A reanalysis of a strong-flow gyrokinetic formalism

We reanalyse an arbitrary-wavelength gyrokinetic formalism [A. M. Dimits, Phys. Plasmas $\bf17$, 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 $\bf31$, 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.


Introduction
Standard gyrokinetic theory [1] uses a gyrokinetic ordering parameter where u is a general E ×B drift speed and v t is the thermal speed. This ordering cannot be satisfied throughout the entirety of a modern tokamak plasma due to the large gradients and flows present in core transport barriers and edge and scrape-off-layer regions. One theory that addresses this problem for electrostatic perturbations in a slab uniform equilibrium magnetic field is that of Ref. [2]. However, in the weak-flow limit, this theory's Poisson equation disagrees with that of standard theory at short wavelengths. In our reanalysis, which focuses on the nuances of this theory compared to standard theory, we obtain a Poisson equation that is consistent with standard theory in the weak-flow limit.
2 Guiding-centre Lagrangian The noncanonical particle Poincaré-Cartan one-form for electrostatic perturbations in a slab uniform equilibrium magnetic field is We transform to a frame moving with a velocity u(x, v, t) such that and noting that the gyroangle is defined with the opposite sign in Ref. [2], which can be expanded as and rearranged as where we have used thatb · u = 0 and φ ≡ 1 2π dθφ(R + ρ), Using integration by parts, Rewriting, Using the vector identity where the last equality follows from Ref. [1].

Gyrocentre Lagrangian
Using the ordering[2] As in standard theory, although the unperturbed Lagrangian γ 0 contains terms of order ǫ −1 and ǫ 0 , computation of the Lagrange matrix components from both order ǫ −1 and ǫ 0 parts of γ 0 yields terms that are of order ǫ 0 . The requirement is equivalent to a restriction on the choice of u given by Two choices that satisfy this condition are As in Ref. [2], u must also satisfy the condition This yields the non-zero Lagrange matrix components Defining and, refactorising, Solving for g 1 in terms of S 1 such that Γ 1 is only composed of a time component, yields the non-zero g 1 components Rearranging where the last equality follows from Ref. [2] and gives Substituting,

Euler-Lagrange equations
The Euler-Lagrange equations, Taking the cross product withb, Using the vector identitŷ Upon substituting and using that Using thatb and that Instead of taking the cross product withb, projecting onto Ω * , Using that and Equation (1), Substituting,μ = 0.
For i = µ, ω µRṘ + ω µθθ = ω tµ . Substituting, The parallel equation of motion has an additional term. Physically, it is a ponderomotive term that typically results from the appearance of a u 2 term in the Lagrangian[3]. Its analogue is present in Ref. [4].

Poisson equation
Defining particle, guiding-centre and gyrocentre coordinates and distribution functions as {z, f }, {Z, F } and {Z,F }, respectively, Note here that the Jacobian to be evaluated is the Jacobian of the transformation from particle to guiding-centre coordinates, not that from particle to gyrocentre coordinates. The Jacobian of either transformation can be written as the square root of the determinant of the relevant Lagrange matrix, ω ij , which is only a function of the symplectic part of the Lagrangian. One implication is that, for standard gyrokinetic theory [1], these two Jacobians are identical, but they are not the same for this extended theory because the symplectic part of the Lagrangian includes a non-gyrotropic term. The Jacobian Alternatively, and, here, the Jacobian Separating the standard and non-standard parts of n(x) as n(x) = n s (x) + n n (x) and takingF (Z) to be a uniform Maxwellian equilibriumF 0 (U, µ), The equations of motion suggest that Taking