Casimir preserving spectrum of two-dimensional turbulence

TL;DR

This paper introduces a structure-preserving numerical method for two-dimensional turbulence on a sphere, accurately capturing the energy spectrum and cascade phenomena with lower computational resolution.

Contribution

It develops a finite-mode, Lie-Poisson structure-preserving integrator for the Navier-Stokes equations on a sphere, demonstrating accurate turbulence spectra and cascades.

Findings

Confirmed the -3 inertial range scaling in the energy spectrum.

Demonstrated robustness of the method at modest resolutions.

Provided evidence for the double energy cascade in 2D turbulence.

Abstract

We present predictions of the energy spectrum of forced two-dimensional turbulence obtained by employing a structure-preserving integrator. In particular, we construct a finite-mode approximation of the Navier-Stokes equations on the unit sphere, which, in the limit of vanishing viscosity, preserves the Lie-Poisson structure. As a result, integrated powers of vorticity are conserved in the inviscid limit. We obtain robust evidence for the existence of the double energy cascade, including the formation of the -3 scaling of the inertial range of the direct cascade. We show that this can be achieved at modest resolutions compared to those required by traditional numerical methods.