Invariant parameterization and turbulence modeling on the beta-plane

TL;DR

This paper develops invariant parameterization schemes for turbulence modeling on the beta-plane using Lie invariance methods, introducing nonlinear hyperdiffusion terms that respect symmetries and analyzing their effects on turbulence spectra.

Contribution

It constructs invariant parameterizations for the vorticity equation on the beta-plane using differential invariants and introduces new nonlinear hyperdiffusion schemes respecting symmetries.

Findings

Invariant hyperdiffusion schemes respect symmetries of the vorticity equation.

Invariant hyperdiffusion approximates the energy spectrum in turbulence simulations.

Invariant and conservative parameterizations are shown to be compatible.

Abstract

Invariant parameterization schemes for the eddy-vorticity flux in the barotropic vorticity equation on the beta-plane are constructed and then applied to turbulence modeling. This construction is realized by the exhaustive description of differential invariants for the maximal Lie invariance pseudogroup of this equation using the method of moving frames, which includes finding functional bases of differential invariants of arbitrary order, a minimal generating set of differential invariants and a basis of operators of invariant differentiation in an explicit form. Special attention is paid to the problem of two-dimensional turbulence on the beta-plane. It is shown that classical hyperdiffusion as used to initiate the energy-enstrophy cascades violates the symmetries of the vorticity equation. Invariant but nonlinear hyperdiffusion-like terms of new types are introduced and then used in the course of numerically integrating the vorticity equation and carrying out freely decaying turbulence tests. It is found that the invariant hyperdiffusion scheme is close to but not exactly reproducing the 1/k shape of energy spectrum in the enstrophy inertial range. By presenting conservative invariant hyperdiffusion terms, we also demonstrate that the concepts of invariant and conservative parameterizations are consistent.