Relaxed Magnetohydrodynamics with Ideal Ohm's Law Constraint

Abstract

Recently, a new magnetofluid dynamics, Relaxed MHD (RxMHD), was constructed using Hamilton's Principle with a phase-space Lagrangian incorporating constraints of magnetic and cross helicity. A key difference between RxMHD and Ideal Magnetohydrodynamics (IMHD) is that IMHD implicitly constrains the magnetofluid to obey the zero-resistivity "Ideal" Ohm's Law (IOL) pointwise whereas RxMHD discards the IOL constraint completely, which can violate the desideratum that all equilibrium solutions of RxMHD form a subset of all IMHD equilibria. The present paper lays the formal groundwork for rectifying this deficiency. In order to impose a weak form of the IOL constraint on RxMHD two forms of the iterative augmented Lagrangian penalty function method are proposed and discussed. It is conjectured this weak-form regularization will allow reconnection and thus avoid the formation of the singularities that plague three-dimensional IMHD equilibria. A unified dynamical formalism is developed that can treat a number of MHD versions. Euler-Lagrange equations and a gauge-invariant momentum equation in conservation form are derived, in which the IOL constraint contributes an external force and internal stress terms until convergence is achieved.

Nature of ideal MHD instabilities as described by multi-region relaxed MHD

“…The equilibrium with q 0 = 0.99 and β p = 0.297 is considered. The normalized SPEC radial eigenfunction, ξ m/n=1/1 s is obtained by solving equation (54) for ξ • n. Hence, it appears to be a qualitative agreement between these two codes.…”

“…As a consequence, equation (33) can also be condensed by taking the MRxMHD in the incompressible limit ∇ • ξ = 0, which will be further discussed in section 3.1.2. More generally, on considering the incompressible limit in Dewar et al [35,36,54] dynamical MRxMHD theories, the ideal interfaces, which act as infinitesimally thin current sheets, supply inertia that specifies the finite frequencies, and only surface waves like the shear-Alfvén wave persist. Thus, it is incompressibility that imparts inertia to the interfaces, because, when we move an interface, incompressibility forces the plasma to move within the interface, requiring a force on the plasma and an equal and opposite reaction force on the interface.…”

“…If there exists an i such that λ i < 0 then an equilibrium is said to be unstable and if all λ i > 0, that is, positive, then an equilibrium is said to be stable. By solving the eigenproblem for the Hessian, we obtain the Illustration of tri-diagonal arrangement of an H 11 j,k,l,l ′ matrix quantity defined in equation (54) where…”

“…Thus, it is incompressibility that imparts inertia to the interfaces, because, when we move an interface, incompressibility forces to move the plasma within the interface, requiring a force on the plasma and an equal and opposite reaction force on the interface. The magneto-sonic acoustic (slow or fast) waves with finite low-frequencies also exist within the Beltrami relaxed sub-regions for compressible MRxMHD plasma 55 , along with coupled surface waves from the interfaces.…”

“…Figure 2. Illustration of tri-diagonal arrangement of an H 11j,k,l,l ′ matrix quantity defined in equation(54) whereA 0 = H 11 j,k,l,l , B 0 = H 11 j,k,l,l+1 , C 0 = H 11 j,k,l−1,l , A 1 = H 11 j,k,l+1,l+1 , B 1 = H 11j,k,l,l+2 and so on.…”