Selective decay by Casimir dissipation in fluids

TL;DR

This paper introduces a mechanism for selective decay in fluid flows by enforcing Casimir dissipation, providing a theoretical framework and examples relevant for modeling turbulent geophysical flows.

Contribution

It develops a general theory of selective decay using Lie-Poisson structures and introduces a scale-selection operator for parameterizing scale interactions.

Findings

Mechanism enforces Casimir dissipation while conserving energy.

Examples demonstrate the application of the theory to fluid equations.

Potential use in modeling large-scale geophysical turbulence.

Abstract

The problem of parameterizing the interactions of larger scales and smaller scales in fluid flows is addressed by considering a property of two-dimensional incompressible turbulence. The property we consider is selective decay, in which a Casimir of the ideal formulation (enstrophy in 2D flows, helicity in 3D flows) decays in time, while the energy stays essentially constant. This paper introduces a mechanism that produces selective decay by enforcing Casimir dissipation in fluid dynamics. This mechanism turns out to be related in certain cases to the numerical method of anticipated vorticity discussed in \cite{SaBa1981,SaBa1985}. Several examples are given and a general theory of selective decay is developed that uses the Lie-Poisson structure of the ideal theory. A scale-selection operator allows the resulting modifications of the fluid motion equations to be interpreted in several examples as parameterizing the nonlinear, dynamical interactions between disparate scales. The type of modified fluid equation systems derived here may be useful in modelling turbulent geophysical flows where it is computationally prohibitive to rely on the slower, indirect effects of a realistic viscosity, such as in large-scale, coherent, oceanic flows interacting with much smaller eddies.