygen Burning
• Zonal flow scaling in rapidly-rotating compressible convection
• Dissipation Layers in Rayleigh-Bénard Convection: A Unifying View
• The compressional beta effect: a source of zonal winds in planets?
• The Breakdown of the Anelastic Approximation in Rotating Compressible Convection: Implications for Astrophysical Systems

Cited by in corpus (15)

• Rayleigh-Bénard convection with a melting boundary
• Implicit large eddy simulations of anisotropic weakly compressible turbulence with application to core-collapse supernovae
• Turbulent transport by diffusive stratified shear flows: from local to global models. Part I: Numerical simulations of a stratified plane Couette flow
• Penetrative Convection in Partly Stratified Rapidly Rotating Spherical Shells
• Spherical-shell boundaries for two-dimensional compressible convection in a star
• 3D Simulations and MLT: I. Renzini's Critique
• Fully developed anelastic convection with no-slip boundaries
• Weakly non-Boussinesq convection in a gaseous spherical shell
• Experimental study of convection in the compressible regime
• Strongly superadiabatic and stratified limits of compressible convection
• Turbulent mesoscale convection in the Boussinesq limit and beyond
• Compressible turbulent convection in highly stratified adiabatic background
• Scale Selection in the Stratified Convection of the Solar Photosphere
• Validity of Sound-Proof Approaches in Rapidly-Rotating Compressible Convection: Marginal Stability vs. Turbulence
• Compressible turbulent convection: The role of temperature-dependent thermal conductivity and dynamic viscosity