Gyrokinetic Simulation of Magnetic Compressional Modes in General Geometry

Abstract

A method for gyrokinetic simulation of low frequency (lower than the cyclotron frequency) magnetic compressional modes in general geometry is presented. The gyrokinetic-Maxwell system of equations is expressed fully in terms of the compressional component of the magnetic perturbation, δB∥, with finite Larmor radius effects. This introduces a “gyro-surface” averaging of δB∥ in the gyrocenter equations of motion, and similarly in the perpendicular Ampere’s law, which takes the form of the perpendicular force balance equation. The resulting system can be numerically implemented by representing the gyro-surface averaging by a discrete sum in the configuration space. For the typical wavelength of interest (on the order of the gyroradius), the gyro-surface averaging can be reduced to averaging along an effective gyro-orbit. The phase space integration in the force balance equation can be approximated by summing over carefully chosen samples in the magnetic moment coordinate, allowing for an efficient numerical implementation.

Gyrokinetic particle simulation of drift-compressional modes in dipole geometry

“…4-point averaging was next implemented by determining Dv, Dw as well as Df from Eq. (15). The frequency in this case was 0:044X i , same as with 2-point averaging, and as in the drift-kinetic limit.…”

“…II B, Dw ¼ Dv ¼ 0 was first imposed in Eq. (15), which amounts to explicitly enforcing the k w =k f ! 0 limit, and compared to results obtained using Bessel functions.…”

“…(10) may be approximated by carefully choosing samples in the l-coordinate and their corresponding weights. 15,21 C. Flux coordinates for dipole geometry It is convenient to use the flux coordinates ðv; w; fÞ, in which the field lines are straight. The flux coordinates are defined by B 0 ¼ rw  rf, where w and f represent two directions perpendicular to the equilibrium magnetic field B 0 .…”

“…(B8)-(B10), since in particle simulations the gyro-orbit and gyro-surface averages are performed explicitly in configuration space by taking averages of the field quantities along the gyro-orbit or the surface enclosed by it, respectively. 15,20 The Poisson's equation (B8), the parallel Ampère's law (B9), and the low frequency perpendicular force balance equation (B10) may be solved in configuration space by combining the methods described in Refs. 15, 20, 21. For purely compressional perturbations, the use of v k formulation was motivated by the fact that unlike a formulation which uses energy as one of its coordinates, it does not contain @ t in the equations of motion.…”

Variational approach to low-frequency kinetic-MHD in the current coupling scheme

“…Note that the variable λ may be interpreted as a normalized radial coordinate on the disc swept out by rotating a particle's gyroradius vector. The equations defining our model are therefore (46)(47)(48)(49)(50)(51)(52) with n h , J h , a gy , and u gy given by (66)(67)(68)(69). It is not difficult to verify that this model agrees with the BDC model when the background magnetic field is uniform.…”