Alladi Ramakrishnan Hall

Conservative regularization of compressible 3D Eulerian flows (Continued)

Govind Krishnaswami

CMI, Chennai

3D ideal systems like Euler flow and MHD may develop singularities in
vorticity. Naiver-Stokes viscosity and resistivity provide dissipative
regularizations. We propose a minimal, local, conservative, nonlinear,
dispersive `twirl' regularization of compressible flow and ideal MHD,
a sort of 3D analogue of the KdV equation. It involves a regularizing
length that is like a position-dependent mean free path. The twirl
regularization preserves space-time symmetries, and with natural
boundary conditions, leads to associated conservation laws. We obtain
a priori bounds on kinetic energy and enstrophy. A Hamiltonian and
Poisson bracket formulation is developed and applied to find
regularizations that bound higher moments of vorticity. A `swirl'
velocity field is identified, into which vorticity and magnetic field
are frozen, generalizing the Kelvin-Helmholtz and Alfven theorems. The
steady regularized equations are used to model a rotating vortex and a
plane vortex sheet. The twirl regularization could facilitate
numerical simulations and statistical treatments of vortices, and may
arise from kinetic theory in an expansion in Knudsen number.

Based on joint work with S. Sachdev and A. Thyagaraja.