M.A. Balikhin1, S.N. Walker1, T. Dudok de Wit2, H.St.C.K. Alleyne1 L.J.C. Woolliscroft1, W.A.C. Mier-Jedrzejowicz3, W. Baumjohann4
1 Dept. of ACSE, University of Sheffield, Sheffield, E-mail:
2 Centre de Physique Theorique, CNRS-Universite de Provence, Marseille
3 Imperial College of Science, Technology and Medicine, London
4 Max Planck Institut für Extraterrestische Physik, Garching bei Munchen
Previous studies have shown that quasi-monochromatic waves in the frequency range 1-15 Hz are usually observed upstream of the ramp of supercritical quasi-perpendicular shocks. A number of models have been proposed to explain the origin of these waves. In order to differentiate between these models, one has to determine both the observed frequencies and also wave vectors of the measured waves. The present paper is devoted to the determination of the dispersion relation of these waves, using simultaneous data from AMPTE UKS and AMPTE IRM.
Low frequency waves in the frequency range 100 - 101 Hz have been observed upstream of the ramp of supercritical quasi-perpendicular shocks (e.g., Greenstadt et al., 1970; Fairfield, 1974; Balikhin et al., 1991; Krasnosel'skikh et al., 1991; Orlowski et al., 1995.) It was shown that these waves are right-handed circularly polarized waves, propagating upstream in the whistler mode.
Various mechanisms have been proposed to explain the observations of these waves upstream of the ramp region of quasi-perpendicular shocks. Some of these proposed models relate the waves to the dynamics of a shock front (e.g. Tidman and Northrop, 1968; Krasnosel'skikh 1985.) Other models treat these waves as a result of different instabilities (Wong and Goldstein, 1988; Orlowski et al., 1995.)
This paper determines the wave vectors and the dispersion relation of these waves using magnetic field data measured by AMPTE-UKS (UKS) and AMPTE-IRM (IRM) data during a crossing of the Earth's bow shock crossing on day 364 of year 1984 at about 04:29. This analysis allows us to determine the correctness of the various proposed models.
The method used in the present paper was described by Balikhin and Gedalin, (1993) and Dudok de Wit et al., (1995.) This method can only be applied to simultaneous, multi-satellite measurements. It is based on the relation between the phase difference at a given frequency f and the projection of the wave vector on the satellite separation vector.
The magnitudes and GSE Y-components of the magnetic field B measured during the Earth's bow shock crossing by UKS and IRM satellites on day 364/1984 are displayed in Figure 1 and Figure 2 respectively. The upstream parameters of the crossing were: Bn 55°, Ma 4.8, ci 0.5 radsec-1, pe 110 103 radsec-1.
1. The magnitude |B| and GSE Y_component of the
magnetic field measured during the Earth's bow shock crossing by AMPTE
UKS on day 364(1984) at 04:29 UT. The X-axis represents time (UT) in seconds
after 04:29:00 UT. The Y-axis represents the field amplitude in nT. At
the start of this time period UKS was in the solar wind, crossing the shock
at around 04:29:48.
Fig. 2. The magnitude |B| and GSE Y_component of the magnetic field measured during the Earth's bow shock crossing by AMPTE IRM on day 364(1984) at 04:29 UT. The X-axis represents time (UT) in seconds after 04:29:00 UT. The Y-axis represents the field amplitude in nT. At the start of this time period UKS was in the solar wind, crossing the shock at around 04:29:47.5.
Quasi-monochromatic oscillations can be seen in Figures 1-2 upstream of the ramp. During the period 04:29:20-04:29:40 oscillations were observed below 3 Hz. As can be seen from Figures 1and 2 during time interval 04:29:23-04:29:34 wave packets of coherent oscillations are very well defined. This time interval was used for the determination of the dispersion relation making use of the method described in Dudok de Wit et al., (1995.)
The joint f1__ksep spectrum (where f1 is the frequency in the satellite frame ksep is the projection of wave vector on the UKS and IRM separation vector ) estimated from UKS-IRM data is shown in Figure 3.
Fig. 3. The joint f1__ksep spectrum estimated from AMPTE UKS-AMPTE IRM data measured during the time interval 04 : 29 : 23-04 : 29 : 34 UT on day 364/1984. The three branches of the spectrum shown correspond to the n=-1, n=0 and n=l solutions to equation (1). Of the three branches, only the n=l branch provides a physical solution.
The method of the spectrum calculation was similar to Dudok de Wit et al., (1995) method for the estimation of the dispersion of "linear waves". The periodicity of this spectrogram with ksep is due to the relation between the phase difference and ksep . An ambiguity 2n in the determination of phase difference leads to the ambiguity in the determination of ksep . Each branch of this infinite spectrum will correspond to some integer value of n in the multi-value expression for ksep :
where We will denote the various solutions of this multi-valued relation (shown in Figure 3) as n = i_branches. The central branch corresponds to n = 0. Negative values of ksep correspond to the waves propagating from IRM to UKS while positive values correspond to propagation from UKS to IRM.
It should be noted that the only branch that can correspond to the real situation is that for which ksep approaches zero as frequency approaches zero. As can be seen from Figure 3, this corresponds to the n = 1 branch which is the only one used for further consideration. The direction of wave propagation was determined using the minimum variance method. Knowledge of the direction of and its projection on the UKS and IRM separation line enables us to calculate the wave vector itself. The plasma frame frequency was determined on the basis of the Doppler shift equation. The resulting dispersion relation is shown in Figure 4.
Fig. 4. Dispersion relation for the waves observed during the time interval 04:29:23-04:29:34 UT on the day 364/1984. The crosses represent the dispersion computed from the data. The solid line represents the theoretical dispersion of whistler mode waves.
The theoretical dispersion relation for whistler waves propagating at the same angle to the magnetic field as the observed waves :
is also drawn in Figure 4. It can be seen from this figure that the experimentally determined dispersion relation is in agreement with the theoretical dispersion curve. As can be seen from Figure 4, the observed whistler waves have frequency in the plasma rest frame in the range
and wave vectors in the range Their phase velocity is directed upstream and is about Vph = 700 km - sec -1. The angles of their phase velocity and the shock normal is Their plasma frame frequency is a few times larger than their observed frequencies, so these waves are almost standing in the satellite frame.
These waves cannot be generated by the using the mechanism proposed by Wong and Goldstein, (1988) because the frequency of the observed waves is much higher than frequencies estimated in the framework of their model.
Since maximal wave length of these waves exceeds the ramp width for this shock (Walker et al., 1996), it is very unlikely that these waves can be a result of an amplification process inside the ramp region as was proposed in (Orlowski et al., 1995.)
In macrodynamic models (e.g. Tidman and Northrop, 1968; Krasnosel'skikh, 1985) the waves which were observed in the upstream region are more likely to be almost standing in the shock frame. The comparison of the frequencies of the waves under investigation in the present paper in the satellite frame and in the plasma rest frame, show that these waves are almost standing in the satellite frame.
In the Tidman and Northrop, (1968) model, generation of the waves takes place continuously. In this case it would be more likely that a continuous set of waves rather than bursts of quasi-monochromatic nonlinear packets of whistler oscillations would be observed.
In the Krasnosel'skikh (1985) model which is based upon ramp overturning theory, the nonlinear structures propagating from the ramp are the result of the ramp evolution and overturning. If these processes occur, the resulting structures will occasionally be observed upstream of the ramp. Another process which can be observed is the steepening of the ramp between the two subsequent overturnings. Both these processes can be observed using multi satellite measurements when the satellite separation distance is small, because the characteristic time and scale of these processes are of the order of hi -1 1 s and a few 10 km respectively (Krasnosel'skikh, 1985). Usually the separation between satellites is too large to provide a sufficiently small time difference in the observations of the same shock to register the ramp steepening. Fortunately, in a few crossings of the Earth's bow shock by UKS and IRM, the separation distance along the shock normal was small enough to permit the study of these processes.
The magnitudes of the magnetic field measured during the bow shock crossings by UKS (line) and IRM (dots) on the day 362/1984 at 11:47:30 UT are displayed in Figure 5. A few nonlinear structures can be observed of upstream in the part adjacent to the ramp region. In the present short report we will consider only two of them - the ramp itself and the nonlinear structure most remote from the ramp which was observed by IRM at 11:47:20. In Figure 6 the absolute values BUKS and BIRM are redrawn but a time shift of 3.375 seconds has been applied to the IRM data set. It is obvious from this figure that the part of the shock measured by UKS at 11:47:31-36 is stationary during the period of time required for both satellites to traverse it. This "stationary" part of a shock can be used to estimate the relative velocity of the shock Vsh 4.15 kms-1 The process of ramp steepening, which could lead to the gradient catastrophe can be seen from this figure. The velocity of the leading edge of the ramp relative to the "stationary" part of the shock is about 1 _ 2 kms-1.
Fig. 5. The magnitude |B| of the magnetic field measured during the bow shock crossings by UKS (line) and IRM (dots) on day 362/1984 at 11:47:30 UT. IRM data are shifted on 20 nT.
Fig. 6. The magnitudes of the magnetic field measured during the bow shock crossings by UKS (line) and IRM (dots) on the day 362/1984 at 11:47:30 UT. IRM data are shifted in time on 3.375 s.
In Figure 7 the absolute values BUKS and BIRM are drawn, with a time shift of 1.875 seconds applied to the IRM data. The shapes of the most remote nonlinear structure are similar in both the IRM and UKS data sets. The correlation between IRM and UKS data for this period is high, with a well defined maximum (max 0.8). The velocity of this structure relative to the "stationary" part of the shock is 3 kms-1.
Fig. 7. The magnitudes of the magnetic field measured during the bow shock crossings by UKS (line) and IRM (dots) on the day 362/1984 at 11:47:30 UT. IRM data are shifted in time on 1.875 s and on 5 nT.
The low frequency turbulence generated upstream of a high mach number shock is characterized by
1) Quasiperiodic overturning of the ramp takes place as a result of the gradient steepening in the shock front.
2) Emission of the nonlinear structures towards upstream occurs quasi-periodically as a result of the ramp overturning.
3) Evolution of these nonlinear structures leads to the low frequency waves observed upstream of the ramp, in agreement with Krasnosel'skikh (1985) theoretical conclusions.
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