High-resolution simulations of compressible convection using the piecewise-parabolic method

We use two-dimensional simulations to study turbulent and fully compressible thermal convection in deep atmospheres. Density contrasts across these convective layers are typically around 50. The fluid model is that of an ideal gas with a constant thermal conductivity. The piecewise-parabolic method (PPM), with thermal conductivity added in, is used to solve the fluid equations of motion. No explicit viscosity is included, and the low numerical viscosity of PPM leads to a very low effective Prandtl number. The resulting flows are temporally and spatially chaotic. Some of the simulations possess sufficiently high spatial resolutions that eddies of widely ranging sizes are allowed to coexist in the same pressure scale heights. We study a variety of convective systems: we vary the mesh resolution, the initial state, the strength with which the convection is driven, and the thermal conductivity. All of our results are subject to the highly idealized model we have adopted. The main point of this study is to illustrate the effects of a wide range of spatial scales on the nature of compressible convection. We describe and characterize the numerical viscosity of PPM and relate this numerical viscosity to the rate of decay of eddies and power spectra of the convective flows. We analyze the statistics of eddies and find a universal eddy profile for our low Prandtl number convective flows. We perform a Fourier analysis of the convective velocities and determine timescales for dynamic relaxation, dependence of the flow with depth, and relations between mesh resolution and global convective flow. The velocity power spectra systematically steepen as the depth increases, so that velocity over long wavelengths increases with depth and relative velocities across small wavelengths decreases with depth. We examine dynamic and thermal relaxation processes, interaction of small and large scales, mechanisms that undermine the stability of smooth flows, global structure of high-resolution, low Prandtl number-convection, compressibility, and small-scale structures in convection. Supersonic flow is present in all of our models, even for the most weakly driven where the temperature gradient along the lower boundary exceeds the adiabatic value by 0.4%. Shocks are ubiquitous and are an important mechanism for damping the flow. Small and large eddies coexist in the largest scale heights. Large eddies occasionally span the depth of the layer, but there are no persistent convection cells. Small eddies exist at all depths independent of local scale height. The largest eddies are predominantly in the largest/lowest scale heights. The correlation lengths of velocity fields and energy fluxes tend to be longer for vertical displacements than for horizontal displacements. Over the span of resolutions tested here, correlation lengths systematically decrease with increasing mesh resolution.