Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T07:16:23.291Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical resolution of the global eigenvalue problem for the gyrokinetic-waterbag model in toroidal geometry

Part of: Focus on Fusion

Published online by Cambridge University Press:  27 March 2017

D. Coulette Open the ORCID record for D. Coulette [Opens in a new window]  and
N. Besse
D. Coulette*
Affiliation:
Institut de Recherche en Mathématiques Avancées, Université de Strasbourg 7 rue René Descartes, 67084 Strasbourg CEDEX, France
N. Besse
Affiliation:
Laboratoire J.-L. Lagrange, Université Côte d’Azur, Observatoire de la Côte d’Azur, Bd de l’Observatoire, CS 34229 06304 Nice CEDEX 4, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we present two codes for the linear stability analysis of the ion temperature gradient instability in toroidal geometry using a gyrokinetic multi-waterbag model for ion dynamics. The first one solves the linearized ion dynamics as an initial value problem, while the second relies on an asymptotic expansion in the so-called ballooning representation allowing us to build a tractable eigenvalue problem. Results from the two codes are presented and compared for equilibria based on modified Cyclone parameters. A good agreement between both codes is found for a class of equilibria with a narrow extent in perpendicular velocity and for which trapped particle orbits are ignored. The local spectrum computed by the eigenvalue is shown to agree remarkably well with previous Cyclone results when trapped particle orbits are included. Lastly we discuss how the equilibrium building procedure for this type of waterbag model requires particular care when dealing with closed equilibrium contours related to the presence of trapped particle orbits.

Keywords

fusion plasmaplasma instabilitiesplasma simulation
Type
Research Article
Information
Journal of Plasma Physics , Volume 83 , Issue 2 , April 2017 , 905830207
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

Abdoul, P. A., Dickinson, D., Roach, C. M. & Wilson, H. R. 2015 Using a local gyrokinetic code to study global ion temperature gradient modes in tokamaks. Plasma Phys. Control. Fusion 57 (6), 065004.CrossRefGoogle Scholar
Abel, I. G., Plunk, G. G., Wang, E., Barnes, M., Cowley, S. C., Dorland, W. & Schekochihin, A. A. 2013 Multiscale gyrokinetics for rotating tokamak plasmas: fluctuations, transport and energy flows. Rep. Prog. Phys. 76 (11), 116201.Google Scholar
Besse, N. & Bertrand, P. 2009 Gyro-waterbag approach in nonlinear gyrokinetic turbulence. J. Comput. Phys. 3973–3995.Google Scholar
Besse, N. & Coulette, D. 2016 Asymptotic and spectral analysis of the gyrokinetic-waterbag integro-differential operator in toroidal geometry. J. Math. Phys. 57 (8), 081518.Google Scholar
Brizard, A. J. & Hahm, T. S. 2007 Foundations of nonlinear gyrokinetic theory. Rev. Mod. Phys. 79, 421–468.Google Scholar
Candy, J. & Waltz, R. E. 2003 An Eulerian gyrokinetic-Maxwell solver. J. Comput. Phys. 186, 545–581.CrossRefGoogle Scholar
Colombi, S. & Touma, J. 2008 Vlasov–Poisson: the waterbag method revisited. Commun. Nonlinear Sci. Numer. Simul. 13, 46–52.CrossRefGoogle Scholar
Colombi, S. & Touma, J. 2014 Vlasov–Poisson in 1d: waterbags. Mon. Not. R. Astron. Soc. 441, 2414–2432.Google Scholar
Connor, J. W. & Taylor, J. B. 1987 Ballooning modes or fourier modes in a toroidal plasma? Phys. Fluids 30, 3180–3185.CrossRefGoogle Scholar
Coulette, D. & Besse, N. 2013a Multi-waterbag models of ion temperature gradient instability in cylindrical geometry. Phys. Plasmas 20, 052107.Google Scholar
Coulette, D. & Besse, N. 2013b Numerical comparisons of gyrokinetic multi-waterbag models. J. Comput. Phys. 248, 1–32.Google Scholar
Davies, B. 1986 Locating the zeros of an analytic function. J. Comput. Phys. 66 (1), 36–49.Google Scholar
Delves, J. L. & Lyness, J. N. 1967 A numerical method for locating the zeros of an analytic function. Maths Comput. 21, 543–560.Google Scholar
Dif-Praladier, G., Grangirard, V., Sarazin, Y., Garbet, X. & Ghendrih, P. 2008 Defining an equilibrium state in global full- $f$ gyrokinetic models. Commun. Nonlinear Sci. Numer. Simul. 13, 65–71.Google Scholar
Dimits, A. M., Bateman, G., Beer, M. A., Cohen, B. I. & Dorland, W. 2000 Comparisons and physics basis of tokamak transport models and turbulence simulations. Phys. Plasmas 7, 969–983.Google Scholar
Dorland, W., Jenko, F., Kotschenreuther, M. & Rogers, B. N. 2000 Electron temperature gradient turbulence. Phys. Rev. Lett. 85, 5579–5582.CrossRefGoogle ScholarPubMed
Dubin, D. H. E, Krommes, J. A., Obermann, C. & Lee, W. W. 1983 nonlinear gyrokinetic equations. Phys. Fluids 26, 3524–3535.CrossRefGoogle Scholar
Frieman, E. A. & Chen, L. 1982 Nonlinear gyrokinetic equations for lowfrequency electromagnetic waves in general plasma equilibria. Phys. Fluids 25, 502–508.Google Scholar
Grandgirard, V., Brunetti, M., Bertrand, P., Besse, N., Garbet, X., Ghendrih, P., Manfredi, G., Sarazin, Y., Sauter, O., Sonnendrucker, E. et al. 2006a A drift-kinetic semi-Lagrangian 4d code for ion turbulence simulation. J. Comput. Phys. 217 (2), 395–423.CrossRefGoogle Scholar
Grandgirard, V., Sarazin, Y., Angelino, P., Bottino, A., Crouseilles, N., Darmet, G., Dif-Pradalier, G., Garbet, X., Ghendrih, P., Jolliet, S. et al. 2007 Global full- $f$ gyrokinetic simulations of plasma turbulence. Plasma Phys. Control. Fusion 49 (12B), B173–B182.CrossRefGoogle Scholar
Grandgirard, V., Sarazin, Y., Garbet, X., Dif-Pradalier, G., Ghendrih, P., Crouseilles, N., Latu, G., Sonnendrücker, E., Besse, N. & Bertrand, P. 2006b Gysela, a full- $f$ global gyrokinetic semi-Lagrangian code for itg turbulence simulations. AIP Conf. Proc. 871, 100–111.Google Scholar
Grandgirard, V., Sarazin, Y., Garbet, X., Dif-Pradalier, G., Ghendrih, P., Crouseilles, N., Latu, G., Sonnendrücker, E., Besse, N. & Bertrand, P. 2008 Computing itg turbulence with a full- $f$ semi-Lagrangian code. Commun. Nonlinear Sci. 13 (1), 81–87.CrossRefGoogle Scholar
Gravier, E., Klein, R., Morel, P., Besse, N. & Bertrand, P. 2008 Gyrokinetic-waterbag modeling of low-frequency instabilities in a laboratory magnetized plasma column. Phys. Plasmas 15 (12), 122103.Google Scholar
Görler, T., Lapillonne, X., Brunner, S., Dannert, T., Jenko, F., Merz, F. & Told, D. 2011 The global version of the gyrokinetic turbulence code gene. J. Comput. Phys. 230 (18), 7053–7071.Google Scholar
Hahm, T. S., Wang, Lu & Madsen, J. 2009 Fully electromagnetic nonlinear gyrokinetic equations for tokamak edge turbulence. Phys. Plasmas 16, 022305.CrossRefGoogle Scholar
Happel, T., Navarro, A. B., Conway, G. D., Angioni, C., Bernert, M., Dunne, M., Fable, E., Geiger, B., Görler, T., Jenko, F. et al. 2015 Core turbulence behavior moving from ion-temperature-gradient regime towards trapped-electron-mode regime in the ASDEX Upgrade tokamak and comparison with gyrokinetic simulation. Phys. Plasmas 22 (3), 032503.Google Scholar
Hastie, R. J. & Taylor, J. B. 1981 Validity of ballooning representation and mode number dependence of stability. Nucl. Fusion 21 (2), 187–191.Google Scholar
Hazeltine, R. D., Hitchcock, D. A. & Mahajan, S. M. 1981 Uniqueness and inversion of the ballooning representation. Phys. Fluids 24, 180–181.Google Scholar
Hazeltine, R. D. & Meiss, J. D. 2003 Plasma Confinement. Dover.Google Scholar
Hazeltine, R. D. & Newcomb, W. A. 1990 Inversion of the ballooning representation. Phys. Fluids B 2, 7–11.Google Scholar
Howard, N. T., White, A. E., Reinke, M. L., Greenwald, M., Holland, C., Candy, J. & Walk, J. R. 2013 Validation of the gyrokinetic model in ITG and TEM dominated l-mode plasmas. Nucl. Fusion 53 (12), 123011.Google Scholar
Idomura, Y., Ida, M., Kano, T., Aiba, N. & Tokuda, S. 2008 Conservative global gyrokinetic toroidal full- $f$ five-dimensional Vlasov simulation. Comput. Phys. Commun. 179 (6), 391–403.CrossRefGoogle Scholar
Idomura, Y. & Jolliet, S. 2011 Performance evaluations of advanced massively parallel platforms based on gyrokinetic toroidal five-dimensional Eulerian code GT5D. Prog. Nucl. Sci. Technol. 2, 620–627.Google Scholar
Idomura, Y., Tokuda, S. & Kishimoto, Y. 2003 Global gyrokinetic simulation of ion temperature gradient driven turbulence in plasmas using a canonical Maxwellian distribution. Nucl. Fusion 43, 234–243.Google Scholar
Jenko, F., Dorland, W., Kotschenreuther, M. & Rogers, B. N. 2000 Electron temperature gradient driven turbulence. Phys. Plasmas 7, 1904–1910.Google Scholar
Jolliet, S., Bottino, A., Angelino, P., Hatzky, R., Tran, T. M., Mcmillan, B. F., Sauter, O., Appert, K., Idomura, Y. & Villard, L. 2007 A global collisionless PIC code in magnetic coordinates. Comput. Phys. Commun. 177 (5), 409–425.Google Scholar
Kaneko, H. & Xu, Y. 1994 Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind. Maths Comput. 62, 739–753.Google Scholar
Klein, R., Gravier, E., Morel, P., Besse, N. & Bertrand, P. 2009 Gyrokinetic waterbag modeling of a plasma column: magnetic moment distribution and finite Larmor radius effects. Phys. Plasmas 16 (8), 082106.CrossRefGoogle Scholar
Ko, K. H., Sakkalis, T. & Patrikalakis, N. M. 2008 A reliable algorithm for computing the topological degree of a mapping in $\mathbb{R}^{2}$ . Appl. Maths Comput. 196 (2), 666–678.Google Scholar
Kravanja, P. & van Barel, M. 2000 Computing the Zeroes of Analytic Functions. Springer.CrossRefGoogle Scholar
Lapillonne, X., Brunner, S., Dannert, T., Jolliet, S., Marinoni, A., Villard, L., Görler, T., Jenko, F. & Merz, F. 2009 Clarifications to the limitations of the $s{-}\unicode[STIX]{x1D6FC}$ equilibrium model for gyrokinetic computations of turbulence. Phys. Plasmas 16 (3), 032308.CrossRefGoogle Scholar
Lehoucq, R.1996 An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices. Preprint MCS-P547–1195, pp. 1–41.Google Scholar
Lehoucq, R. 1999 On the convergence of an implicitly restarted Arnoldi method. SIAM J. Matrix Anal. Applics. 23, 551–562.CrossRefGoogle Scholar
Lehoucq, R., Maschoff, K., Soresen, D. & Yang, C.1996 Parpack. ARPACK is software located at http://www.caam.rice.edu/software/ARPACK/.Google Scholar
Littlejohn, R. G. 1979 A guiding center hamiltonian: a new approach. J. Math. Phys. 20, 2445–2458.CrossRefGoogle Scholar
Littlejohn, R. G. 1982 Hamiltonian perturbation theory in noncanonical coordinates. J. Math. Phys. 23, 742–747.Google Scholar
McMillan, B. F., Lapillonne, X., Brunner, S., Villard, L., Jolliet, S., Bottino, A., Görler, T. & Jenko, F. 2010 System size effects on gyrokinetic turbulence. Phys. Rev. Lett. 105, 155001.Google Scholar
Morel, P., Dreydemy, F., Berionni, V., Coulette, D., Besse, N. & Gurcan, O. 2014 A multi water bag model of drift kinetic electron plasma. Eur. Phys. J. D 68, 220.Google Scholar
Morel, P., Gravier, E., Besse, N., Ghizzo, A. & Bertrand, P. 2008 The water bag model and gyrokinetic applications. Commun. Nonlinear Sci. Numer. Simul. 13, 11–17.Google Scholar
Morel, P., Gravier, E., Besse, N., Klein, R., Ghizzo, A., Bertrand, P., Garbet, X., Ghendrih, P., Grandgirard, V. & Sarazin, Y. 2007 Gyrokinetic modelling: a multi water bag approach. Phys. Plasmas 14 (11), 112109.Google Scholar
Nakata, M., Honda, M., Yoshida, M., Urano, H., Nunami, M., Maeyama, S., Watanabe, T.-H. & Sugama, H. 2016 Validation studies of gyrokinetic ITG and TEM turbulence simulations in a JT-60U tokamak using multiple flux matching. Nucl. Fusion 56 (8), 086010.Google Scholar
Newcomb, W. A. 1990 The balooning transformation. Phys. Fluids B 2, 86–96.Google Scholar
Qi, L., Kwon, J., Hahm, T. S. & Jo, G. 2016 Gyrokinetic simulations of electrostatic microinstabilities with bounce-averaged kinetic electrons for shaped tokamak plasmas. Phys. Plasmas 23 (6), 062513.Google Scholar
Rahbar, S. & Hashemizadeh, E. 2008 A computational approach to the Fredholm integral equation of the second kind. In Proceedings of the World Congress on Engineering 2008, vol. II, Newswood Limited.Google Scholar
Rewoldt, G., Lin, Z. & Idomura, Y. 2007 Linear comparison of gyrokinetic codes with trapped electrons. Comput. Phys. Commun. 177 (10), 775–780.Google Scholar
Saad, Y. 2001 Numerical Methods for Large Eigenvalue Problems, 2nd edn. SIAM.Google Scholar
Sorensen, D. C. 1992 Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Applics. 13, 357–385.Google Scholar
Sousbie, T. & Colombi, S. 2016 ColDICE: a parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation. J. Comput. Phys. 321, 644–697.Google Scholar
Sugama, H. 2000 Gyrokinetic field theory. Phys. Plasmas 7 (2), 466–480.Google Scholar
Taylor, J. B., Wilson, H. R. & Connor, J. W. 1996 Structure of short-wavelength drift modes and transport in a toroidal plasma. Plasma Phys. Control. Fusion 38 (2), 243–250.Google Scholar
Wang, L. & Hahm, T. S. 2010 Nonlinear gyrokinetic theory with polarization drift. Phys. Plasmas 17, 082304.Google Scholar
Zapata, J. L. G. & Martin, J. C. D. 2011 A geometric algorithm for winding number computation with complexity analysis. J. Complexity 28, 320–345.Google Scholar