Quasilinear model for energetic particle diffusion in radial and velocity space

Abstract

A quasilinear model for passive energetic particle (EP) turbulent diffusion in radial and velocity space is fitted and tested against nonlinear gyrokinetic tokamak simulations with the GYRO code [J. Candy and R.E. Waltz, Phys. Rev. Lett. 91, 045001 (2003)]. Off diagonal elements of a symmetric positive definite € 2 × 2 EP diffusion matrix account for fluxes up radial (energy) gradients driven by energy (radial) gradients of the EP velocity space distribution function. The quasilinear ratio kernel of the model is provided by a simple analytic formula for the EP radial and velocity space EP diffusivity relative to radial thermal ion energy diffusivity at each linear mode of the turbulence driven by the thermal plasma. The TGLF [G.M. Staebler, J.E. Kinsey and R.E. Waltz, Phys. Plasmas 14, 0055909 (2007), ibid. 15, 0055908 (2008)] tokamak transport model provides the linear mode frequency and growth rates to the kernel as well as the nonlinear spectral weight for each mode.

Global anomalous transport of ICRH- and NBI-heated fast ions

“…(6) and/or by including the additional term in the numerator of Eq. (5). Furthermore, we note that previous work done on fast ion dilution [17,4] is also consistent with this model.…”

“…A proof of Onsager symmetry for quasilinear radial transport of a Maxwellian species has already been shown [19], but this is not the case for nonlinear turbulence [20]. In this appendix, we'll show that this symmetry holds for quasilinear transport even in r-v phase space, as has been observed here and elsewhere [5]. This is more than coincidence, and here we show that this is rigorous as long as the magnetic drift velocity is dominant over the nonlinear drift velocity in the gyrokinetic equation.…”

“…Note that the experimental scaling of thermal diffusivity with respect to temperature ratio χ i ∝ τ −3 [16] is remarkably consistent with our results when applying Eq. (5). We expect this model to be valid to the extent that the following assumptions hold: the turbulence is dominated by electrostatic dynamics; adiabatic electrons and fast ions remain a good approximation; the temperature ratio continues to obey the scaling of Ref.…”

Energetic ion transport by microturbulence is insignificant in tokamaks

“…One method uses the newly developed code DEP. 74 DEP is a quasilinear model in which the ratio of energetic ion turbulent diffusivity matrix elements to v i is calculated in radial and velocity space. Only matrix elements for radial diffusion driven by radial gradients are used, i.e., velocity gradients are ignored.…”

Development and validation of a critical gradient energetic particle driven Alfven eigenmode transport model for DIII-D tilted neutral beam experiments

“…Going forward, the treatment of EP transport can be improved with the use of kinetic transport codes and the use of velocity space dependent effective diffusivity models for ITG/TEM induced EP transport; e.g. the energy dependent version of the Angioni model [7,8], the DEP model [18], or the Pueschel model [19]. Computationally intensive nonlinear GYRO simulations of this DIII-D discharge with low-n AE modes embedded in high-n ITG/TEM turbulence in the presence of equilibrium ExB shear should provide a more accurate recipe for the critical gradient as well as some insight into the development of a velocity space dependent effective diffusivity for stiff critical gradient transport of EPs induced by AEs.…”