Rotating Convection and Flows with Horizontal Kinetic Energy Backscatter

Numerical simulations of large-scale geophysical flows typically require unphysically strong dissipation for numerical stability. Toward energetic balance various schemes have been devised to reinject this energy, in particular by horizontal kinetic energy backscatter. In a set of papers, some of the authors have studied this scheme through its continuum formulation with momentum equations augmented by a backscatter operator, e.g., in rotating Boussinesq and shallow water equations. Here we review the main results about the impact of backscatter on certain flows and waves, including some barotropic, parallel, and Kolmogorow flows, as well as internal gravity waves and geostrophic equilibria. We particularly focus on the possible accumulation of injected energy in explicit medium-scale plane waves, which then grow exponentially and unboundedly, or yield bifurcations in the presence of bottom drag. Beyond the review, we introduce the rotating 2D Euler equations with backscatter as a guiding example. For this we prove the new result that unbounded growth is a stable phenomenon occurring in open sets of phase space. We also briefly consider the primitive equations with backscatter and outline global well-posedness results.