Abstract
We reanalyse an arbitrary-wavelength gyrokinetic formalism [A. M. Dimits, Phys. Plasmas 17, 055901 (2010)], which orders only the vorticity to be small and allows strong, time-varying flows on medium and long wavelengths. We obtain a simpler gyrocentre Lagrangian up to second order. In addition, the gyrokinetic Poisson equation, derived either via variation of the system Lagrangian or explicit density calculation, is consistent with that of the weak-flow gyrokinetic formalism [T. S. Hahm, Phys. Fluids 31, 2670 (1988)] at all wavelengths in the weak flow limit. The reanalysed formalism has been numerically implemented as a particle-in-cell code. An iterative scheme is described which allows for numerical solution of this system of equations, given the implicit dependence of the Euler-Lagrange equations on the time derivative of the potential.
Highlights
The weak-flow gyrokinetic formalism1,2 uses a gyrokinetic ordering parameter$ x=X $ vEÂB=vt ( 1; (1)with x a characteristic frequency, X the gyrofrequency, vEÂB the E Â B drift speed, and vt the typical thermal speed.The ordering (1) may be poorly satisfied in the core and edge of tokamak plasmas because of either large overall rotation or relatively strong flows in the pedestal
This is a maximal ordering in the sense that a larger vorticity on any scale would lead to breaking of the magnetic moment invariance, as nonlinear frequencies are comparable to the vorticity
It is a ponderomotive term that typically results from the appearance of a u2 term in the Lagrangian;10 the analogue of this term is present in Ref. 3
Summary
With x a characteristic frequency, X the gyrofrequency, vEÂB the E Â B drift speed, and vt the typical thermal speed. Where v0EÂB is the characteristic magnitude of the spatial derivatives of the E Â B drift velocity This is a maximal ordering in the sense that a larger vorticity on any scale would lead to breaking of the magnetic moment invariance, as nonlinear frequencies are comparable to the vorticity. In the weak-flow limit, the gyrokinetic Poisson equation of Ref. 5 disagrees with that of the weak-flow gyrokinetic formalism at wavelengths comparable to the gyroradius. We rederive this theory and explain some minor but important departures from the derivation of the weak-flow theory. We obtain a Poisson equation, via both a variational and direct method, that, in the weak-flow limit, agrees with the weak-flow gyrokinetic Poisson equation at all wavelengths
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