Effect of Resistivity on Hydromagnetic Instabilities in Multipolar Systems

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Abstract
The effect of electrical resistivity on hydromagnetic instabilities is studied with special emphasis on interchange instabilities. A ``stellarator'' expansion is utilized to reduce the stability problem to one of solving two coupled second-order ordinary differential equations. This expansion enables one to treat a general geometry in a simple manner. A low-pressure strongly stabilized pinch is one special case. Further expansions are made in which perturbations vary rapidly in a thin layer the width of which is related to the resistivity. Highly localized interchange instabilities are found with growth rates which are independent of the resistivity in the limit as it goes to zero. These exist whenever the pressure decreases outwards. Interchanges with eddies of finite length grow more slowly, as the cube root of the resistivity. A stability criterion for king modes, which are characterized by vanishing pressure gradients in the resistive layer, is obtained. It depends on the entire plasma configuration and is identical to the one that would be obtained from the energy principle if the resistive layer were replaced with a vacuum. Gyration radius effects which are not included may significantly affect these resistive instabilities because they have slow growth rates and vary rapidly in a localized region.

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