Table of contents

This study is of a review nature. In section 1 is described the method of investigation of the stability of stationary oscillations of not too large amplitude (the ratio of the hydrodynamic wave velocity to phase velocity serves as a small parameter). Instabilities arise because of the presence of a positive feedback between small sinusoidal oscillations propagating in the "background" of the finite-amplitude wave. Instabilities recall quasi-particle decays: frequencies and wave vectors of developing oscillations are related to the frequency and wave vector of the initial wave by relations typical of the laws of energy and momentum conservation. A characteristic peculiarity of growth rates of developing oscillations is their proportionality to the value of the initial wave amplitude.

A concrete example of finding the growth rate of small perturbations is discussed; the stability of a sinusoidal Alfvén wave of small amplitude (section 2A) is investigated. Instability of such a wave is related to compressibility of the medium (for incompressible media instability disappears).

For large-amplitude oscillations there is no general method for stability investigation. However even here a positive feedback between small oscillations can appear. For this reason an Alfvén wave is unstable (in a compressible medium!) if it has an arbitrary amplitude with a sawtooth profile for the magnetic field lines (the chosen profile of the magnetic field lines enables one to solve the problem precisely (section 2B)).

In the concluding portion of the review (sections 3 and 4) the results of investigation of the stability of other oscillation modes are given, such as ion-acoustic oscillations of an non-isothermal plasma, electron Langmuir oscillations, magneto-acoustic oscillations, etc.

Investigation has been made of the effect of thermal plasma electron and ion motion on wave propagation in a non-relativistic, low-pressure plasma across a magnetic field. The dispersion equation for an ordinary wave has the solutions ω_s(k) close to sω_α (s = 1, 2...) for any ratio of wavelength (ƛ = 1/k) to the Larmor radius ϱ_α of the α type particles having thermal velocity (see fig. 1)). In order of magnitude ω_s(k)−sω_α≈−x_αω_α for kϱ_α≈1.

The dispersion equation for an extraordinary wave has also an analogous solution (see fig. 5) in the high-frequency region (ω ≈ ω_e), and also a solution corresponding to longitudinal plasma oscillations for k²ϱ_α² ≫ x_α Longitudinal-oscillation frequencies for the case of a dense plasma, when Ω_e ≫ ω_e (Ω_e is the electron Langmuir frequency, and Ω_e their gyro-frequency) decrease for an increase of kϱ_e, diminishing from the values ω ≈ sω_e (s = 2, 3...) for kϱ_e ≪ 1, down to ω ≈(s − l)ω_e for kϱ_e ≫ 1 (see fig. 5). For a low-density plasma (Ω_e ⪅ ω_e), plasma-frequency behaviour is shown in figs. 2 and 3.

In the low-frequency region (ω ⪅ ω_i) the dispersio equation for an extraordinary wave determines (for the case of a cold plasma) the frequency ω = kF_A corresponding to a magneto-acoustic wave (V_A is the Alfvén velocity). This expression is inapplicable for kϱ₁≈s(s = 1, 2...) when ω≈sω_i. In this region it is necessary to take into account thermal ion motion. Behaviour of the various branch frequencies of an extraordinary hot-plasma wave for ω ≈ ω_i fin coi is comparatively complex (see fig. 6).

Investigation has been carried out of the self-excitation (caused by beam instability) of electron (high frequency) branches of the oscillations discussed, for the passage of an ion cyclotron wave through a plasma.

It is shown that the intensity of cyclotron radiation of electrons in the plasma-magnetic-field boundary layer decreases by mc²/T times (T is plasma temperature) in comparison with radiation in the void, because dielectric permeability is large in a frequency range of the order of cyclotron frequency. Radiation intensity per unit plasma surface equals in order of magnitude, where n₀ is plasma density.

Because of plasma resonance in a non-uniform layer, bremsstrahlung in frequency ranges lower than the maximum Langmuir frequency is mc²/T times as large as cyclotron radiation, even for the case of rare collisions.

If Langmuir oscillations in the layer are unstable because of the presence of electron drift velocity, radiation of the layer exceeds thermal radiation at the Langmuir frequency by (mc²T²/e⁶n₀)^1/2 times.

The spectra and damping of some new branches of hot-plasma oscillations propagating perpendicular to or almost perpendicular to the external magnetic field H₀ are discussed. Investigation has been performed of (a) an oscillation branch adjoining longitudinal ion-cyclotron oscillations, (b) extraordinary ion-cyclotron and magneto-acoustic waves and (c) ordinary ion-cyclotron and hybrid waves. Phase velocity of these waves along the magnetic field V = ω/k_|| is considerably larger than ion thermal velocity (k_|| is the wave-vector projection in the magnetic-field direction).

Waves adjoining longitudinal cyclotron oscillations have a wavelength of the order of the Larmor radius of an ion with thermal velocity and frequencies of the order of ion cyclotron frequency; the phase velocity of these waves along the magnetic field is considerably smaller than thermal electron velocity υ_e. These waves can be separated from the general dispersion queation for the case of a low-pressure plasma when x ≪ 1, where (n₀ is the density; T_i, the plasma ion temperature). For x ≪ cos²θ ≪1 (θ is the angle between H₀ and the propagation direction) this branch changes to longitudinal ion cyclotron oscillations.

The dispersion equation for an ordinary wave for V≪v_e has two solutions, corresponding to the hybrid wave with frequency ω = ω₀ = k_||kV_sV_A/ω_i), where V_s = (T_e/m_i)^1/2, ω_i = eH₀/m_i C and V_A = H₀/(4πn₀)^1/2 and the ion-cyclotron wave with ω≈sω_i (s = 1, 2,...).

For the oscillations pointed out the damping rates which are caused by Cerenkov absorption by plasma electrons and ions and cyclotron absorption by plasma ions have been found. Oscillation excitation by fluxes of low-density charged particles passing through plasma along a magnetic field has also been examined. Cerenkov and cyclotron wave excitation by both a hot and a cold beam, have been investigated.

The trapped reversed flux a 26⋅kJ theta pinch with known initial conditions is measured in as a function of time with a radial multiple probe and compared with numerical calculations. The measured flux agrees with the numerical calculations during the implosion of the plasma but decreases rapidly after the maximum compression whereas the calculated flux remains almost constant because of the increasing plasma temperature.

It is shown that radiation cooling by probe impurities can explain this enhanced field diffusion in the neighbourhood of the probe. The numerical calculations can be fitted to the experimental results by increasing the radiation loss due to hydrogenic continuum radiation by a few times 10⁴. It is estimated that the heat flow to the probe, which is mainly carried by ions (with an energy according to the floating potential of the probe), liberates enough oxygen to account for this factor due to line radiation. Radial framing-camera photographs show a tearing mode resistive instability developing near the probe which seems to be started off by this radiation cooling.

A limit on the reliability of probe measurements is given as a function of plasma density and temperature.

The plasma column of a theta pinch with an antiparallel trapped magnetic field, has a tendency to break up into several discrete plasma rings. With a smaller theta-pinch apparatus experiments were done to search for mechanisms responsible for triggering such macroinstabilities. Probe ports inserted into the plasma and also external side pockets on the discharge tube cause an early and drastic breakup of the plasma column. But even in smooth tubes discrete plasma rings occur. They could be suppressed by having some discharges with a parallel magnetic field preceding those with a reversed one. This "memory" effect could be explained by the contamination of the inner tube wall by water after each discharge. In this way the plasma column with an antiparallel trapped magnetic field was stable until peak magnetic field. This stability proved to be insensitive to adding amounts of oxygen–as much as 4.6%—to the hydrogen gas and also to lowering the degree of pre-ionization. Finally comparison has been made for the observed formation time of the plasma rings and their number with theoretical predictions. With plasma temperatures derived from the appearance of O IV lines the formation time can be explained by the resistive diffusion time whereas the theoretical predictions of Furth, Killeen and Rosenbluth for tearing-mode resistive instabilities seem rather to give too small values.

Oscillations of frequency ν≈100 kHz were detected in a plasma created by an electron beam in a magnetic field. The amplitude of these oscillations depends on the strength of the external field of frequency ω≫ν The high-frequency field is in turn modulated by the low frequency, and this produces combination frequencies (ω±ν). The oscillation amplitudes at the combination frequencies ω+ν and (ω–ν are not equal; as in the case of the Raman effect, this may be related to the effectiveness of the interaction of various frequencies and to the direction of energy transfer.

In this work we have found the depth of penetration of a travelling magnetic field into plasma in the presence of electron increase owing to Hall effect. Since the electrons move in the direction of the field displacement, penetration depth increases in comparison with the case of electrons that are motionless on the average and is determined by the effective frequency Ω = k|V − ν⁰|; V is field velocity, ν⁰ electron velocity (that is, there occurs a peculiar "Doppler effect"). Velocity ν⁰ for a non-linear case is a function of the coordinates and depends on the magnitude of the field, its displacement velocity and also on conduction. The electrons are entrained in the direction of the field motion, which always leads to skin-depth increase. A rough calculation of the penetration depth can be made according to the formula δ = δ₀ (1 – ⟨B⟩²/B₀²)⁻¹ δ₀=C/(4πσkV)^½ is the "normal" skin depth; B₀ is the alternating-magnetic-field amplitude on the plasma surface; ⟨B⟩ is the magnetic-field magnitude on the plasma surface; this field creates a current of entrained electrons. Calculation is applicable in order of magnitude up to ⟨B⟩≈B₀.

For high-frequency plasma investigation the method of atomic-beam probing of kilo-electron-volt energies is the most promising one. Reduction of the beams resulting from atomic interactions enables determination of the density of plasma components and their temperatures over a wide range of values.

In this article probing of a helium plasma of a high-frequency pulsed discharge was performed with helium (8 and 15 keV) and argon (10 keV) beams, enabling determination of ion and neutral distribution in time, as well as electron temperature. The plasma parameters so obtained coincide satisfactorily with microwave and probe measurements.

This article suggests using for plasma-turbulence description a diagram technique analogous to the one suggested by Wyld for hydrodynamic turbulence Such an approach enables one to abandon the perturbation-theory limits used for the study of weakly turbulent pulsations and to obtain approximate equations with the aid of which turbulence with γ ≈ ω can be investigated.

With this purpose in mind, a precise solution of the Vlasov equation has been found for some types of turbulent-plasma pulsations in a magnetic field without assumption of smallness of the pulsation amplitud. This solution has the appearance of an infinite series in powers of the potential of the electric field. Each term of this series has a corresponding graphical picture. The further procedure for obtaining turbulence equations for a plasma is in many respects similar to that of ref. 1: series expansion is performed with respect to a random potential as well as averaging of the products of the random potentials over a Gaussian statistical ensemble and addition of infinite diagram subseries. The resulting equations represent a system of an infinite number of integral equations of infinite order for an infinite number of functions. By way of one or another reasonable dis-continuity of this system of equations, a closed system of a finite number of finite-order equations can be obtained. In this article one of these approximate-equation systems is constructed, representing a certain analogy with the hydrodynamic equations of Kraichnan.

For certain toroidal closed surfaces (M-and-S surfaces) of a β = 1 plasma without resultant azimuthal currents, theory predicts equilibrium. We have produced a toroidal plasma by means of a toroidal theta pinch discharge (B_max = 10 kG, frequency of main discharge, 30 kHz). By adding periodic, additional windings we corrugate the azimuthal field and thus approach M-and-S surfaces. Smear-camera and image-converter investigations of discharges in helium and hydrogen show that in approaching the theoretical surface we approximate equilibrium. This could be observed over time intervals more than three times as long as the drift time in a purely azimuthal field under otherwise identical conditions. The electron densities are in the range of 10¹⁵ to 10¹⁷ cm⁻³, the temperature 5 to 10 eV and beta (β) between 0.1 and 1.

The sequence of experiments resulting in the development of steady-state dense hot-electron plasmas is briefly described. These plasmas are produced in magnetic-mirror machines by radiation at the electron-cyclotron frequency. The electron-cyclotron plasma with the greatest stored energy to date has a volume of ∼ 50 liters, an electron "temperature" of 120 keV, a density of 4 — 7 × 10¹¹/cm³, an d an average beta of ∼ 0.4. This plasma is created in the EPA Facility by a 50-kW c.w. 10.6-gHz microwave power source. The construction and operation of the machine are briefly discussed. The analysis and interpretation of the bremsstrahlung and diamagnetic measurements employed to determine these parameters are given. Diamagnetic and particle decay measurements on a smaller machine, the Physics Test Facility, are also described.

This article discusses in quasi-linear approximation the convective instability of a dilute plasma that occurs when the plasma density exceeds a certain critical value. It is shown that the equations determining the frequency and amplitude of non-linear oscillations enable introduction of an effective potential energy. The oscillations themselves in a certain sense equivalent to the oscillations of a non-linear, two-dimensional, axially symmetric oscillator (rotator) that moves without friction. Consideration of non-linear effects has made possible considerable specification of the instability criterion found in the linear approximation. It is shown that plasma stability with respect to finite-amplitude oscillations depends on the quantity d³n/d(r²)³; here n is plasma density, r the distance from the axis of the system.

Two principal limitations on maximum magnetic fields generated by flux-compression are connected with two fundamental processes, magnetic field diffusion and the conversion of kinetic energy of the conductor compressing the field into stored energy of this field. Criteria for maximum fields are obtained, discussed and applied to typical experimental situations.

A translation of the abstracts of all the articles for this issue into English, French, Russian and Spanish.