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Center for Nonlinear Studies |
We show that small distortions on the boundaries are amplified toward the inner layers of a magnetized plasma if the system is close to marginal stability for the ideal MHD interchange mode. It is also shown that such marginal systems can be nonlinearly unstable. We then show that the combination of these two phenomena of boundary amplification and nonlinear instability can result in an induced nonlinear instability which is highly sensitive to the boundary perturbation. To elaborate, if the fractional deviation from marginality is by a small parameter Dβ/β, then a small perturbation on the boundary, of fractional size δ/a, will amplify interchange displacements in the core of the plasma by an amplification factor (δ/a)/(Dβ/β). This type of amplification has been shown previously for tearing1 and kink-like2 modes, but not for interchange modes. We show further that at marginal stability the system is inherently nonlinearly unstable, i.e. a critical boundary perturbation will destabilize the interchange mode even for systems above the linear stability threshold for interchanges. The boundary perturbations then induce the nonlinear instability, as may be expected; however, this inducement is highly sensitive to the boundary in that the system can go unstable from much smaller fractional boundary distortions, we show, of order [(δ/a)/(Dβ/β)]3/2. This sensitivity to the boundary has the implication that magnetic configurations designed to confine plasma close to the β limit within a tolerance of Dβ/β would necessitate that the design be more sensitive to boundary perturbations; specifically, boundary tolerances need to be better than (Dβ/β) 3/2. Such considerations could be of significant importance in the design of axisymmetric tolerances for advanced tokamaks as well as in the fully 3D design of stellarators for fusion.
1 A. Reiman and D. Monticello, Phys. Fluids B 3, 2230 (1991).
.2 A. H. Boozer, Physical Review Letters 86, 5059 (2001).
Host: Tom Intrator