Quasilinear Effects Research Articles

Energy spectrum of α-particles escaping from a plasma as a result of turbulent diffusion

On the assumption that the main reason for the escape of α-particles from plasma is turbulent diffusion, the authors find the fraction of thermonuclear reaction energy transmitted to the plasma, the flux of α-particles escaping from the plasma and the energy distribution of the particles. The energy distribution depends to a great extent on whether quasi-linear relaxation of the α-particle distribution function is appreciable or not. The case of negligibly small quasi-linear distortion of the α-particle distribution is examined, as well as the case where quasi-linear effects are dominant. It is shown that the energy spectrum of the α-particles leaving the plasma as a result of turbulent diffusion contains both slowed-down and accelerated (ϵ > 3.5 MeV) particles.

A computer simulation of cold plasma effects on the whistler instability for geostationary orbit plasma parameters

This paper presents the results of a computer simulation of the modifications produced in the whistler instability by the addition of varying amounts of cold plasma to an infinite homogeneous warm (w) plasma with parameters typical of those observed in the equatorial magnetosphere beyond the plasmapause (β ≡ 4πnwkT∥,w/B0² = 0.8 and (T⊥/T∥)w = 2). Specifically, the particle-in-cell simulation method for electromagnetic interactions in infinite homogeneous magnetized plasmas has been used. The experiment has been performed for three values of relative cold (c) plasma concentrations, nc/nw = 1, 2, and 5; prior results for nc/nw = 0 are also discussed. The results of the experiment confirm the basic predictions of theoretical linear stability analysis for the prescribed conditions. In the linear stage of evolution the addition of cold plasma (1) increases the instability range in k space and thus enables resonant interactions of electrons having lower parallel velocities and (2) displaces the k location of the maximum rate of growth toward larger k values. Additional quasi-linear effects indicate that as a consequence of particle diffusion in velocity space the thermal anisotropy ratios (T⊥/T∥)w decrease continuously, the result being a decrease of the larger k mode growth rates and a displacement of maximum wave amplitude toward smaller k values. All the systems relax to quasi-stationary states with approximately the same (T⊥/T∥)w value, ≃1.35 (which linearly should be unstable). However, in this stage, the larger the initial nc/nw ratio, the larger the total energy in the electromagnetic waves.

Mode-Coupling and Wave-Particle Interactions for Unstable Ion-Acoustic Waves

A theory for the spatial development of linearly unstable, coupled waves is presented in which both quasilinear and mode-coupling effects are treated in a self-consistent manner. Steady-state excitation of two waves (with frequencies ω1, ω2) is assumed at the boundary x=0, the plasma being homogeneous in the y and z directions. Coupled equations are derived for the x dependence of the amplitudes of the primary waves (ω1, ω2) and the secondary waves, nω1+mω2 (n and m being integers), correct through terms of second order in the wave amplitude, eφ/Te, but without the usual approximation of small growth rates. This general formalism is then applied to the case of coupled ion-acoustic waves driven unstable by an ion beam streaming in the direction of the x axis. If the modifications of the ion beam by the waves (“quasilinear” effects) are ignored, “explosive” instabilities (singularities in all of the amplitudes at finite x) are found even when all of the waves have positive energy. If these wave-particle interactions are included, the solutions are no longer singular, and all of the amplitudes have finite maxima, at locations in reasonable agreement with experimental results of Taylor and Ikezi.

A dynamic theory of type III solar radio bursts

Numerous experimental data indicate that type III solar radio bursts are generated by the streams of fast electrons in the corona. The process of the electron acceleration in the flare region is, in general, of the character of a short time local explosion. As a result, a spatially limited stream with inhomogeneous front and back is formed in the corona. The present paper shows that such a spatial structure of fast electrons radically changes the dynamics of the stream instability development. In particular, for example, despite strong quasilinear relaxation, in this case the electron stream can generate plasma waves in the corona for a long time due to faster electrons escaping out of the front of the stream. The extension of the stream in the outer corona where the collisions are negligible is of similar character. The maximum of the energy density in the packet of excited plasma waves travels in the corona with constant mean velocity which is defined by the fast-electron energy at the moment of their formation in the flare region. Therefore, in spite of the considerable influence of quasilinear effects on the stream motion, the velocity of type III sources found in terms of the drift velocity remains unchanged. This creates the illusion of stream stabilization. The energy dissipation of plasma waves for the low frequency type III bursts is fully determined by Landau damping in the tail of the stream. Because of this, the temperature estimates of the outer corona from time profiles of type III bursts are incorrect. The theoretical curves of the time variation of radiation at different frequencies agree well with the experimental data in the hectometer wave range under the assumption that the electromagnetic wave generation takes place at the second harmonic of the plasma frequency. In this case it is necessary to decrease the electron density in the solar corona at distances of about 5 to 30 R⊙ 4 times as compared with the previous densities previously derived from type III data.

Nonlinear Theory of the Cross-Field Instability—Explosive Mode Coupling

Nonlinear evolution of low frequency drift waves in weakly ionized plasmas is investigated, including the quasilinear and mode coupling effects. An equation governing the development of these waves is presented in terms of an external parameter (electric field) under appropriate assumptions. The equation states that the plasma system experiences two types of transition when one increases the external parameter from zero to infinity. Below a certain value of the parameter any perturbations are stabilized, but at the critical point the system makes a soft-type transition and an unstable drift mode is excited which finally reaches a finite amplitude. When the parameter is increased beyond the critical point, the system sees another transition point beyond which the unstable mode can no longer remain a finite amplitude but diverges within a finite time; this implies an explosive instability. This explosive transition is caused by the mode coupling process which acts to destabilize the initially unstable (linear) mode. It is inferred thereby that in the real plasma a strong turbulence is realized in which a number of modes are strongly excited and interact with each other.

Self-consistent solution of the quasi-linear theory: Application to the spectral shape and intensity of VLF waves in the magnetosphere

During a gyroresonant interaction taking place inside the magnetosphere, the relaxation of the warm electrons anisotropy, which is due to quasi-linear effects, changes the growth rate of the VLF waves. A self-consistent treatment of this process enables us to compute the frequency spectrum of these waves. Assuming that the energy diffusion of electrons occurs in a characteristic time much larger than the time required for pitch-angle diffusion, we solve (in a self-consistent way) the coupled system which consists of the diffusion equation for particles and energy conservation for waves. Such a method is similar to the one used by Drummond and Pines (1962) to compute the quasi-linear spectrum in the electrostatic case. In the specific case of emission by magnetospheric electrons, the results of our computation is in agreement with satellite datas from the point of view of both the spectral shape and of the wave intensity.

Zur nichtlinearen Theorie der Strahl‐Plasma‐Wechselwirkung IV. Numerische Ergebnisse: Elektronenschwingungen

AbstractThe spatial variations of the electric field E and the electron distribution function fa for the case of beam plasma amplification are computed on the basis of a nonlinear approximation developed in the earlier parts of the present paper [1–3]. The essential results are: With increasing distance x from the point, where the beam enters the plasma, the amplitude Ã1 of the first harmonic increases, reaches a maximum value at some point xm and than decreases monotonically without reaching a further maximum. In the region around xm we find a large increase of the higher harmonics, i.e. increasing nonlinearity, and also an increasing phase velocity, which soon approches the beam velocity.The non‐oscillating part of the distribution function of the beam electrons deformes, but not so much as the quasilinear theory predicts, so that amplitude settling must be interpreted as a nonlinear but not as a quasilinear effect. The results show a good agreement with some typical experiments on beam plasma interaction.

Collisional effect on the quasilinear process associated with the ion-wave instability

Using a simple model we show that the quasilinear feedback effect of enhanced ion-wave fluctuations on the average electron distribution in a current-carrying plasma is negligible as far as collisions can effectively prevent oscillation of trapped electrons.

Hypothesis on the origin of the spiral structure of the galaxies

The spiral waves in a model galaxy consisting of differentially rotating and non-rotating subsystems are considered on the basis of participating phenomena. The subsystems involved represent the Populations I and II of normal spirals, respectively. The spiral waves in such a systems are unstable in the Landau sense. Due to this instability they grow up to attain finite amplitude. This growth is stopped by a non-linear effect (the quasilinear effect), the steady state with waves of finite amplitude being established. The hypothesis is proposed that these waves should be identified with the spiral structure of the galaxies.

The quasilinear theory of two-stream cyclotron instability of electron beams

The paper is devoted to a study of cyclotron instability (Ω ≈ Ωe) of two opposite beams of electrons from the aspect of the quasilinear effects. For this case of instability the authors evolve the diffusion equation in the velocity space for an averaged distribution function. They estimate the relaxation timesΤ⊥,Τz of the spreading of the distribution function in the velocity space. At the end of the paper, they compare their results with an experiment described by M. Seidl and P. sunka [Phys. Letters11 (1964), 31].

Solution of the Kinetic Equation for an Unstable Plasma in an Electric Field

The response of a homogeneous plasma to the sudden application of a strong, uniform electric field, E0, is studied by solving, self-consistently, the coupled equations for the one- and two-particle distribution functions. The field is assumed to be large compared to that which produces a ``runaway'' current, so that particle-particle collisions have a negligible effect on the current flow, and particle-wave interactions dominate. The principal approximations are the neglect of intrinsic three-particle correlations (i.e., linearization in fluctuations) and the use of an adiabatic ansatz for the time dependence of the fluctuations. For the case of unequal temperatures, Te ≫ Ti, two different methods of solution are employed: a moment approximation, in which the distribution functions are assumed Maxwellian, with mean velocity and width determined self-consistently; and a more exact treatment in which the distribution function for velocities parallel to the applied field is determined by direct numerical solution of the kinetic equation using on-line computational techniques. In both cases, it is found that the unstable ion acoustic waves grow from the thermal fluctuation level to a magnitude sufficient to cause a sharp and sudden decrease in the current. Examination of the kinetic equation shows this drop to be ascribable to the dynamic friction associated with the unstable waves. Although waves propagating parallel to the applied field are most important at early times, those propagating at finite angles soon become dominant; this casts some doubt on one-dimensional approximations to this problem. The limitation of the exponential growth of unstable waves resulting from the non-linear (or quasilinear) effects is clearly evident, notwithstanding the presence of the external driving force, E0.