Available Energy and Ground States of Convective Hydrodynamic and Hydromagnetic Instabilities

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.

Figures (3)

• Figure 1: A schematic of the restacking-relaxation method applied to the compressible RTI: (a) the unstable initial profiles subject to incompressible restacking; (b) the restacked profiles subject to Lagrangian relaxation; (c) the relaxed profiles, i.e., the ground state. The bars represent discrete fluid elements, whose colors denote the initial position $x_0$.
• Figure 2: Examples of initial (stars), restacked (triangles), and ground state (circles) profiles: the density (a), pressure (b), and specific entropy (c) in the compressible RTI; and the pressure (d), magnetic field (e), and stability indicator $\sigma$ (f) in the sausage instability. The markers represent discrete fluid elements, whose colors denote the initial position ($x_0$ or $r_0$).
• Figure 3: Comparison between the ground states (broken colored) obtained with the proposed method and the settled profiles (solid colored) in viscous simulations: density (a) and pressure (b) profiles in the RTI, and pressure (d) and magnetic field (e) for the sausage instability. The gray broken lines show the initial profiles. The decayed potential energy in viscous simulations (triangles) and the peak kinetic energy in non-viscous simulations (stars) all show proportionality to the available energy for both the RTI (c) and the sausage instability (f). The black broken lines show linear fits.