Hydrodynamic Nambu Brackets derived by Geometric Constraints

TL;DR

This paper introduces a geometric method to derive Nambu brackets for 2D hydrodynamics, encompassing models like surface quasi-geostrophy and Rayleigh-Bénard convection, with potential applications in conservative numerical algorithms.

Contribution

It presents a novel geometric derivation of Nambu brackets for 2D hydrodynamics and extends the formalism to coupled systems like convection, including new conservation law formulations.

Findings

Derivation of Nambu brackets using geometric constraints.

Inclusion of generalized models like surface quasi-geostrophy.

Extension to coupled hydrodynamic-thermodynamic systems like Rayleigh-Bénard convection.

Abstract

A geometric approach to derive the Nambu brackets for ideal two-dimensional (2D) hydrodynamics is suggested. The derivation is based on two-forms with vanishing integrals in a periodic domain, and with resulting dynamics constrained by an orthogonality condition. As a result, 2D hydrodynamics with vorticity as dynamic variable emerges as a generic model, with conservation laws which can be interpreted as enstrophy and energy functionals. Generalized forms like surface quasi-geostrophy and fractional Poisson equations for the stream-function are also included as results from the derivation. The formalism is extended to a hydrodynamic system coupled to a second degree of freedom, with the Rayleigh-B\'{e}nard convection as an example. This system is reformulated in terms of constitutive conservation laws with two additive brackets which represent individual processes: a first representing inviscid 2D hydrodynamics, and a second representing the coupling between hydrodynamics and thermodynamics. The results can be used for the formulation of conservative numerical algorithms that can be employed, for example, for the study of fronts and singularities.