We investigate the linear stability of inviscid flows which are subject to a conservative
body force. This includes a broad range of familiar conservative systems, such as
ideal MHD, natural convection, flows driven by electrostatic forces and axisymmetric,
swirling, recirculating flow. We provide a simple, unified, linear stability criterion valid
for any conservative system. In particular, we establish a principle of maximum action
of the form
formula here
where η is the Lagrangian displacement,e is a measure of the disturbance energy,
T and V are the kinetic and potential energies, and L is the Lagrangian. Here d
represents a variation of the type normally associated with Hamilton's principle,
in which the particle trajectories are perturbed in such a way that the time of
flight for each particle remains the same. (In practice 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 definite and this is achieved if L(η) is a maximum
at equilibrium. This captures many familiar criteria, such as Rayleigh's circulation
criterion, the Rayleigh–Taylor criterion for stratified fluids, Bernstein's principle for
magnetostatics, Frieman & Rotenberg's stability test for ideal MHD equilibria, and
Arnold's variational principle applied to Euler flows and to ideal MHD. There
are three advantages to our test: (i) d2T(η) has a particularly 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 energy principles, such as the energy-Casimir method or the Kelvin–Arnold variational principle, there is no need to
identify all of the integral invariants of the flow as a precursor to performing the
stability analysis. We end by looking at the particular case of MHD equilibria. Here
we note that when 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, however, the
flow invariably violates the energy criterion and so could, but need not, be unstable. In
such cases we identify one mode in which the Lagrangian displacement grows linearly
in time. This is reminiscent of the short-wavelength instability of non-Beltrami Euler
flows.
