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Ideal systems like MHD and Euler flow may develop singularities in vorticity(w = curl v). Viscosity and resistivity provide dissipative regularizations ofthe singularities. In this paper we propose a minimal, local, conservative,nonlinear, dispersive regularization of compressible flow and ideal MHD, inanalogy with the KdV regularization of the 1D kinematic wave equation. Thiswork extends and significantly generalizes earlier work on incompressible Eulerand ideal MHD. It involves a micro-scale cutoff length lambda which is afunction of density, unlike in the incompressible case. In MHD, it can be takento be of order the electron collisionless skin depth c/omega_pe. Ourregularization preserves the symmetries of the original systems, and withappropriate boundary conditions, leads to associated conservation laws. Energyand enstrophy are subject to a priori bounds determined by initial data incontrast to the unregularized systems. A Hamiltonian and Poisson bracketformulation is developed and applied to generalize the constitutive relation tobound higher moments of vorticity. A `swirl' velocity field is identified, andshown to transport w/rho and B/rho, generalizing the Kelvin-Helmholtz andAlfven theorems. The steady regularized equations are used to model a rotatingvortex, MHD pinch and a plane vortex sheet. The proposed regularization couldfacilitate numerical simulations of fluid/MHD equations and provide aconsistent statistical mechanics of vortices/current filaments in 3D, withoutblowup of enstrophy. Implications for detailed analyses of fluid and plasmadynamic systems arising from our work are briefly discussed.

Local conservative regularizations of compressible magnetohydrodynamic and neutral flows