Bernstein-Greene-Kruskal analysis of electrostatic solitary waves observed with Geotail

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. A10, PAGES 22,131-22,139, OCTOBER 1, 1997 Bernstein-Greene-Kruskal analysis of electrostatic solitary waves observed with Geotail V. L. Krasovsky, I H. Matsumoto, and Y. Omura Radio Atmospheric Science Center, Kyoto University, Kyoto, Japan Abstract. Electrostatic solitary waves (ESW) relevant to the high-frequency component of broadband electrostatic noise spectra observed by Geotail spacecraft in the magnetotail are analyzed. A general description of the ESW, as stationary solutions of Vlasov-Poisson equations, is developed on the basis of the Bernstein- Greene-Kruskal (BGK) technique. It is shown that the structure and relationships among basic parameters of a localized perturbation are primarily determined by resonant and nonresonant plasma screenings of bunched electrons trapped by a potential pulse moving in a plasma. The dependence of the typical width of the pulse on its drift velocity and pulse amplitude is found, which can be treated as a nonlinear dispersion relation for the localized perturbation. The theoretical results are compared with the Geotail data and a particle simulation. 1. Introduction One of the most surprising results obtained by the Geotail spacecraft in the distant terrestrial magnetotail is a discovery of waveforms of broadband electrostatic noise (BEN). Matsumoto et al. [1994] found that BEN is composed of a single or a sequence of isolated bipolar electric field pulses called electrostatic solitary waves (ESW). In other words, they pointed out that BEN has waveforms of solitary bipolar pulses instead of the expected continuous random oscillations. A typical waveform of BEN is shown in Figure 1. According to the Geotail data these signatures are a characteristic feature of BEN in the plasma sheet boundary layer where its intensity is generally maximum. Another characteristic property of BEN is well-defined correlation with charged particle flows. These features suggesthat the origin of BEN can be closely related to the interaction of electron (and/or ion) flows in the magnetotail, and the observed solitary pulses can provide an example of an excitation of nonlinear wave structures in a beam- plasma system. It is generally taken that the low-frequency component of BEN spectra is closely connected with the ions flowing along the plasma sheet boundary layer [Gurnett et al., 1976; Gurnett and Frank, 1977]. Accordingly, this portion of the spectra is attributable to ion beam-driven instabilities [ Grabbe and Eastman, 1984; Akimoto and Omidi, 1986; Ashour-Abdalland Okuda, 1986]. There X Also at Space Research Institute, Russian Academy of Sciences, Moscow. Copyright 1997 by the American Geophysical Union. Paper number 97JA /97/97JA ,131 were efforts to explain also the high-frequency component in terms of ion beam-driven modes [Grabbe, 1985] or Doppler-shifted electron acoustic modes [Schriver and Ashour-Abdalla, 1987; Baumjohann et al., 1989]. The contribution of field-aligned electron beams [Parks et al., 1984] to the generation of high-frequency spectra has been discussed by Baumjohann et al. [1989]. Onsager et al. [1993] described the correlation of BEN with the high-energy electron component in the absence of ion flows. Finally, the fact that the ESW pulse width found by the Geotail plasma wave instrument (PWI) is of the order of a period of electron plasma oscillations has indicated a close association between BEN and elec- tron dynamics. This suggests that electrons most likely contribute to the generation mechanism of the highfrequency component of BEN. In the course of recently carried out numerical simu- lations on the behavior of beam-plasma systems it was found out that the long-time evolution of the system leads to formation of localized nonlinear wave structures similar to the BEN wave forms [Matsumoto et al., 1994; Omura et al., 1994, 1996]. It was also found that the localized wave structures have a one-to-one correspondence to the so-called phase space electron holes (EH) [Berk et al., 1970] which comove with corresponding solitary waves. Omura et al. [1994, 1996] have shown that the beam-plasma system evolves very slowly after the saturation of the beam-plasma instability caused by beam trapping. The vortices in the electron phase space coalesce with each other and merge into new, usually more intense, localized perturbations of the same type. Repeated coalescence leads to a generation of isolated solitary potential structure associated with the EH. Figure 2 shows one example of such localized perturbations in the electron phase space (upper panel) and corresponding waveforms of the electric field (lower panel). The figure is produced by a particle simula-

2 22,132 KRASOVSKY ET AL.: ANALYSIS OF ELECTROSTATIC SOLITARY WAVES +64 Date: Time: 15:24:21 to find an allowable functional form of electron velocity distribution for a given ESW waveform, and then we go forward to establish the relation between the amplitude and the spatial scales of a solitary pulse for a model electron hole distribution function. Section 4 presents our conclusions. 2. General Description of a Localized Positive BGK Pulse -64 o 5 Time (msec) Figure 1. Typical BEN waveform most frequently observed by Geotail spacecraft in the plasma sheet boundary layer of the magnetotail. tion as a continuation of a simulation run presented in Figure 7 by Omura et al. [1996]. The repeated interaction of the EHs often leads to formation of well-defined solitary pulses of the electrostatic potential with waveforms looking very similar to the waveforms observed by Geotail. As the EH is very stable and has a long life, the probability of picking up such isolated pulses in a form of the ESW is much higher for Geotail than the chances of picking up its early linear or the beginning of nonlinear stage of the instability. The long-lived wave perturbations under consideration are purely electrostatic in their nature and according to simulation results look like almost stationary structures in the phase space as shown in Figure 2. As a consequence of the phase-mixing process, the electron distribution function tends to the Bernstein-Greene- Kruskal (BGK) [Bernstein et al., 1957] equilibrium in the course of the evolution (see, for example, Thompson [1971]). On this basis, it is of interest to analyze such localized nonlinear waves with the use of the BGK approach. Our goal is to examine the structure of a localized BGK wave in order to find out the general rules of its propagation in a plasma. The Geotail ESW observations provide information on the amplitude and time width of the solitary pulse but do not permit determination of its spatial scale. Therefore to extract further information on its spatial scale from the observed ESW (BEN) waveforms, it is necessary to establish a theoretical relation between its amplitude and spatial width. Once the relation is established, we could estimate the velocity of the ESW pulses observed by Geotail as we have information of its duration in time. In section 2, we give a mathematical description of a localized BGK solitary pulse. In section 3, we attempt The formation of localized BGK waves is fairly common in a nonequilibrium plasma. The nonlinear structures were found even in pioneering computer simulations of a beam-plasma interaction (see, for example, Berk et al., [1970]). Solitary pulses in laboratory plasmas were first described by Ikezi et al. [1971] and Saeki [1973]. A number of theoretical studies have been made to wave structures associated with so-called phase density holes, as a structural element of plasma turbulence [Dupree, 1982]. The theory of negative localized BGK perturbations caused by ion phase density holes has also found fruitful application in space plasma physics [Berman et al., 1985; Tetreault, 1991]. In this section we consider similar in structure but purely electron perturbations yielding a positive localized hump of potential as observed by the Geotail PWI. Similar localized wave opt i i I I I i I I. I ß ".-." -...-'.-i.'.:. '":'.'... :'-,..'." 0.0 t:'-" ':: -'- :- 4,.,- -4.0,,,, I,,,, 0'08I... '... _1 0.00,-E_O vtt / op Figure 2. (Top) Electron holes in x -v phase space and (bottom) the corresponding spatial profile of the electric field E at COpt of a simulation run of an electron bump-on-tail instability by a one-dimensional electrostatic particle code with 1024 grid points and 10,137,472 particles. This is a continuation of a simu- lation run presented in Figure 7 of Omura et al. [1996]. The velocity, length and electric field are normalized in terms of the total plasma frequency and the initial thermal velocity of the majority thermal electrons. The density of the electron beam forming the bump is 5% of the total electron density.

3 KRASOVSKY ET AL.' ANALYSIS OF ELECTROSTATIC SOLITARY WAVES 22,133 perturbations, as applied to a laboratory plasma exper- that the solution of (1) is an arbitrary function f(w) iment in a Q machine, were studied in [$chamel, 1979; of the total electron energy in the wave frame W = Turikov, 1984; Kono et al., 1986] in order to explain v2/2-e (x). For a single positive potential hump shown the structure of phase density electron holes [Lynov et in Figure 3, we can write the Poisson equation in the al., 1985]. However, our treatment is based on a more form general approach and goes beyond their electron hole model. This allows us to study general properties of dx 2 the positive BGK pulse observed by Geotail. - + dw(f+ v/2(w + f_) O) + : dw(f+ v/2(w+o) + f_) ' Consider a localized stationary perturbation of the ( ) electrostatic potential propagating in a collisionless plasma along the external magnetic field. Since the where f+(w) and f_ (W) correspond to particles movtypical duration of the pulses observed by Geotail is ing in the positive and negative directions, respectively, about a period of electron plasma oscillations, we shall in the frame of the moving BGK positive pulse. The restrict ourselves to the case of purely electron waves first, second and third terms in the right-hand side repassuming ions to be immobile. According to our prelimresent the densities of ions, untrapped passing electrons inary results of two-dimensional numerical simulations and trapped electrons, respectively. In Figure 3, we also of long-timevolution of a beam-plasma system, the present a qualitative pattern of the electron phase traone-dimensional model [Omura et al., 1996] describes jectories and a typical perturbation of the total electron adequately the wave structures under study. This is density. We can see that, in essence, the electrostatic because the external magnetic field plays the role of potential perturbation is caused by trapped electrons a contributory factor for the one-dimensional character of electron dynamics. Therefore, we will solve the 4( ) problem in a one-dimensional electrostatic formulation, A using the BGK approach. Let the longitudinal wave be stationary and move with a constant velocity in the direction of the external SOI TARY POS magnetic field x. We start with the Vlasov-Poisson system of equations in the frame of reference moving with the pulse drift velocity u, V x / + Ox /=o Ov ' ( ) = - +, n - dv f(v, x). ( ) dx 2 Hereafter the following units are used to dedimensionalize the physical variables OF POTENTIAL/ TRAPPED L 0 L b w= - (=) [ ]-, It]- ;, [ ]- Iv]- [u] =, [ne] - no, If] - no/ n, [ b] - mvt /2e, [El- mv, = (note/eo) /2, ½rD Te 2 ½2no e2 o eom n - rd p, T -- m, (3) where v and V = v + u are the electron velocities in the moving and the rest frames of reference respectively, no is unperturbed plasma density and Te is the electron temperature. The steady state electron distribution function can be chosen, for example, in the form of the Maxwellian distribution, W=O TOTAL F.,L aon DENSITY L W>0 PASgING fo(v)- V - exp - 2 ' (4) Following the general description of stationary nonlinear waves given by Bernstein et al. [1957] we note Figure 3. (a) A solitary positive'pulse of potentials corresponding to ESW. (b) Electron trajectories on the corresponding phase plane. (c) Qualitative spatial dependence of electron density perturbation.

4 . 22,134 KRASOVSKY ET AL.: ANALYSIS OF ELECTROSTATIC SOLITARY WAVES streamlined by the antrapped plasma electrons in the phase plane. In the rest frame of reference this trapped function in the vicinity of the resonance V0 u (under the condition ua /9' <( i ) to calculate nr, and particle "cluster" moving with the velocity u is screened expand the denominator of the integrand everywhere by plasma particles at rest. The density of antrapped over the domain of integration (under the condition electrons is slightly reduced due to their velocity per- A (( i ) to calculate nn. Then the terms proportional turbation at the center of the pulse, resulting in the to qs(2q+l)/4(q = 0, 1,2,...), associated with the artififormation of an excess positive charge. The availability of this positive charge leads to the compensation of electron-electron repulsive forces of the trapped electrons and thereby to the possibility of a self-consistent BGK equilibrium. The potential pulse is symmetric with respect to the cially introduced boundaries, cancel one another in the sum (7), so that the result does not depend naturally on the choice of the partition of the integration interval. Keeping only the leading terms in the nonresonant and resonant contributions, we finally find the expression for the untrapped electron density as origin of coordinates. It is described by the solution of (5) with the boundary conditions dqb/dx = 0 and q) = np= 1 - Pq)- 2 fo V/, (9) qsma x ---- A at x = 0, and q5 = 0 at x = 4-o0. The passing untrapped electron distribution function is determined where fo = fo(u) is the value of the unperturbed disby its form at infinity where the perturbation equals tribution function at the resonant velocity V = u and zero. In the case of the Maxwellian distribution (4) one P = P(u) is the well-known principal value Vlasov integral can write f(w) - fo[vo(w)] 1 [ 1 -- V exp- (u +, (c) where Vo(W) = u + x/ -W is the untrapped electron velocity at x = :Fo0. Using (6), we can find the plasma response to the perturbation introduced by the moving trapped electrons. Changing the integration variable in (5) allows us to represent the density np of the antrapped electrons as the sum of two terms which can be treated, by analogy with periodic waves, as the contributions of the resonant and nonresonant particles. where 0 øc dw(f+ + f_) =/_x dvofo(vo) v/:(w + 1+ = na + n3, (7) dvofo(vo) T R = j _ /4 / 1 + (Vo_ ). 252 nn -- -]- b + / 2 TM O<) 1 + (Vo_,). In the most interesting case for us, namely that of weak perturbations, the integral (7) can be expanded in half-integer powers of the amplitude A. For this purpose we have divided this integral into the parts (8) 1/4 with the help of the integration boundaries u 4- b. The choice of these boundaries is rather formal and nonunique. However, this partition enables us to find the leading terms in the expansion of the integral (7) at A << i by direct calculations and thereby to extract the contributions of the resonant and nonresonant domains. In doing so, one can expand the distribution P(u) = P.V./7 dv(ofo/ov) (10) Equation (9) plays an important role in the subsequent analysis. It describes the plasma response to a moving localized perturbation of the electrostatic potential and can be employed for any trapped electron distribution function. Both nonresonant and resonant electrons (Vo u) contribute to the screening process, as represented by the second and the last terms in the righthand side of (9), respectively. The second term coincides in appearance with the analogous expression for linear periodic waves and represents the linear plasma response caused by nonresonant electrons. The last term in (9) describes the density perturbation of the resonant electrons in the total absence of particles in the trapping region (see Figure 3). Clearly the density of such an electron "hole" equals to the product of the distribution function in the resonance and the width of the trapping region in velocity space which is proportional to b / '. The behavior of the function P(u) for the Maxwellian distribution can be found, for example, in the work by Fried and Conte [1961]. In the limiting cases of small and large velocities it is approximated by the following expressions, respectively, P(u) - -l + u 2, u << l; P(u) - u 1+ -, u >> l. (11) Taking into account that f+ = f_ = F(W) for trapped electrons, we can represent (5) in the form : dwf(w) d2 = -Pq5-2fo(2qS) / (12) +,) The integration of (12) with the boundary conditions above leads to the "nonlinear oscillator" equation determining the electrostatic potential profile 1 (dqs) 2 +v(,)-0,

5 KRASOVSKY ET AL.: ANALYSIS OF ELECTROSTATIC SOLITARY WAVES 22,135 with the pseudopotential of the 'oscillator' being 1 2 3/2-2 dwf(w)v/2(w + q ). (14) For a given trapped particle distribution function F(W), we can determine the potential profile ½- O(x) from (13) and (14) and, as a consequence of boundary conditions, a relationship among the parameters of the pulse. However, so far as such a solution is not unique owing to the arbitrariness of the choice of F, the derived (for the distribution of a specific form) relationships and restrictions (for example, the restriction on the pulse velocity) are not, strictly speaking, universal. The artificial choice of the trapped electron distribution function seems to be one of the reasons for the contra- dictoriness of the results of BGK theory of phase space electron holes [$chamel, 1979; Kono et al., 1986; $ayal et al., 1994] as a particular case of localized BGK solutions. Another approach is to take advantage of a potential profile ½(x) based on observations and find the function U(½) or the total electron density perturbation h(½) - n du/d½. Then we can solve the formal expression for the distribution function of trapped particles [Bernstein et al., 1957] which, in our case, takes the form - P (-2w) + fo r 1/-w V/-2(qb+ de W) d2u dqb 2 ' +- ß (15) However the right-hand side of (15) may formally take negative values at some W. Since the distribution function cannot be negative, this can also lead to restrictions on the wave parameters for the assumed wave profile. Typical simple calculations of particular solutions of (12) will be described in the next section. Here we shall study some general properties of the BGK pulse structure. To extract the amplitude dependence in an ex- plicit form, it is convenient to introduce more suitable normalized variables ½=Aa, W--Aw, U-A2 I,, (16) where the function a - a(x) (0 < a < 1) describes the spatial dependence of the potential. According to (13) the wave profile is determined by 1 (da)2 i da'.(17) x- ) Equations (14) and (15) give, respectively, ß dwf(w)v/2(a - w) dwg(w)v/a - w, (ls) and i 8fo 2, (19) o-- Pa2 + 3x/ - a s/ F(w) - Fo(w) + vc(w), (20) Fo - P x - w '/2 + fo, (21) 1 /w da _ o d2 do 2 ' (22) (7-2 r v/w where we have separated the terms o and Fo that depend on plasma response. The boundary condition da/dx-0 at x:0 (or (1)-0) leads to the additional restriction on the function G that results from (ls), dwg(w)v/1 - w - O, (23) whence it follows that G is a function with alternating signs. The trapped electron distribution function written in the form (20) consists of the term Fo (which is always positive for small-amplitude perturbations) and the additional term G. Inserting (20) into (18), one can find that the contribution of Fo is compensated by the contribution of the plasma (untrapped) electrons, which is represented by the first term o(a) of (18). A choice of the additional function G enables one to construct various BGK solutions for a given electron distribution function fo(vo) and fixed parameters of the pulse u and A. Clearly, however, the sum (20) has to be positive for any w in the interval 0 < w < i which corresponds to trapped particles. It is quite easy to satisfy this restriction for small values of G and consequently small values of ß as is obvious from (18). It is noted from (17) that the small values of G and ß correspond to large-scale perturbations. Hence, the class of possible solutions is quite wide under the condition of charge compensation F Fo. Thus to construct BGK solutions one can use (17) and (18) for a given G(w) satisfying (23) with the requirement F > 0, or calculate G and F with the help of (20), (21) and (22) for a given wave profile. Depending on the behavior of the pseudopotential ß (a) at small a, the shape of the electrostatic potential a(x) can be spatially confined to a finite region (if the integral (17) converges at a - 0) or can have tails extending to infinity (if this integral diverges). With the aid of (17) and (19) it is easy to see also that since the function (a) has to be negative everywhere at 0 < a < i the solutions in the form of a localized pulse are impossible without trapped particles and also impossible in the absence of trapped electrons within a finite interval adjoining the separatrix w - 0 (see Figure 3). For a given wave profile (or for a given trapped electron distribution function) the pulse width L depends on its velocity u and amplitude A as represented by L - L(u,A). This dependence reflects the effect of boundary conditions and can be treated like an analog of the dispersion relation for periodic waves. If we ig-

6 22,156 KRASOVSKY ET AL.: ANALYSIS OF ELECTROSTATIC SOLITARY 'WAVES nore fine details of the behavior of the functions and C(w), the typical value L can be estimated roughly. It follows from (17) that [ [ L - '. According to (18) and (22) the minimum negative value is also of the order L -2. Hence the positiveness of F results in the inequality L _> Lo, where The spatial scale Lo is closely related to the existence of the minimum time that is necessary for formation of a plasma response to an introduced perturbation. If the pulse velocity u is high enough so that the number of resonant untrapped electrons is negligible (fo(u)/j /2 << P(u)), the minimum response time is of the order of a period of electron plasma oscillations 2 rv: -. In that case the excess charge of trapped electrons is mainly screened by nonresonant plasma particles with the pulse width L _> Lo "" u (Lo "" u/v:p with usual notation). In the opposite limiting case fo(u)/a / ' >> P(u), the excess charge is screened by resonant plasma electrons, and the typical width of perturbation depends also on the amplitude Lo '" AX/4/fo / '(u). The solutions with L Lo are unrealizable. At the same time, very extensive perturbations L Lo seem to be unstable, similar to uniform electron beams moving in a plasma. Hence, we can conclude that the typical pulse width L " L0 is determined by (24). This is also supported by the calculations given below. The relationship between the parameters of the pulse provides a basis for useful data processing. For example, with the help of the relations L "-' uv- and Eo "" A/L one can find a relationship E0 - E0( -) between the amplitude E0 and duration - of ESW from (24). On the other hand, this dependence can be reproduced from the observed waveforms like that depicted in Figure 1. However, the detailed comparison of the theory with the observations requires careful analysis of a great many wave signatures, insofar as the amplitude and duration ranges recorded by Geotail are not too wide. A supplementary difficulty is caused by the absence of direct measurements of the electron distribution function. The energy spectra obtained by the hot plasma analyzer of the comprehensive plasma instrumentation on board Geotail provide, unfortunately, only circumstantial information about the distribution function averaged over the time well above the duration of ESW. Nevertheless, the attempt to analyze the interrelationships of ESW parameters using statistical data processing is of independent interest and could be considered as a prospective work and a next step in studies of the high-frequency portion of BEN. parameters of the solitary wave. We can derive solutions either for a given wave profile or for a given form of the trapped electron distribution function. So far as the waveforms of BEN have been known, the corresponding BGK solutions with a given potential profile can be helpful for the analysis of the Geotail data. Here we demonstrate the search for such solutions by an example of a simple wave profile of the Gaussian form C = exp(-x2/l 2) (25) which is an approximation of the time-scanning data obtained by Geotail as shown in Figure 4. Using (17), we find the function (a) and the spatial dependence of the total electron density perturbation h=n_l=_a d d-- - 2A -5, ( 2x L ) 2.- i exp(-x '/L ') The trapped electron distribution function F is determined by (20) with the function taking, according to (22), the form 2wl/2 G(w) -.L2(1-lnX6w 2). (28) The next step is to check the positiveness of F. In the case of the "fast" pulse screened by nonresonant plasma electrons u 1, P(u) "-' u -2, neglecting the term with small fo, we come to the restriction on the spatial scale L > Lmin -- u(21n16-2) / '. The physical sense of this constraint has been discussed in the previous section. With reference to the expressions (20),(21) and (28), it can be seen that for extensive perturbations L > Lmin 4O0 o o o o o I t I I O.S,, ' ' I ' ' ' I I GSM-X : Y : Z : LIT: 13:10: $.644 : ',, _o_o.0 o o,,, I,,,, l I 1.5 Time (mse, c) 3. BGK Solutions for Solitary Waves In this section, we derive specific BGK solutions for a solitary wave on the basis of the general approach described in the previous section. The BGK solutions give spatial dependencies of physical quantities in an explicit form and show the interrelations between basic Figure 4. Temporal scanning of the ESW potential (in arbitrary units) deduced from the Geotail field measurements, where the ESW is assumed to be moving with a certain unknown drift velocity relative to the spacecraft. The solid line is an approximating Gaus- 2 sian curve b = boexp[-(t- tin) / ' ], bo = 371, tm= 0.798, ; =

7 KRASOVSKY ET AL.: ANALYSIS OF ELECTROSTATIC SOLITARY WAVES 22,137 the trapped particle distribution function F -- Fo(W) is a monotonically increasing function, while at L = Lmin the trapped electron distribution function takes the form of the phase space electron hole, F- (4V /7rLmin)W1/21nw -1, so that F - 0 at the bottom of the potential well w = 1. According to the calculations carried out, this tendency toward electron hole formation applies also to other "bell" - like shapes of the electrostatic potential, such as O -- cos4(71'x/2l), x I_ L and a = cosh-2(x/l). Within the framework of the model described above, the minimum duration of the pulse equals rmin=lmin/u 2. Noting that time is normalized by w; and that wp _ 104 rms- in the plasma sheet boundary layer, we find a good agreement between our theoretical model and the observed pulse width r 0.17 ims shown in Figure 4, provided r Train. According to the measurements [Matsumoto et al., 1994], the electric field amplitude of ESW is of the order of Eo 3 x 102/ V/m ß If the velocity of the solitary pulse is taken to be u m 2 x 104 km/s, which corresponds to the energy Ee m 1 kev, the pulse width and potential amplitude can be estimated as L m 5-20 km and ½o V. Thus, using the observed waveforms as a base, one can find the appropriate BGK solutions describing ESW. The minimum width of the moving BGK pulse corresponds to the minimum duration in the rest frame of reference and, hence, determines the upper bound of the BEN frequency spectra. At the same time, BGK analysis provides a means for a clear knowledge of the ESW structure. In a qualitative sense this structure is comparatively simple. A bunch of moving electrons trapped in the self-consistent potential Well, like an inhomogeneity of charge density, is screened by the plasma. However, the plasma response and thereby the nature of the screening depend on the drift velocity and the electron distribution function of the plasma. If the drift velocity far exceeds the thermal velocity u >> h, as in the above example, and the number of untrapped resonant particles is negligible, the trapped electrons are ma/nly screened by the nonresonant electrons. Under these conditions the screening is dynamic in character rather than thermal, and the typical scale of the screening is far in excess of the Debye length L uwp I >> r 9 = hw -1. The equilibrium distri- bution function of the trapped electrons is bound to be such that a local pressure gradient would be compensated by the electrostatic forces. With a knowl- edge of the distribution function, it is possible to calculate directly all physical parameters of the moving trapped electron bunch and the accompanying electrostatic pulse such as the total charge of the trapped electrons, the energy of ESWs and so on. The above analysis based on the approximation of waveform (25) is also applicable to a comparison between the theoretical and numerical models of ESW. As an example, compare the typical width of the BGK pulse (25) with the spatial scale of the localized perturbations exhibited in the numerical experiment by Omura et al. [1996]. The velocity of the electron holes shown in Figure 2 is about u From Fried and Conte [1961] we find P(u) The estimate (24) yields the value L , which is close to the above stated minimum (for the accepted model (25)) width Lmin On the other hand, with the lower panel of Figure 2 it is an easy matter to find the distance d between the center of the perturbation, where E = 0, and the position corresponding to the maximum field, where cqe/ox = -fi = 0. For example, this distance is about d for the electron hole shown at the middle of the panels in Figure 2. Then, using (27) within the framework of the adopted model, we find the value L 2 /2d 4.9 clos enough to L0 and Lmin ß Thus, with allowance made for the evaluating character of our calculations, the BGK model of the electron phase density hole is in reasonably good agreement with the results of the numerical simulation. Now let us consider the type of localized BGK waves closely related to the formation of a specific trapped electron distribution known as phase space electron holes [Schamel, 1979; Kono et al., 1986; Sayal et al., 1994; Turikov, 1984; Lynov et al., 1985]. Similar BGK structures usually arise in the course of the nonlinear stage of beam-plasma instabilities [Omura et al., 1996; Berk et al., 1970], and their typical feature is some deficiency of trapped particles near the bottom of the potential well where w - 1, and some excess in the vicinity of the separatrix w = 0 (see Figure 3.). One of the simplest models describing such an "inverted" trapped particle population can be chosen in the form F o-const, w < Wo. (29) F(w) - 0, w >_ Insofar as (1) = 0 according to the boundary conditions, with the help of (18) we come to the necessary relationship between quantities Fo and Wo, Fo - (fo + 3Pv/ /16)/a(wo), (30) a(wo) - I - (1- Wo) a/2, so that the pseudopotential takes the form _ 2a b _ (1 + b - a)a 3/2 - aba 2 P where the parameter _{(l+b)(a-wo) 3/2, a>wo (31) 0, o _< Wo, b(u, A) - 3P(u)x/--2-2 /16 fo(u) (32) is small for "slow" perturbations or large for "fast" ones depending on the velocity and the amplitude. Equations (17) and (31) determine the wave profile as well as the width of pulse (33)

8 22,138 KRASOVSKY ET AL.' ANALYSIS OF ELECTROSTATIC SOLITARY WAVES Of special interest is the limiting case of a sharply defined electron hole, where all trapped particles are concentrated near the separatrix Wo << 1. In this situation the electron density perturbation has sharp maxima at the edges of the pulse, which are caused by the deceleration of trapped electrons near the turning points, where they are reflected from the potential well. The spatial dependence of the electrostatic potential at the edges of the pulse is described by the expression a- Wo[(œ- x)/œ2] 4, 0 a Wo, (34) where the corresponding scale length equals L2-2v wo3/4(p + 16fo/32x/ ) - '/2 << L. (35) In the central region, with the exception of the small area near the edge a - Wo, the shape of the "fast" pulse b >> I has the form (36) Using this expression one can find also the profiles of the trapped electron density and the plasma density perturbation at the center of the pulse: nt - (A/4u e) cos-2/a(rrx/2l), p (A/u 2) cos4/a(rrx/2l). (37) In the opposite limit b << 1 the shape and the width of the "slow" electron hole are determined by - L 2E - K ' T- arccos (0 1/4), ;- - 2/2, (38) L - K), (39) where K(T, n) and E(T, n) e elliptic integrals of the first and second kind respectively with modulus n, and K and E are the corresponding complete elliptic int grals with the sine modulus. It should be noted that the dependence of the perturbation width on the velocity and amplitude found above (equations(36) and (39)) is in qualitative agreement with the results of the paper by Tu kov [1984] in spite of rather different appro hes used for different forms of wave profiles. This reflects the fact that the general property of loc ized BGK perturbations sociated with screening of trapped electrons moving in a pl ma is independent of the details of the wave profile d specific fore of the distribution function of the trapped electrons. Finally, we mint remark that there are not any restrictions on the electron hole velocity fo d in some preyiota papers. This result a s well with nmeric simulations where one can reproduce fairly wide v iations of ESW with a wide r ge of u. Furthermore, with the help of generm expressions (17)-(22) it is e y to see that the cl s of possible solutions is rather wide under the condition of charge compensation F Fo. 4. Conclusions Bipolar pulses of electric field recorded by the Geotail spacecraft provide new important information on the high-frequency component of BEN in the magnetotail. According to the extensive experimental data, the observed pulses are typical phenomena in the plasma sheet boundary layer. The short duration of these elec- trostatic perturbations, of the order of co -, indicates that their structures are closely connected to electron dynamics. To account for the waveforms revealed by Geotail, Matsumoto et al. [1994] have proposed the hypothesis of the generation of moving localized waves, called ESW, at the nonlinear stage of an electron beamplasma instability. This proposal has been confirmed by particle simulation [Omura et al., 1996] on the longtime behavior of beam-plasma systems, wherein the ESW were revealed in the form of long-lived electron phase density holes. As a next step in the study of the ESW structure, we have considered localized BGK solutions in the simple form of a solitary positive pulse. These solutions describe a bunch of trapped electrons moving in a plasma with a constant velocity. Since charge bal- ance is a necessary condition for the existence of a selfconsistent equilibrium, parameters of the localized wave are mainly determined by the plasma screening of the solitary potential. We have found a simple expression for the plasma response to the moving perturbation of the electrostatic potential that consists of the contributions of the nonresonant and resonant antrapped plasma electrons. It is shown that a bunch of electrons trapped by the potential can be screened, respectively, by the nonresonant and resonant plasma electrons depending on their distribution function, the velocity of the solitary pulse and its amplitude. The fast pulse is screened usually by nonresonant particles, so that the typical pulse width is of the order of one or several cycles of electron plasma oscillations. This is in agreement with the pulse duration observed by Geotail and, hence, serves as another support for the model of ESW suggested as an explanation of the high-frequency component of BEN spectra. The typical width of a slow pulse depends on the pulse amplitude and the value of distribution function at the resonant velocity. A similar interrelation of the parameters should also be expected for a non-maxwellian electron distribution, when the number of passing untrapped resonant electrons is not too small, as in high-density beam-plasma systems. The derived interrelationships of the pulse parameters and the relation between the shape of a localized perturbation and the trapped electron distribution function allow an understanding of the ESW structure and provide a means of more detailed comparison be- tween the ESW model [Matsumoto et al., 1994] and the Geotail observations of the BEN waveforms. On the other hand, they are also convenient for an interpretation of the numerical experiments. According to the analysis carried out for short stationary potential pulses

9 KRASOVSKY ET AL.: ANALYSIS OF ELECTROSTATIC SOLITARY WAVES 22,139 propagating in a plasma, the distribution function of trapped particles takes the form of a phase space hole, which usually arises at the saturation stages of beamplasma instabilities [Omura et al., 1996]. Finally, it should be noted that the present method can also be applied to studies of slowly evolving quasi-equilibrium electrostatic pulses as well as to an analysis of screening processes and dynamics of localized plasma inhomogeneities moving in space plasmas. waves turbulence on auroral field lines, J. Geophys. Res., 82, 1031, Gurnett, D. A., L. A. Frank, and R. P. Lepping, Plasma waves in the distant magnetotail, J. Geophys. Res., 81, 6059, Ikezi, H., P. J. Barrett, R. B. White and A. Y. Wong, Electron plasma waves and free-streaming electron bursts, Phys. Fluids, 1, 1997, Kono, M., M. Tanaka, and H. Sanuki, A stationary electron hole associated with a Langmuir wave, Phys. Scr., 3, 235, Lynov, J.P., P. Michelsen, H. L. P cseli, J. Juul Rasmussen Acknowledgments. We thank Hirotsugu Kojima and and S. H. Sorensen, Phase space models of solitary elec- Satoko Horiyama for their help in data analysis and discus- tron holes, Phys. $cr., 31, 596, sion of the ESW waveforms observed by Geotail. One of Matsumoto, H., H. Kojima, T. Miyatake, Y. Omura, M. us (V. L. K.) is most grateful to RASC, Kyoto University Okada, I. Nagano and M. Tsutsui, Electrostatic solitary for hospitality and the Ministry of Education, Culture and waves (ESW) in the magnetotail: BEN wave forms ob- Science of Japan for financial support. The computer simserved by Geotail, Geophys. Res. Left., 21, 2915, ulation presented in Figure 2 was done by the KDK com- Omura, Y., H. Kojima and H. Matsumoto, Computer simuputer system of RASC, Kyoto University. 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Eastman, Generation of broadband electrostatic waves in the magnetotail, J. Geophys. Res., 89, 3865, Gurnett, D. A., and L. A. Frank, A region of intense plasma V. L. Krasovsky, H. Matsumoto, and Y. Omura, Radio Atmospheric Science Center, Kyoto University, Uji, Kyoto, 611, Japan. (e-maih omura@kurasc. kyoto-u.ac.jp) (Received August 5, 1996; revised March 31, 1997; accepted July 4, 1997.)