A finite model of two-dimensional ideal hydrodynamics

TL;DR

This paper introduces a finite-dimensional Lie algebra model that approximates 2D ideal hydrodynamics on a torus, comparing numerical results with continuum theory and ensemble averages.

Contribution

It presents a finite N Lie algebra formulation of 2D hydrodynamics that converges to the continuum limit, with numerical validation and comparison to statistical ensembles.

Findings

Finite N model approximates 2D hydrodynamics

Numerical simulations match continuum behavior for large N

Time-averaged vorticity moments agree with ensemble averages

Abstract

A finite-dimensional su($N$) Lie algebra equation is discussed that in the infinite $N$ limit (giving the area preserving diffeomorphism group) tends to the two-dimensional, inviscid vorticity equation on the torus. The equation is numerically integrated, for various values of $N$, and the time evolution of an (interpolated) stream function is compared with that obtained from a simple mode truncation of the continuum equation. The time averaged vorticity moments and correlation functions are compared with canonical ensemble averages.