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Dory-Guest-Harris instability as a benchmark for continuum kinetic Vlasov-Poisson simulations of magnetized plasmas. (English) Zbl 1349.82127

Summary: The Dory-Guest-Harris instability is demonstrated to be a well-suited benchmark for continuum kinetic Vlasov-Poisson algorithms. The instability is a special case of perpendicularly-propagating kinetic electrostatic waves in a warm uniformly magnetized plasma. A complete derivation of the closed-form linear theory dispersion relation for the instability is presented. The electric field growth rates and oscillation frequencies specified by the dispersion relation provide concrete measures against which simulation results can be quantitatively compared. A fourth-order continuum kinetic algorithm is benchmarked against the instability, and is demonstrated to have good convergence properties and close agreement with theoretical growth rate and oscillation frequency predictions. Second-order accurate simulations are also shown to be consistent with theoretical predictions, but require higher resolution for convergence. The Dory-Guest-Harris instability benchmark extends the scope of current standard test problems by providing a substantive means of validating continuum kinetic simulations of magnetized plasmas in higher-dimensional 3D \((x, v_x, v_y)\) phase space. The linear theory analysis, initial conditions, algorithm description, and comparisons between theoretical predictions and simulation results are presented.

MSC:

82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
82D10 Statistical mechanics of plasmas

Software:

Vador; AstroGK
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References:

[1] Birdsall, C.; Langdon, A., Plasma Physics via Computer Simulation (1985), McGraw-Hill: McGraw-Hill New York, NY
[2] Arber, T.; Vann, R., A critical comparison of Eulerian-grid-based Vlasov solvers, J. Comput. Phys., 180, 1, 339-357 (2002) · Zbl 1001.82105
[3] Banks, J.; Hittinger, J., A new class of nonlinear finite-volume methods for Vlasov simulation, IEEE Trans. Plasma Sci., 38, 9, 2198-2207 (2010)
[4] Filbet, F.; Sonnendrücker, E., Comparison of Eulerian Vlasov solvers, Comput. Phys. Commun., 150, 3, 247-266 (2003) · Zbl 1196.82108
[5] Guo, W.; Qiu, J.-M., Hybrid semi-Lagrangian finite element-finite difference methods for the Vlasov equation, J. Comput. Phys., 234, 0, 108-132 (2013) · Zbl 1284.35438
[6] Qiu, J.-M.; Christlieb, A., A conservative high order semi-Lagrangian {WENO} method for the Vlasov equation, J. Comput. Phys., 229, 4, 1130-1149 (2010) · Zbl 1188.82069
[7] Rossmanith, J. A.; Seal, D. C., A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations, J. Comput. Phys., 230, 16, 6203-6232 (2011) · Zbl 1419.76506
[8] Suzuki, A.; Shigeyama, T., A conservative scheme for the relativistic Vlasov-Maxwell system, J. Comput. Phys., 229, 5, 1643-1660 (2010) · Zbl 1186.82078
[9] Besse, N.; Latu, G.; Ghizzo, A.; Sonnendrücker, E.; Bertrand, P., A wavelet-MRA-based adaptive semi-Lagrangian method for the relativistic Vlasov-Maxwell system, J. Comput. Phys., 227, 16, 7889-7916 (2008) · Zbl 1194.83013
[10] Sircombe, N.; Arber, T., VALIS: a split-conservative scheme for the relativistic 2D Vlasov-Maxwell system, J. Comput. Phys., 228, 13, 4773-4788 (2009) · Zbl 1175.82059
[11] Umeda, T.; Miwa, J.-i.; Matsumoto, Y.; Nakamura, T. K.M.; Togano, K.; Fukazawa, K.; Shinohara, I., Full electromagnetic Vlasov code simulation of the Kelvin-Helmholtz instability, Phys. Plasmas, 17, 5 (2010)
[12] Candy, J.; Waltz, R. E., Velocity-space resolution, entropy production, and upwind dissipation in Eulerian gyrokinetic simulations, Phys. Plasmas (1994-present), 13, 3 (2006)
[13] Numata, R.; Howes, G. G.; Tatsuno, T.; Barnes, M.; Dorland, W., AstroGK: astrophysical gyrokinetics code, J. Comput. Phys., 229, 24, 9347-9372 (2010) · Zbl 1204.85021
[14] Thomas, A.; Tzoufras, M.; Robinson, A.; Kingham, R.; Ridgers, C.; Sherlock, M.; Bell, A., A review of Vlasov-Fokker-Planck numerical modeling of inertial confinement fusion plasma, J. Comput. Phys.Special Issue: Computational Plasma Physics, 231, 3, 1051-1079 (2012) · Zbl 1385.76015
[15] Holod, I.; Lin, Z., Statistical analysis of fluctuations and noise-driven transport in particle-in-cell simulations of plasma turbulence, Phys. Plasmas (1994-present), 14, 3 (2007)
[16] Langdon, A. B., Kinetic theory for fluctuations and noise in computer simulation of plasma, Phys. Fluids, 22, 1, 163-171 (1979) · Zbl 0391.76088
[17] Nevins, W.; Dimits, A.; Hammett, G.; Dorland, W.; Shumaker, D. E., Discrete particle noise in particle-in-cell simulations of plasma microturbulence, Phys. Plasmas, 12, 12 (2005)
[18] Datta, K.; Kamil, S.; Williams, S.; Oliker, L.; Shalf, J.; Yelick, K., Optimization and performance modeling of stencil computations on modern microprocessors, SIAM Rev., 51, 1, 129-159 (2009) · Zbl 1160.65359
[19] Cheng, C.; Knorr, G., The integration of the Vlasov equation in configuration space, J. Comput. Phys., 22, 3, 330-351 (1976)
[20] Valentini, F.; Veltri, P.; Mangeney, A., A numerical scheme for the integration of the Vlasov-Poisson system of equations, in the magnetized case, J. Comput. Phys., 210, 2, 730-751 (2005) · Zbl 1083.82032
[21] Umeda, T.; Omura, Y.; Yoon, P.; Gaelzer, R.; Matsumoto, H., Harmonic Langmuir waves. III. Vlasov simulation, Phys. Plasmas, 10, 2, 382-391 (2003)
[22] Ghizzo, A.; Bertrand, P.; Shoucri, M.; Johnston, T.; Fualkow, E.; Feix, M., A Vlasov code for the numerical simulation of stimulated Raman scattering, J. Comput. Phys., 90, 2, 431-457 (1990) · Zbl 0702.76080
[23] Califano, F.; Cecchi, T.; Chiuderi, C., Nonlinear kinetic regime of the Weibel instability in an electron-ion plasma, Phys. Plasmas, 9, 2, 451-457 (2002)
[24] Mangeney, A.; Califano, F.; Cavazzoni, C.; Travnicek, P., A numerical scheme for the integration of the Vlasov-Maxwell system of equations, J. Comput. Phys., 179, 2, 495-538 (2002) · Zbl 1001.78025
[25] Suzuki, A.; Shigeyama, T., Detailed analysis of filamentary structure in the Weibel instability, Astrophys. J., 695, 2, 1550 (2009)
[26] Ghizzo, A.; Bertrand, P.; Shoucri, M.; Fijalkow, E.; Feix, M., An Eulerian code for the study of the drift-kinetic Vlasov equation, J. Comput. Phys., 108, 1, 105-121 (1993) · Zbl 0779.65078
[27] Sonnendrücker, E.; Roche, J.; Bertrand, P.; Ghizzo, A., The semi-Lagrangian method for the numerical resolution of the Vlasov equation, J. Comput. Phys., 149, 2, 201-220 (1999) · Zbl 0934.76073
[28] Dory, R. A.; Guest, G. E.; Harris, E. G., Unstable electrostatic plasma waves propagating perpendicular to a magnetic field, Phys. Rev. Lett., 14, 131-133 (1965)
[29] Byers, J. A.; Grewal, M., Perpendicularly propagating plasma cyclotron instabilities simulated with a one-dimensional computer model, Phys. Fluids, 13, 7, 1819-1830 (1970)
[30] Crawford, F.; Tataronis, J., Absolute instabilities of perpendicularly propagating cyclotron harmonic plasma waves, J. Appl. Phys., 36, 9, 2930-2934 (1965)
[31] Tataronis, J.; Crawford, F., Cyclotron harmonic wave propagation and instabilities: I. Perpendicular propagation, J. Plasma Phys., 4, 231-248 (1970)
[32] Bernstein, I. B., Waves in a plasma in a magnetic field, Phys. Rev., 109, 1, 10-21 (1958) · Zbl 0079.44102
[33] Krall, N.; Trivelpiece, A., Principles of Plasma Physics (1973), McGraw-Hill: McGraw-Hill New York, NY
[34] Gurnett, D. A.; Bhattacharjee, A., Introduction to Plasma Physics: With Space and Laboratory Applications (2005), Cambridge University Press · Zbl 1376.82002
[35] Crawford, F., A review of cyclotron harmonic phenomena in plasmas, Nucl. Fusion, 5, 1, 73 (1965)
[36] Cottrell, G.; Dendy, R., Superthermal radiation from fusion products in JET, Phys. Rev. Lett., 60, 1, 33-36 (1988)
[37] Goede, A.; Massmann, P.; Hopman, H.; Kistemaker, J., Ion Bernstein waves excited by an energeticion beam ion a plasma, Nucl. Fusion, 16, 1, 85 (1976)
[38] Hubbard, R. F.; Birmingham, T. J., Electrostatic emissions between electron gyroharmonics in the outer magnetosphere, J. Geophys. Res., 83, A10, 4837-4850 (1978)
[39] Perraut, S.; Roux, A.; Robert, P.; Gendrin, R.; Sauvaud, J.-A.; Bosqued, J.-M.; Kremser, G.; Korth, A., A systematic study of ULF waves above FH+ from GEOS 1 and 2 measurements and their relationships with proton ring distributions, J. Geophys. Res., 87, A8, 6219-6236 (1982)
[40] Ashour-Abdalla, M.; Kennel, C., Nonconvective and convective electron cyclotron harmonic instabilities, J. Geophys. Res., 83, A4, 1531-1543 (1978)
[41] Kaufmann, R. L.; Dusenbery, P. B.; Thomas, B. J., Stability of the auroral plasma: parallel and perpendicular propagation of electrostatic waves, J. Geophys. Res., 83, A12, 5663-5669 (1978)
[42] Post, R. F.; Rosenbluth, M., Electrostatic instabilities in finite mirror-confined plasmas, Phys. Fluids, 9, 4, 730-749 (1966)
[43] Guest, G., Electron Cyclotron Heating of Plasmas (2009), John Wiley and Sons
[44] Harris, E., Plasma instabilities associated with anisotropic velocity distributions, J. Nucl. Energy, Part C, Plasma Phys. Accel. Thermonucl. Res., 2, 1, 138 (1961)
[45] Lee, J. K.; Birdsall, C. K., Velocity space ring-plasma instability, magnetized, part II: simulation, Phys. Fluids, 22, 7, 1315-1322 (1979) · Zbl 0402.76100
[46] Umeda, T.; Ashour-Abdalla, M.; Schriver, D.; Richard, R. L.; Coroniti, F. V., Particle-in-cell simulation of Maxwellian ring velocity distribution, J. Geophys. Res., 112, A4 (2007)
[47] Colella, P.; Dorr, M.; Hittinger, J.; Martin, D., High-order, finite-volume methods in mapped coordinates, J. Comput. Phys., 230, 8, 2952-2976 (2011) · Zbl 1218.65119
[48] Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, (Quarteroni, A., Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol. 1697 (1998), Springer: Springer Berlin, Heidelberg), 325-432 · Zbl 0927.65111
[49] Harris, E., Unstable plasma oscillations in a magnetic field, Phys. Rev. Lett., 2, 2, 34-36 (1959)
[50] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G.; Bateman, H., Higher Transcendental Functions, California Institute of Technology H. Bateman MS Project, vol. 2 (1953), McGraw-Hill: McGraw-Hill New York, NY · Zbl 0052.29502
[51] Crawford, F., Cyclotron harmonic waves in warm plasmas, J. Res. Natl. Bur. Stand., 69D, 6, 789 (1965)
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