Current-Driven Rotating-Kink Mode in a Plasma Column with a Non-Line-Tied Free End

Abstract

First experimental measurements are presented for the kink instability in a linear plasma column which is insulated from an axial boundary by finite sheath resistivity. Instability threshold below the classical Kruskal-Shafranov threshold, axially asymmetric mode structure and rotation are observed. These are accurately reproduced by a recent kink theory, which includes axial plasma flow and one end of the plasma column that is free to move due to a non-line-tied boundary condition. The current driven kink instability is a magnetohydrodynamic (MHD) instability which affects current carrying plasmas in Nature and laboratory. The kink mode structure and stability condition are strongly dependent on the system geometry and the boundary conditions (BCs). Kruskal[1] and Shafranov [2] (hereafter referred to as KS) considered first the ideal MHD stability of a cylindrical plasma column with magnetic field components (0, B θ , B z ) using cylindrical coordinates (r, θ, z). For an infinitely long (equivalent to periodic axial BCs) column, they obtained a linearly unstable helical kink mode of structure ξ = e i(θ+2πz/L) when the plasma current I p exceeds the Kruskal-Shafranov limitwhere a and L are, respectively, the radius and length of the current channel, and ξ is the displacement of the plasma column from the equilibrium position. The KS theory has been quite successful in predicting the behavior of toroidal plasmas for which the periodic BCs yield a proper accounting for the finite length of the system. In linear systems, however, substantial deviations from KS predictions can result from different axial BCs. The importance of the BCs has long been recognized [3,4] and is of particular relevance to the stability of linetied flux ropes in space physics (c.f. Ref. [5] and survey Ref.[6]), and astrophysical jets [7]. In recent years, there has been a renewed interest in the stability of a line-tied plasma column in laboratory devices (see Refs. [8,9,10] and references therein). The kink stability of a plasma column with line-tied ends has been investigated in linear devices, where line-tying is attributed to the presence of highly conducting end plates [11], and in open systems to a local discontinuity for the Alfvén velocity that forms a virtual boundary around the system [12,13].In this Letter, we experimentally investigate the external kink instability in conditions where one end of the plasma column is line-tied to the plasma source, and the other end is not line-tied and therefore free to slide over the surface of the end-plate. The latter BC is a result of plasma sheath resistance that insulates, at least partially, the plasma from the end-plate. Compared to the (6) external anode. On the right, an axial cut near the external anode. The fast camera (7) is located at the midplane and views the plasma column along the x direction. The bi-dimensional magnetic probe (8) measures (δBx, δBy) at the edge. Also schematically shown is the plasma column whose end rotates at the external anode.line-tied case, we find signi...

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

“…For data near or below the B instability threshold, for instance as in Fig. 3, the plasma motion can still be described by a linear dynamical model [13,14]. The kink threshold is 50% larger for 22:5 than for 90 .…”

“…For the experiment shown in this Letter, there is a conducting boundary partially insulated by a sheath. As shown in previous work [14], the plasma-sheath boundary provides an electric field E and a tangential force that drives a plasma footpoint movement @@ t Ẽ B B 2 , where~is the vector displacement. Thus, in the following we first present the experimental evidence that there exists a sheath and then we argue there is a tangential electric field which is also affected by the boundary shape, so that this constraint leads to the BC.…”

“…In toroidal systems there can be one unidirectional (e.g., Alfvén) wave, but in a finite cylindrical system, there can be reflections. The boundary condition can radically affect the nature of the Alfven spectrum [1], as well as the stability properties [14,15]. For instance field line curvature and thus the B r B MHD restoring force is reduced when the end becomes non-line-tied and is nonzero.…”

“…Figure 1(a) is a perspective schematic of the RSX experimental setup and coordinates. Figure 1(b) shows an end-on, perspective, H image of the whole kinked plasma column which is a helical distortion wrapped around a conical surface [6,14,17], between gun and cone anode positions. The side view image in Fig.…”

“…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].…”