Two-stream instability of electrons in the shock front

Transcription

1 GEOPHYSICAL RESEARCH LETTERS, VOL.???, XXXX, DOI:1.129/, Two-stream instability o electrons in the shock ront M. Gedalin Department o Physics, Ben-Gurion University, Beer-Sheva, Israel 1. Introduction It is widely believed (see Scudder [1995] and reerences therein) that electron heating in the shock ront is mainly due to the electron interaction with the quasi-static electric and magnetic ields in the essentially one-dimensional stationary shock proile. During this interaction each electron, crossing the shock rom upstream to downstream, gets the same amount o energy, namely eϕ HT, where ϕ HT is the cross-shock potential in the de Homan-Teller rame. Adiabatic mechanism [Feldman et al., 1982; Goodrich and Scudder, 1984; Feldman, 1985; Thomsen et al., 1987; Schwartz et al., 1988; Hull et al., 1998], which works in most shock proiles, implies that the electron magnetic moment is conserved across the shock, that is, v 2 /B = const everywhere (index reers to the local magnetic ield direction). In this case the electrons are eiciently accelerated along the magnetic ield. Some electrons are relected because o the magnetic barrier. In very thin shocks, where the ramp width is o the order o the electron inertial length [Newbury and Russell, 1996], electrons may be demagnetized in a part o the ramp and accelerated across the magnetic ield [Balikhin et al., 1993; Gedalin et al., 1995; Gedalin and Balikhin, 1998; Gedalin et al., 1997]. This results in a dierent distribution o the acquired energy among the parallel and perpendicular degrees o reedom. Independently o what is the particular mechanism o the electron energization, the upstream electrons, whose parallel velocities are directed upstream and which never enter the shock, must have come rom the downstream region. A substantial part o the phase space is not accessible to electrons, so that the electron distribution inside the ramp and just behind it has a hole in the distribution [Hull et al., 1998], which should be illed somehow to produce the observed distributions. It was suggested that this hole may be illed with some preexisting electron population [Feldman et al., 1982; Feldman, 1985], in which case the electron distribution orms nonlocally. Only the high energy tail is produced at the shock while the rest o the distribution is ormed at another relecting boundary, which requires very good coordination between the shock and this boundary. This scenario was never analyzed in detail. Another suggestion is the electrostatic and whistler instabilities in the drit (adiabatic) regime [Veltri et al., 199, 1992 ; Veltri and Zimbardo, 1993a, b]. Hull et al. [1998] adopted the phenomenological approach where the hole is illed homogeneously up to the density required by the Rankine-Hugoniot relations, which does not provide physical explanation why such illing should occur. In this paper we consider the consequences o the act that the collisionless electron distribution inside the ramp essentially consists o two counterstreaming beams, which is the classical two-stream instability case. We ind the growth rate o the instability and show that it is capable o smoothing the electron distribution inside the ramp. In what ollows we consider the adiabatic case, since in the demagnetized regime the electron distribution cannot be determined analytically. However, the results should not dier substantially, especially or strong shocks with large de Homan-Teller cross-shock potential. 2. Electron distribution inside the ramp For simplicity we assume that the upstream distribution is Maxwellian u (v u,, v u, ) = n u (2πv 2 T ) 3/2 exp[ (v u, V sh ) 2 + vu, 2 2vT 2 ], (1) where V sh = V u / cos θ is the bulk upstream plasma velocity along the magnetic ield in the de Homan-Teller rame (V u is the plasma velocity in the normal incidence rame, and θ is the angle between the shock normal and upstream magnetic ield). The distribution unction is normalized so that 2π u dv u, vu, dv u, = n u. Let ϕ(x) be the de Homan-Teller potential and B(x) the total magnetic ield throughout the shock. Conservation o energy and magnetic momentum read v 2 + v2 = vu, 2 + v2 u, + 2eϕ, m e (2) v 2 = vu, (B/B 2 u ). (3) 1

2 X - 2 GEDALIN: TWO-STREAM INSTABILITY Collisionless dynamics implies that throughout the shock (v, v ) = u (v u,, v u, ), where the unctional orm o u (v u,, v u, ) is given by (1), while the corresponding v u, and v u, are ound by solving (2) (3): v 2 u, = v 2 (B u /B), v u, = Q sign(v ), (4) Q = v 2 + v2 (1 B u /B) 2eϕ m e. (5) Obviously, the condition Q < deines the region in the velocity space which is not accessible or electrons. In what ollows we need the one-dimensional distribution unction F (v ) = 2π (v, v )v dv. Figure 1 shows.4.3 R=1.5 np=.35 nm= R=2 np=.34 nm= R=2.5 np=.33 nm=.2.2 R=3 np=.32 nm= R=3.5 np=.32 nm=.2.2 R=4 np=.31 nm= Figure 1. Evolution o the distribution unction F (v ) through the ramp. The parameters are as ollows: de Homan-Teller cross-shock potential eϕ HT = 12T eu, V u/v T e cos θ =.2. In the igure R = B/B u, n is the density o transmitted electrons, and n b is the density o the backstreaming electrons normalized on the upstream density. R monotonically increases rom R = 1 (upstream) to R = 4 (downstream). the evolution o the distribution unction F (v ) across the shock ront. The de Homan-Teller cross-shock potential is estimated with the widely used polytropic relation [Schwartz et al., 1988]: eϕ HT = γ γ 1 (T ed T eu ), (6) where we chose γ = 2 and T d /T u = B d /B u ( adiabatic case), and B d /B u = 4. The parameter V sh /v T e =.2, that is, upstream electrons are subsonic in the de Homan-Teller rame. The distribution unction depends on the local magnetic compression ratio R = B/B u, and we assume that the potential ollows the magnetic ield proile [Hull et al., 1998; Gedalin et al., 1998]: ϕ = ϕ (B B u )/(B d B u ). The transmitted and backstreaming electron densities (normalized on the upstream density) are denoted in the igure as n and n b, respectively. 3. Two-stream instability We consider the high-requency electrostatic oscillations with the wavevector k B. The corresponding dispersion relation is ɛ =, where ɛ = 1 ω2 pe dv F (v ), (7) k ω kv where ω 2 pe = 4πn u e 2 /m e is the upstream plasma requency, and the distribution unction F (v ) is qualitatively shown in Figure 1. Qualitative analysis o the stability o this two-humped distribution can be done without solving the dispersion

3 GEDALIN: TWO-STREAM INSTABILITY X - 3 relation. The distribution should be unstable with respect to excitation o Langmuir waves, propagating in both directions. Analysis o the exact distribution unction is diicult. Moreover, in thin shocks or with weak nonstationarity the distribution unction can be expected to be distorted rom (1). We shall, thereore, perorm a semi-quantitative analysis o the model distribution consisting o a waterbag distribution = n θ(v 2 v 2 ) and two beams at the end o it b = n b [δ(v V ) + δ(v + V )]. The distribution F = + b resembles the numerically ound distributions in Figure 1. The beams are taken o equal densities to simpliy the analysis (this does not change qualitative conclusions). It can be seen rom Figure 1 that the beam and waterbag densities are comparable. The corresponding dispersion relation reads: 1 ω 2 ω 2 k 2 V 2 2ω2 b (ω2 + k 2 V 2 ) (ω 2 k 2 V 2 )2 =. (8) Hydrodynamic instability occurs when k 2 V 2 < 2ω 2 b ω2. Figure 2 shows the temporal growth rate o the two-stream g k (a) p o (b) Figure 2. (a) Growth rate Γ and (b) spatial growth rate κ o the two-stream instability or n b /n =.75. instability or n b /n =.75 as a unction o k. The maximum growth rate Γ is.14ω and occurs at k.25ω /V. Taking V v T e and ω.5ω pe, one inds the growth rate o Γ.1ω pe and kr De.1. Spatial development o the instability is described by the spatial growth rate κ (spatial dependence is exp(κx)) as a unction o requency. It is seen that the maximum spatial growth rate occurs at about ω.7ω and the spatial scale 1/κ V /ω is much less than the electron inertial length. I n b <.5n the model distribution is not suitable or the analysis and the instability should be considered kinetically. In the kinetic regime the instability is not aperiodic and Langmuir waves are excited. Analytical consideration does not seem possible or general distribution o the orm (1) but some qualitative conclusions can be made. The total density is.5n u so that the excited wave requency is ω r.7ω pe. The expected resonant wavenumbers lie in the range(.2.5)ω r /v T e

4 X - 4 GEDALIN: TWO-STREAM INSTABILITY so that kr De 1. The growth rate o the instability can be also expected to be Γ.1ω r. The corresponding spatial scale is V u /Γ c/ω pe, and thereore is much smaller than the typical scale o the magnetic ield variation. The instability should be very eicient in relaxation o the distribution unction, this relaxation taking place at the time scale substantially less than the ramp crossing time. The relaxation starts at the edges o the ramp and results in the illing o the gap and lowering o the peaks. I we consider only the parallel instability (k B) the distribution relaxation occurs at constant v and in the inal state a plateau / v = is ormed in the distribution (see, or example, Swansson [1989]), which could be responsible or the observed lattops. The edges o the plateau correspond to larger v than v corresponding to the maximum o F (v ), which means that the Liouville mapping o the maximum in the upstream distribution to the edge o the observed downstream distribution [Schwartz et al., 1988] would result in overestimates o the de Homan-Teller cross-shock potential. This is in agreement with the Schwartz et al. ound discrepancy between this method and the method o the polytropic estimate based on (6): it was ound that the edge mapping gives systematically higher estimates o the cross-shock potential than the polytropic approximation. In reality not only waves with k B are excited, and the relaxation should involve the perpendicular velocity v as well. Respectively, instead o plateau a quasi-plateau is ormed [Swansson, 1989], which, nevertheless, does not change the basic conclusion that the two-stream instability should result in the ast gap illing and lattopped distribution ormation. 4. Conclusions As a result o the collisionless electron dynamics in the shock ront a two-stream unstable distribution is ormed inside the ramp. Typical temporal and spatial scales o the developing instability are much smaller than the ramp crossing time and magnetic ield inhomogeneity scale, respectively. Thereore, the instability may provide ast illing o the gap in the collisionless electron distribution. During the relaxation the distribution tends to acquire a lattop-like shape. The high energy end o the distribution is not involved in the relaxation and may be Liouville mapped onto the upstream distribution. Because o the relaxation the edge o the downstream electron distribution does not correspond to the maximum in the upstream distribution but to incident electrons with higher energies. Thereore, the standard Liouville mapping o the edge, used by Schwartz et al. [1988] or the determination o the de Homan-Teller cross-shock potential, should overestimate the potential. Obviously, comprehensive quantitative analysis o the high requency instabilities requires, in general, consideration o oblique waves too. The above analysis exploited the properties o the distribution ormed due to the acceleration o transmitted electrons and deceleration o electrons leaking rom the downstream region, that is, a particular mechanism o the distribution ormation inside the ramp. Nevertheless, the growth o beams should be the common eature o the all collisionlessly ormed distributions. We expect that these distributions are two-stream (bump-on-tail) unstable. Since these electrostatic instabilities are ast, they may result in eective bringing o the distribution to the marginally stable state, with low or no beams at all. Since the quasilinear diusion can be considered as turbulent collisions, we argue that the ormation o the inner part o the electron distribution is collisional, while the high energy tail is ormed collisionlessly. I the above electrostatic instabilities occur they would be observed as appearance o the electrostatic noise inside the ramp at requencies below the upstream plasma requency. Such electrostatic noise at requencies between the ion and electron plasma requencies has been observed at the shock ront [Rodriguez and Gurnett, 1975] and later tentatively identiied by Onsager et al. [1989] as electron beam mode waves. It has been shown recently [Matsumoto et al., 1997; Bale et al., 1998] that this noise exists in the orm o the short nonlinear structures with the typical spatial scale o several Debye lengths, in agreement with our estimated o the typical scale o the most unstable perturbations. Omura et al. [1994] have shown that electron holes can be ormed as a result o nonlinear evolution o the two-stream instability with large beam to background density ratio. These holes can nonlinearly decay into smaller holes [Saeki and Genma, 1998] and probably result in large amplitude electric ield spikes. This inhomogeneous turbulence should scatter electrons producing eective electron resistivity. Bale et al. [1998] suggested that the electron holes may scatter ions too. More reliable quantitative conclusions require detailed analysis o the behavior o the electron distribution inside the ramp (which in turn requires better knowledge o the shock structure). However, it seems that the two-stream instability o the collisionlessly ormed electron distribution may trigger the electrostatic noise generation in the observed requency band. Acknowledgments. The research was supported in part by Israel Science Foundation under grant No. 261/96-1. The author is grateul to V. Krasnoselskikh and both reerees or useul comments. 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