Conservative regularization of compressible flow and ideal magnetohydrodynamics

Abstract

Ideal systems like MHD and Euler flow may develop singularities in vorticity (w = curl v). Viscosity and resistivity are dissipative regularizations. We propose a minimal, local, conservative, nonlinear, dispersive regularization of compressible flow and ideal MHD, in analogy with the KdV regularization of the 1D Hopf equation. This work significantly extends earlier work on incompressible Euler and ideal MHD. It involves a cut-off lambda inversely proportional to square-root of density rho, which is like a position-dependent mean free path. In MHD, lambda can be taken of order the electron collisionless skin depth. The regularizing `twirl' term is - lambda w x curl w. Such a term could be important in high speed flows with vorticity and arise in an expansion of kinetic equations in Knudsen number. A magnetic analogue of the twirl term - (B x curl B)/(rho mu_0), arises as the Lorentz force in ideal MHD. Our regularization preserves symmetries of the ideal systems, and with appropriate boundary conditions, implies associated conservation laws. Energy and enstrophy are subject to a priori bounds determined by initial data. A Hamiltonian and Poisson bracket formulation is developed and used to generalize the constitutive relation to bound higher moments of w and curl w. A `swirl' velocity field is shown to transport w/rho and B/rho, generalizing the Kelvin-Helmholtz and Alfv\'en theorems. The steady regularized equations are used to model a rotating vortex, MHD pinch, plane vortex sheet, channel flow, plane flow and propagating spherical and cylindrical vortices; solutions are more regular than corresponding Eulerian ones. The proposed regularization could facilitate 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 systems are discussed.