Gyrokinetic ordering and turbulent transport

A representative electrostatic ion-scale ordering uses gyroradius divided by equilibrium gradient length as a small parameter. Fluctuation frequency relative to cyclotron frequency, parallel relative to perpendicular wavenumber, and relative distribution perturbation are comparably small. The perpendicular wavelength can remain comparable to the gyroradius. This combination permits rapid gyromotion to be averaged without treating the fluctuating field as uniform across every orbit. Other gyrokinetic orderings exist, especially for strong flows, electromagnetic fluctuations, or edge regions.

Small frequency alone is insufficient. A fluctuation with a parallel wavelength as short as its perpendicular wavelength, an order-unity distribution perturbation, or an equilibrium varying across one gyroradius can violate the stated expansion. Near sharp boundaries, magnetic nulls, cyclotron resonances, or strong separatrix structures, the appropriate reduction may change. An ordering is an asymptotic hierarchy that organizes retained terms and errors; it is not a fitted label assigned after a simulation produces plausible pictures.

The reduced distribution depends on three gyrocenter position coordinates, parallel velocity, and magnetic moment, giving five phase-space dimensions. Fast gyrophase has been removed as an independent coordinate. Parallel streaming, magnetic trapping, drift motion, and resonant velocity dependence can remain. A gyrokinetic model is therefore not merely a density-and-pressure fluid closure. It can retain Landau phase mixing while filtering the cyclotron oscillations that violate its low-frequency assumptions.

The fluctuating potential acting on a ring is gyroaveraged, and a perpendicular Fourier component receives a Bessel factor depending on the particle's orbit radius. Since perpendicular wavenumber times gyroradius is order one, replacing that factor by unity is not generally allowed. Field equations must retain the corresponding polarization response to maintain consistent charge and energy accounting. Guiding-center reduction and fluctuation gyroaveraging serve different purposes, and a numerical implementation must specify both its geometry and its reduced field model.

A fluctuating radial velocity with zero mean can still carry a net particle flux when density fluctuations are correlated with it. The corresponding energy flux weights the distribution by particle energy; conductive heat additionally requires a stated subtraction of mean convective enthalpy. Fluctuation amplitudes alone cannot determine either flux because their relative phases matter. Two large signals in quadrature can give zero average particle flux, while smaller correlated signals can produce a finite one.

Mean density and temperature profiles evolve on a slower transport timescale than the turbulence in a multiscale formulation. Their gradients supply fluctuation free energy, which can move through spatial and velocity scales before collisions dissipate it. Gyrokinetic free-energy conservation is a valuable numerical audit, but conservation alone does not prove a turbulence prediction converged. Check spatial and velocity resolution, domain, boundary conditions, collision model, field normalization, and ordering validity. Saturated transport depends on nonlinear correlations, not solely on a linear growth rate.