Simulations of coronal type III solar radio bursts: 2. Dynamic spectrum for typical parameters

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi:10.1029/2007ja012958, 2008 Simulations of coronal type III solar radio bursts: 2. Dynamic spectrum for typical parameters Bo Li, 1 Iver H. Cairns, 1 and Peter A. Robinson 1 Received 26 November 2007; revised 13 March 2008; accepted 24 March 2008; published 24 June 2008. [1] Predictions are presented for the dynamic spectrum of a coronal type III burst observed at Earth, using a newly developed simulation model and employing realistic electron release and coronal parameters. The spectrum is studied in detail in association with the dynamics of beam and waves in the source. The frequency drift rate, radio flux, brightness temperature, and temporal profile of the type III burst agree semiquantitatively with typical observations. The simulation model is thus viable. Because of strong freefree absorption and scattering-induced damping, the flux of f p emission is significantly lower than that of 2f p emission and is below the lower thresholds of typical radio instruments. Moreover, the f p emission terminates at frequencies higher than the minimum simulated, and the 2f p emission appears to terminate at higher coronal altitudes that are not simulated because of computational limitations. Further simulations indicate that F-H pairs may exist under favorable conditions (e.g., generally, lower levels and larger length scales of the density fluctuations). Citation: Li, B., I. H. Cairns, and P. A. Robinson (2008), Simulations of coronal type III solar radio bursts: 2. Dynamic spectrum for typical parameters, J. Geophys. Res., 113,, doi:10.1029/2007ja012958. 1. Introduction 1 School of Physics, University of Sydney, Sydney, New South Wales, Australia. Copyright 2008 by the American Geophysical Union. 0148-0227/08/2007JA012958 [2] Type III solar radio bursts show rapid frequency drift, a feature distinct from other types of solar radio bursts [Kundu, 1965; Melrose, 1980; Dulk, 1985; Benz, 1993]. In situ observations show that beam electrons produced during solar flares are the energy source, streaming outward from the Sun to interplanetary (IP) space and driving via the plasma emission mechanism, Langmuir waves at the local plasma frequency f p, and radio emission at the fundamental frequency f p and/or its second harmonic 2f p. [3] Observations have shown that type III bursts have a list of general characteristics [Kundu, 1965; Melrose, 1980; Suzuki and Dulk, 1985; Benz, 1993]: (1) The emission drifts fast from high to low frequencies, and sometimes continues to IP space. (2) The drift rate decreases with decreasing frequency. (3) The observed drift rates correspond to beam speeds between about 0.2c and 0.6c, with a typical speed c/3, where c is the speed of light. (4) The radio fluxes and (5) the brightness temperatures vary widely by orders of magnitude from burst to burst. The brightness temperatures are typically in the range 10 8 Kto10 12 K and change with frequency. (6) The burst duration increases as frequency decreases. (7) The temporal profile of the flux shows, in general, fast rise and slow decay, and the decay is approximately exponential. The decay constant increases as frequency decreases. (8) Harmonic pairs exist sometimes in bursts in the corona. (9) The harmonic ratio of the pair is less than 2, with an average of 1.8. (10) Coronal type III bursts tend to cluster in groups. Detailed study of such characteristics for a simulated coronal type III burst is the subject of this work. [4] In the first paper (paper 1) [Li et al., 2008] of this set of three, we develop a numerical model to simulate coronal type III bursts. The aims of the present paper (paper 2) are (1) to present the simulation results for a coronal type III burst using realistic parameters for the beam acceleration and coronal conditions and (2) to assess the simulation model by performing detailed comparisons of the predicted and observed characteristics of type III dynamic spectra. The effects of varying the coronal and beam parameters on type III characteristics will be presented in paper 3. [5] Paper 2 is structured as follows. In section 2 we briefly recapitulate the simulation model presented in paper 1. Section 3 introduces realistic simulation parameters for beam acceleration and coronal conditions. The simulated type III burst, including the dynamic spectra and dynamics of beam and waves in the source region is presented in sections 4 and 5, with the scattering-induced damping of f p emission being switched off. Detailed comparisons between simulations and observations are made for spectral characteristics such as frequency drift rates, flux levels, trends of flux variation with frequency, and brightness temperature. Semiquantitative agreement is demonstrated and thus the model is established to be viable. Section 6 studies in detail the effects of scattering-induced damping on f p emission. Section 7 discusses briefly the limitations of 1of14

Figure 1. Schematic diagram showing the simulations (not to scale). The source region is a conical frustum, which subtends a half angle q b at the Sun and has length l along its axis and radial distance d from the Sun. Within the source, the dynamics of beam, Langmuir waves and ion-sound waves are simulated in one dimension along the axis of the conical frustum, but radiation is simulated in three dimensions. At the end of the source region, rays subtend uniform cones of half angles q F and q H at the Sun for f p and 2f p emissions, respectively. The observer is at Earth, within the radiation cones, and the Earth Sun distance r is much greater than d (so r d > l). The arrows indicate that rays exiting the source region propagate in straight lines, with the thick arrow reaching the observer. the simulation results. A summary of the results is given in section 8. 2. Overview of the Simulation Model [6] The simulation model for this work was presented in paper 1, so we give here only an outline of the model. [7] In paper 1, we assume that the following plasma emission processes are responsible for coronal type III bursts [Robinson and Cairns, 1998a, 1998b, 1998c; Robinson and Benz, 2000]: step I, a beam generates primary Langmuir waves L by the bump-on-tail instability; step II, L waves undergo electrostatic (ES) decay L! L 0 + S and generate product Langmuir waves L 0 and ion-sound waves S; step III, S waves stimulate the L waves to produce fundamental (f p ) transverse waves F via electromagnetic (EM) decay L! F + S; and step IV, L wave pairs coalesce to generate second harmonic (2f p ) transverse waves H via L + L 0! H. Thus the type III system includes five basic elements: electron beam, ES Langmuir and ion-sound waves, and EM fundamental and second harmonic waves. [8] The simulation model, as illustrated by the schematic diagram in Figure 1, includes the three-dimensional (3-D) structure of the source region, the dynamics in the source of the electron beam, Langmuir waves, ion-sound waves, fundamental and second harmonic radiation, and the propagation of radiation from the corona to IP space. It predicts the radiation dynamic spectrum measured by a remote observer. [9] In the model, the following approximations and assumptions were made: [10] 1. The source region is 3-D and is approximated by a conical frustum with a half angle q b at the Sun. [11] 2. Within the source, beam and ES wave dynamics are simulated in a 1-D box along the axis of the conical frustum, due to computational limitations [Li et al., 2002, 2003]. However, EM radiation is simulated in three dimensions [Li et al., 2005a, 2005b]. The box has length l, with its end a distance d from the Sun. [12] 3. Refraction of Langmuir waves and both refraction and reflection of EM waves on large-scale density variations within the source region are important [Li et al., 2006a, 2006b]. In addition, scattering of EM radiation off smallscale density fluctuations leads to directional isotropization and angular broadening of the radiation [Steinberg et al., 1971; Riddle, 1974], as well as time delay and damping for the f p emission seen by a remote observer [Ginzburg, 1964; Robinson and Cairns, 1998a, 1998b; Robinson and Benz, 2000]. The radiation subtends a uniform cone of half angle q T at the Sun at the end of the source region, where T = F or H. The observer is at Earth, within the radiation cone, and the Earth Sun distance r is much greater than d. The time delay is assumed to follow an exponential decay with a time constant t d, which is defined by equation (42) of paper 1. [13] 4. Radiation propagates at average group speeds (<c) within the source region, and at c once is outside the source region. [14] 5. For 2f p emission only antisunward radiation is considered, and both the sunward and antisunward radiation is included for the f p emission [Robinson and Cairns, 1998a, 1998b; Robinson and Benz, 2000; Li et al., 2006b]. [15] 6. Radiation loss by free-free absorption is taken into account [Benz, 1993], but losses by other wave-wave processes (e.g., F! L + S) are neglected [Robinson and Cairns, 1998a; Robinson and Benz, 2000]. [16] By including the 3-D source character and effects of refraction, reflection, scattering, and free-free absorption, the radio flux F T measured by the observer owing to radiation from a finite layer of the conical-frustum source in Figure 1 was first calculated and is given by equation (17) in paper 1. The dynamic spectrum at the observer was then obtained by summing the above flux over source locations and simulation times, and is given by equation (18) in paper 1. The characteristics of type III bursts calculated are: brightness temperature T b (equation (19) in paper 1), frequency drift rate df/dt, half power time-duration t D, and harmonic ratio R HF (evaluated at the observer for a given time as the ratio of frequencies for the peak harmonic and fundamental radiation fluxes). 3. Simulation Parameters [17] To establish the validity of the simulation model, in sections 4 6 we show simulation results using realistic coronal and beam acceleration parameters from observations, as summarized in Table 1. The coronal conditions are described here via an isothermal atmosphere with Maxwellian distributions at T e = T i = 2 MK, and the number density varies according to the 4 Baumbach-Allen model [Baumbach, 1937; Allen, 1947] to represent an active region in the corona [Hughes and Harkness, 1963; Benz et al., 1983; Paesold et al., 2001]: nðþ¼4 r 10 14 2:99 1:55 þ 16 r r 6 þ 0:036 r 1:5 m 3 ; ð1þ 2of14

Table 1. Simulation Parameters of Coronal Conditions, Beam Acceleration, Radiation Source, Angular Spectra of ES Waves, and Radiation Propagation a Parameter Value Unit Comment Coronal Condition T e 2 MK T i 2 MK Dn/n 7% Value chosen on the basis of the work by Robinson and Cairns [1998a, 1998c] for predicted t d to be consistent with observations Beam Acceleration T h 25 MK F acc 5 10 6 Value chosen similar to that from Li et al. [2006b] and Li [2007] t 0 5 10 2 s dt 10 2 s x 0 0.104 Gm dx 3 10 3 Gm Radiation Source l 0.24 Gm Value chosen for the size of the 1-D simulation box in Figure 1 d 0.34 Gm q b 10 Angular Spectra of ES Waves b 20 Value chosen similar to that from Willes et al. [1996] and Li et al. [2006b] Radiation Propagation q F 30 Value chosen on the basis of the work by Robinson and Cairns [1998a] to be consistent with observations q H 90 Value chosen on the basis of the work by Robinson and Cairns [1998a] to be consistent with observations a Parameters are used in section 3. Observations are based on those by Bougeret et al. [1970], Stewart [1974b], Raoult and Pick [1980], Aschwanden et al. [1995], Aschwanden [2002], and Klein et al. [2005], unless otherwise specified. where T e and T i are electron temperature and ion temperature, respectively, r = r/r, and R is the solar radius. [18] Paper 1 introduces various parameters related to the production of the beam and the angular shapes of the ES wave spectra. On the basis of observations [Aschwanden et al., 1995; Aschwanden, 2002; Klein et al., 2005] we choose the following beam acceleration parameters for the source term given by equation (A7) in paper 1: a fraction F acc of electrons is heated from T e to T h = 25 MK in a region characterized by a central location x 0 = 0.104 Gm and a Gaussian spread dx =3 10 3 Gm, at the central heating time t 0 =5 10 2 s with a Gaussian spread dt =10 2 s. We choose F acc =510 6, similar to that of Li [2007]. For the arc-shaped ES angular spectra described by equation (A8) in paper 1, we choose b = 20 that corresponds to a characteristic angular spread of 18, following our previous work [Li et al., 2006b]. [19] For the radiation source size we use q b =10 from observations [Bougeret et al., 1970; Stewart, 1974b; Raoult and Pick, 1980]. We assume further that the radiation cones at the observer have half angles q F =30 and q H =90, based on theory and analyses of observational data [Robinson and Cairns, 1998a, 1998b; Robinson and Benz, 2000]. These are used in equation (4) in paper 1 via the solid angles W b and W T. [20] Since no observational data are available for density fluctuations in the coronal regions (1.14 r 1.49) of interest here, we assume for the fluctuations the same RMS level Dn/n = 7% and the same scaling of mean length scale hli as from Robinson and Cairns [1998a, 1998c]. However, in the simulations, hli is smaller by a factor about 12 than that of Robinson and Benz [2000] because of the different density model (equation (1)) used here from that (see equation (40) in paper 1) used by Robinson and Benz [2000], such that the calculated time constant t d for f p flux at the observer (see equation (42) in paper 1 and section 4.7 of this paper) is consistent with observations. [21] Note that the actual parameters for a specific type III burst may differ from those above, so our aim here is to demonstrate that a type III burst simulated with realistic parameters yields good agreement with typical observations. Further, the simulations here will only cover the limited source region 1.14 r 1.49; regions with r > 1.49 cannot be included because of computational memory and run time restrictions. 4. Simulated Dynamic Spectra and ES Waves in the Source Region [22] Here we study the radiation dynamic spectra measured by a remote observer at Earth and the evolution of ES waves within the source region. The emphasis here is on detailed study of the characteristics of the simulated dynamic spectra and comparisons of these characteristics with observations. We demonstrate that the simulated results are consistent qualitatively and quantitatively with the typical features of coronal type III bursts. [23] Figure 2 shows the Langmuir energy density W in the source region and EM radiation spectra seen at Earth. Here W is obtained by integrating hw L (k L ) N L (t,x,k L ) over k L, where w L and k L, N L are the Langmuir frequency, wave number, and occupation number, respectively [Li et al., 2008]. Figure 3 shows the corresponding variations of peak 3of14

distribution first to become unstable to the production of Langmuir waves is [Benz, 1993; Bastian et al., 1998] dl 27 v 2 e v e dt; ð2þ v h where v e and v h are the thermal speeds of the background and hot electrons, respectively. Assuming that relation (2) holds approximately for our form of heating (equation (A7) of paper 1) and substituting v e =(k B T e /m e ) 1/2, v h =(k B T h /m e ) 1/2, and the heating parameters in section 3 into (2) yields d L 120 km, where m e is the electron mass. This result is consistent with the estimates of d L 100 1000 km based on observational data of bidirectional coronal type III bursts and hard X-ray pulses [Aschwanden et al., 1995]. [25] The fan shape of the Langmuir energy in tx space occurs because of time-of-flight beam formation, quasilinear interaction between beam and L waves, and ES decay of the L waves, with dominance of the former two factors [Li et al., 2002, 2003]. Figure 3a shows that the Langmuir energy density peaks at f L 159 MHz. This peak occurs because of the beam-driven Langmuir waves, since our previous work showed that ES decay processes are slaved Figure 2. The simulated type III event: (a) Langmuir wave energy density log 10 [W/(1 J m 3 )]. (b) and (c) Dynamic spectra observed at Earth for (b) f p and (c) 2f p radiation, in log 10 [F /(10 22 Wm 2 Hz 1 )]. (d) Variation of f p with height. The dashed curve in Figure 2a shows the trajectory of maximal Langmuir energy density, which evolves at a mean beam speed hv b i0.18c. energy density and peak flux with frequency. (The ionsound waves are essentially thermal, because of weak ES decay and strong damping because T e /T i = 1, and so are not shown.) [24] We see from Figure 2a that after an impulsive, localized heating of the corona (see the term S b in equation (A7) of paper 1), L waves are quickly enhanced near the heating site and fill a fan in coordinate space. (The slightly enhanced but still thermal Langmuir waves for x ] 0.15 Gm are due to the heating [Li et al., 2002].) The immediate generation of L waves near the heating region is due mainly to the short heating duration dt and relatively larger T h than T e. Analyses have shown that for a spatially localized heating event with a Maxwellian distribution and the number of hot electrons increasing exponentially over a time period dt, the distance dl required for the electron Figure 3. The variations with frequency of (a) the maximum Langmuir energy density and of the maximum flux of (b) f p and (c) 2f p emission, corresponding to Figures 2a, 2b, and 2c, respectively. In Figures 3b and 3c the solid curves include free-free absorption, while the dotted curves do not. 4of14

4.1. Frequency Drift [28] The burst drifts down in frequency due to the beam propagation into regions of lower density. In the plasma emission scenario, the variation of solar wind density with heliocentric distance can be derived from the speed of type III electron beam and the frequency drift rate of type III bursts [e.g., Wild, 1950; Hughes and Harkness, 1963; Alvarez and Haddock, 1973; Mann et al., 1999; Klassen et al., 2003]. Equivalently, for a given density model, there exists a quantitative relation between the frequency drift rate and beam speed. In its simplest form, the relation can be written as df dt ¼ v b dn 2n dr f : ð3þ Figure 4. Variation of the peak radio frequency with time for (a) f p and (c) 2f p emission and variation of the frequency drift rate with frequency for (b) f p and (d) 2f p emission. In Figures 4b and 4d, the solid curves are from the simulation, and the dotted curves are from relation (3) with v b = 0.18c. to the primary beam-l wave evolution [Li et al., 2002, 2003]. [26] For convenience of later reference in section 4.1, we now define the mean beam speed hv b i such that it corresponds to a trajectory along the curve of maximal Langmuir energy density in tx space [Li et al., 2002]. Figure 2a shows this trajectory evolves at an approximately constant speed hv b i0.18c, indicating that the mean beam speed remains nearly constant within the frequency range simulated. [27] The f p and 2f p radiation observed at Earth in Figures 2b and 2c are due to propagation of the F and H waves generated in the source via EM decays L 0! F 1 + S q and L 1! F 2 + S q, and coalescences L 0 + L 1! H 1 and L 1 + L q! H 2, respectively. These processes involve beam-driven waves L 0 and product waves L 1 from the ES decay L 0! L 1 + S q, where the subscript q indicates that the corresponding waves are thermal (further details are given in section 5). The minimum time for radiation originating at r = 1.14 (where the f p simulated is the highest) to reach the Earth is about 497 s assuming travel at c. Accordingly, the times in Figures 2b and 2c for radiation are thus different by about this propagation time from those in Figure 2a for the source Langmuir waves. The detailed burst characteristics are studied in sections 4.1 4.9, with the scattering-induced damping of f p emission discussed in section 4 of paper 1 being switched off. This damping effect is included and discussed in detail in section 6. To achieve (3), the following assumptions are made: (1) the beam moves radially at an average, constant speed v b, (2) the radiation is produced at f = f p (r) orf =2f p (r), (3) rays propagate at constant speed (i.e., c) from the source to the observer, (4) propagation effects (e.g., refraction and scattering) are negligible, and (5) the time difference for peak radio emission at any two locations is same as the time difference for the arrival of beam at these locations. [29] Figure 4 shows the variation of peak frequency with time and the variation of the drift rate with frequency, using the f p and 2f p emission spectra in Figures 2b and 2c. Note that in calculating df/dt using the data in Figures 4a and 4c, the curves there are least squares fitted. The drift rates predicted by (3) for the density model (1) and an assumed v b = 0.18c are shown in Figures 4b and 4d. [30] Before studying simulated drift rates in detail, we first discuss the results in Figures 4a and 4c in the context of the relative sequence in time for peaks in the flux of f p and 2f p radiation. Figures 4a and 4c show two features: (1) For 2f p emission at frequencies twice those of f p emission, the former arrives earlier than the latter; for example, at frequencies 200 MHz for 2f p emission and 100 MHz for f p emission, the 2f p emission peaks at t 500.0 s, while the f p flux peaks about 0.7 s later; and (2) for a given frequency f, f p emission arrives earlier than 2f p emission; for instance, at f = 150 MHz, f p emission peaks at t 498.5 s, and 2f p emission is delayed by about 3 s. Both features are consistent with observations [Wild et al., 1954; Suzuki and Dulk, 1985]. [31] For feature 1, both f p and 2f p radiation are generated at the same coronal location r. The earlier arrival of the 2f p emission occurs due mainly to the combined effects of three factors: 2f p emission propagates within the simulated source region at faster group speeds than f p emission, f p emission is time delayed by scattering (more discussion on this aspect is deferred to Figure 9a in section 4.7), and both f p and 2f p emission propagate at c once they leave the source region. For feature 2, f p and 2f p emission are emitted at different radial positions, with 2f p emission at larger heights. Since the beam speed is less than the ray propagation speed c when rays are far from the source regions, for a given frequency f the beam reaches the height r H for 2f p emission at f after f p emission at f would reach r H if it travelled at c. This effect is partially counteracted by f p emission travelling at group speeds less than c, within the source region. 5of14

Figure 5. Variation of the frequency drift rate with frequency for f p emission when (a) scattering is switched off (dashed curve) and (b) scattering is switched off and hv F i = c is assumed (dashed curve). The solid curves are from the simulation in Figure 4, and the dotted curves are from the prediction (3) with v b = 0.18c. See text for details. Further detailed study of the relative time delays, as in earlier work [e.g., Robinson and Cairns, 1998a, 1998b] is beyond the scope of the present study, but will be pursued in future. [32] We see from Figure 4d that for 2f p emission the simulated drift rate agrees quantitatively with the prediction (3) if it is assumed that v b = 0.18c. In fact, the actual beam speed derived from the beam dynamics shown in section 5.1 is 0.18c, and the mean beam speed hv b i remains 0.18c in Figure 2a. Thus the relation (3) holds for 2f p emission with the physical beam speed. However, for f p emission, Figure 4b shows that the simulated drift rate is smaller than that predicted, especially at high frequencies. Consequently, the assumptions associated with the relation (3) are numerically justified for 2f p emission, but not for f p emission. These results occur because (1) the beam propagates at an approximately constant speed hv b i0.18c, (2) near the source the propagation speed of f p emission varies and is much smaller than c, while the propagation speed of 2f p emission there is significantly larger and close to c [Li et al., 2006b], and (3) propagation effects on f p emission are more important than on 2f p emission, due to damping and time delay by scattering and stronger free-free absorption, as will be shown in detail in sections 4.5 4.6. To illustrate points 2 and 3, we explicitly show next the effects of scattering and propagation speeds on the drift rate of f p emission. (We find that for both f p and 2f p emission the effects of free-free absorption on the drift rates are negligible and thus are not shown.) [33] Figure 5a shows the drift rate of f p emission when scattering is switched off for otherwise identical conditions to Figure 4b. We see now that the simulated drift rate agrees better, especially at low frequencies, with the prediction (3) than in Figure 4b. This indicates that scattering of f p emission has strong effects on the drift rate. Figure 5b shows the drift rate when scattering is switching off and it is assumed that hv F i = c, instead of being calculated via equation (2) in paper 1, for otherwise identical parameters to Figure 4b. We see that the simulated drift rate agrees slightly better in Figure 5b than in Figure 5a with the model (3), implying that the drift rate is affected by the propagation speed but to a lesser degree than the scattering effects. Still some differences remain between the simulation and the prediction at high and low frequencies in Figure 5b. These may be caused by other effects, such as more f p emission at higher v b than the value hv b i appropriate for 2f p emission and maximal Langmuir levels. This effect is favored by the nonlinear rate for process 3 in section 2, which increases for higher v b [Robinson and Cairns, 1998b]. 4.2. Radiation Flux [34] From Figures 2c and 3c we see that the harmonic flux varies from about 10 22 Wm 2 Hz 1 (i.e., 1 solar flux unit, or 1 sfu) near the onset at 350 MHz, to a peak of 1.9 10 20 Wm 2 Hz 1 at f H 299 MHz, and to about 1.1 1 20 W m 2 Hz 1 at f H 150 MHz at the lowest frequencies simulated. The range of observed radiation fluxes shows wide scatter from burst to burst. For instance, Bougeret et al. [1970] found that at 169 MHz the maximum flux is about 2.5 10 21 Wm 2 Hz 1, and Elgaroy and Lyngstad [1972] showed that at 225 MHz the peak flux ranges from about 10 21 Wm 2 Hz 1 to 3 10 20 Wm 2 Hz 1. Later observations showed, for example, a strong burst whose flux at 265 MHz reached about 9 10 20 W m 2 Hz 1 [Kane et al., 1982], while another burst has a flux of 10 19 Wm 2 Hz 1 at 163 MHz [Benz et al., 1983]. Therefore, the flux of the simulated 2f p emission is consistent with observations for moderately strong type III bursts. [35] Figures 2b and 3b show that the flux of f p emission is lower than that of the 2f p emission. For instance, the maximum flux is about 5 10 21 Wm 2 Hz 1 at f F 154 MHz. So the f p emission corresponds to weak type IIIs. In fact, the flux of the f p emission will be even lower when further losses due to scattering are taken into account in section 6. 4.3. Brightness Temperature [36] The brightness temperature T b is calculated from equations (19) (21) in paper 1 using the predicted radiation flux and source parameters. For instance, using the maximum flux of 4.8 10 21 Wm 2 Hz 1 at f F 154 MHz, and substituting q b =10 and x = 0.13 Gm (compare Figure 2d) into equations (20) and (21) in paper 1 we have a source distance R 1 AU and a source diameter D = 0.29 Gm. Thus equation (19) in paper 1 yields T b 2 10 8 K. Figure 6 shows the maximum brightness temperature versus frequency for both the f p and 2f p emissions, ranging from 10 7 Kto about 4 10 8 K. [37] We see from Figure 6 that near the onset of both emissions T b rises quickly as frequency decreases. Afterward, for the f p emission T b reaches a peak and falls with decreasing frequency, while for the 2f p emission T b increases very slowly as frequency decreases. So the trends in T b versus frequency are qualitatively consistent with observations [Suzuki and Dulk, 1985, and references therein]. In addition, the above observations showed that at 169 MHz the average T b for 39 bursts was 6 10 7 K with maximum about 3 10 8 K, while other observations showed that for 120 bursts at 160 MHz T b ranged from about 3 10 7 K to 10 10 K [Steinberg et al., 1984]. 6of14

Figure 6. The brightness temperature spectra for the simulated (a) f p and (b) 2f p emission in Figures 2b and 2c, respectively. Therefore the simulated T b are in quantitative agreement with the observations. 4.4. Trends of Radiation Flux Versus Frequency [38] The flux of the f p emission predicted in Figure 3b increases with decreasing frequency before it reaches a peak value at f F 154 MHz, then decreases with decreasing frequency and eventually terminates near 95 MHz (higher than the lowest frequency 74 MHz simulated). The flux of 2f p emission in Figure 3c varies similarly to that of f p emission, except that it reaches a maximum at a much higher frequency f H 300 MHz, and is significantly enhanced within the entire frequency range studied. In addition, 2f p emission appears to continue to lower frequencies (higher altitudes) than simulated, where of course it may terminate. [39] Observations of coronal type IIIs show that for some bursts the radio flux increases with decreasing frequency, while some other bursts exist only within limited frequency ranges and have peak fluxes at intermediate frequencies [Wild et al., 1954; Elgaroy and Lyngstad, 1972; Benz et al., 1982; Aschwanden et al., 1990]. Therefore, the trends of variation of the simulated flux with frequency are qualitatively in agreement with observations. 4.5. Frequencies at Peak Radiation Flux [40] The peak (f F 154 MHz) of the f p emission in Figure 3b is closely related to the peak (f L 159 MHz) in the Langmuir energy density in Figure 3a. This occurs because of the joint effects of the following: (1) The F waves generated in the source are dominated by F 1 waves that are emitted via EM decay of the L 0 waves: L 0! F 1 + S q. (2) The Langmuir energy density is contributed primarily by the L 0 waves, as discussed earlier. (3) The thermal ion-sound energy density decreases monotonically and weakly with height [Li et al., 2006b]. (4) Propagation of f p emission is affected by scattering and free-free absorption. Factors 1 3 indicate that the variation of the source f p emission resembles that of the Langmuir energy density. However, because of the additional propagation effects 4, the peak frequency f F is shifted down slightly from f L. The detailed physics of factor 4 is elaborated in the next three paragraphs. [41] Figure 7 shows the radiation flux when free-free absorption is switched off, and Figures 3b and 3c show further the corresponding peak fluxes. We see from Figure 3b and 7a that under such conditions, the f p emission flux is significantly larger than when free-free absorption is included, e.g., by a factor of 70 at 159 MHz. This demonstrates that free-free absorption is very strong. Moreover, Figure 3b shows that when free-free absorption is not included the flux of the f p emission peaks at a slightly higher frequency f free F 160 MHz, and so is very similar to f L. However, because of the presence of free-free absorption, free the actual peak frequency of the f p flux is shifted from f F down to the slightly lower frequency f F, as seen in Figure 3b. [42] Note that Figure 7a shows the existence of almost nondrifting, relatively low levels of emissions ahead of and behind the normal, fast drifting f p emission. The emission behind the normal f p emission is associated with the weak F 2 waves driven via the process L 1! F 2 + S q by the L 1 waves, whose propagation is much slower than the beam propagation (compare section 5). The emission ahead of the normal f p emission is due to thermal radiation. In addition, the slightly higher levels of emission around f p 180 MHz between t 497.8 s and t 499.2 s is due to the enhanced thermal fundamental waves F q generated by thermal Langmuir waves L q enhanced by the heating (compare Figure 2a). [43] Another factor affecting the flux levels of the f p emission is related to the time delays caused by scattering: the f p emission is spread out in arrival time, so that the f p emission lasts longer at the expense of lower intensity. Figure 7. The simulated dynamic spectra for (a) f p and (b) 2f p radiation in log 10 [F/(10 22 Wm 2 Hz 1 )], when free-free absorption is switched off. The other conditions are identical to those in Figures 2b and 2c. 7of14

Figure 8. Temporal profiles of (a) Langmuir energy density and (b) f p and (c) 2f p flux at specific frequencies. In Figures 8a and 8b, solid curves correspond to f p = 172 MHz, dotted curves correspond to 148 MHz, dashed curves correspond to 124 MHz, dot-dashed curves correspond to 92 MHz, and dot-dot-dashed curves correspond to 79 MHz. In Figure 8c, the frequencies are about twice those in Figures 8a and 8b for the same curves. Therefore, the levels of the f p flux are further reduced by the scattering-induced time delay. Nevertheless, this effect appears to change the maximum f p flux by only a factor 2 (see section 4.7), and so is minimal in comparison with the intrinsic time variations in the source emission and damping by free-free absorption. However, scattering does significantly change the temporal profile of f p emission as shown in section 4.7. So, in summary, the actual peak frequency f F is determined by the joint effects of source emission and radiation propagation. [44] The peak frequency f H 299 MHz of the 2f p emission in Figure 3c is only 1.88 f L, with f L 159 MHz at the maximal Langmuir levels. This occurs mainly because, first, the H waves are primarily H 1 waves generated by the coalescence L 0 + L 1! H 1 between the enhanced L 0 and L 1 waves (compare section 5). Second, for maximal emission of the H 1 waves in the source, optimal requirements on both the levels and the wave numbers of the L 0 and L 1 waves need to be met. Therefore, the frequency for the maximal H 1 emission does not necessarily correspond to 2 f L. This is demonstrated in Figures 7b and 3c when freefree absorption is switched off. (Note that in Figure 7b the low levels of emission near 300 MHz between 498.8 s and 499.2 s are associated with the weak source H 2 waves driven by L 1 + L q! H 2 ). In this case, the 2f p flux still peaks near f free H 299 MHz, and so remains less than 2f L, showing that the source physics and not propagation predominantly determines this peak frequency. [45] By comparing Figures 3b with Figure 3c, we see that during propagation 2f p emission suffers relatively less loss because of free-free absorption than f p emission. This result is in qualitative agreement with analyses for other coronal density models [Benz, 1993; Robinson and Benz, 2000]. 4.6. Termination Frequencies [46] By comparing Figure 2b with Figure 7a and detailed examination of Figure 3b, we see that f p emission terminates at frequencies (i.e., termination frequencies) that are near identical (95 MHz) irrespective of whether free-free absorption is included. This implies that the termination frequency of the f p emission is determined mainly by the source emission, namely, the weak generation of the F waves at large heights. [47] At the greatest heights simulated in the source region, the F waves are driven primarily by relatively low levels of Langmuir waves (see Figure 2a) and thermal ionsound waves (noting that the level of the thermal ion-sound waves decreases with increasing height [Li et al., 2006a]). Thus the Langmuir EM decay processes there weaken significantly till they cease near x 0.27 Gm (compare Figure 2d), where source f p emission becomes negligible. [48] However, for the 2f p radiation, the source emission at the largest heights simulated is still significant even though it decreases with height. So the 2f p emission continues down to the lowest frequencies studied, where losses by free-free absorption are very weak (see Figure 3c). Nevertheless, it appears that eventually the 2f p emission will terminate at even lower frequencies in the upper corona. 4.7. Temporal Profile [49] The temporal profiles of the f p and 2f p radiation display different behaviors. Figure 8 shows the temporal evolution of the Langmuir energy density and the f p and 2f p radio fluxes at several frequencies, obtained as horizontal slices of Figures 2a 2c. Note that the frequencies of the 2f p emission in Figure 8c are about twice those of the corresponding Langmuir energy density and f p emission in Figures 8a and 8b. We see from Figure 8b that the f p emission has a faster rise than decay, and that the decay is exponential. For the 2f p emission, Figure 8c shows that at the highest frequency (344 MHz) near the radiation onset, the emission rises faster than it decays, while at lower frequencies the decay is faster than the rise. We see further that for 2f p emission both types of the profiles are similar to the profiles of the Langmuir energy density in the corresponding source regions in Figure 8a. [50] As discussed in section 2 and in section 4 in paper 1, f p emission is delayed because of scattering and decays with a time constant t d. Further, because of scattering, only a fraction h F of source f p emission can escape to infinity (assuming there are no other loss mechanisms). Figure 9 shows the variations of t d and h F with frequency obtained from equations (36) (38), (42), and (43) in paper 1. We see 8of14

Figure 9. Variations with frequency of (a) the time constant t d and (b) the fraction h F of the escaping f p emission from the source for density fluctuation parameters given in section 3. from Figure 9a that t d increases with decreasing f p, consistent qualitatively with observations [Suzuki and Dulk, 1985; and references therein] and theory [Robinson and Cairns, 1998a]. Observed decay constants t d vary from burst to burst. For instance, for 25 bursts in the range 159 179 MHz on average ht d i = 0.3 ± 0.1 s [Poquérusse, 1977], while ht d i = 0.26 s for bursts that occurred during 4 days at 169 MHz [Caroubalos et al., 1974], and at 76 MHz ht d i = 0.34 ± 0.03 s for 25 bursts [Daigne and Møller-Pedersen, 1974]. Thus the simulated values of t d are in quantitative agreement with observations. (The discussion of h F in Figure 9b is deferred to section 6 where the effects of scattering-induced damping on f p emission are studied in detail.) [51] The effect of the above time delay is further demonstrated in Figure 10, which shows the temporal profiles of the f p flux for the scattering-free case, with all the other conditions the same as in Figure 8b. We see from Figure 10 that under such conditions the f p flux shows faster decay than rise (except at 172 MHz where decay is slower than rise), in contrast to Figure 8b. In addition, Figure 10 shows that the profiles of the scattering-free f p flux are similar to those of the 2f p flux in Figure 8c and also the Langmuir energy density in Figure 8a in the corresponding source regions, indicating that for this case the temporal variation of f p emission depends dominantly on source dynamics rather than on propagation effects. Moreover, Figure 10 shows that the flux levels are slightly higher than in Figure 8c, as discussed in section 4.5. [52] The similarity in the temporal variations of the 2f p flux at a given frequency and of the Langmuir energy density at the related source location indicate that it is the source dynamics not the propagation effects that determines the temporal variation of 2f p emission. This result is consistent with previous suggestions that the decay profiles of some type III bursts are closely related to the temporal profiles of the beam or Langmuir waves in the corona [Poquérusse, 1977; Poquérusse et al., 1984; Abrami et al., 1990]. [53] The temporal evolution of the Langmuir energy density at a given frequency (location) in Figure 8c is due to the following [Li et al., 2002]: For a given location, early on the arriving electrons have very high speeds and thus are very rare, so the L waves are enhanced only slightly owing to relatively small growth rates. With the arrival of the more numerous slower electrons shortly after, the L waves are enhanced to much higher levels and build up gradually. However, as the beam passes this location at larger times, the later arriving very slow electrons can no longer drive L waves in the presence of strong Landau damping caused by background electrons, thus the L waves are rapidly restored to thermal levels. Therefore, Langmuir energy density at a given frequency shows relatively slow rise and fast decay. 4.8. Time Duration [54] The half-power duration t D of both f p and 2f p emission is shown in Figures 11a and 11b, and lies in the ranges 0.2 0.8 s for 90 170 MHz, and 0 0.78 s for 150 340 MHz, respectively. The range of observed t D shows wide scatter from burst to burst. For instance, for f p emission t D varies between about 0.3 s and 0.9 s with ht D i = 0.69 s for bursts that occurred during 4 days at 169 MHz [Caroubalos et al., 1974], and ht D i = 0.66 s at 76 MHz [Daigne and Møller-Pedersen, 1974]. For 2f p emission, ht D i = 1.33 ± 0.42 s at 159 179 MHz [Poquérusse, 1977], and t D varies between about 1.0 s and 2.1 s with ht D i = 1.47 s for bursts that occurred during 4 days at 169 MHz [Caroubalos et al., 1974]. For unidentified emission modes, observations showed that t D varies between about 0.2 s and 3.2 s with a mean of 0.8 s for bursts at 318 MHz, and from 0.3 s to 5.8 s with a mean of 1.5 s at 159 MHz [Elgaroy and Lyngstad, 1972]. Thus the simulated values of t D lie within the observational ranges. Figures 11a and 11b further show the existence of linear relationships between t D and f: t D = 1.59 0.0082f sforf p emission (for f > 100 MHz), and t D = 1.29 0.0038f sfor2f p emission, where f is in MHz. [55] For f p emission, Figure 11c shows the variations of t d versus t D. We find that a linear relation exists between t d and t D, i.e., t d 0.32 t D + 0.82, except at frequencies near the burst termination (where both t d and t D are relatively large compared with near the burst start). Interestingly, the same functional form was established from observations (where Figure 10. Temporal profiles of the f p flux at the same frequencies as in Figure 8b for the scattering-free case. 9of14

the source region so that the 2f p emission driven at a larger height (due to beam propagation) reaches the observer later than f p emission at the same frequency generated nearer the Sun, as discussed in Sec 4.1 regarding the relative time sequence of f p and 2f p emission. The reasons for the lower flux levels of f p emission than 2f p emission have been given in sections 4.2, 4.5, and 4.6. [58] Figure 12 shows the predicted variation of the harmonic ratio R HF versus time. We see that R HF varies between 1.73 and 1.95 and decreases with time. (Note that during the period t = 501 501.6 s (not shown), the f p emission terminates at a constant frequency about 95 MHz, as indicated by the contours in Figure 2b, so R HF is not calculated for t ^ 501 s.) Accordingly, R HF is consistent quantitatively with observations [Wild et al., 1954], where the harmonic ratio varies between 1.85 and 2.00 for observations within the range 40 240 MHz. As a comparison, we have the harmonic ratio R RC HF at the source (see equation (50) in paper 1) lying between 1.72 and 1.76 in the frequency range studied. Again, these results are consistent with the average of 1.80 ± 0.14 for inverted U-type bursts in the range 25 210 MHz [Stewart, 1974a], and R RC HF = 1.74 at about 25 MHz according to a theory [Robinson and Cairns, 1998a]. Figure 11. Variation with frequency of the duration t D for (a) f p and (b) 2f p emission and (c) variation of t D with time constant t d for f p emission. the mode of radiation was conjectured to be f p emission), but with different coefficients: t d 0.34t D +0.03 [Abrami et al., 1990]. Such detailed qualitative consistency with observations suggests that our model for the generation of source F waves and the propagation of these waves captures most of the physics actually involved in some type III bursts. 4.9. Harmonic Pairs [56] From Figures 2b and 2c we see that at a given frequency within the range 150 ] f ] 170 MHz: (1) a fundamental-harmonic (F-H) pair exists, both with the given frequency but emitted from different locations, (2) 2f p emission dominates here, and (3) the f p emission starts earlier than the 2f p emission. These results are all consistent qualitatively and quantitatively with observations [e.g., Wild et al., 1954; Caroubalos et al., 1974; Dulk and Suzuki, 1980]. For instance, observations showed that in the range 40 240 MHz 2f p emission lagged behind f p emission at the same frequencies by between 1.5 s and 5 s [Wild et al., 1954; Kundu, 1965]. Figures 2b and 2c show that at a frequency of 150 MHz, 2f p emission starts about 1.8 s after the f p onset at the same frequency, thus is consistent with the observations. [57] The earlier start of the f p emission than the 2f p emission at the same frequency occurs mainly because the beam speed is less than the ray propagation speed c outside 5. Phase Space Evolution of Source Beam and Waves [59] Here we study the evolution of the electron beam and the ES and EM waves in phase space within the source region. 5.1. Beam and Langmuir Waves [60] Figure 13 shows at two times the phase space electron distribution function f e (x,v) and Langmuir wave occupation number N L (x,v) that are calculated via equations (A1) and (A2) in paper 1, respectively, where v denotes the electron speed, and Langmuir wave phase speed parallel to the beam velocity. Figure 13a shows that at t a = 1.2 s there is a beam within the region x (0.13 0.25) Gm, and fast electrons with speeds about 0.4 c have arrived at x 0.25 Gm. The beam at x (0.15 0.17) Gm has relaxed at low speeds between 0.1 c and 0.2 c, as indicated by the vertical contour lines in Figure 13a. Figure 13b shows that the beam-driven L 0 waves are enhanced between x 0.14 Gm and x 0.21 Gm and peak at phase speed 0.17 c. In addition, the L 0 waves are strongest for x (0.15 0.17) Gm as expected from the flattening of the beam in this region. Figure 12. Variation of the harmonic ratio R HF with time for the simulated f p and 2f p emissions in Figures 2b and 2c. 10 of 14

5.2. EM Waves [63] Figure 14 shows the f p and 2f p emission rates G T as functions of wave numbers k Tx and k Tr, which are parallel and perpendicular to the beam direction, respectively, at x = 0.16 Gm for the times in Figure 13. Here G F and G H are calculated via equations (A18) and (A25) in paper 1, respectively, where G T are functions of the 3-D wave number k T and polar angle c T with respect to the beam direction. At t a, Figure 14a shows that G F has an approximately dipolar radiation pattern and a pair of peaks at k Fx = 0 and k Fr ±0.03m 1. These peaks are produced by EM decay L 0! F 1 + S q of the L 0 waves in Figure 13b. Figure 14b shows that the 2f p radiation rate G H has a nearly quadrupolar emission pattern and two sets of peaks. The primary set with main peaks at k Hx jk Hr j5.03 m 1 are due to the coalescence L 0 + L 1! H 1 between the L 0 and L 1 waves at this location. The much weaker set of peaks visible near k Hx jk Hr j 5.01 m 1 are driven by the second coalescence L 1 + L q! H 2 (compare Figure 13b). The radiation patterns in Figures 14a and 14b are consistent with previous analytic and numerical work [Zheleznyakov and Zaitsev, 1970; Cairns, 1987a, 1987b; Willes et al., 1996; Li et al., 2005a, 2005b]. [64] Figure 14c shows that at the later time t b,thef p emission rate is only weakly enhanced by the decay L 1! F 2 + S q, since at this instant only the L 1 waves are super- Figure 13. Evolutions of (a) and (c) electron distribution function f e (x,v) and (b) and (d) Langmuir wave occupation number N L (x,v) att a = 1.2 s (Figures 13a and 13b) and t b = 4.0 s (Figures 13c and 13d). However, between x 0.21 Gm and x 0.25 Gm the L waves are near thermal, because the beam there is too weak to effectively drive L waves. (The weakly enhanced Langmuir waves at x ] 0.14 Gm are due to beam heating [Li et al., 2002].) [61] Figure 13b also shows that the L 1 waves produced by the ES decay L 0! L 1 + S q are much weaker than the L 0 waves and have wave vectors opposite to the direction of beam propagation. These L 1 waves are superthermal over a much larger region than the L 0 waves, extending closer to the heating site down to x 0.12 Gm. Our previous work [Li et al., 2003] showed that the L 1 waves persist for a while after the beam has passed and the primary L 0 waves have dropped back to thermal. This occurs because the L 1 waves have much larger phase speeds than the L 0 waves, so they are more weakly Landau damped and live longer than the L 0 waves. [62] Figure 13c shows that later, at t b = 4.0 s, the beam has arrived at more distant locations than at t a. In fact, highspeed electrons have exited the simulation box, and the slow beam near x (0.28 0.33) Gm has relaxed. The L 0 waves at x ^ 0.26 Gm in Figure 13d are strongly enhanced in the beam direction, while the L 1 waves are enhanced over a much larger region from x 0.30 Gm down to much smaller x 0.15 Gm (where these waves exist only at relatively large phase speeds due to being more weakly Landau damped than at smaller phase speeds, and so last longer). Figure 14. Emission rates (a) and (c) G F (k Fx,k Fr )forf p emission and (b) and (d) G H (k Hx,k H r)for2f p radiation at source location x = 0.16 Gm at the same times as in Figure 13: t a = 1.2 s (Figures 14a and 14b), and t b = 4.0 s (Figures 14c and 14d). Note that in Figures 14b and 14d the labels ±4.97 indicate the wave numbers at the origins: 4.97 for positive k Hx and k H r and -4.97 for negative k Hx and k Hr. 11 of 14