Ideal systems like MHD and Euler flow may develop singularities in vorticity (w = ∇ × v). Viscosity and resistivity provide dissipative regularizations of the singularities. In this paper we propose a minimal, local, conservative, nonlinear, dispersive regularization of compressible flow and ideal MHD, in analogy with the KdV regularization of the 1D kinematic wave equation. This work extends and significantly generalizes earlier work on incompressible Euler and ideal MHD. It involves a micro-scale cutoff length λ which is a function of density, unlike in the incompressible case. In MHD, it can be taken to be of order the electron collisionless skin depth c/ω pe . Our regularization preserves the symmetries of the original systems, and with appropriate boundary conditions, leads to associated conservation laws. Energy and enstrophy are subject to a priori bounds determined by initial data in contrast to the unregularized systems. A Hamiltonian and Poisson bracket formulation is developed and applied to generalize the constitutive relation to bound higher moments of vorticity. A 'swirl' velocity field is identified, and shown to transport w/ρ and B/ρ, generalizing the Kelvin-Helmholtz and Alfvén theorems. The steady regularized equations are used to model a rotating vortex, MHD pinch and a plane vortex sheet. The proposed regularization could facilitate numerical simulations of fluid/MHD equations and provide a consistent statistical mechanics of vortices/current filaments in 3D, without blowup of enstrophy. Implications for detailed analyses of fluid and plasma dynamic systems arising from our work are briefly discussed.

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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 λl2(wl+(ql/ml)B)×(∇×wl) in the ion/electron velocity equations (l=i,e) where wl are vorticities. The cut-off lengths λl and number densities nl must satisfy λl2nl=Cl, where Cl are constants. A novel feature is that the “flow” current ∑lqlnlvl in Ampère's law is augmented by a solenoidal “twirl” current ∑l∇×∇×λl2jflow,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)λl2ρlwl2. It is shown that the equations admit a Hamiltonian-Poisson bracket formulation. Furthermore, singularities in ∇×B are conservatively regularized by adding (λB2/2μ0)(∇×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.

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Ideal gas dynamics can develop shock-like singularities with discontinuous density. Viscosity typically regularizes such singularities and leads to a shock structure. On the other hand, in 1d, singularities in the Hopf equation can be non-dissipatively smoothed via KdV dispersion. In this paper, we develop a minimal conservative regularization of 3d ideal adiabatic flow of a gas with polytropic exponent γ . It is achieved by augmenting the Hamiltonian by a capillarity energy β(ρ)(∇ρ) 2 . The simplest capillarity coefficient leading to local conservation laws for mass, momentum, energy and entropy using the standard Poisson brackets is β(ρ) = β * /ρ for constant β * . This leads to a Korteweg-like stress and nonlinear terms in the momentum equation with third derivatives of ρ , which are related to the Bohm potential and Gross quantum pressure. Just like KdV, our equations admit sound waves with a leading cubic dispersion relation, solitary waves and periodic traveling waves. As with KdV, there are no steady continuous shock-like solutions satisfying the Rankine-Hugoniot conditions. Nevertheless, in 1d, for γ = 2 , numerical solutions show that the gradient catastrophe is averted through the formation of pairs of solitary waves which can display approximate phase-shift scattering. Numerics also indicate recurrent behavior in periodic domains. These observations are related to an equivalence between our regularized equations (in the special case of constant specific entropy potential flow in any dimension) and the defocussing nonlinear Schrödinger equation (cubically nonlinear for γ = 2 ), with β * playing the role of 2 . Thus, our regularization of gas dynamics may be viewed as a generalization of both the single field KdV and nonlinear Schrödinger equations to include the adiabatic dynamics of density, velocity, pressure and entropy in any dimension.