Online learning of subgrid-scale models for quasi-geostrophic turbulence in planetary interiors

Machine learning approaches to subgrid-scale (SGS) modelling are now well established in atmospheric and oceanic applications. Among these, online end-to-end learning, where the differentiable solver participates in the training, has shown particular promise. Yet, existing studies are largely restricted to idealised periodic domains, with no mechanical boundaries, precluding them from addressing the dynamics of bounded rotating flows relevant to planetary interiors. Here we consider two-dimensional quasi-geostrophic turbulence in a rapidly rotating annular bounded domain. We examine three configurations varying the geometry of the container and the rotation rate. The system exhibit key features such as zonal jets, Rossby waves, and, in the spherical shell geometry, a slow quasi-periodic inward drift of the jets. The spectral properties of the zonal and non-zonal flow can be understood in the framework of zonostrophic turbulence theory. We develop a differentiable solver for this system, which allows us to train SGS models online, over a time span of one turnover time, using coarse-grained data from direct numerical simulations. In all cases, a SGS model trained on a single turnover time accurately reproduce global integrated diagnostics, energy spectra as well as long-term dynamical behaviours ---such as jet migration--- occurring on timescales which exceed the training window by one order of magnitude. The online-trained model further outperforms classical hyperdiffusivity and Leith closure schemes, for which reducing the radial resolution is impractical. The resulting speed-up paves the way to further investigations of long-term geophysical fluid processes beyond the reach of direct numerical simulations.