Turbulent mesoscale convection in the Boussinesq limit and beyond

Mesoscale convection covers an intermediate scale range between highly fluctuating small-scale turbulence and the global organization of the convection flow. It is often characterized by an order of the convection patterns despite very high Rayleigh numbers and strong turbulent fluctuations. In this article, we review previous and discuss several new aspects of mesoscale convection which have been obtained by three-dimensional direct numerical simulations. These numerical studies are performed in the simplest, yet characteristic configuration of a plane layer that is heated from below and cooled from above. The setup is disentangled from other physical processes which enhance the physical complexity of mesoscale convection in real applications. We cover the role of thermal and mechanical boundary conditions for structure formation, the role of constant rotation of the layer, the impact of the domain shape, and the role of Prandtl number Pr. With respect to the last point, we also obtained results for very low Pr values that arise in astrophysical convection, but are inaccessible in controlled laboratory experiments with liquid metals. In addition to the simplest case of Boussinesq convection, we report studies of non-Boussinesq mesoscale convection in the same configuration. To this end, we investigate effects of compressibility and temperature dependence of material properties. The kinetic energy dissipation rate turns out to remain a central quantity for the turbulent mixing in compressible convection. The different components of energy dissipation rate, their intermittent statistics, their connection to turbulent viscosity, and the resulting multifractal properties are analyzed. Finally, we reflect on what can be learned from the presented studies in idealized configurations for realistic mesoscale convection flows in nature with their increased physical complexity.