Magnetohydrodynamic (MHD) instability of current carrying plasmas is a topic of importance to astrophysical and laboratory plasmas. In the Sun, current driven MHD instability is believed responsible for solar flares. The twisting of solar coronal loops caused by photospheric shear flow and vortices results in current flow parallel to the magnetic fields. When the current exceeds a critical value dependent upon the background magnetic field, instability can occur, resulting in a flare or mass ejection; similar phenomena may occur in accretion disks. Considerable theoretical effort has been made to understand the instability onset by modeling these plasmas as line-tied cylindrical plasmas. Current driven instabilities may also be responsible for the sinusoidal form of some astrophysical jets. Laboratory plasmas, such as the tokamak, reversed field pinch and spheromak, are all susceptible to kink instabilities from the same excessive twisting in field lines. When the current in the plasma exceeds a critical value, the plasma deforms into a helical structure which grows exponentially on a time scale similar to the time it takes an Alfven wave to propagate across the plasma.
The stability condition depends in part upon the boundary conditions in the axial direction. For periodic boundary conditions (appropriate to toroidal plasmas) Kruskal and independently Shafranov (hereafter referred to as KS) discovered that the condition for stability is that the safety factor q(a) < 1. In periodic systems, the theory was extended in several ways. First, it was found that a fast growing external kink becomes unstable whenever the value of q is just below a low order rational value qn. Second, the plasma can become unstable when the safety factor drops below 1 somewhere inside the plasma (in this case the boundary of the plasma may not move at all and the mode is classified as an internal kink mode). Finally, finite plasma resistivity can lead to the phenomenon of reconnection. In periodic systems, resistive instabilities occur at resonant surfaces where the wave number of the MHD instability is locally parallel to the equilibrium magnetic field. Moreover, it was found that at finite amplitude, the current profile of the internal kink mode becomes singular at the resonant surface.
While kink instability theory has been quite successful in predicting the behavior of toroidal plasmas, it has not been tested in cylindrical geometries for which the boundary conditions are modified due to line-tying in highly conducting endplates. By line-tying we mean that the plasma is constrained to be stationary at the ends of the cylinder due to the presence of a conducting surface. In the line-tied pinch geometry the safety factor q still governs stability. If the plasma has uniform current density, line-tied plasmas can be analytically shown to be unstable to an external kink when q(a) < 1.
For non-uniform current profiles, the stability analysis is more complicated and depends upon the details of the equilibrium. Several authors have performed numerical simulations of the line-tied kink mode through the linear growth phase and followed it into the non-linear saturated phase. The numerical results are consistent with the requirement that q drop below one somewhere in the plasma, and they show that the plasma tends to kink into a knot-like instability half-way between the conducting plates. If q < 1 everywhere, the entire column distorts into helical external kink. If q is only below one in the central part of the plasma, an internal instability results with formation of strong, possibly singular currents and ultimately reconnection at the q = 1 radius. That is, reconnection in the line-tied plasmas occurs in spite of the fact that there are no resonant surfaces. Whether the nonlinear extension of these modes display current sheets, and whether a current singularity is compatible with the line tied boundary conditions remains an open question.