Algebraic construction of a Nambu bracket for the two-dimensional vorticity equation

TL;DR

This paper presents an explicit algebraic method to construct a Nambu bracket for the 2D vorticity equation, connecting Lie-Poisson structures with structure-preserving discretizations.

Contribution

It introduces an algorithmic approach to derive the Nambu bracket from Lie--Poisson form using the Zeitlin discretization, advancing fluid mechanics theory.

Findings

Explicit construction of Nambu brackets from discretization

Connection between Lie--Poisson and Nambu structures

Continuum limit derivation of Nambu brackets

Abstract

So far fluid mechanical Nambu brackets have mainly been given on an intuitive basis. Alternatively an algorithmic construction of such a bracket for the two-dimensional vorticity equation is presented here. Starting from the Lie--Poisson form and its algebraic properties it is shown how the Nambu representation can be explicitly constructed as the continuum limit from the structure preserving Zeitlin discretization.