Numerical simulation of type III solar radio bursts caused by high-density electron beam

Numerical simulation for the type III solar radio bursts in meter wavelengths was made with the electron beam of a high number density enough to emit fundamental radio waves comparable in intensity with the second harmonic.This requirement is fulfilled if the optical thickness ₁ for the negative absorption (amplification) becomes -23 to -25. Since ₁ is roughly proportional to the time-integral of the electron flux of the beam, the intensity of the fundamental waves depends strongly on the parameters which determine the electron flux. Therefore, it is most unlikely that the harmonic pairs of type III bursts of the first and the second harmonics occur frequently with comparable intensities in a wide frequency range, say 200 MHz to 20 MHz, if we take the working hypothesis that the fundamental waves are caused by the scattering of electron plasma waves by thermal ions and amplified during the propagation along the beam.However, we cannot rule out the possibility that single type III bursts with short durations or group of such bursts are the fundamental waves emitted by the above mechanism, but only if the observed large size of the radio source can be attributed to the radio scattering alone.     

Fundamental wave of type III solar radio bursts and whistler waves

It is demonstrated by a numerical simulation that both the whistler waves and plasma waves are excited by a common solar electron beam. The excitation of the whistler waves is ascribed to the loss-cone distribution which arises at a later phase of the passage of the beam at a given height due to a velocity dispersion in the electron beam with a finite length. It is highly probable that the fundamental of type III bursts are caused by the coalescence of the whistler waves and the plasma waves excited by a common electron beam, although the plasma waves must suffer induce scatterings by thermal ions to have small wave numbers before the coalescence to occur.     

Dynamics of a cloud of fast electrons travelling through the plasma

Numerical analysis has been carried out on the one-dimensional quasi-linear relaxation of a group of fast electrons travelling through the plasma. It is demonstrated that the electron velocity distribution of fast electrons tends to be a plateau form exciting the electron plasma waves and that the plasma waves are almost completely reabsorbed later by electrons arriving later. Both the velocity range and time interval in which quasi-plateau distribution is formed increase with distance from the origin of the fast electrons. There is no net energy loss of the electron cloud during the travel through the plasma if we neglect both the collisional losses and the scattering of plasma waves. Although the present computation is preliminary and limited to rather low beam density, we can see that the characteristics of both the electron beam and the plasma waves tend, with distance, to those of the analytical solution given by Ryutov and Sagdeev; though a modification to set a low velocity cutoff on the plasma waves due to the thermal electrons is necessary.     

Dynamics of electron beams in the inhomogeneous solar corona plasma

Dynamics of a spatially-limited electron beam in the inhomogeneous solar corona plasma is considered in the framework of weak turbulence theory when the temperature of the beam significantly exceeds that of surrounding plasma. The numerical solution of kinetic equations manifests that generally the beam accompanied by Langmuir waves propagates as a beam-plasma structure with a decreasing velocity. Unlike the uniform plasma case the structure propagates with the energy losses in the form of Langmuir waves. The results obtained are compared with the results of observations of type III bursts. It is shown that the deceleration of type III sources can be explained by corona inhomogeneity. The frequency drift rates of the type III sources are found to be in good agreement with the numerical results of beam dynamics.     

Dynamics of a cloud of fast electrons travelling through the plasma

A simulation of normal type III radio bursts has been made in a whole frequency range of about 200 MHz to 30 kHz by the usage of the semi-analytical method as developed in previous papers for the plasma waves excited by a cloud of fast electrons. Three-dimensional plasma waves are computed, though the velocities of fast electrons are assumed to be one-dimensional. Many basic problems about type III radio bursts and associated solar electrons have been solved showing the following striking or unexpected results.Induced scattering of plasma waves, by thermal ions, into the plasma waves with opposite wave vectors is efficient even for a solar electron cloud of rather low number density. Therefore, the second harmonic radio emission as attributed to the coalescence of two plasma waves predominates in a whole range from meter waves to km waves. Fundamental radio emission as ascribed to the scattering of plasma waves by thermal ions is negligibly small almost in the whole range. On the other hand, third harmonic radio emission can be strong enough to be observed in a limited frequency range.If, however, the time integral of electron flux is, for example, 2 × 10¹³ cm^–2 (>5 keV) or more at the height of 4.3 × 10¹⁰ cm ( _p = 40 MHz) above the photosphere, the fundamental may be comparable with or greater than the second harmonic, but an effective area of cross-section of the electron beam is required to be very small, 10¹⁷ cm² or less, and hence much larger sizes of the observed radio sources must be attributed to the scattering alone of radio waves.The radio flux density expected at the Earth for the second harmonic can increase with decreasing frequencies giving high flux densities at low frequencies as observed, if x-dependence of the cross-sectional area of the electron beam is x ^1.5 or less instead of x ², at least at x 2 × 10¹² cm.The second harmonic radio waves are emitted predominantly into forward direction at first, but the direction of emission may reverse a few times in a course of a single burst showing a greater backward emission at the low frequencies.In a standard low frequency model, a total number of solar electrons above 18 keV arriving at the Earth orbit reduces to 12% of the initial value due mainly to the collisional decay of plasma waves before the waves are reabsorbed by the beam electrons arriving later. However, no deceleration of the apparent velocity of exciter appears. A change in the apparent velocity, if any, results from a change in growth rate of the plasma waves instead of the deceleration of individual electrons.Near the Earth, the peak of second harmonic radio flux as emitted from the local plasma appears well after the passage of a whole solar electron cloud through this layer. This is ascribed to the secondary and the third plasma waves as caused in non-resonant regions by the induced scattering of primary plasma waves in a resonant region.     

Stochastic wave growth, power balance, and beam evolution in type III solar radio sources

Energy-balance arguments are combined with the stochastic-growth theory of type III radio sources to determine the properties of the source in average dynamical equilibrium with the beam, and the beam's long-term evolution. Purely linear stochastic-growth theory has previously emphasized that the beam evolves to a state close to marginal stability. Small mean residual deviations from marginal stability are present at dynamical equilibrium and these lead to residual energy flows that feed the waves observed in situ and by remote receivers; consequently the beam energy is depleted. Here, dynamical equilibrium beam and wave levels are estimated for the first time and it is found that the main sink of beam-driven Langmuir waves is either via electrostatic decay into product Langmuir and ion-sound waves or via scattering by short-wavelength density fluctuations, depending on the conditions. Improved estimates of energy branching ratios imply that, at 1 AU from the Sun, typically 20% of the beam energy is converted to Langmuir waves that are scattered off low-frequency density fluctuations and then dissipated, with almost all the remaining waves undergoing electrostatic decay, although as little as one-third of the Langmuir waves may decay in atypical circumstances. Of order 10^–3 of the beam energy is converted into sound waves, which are mostly dissipated, and of order 10^–5 is converted into potentially observable electromagnetic waves. The mean lifetime of the Langmuir waves at 1 AU is 1–40 s, while that of the beam is of order 1000 s. The beam density decreases relative to that of the background as the beam propagates. For most parameters, analysis of energy losses from the beam to the waves shows that the beam velocity decreases at roughly the same rate as the thermal velocity of the background plasma. It is argued from these considerations, and from in situ observations at 1 AU, that these trends imply that only the densest and fastest type III beams will be able to penetrate much past 1 AU from the Sun. This implies a low-frequency cutoff to type III emission at roughly 10 kHz, in good agreement with recent Ulysses remote observations, showing their consistency with in situ measurements.     

Dynamics of a cloud of fast electrons travelling through the plasma

Numerical analysis of quasi-linear relaxation has been made for four models of electron beam with a finite length travelling through the plasma. In Model 4, a model atmosphere of the corona is adopted and also an increase in the cross-section of the electron beam is taken into account. The electron velocity distribution generally becomes a quasi-plateau form in limited velocity and time ranges. If, however, collisional decay of the fast electrons is too strong and the initial beam density is not high enough, the plateau does not appear. Collisional damping of plasma waves cannot be neglected, since the growth rate of the waves is strongly suppressed by the appearance of the quasi-plateau.An approximate formula for the velocity distribution of the solar electrons passing through the corona has been derived analytically taking into account not only the interaction with plasma waves, but also the collisional damping of the plasma waves and collisions with thermal particles. By the use of this formula, we can easily compute the time profile of the plasma waves caused by these solar electrons at any given place in the interplanetary space. The validity of this semi-analytical approach is checked by the numerical analysis of Model 4, showing a satisfactory fit between the numerical and semi-analytical results.The direct application of this method to the problems of type III radio bursts is left to a later paper.     

Low frequency turbulence in the solar corona and fundamental radiation of type III solar radio burst

On the basis of the previous numerical simulations, a new mechanism for the emission of the fundamental radio waves of solar radio type III bursts is presented. This hypothesis is to attribute the fundamental radio emission to the coalescence of the plasma waves with the low frequency turbulence, whistler or ion acoustic waves, pre-existing on the way of the electron beam which excite the plasma waves.It is estimated that ion acoustic waves could be occasionally unstable in the solar corona due to that drifting bi-Maxwellian distribution of electrons as observed in the solar wind, which is probably caused by collision-less heat conduction.It is also suggested that the reduced damping of the ion acoustic waves in such a distorted electron distribution in the corona may decrease the threshold electric current to cause the anomalous resistivity to be the onset of the solar flares.     

Time profile of type III bursts

On the hypothesis that the time profile of a type III burst corresponds directly to the flux of electron beam, the similarity of time profile is shown to be maintained even if the electron velocity decreases with distance provided that the time is normalized to unity at the time of maximum flux. The observed time profiles of type III bursts with simple shape seem to follow the similarity law in almost all frequency range. This evidence may indicate that the time profile, both the rising and decaying phases, of a type III burst should be attributed to a common origin, e.g., the time variation of exciter determined by the initial velocity distribution in the electron beam, instead of attributing the rising time to the beam length and the decay time to the damping of plasma waves after the passage of the electron beam.     

Combination Scattering by Anisotropic Langmuir Turbulence with Application to Solar Radar Experiments

We develop a theory for radar signal scattering by anisotropic Langmuir turbulence in the solar corona due to a t+l ⇄ t process. Langmuir turbulence is assumed to be generated within a cone by a narrow type III burst electron beam. Using wave-kinetic theory we obtain expressions for the frequency shift, scattering cross-section of the turbulence, coefficient of absorption (due to scattering) and optical depth. On the basis of those expressions we give some estimates for an echo spectrum. We show that the minimum radar echo frequency shift is determined by the minimal phase velocity of the Langmuir waves, the maximum shift is determined by the electron beam velocity, but in any case it can not exceed −wt/2 (decay) and wt (coalescence), where wt is the frequency of a radar signal. The angular characteristics of the scattered signal differ dramatically for the cases of coalescence and decay. The signal is scattered into a narrow cone high above the specular reflection point (wp ≪ wt), but in the vicinity of wp ∼ wt/2 the red-shifted echo is scattered isotropically, while the blue-shifted echo is scattered into a even narrower cone. We show that absorption (due to scattering) increases with increasing radar frequency. The dependence of the absorption on the local plasma frequency is strongly determined by the Langmuir turbulence spectrum. Our theory shows that the role of the nonlinear scattering process t+l ⇄ t is essential and that such process can be used for radar studies of the spectral energy density of anisotropic Langmuir turbulence.     

Radio emission of fast cosmic ions

It is usually assumed that the ions of cosmic rays contribute nothing to the observable electromagnetic radiation. However, this is true only when these ions are moving in a vacuum or a quiet (nonturbulent) plasma. In the case of fast ions in a turbulent plasma, there is an effective nonlinear mechanism of radiation which is discussed in this paper. The fast ion (relativistic or nonrelativistic) moving in the plasma creates a polarization cloud around itself which also moves with the particles. The turbulent plasma waves may scatter on the moving electric field of this polarization cloud. In the process of this scattering an electromagnetic wave with frequency (2.7) is generated. Let ₁ and k₁ be the frequency and wave vector of turbulent plasma waves,V is the velocity of the ion, and is the angle between the wave vector of electromagnetic radiation and the direction of the ion velocity. The method of calculating the probability of the conversion of plasma waves (k₁) into electromagnetic waves (k) by scattering on an ion with velocityV is described in detal in Section 2 (Equation (2.14)).The spectral coefficients of spontaneous radiation in the case of scattering of plasma waves on polarization clouds created by fast nonrelativistic ions are given in (3.6) for an ion energy distribution function (3.4) and in (3.8) for more general evaluations. The Equations (3.9)–(3.13) describe the spectral coefficients of spontaneous emission for different modes of plasma turbulence (Langmuir (3.9), electron cyclotron in a weak (3.10) or strong (3.11) magnetic field and ion acoustic (3.12)–(3.13) waves). The coefficients of reabsorption or induced emission are given by Equations (3.14) and (3.16)–(3.19). There is a maser effect in the case of scattering of plasma waves on a stream of ions. The effective temperature of the spontaneous emission is given by Equation (3.15). The spectral coefficients of radiation due to scattering of plasma waves on relativistic ions are calculated in the same manner (Equations (4.14)–(4.15)). The total energy loss due to this radiation is given in Equations (4.23)–(4.25). The coefficients of induced emission are given in (4.26)–(4.28).The results are discussed in Section 5. It is shown that the loss of energy by nonlinear plasma radiation is much smaller than the ionization loss. However, the coefficients of synchrotron radiation of electrons and nonlinear radiation of ions under cosmic conditions may be comparable in the case of a weak magnetic field and fairly low frequencies (5.5)–(5.6). Usually the spectrum of nonlinear plasma radiation is steeper than in the case of synchroton radiation. Equation (5.10) gives the condition for nonlinear radiation to prevail over thermal radiation.Translated by D. F. Smith.     

Towards a theory for type III solar radio bursts

The procedure developed in Smith (1974) to model the radiation source for type III bursts is modified to include scattering of radiation in the source itself. Since the inhomogeneities in the source must have the same statistical properties as the inhomogeneities used in tracing radiation from the source to the observer, these two parts of the type III problem are no longer uncoupled. Thus we use inhomogeneities consistent with the scattering inhomogeneities of Steinberg et al. (1971) and Riddle (1974) and apply the procedure to an archetype ‘fundamental-harmonic’ pair observed at Culgoora on 28 September, 1973 at 0319 UT. We find that it is impossible to model this burst with a source which is homogeneous in the sense that every part of the source has the same energy density in plasma waves. The density inhomogeneities in the source severely hamper amplification of the supposed fundamental. Possible ways out of this dilemma are discussed, including second harmonic pairs and a source with an inhomogeneous distribution of plasma waves. It is concluded that none of the possibilities are completely satisfactory to explain present observations and suggested that critical observations are missing.     

THE POLARISATION OF SECOND HARMONIC CORONAL TYPE III BURSTS

The process of second harmonic plasma emission is considered, where two simplifying approximations made in previous treatments are relaxed. The revised theory can account for strong polarisation observed in some harmonic coronal type III bursts, and predicts that a correlation between these bursts and the fastest electron beam speeds associated with type III emission should be apparent. These high observed degrees of polarisation could not be explained in the earlier theory. In most cases, polarisation in the sense of the magnetoionic o-mode is predicted. The predicted degree of polarisation is shown to be strongly dependent on the form of the Langmuir wave number spectrum, and in particular on the magnitude of the wave number offset between beam-driven and backscattered Langmuir waves.     

Electron acceleration in an array of auroral arcs

Energy spectra of electrons encountered on a rocket flight across an array of auroral arcs are employed to test three related models of electron acceleration. All three are based on a potential difference existing between the source plasma in the magnetosphere and the observation point in the ionosphere. One of the models provides a satisfactory fit to the observed spectra. Two alternative mechanisms are suggested to explain this model. The first possibility is a time-varying potential difference, which results in the accelerated electrons being observed with a statistical distribution of energy gain. The second possibility, which results in the same energy gain distribution, is a constant potential difference operating in conjunction with plasma instabilities generated by the accelerated beam. The energy gain distribution in the second case is therefore a consequence of a constant potential difference and a variable energy loss. In addition it is suggested that electrostatic waves generated by the instabilities could accelerate ambient plasma to suprathermal energies. Application of the model to the complete data set yields a continuous record of the parameters defining the acceleration and source plasma across the array of arcs. Reference is also made to an acceleration mechanism involving resonance with electrostatic waves.     

Generation of solar type III bursts at the second harmonic

Using the data from our experiments on the IMP-6 (Explorer 43) satellite, we have examined over 200 type III bursts at kilometric wavelengths, including 16 bursts which were accompanied by >18 keV electron events with sharp onsets, in a search for the electrostatic waves which, according to theory, should be the primary source of type III bursts. No electrostatic waves of sufficient intensity to generate the type III bursts by any of the wave-wave scattering theories which produce the second harmonic of the plasma frequency, have been found.     

The propagation of electron fluxes in the solar corona

The problem of energetic electron flux propagation in the solar coronal plasma is solved with due regard for the influence of the oppositely directed neutralizing cold electron flux and the kinematic escape effect of the electrons with different velocities. It is shown that the flux electrons are accelerated in the process of propagation, thus forming a beam, whose velocity is constant on rather long time scales. Three regimes can be realized in this case. In the first regime, plasma waves do not have time to be excited because they escape rapidly from resonance with the beam. In the second regime, waves are excited, but the beam does not have time to relax. The third regime is quasi-linear relaxation.The generation of solar type III radio bursts in the second regime of electron flux propagation is considered.     

Acceleration of particles by electron plasma waves in a moderate magnetic field

A general scheme is established to examine any magnetohydrodynamic (MHD) configuration for its acceleration potential including the effects of various types of plasma waves. The analysis is restricted to plasma waves in a magnetic field with electron cyclotron frequency less than, but comparable to, the electron plasma frequency (moderate field). The general role of electron plasma waves is examined in this paper independent of a specific MHD configuration or generating mechanism in the weak turbulence limit. The evolution of arbitrary wave spectra in a non-relativistic plasma is examined, and it is shown that the nonlinear, process of induced scattering on the polarization clouds of ions leads to the collapse of the waves to an almost one-dimensional spectrum directed along the magnetic field. The subsequent acceleration of non-relativistic and relativistic particles is considered. It is shown for non-relativistic particles that when the wave distribution has a negative slope the acceleration is retarded for lower velocities and enhanced for higher velocities compared to acceleration by an isotropic distribution of electron plasma waves in a magnetic field. This change in behavior is expected to affect the development of wave spectra and the subsequent acceleration spectrum.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Plasma waves caused by transient heat conduction in a coronal loop and anomalous resistivity

Numerical simulation is made of the transient heat conduction during local heating in a model coronal magnetic loop with an axial electric current. It is assumed that a segment near the top of the normal coronal loop is heated to above 10⁷ K by a sufficiently small heat input as compared with the total flare energy. A hump appears in the velocity distribution of electrons moving down the temperature gradient with speeds slightly below the thermal one. Consequently, electron plasma waves are excited. The high intensity of the waves persists in the upper region of the loop for more than a second until the termination of the simulation. The energy density of the plasma waves normalized with respect to thermal density is 10^–3.5 at maximum. A theoretical estimate gives an anomalous resistivity 5 orders of magnitude greater than an initial value. Based on the above result, we propose a model for impulsive loop flares.     

Numerical simulation of the weak turbulence excited by a beam of electrons in the interplanetary plasma

The electron distribution functions measured at 1 AU in an electron stream passing the ISEE-3 spacecraft (Lin et al., 1981) are used as input data to a programme which simulates in a one-dimensional model the interaction between fast electrons and plasma waves (quasi-linear relaxation) together with the plasma wave scattering off the background ions. While the computed spectral energy density of the plasma waves excited in resonance with the streaming of electrons is below Zakharov's threshold of strong turbulence, it is sufficiently high to undergo a fast induced scattering off the background ions. The resulting spectrum is concentrated around the wave vector k = 0. Some simple analytic considerations show that this stage leads necessarily to the crossing of Zakharov's threshold and therefore this indirect excitation of strong turbulence seems to play an essential role in understanding the Langmuir turbulence generated by the motion of fast electron beams in the interplanetary plasma.     

Vlasov – Maxwell,Self-consistent Electromagnetic Wave Emission Simulations in the Solar Corona

1.5D Vlasov – Maxwell simulations are employed to model electromagnetic emission generation in a fully self-consistent plasma kinetic model for the first time in the context of solar physics. The simulations mimic the plasma emission mechanism and Larmor-drift instability in a plasma thread that connects the Sun to Earth with the spatial scales compressed appropriately. The effects of spatial density gradients on the generation of electromagnetic radiation are investigated. It is shown that a 1.5D inhomogeneous plasma with a uniform background magnetic field directed transverse to the density gradient is aperiodically unstable to the Larmor-drift instability. The latter results in a novel effect of generation of electromagnetic emission at plasma frequency. The generated perturbations consist of two parts: i) non-escaping (trapped) Langmuir type oscillations, which are localised in the regions of density inhomogeneity, and are highly filamentary, with the period of appearance of the filaments close to electron plasma frequency in the dense regions; and ii) escaping electromagnetic radiation with phase speeds close to the speed of light. When the density gradient is removed (i.e. when plasma becomes stable to the Larmor-drift instability) and a low density super-thermal, hot beam is injected along the domain, in the direction perpendicular to the magnetic field, the plasma emission mechanism generates non-escaping Langmuir type oscillations, which in turn generate escaping electromagnetic radiation. It is found that in the spatial location where the beam is injected, standing waves, oscillating at the plasma frequency, are excited. These can be used to interpret the horizontal strips (the narrow-band line emission) observed in some dynamical spectra. Predictions of quasilinear theory are: i) the electron free streaming and ii) the long relaxation time of the beam, in accord with the analytic expressions. These are corroborated via direct, fully-kinetic simulation. Finally, the interplay of the Larmor-drift instability and plasma emission mechanism is studied by considering a dense electron beam in the Larmor-drift unstable (inhomogeneous) plasma. The latter case enables one to study the deviations from the quasilinear theory.