Wave Kinetic Equation Research Articles

Kinetic wave equations for drift and shear Alfvén waves inaxisymmetric toroidal plasmas

The conductivity tensor operator obtained by integrating the linearized Vlasovequation along the unperturbed orbits in the drift approximation isspecialized to frequencies well below the ion cyclotron frequency. Thediamagnetic contribution and the contribution of the particle drift motion areincluded, and finite Larmor radius effects are retained to all orders. Underappropriate assumptions of broad validity we obtain a set of coupledintegrodifferential wave equations for the amplitudes of the Fouriercomponents of the fields in the poloidal direction. These equations describedrift and shear Alfvén waves in axisymmetric toroidal geometry, and are in aform well suited for efficient numerical discretization. We have verified thatthey reproduce correctly several known results based on the gyrokinetic theoryunder the appropriate conditions. The essential equivalence between thetraditional Vlasov approach and the gyrokinetic formalism is therebyestablished. The range of validity of the approximations required to obtainthis result is explicitly stated, and the physical origin of the contributionsto the oscillating current and charge densities are clearly identified.

Numerical modeling of the constant flux spectra for surface gravity waves in a case of angular anisotropy

The process of constant flux spectra formation is studied numerically, for the case of angular anisotropy. The method of investigation is based on a numerical solution of the nonlinear four-wave kinetic equation for gravity waves, added with the terms of a source and a sink for wave energy. Stationary solutions of such an equation are just the constant flux spectra under investigation. Two principally different situations were considered: the case of a constant energy flux to the high frequencies (the case of “flux up”) and the case of a constant wave action flux to the low frequencies (the case of “flux down”). It was found that, for the case of flux down, the stationary two-dimensional spectra are angularly isotropic, independently of the initial anisotropy of the system. But for the case of flux up, the stationary spectra have specific angular anisotropy. For both cases, corresponding one-dimensional spectra have shapes close to ones analytically derived in [Dokl. Akad. Nauk SSSR, 170 (1996) 1292–1295 (in Russian); Izv. Akad. S. USSR, Atmos. Ocean Phys. 18 (1982) 970–979], for the case of angular isotropy. Numerous features of the process considered are specified, and the mentioned peculiarities of the results are discussed.

Generalized weak turbulence theory

In this article we present the derivation of a generalized weak turbulence kinetic equation for unmagnetized collisionless plasmas in a uniform medium. For the sake of simplicity the present formulation assumes longitudinal electrostatic interaction only, and the effects of spontaneous thermal fluctuations are ignored. In spite of these simplifications, the present formalism represents a generalization of the existing weak turbulence theory in that a nonlinear eigenmode excited in a turbulent plasma with frequency close to twice the plasma frequency is incorporated into the discussion. Traditional weak turbulence theory emphasizes various linear and nonlinear interactions among wave modes in quiescent plasmas (i.e., Langmuir and ion-sound waves). In contrast, the present formalism describes linear and nonlinear interactions among Langmuir, ion-sound, and the new nonlinear eigenmode. Nonlinear wave kinetic equations for these modes are systematically derived, and the particle kinetic equation which generalizes the well known quasilinear diffusion equation, is also derived.

Nonlinearplasmon damping in the quark-gluon plasma

On the basis of the approximate dynamical equations describing thebehaviour of a quark-gluon plasma (QGP) in the semiclassical limit andthe Yang-Mills equation, the kinetic equation for longitudinal waves(plasmons) is obtained. The equality of the matrix elements for nonlinearscattering of a plasmon by QGP particles in covariant and temporal gaugesare established using the Ward identities. The physical mechanismsdefining nonlinear Landau damping are analysed. The problem of theconnection between the nonlinear Landau damping rate of longitudinaloscillations with the damping rate obtained within the framework of thehard thermal loops approximation, is considered. It is shown that thegauge-dependent part of the nonlinear Landau damping rate for theplasmons with zero momentum vanishes on mass-shell.

In search of the elusive zonal flow using cross-bicoherence analysis

We show that the modulational instability growth rate of zonal flows is determined directly from the quasilinear wave kinetic equation. We also demonstrate the relation between zonal-flow growth and the cross bispectrum of the high-frequency drift-wave-driven Reynolds stress and the low-frequency plasma potential by explicit calculation. Experimental measurements of the spatiotemporal evolution of the spectrum integrated bicoherence at the L-->H transition near the edge shear layer indicate a modification in the nonlinear phase coupling, which might be linked to the generation of sheared ExB flows.

Basic Principles of Kinetic Equations for Particles, Waves, and Magnetic Field Lines

In this overview, the main arguments for a kinetic description of a classical, non-relativistic many-body system are reviewed. The need and strategy for a kinetic description of plasma particles are discussed. The Vlasov, the Landau-Fokker-Planck, and the Balescu-Lenard equations are presented as the most useful kinetic equations for the particle distribution functions. In the second part, some simple applications are discussed. First, collision frequencies are derived. Second, it is shown that in the mean field approximation a linearization of the initial value problem can already give interesting insights into the (collective) dynamic behaviors. Third, quasi-linear and weak turbulence theories are discussed. Fourth, it is argued why in many cases a reduction to a plasmadynamic (fluid) description is appropriate, and popular truncations are summarized. Finally, the generality of the statistical methods is demonstrated on the example of magnetic field line diffusion.

Comparative analysis of two formulations of geometrical optics. The effective dielectric tensor

Two apparently diverse formulations of geometrical optics relevant to space- and time-varying dispersive, anisotropic media are shown to be equivalent by virtue of the relationship between the effective dielectric tensor and the plane-wave dielectric tensor. As a counterpart of the wave kinetic equation governing the wave-action density in phase space, the equation for the transport of action density in physical configuration space is obtained, which entails, in particular, a wavepacket adiabatic invariant in the limit of which the effect of both dissipation and (internal and/or external) current-sources is negligible.

On the Kolmogorov–Zakharov spectra of weak turbulence

Zakharov discovered that wave kinetic equations for weak turbulence have exact power-law solutions which are similar to the Kolmogorov spectrum of hydrodynamic turbulence. We present a considerably simplified derivation of these solutions, i.e. Kolmogorov–Zakharov spectra. Even to a greater extent we simplify the derivation of certain “universal” corrections to these spectra, obtained by Kats and Kontorovich. The technique is utilized as well to derive the expressions for the so-called Mellin functions (that describe the behavior of weak turbulence in the vicinity of the Kolmogorov–Zakharov spectra). Using this general approach, we obtain Kolmogorov–Zakharov spectra for a class of weak turbulent media with dissipative nonlinearity. We also find a large family of anisotropic spectra, which are exact solutions of the corresponding kinetic equation. As a physical example, we consider the plasma turbulence when the main nonlinear process is the scattering of plasmons by electrons.

Generalized action invariants for drift waves-zonal flow systems

Generalized action invariants are identified for various models of drift wave turbulence in the presence of the mean shear flow. It is shown that the wave kinetic equation describing the interaction of the small scale turbulence and large scale shear flow can be naturally writen in terms of these invariants. Unlike the wave energy, which is conserved as a sum of small- and large-scale components, the generalized action invariant is shown to correspond to a quantity which is conserved for the small scale component alone. This invariant can be used to construct canonical variables leading to a different definition of the wave action (as compared to the case without shear flow). It is suggested that these new canonical action variables form a natural basis for the description of the drift wave turbulence with a mean shear flow.

Coupled hydromagnetic wave excitation and ion acceleration at interplanetary traveling shocks and Earth's bow shock revisited

A revised version of the self‐consistent theory of ion diffusive shock acceleration and the associated generation of hydromagnetic waves is presented. The theory generalizes and corrects the theory of Lee [1982, 1983]. Lee assumed a linear dependence of the anisotropic part of the ion distribution function on the cosine of the ion pitch angle. Here the wave growth or damping rate is again calculated using linear theory, but a more general ion anisotropy is calculated using the pitch angle diffusion equation. The wave intensity satisfies a wave kinetic equation, and the ion omnidirectional distribution function satisfies the energetic particle transport equation. These coupled equations are solved numerically and compared with an analytical approximation similar to that derived by Lee. The analytical approximation provides an accurate representation of both the proton distribution and the wave intensity. A comparison is made between the predicted wave magnetic power spectral density adjacent to the shock as a function of frequency and the wave spectrum measured by ISEE 3 at the November 11–12, 1978, interplanetary traveling shock. There is excellent agreement between the predicted and measured power spectral density in the frequency range of 0.03–0.3 Hz. A comparison is also made between the predicted total wave energy density and that observed upstream of Earth's bow shock by the AMPTE/IRM satellite for a statistical survey of ∼400 near‐to nose events from late 1984 and 1985. This comparison revises the result presented by Trattner et al. [1994]. The correlation between the observed wave power and that predicted, based on the observed energetic proton energy density, is very good with a correlation coefficient of 0.92. However, the average observed wave magnetic energy density is ∼63% of that predicted, suggesting possible wave dissipation which is not included in the theory.

Turbulence of capillary waves — theory and numerical simulation

An ensemble of weakly interacting capillary waves on a free surface of deep ideal fluid is described statistically by methods of weak turbulence. The stationary kinetic equations for capillary waves have an exact Kolmogorov solution which gives for the spatial spectrum of elevations asymptotics I k = C( P 1/2/ σ 3/4) k −19/4. The Kolmogorov constant C is found analytically together with the interval of locality in K → -space. Direct numerical simulation of the dynamical equations in the approximation of small surface angles confirms the presence of almost istropic Kolmogorov spectrum in the large k → region. Besides, at small amplitudes of the pumping, an esentially new phenomenon is found: “frozen” turbulence, in which, despite the big number of interacting waves (of the order of 100) there is no energy flux toward high k → . This phenomenon is connected with the finiteness of the region (or, in other words, discreteness of the spectrum in Fourier space). This is believed to be universal for different sorts of nonlinear systems.

Electron beam acceleration by Compton scattering of extraordinary waves

Acceleration and heating of a high-energy or relativistic electron beam due to Compton scattering induced by nonlinear Landau damping of almost perpendicularly propagating extraordinary waves are investigated theoretically based on kinetic wave equations and transport equations derived from Vlasov–Maxwell equations. The numerical analysis of nonlinear wave–particle coupling coefficients in these equations has shown that the electron beam can be accelerated efficiently to the phase velocity of the beat wave near the speed of light by Compton scattering of two extraordinary waves with almost the same frequencies. The acceleration or deceleration of the electron beam occurs in accordance with whether the phase velocity of the beat wave is slightly larger or smaller than the velocity of the electron beam, respectively. For the frequencies of two waves lower than the upper-hybrid frequency (ω≲ωh), or for those exceeding the right-hand cutoff frequency (ω≳ωR), the acceleration and deceleration of the electron beam become significantly strong.

Gauge invariance of the Landau nonlinear damping decrement of Bose excitations in a quark-gluon plasma: Part I

On the basis of the approximate dynamical equations describing the behavior of quark-gluon plasma (QGP) in the semiclassical limit and Yang-Mills equation, the kinetic equation for longitudinal waves (plasmons) is obtained. With the Ward identities the gauge invariance of obtained nonlinear Landau damping rate is proved. The physical mechanisms defining nonlinear scattering of a plasmon by QGP particles are analyzed. The problem on a connection of nonlinear Landau damping rate of longitudinal oscillations with damping rate, obtained in the framework of hard thermal loops approximation, is considered. It is shown that the gauge-dependent part of nonlinear Landau damping rate for the plasmons with zero momentum vanishes on mass-shell.

The problem of nonlinear landau damping in quark-gluon plasma

On the basis of the semiclassical kinetic equations for quark-gluon plasma (QGP) and Yang-Mills equation, the generalized kinetic equation for waves with regard to its interaction is obtained. The physical mechanisms defining nonlinear scattering of a plasmon by QGP particles are analyzed. The problem on a connection of nonlinear Landau damping rate of longitudinal oscillation with damping rate, obtained on the basis of hard thermal loops approximation, is considered.

Kinetic model of Alfvén wave light-ion gyroresonance heating

The absorption of Alfvén waves by gyroresonant interaction with the light-ions in an ionospheric oxygen-hydrogen plasma with a parallel magnetic field gradient is reconsidered, taking into account simultaneously the plasma inhomogeneity and the finite temperature of the resonant ions (Ti≠0). A kinetic full wave equation is derived, that is valid in the vicinity where the wave frequency matches the local proton gyrofrequency. It is analytically solved for the case of a Lorentzian distribution function. The energy transmission, reflection and absorption coefficients, for waves incident from the high magnetic field side onto the gyroresonant interaction region, are found to be the same as for the cold plasma case (Ti=0). Conversely, the absorption of waves incident from the low magnetic field side is found to be enhanced and strongly depends on the ionic temperature, whereas their transmission to the high magnetic field side occurs still in the same proportion as for the cold plasma case. Elaborating local dispersion curves and evaluating the full wave solutions enable us to interpret qualitatively these results and to extrapolate them for the case of a Maxwellian distribution function. For realistic ionospheric plasma conditions a sensible increase of the gyroresonant absorption of the waves is thus found.

Basic Principles for a Kinetic Description of Particles and Waves

In this overview, the main arguments for a kinetic description of a classical non-relativistic many particle system are reviewed. First, the need and strategy for a kinetic description of plasma particles is discussed. The Vlasov, the Landau-Fokker-Planck, and the Balescu-Lenard equations are presented as the most useful kinetic equations for the particle distribution functions. It is shown that a linearization of the initial value problem can already give interesting insights into the dynamic behaviors. In many cases a reduction to a plasmadynamic (fluid) description is appropriate, and popular truncations are summarized. Finally, the basic methods for a kinetic description of waves are presented. When some wave excitations are driven unstable and the collective motion of particles dominates, the wave-kinetic equations will be the appropriate dynamical equations. It is shown that spectra of the Kolmogorov-Obukhov type are exact stationary solutions of the latter.

Theory of drift wave turbulence

Toroidal ion temperature gradient (ITG) driven drift mode turbulence has been analyzed analytically and numerically. By using weak nonlinearity arguments and random phase approximation, dynamic and wave kinetic equations are derived. It is found that three different nonlinearities, namely E × B, convective and diamagnetic nonlinearities, play important role in the turbulent spectral transfer. The power spectra of the weak ITG-mode turbulence are obtained analytically for |k| ≫ 1 and |k| < 1 ranges in the wave number space. It is shown that forward energy cascading due to convective and diamagnetic nonlinearities will balance the inverse energy cascading due to E × B nonlinearity at |k| ≈ 1/ρs (k is wave number, ρs = (cs/ωci), cs is the sound velocity and ωci is the ion cyclotron frequency) and results in energy condensation at |k| ≈ 1/ρs.

Ion heating up to 1 MeV range with higher harmonic ICRF wave on JT-60U

The properties of protons under acceleration by an ion cyclotron range of frequency (ICRF) waves with the second to fourth hydrogen harmonics have been investigated in the JT-60U tokamak at the Japan Atomic Energy Research Institute (JAERI). Protons have been accelerated up to 1 MeV in the presence of an ICRF wave of fixed frequency, neutral beams (NB), and a fixed toroidal magnetic field which is scanned through several plasma discharges. The tail temperature of the protons, which is evaluated in the range 0.32 - 0.86 MeV, has been observed to increase in the second to third harmonics, however increase of the tail temperature in the third to fourth harmonics has not been observed clearly. Furthermore, the dependence of tail temperature on the harmonic number has been found to be in qualitative agreement with results from a simulation code analysis based upon the one-dimensional Fokker - Planck equation coupled with the kinetic wave equation. Experimental values for the stored energy of the accelerated ions have shown, however, that the response of stored energy to changes in absorbed ICRF power is much stronger than the response to changes in harmonic number. Also, the incremental energy confinement times for heating discharges matching the third and fourth harmonics ( and ) of hydrogen have been observed to be less than half that for those matching the second harmonic. It has been found that suppression of the absorbed ICRF power accompanied with the occurrence of cavity resonance in the and heating discharges reduces the stored energy of the accelerated ions and the incremental energy confinement time.

Quantum chaos in a stochastic model

An idealized problem in the theory of long-range ocean acoustic propagation through randomly fluctuating mesoscale structure is reformulated as a time-dependent quantum mechanics problem with a time-varying Markovian potential function. The effective Planck constant is ℏ=δε−2/3, where δ=(k0l)−1≪1 measures diffraction, and ε≪1 is the strength of the mesoscale structure. Also k0 is the acoustic wave number and l is the scale length of the mesoscale structure. In the formal classical limit (ℏ→0), the parabolic ray equations are nonintegrable and the relative motion of two particles for small initial separations exhibits chaotic behavior characterized by a positive Lyapunov exponent, ν∼ε2/3/l. It is shown that this physically relates to the exponential proliferation of caustics. A generalized wave kinetic equation (GWKE) is derived for the evolution in phase space of the two-particle Wigner function. The GWKE is analytically examined for ℏ≪1 by a novel boundary layer method, called the ‘‘extended quantum notch method,’’ which yields several important results: First, the ‘‘log breakdown time (range)’’ tb∼ν−1log(ℏ−1) is obtained where semi-classical theory breaks down due to the saturation of caustics, and it is shown that this is the range where the scintillation index saturates to unity; and, second, a wave (quantum) manifestation of classical chaos is found to be the exponential decay of the scintillation index beyond its peak value. [Work supported by ONR.]

Momentum-space diffusion due to resonant wave–wave scattering of electromagnetic and electrostatic waves in a relativistic magnetized plasma

The momentum-space diffusion equation and the kinetic wave equation for resonant wave–wave scattering of electromagnetic and electrostatic waves in a relativistic magnetized plasma are derived from the relativistic Vlasov–Maxwell equations by perturbation theory. The p-dependent diffusion coefficient and the nonlinear wave—wave coupling coefficient are given in terms of third-order tensors which are amenable to analysis. The transport equations describing energy and momentum transfer between waves and particles are obtained by momentum-space integration of the momentum-space diffusion equation, and are expressed in terms of the nonlinear wave—wave coupling coefficient in the kinetic wave equation. The conservation laws for the total energy and momentum densities of waves and particles are verified from the kinetic wave equation and the transport equations. These equations are very useful for the theoretical analysis of transport phenomena or the acceleration and generation of high-energy or relativistic particles caused by quasi-linear and resonant wave—wave scattering processes.