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

The use of machine learning to represent small-scale processes, such as 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. However, most studies have focused on idealised periodic domains or spheres, neglecting mechanical boundaries present in systems such as planetary interiors. To address this issue, we introduce a pseudo-spectral differentiable solver for the study of two-dimensional quasi-geostrophic turbulence in a rapidly rotating, axially symmetric bounded domain. A key advantage of the online learning approach is its implicit correction of the commutation errors arising from the irregular Chebyshev grid used in the radial direction, achieved through the estimation of correction terms for the filtered equations. In addition, since Chebyshev polynomials are not boundary-preserving, we project training data extracted from the high-resolution direct numerical simulation (DNS) from the fine grid onto the coarse grid using a Galerkin approach that ensures compatibility with the boundary conditions. 

We examine three configurations, varying the geometry (between an exponential container and a spherical shell) and the rotation rate. The flow is driven by a prescribed analytical forcing that mimics a network of pumps, allowing precise control over the energy injection scale and an exact estimate of the power input. For each case, we evaluate the accuracy of the online-trained SGS model against the reference DNS 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 geometry, which exhibits a quasi-periodic recurrence time of O(10) turnover times. These results suggest a promising path towards developing SGS models for planetary and stellar interior dynamics, including dynamo processes. They indicate that costly DNS may need to be run only for short durations to generate training data, enabling subsequent long-term simulations with the trained model at a negligible computational cost.