Numerical simulations of turbulent Rayleigh-B\'enard 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 ϵ=ΔT/Tr≪1, where ΔT denotes the superadiabatic temperature drop over the
convective layer and Tr the bottom temperature. Using direct numerical
simulations, a systematic comparison of anelastic and fully compressible
convection is carried out. With decreasing superadiabaticity ϵ, 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 ϵ for numerical
reasons. On the other hand, if the astrophysical system studied contains
ϵ∼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 ϵ
for efficiency reasons remain moderate. If quantitative errors of the order of
10% are acceptable in such low ϵ regions, our work suggests that
fully compressible simulations can indeed be computationally more efficient
than their anelastic counterparts.Comment: 24 pages, 9 figure