Extension of geodesic acoustic mode theory to helical systems

Abstract

The present paper extends the theory of geodesic acoustic mode ͑GAM͒ oscillation, which so far has been applied to tokamaks, to helical systems. By using drift kinetic equations for three-dimensional equilibriums, a generalized dispersion relation is obtained including Landau damping. The oscillation frequency is obtained in terms of the squared sum of Fourier components of the magnetic field intensity expressed by means of magnetic flux coordinates. An analytic form of the collisionless damping rate of GAM is obtained by solving the dispersion relation perturbatively. It is found that the GAM frequency is higher in helical systems than in tokamaks and that damping rate is enhanced in multi-helicity magnetic configurations. However, damping rates are predicted to be small if the temperature of electrons is higher than that of ions.

Collisionless damping of zonal flows in helical systems

“…The GAM oscillations are observed experimentally by measuring the radial electric field in both tokamaks and helical systems. 8,9 Drift kinetic evaluations of frequencies and damping rates of the GAM in tokamaks were done by Lebedev et al 10 and by Novakovskii et al 11 Recently, Watari et al 12 derived the dispersion relation for the GAM in helical systems based on the drift kinetic equation although they neglected a part of the electrostatic potential which depends on the poloidal and toroidal angles so that their dispersion relation shows a slight difference from that of Lebedev et al 10 in the collisionless tokamak limit. Also, all the above-mentioned drift kinetic studies assume the radial widths of ion drift orbits to be negligibly smaller than the radial wavelength of the potential and their local drift kinetic models do not include the magnetic drift term of the perturbed distribution function that the gyrokinetic equation does.…”

Geodesic Acoustic Eigenmodes

“…The extension to a more general profile of temperature T (r) is possible. When T e comes closer to T i , the screening owing to the finite-gyroradius effect also appears inJ D,r as was explained in [12][13][14][15], so that the coefficient to k 2 ρ 2 i in Eq. (1) becomes smaller (i.e., the radial wavelength becomes shorter).…”

“…In the limit of T e T i , the relation v th,i /R ω holds for ω ∼ ω G , andJ D,r is dominated by the electron response (v th,i : ion thermal velocity) [14]. Therefore,J D,r is not significantly influenced by the finite gyroradius effect.…”

“…The dispersion relation of GAMs, ω = ω G , is derived by balancing the cross-field currentJ D,r (due to the magnetic field curvature) and the ion polarization currentJ p,r under the imposition of an electrostatic perturbation that has a formφ exp(ikr − iωt) in the leading order [12][13][14][15]. In order to study the radial eigenmode with analytic transparency, we take a simple collisionless limit with T e T i…”

“…where the lowest order finite-gyroradius correction is included (see [12][13][14][15][16] for a more detailed derivation). We consider the case in which the temperature decreases in radius, and choose the radius r 0 where ω 2 = ω 2 G (r 0 ) holds.…”