  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 2-fluid 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 cut-off 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
2-fluid 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 Hamiltonian-Poisson 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
Kelvin-Helmholtz 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 shock-like singularities in compressible flow generalizing
both the KdV and nonlinear Schrodinger equations to the adiabatic dynamics of a
gas in 3D.
