A Review of Density Holes Upstream of Earth s Bow Shock

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1 /2011/31(6) Chin. J. Space Sci. Ξ ΛΠΠ G K Parks, E Lee, N Lin, J B Cao, S Y Fu, J K Shi. A review of density holes upstream of Earth s bow shock. Chin. J. Space Sci., 2011, 31(6): A Review of Density Holes Upstream of Earth s Bow Shock GKParks 1 ELee 2 NLin 1 JBCao 3 SYFu 4 J K Shi 5 1(Space Sciences Laboratory, University of California, Berkeley, USA) 2(Kyung Hee University, Suwon, Korea) 3(School of Astronautics, Beihang University, Beijing ) 4(Institute for Space Physics, Peking University, Beijing ) 5(State Key Laboratory of Space Weather, Center for Space Science and Applied Research, Chinese Academy of Sciences, Beijing ) Abstract Larmor size transient structures with density depletions as large as 99% of ambient solar wind density levels occur commonly upstream of Earth s collisionless bow shock. These density holes have a mean duration of 17.9±10.4 s but holes as short as 4 s have been observed. The average fractional density depletion (δn/n) inside the holes is 0.68±0.14. The density of the upstream edge moving in the sunward direction can be enhanced by five or more times the solar wind density. Particle distributions show the steepened edge can behave like a shock, and measured local field geometries and Mach number support this view. Similarly shaped magnetic holes accompany the density holes indicating strong coupling between fields and particles. Current densities as large as 150 na m 2 are observed at the leading compressed edge. The waves are elliptically polarized and rotating in the sense of ions (left hand) in the plasma frame. The waves appear to grow and steepen as the density holes convect with the solar wind toward the Earth. The transient nature of density holes suggests that the temporal features could represent the different stages of nonlinear evolutionary processes that produce a shock-like structure. The density holes are only observed with upstream particles, suggesting that back-streaming particles interacting with the solar wind are important. The significance of these observations is still being investigated. Key words Density holes, Solar wind, Upstream of Earth s bow shock 1 Introduction The Earth s bow shock is the best-studied collisionless electromagnetic discontinuity in space. Although much is known about the bow shock, fundamental questions remain unanswered. It is still not known what physics can produce a shock-like discontinuity in collisionless plasmas. Clues to this fundamental problem have been recently obtained from observing the behavior of density holes upstream of Earth s bow shock [1 2]. Density Holes (DH) are the smallest known nonlinear ion structures observed in the upstream region of the bow shock. Density holes have dimensions on The research at UC Berkeley is performed under the auspices of a NASA Grant No. NNG04GF23G. Cluster is a joint project of ESA and NASA and Double Star a joint project of ESA and the Chinese Space Agency. The work at Kyung Hee University was supported by the WCU program through NRF funded by MEST of Korea (R ) Received September 28, parks@ssl.berkeley.edu

2 694 Chin. J. Space Sci. Ξ ΛΠΠ 2011, 31(6) the order of an ion Larmor radius. While they resemble previously studied upstream structures, Hot Flow Anomalies (HFAs) [3], Hot Diamagnetic Cavities (HDCs) [4] and Foreshock Cavities (FC) [5],DHs are shorter in duration and have deeper holes. Density holes also occur much more frequently than HFAs and HDCs that occur rarely. Only about 30 HFs have been observed throughout the lifetime of ISEE and AMPTE (3, 4). Density holes have large flow deviations similar as in HFAs and HDCs. Foreshock Cavities have overshoots in density and magnetic field intensities at their edges but they do not have the strong density depletions that define DHs. Density holes are observed under a variety of solar wind conditions but only in the presence of back streaming bow shock particles. However, not all back streaming particles produce DHs. The mechanisms that can produce DHs still remain unknown and should challenge theorists. This article reviews the main features of DHs and discusses what we know and don t know about them (The material in this review come from references [1,3,6 8] ). 2 Observations Figure 1 shows typical data obtained in the vicinity of the bow shock by the ion and magnetic field experiments on the European Cluster and Chinese Double Star (DS) scientific satellites. Cluster was outbound and DS inbound. The Solar Wind (SW) in the energy flux spectrogram (top panel) is indicated by the narrow red band, at 2 kev/q, while the Magnetosheath Fig. 1 Cluster 1 and Double Star observed nearly identical density hole structures, for example, 08:56 UT. Cluster 1 was outbound and Double Star inbound. The locations of the spacecraft are shown at the bottom (GSE coordinates are used throughout). The panels from top to bottom show energy flux, density, velocity and temperature

3 GKParks,et al.: A Review of Density Holes Upstream of Earth s Bow Shock 695 (MS) is a broad energy band. Cluster 1 exited the bow shock at 08:29 UT while DS crossed the shock 09:22 UT. The bow shock moved in and out and the SW and MS plasmas were detected several times. The weaker fluxes above the few kev/q SW are the backstreaming particles moving toward the Sun. The SW speed on this day was 600 km/s in the V x direction with a density n 2.5cm 3. The sharp dips (panel 2) in the density n that go below the average SW density (n sw ) are DHs. The DHs are followed by overshoots above n sw (see 08:55 UT and 09:01 UT). Examination of data from many crossings indicates that the DHs appear in bunches close to the bow shock, but more in isolation further away. Density holes have been seen all the way to Cluster apogees (about 19 R e ). The V sw and temperatures (T ) accompanying the DHs behave as they do near the bow shock: V x decreases almost to zero, and V y and V z deviate substantially and T increases by an order of magnitude. Figure 2 shows an example of a DH in high time resolution observed on 3 February We see decrease of density from 10 cm 3 to 1.5 cm 3, decrease of the solar wind flow with V x 400 km s 1 to 100 km s 1, V y increasing from 0 to 200 km s 1,and V z from 0 to 400 km s 1, increase of the temperature from T 10 6 K outside the hole to Kin the hole, and overshoot of the density at the edges Fig. 2 A density hole is observed on 3 February From top to bottom are energy spectrogram, density, bulk speeds in GSE, temperature parallel and perpendicular to B, and magnetic field and B-field components. Data shown are 4s averages (spin averaged)

4 696 Chin. J. Space Sci. Ξ ΛΠΠ 2011, 31(6) with a much larger one on the upstream side (50 cm 3 ) than the downstream side (20 cm 3 ). Note the magnetic field B has similar shape as the density hole and that B y and B z change sign across the density hole indicating crossing of a current sheet. Current sheets are associated with most density holes we have studied. 2.1 Lagrangian Representation The right panel of Figure 3 shows spin resolution (4 s) data of a DH detected at 08:55 UT on 2 March 2005 by Cluster 1 (Double Star also detected this hole, not shown). For this hole, n at both edges increased from the solar wind value of 2 to 3.5cm 3, while inside the hole, it decreased to 0.45 cm 1.The bulk T rose from Kto KandV x decreased from about 600 km s 1 to 0 km/s while V y increased from 100 km s 1 to 600 km s 1. The solar wind parameters were, β = 8π nkt/b 2 1.6, T e /T i 2, V A = B/(4πnm i ) 1/2 56 km s 1 and C s =(kt e /m i ) 1/2 90 km s 1. The left panel of Figure 3 shows a Lagrangian view of this DH (magnetic field). This view presents the DH structure as a function of the space coordinate. A timing analysis from the four Cluster spacecraft assuming a planar geometry yielded a normal ( 0.65, 0.75, 0.065) for the downstream edge at 08:55 UT and a speed of 530 km s 1 along the normal. The normal of the upstream edge was ( 0.80, 0.59, 0.11) with an antisunward speed 320 km s 1,indicating a significant relative motion. Detection times for the edges by Double Star downstream indicate that the speed of the edges could have changed (not Fig. 3 A Lagrangian view (left) of a density hole (right) observed by Cluster 1. The E-field data were averaged and high frequency variations were removed. The density holes are accompanied by nearly identically shaped magnetic holes with reduced intensity in the hole and steepened B at the edge. The scale of B is linear while for n it is logarithmic. Temperature T inside the hole is about 100 times the solar wind T. The GSE coordinates of Cluster 1 position are (12.0, 3.2, 4.9) R e

5 GKParks,et al.: A Review of Density Holes Upstream of Earth s Bow Shock 697 shown). The resulting downstream edge acceleration was estimated to be 56 km s 2, and the sunward edge deceleration 23 km s 2. This implies the hole was growing at an increasing rate (Timing errors are about 10%. In the case of acceleration, the errors are larger). The size of the downstream edge at Cluster was estimated to be 1300 km, the upstream edge 2300 km, and the hole 3700 km (Including acceleration would make them larger). The gyroradius of 4 kev proton in B = 4 nt is 1100 km. The growth time of the hole, obtained by dividing the width of the hole by the relative velocity of the two edges yields 10 s (including acceleration will reduce this value). The B variations were in phase with the density but the B intensity at the downstream edge was about 20% larger than the exterior field (4.5nT to 5.5nT), while at the upstream edge it nearly doubled, from 4 nt to 7 nt. Inside the hole, B decreased to 0.5nT. The B field at both edges was compressed and the normal component B n nearly vanished throughout. E included both tangential E t, and normal E n with E t perpendicular to B t. Here E t = E t1 + E t2. Measurements of B inside the hole indicated presence of small and fluctuating fields (Small fluctuations are also seen in E but data have been smoothed). This density hole is a published SLAMS event and the behavior of E and B has been studied [9 11]. SLAMS have been studied in detail using magnetic field data but not the ions associated with them. 2.2 Current Density Two DHs were observed on 2 April 2002 (Figure 4) by Cluster 1 in front of the bow shock located at a distance of (9.8, 2.2, 8.1) R e. Let us focus on the first DH with the minimum at 03:36:20 UT. The density in the hole is 0.4cm 3, which is a factor of 4 less than the solar wind density, n sw 1.6cm 3. Both edges show increase of the density but the overshoot is much larger on the upstream edge. The density here is 8 cm 3 and about five times larger than the n sw. As mentioned earlier, large solar wind decrease is commonly observed with all DHs. In this particular case, the solar wind V x 700 km s 1 was reduced to 0 km s 1. Observations of density holes by four Cluster spacecraft separated by a few hundred kilometers allow us to compute the current density (J) usingthe curlometer technique. Current densities in GSE system and relative to the directions of the B-field are shown in the bottom two panels. The largest J are observed with the overshoot in the upstream edge. The peak J = 150 na m 2. Currents are flowing mostly parallel to B although there are also currents perpendicular to B. The current along B is carried mostly by the electrons (not shown). 2.3 Steepening of the Edges The Cluster mission provides a unique opportunity to make in situ measurements of the temporal development. During the apogee passes in the solar wind in 2003, Cluster spacecraft were aligned predominantly along the Sun-Earth line with the separation as large as 1.6 R e (radius of the Earth). This configuration enabled us to observe the temporal development of structures moving with the solar wind as they passed by the spacecraft. Here we show an example of temporal development of a compressional edge formed at the upstream edge of a DH. Figure 5 shows a DH observed on 12 March At 02:29 UT Cluster SC4 was in the solar wind at (13.2, 5.4, 8.5) R e in GSE coordinates, and separated from SC1 in the X-, Y -, and Z-directions by 1.53, 0.07, and 0.64 R e, respectively (see Figure 5a, b). Initially, SC1 and SC2 were in the magnetosheath while SC3 and SC4 in the solar wind. Because the bow shock moved earthward, SC1 and SC2 moved into a DH, which was in front of the bow shock and in contact with the bow shock. The B-field measurements from the four Cluster spacecraft are shown in Figure 5. A DH was observed on SC4 between 10:47:50 UT and 10:48:20 UT. It is identified by a depression in B and the spacecraft potential, φ sc, which is a proxy for electron density. A compressional pulse was observed at the upstream edge of the DH (10:48:45 UT, marked by the orange bar in Figure 5c). The amplitude of the pulse, δb = B 0, was about 0.84, where B 0 is the magnitude of the upstream B field and δb is the enhancement at the pulse. Subsequent observations from the other spacecraft indicate the DH developed in a complicated way, including large amplitude peaks (e.g., between 10:48:00 UT and 10:49:05 UT in Figure 5d).

6 698 Chin. J. Space Sci. Ξ ΛΠΠ 2011, 31(6) Fig. 4 Spin averaged (4 s) bulk parameters obtained from the distribution function obtained by an ion instrument and magnetometer on Cluster. From top to bottom, energy spectrogram, density, bulk speeds and magnetic field in GSE. The bottom two panels are currents derived from the curlometer technique using the four spacecraft data On SC3 δb/b 0 increased slightly to 0.95, indicating growth without change in shape. On SC2, however, the amplitude has more than doubled, δb = B The estimated average growth rate Δ (δb/b 0 )/Δt between SC4 and SC3 is s 1, while between SC3 and SC2, Δ(δB/B 0 )Δt s 1, about 40 times larger. Thus, the pulse grew impulsively between SC3 and SC2. Moreover, the edge of the pulse facing the solar wind (10:49:45 UT) sharply steepened. Because the separation between SC2 and SC3 was mainly along the X direction (ΔX 0.75, ΔY 0.11, and ΔZ 0.16 R e ) and the structure was moving in the X direction embedded in the solar wind, it is reasonable to assume that a similar plasma region was sampled by SC2 and SC3. Thus, the variations observed from SC3 to SC2 can be interpreted consistently as growing and breaking of a nonlinear pulse. On SC1 it is possible to see that the pulse had developed into a shock-like structure with a ramp,

7 GKParks,et al.: A Review of Density Holes Upstream of Earth s Bow Shock 699 Fig. 5 Orbit of the Cluster spacecraft projected onto the (a) xy and (b) xz planes in the geocentric solar ecliptic coordinates on 16 February 2003 (blue line). Shown in included boxes is the configuration of the Cluster spacecraft at 10:50 UT. The gray curve represents the model bow shock location. Magnetic field measurements are shown from (c) to (f). Full resolution (22.5 Hz) data were used. The compressional pulse and shocklike structure are marked by the orange bar. Spacecraft potential, φ sc, is also plotted in (c). Bottom panels show (g) magnetic and (h) electric fields measured by SC1 (solid lines) and SC2 (dotted lines). The time on SC2 was shifted by 30 s to match the edges overshoot, and a magnetosheath-like downstream region. The amplitude was comparable to that on SC2, indicating that after steepening the amplitude ceased to grow. Figures 5(g) and 1(h) show an expanded view of B and electric (E) fields measured on SC1 (solid lines) and SC2 (dotted lines; shifted by 30 s to match the edge) across the edge. On SC1, a few Hz frequency oscillations occurred across the edge (within vertical bars) in both B and E fields, while the transition on SC2 was smooth. These embedded oscillations may be an early phase of whistler mode waves or electromagnetic oscillations that could propagate to the upstream region of shocks. 2.4 Waves Power spectra obtained from two wave experiments [12 13] for the density hole on 2 March 2005 are complex and not all understood (Figure 6). Wave power for δe and δb was considerably enhanced inside and at the edges of the density hole. For δb, power inside the hole was enhanced from a few Hz to close to the ion cyclotron frequency ( Hz) while enhancement for δe was observed at a few khz. These waves are in the same frequency range as the Doppler shifted ion acoustic waves frequently observed in the solar wind. We also see enhanced power for δe above the electron cyclotron frequency

8 700 Chin. J. Space Sci. Ξ ΛΠΠ 2011, 31(6) Fig. 6 Top three panels show power spectra of δe (mv 2 m 2 Hz 1 ) from 2 to 50 khz (white line is estimated plasma frequency), and sum of δe in spin plane from 8 Hz to 4 khz and sum of δb (nt 2 Hz 1 ). White line is calculated electron cyclotron frequency. The bottom three panels show magnetic field components and magnitude and the spacecraft potential that is proxy for electron density inside the hole. These waves are not yet all identified since electrostatic waves above the electron cyclotron frequency have not been seen previously in the solar wind (to our knowledge). We now discuss polarization of the waves in DHs. For this analysis, we have used the high time resolution B-field (22.5Hz) and E-field (25Hz) measurements to determine the phase velocity and the sense of polarization for this DH. We assume that the wave front is planar and uniform on the scale of the spacecraft separation (a few hundred kilometers) and that the waves are propagating with a constant phase velocity. Then the phase speed in the plasma frame can be computed from V pl = V ph k V sw where V pl is the phase speed in the plasma frame, V ph is the phase speed in the spacecraft frame, k is the wave normal, and V sw is the average bulk speed. The normal k is determined from minimum (maximum) variance analysis for δb(δe). We use Faraday s law and obtain k δe = ωδb where ω is the frequency and δe and δb are fluctuations of E and B fields. The phase speed in the spacecraft frame is obtained from V ph =(ω/k) =k (δe δb)/δb 2. Doppler shifting it yields the phase speed in the plasma frame. We will examine the low frequency waves close to the ion Larmor frequency that are seen with DHs. A detailed discussion of the timing method and wave properties is also given in Ref.[7]. Minimum variance analysis yielded k =( 0.89, 0.43, 0.18) and V sc, 281 km s 1. Noting that the solar wind speed is V sw =( 350, 70, 20)km s 1,the phase velocity in the plasma rest frame in the direction k is V pl = V sc k V = 62 km s 1. The wave is propagating sunward with a speed 62 km s 1 but is

9 GKParks,et al.: A Review of Density Holes Upstream of Earth s Bow Shock 701 convected earthward with the solar wind. The plasma frame velocity is approximately the Alfvén speed, 47 km s 1 of the ambient solar wind, where we used B 8.3 nt and solar wind density of n 15 cm 3. The angle between k and the B-field is The wave is thus an obliquely propagating wave. Knowing k, we can project the δb and δe into the plane of the wave front. Figure 7 shows hodogra- Fig. 7 Sense of polarization for magnetic (a) and electric (b) fields. The left column of each shows hodograms in the plane normal to k. The second and third columns are hodograms in the other planes. Observations are shown from the four Cluster spacecraft. The asterisk indicates the start point. Both electric and magnetic fields are right hand elliptically polarized in the spacecraft frame. In the plasma frame, they are left-hand elliptically polarized

10 702 Chin. J. Space Sci. Ξ ΛΠΠ 2011, 31(6) ms of the δe and δb components in this plane from which one sees the waves are right hand elliptically polarized in the spacecraft frame and thus they are left handed in the plasma frame. The δe-field is perpendicular to the δb, indicating these are plane electromagnetic waves. 2.5 Ion Distributions An ion distribution associated with a DH at 08:55 UT (2 March 2005) is shown in Figure 8. This velocity space plot is a cut normal to B through the main solar wind beam (centered) and shows the non-gyrotropic character of the plasma (B points out of the page.) This 3-spin (12 s) integration overlaps the upstream edge of the structure and is time-aliased, but still reveals an arc of ions in the upper left quadrant that appears to be a gyro-reflected component similar to those seen at the foot of collisionless shocks. Using the upstream sunward edge normal direction and the speed from timing analysis, we obtained a normal incidence Alfvén Mach number M A =3.3. The upstream exterior B was about ( 1, 1, 3) nt, which was > 80 from the normal direction, indicating the solar wind flow into this structure was supercritical with a strongly perpendicular geometry. We thus have good reason to expect that a shock-like boundary might form, and the presence of reflected ions suggests an associated cross-boundary E-field as well. A feature not apparent in the left panel of Figure 7 is revealed in a distribution (right panel) obtained interior to a DH observed at 05:16 UT on 3 February 2002, identified as a SLAMS event (11). In this plot, the slice is through a plane defined by V sw B with B pointing to the right, and again the main solar wind beam is centered. Comparisons with upstream distributions obtained a few spins later show that the main solar wind beam component is reduced in phase space density by more than an order of magnitude, consistent with the large reduction in Fig. 8 Ion phase space distributions of density holes. Left panel: a B-perpendicular cut through the main solar wind beam in the upstream edge of the 08:55 UT, 2 March 2005 event. The field points out of the page and the horizontal axis points nearly anti-sunward. Most particles have relatively modest energies, and the velocity scale includes only ±1250 km s 1 to highlight their behavior. Right panel: a V sw B slice through a distribution obtained within the interior of the 05:16 UT, 3 February 2002 density hole. B points to the right (Reprinted from Ref. [1])

11 GKParks,et al.: A Review of Density Holes Upstream of Earth s Bow Shock 703 overall density n. Also present is a significant suprathermal population that appears much like an upstream intermediate distribution. The temperature in DHs rose to > 10 7 K, and can be accounted for by the presence of this energetic component and the reduction of the colder solar wind population. Note that while the solar wind beam is observed to slow down and change direction within and at the edges of the density hole, the measured beam velocity differs somewhat from the computed moments, indicating relative motion of the energetic component. The latter ions may originate in backstreaming particles observed exterior to the structure, retaining some of their initial field-aligned momentum. 3 Discussion We have reviewed observations of DHs including currents and low frequency electromagnetic waves observed in the upstream region of the bow shock. The currents in the upstream edge of DHs can be large and they are associated with waves that are left hand elliptically polarized in the plasma frame. These waves are most likely to be ion cyclotron waves propagating sunward in the solar wind frame but convected earthward by the solar wind and the spacecraft observes them as a right handed polarized wave. These waves appear to be growing nonlinearly into solitary pulses forming a shock-like structure. Density holes have features similar to SLAMS. Although the exact relationship between DHs and SLAMS is yet to be established, a few cases of the SLAMS events that have been published have DHs. SLAMS, like DHs, have nearly vanishing magnetic field at the center and the field is steepened 2 4times the ambient solar wind values at the edge. The mechanisms responsible for the nonlinear growth are still unknown. Several characteristics of DHs are similar to those of HFAs, HDCs and SLAMS (Table 1). They all have significant bulk flow deflections, are filled with heated plasma, have enhanced edge densities, and one or both edges have compressed magnetic field. Density holes have durations similar to SLAMS but much shorter than HFAs and HDCs, and correspondingly smaller dimensions. The order of magnitude density depletions have not been reported previously for HFAs. HDCs or SLAMS. Density holes are common while HFAs and HDCs occur rarely. Only eight HFA events had been reported in ISEE observations spanning more than two years, and the combined number including HDCs observed by ISEE, AMPTE-IRM and AMPTE-UKS was about 30. We have shown an example of SLAMS event that Table 1 Upstream transient structures property DH HFA/HDC SLAMS FC duration about 18 s few minutes s > few minutes scale length R R e R e >R e δn/n 0.7 < 0.2 no report < 0.2 Bulk V /(km s 1 ) V x 0 V x 100 no report V sw T (hole)/mk no report > 103 overshoot yes yes yes yes shock-like yes yes yes no occurrence frequent rare frequent rare E-field yes no report yes no report upstream ions yes yes no report no report electron hole yes no report no report no report waves yes no report yes no report currents yes no report inferred no report

12 704 Chin. J. Space Sci. Ξ ΛΠΠ 2011, 31(6) has a density hole with heated back-streaming plasmas associated with it. In addition, the DHs near the shock tend to have elevated edges that account for more particles than those depleted within the hole. The heating plus the net gain in particles is consistent with what is observed across the shock transition, and reflects as expected net density increases within structures moving relative to the solar wind flow. Density holes are observed only when backstreaming energetic particles are present although the latter can exist without the former. These intervals usually have B perturbations. However, the details involved in creating the density holes are not known. Density holes have been observed for solar wind velocities 400 to 800 km s 1, n 1 10 cm 3, and T K. While B and n are correlated, deep density holes with large edge enhancements can have shallow magnetic holes and weak B overshoots, while shallow density holes with weak edge enhancements can have deep magnetic holes and large overshoots. The similarities of the larger HFAs and HDCs with DHs suggest HFAs may be several smaller density holes piling up. But this suggestion needs to be further studied. We will continue to study the DHs, HFAs and SLAMS. We will study the behavior of electromagnetic and electrostatic wave data and particle observations in order to establish the roles of wave particle interactions. Improved statistics on upstream plasma and beam parameters (density, speed, temperature, Mach number, plasma and shock geometry) may reveal factors controlling their origin and development. References [1] Parks G K, Lee E, Lin N, et al. Larmor radius size density holes discovered in the solar wind upstream of Earth s bow shock [J]. Phys. Plasma., 2006, 13: [2] Parks G K, Lee E, Lin N, et al. Density holes in the upstream solar wind [C]//AIP Conference Proceedings, 2007, 932:9-15 [3] Schwartz S, Chaloner C, Christiansen P, et al. Anactive current sheet in the solar wind [J]. Nature, 1985, 318: [4] Thomsen M, Gosling J, Fuselier S, et al. Hot, diamagnetic cavities upstream from the Earth s bow shock [J]. J. Geophys. Res., 1986, 91: [5] Sibeck D, Phan T, Lin R, et al. Wind observations of foreshock cavities: A case study [J]. J. Geophys. Res., 2002, 107: [6] Parks G K, Lee E, Lin N, et al. Current density and wave polarization observed in density holes upstream of earth s bow shock [C]//AIP Conference Proceedings, 2008, 1039: [7] Lin N, Lee E, Mozer F, et al. Nonlinear low-frequency wave aspect of foreshock density holes [J]. Anales Gephys., 2008, 26: [8] Lee E, Parks G K, Wilber M, et al. Nonlinear development of shock like structure in the solar wind [J]. Phys. Rev. Lett., 2009, 103: [9] Lucek E A, et al. Cluster magnetic field observations at a quasi-parallel bow shock [J]. Ann. Geophys., 2002, 20: [10] Lucek E A, et al. Cluster observations of structures at quasi-parallel bow shocks [J]. Ann. Geophys., 2004, 22: [11] Lucek E, Horbury T S, et al. Cluster observations of hot flow anomalies [J]. J. Geophys. Res., 2004, 109:A06207 [12] Cornilleau-Wehrlin N, et al. First results obtained by the Cluster STAFF experiment [J]. Ann. Geophys., 2003, 21: [13] Decreau P, et al. Early results from the Whisper instrument on Cluster: an overview [J]. Ann. Geophys., 2001, 19:1241