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Landau damping in a turbulent setting
1. L. Landau, J. Exp. Theor. Phys. 16, 574 (1946);
1. L. Landau, J. Phys. USSR 10, 26 (1946).
5. L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators (Princeton University Press, 2005).
9. G. G. Howes, S. C. Cowley, W. Dorland, G. W. Hammett, E. Quataert, and A. A. Schekochihin, J. Geophys. Res. 113, A05103, doi:10.1029/2007JA012665 (2008).
10.Actually, the model is quite generic, as it merely assumes the local conservative flow of energy in k-space, corrected by scale-dependent damping.
13.A refined model was proposed by Howes, TenBarge, and Dorland (Ref. 11) that already identifies a mechanism to explain the modest effect of Landau damping on the cascade. The present work describes a distinct mechanism; we note that these mechanisms are not mutually exclusive.
16.The term “irreversible” is used here in the broader sense of statistical mechanics, rather than the restrictive sense of collisional entropy production in plasmas. See Sec. VII.5 of Kampen and Felderhof (Ref. 19) for a detailed discussion.
18. D. Swanson, Plasma waves (Institute of Physics, Bristol, 2003).
19. N. G. Van Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics (North-Holland, Amsterdam, 1967).
23.The transformation used here is expressed in slightly different notation than that used by Morrison and Pfirsch (Ref. 22), but is otherwise equivalent; see Morrison (Ref. 20) for a rigorous mathematical treatment.
25. E. Mazzucato, R. Bell, S. Ethier, J. Hosea, S. Kaye, B. LeBlanc, W. Lee, P. Ryan, D. Smith, W. Wang, J. Wilson, and H. Yuh, Nucl. Fusion 49, 055001 (2009).
27.For example, for and , the free energy is .
28. R. J. Goldston and P. H. Rutherford, Introduction to Plasma Physics (Institute of Physics, Bristol, 1995).
29. U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, 1995).
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To address the problem of Landau damping in kinetic turbulence, we consider the forcing of the linearized Vlasov equation by a stationary random source. It is found that the time-asymptotic density response is dominated by resonant particle interactions that are synchronized with the source. The energy consumption of this response is calculated, implying an effective damping rate, which is the main result of this paper. Evaluating several cases, it is found that the effective damping rate can differ from the Landau damping rate in magnitude and also, remarkably, in sign. A limit is demonstrated in which the density and current become phase-locked, which causes the effective damping to be negligible; this result offers a fresh perspective from which to reconsider recent observations of kinetic turbulence satisfying critical balance.
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