Large-scale perturbations due to a small-scale instability in a finite-conductivity plasma
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MA INSTABILITY
Large-Scale Perturbations due to a Small-Scale Instability in a Finite-Conductivity Plasma V. V. Arsenin Nuclear Fusion Institute, Russian Research Centre Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia Received July 10, 2006
Abstract—By considering kink modes in a plasma cylinder in a strong axial magnetic field as an example, it is demonstrated that, because of the finite plasma conductivity (the finite longitudinal plasma permittivity ε||), large-scale perturbations can grow with time due to a small-scale instability that develops near a certain magnetic surface. PACS numbers: 52.55.Tn, 52.35.Qz DOI: 10.1134/S1063780X07010023
1. INTRODUCTION
a certain range of values of the axial magnetic field or, equivalently, of the safety factor q (the existence of such a range, as well as its length, depends on the current profile and on the azimuthal number m), the corresponding ideal mode can be unstable, with the maximum growth rate γMHD ~ |k|| |vA, where vA is the Alfvén velocity. The infinite conductivity approximation is, however, inapplicable to the vicinity of a resonance magnetic surface at which k|| = 0. In this case, the radial profile of the radial component Br of the magnetic perturbation in a large-scale wave has a break (the derivative ∂Br /∂r varies sharply across a thin layer). Depending on the shape of the current profile, the break may have such a sign that another type of instability develops, namely, a tearing-mode instability, whose growth rate is much lower than |k|| |vA. In what follows, it will be shown that, because of the finite plasma conductivity, the growth of large-scale perturbations due to their coupling with small-scale unstable perturbations is also possible when neither ideal instability nor a tearing mode develops.
MHD oscillations of a finite-conductivity magnetized plasma are described by differential equations containing spatial derivatives of higher order than those in the equations for a perfectly conducting plasma. In a plasma with a high but finite conductivity such that the skin depth D is much less than the plasma size a across the magnetic field, two types of perturbations can be distinguished: large-scale perturbations with a characteristic wavelength of λ ~ a, which are very similar to ideal modes (and, accordingly, for which the higher derivatives play an insignificant role), and small-scale perturbations with λ ~ D, which may be localized. To first order in the ratio D/a, large- and small-scale modes are independent of one another. However, in the higher order approximations, there is a weak coupling between the modes occurring on such scales. The present paper focuses on the situation in which a largescale mode, although stable, can nevertheless grow in time because of its coupling with an unstable smallscale mode, even when the latter is spatially localized. This effect is demonstrated by considering a plasma cylinder in a strong axial magnetic field as an example. Oscillations of a perfectly conducting plasma are described by a second-order
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