FLUCTUATING
MAGNETIC
FIELDS
IN THE MAGNETOSPHERE*
II. U L F Waves
R. L. M c P H E R R O N , C. T. R U S S E L L , and P. J. C O L E M A N , Jg.
Dept. of Planetary and Space Science, and Institute of Geophysics and Planetary Physics,
University of California, Los Angeles, Calif., U.S.A.
(Received 21 January, 1972)
Abstract. The study of ULF waves in space has been in progress for about 12 years. However, because
of numerous observational difficulties the properties of the waves in this frequency band (10-3 to
1 Hz) are poorly known. These difficultiesinclude the nature of satellite orbits, telemetry limitations on
magnetometer frequency response and compromises between dynamic range and resolution. Despite
the paucity of information, there is increasing recognition of the importance of these measurements in
magnetospheric processes. A number of recent theoretical papers point out the roles such waves play
in the dynamic behavior of radiation belt particles.
At the present time the existing satellite observations of ULF waves suggest that the level of geomagnetic activity controls the types of waves which occur within the magnetosphere. Consequently,
we consider separately quiet times, times of magnetospheric substorms and times of magnetic storms.
Within each of these categories there are distinctly different wave modes distinguished by their
polarization: either transverse or parallel to the ambient field. In addition, these wave phenomena
occur in distinct frequency bands. In terms of the standard nomenclature of ground micropulsation
studies ULF wave types observed in the magnetosphere include quiet time transverse - Pc 1, Pc 3, Pc 4,
Pc 5 quiet time cornpressional - Pc 1 and Pi 1; substorm compressional Pi 1 and Pi 2; storm transverse - Pc 1 ; storm compressional Pc 4, 5. The satellite observations are not yet sufficient to determine
whether the various bands identified in the ground data are equally appropriate in space.
1. Introduction
A l t h o u g h low frequency waves have been studied on the surface of the earth for m a n y
years, the i m p o r t a n c e of such waves in the physical processes occurring within the
magnetosphere was n o t generally realized until quite recently. However, in the past
five years, m a n y theories of the dynamics of the magnetospheric particle p o p u l a t i o n
have been f o r m e d in which these waves play a f u n d a m e n t a l role. F o r example, Swift
(1967) a n d Liu (1970) have b o t h suggested drift wave instabilities in the ring current
m a y be i m p o r t a n t in magnetospheric substorms. Hasegawa (1969) a n d Lanzerotti et al.
(1969) have considered the drift m i r r o r instability of the storm-time ring current a n d
also its effect on precipitation. S o n n e r u p et al. (1969) examined several different models
of wave generation in a n a t t e m p t to explain m o d u l a t i o n of energetic p r o t o n s d u r i n g
magnetic storms. S o u t h w o o d et al. (1969) studied the b o u n c e r e s o n a n t interaction
between pulsations a n d trapped particles. Coroniti a n d K e n n e l (1970a) have examined
the drift wave instability of the i n n e r edge of the electron p l a s m a sheet, a n d in addition
* Publication No. 982. Institute of Geophysics and Planetary Physics, University of California,
Los Angeles, Calif. 90024.
Space Science Reviews 13 (1972) 411--454. All Rights Reserved
Copyright 9 1972 by D. Reidel Publishing Company, Dordrecht-Holland
412
I~. L. M C P H E R R O N ET A L .
how waves produced by this or other mechanisms can cause modulated precipitation
of energetic electrons (Coroniti and Kennel, 1970b). Cornwall et al. (1970) have shown
how ion cyclotron turbulence in the region of overlap of the ring current and plasmasphere contributes to the decay of the stormtime ring current. Dungey and Southwood (1970) find the characteristics of magnetic fluctuations near the magnetopause
strong support for their theory of the Kelvin-Helmholtz instability. These theories all
emphasize the importance of knowing the amplitudes, the polarizations, and the
regions of occurrence of low frequency waves in the magnetosphere.
Ground based studies provide some clues as to the magnetospheric population of
waves, but to test the theories and determine actual character and morphology of the
waves, these waves must be measured in situ, where the wave-particle interactions are
taking place. Ground based observations are modified in a very complex way by both
ionospheric and ground currents and may be altered or reflected in propagating from
the equatorial regions through the ionosphere to the ground (Prince and Bostick,
1964; Field and Greifinger, 1967 and references therein). Furthermore, once these
waves reach the ground-ionospheric wave guide, they may propagate parallel to the
surface of the Earth (Tepley and Landshoff, 1966; Manchester, 1966). The only way
to avoid these problems is to measure the waves on satellites deep in the magnetosphere. However, this, too, poses several problems.
The first problem is that satellites move. They usually move both in radial distance,
magnetic latitude, and local time simultaneously. Synchronous orbiters in circular
orbits, however, restrict this movement to only a local time motion. Thus, the analysis
of such data is much simpler. The synchronous orbiters can map out the diurnal
variation of phenomena every day and thus supply a good statistical description of
this region of space. Eccentric orbiters, although they map out a much larger volume
of the magnetosphere, require a year's complete data to map out the local variation of
a phenomenon and it requires many years' data to determine an accurate statistical
map. Furthermore, the radial motion of an eccentric orbiter can make the detection
of L-shell restricted phenomena quite difficult. Finally, we note the near impossibility
of detecting ULF wave phenomena in a low altitude orbit. First, there is the difficulty
of measuring the waves in a high background field (,-~89G), which is compounded by
the fact that the spin period of spinning satellites and the boom vibration period of
stabilized spacecraft are usually in the ULF range. Second, the motion across field
lines in a polar orbit, the most common low altitude orbit for scientific studies, is too
rapid to identify low frequency waves. Typically, such a satellite moves 4 ~ in latitude
per minute. Thus, for example, it would move from L = 6 to L---7 in less than 30 s.
The second problem with satellite studies is the limitations of telemetry systems. In
order to determine the frequency and polarization of a wave it must be sampled at
least twice as fast as the wave frequency. Early satellites often had low sample rates.
Another associated problem is the size of the digitization interval. The number of
telemetry bits required for a digital sample governs the number of digital windows
possible over the dynamic range of the instrument. For example if 8 bits are available
for a sample and a 1 7 digital window is desired, then the dynamic range is limited to
4l 3
FLUCTUATING MAGNETIC FIELDS IN TIlE MAGNETOSPHERE, II
__ 128 ?. However, it is desirable to have a digital w i n d o w smaller t h a n 1 ? a n d to
m e a s u r e waves in b a c k g r o u n d fields m u c h larger t h a n 128 7. The solution to this
p r o b l e m is either to increase the n u m b e r o f telemetry bits per w o r d or to sacrifice the
m e a s u r e m e n t s o f the b a c k g r o u n d field. The f o r m e r solution has been used on several
recent spacecraft b y using sets o f offset fields a r o u n d the m a g n e t o m e t e r , the m a g n i t u d e
o f which are telemetered separately. The latter solution is o b t a i n e d in the use o f a
search coil which measures the field derivative.
Because o f these limitations, relatively few spacecraft have p r o v i d e d ideal measurements o f m i c r o p u l s a t i o n s within the m a g n e t o s p h e r e . In fact, all the observations o f
U L F waves in space have been m a d e with only 11 o f the m o r e t h a n 40 spacecraft which
have carried m a g n e t o m e t e r s in the m a g n e t o s p h e r e . The characteristics o f these 11
m a g n e t o m e t e r s are s u m m a r i z e d in T a b l e I. Therefore, despite the fact that space exTABLE I
Characteristics of various magnetometers used for the study of ULF waves in space
Spacecraft
Magnetometer
Axes Spin period
Dynamic range
Quantization
Sample rate
Pioneer 1, 5
Explorer 6
Explorer 12
Explorer 14
Explorer 26
Explorer 33
ATS 1
Dodge
OGO 3, 5
OGO 5
Search coil
Search coil
Fluxgate
Fluxgate
Fluxgate
Fluxgate
Fluxgate
Fluxgate
RB Vapor
Fluxgate
1
1
3
3
2
3
2
3
T
3
0.6-1220 7
0.6-1200 ~
=:1000 7
=:500 7
-4-2000 ,/
•
=:60, =:100 7
:5200 7
=:250 7
3-4000 ~
•
y
V
V
24 7
10 ~
40 7
0.4, 1.2, 2 7
17
1/2 7
A
z8
A
A
3.13 Hz
3.13 Hz
3.13 Hz
0.16 Hz
6.25 Hz
2.0 Hz
A
0.87
6.9
56 Hz
0.5 s
0.37 s
0.5 s
~5 s
2.5 s
0.6 s
0
0
0
V - Varies with background field strength.
A - Analog data whose sampling resolution can be varied in different analyses.
T - Total field instrument.
p l o r a t i o n is n o w in its second decade, in situ U L F wave studies are still in their infancy,
a n d we have d e t e r m i n e d only a r o u g h picture o f the m a g n e t o s p h e r i c U L F wave p o p u lation.
I n c o n t r a s t to the study o f waves in space, the study o f g r o u n d m i c r o p u l s a t i o n s has
p r o c e e d e d at a r a p i d rate over the last decade. I n a recent review, Saito (1969) shows
t h a t the c u r r e n t w o r l d o u t p u t is o f the o r d e r o f 50 p a p e r s per year with m o r e t h a n
1000 p a p e r s p u b l i s h e d to date. Saito's p a p e r p r o v i d e s one o f the m o s t recent reviews
o f the o u t s t a n d i n g results of these papers. Even if U L F waves were n o t f u n d a m e n t a l
to the d y n a m i c s o f the m a g n e t o s p h e r e , the existence o f this extensive w o r k on g r o u n d
m i c r o p u l s a t i o n s alone w o u l d justify the e x a m i n a t i o n o f these waves in space.
The r e a d e r interested in g r o u n d observations o f m i c r o p u l s a t i o n s is also referred to
the recent b o o k , Geomagnetic Micropulsations, b y Jacobs (1970); to the reviews on
414
R.L. MCPHERRON ET AL.
the use of micropulsations as diagnostics of the magnetosphere by Troitskaya and
Gul'elmi, (1967) and more recently by Aubry (1970); and to the review of some theoretical aspects of micropulsations by Dungey and Southwood (1970).
To date, reviews of satellite observations of U L F waves have either concentrated
on one type of wave phenomena or have presented only a brief overview of the waves.
Recent reviews of this type include those by Coleman (1970); Coleman and McPherron
(1970); Jacob s (1970); McPherron and Coleman (1970a); Russell and H olzer (1970);
McPherron (1971); and Russell (1971 a). Examining these earlier reviews, we find that
while U L F waves are observed in all regions of space, their characteristics are markedly different in each region. These distinct regions include the interplanetary medium,
the solar wind upstream of the bow shock, the magnetosheath, the magnetopause, the
magnetosphere, and the tail. In this review we limit ourselves to discussing those waves
which may have a direct bearing on ground observations, i.e. ,we restrict our attention
to the region enclosed by the magnetopause, the magnetosphere and the magnetotail.
The organization used for ELF and VLF waves in paper 1 (Russell et al., 1972), i.e.,
a division into whistler phenomena, high altitude emissions and low altitude emissions
obviously cannot be used for U L F waves. There is no strict U L F analogue of the
whistler, and for the reasons stated above there are no low altitude satellite observations of U L F waves. However, the magnetospheric U L F wave phenomena appear to
fall within certain categories based on the degree of magnetic disturbance. They can
be further divided on the basis of being transverse or compressional waves, and finally,
on the basis of their characteristic spectrum. We, therefore, have used the degree of
magnetic disturbance and the wave polarization to organize this review. Since there
is yet little direct evidence linking the satellite observations to ground micropulsations,
we have avoided using the usual Pc-Pi classification to order the data. However, to
facilitate a comparison with ground based work, Table II indexes the sections of this
review which contain discussions of wave phenomena in the various Pc and Pi bands.
In the following four sections we consider separately initial observations, magnetospheric wave phenomena occurring during quiet times, during magnetospheric substorms, and during magnetic storms. Section 6 discusses the measurement of the transfer function of a field line. Finally, Appendix I discusses the techniques used in the
analysis of U L F waves.
TABLE II
Relation of standard micropulsation nomenclature to classification used in this paper.
Table entries are the section where phenomenon is discussed
Type
Polarization
Pc 1
Pc 2, 3
Pc 4, 5
Pi 1
Pi 2
Quiet t i m e
Transverse
Compressional
Transverse
Compressional
Transverse
Compressional
3. l
3.2
4.1
.
5.1
-
3.1
3.2
.
.
-
3.1
3.2
-
4.1
4.2
-
-
Substorm
Storm
.
.
.
.
5.2
.
FLUCTUATING MAGNETIC FIELDS IN THE MAGNETOSPHERE~ II
415
2. Initial Observations
The first observations of U L F waves in space were reported by Sonett et al. (1960).
Using a search coil magnetometer on Pioneer 1 they measured two components of the
magnetic field perpendicular to the spin axis of the spacecraft. Power spectral analysis
of the fluctuations revealed discrete lines superimposed on a 1/ffalloff towards higher
frequencies. The compressional nature of these fluctuations and their association with
a sharp drop in field magnitude suggested they were turbulence associated with the
termination of the geomagnetic field on the dayside. The sudden decrease in magnitude
at extreme distances indicates the spacecraft passed into the undisturbed solar wind
and identifies the fluctuations as magnetosheath turbulence.
Similar observations were reported by Coleman et al. (1960) from search coil
measurements on Pioneer 5. The radial distance of the observations and the character
of the fluctuations again identify these as magnetosheath turbulence.
The first observations of U L F waves inside the magnetopause are those of Sonett
et al. (1962). The search coil measurements of Pioneer 1 were spin demodulated,
obtaining amplitude and phase of the fluctuations between 4 and 7 R e on the dayside.
Spectral analysis was performed on these two quantities for intervals of 30-100 s
duration. This analysis revealed rms power from 3-8 7 between 0 and 1.0 Hz. Some
spectra had peaks around 0.5 and 1 Hz. An attempt was made to interpret these
results in terms of wave modes, mode coupling, etc. However, the lack of the third
component of the vector field made this exceedingly difficult. No clear association
between these fluctuations and later satellite observations has been made.
Long period waves within the magnetosphere were first reported by Judge and
Coleman (1962). Search coil data from Explorer 6 were spin demodulated and expressed in terms of two orthogonal components. The field variations, which were observed at about 6 R~ in the morning sector, were compared to simultaneous variations
in energetic electrons. The event chosen for detailed study was one in which regular
field variations were in phase with variations in the particles. By assuming different
types of wave modes present and subtracting out their contributions to the observations, the authors concluded the variations were due to a mixed transverse and compressional mode. They assigned a period of 100 s to the compressional part, and 200 s
to the transverse part. The transverse wave was right hand elliptically polarized with
respect to the ambient magnetic field (in the sense of electron gyration). The wave
appeared to be damped with a time constant of approximately 500 s.
In this case, also, the authors were hampered by the lack of the third component of
the field. Despite this, however, the characteristics of the wave fall in the category of
quiet time, transverse Pc 4 waves, to be discussed below.
A report of a long period oscillation of period greater than 100 s was also made by
Sonett (1963). Using data from Pioneer 1, he concluded that the oscillations were
either a radial perturbation or the projection in the meridian plane of an elliptically
polarized wave. Although the primitive apparatus precluded thorough analysis, the
observation substantiated the earlier report by Judge and Coleman (1962).
416
R.L.
MCPHERRON
ET AL.
The first complete vector field observations of U L F waves in the magnetosphere
were reported by Patel and Cahill (1964). A three component ttuxgate on Explorer 12
recorded two long period wave events between 7 and 8 R~ around 1100 LT. These
events had periods of 120 and 180 s and amplitudes of 6 to 8 7- They were transverse
and right elliptically polarized with respect to the ambient field.
These authors also reported the first correlation between satellite observations of
waves and ground micropulsations. They observed that during both events waves of
similar amplitude, period, and polarization were present on the ground near the subsatellite point. The travel time between the satellite and the ground appeared to be
about 90 s. These waves are clearly identifiable as quiet time Pc 4 pulsations.
3. Quiet Time Wave Phenomena
3.1. TRANSVERSEWAVES
A more detailed examination of quiet time transverse waves on Explorer 12 was reported by Patel (1965). He found eight examples of transverse waves between 6.8 and
10.6 R e and 0700-1200 LT. These events were selected on the basis of having a quasisinusoidal waveform and 1-2 complete cycles. The amplitudes ranged from 6-11 7
and periods between 100 and 180 s. All events were elliptically polarized transverse to
the ambient field. Four events before 1045 LT were right hand elliptically polarized
with respect to the ambient field while the four events after this time were left hand
polarized (in the sense of proton gyration). This result is in agreement with ground
observations. Figure 1 is an example of the waveform observed by Patel; Figure 2
illustrates their elliptical polarization. Also shown are the simultaneous ground observations near the subsatellite point.
I
I
49065
48152 km
Q
0
Br
-34
B~
-36
-38
I
1
2118
I
~
I
L
'
::)120
I
I
I
1
2122
l
I
I
I
/
2124 UT
Figs. la-b. Waveform of transverse, Pc 4 waves recorded during quiet times by triaxial fluxgate on
Explorer 12 at noon LT. Coordinates used are geocentric spherical (Patel, 1965).
417
FLUCTUATING MAGNETIC FIELDS IN THE MAGNETOSPHERE, l I
I
I
52301
51523 km
7
-2
-4
Br - 6
-8
m
-I0
9
J
@
-12
i
-28 i
9
9
- 36
9
9
9
B~-32
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-34
-36
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|
2053
t
I
f
I
J
I
2055
I
I
2057
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2059
I
UT
Fig. lb.
As pointed out by Patel, these observations were limited by the 24 y quantization
error of the original data. Because the spacecraft was spinning with its axis occasionally transverse to the ambient field, it was possible to reduce the effective quantization
error to about 6 7. Additional improvements in accuracy were obtained by taking 15 s
averages and low pass filtering with a one minute sliding average. The final data had
uncertainties of approximately 3 7 and no variation with periods shorter than 1 min.
Clearly this processing limited the possible wave observations to large amplitude
( > 3 7) long-period ( > 60 s) waves.
A study by Patel (1966a) of Explorer 14 fluxgate data gave similar results. Explorer
14 probed the magnetosphere in the 0400-05 00 LT sector, 20-26 ~ below the magnetic
equator from 6-10 R e. Eight events with amplitudes ranging from 5-25 7, periods
3-30 rain were observed. All events but one were left hand elliptically polarized. One
additional event was found to correlate with ground micropulsations. Some events
appeared to be coupled with compressional waves.
In a short note, Patel (1966b) pointed out that the Pc 4 waves must be quite localized
within the magnetosphere. He was unable to find any events at Explorer 12 and 14
which corresponded to large micropulsation events chosen from the ground magnetograms. On the other hand, the three ground events which correlated with the satellite
were all within 15 ~ LT of the conjugate point of the satellite and near the same L-shell.
The first observations of quiet time transverse waves with high time resolution and
high sensitivity were reported by Cummings et al. (1969). These measurements were
made by the biaxial, spinning fluxgate magnetometer on the synchronous satellite
ATS I with a sample rate of 0.16 s -1 and approximate sensitivity of 0.3 7-
418
R. L. MCPHERRON
ET AL.
Nm Col iege, Alaska
t
Explorer XII
II
i
i IT i
(b)
(o)
Nm College, Alosko
Explorer. XII
5
'~
I
t
2
6
14
I
12
il
4
5
(c)
Fig. 2.
(d)
Hodograms showing polarization of Pc 4 waves presented in Figure 1 and also their correlation with ground observations (Patel, 1965).
An example of the waveform of a typical event is shown in Figure 3 where 15 s
averages have been plotted with a resolution of about 0.5 y. Twenty-five events were
observed during January 1967. Their amplitudes ranged from 1-20 7 peak to peak
with periods from 80 to 240 s. The events were frequently of several hours duration
and appeared to be amplitude modulated. The waves were polarized transverse to the
ambient magnetic field and nearly linear. The most probable direction of polarization
was 30 ~ east of a magnetic meridian plane (in the quadrant between V and D).
The diurnal occurrence and Kp dependence of the 25 events is shown in Figure 4.
Note in this figure that each 10 min interval is defined as an event so that considerably
more than 25 points occur. The waves appear to maximize after dawn and occur primarily in the daylight hours. No events were seen between 17 and 22 LT. The periods
of the waves were grouped around 102 and 190 s. The longer periods were associated
with greater magnetic disturbance.
In recent work, Cummings et al. (1971) studied the statistical characteristics of these
waves by visual analysis of two years of data. They find the probability of occurrence
increases gradually from dawn through the day to a maximum at 1400 LT and then
drops rapidly at dusk. As a function of season, the maximum occurrence (11~ of the
FLUCTUATING-
MAGNETIC
DAY
~
FIELDS
IN
005
--
THE
MAGNETOSPHERE~
DATE
,., P , . , : ' - . . . ~ - . . j - . ~ - - . , . . / . - . ~
.,~..~..f~v
O1 0 5
r
...~,.~.,,f~
419
lI
19 6 7
~
i O~"
_
_
V
0
i0
20
30
40
MINUTES
50
80
Fig. 3. Waveform of transverse Pc 4 wave observed during quiet times by biaxial, spinning fluxgate
magnetometer on synchronous satellite ATS 1 at 0900 LT. H is parallel to Earth's rotational axis,
V radially outward in the equatorial plane and D completes right handed system
(Cummings et al., 1969).
__
~
t 20
__
NUMBER
t,0 E~%,s
I0
4.0
0 §
5.5
I
....
A
~
.....
=
~ ~ d=
2 ..... 9
2 .... 9
2* .... 9
&O
20
o
Kp
== -
oo
,,
ATS - I
dAN
e
o
1967
9
iiiii
N'~
T (min) 2.5
20
I9
=
-?.
.
9
9
04
. . . . ..,,
:~$i:~:i:~,%~
~H~:.~!E!~!~
......
~,,',~
08
12
L.T.
Fig. 4.
,
9 9
9
9 ,,',,
o
O0
99
o co
o
1.0
,
9 9 99
.
16
20
24
(HOURS)
Dependence of period of transverse Pc 4 waves such as shown in Figure 3 on local time and
Kp (Cummings et al., 1969).
420
l~. L . M C P H E R R O N
ET AL.
time) is about January. The frequency of the wave events is distributed almost uniformly over the band 5-15 x 10-3 Hz with an average frequency of 10 .2 Hz (100 s period).
The average maximum amplitude is 2.8 7. The authors find no evidence that the lower
frequency (Pc 5) and the higher frequency (Pc 4) waves are different in any way.
In a recent report, Dwarkin e t al. (1971) described shorter period Pc 3 waves recorded
at the almost synchronous satellite, D O D G E , at 6.3 R E in the equatorial plane. The
most common period was 30_+ 10 s with peak to peak amplitudes of 2 7. The waves
were linearly polarized transverse to the ambient field and oriented azimuthally. They
note that Pc 3 oscillations occurred simultaneously on the ground within a sector of
about 60 ~ The waves were recorded during quiet conditions. A typical waveform is
shown in Figure 5.
(~) ANALOGUERECORDOF'MAGNETOMETER OUTPUT
.
eL
.
.
.
.
.
.
.
~- 1100 LT
:--
2
(b) TRANSFORMED DIGITAL DATA
Y
I
1720
I
1725
I
1730
U.T.
[
1735
I
1740
Z
I
1745
JULY27, 1968
Fig. 5. A transverse linearly polarized Pc 3 wave of 40 s period and 2 7 PP amplitude observed
on the DODGE satellite (Dwarkin et al., 1971).
Similar Pc 3 oscillations are observed at the synchronous satellite, ATS 1. Figure 6
is an example of previously unpublished data showing a 40 s wave of 2 7 peak to peak
amplitude. The wave is nearly linearly polarized transverse to the ambient field. Such
waves were not studied by Cummings et al. (1969) because of the limitations imposed
by 15 s averages and 0.5 7 resolution of the plots.
In another report, Dwarkin e t al. (1970) discussed a Pc 1 event observed at D O D G E .
The event had a peak to peak amplitude of 6 7, a period of 3 s, and it was left hand
circularly polarized transverse to the ambient field. The event was seen simultaneously
at two ground stations nearly conjugate to the foot of a field line passing through
D O D G E . A segment of the waveform of this event and a hodogram of the variation
in the equatorial plane are shown in Figure 7.
This hodogram provides a clear example of the effects of spectral folding, or aliasing.
Although this event was left hand polarized in the analog data as expected for a Pc 1
421
F L U C T U A T I N G MAGNETIC FIELDS I N THE MAGNETOSPHERE, II
ATS-I MAGNETIC FIELD
HIGH-PASS FILTERED DATA
OCTOBER 5, 196g i9:00
eIL ['IF'
I','''I
........
I.......
I.......
i
....
4
Bx
-4
-B ,,ll,l . . . . . .
lllll
l,,l,
........ I . . . . . .
I ....
B
4
By
0
-4
-B
tlll[llllIlI[~lI
I
.... I .... .... I .... . . . . .
Ill......
Iiii
8
4
Bz
o
-4
-@
O
I0
20
30
40
50
BO
MINUTES
Fig. 6. Transverse Pc 3 wave event observed at the synchronous satellite ATS l (1.28 s averages).
Z axis parallel to Earth's rotation axis, Sun vector in X - Z plane, Y axis towards dusk.
event, the Nyquist frequency (half the sampling frequency) of the digitized data is less
than the frequency of the wave. Thus, the successive points on the hodogram appear
to rotate in a right handed sense, while the major axis of this apparently very elliptical
wave rapidly precesses in a left-handed sense. If the data had been digitized at a much
higher rate, this hodogram would have revealed a left hand circularly polarized wave
with constant amplitude.
3.2. COMPRESSIONAL WAVES
Compressional waves have been less frequently reported in satellite observations. As
mentioned previously, Judge and Coleman (1962) concluded that their Pc 4 event was
a mixed transverse and compressional mode. Patel (1965) found three examples of
compressional waves in the Pc 4 band. Lindsey (1968) examined Explorer 26 data
422
R . L . MCPHERRON ET AL.
DEC. 22, 1968
(a) ANALOGUE RECORD OF MAGNETOMETER OUTPUT
~i!i~!tiii!!
iii!~!'i!.iiii..!$i
i iii i i i ! i i i i i :::iii I !~i~!iii::i~
(b) TRANSFORMED DIGITAL DATA
N~
~
x
i-
>
I-
Y
Z
1550
I
I
1555
1600
I
I
I
1605
1610
1615
U.T.
(dl POLARIZATION DIAGRAM OF A SEGMENT OF (c)
(c) ENLARGEMENT OF A SEGMENT OF (b)
=E
x
16:00:51-16:01:07 U.T.
(--10:30 S.L.T.)
At = 1.26 SEC
1
Q
Y
1 East
!
I
I
1600
1601
1602
U.T.
1
1603
Downword
11
SENSEOF ROTATION
0
ELECTRONS
O
PROTONS
Fig. 7. A transverse, left hand polarized Pc 1 wave of 3 s period and 6 7 PP amplitude observed
on the DODGE satellite (Dwarkin et al., 1971).
during both quiet and disturbed times. He f o u n d b o t h compressional and transverse
waves similar to those o f Patel. Power spectral analysis suggested their energy was
limited to a rather narrow frequency band.
Barfield e t al. (1971a) have studied a purely compressional wave of 107 s period and
i0 7 peak to peak amplitude recorded on ATS 1 at 0800 LT. Coherent oscillations o f
energetic electrons (0.3-1.9 MeV) in phase with the magnetic variations were recorded
simultaneously on the same satellite. Figure 8 shows the waveform of the fluctuations
and Figure 9 the auto spectra of particles and fields.
This same event has been studied by Lanzerotti and Tartaglia (1971) and c o m p a r e d
FLUCTUATING
MAGNETIC
FIELDS IN THE MAGNETOSPHERE~
423
II
5E
~
I0{E
5z
>l.05MeV )
~
..................
....
~5~
~..~>~',,V~r
. . . .
.~,
"~'
~, r r ' ~ "
-~
.....,~
: t R,~ .,,,,~~ ~ i: ' !"t ~,(ii!,'V~,~',f '/~i/'r/. ''''
v ,~ljl,/V'~ttd, B'~ ',, ~ ~' '~
-
R2
i
o
.........
.........................................................................................................................................................
9
~.
:
•
r,~o~-
a'.<'0................
'
~ ,3c -
~ .~,r,(,@\A,%,V\,\f "<v"
L,E
~-
2
17
19
r8
HOURS UI, FEBRUARY14, 1967
Fig. 8. Compressional Pc 4 wave event observed at the synchronous satellite 0800 LT with
simultaneous in phase fluctuations of energetic electrons. Two successive hours are shown. Magnetic
field data have been transformed to a field aligned current system with Bp radial, B~ azimuthal
and Be along the field (Barfield et al., 1971a).
i0-1
....
I ....
i ....
I ....
i ....
I ....
I0 2
-~
Specirurn 0fLN
J•
(E = i.I MeV)
-
T
10-3:o
__
O0
o~
>_-
~o~%
%
10-4 -
-.
-"- 9
"
O'3
10-5 =
oO
So~
o
o~
."-
ooo
o_
%%0
" ~ 1d~ 7oo6 1 7 6o~
1 7o 6 .LI 1~,
]'
.
"~" ~ 9
10-6
i0- 07
i
9
I I
~
9
~
fc
.!
....
~
,..
o
~o
..
_-
o
-
~
o
~o~
~
30
c
'S>~
.'...,o~
of LN BrS
~
o
-
. ..:'.-.
*
;
~176
~
60
FREQUENCY, cph
Fig. 9.
Auto spectra of particles and fields for compressional Pc 4 wave event shown in Figure 8.
Note peak of 34 cycles per hour (106 s) (Barfield et al., 1971a).
424
R . L . M C P H E R R O N E T AL.
10-2
L l l
- -
2
------COLLEGE
io-S
I
ATS-L
JbHG+
2
2
bbG+bzG
t
\
~o-4
N.
o
10.5
10 6
',i L_
l
rG 7
I
O.O01
~
~
~ I
0.005
~
,
~ p
I
0.01
,
J
I
0.015
FREQUENCY, cps
F i g . 10. P o w e r s p e c t r a o f a P c 4 w a v e e v e n t s i m u l t a n e o u s l y o b s e r v e d a t A T S 1 a n d n e a r its n o r t h e r n
conjugate point. The satellite signal was purely compressional
while the ground signal was almost
entirely transverse (Lanzerotti and Tartaglia, 1971).
to simultaneous ground observations. Rapid run magnetograms from College, Alaska,
approximately 600 miles ENE of the ATS 1 conjugate point were digitized and subjected to spectral analysis. The results shown in Figure 10 indicate the power in the
ground signal was about 50 times smaller than the satellite signal although a clear
peak was evident at the ground. The plane of polarization of the ground signal was
tilted 12 ~ below the ground plane. The event was left hand polarized with an ellipticity
(defined in appendix) of approximately 0.7. The major axis of the perturbation ellipse
was oriented at 45 o to true north (NE to SW). Essentially no Z component was observed on the ground despite the fact the signal was compressional at ATS 1.
A survey of the occurrence of compressional micropulsations throughout the magnetosphere has been reported by Heppner et el. (1970). The total magnetic field as
measured by rubidium vapor magnetometer on OGO-3 and OGO-5 was plotted on
microfilm at two minutes per frame. These plots were scanned visually for sinusoidal
fluctuations of the total field. The sensitivity threshold was of order 0.3 ~ depending
somewhat on the plotting scale and the wave period. To organize their results they
425
FLUCTUATING MAGNETIC FIELDS IN THE MAGNETOSPHERE, II
OGO E RUBIDIUM RAW AND DIFF PLOT
START TIME 68114211015610.0
A
100.0
B
30.0
90.0
24.0
80.0
18.0
70.0
12.0
60.0
6.0
V~
<
E
<(
o
Q
.J
ILl
E
MEAN RB IS 66.4
50.0
0.0
E
40.0
E
i
30.0
12.0
20.0
18.0
10.0
24.0
0
MEAN DIFF - 8 . 9
0
30.
60.
TIME IN SECONDS
90.
30.0
120.
RB FIELD IS ~ a ~
DIFF FIELD IS . . . .
DIFFERENCE FIELD = RUBIDIUM-ORBIT
Fig. 11. Typical microfilm plot of rubidium magnetometer data obtained by OGO 3 and 5 and
used to survey occurrence of micropulsations in the magnetosphere (Heppner et al., 1969).
separated the magnetosphere into volume elements of roughly 3 units of L, 3 hours of
LT and 40 ~ magnetic latitude centered about the equator. All data for Kp > 5 were
excluded. It should be noted that although their instrument detects only the compressional component the authors feel that most waves are not 100 K transverse enabling
them to detect the occurrence of predominantly transverse waves as well. A typical
example of a Pc 1 wave is shown in Figure 11.
Their results show Pc 4, 5 (45-240 s) have a maximum of occurrence on the dayside.
Further, these pulsations are observed more frequently at L > 8. Pc 3 (10-45 s) also
have a maximum on the dayside, but are most frequent around the auroral zone (L =
426
R . L . MCPHERRON ET AL.
BAND LIMITED PULSATION EVENT
UCLA MAGNETOMETER 1700-1706 UT JUNE 8, 1968
PRINCIPAL
+
AXoSDINATES
~
A
~_
By ~
-y
Bz~ ' / ~ - / Y ~ ' ~ - ~ / ~ x V ~ f
I
t
1700
I
1702
i
[
~Jx z
i
1704
I
I706
Fig. 12. Band pass filtered (0.05~0.05 Hz) segment of transverse waves recorded at OGO 5
during substorm recovery. Satellite was inward at 6 RE, just below magnetic equator on dawn meridian. The principal axis coordinate system has Z axis parallel to local field, Y radially inward and
X towards the Sun (McPherron and Coleman, 1970).
OGID-5 BLP NAVE EVENTDURINGSUBSTI3RM
B JUNE 1968
10~
\
10 2
LU
Z: 101
10 ~
I0 ~i
a_
"""-.~1,
1Q-a
L 0 -3.
"'-,I
IQ-
I[
FREQUENCY {HERTZ)
FREQUENCY (HERTZ]
1
~,~ ~
0
BZ-BX
,
,
,$3,~.,,~
180
l~
o_
BY-BZ
u
-180
1
180
u
-a'180
Flg. 13
0.00
O , l O 0 , 2 0 0.30
FREQUENCY (HERTZ)
o.oo o,10 o:2o 0.30
FREQUENCY (HERTZ)
Spectral analysis of the wave event of Fig. 12. Note peak appears on top of rapidly
falling background spectrum.
FLUCTUATING MAGNETIC FIELDS IN THE MAGNETOSPHERE, II
427
5-8). Also, Pc 3 oscillations are more frequent than Pc 4. Pc 1, 2 ( T < 10 s) waves are
most frequent on the dawn meridian inside the magnetopause. They seem to be closely
associated with irregularities in the main field.
4. S u b s t o r m A s s o c i a t e d W a v e P h e n o m e n a
4.1. TRANSVERSEWAVES
Quasi-sinusoidal transverse waves occurring during substorms have been reported by
McPherron and Coleman (1971a). Pulsations of 20 s period and 0.3 7 rms amplitude
were observed on the OGO-5 spacecraft just below the magnetic equator as it was inbound on the dawn meridian between 7 and 5 R~. The average polarization was
transverse to the local field with the major axis azimuthal in the equatorial plane. The
average ellipticity was 0.3 and predominantly right handed with respect to the local
field. The waves appeared to consist of bursts propagating within a cone of about 30 ~
about the local field. Spectra of the pulsations showed a large peak superimposed on
a rapidly falling background spectrum. Bursts of whistler chorus at twice the wave
period were observed simultaneously on the satellite, but no modulation of energetic
electrons could be detected. Pc 2, 3 type pulsations were simultaneously observed
during the recovery phase of a polar substorm at two stations near the subsatellite
point. Figures 12 and 13 illustrate the waveform and spectra in the principal axis
ATS MAGNETIC FIELD VARIATIONS
--
BX
AUGUST
1645
....
I
,
17
'
[967646
I
J
/\'\
//
,
"
i-\.,/~,-~ /\
'-\/
.,\/
/\\
, I
9
I
9
\~\~. ~.
9 /\
..
i
]
[
I
//\
\ 0)"
/"\.,._;
\ /t
23see .J
///
9
? L./
i
/
I
/
I
I
]
I
I
I
I
I
1645
1646
UNIVERSAL TIME
Fig. 14. Waveform of substorm associated transverse wave recorded at ATS 1 near the dawn
meridian. The wave is nearly linearly polarized transverse to the ambient field. The Y axis is roughly
azimuthal (within 30 ~ (Coleman and McPherron, 1970).
428
R . L . M C P H E R R O N ET AL.
coordinate system of the perturbation. (See appendix for definition of principal axis
system.) Figure 12 of Paper I shows the simultaneous dynamic spectrum of the ELF
chorus and the waveform during this event.
A similar wave event on ATS 1 has been reported by Coleman and McPherron
(1970). This event was also observed during the recovery phase of a substorm around
the dawn meridian. The wave period was 25 s and peak to peak amplitude 2 7. It was
nearly linearly polarized transverse to the ambient field. The waveform of this event
in a field aligned coordinate system is shown in Figure 14. A previously unpublished
power spectrum for this event is shown in Figure 15.
This same event has been studied by Parks and Winckler (1969) using observations
of trapped energetic electrons at ATS 1, and precipitating electrons near its conjugate
point. They note that both in the equatorial plane and the auroral zone, modulation
with 11 s period was present. This period is roughly half the period of the waves observed at the same satellite. These results should be compared to the OGO-5 observations of McPherron and Coleman (1971a) mentioned above, that whistler chorus was
modulated at roughly half the wave period, i.e., 10 s. Figures 16 and 17 show the
analysis of the ATS 1 event by Parks and Winckler.
Substorm associated irregular fluctuations of 10-30 s period at ATS 1 have also
BAND LIMITED PULSATIONS
1 6 0 8 - 1 7 1 6 UT, AUGUST 17, 1967
UCLA FLUXGATE MAGNETOMETER
ATS- I
E IGENVALUES
10 '~
103
tO2
I ....
EIGENVALUES
10 ~ :
i03 ]
26SEC
0.4~'RM~
~: I02 .
tO 1
~
to~
~: lo o
i0-I
~
L
~
lot
10-1
~: 10 -~
~
oc 10 -2.
aJ
3~
~o-3
-
~ io -3_
i0 -~
i
i0 -~
01
0
0,0
FREQUENCY ~HERTZ)
-
~
_
~
0.2
0.3
O.q
FREQUENCY (HERTZ)
O.l
POLARIZATION AND ELLIPTICITY
/
,.=, _,IW , ,'r'~ W ~ ~ V l ~
S
SPECTRAL ANALYSIS PARAMETERS
NUMBER OF POINTS = 4 0 9 6
SAMPLE INTBRVAL = 1.0 SEC
DEGREES OF FREEDOM 26
BAND WIDTH = .00317 HZ
BANDSE}:~RATION=.OOIg5HZ
0.5
'-
'v ~
i00__ /
I I1
o-
~V--4tr,4~'4V/-'w&Fu
Of'
~'V'~ " "1 ' ~ '
~[v,
'1 '
0.0
0. i
0.2
0.3
O.LI
FREQUENCY (HERTZ)
BX-BY
RANDOM
I 90%
0.5
Fig. 15. Eigen analysis of spectral matrix (see Appendix) for wave event of Figure 14.
FLUCTUATING
I
'
'
'
'
t
'
MAGNETIC
'
'
'
FIELDS IN THE MAGNETOSPHERE,
I
1
DYNAMIC FOURIER
SPECTRAL ANALYSISOF
6,6Re EQUATORIALELECTRON FLUX
(50~E~ 150 KeV)
AUGUSTI~', 1967
1655-1655 UT
v
++L+6.6 Re EQUATORIAL ELECTRONS
:~ .....
~
429
II
~
~
~RELATIVE
ACCUMULATED
+
............................
..................
:
......+!
~1635
16
25.0
50
UNIVERSAL TIME
Fig. 16. Dynamic spectra] analysis of fluctuations in eguatoria] electrons measured by ATS I
during wave event of Figure 14 (Parks and Winckler, 1969).
i
[
,
i
J
i
I
I
i
i
I
I
I
i
I
i
[
i
i
,
i
I
i
i
)YNAMIC FOURIER
SPECTRAL ANALYSISOF
BALLOON X-RAYS
AUGUST 17, 1967
1634 - ]658 UT
BALLOON X- RAYS
o3m
>......
<m
ocz
= o
•
RELATIVE
ACCUMULATED
POWER
o • 0.155-WI~
0.144'
ul
u._
~1
~4~;, ,.~=~
4+=
1655
"~.
1640
!~
.
.
.
....
.
1645
1650
1655
UNIVERSAL TIME
Fig. 17.
Dynamic spectral analysis of modulation of precipitating electrons in auroral zone near
ATS 1 conjugate point during wave event of Figure 14 (Parks and Winckler, 1969).
430
R. L. M C P H E R R O N ET AL.
TUNGSTEN DYN
3.3 DB CC
,5
~
.2-
.I-
O,
I
/
ol2o
I
0200
0230
0500
0550
UNIVERSAL TIME
Fig. 18.
Dynamic spectrum of Pc 1 event on October 10, 1969 at conjugate point of ATS 1 (Tungsten, N.W.T., Canada). Pure white is highest power, pure black next highest, etc.
ATS-I
DYNAMIC PO/ER (dD/dt)
5,3 DB CONTOURS
.4.
0=
01~o
0200
0230
UNIVERSAL TIME
0300
0330
Fig. 19. Dynamic spectrum of Pc 1 event of Figure 18 as recorded simultaneously at ATS 1. High
background noise evident in time series is spacecraft interference. This interference produces a ramp
function of 5.12 s period. Fourier components of this ramp are apparent at 0.095 H z and 0.390 Hz
as interference lines.
FLUCTUATING
MAGNETIC
FIELDS
IN THE MAGNETOSPHERE~
431
II
been correlated with simultaneous X-ray pulsations in the auroral zone by Pierson
and Cummings (1969). No pronounced peaks could be found in the spectra of the
pulsations at ATS 1 or the ground.
McPherron and Coleman (1971b) have recently reported a substorm associated Pc 1
event observed simultaneously at ATS 1 and its conjugate point at Tungsten, N.W.T.,
Canada. Figure 18 shows a dynamic spectrum of the signal recorded on the ground
and Figure 19 at the satellite. (For a discussion of this technique see appendix.) The
ground signal was clearly more complex than the signal at the satellite. This difference
was attributed to the effects of horizontal propagation of Pc 1 causing signals other
than those vertically incident at the ground station to be recorded. Power spectra were
calculated for the two portions of the dynamic spectra which appear to overlap. The
results are summarized in Figure 20.
SATELLITE- GROUND TRANSFER FUNCTION
UCLA MAGNETOMETERS
ATS-I AND TUNGSTEN, NWT, CANADA
IPDP EVENT OCT
1969
0 2 4 2 - 0516
UT
0516- 0350
UT
t~
"1-
i (A/T) 2 = (,62rl.217)2=8.6
I
i0 I
i (A/T)2= (,237/,i5 •
2.6
I
(.9
I
jO-I
~
II
^
l
f
I
rr"
Ud
,
1
I
ATS-I
10.2
ta
TU~3STEN
,
FO-3 .
,
, \
0
FREQUENCY (HERTZ)
Fig. 20.
FREQUENCY (HERTZ)
Comparison of power spectra at ATS 1 and Tungsten for two intervals when similar signals
were present in the dynamic spectra of Figures 18 and 19.
During the first interval the spectra corresponded best in a 0.2 Hz band centered at
0.27 Hz with the power at the satellite about 2.6 times as large as the ground. In the
second interval the satellite power peaked at 0.18 Hz and was about 8.6 times as great
as the ground. The interpretation suggested by the authors for this enhancement in the
power ratio is motion of the satellite with respect to the field line on which the Pc 1
signal was propagating, i.e., a change in relative conjugacy.
4.2. COMPRESSIONALWAVES
Low frequency waves predominantly compressional in nature have been reported at
ATS 1 during substorms by McPherron and Coleman (1970b). Large amplitude irregular fluctuations were found to be associated with the expansion phase of a magnetospheric substorm when the synchronous satellite was close to local midnight. Spectra
432
R.L. MCPHERRON
ET AL.
of these fluctuations are steep with no significant peaks. Near midnight the waves
begin shortly after the onset of the recovery in the H component usually associated
with a substorm at the synchronous orbit (Cummings et al., 1968). Near dusk the
waves are observed in association with a depression in H, but after the substorm
expansion has begun at midnight. Figure 21 is an example of an unusually large
event. Figure 22 shows auto spectra of all three components of the field during this
event. The diurnal occurrence pattern for these events begins in the late afternoon
and is a maximum at local midnight. The events are rare in the early morning hours
and are not seen during day time. Coleman and McPherron (1970) point out that
fiTS MflGNET@GRflM
SUBST@RM
OURING
MRRCH
MRGNETOSPHERIC
3,1967
11-12
UT
Z
F-'- LLI
-9
~.
r'r- I--EE CZ:
V
'~
I~l
5Z0
(/')0
Z
~"
UJ
Ok.~
--
H
j
CE
rU
(F)
o
'o
'o
TIME
;o
o
(MINUTES]
Fig. 21. Waveform of predominantly compressional fluctations associated with substorm expansion
phase at ATS 1, 01-02, LT. Same coordinate system as Figure 3 (McPherron and Coleman, 1970b).
PONER $ P E C T R R
OF MRG F I E L D
AT
RT5 D U R I N G
iO ~
10 3
10 2
~z
101
10 o
io -~
n_
i
10 -2 .40
FREQUENCY [HERTZ]
FREOUENCT ~HERTZ]
Fig. 22. Auto spectra of the three field components for the first wave event of Figure 21 (McPherron
and Coleman, 1970b).
FLUCTUATING
MAGNETIC
FIELDS IN THE MAGNETOSPHERE,
II
433
D SPIKE EVENTS AT ATS-SPRING 1967
p
~
~
MAR 3
H ~
iI
~ . - . --~jI
i
:F 5 0 GAMMA
l
i
,~:~iN~i~.~
MAR I
i
MAR
r
i
i
i
APR
r
- 50
,
t
i
~
p
+30
SUBSTORM
EXPANSION
i
i
I
i
+60
i
i
i
i
J
+90
TIME (MINUTES)
Fig. 23. Associationof predominantly compressional, irregular fluctuations at ATS 1 with recovery
in H component and trailing edge of D spikes (shading). Note also rather sudden termination of
fluctuations after interval of 30-120 rain (Coleman and McPherron, 1970).
when negative D spikes (sharp decreases in declination probably caused by field
aligned currents) are observed, the irregular fluctuations tend to follow the trailing
edge of the D spike. Further, the events tend to terminate suddenly after about one
hour. Figure 23 illustrates both these points.
Similar irregular fluctuations are observed in the near geomagnetic tail during
substorms (Russell et al., 1970, 1971a; Russell, 1971b). An expansion of the plasma
sheet on the midnight meridian is associated with the expansion phase of an auroral
substorm. The region inside the expanding boundary is quite turbulent. The waveform
of the fluctuations is highly irregular with no particular preference for any field component. Spectra are smooth with no significant peaks. Figures 24 and 25 illustrate
the waveform and spectra of these fluctuations.
In addition to these substorm associated waves which are directly observed, there
have been several reports of quasiperiodic boundary motions in the tail. The existence
of these boundary motions is evidence for the existence of hydromagnetic waves in
the tail of the same period. Mihalov et al. (1970) show a multiple neutral sheet crossing
exhibiting reversals of the field at intervals of 1 to 2 rain and lasting for 25 min at a
geocentric distance of 73 R e . These data are shown in Figure 26. Note the similarity
of these data to the ATS data shown in Figure 21. In studying many such multiple
neutral sheet crossings Mihalov et al. find that the interval between successive crossings
ranges from 30 to 1000 s with a most probable value near 100 s.
434
R . L . M C P H E R R O N ET AL.
MAGN ETOTAIL FLUCTUATIONS
UCLA O G O - 5
AUGUST
FLUXGATE MAGNETOMETER
20,
1968
0827-0832
UT
4O
IBI 50
B•
2O
5O
2o
o
By
-Io
Bz
I
c.
W
.....
~+~"~--~ ~'.. " t,.........~.-.~i
1o
o
\.,
w",
,'
............,
~ ...........
,
"~-~"+~'~"~"~A/~"~',
"i~"~-~, "~'+''-'~
0827
....
,4""\ /
0828
0829
0850
UNIVERSAL
S
0851
. ,. . .
"
0832
TIME
Fig. 24. Irregular fluctuations inside expanding plasma sheet. Observations made by OGO 5 at
position in GSM coordinates (--16.2, 1.0, 7.3 R~) during expansion phase of substorm. Approximate
distance above neutral sheet is 2.3 RE (Russell et al., 1970).
MAGNETOTAIL
FLUCTUATIONS
U C L A OGO-5 FLUXGATE
AUGUST
2_0, 1968
MAGNETOMETER
0827-0852 UT
ao L
I-
z ~i
w
a
-r
I--
IaJ
on fi
I 0 "z
I 0 -I
I
I0 I
FRE QUENCY (HERTZ)
Fig. 25.
Auto spectra of magnetic field for fluctuations shown in Figure 24 (Russell, 1971).
FLUCTUATING
MAGNETIC
FIELDS
IN
THE
MAGNETOSPHERE,
U
#!
90-
6-.
4-
2I
-
A
B,~,y_~_
_
[
i
L
L
l
,
L) + !> t~- (
l
+
1
l
L
l
I
i
+
+
I
t
;
l
;
l
l
l
l
l
+
+
I
k
t
l
l
1
1
1
1
+
t
l
!>
+
l
1
tJ
1
+
435
II
1
+
i
l
Z
3
3
I
I
5-
--5-6
l
l
l
t
l
l
i
,
l
l
l
l
l
l
l
(
I
I [ 1 1
I
t
l
l
l
l
l
6-
Bz,)"
I -
-]-
-4
L
2540
I
i
i
2345
~
~
i
~
t
2350
i
i
i
~
i
,
2355
i
i
,
t
t
i
0000
TIME, U T, SEPTEMBER 8 - 9 ,
i
I
i
0005
:
J
:
~
i
OOrO
I
1
]
i
I
0015
I
1
I
f
0020
IgES
Fig. 26. The three solar magnetospheric vector components of the magnetic field during a crossing
of the neutral sheet by Explorer 33 spacecraft 73 RE behind the Earth (Mihalov et al., 1970).
A similar oscillation in the boundary of the plasma sheet has been reported by
Russell (1971a) at 12.5 RE behind the Earth. Figure 27 shows the successive entries
into and exits from the plasma sheet which occurred during a substorm expansion
of the plasma sheet. The period of this boundary oscillation was roughly two minutes,
in good agreement with the observations of Mihalov et aI. at 73 R E and of ATS-1
at 6.6 RE.
5. Geomagnetic Storm Associated Wave Phenomena
5.1. TRANSVERSEWAVES
Predominantly transverse waves have been reported on ATS-1 during magnetic
436
R.L.MCPHERRON
ET
AL.
storms with frequencies in the Pc 1 band. The first report, by Barfield et al. (1970),
described an event with 5 s period and 1-2 7 amplitude. This event was associated
with the longer period mixed mode stormtime waves described below, and may have
been amplitude modulated by them (see Figure 28). Maxima of the Pc 1 wave amplitude occurred at the extrema of the long period waves. The Pc 1 waves were left
hand elliptically polarized with the orientation of the major axis rather variable.
tBI 20
io
Bx
o[
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20
io
o
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x 3o
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.
.
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UNIVERSAL T I M E
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.
.
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AUGUST20, 1968
Fig. 27. The three solar magnetospheric vector components of the magnetic field and the total
field during a crossing of the plasma sheet boundary by the OGO-5 spacecraft. The depressions of
the X component upon entry into the plasma sheet have been shaded for
easy identification (Russell, 1971c).
It was noted that the average flux of 600 to 1000 keV protons was unusually high,
presumably due to the direct access of solar wind protons. A similar Pc 1 event was
seen in association with the long period waves and solar flare protons of another storm.
It was noted however that Pc 1 waves also occurred in association with the protons
when no long period waves were present.
Another report of transverse waves during a magnetic storm made by Russell
et al. (1970) used the U C L A magnetometer on OGO-5. At the time the waves were
observed the satellite was outbound at 45 ~ magnetic latitude, at a distance of 5 R~
FLUCTUATING
437
M A G N E T I C F I E L D S I N T H E M A G N E T O S P H E R E ~ I1
on the noon meridian. The presence of magnetosheath electrons during the wave
events indicates that the satellite was within the polar cusp (Russell et al., 1971b).
The frequency of the waves was about 7 Hz or about 80% of the local proton gyro
frequency. Their waveform in the principal axis coordinate system is shown in Figure
29. The peak to peak amplitude of this event at its maximum was 10 7. The polarization of this segment of data was highly elliptical as shown in Figure 30. It should
UCLA MAGNETOMETER EXPERIMENT A T S - I
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I
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Fig. 28. Waveform of storm associated transverse Pc 1 wave correlated with longer period, mixed
mode waves (Barfield et al., 1970).
be noted that the average direction of the ambient field was along the Y axis of the
principal axis coordinate system; also, the direction of propagation of the wave
should be in the direction of least variation, i.e., the Z principal axis. Thus, the direction of propagation, the ambient field, and the major axis of polarization are mutually
orthogonal. This is the characteristic of the Alfv6n wave (left hand mode) propagating perpendicular to the ambient field. Power spectra of the entire wave event in
the principal axis coordinate system of the previous figure are shown in Figure 31.
The complex structure of the peaks in each component is consistent with the occurrence of several different bursts of waves during the interval.
438
R.L.MCPHERRON
ET
AL.
5.2. COMPRESSIONALWAVES
Waves with a large compressional component but having a coupled transverse variation have been discussed by Sonnerup e t al. (1969). Fluxgate measurements on Explorer 26 at 5 RE, 1330 LT and 6 ~ magnetic latitude during a magnetic storm revealed fluctuations of 5 min period in magnitude, declination, and inclination.
Simultaneous modulation of energetic protons was observed with minima in the field
magnitude corresponding to maxima in the proton fluxes at all energies. No close
Pc I MAGNETIC MICROPULSATIONS
1254 UT NOVEMBER I, 1968
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Fig. 29. Waveform of high frequency Pc 1 wave event occurring during magnetic storm (Russell
et al., 1970).
correlation between the satellite observations and ground micropulsations could be
established although it was noted that the wave event was associated with a magnetospheric substorm. The polarization of the wave was determined by eigenanalysis
and hodograms in the principal axis coordinate system. The wave was elliptical with
a ratio of major to minor axis of 16.9 7/6.1 7- The orientation of the major axis was
almost in a magnetic meridian plane, but tipped inwards towards the Earth from the
ambient field by 30 ~. The normal to the polarization ellipse, i.e., the axis of wave
propagation, was pointed outwards from the Earth 60 ~ above the magnetic equatorial
plane and 34 ~ east of a magnetic meridian plane (towards dusk). The polarization
appeared to be left handed with respect to the ambient field. Spectral analysis gave
FLUCTUATING
< B>ErGEN
(3.4,
M A G N E T I C F I E L D S I N T H E M A G N E T O S P H E R E , II
439
NOVEMBER l, 1968
1254 : 23.[8-23.68
513.2,6.5 )
9
X
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-5
EIGEN VALUES
38.4:7.1 : I
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-
5
fwAvr = 6.5 hz
~
1254: 23.68 -24.15
57"
-5
FILTER CORNERFREQUENCY=2.TBhz
fPROTON6YRO= 7.8 hz
fw / fp.G. = .83
Fig. 30.
Hodograms for Pc 1 event of Figure 29. Note average direction of main magnetic field
is along //axis (Russell et al., 1970).
102
c~
10
--.
10 o
LO
Z
Od
~>~
i@ -I
~:
10-2
tD
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Fig. 31.
I0 ~
I01
FREQUENCY (HERTZ)
t02
Auto spectra of Pc 1 event in principal axis coordinate system of data shown in Figure 29
(Russell et al., 1970).
440
R . L . MGPHERRON ET AL.
-20
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Fig. 32a-b. Panel A shows the simultaneous fluctuations in the magnitude of the field and energetic
protons observed in association with a substorm occurring during a magnetic storm. Explorer 26 was
at 5 R E 1330 LT, and 6 ~ magnetic latitude. Pane] B shows details of the variations in components
of the field (Sonnerup et al., 1969).
~,20
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Fig. 32h.
.,,
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441
FLUCTUATING MAGNETIC FIELDS IN THE MAGNETOSPHERE, n
a sharp peak at 315 s in all three components. The particle and field observations
are shown in Figure 32. Hodograms and spectral analysis are shown in Figure 33.
The same phenomenon has been reported at ATS-1 by Barfield and Coleman (1970).
Quasi-sinusoidal fluctuations primarily in the magnetic meridian plane were observed
with 1 to 4 h duration during four of the first six magnetic storms of 1967. Wave
periods from 2-15 rain and average amplitudes of 10 7 were typical. The polarization
7o-~.B~2 (o')
60
50
I
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Fig. 33a-b. Panel A shows hodogram for event of Figure 32. Orientation of principal axis explained
in text. Panel B shows power spectra of same event (Sonnerup et al., 1969).
5
I
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;
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Fig. 33b.
(2. 3 s
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w.=-..
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0.4
442
R . L . M C P H E R R O N ET AL.
was nearly linear with both transverse and compressional components. During one
event electrons were modulated in phase and protons out of phase with the waves
(Parks et al., 1969; Lanzerotti et al., 1969). All events occurred around the main
phase maximum and in afternoon local time.
In more recent work, Barfield et al. (1971b), detailed properties of these waves
were examined. Spectral analysis showed that a frequent characteristic of these waves
is a harmonic structure. An example is shown in Figure 34, where the fundamental,
second, and third harmonics are evident as significant peaks in one of three components.
The various harmonics were found to be elliptically polarized. This is particularly
ATS-I
WAVE EVENT DURING
MAGNETIC STORM
104
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103
103
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102
[01
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0550-0450UT
MAY 7, 1 9 6 7
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0,009
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0. 027
FREQUENEY (HEFiTZI
0,036
Spectral analysis of storm time, mixed mode waves. Note the harmonic structure of the
signal shown by upper right panel (peaks occur at 0.0052, 0.0104 and 0.0156 Hz). Also, note in the
lower right panel, the high coherency in the X - Y plane at the fundamental. The coordinate system
is determined by the principal axis of the perturbation at the fundamental frequency
Fig. 34.
(Barfield et al., 1971b).
FLUCTUATING MAGNETIC FIELDS IN THE MAGNETOSPHERE, II
443
evident in the example from the high coherency at the fundamental in the X - Y plane.
Eigenanalysis was used at each harmonic to determine the characteristics of the polarization. The results were found to be consistent with the assumption that the major
axis and direction of propagation of all three harmonics lay in a swept back magnetic
meridian plane tilted approximately 45 ~ with respect to the Earth's dipole axis.
Actual orientations of the principalaxes of the perturbation ellipsoids for the three
harmonics in the above example are summarized in Figure 35. Ellipticities of the
three harmonics were quite high, the ratio of power in the J( and Y principal axes
were respectively: ~ 16, ~ 8, and ~ 4.
The sense of rotation of the polarization ellipse was examined by the cumulative
area technique (see appendix for description). The predominant polarization of the
three harmonics were respectively right, left, and right handed with respect to the
ambient magnetic field. The time behavior of the polarization of each harmonic,
however, was quite complex, Figure 36. The various harmonics did not occur simultaneously, and the second and third harmonics reversed their sense of rotation at
several apparently significant times.
A statistical examination of the occurrence of these wave events revealed they
occurred primarily in local afternoon near the maximum of the main phase. Further,
ORIENTATION
OF PRINCIPAL AXIS SYSTEMS, MAY 7, t 9 6 7
~ FUNDAMENTAL
X,,Y,,Z,
)~2,Y2,Z2 ~ 2nd HARMONIC
)(3,'~3,Z3 ~ 3rd HARMONIC
r
-t.o
-~
4
\\
-.5
)<
ANALYSIS PERIOD
0548:15-0455:00
UT
Fig. 35. Orientation of the principal axes of each harmonic of the wave event of Figure 34 in the
dipole coordinate system. Note approximate alignment of X and Z axes with the expected direction
of a swept back magnetic meridian plane (Barfield et al., 1971b).
MAY 7, 1967
444
oo
TIME {~s
w~'"
T ] ME {~ECON05)
9 10"
O0
~')OJ
O0
O0
Fig. 36. Instantaneous magnitude and cumulative area swept out by perturbation vector as it
rotates in major plane of the principal axis coordinate system of the fundamental. Data shown from
the bottom to top for first three harmonics (Barfield et al., 1971b).
FLUCTUATINGMAGNETICFIELDSIN THEMAGNETOSPHERE,II
445
nearly every wave event was found to be associated with a magnetospheric substorm
at midnight.
6. The Measurement of the Transfer Function of a Field Line
As mentioned in the introduction, the knowledge of the transfer function of a field
line is very important in studies of micropulsation sources. At the present time there
exists a wide variety of ground measurements of U L F waves which could provide
detailed information on magnetospheric processes if the effect or propagation of the
waves along field lines from their presumed magnetospheric sources were known.
Various analytical treatments of this problem quoted in the introduction show the
characteristics of U L F waves at the Earth's surface are highly dependent on the
ionospheric models and approximations used. Thus, in order to have a transfer function that can be used with confidence, it must be measured.
6.1. OBSERVATIONS
Conceptually, measuring the transfer function can be performed by simultaneously
measuring the field variations at the equatorial crossing point of a field line and at its
conjugate point. Several authors have attempted to do this. For example, Patel and
Cahill, (1964); Patel (1965) and Patel (1966a, b) attempted to correlate quiet time
transverse Pc 4, 5 waves with ground observations near the subsatellite point. In
this work they found the correlation existed only when the satellite was near the same
L shell and within one hour LT of the ground station. They did find for the one or
two wave cycles examined, that the sense of rotation with respect to the magnetic
field in the ground records was consistent with that observed at the satellite.
A recent ground-satellite correlation for quiet time compressional waves has been
reported by Lanzerotti and Tartaglia (1971). These authors show a purely compressional wave at ATS-1 was observed near its conjugate point as an ellipticallypolarized
transverse wave. They show that although peaks were present in the spectra at both
locations, the power at the ground was more than 50 times smaller.
Ground-satellite correlations for substorm associated Pi 1 micropulsations have
been reported by Pierson and Cummings (1969), and more recently by McPherron
and Coleman (1971a). In the earlier report the authors did not find clear spectral
peaks and no definite conclusions were drawn. In the work by McPherron and Coleman, two satellites, OGO-5 and ATS-1, and two ground stations, College, Alaska
and Tungsten, N.W.T., Canada were used. These authors reported that the signal
was seen at all locations except ATS-1, and concluded therefore that the signal was
spatially localized. A more recent examination of the ATS-1 data by the authors of
this review indicates that the signal was also present at ATS- 1. This result was obtained
using the more sensitive techniques of eigen analysis of the spectral matrix described
in Appendix I. The conclusion about spatial location still appears to be justified,
however. One month of ground micropulsation data from College, Alaska (near the
ATS-1 conjugate point) have been compared to ATS-1 data. Only twice during this
446
R . L . MCPHERRON ET AL.
month did any of the frequent substorm associated Pi 1 events at the ground have
associated variations at the satellite. In these two cases the ATS-1 satellite was past
dawn as it was for the two other Pi 1 events discussed above. This lack of correlation
between the ground and the satellite for events prior to dawn suggests that the signals
are spatially localized.
Another attempt to study this problem was made for the stormtime Pc 5, mixed
mode waves by Sonnerup et al. (1969), and more recently by Barfield et al. (1971b).
While no digital data were available at the ground for detailed spectral analysis,
the analog records indicate that oscillations with similar characteristics were absent.
The most recent report of a ground-satellite correlation is the work of McPherron
(1971), on a substorm associated Pc 1 event. Comparison of dynamic spectra at the
two points showed the ground signal had the more complex structure. For two time
intervals that appeared to correlate well, the measured transfer functions were quite
different. For the first of these two intervals the signal on the ground was highly
polarized and left elliptical over the entire Pc 1 band. For the second interval, the
polarization was low and erratic. During both events the signal was left elliptical
at the satellite.
The most likely interpretation of these results is that most of the signal in the Pc 1
band received at the ground station was propagating horizontally in the ionospheric
wave guide. Only when the signal was systematically polarized in a left handed sense
at the ground, is it reasonable to assume that the signal was vertically incident. Unfortunately it is unlikely that ATS-1 and Tungsten, N.W.T., Canada (69.96~
127.750 ~W) were exactly conjugate at this time. Calculations of the northern conjugate
points for ATS-1 by Barish and Roederer (1970) show that Tungsten is close to real
conjugacy only at local midnight, while these observations were made at 1800 LT.
Diurnal, seasonal and solar wind effects can move the calculated conjugate point as
much as 200 km north and/or east and west of Tungsten.
6.2. DISCUSSION
A number of questions are raised by the preceding observations. Perhaps the most
significant is whether the concept of a field line transfer function is meaningful at all.
It appears to be appropriate only where it is reasonable to assume that plane waves
are observed at both the satellite and the ground. If waves were truly plane, conjugacy
would be unimportant since measurements could be made anywhere in the planes
of constant phase. However, in the real situation, it is likely that waves such as Pc 1
are collimated between closely adjacent L shells. In this case it becomes very important
where the satellite is located with respect to these shells. Clearly if the wave is collimated its amplitude will be a significant function of position across these L shells.
Even more serious problems arise at the ground station where ionospheric effects
can convert a vertically incident wave into a wave propagating horizontally in the
ionospheric cavity. In this case the wave will be significantly modified by the mode
conversion and horizontal propagation.
For longer period oscillations than Pc 1 it is not even obvious that the concept
FLUCTUATINGMAGNETICFIELDSIN THEMAGNETOSPHERE,II
447
of a propagating wave is useful. Since at low frequencies wavelengths become long
compared to the length of a field line, it seems more appropriate to think in terms of
oscillations of a resonant cavity. In this case a ground-satellite correlation reduces
to a simultaneous pair of measurements at an unknown point on the boundary and
an unknown point within the cavity.
Clearly measurements at more than two points are required to obtain definite
conclusions. For example, location of the conjugate point of the micropulsations
source field line might be accomplished using a chain of ground observatories. An
example of this technique is the work of Sampson et al. (1971), who showed for Pc
micropulsations that the latitude of maximum signal amplitude and the latitude of
transition from left to right handed polarization are the same.
To determine the location of the satellite with respect to such a source field line
is not an easy task. I f the signals are of sufficient duration it may be possible to use the
micropulsations themselves as field line tracers. Thus, if it is found that Pc micropulsations change their sense of polarization near local noon, as in the case for the
ground signals reported by Sampson et al., they may also change their polarization
sense as a function of radial distance. Since the micropulsation signals are presumed
to be confined to fixed field lines it is possible for a synchronous satellite to cut through
such a shell as it rotates in a distorted magnetosphere. Simultaneous observations
at the satellite and the appropriate ground station determined as discussed previously,
could then be used to define the transfer function.
An alternative to this procedure would be the use of a second magnetometer on an
eccentric satellite in conjunction with the chain of ground stations near the conjugate
point of a synchronous satellite. The eccentric satellite could establish the radial
properties of the micropulsations while the synchronous satellite acted as a monitor
of time variations in space.
It is obvious from the preceding discussion that the goal of determining the transfer
function of a field line will not be easily reached. In fact, it appears likely that fluctuations in the magnetosphere and ionosphere are so closely coupled that such a simple
concept as the transfer function of a field line will be of limited value. Consequently,
our only method to attack this problem at present or in the near future is to determine
average magnetospheric properties such as amplitude, region of occurrence and polarization, and then compare these with the average properties measured on the ground.
Chance near approaches of spacecraft during wave events may give some measure
of the spatial extent of these events but these near approaches are very rare events.
7. Summary
As might be expected from the ground observations, a wide variety of low frequency
waves occurs throughout the magnetosphere. However, at the present time it is not
possible to make other than a tentative association for most of these waves with
distinct phenomena observed on the ground. In fact, the limited satellite observations
make it difficult to separate the magnetospheric waves into distinct phenomena.
448
R.L.MCPHERRON ET AL.
If we attempt to do this without recourse to the ground observations, we conclude
there are at least seven different types of wave phenomena observed in space.
During magnetically quiet times, transverse waves are observed across a wide band
of frequencies from Pc 1 to Pc 5. From the work of Heppner et al. (1970) the probability of observing a given frequency wave depends somewhat on spatial location
but not sufficiently to suggest the divisions derived from ground observations. Other
satellite observations are more limited and presently provide no basis for separating
this broad band of quiet time transverse waves into distinct phenomena.
Quiet time compressional waves have been reported in space but since the observations are limited to several events, it is not known whether they occur throughout
the full spectrum of Pc 1 to Pc 5.
In times of moderate magnetic disturbance (i.e., substorms) two types of transverse
waves have been reported: band limited pulsations and Pc 1. Also, a distinct type of
compressional wave is seen during substorms. In contrast to the transverse waves,
however, its spectrum falls rapidly toward high frequencies without spectral peaks.
Close to the Earth (at synchronous orbit) the disturbance is predominantly parallel
to the ambient field. However, further away in the geomagnetic tail, it appears independent of direction.
During magnetic storms two transverse wave events have been reported, one in
the Pc 1 band and one at higher frequencies (7 Hz). Mixed mode, predominantly
compressional waves have also been reported with 3 to 15 min periods.
The above wave phenomena will undoubtedly be further subdivided as more
satellite observations appear in the literature. However, since there are so few instruments in space capable of measuring these events, it is not likely that understanding
of their origin will come in the near future.
Acknowledgements
We thank the many authors of original papers for permission to use their figures.
We especially thank Drs Patel and Winckler for their helpful comments on the
manuscript. This work was supported, in part, by the National Aeronautics and
Space Administration under research grants N G L 05-007-004 and research contract
NAS 5-9098, and by the National Science Foundation under research grant GA-28907.
Appendix I. The Analysis of ULF Waves
I-1. CLASSIFICATIONOF WAVEPHENOMENA
An electromagnetic wave produces fluctuations in both the magnetic and electric
field. At any instant of time these fluctuations may be represented by two vectors
B(t) and E(t) whose magnitude is the amplitude of the perturbing field and whose
orientation is parallel to the direction of this field. A series of observations of three
components of B(t) or E(t) produces what is known as a vector time series. The
analysis of this signal may be carried out on the original time series (in the time domain)
FLUCTUATING MAGNETIC FIELDS IN THE MAGNETOSPHERE, II
449
or on the Fourier transform of the time series (in the frequency domain) or a combination of both (dynamic spectral analysis).
When a wave is measured, one of the first tasks to be performed is to identify to
which classification of distinct wave phenomena it belongs or to identify it as a new
phenomenon. In order to do this we examine its various characteristics in the time
domain and the frequency domain. The original waveform permits easy identification
of the amplitude, and amplitude modulation. If only one wave is present, or there is
one dominating wave train, other characteristics such as frequency and polarization
may be determined from the original time series. However, often the frequency and
polarization must be determined in the frequency domain especially if there is a broad
band of signals present. Some signals are characterized by their change in frequency
with time. In order to identify these signals, often neither the original waveform
or the Fourier transform of the entire time series is sufficient. In these cases, we use
dynamic spectral analysis, which Fourier transforms short segments, usually overlapping segments, of the time series so that temporal changes in the frequency of the
wave can be resolved.
As stated in this review, the identification of distinct U L F wave phenomena,
particularly in space, is a continuing process. Once a distinct phenomenon has been
identified other properties such as local time dependence, spatial location and correlations with other phenomena serve to characterize the signal. Since most observations
of U L F phenomena have been made with magnetometers we shah restrict our discussion of analysis techniques to those associated with the study of magnetic time
series.
I-2.
T I M E D O M A I N ANALYSIS
In the past, the classification of ground micropulsations has relied primarily on
analysis of data in the time domain. This is evident from the present system of classification of these waves (Jacobs et al., 1964) which is based on the waveform and the
frequency. If the waveform is regular, it is classified as a Pc (pulsations continuous).
If it is irregular, it is called a Pi (pulsations irregular). It is then assigned a number 1
to 5 for Pc phenomena and 1 or 2 for Pi phenomena, according to its frequency.
The use of this visual classification of time series although expedient in using analog
records, is undesirable because the distinction between Pc and Pi phenomena is
subjective and the measurement of periods is rather imprecise. Furthermore, there
are often subsets of phenomena within each classification and some of these phenomena
extend into two or more of the assigned frequency bands.
One of the more fruitful auxiliary techniques of analysis in the time domain is the
use of hodograms. A hodogram is a plot of one of the vector components of a wave
versus another. If three components of a wave have been measured, we can construct
three orthogonal hodograms. From the hodogram we can determine the sense of
rotation. For plasma waves, this sense of rotation should be determined relative to
the direction of the background magnetic field. Waves whose perturbation vector
rotates about the magnetic field in the same sense as protons are called left handed
450
R.L. MCPHERRON ET AL.
waves. Those which rotate about the field in the same sense as electrons are called right
handed.
The hodogram can also show whether a wave is circularly, elliptically or linearly
polarized. If it is linearly polarized, it will appear linearly polarized on all three
orthogonal hodograms. However, unless we are fortunate enough to have one of
the hodogram planes lie in the wavefront we cannot easily tell an elliptical from a
circular wave. However, if one of these planes (called the principal plane) does lie
in the wavefront, or if by digital techniques we rotate the data into this plane, we
can use the hodogram to measure quantitatively how elliptical the wave is. The most
simple measure of this is the ellipticity, the ratio of the minor to major axis. We note
that that the ellipticity may be more generally defined, as the ratio of the perturbation
in the wavefront along the direction defined by the projection of the magnetic field in
the wavefront (Russell, 1968; Russell et al., 1971) to the perturbation perpendicular
to this direction. In this case, either direction may be the major or minor axis and the
ellipticity ranges from 0 to oo.
However, in this review we have used the simpler definition where 0 refers to a
linearly polarized wave and 1 refers to a circularly polarized wave. On the other hand,
it is convenient to assign a sign to the ellipticity, negative for left hand polarized
waves and positive for right hand polarized waves. This can be done for either definition of ellipticity.
If the vector time series is recorded on magnetic tape, we may reprocess the data
at will, any number of times and with a variety of techniques. Any analogue technique
has its counterpart on a digital computer and vice versa. However, since most satellite
data, especially those on U L F waves, are digitized at some stage we shall discuss these
operations in terms of digital techniques. The first of these operations, bandpass
filtering, is a modest advance over the examination of raw data. It allows the study
of waves in a particular frequency range without interference from other phenomena.
This technique has been used on analog data in the past and is still quite useful in
studying digital data, (cf. Figures 6, 12, 27). (For detailed information on digital
filtering see the special issue on digital filters IEEE, 1968.)
Upon isolating a wave by bandpass filtering, we can actually find the wavefront,
or principal plane, of the oscillation if we have a plane wave. This is done by finding
the eigenvalues and eigenvectors of the variance ellipsoid created by the end point
of the perturbation vector as it sweeps out some volume in space. The eigenvector
associated with the minimum eigenvalue is parallel to the wave normal or equivalently
perpendicular to the wavefront. The eigenvector associated with the maximum eigenvalue is the direction of the major axis of the perturbation ellipse. If we have isolated
a single plane wave by our bandpass operation, the ellipticity is simply the square
root of the intermediate eigenvalue to the maximum eigenvalue. The third eigenvalue
may be used as a measure of the noise in the analysis. This technique is entirely analogous to the method of determining the normal to the magnetopause pioneered by
Sonnerup and Cahill (1967, 1968). Finally, we can transform the original data into
a new coordinate system, called the principal axis system with the rotation matrix
FLUCTUATING MAGNETIC FIELDS IN THE MAGNETOSPHERE, II
45t
formed by these three eigenvectors. In practice care must be exercised in this step
not to change the handedness of the coordinate system. Once this rotation has been
performed, hodograms may be constructed. Since the perturbation now, except for
noise, is confined to a single plane, these hodograms provide a very convenient and
useful display of the data, (cf. Figures 12, 30, 33a).
An alternative approach to the analysis of bandpass filtered data may be used if
we are willing to sacrifice our knowledge of the ellipticity and desire only the direction
of the wave normal and the sense of rotation of the wave. In this method, cross
products of successive perturbation are performed (McPherron and Coleman, 1971).
To minimize errors, these cross products should use perturbation vectors spaced
about 88wave period apart. Since each perturbation vector is in the wavefront, the
resulting cross product is along the wave normal. Further it is easy to show that the
dot product of this cross product with the magnetic field is positive for a right hand
polarized wave and negative for a left hand wave.
This technique is very similar to the cumulative area technique used by Dungey and
Southwood, (1970) and McPherron and Coleman, (1971). The cumulative area
technique sums up the area swept out by the perturbation vector with time in some
plane, for example, a plane perpendicular to the magnetic field direction or the
principal plane. This area will decrease or increase depending on the polarization
of the wave, and the direction of the magnetic field. The similarity of the two techniques is, of course, that the area swept out between two successive samples of the
wave is equal to the cross product of the components perpendicular to the background field. These latter two techniques are most useful in studying time series consisting of many short duration wave packets from a variety of sources, (cf. Figure 36).
Analyses in the time domain such as those listed above have the disadvantage that
they can be used on only one frequency band at a time. It is often desirable to analyze
a wide band of frequencies simultaneously. This can be done in the frequency domain.
In the following section, we will outline how this is done.
Analysis of the Spectral Matrix
For a given time interval we calculate the full 3 x 3 spectral matrix where elements
consist of all possible cross spectra between any pair of field components. We diagonalize the real part of this matrix determining its eigenvectors and eigenvalues at
each frequency. These eigenvectors are then used to transform both the real and
imaginary parts of the spectral matrix to the principal axis coordinate system. In
this coordinate system, we take the 2 x 2 submatrix corresponding to the maximum
and intermediate eigenvalues (the principal plane) and perform coherency analysis.
This process separates the submatrix into two parts corresponding to the coherent
and incoherent power in the principal plane. For the coherent portion of the signal
we determine the ellipticity and azimuth of the polarization ellipse.
The physical basis of the foregoing procedure is the following assumption. During
a limited time interval the signal in a narrow frequency band is due to a single plane
wave propagating in a fixed direction with respect to the ambient magnetic field.
452
R.L.MCPHERRON
ET AL.
Then from Maxwell's equation, V . B = 0 and therefore k . B = 0 . Thus, the magnetic
perturbation lies in a plane perpendicular to the direction of propagation. Consequently magnetic field measurements made in an arbitrary coordinate system show
variations in all three field components only because of an inappropriate choice of
coordinate axis. Diagonalization of the real part of the spectral matrix in a frequency
band including the perturbation due to the wave is equivalent to diagonalization
of the variance matrix and determines the plane of polarization. Since the fluctuations
in this plane may be only partially polarized it is further necessary to separate coherent
and incoherent power in the plane of polarization. For coherent power, one may
determine the ratio of the minor to major axis of the polarization ellipse and the
sense of rotation.
The technique of coherency analysis of a signal measured in the plane of polarization of an electromagnetic wave has been discussed several times in recent literature.
The earliest report is that of Fowler et al. (1967). Application of this technique has
been made by Rankine and Kurtz (1970), and Sampson et al. (1971).
Since most geophysical phenomena change rapidly with time, the foregoing procedure is best applied in a dynamic fashion. To do this we break the time series into
a number of overlapping segments. For each segment we carry out the eigen and
coherency analysis described above. The results are stored in two dimensional arrays
with the row index corresponding to the frequency of a given spectral estimate and
the column index to the time of the center of the segment. The results are displayed
as contour maps of a given parameter in the frequency - time plane, i.e., as digital
sonagrams.
Because this dynamic analysis technique yields so many parameters we have found
it convenient to apply it in a more limited fashion. For example, suppose that this
procedure is applied to the entire time interval including the event. It is often found
that the perturbation of interest is confined to a single plane. We then transform the
original time series to the coordinate system having this plane as the X - Y plane.
Application of the dynamic spectral analysis procedure then yields the following
quantities as a function of frequency and time: total power, percent polarization,
polarized power, ellipticity and azimuth. Since ellipticity and azimuth have no meaning unless the percent polarization is high, we do not usually display contour maps of
these two quantities.
If more than one wave is simultaneously present in a given frequency band, and
the waves are propagating in different directions, the above procedure will be in
error. Similarly if the signal cannot be represented by a plane wave, difficulties can
arise. Determination of spectra over intervals longer than the duration of a typical
wave also produces ambiguous results. Despite these difficulties we find the basic
assumption is frequently justified and consistent results obtained. In the final analysis
the merit of any technique is whether it yields new insight into the nature of the physical processes.
FLUCTUATING MAGNETIC FIELDS IN THE MAGNETOSPHERE, II
453
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