Abstract
Geodesic acoustic modes (GAMs) are a fundamental part of turbulence and zonal-flow dynamics in tokamaks. They exhibit simple yet non-trivial dispersive and dissipative properties. In linear numerical simulations, they are often initialized in the form of (e.g., Gaussian) packets that evolve in time. Depending on the parameters, dispersion and damping can act on comparable time scales during the GAM evolution. Wigner-function methods developed in the frame of non-Hermitian quantum mechanics are shown to be applicable to damped geodesic oscillations. In this approach, the standard approximation of “weak damping,” often introduced for the treatment of plasma waves, is not needed. The method requires that the properties of the plasma do not vary significantly across the width of the packet (i.e., in the radial direction), so that a paraxial expansion of the underlying equations around the center of the packet can be applied. For a quadratic Hamiltonian, the equations for the Wigner function governing the packet in the paraxial limit are shown to be equivalent to the equations of paraxial WKB theory (usually applied to the description of high-frequency wave beams in plasmas), with the real Hamiltonian replaced by the corresponding complex one. Analytic solutions are derived in particular cases and shown to agree with the results of global gyrokinetic simulations.
Highlights
Geodesic Acoustic Modes (GAMs) are axisymmetric plasma oscillations originating from the fact that zonal E Â B flows are not divergence-free in tokamak geometry
In the case of homogeneous plasmas, simple analytic solutions can be obtained for the time evolution of the Geodesic acoustic modes (GAMs) electric field, which highlight in particular the effect of selective dissipation of higher wavenumbers on the shape of a GAM packet
It is noted that the equivalence between the Wigner-function approach and the complex paraxial WKB (pWKB) method has been proved in Ref. 54 only for the case of Hamiltonians that are quadratic in the position and momentum variables
Summary
Geodesic Acoustic Modes (GAMs) are axisymmetric plasma oscillations originating from the fact that zonal E Â B flows are not divergence-free in tokamak geometry. Their compression leads to an oscillation, first described by Winsor et al. using the equations of magnetohydrodynamics, which has a typical frequency of the order of the sound speed divided by the major radius of the tokamak Since this seminal work, the interest in GAMs increased considerably, in particular, in the context of the dynamics of turbulence and zonal flows, as reviewed in Ref. 6. 40–42 and where an approach very close to that of this paper is adopted In this respect, it should be stressed that low-frequency eigenmodes in tokamaks (which include non-Hermitian dispersion functions of interest in this paper, see below) can be described in terms of propagating wave packets as well, see Ref. 47 and references therein, a method that can be applied when the group-velocity. Appendix A provides a summary of some basic definitions related to the Wigner–Weyl approach, while the derivation of the paraxial equations is reviewed in Appendix B and the relation between the Gaussian-envelope descriptions in both methods is briefly discussed in Appendix C
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