Solar Wind Sources of Magnetospheric Ultra-Low-Frequency Waves,
edited by M. J. Engebretson, K. Takahashi, and M. Scholer, Geophysical Monograph 81, p.81-98, AGU, Washington DC, 1994
© Copyright 1994 by the American Geophysical Union
G. Le and C. T. Russell
Institute of Geophysics and Planetary Physics, University of California at Los Angeles
Abstract. The Earth's foreshock is the region upstream from the bow shock where the interplanetary magnetic field intersects the bow shock. It is characterized by backstreaming electrons and ions, as well as associated electrostatic and electromagnetic waves over a wide frequency range. One class of upstream electromagnetic waves, large-amplitude compressional ULF waves in the ion foreshock region, has long been postulated as a major source of magnetospheric ULF waves. In this paper, we discuss recent observations of properties of ULF waves. First the general morphology of the foreshock at different IMF orientations and the different types of ULF waves are reviewed. Then observations of critical wave properties of importance to the Earth's magnetosphere, such as frequency, bandwidth, coherence length and spatial evolution in the foreshock, are discussed.
The generation of upstream waves, especially the ultra-low-frequency (ULF) waves, in planetary foreshocks is both an active and an important topic of space plasma physics investigations. These waves are ubiquitous, having been observed in front of the bow shock of all of the planets and in front of comets Halley and Giacobini-Zinner. They modify both the solar wind and the turbulence spectrum convected to the Earth's magnetosphere, altering the nature of the shock jump, the pressure fluctuations on the magnetopause and ultimately the spectrum of waves in the magnetosphere. Moreover, these foreshock waves lead to the acceleration of some particles to very high energies by means of wave-particle interactions and provide us with a testbed in which to study processes thought to be responsible for the acceleration of cosmic ray particles. Finally, the instabilities which drive these waves and the processes associated with the waves provide an excellent test of the basic theories of waves and instabilities in plasma physics.
Upstream waves have been found in front of the bow shocks of all of the planets visited to date. Terrestrial upstream waves have been studied extensively by many authors and much of the early work has been well documented in the 1 June, 1981 special issue of J. Geophys. Res.. Upstream waves have also been investigated at Mercury by Fairfield and Behannon  and Orlowski et al. ; at Venus by Hoppe and Russell  and Orlowski and Russell ; at Mars by Russell et al. [1990a]; at Jupiter by Smith et al.  and Goldstein et al. ; at Uranus by Russell et al. [1990b], Smith et al.  and Zhang et al. [1991a]; and at Neptune by Zhang et al. [1991b]. Based on the early work, as reviewed by Russell and Hoppe , we have developed a picture of the morphology of the upstream wave region. In that picture, wave and particle phenomena upstream of the shock arise from kinetic effects occurring in the shock transition region (see Sonnerup ; Paschmann et al. ; Armstrong et al. ; Scholer ; Gosling and Robson  and references therein]. These effects include: reflection, shock drift acceleration, and heating of post-shock plasma. The effectiveness of the above mechanisms as well as the characteristics of the backstreaming populations (at the shock) are strongly dependent on the direction of the shock normal relative to the plasma flow and the interplanetary magnetic field (IMF) orientation.
For a stationary IMF we are able to define regions (in the shock frame) with characteristic backstreaming populations. Figure 1 shows a schematic of the foreshock geometry. First we can divide the unshocked plasma into two regions, one magnetically unconnected where no IMF lines connect to or cross the shock and one so-called magnetic foreshock region where all IMF lines are connected to the bow shock. The IMF lines tangent to the bow shock form the boundary between those two regions. The superposition of the upstream motion along the magnetic field line and the convection associated with the interplanetary electric field, results in the formation of electron and ion foreshocks within the region of unshocked solar wind connected to the bow shock. Because of the finite velocity of these backstreaming particles and the convection associated with the interplanetary electric field, the electron and proton foreshocks do not fill the entire magnetically connected region, nor are they fully coincident. Moreover, the properties of the waves and plasma are position dependent within the foreshock.
The electron foreshock boundary is defined by the fastest electrons accelerated along the tangent magnetic field lines. Since the convection velocity is small compared to the typical bulk velocity of escaping electrons (which can be many times of their thermal velocity) the electron foreshock boundary is only slightly displaced relative to the IMF tangent lines. A variety of associated waves has been found in the electron and ion foreshocks [Hoppe et al., 1981]. In the electron foreshock in addition to electrostatic (Langmuir) mode [Gurnett, 1985] an electromagnetic mode at ULF frequencies has also been identified: the so-called one-Hertz whistlers [Russell et al., 1971; Fairfield, 1974; Hoppe et al., 1981, 1982]. One example of one-Hz waves is shown in Figure 2(a).
The ion foreshock boundary in contrast appears to be defined by position-dependent acceleration processes that do not allow protons to move upstream (in the shock frame) until some distance downstream of the electron foreshock [Thomsen, 1985]. It is located much deeper in the magnetic foreshock, beginning close to the region called the quasi-parallel part of the bow shock. Studies of the ISEE 1 and 2 data have revealed that the ion foreshock in turn can be divided into three regions characterized by beam-like, intermediate and/or gyrating, and diffusive ion distributions depending on the local Bn, the angle between the IMF and the bow shock normal [Gosling et al., 1978; Paschmann et al., 1979, 1981]. ULF waves in the ion foreshock have been observed at frequencies below the proton gyrofrequency [Greenstadt et al., 1968; Russell et al., 1971; Hoppe et al., 1981; Le et al., 1992]. These observations revealed several types of ULF waves in the ion foreshock: sinusoidal waves, shocklets and discrete wave packets, and three-second waves. Figure 2(b) shows an example of low-frequency waves which are nearly sinusoidal. They typically have amplitudes of a few nT, are primarily transverse, and exhibit predominantly left-handed polarization in the spacecraft frame. ULF waves in the ion foreshock are most frequently observed in large-amplitude, highly compressional forms. They steepen into small shocklike waveform and have been called shocklets. Discrete wave packets, which have higher frequency, are often associated with the steepening edges of the shocklets, as shown in Figure 2(c). Another type of foreshock waves has frequency near three seconds and is always right-handed and nearly circularly polarized in the spacecraft frame. One example of three-second waves is shown in Figure 2(d).
One type of upstream ULF waves, the large-amplitude compressional waves, is particularly important to the Earth's magnetosphere. These waves must propagate obliquely with respect to the magnetic field since they are highly compressional. Since they were first observed by Greenstadt et al.  and Fairfield , several generation mechanisms related to electromagnetic ion/ion instabilities from linear theory have been proposed to explain the source of these waves [Gary, 1991]. The backstreaming ions have a density of ~ 1% of the solar wind density. Their bulk velocities range from the order of the Alfven velocity (~ 50 km/sec at 1 AU) for diffuse ions to the order of solar wind speed for intermediate, gyrating and beam-like ions [cf. Thomsen, 1985 for a review]. Under these conditions, the most obvious instability is the anomalous Doppler-shifted ion/ion resonant instability, in which the right-handed polarized waves (fast magnetosonic mode) resonate with the backstreaming ion beam [Barnes, 1970; Gary, 1978; Sentman et al., 1981]. This instability generates right-handed waves in the upstream region which propagate in the same direction as the beams (upstream) in the solar wind frame. Since they are convected downstream by the solar wind flow, they will be Doppler-shifted to left-handed waves in the spacecraft frame. When the backstreaming ions are sufficiently fast (Vb > 10 VA) and dense (Nb ~ 10% of solar wind density), besides the above resonant instability, the non-resonant firehose-like instability will excite ULF waves in the right-handed magnetosonic branch which propagate in the direction opposite to the ion beam, i.e., downstream toward the bow shock. These waves are also right-handed in the spacecraft reference frame. Gary et al.  have shown that the non-resonant instability has larger growth rate than the resonant instability if Vb/VA and Nb/Nsw are sufficiently large. In another extreme case when the backstreaming ions are extremely hot with thermal speeds greater than their streaming velocity (Vth > Vb), the left-handed Alfven/ion resonant mode is also unstable. The growth rate for the two resonant instabilities can be comparable [Sentman et al., 1981; Gary, 1985]. This left-handed resonant instability generates waves propagating upstream from the bow shock and will be Doppler-shifted to right-handed waves in the spacecraft frame as they are convected downstream by the solar wind. Thus, according to linear theory, the upstream ULF waves will be more often left-handed in the spacecraft frame under typical conditions and also can be right-handed in the spacecraft frame under extreme conditions of fast, dense beams or hot beams. However, when the waves are right-handed in the spacecraft frame, they can be intrinsically either right-handed magnetosonic mode or left-handed Alfven mode in the solar wind frame. Observations have shown that the left-handed ULF waves (in the spacecraft frame) are indeed the dominant mode in the foreshock region [Hoppe et al., 1981; Hoppe and Russell, 1983]. They have frequencies of ~ 0.1 ( is the local ion cyclotron frequency) and wavelengths of the order of 1 Re. For the right-handed waves (in the spacecraft frame), their intrinsic modes have not been identified.
One important fact of obliquely propagating magnetosonic waves is that there is a first-order density perturbation associated with the magnetic perturbation. In the MHD limit, the density and magnetic perturbation for a magnetosonic waves propagating at an angle are related by [Siscoe, 1983]:
Observations show that the ULF waves generated in the region upstream of the bow shock are convected downstream by the solar wind flow because their group velocity is much smaller than the solar wind flow speed [Hoppe et al., 1981; Hoppe and Russell, 1983]. The downstream convection of compressional ULF waves can have a profound impact on the Earth's magnetosphere. The upstream fluctuations associated with these waves are carried downstream along solar wind streamlines into the magnetosheath. The streamlines in the magnetosheath which approach most closely to the magnetopause pass through the subsolar region of the bow shock. During intervals when the IMF cone angle is small ( < 45 deg), these waves fill the subsolar upstream region, as illustrated in Figure 3 (adapted from Russell et al. ). Under the condition of small IMF cone angle, the solar wind carries the upstream ULF fluctuations to the magnetopause boundary. The magnetopause responds to these pressure fluctuations and transfers wave energy into the dayside magnetosphere. The waves on the magnetopause can excite field line resonances in the magnetosphere, as described by Southwood  and Chen and Hasegawa . Although the process by which the energy enters into the magnetosphere is still not well understood, it has been generally agreed that upstream waves are a major source for dayside Pc 3 and 4 magnetic pulsations in the Earth's magnetosphere (cf. Odera,  for a review).
There is much observational evidence to support this idea. Many studies of the magnetic pulsations in the dayside magnetosphere have indicated that their occurrence is controlled by the IMF cone angle [eg. Troitskaya et al., 1971; Russell et al., 1983; Yumoto et al., 1985; Engebretson et al., 1986, 1987; Yumoto, 1988]. Luhmann et al.  found that magnetosheath turbulence is sensitive to the IMF cone angle and is enhanced in the subsolar region during periods of small cone angles. By using multiple spacecraft observations, Engebretson et al.  found on a case-by-case basis that small cone angles are well correlated with large turbulence in the magnetosheath and the simultaneous excitation of Pc 3 and 4 pulsations in the dayside outer magnetosphere. Other evidence is that the frequencies of both upstream and magnetospheric ULF waves have a similar dependence on the IMF strength, as shown in Figure 4 (adapted from Russell and Hoppe ). In Figure 4, the solid circles and solid lines are from observations of upstream ULF waves. Dashed, dash-dot, and dotted lines are from observations of ground-based Pc 3-4 waves as given by Gul'yel'mi et al. , Gul'yel'mi , and Gul'yel'mi and Bol'shakova , respectively.
In this paper, we discuss recent advances in the understanding of the foreshock ULF morphology and wave properties, as well as remaining problems. We present our studies of critical upstream ULF wave properties of importance to the magnetospheric ULF waves.
The schematic of the foreshock in Figure 1 shows that its geometry depends mainly on the IMF cone angle. The first requirement in studying the general morphology of the foreshock is to quantitatively determine the upstream boundary of the ULF foreshock for different IMF cone angles. This boundary represents the motion of backstreaming ions in the foreshock region since the ULF waves are the consequences of instabilities generated by these backstreaming ions. If the ions originating in the bow shock have velocity components upstream away from the bow shock, the ions will leave the bow shock with the equation of motion to the first order as:
The pioneering work of locating the ULF foreshock boundary can be found in Greenstadt et al.  and Greenstadt , in which this boundary was determined based on detailed case studies. Later this boundary was determined statistically by Greenstadt and Baum , in which they used the ISEE 1 magnetometer data to find actual crossings of the ULF foreshock boundary. Their study clearly showed the IMF cone angle control of the ULF foreshock boundary. They displayed the locations of ISEE 1 at ULF foreshock boundary crossings in the V-B plane for moderate cone angles of 40-50 deg and for small cone angles of 20-30 deg. They found that the patterns of the scatter plots of the crossings defined a boundary for each of the two subsets, but the slopes of the two boundaries are different. The ULF wave foreshock boundary determined in this study also inferred the backstreaming ion velocity of 1.6Vsw.
In our recent study of determining the ULF foreshock boundary, we identified many foreshock boundary crossings in the upstream region when the IMF cone angle was nearly constant for extended time periods and thus the foreshock boundary was steady in space [Le and Russell, 1992a]. The work consisted of two steps, first to determine of the ULF foreshock on the bow shock, and second to determine the slope of the ULF foreshock boundary. In the first step, we examined ISEE bow shock crossings at various positions to determine the source point on the bow shock which separated disturbed (with ULF waves) and undisturbed (without ULF waves) upstream magnetic field. The statistical study of the bow shock crossings showed that the ULF foreshock started at ~ 50 deg . In the second step, we found that the ULF foreshock boundary was less sensitive to small changes of the IMF direction at larger cone angle. The ULF foreshock boundary was well defined in the V-B plane for cone angles > 40 deg. Figure 5 shows the ISEE positions in the V-B plane for five ULF foreshock boundary crossings identified at 50 +/- 5 deg IMF cone angles. The bow shock was scaled by the solar wind dynamic pressure and Mach number, and then, normalized to the same size for each crossing. The spacecraft positions form a clear boundary in the V-B plane. From this boundary, we infer that the backstreaming ions had a velocity of 1.3Vsw along the IMF and a net guiding center velocity of 1.5Vsw in the Earth's frame. On average, the ULF foreshock boundary corresponds to the trajectory of backstreaming ions with a streaming velocity of ~ 1.4Vsw along the IMF in the Earth's frame and a source point at ~ 50 deg when the IMF cone angle is moderate ( > 40 deg). When the IMF cone angle is small (20 < < 30 deg), the ULF foreshock boundary is not well defined.
The foreshock geometry for nearly radial IMF is different from that at moderate and large cone angles. The solar wind convection electric field is very small and the backstreaming ions move along the magnetic field in a nearly flow-aligned IMF condition. It is the most favorable condition for the generation of upstream waves since the particles can go upstream more easily from the bow shock. In this case, most of the day side upstream region is inside the ion foreshock region, although it is still not clear if there is a distinct boundary which separates the ULF foreshock from the undisturbed solar wind. We emphasize that the large-amplitude ULF waves for nearly radial IMF can be convected close to the Earth's magnetosphere under this IMF configuration. Observations also show that it is the most favorable geometry for the occurrence of magnetospheric ULF waves [Russell et al., 1983].
ULF waves observed for nearly radial IMF are typically in the form of steepened shocklets which sometimes have discrete wave packets at the steepening edges. Figure 6 shows examples of ULF waves for nearly radial IMF where each panel consists of 10 minutes of high resolution data within an interval of cone angle < 10 deg which lasted at least one hour. These waves are similar in form to those observed well downstream from the foreshock boundary at moderate and large cone angle. As shown in Figure 7, the ULF wave region extends upstream with a scale length of ~ 23 Re for this geometry. The top panel of Figure 7 shows the normalized ULF wave spectral amplitude as a function of distance from the bow shock along the IMF, which is roughly the same as distance from bow shock along Sun-Earth line for nearly radial IMF. The bottom panel of Figure 7 shows the spatial coverage of these data in the plane which contains the spacecraft and the Sun-Earth line (there is no meaningful V-B plane for nearly radial IMF). Although there is a tendency of decreasing wave amplitude with increasing distance from the bow shock, the decrease of the amplitude is very slow with a scale of ~ 23 Re. Ipavich et al.  found that the upstream 30 keV proton intensity varied exponentially with the radial distance from the bow shock and had a scale length of 7 +/- 2 Re. The IMF direction during the time of peak intensity was within ~ 15 deg of the radial direction for 90% of their events. The two scale lengths are qualitatively consistent to the extent that the upstream particles may have different scale lengths at different energies and the correlation between the beam density and wave amplitude is not exactly linear.
Right-handed polarized waves (in the spacecraft frame) are more frequently observed at nearly radial IMF than at moderate and large cone angles. Under conditions typical of the ion foreshock, the most unstable mode is the right-handed polarized magnetosonic wave due to the resonant instability [Barnes, 1970; Gary, 1978]. These waves will be Doppler-shifted to left-handed polarizations in the spacecraft frame. Left-handed ULF waves (in the spacecraft frame) are indeed the dominant mode observed in the upstream region [Hoppe et al., 1981; Hoppe and Russell, 1983]. However both left-handed and right-handed modes are observed with equal probability for nearly radial IMF and their ellipticity seems to be correlated with the wave amplitude. Figure 8 shows the wave ellipticity versus the normalized wave amplitude. Negative ellipticity indicates left-handed polarization in the spacecraft frame and positive ellipticity corresponds to right-handed polarization in the spacecraft frame. There is a positive correlation between the ellipticity and the wave amplitude with a correlation coefficient of 0.57. A similar correlation between ellipticity and wave amplitude has been reported by Russell et al.  for moderate IMF cone angles. The right-handed waves are stronger than the left-handed waves (in the spacecraft frame).
The correlation between polarization and amplitude suggests that different mechanisms generate waves with different amplitudes. As we discussed in the introduction, according to linear theory, the left-handed polarized waves in the spacecraft frame are favored under typical backstreaming ion condition (Nb ~ Nsw, Vb < 10 VA) via the ion/ion right-handed resonant instability. When the backstreaming ions are very dense and fast (Nb ~ 10 Nsw, Vb > 10 VA) or very hot (Vth > Vb), right-handed polarized waves are observed in the spacecraft frame, that are generated either by the ion/ion nonresonant instability or the ion/ion left-handed resonant instability. The facts 1) that right-handed waves in the spacecraft frame are more often observed and 2) that these waves are stronger for nearly radial IMF suggest that this upstream configuration is a favorable condition for faster and denser ion beams (nonresonant instability) or hotter ion beams (resonant instability). Energetic particle observations have shown that the upstream ions are of diffuse type and have small bulk velocity under radial IMF [Ipavich et al., 1981]. It seems that the nonresonant instability can be dismissed. However, this is just speculation since little has been done to identify the intrinsic mode for the waves which have right-handed polarization in the spacecraft frame.
Compressional ULF waves are an intrinsic feature of quasi-parallel shocks. Their existence in front of the bow shock modifies both the solar wind and the turbulence spectrum convected downstream to the magnetopause. In this section, we review recent observations of the wave properties, especially those of importance to the Earth's magnetosphere, including the magnitude of the pressure fluctuations associated with the waves, the coherence length, and the bandwidth.
Pressure Fluctuations Associated with the Waves
MHD theory predicts that ULF waves cause significant density fluctuations, and thus dynamic pressure fluctuations as well in the unshocked solar wind. Although the fluctuating magnetic field of these waves is the most readily measured aspect of the waves, it is the associated pressure fluctuations which can cause the magnetopause to oscillate in response to the waves. Observations show that density and dynamic pressure fluctuations have been significantly enhanced in the ULF foreshock [Paschmann et al., 1979; Bame et al., 1980; Le, 1991]. The density and dynamic pressure fluctuations associated with the ULF waves are ~ 20% of the average background value. In comparison, fluctuations in the undisturbed solar wind at these frequencies are only ~ 5% of the background average.
Figure 9 shows an example of how the ULF foreshock can modify the solar wind itself. It contains 5 hours data from the ISEE 1 magnetometer and Cross-Fan Solar Wind Experiment, including magnetic field, solar wind ion density, bulk velocity and dynamic pressure in the upstream region. As already noted, the foreshock geometry is very sensitive to the IMF direction. When the IMF changes its direction, the spacecraft may suddenly find itself located inside the foreshock region. In Figure 9, the IMF changes its direction as well as magnitude near 0825UT. Following the onset of the ULF waves, enhanced fluctuations in solar wind ion density, ion bulk velocity and dynamic pressure () are present that are clearly associated with the ULF waves. The bulk of the solar wind is also slowed down (by ~ 38 km/sec in this example) in the foreshock region. Such decelerations are common within the foreshock and are caused by the interaction of the solar wind with backstreaming ions which slow down the incoming solar wind by transferring the momentum flux [Bame et al., 1980; Bonifazi et al., 1980; see also the review by Thomsen, 1985]. Thus varying IMF direction can modify the geometry of the foreshock and alter the distribution of pressure on the magnetopause. In this way IMF directional fluctuations can cause compressions of the magnetosphere and may explain some pressure pulses seen there.
Coherence Length of ULF Waves
ULF waves are carried downstream towards the bow shock and magnetopause along solar wind streamlines and modulate the structures of both the bow shock and the magnetopause if they have sufficient amplitude and scale size. The coherence length of the ULF waves has been investigated using simultaneous observations from the dual ISEE 1 and 2 spacecraft [Le and Russell, 1990a]. In that study, we examined the correlation between these simultaneous observations for different separations of the two spacecraft. Figure 10 shows the cross-correlation coefficients as a function of separation distance perpendicular to the solar wind flow where each line corresponds to a different separation parallel to the solar wind flow. The cross-correlation coefficients decrease as the separation perpendicular to the flow increases. However the cross-correlation coefficients are similar despite different separations along the solar wind flow. The coherence length is on the order of an Earth radius transverse to the solar wind flow, a scale similar to the wavelength. This result is consistent with that estimated from the bandwidth of the power spectra. The limited coherence length in the direction transverse to the flow is mainly due to the solar wind convection effect. In the direction along the solar wind flow, the coherence length is at least several Earth radii. Thus, ULF waves are large-scale coherent structures, and should induce similar scale-size coherent oscillations in the bow shock and magnetopause, and in turn in the magnetosphere itself.
Bandwidth of ULF Waves
The ULF foreshock modifies the turbulence spectrum convected downstream to the magnetopause. Thus, in order to predict the magnetospheric effects of these waves it is of interest to determine the frequency range of enhanced power of magnetic fluctuations in the foreshock region. We have compared the ULF wave power spectrum with the background solar wind spectrum and found that the enhanced wave power has limited bandwidth [Le and Russell, 1990b]. In that study, simultaneous observations from two largely separated spacecraft that are located on either side of the ULF foreshock were examined. The data indicate that the foreshock ULF wave power spectrum has a clear low-frequency cutoff, below which the power spectra are similar in the undisturbed solar wind and in the foreshock. This low-frequency cutoff occurs above ~ 5 mHz. Figure 11 shows one example in which ISEE 1 is inside the foreshock and IMP-8 is in the undisturbed solar wind. From this figure it is apparent that ISEE 1 observes enhanced power at frequencies higher than 7 mHz. There is no significant power enhancement or damping below 7 mHz. This result does not support the suggestion that upstream shock-related pressure oscillations drive magnetopause surface waves and magnetospheric oscillations with periods of ~ 200-600 seconds [Sibeck et al., 1989].
In short the solar wind flow is significantly perturbed in the foreshock region. The perturbation is unsteady due to frequent variations of IMF orientations, perhaps explaining some pressure pulses in the magnetosphere. Wave processes in the foreshock are however band limited and thus can be directly responsible for only a fraction of the waves in the magnetosphere.
It is well known that the properties of upstream ULF waves are position dependent within the foreshock and the waveform varies from nearly sinusoidal to highly steepened as the increases [Hoppe et al., 1981; Russell et al., 1987]. The region populated by the intermediate ions is the forward boundary of the ULF wave foreshock. The waves there are fairly monochromatic and typically last many cycles. They are left-handed polarized in the spacecraft frame. Further downstream from the ULF foreshock boundary in the region populated by diffuse ions, the ULF waves often appear in the form of the steepened shocklets. They exhibit both left-handed and right-handed polarization in the spacecraft frame, and the left-handed and right-handed waves are similar in form, frequency, and wavelength.
Russell et al.  used simultaneous observations from two well separated spacecraft to show that the properties of the ULF waves depend on the their location in the foreshock region. The properties include waveform, amplitude, polarization and power spectrum. Early observations of backstreaming ions also revealed that the properties of backstreaming ions varied with respect to point of origin along the bow shock [Paschmann et al., 1981; Ipavich et al., 1981; Thomsen, 1985].
Since solar wind conditions, especially the IMF direction, are highly variable, early observations of the ULF waves were made under various solar wind conditions and did not reveal the structure of the region behind the foreshock boundary, or spatial variation of ULF waves within this region. When the IMF cone angle changes, the foreshock geometry changes accordingly. Thus it is essential to study the evolution of ULF waves under similar conditions of IMF and bow shock strength, which is another controlling factor. During a long interval of nearly constant solar wind conditions, the motion of backstreaming ions can reach a nearly steady state, as do the waves generated by them. In this case, the changes of wave properties are primarily related to the geometry in the foreshock, i.e., the different depth from the foreshock boundary and distance from the bow shock.
We have examined cases in which the IMF cone angle was nearly constant over an extended time period and ISEE 1 and 2 spacecraft were traveling a large distance within the foreshock [Le and Russell, 1992b]. Figure 12 shows one of the cases examined. The top panel shows the foreshock geometry and ISEE trajectory in the V-B plane. The lower panels show the magnetic field data during this interval. Large-amplitude ULF waves with periods near 40 seconds were present throughout this interval and these ULF waves exhibited predominantly a steepened form (shocklets). The amplitudes of the waves increase gradually as the spacecraft travels to large depth from the foreshock boundary and closer to the bow shock. Following the increase of wave amplitude, the waves vary from elliptically polarized to more linearly polarized. The wave power spectra (Figure 13) shows that the peak of power spectrum became broader towards both lower and higher frequencies while the peak frequency stayed nearly the same as the bow shock is approached. In addition the spectra had a limited bandwidth and a clear low frequency cutoff throughout the interval.
From the bottom panels of Figure 12, it is evident that the discrete wave packets found at the steepening edges of the lower-frequency waves, or shocklets, become more intense and develop more cycles associated with the increasing intensity of the shocklets. Figure 14 shows the wave duration in cycles and the peak-to-peak amplitude of the discrete wave packets (the largest cycle) as a function of distance from the bow shock along a solar wind streamline. The scale length given by the exponential fit is 1.9 Re for the wave duration and 2.8 Re for the wave amplitude. Based on the facts that these wave packets were convected downstream with the solar wind and that their phase velocity was much less than the solar wind velocity, the estimated growth rate for the discrete wave packets was ~ 0.035 for the particular geometry and solar wind conditions of this interval.
Despite intensive studies of the upstream waves over the past two decades, there are still many remaining problems which need further investigation. First of all, the present theoretical models for wave generation are not successful in predicting all the properties of observed ULF waves. From the linear theories the maximum growth rates for all three instabilities always occur for parallel propagation (k//B) over all parameter space, although the growth rate for oblique propagation waves can be significant [Montgomery et al., 1976; Gary, 1991]. However, the large-amplitude ULF waves are observed as highly compressional and steepening. Thus they propagate obliquely to the magnetic field. Hada et al.  proposed a refraction mechanism to account for the observed obliquely propagating waves. In their mechanism the parallel propagating right-handed and left-handed waves are excited by the backstreaming ions in the upstream region. These waves are refracted to oblique propagation as they are convected downstream by the solar wind due to the non-uniform index of refraction caused by the spatial variation of the backstreaming ions. As a result, the waves become compressional and steepen into the shocklets. In this mechanism, geometry is very important and shocklets should not be observed in the nearly radial IMF geometry because the effect of refraction is minimum for parallel propagating waves. However, we have shown evidence that both shocklets and discrete wave packets are observed when the IMF is nearly radial (Figure 6). The linear theories also predict that ULF waves exhibit both left-handed and right-handed polarization in the spacecraft frame. Observations have shown that polarization and wave amplitude are correlated. For the right-handed waves in the spacecraft frame, their intrinsic mode can be either right-handed or left-handed. Thus the identification of the intrinsic mode and conditions for which they occur will help us to understand their generation mechanisms.
Solving these problems also relies largely on a better understanding of upstream particles. Combined data sets from many instruments (both field and plasma) will provide detailed information on the physics of the underlying processes. This is still a weak point in the foreshock study despite of the fact that existing ISEE data sets with high time resolution are available for over a decade. Previous work used either primarily magnetic field data or primarily plasma data to study the morphology of the foreshock region and the spatial variation of waves properties. For example, previous attempts to determine the ULF foreshock boundary were mainly based on field observations and properties of the backstreaming ions were inferred from these observations. However, we do not know the behavior of the backstreaming ions near the foreshock boundary identified from field data. Do the backstreaming ions move upstream at a velocity inferred from magnetic field observations? What is the difference, if any, between the boundaries of backstreaming ions and ULF waves and how does the backstreaming ion distribution function change across the ULF wave boundary? We note that the growth of waves in the unstable backstreaming ions is not instantaneous and in fact waves grow in a parcel of plasma as it convects towards the bow shock. The onset of waves may not simply reflect the variation in backstreaming ion properties. We studied the spatial variation of ULF waves under steady solar wind conditions, but we do not know the spatial variation of the backstreaming ions under these conditions. We do not know how the change of wave properties depends on the change of streaming velocity, density, and distribution function of the backstreaming ions.
The importance of upstream phenomena is not limited to the Earth's upstream region. The comparative study of a variety of foreshocks is very important for understanding the underlying physical processes. Waves have been observed upstream from bow shocks of Mercury, Venus, Mars, Jupiter, Saturn and Uranus. Their similarities and differences are not currently very well understood. First, the solar wind Mach number increases with increasing heliocentric distance. The significant increases in the strength of planetary shocks with increasing distance from the sun may induce considerable changes in the relative efficiencies of the various processes such as leakage and reflection that generate the backstreaming ions. Another important factor that may influence the wave and particle signatures observed in the upstream region is the varying time of connection of the field lines to the planetary shock and the varying radius of curvature of the bow shock relative to characteristic scale lengths, (eg., the proton gyroradius and ion inertial length). For the above reasons it is essential to study a variety of foreshocks and to determine the dependence of the microphysics of the foreshock phenomena on varying boundary conditions.
Acknowledgments. This work was supported by the National Aeronautics and Space Administration under research grant NAGW-2067.
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Figure 1. Schematic of foreshock geometry.
Figure 2. Observations of different types of ULF waves in the Earth's foreshock region. (a) One-Hz waves; (b) Sinusoidal waves; (c) Shocklets and discrete wave packets; (d) Three-second waves.
Figure 3. Foreshock geometry for various IMF cone angles. (Adapted from Russell et al. )
Figure 4. Relationship between the wave frequency and the IMF strength for upstream ULF waves (solid circles and solid line) and for ground-based Pc 3--4 waves (dashed, dash-dotted and dotted lines). (Adapted from Russell and Hoppe )
Figure 5. The spacecraft positions at the ULF foreshock boundary crossings for 50 +/- 5 cone angles are plotted in the V-B plane. The bow shock is scaled by the solar wind dynamic pressure and the magnetosonic Mach number, and then, normalized to the same size for each crossings. (Adapted from Le and Russell [1992a])
Figure 6. Examples of ULF waves for nearly radial IMF. (Adapted from Le )
Figure 7. The upper panel is the normalized ULF wave spectral amplitude as a function of the distance from the bow shock for nearly radial IMF. The solid line is a linear fit to the data and the dotted line is an exponential fit. The lower panel is the data coverage in the plane containing the spacecraft and the Earth-Sun line. (Adapted from Le )
Figure 8. The wave ellipticity versus the normalized amplitude for nearly radial IMF. The correlation coefficient is 0.57, which is significant at almost 100% level. (Adapted from Le )
Figure 9. The time series of magnetic field components, magnetic field strength, ion density, bulk velocity, and dynamic pressure from 0630 to 1130 UT on December 6, 1977. The IMF changes its direction near 0825 UT. There are enhanced fluctuations in ion density and dynamic pressure associated with the ULF waves. (Adapted rom Le )
Figure 10. The cross-correlation coefficients as a function of spacecraft separation perpendicular to the solar wind flow for each 0.1 RE separation parallel to the solar wind flow. (Adapted from Le and Russell [1990a])
Figure 11. The magnetic field time series, power spectra, and foreshock geometry from simultaneous observations of ISEE 1 and IMP-8 on November 1, 1981. (Adapted from Le and Russell [1990b])
Figure 12. The upper panel shows the ISEE trajectory in the V-B plane from 1400 to 2100 UT on September 4, 1983 when the IMF cone angle was nearly constant over an extended time period. The lower panels are magnetic field observations. (Adapted from Le and Russell [1992b])
Figure 13. The power spectra of 32--minute magnetic field data with starting times indicated in the figure on September 4, 1983. Each spectrum is the total power summed over three components, and is plotted by shifting one decade upward from the previous one. (Adapted from Le and Russell [1992b])
Figure 14. The spatial variation of the duration and the peak-to-peak amplitude of discrete wave packets on September 4, 1983. (Adapted from Le and Russell [1992b])