Complex-Hamiltonian paraxial description of damped geodesic acoustic modes

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.