Observations

The knowledge of the phase of wave signals is essential in this study. In order to calculate the instantaneous phase of the wave signals and the Poynting flux, the data are first band-passed filtered as described in the previous section. The instantaneous phase is estimated by the least squares fitting of a sinusoid [Bloomfield, 1976]. For each estimate of phase, the length of the sinusoid fitted to the data is one wave cycle. For the event on day 307, 1977, the phase of the signals relative to that of the sinusoid is plotted in Figure 3, where the results for E_x, E_y, B_x, and B_y in spacecraft coordinates are shown. Because the reference sinusoid has a constant frequency, the phase of the signals usually appears to be gradually drifting due to slight frequency differences with the observed waves. Nevertheless, rapid changes in phase can still be found frequently. The arrows in Figure 3 represent the phase skips that are comparatively clear. Although the phase skipping phenomenon in pulsation signals has been seen in the magnetic field data both in space and on the ground, Figure 3 presents the first time that the phase skips of pulsations are demonstrated in both the electric field and magnetic field data in space. It can also be seen from Figure 3 that most phase skips seen by one instrument have corresponding phase skips seen by the other instrument. Since the two instruments perform independent measurements, this suggests that the phase skips are caused by some physical process in space rather than some instrumental artifact. It should be mentioned that the sinusoid fitting method and the filtered signals produce a smoother shift in phase even though the reality could have a sharp phase skip. The time when phase skips occur may also have some uncertainty of the order of one wave cycle. This is an intrinsic problem that also applies to other published techniques for phase skip identification.

Figure 3. The relative phase of the filtered signals to the phase of the modeled sinusoid. The phase skips are indicated by arrows.

In the following, all the vectors will be expressed in the field-aligned coordinates, which are defined as $\mathbf{b} = \mathbf{B} / \left\vert \mathbf{B} \right\vert$(field aligned), $\mathbf{e} = \mathbf{b} \times \mathbf{r} / \left\vert \mathbf{b} \times \mathbf{r} \right\vert$(eastward), and $\mathbf{n} = \mathbf{e} \times \mathbf{b}$(outward), where $\mathbf{r}$ is in the radial direction. A schematic diagram of this field-aligned coordinate system is shown in Figure 4a.

The Poynting flux $\mathbf{S}$ can be calculated in a straightforward way by having the full 3-D vectors of electric fields and magnetic fields.

Figure 4. (a) Schematic diagram of the field aligned coordinate system. (b) Angles for representing the orientation of the time average Poynting flux.

The top three panels of Figure 5 show the Poynting vectors in the field-aligned coordinates for the Pc3-4 event on day 328, 1977. The Poynting vectors oscillated at the rate twice of the wave frequency, and they appeared to have a bursty nature. From the viewpoint of energy propagation it is customary to use the time average value of the Poynting flux, which is sometimes called the wave intensity in optics.

Figure 5. The Poynting vectors in field aligned coordinates, time average Poynting flux, and the magnetic perturbations for the Pc3-4 event on day 328, 1977.

If the perturbations in the electric field and magnetic field are $d\mathbf{E}$ and $d\mathbf{B}$, respectively, the time average Poynting flux $\langle \mathbf{S} \rangle$ is

$$\langle \mathbf{S} \rangle \mathnormal{} = \frac{1}{T} \int_... ... \frac{d\mathbf{E} \mathnormal{} \times d\mathbf{B}} {\mu_{0}}$$ (1)

where T is the wave period. The magnitude and orientation of the time average Poynting vector $\langle \mathbf{S} \rangle$ and the magnetic perturbations for the same event (day 328, 1977) are plotted in the middle three panels of Figure 5. The orientation of the Poynting flux is represented by the two angles $\alpha$ and $\beta$, whose definitions are shown in Figure 4b. When $\alpha = 0\deg$,the Poynting flux $\langle \mathbf{S} \rangle$ is field aligned; when $\alpha = 90\deg$, $\langle \mathbf{S} \rangle$ is perpendicular to the field. The direction of the perpendicular component of $\langle \mathbf{S} \rangle$ is defined by the angle $\beta$. The rapid modulation of $\left\vert \langle \mathbf{S} \rangle \right\vert$indicates again that the energy of the Pc3-4 waves flows in a bursty fashion, and the direction of $\langle \mathbf{S} \rangle$ changes after several wave cycles. The magnetic perturbations in the field-aligned coordinates are plotted in the bottom three panels. When the change of Poynting flux direction is examined in detail, we find that the phase skips in the wave signals occur when the Poynting flux rapidly changes its direction. For example, at 2057 UT the Poynting flux $\langle \mathbf{S} \rangle$rapidly rotated from the direction antiparallel to the field to the direction toward the field. A phase skip in dB_b is found at the same time. More examples are indicated in Figure 5. Here we see the directional change of Poynting vectors, and the phase skips have a one-to-one relation.

Figure 6. Same as Figure 5 except for the Pc3-4 event on day 307, 1977.

Figure 6 shows the simultaneous and time average Poynting vectors ($\mathbf{S}$ and $\langle \mathbf{S} \rangle$,respectively) for the Pc3-4 event on day 307, 1977. Again the Poynting flux changes its directions every several wave cycles. The phase skips also occurred when the Poynting flux changed its direction.

As seen in a href="fig5.gif">Figures 5 and a href="fig6.gif">6, Poynting vectors can be in many directions. In order to understand the statistical features of Poynting flux directions, 29 clear Pc3-4 events observed by ISEE 1 have been examined. Table 1 lists those events and the corresponding interplanetary magnetic field (IMF) conditions observed by either the IMP 8 or the ISEE 3 spacecraft. The ISEE 3 spacecraft was generally located close to the sunward libration point at $\simeq 200\,R_E$ upstream of the Earth, and the IMP 8 spacecraft orbits the Earth with an apogee $\simeq 40\,R_E$.The delay of the solar wind travel time from the spacecraft to the magnetopause is considered. In the 23 events that the IMF data are available, 16 of them occurred when the IMF cone angle $\cos^{-1} (B_x / B_t) \leq$ 45, and none occurred when the IMF cone angle was greater than $60\deg$.This is consistent with the scenario that the upstream of the bow shock provides an important energy source for Pc3,4 waves [Troitskaya et al., 1971]. In addition, only 6 of the 23 events occurred under southward IMF conditions, which suggests that the reconnection process on the dayside magnetopause is not an important energy provider for these wave activities.

The Poynting vectors of the energy impulses are studied statistically to understand the propagation of wave energy. The samples are selected when the magnitude of the time average Poynting flux is at its maximum, for example, the peak of $\left\vert \langle \mathbf{S} \rangle \right\vert$ at 2058 UT in Figure 5. Totally, 194 Poynting flux samples are obtained from the events listed in Table 1. Figure 7 shows the distribution of their orientations in terms of the $\alpha$ and $\beta$ angles.

Figure 7. Distribution of the $\alpha$ and $\beta$ angles of 194 time average Poynting vectors.

The occurrence rates of the $\alpha$ and $\beta$ angles of the Poynting flux are also shown alongside the $\alpha$-$\beta$ plot. It can be seen that most of the Poynting vectors are oriented in the direction close to the ambient magnetic field. Approximately 73% of the vectors have an $\alpha$ angle less than 45(or larger than 135). In addition, the perpendicular component of $\langle \mathbf{S} \rangle$is more likely to be close to the east-west direction. The Poynting vectors with outward energy flow ($\vert \beta \vert < 90\deg$)slightly outnumber those with inward energy flow ($\vert \beta \vert \gt 90\deg$).

Poynting vectors in the n-e plane, which is perpendicular to the background magnetic field, and their locations of measurements are plotted in Figure 8 for further visualizing the energy propagation in the magnetosphere. Figure 8a shows the perpendicular component of $\langle \mathbf{S} \rangle$, or $\langle \mathbf{S} \rangle_\perp$,of the 194 Poynting flux samples as in Figure 7. The length of the arrows is proportional to the logarithm of $\vert\mathbf{S}\vert$.The arrows form into subgroups in Figure 8 since each wave event has several Poynting flux samples due to multiple impulses of wave energy. It is apparent that the directions of $\langle \mathbf{S} \rangle_\perp$may vary dramatically in a relatively small region in the magnetosphere as seen previously in Figures 5 and 6. In contrast to Figure 8a, Figure 8b shows the mean Poynting vectors for the 29 Pc3-4 events. In the morning sector, the wave energy appears to flow inward and from dawn to noon. In the afternoon sector, most of the wave energy still propagates eastward although the picture is less clear. Overall, we find that the mean Poynting flux in the n-e plane flows eastward and inward. However, each event contains several impulses of wave energy which may have very different directions of propagation.

Figure 8. (a) The Poynting vectors in the n-e plane and the locations of observations. The length of the arrows is proportional to the logarithm of $\vert\mathbf{S}\vert$. (b) As Figure 8a except that each arrow represents the mean Poynting vector for an event in Table 1.