Stability Results for Idealised Shear Flows on a Rectangular Periodic Domain

Abstract

We present a new linearly stable solution of the Euler fluid flow on a torus. On a two-dimensional rectangular periodic domain $[0,2π)\times[0,2π/ κ)$ for $κ\in\mathbb{R}^+$, the Euler equations admit a family of stationary solutions given by the vorticity profiles $Ω^*(\mathbf{x})= Γ\cos(p_1x_1+ κp_2x_2)$. We show linear stability for such flows when $p_2=0$ and $κ\geq |p_1|$ (equivalently $p_1=0$ and $κ{|p_2|}\leq{1}$). The classical result due to Arnold is that for $p_1 = 1, p_2 = 0$ and $κ\ge 1$ the stationary flow is {nonlinearly} stable via the energy-Casimir method. We show that for $κ\ge |p_1| \ge 2, p_2 = 0$ the flow is linearly stable, but one cannot expect a similar nonlinear stability result. Finally we prove nonlinear instability for all equilibria satisfying $p_1^2+κ^2{p_2^2}>\frac{3(κ^2+1)}{4(7-4\sqrt{3})}$. The modification and application of a structure-preserving Hamiltonian truncation is discussed for the $κ\neq 1$ case. This leads to an explicit Lie-Poisson integrator for the truncated system.