Weak Turbulence Theory Research Articles

Beam-Plasma Structures at Propagation of Electron Beams inPlasma

In the frame of the theory of weak turbulence the propagation ofelectron beams is considered. Using the smallness of quasilinear timetransition from kinetic equations to gas-dynamic ones has been done.The obtained gas-dynamic equations are solved for one, two andN monoenergetic beams. It is shown that each beam generallypropagates as a beam-plasma structure consisting of electrons andLangmuir waves. Interaction of the structures at their propagationleads to electron exchange and spatial shape changing. Differentsituations are discussed in dependence on initial beam velocities anddensities.

Nonlinear particle dynamics in a broadband turbulence wave spectrum

In the statistical quasilinear theory of weak plasma turbulence, charged particles moving in electrostatic fluctuations diffuse in velocity, i.e. the velocity variance 〈Δv2(t)〉 increases linearly with time t, for times long compared with the auto-correlation time τac of the field, which may be estimated as the reciprocal of the spectral width of the fluctuations. Recent test-particle simulations have revealed a new regime at very long timescales t[Gt ]τac where quasilinear theory breaks down, for intermediate field amplitudes. As this behaviour is not consistent with a diffusion on quasilinear timescales, the problem of the motion of particles in a broadband wave field, for the case of a slowly growing field, is considered here from a purely dynamical point of view, introducing no statistics on the field and no restriction on the amplitude of this field. By determining, on a given timescale, and in the frame of wave–particle interaction, the spectral width over which waves interact efficiently with a particle, a new timescale is found: the nonlinear time of wave–particle interaction τNL∝ (spectral density of energy)−1/3[Gt ]τac. This is the correlation time of the dynamics. For times shorter than τNL, the particles trajectories remain globally regular, and do not separate: they follow a quasifractal set of dimension 2. For times long compared with τNL, there appears a ‘true’ diffusive regime with mixing and decorrelation, due to nonlinear mixing in phase space and the localization of the wave–particle interaction. These theoretical results are confirmed by a numerical study of the velocity variance as a function of time. In particular, the particle dynamics really do become diffusive on timescales several orders of magnitude longer than that predicted by quasilinear theory (namely [Gt ]τNL[Gt ]τac). Finally, deviations from the quasilinear value of the diffusion coefficient and wave growth rate, discussed in the literature, are explained.

Gasdynamic description of electron-beam flying-off in a plasma

The propagation of one and two electron beams in a plasma is considered in the case when the source is time-dependent. When the electron-plasmon interaction time is assumed to be much less than the electron flying-off time, the main gasdynamic equations are obtained on the basis of the quasilinear theory of weak turbulence. The solution for a monoenergetic beam is a beam–plasma structure moving with constant velocity. In general, two beams propagate as two beam–plasma structures with constant velocities. However, the second electron beam influences the flying-off dynamics of both beams. This peculiar interaction modifies the shapes of the structures.

Generalized plasma dispersion functions

Generalized plasma dispersion functions have recently arisen in the derivation of an exact expression for the quadratic response tensor for a plasma with constituent species having Maxwellian velocity distributions. A range of mathematical properties satisfied by the generalized plasma dispersion functions are derived here, including recursion relations, series expansions, and approximate analytic forms, including Padé approximants. These properties are of central importance to the application of the exact quadratic response tensor to descriptions of second-order processes in weak and strong turbulence theory, such as three-wave interactions in warm plasmas.

Velocity correlations in weakly turbulent surface waves

The Hamiltonian theory of weak turbulence for surface waves has been developed quite extensively @1#. The focus has been centered on the Kolmogorov spectrum n(k), but the experimental attempts to confirm the theory have been very limited. For capillary surface waves, elaborate experimental techniques have been used in order to test the scaling properties derived for the spectrum n(k) @2#. Numerical calculations of the spectrum have also been carried out for capillary waves @3#. In this Brief Report, we consider velocity correlations rather than the spectrum itself. We derive an expression for the variance ^udv(R)u 2 & of the distribution of velocity differences over a ~horizontal! distance R. Unlike the spectrum n(k), ^udv(R)u 2 & can rather easily be studied experimentally by measuring the separation of tracers launched on the fluid surface @4,5#. Experiments on relative particle motion have previously been performed for ~deep-water! capillary waves @5#. For the weakly turbulent regime, it was argued that

The Interaction of Shocks with Dispersive Waves II. Incompressible‐Integrable Limit

This is the second in a two‐part series of articles in which we analyze a system similar in structure to the well‐known Zakharov equations from weak plasma turbulence theory, but with a nonlinear conservation equation allowing finite time shock formation. In this article we analyze the incompressible limit in which the shock speed is large compared to the underlying group velocity of the dispersive wave (a situation typically encountered in applications). After presenting some exact solutions of the full system, a multiscale perturbation method is used to resolve several basic wave interactions. The analysis breaks down into two categories: the nonlinear limit and the linear limit, corresponding to the form of the equations when the group velocity to shock speed ratio, denoted by ε, is zero. The former case is an integrable limit in which the model reduces to the cubic nonlinear Schrödinger equation governing the dispersive wave envelope. We focus on the interaction of a “fast” shock wave and a single hump soliton. In the latter case, the ε=0 problem reduces to the linear Schrödinger equation, and the focus is on a fast shock interacting with a dispersive wave whose amplitude is cusped and exponentially decaying. To motivate the time scales and structure of the shock‐dispersive wave interactions at lowest orders, we first analyze a simpler system of ordinary differential equations structurally similar to the original system. Then we return to the fully coupled partial differential equations and develop a multiscale asymptotic method to derive the effective leading‐order shock equations and the leading‐order modulation equations governing the phase and amplitude of the dispersive wave envelope. The leading‐order interaction equations admit a fairly complete analysis based on characteristic methods. Conditions are derived in which: (a) the shock passes through the soliton, (b) the shock is completely blocked by the soliton, or (c) the shock reverses direction. In the linear limit, a phenomenon is described in which the dispersive wave induces the formation of a second, transient shock front in the rapidly moving hyperbolic wave. In all cases, we can characterize the long‐time dynamics of the shock. The influence of the shock on the dispersive wave is manifested, to leading order, in the generalized frequency of the dispersive wave: the fast‐time part of the frequency is the shock wave itself. Hence, the frequency undergoes a sudden jump across the shock layer.In the last section, a sequence of numerical experiments depicting some of the interesting interactions predicted by the analysis is performed on the leading‐order shock equations.

Weak Plasma Turbulence Theory and Some Other Items of Plasma Kinetics

A new approach to a plasma kinetic description is discussed, the beginnings of which were published recently (Erofeev 1997a). It is shown that calculations of the three-wave collision integral following this approach confirm the intensity and structure of the three-wave collision integral obtained in the traditional theory. The reported kinetics extend the area of applicability for the weak plasma turbulence theory: apart from waves it properly accounts for the effect of various other plasma nonlinear structures of the type of solitons, drift vortices, collapsing cavities and so on. Some directions for further studies are also discussed.

Kinetics of Langmuir Turbulence: An Equation for the Two‐time Correlation Function

AbstractThe two‐point correlation function is reduced to expansion in terms of the two‐time correlation function with improved accuracy. The ideas reported extends the applicability area for calculations typical to the weak plasma turbulence theory on the “strong plasma turbulence”. Some implications of the results obtained for the dynamics of Langmuir turbulence are discussed.

The asymptotic method developed from weak turbulent theory and the nonlinear permeability and damping rate in QGP

With asymptotic method developed from weak turbulent theory, the kinetic equations for QGP are expanded in fluctuation field potential $A^T_\mu $. Considering the second-order and third-order currents, we derive the nonlinear permeability tensor function from Yang-Mills field equation, and find that the third-order current is more important in turbulent theory. The nonlinear permeability formulae for longitudinal color oscillations show that the non-Abelian effects are more important than the Abelian-like effects. To compare with other works, we give the numerical result of the damping rate for the modes with zero wave vector.

Nonlinear wave structures in collisional plasma of auroral E-region ionosphere

Abstract. Studies of the auroral plasma with small-scale inhomogenieties producing the VHF-radar reflections (radar aurora) when observed in conditions of the saturated Farley-Buneman instability within the auroral E region, show strong nonlinear interactions and density fluctuations of 5–15%. Such nonlinearity and high fluctation amplitudes are inconsistent with the limitations of the weak turbulence theory, and thus a theory for arbitrary amplitudes is needed. To this end, a nonlinear theory is described for electrostatic MHD moving plasma structures of arbitrary amplitude for conditions throughout the altitude range of the collisional auroral E region. The equations are derived, from electron and ion motion self-consistent with the electric field, for the general case of the one-dimensional problem. They take into account nonlinearity, electron and ion inertia, diffusion, deviation from quasi-neutrality, and dynamical ion viscosity. The importance of the ion viscosity for dispersion is stressed, while deviation from the quasi-neutrality can be important only at rather low plasma densities, not typical for the auroral E region. In a small amplitude limit these equations have classical nonlinear solutions of the type of "electrostatic shock wave" or of knoidal waves. In a particular case these knoidal waves degrade to a dissipative soliton. A two-dimensional case of a quasi-neutral plasma is considered in the plane perpendicular to the magnetic field by way of the Poisson brackets, but neglecting the nonlinearity and ion inertia. It is shown that in these conditions an effective saturation can be achieved at the stationary turbulence level of order of 10%.

A one-dimensional model for dispersive wave turbulence

A family of one-dimensional nonlinear dispersive wave equations is introduced as a model for assessing the validity of weak turbulence theory for random waves in an unambiguous and transparent fashion. These models have an explicitly solvable weak turbulence theory which is developed here, with Kolmogorov-type wave number spectra exhibiting interesting dependence on parameters in the equations. These predictions of weak turbulence theory are compared with numerical solutions with damping and driving that exhibit a statistical inertial scaling range over as much as two decades in wave number.

Nonlinear response of driven systems in weak turbulence theory

A method is presented for predicting the saturation levels and particle transport in weakly unstable systems with a discrete number of modes. Conditions are established for either steady-state or pulsating responses when several modes are excited for cases where there is and there is not resonance overlap. The conditions for saturation and the associated transport are discussed. Depending on parameters, the saturation level can be low, with only a small fraction of the available free energy released to waves and with no global transport, or the saturation level can be quite high, with almost complete conversion of free energy to wave energy coupled with rapid transport.

On the stationary energy spectra for unidirectionally propagating internal gravity waves

Starting from the Euler equations in the Boussinesq approximation for a stratified fluid in two dimensions we establish a kinetic equation for a statistical ensemble of weakly non-linear internal gravity waves propagating in the same horizontal direction. We find a transformation of this equation to a standard form adopted in the weak turbulence theory and then, limiting ourselves to high vertical-low horizontal wavenumber waves, get equilibrium (Rayleigh-Jeans) and non-equilibrium (Kolmogorov) stationary energy spectra, among them a k −3 vertical wavenumber spectrum.

Turbulence of capillary waves.

A numerical model for direct simulation of the surface of ideal fluid based on the expansion of theHamiltonian of the surface up to terms of fourth order is developed. For the case of capillary wavewe observe the formation of powerlike spectrum of spatial elevations close to one predicted by weak-turbulent theory

The Interaction of Shocks with Dispersive Waves I. Weak Coupling Limit

We introduce and analyze a model for the interaction of shocks with a dispersive wave envelope. The model mimicks the Zakharov system from weak plasma turbulence theory but replaces the linear wave equation in that system by a nonlinear wave equation allowing the formation of shocks. This paper considers a weak coupling in which the nonlinear wave evolves independently but appears as the potential in the time‐dependent Schrodinger equation governing the dispersive wave. We first solve the Riemann problem for the system by constructing solutions to the Schrodinger equation that are steady in a frame of reference moving with the shock. Then we add a viscous diffusion term to the shock equation and by explicitly constructing asymptotic expansions in the (small) diffusion coefficient, we show that these solutions are zero diffusion limits of the regularized problem. The expansions are unusual in that it is necessary to keep track of exponentially small terms to obtain algebraically small terms. The expansions are compared to numerical solutions. We then construct a family of time‐dependent solutions in the case that the initial data for the nonlinear wave equation evolves to a shock as t → t* < ∞. We prove that the shock formation drives a finite time blow‐up in the phase gradient of the dispersive wave. While the shock develops algebraically in time, the phase gradient blows up logarithmically in time. We construct several explicit time‐dependent solutions to the system, including ones that: (a) evolve to the steady states previously constructed, (b) evolve to steady states with phase discontinuities (which we call phase kinked steady states), (c) do not evolve to steady states.

Friction coefficient and correlation length of stochastic magnetic field lines

An averaged transverse velocity of stochastic magnetic field lines between two tearing modes is presented by using the renormalized weak turbulence theory. The averaged velocity gives the friction coefficient of moving magnetic field lines which is expressed in terms of a renormalized propagator. A correlation length (time) of magnetic field lines, which is the time integrated renormalized propagator, is presented with respect to various parameters. The diffusion coefficient given in a previous paper (Sugimoto et al 1994 Plasma Phys. Control. Fusion 36 383) and the friction coefficient obtained here are related to each other by the so-called Einstein relation. Although the renormalized propagator given here involves more effects such as collision and friction types than that in the previous paper, which has the term of the diffusion type effect alone, the former effects do not greatly modify the renormalized propagator. Effects of collision and friction types do not play an important role in the propagator and the diffusion type effect alone is essential.

Stability of weak turbulence spectra in superfluid helium

We investigate turbulence phenomenon in superfluid helium. The kinetic theory of weak turbulence is applied to examine the stability problem by the Mellin transformation method. We show that the instability of the well-known power distribution appears at large k as well as at small k .

Stochastic properties of the modulational interaction in packets of random waves

The stochastic properties of turbulent plasma waves are taken into account in the consideration of the modulational instability. The limits of applicability of the theory of weak plasma turbulence in the description of the modulational processes are found. It is shown that this theory can be applied for the description of the modulational interaction only for sufficiently wide wave spectra (and/or sufficiently low levels of plasma turbulence) when the modulational interaction of different modes in the packet is small. It is demonstrated that the development of the modulational interaction processes results in a rapid decrease of the wave stochasticity.

Modulational processes and limits of weak turbulence theory.

Modulational instability of a broad spectrum of Langmuir waves in unmagnetized plasmas is considered. The problem of thresholds of the modulational instability is studied in detail. Applicability limits (which are associated with stochastic properties of turbulent plasma waves) of the theory of weak plasma turbulence are discussed. It is demonstrated that the weak turbulence theory is appropriate for the description of modulational processes developed only for sufficiently wide wave spectra (and/or sufficiently low levels of plasma turbulence). Instability thresholds found on the basis of the theory of weak plasma turbulence are compared with those obtained on the basis of the WKB ansatz, which does not use the random phase approximation. It is demonstrated that for isotropic three-dimensional as well as for one-dimensional Langmuir wave spectra both these approaches result in similar instability conditions and it is shown that for some special spectra of Langmuir waves those are necessary and sufficient (in the mathematical sense) conditions for the development of the modulational instability. Applicability limits of the WKB approximation for the description of the modulational processes are considered in detail and compared with those of the weak turbulence theory.

A model for the three‐dimensional magnetic field correlation spectra of low‐frequency solar wind fluctuations during Alfvénic periods

Using statistical homogeneity, magnetic field fluctuations in the solar wind during Alfvénic periods are analyzed in terms of the two independent polarizations allowed for each wave vector k. It is shown that the energy spectra of the two polarizations can be related both to the correlation tensor and to the variance matrix, which is generally used to characterize the anisotropy of the turbulence. Assuming simple anisotropic power law models, the parameters defining the spectra of the two polarizations are determined by fitting the eigenvalues of the variance matrix on the corresponding eigenvalues evaluated by Bavassano et al. (1982) for Helios 2 data. Then the corresponding form of the correlation tensor is obtained. In particular, we found that the spectrum of polarization [1] fluctuations (corresponding, in a weak turbulence theory approach, to the Alfvén mode) is steeper than the polarization [2] spectrum (corresponding to the magnetosonic modes). While the former spectrum is dominated by wave vectors parallel to B0, the latter is strongly flattened on the plane containing the radial and the mean magnetic field B0 directions. A discussion of these results in connection with other observational and theoretical issues is outlined.