Ballooning instabilities in tokamaks with sheared toroidal flows

Abstract

The stability of ballooning modes in the presence of sheared toroidal flows is investigated. The eigenmodes are shown to be related by a Fourier transformation to the nonexponentially growing Floquet solutions found by Cooper [Plasma Phys. Controlled Fusion 30, 1805]. It is further shown that the problem cannot be reduced further than to a two-dimensional partial differential equation. Next, the generalized ballooning equation is solved analytically for a circular tokamak equilibrium with sonic flows, but with a small rotation shear compared to the sound speed. With this ordering, the centrifugal forces are comparable to the pressure gradient forces driving the instability, but coupling of the mode with the sound wave is avoided. A new stability criterion is derived that explicitly demonstrates that flow shear is stabilizing at constant centrifugal force gradient.

Flow shear stabilization of rotating plasmas due to the Coriolis effect

“…Because of its dependence on the sign of the flow shear, the pressure-Coriolis effect can be discriminated from other effects like that of Ref. [23,24], the parallel velocity gradient or Kelvin-Helmholtz instability, and nonlinear flow shear suppression.…”

“…This may very well be due to the shearing effect investigated for ballooning modes in Refs. [23,24], which is always stabilizing. Indeed, for a similar equilibrium with a higher value of β ≈ 0.05 on axis, the growth rate of the higher n modes was found to decrease with increasing positive flow shear k.…”

Sheared flow effects on ballooning instabilities in three-dimensional equilibria

“…We approach this problem by using the asymptotic technique developed by Waelbroeck and Chen for tokamak plasmas. 8 Before preceding with the problem with flow, we review the threory for static plasmas. In this limit, the eikonal is no longer time dependent; the eikonal solution is the eigenmode solution and the growth rate is identified with a local eigenvalue.…”

“…IV, a calculation of the growth rate is made using a procedure similar to that used in Ref. 8. A summary of the results in given in Sec.…”

“…[5][6][7][8][9][10][11][12] These calculations rely on using a ballooning mode Wentzel-Kramers-Brillouin ͑WKB͒-like formalism. 13,14 In the static limit ͑no equilibrium flow͒, the ballooning transformation produces a lowest order solution in the form of an ordinary differential equation along the field line.…”