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In this paper, we study geostrophic turbulence without external forcing or dissipation, using a Casimir-preserving numerical method. The research examines the formation of large zonal jets, common in geophysical flows, especially in giant gas planets. These jets form due to the east-west stretching of vortices, influenced by the gradient of the Coriolis parameter, leading to a critical latitude beyond which jets do not form. Using a global quasi-geostrophic model with a fully latitude-dependent Coriolis parameter, we investigate this critical latitude, which is theorized to depend only on the product of the Rossby number and the Lamb parameter. By simulating random flow fields, the critical latitude was identified through zonally averaged zonal velocity profiles. Results align with geostrophic theory, especially near typical Rossby and Lamb parameter values for Earth's atmosphere. However, in the regime of weak rotation (high Rossby numbers) and strong stratification (high Lamb values), no clear critical latitude emerges; instead, zonal jet amplitude and width decrease gradually towards the poles. This research paves the way for further study of jet dynamics under a fully latitude-dependent Coriolis parameter.

Accurate long-term predictions of large-scale flow features on planets are crucial for understanding global atmospheric and oceanic systems, necessitating the development of numerical methods that can preserve essential physical structures over extended simulation periods without excessive computational costs. Recent advancements in the study of global single-layer barotropic models have led to novel numerical methods based on Lie-Poisson discretization that preserve energy, enstrophy and higher-order moments of potential vorticity. This work extends this approach to more complex stratified quasi-geostrophic (QG) systems on the sphere. In this work, we present a formulation of the multi-layer QG equations on the full globe. This allows for extending the Lie-Poisson discretization to multi-layer QG models, ensuring consistency with the underlying structure and enabling long-term simulations without additional regularization. The numerical method is benchmarked through simulations of forced geostrophic turbulence and the long-term behaviour of unforced multi-layered systems. These results demonstrate the structure-preserving properties and robustness of the proposed numerical method, paving the way for a better understanding of the role of high-order conserved quantities in large-scale geophysical flow dynamics.

An integrator for a class of stochastic Lie-Poisson systems driven by Stratonovich noise is developed. The integrator is suited for Lie-Poisson systems that also admit an isospectral formulation, which enables scalability to high-dimensional systems. Its derivation follows from discrete Lie-Poisson reduction of the symplectic midpoint scheme for stochastic Hamiltonian systems. We prove almost sure preservation of Casimir functions and coadjoint orbits under the numerical flow and provide strong and weak convergence rates of the proposed method. The scalability, structure-conservation, and convergence rates are illustrated numerically for the (generalized) rigid body, point vortex dynamics, and the two-dimensional Euler equations on the sphere., Comment: 29 pages, 11 figures, all comments are welcome!

A framework for deriving probabilistic data-driven closure models is proposed for coarse-grained numerical simulations of turbulence in statistically stationary state. The approach unites the ideal large-eddy simulation model and data assimilation methods. The method requires a posteriori measured data to define stochastic flow perturbations, which are combined with a Bayesian statistical correction enforcing user-specified statistics extracted from high-fidelity flow snapshots. Thus, it enables computationally cheap ensemble simulations by combining knowledge of the local integration error and knowledge of desired flow statistics. A model example is given for two-dimensional Rayleigh-B\'enard convection at Rayleigh number $\mathit{Ra}=10^{10}$, incorporating stochastic perturbations and an ensemble Kalman filtering step in a non-intrusive way. Physical flow dynamics are obtained, whilst kinetic energy spectra and heat flux are accurately reproduced in long-time ensemble forecasts on coarse grids. The model is shown to produce accurate results with as few as 20 high-fidelity flow snapshots as input data., Comment: 18 pages, 12 figures. The data and adopted implementations that support the findings of this study are publicly available in Zenodo at http://doi.org/10.5281/zenodo.13353273

We present a geometric derivation of the quasi-geostrophic equations on the sphere, starting from the rotating shallow water equations. We utilise perturbation series methods in vorticity and divergence variables. The derivation employs asymptotic analysis techniques, leading to a global quasi-geostrophic potential vorticity model on the sphere without approximation of the Coriolis parameter. The resulting model forms a closed system for the evolution of potential vorticity with a rich mathematical structure, including Lagrangian and Hamiltonian descriptions. Formulated using the Lie-Poisson bracket reveals the geometric invariants of the quasi-geostrophic model. Motivated by these geometric results, simulations of quasi-geostrophic flow on the sphere are presented based on structure-preserving Lie-Poisson time-integration. We explicitly demonstrate the preservation of Casimir invariants and show that the hyperbolic quasi-geostrophic equations can be simulated in a stable manner over long time. We show the emergence of longitudonal jets, wrapped around the circumference of the sphere in a general direction that is perpendicular to the axis of rotation., Comment: 16 pages, 12 figures, first version - all comments are welcome!

A closure model is presented for large-eddy simulation (LES) based on the three-dimensional variational data assimilation algorithm. The approach aims at reconstructing high-fidelity kinetic energy spectra in coarse numerical simulations by including feedback control to represent unresolved dynamics interactions in the flow as stochastic processes. The forcing uses statistics obtained from offline high-fidelity data and requires only few parameters compared to the number of degrees of freedom of LES. This modeling strategy is applied to geostrophic turbulence on the sphere and enables simulating indefinitely at reduced costs. It proves to accurately recover the energy spectra and the zonal velocity profiles in the coarse model, for three generic situations., Comment: 18 pages, 17 figures

A data-driven turbulence model for coarse-grained numerical simulations of two-dimensional Rayleigh-B\'enard convection is proposed. The model starts from high-fidelity data and is based on adjusting the Fourier coefficients of the numerical solution, with the aim of accurately reproducing the kinetic energy spectra as seen in the high-fidelity reference findings. No assumptions about the underlying PDE or numerical discretization are used in the formulation of the model. We also develop a constraint on the heat flux to guarantee accurate Nusselt number estimates on coarse computational grids and high Rayleigh numbers. Model performance is assessed in coarse numerical simulations at $Ra=10^{10}$. We focus on key features including kinetic energy spectra, wall-normal flow statistics, and global flow statistics. The method of data-driven modeling of flow dynamics is found to reproduce the reference kinetic energy spectra well across all scales and yields good results for flow statistics and average heat transfer, leading to computationally cheap surrogate models. Large-scale forcing extracted from the high-fidelity simulation leads to accurate Nusselt number predictions across two decades of Rayleigh numbers, centered around the targeted reference at $Ra=10^{10}$., Comment: 29 pages, 15 figures

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 2-sphere, 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.

In this paper, we propose and assess several stochastic parametrizations for data-driven modelling of the two-dimensional Euler equations using coarse-grid SPDEs. The framework of Stochastic Advection by Lie Transport (SALT) [Cotter et al., 2019] is employed to define a stochastic forcing that is decomposed in terms of a deterministic basis (empirical orthogonal functions, EOFs) multiplied by temporal traces, here regarded as stochastic processes. The EOFs are obtained from a fine-grid data set and are defined in conjunction with corresponding deterministic time series. We construct stochastic processes that mimic properties of the measured time series. In particular, the processes are defined such that the underlying probability density functions (pdfs) or the estimated correlation time of the time series are retained. These stochastic models are compared to stochastic forcing based on Gaussian noise, which does not use any information of the time series. We perform uncertainty quantification tests and compare stochastic ensembles in terms of mean and spread. Reduced uncertainty is observed for the developed models. On short timescales, such as those used for data assimilation [Cotter et al., 2020], the stochastic models show a reduced ensemble mean error and a reduced spread. Particularly, using estimated pdfs yields stochastic ensembles which rarely fail to capture the reference solution on small time scales, whereas introducing correlation into the stochastic models improves the quality of the coarse-grid predictions with respect to Gaussian noise., Comment: 18 pages, 14 figures

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 and the stochastic sine-Euler equations., Comment: 27 pages, 9 figures, fifth version, all comments are welcome!

Small-scale features of shallow water flow obtained from direct numerical simulation (DNS) with two different computational codes for the shallow water equations are gathered offline and subsequently employed with the aim of constructing a reduced-order correction. This is used to facilitate high-fidelity online flow predictions at much reduced costs on coarse meshes. The resolved small-scale features at high resolution represent subgrid properties for the coarse representation. Measurements of the subgrid dynamics are obtained as the difference between the evolution of a coarse grid solution and the corresponding DNS result. The measurements are sensitive to the particular numerical methods used for the simulation on coarse computational grids and can be used to approximately correct the associated discretization errors. The subgrid features are decomposed into empirical orthogonal functions (EOFs), after which a corresponding correction term is constructed. By increasing the number of EOFs in the approximation of the measured values the correction term can in principle be made arbitrarily accurate. Both computational methods investigated here show a significant decrease in the simulation error already when applying the correction based on the dominant EOFs only. The error reduction accounts for the particular discretization errors that incur and are hence specific to the particular simulation method that is adopted. This improvement is also observed for very coarse grids which may be used for computational model reduction in geophysical and turbulent flow problems.