Nonlinear Plasma Oscillations Research Articles

Multiple water-bag model and nonlinear plasma oscillations: appearance of a new pole

The complete second-order theory of the multiple water-bag model is developed for one-dimensional collisionless electrostatic waves, in terms of the initial value problem. Nonlinear frequency spectra are obtained in Van Kampen's formulation, and a new pole, already predicted by R. W. B. Best, appears for hot plasmas. This theory is successfully checked by a full numerical experiment of the Vlasov–Poisson system.

Non-linear electron plasma oscillations: comments on recent developments, and non-linear frequency shift for the water bag model

A recent paper (see Numano, ibid., vol.17, p.961 (1975)) on the application of the Von Mises transformation to the non-linear one-dimensional electron plasma oscillation problem is commented on, completed and corrected. The frequency shift is computed for a water bag model, valid for all k, and found in excellent agreement with the results of computer experiments.

Nonlinear plasma oscillations in terms of multiple-water-bag eigenmodes

The problem of nonlinear electron plasma oscillations is considered, using the water-bag model with N bags, and involving second-order expansions of the electric field. The solutions are readily obtained and are successfully checked by computer experiment.

Nonlinear plasma oscillations in Josephson tunnel junctions

Singularities of the Josephson critical current in tunnel junctions made of tin and lead resulting from resonance excitation of plasma oscillations are investigated. Data are presented on the effect of the frequency of the external radiation, temperature, and strength of the de magnetic field on these singularities. The results agree excellently with calculations for a simple model. It is discovered that the threshold microwave power varies at different temperatures and magnetic field strengths, and this dependence is used to estimate the resonance Q. The effect of an interference current proportional to cos φ on this dependence is discussed.

Application of a von Mises transformation to a problem of non-linear one-dimensional electron plasma oscillations

A von Mises transformation is applied to a non-linear boundary-value problem on electrostatic plasma oscillations. The results are found in good agreement with those of Davidson et al. (1968).

Application of a von mises transformation to a problem of non-linear one-dimensional electron plasma oscillations

A von Mises transformation is applied to a non-linear boundary-value problem on electrostatic plasma oscillations. The results are found in good agreement with those of Davidson et al. (1968).

Cold plasma wavebreaking: Production of energetic electrons

Nonlinear electron plasma oscillations are investigated in the wavebreaking regime where collective energy is converted into random energy. The Lagrangian description of the electrons is extended beyond fluid element orbit crossing, thereby exposing the essential physics of wavebreaking and permitting simple analytical interpretation of it. In driven cold inhomogeneous plasma a region of large resonant oscillations decays by producing energetic electrons which escape into the less dense plasma. The maximum energy of the fast electrons produced during the driver period in which the first wave breaks is much larger than the energy of an electron oscillating freely in the driver field. The total energy of the fast electrons exceeds that supplied the system by the driver in a period and can be a significant fraction of the total energy in the oscillations at the onset of wavebreaking.

Solution of a Cauchy problem for nonlinear plasma oscillations

Solution of a Cauchy problem for nonlinear plasma oscillations

Nonlinear plasma oscillations in terms of Van Kampen modes: The initial-value problem

The Landau-Vlasov initial-value problem is solved to second order in the field amplitude, using a secularity-free expansion method and neglecting no terms except of third and higher order. Apart from the Landau poles near the plasma frequency ω p, and the poles at the sum and difference frequencies near 2ω p and 0, also some new poles are found between 0 and ω p corresponding to second-order waves with the same phase velocity as one of the generating first-order waves, and the difference of their wave numbers.

Parametric instability in a plasma placed in nonhomogeneous magnetic field

It is shown that nonlinear longitudinal plasma oscillations in a weakly inhomogeneous magnetic field at the vicinity of upper and lower hybrid resonance points are unstable with respect to the build-up of sub- and ultra-harmonic oscillations. The conditions of the build-up and the growth rate of the oscillations are determined.

Nonlinear plasma oscillations in terms of Van Kampen modes

This paper deals with a solution of the equations of Vlasov and Poisson describing one- dimensional plasma motion. The solution involves expansions to second order in the field amplitude. At first order the Van Kampen modes are recovered. The second-order calculations are free from the usual secular terms and squares of delta functions. Finally, an expression is found for the distribution function in which Van Kampen modes also dominate the second-order term.

On nonlinear plasma oscillations near the breaking boundary

On nonlinear plasma oscillations near the breaking boundary

The effect of spatial dispersion on the stability of the plasma longitudinal oscillations

It had been demonstrated that the allowance for plasma spatial dispersion leads to the instability of harmonic and subharmonic nonlinear longitudinal plasma oscillations

A review of the nonlinear theory of plasma oscillations

This is a coherent review of various aspects of nonlinear longitudinal plasma oscillations. The two cases of a single mode and of a continuum of modes are treated separately. In the monochromatic case, the important role of resonant particles is discussed. It is shown that secular terms arise when the Vlasov and Poisson equations are solved by a regular perturbation method in powers of the electric field. The application of singular perturbation methods to remove secularities is treated. In the second case the quasi-linear theory is reviewed and Dupree's extension of it is also discussed. After Al'tshul and Karpman's method of summing the principal secular terms to all orders, the kinetic equation of waves, wave-particle interactions and resonant wave interactions are discussed. Finally the plasma echo phenomenon is treated briefly.

Third-Order Oscillations in a Maxwellian Plasma

Nonlinear plasma oscillations in a classical, nonrelativistic, collisionless, Maxwellian electron gas are considered. There is assumed a small, sinusoidal variation in the spatial part of the initial distribution function, corresponding to excitation of wavenumber modes ±k0. The nonlinear Vlasov equation is solved to third order in the long time limit via the Montgomery-Gorman perturbation expansion, where the expansion parameter is ε, the amplitude of the initial perturbation. The linear, first-order k0 mode of the electric field is dominated by the Landau solution with (negative) damping decrement γL. The third-order k0 mode is modified, however, by singling out the spatially uniform part of the distribution function for special treatment, much in the manner of the quasi-linear theory. A nonlinear damping decrement results such that, for many values of k0 and ε, γL < γN. Thus at sufficiently long times, the modified third-order mode dominates the solution. For certain ε and k0 this behavior resembles the results of Knorr and Armstrong, obtained by numerical integration of the nonlinear Vlasov equation.

Stability of non-linear stationary plasma oscillations

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.

Higher Order Analysis of Non-linear Plasma Oscillations†

ABSTRACT A more careful investigation of non-linear oscillations in a cold plasma as first posed by Amer in terms of the fluctuating charge density discloses the existence of a charge gradient term which leads to a somewhat modified behaviour. An altered second-Order differential equation is encountered whose integration to a first-order equation which is intractable none the lees permits establishment of the occurrence of non-linear periodic motion of the charge. The associated phase space plot is more or less in qualitative Agreement with that of Amer, but significant numerical differences arise.

Relativistic Non-linear Theory of Plasma Oscillations†

ABSTRACT Non-linear theory of plasma oscillations is extended into the threshold relativistic region where radiation damping can be neglected. A general characterization of the stationary oscillatory modes is performed in terms of a phase-space analysis with a comparison of the non-relativistic behaviour. Two procedures are offered for delineation of the amplitude-dependent frequency of the fundamental vibration. One of these is readily adopted to the zero-current limit, allowing a development already featured for the relativistic harmonic oscillator. Thus regimes requiring relativistic classical or quantum mechanical treatment are identified. In general, very high density plasmas require classical relativistic treatment, although tenuous astrophysical plasmas, where extremely large amplitudes of ion-electron motion can be accommodated, also may be included; the relativistic quantum analysis enters when the plasmon energy 𝒦ωp approaches the rest mass energy of the vibrating charge. A critique of recent disclosures on non-linear plasma oscillations is set forth and a novel method for solution of a related class of non-linear differential equations presented.

Relativistic limits to non-linear plasma oscillations

Some results are reported concerning one-dimensional oscillations of finite amplitude in a cold collisionless plasma. The finite-amplitude traveling waves are shown to exist for non-linear electron oscillations; however, their frequency depends on the maximum speed of electrons relative to ions. The finite traveling waves vanish when the frequency is essentially the plasma frequency corresponding to the effective mass of electrons at maximum speed.

The Frequency of Non-linear Plasma Oscillations†

The frequency of plane standing waves of electron moving through a background of infinitely heavy ions is strictly independent of amplitude as long as the electron velocity is a unique function of position. This result is contrary to those presented in recent publications by Amer (1958) and Gold (1959). The frequencies of spherical and cylindrical plasma oscillations, however, do depend upon the amplitude of the oscillation. As a result any phase relationship between electrons eventually will be destroyed and the electron velocity becomes multivalued. Therefore, steady free oscillations do not persist even in a cold plasma. (auth)