Arnol'd developed two distinct yet closely
related approaches to the linear stability of
Euler flows. One is widely used for two-dimensional flows
and involves constructing
a conserved functional whose first variation vanishes and
whose second variation
determines the linear (and nonlinear) stability of the motion.
The second method
is a refinement of Kelvin's energy principle which states
that stable steady Euler
flows represent extremums in energy under a virtual
displacement of the vorticity
field. The conserved-functional (or energy-Casimir) method has
been extended by
several authors to more complex flows, such as planar MHD
flow. In this paper
we generalize the Kelvin–Arnol'd energy
method to two-dimensional inviscid flows
subject to a body force of the form −ϕ∇f.
Here ϕ is a materially conserved quantity
and f an arbitrary function of position and of ϕ.
This encompasses a broad class of
conservative flows, such as natural-convection planar and
poloidal MHD flow with the
magnetic field trapped in the plane of the motion, flows
driven by electrostatic forces,
swirling recirculating flow, self-gravitating flows and
poloidal MHD flow subject to
an azimuthal magnetic field. We show that stable steady
motions represent extremums
in energy under a virtual displacement of ϕ and of
the vorticity field. That is, d1E=0
at equilibrium and whenever d2E is
positive or negative definite the flow is (linearly)
stable. We also show that unstable normal modes must have
a spatial structure which
satisfies d2E=0. This provides a single
stability test for a broad class of flows, and
we describe a simple universal procedure for implementing
this test. In passing, a new
test for linear stability is developed. That is, we
demonstrate that stability is ensured
(for flows of the type considered here) whenever the
Lagrangian of the flow is a
maximum under a virtual displacement of the particle
trajectories, the displacement
being of the type normally associated with Hamilton's
principle. A simple universal
procedure for applying this test is also given. We apply our
general stability criteria
to a range of flows and recover some familiar results. We
also extend these ideas to
flows which are subject to more than one type of body force.
For example, a new
stability criterion is obtained (without the use of Casimirs)
for natural convection in
the presence of a magnetic field. Nonlinear stability is
also considered. Specifically,
we develop a nonlinear stability criterion for planar MHD
flows which are subject
to isomagnetic perturbations. This differs from previous
criteria in that we are able
to extend the linear criterion into the nonlinear regime.
We also show how to extend
the Kelvin–Arnol'd method to finite-amplitude
perturbations.
Overstable rotating convection in the presence of a vertical magnetic field