Gyrokinetic Theory

During the last grant period, we have explored the radial propagation of small-scale radial electric field perturbations ("zonal flows"). These are known to be generated nonlinearly by ITG turbulence, and to strongly regulate the turbulence. Radial propagation would couple different regions of the plasma in a way not previously studied, and could affect perturbative transport and global transport scaling. The group velocity, which measures the rate of radial propagation, is determined by the frequency dependence of the polarizability. The Rosenbluth-Hinton polarization calculation [Rosenbluth 98,Hinton 99] gives no frequency dependence, and hence zero group velocity. We therefore generalized the Rosenbluth-Hinton calculation to include terms previously neglected. We included diamagnetic and electromagnetic terms, but these resulted in only small corrections to the polarization and again gave no frequency dependence. Thus, the group velocity still turned out to be zero, indicating that zonal flows simply don't propagate radially. What we have learned from GYRO simulations, however, is that the ( ) correlation lengths are quite long (comparable to the plasma minor radius) and, even though they do not propagate, may be the mechanism for nonlocal phenomena (local transport flows driven by strong gradients at a large distance). An Ohm's law for tokamak plasmas was derived, which includes the effect of electromagnetic turbulence as well as neoclassical conductivity and bootstrap current [Hinton 04]. Expressions for the turbulence-driven current were derived analytically and then evaluated using turbulence results from the GYRO code. We found that the driven current density has positive peaks on low-order rational surfaces, whose widths are a few ion gyroradii. This could have important effects on tearing-mode stability. For the DIII-D L-mode plasma which we studied, these peak values are comparable to or larger than the Ohmic current density. Because of the errors inherent in measurements of current density, these predicted peak current density values may not be observable. The mechanism for the turbulence-driven current is new, but was shown to be related to the alpha-dynamo current of MHD dynamo theory, using results from a massless electron model [Hinton 03]. We continued our collaboration with W.X. Wang, now at PPPL, on nonlocal neoclassical transport and bootstrap current with finite orbit effects. Recently we found that, near the magnetic axis, the ion heat flux is almost independent of local temperature gradient, and can be larger or smaller than the standard neoclassical heat flux. We also found that bootstrap current is driven by shear in toroidal rotation. It is larger near the magnetic axis than in standard neoclassical theory; it is increased over the small orbit theory by a large temperature gradient but not by a large density gradient.

The conventional gyrokinetic formulation [Antonsen 80,Frieman 82] is a consistent expansion of the full kinetic equation in the small parameter , ordering

and . It is known that may become large in the tokamak edge where steep gradients are observed, and there is a worry that conventional gyrokinetics may break down. Flux-tube codes using the gyrokinetic equations have no radial variation of the equilibrium and exhibit no dependence ( scales out of the equations). Thus, let us speak of the flux-tube limit as the theory. GYRO includes finite- corrections to the latter theory in an implicit way through radial profiles, but does not contain any explicit finite- terms (for example, the parallel nonlinearity). The distinction can be seen by example through a breakdown of the drift effect:

Although the middle term is formally , the variation of the profile gradients over the width of the coherent modes induces a relatively large implicit -dependence: . Nevertheless, since a tractable formulation for complete gyrokinetics does not exist, it makes little sense to include the explicit- terms piecemeal. Moreover, even if a complete theory of the explicit- correction to gyrokinetics was developed and it produced results which were significantly different than the existing theory, it would almost certainly mean that the expansion is diverging. This would not be an improvement. As we see it, the essential problem is to estimate the limitations of the standard gyrokinetic expansion. We propose to directly simulate the full 6-D Vlasov equation and compare it in a precise way to a 5-D GYRO simulation, starting with a simple problem like adiabatic-electron ITG modes in flux-tube geometry. For the Vlasov equation, flux-tube simulations will include finite- effects. At some sufficiently small we expect to recover the GYRO simulations exactly, thus confirming the fidelity of our Vlasov solution. At a larger , and perhaps steeper gradient, we expect to see a deviation from GYRO. It is important to determine if this deviation is significant for conditions in real edge plasma (or in core NSTX plasmas for which is more than larger than in DIII-D). If the deviation is significant, then any theory based on the gyrokinetic expansion (to any order) will be divergent and thus unreliable. The hope is that we will not see a significant breakdown of conventional gyrokinetics for physically realistic fusion plasmas. We believe there are both numerical advantages and disadvantages to solving the 6-D Vlasov equation numerically. While the dimensionality is higher and the eventual code will certainly be more expensive to run than GYRO, in solving the Vlasov equation we actually avoid the major annoyances of gyrokinetics: the nonlocal operators connected with gyroaveraging, and the complicated mapping (or "pullback transformation" [Qin 04]) from gyrocenter to laboratory coordinates. By considering adiabatic electrons, we avoid pathological electron modes (electrostatic Alfvén waves, for example) and expect no phenomena faster than ion cycloton waves. It is likely that a pseudo-spectral semi-implicit scheme can be used to step over the cyclotron motion when required and do a "numerical gyroaveraging" in the stiff limit , in much the same way that the current GYRO can step over the electron motion (linearly) and do numerical bounce-averaging.