Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks

Abstract

The nonlinear gyrokinetic Vlasov equation is derived for an arbitrary magnetized plasma in a local reference frame moving with the nonuniform equilibrium fluid velocity u(r). The derivation of the guiding-center and gyrocenter Hamilton equations, which appear as the characteristics of the gyrokinetic Vlasov equation, is based on the use of Lie-transform perturbation techniques. Although a general form for u is initially used, attention is later focused on an incompressible toroidal equilibrium flow when considering axisymmetric tokamak geometry.

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

“…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.…”

“…To avoid a secular deviation of the particle position from the gyrocenter in a long time gyrokinetic simulation, Wang and Hahm (2010) considered the correction due to the fluctuating E Â B velocity in the definition of the gyrocenter position and included the polarization drift in the gyrocenter equations of motion, which are not retained in this work either. The gyrocenter Hamiltonian for describing turbulent transport in toroidally rotating plasmas first appears in Brizard (1995) …”

“…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.…”

Monte Carlo method and High Performance Computing for solving Fokker–Planck equation of minority plasma particles

“…His work was subsequently extended on many occasions. We encourage the reader especially to the paper by Brizard (1995), and to the paper by Cary & Brizard (2009) (and the references therein) where also the history of the guiding-center theory is outlined in detail. In this contribution, we are mostly following these aforementioned works.…”

On Kinetic Resistive Wall Mode Theory with Sheared Rotation

“…The previous formulation employs the Lagrangian without rotation. In this study, we reformulate δW k by starting from the guiding-center Lagrangian with sheared equilibrium rotation [5] to investigate how the formulation is affected by including the rotational modification to the Lagrangian. The remainder of this paper is organized as follows.…”