Wave Kinetic Equation Research Articles

Stationary turbulent spectra of toroidal ηi mode

Toroidal ion temperature gradient (ITG) driven drift mode turbulence has been analyzed analyti- cally and numerically. By using weak nonlinearity arguments and random phase approxima- tion, dynamic and wave kinetic equations are derived. Three different nonlinearities, namely E×B, convective, and diamagnetic nonlinearities, play important roles 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. 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 the wave number, ρs=cs/ωci, where cs is the sound velocity and ωci is the ion cyclotron frequency) and results in energy condensation at |k|≊1/ρs.

Spectral evolution of two-layer weak geostrophic turbulence. Part I: Typical scenarios

Abstract. Long-time evolution of large-scale geophysical flows is considered in a β-plane approximation. Motions in an infinite 2-layer model ocean are treated as a system of weakly nonlinear Rossby waves (weak geostrophic turbulence). The evolution of the energy spectrum of the barotropic and the baroclinic modes is investigated on the basis of numerical experiments with the kinetic equation for baroclinic Rossby waves. The basic features of free (nonforced inviscid) spectral evolution of baroclinic flows are similar to those of the barotropic motions. A portion of the energy is transferred to a sharp spectral peak while the rest of it is isotropically distributed. The peak corresponds to an intensive nearly zonal barotropic flow. Typically, this well-defined barotropic zonal anisotropy inhibits the reinforcement of its baroclinic analogy. For a certain set of initial conditions (in particular, if the barotropic zonal flow is not present initially), a zonal anisotropy of both modes is generated. The interplay between the multimodal nearly zonal flow components leads to the excitation of large-scale (several times exceeding the scale of the initial state), mostly meridional, baroclinic motions at the expense of the barotropic nearly zonal flow. The underlying mechanism is explained on the level of elementary mixed-triad interaction. The whole wave field retains its essentially baroclinic as well as spectrally broad nature. It evidently tends towards a thermodynamically equilibrated final state, consisting of the superposition of a (usually barotropic, but occasionally multimodal) zonal flow and a wave system with a Raleigh-Jeans spectrum. This evolution takes place as a multi-staged process, with fast convergence of the modal spectra to a local equilibrium followed by a more gradual adjustment of the energy balance between the modes.

Relativistic electron beam acceleration by cascading nonlinear Landau damping of electromagnetic waves in a plasma

Acceleration and heating of a relativistic electron beam by cascading nonlinear Landau damping involving three or four intense electromagnetic waves in a plasma are studied theoretically based on kinetic wave equations and transport equations derived from relativistic Vlasov–Maxwell equations. Three or four electromagnetic waves excite successively two or three nonresonant beat-wave-driven relativistic electron plasma waves with a phase velocity near the speed of light [vp=c(1−γ−2p)1/2, γp=ω/ωpe]. Three beat waves interact nonlinearly with the electron beam and accelerate it to a highly relativistic energy γpmec2 more effectively than by the usual nonlinear Landau damping of two electromagnetic waves. It is proved that the electron beam can be accelerated to more highly relativistic energy in the plasma whose electron density decreases temporally with an appropriate rate because of the temporal increase of γp.

Evolution of surge in the sea of finite depth under the effect of wave-wave interactions

The wave-wave kinetic equation for surface gravity waves occurring in a deep sea and a sea of finite depth is solved numerically, using the Runge-Kutta technique. Spectral evolution was the result only of the waves’ being non-linear, without any contribution from wave generation and dissipation. The JONSWAP frequency spectra and the angular spectra of various widths were used to perform calculations. The existence of a steady angular wave spectrum, following its long evolution in a deep sea, has been confirmed here. For the sea of finite depth, a new result has been obtained. It exhibits the ‘focusing’ of the frequential and the angular spectra when the wavelength of a harmonic from the spectral maximum equals 2π, of the depth of the sea. With the depth further decreasing, the wave spectrum swiftly expands. The three-dimensional wave field in a deep sea becomes two-dimensional in the shallows.

Spectral evolution of surface waves in a deep sea due to the effect of wave-wave interactions

The wave-wave kinetic equation for surface gravity waves in a deep sea is solved numerically, using the Runge-Kutta technique. Spectral evolution of waves resulted only from their being non-linear, with no wave generation and decaying taking place. To perform computations the JONSWAP-type frequency spectra and a variety of angular wave spectra were used. The angular spectrum of waves turned out to be stable. The frequency spectrum differed from the JONSWAP spectrum in that it had a high-frequency part, which was not similar to the Phillips spectrum. The form of the high-frequency spectral slope was determined as a result of spectral evolution and proved to have the form of the ‘−6’ law.

On the Hasselmann and Zakharov Approaches to the Kinetic Equations for Gravity Waves

Abstract It is shown for the first time analytically that two different approaches describing gravity wave turbulence introduced by Hasselmann in 1962 and Zakharov in 1968 result in the same kinetic equation for the second-order correlator.

Nonlinear ion acoustic waves in a strongly coupled plasma subject to external electric field

Instability of ion acoustic waves is studied in a collision-dominated two-component plasma immersed in a strong electric field. Effects of nonlinear mode coupling can cause the instability saturation. These effects are studied in the frame of the kinetic equation for sound waves. The turbulence spectrum is found to have a stream-like space distribution. The result is compared with the Kadomtsev-Petviashvili formula. Applications to the problem of nonlinear skin effect are discussed.

Quasilinear evolution of cyclotron maser instability.

A quasilinear analysis of the relativistic electron cyclotron maser instability is presented. A background plasma is assumed to support the wave motion, while the instability is driven by a tenuous population of energetic electrons possessing a loss-cone feature. The analysis makes use of an efBcient moment method. In this approach, evolution equations for the moments of particle distribution function are derived from the particle kinetic equation. Then, a self-similar model of the loss-cone electron distribution function is imposed. Simultaneously, the wave kinetic equation is solved. The resulting fully self-consistent set of equations that governs the evolution of the particles and unstable waves is solved numerically under physical parameters that represent typical solar microwave burst sources.

Quasilinear analysis of loss-cone driven weakly relativistic electron cyclotron maser instability

This paper presents a quasilinear analysis of the relativistic electron cyclotron maser instability. Two electron populations are assumed: a low-temperature background component and a more energetic loss-cone population. The dispersion relation is valid for any ratio of the energetic to cold populations, and includes thermal and relativistic effects. The quasilinear analysis is based upon an efficient kinetic moment method, in which various moment equations are derived from the particle kinetic equation. A model time-dependent loss-cone electron distribution function is assumed, which allows one to evaluate the instantaneous linear growth rate as well as the moment kinetic equations. These moment equations along with the wave kinetic equation form a fully self-consistent set of equations which governs the evolution of the particles as well as unstable waves. This set of equations is solved with physical parameters typical of the earth’s auroral zone plasma.

The ergodic limit of multipass absorption for fast wave current drive in tokamaks

In many parameter regimes of interest for fast wave current drive (FWCD) in tokamaks, direct absorption of the fast wave by resonant electrons is a weak process and multipass absorption is an important issue. Although both full wave codes and ray tracing codes have been developed to model FWCD, in the multipass regime these tools are computationally intensive, and yield little insight into the nature of the solutions. In this work, an alternative approach is considered. Based on the wave kinetic equation, a natural limit emerges for the multipass regime, where wave energy density, convected along stochastic ray trajectories, uniformly fills the entire accessible phase space. In this ergodic, weak damping limit, the absorbed power density and corresponding wave-driven current density are readily obtained by calculating the appropriate set of one-dimensional k-space integrals at every point in configuration space. The method is used here to model FWCD on the DIII-D tokamak [R. I. Pinsker and the DIII-D Team, Plasma Physics and Controlled Nuclear Fusion Research 1992 (International Atomic Energy Agency, Vienna, 1993), Vol. 1, p. 683]. An example for reactor-grade plasma parameters is also considered.

Numerical modelling of flux spectra formation for surface gravity waves

By means of direct numerical solution of the kinetic equation for surface gravity waves, it is shown that under certain conditions the constant flux spectra of nonlinear waves, first predicted by Zakharov & Filonenko (1966) for an infinite frequency domain, can be formed in a finite frequency interval. For the case of angular isotropic spectra the conditions and timescales of this flux spectra formation are evaluated.

Five-wave kinetic equation for surface gravity waves

A kinetic equation for the spectrum of random surface gravity waves, describing the spectral energy transfer over the spectrum, due to the combined effects of four- and five-wave weakly non-linear interactions has been developed. Equation kernels are given in the explicit form for a fluid of finite depth. It is noted that five-wave interactions break the wave conservation law inherent to the four-wave kinetic equation.

Electron beam acceleration by nonresonantly driven relativistic electron plasma waves

Acceleration and heating of a relativistic electron beam induced by nonlinear Landau damping of intense electromagnetic waves in a plasma are investigated theoretically based on kinetic wave equations and momentum-space diffusion equations derived from relativistic Vlasov–Maxwell equations. Two electromagnetic waves excite nonresonantly a beat-wave driven relativistic electron plasma wave with a phase velocity near the speed of light [vp=c(1−γ−2p)1/2, γp=ω/ωpe]. This wave interacts nonlinearly with the electron beam and accelerates effectively it to a highly relativistic energy γpmec2. When the beat-wave frequency equals the electron plasma frequency, the beam acceleration and the beat-wave energy density become maximum, and they are equal to those by stimulated Raman scattering.

Wave systems with an infinite number of invariants

Can some extra quantities (besides energy and momentum) be conserved in 3-wave (resonance) interactions? The paper describes all 3-wave interactions in which infinitely many independent quantities are conserved. The results are obtained due to the application of web geometry. The correspondence between 3-wave interactions and 3-webs is established. All wave systems, related to the inverse scattering tranform method, correspond to the simplest (the so called coordinate) web. Each extra invariant of a 3-wave interaction defines new conservation law of the corresponding wave kinetic equation (which describes turbulence of the waves). The results are applied to the nonlinear theory of decay instability.

Statistical theory of wave propagation and multipass absorption for current drive in tokamaks

In this work the effect of ray stochasticity on the multipass absorption of lower-hybrid waves, used to drive current in tokamaks, is considered. In toroidal geometry, stochasticity arises as an intrinsic property of the Hamiltonian ray trajectories for lower-hybrid waves. Based on the wave kinetic equation, a diffusion equation is derived, with damping and sources, for the wave energy density in the stochastic layer. This equation is solved simultaneously with the electron Fokker–Planck equation to describe the quasilinear flattening of the electron distribution function and the subsequent modification of the wave damping. Steady-state solutions of this system (obtained numerically) indicate that the spectral gap is filled in a self-regulating manner, so that the boundaries of the diffused wave spectrum are independent of the level of ray stochastic diffusion. This allows the development of a simple (semianalytic) model for the self-consistent wave spectrum and the radial profile of absorbed power.

One-Dimensional Calculation Including the Space Charge Effects of FEL Amplifiers

Numerical calculations are performed based on the one-dimensional model for a tapered FEL amplifier with the combined helical wiggler and axial guide magnetic field operating in Raman regime. The longitudinal electric field due to space charge bunching can be formulated either by solving the coupled Maxwell and linear fluid equations or by solving the coupled kinetic and longitudinal wave equations. Both formulations are incorporated into the numerical code and the simulation results are compared. The criterion for the consideration of the space change effect depends on the value of the electron beam parameter ξ(ξ 2 = e 2 n b /γ 0 m c 2 k 2 w ). Besides, the perspective parameter studies are performed.

Simulation of tapered FEL amplifiers in millimeter and infrared regions

Numerical calculations are performed based on the one-dimensional model for a tapered FEL amplifier with an axial guide field operating in the Raman regime and an IR-FEL amplifier based on a test linac of the Pohang Accelerator Laboratory (PAL). The longitudinal electric field can be formulated either by solving the coupled Maxwell and linear fluid equations or by solving the coupled kinetic and longitudinal wave equations. Both formulations are incorporated into the numerical code and the simulation results are compared. The criterion whether the space charge effect should be considered or not can be described in terms of the electron beam parameter ξ(ξ 2 = e 2n b ϵ 0mγ 0c 2k w 2) . We have also theoretically investigated the possibility of a tunable IR-FEL with a test linac of energy 20–60 MeV of PAL with effects of energy spread and space charge. The current density of the test linac, if properly enhanced, makes it feasible to amplify a 14 kW signal of 10.6 μm radiation to a 2 GW level.

Invariants of 4-wave interactions

We give a complete description of one-dimensional 4-wave resonance interactions in which some extra quantities (besides momentum, energy, number of quasiparticles) are conserved. In this way we obtain new consideration laws for the kinetic equations for waves. In particular, we consider waves in optical fibers, the system of four resonantly interacting wave packets, long wave interactions of annihilation-creation type, various wave systems with quadratic dispersion laws. The results can be important for various problems concerning nonlinear wave dynamics, e.g. for nonlinear optics of waveguides.

A two-nonlinearity model of dissipative drift wave turbulence

A simple, one-field, two-nonlinearity, drift wave model equation is derived to describe the dynamics of a nonuniform magnetized plasma by taking into account the effects of dissipative trapped electron response in the turbulence dynamics. Because of the nonadiabatic response of trapped electrons, mode couplings by both the E×B drift and the polarization drift nonlinearities are present. In this work, the statistical dynamics for this dissipative drift wave turbulence is investigated using the EDQNM (eddy-damped quasinormal Markovian) closure scheme. In particular, apart from the eddy viscosity, a large nonlinear frequency shift is shown to be induced by cross coupling of the two nonlinearities. Thus instability drive is modified by this turbulent back reaction. By taking into account this self-consistency effect, a wave kinetic equation is derived, and the density fluctuation spectrum is obtained in different parameter ranges. The results show that the dynamics of dissipative drift wave turbulence is fundamentally different from that of the familiar Hasegawa–Mima model, because E×B drift nonlinearity blocks the low-k condensation of fluctuation energy. It is shown that both the E×B drift nonlinearity and the nonlinear frequency shift effect transfer energy nonlocally from large to small scales and, in contrast to the predictions of dimensional analysis, their contribution to the nonlinear transfer processes are actually of the same order as that of the polarization drift nonlinearity, even within the Hasegawa–Mima regime. This results in a significant modification of the Hasegawa–Mima spectrum for the short-wavelength drift waves.

Effect of magnetic and density fluctuations on the propagation of lower hybrid waves in tokamaks

Lower hybrid waves have been used extensively for plasma heating, current drive, and ramp-up as well as sawteeth stabilization. The wave kinetic equation for lower hybrid wave propagation is extended to include the effects of both magnetic and density fluctuations. This integral equation is then solved by Monte Carlo procedures for a toroidal plasma. It is shown that even for magnetic/density fluctuation levels on the order of 10−4, there are significant magnetic fluctuation effects on the wave power deposition into the plasma. This effect is quite pronounced if the magnetic fluctuation spectrum is peaked within the plasma. For Alcator-C-Mod [I. H. Hutchinson and the Alcator Group, Proceedings of the IEEE 13th Symposium on Fusion Engineering (IEEE, New York, 1990), Cat. No. 89CH 2820-9, p. 13] parameters, it seems possible to be able to infer information on internal magnetic fluctuations from hard x-ray data—especially since the effects of fluctuations on electron power density can explain the hard x-ray data from the JT-60 tokamak [H. Kishimoto and JT-60 Team, in Plasma Physics and Controlled Fusion (International Atomic Energy Agency, Vienna, 1989), Vol. I, p. 67].