  Thermal conduction is an important energy transfer and damping mechanism in
astrophysical flows. Fourier's law - the heat flux is proportional to the
negative temperature gradient, leading to temperature diffusion - is a
well-known empirical model of thermal conduction. However, entropy diffusion
has emerged as an alternative thermal conduction model, despite not ensuring
the monotonicity of entropy. This paper investigates the differences between
temperature and entropy diffusion for both linear internal gravity waves and
weakly nonlinear convection. In addition to simulating the two thermal
conduction models with the fully compressible Navier-Stokes equations, we also
study their effects in the reduced, "sound-proof" anelastic and
pseudo-incompressible equations. We find that in the linear and weakly
nonlinear regimes, temperature and entropy diffusion give quantitatively
similar results, although there are some larger errors in the
pseudo-incompressible equations with temperature diffusion due to inaccuracies
in the equation of state. Extrapolating our weakly nonlinear results, we
speculate that differences between temperature and entropy diffusion might
become more important for strongly turbulent convection.
