A major challenge in geophysical and observational sciences is the representation and quantification of uncertainty in numerical predictions. Uncertainty stems from various sources, most relevantly from incomplete inclusion of all relevant physical mechanisms in the models and uncertainty in the initial and boundary conditions (T. N. Palmer, 2000). Important models for geophysical fluid dynamics, such as the two-dimensional Euler equations, quasi-geostrophic equations or rotating shallow water equations are derived from the three-dimensional Navier-Stokes equations. A sequence of simplifying assumptions is applied in order to reduce the complexity of the model to a more manageable level, while retaining main flow physics (Zeitlin, 2018). Stochastic extensions to these models have also been derived (Holm & Luesink, 2021). These approximate models are nevertheless rich in dynamics and contain a wide range of spatial and temporal scales. Numerically resolving the entire spectrum of scales is often not computationally feasible, meaning that either the complexity of the model should be reduced even further such that the resulting model is simple enough the be solvable numerically, or the complex model is represented on a coarse computational grid and unresolved scales are replaced by a sub-grid model. The latter option may be combined with stochastic forcing, which provides an effective way to represent unresolved scales in numerical simulations (Buizza et al., 1999;Frederiksen & Davies, 1997;Majda et al., 2001). The use of stochasticity as a means to represent the unresolved scales serves to restore some of the missing small-scale
A resolution-independent data-driven stochastic parametrization method for subgrid-scale processes in coarsened fluid descriptions is proposed. The method enables the inclusion of high-fidelity data into the coarsened flow model, thereby enabling accurate simulations also with the coarser representation. The small-scale parametrization is introduced at the level of the Fourier coefficients of the coarsened numerical solution. It is designed to reproduce the kinetic energy spectra observed in high-fidelity data of the same system. The approach is based on a control feedback term reminiscent of continuous data assimilation. The method relies solely on the availability of high-fidelity data from a statistically steady state. No assumptions are made regarding the adopted discretization method or the selected coarser resolution. The performance of the method is assessed for the two-dimensional Euler equations on the sphere. Applying the method at two significantly coarser resolutions yields good results for the mean and variance of the Fourier coefficients. Stable and accurate large-scale dynamics can be simulated over long integration times.
The 2D Euler equations are a simple but rich set of non-linear PDEs that describe the evolution of an ideal inviscid fluid, for which one dimension is negligible. Solving numerically these equations can be extremely demanding. Several techniques to obtain fast and accurate simulations have been developed during the last decades. In this paper, we present a novel approach which combines recent developments in the stochastic model reduction and conservative semi-discretization of the Euler equations. In particular, starting from the Zeitlin model on the 2sphere, we derive reduced dynamics for large scales and we close the equations either deterministically or with a suitable stochastic term. Numerical experiments show that, after an initial turbulent regime, the influence of small scales to large scales is negligible, even though a non-zero transfer of energy among different modes is present.
Casimir preserving integrators for stochastic Lie-Poisson equations with Stratonovich noise are developed extending Runge-Kutta Munthe-Kaas methods. The underlying Lie-Poisson structure is preserved along stochastic trajectories. A related stochastic differential equation on the Lie algebra is derived. The solution of this differential equation updates the evolution of the Lie-Poisson dynamics by means of the exponential map. The constructed numerical method conserves Casimir-invariants exactly, which is important for long time integration. This is illustrated numerically for the case of the stochastic heavy top.
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