Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometric derivation and structure-preserving simulation of quasi-geostrophy on the sphere

Erwin Luesink, Arnout Franken, Sagy Ephrati, Bernard Geurts

TL;DR

This work derives a global quasi-geostrophic model on the sphere from the rotating shallow water equations using asymptotic perturbation in vorticity and divergence, producing a closed potential vorticity evolution $\partial_t q + \{\psi, q\} = 0$ with $q = (\Delta - \gamma \mu^2)\psi + \frac{2\mu}{\mathrm{Ro}} + 2\mu h$. It casts the model in a Lie-Poisson Hamiltonian framework, identifying a rich geometric structure and an infinite family of Casimirs, and provides a structure-preserving numerical scheme based on an isospectral Lie-Poisson integrator that exactly preserves Casimirs and closely preserves energy. The simulations on the unit sphere exhibit long-time, unforced development of equatorial jets and coherent zonal structures, enabled by the geometric discretization compatible with the sphere’s topology. This approach yields globally valid QG dynamics on the sphere and offers a robust platform for incorporating stochastic transport and thermal effects in future work, with potential applications to Jovian atmospheres and other geophysical flows.

Abstract

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.

Geometric derivation and structure-preserving simulation of quasi-geostrophy on the sphere

TL;DR

This work derives a global quasi-geostrophic model on the sphere from the rotating shallow water equations using asymptotic perturbation in vorticity and divergence, producing a closed potential vorticity evolution with . It casts the model in a Lie-Poisson Hamiltonian framework, identifying a rich geometric structure and an infinite family of Casimirs, and provides a structure-preserving numerical scheme based on an isospectral Lie-Poisson integrator that exactly preserves Casimirs and closely preserves energy. The simulations on the unit sphere exhibit long-time, unforced development of equatorial jets and coherent zonal structures, enabled by the geometric discretization compatible with the sphere’s topology. This approach yields globally valid QG dynamics on the sphere and offers a robust platform for incorporating stochastic transport and thermal effects in future work, with potential applications to Jovian atmospheres and other geophysical flows.

Abstract

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.
Paper Structure (10 sections, 1 theorem, 45 equations, 3 figures, 1 table)

This paper contains 10 sections, 1 theorem, 45 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Evolution equation for the quasi-geostrophic potential vorticity
  3. Euler-Poincaré theorem
  4. Conventional geostrophic balance
  5. Geostrophic balance using vorticity and divergence variables
  6. Lagrangian and Hamiltonian formulation
  7. Energy functionals
  8. Geometry of two-dimensional fluids
  9. Structure-preserving numerical simulation of quasi-geostrophic dynamics on a rotating sphere
  10. Conclusion and outlook

Key Result

Theorem 1

$\,$ Let $\mathfrak{X}(M)$ denote the space of vector fields on a compact finite-dimensional Riemannian manifold $M$. The following two statements are equivalent:

Figures (3)

  • Figure 1: Potential vorticity anomaly (left) and zonal component of the velocity field (right) after 2000 days.
  • Figure 2: A sequence of solutions of freely evolving quasi-geostrophic flow on the sphere. Shown are the instantaneous potential vorticity anomaly (left column) and zonal component of the horizontal velocity field (right column) at the initial time (top row), after 100 days (second row), after 400 days (third row) and after 2000 days (bottom row).
  • Figure 3: Time evolution in the relative deviation of the conserved quantities of the discrete system. On the left, the relative deviation of the discrete Hamiltonian from its initial value is shown. The right figure shows the relative errors in the first 16 Casimirs compared to their respective initial values. The different Casimirs are shown at a lighter color as the index increases. Notably, the Casimirs appear clustered according to the parity of their indices, with even indices being in the range $10^{-16}$-$10^{-13}$, and the odd indices in the range $10^{-12}$-$10^{-10}$.

Theorems & Definitions (1)

  • Theorem 1: Euler-Poincaré