On ion temperature gradient and parallel velocity shear instabilities

Abstract

The local dispersion relation for waves with frequencies in the range of the diamagnetic frequencies ω^∗_j and parallel wave numbers satisfying the conditions k_||c_s/ω^∗_e ~ 1 and qRk_||»1 has been obtained in the framework of kinetic theory keeping the equilibrium density, temperature, and parallel velocity gradients into account (j is the species index, qR the connection length, and c_s the speed of sound). The analysis applies to the cases where the radial scale of the oscillations is comparable to or smaller than the equilibrium length scale. As the velocity-space integral appearing in the dispersion relation can be calculated only in asymptotic limits, exact instability criteria are obtained by means of the Nyquist diagram. Defining T_i = T_i/T_e, η_i = ∂_rlnT_i/∂_rlnN_i, and ζ = ∂_rU_||,i/c_s∂_rlnN_i, it is found that unstable modes appear for η_i>1+ τ_i(which agrees with the standard ion temperature gradient instability condition η_i>2 if ζ_i = 0) and 0<η_i<1-√1- ζ²/(1+τ_i(the case η_i<0 has not been analyzed), i.e., for ζ² ≥ η_i(2-η_i)(1+τ_i) (which does not agree with the standard parallel velocity shear instability condition |τ|>√2 if η_i = 0). The center of the unstable range is characterized by the relation k_||c_s/ω^∗_e = -ζ/2(1+τ_i) from which it follows that qRk_||»1 is verified if [k _βa_s/2(1+τ_i)]qR∂_rU_||,i/c_s»1 (k β is the wave vector component in the direction of the binormal). The oscillations are not tied, under those conditions, to any particular rational surface; the roles of magnetic shear, trapped electrons, ion gyroradius and torus curvature are moreover negligible. The growth/decay rate of the oscillations has been calculated in the neighborhood of marginal (in)stability; the excitation/damping mechanism is (inverse) ion Landau damping. The wave frequency is a function of position so that localization of a wave packet results from a competition between linear growth and distortion (wave breaking in smaller eddies). Applications of the theory include the transition from the edge localized mode-free to enhanced D alpha high confinement regime and intermittency.