Online learning of stochastic closures of quasigeostrophic turbulence

Stochastic parameterisations are essential for representing the uncertainty introduced when numerical models neglect certain scales or components of the Earth system. Moreover, the specific structure of stochastic parameterisations is critical for representing this uncertainty accurately. A ubiquitous (though generally invalid) assumption is that of Markovianity. Computational constraints mean that Markovian parameterisations are much preferred in practice, but identifying optimal Markovian approximations is far from trivial. We propose an "online" data-driven approach to learning Markovian parameterisations, wherein the dynamics of the parameterised model feature explicitly in the loss function, which is based on a proper scoring rule. We apply the method to the problem of sub-grid closure of quasigeostrophic turbulence.