Plasma relaxation and topological aspects in electron magnetohydrodynamics

Abstract

Parker's formulation of isotopological plasma relaxation process toward minimum magnetics energy states in magnetohydrodynamics (MHD) is extended to electron MHD (EMHD). The lower bound on magnetic energy in EMHD is determined by both the magnetic field and the electron vorticity field topologies, and is shown to be reduced further in EMHD by an amount proportional to the sum of total electron-flow kinetic energy and total electron-flow enstrophy. The EMHD Beltrami condition becomes equivalent to the potential vorticity conservation equation in two-dimensional (2D) hydrodynamics, and the torsion coefficient and turns out to be proportional to potential vorticity. The winding pattern of the magnetic field lines appears to evolve therefore in the same way as potential vorticity lines in 2D hydrodynamics.1

Topological implications of the total generalized electron-flow magnetic helicity invariant in electron magnetohydrodynamics

“…This result appears to support a change in the magnetic field topology and hence pave the way for magnetic reconnection in EMHD via a change in the concomitant electron-flow vorticity topology. A related result is the reduction of the lower bound on the magnetic energy in EMHD (Shivamoggi [16]) by an amount proportional to the sum of total electron-flow kinetic energy and total electron-flow enstrophy. This lower bound is produced by the topological barrier provided by the linkage of generalized magnetic field lines underscored by H e = 0.…”

The Okubo–Weiss-type topological criteria in two-dimensional magnetohydrodynamic flows

“…In EMHD, with , being the gyro-radius, the dynamics is dominated by magnetized electrons with the demagnetized ions serving to provide the neutralizing static background (Gordeev, Kingsep & Rudakov 1994). The assumptions underlying the EMHD model are , where is the skin depth, and that the frequencies involved are greater than and , being the cyclotron frequency, , being the plasma frequency, (Shivamoggi 2015 b , 2016; Shivamoggi & Michalak 2019). The equation governing the transport of the generalized magnetic field or a renormalized magnetic field incorporating the effect of electron inertia (or generalized Ohm's law) may be rewritten as follows: or or where and is proportional to the in-plane current density, so also represents the out-of-plane magnetic field.…”

“…In EMHD, with , being the gyro-radius, the dynamics is dominated by magnetized electrons with the demagnetized ions serving to provide the neutralizing static background (Gordeev, Kingsep & Rudakov 1994). The assumptions underlying the EMHD model are , where is the skin depth, and that the frequencies involved are greater than and , being the cyclotron frequency, , being the plasma frequency, (Shivamoggi 2015 b , 2016; Shivamoggi & Michalak 2019).…”