Available Energy and Ground States of Convective Hydrodynamic and Hydromagnetic Instabilities

TL;DR

This paper introduces a novel method combining Gardner's restacking algorithm and Lagrangian relaxation to predict the nonlinear saturation levels of convective instabilities in fluids, with applications to plasma and fusion contexts.

Contribution

The paper presents a new framework that effectively predicts the nonlinear saturation of convective instabilities using a combination of existing algorithms, extending their applicability.

Findings

Excellent agreement with numerical simulations for Rayleigh-Taylor instability.

Successful extension to Z-pinch interchange instability.

Potential for aiding fusion reactor design and operation.

Abstract

We propose a method for predicting the nonlinear saturation level of convective instabilities in neutral and magnetized fluids. The method combines Gardner's restacking algorithm, which computes the available energy and ground states of collisionless plasmas in phase space, and Lagrangian relaxation, where fluid elements find lower-energy equilibria while preserving local invariants. For the incompressible Rayleigh-Taylor instability, the problem is formally equivalent to Gardner's and the restacking algorithm directly applies in configuration space. To treat compressibility, we follow restacking with Lagrangian relaxation to obtain the ground state, and the results show excellent agreement with direct numerical simulations. Successful extension to the $m=0$ interchange instability in a Z-pinch demonstrates the method's potential as a general framework for estimating the nonlinear extent of convective instabilities, which can facilitate the design and operation of fusion reactors.