Data-driven spectral modeling for the (thermal) quasi-geostrophic equations
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Two-dimensional incompressible hydrodynamics models are fundamental for studying physical phenomena in atmospheric and oceanic flows. A characteristic feature of these flows is the formation of vorticity structures over several orders of magnitude. Therefore, one requires a very large computational grid to fully resolve all scales of motion, making direct numerical simulations infeasible. One might choose to represent only the dominant large-scale features of the flow, using a coarse computational grid. A promising way to compensate for accuracies due to coarsening is by means of data-driven modeling. In this work we will present a methodology leads to an online forcing term for coarse-grid numerical simulations, based on measurements obtained from an offline fine-grid reference simulation. The forcing is designed to reproduce the kinetic energy spectrum of the reference solution in the coarse numerical simulations. It is found that this method produces qualitatively accurate large-scale dynamics for the two-dimensional Euler equations on the sphere. It is stable by design and computationally cheap, making it a suitable surrogate for long-time simulations. We will present results of this method extended to the (thermal) quasi-geostrophic equations on the sphere.
