Rotation shear and drift wave stability

The stability of short wavelength (high n, where n is the toroidal mode number) drift eigenmodes in toroidally confined plasma is conventionally analysed using the ballooning transformation. In lowest order in 1/n there is a local eigenvalue, λ(x, k), where k is a parameter representing the radial wave-number. Usually profile variation defines a radial position where the growth rate is a maximum. In next order one finds that this position determines the mode's radial location and that the parameter k is such as to maximise the growth rate. However, if the effects of sheared plasma rotation, dΩ/dq, dominate other profile variation, the growth rate is smaller and, instead, involves an average over a period of k. In this paper we consider a generic drift wave model that generates a local eigenvalue having quadratic radial variations of frequency, ω(x), and growth rate, γ(x), and a periodic variation with k. We derive an analytic dispersion relation for the global eigenvalue, ω. Although requiring numerical solution, this shows that there is a continuous evolution between these two limits as dΩ/dq increases, the transition being quite sharp for high n. The transition can be associated with a critical rotation shear, dΩcrit/dq ∼ O(1/n). The detailed character of the results depends on which of the radial variations, ω(x) or γ(x), dominates.