Geophysical flows are characterized by a wide range of time and space scales, with the location of the smallest dynamically active scales changing incessantly. Thus, geophysical flows might be more efficiently simulated using a dynamically adaptive grid that track small scale features. The presentation will focus on a wavelet-based approach towards this goal, where the flexibility offered by the design of second-generation wavelets is exploited in order to ensure across-scale consistency of mass and vorticity budgets and to preserve discrete conservation properties. The latter are especially desirable in view of long climate simulations.

 

When and where to dynamically refine and coarsen the mesh is a key determinant of efficiency. Often, crude, heuristic criteria based on gradients and tunable thresholds are used. Recent results suggest that a more systematic approach based on local estimators of numerical truncation errors is adequate for idealized simulations, including challenging setups such as statistically homogeneous turbulence. 

 

However, an adequate refinement strategy for realistic simulations remains to be defined. Indeed such simulations are expected to remain under-resolved, so that subgrid-scale models (of turbulence, convection, gravity waves, …) play a crucial role. Their interplay with adaptivity, and even their dependence on resolution, is a non-trivial issue.

 

Joint work with Nicholas Kevlahan and Matthias Aechtner