Abstract
A method has been developed to derive the electric potentials in the
high-latitude ionosphere resulting from any arbitrary combination of the
interplanetary magnetic field (IMF) magnitude and orientation, solar wind
velocity, and dipole tilt angle. This model is based on spherical harmonic
coefficients that were derived by a least error fit of measurements from
multiple satellite passes. These harmonic coefficients have been found to have
systematic variations that can be reproduced by a combination of a Fourier
series and a multiple linear regression formula. Examples of the model output
are shown. In principal, this technique could be used as a fundamental
building block to forecast geomagnetic disturbances, or space weather , from
satellite measurements in the upstream solar wind.
Introduction
Models of the ionospheric electric potentials, or plasma convection patterns ,
have many uses in space physics research. In addition, a capability to predict
ionospheric electric fields and currents from the solar wind/IMF would be an
essential component of space weather forecasts. There has been a steady
evolution in the production of maps of the electric potential patterns that
are associated with various orientations of the IMF. These maps have consisted
of both empirical and theoretical models, often in the form of pictorial
sketches, and limited to a few generic orientations of the IMF. While useful
for the knowledge that they provide, the lack of a flexible, numerical
representation has often limited the previous models utility.
In a recent development [Weimer, 1995] a set of electric potential maps
were produced which show the ionospheric potential variations as a function of
the of IMF angle in the GSM Y-Z plane. These maps were shown for two different
groupings, according to the magnitude of the IMF or the dipole tilt angle.
These electric potential maps were derived from direct, double-probe
measurements of the electric field on the DE-2 satellite [Maynard et al.
1981], using all polar cap passes during which there were IMF
measurements available from the ISEE-3 or IMP-8 satellites. All measurements
of the electric potential along the multiple, random paths in sorted groups
were then used to derive a representation of the potential for the given
conditions in terms of a spherical harmonic expansion
where thetais a function of the geomagnetic colatitude, phi
represents the magnetic local time (MLT), and Plm
is the associated Legendre function. The coefficients Alm
and Blm are determined by a least error fit of the 2,000 to
9,000 measurement points in each group there were obtained at random
locations, rather than averaging the data in latitude-MLT bins. The number of
terms in the expansion is limited in order to filter out high-frequency noise.
The potential maps in Weimer [1995] have provided new insight about the
effects of the IMF on the magnetosphere, particularly in the matter of how the
convection evolves from a standard two-cell circulation pattern for southward
directed IMF (-Z in GSM coordinates) into a four-cell pattern for northward
IMF. As the maps are easily computed by numerical means, the potential
patterns and derived electric fields can be useful in other research, such as
numerical modeling of ionospheric plasma dynamics. However, the utility of
these maps is limited by the fact that they were originally derived only for a
few discrete combinations of IMF magnitude, angle, and season. In a new
improvement described here, a method has been developed to derive the
ionospheric electric potential patterns for any arbitrary combination of the
IMF, dipole tilt angle, and solar wind velocity.
Technique
As the original potential functions are based on spherical harmonics, it is
natural that a more flexible version be based on an interpolation of the
harmonic coefficients from the derived patterns. The critical key to creating
a flexible representation of the electric potentials is based on the fact that
each of the Alm and Blm coefficients in
(1) has a smooth and continuous variation as a function of the IMF angle in
the Y-Z plane, for a given combination of IMF magnitude, dipole tilt angle,
and solar wind velocity. This continuity is illustrated by the example in
Figure 1, which shows only the B11
coefficient as a function of the so-called clock angle for six different
conditions. The headings over each graph show the ensemble averages of the IMF
magnitude, the dipole tilt angle, and the solar wind velocity for each of the
six groups. The + marks show the values of B11 that were
determined directly from the least- error-fit routine at 16 different angles.
As the variation of the spherical harmonic coefficient B11
is periodic with respect to the angle of the IMF vector in the Y-Z plane, a
flexible representation of the electric potential patterns must also be smooth
and periodic, particularly where the IMF angle passes through zero from -Y to
+Y values. This continuity is enforced by deriving a Fourier series
representation for the harmonic coefficients, having the form:
where omega is the IMF clock angle. The Cn and D
n coefficients are derived by a least error fit, or a Fourier
transform can be used. The results of the fit are shown with the solid lines
in Figure 1. This procedure is repeated for each and every one of the A
lm and Blm coefficients for a given group of
patterns, so that each of the 51 significant coefficients in (1) is
represented by its own set of nine Fourier coefficients, as in (2).
After deriving the variations of the spherical harmonic coefficients as a
function of the angle , attention is given to the other variable parameters.
The left column in Figure 1 shows that the Blm(omega) curve
exhibits an orderly evolution as BT increases. The right
column shows a similar evolution as the solar wind velocity increases, and
there are other changes under the influence of variations in the dipole tilt
angle (not shown). These changes in the curves can be reproduced by varying
the Fourier coefficients Cn and Dn. A
multiple linear regression is used to find how each of the Fourier
coefficients varies as a function of these three controlling parameters. The
result is an expression for each Fourier coefficient in the form
where BT is the total magnitude of the IMF in the Y-Z plane,
mu is the dipole tilt angle,VSW is the solar wind
velocity, and Ri are the derived regression coefficients.
In order to derive the regression coefficients, it is necessary to have
measurements of the Fourier coefficients at a number of different combinations
of BT, mu, and VSW. Therefore, the
satellite passes were grouped into 26 separate combinations of IMF magnitude,
dipole tilt angle, and solar wind velocity. As there are a limited number of
useable satellite passes, it is not possible to sort or group the orbits by
all three variables at the same time and still have a sufficient number of
points in the 16 angular bins in order to use in the surface-fitting
procedure. Therefore, each group would be sorted according to just one or two
variables at a time. For example, one set would be sorted by the IMF magnitude
and solar wind velocity while the tilt angle was ignored in the sorting, and
another set would be sorted by tilt angle and solar wind velocity, while the
IMF magnitude had free variation. For each group all of the spherical harmonic
coefficients in (1) were derived at 16 angles, and then all of the Fourier
coefficients in (2) are calculated. These coefficients are then associated
with the ensemble average of the three parameters for their particular group,
for use in the multiple linear regression.
The resulting regression coefficients are found to be capable of faithfully
reproducing the original input data. For example, the dotted line in Figure 1
shows the values of B11 as a function of the IMF clock angle
that have been re-created from the regression coefficients by inserting the
average values of BT, mu, and VSW
into (3) and then using the derived Cn and Dn
in (2). These dotted lines are practically indistinguishable from the
original fits.
By repeating this procedure for each of the Alm and B
lm, the polar cap electric potentials are obtained from (1)
for any arbitrary combination of IMF, dipole tilt angle, and solar wind
velocity. The potential patterns that are derived in this manner are found to
be nearly identical to the originals that are shown in Weimer [1995].
An example of the new model output is shown here in
Figure 2, which illustrates nine different potential patterns as the IMF
angle varies from -90ø (-Y ) through 0 (+Z) to
+90ø (+Y) , for a moderately strong IMF (+10 nT) and solar wind
velocity (500 km/sec) at zero tilt angle. The electric potential undergoes an
orderly evolution from two cells, into four cells for northward IMF (positive
BZ), and back to the more normal two-cell pattern.
Summary
A technique has been developed which has the capability to model the
high-latitude, ionospheric electric potentials for any reasonable combination
of IMF magnitude and orientation, solar wind velocity, and dipole tilt angle.
Although the data which were used to derive this model had group-averaged IMF
magnitudes only up to 11 nT, the linear regression coefficients are capable of
producing reasonable patterns at much greater magnitudes. For example, a
southward IMF of 30 nT produces a pattern that appears to be very realistic,
having a cross-polar potential drop of 304 kV. However, caution is advised
when using or interpreting the results of this model when driven with input
conditions that exceed the limits of the original data set.
Nevertheless, this model has potential for use in space weather applications.
For example, given a measurement of the IMF upstream in the solar wind, it
would be possible to predict in advance the resulting electric fields and
currents in the ionosphere. For purpose of illustration,
Figure 3 shows the electric potentials that are obtained from this model
for the actual conditions that were measured in the solar wind by NASA s WIND
spacecraft prior to 6.7 UT on June 18, 1995. In order to emphasize the space
weather aspect, this potential pattern has been converted from the
research-oriented, geomagnetic latitude-MLT coordinate system into the
geographic coordinate system, which ultimately must be used in an application
which might forecast geomagnetic disturbances at specific locations. Such
predictions would require the use of other model calculations of ionospheric
conductivities in order to derive the currents, which is beyond the scope of
this paper.
Naturally, the question arises about how well this model reproduces the real
electric fields. As a simple, preliminary test of the accuracy, the exact time
for the example in Figure 3 had been selected from magnetometer measurements
at Fort Simpson, Canada. At 6.7 UT this station had passed under the so-called
Harang discontinuity, where the north-south electric field reverses direction
in the pre-midnight region, as indicated by a change in the magnetic field
from a positive to negative bay [Maynard et al., 1977]. As demonstated
by the location of Fort Simpson that is marked in Figure
3, the model electric potential calculations also put Fort Simpson
precisely under the Harang discontinuity at this time. More comprehensive
tests of the accuracy of this model are to be conducted in the future. There
are also plans to add a substorm component, in order to reproduce the changes
in the potential patterns that are observed during substorms.
The electric potential model described here also has applications in basic
research as well, and it already is being used by other researchers in various
numerical simulations of ionospheric plasma structure or electrodynamics. The
coefficient tables and computer codes are available from the author.
Acknowledgments. This research was supported by NSF grant ATM-9506169.
The author thanks Nelson Maynard for helpful discussions and for use of the
electric field data from the DE 2 satellite. The data from IMP 8 and ISEE 3
were provided by the National Space Science Data Center. Ron Lepping and Alan
Lazarus are the Principal Investigators for the magnetometer and plasma
instruments on IMP 8. Edward Smith and John Gosling are the Principal
Investigators for the magnetometer and plasma instruments on ISEE 3. The WIND
velocity data are from Keith Ogilvie and the magnetic field are provided by
the WIND MFI data processing team at Goddard Space Flight Center. The Fort
Simpson magnetometer data were obtained from the National Geophysical Data
Center, and provided by Gordon Rostoker.
References
Maynard, N. C., D. S. Evans, B. Maehlum, and A. Egeland, Auroral vector
electric field and particle comparisons 1. Premidnight convection topology,
J. Geophys. Res., 82, 2227-2234, 1977.
Maynard, N. C., E. A. Bielecki, and H. F. Burdick, Instrumentation for vector
electric field measurements from DE-B, Space Sci. Instrum., 5, 523-534,
1981.
Weimer, D. R., Models of high-latitude electric potentials derived with a
least error fit of spherical harmonic coefficients, J. Geophys. Res., 100,
19,595, 1995.
Figure Captions
Figure 1. Variations of the spherical
harmonic coefficient B11 as a function of the IMF angle in
the GSM Y-Z plane. The + symbols show the values of this coefficient that were
derived by a least error fit of satellite measurements that were grouped by
various criteria. The headings over each graph show the ensemble average of
the IMF magnitude, dipole tilt angle, and solar wind velocity corresponding to
the set of satellite passes in each group. The solid lines show Fourier series
fits to these data. The dotted lines show the curves that were recreated from
a multiple linear regression of each Fourier coefficient as a function of the
three variables.
Figure 2. Electric potentials derived at
nine IMF angles at a fixed IMF magnitude, tilt angle, and solar wind velocity.
The angle is stepped from -90ø (-Y) through 0ø (+Z) to
+90ø (+Y) in 22.5ø increments, as noted in the upper left corner
of each graph. The numbers at the lower left and right corners show the
minimum and maximum potentials in unit of kV.
Figure 3. An example of electric potentials that are obtained for real conditions that were measured in the solar wind by NASA s WIND spacecraft on June 18, 1995, shown in geographic coordinates. During the 40 minutes prior to 5.6 UT the WIND satellite measured an average IMF BY =-1.9 nT and BZ=-7.9 nT, and the solar wind velocity was 350 km/sec. There is a 69 min propagation delay to 6.7 UT. The arrow indicates the longitude of the sub- solar point at this time. One + mark shows the location of the geomagnetic pole and the other one marks the location of Fort Simpson, Canada.
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