Log-Lattices for Atmospheric Flows

We discuss how the projection of geophysical equations of motion onto an exponential grid allows the determination of realistic values of parameters at a moderate cost. This allows us to perform many simulations over a wide range of parameters, thereby leading to general scaling laws of transport efficiency that can then be used to parametrize the turbulent transport in general climate models for Earth or other planets. We illustrate this process using the equation describing heat transport in a dry atmosphere to obtain the scaling laws for the onset of convection as a function of rotation. We confirm the theoretical scaling of the critical Rayleigh number, Rac∼E−4/3, over a wide range of parameters. We have also demonstrated the existence of two regimes of convection: one laminar regime extending near the convection onset, and one turbulent regime occurring as soon as the vertical Reynolds number reaches a value of 104. We derive general scaling laws for these two regimes, both for the transport of heat and the dissipation of kinetic energy, and values of anisotropy and temperature fluctuations.