Cylindrical lower-hybrid electron holes at the Earth s dayside magnetopause

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL.,, doi:0.09/005ja07, 006 Cylindrical lower-hybrid electron holes at the Earth s dayside magnetopause D. Jovanović, P. K. Shukla, and G. E. Morfill 3 Received May 005; revised 9 October 005; accepted 8 November 005; published 0 March 006. [] The analytic model of lower-hybrid electron holes (Jovanović and Shukla, 004) is applied to analyze the strong coherent unipolar electric field signals perpendicular to the ambient magnetic field with a characteristic frequency in the lower-hybrid frequency range, recorded at the dayside magnetopause by the Polar mission (Mozer et al., 004). On the basis of a good agreement with theoretical predictions, these structures are identified as the oblique cylindrical electron holes. Their localization results from the balance of nonlinearity associated with the electrons that are trapped along the magnetic field direction and undergo the ~E ~B drift in the perpendicular direction and the dispersion provided by the electron polarization drift. It is suggested that electron holes can be related to the nonlinear evolution of the Buneman instability of lower-hybrid waves, driven by the parallel electric field that develops in the course of collisionless reconnection. Citation: Jovanović, D., P. K. Shukla, and G. E. Morfill (006), Cylindrical lower-hybrid electron holes at the Earth s dayside magnetopause, J. Geophys. Res.,,, doi:0.09/005ja07.. Introduction [] The magnetopause is a relatively thin layer ( km) that separates the compressed (shocked) solar wind plasma of the magnetosheath from the Earth s magnetosphere. On the upstream (day) side, it is located around 0 Earth radii from the Earth where the dynamic pressure of the solar wind equals the magnetic pressure of the geomagnetic field, while on the downstream (night) side it extends into the elongated outer boundary of the magnetotail. Owing to the strong inhomogeneity of the magnetic field that abruptly changes from the IMF (interplanetary magnetic field) to Earth s magnetic field, the reconnection of magnetic field lines takes place within the magnetopause. In the case of a southward IMF, a strong-shear magnetopause is formed on the dayside, which is where the reconnection takes place. Conversely, for a northward IMF and a low-shear dayside magnetopause the reconnection occurs further downstream at the nightside, beyond the auroral region. As the IMF and the Earth s magnetic field are never exactly antisymmetric, a sizable fraction of the magnetic field does not participate in the reconnection, as noted by Mozer et al. [004]. In other words, the reconnection in the magnetopause occurs in the presence of a guide magnetic field. The reconnection occurs also at the central line of the magnetotail due to the opposite polarity of the Earth s magnetic field stretched out by the solar wind. The densities and temperatures of the magnetosheath and magnetosphere are vastly different from each other, ranging Institute of Physics, Belgrade, Yugoslavia. Institut für Theoretische Physik IV, Ruhr Universität Bochum, Bochum, Germany. 3 Max-Planck-Institut für Extraterrestrische Physik, Garching, Germany. Copyright 006 by the American Geophysical Union /06/005JA07 from T e ] T i 0. KeV and n e = n i 5 5 cm 3, sometimes as much as 40 cm 3 in the magnetosheath to T e ] T i KeV and n e = n i cm 3 in the magnetosphere [Phan and Paschmann, 996; Russell, 990; Mozer et al., 004] ( cfm?fobjectid=363). The magnetic field strength varies typically [see, e.g., Cattell et al., 003] from B 0 5 nt in the magnetosheath to B 70 nt in the magnetosphere. Because of the variations in the solar wind and in the direction and strength of the IMF, as well as due to magnetic reconnection, the magnetopause is constantly in motion, with variable velocities in the range 300 km/s, typically 40 km/s, both perpendicular and parallel to the magnetic field direction [see Phan and Paschmann, 996, and references therein]. As a consequence, the magnetopause is a turbulent environment that accommodates a broad range of nonlinear phenomena. [3] In a recent paper [Mozer et al., 004], coherent nonlinear structures have been described, observed by the Polar spacecraft at the magnetospheric side of the dayside magnetopause. They feature a strong unipolar electric field of order 50 0 mv/m, typically 75 mv/m that is purely electrostatic and predominantly perpendicular to the magnetic field direction, with a possible parallel component not exceeding 0% that could not be accurately measured with the available diagnostics. The properties of 8 such events are listed in Table of Mozer et al. [004], from which one easily infers that their duration is in the lower-hybrid period domain, ranging between 0.8 and 0.46 T LH, where T LH = p(jw e jw i ) is the period of the lower-hybrid oscillations, and W e (W i ) is the electron (ion) gyrofrequency. Unfortunately, with the existing tools on board Polar the propagation velocity of the structures and the relevant particle data (such as the electron and proton distribution functions) could not be established. Mozer et al. [004] speculate that these coherent structures result neither from the spatial of9

2 gradients of the plasma properties, which are known to produce bipolar electric fields, nor from solitary Langmuirwave-type electron phase-space structures (holes), whose electric fields are mostly parallel to the ambient magnetic field. The latter, very rapid nonlinear structures whose duration is in the range (typically 5 0 times longer) of the Langmuir wave period T p,e (T p,e =p/w p,e, where w p,e = (n 0 e / 0 m e ) / is the electron plasma frequency, n 0 is the unperturbed plasma number density, and e, m e are the magnitude of the electron charge and the mass, respectively), have been often observed to accompany the magnetic reconnection events, both at the Earth s dayside magnetopause with southward IMF (observed by Polar [Cattell et al., 00] and Geotail [Deng et al., 003] spacecrafts) and in the Earth s magnetotail by Cluster ([Cattell et al., 005]). Noticeably, Cattell et al. [005] commonly recorded also very large amplitude lower-hybrid waves, having electric fields up to 50 mv/m, perpendicular to the ambient magnetic field, as well as the holes in the lower-hybrid time range, accompanied by the bursts of the upper-hybrid waves. Computer simulations by Deng et al. [003] and Drake et al. [003] revealed that large-amplitude Langmuir and lower-hybrid waves, respectively, are produced by two types of Buneman instability (parallel and oblique) of the magnetic field-aligned electron beams that come from electron acceleration by the reconnection electric fields. The development of thin current sheets parallel to the guide magnetic field, whose thickness is of the order of the electron skin depth c/w p,e, is well known from the simulations of collisionless reconnection [see, e.g., Del Sarto et al., 003, and references therein]. These electron flows, conceivably susceptible to the Buneman instability, are localized in the vicinity of the X-point and along the magnetic separatrices, where they are driven by the parallel ambipolar electric field and/or parallel electron stress. They have been recorded both in the subsolar magnetopause [Scudder et al., 00; Mozer et al., 003] and in the magnetotail [Cattell et al., 005], and they are regarded as the signature of the electron diffusion regions associated with the reconnection. Although the role of these current sheets in the reconnection and in the generation of electron holes is still not quite clear, the analysis of Deng et al. [006] showed that the electron holes tend to be observed along the plasma sheet boundary layer and near the X-line region when an intense narrow electron beam or narrow counterstreaming beams occurred. Conversely, when the wave train of solitary structures was observed by Geotail, also the enhanced fluxes along the ambient magnetic field were recorded. Furthermore, Deng et al. [006] showed that the timing of electron holes observed by Cluster yields the speeds that are consistent with the Buneman instability. [4] The emergence of new observational data [Mozer et al., 004] brings about the need to check the validity of the existing physical models and also provides the means for their refinement. Namely, nonlinear structures with predominantly unipolar perpendicular electric fields and a small parallel component that is bipolar have already been described in the literature. They were extensively studied analytically and numerically, both in the frequency range of drift waves, i.e., below the ion gyrofrequency [Jovanović and Shukla, 000] and of the lower-hybrid waves [Jovanović et al., 00; Jovanović and Shukla, 004]. A simple geometric analysis by Jovanović et al. [00], as well as the numerical experiments of Jovanović and Shukla [000, 004] revealed that such characteristic signature could be produced by electron holes that are elongated along the ambient magnetic field, being either cigar-shaped and parallel to the magnetic field or cylindrical and slightly oblique to it, and move along the magnetic field with the velocity in the electron thermal range. Similar cigar-shaped phase-space vortices produced by the electron trapping by a linearly unstable lowerhybrid wave were observed in three-dimensional (3-D) numerical simulations of collisionless magnetic reconnection by Drake et al. [003]. [5] Using the physical model of Jovanović and Shukla [004], under the realistic magnetopause conditions with respect to the magnetic field intensity, plasma density, and temperature, as well as the anisotropy of the electron distribution function in the parallel direction, we obtain a good agreement with the observations of Mozer et al. [004]. Thus we positively identify those events as the oblique lower-hybrid electron holes. In contrast to our earlier papers, where the properties of electron holes were studied in simple cases when the effects of ion dynamics could be completely neglected, either due to the short duration of the structure [Jovanović et al., 00] or due to its shape which precluded any significant ion motion perpendicular to the direction of propagation of the hole, we fully include the effects of the ion mobility. The latter is known to be of crucial importance for the Buneman instability of the lower-hybrid waves, which saturates eventually by trapping electrons in the parallel direction and deforming their distribution function into a double-hump shape, i.e., to an electron hole [Drake et al., 003]. We demonstrate that perpendicular perturbations arising from the ion dynamics within a lower-hybrid oblique-slab electron hole, produced by the electron trapping by a plane lower-hybrid wave, have the form of a kink. As a result, any instability of a -D slab is likely to break it into a sequence of more or less cylindrical structures immersed in the bath of lower-hybrid radiation with long wavelength. [6] Recently, a similar theory [Jovanović et al., 005] has been successfully applied in the deep Earth s magnetotail to explain the simultaneous occurrence of the upper-hybrid bursts and electron holes, observed by the Wind [Farrell et al., 00] and CLUSTER [Cattell et al., 005] spacecrafts in the vicinity of the X-point during the collisionless reconnection. Jovanović et al. [005] demonstrated that the explosive instability, resulting from the interaction between the upper-hybrid and linearly unstable lower-hybrid waves (due to an oblique Buneman instability), may be saturated by the electron trapping in the combined lowfrequency and ponderomotive potentials, yielding an oblique lower-hybrid hole coupled with a nonlinear Schrödinger soliton. [7] The manuscript is organized in the following fashion. Section contains the governing equations for the electrons and ions, controlling the electron holes in the lower-hybrid frequency range. The nonlinear equations for the slab and cylindrical electron holes are presented in section 3. In section 4 we discuss the effects of the perpendicular ion motion on the oblique electron holes, and the application of our work to to the Earth s magnetopause is presented of9

3 in section 5. Section 6 contains a brief summary and conclusions.. Governing Equations [8] First, following Jovanović and Shukla [004], we briefly present the basic equations. They are derived using the standard description of electron guiding centers kinetics in a magnetized plasma [see D Ippolito and Davidson, 975] obtained by integrating the Vlasov equation with respect to particle gyroangles. The underlying iterative procedure is applicable for low-amplitude phenomena, ef/ T e, that are slowly varying compared to the electron gyrofrequency, jd/dtj jw e j, provided that the curvature radius of magnetic field lines is larger than the Larmor radius, jr (~B/B)j r L,e (where r L, e = v T,e /jw e j and v T,e is the electron thermal speed). These rather lengthy expression can be simplified under the following conditions that are characteristic for the dayside magnetopause regions of Mozer et al. [004]. [9]. Using the data of Mozer et al. [004] and the fact that the temperature at the magnetopause does not exceed that in the magnetosphere, T e ] kev, we find that the ratio of the thermodynamic and magnetic pressures is small, n e T e /(c 0 B ) < 0., which permits us to decouple the compressional and torsional components of the magnetic field perturbations. We will consider only the torsional component. Furthermore, the curvature and inhomogeneity of the background magnetic field can be neglected on the spatial scales of our interest, and we may express the electric and magnetic fields as ~E = rf ~e z (@/@t)a z and ~B = B z,0 ~e z ~e z ra z, respectively, where B z,0 ~e z is the homogeneous part of the unperturbed magnetic field, f and A z are the electrostatic- and the z-component of the vector potential, and ~e z is the unit vector in the z direction. [0]. The electron temperature is usually anisotropic in the magnetopause, T ek T e? (the subscripts k and? denote the directions parallel and perpendicular to the ambient magnetic field, respectively), both in the strong shear dayside region and in the low latitude boundary layer at its dawn/dusk flanks [Phan and Paschmann, 996; Deng et al., 003; Nishida, 000]. This is true also in other regions where the magnetic reconnection takes place, e.g., in the magnetotail [Mozer et al., 004; Wygant et al., 00] or which are traversed by parallel electron beams, such as the auroral region [Janhunen et al., 004; Ergun et al., 999]. The temperature anisotropy arises due to the turbulence created by parallel electron beams. Although it is not crucial for the creation of electron holes, it permits us to simplify the algebra by neglecting the finite electron Larmor radius effects, while keeping those of the electron polarization. [] For perturbations whose perpendicular and parallel length scales are l? r L,ek and l k l D,ek (where r L,ek = v T,ek /W e and l D,ek = v T,ek /w p,e are the electron Larmor radius and Debye length associated with parallel temperature, respectively), the electron drift-kinetic equation has þ ~V r g E k ¼ 0; m k ðþ where ~V = v k ~e z + (/B z,0 ) ~e z r(f v k A z ) is the guiding center velocity, E k = [@/@z (/B z,0 )(~e z ra z ) r]f (@/@t)a z is the parallel component of the electric field, v k is the parallel particle velocity, and g = log f(v k )+r L,ekr? (ef/t ek ). Here f(v k ) is the electron distribution function integrated over the velocity components perpendicular to the magnetic field, and the second term in the definition of g is a small correction coming from the divergence of the electron polarization drift. [] In the same ordering for perturbations whose frequency is much larger than the ion gyrofrequency W i = eb z,0 /m i, and the phase velocity is much larger than the ion thermal speed, the ion dynamics is governed by the linearized hydrodynamic equations for unmagnetized two-dimensional cold ions, i /@t + n 0 r? ~v i,? = 0 i,? = (e/m i )r? f, n ¼ en 0 r? f: ðþ m i Here n i and m i are the ion density and the ion mass, respectively. [3] We seek the electron hole as a stationary traveling solution of equations () and (). = u where u z is the parallel phase speed, the nonlinear characteristics of the drift-kinetic equation () are readily found from dx ¼ dy ¼ V x V y dz v k u z ¼ ð m e=eþdv k ^d k ðf u z A z Þ ; ð3þ where ^d k (/B z,0 )(~e z ra z ) r. [4] It is well known from the nonlinear dynamics that a nonlinear system possesses stable solutions only if the number of conserved quantities equals the number of degrees of freedom. Thus it is necessary to calculate all nonlinear characteristics (conserved quantities) described by equation (3). That is practically impossible in general, and we may proceed only in certain simplified cases with the reduced number of physical dimensions. Two obvious candidates are as follows: [5]. Parallel solution depends only on z and some combination of x and y. Then, there exists only one conserved quantity, the electron energy W, where W = (m e /)(v k u z ) e(f u z A z ). [6]. Oblique solution depends only on x and a linear combination of y and z, written as y 0 = y + z tan q, where q is an arbitrary angle. The oblique solution possesses two conserved quantities, the energy W and the canonical momentum P, where P = m e v k e(a z xb z,0 ). [7] In principle, the steady-state electron distribution function can be expressed as an arbitrary function of the conserved quantities W and P. The actual functional dependence needs to be determined from the given boundary conditions in the real and velocity spaces. Conveniently, instead of the integrals of motion W, P, we use their linear combinations v (0) k and x (0), defined as v (0) k = u z + sign(v k u z )(W/m e ) and x (0) =(P/m e v (0) k )/(W e tan q). Using the (0) explicit expressions for W and P, we have v k = u z + sign(v k u z )[(v k u z ) (e/m e )(f u z A z )] / and x (0) = x +(v k v (0) k ea z /m e )/(W e tan q), and we readily note that in the absence of the potentials, namely f = A z = 0, we have 3of9

4 v (0) k = v k and x (0) = x. Thus if f and A z were switched on at t =, and the evolution of the system was infinitely slow (i.e., adiabatic) throughout its history, the conserved (0) quantities v k and x (0) can be interpreted as the initial velocity and the initial position of the particle that at the time t had the velocity v k and the position x. In the initial moment t = the energies of all particles were positive definite. This means that the particles which at the present time have a negative energy, W < 0, must have undergone some nonadiabatic process during their history. As a consequence, their dynamics did not obey the simple law W = const., and the quantities v (0) k and x (0) are not their initial velocity and initial position. 3. Effects of Resonant Electrons [8] The unperturbed distribution function is adopted in the form of a shifted Maxwellian, corresponding to an electron beam moving with the velocity v z,0 ~e z, while for the perturbed function we use the standard Schamel s model [see, e.g., [Schamel, 000; Luque and Schamel, 005], modified by the effects of the electron polarization. Implicitly, this model assumes that the dynamics of the particles whose energy is positive has been purely adiabatic and that the quantities v (0) k and x (0) (which are purely real) represent their initial velocity and position. Thus their distribution is found simply by replacing in the unperturbed distribution function the particle velocity and position by their initial values. Conversely, the particles with negative energies were involved in some nonadiabatic process at some point of their history, and a detailed knowledge of their dynamics would be necessary to construct their distribution function. Most of these adiabatic processes (such as the electron-ion collisions, turbulent diffusion in the real and velocity spaces, etc.) are very weak under the magnetospheric conditions and are not even included in the basic equation (). At the present time, the particles with negative energies must be confined (or trapped) on closed orbits so that they never reach the asymptotic region of space where f = A z = 0 and the energy is positive. Instead of performing a very detailed study of their history, we adopt the plausible model equation (4) in which the distribution of trapped particles is taken to be Gaussian, resulting from the large number of their oscillations on the closed orbits, but their effective temperature can be different (possibly negative) that that of the free particles. We have f v n k ¼ pffiffiffiffiffi 0 exp =v T;ek Re v ðþ 0 k v z;0 p vt;ek þ b e Im v ðþ 0 ð e=me Þr L;ek r? : f ð4þ k It is worth noting that due to the assumption of an initially homogeneous distribution function (independent of x), in the above we did not make use of the characteristic x (0), i.e., of the canonical momentum P. The trapping parameter b e is related with the number of trapped particles, and it will be regarded as a free parameter. For electron holes (i.e., nonlinear density depletions) we have b e < 0. The density and parallel fluid velocity of the electrons are now readily found by the appropriate integration of equation (4), which with the accuracy to (ef/t ek ) 3/ gives and h n e ¼ n 0 r L;ek r? ef=t i ek þ a F ð4=3þbn F 3= ; ð5þ h i v z;e ¼ u z v z;0 a F ð4=3þbv F 3= ; ð6þ where F =(e/t ek )(f u z A z ), a is the real Z part of the plasma dispersion function, a =(pv T,ek ) / dvkv 0 k(v 0 k 0 u z + v z,0 ) exp ( v 0 k /v T,ek ), and the parameters b n and b v come from the contribution of the resonant particles, both free and trapped, b n = p / ( b e ) exp[ (u z v z,0 ) /v T,ek ] and b v = p / (3/ b e ) exp[ (u z v z,0 ) /v T,ek ]. Substituting (5) and (6), together with the ion density (), into Posson s equation and Ampere s law, and after some tedious but straightforward algebra, for nonlinear modes whose parallel phase velocity satisfies u z /c min(, a l D,ek r? ), our basic equation is obtained which describes small amplitude (F ) electron holes in a magnetized plasma ( "! l D;ek þ w p;e W e F af þ 4 3 b nf 3= þ w p;i u z l D;ek F ¼ 0: One should bear in mind that equation (7) is based on only one characteristic of the drift-kinetic equation (), i.e., on the energy conservation, and that it holds only in special cases labelled as parallel and oblique solutions discussed above. [9] In the linear case, i.e., in the absence of trapped electrons, b n = 0, and for large phase speeds ju z v z,0 j v T,ek yielding a v T,ek /(u z v z,0 ), equation (7) reduces to the textbook dispersion relation for electrostatic waves in a magnetized plasma, in which electrons are streaming and the ions are at rest. We have þ w p;e W e k? w p;e k kz w k z v z;0 k w p;i w ¼ 0; where w = k z u z. The solutions of equation (8) for the parallel and almost perpendicular wave propagation are displayed in Figures and, respectively (note different normalizations in these two figures). They are found using the typical plasma parameters from Mozer et al. [004, Table ], indicating that the magnetopause region where the electron holes were observed was relatively weakly magnetized, with the parameter w p,e /W e ranging between and 00, typically w p,e /W e 35. For the Buneman instability of the waves propagating parallel to the magnetic field (k? = 0), the growth rate g and the real part of the frequency w R of the fastest growing mode scale as g w R 0.05 w p,e. The fastest growing mode is generated when the unperturbed electron beam speed matches the phase velocity of the corresponding Langmuir wave, k z v z,0 w p,e. These data are ð7þ ð8þ 4of9

5 Figure. Parallel propagation, k? = 0. The real and imaginary parts of the frequency (solid and dashed lines, respectively) of a linearly unstable mode described by the dispersion relation, equation (8), as a function of the zeroorder Doppler shift k z v z,0. For the parallel mode, the frequency and the Doppler-shift are normalized by the electron plasma frequency w p,e. The relevant plasma parameter is adopted to match the magnetopause p ffiffiffiffiffi conditions of Mozer et al. [004], w p,e /jw e j = 35. consistent with the satellite observations of large amplitude Langmuir waves following the fast reconnection and the emergence of electron holes [Deng et al., 003; Cattell et al., 00, 005], whose duration is 5 0 Langmuir wave periods. Conversely, for the oblique Buneman instability of lower-hybrid waves with tan q = k z /k? = /5, we have g w R 0.7 w LH, for a mode that is driven by an electron beam whose velocity is somewhat larger than the parallel phase velocity of lower-hybrid waves, namely k z v z,0 3 w LH. These two linearly unstable modes are likely to saturate by the trapping of resonant electrons, yielding parallel (or Langmuir-type) and oblique (or lower-hybrid) electron holes, respectively. The Langmuir-type electron holes have been known from satellite observations [Deng et al., 003; Cattell et al., 00, 005], and they have been extensively investigated both analytically and numerically. In the following we will study only the oblique structures in order to address the observations of perpendicular electric fields by Mozer et al. [004]. [0] Although in relatively weakly magnetized plasmas with jw e j < w p,e, e.g., in the magnetopause, the growth rate of the linear modes propagating parallel to the magnetic field exceeds that of the lower-hybrid waves (which can be seen also from our Figures and ), the growth of the lower hybrid is often unsuppressed. Recently, the presence of the oblique Buneman instability in a plasma with jw e j/w p,e = 0.5 was implied in the 3-D simulations of Singh [004] and Drake et al. [003], but due to the numerical limitations they used nonphysical ion-to-electron mass ratios m i /m e = 6 and m i /m e = 00, respectively, which made the lower-hybrid mode less distinct. [] In the long-standing debate on how the electron beam can drive the lower-hybrid waves when the highfrequency waves propagating parallel to the magnetic field have a bigger growth rate, several mechanisms have been put forward. One possibility [see Singh et al., 00, and references therein] is that the high-frequency waves with relatively large perpendicular group velocity quickly escape from the electron beam of finite perpendicular dimensions, leaving the lower-hybrid waves to interact over longer distances. Alternatively, it has been suggested that in the presence of a hot electron population in the beam, the electron distribution function does not get completely plateaued by the high-frequency waves, leaving the lower-hybrid waves an opportunity to grow during a later phase. This mechanism was confirmed in the numerical simulations [Omura et al., 003], which demonstrated that when the parallel electron temperature exceeds the ion temperature, T ek T i, the Buneman instability of the high-frequency waves propagating parallel to the magnetic field gets suppressed by the excitation of acoustic waves, and a bump-on-tail-like electron distribution is produced. The latter is known to yield the lowerhybrid electron holes, as shown already in the early -D simulations by Dum and Nishikawa [994]. At first, the broad spectrum of parallel-propagating waves is excited, which saturates at a relatively low level and without particle trapping. At later times, and in a plasma with jw e j/w p,e =, Dum and Nishikawa [994] observed the electron trapping by a lower-hybrid wave driven by the oblique bump-on-tail instability. Thus we can envisage a two-stage scenario for the creation of electron holes in a weakly magnetized current carrying plasma, subject to both types of the Buneman instability, in which the parallel instability is quenched at a low level, possibly without any particle trapping, and in the second stage the lower-hybrid waves are excited by the residual energy of the beam. [] An alternative scenario for the creation of electron holes aligned with the lower-hybrid waves was suggested by Umeda et al. [00], which does not rely upon the direct particle trapping in a rapidly growing lower hybrid wave. It was suggested that the lower-hybrid structures arise in a three-stage process that starts with the creation Figure. Oblique propagation. The propagation angle is adopted within the domain that is allowed for the Mozer et al. [004] structures, tan q = k z /k? = /5. The real and imaginary parts of the frequency (solid and dashed lines, respectively) of a linearly unstable mode described by the dispersion relation, equation (8), as a function of the zeroorder Doppler shift. For the oblique mode, the frequency and the Doppler shift are normalized by the lower-hybrid frequency w LH =(jw e jw i ) p ffiffiffiffiffi. The relevant plasma parameter is adopted as w p,e /jw e j = 35. 5of9

6 of -D electron holes by the electron trapping in the fastest growing (i.e., parallel propagating) wave. In the next stage, such Langmuir-type electron holes are bent and modulated by the lower-hybrid waves, which leads to their decay into a gas of localized -D structures, which in the final stage coalesce into larger structures aligned with the lower-hybrid wave. Such scenario was supported by the -D particle-in-cell simulations of the Buneman instability in a space plasma [Umeda et al., 00], but due to the low dimensionality certain relevant physical effects (e.g., ~E ~B convective nonlinearity) were left out. [3] It should be noted that the particle trapping by a rapidly growing, linearly unstable monochromatic wave is not the only physical mechanism that may produce the phase space vortices. They may arise also due to the selforganization in a fully developed turbulence. Besides the three-stage mechanism of Umeda et al. [00], a very illustrative example is the emergence of phase-space vortices (in this case, in the ion phase space) in the fluid turbulence, resulting from a plasma flow around an obstacle [Guio and Pécseli, 005]. Thus different alternative scenarios for the creation of lower-hybrid electron holes may be considered that do not depend solely on the oblique Buneman instability. However, that would require extensive three-dimensional numerical simulations of a magnetized plasma and is beyond the scope of the present paper which is devoted to the analytic prediction of the possible stationary nonlinear states. [4] For an oblique structure we = tan and introduce the scalings x! A LH (+ w p,e /W e ) (x/l D,ek ), y! A LH ( + w p,e /W e + tan q) (y/l D,ek ), and F! (4b n /3A LH ) F, where A LH = a (m e /m i )(v T,ek/u z ) ( + tan q), which permits us to rewrite equation þ F F þ F 3 þ ¼ 0; where h =(m e /m i )(v T,ek/u z )[( + w p,e /W e ) + tan q] A LH. As for the typical electron holes in the magnetopause we have v T,ek u z and a (ef/t e ) , the parameter h in equation (9) is small, h, when the angle between the cylindrical hole and the magnetic field satisfies tan q > (m e /m i )(v T,ek/au z ) m e /m i. This angle coincides with the domain of k z /k? for which the linear Buneman instability of the lower-hybrid waves takes place. [5] In a strictly -D case, = 0, equation (9) is readily integrated in the form of an oblique slab electron hole [see, e.g., Jovanović and Shukla, 004; Schamel, 000] ð9þ F sl ¼ 5 y 6 sech4 ; ð0þ 4 that emerges from an unstable lower-hybrid plane wave with k x = 0. Alternatively, an oblique, cylindrically symmetric solution displayed in Figure 4 is found when the tilt of the structure, q, is in the range of the propagation angles of linearly unstable lower-hybrid waves, when we may set h! 0 and neglect the last term in equation (9). Then, /@x /@y + (/r)(@/@r), where r = (x + y ), we solve equation (9) in the form a bell-shaped structure that can be approximated by F cyl :588 þ 0:00556 r 0:65 I :6 0 r :6 ; ðþ where I 0 is a modified Bessel function of the order zero. 4. Effects of Perpendicular Ion Motion [6] Both oblique solutions, equations (0) and () described in the section 3, are obtained after neglecting the ion dynamics perpendicular to the direction of their propagation, i.e., taking h@ /@x! 0. A simple inspection reveals that when there exists an electric field along the x axis, the full equation (9) does not provide a mechanism that can restore the ions in their initial positions. In other words, a propagating x-dependent electron hole that initially was well-localized along y, would leave behind a tail of displaced ions. The consecutive oscillations of those ions, together with ambipolar electrons, eventually produce longwavelength lower-hybrid waves. Thus it is necessary to investigate the stability of -D electron holes, equation (0), in the presence of such oscillations. As an important example, we present a solution which propagates with the same phase velocity as the waves created due to the ion displacement within the hole, i.e., whose tail is stationary in the reference frame moving with the nonlinear structure. [7] First, we seek a modified slab solution that is weakly x-dependent, in the form F(x, y) =F sl (y) +df sl (y) sin k x x, where df(y) F sl (y). A linearized equation for the perturbation df, deduced from equation (9), has k x þ 3 F slðyþ df sl h kx df sl ¼ 0: ðþ Equation () is readily solved numerically and the solution is displayed in Figure 3. It is important to note that for any given value of the parameter h, which determines the ion mobility, there exists only one value of the perpendicular wave number k x for which the solution df is finite at x = ±. The perturbation df sl (y) is an odd function of y. Asa consequence, the ions are almost completely restored to their initial position and the tail is minimized, with the amplitude much less than 0% of the maximum value that df sl (y) reaches inside the zero-order solution F sl (y). Stationary solution F(x, y), described by equations (0) and (), is expected to be the most stable among the tailed structures, since their nonlinear core, equation (0), does not leak the energy by emitting the lower-hybrid waves. An instability of such stationary solution might result only from the resonant interaction between trapped electrons and the ambipolar electric field. Namely, the electrons that are bouncing in the direction of the z axis, simultaneously undergo oscillations also in the x-direction due to the ~E? ~B drift, where ~E? = ~e y (@/@y) F sl (y + z tan q). Thus any asymmetry in the distribution of trapped electrons produces an electric charge that is oscillating in the x direction, which may be in resonance with the oscillations of the ion fluid. A similar resonant mechanism for the instability of Langmuir-type electron holes, due to the interaction between electron bounce motion and their 6of9

7 transformation along x, equation (4) can be integrated twice, yielding Z y ^p df cyl ¼ h k x sin Ky dy 0 F cyl ðk x ; y 0 Þ cos Ky 0 ; ð5þ K y 0 where y 0 is a constant of integration, K =[hk x /(k x + )], and we used the fact that F cyl is an even function of y. Equation (5) is further simplified since the zero-order solution F cyl is localized p ffiffiffi to a distance, i.e., much shorter than /K / h. Then, within the adopted accuracy in the small parameter h, in the integral on the right-hand side we can set cos Ky 0. Finally, in the tail of the structure, i.e., at distances y >, equation (5) can be further integrated, yielding df cyl ðk x ; yþ ¼ K sin Ky c sign y Z 0 dy 0 F cyl ðk x ; y 0 Þ ; ð6þ Figure 3. Contour plot of perturbed oblique slab lowerhybrid electron hole, F(x, y) =F sl (y) +df(y) sin(k x x), with the parameters k x = 0. and h = 0.. Normalizations are given in the text, see equation (9). cyclotron rotation, was demonstrated by Jovanović and Schamel [00]. Figure 3 indicates that the temporal growth of the perturbation df sl, superimposed on the oblique slab hole (0), corresponds to the bending, or kink, of such structure (at least in the linear phase). Eventually, this may lead to the breakup of the slab into a sequence of isolated structures immersed in the bath of long-wavelength lowerhybrid oscillations, plausibly similar to the cylindrical holes, equation (). [8] To study the effects of the transverse ion motion on a cylindrical hole, in a stationary regime, we linearize equation (9) around the zero-order solution F cyl, equation r? þ 3 F cyl df cyl F cyl þ df cyl : ð3þ In the spatial region where F cyl > h, i.e., for r, the inhomogeneity in equation (3) is strong, the spatial scales in the x and y directions are of the and we can drop the term (@ /@x ) df cyl on the right-hand side. Conversely, at larger distances where F cyl ] h, we keep this term but neglect the inhomogeneities. Then, with the accuracy to leading order in the small parameter h, equation (3) can be @ df cyl ¼ F ; ð4þ where ^p = r? + (3/) F cyl. Obviously, with the accuracy to first order in h, equation (4) is applicable also in the region r ], where the term h(@ /@x )^pdf cyl on the left-hand side in negligible. Using the Fourier where c is a constant of integration. Expression (6) describes an oscillating tail created by the transverse ion motion trailing a cylindrical p hole, equation (). The tail has a small amplitude, ffiffiffi h Fcyl, and p ffiffiffi the characteristic length in the y direction of order / h, which is much longer than its characteristic scale along x. 5. Application to the Earth s Magnetopause [9] As shown by Jovanović and Shukla [004], the oblique slab features bipolar pulses both in the parallel and perpendicular directions. Conversely, the oblique cylinder features a bipolar pulse in the parallel direction, while in the perpendicular direction the signal is bipolar in the plane determined by the magnetic field and the axis of the structure and unipolar perpendicular to it. As the satellite coordinate system is at an arbitrary angle to this plane, both x and y components of the recorded signal are an admixture of unipolar and bipolar pulses that is visually similar to a simple unipolar pulse. Actually, the small asymmetry of perpendicular signals in the form of a shoulder on one side, see Figure 4, as well as a rare occurrence of bipolar perpendicular signals, has been recorded. In Figure 4. the results of a numerical experiment are displayed, in which we calculated the characteristic electric field signal that a rapidly moving cylindrical structure () produces on a spacecraft, under the physical conditions that are typical for the Earth s dayside magnetopause close to its boundary with the magnetosphere. Following Mozer et al. [004], we adopted the electron temperature to be T ek = kev, which corresponds to the electron thermal speed v T,ek = km/s. Then, for the events in Table of Mozer et al. [004], the electron Debye length ranges between m, typically l D,ek = 0 m, while w p,e /W e ranges between and 00, typically w p,e /W e = 35. The velocity of the structure u z and the angle q are adopted so as to be in the domain of the propagation velocities and propagation angles of the linearly unstable lower-hybrid waves, tan q = /5 and u z =. v T,ek. We also adopt physically reasonable values A LH = 0.4 and b n = 0.65, which are governed by the still unknown quantitative features of the electron distribution function anisotropy, such as its elongation and the 7of9

8 Figure 4. Three components of the electric field signal that an oblique cylindrical electron hole produces on a spacecraft under typical [Mozer et al., 004] conditions. The hole s parameters are tan q = /5, u z =. v T,ek, A LH = 0.4, and b n = 0.4, the satellite is 5l D,ek far from the axis, and the angle between the perpendicular velocities of the satellite and of the structure is 40. The electric fields are expressed in mv and the time in ms. depth of its trough, i.e., by the initial streaming velocity of the electrons, v e,0, and the trapping parameter b e. Under these assumptions, the perpendicular size of the structure is l? 70 l D,ek 5 km, the ion mobility parameter h remains small, h = 0., and the peak electrostatic potential in the center of the structure is sufficiently small compared to the electron temperature for the theory to be valid, ef cyl /T ek < 0.5. [30] For a given spatial structure of the electron hole, the observational signature largely depends on the direction of movement and the relative positions between the structure and the satellite. Unfortunately, because of the scarcity of the observational data, it is possible to make comparisons only on a small sample. As an illustrative example, for a satellite that is 3300 m (i.e., 5 l D,ek ) away from the axis of the cylinder, and for a typical angle between the perpendicular velocity components of the satellite and of the structure j =40, we obtain an excellent agreement with the observational data. The perpendicular and perpendicular electric fields magnitudes presented in our Figure 4, as well as their duration, coincide with the typical Polar values listed in Table of Mozer et al. [004]. While the shape of the two perpendicular pulses is dependent on the arbitrary angle, their overall form is rather insensitive to it. It remains unipolar, with a varying size of the shoulder which very rarely becomes comparable to the main pulse. It should be noted that the signal rapidly decreases with the distance of the satellite from the axis, and the small variations of the electric field, including the asymmetric shoulders, can be masked by the noise. 6. Summary and Conclusions [3] In this paper we have used the theoretical model of oblique lower-hybrid electron holes to investigate the coherent electric field signals recorded recently in the strong shear, dayside Earth s magnetopause. It is demonstrated that nonlinear structures with very similar signatures may be constructed analytically as self-consistent stationary solutions of the equations describing nonlinear lower-hybrid waves, including the effects of the electrons that are trapped in the direction parallel to the magnetic field. The timing of such structures, as well as their tilt relative to the magnetic field, are consistent with the characteristic period and the angle of propagation, k z /k y, of a linearly unstable lowerhybrid wave, due to the oblique Buneman instability driven by a parallel electron beam. Such beams are known to emerge in the course of the fast collisionless reconnection in the presence of a guide magnetic field, both from the computer simulations [Del Sarto et al., 003] and from the observations of the electron diffusion regions, associated with the reconnection between the southward interplanetary magnetic field and the Earth s magnetic field in the dayside magnetopause [Scudder et al., 00; Mozer et al., 003]. Using the physical model of a cylindrical hole that is found as the stationary solution of the drift-kinetic equation with the Schamel s distribution function and under the realistic dayside magnetopause conditions, we obtained an analytic nonlinear solution whose main features their duration, intensity and three-dimensional shape coincide with those observed by the Polar mission [Mozer et al., 004]. [3] A possible scenario of creation of cylindrical structures starts with the oblique Buneman instability which produces a lower-hybrid wave propagating almost perpendicularly to the magnetic field, whose parallel phase velocity is in the electron thermal range. Eventually, the linear instability saturates by the trapping of resonant electrons in the direction parallel to the guide magnetic field. Initially, the electron hole is predominantly one-dimensional in the form of an oblique slab. 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Geophys. Res., 07(A8), 0, doi:0.09/00ja9003. D. Jovanović, Institute of Physics, P. O. Box 57, 00 Belgrade, Serbia and Montenegro, Yugoslavia. (djovanov@phy.bg.ac.yu) G. E. Morfill, Max-Planck-Institut für Extraterrestrische Physik, D Garching, Germany. (gem@mpe.mpg.de) P. K. Shukla, Institut für Theoretische Physik IV, Ruhr Universität Bochum, D Bochum, Germany. (ps@tp4.rub.de) 9of9