Conservative regularization of compressible dissipationless two-fluid plasmas

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This paper extends our earlier approach [cf. A. Thyaharaja, Phys. Plasmas 17, 032503 (2010) and Krishnaswami et al., Phys. Plasmas 23, 022308 (2016)] to obtaining à priori bounds on enstrophy in neutral fluids and ideal magnetohydrodynamics. This results in a far-reaching local, three-dimensional, non-linear, dispersive generalization of a KdV-type regularization to compressible/incompressible dissipationless 2-fluid plasmas and models derived therefrom (quasi-neutral, Hall, and ideal MHD). It involves the introduction of vortical and magnetic “twirl” terms λ l 2 ( w l + ( q l / m l ) B ) × ( ∇ × w l ) in the ion/electron velocity equations ( l = i , e) where w l are vorticities. The cut-off lengths λl and number densities nl must satisfy λ l 2 n l = C l, where Cl are constants. A novel feature is that the “flow” current ∑ l q l n l v l in Ampère's law is augmented by a solenoidal “twirl” current ∑ l ∇ × ∇ × λ l 2 j flow , l. The resulting equations imply conserved linear and angular momenta and a positive definite swirl energy density E * which includes an enstrophic contribution ∑ l ( 1 / 2 ) λ l 2 ρ l w l 2. It is shown that the equations admit a Hamiltonian-Poisson bracket formulation. Furthermore, singularities in ∇ × B are conservatively regularized by adding ( λ B 2 / 2 μ 0 ) ( ∇ × B ) 2 to E *. Finally, it is proved that among regularizations that admit a Hamiltonian formulation and preserve the continuity equations along with the symmetries of the ideal model, the twirl term is unique and minimal in non-linearity and space derivatives of velocities.