On the motion of a z-pinch in an inhomogeneous transverse magnetic field

An analytical treatment of the large scale axisymmetric motion of a z-pinch in an arbitrary transverse magnetic field is given. The analysis is based on an expansion in the non-circularity of the plasma cross-section, but does not involve any assumption of small displacements of the plasma from some equilibrium position. In order to include the effect of a possible internal rotation of the plasma, the pinch is initially modeled by an incompressible, surface current ideal MHD plasma with constant vorticity. The presence of homogeneous, axial magnetic fields B i and B e (inside and outside of the pinch, respectively) is shown to have no effect on the axisymmetric motion. One particular application concerns the axisymmetric m = 1 instability (slip mode) of non-circular pinches, relevant for a number of Tokamaks and for the Extrap z-pinch configuration. It is shown that the plasma rotation increases the linear as well as the nonlinear growth of this instability. In a "quasi-static" approximation the method is readily extended to volume current plasmas, and is found to greatly simplify the calculation of linear and nonlinear stability of non-circular pinches, as compared to previous investigations. Some illustrative examples are given. The role of inertia in the time evolution of the plasma non-circularity, leading to the possibility of axisymmetric stability of a z-pinch in a time-varying magnetic field, as well as to the possible existence of "precessing equilibria" of a z-pinch in Extrap geometry, is briefly discussed.