A minimal-variable symplectic method for isospectral flows

2019

J. Fluid Mech.

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The incompressible Euler equations on a sphere are fundamental in models of oceanic and atmospheric dynamics. The long-time behaviour of solutions is largely unknown; statistical mechanics predicts a steady vorticity configuration, but detailed numerical results in the literature contradict this theory, yielding instead persistent unsteadiness. Such numerical results were obtained using artificial hyperviscosity to account for the cascade of enstrophy into smaller scales. Hyperviscosity, however, destroys the underlying geometry of the phase flow (such as conservation of Casimir functions), and therefore might affect the qualitative long-time behaviour. Here we develop an efficient numerical method for long-time simulations that preserve the geometric features of the exact flow, in particular conservation of Casimirs. Long-time simulations on a non-rotating sphere then reveal three possible outcomes for generic initial conditions: the formation of either 2, 3, or 4 coherent vortex structures. These numerical results contradict both the statistical mechanics theory and previous numerical results suggesting that the generic behaviour should be 4 coherent vortex structures. Furthermore, through integrability theory for point vortex dynamics on the sphere, we present a theoretical model which describe the mechanism by which the three observed regimes appear. We show that there is a correlation between a first integral γ (the ratio of total angular momentum and the square root of enstrophy) and the long-time behaviour: γ small, intermediate, and large yields most likely 4, 3, or 2 coherent vortex formations. Our findings thus suggest that the likely long-time behaviour can be predicted from the first integral γ.

“…The matrix W is an auxiliary variable implicitly defined (together with W n+1 ) by the two equations in (2.11). For further details on the method (2.11) we refer to Viviani (2019).…”

Section: Isospectral Midpoint Methodsmentioning

confidence: 99%

“…In the numerical scheme W is an auxiliary N × N matrix variable implicitly defined in the first line of (9). For the derivation and theory behind (9) we refer to [42].…”

Section: Time Discretisationmentioning

confidence: 99%

A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics

Modin

2019

J. Fluid Mech.

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“…In this section, we present various examples of the results shown above. More precisely, we integrate equations (5) with the isospectral midpoint method [24], for randomly generated 4 initial conditions with zeromomentum M and zero-diagonal components in D 0 (N, k), for N = 257 and different k. We always take time-step h = 0.1 and normalized initial vorticity, with respect to the spectral norm.…”

Section: Perturbation and Interaction Of Quantized Rossby-haurwitz Wavesmentioning

confidence: 99%

An algebraic approach to the spontaneous formation of spherical jets

2022

JCD

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<p style='text-indent:20px;'>The global structure of the atmosphere and the oceans is a continuous source of intriguing challenges in geophysical fluid dynamics (GFD). Among these, jets are determinant in the air and water circulation around the Earth. In the last fifty years, thanks to the development of more and more precise and extensive observations, it has been possible to study in detail the atmospheric formations of the giant-gas planets in the solar system. For those planets, jets are the dominant large scale structure. Starting from the 70s, various theories combining observations and mathematical models have been proposed in order to describe their formation and stability. In this paper, we propose a purely algebraic approach to describe the spontaneous formation of jets on a spherical domain. Analysing the algebraic properties of the 2D Euler equations, we give a characterization of the different jets' structures. The calculations are performed starting from the discrete Zeitlin model of the Euler equations. For this model, the classification of the jets' structures can be precisely described in terms of reductive Lie algebras decomposition. The discrete framework provides a simple tool for analysing both from a theoretical and and a numerical perspective the jets' formation. Furthermore, it allows to extend the results to the original Euler equations.</p>

“…[13]). For the hyperbolic plane, we use the hyperbolic midpoint method [44] (see also [35]). All these methods are second order approximations of the exact flow map.…”

Section: Outlook: Long-time Predictions For 2d Euler Equationsmentioning

confidence: 99%

Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey

Modin

2020

Arnold Math J.

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Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. Integrability results for few point-vortices on various domains is a vivid topic, with many results and techniques scattered in the literature. Here, we give a unified framework for proving integrability results for $$N=2$$ N = 2 , 3, or 4 point-vortices (and also more general Hamiltonian systems), based on symplectic reduction theory. The approach works on any two-dimensional manifold with a symmetry group; we illustrate it on the sphere, the plane, the hyperbolic plane, and the flat torus. A systematic study of integrability is prompted by advances in two-dimensional turbulence, bridging the long-time behaviour of 2D Euler equations with questions of point-vortex integrability. A gallery of solutions is given in the appendix.