Finite-Larmor-radius-induced global geodesic acoustic modes—a two-fluid model

Abstract

A linear theory of ion finite-Larmor-radius (FLR)-induced global geodesic acoustic modes (GAMs) based on the electrostatic two-fluid model is developed, in which modest ion FLR effects are encompassed through polarization drifts. The radial differential equation obtained for the eigenmode is a type of generalized Schrödinger equation, in which the eigenfrequency is mixed with the potential term in a complex manner. By numerically solving this equation as a genuine boundary value problem, it is found that for typical equilibrium profiles, a series of global GAMs exist, with relatively higher frequencies that intersect with the GAM continuum. Those with frequencies in the intermediate range are forbidden by the singularity of the local dispersion function. When we restrict the poloidal θ harmonics up to m = ±1, we successfully interpret the bound-state profiles of global GAMs discovered in the simulation performed by Miyato et al (2006 Plasma Phys. Control. Fusion 48 A335). We also point out the limitation of their model.

A gyro-fluid model of global geodesic acoustic modes supported by finite ion Larmor radius

“…This model has its own merits: (1) in principle, it is capable of collecting the types of ion FLR effects as many as the gyro-kinetic formalism [24]; (2) while slightly complicated than the usual two-fluid models, it still bears the same intuitive physical meanings as those two-fluid models; (3) it dramatically saves the simulation time as the ion gyration has been 'frozen' by the gyro-phase-average; (4) it can yield an analytical expression of the local dispersion function that is tractable yet accurate enough. In fact, we will see that the local GAM dispersion function out of this gyro-fluid model is almost the same as that from the gyrokinetic approach, while is more accurate than the MHD [6] or two-fluid [23] counterparts, particularly in that the gyro-fluid model takes into account the anisotropy of the ion pressure perturbation, thus correctly recovers the kinetic ion adiabatic index 7/4. Similar to the two-fluid formulation [23], the other minor advantage of the gyro-fluid formulation lies in that, in deriving the radial differential equation of the eigen-mode, the necessary radial derivatives associated with FLR of ion are retained, so the cyclic replacements ∂/∂r → ik r → ∂/∂r are avoided, hence the essence of the eigenfunction becomes unambiguous.…”

“…Itoh [21] and Sasaki [22] et al came up with the eigendifferential equations for the GGAMs supported by ion Larmor radius ρ i , that essentially take the form Cρ 2 i d 2 φ(r)/dr 2 + (ω − ω GAM ) φ(r) = 0, where C is a constant, and ω G is the local GAM dispersion relation obtained from some fluid or gyrokinetic models. However, as have demonstrated in [23] based on a two-fluid model and in [24] on a gyro-kinetic model, although the structures of the eigen-mode equations in these three models are roughly similar, the factor C is far more complicated than a constant as it is a function of the eigenfrequency ω and the equilibrium profiles, i.e. C = C(ω, r), whose concrete form depends on the plasma model adopted.…”

“…Although a series of GGAMs have been obtained by solving a boundary value problem in [23], several sorts of ion FLR effects other than the polarization and diamagnetic drifts are not captured due to the limitation of the two-fluid model, as already noticed by the authors. One serious issue of that two-fluid model is that the issue of singularity may be exaggerated as a consequence of the incomplete ion FLR effects.…”

“…C = C(ω, r), whose concrete form depends on the plasma model adopted. Moreover, in [23] it has been shown that the function C(ω, r) may become vanishing at some radial location for a wide range of constant ω values, which makes (ω − ω GAM )/C(ω, r) singular. Hence, the models [21,22] are oversimplified.…”

“…In order to leave a non-singular frequency gap that can accommodate the allowable eigen-frequencies, the equilibrium profiles have to be tuned carefully, e.g., the edge ion temperature has to be taken to be relatively high. Thus, extending the model [23] to overcome its incompleteness becomes the main purpose of this paper.…”