Revised time-of-flight calculations for
high-latitude geomagnetic pulsations using a
realistic magnetospheric magnetic field model
J. A. Wild
1
and T. K. Yeoman
Department of Physics and Astronomy, University of Leicester, Leicester, UK
C. L. Waters
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, New South Wales, Australia
Received 7 December 2004; revised 9 May 2005; accepted 18 August 2005; published 11 November 2005.
[1] We present a simple time-of-flight analysis of Alfve´n pulsations standing on closed
terrestrial magnetic field lines. The technique employed in this study in order to calculate
the characteristic period of such oscillations builds upon earlier time-of-flight estimates
via the implementation of a more recent magnetospheric magnetic field model. In this case
the model employed is the Tsyganenko (1996) field model, which includes realistic
magnetospheric currents and the consequences of the partial penetration of the
interplanetary magnetic field into the dayside magnetopause. By employing a simple
description of magnetospheric plasma density, we are therefore able to estimate the period
of standing Alfve´n waves on geomagnetic field lines over a significantly wider range of
latitudes and magnetic local times than in previous studies. Furthermore, we investigate
the influence of changing season and upstream interplanetary conditions upon the
period of such pulsations. Finally, the eigenfrequencies of magnetic field lines computed
by the time-of-flight technique are compared with corresponding numerical solutions to
the wave equation and experimentally observed pulsations on geomagnetic field
lines.
Citation: Wild, J. A., T. K. Yeoman, and C. L. Waters (2005), Revised time-of-flight calculations for high-latitude geomagnetic
pulsations using a realistic magnetospheric magnetic field model, J. Geophys. Res., 110, A11206, doi:10.1029/2004JA010964.
1. Introduction
[2] Since Dungey [1954] proposed for the first time that
the long-period geomagnetic pulsations observed on the
ground might be the result of standing Alfve´n waves being
excited on geomagnetic field lines, a wealth of ground-
based and satellite observations have established the impor-
tance of field line oscillations, and it has become recognized
that such waves transfer both energy and momentum
through the coupled magnetosphere-ion osphe re system.
These processes are most significant in the high-latitude
ionosphere, where the magnetosphere-ionosphere interac-
tion is strongest. The waves also act as an important
diagnostic of magnetospheric morphology and dynamics.
[
3] In general, ultra-low frequency (ULF) waves have an
energy source external to the Earth, such as an impulse in
the solar wind, solar wind buffeting, or the Kelvin-Helm-
holtz instability on the magnetopause. However, the se-
quence of events which connect such broadband boundary
oscillations to the resonant driving of an individual field line
at its natural eigenfrequency is not straightforward. Initially,
a direct coupling between the wave source on the boundary
and a local field line was proposed, which successfully
explained many observed features of field line resonances
[e.g., Southwood, 1974]. The observed discrete frequency
spectrum remained a puzzle, however, and subsequently
global compressional oscillations of the magnetosphere,
setting the cavity formed by magnetospheric boundaries
into oscillation, were suggested as a cause of a local field
line resonating at its characteristic frequency by Kivelson et
al. [1984]. In this scenario the field line driven to resonance
was one whose eigenfrequency matched a natural frequency
of the magnetospheric cavity. More recently, it has become
accepted that the magnetospheric cavity oscillation is better
described as a magnetospheric waveguide, with the charac-
teristic frequencies of the waveguide coupling to field line
oscillations [e.g., Harrold and Samson, 1992; Samson et al.,
1992; Rickard and Wright, 1994]. Such a waveguide mode
may be excited directly by buffeting of the magnetosphere
or through a broadband wave source on the magnetopause
[e.g., Mann et al., 1999]. However, experimental evidence
for plasmatrough cavity/waveguide modes remains sparse
[e.g., Waters et al., 2002].
[
4] In understanding observations of field line oscilla-
tions, a knowledge of the expected eigenperiods of the field
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, A11206, doi:10.1029/2004JA010964, 2005
1
Now at Department of Communication Systems, Lancaster University,
Lancaster, UK.
Copyright 2005 by the American Geophysical Union.
0148-0227/05/2004JA010964$09.00
A11206 1of10
lines as a function of magnetic latitude, local time, and
season is clearly of vital importance. At middle and low
latitudes, where the magnetic field is close to dipolar,
theoretical estimates of field line eigenperiods have been
computed by Cummings et al. [1969] and Sinha and
Rajaram [1997], for both the toroidal and polodial modes
of oscillation. In such calculations a hydrogen plasma is
assumed. At low and equatorial latitudes, where heavier
species such as oxygen and helium take over, similar
calculations have been presented by Poulter et al. [1988].
At high latitudes, deviations from a dipolar field become
significant. Such complexities in the field geometry prevent
an analytical treatment of the eigenperiods, and this problem
was addressed via a simple time-of-flight approach by
Warner and Orr [1979]. Warner and Orr [1979] used the
Mead and Fairfield [1975] model of the terrestrial magnetic
field, which allowed calculations of field line eigenperiods
up to latitudes of 75 (68) in the noon (midnight) sector
and also allowed the effects of magnetic activity and dipole
tilt to be investigated. The Warner and Orr [1979] time-of-
flight approach to eigenmode calculations at high latitude
has been used extensively in the interpretation of ULF wave
observations. Their results were invoked in the interpreta-
tion of early results from auroral zone and subauroral VHF
coherent radar measurements of ULF waves [e.g., Villain,
1982; Yeoman and Lester, 1990]. The results have also
provided a baseline for the explanation of the local time
variations of magnetic field oscillations at Pc5 frequencies
observed in ground-based data [Yumoto and Saito, 1983;
Glassmeier and Stellmacher, 2000]. The technique has been
equally important in the interpretation of the observed
periods in ground-based data at very high magnetic latitudes
near the cusp region [Ables et al., 1988; Waters et al., 1995;
Clauer et al., 1997] and similarly in high-latitude spacecraft
observations out to L shells up to L = 9 [e.g., Anderson et
al., 1989, 1990]. In addition to these experimental studies,
the modeling of Warner and Orr [1979] has also been used
to provide baseline eigenfrequencies for ULF wave model-
ing efforts in realistic geometries [e.g., Allan et al., 1986;
Lee and Lysak, 1990; Wright, 1992].
[
5] The eigenfrequencies of ULF waves provide an
important diagnostic of the plasma loading along the field
line, both within the plasmasph ere and the plasmatrough
[e.g., Poulter et al., 1984; Waters et al., 1996; Loto’aniu et
al., 1999; Dent et al., 2003]. At higher latitudes the
interpretation of such data in terms of plasma loading
clearly requires a realistic model of the magnetic field
geometry as a prerequisite for any accurate inference of
the plasma density.
[
6] Since the original calculations of Warner and Orr
[1979], great advances have been made in magnetic field
modeling, and the limitations of the Mead and Fairfield
[1975] model have become clear. The location of the open/
closed field line boundary and magnetic field stretching in
the midnigh t sector are two particularly clear instances
where the Mead and Fairfield [1975] model requires
updating. Recently, Rankin et al. [2000] solved the funda-
mental eigenmode equations under the magnetic field ge-
ometry of the Tsyganenko [1996] magnetic field model
[Tsyganenko, 1995, 1996] in the midnight sector, demon-
strating that frequencies could result which were an order of
magnitude lower than those expected for a dipole field.
Here we present a refinement of the simpler time-of-flight
approximation of Warner and Orr [1979], based upon the
Tsyganenko [1996] magnetic field model in order to explore
such effects under a wide variety of latitude, local time,
dipole tilt, and magnetic activity conditions.
2. Models
[7] Approximating the oscillations on terrestrial field
lines as Alfve´n waves standing on individual field lines,
the period of oscillation, t, can be approximated by the
time-of-flight approximation
t ¼ 2
Z
ds
V
A
; ð1Þ
where V
A
is the Alfve´n velocity, and the integration is carried
out over the entire length of the field line. The Alfve´n speed at
some position on the field line s is described by
V
2
A
¼
B
2
m
0
r
; ð2Þ
where B and r denote the local magnetic field strength and
plasma density, respectively. Therefore in order to calculate
the period of an Alfve´nic oscillation on a terrestrial field line,
it is necessary to employ realistic descriptions of the
magnetospheric magnetic field and plasma environment.
2.1. Magnetic Field Model
[
8] For the purposes of this investigation, we have
implemented the Tsyganenko [1996] magnetic field model
[Tsyganenko, 1995, 1996], hereafter referred to as the
‘‘T96’’ model. This model has several significant advan-
tages over the magnetospheric field model of Mead and
Fairfield [1975] employed by Warner and Orr [1979]. Most
importantly for the current study, the T96 model describes
the Earth’s magnetotail much more realistically than that of
Mead and Fairfield [1975] which extended to only 17 R
E
downstream of the planet. Furthermore, the magnetic field
described by the T96 model is derived from several modular
elements representing magnetic field sources within the
magnetopause current system, the ring current, the cross-
tail current sheet, and Region 1 and 2 Birkeland current
systems, in addition to the Earth’s internal field. As such the
T96 model contains none of the built-in dawn-dusk and
north-south symmetries found within the earlier model and
allows for a more realistic description of local time effects.
An additional key difference between the field models of
Tsyganenko [1995, 1996] and Mead and Fairfield [1975] is
that the former is parameterized by the Dst index, the solar
wind dynamic pressure, and the B
Y
and B
Z
components of
the IMF incident upon the magnetopause, whereas the latter
relies only upon a crude parameterization by the K
P
index.
2.2. Plasma Model
[
9] In order to investigate the implementation of a more
realistic magnetospheric magnetic field model upon the
results of Warner and Orr [1979], we have retained the
plasma density model utilized in that study. This model,
based upon data reviewed by Chappell [1972], is summa-
rized in Table 1. The in situ magnetospheric plasma densi-
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ties are drawn from OGO 5 mass spectrometer measure-
ments during 1968 –1969. The distribution of magneto-
spheric plasma along individual field lines at distances
greater than 2 R
E
can then be represented as proportional
to r
n
, with the exact value of n depending upon the chosen
region of the magnetosphere We have departed from this is
in only two regards. First, the description of the plasma
density within the plasmatrough given by Warner and Orr
[1979], which is local time dependant due to filling from the
ionosphere during the day, is insufficiently detailed to allow
full implementation in this model. We have therefore
reverted to the plasmatrough plasma density distribution
predicted by Chappell et al. [1971] assuming a constant
upward flux of 3 10
8
ions cm
2
s
1
with a collisionless
distribution of ionization along the field line. Second, due to
the extended tail of the T96 model, compared to that of
Mead and Fairfield [1975], it has been necessary to
introduce a minimum magnetospheric plasma density in
order to avoid excessively large Alfve´n speeds in the distant
tail. For this study, we have therefore set the minimum
plasma number density to be 0.01 ions cm
3
.
3. Results
[10] Figure 1 presents an overview of the estimated
variationinAlfve´n pulsation period as a function of
geomagnetic latitude under a variety of conditions. In order
to gauge the accuracy of the time-of-flight method
employed in this study, we compare the results with those
from a numerical solution to the wave equations approach
under identical geomagnetic conditions. Each panel of
Figure 1 presents a pulsation period-latitude profile calcu-
lated using the time-of-flight approximation applied to the
T96 model (solid line), the same approach applied to a
purely dipolar field (dotted line), and a numerical solution
Table 1. Plasma Density Models for Various Magnetospheric
Regions (Adapted From Warner and Orr [1979])
a
Region
Typical Density
at L = 4, ions cm
3
Equatorial
Density Variations
Extended plasmasphere 450 R
3
Dusk plasmasphere
b
188 R
4
Plasmatrough
c
0800 MLT 15 R
4
1400 MLT 60 R
3
Detached plasma
Upper limit 400 R
4
Lower limit 100 R
4
a
Minimum density = 0.01 ions cm
3
.
b
Limited to 1500 – 2100 MLT.
c
MLT dependence described by Chappell et al. [1971]; see text for
details.
Figure 1. The variation of pulsation period with geomag-
netic latitude, calculated using the time-of-flight approx-
imation (solid lines) and the Singer et al. [1981] numerical
technique (dashed lines) based upon the T96 magneto-
spheric model. In each case the model is parameterized as
indicated. An equivalent period-latitude profile, computed
using a time-of-flight analysis based upon a simple dipolar
field and an identical magnetospheric plasma distribution, is
shown for comparison (dotted line).
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of the wave equations based upon the application of the
equations of Singer et al. [1981] to the T96 model (dashed
line). The wave equation described by Singer et al. [1981]
was numerically solved using a fourth-order Runge-Kutta
algorithm as described by Waters et al. [1995]. In order to
numerically solve the Singer equation, it is necessary to
compute the geometry of adjacent field lines to obtain the
h
a
parameter. At high latitudes, the field tracing for one
field line may track to the opposite hemisphere, while the
other tracks an open field line. Furthermore, the separation
between adjacent field lines can become large along some
sections with questionable results from the numerical inte-
gration. Any results that exhibited field tracing difficulties
were rejected.
[
11] In each panel of Figure 1, the plasma model
employed is identical in order to emphasize the variations
due to differing magnetic field configurations. More specif-
ically, we have defined the plasma density at L =4tobe
100 ions cm
3
, decreasing with radial distance in the
equatorial plane with a r
4
dependence. In all panels, three
of the four T96 input parameters remain constant (Dst =
0 nT, IMF B
Y
=IMFB
Z
= 0 nT).
[
12] Figure 1a corresponds to the pulsation period in the
0200 MLT magnetic meridian under spring equinox con-
ditions (equivalent to 0 dipole tilt) with the solar wind
dynamic pressure (P
SW
) set at 2 nPa. In this configuration,
the time-of-flight (solid line) and Sing er et al. [1 981]
(dashed line) estimates of field line eigenfrequency are in
almost exact agr eement in the region where both are
calculated. Furthermore, both approaches based upon the
T96 model estimate the eigenperiod to be significantly (in
excess of 300%) larger than the time-of-flight estimate for a
dipolar field geometry (dotted line). This is as expected
given that the dipolar magnetic field is a wholly unrealistic
description of the midlatitude/high-latitude magnetic field
geometry in the magnetotail.
[
13] Figure 1b corresponds to the 1200 MLT (noon)
meridian at northern hemisphere summer solstice (equiva-
lent to 34 dipole tilt) with P
SW
= 6 nPa, while Figure 1c
corresponds to the 0800 MLT magnetic meridian under
winter solstice conditions (equivalent to 34 dipole tilt)
with P
SW
= 1 nPa. In both cases, the agreement between the
two approaches is generally excellent at magnetic latitudes
equatorward of 70 Mlat (the numerically calculated
figure being 75% of the time-of-flight equivalent). At
latitudes poleward of this, the difference between the values
increases (the equivalent comparison being 65% in
Figure 1b at 75 Mlat). However, we note that in the
case of Figure 1b, the estimates then converge at 78.
Therefore the time-of-flight approach (when applied to the
T96 model field) yields estimates tha t are broadly in
agreement with the Singer et al. [1981] numerical tech-
nique. While at very high latitudes (poleward of 75 Mlat)
the difference between the two estimates may be significant
(several tens of percent), it is difficult to determine which
technique is the most accurate. Given that the time-of-flight
approach allows the rapid investigation of Alfve´n pulsations
over a larger range of latitudes than the Singer et al. [1981]
technique (subject of course to the accuracy of the magnetic
field model employed), we shall use it to investigate the
latitudinal, diiurnal, seasonal variations in the frequency/
period of Alfve´nic pulsations on geomagnetic field lines. In
the final section our estimates will be compared to field line
oscillations presented in investigations by other authors.
3.1. Variation of Period With Latitu de
[
14] Figures 2a, 2b, and 2c present the latitudinal varia-
tions in Alfve´n pulsation period under equinox conditions
between 57–80 magnetic latitude along the 0800 MLT,
1600 MLT, and 0000 MLT magnetic meridians, respectively.
In all cases the T96 input parameters were set to P
SW
=
2 nPa, Dst = 0 nT, IMF B
Y
and IMF B
Z
= 0 nT such that the
configuration of the magnetospheric field remained con-
stant. In each panel the results of time-of-flight analysis for
the plasma regions listed in Table 1 are presented.
[
15] As expected, the characteristic period of Alfve´nic
micropulsations generally increases with magnetic latitude
in all local time sectors, irrespective of the plasma distri-
bution selected. Figure 2a is broadly consistent with previ-
ous results under similar geomagnetic conditions. However,
our calculations improve upon those of Warner and Orr
[1979] in several regards. Most significantly, the more
realistic magnetic field model employed allows us to extend
our time-of-flight calculations to magnetic field lines that
originate at higher latitudes. For example, we note that in
this local time sector, the calculations of Warner and Orr
[1979] were limited to magnetic latitudes <73. Comparison
of Figures 2a and 2b reveals slight differences between the
postdawn and predusk sectors (in addition to the extra dusk
plasmasphere trace), the consequence of local time asym-
metries contained within the T96 magnetic field model but
omitted from the Mead and Fairfield [1 975] model
employed by Warner and Orr [1979]. Furthermore, merid-
ional latitude-period profiles originating from the southern
hemisphere (not shown) indicate small north-south asym-
metries, also not included in the Mead and Fairfield [1975]
model.
[
16] The final latitude-period profile presented here cor-
responds to the midnight meridian (Figure 2c). These
profiles represents a significant departure from the estimated
pulsation period-latitude relationships proposed by Warner
and Orr [1979] due to the extended nature of the magneto-
tail described by the T96 model. While the Mead and
Fairfield [1975] model was valid only within 17 R
E
of the
Earth, the equatorial crossing points of field lines in the
midnight sector are predicted to be located much further
down tail at latitudes poleward of 68 Mlat . For the
purposes of this investigation , we have designated any
magnetic field lines that extend further than 300 R
E
from
the Earth as ‘‘open.’’ The local maximum in each latitude-
period profile (located at 68 Mlat) is the result of the
rapidly increasing field line length at increasingly northward
positions on the midnight meridian. This effect, a conse-
quence of T96 model configuration, is indicated i n
Figure 2c: the variation of field line length in the midnight
sector, plotted as a function of magnetic latitude, is indicated
by the dashed line, according to the scale on the right-hand
axis. This effect is limited in azimuth to approximately
±2 hours MLT either side of midnight.
3.2. Variation of Period With MLT
[
17] Figure 3 indicates the variations in micropuslation
period for the different plasma regions at equinox as a
function of MLT at 68 (Figure 3a), 72 (Figure 3b), and 76
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A11206
(Figure 3c) Mlat . As in previous cases, the T96 input
parameters were set to P
SW
= 2 nPa, Dst = 0 nT, IMF B
Y
=
IMF B
Z
= 0 nT. The results shown in Figure 3a are
consistent with those presented by Warner and Orr
[1979]. In the extended plasmasphere, dusk plasmasphere,
Figure 3. The variation of pulsation period as a function
of magnetic local time under equinox conditions for
terrestrial field lines at (a) 68, (b) 72, and (c) 76 Mlat.
Figure 2. Magnetic latitude-period profiles for various
plasma distributions at (a) 0800 MLT, (b) 1600 MLT, and
(c) 0000 MLT under equinox conditions. The radial
displacement of the equatorial crossing point of magnetic
field lines are indicated in each case. In Figure 2c the
variation of magnetic field line length in the midnight sector
described by the T96 model is also plotted as a function of
magnetic latitude under equinox conditions (dashed line).
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![Table 1. Plasma Density Models for Various Magnetospheric Regions (Adapted From Warner and Orr [1979])a](/figures/table-1-plasma-density-models-for-various-magnetospheric-15v5mcn2.webp)
![Figure 7. (a) Quiet-time field line resonance frequencies as a function of L value for the interval 0700–0800 UT on 1 January 1998, as presented by Menk et al. [2004]. These frequencies were determined via the twin-station crossphase (filled circles) and single-station power ratio analysis (unfilled circles) of ground-magnetometer data. (b) The diurnal variation of the Alfvén pulsation continuum (solid lines) derived via cross-phase analysis for three station pairs of the IMAGE ground magnetometer array, as presented by Mathie et al. [1999]. The symbols indicate the station pair used according to the key. Eigenfrequencies derived via the T96 time-of-flight technique are overlaid at the appropriate latitude (dotted lines).](/figures/figure-7-a-quiet-time-field-line-resonance-frequencies-as-a-26wsbwec.webp)