7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Extended magnetohydrodynamic approach to plasma-vacuum interface
Makoto Hirota
Institute of Fluid Science, Tohoku University
e-mail (speaker): makoto.hirota.d5@tohoku.ac.jp
The magnetohydrodynamic (MHD) equations arewidely used as the macroscopic fluid model for plasmaand have been solved analytically and numerically forvarious applications. The MHD equations are originallyderived from the two-fluid equations (for ions andelectrons) by assuming quasi-neutrality and neglecting
the Hall and electron-inertia effects. However, this MHD
approximation is invalid when plasma density decreases
to a certain level. The plasma-vacuum interface (i.e., thefree boundary of plasma) is therefore difficult to treat in
the standard MHD. For example, the phase speed of
Alfvén wave diverges unphysically in the low-densitylimit.
In fact, the Hall and electron-inertia effects arenonnegligible in the scales of ion and electron’s inertial
lengths, respectively, and they are inversely proportional
to the square root of density. While the dense plasma
region is well described by MHD, such the two-fluid
effects should be taken into account in the region wherethe density decays to zero (i.e., vacuum). For this reason,
we are naturally led to study the extended MHD(XMHD) model [1], in which the Hall andelectron-inertia terms are restored while quasi-neutrality
is still assumed (by taking the light speed infinite). The
collisional effects such as resistivity and viscosity arealso taken into account in this study. The XMHD
equations are, then, well-posed in both plasma andvacuum regions.
To demonstrate it numerically, we consider a cylindricalplasma column which has only one-dimensional radial
distribution. An equilibrium solution (well-known as theZ pinch) may exist by imposing a constant electric field(𝐸𝑧0 ) along the axial (𝑧) direction. The corresponding
XMHD equations in the cylindrical coordinates are
reduced to a nonlinear ordinary differential equation.
This order is increased by 2 due to the electron-inertia
effect, indicating that it is a kind of singular perturbation
for MHD. In this symmetric geometry, the Hall term
becomes automatically zero and has no impact on theequilibrium solution.
Since the cylindrical plasma is pinched byelectromagnetic force, the plasma region is surroundedby vacuum at the equilibrium state if the plasma beta issufficiently low. The density profile ρ of the MHD
solution is shown by the dashed line in Fig. 1. In XMHD,
the density becomes exactly zero at a position 𝑟 = 𝑎 if
adiabatic change is assumed with a specific heat ratio γ >1. The XMHD solution is not smooth at 𝑟 = 𝑎 and thefinite difference scheme fails there. This singularity
needs to be regularized appropriately in numericalanalysis. The resultant XMHD solution is shown by thesolid lines in Fig. 1, where the resistivity and kinematic

viscosity are constant everywhere for simplicity.
The numerical solution indicates that there exists a“boundary layer” that connect the inner MHD solution
and the outer vacuum magnetic field. Within the layer, a
localized surface electric current is driven due to theelectron-inertia effect. Although the collisional effecttends to attenuate this surface current, the constantresistivity is insufficient for that and, hence, electrons areaccelerated within the low-density layer. Because of thisboundary layer, the XMHD equilibrium solution does notconverge to the MHD one in the limit of zero
electric-inertia effect. In particular, the equilibrium
position of the plasma-vacuum interface shifts inward
depending on the magnitude of surface current.
In XMHD, we need to solve the relative velocity 𝒖 =𝒗𝑖 − 𝒗𝑒 of ion and electron even in the vacuum region.
Then, the kinematic viscosity and the no-slip boundary
condition are necessary for well-posedness andnumerical stability. In the MHD limit, the width 𝛿 ofthe boundary layer tends to zero, but the surface current
remains finite. By considering matching conditions atplasma-vacuum interface, we can obtain its position 𝑟 =𝑎 analytically, which depends on only the plasma beta.
This work was supported by JSPS KAKENHI Grant
Number 21K03498 and 21H01056.
References[1] K. Kimura and P. J. Morrison, Physics of Plasmas 21,082101 (2014).

Figure 1. Radial profile of a cylindrical equilibrium ofXMHD under axial electric field (Z pinch).