An energy criterion for the linear stability of conservative flows

We investigate the linear stability of inviscid flows which are subject to a conservative body force. This includes a broad range of familiar conserva tive systems, such as ideal MHD, natural convection, flows driven by electr ostatic forces and axisymmetric, swirling, recirculating flow. We provide a simple, unified, linear stability criterion valid for any conservative sys tem. In particular, we establish a principle of maximum action of the form e=1 /2 integral eta(2) dV - d(2)L(eta) = constant, d(1)L(eta) = 0, -d(2)L(eta) = d(2)T(eta)- d(2)V(eta), where eta is the Lagrangian displacement, we is a measure of the disturbanc e energy, T and V are the kinetic and potential energies, and L is the Lagr angian. Here d represents a variation of the type normally associated with Hamilton's principle, in which the particle trajectories are perturbed in s uch a way that the time of flight for each particle remains the same. (In p ractice this may be achieved by advecting the streamlines of the base flow in a frozen-in manner.) A simple test for stability is that e is positive d efinite and this is achieved if L(eta) is a maximum at equilibrium. This ca ptures many familiar criteria, such as Rayleigh's circulation criterion, th e Rayleigh-Taylor criterion for stratified fluids, Bernstein's principle fo r magnetostatics, Frieman & Rotenberg's stability test for ideal MHD equili bria, and Arnold's variational principle applied to Euler flows and to idea l MHD. There are three advantages to our test: (i) d(2)T(eta) has a particu larly simple quadratic form so the test is easy to apply; (ii) the test is universal and applies to any conservative system; and (iii) unlike other en ergy principles, such as the energy-Casimir method or the Kelvin-Arnold var iational principle, there is no need to identify all of the integral invari ants of the flow as a precursor to performing the stability analysis. We en d by looking at the particular case of MHD equilibria. Here we note that wh en u and B are co-linear there exists a broad range of stable steady flows. Moreover, their stability may be assessed by examining the stability of an equivalent magnetostatic equilibrium. When u and B are non-parallel, howev er, the flow invariably violates the energy criterion and so could, but nee d not, be unstable. In such cases we identify one mode in which the Lagrang ian displacement grows linearly in time. This is reminiscent of the short-w avelength instability of non-Beltrami Euler flows.