D. R. Weimer

ONE TARA BOULEVARD, SUITE 302

NASHUA, NEW HAMPSHIRE 03062-2801

dweimer@mrcnh.com

** 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* theta*is a function of the geomagnetic colatitude,* phi*
represents the magnetic local time (MLT), and* P _{l}^{m}*
is the associated Legendre function. The coefficients

The potential maps in

** 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* A _{lm}* and

As the variation of the spherical harmonic coefficient

where* omega* is the IMF clock angle. The* C _{n}* and

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

where* B _{T}* is the total magnitude of the IMF in the Y-Z plane,

In order to derive the regression coefficients, it is necessary to have measurements of the Fourier coefficients at a number of different combinations of

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

By repeating this procedure for each of the

** 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* B _{11}* 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* B _{Y}* =-1.9 nT and

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