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

Klas Modin, Milo Viviani

TL;DR

This work addresses the long-time behavior of the incompressible 2D Euler equations on the sphere by introducing a Casimir-preserving discretization built from geometric quantization and a Lie–Poisson time integrator. The proposed isospectral scheme exactly conserves Casimirs and nearly conserves the Hamiltonian, enabling long-time simulations without hyperviscosity. Across zero-momentum and nonzero-momentum initial data, the authors observe three possible quasi-stationary vortex formations (2, 3, or 4 blobs), with the regime determined by the ratio $\gamma=\|\mathbf{L}\|/(R\sqrt{\mathcal{C}_2})$, challenging statistical-mechanics predictions. They connect these regimes to integrability properties of point vortex dynamics on the sphere and use KAM theory to explain regime transitions, offering a principled mechanism for long-time dynamics and suggesting directions for further statistics, rotating-sphere cases, and viscosity-augmented models.

Abstract

The incompressible 2D Euler equations on a sphere constitute a fundamental model in hydrodynamics. 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 the statistical mechanics theory and show that previous numerical results, suggesting 4 coherent vortex structures as the generic behaviour, display only a special case. Through integrability theory for point vortex dynamics on the sphere we present a theoretical model which describes 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 $γ$.

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

TL;DR

This work addresses the long-time behavior of the incompressible 2D Euler equations on the sphere by introducing a Casimir-preserving discretization built from geometric quantization and a Lie–Poisson time integrator. The proposed isospectral scheme exactly conserves Casimirs and nearly conserves the Hamiltonian, enabling long-time simulations without hyperviscosity. Across zero-momentum and nonzero-momentum initial data, the authors observe three possible quasi-stationary vortex formations (2, 3, or 4 blobs), with the regime determined by the ratio , challenging statistical-mechanics predictions. They connect these regimes to integrability properties of point vortex dynamics on the sphere and use KAM theory to explain regime transitions, offering a principled mechanism for long-time dynamics and suggesting directions for further statistics, rotating-sphere cases, and viscosity-augmented models.

Abstract

The incompressible 2D Euler equations on a sphere constitute a fundamental model in hydrodynamics. 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 the statistical mechanics theory and show that previous numerical results, suggesting 4 coherent vortex structures as the generic behaviour, display only a special case. Through integrability theory for point vortex dynamics on the sphere we present a theoretical model which describes the mechanism by which the three observed regimes appear. We show that there is a correlation between a first integral (the ratio of total angular momentum and the square root of enstrophy) and the long-time behaviour: 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 .

Paper Structure

This paper contains 18 sections, 1 theorem, 38 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Numerical integration algorithm
  3. Spatial discretization via geometric quantization
  4. $L_\alpha$-approximation
  5. The quantized system
  6. Time discretisation
  7. Isospectral midpoint method (IsoMP)
  8. Heun's method
  9. Complexity
  10. Time scaling
  11. Simulation results
  12. Initial data with zero momentum
  13. Generic initial data
  14. Relation with point vortex dynamics
  15. Zero momentum case
  16. ...and 3 more sections

Key Result

Theorem 1

Consider the Poisson algebra $(C^\infty_0(\mathbb{S}^2,\mathbb{C}),\lbrace\cdot,\cdot\rbrace)$ with Poisson bracket defined by bracket1. Then, for the projections $p_N$ and any choice of matrix norms $d_N$, $(\mathfrak{sl}(N,\mathbb{C}),[\cdot,\cdot]_N)_{N\in\mathbb{N}}$ is an $L_\alpha$-approximati

Figures (10)

  • Figure 1: Simulation with two different time integration methods: IsoMP \ref{['IRK_mid']} and Heun \ref{['Heun']}. Vorticity $\omega(x,t)$ clockwise from the top-left at $t=0s,4s,40s,400s$, for the initial data in DQM. The horizontal axis is the azimuth $\varphi\in[0,2\pi]$ and the vertical axis is minus the inclination $\theta\in[0,\pi]$. The results are visually indistinguishable up to $t=40$. At $t=400$ there are some differences in the positions of the vortex blobs. See also Movie 1, Movie 2, and Movie 3 of the supplementary material.
  • Figure 2: Hamiltonian variation $|H-H_0|$ (left column) and enstrophy variation $|E-E_0|$ (right column) with the IsoMP (upper row) and Heun (lower row) time integrators.
  • Figure 3: Scatter plots of vorticity $\omega$ versus stream function $\psi$ at $t=400$. (left) Using the IsoMP time integrator (\ref{['IRK_mid']}), and (right) using the Heun time integrator (\ref{['Heun']}).
  • Figure 4: Pairs of initial (upper) and final (lower) vorticities for the 16 generic simulations with $L^2(\mathbb{S}^2)$ random initial data. The numbers labelling the simulations correspond to those in \ref{['fig:ratio_momentum_enstrophy']}.
  • Figure 5: Behaviour of the vortex blobs for the same initial data as in \ref{['fig:sim_1_isoMP']}. The quasi-periodic motion of the blobs is tracked in the scatter plots.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Theorem 1: bhssbms
  • Remark 1
  • Remark 2