Damping of toroidal ion temperature gradient modes

Abstract

The temporal evolution of linear toroidal ion temperature gradient ͑ITG͒ modes is studied based on a kinetic integral equation including an initial condition. It is shown how to evaluate the analytic continuation of the integral kernel as a function of a complex-valued frequency, which is useful for investigating the asymptotic damping behavior of the ITG mode. In the presence of the toroidal magnetic drift, the potential perturbation consists of normal modes and a continuum mode, which correspond to contributions from poles and from an integral along a branch cut, respectively, of the Laplace-transformed potential function of the frequency. The normal modes have exponential time dependence while the continuum mode, which has a ballooning structure, shows a power law decay ϰt Ϫ2 , where t is the time variable. Therefore, the continuum mode dominantly describes the long-time asymptotic behavior of the perturbation for the stable system. By performing proper analytic continuation for the dispersion relation, the normal modes' growth rate, real frequency, and eigenfunction are numerically obtained for both stable and unstable cases.

Oscillations of zonal flows in stellarators

“…A similar algebraic decay has earlier been established for ion-temperaturegradient-driven modes [9,10].…”

Study of electromagnetic microinstabilities in helical systems with the stellarator expansion method

“…͑22͒-͑24͒ are numerically solved to obtain linear growth rates, real frequencies, and mode structures of electromagnetic microinstabilities in a helical system. Employing procedures by Sugama 13 for proper analytic continuation of the dispersion relation in the complex frequency plane, our numerical code can calculate both positive and negative growth rates, which is useful for accurately determining the critical condition for the marginal stability.…”

“…Here, in order to fully take account of the finite-gyroradius effect on the electromagnetic mode, we use the kinetic integral eigenmode equations similar to those by Dong et al for tokamaks, 10 which are derived from the ion gyrokinetic equation, 11,12 the quasineutrality condition, Ampère's law, and the massless electron approximation. Our numerical solution to the kinetic integral eigenmode equations utilizes procedures by Sugama 13 for proper analytic continuation of the dispersion relation in the complex frequency plane, by which we can calculate both positive and negative growth rates so as to accurately determine the critical condition for the marginal stability.…”

Toroidal ion temperature gradient driven instability in plasma with shear flow

“…6 The other method is to solve the initial-value problem by use of a Laplace transform in time. 7 The important aspect of the initial-value problem is that, owing to the existence of continuous spectrum of normal modes some disturbances may grow or decay as the nonmodal ones with a power-like dependencies of amplitudes with time ͑see, for example, Refs. 7, 8͒.…”