Type III bursts produced by power law injected electrons in Maxwellian background coronal plasmas

Transcription

1 JOURNAL O GEOPHYSICAL RESEARCH: SPACE PHYSICS, VOL. 118, , doi: /jgra.50445, 2013 Type III bursts produced by power law injected electrons in Maxwellian background coronal plasmas Bo Li 1 and Iver H. Cairns 1 Received 24 ebruary 2013; revised 12 May 2013; accepted 7 July 2013; published 2 August [1] Simulations are presented for coronal type III bursts produced by injection of energetic electrons with power law speed spectra onto open magnetic field lines embedded in an otherwise unmagnetized Maxwellian background coronal plasma, including quasi-linear wave-particle interactions and nonlinear wave-wave processes. The simulations show that although fast electrons with speeds > 0.3c are injected, they are important only to the onset and not to the peak of f p emission, where f p is the local electron plasma frequency. Instead, slower beam electrons are the major drivers of the peak f p emission. Therefore, the type III beam speeds derived from the drift rates of peak f p emission are less than the typical speeds of c/3 observed for coronal type III bursts. This occurs mainly because the number of fast beam electrons with speeds > 0.3c is much less than the slower ones, causing weaker f p emission from these fast beam electrons. Comparisons are made with injected electrons having Maxwellian spectra. We find that type III beams are faster when the injection has power law spectra, since there are more fast electrons injected than for Maxwellian spectra. These results suggest that type III beams produced in the corona with Maxwellian background particle distributions and either power law or Maxwellian spectra can account only for the lower half of the observed range c of type III beam speeds but not for the upper half. Citation: Li, B., and I. H. Cairns (2013), Type III bursts produced by power law injected electrons in Maxwellian background coronal plasmas, J. Geophys. Res. Space Physics, 118, , doi: /jgra Introduction [2] Energetic electrons accelerated during flares can produce type III solar radio bursts, which drift rapidly from high to low frequencies [Suzuki and Dulk, 1985; Bastian et al., 1998]. On propagating upward into the solar wind, fast electrons outrun slow ones and form beams, which generate Langmuir waves and radio emission near f p and/or 2 f p,wheref p is the local electron plasma frequency. In the corona, type III beams have speeds v b c, with typical speeds c/3 [Suzuki and Dulk, 1985; Klassen et al., 2003]. The speed v b of a type III beam is often derived from the well-known relation [Wild, 1950]: df dt = v b d(ln n e ) f. (1) 2 dr Here df /dt is the drift rate at frequency f of the peak intensity of the type III burst, n e (r) is the radial density profile along the path of the beam that propagates at a constant speed v b, 1 School of Physics, University of Sydney, Sydney, New South Wales, Australia. Corresponding author: B. Li, School of Physics, University of Sydney, Sydney, NSW 2006, Australia. (boli@physics.usyd.edu.au) American Geophysical Union. All Rights Reserved /13/ /jgra and r is the radial distance from the Sun s center. Sometimes, even faster type III beams in the corona are observed [Wild et al., 1959; Raoult et al., 1989], including those associated with a subclass of type III bursts type IIId bursts [Poquérusse, 1994; Klassen et al., 2003; Liu et al., 2009]. Type IIId bursts have much larger drift rates (or much shorter drift times that may be instrumentally unresolvable) than normal type III bursts. It is commonly believed that type IIId bursts are produced by fast coronal beams with v b >0.5c, even approaching c, where the beams propagate nearly along the line-of-sight toward the observer [Poquérusse, 1994; Klassen et al., 2003]. In contrast, v b is smaller in the solar wind. Specifically, v b c with an average of 0.06c when using the observed drift rates of interplanetary type III bursts [Dulk et al., 1987]. On the other hand, v b is larger, ranging from 0.05c to 0.25c, when a radio tracking approach [ainberg et al., 1972] or time-of-flight technique [Malaspina et al., 2011; Graham et al., 2012] is used. [3] In situ measurements show that electrons leaving the Sun have broken power law spectra in energy or equivalently in speed [Lin, 1974, 2011]. These escaping electrons are commonly believed to produce type III bursts [Lin, 1974; Reiner, 2001; White et al., 2011] and may also produce very weak X-rays in the corona [Krucker et al., 2008; Bain and letcher, 2009; Glesener et al., 2012]. It remains elusive whether the observed electron spectra are direct signatures of the acceleration process or of

2 the injection of upward moving electrons in the corona [Bastian et al., 1998; Aschwanden, 2002; Krucker et al., 2009]. This is because during propagation various transport effects, such as wave-particle and wave-wave interactions in inhomogeneous plasmas [e.g., Li et al., 2006; Reid and Kontar, 2010; Ziebell et al., 2011], can modify the electron spectra. On the other hand, electrons propagating downward toward the Sun are also observed to have power law or broken power law spectra, as inferred from X-rays emitted in the chromosphere during flares [Lin, 1974, 2011]. The downgoing electrons can also produce type III bursts that drift from low to high frequencies, commonly called reverse-slope-drifting type III bursts [e.g., Aschwanden et al., 1993; Xie et al., 2000]. Often, similar spectral forms are used for electrons that produce either type III bursts or chromospheric X-rays, implicitly assuming approximately symmetric magnetic reconnection yet with greatly different numbers of upward-accelerated and downward-accelerated electrons. or example, Reid et al. [2011] assumed a common acceleration site for X-ray-emitting and type IIIemitting electrons and the same spectral index for both groups of electrons. [4] Type III beam electrons can also evolve from hot, thermal flare plasmas, which have typical temperatures MK, much higher than the coronal background [Benz, 2002]. Sometimes, flare plasmas show a second, superhot component, which has temperature & MK [Lin et al., 1981; Christe et al., 2008; Caspi and Lin, 2010]. Observations show that some of the superhot plasmas are associated with acceleration of type III electrons [Christe et al., 2008]. Often, for simplicity, the spectra of both the cold and hot populations are assumed to have Maxwellian speed distributions [e.g., Grognard, 1985; Ziebell et al., 2001; Benz, 2002; Kontar and Pecseli, 2002; Li et al., 2008a]; yet generalized Lorentzian (or kappa) distributions for both were adopted in a recent type III theory [Robinson and Cairns, 1998b; Robinson and Benz, 2000]. [5] Recently, significant progress has been made in simulating coronal type III bursts, both in the radiation source and for radiation observed remotely [e.g., Li et al., 2008a, 2011a]. The predicted type III spectral characteristics, e.g., drift rate and brightness temperature, agree quantitatively with typical observations, for metric or decimetric type III bursts with unidirectional [Li et al., 2008b, 2009, 2011a] or bidirectional [Li et al., 2008c, 2011b] drift morphologies. urthermore, the simulations have been extended to predict type III bursts perturbed by shocks [Li and Cairns, 2012], and the predictions are consistent with the limited observations available. However, the simulated type III beams have v b. 0.2c, which are obtained from the analytic prediction (1) for f p and/or 2 f p emission and also directly from the beam dynamics in the source. In these simulations the beams are produced in coronal background plasmas with Maxwellian (or thermal) distributions, by heating a small fraction ( ) of background electrons to typical flare temperatures ( MK) with Maxwellian spectra. These simulations also show that the values of v b are insensitive to variations in coronal plasma parameters and electron acceleration within the ranges inferred from observations [see Li et al., 2009, and references therein]. or example, v b 0.19c shows insignificant (< 2%) changes when the flare temperature increases from 12 to 30 MK [Li et al., 2009]. The predicted speeds are then at the lower end of the observed range. These simulations thus suggest that the combination of Maxwellian electron heating and a Maxwellian background plasma can only produce relatively slow type III beams and is unable to generate the often observed beams with v b & 0.2c. [6] Motivated by the observations and suggestions that the injected electrons have (broken) power law distributions, and by the need to address the observed fast type III beams, in this paper we simulate f p emission for coronal type III bursts that are produced by injection of nonthermal electrons with power law speed spectra into the corona, where the ambient particles are unmagnetized and have Maxwellian distributions. Note that the corresponding 2 f p emission is not simulated, due to model limitations discussed further in section 5. We find that the newly predicted f p emission drifts faster for injected electrons with power law spectra than with Maxwellian speed spectra. Nevertheless, the predicted beams are only moderately fast, with v b < 0.3c in general, and so are still unable to reproduce the observed fast-drifting type III bursts with v b > 0.3c. Consequently, the results from this paper and previous work [e.g., Li et al., 2009, 2011a] show that, in general, fast type III beams with v b >0.3ccannot be produced in the corona if the coronal background particles are Maxwellian distributed and the injected electrons have either power law or thermal injection spectra. In contrast, our other recent simulations [Li and Cairns, 2013a;Liand Cairns, 2013b] show that coronal type III beams with v b >0.3ccan be naturally produced in background plasmas that are non-maxwellian (specifically, with kappa distributions) by injected electrons with either power law or Maxwellian spectra. [7] The paper is organized as follows. Section 2 first introduces the two types of speed spectra, i.e., power law and Maxwellian, for electrons injected onto an open magnetic field line. It then outlines the type III simulation model, which was originally developed for Maxwellian electron heating [Li et al., 2008a, 2008c]. Section 3 predicts for a representative power law injection the f p emission observable at Earth and the electrons and waves in the coronal source region. These results are then compared with those for a Maxwellian injection. We find that the fast yet rare electrons from the power law injection contribute primarily to the radiation onset, while the much slower yet more numerous injected electrons are the principal drivers of the peak f p emission. Moreover, the drift rates of the f p emission for the power law injection agree quantitatively with the analytic prediction (1) for v b 0.2c, which is the speed of the beam and waves found in the simulated source. Section 4 studies how the spectral characteristics of f p emission and the beam speed scale with the parameters of the power law injection spectrum. We find that v b varies weakly but is restricted to v b. 0.25c for realistic ranges of these parameters. Section 5 discusses further the simulation results and proposes future improvements to the simulations. The conclusions are drawn in section Simulation Model [8] We assume that type III bursts are produced by quasilinear and nonlinear processes involved in plasma emission [e.g., Melrose, 1980; Cairns, 1987a, 1987b; Bastian et al.,

3 1998;Robinson and Cairns, 1998a]. Other mechanisms such as linear mode conversion [Kim et al., 2008; Schleyer et al., 2013], antenna radiation from Langmuir eigenmodes [Malaspina et al., 2010, 2012], and nongyrotropic beamdriven emission [Tsiklauri, 2011; Pechhacker and Tsiklauri, 2012a, 2012b] are assumed to be less important and are not included. The injection of energetic electrons produced during a flare onto an open magnetic field line embedded in a plasma whose waves are treated using unmagnetized theory is represented by adding a source term G to the quasi-linear electron transport equation [e.g., Li et al., 2003, 2011a]: + @v G(t, x, v) = + G(t, x, v), (2) inj p I(v) exp (t t 0) 2 (x x 0) 2. (3) ıt (ıt) 2 (ıx) 2 Here f e = f e (t, x, v) is the electron distribution function at time t and radial distance x = r Rˇ above the photosphere (r = Rˇ), v is the electron speed, and f e is normalized to the plasma density n e. The first and second terms on the right-hand side of equation (2) represent spontaneous and induced emission, respectively, where the diffusion coefficient D describes the coupling between beam electrons and Langmuir waves. [9] The injection via the G term in equation (2) is localized in time and position, taking place at a height centered at x = x 0 and at a central time t = t 0 after the start of the simulations, and with characteristic spatiotemporal domains ıx and ıt. The total number of injected electrons is a fraction inj of the background electrons in the source. The speeddependent function I(v) assumes two forms, representing either a power law injection: I(v) = p 1 1, v < v 0, pv 0 (v 0 /v) p (4), v v 0, or a Maxwellian injection: I(v) =f Max h (v, T h ), (5) and R I(v)dv =1in both cases. or the power law injection, the spectrum (4) has a break speed v 0 and a spectral index p for v v 0, and is flat for v < v 0. or the Maxwellian injection, f h Max in (5) is Maxwellian with flare electron temperature T h. The injected electrons subsequently form a beam due to time-of-flight effects. The time t min required to form a beam above the background at a location x (> x 0 + ıx) away from the injection site is approximately (x x 0 ıx)/v max, where v max is the maximum speed of injected electrons with nonnegligible number density. It is expected that t min is smaller for a power law injection than for a Maxwellian injection, because for the Maxwellian injection I(v) falls off much more rapidly with increasing v and so v max is smaller. The duration of the beam at x is approximately = t max t min =[(v max v min )(x x 0 )+(v max +v min )ıx)](v min v max ) 1, where t max =(x x 0 + ıx)/v min is the time for the slowest electrons (with speed v min ) from the injection to reach the location x. The characteristic injection duration ıt is small compared with the times t min, t max,, and the quasi-linear relaxation time [cf. Li et al., 2002] for all x of interest here The simulations below show that a beam at a given location exists for a macroscopic time (e.g., 1 s in igure 3c) before being quasi-linearly relaxed. [10] The detailed type III processes included in our model are the following [Li et al., 2008a]: (i) growth of Langmuir (L) waves by the bump-on-tail beam instability and quasilinear interaction with the beam; (ii) production of Langmuir (L 0 ) and ion-sound (S) waves by electrostatic (ES) decay L! L 0 + S of L waves, where the L 0 and S waves are in the forward and backward hemispheres, respectively, relative to the beam velocity direction; (iii) generation of f p radiation ( waves) and ion-sound (S 0 ) waves by electromagnetic (EM) decay L! + S 0 of L waves; and (iv) production of 2 f p radiation (H waves) by coalescence L + L 0! H of the L and L 0 waves. Our model [Li et al., 2008a, 2008c] considers a 3-D source region with stratified 2-D layers that vary with x along a magnetic field line (idealized to be radial). The model predicts the temporal evolution of a type III beam and the associated L and S waves in phase space as functions of x, v, and wave number (k), using equation (2) for electrons, and the coupled equations (A2) and (A14) in Li et al. [2008a]for L and S waves, respectively. These equations are quasi-linear equations for wave-particle interactions that are supplemented with source terms corresponding to the above nonlinear wave-wave processes. The model then predicts the emission rates for and H waves, using equations (A18) and (A25) in Li et al. [2008a], respectively. The model includes the effects of propagation of radiation in order to predict the dynamic spectra of radiation observed at Earth, i.e., the radio flux ˆT(t, f ), wheret = or H are for f p or 2 f p emission, respectively. In further detail, ˆT(t, f ) is related to the radiation emission rate T (k T, ) via [Li et al., 2008a] X Z Z d sin C T (t, f; ) ˆT(t, f )= X t s x s (6) where x s is the source location at time t s, k T is the emission wave number, and is the emission angle relative to the beam direction. Qualitatively, the factor C T (t, f; ) in equation (6) is independent of k T, but dependent on t, f, the details of the source (e.g., x s and ), and radiation propagation effects (e.g., scattering and free-free absorption) [Li et al., 2008a]. Quantitatively, C T is defined through equations (1), (4), (17), and (18) in section 3 of Li et al. [2008a]. Note that for nonthermal f p emission with T = in equation (6), k varies significantly. This is in contrast to k H for 2 f p emission with T = H in equation (6), where k H is approximately constant with k H p p 3! L /c k 3ve k L /c, according to the dispersion relations! T 2 =!2 L + k 2 T c2 and! L 2 =!2 p +3k2 L v2 e for EM and Langmuir waves, dk T [k 2 T T(t s, x s, f, k T, )], respectively [Cairns, 1987a, 1987b; Li et al., 2005a, 2005b]. Here k L is the Langmuir wave number in the beam velocity direction and is related to the beam speed v via the resonance condition! L = k L v,! 2 p = n ee 2 /m e " 0, v 2 e = k BT e /m e,and m e, e,andt e are the electron mass, charge, and temperature, respectively. Therefore, ˆ implicitly depends on k through k 2 via the k T integral in equation (6). The weighted emission rate k 2 is then studied below, not the emission rate T itself, in order to quantitatively link the source with the radiation observed remotely.

4 [11] The f p emission rate in equation (6) due to EM decay L! + S 0 is calculated as the sum of two parts [Li et al., 2005b]: (k, ) = f (k, )+ b (k, ), (7) where the superscripts f and b indicate contributions from the forward and backward hemispheres, respectively, relative to the beam velocity direction, of the L and S 0 pair. In this work, (k, ) is evaluated using equations (53) and (54) in Li et al. [2005b]. 3. Representative Simulations [12] Simulation predictions are presented here for the f p emission of a coronal type III burst that is generated by accelerated electrons with a representative power law injection spectrum in a Maxwellian background corona. The predictions are then compared with those for a Maxwellian injection spectrum. [13] The two simulations presented in this section have different electron injection profiles I(v) but otherwise identical parameters. In Simulation 1 (S1) the injected electrons follow the power law spectrum given by equation (4). We assume that p =8and v 0 =710 7 ms 1 (corresponding to a break energy of 14 kev), within the ranges used in previous modeling [e.g., Harvey, 1976; Hannah and Kontar, 2011; Reid et al., 2011]. In Simulation 2 (S2) the injected electrons have a Maxwellian spectrum (5), with a flare temperature T h =25MK. Based on observations [Klein et al., 2005; Aschwanden, 2002], we choose for both S1 and S2 that the injection obeys equation (3) with a central height x 0 = Gm and a central time t 0 = 50 ms after the start of the simulations. The injection has a Gaussian profile with characteristic spatiotemporal scales of ıx =3Mm and ıt = 2 ms. The parameter inj is chosen to be 10 5, based on our work for Maxwellian injection spectra [e.g., Li et al., 2008b]. [14] The coronal plasma is assumed to be isothermal with T e = T i =2MK, where T i is the ion temperature. On large spatial scales, the background electron number density n e is assumedtovarywithx according to the fourfold Baumbach- Allen profile [Allen, 1947]. At the central height (x 0 )ofthe injection n e = m 3,andsof p = 178 MHz. This n e (x) variation is implemented self-consistently using the technique developed in Li et al. [2011a]. Besides the largescale regular n e (x) variation, small-scale density fluctuations are ubiquitous in the corona [Coles and Harmon, 1989]. Here we consider the effects of these fluctuations on f p emission leaving its source. The fluctuations scatter f p emission, causing time delays for and losses of, the f p emission reaching the observer [e.g., Riddle, 1974; Robinson and Cairns, 1998a]. We model these effects, through the coefficient C (t, f; ) for f p emission in equation (6), in terms of two parameters characterizing the fluctuations: the RMS level ın e /n e and the mean length scale l [Robinson and Cairns, 1998a; Li et al., 2008a]. Since observational data are available rarely for r < 2Rˇ [cf. Coles and Harmon, 1989], the coronal region of interest here, we assume similar parametrizations to those in previous work. Specifically, we assume spatially uniform ın e /n e =1%[e.g., Steinberg et al., 1971; Thejappa and MacDowall, 2008]. The length scale l(x) is assumed to increase with x [e.g., Steinberg et al., 1971; Subramanian and Cairns, 2011] and is given by l(x) =10 5 [(x + Rˇ)/1 AU] 1.61 m[cf.robinson and Cairns, 1998a]. The remote f p emission depends sensitively on both parameters, and further details can be found in Robinson and Cairns [1998a] and Li et al. [2008b, 2009] Remote Radiation [15] igures 1a 1e show the predicted f p dynamic spectra for S1 and S2, as well as the variations with f of the maximum flux ˆmax and the corresponding frequency drift rate df /dt. We see from igures 1a and 1b that both bursts are weak yet are observable for typical instruments with thresholds 1 sfu (solar flux unit) (= Wm 2 Hz 1 ). This result of weak f p emission is consistent with our previous simulations that have Maxwellian spectra for the injected electrons [e.g., Li et al., 2008b, 2009]. igures 1a and 1b show that at a given f, the burst in S1 has a much earlier onset and slightly earlier termination than in S2. or example, at the lowest simulated frequency f =50MHz the burst in S1 starts at t 500 s, while the one in S2 starts about 2 s later. Besides, the burst in S1 has a slightly higher (by 3%) onset frequency of 170 MHz than in S2. We see also that at a given time, the burst in S1 has a broader bandwidth than in S2, especially at low frequencies. igures 1a and 1b also show that both bursts grow stronger as they drift down to lower frequencies and that ˆmax at a given f in igure 1c is similar for both cases, except near the onset frequencies. [16] We have also studied the brightness temperatures T b for both cases (not shown). We found that the profile of T b (t, f ) is similar to ˆ(t, f ) [e.g., Li et al., 2008b] and that T b falls within the range, yet near the lower bound, of observations [Suzuki and Dulk, 1985, and references therein]. or example, in S1, the maximum brightness temperature T max b increases with decreasing f [cf. Li et al., 2008b, 2008c], with T max b = Kand Katf = 165 MHz and f =50MHz, respectively. [17] igure 1d shows the drift rate df /dt corresponding to the maximum flux ˆmax.Thedf/dt results are obtained in two ways: from the simulated bursts in igures 1a and 1b and from the analytic relation (1), as shown by solid and dashed curves, respectively, in igure 1d. We see that the burst in S1 drifts faster, but not by a large amount, than that in S2. or example, igure 1d shows that for S1 df /dt 23 MHz s 1 at f = 100 MHz, about 25% larger in magnitude than for S2. We also see that the analytic df /dt predictions for v b =0.20c and 0.16c for S1 and S2, respectively, agree well with the simulated df /dt results. These predicted v b values are consistent with the beam dynamics in the source described in section 3.2. Thus, the relation (1) holds approximately for the f p emission produced in S1, where the accelerated electrons have the power law injection spectrum, as found for Maxwellian injection spectra in S2 and previously [Li et al., 2008b]. The drift rate results further indicate that the type III beam produced in S1 is faster than in S2, yet only by 25%. [18] Although the intensity peak of the f p emission for S1 drifts only slightly faster than that for S2, this is not the case for the radiation onset. This result is explicitly demonstrated in igure 1b, by superposing the locus of the onset of the burst in igure 1a. We see from igure 1b that the onset of the radiation for S1 drifts much faster, thus indicating the presence of a significantly larger number of fast beam electrons than for S2. On the other hand, we also 4751

5 igure 1. Predicted dynamic spectra of f p emission and the corresponding profiles of spectral characteristics for S1 and S2. Dynamic spectra for electron injection spectra of (a) S1: power law with p = 8 and (b) S2: Maxwellian with T h = 25 MK. White dotted and white solid curves in igures 1a and 1b trace the trajectories of the maximal flux ˆmax for S1 and S2, respectively, along which df/dt are calculated. The black dashed curve in igure 1b shows the outline of the f p emission in igure 1a. requency profiles of (c) ˆmax and (d) df/dt for p = 8 (red) and Maxwellian (black). The corresponding dashed curves in igure 1d are predictions of equation (1) for v b = 0.20c and 0.16c, respectively. (e) Temporal profile of ˆ at f = 82.5 MHz for p =8(red) and the Maxwellian (black). overplot in igure 1b the locus of the end times of the radiation in igure 1a. We see that for S1 the end time at a given f is also earlier; yet the corresponding time advance is less than for the onset time. The end time results thus indicate that in S1 there are fewer slow beam electrons that are able to produce observable f p emission. igure 1e demonstrates further such time differences, by showing the temporal profiles of ˆ( f = 82.5 MHz) for both S1 and S2. [19] We also estimated the onset frequency drift rate for the f p emission in igure 1a, which corresponds to the onset time at a given f [e.g., ainberg et al., 1972]. We found that the onset drift rate is approximately constant with varying f, with values varying between 68 MHz s 1 at f = 165 MHz and 53 MHz s 1 at f = 50MHz. So the rate changes here are much smaller than those in igure 1d for df /dt at the radiation peak, yet the magnitudes of the onset rates are larger than df /dt, especially at low f. The relatively small changes in the onset drift rates with f are easily understandable from igure 1a since there the trajectory of the onset time is almost linear with respect to f. The relation (1), which predicts strong variation of the drift rate with the density profile, thus does not apply to these onset drift rates. [20] Because the radiation propagates the same in S1 and S2, independent of the details of the injection profiles of the accelerated electrons [Li et al., 2008a], the differences in the remote radiation are thus manifestations of changes in the source. Therefore, we study next the source region in order to understand the origins of such differences, e.g., the origin of the slightly faster beam for S Radiation Source [21] igure 2 shows the phase space electron distribution function f e (x, v), Langmuir wave occupation number N L (x, v), and the weighted emission rate k 2 (x a, k, ) from equation (6) in the source region at x a =0.30Gm at different times t a =3.3sand4.2 s for S1 and S2, respectively. (The ion-sound waves remain closely thermal, labeled as S below, due to the strong damping because of T e /T i =1and so are not shown.) These choices of t a show k 2 near its maximum at the chosen frequency f p (x a ) = 82.5 MHz, and thus relate directly to the observables ˆmax [via equation (6)] and the associated drift rate df /dt, and to the beams and waves that produce the dominant radiation. At a given location, the value of k for the maximum varies significantly with time [Li et al., 2005b]. or example, for S1 at t =1.4sthe peak has log 10 ( /s 1 ) = 14.7 with k 0.08 m 1, whereas 4752 the corresponding k 0.27 m 1 at t = 3.5 s when the peak is similar. Thus, when linking source emission to remote radiation, the weighted emission rate k 2 represents the source emission better than the actual rate, especially when considering ˆmax and the associated df /dt. [22] We see from igures 2a and 2b that both qualitative and quantitative differences in f e (x, v) exist between S1 and S2: (i) The beam in igure 2a extends further away from the injection site even though the time t a is about 1 s earlier than in igure 2b. (ii) The number of beam electrons with relatively low speeds (. 0.2c) is less for S1 than for S2. (iii) At a given location, the beam in S1 is faster than in S2. or example, at x = x a the beam speed for e /@v >0ranges from v min 0.13c to v max 0.23c in S1 and from 0.13c to 0.18c in S2. So v max is larger for S1 while v min is similar for both. Since v b R f e (v)vdv/ R f e (v)dv (v max + v min )/2, the

6 igure 2. Beam and waves in source regions in (left) S1 and (right) S2 at times t a =3.3and 4.2 s for S1 and S2, respectively, when k 2 maximizes at f = f p (x a =0.30Gm)=82.5MHz. (a d) Phase space distributions f e (x, v) and N L (x, v) of electrons and Langmuir waves, respectively. Dashed lines in igures 2a 2d indicate x = x a. (e and f) Weighted f p emission rate k 2 at x a as a function k and. v b values of 0.20c and 0.16c obtained in igure 1d using the simulated df /dt and the relation (1) for S1 and S2, respectively, are consistent with the source values of 0.18c and 0.16c, respectively. The larger v max for S1 occurs because of two effects. irst, significantly larger numbers of faster electrons are injected into the background corona (at the expense of relatively slow ones) in S1 than in S2, due to the power law form of the injected electrons in S1 compared to the Maxwellian form in S2. Second, these faster electrons in S1 produce a faster beam that reaches x a and locations further away at the time t a = 3.3 s, earlier than the slower beam observed at the later time t a =4.2sinS2. [23] igures 2c and 2d show that beam-driven Langmuir (L 0 ) waves are generated at the phase speeds v corresponding to the beams in igures 2a and 2b, respectively. These waves are produced due to competition between quasi-linear growth and Landau damping [e.g., Grognard, 1985]. We also found that for both cases, the products of Langmuir ES decays [Li et al., 2003]: L 1 waves (with v < 0) from the ES decay of the beam-driven L 0 waves, i.e., L 0! L 1 + S, and L 2 waves (with v >0) from the second-generation ES decay L 1! L 2 + S. We see from igure 2d for S2 that the maximum levels of L 0 waves are higher than for S1, due primarily to the larger numbers of beam electrons in igure 2b. Nonetheless, the loci of the peaks in the waves in igure 2d, either with positive or negative v, are limited to smaller v than in igure 2c. This is because the beam for S2 is limited to smaller v, leading to larger k L for the L 0 waves and subsequent products of the ES decays, and so smaller v =! L /k L [Ziebell et al., 2001; Kontar and Pecseli, 2002; Li et al., 2003]. [24] igures 2e and 2f show that the weighted emission rate k 2 peaks at =90 ı,asfor [e.g., Li et al., 2005b, 2008b]. The dominant peaks in igures 2e and 2f are produced by the EM decay L 0! 1 + S, and the weaker peaks at smaller k are generated by the EM decay L 1! 2 + S. We see also when comparing igures 2e and 2f that the main emission peak shifts to smaller k, i.e., k =0.26m 1 for S1, smaller by about 30% than for S2. The smaller k in igure 2f originates from the smaller k L (L 0 ) [or larger v(l 0 )] in igure 2c as discussed above and from the relation k / k L [Cairns, 1987a; Li et al., 2005b]. We see also that the source emission is stronger for S2, due primarily to the stronger L 0 waves in igure 2d. [25] igure 3 further contrasts the temporal evolution for S1 and S2 of f e (v) and N L (v) at the location x = x a at various times. igure 3 also shows f e (v) at x = x 0 and t = t 0, the central location and central time, respectively, for the electron injection. We see on comparing igure 3a with igure 3c that there are three different evolutionary stages for the two 4753

7 igure 3. Snapshots at specific times of phase space distributions of (left) electrons and (right) Langmuir waves at x a =0.30Gm for (top) S1 and (bottom) S2. Solid curves are for t =1.8to 4.2 s and dotted curves for t =4.5and 4.8 s. (b and d) The relative wave number k L /k 0 corresponding to v is shown on the top axes, where k 0 =2! p c s /3v 2 e =2.2m 1. (a and c) The dashed curves are for the electron distribution at t 0 =0.05sandx 0 = Gm, which are the central time and location of the electron injection given by equation (3). beams at x = x a in S1 and S2, where the stages are directly related to the different electron distribution f e (t 0, x 0, v) due to the different injections between S1 and S2. [26] 1. At early times t <3.0s the beam in S1 is more pronounced, i.e., has larger height relative to the background and wider width, and extends to larger v. or example, at t = 1.8 s the beam in S1 has peak height log 10 [f e /(m 4 s)] 2.9 at v 0.36c, while at the same time, the beam in S2 is very weak with the peak height 2.5 orders of magnitude lower at similar v. The stronger beam in S1 occurs because there is a larger fraction of fast electrons for the power law injection than for the Maxwellian injection in S2, as shown by comparing f e (t 0, x 0, v) between the two cases. The corresponding beam electrons in S1 drive significantly stronger L 0 waves in igure 3b than in igure 3d for S2, where the Langmuir waves are almost thermal for t. 2.1 s. [27] 2. igures 3a and 3c show that the later-arriving (3.0 s. t <4.2s), moderately fast beam electrons are comparable between the two cases. This occurs because similar amounts of moderately fast electrons are injected for the two simulations, as shown by the corresponding f e (t 0, x 0, v). The corresponding L 0 waves in igures 3b and 3d are also similar, yet those in S1 are still stronger. We see further from igure 3b that the levels of L 0 waves peak at t 3.3 sand 4754 v 0.19c, while those in igure 3d are still growing with time. [28] 3. The beam (if it exists at all) in S1 for t 4.2 sis much weaker than that in S2. We see that at t =4.2sthe number of beam electrons in S1 is much lower than in S2 and the beam in S1 is narrower. At even later times we see that there is no beam in S1, while the beam in S2 is still quite strong at t =4.5s it is no longer found at this location for t & 4.8 s. [29] These differences occur because the number of relatively slow electrons injected in S1 is less than in S2, as expected since the total number of injected electrons is the same in both cases but more fast electrons are injected in S1, as again shown by the different f e (t 0, x 0, v) for the two cases. We see from igure 3b that in this last stage the L 0 waves in S1 decrease greatly with t and are only slightly nonthermal at t =4.5s. In contrast, igure 3d shows that the L 0 waves in S2 reach peak levels at t 4.2 sandv 0.15c and decrease only slightly by the time 4.5 s. [30] The results above demonstrate that the L 0 waves at a given location peak at smaller t and larger v (or smaller k L ) for the power law injection than for the Maxwellian injection, with phase speeds v 0.19c and 0.15c for the two types of injection, respectively. This occurs due to the

8 igure 4. Effects of varying p on (a) ˆmax and (b) df/dt at 120 MHz (stars) and 82.5 MHz (circles), and on (c) v b.the other simulation parameters are as in simulation S1. much greater number of fast electrons and much smaller number of slow electrons for the power law injection. Consequently, and so k 2 in S1 peak at smaller t and k, as in igure 2e. The remote radiation in igure 1a thus drifts faster than in igure 1b. Moreover, early in their evolution, the L 0 wavesins1havelargerv, sothecorresponding k L are smaller; e.g., k L & 2k 0 at t = 1.8 s in igure 3b. The parameter k 0 = 2! p c s /3v 2 e [Melrose, 1986], with k 0 =2.2m 1 here, is the maximum difference in wave number between the L and L 0 waves involved in the ES decay L! L 0 + S. It also strongly affects the kinematics and rates of the ES and EM decays [Melrose, 1986; Cairns, 1987a; Robinson et al., 1994], and hence the f p and 2 f p emission rates [Cairns, 1987a, 1987b; Robinson and Cairns, 1998b; Li et al., 2005a, 2005b]. or example, for the ES decay L 0! L 1 + S here, k 0 directly controls k(l 1 ) via the relation k(l 1 )= k(l 0 )+k 0, as demonstrated previously [Melrose, 1986; Cairns, 1987a; Robinson and Cairns, 1998b; Ziebell et al., 2001; Kontar and Pecseli, 2002; Li et al., 2003]. urther discussion regarding prediction of 2 f p emission rate is deferred to section Scalings With the Parameters of Power Law Injection [31] In situ observations of energetic electrons and X- ray observations during flares suggest that both the power law index p and the break speed v 0 in equation (4) vary from event to event [Lin, 1974, 2011; White et al., 2011]. This section studies the effects on remote f p emission of varying p and v 0 separately. The ranges of p and 4755 v 0 assumed here follow those inferred from observations [Krucker et al., 2008; Bain and letcher, 2009; Glesener et al., 2012] and used in previous modeling [Harvey, 1976; Hannah and Kontar, 2011; Reid et al., 2011; Zharkova and Siversky, 2011] Scalings With the Power Law Index p [32] We choose p to vary between 6 and 16. This range is suggested by recent observations of X-rays associated with type III bursts [Krucker et al., 2008; Bain and letcher, 2009; Glesener et al., 2012]. It is also used in previous modeling for type III bursts [Harvey, 1976], for both type III bursts and X-rays [Reid et al., 2011], and for Langmuir waves in the lower corona [Zharkova and Siversky, 2011]. [33] igure 4 shows the variations with p of ˆmax and df /dt at two frequencies f = 120 and 82.5 MHz, and of v b, for otherwise identical parameters to S1. We see that as p increases, all the quantities decrease in magnitude. In more detail, the decreases are quick for p. 8 but are much slower for larger p, and all the quantities appear to approach their asymptotic limits near the upper end of p. igure 4 shows that the maximum decreases for all the quantities are by factors <2between p =6and 16. The decreases occur because alargerp corresponds to fewer fast electrons and more slow electrons being injected for a given inj, and so a smaller beam speed, less free energy in the beam, and consequently weaker radiation. [34] Nonetheless, igure 4b also shows that the df /dt results at f = 120 MHz change more slowly and smoothly than at f = 82.5 MHz when p decreases from 7 to 6. Thatis, the relation (1) predicts the simulated df /dt results between f = 165 and 50 MHz less well for p =6than for the other p values. In fact, for p = 6 the variations in df /dt with f are not closely linear or well fitted with a single v b [unlike igure 1d]; however, the value plotted in igure 4c is the best estimate, with v b 0.21c and 0.35c at f 165 and 50 MHz, respectively. uture work will address simulations for p <6if smaller p values than used here are observed to be relevant. [35] igure 4c shows that 0.19c v b 0.25c for the simulated range of p, larger than the result v b 0.16c for the Maxwellian injection in S2. So the f p emission produced by the power law injections studied here always drifts faster than that in S2. However, these v b values are still near the lower end of the observed range c of type III beam speeds v b [Suzuki and Dulk, 1985; Klassen et al., 2003] Scalings With the Break Speed v 0 [36] ollowing previous modeling [Hannah and Kontar, 2011; Zharkova and Siversky, 2011], the break speed v 0 is set to vary between ms 1 and ms 1.Thetwo bounds correspond to break energy of 2.6 and 23 kev, respectively, and have v 0 /v e 5.5 and 16.4, respectively. [37] igures 5a 5c show the predicted variations with v 0 of ˆmax and df /dt at two frequencies f = 120 and 82.5 MHz, and of v b, for conditions otherwise identical to S1. We see that an increase in v 0 produces increases in the magnitude of all three quantities. This occurs because, according to the spectrum (4), for a given ratio inj of injected electrons to background electrons more fast electrons are injected for a larger v 0, at the expense of the number of slow electrons. The resulting faster beam electrons subsequently generate

9 igure 5. Effects of varying v 0 on (a) ˆmax and (b) df/dt at 120 MHz (stars) and 82.5 MHz (circles), and on (c) v b.the other simulation parameters are as in simulation S1. stronger Langmuir waves and so stronger f p emission, which drifts faster. We see from igure 5c that v b =0.25cfor v 0 = ms 1. Therefore, the predicted type III beam is still slower than typical type III beams with v b c/3 [Suzuki and Dulk, 1985; Klassen et al., 2003]. [38] Moreover, igure 5a shows ˆmax. 1 sfu at the two frequencies for v ms 1. Thus, the f p radiation is hardly observable for small v 0, even at the lowest simulated frequency f =50MHz (e.g., where ˆmax =2.3sfu when v 0 =410 7 ms 1 ). or such small v 0 values the injected electrons are mostly slow, the beam-driven Langmuir waves thus have small phase speeds, and so these waves are strongly Landau damped. Consequently, the radiation is insignificant. 5. Discussion [39] Sections 3.1 and 3.2 show that for both S1 and S2, the simulated drift rates df /dt of the peak f p emission agree quantitatively with the analytic prediction (1) for the beam speeds v b found in the simulated radio sources. This agreement is consistent with our previous quasi-linear-based simulations with different (Maxwellian or kappa) background plasma conditions, different (Maxwellian or power law) spectra of injected particles, and nonlinear processes [e.g., Li et al., 2008b, 2008c, 2011a; Li and Cairns, 2013a]. This suggests that the relation (1) holds well for type III bursts, at least when the quasi-linear approximation applies. The v b values obtained from the prediction (1) and from the source beam dynamics are 0.20c and 0.16c for S1 and S2, respectively. Thus, the type III beam for S1 is faster, due to the 4756 injected electrons having a power law spectrum rather than a Maxwellian spectrum in S2. Nevertheless, the quantitative increase in v b here is only moderate, i.e., by 25% for the specific power law parameters (p =8and v 0 = ms 1 )in S1 and flare temperature (T h =25MK) in S2. [40] urther, scaling studies in section 4 show that varying the power law parameters p and v 0 affect v b, in addition to the fluxes and drift rates of f p emission. Specifically, v b increases with decreasing p or increasing v 0. The increases are by factors < 1.6 for the values of p and v 0 inferred from observations [e.g., Krucker et al., 2008; Glesener et al., 2012] and used in previous modeling [e.g., Hannah and Kontar, 2011; Reid et al., 2011], with the largest v b =0.25c for p =6or v 0 =910 7 ms 1. Note that igures 4 and 5 suggest that v b will be >0.25c if p and v 0 are extended to smaller and larger values, i.e., < 6 and > ms 1, respectively. Therefore, our simulations suggest that occasionally, type III beams can have v b >0.25c if they are produced by power law injections with such small p and/or large v 0 values. However, for the ranges of p and v 0 values inferred from observations, the beam speeds are still near the lower end of the observed range c of type III beam speeds [Suzuki and Dulk, 1985; Klassen et al., 2003]. [41] On the other hand, the earlier onset of the f p emission in S1 than in S2 especially at lower frequencies (see igures 1a and 1b) is caused by the fact that the driving beam has a significantly larger number of fast beam electrons with v >0.25c (see igures 3a and 3b). In contrast, the radiation onset in igure 1b for S2 is generated primarily by slower beam electrons with v <0.25c (see igures 3c and 3d), as expected for the Maxwellian injection spectrum in S2. urthermore, the speed v f of the beam electrons responsible for the radiation onset can be estimated by using the time-offlight approximation: d/v b d/v f = t.hered is the distance traveled by the beam electrons to reach a given coronal layer, t is the time delay between radiation onset and peak emission at the frequency corresponding to the layer, and v b is derived from the df /dt results. or example, igure 1e for S1 shows that t 1.5 s for the radiation at f = 82.5 MHz, and d x a x Gm, the distance between the injection site and the coronal layer corresponding to f p = 82.5 MHz. We thus have v f 0.36c, consistent with the results of igure 3a at t =1.8s, which is earlier by t than the time t =3.3s when the weighted emission rate k 2 is near maximum as in igure 2e. [42] Thus, the predicted modifications to f p emission at Earth for power law versus Maxwellian electron injection spectra are observable manifestations of changes in electron injection occurring in the corona. Type III processes for power law injection spectra take place at both small and large values of phases speed v and wave number k, but at small v and large k for Maxwellian injection spectra. In addition, for power law injection, the dominance of strong waves at slightly larger v and slightly smaller k due to the injection of faster electrons produces slightly fasterdrifting type III bursts and so slightly faster beams than for Maxwellian injection. Moreover, the much earlier radiation onset for the power law injection is an observable manifestation of the presence of even faster beam electrons due to the injection. [43] Qualitatively, the simulation results here for coronal situations should also be applicable to type III bursts in

10 the solar wind, suggesting that interplanetary type III beams will be slightly faster for power law than for Maxwellian injection spectra. Nevertheless, losing energy to waves along their paths will gradually decelerate the beams somewhat, leading generally to slower type III beams in the solar wind than in the corona [ainberg et al., 1972; Dulk et al., 1987; Malaspina et al., 2011]. [44] The present work predicts only f p emission but does not predict 2 f p emission. This is because for 2 f p emission, an approximation made in our model [Li et al., 2008a, 2008c] is not satisfied when very fast beam electrons are present due to the power law injection. Specifically, our type III model considers that H waves (2 f p emission) are produced by the coalescence L + L 0! H coupling forward ( f ) and backward ( b ) Langmuir waves via fb and bf processes, and assuming k L k H or equivalently, k L k 0 [Willes et al., 1996]. This assumption is valid for the Maxwellian injection spectra considered since the Langmuir waves satisfy k L k 0, as demonstrated in igure 3d. However, for power law injection spectra, the simulated k L associated with fast beam electrons is small and comparable to k 0 (see igure 3b), and so the model assumption is inappropriate. Thus, the other coupling processes ( ff and bb ) linked to coalescence of Langmuir waves with small wave numbers [Willes et al., 1996] need to be included in order to model the production of 2 f p emission properly. [45] The ff and bb processes are expected to have important contributions to the onset of 2 f p dynamic spectra, since these processes are associated with the fastest type III beam electrons. urther, these new processes may even contribute significantly to peak emission under specific coronal and beam conditions, because the associated 2 f p emission rate can peak sharply at k L /k H 1 for very narrow Langmuir spectra N L (k L ) [Willes et al., 1996]. That being the case, the v b values derived from the drift rates df /dt for the peak in 2 f p emission may differ from those obtained here from the corresponding f p emission. Moreover, relativistic effects may be important for the very fast beam electrons and associated waves; e.g., the electrons with v c in S1 when the beam is first formed, although their contributions to the f p emission in S1 is negligible. [46] The results here and in previous work [Li et al., 2008b, 2008c, 2009, 2011b; Li and Cairns, 2012] show that, in general, type III beams produced in the corona with Maxwellian background particle distributions can only have relatively small speeds v b <0.3c, irrespective of whether the injected energetic electrons have power law or Maxwellian spectra. Consequently, the often observed moderate to fast type III beams with v b >0.3c [e.g., Suzuki and Dulk, 1985; Klassen et al., 2003] cannot be produced in coronal plasma whose background particles are Maxwellian distributed. However, the coronal background particles may have non- Maxwellian distributions, e.g., kappa () distributions, as inferred from solar wind data [Ko et al., 1996; Maksimovic et al., 1997b] and proposed theoretically [Scudder, 1992; Maksimovic et al., 1997a]. Our other recent simulations [Li and Cairns, 2013a, 2013b] show that distributions of background electrons have significant effects on type III bursts. Importantly, we found that fast type III beams with v b c can be naturally produced in coronal plasmas with <8by injected electrons with either power law or Maxwellian spectra [Li and Cairns, 2013a, 2013b]. 6. Conclusions [47] Predictions from simulations including quasi-linear and nonlinear interactions are presented for f p emission of coronal type III bursts that result from injection of energetic electrons with power law speed spectra onto open magnetic field lines in an otherwise unmagnetized Maxwelliandistributed background corona. The simulations generalize our previous model [Li et al., 2008a, 2008c] for type III bursts produced by energetic electrons heated during flares with Maxwellian speed spectra, by allowing injection of very fast power law electrons, as indicated by in situ electron data and X-ray data [e.g., Lin, 2011; White et al., 2011]. We found that although the power law injection introduces a wider range and larger number of faster electrons into the electron beam than for a Maxwellian injection, these electrons are important, in general, only to the onset and not to the peak of f p emission, because there are not many fast electrons. In contrast, the far more numerous slower electrons contribute significantly to the peak f p emission. Consequently, the type III beam speeds v b obtained from the drift rate df /dt of the peak f p emission by using the analytic relation (1) are larger, but not significantly larger in general, than the values of v b for Maxwellian injection spectra. Nevertheless, for power law injection the v b values obtained from df /dt (for the peak f p emission) are consistent with the beam speeds in the radiation sources, as found for Maxwellian injection [e.g., Li et al., 2008b]. [48] Specifically, we find that v b. 0.25c for a Maxwellian background plasma and electron injections with the power law spectrum (4), for parameters currently considered realistic. The upper limit of v b 0.25c here is greater by only 25% than that ( 0.2c) for Maxwellian injection [e.g., Li et al., 2008c, 2009]. Thus, the predicted speeds of type III beams produced in the corona with Maxwellian background particle distributions can account only for the lower half of the observed range c of type III beam speeds v b [Suzuki and Dulk, 1985; Klassen et al., 2003]. Nevertheless, the simulations here indicate that observations of such relatively slow-drifting type III bursts do not necessarily suggest that all the beam electrons are slow. Instead, some beam electrons can be fast with v > 0.3c and contribute significantly to the radiation onset. These fast beam electrons are generally present in a type III source for power law injection. [49] We simulated here only the f p emission, but not the corresponding 2 f p emission. This limitation is imposed by our simulation model [Li et al., 2008a, 2008c], which for 2 f p emission assumes large Langmuir wave numbers k L with k L k 0 =2! p c s /3v 2 e,wherek 0 is a parameter directly relevant to ES and EM decays [e.g., Cairns, 1987a, 1987b; Robinson et al., 1994]. So the model includes only the processes involving the coupling between pairs of Langmuir waves in opposite (forward and backward) hemispheres relative to the beam direction. Although k L k 0 holds for the Maxwellian injection spectra [e.g., Li et al., 2008b], this condition is not always met in the simulations for power law injection. We found that for power law injection spectra, Langmuir waves are significantly enhanced at small wave numbers, where k L k 0 due to the presence of fast beam electrons. uture work should include the other coupling processes relevant to small k L (or large v)[willes et al., 1996] in order to predict the 2 f p emission properly. 4757