PUBLICATIONS. Journal of Geophysical Research: Space Physics

PUBLICATIONS Journal of Geophysical Research: Space Physics RESEARCH ARTICLE Key Points: Saturated beam instabilities lead to plateau distributions with finite current Current-driven Langmuir oscillations produce amplitude modulations Mechanism for double-peak spectra observed in solar wind, planetary foreshocks Correspondence to: R. D. Sydora, rsydora@ualberta.ca Citation: Sauer, K., and R. D. Sydora (2015), Current-driven Langmuir oscillations and amplitude modulations Another view on electron beam-plasma interaction, J. Geophys. Res. Space Physics, 120, 235 252, doi:10.1002/ 2014JA020409. Received 23 JUL 2014 Accepted 28 NOV 2014 Accepted article online 3 DEC 2014 Published online 19 JAN 2015 Current-driven Langmuir oscillations and amplitude modulations Another view on electron beam-plasma interaction K. Sauer 1 and R. D. Sydora 1 1 Department of Physics, University of Alberta, Edmonton, Alberta, Canada Abstract The origin of Langmuir amplitude modulations and harmonic waves observed in the solar wind and in planetary foreshock regions is investigated in beam plasmas where the saturation process of the beam instability is accompanied with the formation of a plateau distribution. This saturated state represents a current which is shown to drive homogeneous electric field oscillations at the plasma frequency. This simple mechanism has been ignored in most numerical studies based on Vlasov or particle-in-cell simulations because of the use of the Poisson equation which is not suitable to describe the mechanism of current drive in plasmas with immobile ions; instead, Ampere s law must be used. A simple fluid description of stable plateau plasmas, coupled with Ampere s law, is applied to illustrate the basic elements of current-driven Langmuir oscillations. If beam-generated Langmuir/electron-acoustic waves with frequencies aboveorbelowtheplasmafrequencyare simultaneously present, beating of both wave modes leads to Langmuir amplitude modulations, thus providing an alternative to parametric decay. Furthermore, very important implications of our studies (presented separately) concern the electrostatic and electromagnetic second harmonic generation by nonlinear interaction of Langmuir oscillations with finite wave number modes which are driven by the plateau current as well. 1. Introduction The present study represents an extension of previous theoretical work on mode crossing effects and related phenomena in beam-plasma interaction [Sauer and Sydora, 2012]. In nonmagnetized plasmas our main interest was to search for mechanisms which may explain the appearance of Langmuir amplitude modulations that have been observed in several spacecraft measurements under a variety of different plasma conditions [Gurnett et al., 1981, 1993; Hospodarsky et al., 1994; LaBelle et al., 2010; Sigsbee et al., 2010; Graham and Cairns, 2013]. These modulations show no unique signature concerning amplitude, coherence, and overall structure; however, occasionally very regular wave packets have been measured which are manifested in the power spectrum of the electric field as pronounced double peaks close to the electron plasma frequency ω e [e.g., Gurnett et al., 1993; Sigsbee et al., 2010]. The relative frequency shift between the peaks δω/ω e is typically near 0.02 [LaBelle et al., 2010] but may also reach values larger than 0.1 [Soucek et al., 2005]. With regards to the origin of one of the two peaks, there is general consensus that they are produced by beam instability. The source of the second peak remains controversial and in the literature several mechanisms have been discussed, such as the modulational instability, parametric decay, and particle trapping [Hasegawa, 1975]. For all of these, one of the main difficulties is their failure to explain why the two waves generating the double peak have nearly the same amplitude. This signature is inconsistent with any kind of interaction process that finally produces a secondary (daughter) wave of much lower amplitude compared with the primary (mother) wave. The other serious argument against the common interpretation that the second Langmuir wave results from parametric decay is the observation of frequency shifts which are larger than the ion plasma frequency, whereby ion-acoustic waves as decay product are excluded. Our view is especially supported by cusp measurements [LaBelle et al., 2010] where contributions by Doppler shifts are absent which, on the other side, are of crucial importance in parametric theory to explain the observed double-peak spectra. A new perspective on wave generation in beam-plasma systems has been established in the context of our particle-in-cell (PIC) simulations in magnetized plasmas [Sauer and Sydora, 2012]. An important feature of these simulations, in both over-dense (Ω e /ω e < 1) and underdense plasmas (Ω e /ω e > 1), where Ω e is the electron cyclotron frequency, for the case of oblique wave propagation (with respect to the external SAUER AND SYDORA 2014. American Geophysical Union. All Rights Reserved. 235

magnetic field), we have found that additional wave modes besides the modified Langmuir mode become active, namely, in wave number ranges which are well separated from maximum beam instability, k max c/ω e ~ c/v b (V b is the beam speed). Whistler waves were found to appear at kc/ω e 1in plasmas with ω e > Ω e even for the case where the beam is out of resonance with these waves. Examples are illustrated in Sauer and Sydora [2012]; see Figures 5 and 6 there. Relevant space observations have been described by Kellog et al. [1992], Moullard et al. [1998], and in laboratory by Stenzel [1977]. Similar effects have been found in underdense plasmas where beam-plasma interaction is accompanied by both enhanced wave activity at the mode crossing point at ω ~ ω e and kc/ω e ~ 1 and for the R-mode waves around Ω e ;see Figure 7 in the paper above. Efforts have been made to discover the origin of this unexpected wave excitation beyond the range of beam instability. Wave-wave interaction as usually assumed has been excluded since the frequency and wave number matching conditions cannot be satisfied. So, we have come to the conclusion that the plateau distribution of the saturated state acts as driver of the observed wave modes outside regions of beam resonance. Hereby, one has to note that velocity components transverse to the magnetic field may arise if oblique propagation is considered. Evidence that the plateau distribution of the saturated beam-plasma state may drive other wave modes in the form of a current has led us to the general problem of current-driven waves in plasmas. The simplest case already appears in isotropic plasmas and motivated us to use Ampere s law (1/c 2 ) E x / t = μ 0 j x instead of the Poisson E x / x = ε 0 ρ in order to take into account the plateau current. In so doing, by means of fluid or PIC simulations, it is shown that in stable plateau plasmas Langmuir oscillations (ω = ω e, k = 0) are generated whose amplitude is proportional to the driving plateau current j p ~ n p V p, where n p and V p are the plateau density and velocity, respectively. In this context, we refer to the recent paper of Baumgaertel [2013] in which beam-aligned electric field oscillations (ω = ω e ) are discussed and their beating with beam-generated Langmuir waves is considered to be the origin of Langmuir amplitude modulations. Corresponding hybrid code simulations in plasmas with immobile ions have also been done without noting that the application of Ampere s law instead of the Poisson equation is the essential difference between most of the previous simulations by other authors in which no Langmuir oscillations (at k = 0) occur. The content of the paper is as follows. In section 2 Langmuir oscillations at the plasma frequency ω e are described by a fluid approach. Simple dispersion theory of a plasma consisting of two electron populations (main plasma and beam) with finite temperature is used (section 2.1) to illustrate the transition from an unstable beam to a stable plateau when the beam temperature increases. In section 2.2 the temporal evolution of the electric field and of the other plasma parameters is determined by solving the system of fluid equations (continuity equation and equation of motion for the two electron populations) together with Ampere s law which is required to get Langmuir oscillations. Support comes from particle-in-cell (PIC) simulations presented in section 2.3, where the results of using Poisson equation and Ampere s law are compared. The formation of Langmuir amplitude modulations due to the superposition of Langmuir oscillations and beam-generated Langmuir waves is considered in section 3. Here it is shown that the simultaneous presence of Langmuir oscillations and Langmuir waves with finite wave number, which are excited by a weakly unstable beam, generates a beating structure that depends on the frequency of the growing wave. Results of PIC simulations of beam and plateau plasmas are discussed in section 4, thus confirming results from the fluid model. Ignoring nonlinear effects, response theory, in which the electric field is expressed by a linear relation to a given current, is applied to describe Langmuir amplitude modulations (section 5). For this model, the current must contain two contributions, one constant due to the plateau current and the other which results from beam-excited waves at saturation. In section 6, Langmuir amplitude modulations are presented in more detail, starting with a discussion of space measurements followed by predictions of Vlasov dispersion theory. In section 7, summary and conclusions are given. 2. Langmuir Oscillations in Plasmas With Plateau Distributions 2.1. Transition From Beam Instability to a Stable Plateau The saturation of the beam instability has recently been studied in great detail by many authors using PIC and Vlasov simulations [e.g., Kasaba et al., 2001; Matsukyio et al., 2004; Silin et al., 2007; Umeda, 2007]. The characteristic signatures of the transition from beam instability to a saturated state can be described as follows: Langmuir waves are excited if the velocity of the beam electrons is in resonance with the phase SAUER AND SYDORA 2014. American Geophysical Union. All Rights Reserved. 236

Figure 1. Sketch of the transition from (a) pronounced beam instability to (d) a stable state. Distribution function versus velocity normalized to thermal speed (top), real (ω; middle), and imaginary (γ; bottom) parts of frequency normalized to the electron plasma frequency ω e versus wave number k normalized to the Debye length λ D. Maximum instability appears in Figure 1a kλ D ~ 0.07, ω/ω e ~ 0.95; Figure 1b kλ D ~ 0.13, ω/ω e ~ 0.95; and Figure 1c kλ D ~ 0.27, ω/ω e ~ 1.1. velocity of the waves. This happens on the lower velocity side of the beam (lower beam mode) where the electron distribution function has a positive velocity gradient. Subsequent wave-plasma interaction leads to beam heating by which the beam becomes broader in the direction of smaller velocities. Expressed by macroscopic quantities, this means an enhancement of the beam thermal velocity and a reduction of its drift speed. During this evolution, the gap in velocity space between the beam and the main plasma diminishes in time up to saturation where the beam instability is completely quenched and a plateau distribution is formed. To illustrate the transition from the onset of beam instability up to saturation in a simpler form, we use a fluid approach by means of water-bag distributions which allows for an easier representation of the temporal evolution of the distribution functions and a simple calculation of the dispersion characteristics of the associated unstable Langmuir waves. For this purpose, an electron plasma at rest with density n eo and thermal speed V Te =(kt e /m e ) 1/2 is considered. Ions are taken as immobile forming the charge-neutralizing background. The inflowing beam is described by its density n po, drift velocity V Dp, and initial thermal speed V Tp =(kt p /m e ) 1/2.(T e and T p are the temperatures of the main plasma and the beam (plateau), respectively; the index p is used here in order to indicate its simultaneous marker of plateau parameters.) The dispersion relation for such a plasma configuration is composed of the contributions from both the main plasma and the beam (plateau) and can be written in the common form as εω; ð k ω 2 e Þ ¼ 1 ω 2 γk 2 V 2 Te ω 2 p 2 ¼ 0 (1) ω kv Dp γk 2 V 2 Tp This expression, which one can find in classical textbooks, can easily be derived from the Vlasov formula by using water-bag distributions or by the usual procedures (linearization and periodic ansatz) if one starts from the full set of fluid equations which are given later on in section 2.2. ω e =(n eo e 2 /ε o m e ) 1/2 and ω p =(n po e 2 /ε o m e ) 1/2 are the plasma frequencies of the two populations, main plasma and beam (plateau), respectively, and γ is the adiabatic constant (= 3/2). Figures 1a 1d (top) show envisaged beam-plasma distribution functions, expressed by water-bag distributions, at 4 times during the evolution of beam-plasma interaction, starting from the initial beam SAUER AND SYDORA 2014. American Geophysical Union. All Rights Reserved. 237

(t = t 0 ) up to the formation of a full plateau (t = t 3 ). To illustrate how the related beam instability develops in time, a relatively arbitrary variation of the beam parameters has been chosen, but based on our experience from kinetic simulations [Sauer and Sydora, 2012]. For the initial beam shown in Figure 1a the following parameters have been taken: density α = n po /n eo = 0.01, beam velocity V p = V Dp /V Te = 16, and thermal velocity U p = V Tp /V Te =1(T p /T e = 1). At later times the beam is heated and a lowering of its drift velocity takes place. Thus, at t = t 1, t = t 2, and t = t 3 for the beam, the following parameters have been chosen: (Figure 1b) V p = 13, U p =3(T p /T e = 9), (Figure 1c) V p = 11, U p =4(T p /T e = 16), and (Figure 1d) V p = 10, U p =5(T p /T e = 25). As clearly seen, during the transition from maximum beam instability to full plateau formation, the maximum growth rate of the beam instability goes down and shifts to larger wave numbers. Whereas the wave number of maximum instability is initially at kλ D ~ 0.07, it varies to about kλ D ~ 0.3 just before the beam saturates. For later discussion it is important to note that the related frequency shifts simultaneously from values below the plasma frequency ω e to values above. The frequencies at maximum instability for the three configurations shown in Figures 1a 1c are ω ~ 0.95, 1.02, and 1.1. The frequency of the first instability (ω ~ 0.95ω e ) belongs to the electron-acoustic mode which intersects the Landau mode resulting in mode splitting near the plasma frequency. Finally, it should be mentioned that the above fluid approach has been compared with Vlasov dispersion theory using plateau-type distribution functions that have been modeled by a soft-corner distribution function, first introduced in kinetic dispersion analysis by Baumgaertel and Sauer [1976] or by a superposition of Maxwellians [Silin et al., 2007]. It has been found that the general trend of our simple fluid approach with respect to the beam instability remains nearly the same. There are only modifications due to Landau damping if effects in the range of larger wave numbers (kλ D > 0.2) are analyzed. Indeed, more significant differences are found in the electron-acoustic (beam) modes which appear in the fluid approach but are generally strongly damped in kinetic theory beyond the direct vicinity of the electron plasma frequency. 2.2. Langmuir Oscillations Driven by Plateau Electrons In this section we return to our initial discussion on wave activity after plateau formation which appears in wave number ranges often far beyond maximum linear instability. In our recent PIC simulations of beam-plasma interaction in magnetized plasma, Sauer and Sydora [2012], the onset of large-amplitude waves at small wave numbers (kλ D 1) have been observed which are well separated from the originally unstable Langmuir waves. Since ion motion has been ignored and the frequency and wave number matching relations between both regions of wave activity could not be satisfied, parametric decay and other types of nonlinear wave-wave interaction processes as possible explanation have been excluded. Since this long-wavelength wave activity always appeared after beam saturation, the idea was formulated that the plateau plasma acts as a driver of wave modes which are not directly related to the beam instability. In this sense, the so-called antenna concept has been introduced by us. Applying the antenna concept to the well-studied case of beam-plasma interaction in nonmagnetized plasmas, the natural question arises: what is the electric response of a plasma with a plateau distribution as shown in Figure 1d, which is stable with respect to the excitation of Langmuir waves? Since the plateau distribution represents a current (j p = en po V Dp ), we can also ask if there is any plasma response to this current. To answer this question, we apply the fluid approach and consider a simple one-dimensional box of length L which is continuously filled by the incoming plateau plasma having the (normalized) density α = n po /n eo, drift velocity V p = V Dp /V Te and thermal velocity U p = V Tp /V Te. The system of fluid equations in which both populations, the main plasma (index e) and the plateau plasma (index p) are coupled by the longitudinal electric field E x, is easily written as t n e þ ð x n ev ex Þ ¼ 0 (2) ð t n ev ex Þþ x n ev 2 ex þ γ U 2 e x n e ¼ q e n e E x (3) t n p þ x n pv px ¼ 0 (4) n p v px þ t x n pv 2 px þ γ U 2 p x n p ¼ q e n p E x (5) SAUER AND SYDORA 2014. American Geophysical Union. All Rights Reserved. 238

Figure 2. (a) Space-time evolution of the electric field in units of E 0 =(m e /e)ω e V Te ; (b) temporal evolution of both the electric field (top) and the deviation of the (normalized) beam speed from its initial value at one point within the box (x 0 = 250λ D ), using the parameters of Figure 1d: α = 0.01, V p = 10, and U p =5. t E x ¼ q e n e v ex þ n p v px (6) The densities and velocities are normalized to the background electron density (n eo ) and to the corresponding thermal velocity (V Te ) of the main population, respectively, i.e., U e = 1. Time is normalized to the reciprocal plasma frequency ω e =(n eo e 2 /ε o m e ) 1/2 and spatial coordinate x to the Debye length λ D = V Te /ω e. The electric field amplitude is normalized to E 0 =(m/e) ω e V Te. It is important to point out that the x component of Ampere s law is taken here instead of the normally used Poisson equation. The flowing plasma (minor component-plateau plasma) of given density penetrates the domain at x = 0 and at the right boundary (x = L) free boundary conditions are used. The fluid equations have been solved numerically by means of discretization using the flux-corrected algorithm following the code SHASTA (Sharp And Smooth Transport Algorithm) of Boris and Book [1973]. Results for the parameters of the stable plateau configuration in Figure 1d, with α =(n po /n eo )=0.01, V p =(V Dp /V Te )=10andU p =(V Tp /V Te ) = 5, are shown in Figure 2 where (a) the x-t variation of the electric field and (b) the time variation at a fixed point within the box (x o =250λ D ) are plotted. As seen there, uniform Langmuir oscillations appear whose frequency (according to simple analysis that follows) is just the electron plasma frequency which belongs to the total density of the plasma-plateau configuration. Since these oscillations are driven by the plateau current ( j p ~ αv p ), we use the term current driven. The frequency and amplitude of the Langmuir oscillations can be analytically calculated from the governing equations (2) (6) if uniform conditions ( / x = 0) are assumed for all quantities. The system reduces to a simple equation of oscillation for the electric field d 2 dt 2 E x ω 2 e E x ¼ 0 (7) where ω e is the plasma frequency of the whole system with the total density n o = n eo + n po. If the initial conditions E x (0) = 0, v ex (0) = 0, and v px (0) = V p are imposed, one finds the solutions E x ðþ¼ E t m sinðω e tþ with E m ¼ αv p (8) for the electric field and v ex ðþ¼ t v px ðþ V t p ¼ E m ½cosðω e tþ 1Š (9) for the electron velocity v ex (t) which is identical with the change of the plateau drift velocity v px (t) V p. Taking the parameters used in the numerical solution of the full system of equations (2) (6), α = 0.01, V p = 10, the analytic expressions in equations (8) and (9) yield electric field and velocity variations that are in very good agreement with the numerically obtained solutions shown in Figure 2b. At first glance, the appearance of the described Langmuir oscillations seems to be a curious phenomenon that has not been previously mentioned. There is only the recent paper by Baumgaertel [2013] in which he pointed out the importance of this oscillating solution with respect to the interpretation of Langmuir amplitude modulations. He argued that the absence of Langmuir oscillations in the great majority of beam-plasma simulations is traced back to the realization of current-free conditions. Of course, the presence SAUER AND SYDORA 2014. American Geophysical Union. All Rights Reserved. 239

Figure 3. Langmuir/electron-acoustic waves in plasmas with plateau distribution. (a) Electron distribution function composed of main plasma and plateau which is shaped by superposition of shifted Maxwellians. Different from Figure 1d, the plateau density here is α = 0.05. The dashed line marks the plateau drift velocity V p. (b) frequency versus wave number of the Langmuir/electron-acoustic waves, and (c) related damping γ/ω e. There is a broad wave number range (0.07 kλ D 0.3) over which the waves are nearly undamped. PIC simulation results which demonstrate the different evolution of the electric field according to which equation has been used for its calculation; (d f) Poisson equation or (g i) Ampere law. In both cases, the same electron distribution function of Figure 3a has been used initially (t = 0). The x-t evolution of the electric field is shown in Figures 3d and 3g; pronounced differences become evident. The temporal evolution of both the total electric energy and of the electric field at x o /λ D = 250 are plotted in Figures 3e and 3h and Figures 3f and 3i, respectively. of a current in a plasma is a necessary requirement for the existence of Langmuir oscillations; however, whether Langmuir oscillations appear in any kind of simulations of current-driven plasmas crucially depends on the choice of the equation that is used to determine the electric field. After our fluid studies (supplemented by PIC simulations as seen later) we can clearly state that in plasmas with immobile ions current-driven Langmuir oscillations are not contained in the Poisson equation as taken in most of the beam-plasma studies. One has to use Ampere s law with the inclusion of a time derivative. A general treatment based on the Vlasov approach, of how the solution of the Poisson equation and Ampere s law are related has been done by E. Sonnendruecker, Max Planck Institute for Plasma Physics (private communication, 2014). He has shown that the electric field solution of the Vlasov-Ampere equation is the sum of two terms: E = E ~ + E where E ~ is the solution of the Poisson equation and E can be computed analytically at any time according to E ðþ¼ t J ð0þ sinðω e tþ (10) ε o ω e J (0) is the average electron current which is determined by the initial condition of the distribution function f 0 by J ð0þ ¼ e L L 0 vf 0ðx; vþdxdv: (11) For a plasma with plateau distribution of density n p0 and drift velocity V Dp the electric field of equation (10) is identical with the expression (8) which is written there in normalized units. Ponno et al. [2008] pointed out that the simple relation (10) is only valid if the ions are assumed to be immobile. If ions are mobile the situation is more complex and will be a topic of future investigation. 2.3. Current-Driven Langmuir Oscillations in PIC Simulations In order to demonstrate that essential differences in PIC simulations arise depending on what equation we chose to calculate the electric field E x, Poisson equation (Figures 3d 3f) or Ampere s law (Figures 3g 3i), we SAUER AND SYDORA 2014. American Geophysical Union. All Rights Reserved. 240

start the PIC simulations with a stable plasma-plateau configuration, similar to that in Figure 1d. The selected two-component electron plasma consists of the main population with Maxwellian distribution function and a plateau which is formed by superposition of multiple shifted Maxwellian distribution functions (Figure 3a). The corresponding Vlasov dispersion relation in its common form has been solved numerically. As seen in Figure 3b and discussed in several previous papers, mode splitting between the Langmuir mode of the main plasma and the electron-acoustic mode of the plateau plasma takes place. In contrast to the usual two-temperature plasma, however, the Langmuir wave which belongs to the plateau is nearly undamped over a relatively broad wave number range (Figure 3c). Starting from the stable plasma-plateau configuration as described above, its further space-time evolution is calculated by means of PIC simulations. We mainly focus on the differences obtained using the Poisson equation and Ampere s law to solve for the electric field. Periodic boundary conditions for fields and particles are used. Simulation parameters are L = 512 λ D, Δx = λ D, N = 1000 particles/cell, and time step ω e Δt = 0.05. For comparison, the main results are illustrated in Figures 3d 3i. Figures 3d 3f are from simulations including the Poisson equation; Figures 3g 3i show corresponding results when Ampere s law is used. Clear differences are visible, particularly in the x-t evolution of the electric field (Figures 3d and 3g). Whereas the picture on the left exhibits the typical stripe pattern of a propagating wave with finite phase velocity (V ph /V Te 6), the horizontal colored lines in Figures 3g 3i correspond to Langmuir oscillations (ω ~ ω e, k ~0)which occur if Ampere s law is taken, as already described in the fluid approach of section 2.2. In order to understand the physical origin of the wave pattern in Figure 3d, which is obtained if the Poisson equation is used, we refer to investigations on the thermal fluctuation spectrum in plasmas with two electron components. In a paper by Lund et al. [1996] it has been shown how the wave number of maximum thermal activity depends on different parameters of the second electron population, such as density, drift velocity, and temperature. In this sense, the plasma with plateau distribution represents a particular kind of multicomponent plasma for which the general formula for electrostatic fluctuations in plasmas with drifting Maxwellian components can be applied. Finally, we want to point out clear differences that appear if the temporal evolution of the electric field at a certain spatial point for both cases is compared. In particular, the periodic variations of the electric field in Figure 3i are a clear manifestation of current-driven Langmuir oscillations at the plasma frequency ω e which only occur if the plasma is described by Ampere s law. 3. Fluid Approach to Langmuir Amplitude Modulations 3.1. Modulations due to a Weakly Unstable Wave In the previous section we have shown that Langmuir oscillations at the plasma frequency are driven by the current which exists in plasmas with a plateau-type second population. Up to now, only the case of a completely stable plateau distribution function has been presented. As a next step, we want to consider the case of a weakly unstable plasma-plateau configuration. For this purpose we select the situation which is shown in Figure 1c where the plateau parameters α = 0.01 (relative density), V p = 11 (normalized drift velocity) and U p = 4 (normalized thermal velocity) have been used. For these parameters there is a weak beam instability with a maximum growth rate of γ/ω e ~ 0.02 at k max λ D ~ 0.28 and corresponding frequency of ω ~ 1.1ω e, see Figure 1c. Since our fluid approach contains no mechanism of beam saturation, the temporal evolution of the beam instability is limited to its initial phase. Results of numerical integration of the basic equations (1) (5) are summarized in Figure 4. Figure 4a (top) represents the x-t evolution of the electric field up to ω e t = 250 and clearly reveals how the electric field develops. At the beginning when the instability begins to grow exponentially, almost pure Langmuir oscillations (k ~ 0) are seen as straight lines. At later time (ω e t 200), the unstable Langmuir wave with finite wavelength λ ~2π/k max reaches comparable amplitudes and leads to amplitude modulations due to the superposition (beating) of both electric field contributions. This becomes more evident in Figure 4b which shows the electric field amplitude at ω e t = 220 which varies with λ ~ 22.5 λ D. Naturally, the plateau velocity V px in Figure 4c displays the same spatial dependence. The time variation of the electric field at a selected point within the box (x 0 /λ D = 300) is shown in Figure 4d. The amplitude modulation arises due to the superposition of the purely oscillating electric field with the field of the Langmuir wave. Complete field annihilation, as seen there, is possible if both contributions have comparable amplitudes. Since the amplitude of Langmuir oscillations is fixed by E m = αv p ~ 0.1, from SAUER AND SYDORA 2014. American Geophysical Union. All Rights Reserved. 241

Figure 4. Space-time variation of the electric field E x and of the beam (plateau) velocity v px for a weakly unstable beam-plasma configuration, calculated with the parameters of Figure 1c: α =0.01,V p =11,andU p =4.(a)Thecolorplotshowsthatthe electric field is composed of two contributions, one is due to the Langmuir oscillations (k = 0) as in Figure 1a and the other is the electric field due to the weakly unstable wave (k = k max )withfinite wave length λ =2π/k max.(bandc) The spatial variation of the electric field E x and the plateau velocity V px at the selected time ω e t = 220. (d) Almost complete annihilation of the electric field seen on top results from the beating of both electric field contributions in the case that they are of nearly equal amplitudes. (e) The frequency spectrum of the time period up to ω e t = 300. equation(7), electric field nodes may only appear if the maximum electric field amplitude is in the range of twice that value, that means, about 0.2 which is the case. Finally, the fact that the electric field of a (beam) plateau-plasma is composed of two contributions of slightly different frequencies can nicely be seen in its frequency spectrum. It is plotted in Figure 4e for the time period up to ω e t = 300. From the preceding discussion it is evident that the double peak is a consequence of the two mechanisms of wave generation which are involved here. The first peak at ω 1 = ω e is due current-driven Langmuir oscillations, and the second one results from the unstable wave at frequency ω 2 which corresponds to maximum instability. For the instability considered here with the parameters of Figure 1c, this frequency is above the plasma frequency (ω 2 ~ 1.1), in agreement with the spectral plot in Figure 4e. Finally, it bears mentioning that the use of Ampere s law instead of the Poisson equation taken in most of beam-plasma-studies leads to significant differences. Whereas Langmuir oscillations only occur if Ampere s law is applied, the unstable Langmuir wave due to beam instability is contained in both descriptions. That is obviously the reason for the absence of Langmuir oscillations in most of the publications of beam-plasma interaction. We return to this point when corresponding results of PIC simulations are discussed. 3.2. Modulations by a Given Langmuir Wave Since our simple fluid approach contains no beam saturation mechanism, its application to Langmuir amplitude modulations is limited to short times. Nevertheless, in order to model a saturated state, we introduce a simple technique of prescribing a beam-saturated Langmuir wave in a stable plateau-plasma by SAUER AND SYDORA 2014. American Geophysical Union. All Rights Reserved. 242

Figure 5. Generation of Langmuir amplitude modulation by the superposition of Langmuir oscillations in a plateau plasma (α = 0.01, V p = 10, U p = 5) with a prescribed electron-acoustic wave of given frequency ω = 0.95ω e and an amplitude which was chosen such that both contributions to the electric field are almost the same. (a) The beating structure of the electric field E x (x,t) is clearly seen as large-scale amplitude variation with a time period of ω e T ~ 150. (b) The spatial electric field variation at ω e T 0 = 430. Its wavelength of about 65λ D corresponds to the wave number kλ D ~ 0.1 of the given wave which propagates in the plateau plasma. (c) The temporal evolution of both the electric field E x and the velocity v px V p, respectively, are plotted at x 0 = 250λ D. Pronounced Langmuir amplitude modulations are observed which give rise to the clear double-peak spectrum of electric field, shown in Figure 5d. The insert shows an expanded view of the electric waveform with the phase jump by 180 at the node of the envelope. adding an additional term on the right side of Ampere s law, equation (6). This external current (j ~ ), which may generate an electrostatic (Langmuir or electron-acoustic) wave of given amplitude (A ~ ), frequency(ω ~ ), and wave number (k ~ ), is simply written as j ~ = A ~ cos(ω ~ t k ~ x) where ω ~ and k ~ are related by the dispersion relation of the stable plasma-plateau system; see Figure 1d. In Figure 5 numerical solutions of the fluid equations (2) (6) including the external current j ~ are presented. For the plasma-plateau configuration which drives the Langmuir oscillations, the parameters of Figure 1d are used again. In addition, we assume that the electron-acoustic wave, with frequency ω ~ =0.95ω e and related wave number k ~ λ D = 0.1, is excited by the corresponding external current j ~. Its amplitude A ~ has been varied to find the value which produces a wave electric field that is nearly the same as that of the Langmuir oscillation (E m = 0.1). The value of A ~ = 0.006 was found to be the appropriate amplitude. Figure 5a shows the space-time evolution of the electric field. Again, amplitude modulations are clearly seen in form of inclined colored stripes, whereby a large-scale periodicity of ω e T ~ 150 can be estimated. The spatial dependence of the electric field at ω e t ~ 430 is plotted in Figure 5b. From there, a wave length of about λ ~62λ D is found which corresponds to the original wave number k ~ λ D ~ 0.1 of the given current. Clear Langmuir mode modulations are visible in Figure 5c (top and bottom) where the temporal variation of the electric field E x and the velocity difference v px V p are plotted. Of course, the amplitude of the prescribed electron-acoustic wave has been selected in such a way that it competes with the electric field SAUER AND SYDORA 2014. American Geophysical Union. All Rights Reserved. 243

Figure 6. Langmuir amplitude modulation at beam-plasma interaction; results from PIC simulation. (a) The initial bump-on-tail distribution function and the related dispersion of Langmuir/ electron-acoustic waves, where (b) frequency and (c) growth rate versus wave number are shown. (d) The color-coded ω-k dependence of the electric field at the final time level (ω e T end =200). Main features are marked there: The occurrence of Langmuir oscillations at k =0andω = ω e, the beam instability with its maximum at k max λ D ~0.3andω max ~0.87ω e, two contributions to the second harmonic ω ~2ω e and finally a weak indication of a low-frequency (beat) mode. (e) The Langmuir amplitude modulation arising from the beating of Langmuir oscillations with electron-acoustic waves of maximum instability. of the Langmuir oscillations E m = αv p = 0.1. Only in this case the electric fieldspectrum,showninfigure5d, has a double peak with nearly equal amplitudes of both contributions. The frequency difference between both peaks is about 0.05 according to the beating of Langmuir oscillations (ω = ω e )withthe electron-acoustic mode (ω ~ =0.95ω e ). 4. PIC Simulation of Beam-Plasma Interaction In this section, PIC simulations are used to demonstrate the occurrence of Langmuir amplitude modulations at beam-plasma interaction. Through our fluid simulations, it has been shown that Langmuir amplitude modulations result from the beating of Langmuir oscillations (ω = ω e, k = 0) and beam-excited Langmuir/electron-acoustic waves (ω ~ ω e, k ~ ω e /V b ) which have been considered as given. For a realistic beam situation, however, the problem of using a fluid approach is that beam saturation cannot be described; a kinetic model is required. The parameters which we take are α = 0.1, V p = 4, and U p = 0.3. The use of a relatively high beam density makes it easier to overcome the thermal and numerical noise level which may disturb the Langmuir oscillations. Results of the PIC simulation extended up to ω e t = 200 are shown in Figure 6. In the left three panels, the initial distribution function (Figure 6a), frequency (Figure 6b), and growth rate versus wave number (Figure 6c) are illustrated. Maximum growth rate appears at k max λ D ~ 0.3 with a related frequency of ω max ~ 0.87ω e. Figure 6d represents the color-coded ω-k dependence of the electric field at the end of integration time. The main features marked there include the following: (i) the occurrence of Langmuir oscillations at k = 0 and ω = ω e, (ii) the beam instability with its maximum at k max λ D ~ 0.3 and ω max ~ 0.87ω e, (iii) two contributions to the second harmonic ω ~2ω e, and finally (iv) a weak indication of a low-frequency mode likely caused by beating of both high-frequency waves near ω e. Since second harmonic generation is a topic of great relevance for space observations, a detailed discussion of underlying interaction processes in stable plateau plasmas and at beam-plasma instability is beyond the scope of this paper and will be reported elsewhere. Figure 6e, finally, clearly shows the amplitude modulation in form of wave packets which arise SAUER AND SYDORA 2014. American Geophysical Union. All Rights Reserved. 244

dueto the superposition of Langmuir oscillations with the beam-excited waves of maximum instability. The beating period of ω e T ~ 50 corresponds to the related frequency difference of δω/ω e ~ 0.13. Parts of these results are similar to those presented in the paper by Baumgaertel [2013]. He pointed out the importance of Langmuir oscillations without recognizing, however, that the essential difference to previous simulations is caused by using Ampere s law instead of the Poisson equation. 5. Response Theory of Amplitude Modulations Response theory has been proven to be a suitable tool that assists in understanding and interpreting some of the effects found in simulations of Langmuir oscillations and related amplitude modulations. Under the assumption of small-amplitudes, the linearized version of the Maxwell-fluid equations (2) (6) can be used to calculate the electric field in response to a current. Thus, linearization and subsequent Fourier transform of these equations leads us to the simple expression j je x ðk; ωþj ¼ ext ðþ k εω; ð kþ where ε(ω,k) is the dielectric function of the plateau plasma according to equation (1) and j ext is an external current. It may represent the constant plateau current of the incoming plasma flow or it may be adapted by the current which belongs to the spectrum of beam-saturated waves or, maybe, both. Thus, different models of driving currents can be used to easily test their response on the ω-k dependence of the electric field. In the following, we illustrate the response theory by considering the example of Langmuir amplitude modulations which, as previously discussed, arise from the superposition of Langmuir oscillations and beamgenerated waves. Accordingly, the current j ext is composed of two contributions, one due to the plateau current j p ~ αv p and the other is the current (j wave ) which belongs to the spectrum of saturated waves. Dispersion analysis and results of kinetic simulations have been used in order to constitute a reasonable approximation of its wave number dependence, especially regarding maximum position and saturation amplitude. As a stable plasma configuration we take a plateau plasma similar to Figure 1d with α = 0.01, V p = 7 and U p =3. The application of the response theory to the plasma system above is illustrated in Figure 7. In Figure 7a the dispersion relation ω(k) of the stable plateau plasma is plotted. The assumed growth rate of beam instability, which determines the dominant spectrum of Langmuir waves before its saturation takes place, versus k is shown in Figure 7b. Beam saturation generates the related current which has been adapted in Figure 7c as a peak at the point where the growth rate has a maximum. Lastly, the constant contribution comes from the plateau current j p ~ αv p. Further, in Figure 7c 1/ε(k) for the two Langmuir/electron-acoustic modes is shown. According to equation (12), its wave number dependence is the decisive factor in the E x (ω,k) calculation. Finally, the singularity at k =0(ω = ω e ) leads to the electric field peak and is therefore responsible for the occurrence of Langmuir oscillations as seen in Figure 7d where the ω k dependence of the electric field is presented. There are two amplitude maxima. One at ω = ω e and k = 0 which represents Langmuir oscillations due to the drift of the plateau plasma. The other peak is at kλ D 0.1 and results from the current contribution of the beam-excited waves after saturation. Here the amplitude of this contribution has been selected in such a way that both the Langmuir oscillations and the beam-related waves have nearly the same amplitudes. That is clearly expressed in Figure 7e which shows the frequency spectrum after the k integration of E x (ω,k). A pronounced double peak appears whose relative frequency shift for the selected set of parameters is δω/ω e ~ 0.02. If the waves of slightly different frequency act together, Langmuir amplitude modulations arise due to their beating. Of course,onlyinthecasethatbothwaveshavecomparableamplitudes, modulations with distinct nodes are formed. The simple formulation of response theory has proven to be a useful tool for our Langmuir oscillation and amplitude modulation application in two respects. First, for cases where the dielectric function and external current contain a large set of parameters it offers the possibility of quickly obtaining conditions under which double-peak spectra and associated frequency shifts occur for select parameters. Second, it is an efficient way to help establish parameters in more complex models such as kinetic simulations. ω ω e (12) SAUER AND SYDORA 2014. American Geophysical Union. All Rights Reserved. 245

Figure 7. Langmuir amplitude modulation described by response theory of nonmagnetized plateau plasmas. (a) Frequency versus wave number of the two Langmuir/electron acoustic modes for a plateau plasma with α =0.01,Vp =7,andU p =3; (b) growth rate of simulated beam instability before saturation; (c) dotted and dashed curves: 1/ε(k) for the two wave modes of Figure 7a; solid curve: adapted current distribution q/q 0 consisting of the constant plateau current superimposed by a localized peak for the simulated source of beam-driven Langmuir waves at k max λ D ~ 0.12; (d) color-coded ω-k dependence of the electric field, dashed circles mark the two regions of maximum amplitude; and (e) frequency spectrum of the electric field. The sharp peak at ω e represents the Langmuir oscillations, the other one at ω ~ 0.98ω e results from the beam-driven Langmuir waves at k ~ ω e /V b,wherev b is the original beam velocity. 6. Langmuir Amplitude Modulations Observed in Space 6.1. Description of Events Numerous publications on spacecraft and rocket measurements of Langmuir amplitude modulations can be found in the literature [e.g., Gurnettetal., 1981, 1993; Hospodarsky et al., 1994; Hospodarsky and Gurnett, 1995; Sigsbee et al., 2010; LaBelle et al., 2010; Malaspina et al., 2011; Graham and Cairns, 2013; see also Malaspina et al., 2010, 2013; Graham et al., 2014]. In the following, we discuss four examples of events in which the measured Langmuir wave spectrum exhibit distinct double peaks close to the plasma frequency. Characteristic signatures of these events are the variation of their beat frequency and electric field amplitude E over a broad range. Modulated Langmuir waves with amplitudes from mv/m up to V/m have been detected. An essential parameter for event classification is the ratio E/E 0 where E 0 is given by E 0 =(m/e)ω e V Te. To begin with, events with E/E 0 10 3 are considered. First measurements of Langmuir bursts in the solar wind with a resolution high enough to detect the fine structures of the beat-type waveforms and the spectrum have first been published by Gurnett et al. [1993]. In association with type III radio bursts the plasma wave instrument aboard the Galileo spacecraft has measured Langmuir wave activity with waveforms of considerable variability. Among isolated and irregularly spaced wave packets, regular Langmuir amplitude modulations with double-peak spectrum have been observed in the amplitude range of mv/m and smaller. The separation of the two spectral peaks generally varied between 200 and 500 Hz, which corresponds to a relative frequency splitting of δf =(f f e )/f e ~ 0.01 0.02, where f e is the electron plasma frequency (~25 khz). In Figure 8a corresponding spectrum and waveform measurements by Gurnett et al. [1993] are shown. With nominal solar wind parameters, n e ~7cm 3 and T e ~ 10 ev, one gets E 0 ~ 1 V/m, meaning that the normalized amplitude of the modulated Langmuir waves is in the range E/E 0 10 3. SAUER AND SYDORA 2014. American Geophysical Union. All Rights Reserved. 246

Figure 8. Measured spectra and related waveforms of Langmuir amplitude modulations with different beat frequencies δf/f e =(f f e )/f e in the range 0.03 δf/f e +0.1. (a) Representative solar electron event aboard Galileo by Gurnett et al. [1993], δf/f e = (0.01 0.02). The amplitude of the modulated Langmuir waves is below 1 mv/m. (b) Rocket measurement of a Langmuir burst in the cusp region by LaBelle et al. [2010]. The observed beat frequency is δf/f e ~ 0.03 according to f e ~ 500 khz and a frequency separation of δf ~ 15 khz. Main lines are indicated by arrows. (c) Langmuir power spectrum and waveform from Earth s foreshock from Cluster measurements by Sigsbee et al. [2010]. The beat frequency is δf/f e ~ 0.03. This selected event exhibits double peaks at the plasma frequency f e and its second harmonic 2f e as well as a peak at the beat frequency itself. (d) Double peak of modulated Langmuir waves with large separation (δf/f e ~ 0.1) from Cluster terrestrial foreshock observations by Soucek et al. [2005] and related waveform. In the spectrum there is also a visible weak peak at the beat frequency of δω ~ 2 khz. Modulated Langmuir waves of higher amplitude (E/E 0 0.02) have been observed by LaBelle et al. [2010] on rocket flights in the cusp region. The advantage of these wave measurements is that they are not affected by Doppler effects due to plasma motion, as in the solar wind or planetary foreshock regions. The high-frequency electric field instrument on board the twin rockets continuously sampled the electric field wave forms up to 10 MHz. Intense Langmuir waves up to about 800 mv/m have been measured during underdense conditions (2 < Ω e /ω e 6), whereas the modulation frequency varied between 1 khz (about 10% of the time) up to 50 khz. Of particular interest to us is one event, called burst #1, which has been analyzed in greater detail. Its main signatures, i.e., frequency spectrum and associated waveform, are shown in Figure 8b. In the spectral plot one clearly recognizes the double peak with a relative frequency shift of δf/f e ~ 0.03 between both peaks. The maximum electric field amplitude for this event is approximately 30 mv/m. With ω e ~3 10 6 s 1 and V Te ~10 5 m/s, i.e., E 0 =(m/e)ω e V Te ~1.5V/m, this corresponds to a relative electric field amplitude of E/E 0 ~ 0.02. Further Langmuir burst events, which are described in the paper by LaBelle et al. [2010], have higher amplitudes, 80 100 mv/m for burst#2 and 150 300 mv/m for burst#3. SAUER AND SYDORA 2014. American Geophysical Union. All Rights Reserved. 247

Measurements by Sigsbee et al. [2010] with the Wideband Data Plasma Wave Receiver aboard Cluster in the Earth s foreshock provide us with the third class of Langmuir electric wave forms. The goal of their analysis was to find out the occurrence rate of different types of power spectra, as single- or double-peaked ones, depending on the Langmuir wave amplitude. A remarkable feature in the context of our studies is that the double-peaked category belongs to low electric field amplitudes that are between 0.1 and 22 mv/m. Figure 8c shows a representative example for this category. At higher amplitudes, greater than 22 mv/m, most power spectra fell into the single peaked f e and 2f e categories. With the measured parameters ω e ~210 5 s 1 and V Te ~ 1.5 10 6 m/s for the Earth foreshock, one gets E 0 ~ 1.5 V/m. Thus, the electric field amplitude of E = 22 mv/m corresponds to the normalized value E/E 0 ~ 0.015. Lastly, STEREO observations by Graham and Cairns [2013] should be mentioned which show two-peaked Langmuir wave spectra where the Langmuir waves are of higher amplitude than 22 mv/m, e.g., 40 mv/m (see Figure 3 there). The reason for these differences is not clear. A class of observations of modulated Langmuir waves with particularly large beat frequency has been made using Cluster measurements, by Soucek et al. [2005] and J. Soucek, Department of Space Physics, Institute of Atmospheric Physics, Academy of Sciences of the Czech Republic (private communication, 2013), who found very pronounced Langmuir amplitude modulations with beating frequencies of δω/ω e 0.1, but relatively small amplitudes of less than 10 mv/m. In Figure 8d an example of a typical spectrum and Langmuir waveform taken from the Soucek et al. [2005] paper is shown. Our interpretation of the obtained spectrum is that the lower frequency peak, which in many cases is more intense than the higher-frequency one and often marked by greater spectral sharpness, is caused by current-driven Langmuir oscillations at ω e. The other, mostly broader peak obviously results from Langmuir waves which are excited by a moderate dense beam ( 1%) of low speed (V b 5V Te ). 6.2. Predictions of Vlasov Dispersion Theory and Interpretation In this section, it is assumed that the electric field spectrum exhibits a double peak which results from two equal contributions, meaning Langmuir oscillations and beam-generated Langmuir waves have the same amplitudes. Then, the procedure starts with determining the normalized electric field E 0 =(m/e) ω e V Te,an essential quantity which is only defined by the parameters of the background plasma (density n e0 and temperature T e ). For instance, in the solar wind near Earth this value is E 0 ~(1 2)V/m. On the other hand, the electric field E/E 0 of current-driven Langmuir oscillations is, according to the relation in equation (8), directly proportional to the current of the plateau (beam) plasma, i.e., E/E 0 = αv p where α = n p0 /n e0 and V p = V Dp /V Te are the corresponding normalized density and drift velocity, respectively. Thus, if E/E 0 is known from measurements and α is varied, the related velocity V p is simply given by V p =(E/E 0 )/α; that means both beam parameters are fixed. With the additional assumption that plasma and beam temperature are the same (T p = T e ), these parameters are used now to solve the Vlasov dispersion relation of beam-generated Langmuir waves. Following these assumptions, the diagrams in Figure 9 have been constructed for five values of E/E 0 (0.001, 0.005, 0.02, 0.05, and 0.1). From top to bottom, the relative frequency shift δω/ω e =(ω-ω e )/ω e where ω is the frequency of the most unstable wave (Figure 9a), the related normalized wave number k max λ D (Figure 9b), the normalized growth rate γ/ω e (Figure 9c), and the beam velocity V p versus the normalized beam density α (Figure 9d) are plotted. Thus, one can draw the following conclusion: if one knows from measurements both amplitude E/E 0 and frequency shift δω/ω e, the associated range of beam velocity and density can be obtained from Figure 9 by going down to the density axis. For example, using E/E 0 = 0.02 and δω/ω e = 0.02, one would read out beam densities in the range of α ~10 3 to which beam velocities of V p ~ 20 belong. We mention here that predictions for small beam velocities (V p < 5), and correspondingly small growth rates, have to be considered carefully since the fluid behavior, valid in case of large beam velocities, becomes more and more modified by kinetic effects. In this case, the simple expression for the electric field of Langmuir oscillations, E/E 0 = αv p, is no longer valid and the electric field dependence on the plasma drift has to be determined by means of kinetic methods in which the particular velocity distributions have to be taken into account. Of course, Figure 9 is modified if colder or warmer beams, with respect to the background electron plasma, are considered. SAUER AND SYDORA 2014. American Geophysical Union. All Rights Reserved. 248