Landau Resonance Research Articles

Classical single-particle dynamics of the anomalous Doppler resonance

A gyroaveraged Lagrangian is used to describe the motion of a single electron undergoing anomalous Doppler resonance with a wave that has both electrostatic and electromagnetic components, propagating at arbitrary inclination to the magnetic field. It is shown that the flows of parallel and perpendicular energy are oppositely directed, and have magnitudes in the ratio 1+ω/(Ω/γ):1. The change in perpendicular kinetic energy, or equivalently Larmor radius, is related very simply to the E×B drift occurring at Landau resonance. The classical single-particle description of the anomalous Doppler resonance underlies the existing classical collective descriptions, and is the classical counterpart to the well-known quantum single-particle description.

Resonance between coherent whistler mode waves and electrons in the topside ionosphere

Landau resonance and cyclotron resonance of coherent whistler mode waves and energetic electrons are explored for magnetoplasmas with appreciable gradients in the plasma density and magnetic field strength. It is shown that in the topside ionosphere of the earth near the ion transition height the gradients in plasma density and magnetic field strength along a magnetic field line may match in a way which enhances both Landau and cyclotron interactions between waves and electrons at the loss cone pitch angle. The pitch angle scattering induced by a signal from a ground‐based VLF transmitter in the ionosphere above the transmitter has been estimated and compared to the pitch angle scattering induced by naturally occurring ELF hiss through cyclotron resonance. It is found that the expected scattering due to plasmaspheric hiss is an order of magnitude larger than that due to Landau resonance in the topside ionosphere. Pitch angle scattering due to cyclotron resonance in the topside ionosphere, however, may be larger by a factor of 2. We suggest that the “fast Trimpi” effect may be caused by a cyclotron resonance interaction in the topside ionosphere.

Simulation model for lower hybrid current drive

A simulation model for steady-state, lower hybrid current drive is described which incorporates a relativistic, one-dimensional Fokker–Planck calculation and a toroidal ray tracing code. Two-dimensional (v⊥) effects are included in the Fokker–Planck analysis in the form of a large perpendicular electron temperature that results from pitch angle scattering. An increase in the parallel refractive index of the lower hybrid waves, arising from toroidal geometry effects, is proposed as a physical mechanism whereby injected rf waves at high phase velocity (ve≪v∥ ≲c) can interact via the Landau resonance with electrons at low phase velocity (v∥ ≲3ve). Numerical results relevant to the Alcator C [Phys. Rev. Lett. 53, 450 (1984)] and PLT [Phys. Rev. Lett. 49, 1255 (1982)] experiments are presented which demonstrate the dependency of the current drive efficiency on various plasma parameters.

The wave magnetic field amplitude threshold for nonlinear trapping of energetic gyroresonant and Landau resonant electrons by nonducted VLF waves in the magnetosphere

Using a recently developed theory, we calculate the wave magnetic field amplitude threshold Bτ necessary to allow the nonlinear trapping of energetic gyroresonant and Landau resonant electrons by VLF whistler mode waves in the magnetosphere, propagating at an arbitrary angle, ψ, with respect to the earth's magnetic field, B0. This theory predicts that the amplitude threshold is governed near the magnetic equatorial plane by the gradient of ψ with respect to distance, z, along B0. Using commonly accepted models of the magnetosphere and computer raytracing techniques, we determine the function ψ(z) for magnetic shells in the range 2 ≤ L ≤ 5 and for frequencies 5 kHz ≤ f ≤ 17.8 kHz. We then use the functions ψ(z) to calculate Bτ. It is found that the minimum values of Bτ along each L shell generally occur at points of “second‐order” resonance where both υz = υR and , where υz is the particle velocity along the z axis and υR is the resonance velocity. In the case of gyroresonance, for a given frequency it is found that for most L shells accessible to the input rays, there is a single point of second‐order resonance near the magnetic equator, and Bτ there is larger than that associated with ducted waves of the same frequency on the same L shell. However, over a narrow range of L shell, as many as three points of second, order resonance can exist on each magnetic shell for the case of nonducted waves, suggesting that as many as three regions of VLF emission generation can exist on the same magnetic field line. The results of the calculations agree qualitatively with experimental data concerning triggered VLF emissions, which indicates that Bτ is generally larger for nonducted waves than for ducted waves and that nonducted waves tend to trigger multiple VLF emissions. Calculations also indicate that as many as three points of second order resonance can exist for energetic electrons experiencing a nonlinear Landau resonance with nonducted waves on certain L shells. Minimum threshold values of Bτ for the nonlinear Landau resonance can be lower than that for the nonlinear gyroresonance interaction, suggesting the possibility that nonducted waves may sometimes trigger VLF emissions via the Landau resonance mechanism.

Beam‐driven electron acoustic waves upstream of the Earth's bow shock

Intense electrostatic waves of short wavelength are frequently observed upstream of the earth's bow shock. These waves are very impulsive, and their peak amplitudes correlate strongly with the temporal flux profiles of simultaneously observed energetic ions and electrons which are reflected from the shock. Our paper suggests an explanation of this broadband electrostatic noise in terms of electron acoustic waves driven unstable by the ion and/or electron beams. These waves do not exist in thermodynamic equilibrium plasma and therefore require nonthermal distributions for their occurrence. A simple model is developed for finite temperature beam and background particles with Maxwellian or Lorentzian distributions. The beam‐associated instability can be classified as being due to Landau resonance or negative wave energy. Depending on the plasma parameters the mode discussed here naturally extends and links to the ion acoustic wave or electron plasma oscillation. A numerical parameter study is provided for various beam plasma situations. The frequency theoretically obtained ranges between the ion and the electron plasma frequency, and the wavelength amounts to several Debye lengths for maximum growth. Since the beam mode has a linear dispersion relation, the Doppler‐shifted frequency in the spacecraft frame should depend linearly on the beam velocity and on the wave vector as well. It is anticipated that these characteristics may allow parts of the observed broadband high‐frequency noise to be identified as beam‐driven electron acoustic waves.

The propagation and growth of whistler mode waves generated by electron beams in Earth's bow shock

In this study the propagation and growth of whistler mode waves generated by electron beams within the earth's bow shock are investigated using a planar model for the bow shock and a model electron distribution function. Within the shock, the model electron distribution function possesses a field‐aligned T⊥ > T∥ beam that is directed toward the magnetosheath. Waves with frequencies between about 1 and 100 Hz with a wide range of wave normal angles are generated by the beam via Landau and anomalous cyclotron resonances. However, because the growth rate is small and because the wave packets traverse the shock quickly, these waves do not attain large amplitudes. Waves with frequencies between about 30 and 150 Hz with a wide range of wave normal angles are generated by the beam via the normal cyclotron resonance. The ray paths for most of these waves are directed toward the solar wind, although some wave packets, because of plasma convection, travel transverse to the shock normal. These wave packets grow to large amplitudes, because they spend a long time in the growth region. The results suggest that whistler mode noise within the shock should increase in amplitude with increasing upstream θBn. The study provides an explanation for the origin of much of the whistler mode turbulence observed at the bow shock.

Nonlinear pitch angle scattering and trapping of energetic particles during Landau resonance interactions with whistler mode waves

The time averaged equations of motion derived by Inan and Tkalcevic (1982) are used in a test particle study of Landau resonant interactions of whistler mode waves and energetic particles in the magnetopshere. By computing individual particle trajectories it is shown that the pitch angle scattering and energy exchange is significantly different for the two classes of particles, trapped and untrapped. The trapped particles are characterized by a bounded phase variation, whereas the nontrapped particles exhibit unbounded phase variation. The threshold wave electric field intensities necessary for trapping are determined. Full distribution test particle simulation is carried out to determine the precipitated electron flux that would be induced as a result of these interactions. It is shown that for typical parameters the resulting precipitation fluxes for Landau resonance interactions are much smaller than those induced in gyroresonance interactions, even for wave field intensities that are much higher than the trapping threshold.

Chorus‐related electrostatic bursts at Jupiter and Saturn

Analyses of the wide band plasma wave data obtained by Voyager 1 and 2 at Jupiter and Saturn have revealed electrostatic bursts similar to those recently discovered at earth in association with whistler mode chorus. In all three magnetospheres the bursts are characterized by sporadic emissions near or slightly below the electron plasma frequency with bandwidths ranging from 10% to more than 50% of the center frequency. The events found at Jupiter occur in the middle magnetosphere during both the dayside as well as the early morning passes. At Saturn the bursts occurred in the outer regions of the magnetosphere during the dayside pass. In each of the events analyzed, evidence exists for modulation of the electrostatic bursts by a low‐frequency wave, presumably chorus. One of the observations gained at Jupiter includes the detection of a low‐frequency band at the proper frequency for chorus. Detailed waveform analysis confirms that this band does, indeed, modulate the electrostatic bursts. On the basis of the present understanding of the terrestrial observations, it is believed that the electrostatic bursts are generated by an electron beam trapped in Landau resonance with the chorus.

Whistler mode turbulence generated by electron beams in Earth's bow shock

In this study, the Landau and cyclotron growth rates of whistler mode waves in earth's bow shock are calculated by using electron distribution functions obtained with the fast plasma experiment on ISEE 2. Three electron distribution functions measured within the transition region of the shock are analyzed. An important feature of these electron distribution functions is the presence of a field‐aligned beam with a T⊥ > T∥ anisotropy. The beam velocity vector is directed toward the magnetosheath. The calculations show that the electron distribution functions spontaneously generate whistler mode waves with plasma rest frame frequencies between about 0.1 and 100 Hz. The wave normal angles of the generated whistlers range from 0° to the resonance cone angle. Electromagnetic Landau resonance and/or cyclotron resonance contribute to wave growth over the range of velocity distributions observed. The waves that are generated by the normal cyclotron resonance have wave vectors directed toward the solar wind, while those that are generated by the Landau and the anomalous cyclotron resonances have wave vectors directed toward the magnetosheath. The predictions of the study are in qualitative agreement with observations of whistler mode waves near earth's bow shock.

Electron acceleration by Landau resonance with whistler mode wave packets

Recent observations of electrostatic waves associated with whistler mode chorus emissions provide evidence that electrons are being trapped by Landau resonance interactions with the chorus. In this paper we discuss the trapping, acceleration and escape of electrons in Landau resonance with a whistler mode wave packet. We show that acceleration can occur by both inhomogeneous and dispersive effects. The maximum energy gained is controlled by the points where trapping and escape occur. Large energy changes are possible if the frequency of the wave packet or the magnetic field strength increase between the trapping and escape points. Various trapping and escape mechanisms are discussed.

Ray‐tracing studies and path‐integrated gains of ELF unducted whistler mode waves in the Earth's magnetosphere

Gyroresonance and Landau resonance interactions between unducted low‐frequency whistler waves and trapped electrons in the earth's plasmasphere have been studied. Ray paths for waves launched near the plasmapause have been traced. In agreement with recent findings by Thorne et al. (1979), we have found waves which return through the equatorial zone with field‐aligned wave normal angles. However, when the growth along the ray path is calculated for such waves, assuming an electron distribution function of the form E−n sinm α, we find that for all the waves considered, the local growth rate becomes negative before plasmapause reflection, limiting the total gain to small values. Most waves reach zero gain before reflection. This is the result of Landau damping at oblique propagation angles, which necessarily occurs before reflection can take place. We conclude that the concept of cyclic ray paths does not provide an explanation for the generation of unguided plasmaspheric hiss.

Electrostatic bursts generated by electrons in Landau Resonance with whistler mode chorus

Recent studies of wideband plasma wave data from the ISEE 1 and ISEE 2 spacecraft have revealed that whistler mode chorus emissions in the earth's outer magnetosphere are often accompanied by high‐frequency bursts of electrostatic waves with a frequency slightly below the electron plasma frequency. Investigations have shown that in some cases the electrostatic waves are modulated at the chorus frequency. Further studies using the plasma analyzer (LEPEDEA) data on ISEE 1 indicate that these bursts are produced by a ‘beam’ of electrons in Landau (longitudinal) resonance with the chorus wave and thus moving at the chorus phase velocity. A threshold exists in the chorus intensity below which the electrostatic bursts do not appear. The high‐frequency electrostatic waves are believed to be caused by a type of two‐stream instability called the resistive medium instability. The resistive medium instability is characterized by a reduction in the electrostatic burst frequency below the electron plasma frequency, The instability is applicable only in the regime where V0/VT is on the order of 1, where V0 is the velocity of the beam and VT is the average thermal velocity of the plasma electrons. Our derivation assumes cold ions but warm electrons. The instability requires Landau damping to operate. Thus the beam velocity must be in the region of steep slope on the electron distribution function rather than in the high‐velocity tail region. In the cases examined from the LEPEDEA data the electron thermal energies are on the order of a few hundred eV. The beam velocities in the observed cases were ≈ 400 eV and ≈ 630 eV, thus verifying that the electrostatic bursts are in the proper regime for the resistive medium instability.

The oblique whistler instability in the Earth's foreshock

The linear Vlasov stability properties of electron velocity distributions, similar to those observed in the upstream foreshock region in association with obliquely propagating whistler waves at approximately 1 Hz, are studied. These distributions are modeled by a sum of bi‐Maxwellians with drift speeds parallel to the magnetic field B. We find such distributions to be stable to modes with wavevectors k parallel to B but unstable to whistler waves propagating obliquely to the magnetic field. The frequencies and wavelengths of these unstable modes agree well with those of whistlers observed upstream of the earth's bow shock. The free energy source driving the instability is a region of positive parallel slope ∂fe/∂υ∥ > 0 at large pitch angles (about 85°) and intermediate energies (about 20 eV), probably corresponding to solar wind electrons magnetostatically reflected from the magnetic ramp of the bow shock. The whistlers grow via electromagnetic Landau resonance with this free energy source.

Kinetic Alfven wave propagation in cylindrical Tokamak

Propagation and absorption of kinetic shear Alfven wave, which is launched by a helical oscillating current sheet, are examined in a collisionless, current-carrying cylindrical plasma in sheared magnetic field. Kinetic effects, such as effects of the finite gyroradius, Landau resonance and collisionless resistivity, as well as finite- beta effects, density, temperature and current inhomogeneity effects are included. The authors find that the propagating shear Alfven wave excites drift wave around the resonant layer, and the wave is absorbed through Landau resonance. The propagation is strongly affected by the magnetic shear, and the spatial distribution of the absorbed power is determined by the kinetic effects near the resonance layer. The calculated coupling coefficient to the antenna and heating efficiency are found to have strong dependence on locations of antenna and shell.

Heating of and force acting on the electrons of the magnetic plasma by the electromagnetic wave

The heating of and the force acting on the electrons of the magnetic plasma by the externally launched electromagnetic wave are investigated from first principles. The quantum-mechanically extended Vlasov equation is first derived to treat properly the inverse bremsstrahlung, and then a quasilinear equation is derived from this Vlasov equation to consider properly the electron velocity-space diffusion by the turbulence induced by the electromagnetic wave. The heating rate of and the force acting on the electrons are explicitly expressed in the forms of hydrodynamic energy and momentum equations derived from the quasilinear equation. The heating is shown to be due to the collisional and collisionless inverse bremsstrahlen of the electromagnetic wave in the magnetic field under the influence of the plasma field, and the Landau and cyclotron resonance damping of the plasma waves induced by the electromagnetic wave while the force is due to only the former. It is shown that the force is negligible in the vicinity of the electron cyclotron frequency except for the ordinary wave propagating in the direction perpendicular to the magnetic field.

Nonlinear equations of motion for Landau resonance interactions with a whistler mode wave

A simple set of equations are derived for describing the cyclotron averaged motion of Landau resonant particles in a whistler mode wave propagating at an angle to the static magnetic field. It is argued that for the parameters of the inner magnetosphere, the equations would be an accurate description of the near‐resonant particles' motion. The relative magnitudes of the wave magnetic and electric field forces are discussed and the parameter ranges for which the wave magnetic field effects can be neglected are specified.

The high-β universal drift mode

A linear kinetic analysis of the universal drift mode, valid for arbitrary values of β( = 8πp/B2), is presented. The three high-β effects on the universal, ∇B drifting particles, coupling to the drift Alfvén wave and compressional magnetic fluctuations are examined in detail. For small values of β, the drift wave behavior is dominated by coupling to the Alfvén wave, while for β∼O(1) the wave behavior is determined by ∇B effects. For most plasma and wave parameters, the mode is stable for βi⩾7%. The stabilization mechanism is the Alfvén induced frequency downshift which reduces the parallel phase velocity of the wave until the (slightly) ∇B broadened ion Landau resonance damps the wave. For ηe( = ∂ ln Tc/∂ ln N)≳0, a peculiar case is found where the mode remains unstable for βi⩽0(50%). This is caused by the ∇B frequency upshift coupled with inverse transit time damping.

Generation of ion‐conic distribution by upgoing ionospheric electrons

Downward currents in auroral regions are commonly measured with amplitudes of 1–5 µ A/m². Such currents are likely the result of upgoing thermal ionospheric electrons falling through a field‐aligned potential drop on the order of their thermal energy. Similar distributions of upgoing ionospheric electrons may also occur in regions of diffuse auroral electron precipitation to preserve current continuity in the presence of the loss of the precipitating electrons to the ionosphere. The drift velocity of the upgoing electrons is sufficient to excite electrostatic ion‐cyclotron waves. In addition to being in Landau resonance with the upgoing electrons, these waves cyclotron resonate with the upgoing thermal ions with a parallel energy of several eV. Calculated quasi‐linear diffusion rates using measured wave spectra indicate the resonant ions can be heated to perpendicular energies on the order of 100 times the initial ion thermal energy. Comparison with auroral particle observations, both at low altitude (∼1500 km) and higher altitude (∼6000 km) shows that ion‐conic distributions can readily be explained by such quasi‐linear diffusion. The results imply that conic distributions should not occur simultaneously with the keV electron precipitation associated with discrete arcs in regions of upward current but should occur in regions of upgoing ionospheric electrons which may include regions near the edges of auroral arcs and regions of diffuse auroral electron precipitation.

Destabilitization of low mode number Alfvén modes in a tokamak by energetic or alpha particles

With the inclusion of finite Larmor radius effects in the shear Alfvén eigenmode equation, the continuous Alfvén spectrum, which has been extensively discussed in ideal magnetohydrodnamics, is removed. Neutrally stable, discrete radial eignmodes appear in the absence of sources of free energy dand dissipation. Alpha (or energetic) particle toroidal drifts destabilize these modes, provided the particles are faster than the Alfvén speed. Although the electron Landau resonance contributes to damping, a stability study of the parametric variation of the energy and the density scale length of the energetic particles shows that modes with low radial mode numbers remain unstable in most cases. Since the alpha particles are concentrated in the center of the plasma, this drift-type instability suggests anomalous helium ash diffusion. Indeed, it is shown that stochasticity of alpha orbits due to the overlapping of radially neighboring Alfvén resonances is induced at low amplitudes, eiφ/Ti≳0.05, implying a diffusion coefficient Drα≳4.4×103 cm2/sec.

Eigenmode Analysis of the Electromagnetic Wave in the Nonuniform Vlasov Plasma. II. Quantization Condition

A new method for eigenmode analysis of high mode number electromagnetic waves in an inhomogeneous medium is presented. The method is based on an integral equation in the wavenumber srace and is applicable to the case where kinetic effects such as finite gyroradius effects, Landau and cyclotron resonance effects in high temperature magnetically confined plasmas are important. A quantization condition useful to numerically determine the eigenfrequencies is derived which is different from the conventional Bohr-Sommerfeld condition because of the electromagnetic nature of the wave. A systematic computational scheme of the quantization condition is presented and is shown to be highly efficient by a simple example.