Institute of Geophysics and Planetary Physics and Department of Earth and Space Sciences, University of California, Los Angeles, CA 90024, U.S.A.
Originally published in:
Adv. Space Res., Vol. 8, No. 9-10, pp. (9)147-(9)156, 1988.
Two wave MHD populations are seen at collisionless shocks: precursor waves standing in the shock ramp which form an integral part of the shock and upstream waves which are usually attempting to propagate upstream but are carried back toward the shock by the solar wind flow. Both types of waves are observed at interplanetary shocks and planetary bow shocks. In general, interplanetary shocks are weaker than planetary shocks. The difficulty in studying interplanetary shocks is that the shock normal is hard to determine accurately but multiple spacecraft measurements are of some assistance in this regard. To study strong shocks we can use planetary bow shocks whose normal directions are constrained by geometry. Two types of multispacecraft studies have been used, closely spaced ones such as with ISEE-1 and -2 and more distantly separated ones such as with ISEE and UKS. These studies suggest that our paradigm for the evolution of large amplitude or "fully developed" turbulence needs some revision.
Waves appear at and in front of collisionless shocks for essentially two reasons. First, waves are an integral part of the shock structure and can provide some of the necessary dissipation. We call these precursor waves /1,2/. Secondly, waves appear far upstream of collisionless shocks and are overtaken by the shock (or swept into the shock) even though they are attempting to propagate upstream away from the shock. We call these waves upstream waves. They are generated by particles either reflected from the shock front /3,4/ or leaking from behind the shock /5,6/.
The structure of collisionless shocks depends on several factors such as the strength of the shocks, the beta of the plasma and the magnetic field direction. The strength of the shock is measured in terms of its Mach number, the ratio of the velocity of the shock relative to the plasma ahead of the shock normalized by the velocity of small amplitude waves of the same type as the shock. In this review we will concern ourselves only with fast magnetosonic shocks.The beta of the plasma is the ratio of the thermal pressure to the magnetic pressure in the upstream plasma. In the solar wind both electrons and ions generally make significant contributions to beta. The orientation of the magnetic field that is important is the orientation of the upstream magnetic field relative to the shock normal. In many situations such as when studying interplanetary shocks the direction of the normal is difficult to determine. Here observations from multiple spacecraft can be useful. Shocks whose normals lie about 45^{o} or more from the field direction are called quasi- perpendicular. Those within 45^{o} are called quasi-parallel.
To study upstream waves we need to examine both interplanetary shocks and planetary bow shocks. Interplanetary shocks provide us with examples of weak shocks generally of Mach number 3 and below. Planetary bow shocks are usually stronger than Mach 3.
The purpose of this review is to examine some of the properties of upstream waves that we have learned through the use of multiple satellites. We will first examine the simplest shocks, the "weak" interplanetary shocks as examined using principally ISEE-1, -2 and -3. Since determining the orientation of these shocks is critical to being able to understand their properties we will also review techniques for determining their orientation. Next we examine the waves upstream of the terrestrial bow shock as determined from widely-spaced observations with ISEE and UKS and then review the observations of ISEE-1 and -2 at much closer distances.
In order to determine the direction of the normal it is usual to determine two directions that are perpendicular to the shock normal and obtain the normal from the normalized vector cross product of these two directions. Two directions which are perpendicular to the shock normal are the direction of change in field across the shock since the normal component is conserved and the direction of the cross product of the upstream and downstream fields because these two field directions and the shock normal are coplanar. The normal derived in this way is usually called the coplanarity normal. It is not accurate when the two directions whose cross product is taken are parallel which occurs for both parallel and perpendicular shocks.
Another vector that is perpendicular to the shock normal is the cross product of the upstream field (or the downstream field) with the vector change in velocity across the shock. The difference in field strength can also be used in place of the upstream or downstream fields. This normal is called the mixed mode normal /7/. Variants of these techniques have recently been proposed both simple /8/ and complex /9/.
If one has the unusual luxury of four simultaneous, closely-spaced measurements as will be available from the Cluster mission one can use the relative timing and spacing of the measurements to derive the normal, by solving the matrix equation:
where X_{i0} and t_{i0} are the separation vectors and time lags between satellite 'i' and satellite '0'. If there are more than 4 spacecraft, all observations can be incorporated. The solution will then be over-constrained and will be a 'best-fit' solution. The other constraints discussed above can also be added in an analogous way to the handling of the fifth spacecraft. This process was followed for a group of 16 interplanetary shocks /2/ observed with multiple spacecraft to give the first set of well characterized interplanetary shocks. We next examine some of the results determined from this study.
Fig. 1. Whistler precursors for 4 interplanetary shocks as observed by ISEE-2. The component of the magnetic field shown is along the projection of the interplanetary magnetic field on the shock plane and contains the jump in the magnetic field across the shock /after 2/. |
Fig. 2. Turbulence seen upstream of 5 interplanetary shocks. The component shown is perpendicular to the upstream magnetic field and the shock normal /after 2/. |
As discussed above there are two types of waves observed at interplanetary shocks, precursor waves on the shock ramp and upstream waves well ahead of the shock. Figure 1 shows whistler precursors observed for four shocks in our interplanetary shock study group. Figure 2 shows five examples of upstream turbulence. The 16 shocks yielded 32 independent shock observations at which the shock precursor could be studied. (For these purposes ISEE-1 and -2 observations were too close to be considered independent). Of these 32 observations precursor waves were seen on only 9 shocks as illustrated in Figure 3 /10/. These shocks were quasi-parallel and weak. When the Mach number increased above about 1.6 the precursors disappeared. However, it appears that the reason for the disappearance was that the precursor wave frequency in the spacecraft frame became too rapid to resolve.
Fig. 3. The parameters of the interplanetary shocks studied showing which ones had precursors. Plotted is the magnetosonic Mach number of each shock versus the cosine of the angle between the shock normal and the upstream magnetic fields. The solid circles indicate which shocks had precursor waves /after 2/. |
The availability of data from 2 very closely-spaced spacecraft, such as from ISEE-1 and -2 which were usually stationed only 100's of km apart, allows the standing wave nature of the precursor to be probed. Only three cases in Figure 3 were amenable to such study events 1, 3 and 4. The top three traces of Figure 1 correspond to cases 1, 3 and 4, respectively. Knowing the solar wind velocity, the separation between spacecraft and the time delay between spacecraft, we can determine the wavelength and velocity of the precursor waves. For these three cases we find that within experimental error the waves are phase standing with respect to the shock front and that the waves are propagating nearly along the magnetic field direction. Figure 4 shows the geometry of case 1. The wavelength of the wave is 871 km and it is propagating at 91 km s-1 along the wave normal which corresponds to a velocity of 109 km s^{-1} along the shock normal which within experimental error equals the 120 km s^{-1} velocity of the shock relative to the upstream plasma.
Fig. 4. The geometry of the precursor waves seen on November 20, 1979 by ISEE-1 and - 2. The plane shown is the LN shock plane which contains the upstream magnetic field. Vectors B, k, N, and S are the directions of the upstream magnetic field, the precursor wave normal, the shock wave normal and the separation vector between the 2 spacecraft, respectively. The times listed are the times for one wavefront to cross both spacecraft (0.65s) and for the second wavefront to be crossed by a single spacecraft (4.8 sec). |
These precursor waves thus sit fixed with respect to the shock. The group velocity, however, exceeds the phase velocity and thus energy is flowing from the shock into the upstream plasma via these waves and they participate in the overall dissipation process at the shock. A similar set of precursors has been studied for laminar terrestrial bow shocks /11/. These shocks were all quasi- perpendicular rather than quasi-parallel. Two precursor waves were found, one propagating parallel to the magnetic field as for the quasi-parallel shocks and also one propagating along the shock normal. Only the latter wave appeared to be phase standing.
Fig. 5. Power spectra of the waves shown in Fig. 2 /after 2/. |
Further upstream of interplanetary shocks are observed irregular waves with rather featureless spectra. Figure 2 showed examples of some of these waves. Figure 5 shows power spectra for these waves for the component of the magnetic field perpendicular to both the shock normal and the upstream magnetic field. Analysis of these waves show that they are propagating in the direction of the magnetic field and have both left and right polarization. Sometimes they are very nearly linearly polarized. They are mainly transverse waves with little compressional power. This observation is in contrast to the waves upstream of the bow shock which we will find are quite compressional.
Fig. 6. The logarithm of the wave amplitude of upstream turbulence as a function of the cosine of the angle between the interplanetary field and the shock normal. The amplitude is the square root of the trace of the spectral matrix integrated over the frequency band from 0.03 to 0.3 Hz. The straight line is the least squares fit to the points ignoring the lowest two points /after 2/. |
The amplitude of waves upstream of interplanetary shocks depends strongly on the direction of the magnetic field relative to the shock normal. This is shown in Figure 6 which plots the wave amplitude for all shocks in our study with magnetosonic Mach numbers greater than 1.5. Waves are weakest in front of the perpendicular shock and strongest in front of the parallel shock.
Fig. 7. Ion distributions in the terrestrial foreshock. The narrow peak in each distribution is the solar wind and the foreshock ions appear as a narrow 'reflected' beam on the left, as a more kidney-bean shaped structure in the middle and as a more nearly isotropic 'diffuse' distribution on the right /after ll/. |
While the curvature of interplanetary shocks may be quite small on the scale of the upstream waves in front of them, the curvature of the bow shock is quite large so that the magnetic field geometry and the properties of backstreaming beams are quite position dependent. Figure 7 shows the current paradigm for the distribution function of upstream ions /11/. Electrons and ions stream along the first tangent field line but ions do not move as swiftly as the electrons and so the ions are swept anti-sunward perpendicular to the IMF by the inter-planetary electric field causing the ion foreshock to be displaced downstream from the electron foreshock. The ion beam properties are thought to be a function of position as illustrated. Near the fore- shock boundary are narrow "reflected beams". In front of the quasi-parallel shock are broad "diffuse beams" and in the middle are intermediate beams. Figure 8 shows the current paradigm for the nature of MHD waves in the foreshock region. Near the ion foreshock possibly caused by ion beams or perhaps by the electrons are approxmately 1 Hz waves. Further back from the ion foreshock transverse low frequency waves are seen. Well behind the foreshock are steepened waves we call shocklets and their precursors we call discrete wave packets. We will see that this paradigm needs some revision.
Fig. 8. Low-frequency waves in the terrestrial foreshock /after ll/. |
Fig. 9. Magnetic power spectra, time series and foreshock geometry on October 23, 1984 as observed by UKS and ISEE-2 when UKS was just ahead of the expected ion foreshock and ISEE-2 was well behind it /after 12/. |
We can test this picture using the availability of widely-spaced measurements from ISEE and UKS in 1984. Figure 9 shows the magnetic field measured by UKS and ISEE-2 on October 23, 1984 when ISEE-2 was behind the foreshock and UKS just ahead of it /12/. From the spectrum we see that the power at ISEE is enhanced over almost the entire frequency band measured from 4 x 10^{-2} Hz to about 4 x 10^{-l} Hz. There is a dominant period for the upstream waves but most of the ower does not reside in the spectral peak.
Fig. 10. Magnetic power spectra, time series, and foreshock geometry on October 30, 1984 by UKS and ISEE-2 when ISEE-2 was just behind the expected ion foreshock boundary /after 12/. |
The next example shown in Figure 10 shows a situation with ISEE-2 near the foreshock and UKS well downstream of the foreshock on October 30, 1984. At ISEE-2 there is an enhancement of the highest frequencies only, above about 1 x 10^{-1} Hz. At UKS the waves at both high and low frequencies are present.
Fig. 11. Magnetic power spectra, time series and foreshock geometry on November 5, 1984 when ISEE-2 and URS had a very similar fore-shock geometry in the B-V plane but were separated by 8.3 earth radii perpendicular to this plane /after 12/. |
Fig. 12. Magnetic power spectra, time series and foreshock geometry on November 26, 1984 when ISEE and UKS are nearly upstream of one another and the magnetic field is almost aligned with the flow. ISEE-2 and UKS are separated by only 0.83 earth radii in the direction perpendicular to the B-V plane /after 12/. |
The next example shown in Figure 11 shows ISEE and UKS at a very similar location in the foreshock but separated by 8.3 Re perpendicular to the plane of the page. The spectra are very similar. In contrast Figure 12 shows a case when ISEE and UKS are close but the magnetic field is nearly aligned with the flow. Close to the shock there are large pulsations but a short distance upstream the magnetic field is much more quiet. This example is the first clue that the orientation of the magnetic field to the flow is perhaps more important in determining wave properties than originally thought.
Fig. 13. The ellipticity of upstream waves near the peak of the power spectrum as a function of the fractional amplitude of the waves /after 12/. |
An interesting result of this study is shown in Figure 13 which presents the ellipticity of waves as a function of amplitude. The strongest waves are right-handed and the weakest left-handed. We interpret this to imply that at low beam strengths the resonant ioncyclotron instability occurs in which a backstreaming ion beam overtakes and resonates with a right-handed whistler-mode wave traveling upstream. The right- handed wave is swept downstream and seen as a left-handed wave in the spacecraft frame. When the beam intensity is very strong the beam is firehose unstable generating a right-handed wave traveling toward the shock. These waves are stronger than those generated by the ion cyclotron instability and do not suffer polarization reversal because the solar wind sweeps them in the direction of their motion. Thus, beam strength, or perhaps relative streaming velocity, is an important factor in generating the waves.
When the two measurement sites are closely spaced we can use the relative delay from one spacecraft to the next, plus the solar wind velocity to determine which direction and how fast the waves are travelling in the plasma frame. This was first used on the whistler mode wave packets which are often found on the steepened edge of shocklets in the foreshock region /13/. In such a study the wave normal is calculated as the direction of minimum variance. This is particularly easy for these discrete wave packets because the waves are quite circularly polarized. The time delays and the spacecraft separation along the wave normal are then calculated and the wave velocity in the spacecraft frame calculated. The solar wind velocity is then projected along the wave normal and the velocity in the solar wind frame can be calculated. The wave period in the spacecraft frame allows the wavelength to be calculated as well as the rest frame wave frequency. This study showed that as had been speculated earlier the waves were right-hand polarized in the plasma frame, were trying to propagate sunward against the solar wind flow but were being blown backward toward the shock by the supersonic solar wind.
Fig. 14. Examples of the waveforms for intermediate or transverse waves in the lower panel for both ISEE-1 (thick line) and ISEE-2 (thin line). The top panel shows the cross-correlation as a function of time delay. The Bi component is the component along the direction of maximum variance /after 14/. |
This technique can also be used on the lower frequency less circularly polarized waves /14/. Figure 14 shows an example of transverse waves together with the cross- correlation coefficient between the two spacecraft. Not illustrated here are the I Hz waves most probably associated with upstreaming electrons /11/. The dual spacecraft measurements of ISEE-1 and -2 show that these waves are in general right-handed waves being swept across the spacecraft. However, it is not clear from the studies to date whether there is some Alfvenic or left-handed turbulence attempting to propagate toward the sun and being swept earthward or whether the observed right-handed waves are due to waves generated in a righthanded sense and propagating toward the earth. More work needs to be done here. We note that when the waves propagate at a large angle to the solar wind flow no polarization reversal occurs. Thus left-handed waves propagating (slightly) upstream will remain lefthanded. Figure 15 shows the frequency and wave number of ULF waves observed in the foreshock /11/. The so-called beam associated waves are the approximately 1 Hz waves often seen in conjunction with backstreaming electrons. These waves seem to be all part of the same right-handed dispersion relation.
Fig. 15. Normalized rest frame frequencies of ULF waves studied in the terrestrial foreshock using the dual satellite pair ISEE-1 and -2 to measure wavelengths and remove Doppler shifts /after ll/. |
Fig. 16. Projections into the ecliptic plane of spacecraft locations indicated by dots and ecliptic plane components of the interplanetary magnetic field during intermediate or transverse waves (top panel) and during shocklets or fully developed turbulence (bottom panel). All field directions and positions have been rotated about the X axis so that all point more along the usual spiral direction than orthogonal to it /after 14/. |
The angle between the magnetic field and the solar wind flow seems to be very important for upstream wave generation. Figure 16 shows the direction of the magnetic field during intermediate or transverse wave events in the top panel and shocklets or fully developed turbulence in the bottom panel /14/. Shocklets appear when the magnetic field is most nearly aligned with the flow. This is in fact contrary to the usual paradigm for upstream waves which has shocklets occurring deep in the foreshock region near the flanks.
Fig. 17. High resolution magnetic field measurements in spacecraft coordinates obtained by ISEE-1 (heavy trace) and ISEE-2 (light trace) on September 1, 1979. The ISEE-1 trace has been offset from the ISEE-2 trace by 2-seconds to compensate for the solar wind travel time from ISEE-2 to ISEE-1. ISEE-2 upstream of ISEE-1 sees all features first. Note the large compression of the magnetic field at ISEE-1 coincident with smaller shocklet at ISEE-2 at 1714:15 UT. |
Fig. 18. High resolution magnetic field measurements in spacecraft coordinates obtained by ISEE-1 (heavy trace) and ISEE-2 (light trace) on September 1, 1979. ISEE-1 measurements have been offset from ISEE-2 measurements by 1.5 seconds to account for the solar wind travel time. ISEE-2 sees all features first including shock-like field increases despite the fact that ISEE-1 is closer to the nominal shock. However, the perturbations at ISEE-1 are generally larger than at ISEE-2. |
Shocklets also have an important role in determining the structure of the quasi-parallel shock. Figures 17 and 18 show measurements obtained by ISEE-1 and -2 when the field was nearly parallel to the solar wind flow on average and ISEE-2 was nearly precisely upstream from ISEE-1 and both were near the nose. The intervals clearly show shocklets which reach ISEE-2 the upstream spacecraft first and then ISEE-1. The shocklets are being swept downstream. Interestingly, not only are shocklets being swept downstream but when ISEE-2 encounters what we would consider to be the actual shock front, it is also being swept downstream. However, the shock effects are always equal to or greater in strength at the downstream vehicle. This leads us to the conclusion that the shock begins upstream as a shocklet and as it is swept downstream it grows until it reaches the main shock front at which time it is stationary and strong and merges in with the rest of the shock.
We note that upstream of the shock the shocklets appear to have large (-1000's of km) scale sizes but behind the shock the scale size for turbulence in the magnetic field is much less than 1000 km. This observation is consistent with the work of Greenstadt and coworkers on the pulsation shock /15/.
Two populations of waves exist upstream of interplanetary shocks: precursors which are part of the shock structure and stand in the solar wind flow and turbulent upstream waves that are generated well upstream of the shock by particles backstreaming from the shock into the upstream plasma. Precursor waves are seen both upstream of interplanetary shocks and planetary shocks. However, the relationship between these waves is not yet clear.
Upstream waves are also found in front of both interplanetary and planetary shocks. In front of interplanetary shocks the amplitude of these waves increases as the angle between the magnetic field and the shock normal decreases. In front of planetary bow shocks the properties of the waves appear to evolve as the angle between the magnetic field and flow changes. When the field is nearly aligned with the flow the steepened wave packets known as shocklets appear. These waves are swept back into the quasi-parallel shock and interact with it and may be responsible for the pulsating nature of such shocks. In fact, for the quasi-parallel shock the upstream waves may be thought of as an extension of the shock itself.
This work was supported by the National Aeronautics and Space Administration under research contract NAS5-28448.
1. C. T. Russell, Planetary bow shocks, in Collisionless Shocks in the Heliosphere: Reviews of Current Research, (ed. by B. Tsurutani and R. G. Stone), 109-130, American Geophysical Union, Washington, D. C. (1985).
2. C. T. Russell, E. J. Smith, B. T. Tsurutani, J. T. Gosling and S. J. Bame, Multiple spacecraft observations of interplanetary shocks: Characteristics of the upstream turbulence, in Solar Wind Five NASA Conf. Publ., 2280, edited by M. Neugebauer, pp. 385-400, Washington, D. C. (1983).
3. B. U. 0. Sonnerup, Acceleration of particles reflected at a shock front, J. Geophys. Res., 74, 1301 (1969).
4. G. Paschmann, N. Sckopke, J. R. Asbridge, S. J. Bame and J. T. Gosling, Energization of solar wind ions by reflection from the Earth's bow shock, J. Geophys. Res., 85, 4689 (1980).
5. D. A. Tidman and N. A. Krall, Shock Waves in Collisionless Plasmas, 175pp, Wiley-Interscience, New York (1971).
6. J. P. Edmiston, C. F. Kennel and D. Eichler, Escape of heated ions upstream of quasiparallel shocks, Geophys. Res. Lett., 9, 531 (1982).
7.B. Abraham-Shrauner and S. H. Yun, Interplanetary shocks seen by Ames plasma probe on Pioneer 6 and 7, J. Geophys. Res., 81, 2097-2102 (1976).
8.E. J. Smith and M. E. Burton, Shock analysis: Three useful new relations, J. Geophys. Res., 93, 2730-2734 (1988).
9. A. Vinas and J. D. Scudder, Fast and optimal solution to the "Rankine- Hugoniot" problem, J. Geophys. Res., 91, 39-58 (1986).
10. M. M. Mellott and E. W. Greenstadt, ISEE-1 and -2 observations of laminar bow shocks: Whistler precursors and shock structure, J. Geophys. Res., 89, 2151-2161 (1984).
11. C. T. Russell and M. M. Hoppe, Upstream waves and particles, Space Sci. Rev., 34, 155-172 (1983).
12. C. T. Russell, J. G. Luhmann, R. C. Elphic, D. J. Southwood, M. F. Smith and A. D. Johnstone, Upstream waves sinultaneously observed by ISEE and UKS, J. Geophys. Res., 92, 7354-7362 (1987).
13. M. M. Hoppe and C. T. Russell, Whistler mode wave packets in the earth's foreshock region, Nature, 287, 417-420 (1980).
14. M. M. Hoppe and C. T. Russell, Plasma rest frame frequencies and polarizations of low-frequency upstream waves: ISEE-1 and -2 observations, J. Geophys. Res., 88, 2021-2028 (1983).
15. E. W. Greenstadt, M. M. Hoppe and C. T. Russell, Large-amplitude magnetic variations in quasi-parallel shocks: Correlation lengths measured by ISEE-1 and - 2, Geophys. Res. Lett., 9, 781-784 (1982).