R. E. Denton
Physics and Astronomy Department, Dartmouth College, Hanover, New Hampshire
The magnetosheath is the region of interaction between the solar wind plasma and the Earth's magnetic field. The magnetosheath exhibits large inhomogeneity due to the stress of the solar wind dynamic pressure. Since this pressure is pushing nearly continuously on the magnetopause, the structure of the magnetosheath is in the roughest approximation time stationary. This makes it an ideal region for studying the effects of such inhomogeneity. Inhomogeneity has a crucial effect on the generation and propagation of ultra low frequency (ULF) waves. In turn, ULF waves have significant feedback on the macroscopic plasma.
The purpose of this paper is to review what we know about magnetosheath ULF waves. Our presentation is not totally comprehensive, but rather expresses our view of major issues in ULF wave studies as presented at the 1999 IAGA meeting [Denton, 1999]. Two recent reviews of a similar nature have been given by Schwartz et al.  and Song and Russell . An outline of our paper is as follows: In section 2, we discuss methods of mode identification. In section 3, we discuss some important properties of the magnetosheath which relate to the generation and propagation of ULF waves. In section 4, we discuss ULF waves in the magnetosheath when the upstream conditions are quasi-parallel, and in section 5, we discuss ULF waves in the magnetosheath when the upstream conditions are quasi-perpendicular. Finally, we have a summary in section 6.
Since the mode properties can be strongly modified by the kinetic nature of a plasma, it is best to use kinetic theory when making comparisons with observation for the purpose of mode identification [Denton et al., 1995a; Krauss-Varban et al., 1994]. At frequency f Fcp, where Fcp is the proton cyclotron frequency, up to four distinct modes can propagate in a relatively isotropic plasma: the three waves corresponding to the normal modes of MHD theory and the mirror mode. The mirror mode has zero phase velocity in a homogeneous plasma (though it can acquire a finite phase velocity when there is inhomogeneity [Johnson and Cheng, 1997]). The three modes with finite phase velocity are the magnetosonic/fast/whistler mode, the Alfvén/ion-cyclotron mode, and the ion-acoustic/slow/sound mode. The first term listed (i.e., "magnetosonic,'' "Alfvén,'' and "ion acoustic'') represents our preferred notation [Denton et al., 1995a] and the other terms have been used by other authors [for example, Krauss-Varban et al., 1994]. The terms "fast'' and "slow'' here are appropriate descriptions of the phase speed of the respective waves at low parallel beta b||p 8 pp||p/B02 in CGS units, where p||p is the proton pressure parallel to B, and B is the magnetic field. At high beta, what we call the "slow'' mode sometimes has the largest phase velocity [Denton et al., 1995a; Krauss-Varban et al., 1994]. Our use of the terms "slow'' and "fast'' here agrees with the nomenclature of Krauss-Varban et al. , which was designed to use the same term to refer to the wave with similar fluctuations (transport ratios), regardless of phase speed. On the other hand, Gary and Winske  and Lacombe et al. [1992, 1995] have retained the use of "slow'' and "fast'' to refer explicitly to phase speed. (It is easy to see why we would prefer not to use these terms.)
The linear modes are defined on dispersion surfaces as a function of the wave vector k. One can distinguish whether the modes are approximately quasi-parallel ( k approximately parallel to B ) or quasi-perp ( k approximately perpendicular to B ) [Denton et al., 1995a]. Both the magnetosonic/fast/whistler and Alfvén/ion-cyclotron modes are Alfvén-like for parallel propagation. In that case, the magnetosonic/fast/whistler mode is right hand polarized while the Alfvén/ion-cyclotron mode is left hand polarized. Both of these waves become more and more linearly polarized as k turns away from B. As discussed by Schwartz et al. , identification of wave modes is hampered by a number of factors. Most of the theory is based on linear homogeneous theory, whereas the observed modes are observed in an inhomogeneous medium and are often significantly nonlinear. Perhaps the greatest problem is that there can be a mixture of different modes. Denton et al.  found that in a number of cases, the identification of the wave mode was not certain.
Methods of mode identification use transport ratios, phase angle relations, and polarization information (see review by Schwartz et al.  for an extensive list of transport ratios with references). Gary and Winske  introduced the term "transport ratios'' to denote dimensionless ratios of the squares of fluctuating field and plasma quantities. One of the most commonly used transport ratios is the magnetic compressibility, the ratio of parallel to total magnetic fluctuations
where the angular bracket indicates an average (often calculated using the power spectrum at a particular frequency). This is expected to be large for the quasi-perpendicular magnetosonic (fast) mode and mirror mode, but small for the Alfvén mode. A quantity functionally equivalent to CB was used by Anderson and Fuselier  to distinguish Alfvén (ion cyclotron) waves from the mirror mode.
Two other useful transport ratios are the compressibility Cp
where dnp and np0 are the fluctuating and equilibrium (slow time scale) proton density, respectively, and B0 is the equilibrium magnetic field, and the Alfvén ratio RAp
where d vp and VA are the fluctuating proton velocity and equilibrium Alfvén speed B0 / 4 pmp np in CGS units, where mp is the proton mass. Figure 1 is a dramatic demonstration of the effectiveness of these transport ratios for mode identification. Figure 1 shows the transport ratios Cp and RAp obtained for waves observed near the magnetopause (crosses). These waves were thought to be either the mirror mode or the ion acoustic (slow) mode based on the relative phase in dnp and dB|| (not shown). Based on the transport ratios in Figure 1, it is clear that these waves should be identified as the mirror mode.
A second important determining factor for wave identification is based on the phase difference between various fluctuating quantities. For the magnetosheath data of Figure 1, dnp and dB|| were shown to be 180 o out of phase. Such a phase relation exists for the quasi-perpendicular mirror and ion acoustic (slow) modes, but dnp and dB|| should be in phase for the quasi-perpendicular magnetosonic (fast) mode.
Polarization information can be very useful for determining the wave mode. In the Earth's ion foreshock (the region just outside the bowshock where ions reflected off the bowshock are observed), Alfvén-like waves are often observed. These may be left or right hand polarized (the latter is the quasi-parallel magnetosonic mode in our nomenclature), where the handedness indicates the direction of rotation of fluctuating quantities about the equilibrium magnetic field. In order to determine polarization, it is most beneficial to have multi-spacecraft observations. Russell et al.  used simultaneous observations by the ISEE 2 and AMPTE/UKS spacecraft to the determine polarization of foreshock waves. Blanco-Cano and Schwartz , using single spacecraft data, measured a number of transport ratios to identify these waves; unfortunately these transport ratios did not enable them to distinguish the waves. Their identification was based mainly on a determination of polarization which depended on minimum variance analysis (to determine the direction of k ) and properties of the cross-helicity which enabled them to eliminate the sign ambiguity ( ) [Blanco-Cano and Schwartz, 1997]. Using the minimum variance direction for magnetic fluctuations is a common method to infer the direction of k (based on Faraday's law), but can be misleading if there is a mixture of modes (see [Denton et al., 1996]). Use of multi-spacecraft data is therefore highly desirable for determining wave polarization.
The particular properties of the magnetosheath play a large role in the generation, propagation, and modification of magnetosheath ULF waves. Whether the upstream conditions are "quasi-perpendicular'' or "quasi-parallel'' depends on the direction of the interplanetary magnetic field (IMF) when the solar wind plasma reaches the bowshock. (The use of "quasi-perpendicular'' and "quasi-parallel'' here should not be confused with their use to describe the relative orientation of k and B in "quasi-perpendicular'' and "quasi-parallel'' wave propagation.) By upstream conditions, we mean then that if we were to follow the convecting plasma backward in time, the IMF would be "quasi-perpendicular'' or "quasi-parallel'' at the time that the plasma crossed the bowshock. The plasma convects through the magnetosheath to its present position maintaining some general characteristics resulting from that orientation. The plasma with quasi-perpendicular upstream conditions (when the IMF is roughly aligned with the bowshock, or perpendicular to the bowshock normal) exhibits the most regular structure. The value of B0 typically increases from the bowshock to the magnetopause. Sometimes, particularly for low magnetic shear [Phan et al., 1994], the density np0 decreases near the magnetopause; this region of decrease is sometimes called the plasma depletion layer. Field lines in the magnetosheath drape around the magnetopause leading to field line curvature.
When the upstream conditions are quasi-perpendicular, the plasma
typically exhibits temperature anisotropy
When the upstream conditions are quasi-parallel,
(so that the IMF is roughly
aligned with the bowshock normal), the plasma typically has large
plasma beta and is very turbulent.
The high beta conditions arise from the fact that
the shocked solar wind plasma is compressed, leading to
large increase in plasma pressure, but the quasi-parallel magnetic
field is not (this is precisely true only when the IMF is
exactly quasi-parallel). When the upstream conditions are
the magnetic field exhibits large amplitude fluctuations which
accompany the fluctuations in thermal plasma pressure, the latter
of which lead to the dominant force density in the high beta
The direction of the magnetic field is then ill-defined, as are
the separate values of
Typically, broad band turbulent waves are observed when the upstream conditions are quasi-parallel, with power peaking in the Pc3-4 range (period from 10-150 s) [Engebretson et al., 1991, and references therein; see their Plate 4 for an example of broad band turbulence]. These waves are thought to be caused, at least in part, by the ion foreshock waves mentioned in Section 2 (see papers in [Engebretson et al., 1994], particularly [Krauss-Varban, 1994]). They are also generally thought to be the source of the toroidal Alfvén wave (with magnetic fluctuation in the azimuthal direction) harmonics observed in the magnetosphere [Anderson, 1993, and references therein], though the path the wave power takes to get to the magnetosphere is still a matter of debate (Mark Engebretson, private communication, 1999).
A statistical survey by Anderson and Fuselier  showed that Alfvén/ion cyclotron and mirror waves could be observed when the upstream conditions are quasi-parallel, though the most likely situation was to observe broadband waves. Even when there is no explicit evidence for the existence of ion cyclotron waves amid the broad band turbulence, there is indirect evidence that ion cyclotron waves are present (anisotropy-beta correlation; see Section 5.2) [Fuselier et al., 1994].
Little has been done theoretically to describe the high beta waves. A recent paper applies an approach based on magnetohydrodynamics (MHD) [Song et al., 1998], though it may be difficult to justify the use of MHD and the assumption that the equilibrium magnetic field is steady.
When the upstream conditions are quasi-perpendicular, the magnetosheath exhibits more regularity in macroscopic structure and in the ULF waves observed. We consider the plasma as it convects from the bowshock to the magnetopause and the characteristic waves of each region. One should keep in mind that not all the waves described here may be present at any one time.
Since the bowshock is itself a manifestation of the quasi-perpendicular magnetosonic (fast) mode reflecting off the magnetopause, it is not too surprising that Lacombe et al.  identified this wave within the bowshock structure. They also identified both quasi-parallel and quasi-perpendicular Alfvén waves; these may be due to fluctuations at the bowshock.
A large proton temperature anisotropy
demonstrates the growth of ion cyclotron waves in the post bowshock
region. There are two bowshock crossings in Figure 4;
we concentrate on the first one at 0655 UT.
Following this crossing,
Sckopke et al.  observed a "double humped,'' or bifurcated structure in wave power as a function of frequency. Brinca et al.  interpreted the bifurcated spectra as being due to the combination of a core and beam component of protons, but it is possible that the bifurcated spectra result from the presence of He2+ as discussed in Section 5.4.3.
At the same time that the ion cyclotron waves grow large in
(0657 UT), there is a transfer of energy
and that this curve is not far above the marginal stability
condition for ion cyclotron waves
[Anderson et al., 1994;
Denton et al., 1994c;
Gary et al., 1993, 1994].
While the value of
While it seems clear that the transverse waves observed in
are caused by the ion cyclotron
instability, Alfvén waves are sometimes
observed in the outer
Lacombe et al.  identified both Alfvén (ion cyclotron) waves and mirror mode waves (see Section 5.3) in the outer magnetosheath.
Due to the increase in
T||p caused by the ion cyclotron waves
in the outer magnetosheath (Figure 4), the value of
T||p increases. As a result of the increased
b||p and decreased value
Alfvén (ion cyclotron) waves are also sometimes observed in the middle magnetosheath. Some evidence suggests that the ion cyclotron waves may be the dominant cause of temperature anisotropy regulation even when the mirror mode is the dominant wave [McKean et al., 1992, 1994]; however, except for very low b||p, the mirror mode marginal stability criterion is similar to (4) indicating that the mirror mode could also possibly play a role.
Lacombe et al.  claims to identify the He2+ cutoff mode (see also [Denton et al., 1994a] for pictures of the dispersion surface) which exists due to the presence of the heavy ion.
Figure 7 shows a density enhancement observed by the ISEE 1 spacecraft in front of the magnetopause on September 5, 1978. Based on such observations, Song et al. [1990b, 1992a] hypothesized that the enhancement was due to an ion acoustic (slow) mode shock in front of the magnetopause. Song et al. [1992b, 1994] went on to identify ion acoustic (slow) waves in the region of the magnetosheath close to the magnetopause. The observed waves had magnetic fluctuations out of phase with density fluctuations. The waves could thus be the mirror mode (Figure 6) or the ion acoustic mode. The identification as the ion acoustic mode was based on the fact that the observed waves had a finite upstream phase velocity in the rest frame of the plasma. (Recall that the mirror mode has zero frequency in a homogeneous plasma.)
Whether or not density enhancements like that in Figure 7 are related to a standing ion acoustic (slow mode) front or shock [Song et al., 1990b, 1992a] is still a matter of debate; however, the weight of evidence is that the waves observed by Song et al.  are the mirror mode, not the ion acoustic mode. The linear ion acoustic mode is strongly damped, as is well known, but it was argued that nonlinear effects might allow the mode to persist [Song et al., 1994]. Denton et al. [1995a] showed that the values of the transport ratios strongly indicated the mirror mode (see Figure 1). The problem remained, however, to explain the finite phase velocity of the waves. Omidi and Winske  showed using a one dimensional hybrid code simulation of the magnetosheath that mirror modes convecting from the bowshock pile up at the magnetopause; they are unable to penetrate to the low beta magnetosphere (see Figure 8).
Subsequently, Johnson and Cheng  solved for the mirror eigenmodes in a model magnetosheath and found that due to the inhomogeneity, the mirror mode acquires a finite phase velocity near the magnetopause. (Their theory did not include coupling to the ion acoustic mode; it is possible that there is coupling between the two modes.) Finally, Lin et al.  examined additional phase relations between field components that indicated that the waves of Song et al.  were really the mirror mode. (The linear theory of Lin et al.  is at best of doubtful relevance to the mirror mode phase relations they use, but the mirror mode phase relations are verified using hybrid code simulations. We are forced, however, to accept their linear fluid theory for the ion acoustic mode phase relations.)
In addition to identifying the ordinary quasi-perpendicular mirror mode ( k approximately B ), Denton et al. [1995a] identified the quasi-parallel mirror mode ( k approximately B ). Denton et al.  reexamined the identification of these waves and concluded that "the quasi-parallel mirror mode may be observed in the inner magnetosheath, but that identification is not certain.'' The technique of Lin et al.  was used to examine event number 7 of Denton et al.  for a third time (J. K. Chao, private communication, 1998). There was mixed evidence for the mirror mode and the Alfvén wave. It appears that both of these modes may be present simultaneously. This makes the identification of the quasi-parallel mirror mode very uncertain.
As mentioned earlier,
the magnetic field generally increases toward the magnetopause.
In addition, a region of depleted density
np, called the depletion
layer, may be present adjacent to the magnetopause.
If there is such a decrease in
np, there is usually also
a decrease in the parallel temperature
T||p. All these
factors lead to a decrease in
b||p as the magnetopause
and the decrease in
T||p leads to an increase in
Figure 9 shows magnetic power spectra measured by the AMPTE/CCE spacecraft and magnetic field and density data measured by CCE and AMPTE/IRM after a magnetopause crossing at ~1300 UT observed by CCE. At that time, the AMPTE/IRM spacecraft was upstream in the solar wind. The IRM density and magnetic field data shows that the decreased density and increased magnetic field measured by CCE from 1300-1340 UT cannot be explained by upstream temporal variations; CCE is therefore in the plasma depletion layer at that time. To the left in Figure 9, the value of b||p is lower (due to lower proton density and T||p (not shown) and larger B0 ). At the lowest values of b||p, there is a gap in the transverse power spectra at about 0.7 Hz; the gap is close to the He 2+ (alpha particle) gyrofrequency Fa. Moving from lower to higher b||p, Figure 9 reveals a progression from bifurcated transverse spectra above and below Fa (B), to continuous transverse spectra extending across Fa (C), to transverse spectra below Fa (L), to the simultaneous presence of low frequency transverse power and parallel power assumed to be associated with the mirror mode (LM), to just parallel power (M) [Anderson et al., 1994]. (There is no stop He2+ stop band due to the high temperature of the He2+ ; see [Denton et al., 1994a].)
The structure of the wave spectra displayed in Figure 9 was accounted for by Denton et al. [1994c]. Their work was based on a simple model of the magnetosheath which assumed 4% He2+ and incorporated the anisotropy-beta relation (4); a Vlasov dispersion solver was used to solve for the real and imaginary parts of the wave frequency. Figure 10 shows the main results of their model. Figure 10a shows that the ion cyclotron instability splits into two branches at low parallel beta b||p, one driven by the temperature anisotropy of the protons (marked "p'') and one driven by the temperature anisotropy of the He2+ (alpha particles) (marked " a ''). These two modes merge at high b||p. Figure 10b shows the normalized growth rate for these two modes and also for the mirror mode (which has zero real frequency, so it does not appear in Figure 10a). The letters "B,'' "C,'' "L,'' "LM,'' and "M'' in Figure 10a stand for bifurcated, continuous, low frequency, low frequency and mirror, and mirror mode spectra, respectively, as described above in reference to Figure 9. Anderson et al.  used a collection of 102 magnetosheath intervals observed by the AMPTE/CCE spacecraft and sorted the observed spectra into these types. The letters ("B,'' "C'' etc.) in Figure 10a are placed at a value of b||p corresponding to the average value of observed b||p for each spectral type.
It is clear that the theoretical curves in Figure 10a indicate bifurcated spectra (distinct proton and alpha particle branches), continuous spectra (with frequency extending above wr / Wp = 0.25 ), and low frequency ( wr / Wp 0.25 ) spectra at the respective b||p values indicated by the letters "B,'' "C,'' and "L'' and "LM.'' Furthermore, the growth rate of the mirror mode becomes comparable than that of the ion cyclotron waves at a value of b||p corresponding to the "LM'' (low frequency and mirror) spectra, and becomes greater than that of the ion cyclotron waves at a value of b||p corresponding to the "M'' (mirror) spectra, Note the remarkable similarity between the frequency spread of transverse power in Figure 9 and the spread of ion cyclotron real frequency in Figure 10. Based on the detailed agreement between the observations and model, we regard the identification of these ion cyclotron and mirror waves as quite solid.
Broadband transverse waves are observed in the magnetopause current layer (region of large magnetic shear), especially during periods of southward IMF leading to large magnetic shear at the magnetopause [Rezeau et al., 1989, 1993; Song et al., 1993]; see also [Anderson and Fuselier, 1993]. All of these authors assume that the broadband waves are associated with reconnection. Rezeau et al.  describe the waves as Alfvénic, while Song et al.  describe the waves with peak power as ion cyclotron waves. Song et al.  states also that the wave power extends up to the electron gyrofrequency. This high frequency turbulence is also highly correlated with the magnetic shear [Zhu et al., 1996] and is now thought to be associated with whistler waves resulting from reconnection [Mandt et al., 1994; Shay et al., 1999].
Johnson and Cheng  describe these waves as kinetic Alfvén waves resulting from mode conversion of compressional (mirror) modes convecting toward the magnetopause. The mode conversion occurs in the region of magnetic shear due to finite Larmour radius effects. They argue that particle transport resulting from these waves can be quite substantial. While there are appealing features to this theory, it seems to us that more work needs to be done to verify observationally that such mode conversion is actually occurring (some recent progress in this direction has been made [Park et al., 1999]).
There are four basic ultra-low-frequency (ULF) modes, the Alfvén/ion cyclotron mode, the magnetosonic/fast/whistler mode, the ion acoustic/slow/sound mode, and the mirror mode (Section 2). These can in principle be identified with the help of transport ratios, phase relationships, and polarization information. In many cases, however, a unique identification is difficult to make. Problems are that the theories are usually based on the linear homogeneous assumption, and that there may be a superposition of observed modes.
Whether the upstream conditions are quasi-parallel or quasi-perpendicular depends on the orientation of the interplanetary magnetic field (IMF) at the bowshock (quasi-parallel indicates that the IMF is roughly parallel to the shock normal; see Section 3). When the upstream conditions are quasi-parallel, the magnetosheath plasma usually has a large value of beta, and is characterized by a broad band turbulent spectrum of waves with little ordering by the ill-defined (for high beta) ambient magnetic field (Section 4). The observed wave power is generally thought to be transmitted, with some modification, through the magnetosheath to the magnetosphere from the foreshock.
When the upstream conditions are
quasi-perpendicular, the dominant waves
(Section 5) appear to be caused by
temperature anisotropy ( T
Inhomogeneity plays a crucial role in the generation and evolution of all these waves. For instance, temperature anisotropy (which drives the ion cyclotron and mirror waves) is caused by the presence of the bowshock and the compression and lengthening of flux tubes as they convect across the magnetosheath. Further examples are the modification of the mirror mode near the magnetopause due to the decreasing beta and the generation of waves in the magnetopause current layer due to large magnetic shear. In the quasi-parallel magnetosheath, the broad-band turbulence is in part generated by foreshock waves which come about due to the presence of the bowshock.
The waves also appear to have an appreciable feedback to the
plasma. The most certain effect we know about is the regulation
of the temperature ratio
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