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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 Bi and Be (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.