Multiscale gyrokinetics for rotating tokamak plasmas: fluctuations, transport and energy flows

Abstract

This paper presents a complete theoretical framework for studying turbulence and transport in rapidly rotating tokamak plasmas. The fundamental scale separations present in plasma turbulence are codified as an asymptotic expansion in the ratio ε = ρi/α of the gyroradius to the equilibrium scale length. Proceeding order by order in this expansion, a set of coupled multiscale equations is developed. They describe an instantaneous equilibrium, the fluctuations driven by gradients in the equilibrium quantities, and the transport-timescale evolution of mean profiles of these quantities driven by the interplay between the equilibrium and the fluctuations. The equilibrium distribution functions are local Maxwellians with each flux surface rotating toroidally as a rigid body. The magnetic equilibrium is obtained from the generalized Grad-Shafranov equation for a rotating plasma, determining the magnetic flux function from the mean pressure and velocity profiles of the plasma. The slow (resistive-timescale) evolution of the magnetic field is given by an evolution equation for the safety factor q. Large-scale deviations of the distribution function from a Maxwellian are given by neoclassical theory. The fluctuations are determined by the 'high-flow' gyrokinetic equation, from which we derive the governing principle for gyrokinetic turbulence in tokamaks: the conservation and local (in space) cascade of the free energy of the fluctuations (i.e. there is no turbulence spreading). Transport equations for the evolution of the mean density, temperature and flow velocity profiles are derived. These transport equations show how the neoclassical and fluctuating corrections to the equilibrium Maxwellian act back upon the mean profiles through fluxes and heating. The energy and entropy conservation laws for the mean profiles are derived from the transport equations. Total energy, thermal, kinetic and magnetic, is conserved and there is no net turbulent heating. Entropy is produced by the action of fluxes flattening gradients, Ohmic heating and the equilibration of interspecies temperature differences. This equilibration is found to include both turbulent and collisional contributions. Finally, this framework is condensed, in the low-Mach-number limit, to a more concise set of equations suitable for numerical implementation.

Modern gyrokinetic formulation of collisional and turbulent transport in toroidally rotating plasmas

“…(166). Besides, consistency between the conventional and present formulations of transport theories is clarified further by showing that classical, neoclassical, and turbulent transport fluxes of particles, heat, and toroidal momentum separately defined in the previous works (Braginskii 1965;Hinton and Wong 1985;Catto 1987;Sugama and Horton 1997b;Artun and Tang 1994;Sugama and Horton 1998;Abel et al 2013) are all included in the particle, energy, and toroidal momentum balance equations which are derived in Sects. 4.3, 4.5, and 4.6, respectively.…”

“…5.3, the turbulent part of Eq. (87) coincides with Poisson's equation used in the conventional recursive formulation of the gyrokinetic turbulence theory for toroidally rotating plasmas (Sugama and Horton 1998;Abel et al 2013). …”

“…Historically, using the assumptions mentioned above and classical perturbation methods such as recursive techniques, drift kinetic and gyrokinetic equations for perturbed distribution functions (df ) in the case of the low-flow ordering were individually derived as governing equations for neoclassical and turbulent transport processes, respectively (Hazeltine and Meiss 1992;Rutherford and Frieman 1968;Taylor and Hastie 1968;Antonsen and Lane 1980;Catto et al 1981;Frieman and Chen 1982). In the same way, the df drift kinetic and gyrokinetic equations in the high-flow ordering are derived for toroidally rotating plasmas (Hinton and Wong 1985;Catto 1987;Sugama and Horton 1997b;Artun and Tang 1994;Sugama and Horton 1998) and the derivations of these equations based on the classical methods are comprehensively reviewed by Abel et al (2013) On the other hand, the modern gyrokinetic equations derived from the Lie-transform techniques (Brizard and Hahm 2007;Hahm 1988;Brizard 1989Brizard , 1995Hahm 1996) govern behaviors of the full distribution function (full-F), and if the collision term is included, they should, in principle, simultaneously describe collisional and turbulent processes. Actually, in Sects.…”

“…Here, v Ti is the ion thermal velocity and d $ q Ti =L ( 1 represents the ordering parameter defined by the ratio of the ion thermal gyroradius q Ti to the background gradient scale length L. On the other hand, under the high-flow ordering V 0 ¼ Oðv Ti Þ (Hinton and Wong 1985;Catto 1987;Sugama andHorton 1997a, b, 1998;Artun and Tang 1994;Brizard 1995;Hahm 1996;Miyato 2009;Abel et al 2013), the toroidal momentum transport equation which determines the background radial electric field profile can be derived with the same-order accuracy as the particle and energy transport equations. In this paper, we present a novel formulation of collisional and turbulent transport in toroidal plasmas under the high-flow ordering (Sugama et al 2017) by generalizing the previous study to derive governing equations for background and turbulent electromagnetic fields and gyrocenter distribution functions which satisfy conservation laws for particles, energy, and toroidal momentum.…”

“…In the future magnetic confinement fusion devices such as ITER (Horton and Benkadda 2015), the toroidal flow velocity V 0 is expected to be lower than v Ti . We expected that, in ITER, the Mach number M V 0 =v Ti $ 0:05 while that number is still much larger than the normalized gyroradius d $ q Ti =a $ 2 Â 10 À3 [see, for example, Table 1 in Abel et al (2013)]. …”