Turbulent Relaxation Research Articles

The Formation of Dissipative Structures and Chaotic Beam Relaxation in the Strongly Dissipative Beam‐Plasma Instability

AbstractNumerical solutions of a nonlinear system of equations describing the evolution of the strongly dissipative beam‐plasma instability are presented (time or space dependence of the unstable waves, the evolution of the beam distribution function and the corresponding phase space representations are given). Such an instability establishes if the linear growth rate of the unstable waves is small in comparison to the effective collision frequency of the electrons of the dissipative plasma. It is shown that after saturation of the instability there is an efficient energy transfer from the directed motion of the beam electrons to the thermal motion of the plasma electrons. Arguments are given that the strongly dissipative beam‐plasma instability is possibly the basic plasma heating mechanism which operates in the quasi‐stationary state of the beam plasma discharge. Self‐consistent initial conditions are used for the numerical calculations. Formation and evolution of dissipative structures (vortex structure, chaotic structure) were observed. A profound investigation of the chaotic or turbulent beam relaxation mechanism is presented. Such a relaxation mechanism develops in the case of the nonresonant dissipative instability (e.g. in resistive wall‐electron beam amplifiers). The method of investigation of dissipative structures presented in this paper (application of Shannon's information theory and results of the theory of dynamical systems with chaotic behaviour) can be extended to studies of other beam‐plasma systems within the framework of partial kinetic description.

Interaction of counter-streaming plasmas

The possibility of hot dense plasma creation for the filling of open traps by thermalization of the direct energy of supersonic plasma clusters accelerated by electrodynamic accelerators has been studied theoretically and experimentally. The mean free path of particles in these traps, as a rule, exceeds typical system size, therefore, it is of great interest to study the turbulent mechanisms of counter-streaming plasmas relaxation. A calculation of relaxation length is very important because such turbulent relaxation of counter-streaming plasmas is observed experimentally. Investigation of counter-streaming plasma interaction is of general physical concern. Such situations occur, for example, in astrophysics: the shock wave structure problem, the relaxation of ionosphere plasma flow behind satellites, the spreading of laser-produced plasma in surrounding gas, etc. A detailed discussion is given of the relaxation of counter-streaming plasma due to turbulent mechanisms: the two-stream ion-acoustic instability.

Turbulent relaxation of compressible plasmas with flow

Relaxation of compressible plasmas to an equilibrium with flows is studied. The magnetohydrodynamic (MHD) equations with large parallel thermal conductivity and ergodic field lines show that ∫v⋅B is an invariant even with compression. Also, S=∫ρ ln( p/ργ) is the only entropy-like invariant of the MHD equations with infinite parallel thermal conductivity; S increases in time with finite thermal conductivity. The other invariants are energy, helicity ∫A⋅B, mass ∫ρ, and possibly angular momentum. Equilibria are found by extremizing energy while conserving these invariants or by maximizing entropy with the energy and other invariants as constraints. These invariants are complete in the sense of generating all equilibria that form after relaxation with ergodic field lines. For parallel flows, there are three classes of solutions characterized by the sign of dρ/dB2 and the mirror mode parameter. A sufficient condition for stability is derived. This condition is never satisfied by the class with dρ/dB2>0, indicating the possibility of unstable resistive interchanges. The class with dρ/dB2<0 is stable if generalizations of the local firehose and mirror criteria are satisfied, and if generalized Taylor helicity eigenvalues exceed those of the equilibrium. A comparison of these conditions with those of Frieman and Rotenberg is discussed.

Turbulent spreading of artificial periodic inhomogeneities in the lower ionosphere

We discuss the turbulent relaxation of artificial periodic electron density inhomogeneities (lattices) under conditions characteristic of the D region of the ionosphere. For uniform, isotropic turbulence, the relaxation time is determined by the mean square velocity of turbulent motion and the scale of the inhomogeneity.

Energy principle with global invariants: Applications

Minimum-energy equilibrium states of a pressureless cylindrical plasma are constructed subject to a recently proposed set of constraints appropriate to turbulent relaxation dominated by a tearing mode of single helicity. Stability to ideal and resistive modes is examined. It is found that the relaxed states of minimum energy are always stable to the dominant mode, as well as a class of other low m and n modes, but that there are typically high n residual instabilities in the case of the pitch branch, and a residual m=3, n=2 resistive instability is found for an otherwise stable narrow window in the tokamak branch. It is argued that the residual instabilities may be compatible with experimental observations.

Russian Text Ignored

AbstractThe plasmon turbulence parametrically excited by two strong external waves with a frequency difference close to twice the electron plasma frequency is investigated in underdense plasmas. The nonlinear stabilization of the growing plasmons due to scattering by ions is described by a nonlinear integro‐differential equation for the spectral plasmon energy density. The dependence of the turbulent relaxation of the plasmon energy and its stationary level on the parameters of the plasma and the pump waves is given by analytical and numerical solutions of this equation.

Soliton flash, turbulent ion heating and relaxation oscillations in strong Langmuir turbulence

The mode coupling equations for Langmuir and ion acoustic waves are derived including an external electric field Eext(t), nonlinear ion terms. Landau damping, and (turbulent) ion viscosity. A numerical scheme for integrating these equations shows excellent conservation of invariants and of stationary solutions for Eext(t)=0 and zero dissipation. Numerical solutions show (a) a transient solitary wave structure ('soliton flash') 65, (b) a turbulent quasi-stationary state in which ions are heated predominantly, and (c) two kinds of relaxation oscillations depending upon the ion viscosity. Comparison with recent experiments shows qualitative agreement.

A probabilistic approach to turbulent flame theory

A given statistically stationary field of wind-tunnel grid-generated turbulence acting as the carrier flow of a combustible species is considered. The turbulence is assumed dynamically unaffected by the chemical reaction, i.e. the heat release is taken to be small compared with the initial thermal enthalpy of the mixture. Consequently, the density may be treated as a constant. Reaction rates with explicit scalar-property dependence are used but no restriction on their functional form is imposed, i.e. linear, nonlinear or transcendental reaction rates can be exactly studied. A hierarchy of turbulent probability density functions is presented. The closure problem is discussed. The advantages of working with probabilistic equations rather than in terms of traditional moment formulations are outlined. A closure is proposed at the second order level for the joint probability density function of the scalar and fluctuating velocity equation. After some simplifying assumptions a second Damköhler number based on the diffusive time computed using the turbulence microscale and a dimensionless relaxation parameter, namely, the ratio of a turbulence relaxation time and a characteristic chemical time are seen to be the two controlling parameters. A linear reaction rate is finally chosen for possible comparison with previous results. The limit of very low turbulence intensity can be approximated by the statistically homogeneous study and is analytically solved for three different initial conditions. The general inhomogeneous case is numerically solved for different values of the two parameters mentioned above and various sets of initial scalar fluctuations and turbulence intensities. The effect of the initial scalar-fluctuating velocity correlation is also explored. The homogeneous turbulence limit is for all purposes identical to the numerical solution for turbulence intensity equal to 1%.

Relaxation of a parametric turbulence in a plasma

A theoretical (analytic and numerical) study is made of a nonstationary (transient) plasma turbulence initiated by a parametric instability involving decay of a pump wave into two plasmons and saturated by the stimulated scattering of plasmons on ions. The main features of the relaxation of such a parametric turbulence are governed by the characteristics of the noise saturation mechanism and are in this sense typical of many other instabilities which can be stabilized by stimulated scattering. A solution of a nonlinear integro-differential equation for the spectral density of the noise energy reveals the complete evolution pattern of the appearance and growth of a turbulent noise, which oscillates in time, has a modulated spectrum, and finally attains a smooth stationary (steady-state) distribution. A nonstationary parametric turbulence can develop fully in a dense laser plasma during a nanosecond pulse emitted by a high-power neodymium laser.

Irreversible interaction between cumulants in homogeneous, isotropic, two-dimensional turbulence theory

The hierarchy describing homogeneous and isotropic two-dimensional turbulence is used to show that one of the effects of (n + 1)-th order cumulants is to introduce a turbulent relaxation frequency (or eddy-damping rate) in the equation for the rate of change of n-th order cumulants. As a result, it is possible to construct a Markovian eddy-damped (n + 1)-th order cumulant-discard approximation in which the eddy-damping rate simulates the irreversibility due to the infinite serie of cumulants with order greater than n. In the case of our Markovian eddy-damped quasi-normal approximation the governing equations for the evolution of the energy spectrum are compared to those of stochastic models like the generalized random phase approximation or the self-consistent field approximation.