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Fourier–Hermite spectral representation for the Vlasov–Poisson system in the weakly collisional limit

Part of: Collisions in Collisionless Plasmas

Published online by Cambridge University Press:  03 February 2015

Joseph T. Parker  and
Paul J. Dellar
Joseph T. Parker*
Affiliation:
OCIAM, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
Paul J. Dellar
Affiliation:
OCIAM, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
*
†Email address for correspondence: [email protected]
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Abstract

We study Landau damping in the 1+1D Vlasov–Poisson system using a Fourier–Hermite spectral representation. We describe the propagation of free energy in Fourier–Hermite phase space using forwards and backwards propagating Hermite modes recently developed for gyrokinetic theory. We derive a free energy equation that relates the change in the electric field to the net Hermite flux out of the zeroth Hermite mode. In linear Landau damping, decay in the electric field corresponds to forward propagating Hermite modes; in nonlinear damping, the initial decay is followed by a growth phase characterized by the generation of backwards propagating Hermite modes by the nonlinear term. The free energy content of the backwards propagating modes increases exponentially until balancing that of the forward propagating modes. Thereafter there is no systematic net Hermite flux, so the electric field cannot decay and the nonlinearity effectively suppresses Landau damping. These simulations are performed using the fully-spectral 5D gyrokinetics code SpectroGK, modified to solve the 1+1D Vlasov–Poisson system. This captures Landau damping via Hou–Li filtering in velocity space. Therefore the code is applicable even in regimes where phase mixing and filamentation are dominant.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 2 , April 2015 , 305810203
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Copyright
Copyright © Cambridge University Press 2015 

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References

REFERENCES

Abramowitz, M. and Stegun, I. A. 1972 Handbook of Mathematical Functions, 10th edn.New York: Dover.Google Scholar
Armstrong, T. P. 1967 Numerical studies of the nonlinear Vlasov equation. Phys. Fluids 10, 1269–1280.Google Scholar
Balescu, R. 1963 Statistical Mechanics of Charged Particles. New York: Interscience.Google Scholar
Bardos, C., Golse, F. and Levermore, C. D. 1993 Fluid dynamic limits of kinetic equations II convergence proofs for the Boltzmann equation. Commun. Pure Appl. Math. 46, 667–753.Google Scholar
Bernstein, I. B., Greene, J. M. and Kruskal, M. D. 1957 Exact nonlinear plasma oscillations. Phys. Rev. 108, 546–550.Google Scholar
Birdsall, C. K. and Langdon, A. B. 2005 Plasma Physics via Computer Simulation. Bristol: Institute of Physics.Google Scholar
Bouchut, F. 1993 Existence and uniqueness of a global smooth solution for the Vlasov–Poisson–Fokker–Planck system in three dimensions. J. Funct. Anal. 111, 239–258.Google Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods. Mineola, NY: Dover.Google Scholar
Brunetti, M., Califano, F. and Pegoraro, F. 2000 Asymptotic evolution of nonlinear Landau damping. Phys. Rev. E 62, 4109–4114.Google Scholar
Burnett, D. 1935 The distribution of velocities in a slightly non-uniform gas. Proc. Lond. Math. Soc. (2) 39, 385–430.Google Scholar
Burnett, D. 1936 The distribution of molecular velocities and the mean motion in a non-uniform gas. Proc. Lond. Math. Soc. (2) 40, 382–435.Google Scholar
Camporeale, E., Delzanno, G. L., Bergen, B. K. and Moulton, J. D. 2013 On the velocity space discretization for the Vlasov–Poisson system: comparison between Hermite spectral and Particle-in-Cell methods. Part 1: semi-implicit scheme. arXiv:1311.2098v2.Google Scholar
Case, K. M. 1959 Plasma oscillations. Ann. Phys. NY 7, 349–364.Google Scholar
Cheng, C. Z. and Knorr, G. 1976 The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22, 330–351.Google Scholar
Danielson, J. R., Anderegg, F. and Driscoll, C. F. 2004 Measurement of Landau damping and the evolution to a BGK equilibrium. Phys. Rev. Lett. 92, 245 003.Google Scholar
Dawson, J. M. 1983 Particle simulation of plasmas. Rev. Mod. Phys. 55, 403–447.Google Scholar
Denavit, J. and Kruer, W. L. 1971 Comparison of numerical solutions of the Vlasov equation with particle simulations of collisionless plasmas. Phys. Fluids 14, 1782–1791.Google Scholar
Dolbeault, J. 1999 Free energy and solutions of the Vlasov–Poisson–Fokker–Planck system: external potential and confinement (Large time behavior and steady states). J. Math. Pures Appl. 78, 121–157.Google Scholar
Dorland, W.et al. 2009 Gyrokinetic simulations project. See http://gyrokinetics.sourceforge.net/.Google Scholar
Durran, D. R. 1991 The third-order Adams-Bashforth method: an attractive alternative to leapfrog time-differencing. Mon. Weather Rev. 119, 702–720.Google Scholar
Durran, D. R. 1999 Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. New York: Springer.Google Scholar
Fahey, M. R. and Candy, J. 2004 Gyro: A 5-D gyrokinetic-Maxwell solver. In: Proc. 2004 ACM/IEEE Conf. on Supercomputing, Washington, DC, USA: IEEE Computer Society, pp. 26–33.Google Scholar
Fan, J. and Shen, C. 2001 Statistical simulation of low-speed rarefied gas flows. J. Comput. Phys. 167, 393–412.Google Scholar
Filbet, F., Sonnendrücker, E. and Bertrand, P. 2001 Conservative numerical schemes for the Vlasov equation. J. Comput. Phys. 172, 166–187.Google Scholar
Fried, B. D. and Conte, S. D. 1961 The Plasma Dispersion Function: The Hilbert Transform of the Gaussian. New York; London: Academic Press.Google Scholar
Frieman, E. A. and Chen, Liu 1982 Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria. Phys. Fluids 25, 502–508.Google Scholar
Frigo, M. and Johnson, S. G. 2005 The design and implementation of FFTW3. Proc. IEEE 93, 216–231.Google Scholar
Gagné, R. R. J. and Shoucri, M. M. 1977 A splitting scheme for the numerical solution of a one-dimensional Vlasov equation. J. Comput. Phys. 24, 445–449.Google Scholar
Glassey, R. T. 1996 The Cauchy Problem in Kinetic Theory. Philadelphia: Society for Industrial and Applied Mathematics.Google Scholar
Golse, F. and Saint-Raymond, L. 2004 The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155, 81–161.Google Scholar
Grad, H. 1949a Note on N-dimensional Hermite polynomials. Commun. Pure Appl. Math. 2, 325–330.Google Scholar
Grad, H. 1949b On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331–407.Google Scholar
Grad, H. 1958 Principles of the kinetic theory of gases. In: Thermodynamik der Gase, Vol. 12 (ed. Flügge, S.), Handbuch der Physik, Berlin: Springer, pp. 205–294.Google Scholar
Grant, F. C. and Feix, M. R. 1967 Fourier–Hermite solutions of the Vlasov equations in the linearized limit. Phys. Fluids 10, 696–702.Google Scholar
Hallatschek, K. 2004 Thermodynamic potential in local turbulence simulations. Phys. Rev. Lett. 93, 125 001.Google Scholar
Hammett, G. W., Beer, M. A., Dorland, W., Cowley, S. C. and Smith, S. A. 1993 Developments in the gyrofluid approach to tokamak turbulence simulations. Plasma Phys. Control. Fusion 35, 973.Google Scholar
Hatch, D. R., Jenko, F., Bañón Navarro, A. and Bratanov, V. 2013 Transition between saturation regimes of gyrokinetic turbulence. Phys. Rev. Lett. 111, 175 001.Google Scholar
Heath, R. E., Gamba, I. M., Morrison, P. J. and Michler, C. 2012 A discontinuous Galerkin method for the Vlasov–Poisson system. J. Comput. Phys. 231, 1140–1174.Google Scholar
Helander, P. and Sigmar, D. J. 2002 Collisional Transport in Magnetized Plasmas. Cambridge: Cambridge University Press.Google Scholar
Highcock, E. G., Barnes, M., Parra, F. I., Schekochihin, A. A., Roach, C. M. and Cowley, S. C. 2011 Transport bifurcation induced by sheared toroidal flow in tokamak plasmas. Phys. Plasmas 18, 102 304.Google Scholar
Hockney, R. W. and Eastwood, J. W. 1981 Computer Simulation using Particles. New York; London: McGraw–Hill.Google Scholar
Holloway, J. P. 1996 Spectral velocity discretizations for the Vlasov–Maxwell equations. Transp. Theory Stat. Phys. 25, 1–32.Google Scholar
Hou, T. Y. and Li, R. 2007 Computing nearly singular solutions using pseudo-spectral methods. J. Comput. Phys. 226, 379–397.Google Scholar
Howes, G. G., Cowley, S. C., Dorland, W., Hammett, G. W., Quataert, E. and Schekochihin, A. A. 2006 Astrophysical gyrokinetics: basic equations and linear theory. Astrophys. J. Suppl. 651, 590–614.Google Scholar
Jenko, F., Dorland, W., Kotschenreuther, M. and Rogers, B. N. 2000 Electron temperature gradient driven turbulence. Phys. Plasmas 7, 1904–1910.Google Scholar
Joyce, G., Knorr, G. and Meier, H. K. 1971 Numerical integration methods of the Vlasov equation. J. Comput. Phys. 8, 53–63.Google Scholar
Kanekar, A., Schekochihin, A. A., Dorland, W. and Loureiro, N. F. 2014 Fluctuation-dissipation theorems for a plasma-kinetic Langevin equation. J. Plasma Phys. 81, 305810104.Google Scholar
Kirkwood, J. G. 1946 The statistical mechanical theory of transport processes I. General theory. J. Chem. Phys. 14, 180–201.Google Scholar
Klimas, A. J. and Farrell, W. M. 1994 A splitting algorithm for Vlasov simulation with filamentation filtration. J. Comput. Phys. 110, 150–163.Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 301–305 (English translation: 1991 Proc. R. Soc. Lond. A 434, 9–13).Google Scholar
Krommes, J. A. 2012 The gyrokinetic description of microturbulence in magnetized plasmas. Annu. Rev. Fluid Mech. 44, 175–201.Google Scholar
Krommes, J. A. and Hu, G. 1994 The role of dissipation in the theory and simulations of homogeneous plasma turbulence, and resolution of the entropy paradox. Phys. Plasmas 1, 3211–3238.Google Scholar
Landau, L. D. 1946 On the vibrations of the electronic plasma. J. Phys.-U.S.S.R. 10, 25–34.Google Scholar
Le Bourdiec, S., de Vuyst, F. and Jacquet, L. 2006 Numerical solution of the Vlasov–Poisson system using generalized Hermite functions. Comput. Phys. Commun. 175, 528–544.Google Scholar
Lenard, A. and Bernstein, I. B. 1958 Plasma oscillations with diffusion in velocity space. Phys. Rev. 112, 1456–1459.Google Scholar
Lifshitz, E. M. and Pitaevskii, L. P. 1981 Physical Kinetics. Oxford: Pergamon Press.Google Scholar
Lions, P.-L. and Masmoudi, N. 2001 From the Boltzmann equations to the equations of incompressible fluid mechanics, I. Arch. Ration. Mech. Anal. 158, 173–193.Google Scholar
Loureiro, N. F., Schekochihin, A. A. and Zocco, A. 2013 Fast collisionless reconnection and electron heating in strongly magnetized plasmas. Phys. Rev. Lett. 111, 025 002.Google Scholar
Manfredi, G. 1997 Long-time behavior of nonlinear Landau damping. Phys. Rev. Lett. 79, 2815–2818.Google Scholar
Mesinger, F. and Arakawa, A. 1976 Numerical Methods used in Atmospheric Models, Global Atmospheric Research Program, Vol. 1, Geneva: World Meteorological Organization.Google Scholar
Mouhot, C. and Villani, C. 2011 On Landau damping. Acta Math. 207, 29–201.Google Scholar
Nakamura, T. and Yabe, T. 1999 Cubic interpolated propagation scheme for solving the hyper-dimensional Vlasov–Poisson equation in phase space. Comput. Phys. Commun. 120, 122–154.Google Scholar
Ng, C. S., Bhattacharjee, A. and Skiff, F. 1999 Kinetic eigenmodes and discrete spectrum of plasma oscillations in a weakly collisional plasma. Phys. Rev. Lett. 83, 1974–1977.Google Scholar
Ng, C. S., Bhattacharjee, A. and Skiff, F. 2004 Complete spectrum of kinetic eigenmodes for plasma oscillations in a weakly collisional plasma. Phys. Rev. Lett. 92, 065 002.Google Scholar
Ng, C. S., Bhattacharjee, A. and Skiff, F. 2006 Weakly collisional Landau damping and three-dimensional Bernstein–Greene–Kruskal modes: new results on old problems. Phys. Plasmas 13, 055 903.Google Scholar
Numata, R., Howes, G. G., Tatsuno, T., Barnes, M. and Dorland, W. 2010 AstroGK: astrophysical gyrokinetics code. J. Comput. Phys. 229, 9347–9372.Google Scholar
O'Neil, T. 1965 Collisionless damping of nonlinear plasma oscillations. Phys. Fluids 8, 2255–2262.Google Scholar
Orszag, S. A. 1971 On the elimination of aliasing in finite-difference schemes by filtering high-wavenumber components. J. Atmos. Sci. 28, 1074.Google Scholar
Parker, J. T. and Dellar, P. J. 2014 Hermite expansions with hypercollisionality for velocity space degrees of freedom in ion-temperature-gradient driven instabilities. In preparation.Google Scholar
Parker, J. T., Highcock, E. G. and Dellar, P. J. 2014 SpectroGK: a fully spectral astrophysical gyrokinetics code. In preparation.Google Scholar
Parker, S. E. and Carati, D. 1995 Renormalized dissipation in plasmas with finite collisionality. Phys. Rev. Lett. 75, 441–444.Google Scholar
Pauli, W. 1973 Statistical Mechanics. Cambridge, MA; London: MIT Press.Google Scholar
Peeters, A. G., Camenen, Y., Casson, F. J., Hornsby, W. A., Snodin, A. P., Strintzi, D. and Szepesi, G. 2009 The nonlinear gyro-kinetic flux tube code GKW. Comput. Phys. Commun. 180, 2650–2672.Google Scholar
Plunk, G. G. and Parker, J. T. 2014 Irreversible energy flow in forced Vlasov dynamics. Eur. Phys. J. D 68, 296.Google Scholar
Pohn, E., Shoucri, M. and Kamelander, G. 2005 Eulerian Vlasov codes. Comput. Phys. Commun. 166, 81–93.Google Scholar
Pueschel, M. J., Dannert, T. and Jenko, F. 2010 On the role of numerical dissipation in gyrokinetic Vlasov simulations of plasma microturbulence. Comput. Phys. Commun. 181, 1428–1437.Google Scholar
Rosenbluth, M. N., MacDonald, W. M. and Judd, D. L. 1957 Fokker–Planck equation for an inverse-square force. Phys. Rev. 107, 1–6.Google Scholar
Rutherford, P. H. and Frieman, E. A. 1968 Drift instabilities in general magnetic field configurations. Phys. Fluids 11, 569–585.Google Scholar
Schekochihin, A. A., Kanekar, A., Hammett, G. W., Dorland, W. and Loureiro, N. F. 2014 Stochastic advection and phase mixing in a collisionless plasma. In preparation.Google Scholar
Schumer, J. W. and Holloway, J. P. 1998 Vlasov simulations using velocity-scaled Hermite representations. J. Comput. Phys. 144, 626–661.Google Scholar
Siminos, E., Bénisti, D. and Gremillet, L. 2011 Stability of nonlinear Vlasov–Poisson equilibria through spectral deformation and Fourier–Hermite expansion. Phys. Rev. E 83, 056 402.Google Scholar
Strikwerda, J. C. 2004 Finite Difference Schemes and Partial Differential Equations, 2nd edn.Philadelphia: Society for Industrial and Applied Mathematics.Google Scholar
Tang, T. 1993 The Hermite spectral method for Gaussian-type functions. SIAM J. Sci. Comput. 14, 594–606.Google Scholar
Taylor, J. B. and Hastie, R. J. 1968 Stability of general plasma equilibria - I formal theory. Plasma Phys. 10, 479–494.Google Scholar
van Kampen, N. G. 1955 On the theory of stationary waves in plasmas. Physica 21, 949–963.Google Scholar
Zaki, S. I., Boyd, T. J. M. and Gardner, L. R. T. 1988 A finite element code for the simulation of one-dimensional Vlasov plasmas. II. Applications. J. Comput. Phys. 79, 200–208.Google Scholar
Zocco, A. and Schekochihin, A. A. 2011 Reduced fluid-kinetic equations for low-frequency dynamics, magnetic reconnection, and electron heating in low-beta plasmas. Phys. Plasmas 18, 102 309.Google Scholar