Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-24T02:37:02.207Z Has data issue: false hasContentIssue false
  • English
  • Français

A quasilinear operator retaining magnetic drift effects in tokamak geometry

Part of: Focus on Fusion

Published online by Cambridge University Press:  04 December 2017

Peter J. Catto ,
Jungpyo Lee  and
Abhay K. Ram
Peter J. Catto*
Affiliation:
Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Jungpyo Lee
Affiliation:
Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Abhay K. Ram
Affiliation:
Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The interaction of radio frequency waves with charged particles in a magnetized plasma is usually described by the quasilinear operator that was originally formulated by Kennel & Engelmann (Phys. Fluids, vol. 9, 1966, pp. 2377–2388). In their formulation the plasma is assumed to be homogenous and embedded in a uniform magnetic field. In tokamak plasmas the Kennel–Engelmann operator does not capture the magnetic drifts of the particles that are inherent to the non-uniform magnetic field. To overcome this deficiency a combined drift and gyrokinetic derivation is employed to derive the quasilinear operator for radio frequency heating and current drive in a tokamak with magnetic drifts retained. The derivation requires retaining the magnetic moment to higher order in both the unperturbed and perturbed kinetic equations. The formal prescription for determining the perturbed distribution function then follows a novel procedure in which two non-resonant terms must be evaluated explicitly. The systematic analysis leads to a diffusion equation that is compact and completely expressed in terms of the drift kinetic variables. The equation is not transit averaged, and satisfies the entropy principle, while retaining the full poloidal angle variation without resorting to Fourier decomposition. As the diffusion equation is in physical variables, it can be implemented in any computational code. In the Kennel–Engelmann formalism, the wave–particle resonant delta function is either for the Landau resonance or the Doppler shifted cyclotron resonance. In the combined gyro and drift kinetic approach, a term related to the magnetic drift modifies the resonance condition.

Keywords

fusion plasmaplasma heatingplasma nonlinear phenomena
Type
Research Article
Information
Journal of Plasma Physics , Volume 83 , Issue 6 , December 2017 , 905830611
Copyright
© Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Antonsen, T. M. & Manheimer, W. M. 1978 Electromagnetic wave propagation in inhomogeneous plasmas. Phys. Fluids 21, 2295–2305.Google Scholar
Bajaj, N. K. & Krall, N. A. 1972 Effect of finite $\unicode[STIX]{x1D6FD}$ on the drift cyclotron instanility. Phys. Fluids 15, 657–661.Google Scholar
Becoulet, A., Gambier, D. J & Samain, A. 1991 Hamiltonian theory of the ion cyclotron minority heating dynamics in tokamak plasmas. Phys. Fluids B 3, 137–150.CrossRefGoogle Scholar
Belikov, V. S. & Kolesnichenko, Y. I. 1994 Quasilinear theory for a tokamak plasma in the presence of cyclotron resonance. Plasma Phys. Control. Fusion 36, 1703–1718.Google Scholar
Brambilla, M. 1999 Numerical simulation of ion cyclotron waves in tokamak plasmas. Plasma Phys. Control. Fusion 41, 1–34.CrossRefGoogle Scholar
Calvo, I. & Parra, F. I. 2012 Long-wavelength limit of gyrokinetics in a turbulent tokamak and its intrinsic ambipolarity. Plasma Phys. Control. Fusion 54, 115007.Google Scholar
Catto, P. J. 1978 Linearized gyrokinetics. Plasma Phys. 20, 719–722.Google Scholar
Catto, P. J., Lashmore-Davies, C. N. & Martin, T. J. 1993 A kinetic model of fast wave propagation in the vicinity of the minority ion cyclotron resonance in a toroidal magnetic field. Phys. Fluids B 5, 2909–2921.Google Scholar
Catto, P. J. & Myra, J. R. 1992 A quasilinear description for fast wave minority heating permitting off magnetic axis heating in a tokamak. Phys. Fluids B 4, 187–199.CrossRefGoogle Scholar
Catto, P. J., Myra, J. R. & Russell, D. A. 1994 A quasilinear, Fokker–Planck description of fast wave minority heating permitting off-axis tangency interactions. Phys. Plasmas 1, 52–63.Google Scholar
Catto, P. J., Tang, W. M. & Baldwin, D. E. 1981 Generalized gyrokinetics. Plasma Phys. 23, 639–650.Google Scholar
Chapman, I. T., Graves, J. P., Lennholm, M., Faustin, J., Lerche, E., Johnson, T. & Tholerus, S. 2015 The merits of ion cyclotron resonance heating schemes for sawtooth control in tokamak plasmas. J. Plasma Phys. 81, 365810601.CrossRefGoogle Scholar
Eriksson, L.-G. & Helander, P. 1994 Monte Carlo operators for orbit-averaged Fokker–Planck equations. Phys. Plasmas 1, 308–314.CrossRefGoogle Scholar
Eriksson, L.-G., Mantsinen, M., Borba, D., Fasoli, A., Heeter, R., Sharapov, S., Start, D. F. H., Carlsson, J., Gondhalekar, A., Hellsten, T. et al. 1998 Evidence for a wave-induced particle pinch in the presence of toroidally asymmetric ICRF waves. Phys. Rev. Lett. 81, 1231–1234.Google Scholar
Faulconer, D. W. 1987 Echo-like nonlocal effect due to parallel magnetic inhomogeneity. Plasma Phys. Control. Fusion 29, 433–435.Google Scholar
Helander, P. & Catto, P. J. 2001 Neoclassical current drive by waves with a symmetric spectrum. Phys. Plasmas 8, 1988–1994.CrossRefGoogle Scholar
Hellsten, T., Carlsson, J. & Eriksson, L.-G. 1995 Minority ion cyclotron current drive in tokamaks. Phys. Rev. Lett. 74, 3612–3615.Google Scholar
Jaeger, E. F., Berry, L. A., Dazevedo, E., Batchelor, D. B., Carter, M. D. & White, K. F. 2002 Advances in full-wave modeling of radio frequency heated, multidimensional plasmas. Phys. Plasmas 9, 1873–1881.Google Scholar
Jaeger, E. F., Berry, L. A., Myra, J. R., Batchelor, D. B., Dazevedo, E., Bonoli, P. T., Phillips, C. K., Smithe, D. N., D’IPPOLITO, D. A., Carter, M. D. et al. 2003 Sheared poloidal flow driven by mode conversion in tokamak plasmas. Phys. Rev. Lett. 90, 195001–195004.Google Scholar
Kagan, G. & Catto, P. J. 2008 Arbitrary poloidal gyroradius effects in tokamak pedestals and transport barriers. Plasma Phys. Control. Fusion 50, 085010–085025.Google Scholar
Kapper, G., Kasilov, S. V., Kernbichler, W., Martitsch, A. F., Heyn, M. F., Marushchenko, N. B. & Turkin, Y. 2016 Electron cyclotron current drive simulations for finite collisionality plasmas in Wendelstein 7-X using the full linearized collision model. Phys. Plasmas 23, 112511–112514.Google Scholar
Kaufman, A. N. 1972 Quasilinear diffusion of an axisymmetric toroidal plasma. Phys. Fluids 15, 1063–1069.Google Scholar
Kennel, C. F. & Engelmann, F. 1966 Velocity space diffusion from weak plasma turbulence in a magnetic field. Phys. Fluids 9, 2377–2388.Google Scholar
Lee, X. S., Myra, J. R. & Catto, P. J. 1983 General frequency gyrokinetics. Phys. Fluids 26, 223–229.Google Scholar
Lin, Y., Wukitch, S. J., Bonoli, P. T., Marmar, E., Mossessian, E. D., Nelson-Melby, E., Phillips, P., Porkolab, M., Schilling, G., Wolfe, S. et al. 2003 Ion cyclotron range of frequencies mode conversion electron heating in deuterium–hydrogen plasmas in the Alcator C-Mod tokamak. Plasma Phys. Control. Fusion 45, 1013–1026.Google Scholar
Mantsinen, M. J., Ingesson, L. C., Johnson, T., Kiptily, V. G., Mayoral, M.-L., Sharapov, S., Alper, B., Bertalot, L., Conroy, S., Eriksson, L.-G. et al. 2002 Controlling the profile of ion-cyclotron-resonant ions in JET with the wave-induced pinch effect. Phys. Rev. Lett. 89, 115004.Google Scholar
Parra, F. I. & Calvo, I. 2011 Phase-space Lagrangian derivation of electrostatic gyrokinetics in general geometry. Plasma Phys. Control. Fusion 53, 045001, 49 pp.Google Scholar
Parra, F. I. & Catto, P. J. 2008 Limitations of gyrokinetics on transport time scales. Plasma Phys. Control. Fusion 50, 065014, 23 pp.Google Scholar
Perkins, F. W. 1977 Heating tokamaks via the ion-cyclotron and ion–ion hybrid resonances. Nucl. Fusion 17, 1197–1224.Google Scholar
Smithe, D., Colestock, P., Kammash, T. & Kashuba, R. 1988 Effect of parallel magnetic field gradients on absorption and mode conversion in the ion-cyclotron range of frequencies. Phys. Rev. Lett. 60, 801–804.Google Scholar
Wright, J. C., Bonoli, P. T., Brambilla, M., Meo, F., Dazevedo, E., Batchelor, D. B., Jaeger, E. F., Berry, L. A., Phillips, C. K. & Pletzer, A. 2004 Full wave simulations of fast wave mode conversion and lower hybrid wave propagation in tokamaks. Phys. Plasmas 11, 2473–2479.Google Scholar