We consider the stability of a configuration consisting of a vertical
magnetic field in a planar flow on elliptical streamlines in ideal
hydromagnetics. In the absence of a magnetic field the elliptical flow is
universally unstable (the ``elliptical instability''). We find this universal
instability persists in the presence of magnetic fields of arbitrary strength,
although the growthrate decreases somewhat. We also find further instabilities
due to the presence of the magnetic field. One of these, a destabilization of
Alfven waves, requires the magnetic parameter to exceed a certain critical
value. A second, involving a mixing of hydrodynamic and magnetic modes, occurs
for all magnetic-field strengths. These instabilities may be important in
tidally distorted or otherwise elliptical disks. A disk of finite thickness is
stable if the magnetic fieldstrength exceeds a critical value, similar to the
fieldstrength which suppresses the magnetorotational instability.Comment: Accepted for publication in Astrophysical Journa
We construct the noncanonical Poisson bracket associated with the phase space
of first order moments of the velocity field and quadratic moments of the
density of a fluid with a free- boundary, constrained by the condition of
incompressibility. Two methods are used to obtain the bracket, both based on
Dirac's procedure for incorporating constraints. First, the Poisson bracket of
moments of the unconstrained Euler equations is used to construct a Dirac
bracket, with Casimir invariants corresponding to volume preservation and
incompressibility. Second, the Dirac procedure is applied directly to the
continuum, noncanonical Poisson bracket that describes the compressible Euler
equations, and the moment reduction is applied to this bracket. When the
Hamiltonian can be expressed exactly in terms of these moments, a closure is
achieved and the resulting finite-dimensional Hamiltonian system provides exact
solutions of Euler's equations. This is shown to be the case for the classical,
incompressible Riemann ellipsoids, which have velocities that vary linearly
with position and have constant density within an ellipsoidal boundary. The
incompressible, noncanonical Poisson bracket differs from its counterpart for
the compressible case in that it is not of Lie-Poisson form
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