Velocity Gradient Instability

Abstract

A recently formulated variational definition of stability for a guiding center plasma confined by mirrors was found to yield as a necessary condition for stability that the equilibrium distribution function f0(ε,μ,α,β) be monotone in energy on each magnetic line, ∂f0/∂ε ≤ 0. We now show, conversely, that if ∂f0/∂ε has too large a negative value, the plasma is always unstable. A quantitative formulation of this condition yields two inequalities that are necessary for stability of an arbitrary contained plasma. The same inequalities are sufficient for stability against all localized disturbances. Thus any remaining instabilities can only involve large scale disturbances of the plasma (e.g., interchanges). The localized (small wavelength) instabilities are found to be micro-instabilities even though the latter are usually thought to be missing in a theory with zero gyro radius and zero Debye radius. These micro-stability criteria are related to conditions for the stability of an infinite homogeneous plasma, and they serve to correct and generalize certain estimates of the maximum pressure that can be stably contained in a magnetic well.