Phenomenological theory of the kink instability in a slender plasma column

Abstract

When one deals with a plasma column whose radius a is much smaller than its length L, one can think of it as of a thin filament whose kink instability can be adequately described simply by a 2D displacement vector, ξ x =ξ x (z,t); ξ y =ξ y (z,t). Details of the internal structure of the column such as the current, density, and axial flow velocity distribution would be lumped into some phenomenological parameters. This approach is particularly efficient in the problems with non-ideal (sheath) boundary conditions (BC) at the end electrodes, with the finite plasma resistivity, and with a substantial axial flow. With the sheath BC imposed at one of the end-plates, we find instability in the domain well below the classical Kruskal-Shafranov limit. The presence of an axial flow causes the onset of rotation of the kink and strong axial "skewness" of the eigenfunction, with the perturbation amplitude increasing in the flow direction. We consider the limitations of the phenomenological approach and find that they are related to the steepness with which the plasma resistivity increases at the plasma boundary with vacuum.

Transition of MHD Kink-Stability Properties between Line-Tied and Non-Line-Tied Boundary Conditions

“…The first order perturbed current density equation can be easily found to be: j z enu ÿ j 0 e T e , and j 0 is the zeroth order quantity from Eq. (1) [13]. Thus a perturbation in current density can yield a perturbed sheath potential .…”

“…If the magnetic field is normal to the boundary ( 90 ) [13,19] one can write the incident current density from plasma to sheath as…”

“…If there is a fixed end z 0 plus a PNLT end, then the BC becomes a weighted combination of the aforementioned BC's: @ =@z i k 0 =2 0 z l [13,14].…”

“…In other words, a tangential sheath electric field appears that can drive the plasma motion at the boundary. For a flat endplate ( 90 ) without any constraints on the plasma movement, the plasma in the sheath region can move independently of the magnetic field lines (PNLT) if the sheath resistive decay time (column inductance or sheath resistance) is shorter than the Alfven time [13].…”

“…where V b ÿ V p , V p is the plasma potential, u and v Te are the ion flow speed (on the order of ion sound speed) and electron thermal speed [13], e > 0 is the absolute value of the electron charge, and T e is the electron temperature. The first order perturbed current density equation can be easily found to be: j z enu ÿ j 0 e T e , and j 0 is the zeroth order quantity from Eq.…”