We develop a variational integrator for the shallow‐water equations on a rotating sphere. The variational integrator is built around a discretization of the continuous Euler–Poincaré reduction framework for Eulerian hydrodynamics. We describe the discretization of the continuous Euler–Poincaré equations on arbitrary simplicial meshes. Standard numerical tests are carried out to verify the accuracy and excellent conservational properties of the discrete variational integrator.

Numerical simulations of the Earth's atmosphere and ocean play an important role in developing our understanding of weather forecasting. A major focus lies in determining the large-scale flow correctly, which is strongly related to the parameterizations of sub-grid processes (Frederiksen et al., 2013). The non-linear and non-local nature of the dynamics of geophysical fluid flows make the large-scale flow structures interact with the smaller components. Solving the Kolmogorov scales (Pope, 2000) of geophysical flows is today, and likely for a foreseeable future, completely out of reach. This is due, in the first place, to the formidable computational expense that would be necessary, but also to the complexity of the many fine-scale physical or bio-chemical processes involved. Truncating the fine scales and simply ignoring their actions is highly detrimental to a reliable simulation of the large-scale components of the flow. Yet, an accurate modeling of the fine-scale processes' effects is an excruciatingly difficult task and the idea of a stochastic modeling has strongly attracted the geophysical community since the seminal works of Hasselmann, 1976;and Leith, 1975. For several years, this interest has been strongly strengthened with the emergence of ensemble methods for probabilistic forecasting and data assimilation

<p>We introduce a new representation of the rotating shallow water equations based on a stochastic transport principle. The derivation relies on a decomposition of the fluid flow into a large-scale component and a noise term that models the unresolved small-scale flow. The total energy of such a random model is demonstrated to be preserved along time for any realization. Thus, we propose to combine a structure-preserving discretization of the underlying deterministic model with the discrete stochastic terms. This way, our method can directly be used in existing dynamical cores of global numerical weather prediction and climate models. For an inviscid test case on the f-plane we use a homogenous noise and illustrate that the spatial part of the stochastic scheme preserves the total energy of the system. Finally, using an inhomogenous noise, we show  that the proposed random model better captures the structure of a large-scale flow than a comparable deterministic model for a barotropically unstable jet on the sphere.</p>

Numerical models of weather and climate critically depend on the long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical fidelity of such simulations also depends on properly preserving conserved quantities, such as energy, of the system. To address this apparent paradox, we develop a variational integrator for the shallow water equations that conserves energy but dissipates potential enstrophy. Our approach follows the continuous selective decay framework [F. Gay-Balmaz and D. Holm. Selective decay by Casimir dissipation in inviscid fluids. Nonlinearity, 26(2), 495 (2013)], which enables dissipating an otherwise conserved quantity while conserving the total energy. We use this in combination with the variational discretization method [Pavlov et al., "Structure-preserving discretization of incompressible fluids," Physica D 240(6), 443-458 (2011)] to obtain a discrete selective decay framework. This is applied to the shallow water equations, both in the plane and on the sphere, to dissipate the potential enstrophy. The resulting scheme significantly improves the quality of the approximate solutions, enabling long-term integrations to be carried out.

To accurately predict weather and climate numerical weather prediction (NWP) models are being used (Bauer et al., 2015). Recently, machine learning methods have received more attention as an alternative approach for weather and climate prediction. For example, in Bihlo (2021), Bauer (2018), andWeyn et al. (2021) neural networks are trained on reanalysis data to produce purely data-driven weather forecasts. While still in its infancy, if successful these data-driven approaches would enable issuing weather forecasts at several orders of magnitude faster than conventional NWP models (Pathak et al., 2022).Owing to the inherent uncertainty in the atmospheric dynamical system (Lorenz, 1963) it has been recognized early on in the development of NWP that a measure of uncertainty of a numerical weather forecast can substantially enhance the value of these forecasts. This gave rise to the field of ensemble weather prediction (Leutbecher & Palmer, 2008), which aims to quantify the various sources of uncertainty in NWP models, chief of which are uncertainty in the initial conditions and errors in the numerical model formulation. To overcome these uncertainties, in addition to the single deterministic weather forecast, an ensemble of perturbed forecasts is generated whose overall divergence, or spread, ideally will provide a measure of the uncertainty in the deterministic prediction. In some applications one is interested in the potential worst cases scenarios . Here, the ensemble members themselves are needed as the spread only would not provide the spatial correlation between the different points. The main limiting factor in generating the ensemble is still computational in nature as each ensemble run takes up computational resources thereby limiting the total number of such ensembles, typically less then 100, that can be computed on an operational basis.