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Conservative regularization of neutral fluids and plasmas
Chennai Mathematical Institute.
Ideal systems like the Euler and magnetohydrodynamic (MHD) equations may develop singular structures like shocks, vortex sheets and current sheets. Among these, vortical singularities arise due to vortex stretching which can lead to unbounded growth of enstrophy. Viscosity and resistivity provide dissipative regularizations of these singularities.
In analogy with the dispersive KdV regularization of the 1D inviscid Burgers' equation, we propose a local conservative regularization of ideal 3D compressible flows, MHD and two-fluid plasmas. They have potential applications to high vorticity flows with low dissipation. The regularization involves introducing a vortical `twirl' term lambda^2 w x (curl w) in the velocity equation. The cut-off length lambda must be inversely proportional to the square root of density to ensure the conservation of a `swirl' energy and may be taken to be the Debye length or collisionless skin depth in plasmas. The swirl energy includes positive kinetic, compressional, magnetic and vortical contributions, thus leading to a priori bounds on enstrophy. A Hamiltonian-Poisson bracket formulation is developed and used to establish a minimality property of the twirl regularization. Generalizations of the Kelvin-Helmholtz and Alfven theorems are obtained. We have also extended our regularization to electron-ion two-fluid plasmas. The steady regularized equations are used to model a vortex sheet, rotating vortex etc. Our regularization could facilitate numerical simulations of neutral and charged fluids and a statistical treatment of vortex and current filaments in 3D.
Finally, we briefly describe a conservative regularization of shock-like singularities in compressible flow generalizing both the KdV and nonlinear Schrodinger equations to the adiabatic dynamics of a gas in 3D.