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Casimir preserving stochastic Lie-Poisson integrators

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arXiv 2021年

作者： Luesink, Erwin Ephrati, Sagy Cifani, Paolo Geurts, Bernard Multiscale Modelling and Simulation Department of Applied Mathematics Faculty EEMCS University of Twente PO Box 217 Enschede7500 AE Netherlands Gran Sasso Science Institute Viale F. Crispi 7 L’Aquila67100 Italy Multiscale Physics Center for Computational Energy Research Department of Applied Physics Eindhoven University of Technology PO Box 513 Eindhoven5600 MB Netherlands

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. Copyright © 2021, The Authors. All rights reserved.

关键词： Integration

Zeitlin truncation of a Shallow Water Quasi-Geostrophic model for planetary flow

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arXiv 2023年

作者： Franken, Arnout Caliaro, Martino Cifani, Paolo Geurts, Bernard Multiscale Modelling and Simulation Department of Applied Mathematics Faculty EEMCS University of Twente PO Box 217 Enschede7500 AE Netherlands Gran Sasso Science Institute Viale F. Crispi 7 L’Aquila67100 Italy Multiscale Physics Center for Computational Energy Research Department of Applied Physics Eindhoven University of Technology PO Box 513 Eindhoven5600 MB Netherlands

In this work, we consider a Shallow-Water Quasi Geostrophic equation on the sphere, as a model for global large-scale atmospheric dynamics. This equation, previously studied by Verkley [2009] and Schubert et al. [2009], possesses a rich geometric structure, called Lie-Poisson, and admits an infinite number of conserved quantities, called Casimirs. In this paper, we develop a Casimir preserving numerical method for long-time simulations of this equation. The method develops in two steps: firstly, we construct an N-dimensional Lie-Poisson system that converges to the continuous one in the limit N → ∞;secondly, we integrate in time the finite-dimensional system using an isospectral time integrator, developed by Modin and Viviani [2020]. We demonstrate the efficacy of this computational method by simulating a flow on the entire sphere for different values of the Lamb parameter. We particularly focus on rotation-induced effects, such as the formation of jets. In agreement with shallow water models of the atmosphere, we observe the formation of robust latitudinal jets and a decrease in the zonal wind amplitude with latitude. Furthermore, spectra of the kinetic energy are computed as a point of reference for future studies. Copyright © 2023, The Authors. All rights reserved.

关键词： Kinetic energy

Critical latitude in global quasi-geostrophic flow on a rotating sphere

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arXiv 2024年

作者： Franken, Arnout D. Luesink, Erwin Ephrati, Sagy R. Geurts, Bernard J. Mathematics of Multiscale Modelling and Simulation Department of Applied Mathematics Faculty EEMCS University of Twente PO Box 217 Enschede7500 AE Netherlands Korteweg-de-Vries Institute University of Amsterdam PO Box 94248 Science Park 107 Amsterdam1090 GE Netherlands Department of Mathematical Sciences Chalmers University of Technology Gothenburg412 96 Sweden Multiscale Physics Center for Computational Energy Research Department of Applied Physics Eindhoven University of Technology Eindhoven Netherlands

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. Copyright © 2024, The Authors. All rights reserved.

关键词： Miocene

Casimir preserving numerical method for global multilayer geostrophic turbulence

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arXiv 2024年

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. Copyright © 2024, The Authors. All rights reserved.

关键词： Atmospheric movements