Langmuir Wave Packets Research Articles

Transit-time scattering and heating of a relativistic electron beam in strong Langmuir turbulence

A Fokker–Planck theory is developed to describe the diffusion in momentum space of a beam of relativistic electrons due to multiple transit-time interactions with an ensemble of coherent Langmuir wave packets. The theory incorporates two ingredients: a perturbed-orbit calculation of the momentum change of a test particle during a single transit-time interaction, and an ensemble average of the resulting Fokker–Planck coefficients based on the statistical properties of strong Langmuir turbulence. An approximate analytic solution of the Fokker–Planck equation is obtained for the case of a strongly collimated beam, and is used to interpret measurements of energy and pitch-angle scattering in relativistic-electron-beam (REB) experiments. Fokker–Planck coefficients are also calculated for a weakly collimated beam. It is shown that the theory correctly predicts the amount of energy scattering in REB experiments, but underestimates the pitch-angle scattering regardless of the distribution of wave packet orientations and the degree of collimation of the beam. This discrepancy may be a product of the approximate wave-packet structure assumed in the analysis, or of systematic errors in the experimental data; alternatively, it may imply that a non-transit-time process is responsible for part of the pitch-angle scattering observed.

Phase coupling in Langmuir wave packets: Possible evidence of three‐wave interactions in the upstream solar wind

Some theories of the generation of 2ωpe radiation, in the upstream solar wind, invoke the notion of nonlinear wave‐wave interactions. We present observations, from the WIND spacecraft, of an event in the upstream solar wind with first and second harmonic emissions. The bicoherence spectrum of this event shows coherence amongst components that suggest three‐wave interactions of the type described in (Cairns and Melrose, 1985). We discuss this result and its theoretical context.

Inconsistency of Ulysses millisecond Langmuir spikes with wave collapse in type III radio sources

Recent Ulysses observations of millisecond spikes superposed on broader Langmuir wave packets in type III radio sources are compared quantitatively with constraints from the theory of wave collapse. It is found that both the millisecond spikes and the wave packets have fields at least 10 times too small to be consistent with collapse, contrary to previous interpretations in terms of this process. Several alternative explanations are considered and it is argued that the spikes should be interpreted as either non‐collapse phenomena or observational artifacts. To the extent the observations are representative, this rules out theories for type III bursts at ∼ 1–4 AU that rely on collapse.

Modulational instability of Langmuir wave packets

The broad spectra of random waves in unmagnetized plasmas are considered. A universal nonlinear formalism is developed for description of the nonlinear effects (modulational instability, etc.) in which random plasma waves take part. The integral equations for perturbations of wave field correlation functions are obtained. In a description of the modulational instability of random wave packets these equations play the same role as the set of coupled equations for the fields of modulational perturbations in the case of a single monochromatic pump wave. On the basis of these integral dispersion equations the modulational instability of broad wave spectra is investigated for the one-dimensional situation (when directions of propagation of the random waves coincide with that of the modulational perturbations) and for isotropic wave turbulence.

Interaction of Langmuir wave packets with streaming electrons: Phase-correlation aspects

An analytical model of the interaction between a localized wave packet and energetic electrons is presented. Electrostatic packets of tens to a hundred wavelengths are considered in order to emulate the Langmuir waves observed in the auroral zone and in the solar wind. The phase information is retained, so the results can be applied to wave–particle correlator measurements. The perturbed distribution function is explicitly calculated and is shown to be bounded over all phase space due to a broadening of resonance ascribable to the finite extent of the packet. Its resistive part (in phase or 180° out of phase with the electric field) maximizes for v=ω/k, so that the associated bunching of electrons enables assessment of the characteristic wavelength. The changes in the wave profile due to the interaction with the energetic electrons are calculated. Broad wave packets grow or decay ‘‘self-similarly’’ with a rate given by the standard expression for a plane wave. Narrow, growing packets, on the other hand, quickly widen to sizes determined by the local distribution function. This sets a lower bound to the sizes of observed packets. Present results are supported by test-particle simulations and are in accord with recent correlator data of intense, localized Langmuir waves in the auroral zone.

On Modulational Instability of Turbulent Spectra

AbstractThe character of the modulational instability of a Langmuir wave packet is usually assumed to be similar to that of the instability of a monochromatic pump Langmuir wave if the width of the packet is small when compared to the rate of the instability. This assumption has been confirmed only for the case corresponding to the “hydrodynamical” development of the modulational instability, i.e., for the case when the phase velocity of the modulational perturbations exceeds the ion thermal velocity. Here we consider the opposite case (when the phase velocity of the modulational perturbations is less than the ion thermal velocity) which corresponds to the “static” development of the instability. We demonstrate that the above assumption is valid for the situation considered.

Relative Diffusion and Two-Point Microscale Correlation in Phase Space of Electrons in Localized Langmuir Fields11The project partially supported by National Natural Science Foundation of China through Grant No. 1880109.

We discuss both relative spatial diffusion coefficient and relative velocity diffusion coefficient for electrons interacting with coherent, localized Langmuir wave packets. We find that and have nearly the same positions in which they get a nonzero minimum for small relative position and relative velocity and the diffusion coefficients are greatly reduced dose to minima. We also find that the position in which has a minimum shifts when and vary, while the position for does not move. So we show that the coherent, localized wave packets can drive the formation of strong two-point microscale correlation in phase space only when r and u are small. We also discuss the lifetime of clump.

Theory for low‐frequency modulated Langmuir wave packets

Langmuir wave packets with low frequency modulations (or beats) have been observed in the Jovian foreshock. Here it is argued that these modulations are direct evidence for the Langmuir wave decay L → L′ + S. In this decay, ‘pump’ Langmuir waves L, driven by an electron beam, produce backscattered product Langmuir waves L′ and ion sound waves S. The L and L′ waves beat at the frequency and wavevector of the S waves, thereby modulating the wave packets. Electron beam speeds are deduced from the observed modulations, thereby providing testable predictions for the theory. Beam speeds calculated using the modulated Jovian wave packets (1) are reasonable, at 4 – 10 times the electron thermal speed, (2) are consistent with theoretical limits on the decay process, and (3) decrease with increasing foreshock depth, as expected theoretically. These results strongly support the theory. The modulation depth of some wave packets suggests saturation by the decay L → L′ + S. Applications to modulated Langmuir packets in the Venusian and terrestrial foreshocks and in a type III radio source are proposed, and future tests are described.

Dynamic properties and the two-point correlation function for electrons in localized coherent Langmuir fields.

The dynamic properties for electrons in the localized coherent Langmuir wave fields have been investigated numerically. The Poincar\'e map and the absolute diffusion coefficient account for the existence of regular islands embedded within stochastic regions of phase space. The relative diffusion behavior illustrates that the localized coherent Langmuir wave packets can lead to the formation of a microscale correlation for electrons in phase space. The time evolution of relative quantities further shows that such a microscale correlation has the characteristic of phase-space particle-density granulations, although the electric fields are regular. Finally, the two-point correlation function has been measured by use of the particle simulation technique. That the strong peak behavior of the correlation function can occur only for sufficiently small ${\mathit{r}}^{0}$ and ${\mathit{u}}^{0}$ has been displayed by the simulated results. All these phenomena indicate that the microscale correlation effects (defined also as ``clumps'') indeed exist in our dynamic system.

Correlation Function of Clumps for Electrons in Localized Coherent Langmuir Fields

A correlation function of electrons interacting with localized coherent Langmuir wave packets is obtained by particle simulation. It is shown that the peak behavior of the correlation function for sufficiently small relative coordinates r0 and u0 in phase space also occurs in a spatially inhomogeneous system, and the neighboring particles in phase space are correlated with each other for a lifetime exceeding the coherent time. Our numerical results display the clump effects, which are consistent with previous theoretical prediction of the relative diffusion behavior.

Evidence for Langmuir wave collapse in the interplanetary plasma

With the Fast Envelope Sampler part of the URAP experiment on Ulysses, we have observed much rapidly varying structure in plasma waves in the solar wind. Here we discuss extremely narrow (1 ms) structures observed together with electrostatic Langmuir waves, as well as some broader Langmuir wave packets

Strong Langmuir turbulence at Jupiter?

Langmuir wave packets with short scale lengths l ≲ 100λe have been observed in Jupiter's foreshock. We assess whether the observations are consistent with the relatively new nucleation mechanism for nonlinear collapse of Langmuir waves. Theoretical constraints on the electric fields and scale sizes of collapsing wave packets are summarized, extended and placed in a form suitable for easy comparison with Voyager and Ulysses data. The published data are reviewed and possible instrumental underestimation of fields discussed. New upper limits for the fields of the published wave packets are estimated. Our results are: (1) Wave packets formed at the nucleation scale from the observed large‐scale fields cannot collapse because they are disrupted before collapse occurs. (2) The published wave packets are quantitatively inconsistent with strong turbulence collapse. (3) Strict constraints exist for more intense wave packets to be able to collapse: E ≳ 1 – 8 mV m−1 for scales l ≲ 1000λe. Means for testing these conclusions using Voyager and Ulysses data are suggested.

Relative diffusion and formation of clumps in phase space of electrons in localized Langmuir fields.

A different relative diffusion coefficient for electrons interacting with coherent, localized Langmuir wave packets is proposed. It is shown that the coherent, localized wave packets can drive the formation of clumps in phase space, and only when the relative velocity is zero does the lifetime of clumps diverge logarithmically, with the relative positions of the constituents getting closer to one another.

Two-component model of strong Langmuir turbulence: Scalings, spectra, and statistics of Langmuir waves

A two-component model of strong Langmuir turbulence is developed, in which intense coherent Langmuir wave packets nucleate from and collapse amid a sea of low-level background waves. Power balance between these two components determines the overall scalings of energy density and power dissipation in the turbulence, and of the rate of formation, number density, volume fraction, and characteristic nucleation time of collapsing wave packets. Recent insights into the structure and evolution of collapsing wave packets are employed to estimate the spectra and field statistics of the turbulence. Extensive calculations using the Zakharov equations in two and three dimensions demonstrate that the predictions of the model are in excellent agreement with numerical results for scalings, spectra, and the distribution of fields in the turbulence in isotropic systems; strong support is thus found for the nucleation model. The scaling behavior proves to be insensitive to the form of the damping of the waves at large wave numbers. Wave collapse is approximately inertial between the nucleation and dissipation scales, yielding power-law energy spectra and field distributions in this range. The existence of a fixed arrest scale manifests itself in exponentially decreasing energy and dissipation spectra at high wave numbers and exponentially decreasing field distributions at high field strengths. It is suggested that such an exponential decrease may explain the field distributions seen in recent beam–plasma experiments. Generalizations to turbulence driven anisotropically by beams or governed by equations other than the Zakharov equations are outlined. It is shown that a previously unrecognized scaling observed in beam-driven systems is correctly predicted by the generalized model.

Simulation of the collapse and dissipation of Langmuir wave packets

The collapse of isolated Langmuir wave packets is studied numerically in two dimensions using both particle-in-cell (PIC) simulations and by integrating the Zakharov partial differential equations (PDE’s). The initial state consists of a localized Langmuir wave packet in an ion background that either is uniform or has a profile representative of the density wells in which wave packets form during strong plasma turbulence. Collapse thresholds are determined numerically and compared to analytical estimates. A model in which Langmuir damping is significantly stronger than Landau damping is constructed which, when included in the PDE simulations, yields good agreement with the collapse dynamics observed in PIC simulations for wave packets with initial wave energy densities small compared to the thermal level. For more intense initial Langmuir fields, collapse is arrested in PIC simulations at lower field strengths than in PDE simulations. Neither nonlinear saturation of the density perturbation nor fluid electron nonlinearities can account for the difference between simulation methods in this regime. However, at these wave levels inhomogeneous electron heating and coherent jets of transit-time accelerated electrons in phase space are observed, resulting in further enhancement of wave damping and the consequent reduction of fields in the PIC simulations.

Magnetized Langmuir wave packets excited by a strong beam-plasma interaction.

The physics of beam-plasma interaction, which has been investigated for a long time mostly in relation with solar bursts, is now more widely invoked in various astrophysical contexts such as pulsars, active galactic nuclei, close binaries, cataclysmic variables, \ensuremath{\gamma} bursters, and so on. In these situations the interaction is more likely in the spirit of strong Langmuir turbulence rather than in the spirit of quasilinear theory. Many investigations have been done for two opposite extremes, namely, in very weak and in very strong magnetic fields. Very few properties of the strong Langmuir turbulence are known in the most usual astrophysical situation where the magnetic field plays a significant role but is not strong enough to force the electrons into one-dimensional motion. For this case, we analyze the dynamics of Langmuir wave packets and provide new results about the stability of the solitons against transverse perturbations. It turns out that both the averaged Lagrangian method and the adiabatic perturbation method derived from the inverse scattering transform give exactly the same results (which is not obvious in soliton perturbation theory). In particular, they predict the stability of the solitons as long as the electron gyrofrequency is greater than the plasma frequency (strong magnetic field) and their instability against transverse self-modulation in the opposite case (weak magnetic field); moreover, they allow one to deduce the self-similar collapsing oblate cavitons in the latter case. The laws governing the collapse of the wave packets determine the relaxation of the beam in the surrounding medium and we derive a useful formula giving the power loss of the beam. We outline the astrophysical consequences of this investigation.

Diffusion of electrons in coherent Langmuir wave packets

A new diffusion coefficient for electrons repeatedly interacting with coherent Langmuir wave packets is proposed. The model accounts for regular islands embedded in a region of connected stochasticity, since these islands have a profound effect on the diffusion process. The diffusion coefficient has a resonant form which has been explicity calculated incorporating the finite decay times of the correlation functions. Theoretical predictions compare well with numerical results derived from solutions of the equation of motion.

Breakup and reconstitution of Langmuir wave packets

Recursive behavior has been observed in a two-dimensional electrostatic particle simulation of a coherent intense Langmuir wave packet. The recursion may be associated with the fact that the plasma frequency has a spatial variation in the density depression created by the ponderomotive force.

Nonlinear Landau damping of large-amplitude Langmuir waves and formation of standing nonlinear waves

The interaction of nonlinear Langmuir wave packets corresponding to periodic solutions of the nonlinear Schrödinger equation with thermal plasma particles is considered. The mechanism of interaction is nonlinear Landau damping. The wave packets are slowed down owing to interaction but their amplitudes remain essentially unchanged. The characteristic scale lengths of the slowing down are determined.

Radiation from a strongly turbulent plasma: Application to electron beam-excited solar emissions

The emission of radiation at the plasma frequency and at twice the plasma frequency from beam-excited strong Langmuir turbulence, for the case of low-density high-velocity warm beams, is considered. Under these conditions, Langmuir wave packets undergo (direct) collapse in a time short compared with one e folding of a beam mode. The wave packet energy density threshold for collapse depends only on the beam temperature and velocity, not on the beam density. Upper and lower limits on the volume emissivity for harmonic emission from these collapsing wave packets are found. Within most of this range, the emissivity is large enough to account for observations of second harmonic radiation during type III solar radio wave bursts. The radiation at the fundamental is many orders of magnitude larger than predicted by weak turbulence theory.