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A solvable model of Vlasov-kinetic plasma turbulence in Fourier–Hermite phase space

  • T. Adkins (a1) (a2) and A. A. Schekochihin (a1) (a2)

A class of simple kinetic systems is considered, described by the one-dimensional Vlasov–Landau equation with Poisson or Boltzmann electrostatic response and an energy source. Assuming a stochastic electric field, a solvable model is constructed for the phase-space turbulence of the particle distribution. The model is a kinetic analogue of the Kraichnan–Batchelor model of chaotic advection. The solution of the model is found in Fourier–Hermite space and shows that the free-energy flux from low to high Hermite moments is suppressed, with phase mixing cancelled on average by anti-phase-mixing (stochastic plasma echo). This implies that Landau damping is an ineffective route to dissipation (i.e. to thermalisation of electric energy via velocity space). The full Fourier–Hermite spectrum is derived. Its asymptotics are $m^{-3/2}$ at low wavenumbers and high Hermite moments ( $m$ ) and $m^{-1/2}k^{-2}$ at low Hermite moments and high wavenumbers ( $k$ ). These conclusions hold at wavenumbers below a certain cutoff (analogue of Kolmogorov scale), which increases with the amplitude of the stochastic electric field and scales as inverse square of the collision rate. The energy distribution and flows in phase space are a simple and, therefore, useful example of competition between phase mixing and nonlinear dynamics in kinetic turbulence, reminiscent of more realistic but more complicated multi-dimensional systems that have not so far been amenable to complete analytical solution.

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Antonsen, T. M. Jr., Fan, Z., Ott, E & Garcia-Lopez, E. 1996 The role of chaotic orbits in the determination of power spectra of passive scalars. Phys. Fluids 8, 3094.
Bañón Navarro, A., Morel, P., Albrecht-Marc, M., Carati, D., Merz, F., Görler, T. & Jenko, F. 2011 Free energy cascade in gyrokinetic turbulence. Phys. Rev. Lett. 106, 055001.
Batchelor, G. K. 1959 Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5, 113.
Beer, M. A. & Hammett, G. W. 1996 Toroidal gyrofluid equations for simulations of tokamak turbulence. Phys. Plasmas 3, 4046.
Bernstein, I. B., Greene, J. M. & Kruskal, M. D. 1957 Exact nonlinear plasma oscillations. Phys. Rev. 108, 546.
Bhat, P. & Subramanian, K. 2015 Fluctuation dynamos at finite correlation times using renewing flows. J. Plasma Phys. 81, 395810502.
Boldyrev, S. & Cattaneo, F. 2004 Magnetic-field generation in Kolmogorov turbulence. Phys. Rev. Lett. 92, 144501.
Cook, I. 1978 Application of the Novikov-Furutsu theorem to the random acceleration problem. Plasma Phys. 20, 349.
Davidson, P. A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.
Dorland, W. & Hammett, G. W. 1993 Gyrofluid turbulence models with kinetic effects. Phys. Fluids B 5, 812.
Dupree, T. H. 1972 Theory of phase space density granulation in plasma. Phys. Fluids 15, 334.
Falkovich, G., Gawȩdzki, K. & Vergassola, M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913.
Furutsu, K. 1963 On the statistical theory of electromagnetic waves in a fluctuating medium (I). J. Res. Natl Bur. Stand. 67D, 303.
Goldman, M. V. 1984 Strong turbulence of plasma waves. Rev. Mod. Phys. 56, 709.
Goswami, P., Passot, T. & Sulem, P. L. 2005 A Landau fluid model for warm collisionless plasmas. Phys. Plasmas 12, 102109.
Gould, R. W., O’Neil, T. M. & Malmberg, J. H. 1967 Plasma wave echo. Phys. Rev. Lett. 19, 219.
Hammett, G. W., Beer, M. A., Dorland, W., Cowley, S. C. & Smith, S. A. 1993 Developments in the gyrofluid approach to tokamak turbulence simulations. Plasma Phys. Control. Fusion 35, 973.
Hammett, G. W., Dorland, W. & Perkins, F. W. 1992 Fluid models of phase mixing, Landau damping, and nonlinear gyrokinetic dynamics. Phys. Fluids B 4, 2052.
Hammett, G. W. & Perkins, F. W. 1990 Fluid moment models for Landau damping with application to the ion-temperature-gradient instability. Phys. Rev. Lett. 64, 3019.
Hatch, D. R., Jenko, F., Bratanov, V. & Bañón Navarro, A. 2014 Phase space scales of free energy dissipation in gradient-driven gyrokinetic turbulence. J. Plasma Phys. 80, 531.
Howes, G. G., Klein, K. G. & Li, T. C. 2017 Diagnosing collisionless energy transfer using field-particle correlations: Vlasov–Poisson plasmas. J. Plasma Phys. 83, 705830102.
Kanekar, A., Schekochihin, A. A., Dorland, W. & Loureiro, N. F. 2015 Fluctuation-dissipation relations for a plasma-kinetic Langevin equation. J. Plasma Phys. 81, 305810104.
Kanekar, A. V.2015 Phase mixing in turbulent magnetized plasmas. PhD thesis, University of Maryland, College Park (URL:
Kawamori, E. 2013 Experimental verification of entropy cascade in two-dimensional electrostatic turbulence in magnetized plasma. Phys. Rev. Lett. 110, 095001.
Kazantsev, A. P. 1968 Enhancement of a magnetic field by a conducting fluid. Sov. Phys. JETP 26, 1031.
Kingsep, A. S. 2004 Introduction to Nonlinear Plasma Physics. MZ Press (in Russian).
Klein, K. G., Howes, G. G. & Tenbarge, J. M. 2017 Diagnosing collisionless energy transfer using field-particle correlations: gyrokinetic turbulence. J. Plasma Phys. 83, 535830401.
Knorr, G. 1977 Time asymptotic statistics of the Vlasov equation. J. Plasma Phys. 17, 553.
Kosuga, Y. & Diamond, P. H. 2011 On relaxation and transport in gyrokinetic drift wave turbulence with zonal flow. Phys. Plasmas 18, 122305.
Kosuga, Y., Itoh, S.-I., Diamond, P. H., Itoh, K. & Lesur, M. 2014 Ion temperature gradient driven turbulence with strong trapped ion resonance. Phys. Plasmas 21, 102303.
Kosuga, Y., Itoh, S.-I., Diamond, P. H., Itoh, K. & Lesur, M. 2017 Role of phase space structures in collisionless drift wave turbulence and impact on transport modeling. Nucl. Fusion 57, 072006.
Kraichnan, R. H. 1968 Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11, 945.
Kraichnan, R. H. 1974 Convection of a passive scalar by a quasi-uniform random straining field. J. Fluid Mech. 64, 737.
Kraichnan, R. H. 1994 Anomalous scaling of a randomly advected passive scalar. Phys. Rev. Lett. 72, 1016.
Krommes, J. A. 1997 The clump lifetime revisited: exact calculation of the second-order structure function for a model of forced, dissipative turbulence. Phys. Plasmas 4, 655.
Krommes, J. A. 2015 A tutorial introduction to the statistical theory of turbulent plasmas, a half-century after Kadomtsev’s plasma turbulence and the resonance-broadening theory of Dupree and Weinstock. J. Plasma Phys. 81, 205810601.
Landau, L. 1946 On the vibration of the electronic plasma. Zh. Eksp. Teor. Fiz. 16, 574.
Laval, G., Pesme, D. & Adam, J.-C. 2016 Wave-particle and wave–wave interactions in hot plasmas: a French historical point of view. Eur. Phys. J. H. doi:10.1140/epjh/e2016-70050-2.
Lenard, A. & Bernstein, I. B. 1958 Plasma oscillations with diffusion in velocity space. Phys. Rev. 112, 1456.
Lesur, M., Diamond, P. H. & Kosuga, Y. 2014a Nonlinear current-driven ion-acoustic instability driven by phase-space structures. Plasma Phys. Control. Fusion 56, 075005.
Lesur, M., Diamond, P. H. & Kosuga, Y. 2014b Phase-space jets drive transport and anomalous resistivity. Phys. Plasmas 21, 112307.
Malmberg, J. H., Wharton, C. B., Gould, R. W. & O’Neil, T. M. 1968 Plasma wave echo experiment. Phys. Rev. Lett. 20, 95.
Mandell, N. R., Dorland, W. & Landreman, M. 2017 Laguerre-Hermite pseudo-spectral velocity formulation of gyrokinetics. J. Plasma Phys. (in press) arXiv:1708.04029.
Manheimer, W. M. 1971 Strong turbulence theory of nonlinear stabilization and harmonic generation. Phys. Fluids 14, 579.
Manheimer, W. M. & Dupree, T. H. 1968 Weak turbulence theory of velocity space diffusion and the nonlinear Landau damping of waves. Phys. Fluids 11, 2709.
Mattor, N. 1992 Can Landau-fluid models describe nonlinear Landau damping? Phys. Fluids B 4, 3952.
Mazitov, R. K. 1965 Damping of plasma waves. J. Appl. Mech. Tech. Phys. 6, 22.
Musher, S. L., Rubenchik, A. M. & Zakharov, V. E. 1995 Weak Langmuir turbulence. Phys. Rep. 252, 177.
Novikov, E. A. 1965 Functionals and the random-force method in turbulence theory. Sov. Phys. JETP 20, 1290.
O’Neil, T. 1965 Collisionless damping of nonlinear plasma oscillations. Phys. Fluids 8, 2255.
O’Neil, T. M., Winfrey, J. H. & Malmberg, J. H. 1971 Nonlinear interaction of a small cold beam and a plasma. Phys. Fluids 14, 1204.
Orszag, S. A. & Kraichnan, R. H. 1967 Model equations for strong turbulence in a Vlasov plasma. Phys. Fluids 10, 1720.
Parker, J. T. & Dellar, P. J. 2015 Fourier–Hermite spectral representation for the Vlasov–Poisson system in the weakly collisional limit. J. Plasma Phys. 81, 305810203.
Parker, J. T., Highcock, E. G., Schekochihin, A. A. & Dellar, P. J. 2016 Suppression of phase mixing in drift-kinetic plasma turbulence. Phys. Plasmas 23, 070703.
Passot, T. & Sulem, P. L. 2004 A Landau fluid model for dispersive magnetohydrodynamics. Phys. Plasmas 11, 5173.
Passot, T., Sulem, P. L. & Tassi, E. 2017 Electron-scale reduced fluid models with gyroviscous effects. J. Plasma Phys. 83, 715830402.
Plunk, G. G., Cowley, S. C., Schekochihin, A. A. & Tatsuno, T. 2010 Two-dimensional gyrokinetic turbulence. J. Fluid Mech. 664, 407.
Plunk, G. G. & Tatsuno, T. 2011 Energy transfer and dual cascade in kinetic magnetized plasma turbulence. Phys. Rev. Lett. 106, 165003.
Robinson, P. A. 1997 Nonlinear wave collapse and strong turbulence. Rev. Mod. Phys. 69, 507.
Rudakov, L. I. & Tsytovich, V. N. 1978 Strong Langmuir turbulence. Phys. Rep. 40, 1.
Schekochihin, A. A.2017 Lecture Notes on Kinetic Theory and Magnetohydrodynamics of Plasmas.
Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Plunk, G. G., Quataert, E. & Tatsuno, T. 2008 Gyrokinetic turbulence: a nonlinear route to dissipation through phase space. Plasma Phys. Control. Fusion 50, 124024.
Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Quataert, E. & Tatsuno, T. 2009 Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas. Astrophys. J. Suppl. 182, 310.
Schekochihin, A. A., Cowley, S. C., Taylor, S. F., Maron, J. L. & McWilliams, J. C. 2004a Simulations of the small-scale turbulent dynamo. Astrophys. J. 612, 276.
Schekochihin, A. A., Haynes, P. H. & Cowley, S. C. 2004b Diffusion of passive scalar in a finite-scale random flow. Phys. Rev. E 70, 046304.
Schekochihin, A. A., Iskakov, A. B., Cowley, S. C., McWilliams, J. C., Proctor, M. R. E. & Yousef, T. A. 2007 Fluctuation dynamo and turbulent induction at low magnetic Prandtl numbers. New J. Phys. 9, 300.
Schekochihin, A. A., Parker, J. T., Highcock, E. G., Dellar, P. J., Dorland, W. & Hammett, G. W. 2016 Phase mixing versus nonlinear advection in drift-kinetic plasma turbulence. J. Plasma Phys. 82, 905820212.
Servidio, S., Chasapis, A., Matthaeus, W. H., Perrone, D., Valentini, F., Parashar, T. N., Veltri, P., Gershman, D., Russell, C. T., Giles, B. et al. 2017 Magnetospheric multiscale (MMS) observation of plasma velocity-space cascade: Hermite representation and theory. Phys. Rev. Lett. 119, 205101.
Smith, S. A.1997 Dissipative closures for statistical moments, fluid moments, and subgrid scales in plasma turbulence. PhD thesis, Princeton University;
Snyder, P. B., Hammett, G. W. & Dorland, W. 1997 Landau fluid models of collisionless magnetohydrodynamics. Phys. Plasmas 4, 3974.
Sturrock, P. A. 1966 Stochastic acceleration. Phys. Rev. 141, 186.
Sutton, W. G. L. 1943 On the equation of diffusion in a turbulent medium. Proc. R. Soc. Lond. A 182, 48.
Tassi, E., Sulem, P. L. & Passot, T. 2016 Reduced models accounting for parallel magnetic perturbations: gyrofluid and finite Larmor radius-Landau fluid approaches. J. Plasma Phys. 82, 705820601.
Tatsuno, T., Dorland, W., Schekochihin, A. A., Plunk, G. G., Barnes, M., Cowley, S. C. & Howes, G. G. 2009 Nonlinear phase mixing and phase-space cascade of entropy in gyrokinetic plasma turbulence. Phys. Rev. Lett. 103, 015003.
Teaca, B., Bañón Navarro, A., Jenko, F., Brunner, S. & Villard, L. 2012 Locality and universality in gyrokinetic turbulence. Phys. Rev. Lett. 109, 235003.
Teaca, B., Banón Navarro, A., Told, D. & Jenko, F.2016 Kinetic intermittency in magnetized plasma turbulence. E-print arXiv:1607.03421.
Thornhill, S. G. & ter Haar, D. 1978 Langmuir turbulence and modulational instability. Phys. Rep. 43, 43.
Tsytovich, V. N. 1995 Lectures on Non-linear Plasma Kinetics. Springer.
Vedenov, A. A., Velikhov, E. P. & Sagdeev, R. Z. 1962 Quasilinear theory of plasma oscillations., Nucl. Fusion Suppl. Part 2 465.
Watanabe, T.-H. & Sugama, H. 2004 Kinetic simulation of steady states of ion temperature gradient driven turbulence with weak collisionality. Phys. Plasmas 11, 1476.
Weiland, J. 1992 Nonlinear effects in velocity space and drift wave transport in tokamaks. Phys. Fluids B 4, 1388.
Zakharov, V. E. 1972 Collapse of Langmuir waves. Sov. Phys. JETP 35, 908.
Zakharov, V. E., L’vov, V. S. & Falkovich, G. 1992 Kolmogorov Spectra of Turbulence I: Wave Turbulence. Springer.
Zakharov, V. E., Musher, S. L. & Rubenchik, A. M. 1985 Hamiltonian approach to the description of non-linear plasma phenomena. Phys. Rep. 129, 285.
Zeldovich, Ya. B., Ruzmaikin, A. A. & Sokoloff, D. D. 1990 The Almighty Chance. World Scientific.
Zocco, A. & Schekochihin, A. A. 2011 Reduced fluid-kinetic equations for low-frequency dynamics, magnetic reconnection, and electron heating in low-beta plasmas. Phys. Plasmas 18, 102309g.
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