Nonequilibrium gyrokinetic fluctuation theory and sampling noise in gyrokinetic particle-in-cell simulations

Physics of Plasmas

Volumn 14, Issue 9, 2007, Pages

  Krommes, John A ^a  

^a PRINCETON PLASMA PHYSICS LABORATORY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ACOUSTIC NOISE; PARTIAL DIFFERENTIAL EQUATIONS; PLASMA SIMULATION; PLASMA TURBULENCE;

GYROKINETIC FLUCTUATION THEORY; GYROKINETIC PARTICLE IN CELL SIMULATION; GYROKINETIC PLASMAS; SAMPLING NOISE;

PLASMA THEORY;

EID: 34848831432     PISSN: 1070664X     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2759879     Document Type: Article

References (106)

1. W. M. Nevins, G. W. Hammett, A. M. Dimits, W. Dorland, and D. E. Shumaker, Phys. Plasmas 12, 122305 (2005).
2. A. M. Dimits, G. Bateman, M. A. Beer, B. I. Cohen, W. Dorland, G. W. Hammett, C. Kim, J. E. Kinsey, M. Kotschenreuther, A. H. Kritz, Phys. Plasmas 7, 969 (2000).
3. W. M. Nevins, J. Candy, S. Cowley, T. Dannert, A. Dimits, W. Dorland, C. Estrada-Mila, G. W. Hammett, F. Jenko, M. J. Pueschel, and D. E. Shumaker, Phys. Plasmas 13, 122306 (2006).
4. J. A. Krommes, Phys. Fluids B 5, 1066 (1993).
5. G. Hu and J. A. Krommes, Phys. Plasmas 1, 863 (1994).
6. J. A. Krommes, Bull. Am. Phys. Soc. 51, 325 (2006).
7. J. A. Krommes, Phys. Rep. 360, 1 (2002).
8. C. Cercignani, Ludwig Boltzmann: The Man Who Trusted Atoms (Clarendon, Oxford, 1998).
9. A. Einstein, Ann. Phys. 17, 549 (1905); English translation in A. Einstein, Investigations on the Theory of the Brownian Movement, edited by, R. Fürth, translated by A. D. Cowper (Dover, New York, 1956), p. 1.
10. A. Einstein, Ann. Phys. 17, 549 (1905); English translation in A. Einstein, Investigations on the Theory of the Brownian Movement, edited by, R. Fürth, translated by A. D. Cowper (Dover, New York, 1956), p. 1.
11. A. Einstein, Z. Elektrochem. Angew. Phys. Chem. 14, 235 (1908), English translation in A. Einstein, Investigations on the Theory of the Brownian Movement, edited by, R. Fürth, translated by A. D. Cowper (Dover, New York, 1956), p. 68.
12. A. Einstein, Z. Elektrochem. Angew. Phys. Chem. 14, 235 (1908), English translation in A. Einstein, Investigations on the Theory of the Brownian Movement, edited by, R. Fürth, translated by A. D. Cowper (Dover, New York, 1956), p. 68.
13. G. I. Taylor, Philos. Trans. R. Soc. London, Ser. A 215, 1 (1915).
14. G. I. Taylor, Proc. London Math. Soc. 20, 196 (1921); reprinted in Turbulence: Classic Papers on Statistical Theory, edited by, S. K. Friedlander, and, L. Topper, (Interscience, New York, 1961), p. 1.
15. G. I. Taylor, Proc. London Math. Soc. 20, 196 (1921); reprinted in Turbulence: Classic Papers on Statistical Theory, edited by, S. K. Friedlander, and, L. Topper, (Interscience, New York, 1961), p. 1.
16. L. D. Landau, Phys. Z. Sowjetunion 0369-9811 10, 154 (1936) L. D. Landau [Zh. Eksp. Teor. Fiz. 7, 203 (1937)].
17. L. D. Landau, Phys. Z. Sowjetunion 0369-9811 10, 154 (1936) L. D. Landau [Zh. Eksp. Teor. Fiz. 7, 203 (1937)].
18. Y. L. Klimontovich, The Statistical Theory of Nonequilibrium Processes in a Plasma, edited by, D. ter Harr, translated by H. S. H. Massey and O. M. Blunn (MIT, Cambridge, MA, 1967).
19. C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation (McGraw-Hill, New York, 1985).
20. A. B. Langdon and C. K. Birdsall, Phys. Fluids 13, 2115 (1970).
21. R. L. Morse and C. W. Nielson, Phys. Fluids 12, 2418 (1969).
22. A. B. Langdon, Phys. Fluids 22, 163 (1979).
23. E. A. Frieman and L. Chen, Phys. Fluids 25, 502 (1982).
24. W. W. Lee, Phys. Fluids 26, 556 (1983).
25. D. H. E. Dubin, J. A. Krommes, C. R. Oberman, and W. W. Lee, Phys. Fluids 26, 3524 (1983).
26. J. A. Krommes, W. W. Lee, and C. Oberman, Phys. Fluids 29, 2421 (1986).
27. J. A. Krommes, Phys. Rev. Lett. 70, 3067 (1993).
28. M. Kotschenreuther, in Proceedings of the 14th International Conference on the Numerical Simulation of Plasmas (Office of Naval Research, Arlington, Virginia, 1991), paper PT20.
29. M. Kotschenreuther, Bull. Am. Phys. Soc. 34, 2107 (1988).
30. A. M. Dimits and W. W. Lee, J. Comput. Phys. 107, 309 (1993).
31. S. E. Parker and W. W. Lee, Phys. Fluids B 5, 77 (1993).
32. A. Y. Aydemir, Phys. Plasmas 1, 822 (1994).
33. W. W. Lee and W. M. Tang, Phys. Fluids 31, 612 (1988).
34. J. A. Krommes and G. Hu, Phys. Plasmas 1, 3211 (1994).
35. T.-H. Watanabe and H. Sugama, Phys. Plasmas 11, 1476 (2004).
36. J. Candy and R. E. Waltz, Phys. Plasmas 13, 032310 (2006).
37. J. A. Krommes, Phys. Plasmas 6, 1477 (1999).
38. S. Brunner, E. Valeo, and J. A. Krommes, Phys. Plasmas 6, 4504 (1999).
39. S. Vadlamani, S. E. Parker, Y. Chang, and C. Kim, Comput. Phys. Commun. 164, 209 (2004).
40. Y. Chen and S. E. Parker, Phys. Plasmas 14, 082301 (2007).
41. R. H. Kraichnan, Phys. Fluids 7, 1163 (1964).
42. In discussions of Navier-Stokes turbulence, dielectric functions are not usually mentioned, although they should be. J. A. Krommes and P. L. Similon [Phys. Fluids 23, 1553 (1980)] discussed the special case of the guiding-center plasma, which shows very clearly how R is related to D-1 in a fluid-like model. In kinetic problems with a velocity variable, R most fundamentally describes the kinetic response, while D-1 emerges after integration of R over velocity; the key result is the shielding relation.
43. N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics (McGraw-Hill, New York, 1973).
44. T. G. Jenkins and W. W. Lee, Phys. Plasmas 14, 032307 (2007).
45. Equation can be derived directly by integrating Eq. over ω with the assumption Fk (ω) =2 Fk. The discussion in this section is intended to be pedagogical, not overly technical. A much more thorough and systematic derivation of the equal-time balance equation was given by G. F. Carnevale and P. C. Martin [Geophys. Astrophys. Fluid Dyn. 0309-1929 10.1080/03091928208209002 20, 131 (1982)]. Some aspects of that paper were discussed by J. Krommes and C.-B. Kim [Phys. Rev. E 62, 8508 (2000)]; see also Appendices F and G of Ref..
46. Equation can be derived directly by integrating Eq. over ω with the assumption Fk (ω) =2 Fk. The discussion in this section is intended to be pedagogical, not overly technical. A much more thorough and systematic derivation of the equal-time balance equation was given by G. F. Carnevale and P. C. Martin [Geophys. Astrophys. Fluid Dyn. 0309-1929 10.1080/03091928208209002 20, 131 (1982)]. Some aspects of that paper were discussed by J. Krommes and C.-B. Kim [Phys. Rev. E 62, 8508 (2000)]; see also Appendices F and G of Ref..
47. S. A. Orszag, in Fluid Dynamics, edited by, R. Balian, and, J.-L. Peube, (Gordon and Breach, New York, 1977), pp. 235-374.
48. H. A. Rose, J. Stat. Phys. 20, 415 (1979).
49. B. B. Kadomtsev, Plasma Turbulence (Academic, New York, 1965), translated by L. C. Ronson from the 1964 Russian edition, Problems in Plasma Theory, edited by, M. A. Leontovich, translation edited by, M. C. Rusbridge,.
50. J. A. Krommes and C.-B. Kim, Phys. Fluids 31, 869 (1988).
51. P. C. Martin, E. D. Siggia, and H. A. Rose, Phys. Rev. A 8, 423 (1973).
52. L. D. Landau and E. M. Lifshitz, Statistical Physics, Part I, 3rd ed. (Pergamon, Oxford, 1980), revised and enlarged by E. M. Lifshitz and L. P. Pitaevskii, translated from the Russian by J. B. Sykes and M. J. Kearsley, Chap..
53. P. C. Martin, in Many Body Physics, edited by, C. deWitt, and, R. Balian, (Gordon and Breach, New York, 1968), pp. 37-136.
54. G. Ecker, Theory of Fully Ionized Plasmas (Academic, New York, 1972), Chap..
55. Strictly speaking, this result is valid only for the Vlasov dielectric D(0). When higher-order terms in p are included, the Debye length is slightly modified [see F. Tappert, Ph.D. thesis, Princeton University (1967)]. There are also corrections for scales of the order of the distance of closest approach.
56. J. A. Krommes, Phys. Fluids 19, 649 (1976).
57. N. Rostoker, Phys. Fluids 0031-9171 10.1063/1.1711227 7, 479 (1964); N. Rostoker, Phys. Fluids 7, 491 (1964).
58. N. Rostoker, Phys. Fluids 0031-9171 10.1063/1.1711227 7, 479 (1964); N. Rostoker, Phys. Fluids 7, 491 (1964).
59. A. J. Brizard and T. S. Hahm, Rev. Mod. Phys. 79, 421 (2007).
60. J. A. Krommes, in Turbulence and Coherent Structures in Fluids, Plasmas and Nonlinear Media, edited by, M. Shats, and, H. Punzmann, (World Scientific, Singapore, 2006), pp. 115-232.
61. T. H. Stix, Waves in Plasmas (AIP, New York, 1992).
62. T. G. Jenkins, Ph.D. thesis, Princeton University (2007).
63. G. W. Hammett (private communication, 2007).
64. M. Kotschenreuther, G. Rewoldt, and W. M. Tang, Comput. Phys. Commun. 0010-4655 10.1016/0010-4655(95)00035-E 88, 128 (1995); W. Dorland, F. Jenko, M. Kotschenreuther, and B. N. Rogers, Phys. Rev. Lett. 85, 5579 (2000).
65. M. Kotschenreuther, G. Rewoldt, and W. M. Tang, Comput. Phys. Commun. 0010-4655 10.1016/0010-4655(95)00035-E 88, 128 (1995); W. Dorland, F. Jenko, M. Kotschenreuther, and B. N. Rogers, Phys. Rev. Lett. 85, 5579 (2000).
66. F. Jenko, Comput. Phys. Commun. 125, 196 (2000).
67. J. Candy and R. Waltz, J. Comput. Phys. 186, 545 (2003).
68. M. H. Kalos and P. A. Whitlock, Monte Carlo Methods (Wiley, New York, 1986), Vol. I.
69. Here and elsewhere in the article linear instability appears to be significant [J. A. Krommes, Plasma Phys. Controlled Fusion 41, A641 (1999)]. The discussion must be appropriately modified if nonlinear instability (submarginal turbulence) is important. The common feature is that both linear and nonlinear instability are driven by the same sources of free energy (e.g., profile gradients). Note that there always exists a drive level below which even nonlinear instability is suppressed, although that may not coincide with the threshold for linear instability.
70. D. Ruelle and F. Takens, Commun. Math. Phys. 20, 167 (1971).
71. A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics, 2nd ed. (Springer, New York, 1992).
72. W. D. McComb, The Physics of Fluid Turbulence (Clarendon, Oxford, 1990).
73. M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945); reprinted in Selected Papers on Noise and Stochastic Processes, edited by, N. Wax, (Dover, New York, 1954), p. 113.
74. M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945); reprinted in Selected Papers on Noise and Stochastic Processes, edited by, N. Wax, (Dover, New York, 1954), p. 113.
75. F. J. Dyson, Phys. Rev. 75, 486 (1949).
76. R. H. Kraichnan, J. Fluid Mech. 5, 497 (1959).
77. C. E. Leith, J. Atmos. Sci. 28, 145 (1971).
78. R. H. Kraichnan, J. Fluid Mech. 41, 189 (1970).
79. R. H. Kraichnan, in Nonlinear Dynamics, edited by, R. H. G. Helleman, (New York Academy of Sciences, New York, 1980), p. 37.
80. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).
81. J. C. Bowman, J. A. Krommes, and M. Ottaviani, Phys. Fluids B 5, 3558 (1993).
82. G. F. Carnevale, U. Frisch, and R. Salmon, J. Phys. A 14, 1701 (1981).
83. T. Carleman, Acta Vet. Acad. Sci. Hung. 174, 1680 (1922).
84. R. Kubo, J. Phys. Soc. Jpn. 17, 1100 (1962).
85. R. H. Kraichnan, J. Math. Phys. 0022-2488 10.1063/1.1724206 2, 124 (1961); R. H. Kraichnan, J. Math. Phys. 3, 205 (E) (1962).
86. R. H. Kraichnan, J. Math. Phys. 0022-2488 10.1063/1.1724206 2, 124 (1961); R. H. Kraichnan, J. Math. Phys. 3, 205 (E) (1962).
87. J. C. Bowman, Ph.D. thesis, Princeton University (1992).
88. G. Hu, J. A. Krommes, and J. C. Bowman, Phys. Lett. A 202, 117 (1995).
89. G. Hu, J. A. Krommes, and J. C. Bowman, Phys. Plasmas 4, 2116 (1997).
90. J. C. Bowman and J. A. Krommes, Phys. Plasmas 4, 3895 (1997).
91. H. A. Rose, Ph.D. thesis, Harvard University (1974).
92. M. Doi, J. Phys. A 9, 1465 (1976).
93. D. F. DuBois and M. Espedal, Plasma Phys. 20, 1209 (1978).
94. J. A. Krommes, in Theoretical and Computational Plasma Physics (International Atomic Energy Agency, Vienna, 1978), pp. 405-417.
95. M. Kruskal, in Plasma Physics (International Atomic Energy Agency, Vienna, 1965), p. 373.
96. R. B. White, Asymptotic Analysis of Differential equations (Imperial College, London, 2005).
97. A. Rogister and C. Oberman, J. Plasma Phys. 0022-3778 2, 33 (1968); A. Rogister and C. Oberman, J. Plasma Phys. 3, 119 (1969).
98. A. Rogister and C. Oberman, J. Plasma Phys. 0022-3778 2, 33 (1968); A. Rogister and C. Oberman, J. Plasma Phys. 3, 119 (1969).
99. W. W. Lee, Bull. Am. Phys. Soc. 51, 111 (2006).
100. T. H. Dupree, Phys. Fluids 9, 1773 (1966).
101. T. H. Dupree, Phys. Fluids 10, 1049 (1967).
102. W. W. Lee, S. Ethier, T. G. Jenkins, W. X. Wang, J. L. V. Lewandowski, G. Rewoldt, W. M. Tang, S. E. Parker, and Y. Chen, in Proceedings of the 21st IAEA Fusion Energy Conference, Chengdu, China, (International Atomic Energy Agency, Vienna, 2006).
103. I. Holod and Z. Lin, Phys. Plasmas 14, 032306 (2007).
104. W. W. Lee, J. L. V. Lewandowski, T. S. Hahm, and Z. L. Lin, Phys. Plasmas 8, 4435 (2001).
105. A. Mishchenko, R. Hatzky, and A. Könies, Phys. Plasmas 11, 5480 (2004).
106. R. H. Kraichnan, in Second Symposium on Naval Hydrodynamics (Office of Naval Research, Department of the Navy, Washington, D.C., 1958), p. 29.