  Ideal systems like MHD and Euler flow may develop singularities in vorticity
(w = curl v). Viscosity and resistivity provide dissipative regularizations of
the singularities. In this paper we propose a minimal, local, conservative,
nonlinear, dispersive regularization of compressible flow and ideal MHD, in
analogy with the KdV regularization of the 1D kinematic wave equation. This
work extends and significantly generalizes earlier work on incompressible Euler
and ideal MHD. It involves a micro-scale cutoff length lambda which is a
function of density, unlike in the incompressible case. In MHD, it can be taken
to be of order the electron collisionless skin depth c/omega_pe. Our
regularization preserves the symmetries of the original systems, and with
appropriate boundary conditions, leads to associated conservation laws. Energy
and enstrophy are subject to a priori bounds determined by initial data in
contrast to the unregularized systems. A Hamiltonian and Poisson bracket
formulation is developed and applied to generalize the constitutive relation to
bound higher moments of vorticity. A `swirl' velocity field is identified, and
shown to transport w/rho and B/rho, generalizing the Kelvin-Helmholtz and
Alfven theorems. The steady regularized equations are used to model a rotating
vortex, MHD pinch and a plane vortex sheet. The proposed regularization could
facilitate numerical simulations of fluid/MHD equations and provide a
consistent statistical mechanics of vortices/current filaments in 3D, without
blowup of enstrophy. Implications for detailed analyses of fluid and plasma
dynamic systems arising from our work are briefly discussed.
