Conservative regularization of compressible dissipationless two-fluid plasmas

Abstract

This paper extends our earlier approach [cf. Phys. Plasmas 17, 032503 (2010), 23, 022308 (2016)] to obtaining a priori bounds on enstrophy in neutral fluids (R-Euler) and ideal magnetohydrodynamics (R-MHD). This results in a far-reaching local, three-dimensional, non-linear, dispersive generalization of a KdV-type regularization to compressible/incompressible dissipationless two-fluid plasmas and models derived therefrom (quasi-neutral, Hall and ideal MHD). It involves the introduction of vortical and magnetic `twirl' terms $λ_l^2 ({\bf w}_l + \frac{q_l}{m_l} {\bf B}) \times (\nabla \times {\bf w}_l)$ in the ion/electron velocity equations ($l = i,e$) where ${\bf w}_l = \nabla \times {\bf v}_l$ are vorticities. The cut-off lengths $λ_l$ must be inversely proportional to the square-roots of the number densities $(λ_l^2 n_l = C_l)$ and may be taken as Debye lengths or skin-depths. A novel feature is that the `flow' current $\sum_l q_l n_l {\bf v}_l$ in Ampere's law is augmented by a solenoidal `twirl' current $\sum_l \nabla \times \nabla \times λ_l^2 {\bf j}_{{\rm flow},l}$. The resulting equations imply conserved linear and angular momenta and a positive definite swirl energy density ${\cal E}^*$ which includes an enstrophic contribution $\sum_l (1/2) λ_l^2 ρ_l {\bf w}_l^2$. It is shown that the equations admit a Hamiltonian-Poisson bracket formulation. Furthermore, singularities in $\nabla \times {\bf B}$ are conservatively regularized by adding $(λ_B^2/2 μ_0) (\nabla \times {\bf B})^2$ to ${\cal E}^*$. 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.