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 (field aligned), (eastward), and (outward), where is in the radial direction. A schematic diagram of this field-aligned coordinate system is shown in Figure 4a.

The Poynting flux 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 and , respectively, the time average Poynting flux is

(1) |

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 ( and ,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 upstream of the Earth, and the IMP 8 spacecraft
orbits the Earth with an apogee .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 45,
and none occurred when the IMF cone angle was
greater than .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 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 and angles.

Figure 7. Distribution of the and angles of 194 time average Poynting vectors. |

The occurrence rates of the and angles of the Poynting flux are also shown alongside the - 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 angle less than 45(or larger than 135). In addition, the perpendicular component of is more likely to be close to the east-west direction. The Poynting vectors with outward energy flow ()slightly outnumber those with inward energy flow ().

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
, or ,of the 194 Poynting flux samples as in Figure 7.
The length of the arrows is proportional
to the logarithm of .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
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 . (b) As Figure 8a except that each arrow represents the
mean Poynting vector for an event in Table 1. |