Guillaume Roullet (LOPS, Brest)

Discrete differential geometry, transport and irreversibility

Guillaume Roullet LOPS, Univ. Brest, CNRS, IRD, Ifremer

In large eddy simulations (LES) the transfer of energy and tracer variance across scales is explicitly resolved down to the grid scale, where these quantities are dissipated. The dissipation can be explicit, with a closure at the sub-grid scale, or implicit, by embedding the dissipation into the discretized transport terms, resulting in what is known as implicit LES (ILES). The two main numerical ingredients of ILES are upwinding and nonlinear interpolations for the transport terms. However, it gives an essential role to the numerical methods employed, which may frighten the physicist. Moreover, as the dissipation is hidden in the transport, there is no parameter controlling it, such as the Reynolds number, and it is difficult to diagnose its intensity. In this talk, I will show how the discrete differential geometry offers a unifying framework to discretize the transport terms of fluid equations. The main idea is to consider the model variables (momentum, buoyancy, pressure, etc.) as differential forms, and discretize the transport terms as Lie derivatives. The upwinding and the nonlinear interpolations are naturally performed on the interior product. The second point is to show how, by extracting the irreversible part of the discretized transport, one can compute the local dissipation rate of energy, tracer variance, enstrophy etc.