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Finite‐Resistivity Instabilities of a Sheet Pinch

### Abstract

The stability of a plane current layer is analyzed in the hydromagnetic approximation, allowing for finite isotropic resistivity. The effect of a small layer curvature is simulated by a gravitational field. In an incompressible fluid, there can be three basic types of ``resistive'' instability: a long‐wave ``tearing'' mode, corresponding to breakup of the layer along current‐flow lines; a short‐wave ``rippling'' mode, due to the flow of current across the resistivity gradients of the layer; and a low‐*g* gravitational interchange mode that grows in spite of finite magnetic shear. The time scale is set by the resistive diffusion time τ_{R} and the hydromagnetic transit time τ_{H} of the layer. For large *S* = τ_{R}/τ_{H}, the growth rate of the ``tearing'' and ``rippling'' modes is of order τ_{R} ^{−3/5}τ_{H} ^{−2/5}, and that of the gravitational mode is of order τ_{R} ^{−1/3}τ_{H} ^{−2/3}. As *S* → ∞, the gravitational effect dominates and may be used to stabilize the two nongravitational modes. If the zero‐order configuration is in equilibrium, there are no overstable modes in the incompressible case. Allowance for plasma compressibility somewhat modifies the ``rippling'' and gravitational modes, and may permit overstable modes to appear. The existence of overstable modes depends also on increasingly large *zero‐order* resistivity gradients as *S* → ∞. The three unstable modes merely require increasingly large gradients of the *first‐order* fluid velocity; but even so, the hydromagnetic approximation breaks down as *S* → ∞. Allowance for isotropic viscosity increases the effective mass density of the fluid, and the growth rates of the ``tearing'' and ``rippling'' modes then scale as τ_{R} ^{−2/3}τ_{H} ^{−1/3}. In plasmas, allowance for thermal conductivity suppresses the ``rippling'' mode at moderately high values of *S*. The ``tearing'' mode can be stabilized by conducting walls. The transition from the low‐*g* ``resistive'' gravitational mode to the familiar high‐*g* infinite conductivity mode is examined. The extension of the stability analysis to cylindrical geometry is discussed. The relevance of the theory to the results of various plasma experiments is pointed out. A nonhydromagnetic treatment will be needed to achieve rigorous correspondence to the experimental conditions.

© 1963 The American Institute of Physics

Received Mon Sep 17 00:00:00 UTC 1962
Published online Thu Dec 09 15:30:32 UTC 2004

/content/aip/journal/pof1/6/4/10.1063/1.1706761

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