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

The use of machine learning to represent subgrid-scale (SGS) dynamics is now well established in weather forecasting and climate modelling. Recent advances have demonstrated that SGS models trained via ``online'' end-to-end learning -- where the dynamical solver operating on the filtered equations participates in the training -- can outperform traditional physics-based approaches. Most studies, however, have focused on idealised periodic domains, neglecting the mechanical boundaries present e.g. in planetary interiors. To address this issue, we consider two-dimensional quasi-geostrophic turbulent flow in an axisymmetric bounded domain that we model using a pseudo-spectral differentiable solver, thereby enabling online learning. We examine three configurations, varying the geometry (between an exponential container and a spherical shell) and the rotation rate. Flow is driven by a prescribed analytical forcing, allowing for precise control over the energy injection scale and an exact estimate of the power input. We evaluate the accuracy of the online-trained SGS model against the reference direct numerical simulation using integral quantities and spectral diagnostics. In all configurations, we show that an SGS model trained on data spanning only one turnover time remains stable and accurate over integrations at least a hundred times longer than the training period. Moreover, we demonstrate the model's remarkable ability to reproduce slow processes occurring on time scales far exceeding the training duration, such as the inward drift of jets in the spherical shell. These results suggest a promising path towards developing SGS models for planetary and stellar interior dynamics, including dynamo processes.