Stability Results for Idealised Shear Flows on a Rectangular Periodic Domain

TL;DR

This paper identifies new linear stability conditions for idealized shear flows on a rectangular periodic domain and explores their nonlinear stability and instability regimes, extending classical results and developing structure-preserving numerical methods.

Contribution

It introduces new linear stability results for specific shear flows on a torus and analyzes their nonlinear stability and instability, extending classical stability theory.

Findings

Linear stability for flows with certain wave numbers and aspect ratios.

Nonlinear instability for equilibria beyond a specific parameter threshold.

Development of a Lie-Poisson integrator for the truncated Hamiltonian system.

Abstract

We present a new linearly stable solution of the Euler fluid flow on a torus. On a two-dimensional rectangular periodic domain $[0,2\pi)\times[0,2\pi / \kappa)$ for $\kappa\in\mathbb{R}^+$, the Euler equations admit a family of stationary solutions given by the vorticity profiles $\Omega^*(\mathbf{x})= \Gamma \cos(p_1x_1+ \kappa p_2x_2)$. We show linear stability for such flows when $p_2=0$ and $\kappa \geq |p_1|$ (equivalently $p_1=0$ and $\kappa{|p_2|}\leq{1}$). The classical result due to Arnold is that for $p_1 = 1, p_2 = 0$ and $\kappa \ge 1$ the stationary flow is {nonlinearly} stable via the energy-Casimir method. We show that for $\kappa \ge |p_1| \ge 2, p_2 = 0$ the flow is linearly stable, but one cannot expect a similar nonlinear stability result. Finally we prove nonlinear instability for all equilibria satisfying $p_1^2+\kappa^2{p_2^2}>\frac{{3(\kappa^2+1)}}{4(7-4\sqrt{3})}$. The modification and application of a structure-preserving Hamiltonian truncation is discussed for the $\kappa\neq 1$ case. This leads to an explicit Lie-Poisson integrator for the truncated system.