Conservative regularization of neutral fluids and plasmas
Abstract
Ideal systems of equations such as Euler and MHD may develop singular structures like shocks, vortex/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 2fluid plasmas (with 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 cutoff length lambda must be inversely proportional to square root of density to ensure the conservation of a `swirl' energy. The latter includes positive kinetic, compressional, magnetic and vortical contributions, thus leading to a priori bounds on enstrophy. The extension to 2fluid plasmas involves additionally magnetic `twirl' terms in the ion and electron velocity equations and a solenoidal addition to the current in Ampere's law. A HamiltonianPoisson bracket formulation is developed using the swirl energy as Hamiltonian. We also establish a minimality property of the twirl regularization. A swirl velocity field is shown to transport vortex and magnetic flux tubes as well as w/rho and B/rho, thus generalizing the KelvinHelmholtz and Alfven theorems. The steady regularized equations are used to model a rotating vortex, MHD pinch and vortex sheet. Our regularization could facilitate numerical simulations and a statistical treatment of vortex and current filaments in 3D. Finally, we briefly describe a conservative regularization of shocklike singularities in compressible flow generalizing both the KdV and nonlinear Schrodinger equations to the adiabatic dynamics of a gas in 3D.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.14086
 Bibcode:
 2020arXiv200714086S
 Keywords:

 Physics  Fluid Dynamics;
 Physics  Plasma Physics
 EPrint:
 PhD thesis, Chennai Mathematical Institute (2020). 110 pages, 17 figures, Based on arXiv:1510.01606, arXiv:1602.04323, arXiv:1711.05236, arXiv:1910.07836