Pages 895-905 |

D. J. Knipp^{1} and B. A. Emery^{2}

^{1}Department of Physics, Suite 2A6,
Fairchild Hall, U. S. Air Force Academy, CO 80840, USA,

E-mail: knipp@ncar.ucar.edu

^{2}High Altitude Observatory, National Center
for Atmospheric Research, Boulder CO 80307-3000, USA

ABSTRACT

In this article we briefly review the Assimilative Mapping of Ionospheric
Electrodynamics (AMIE) procedure and discuss responses to inferred dayside
and nightside reconnection events from an ionospheric viewpoint. Output
from the AMIE procedure is used to illustrate our discussion. We focus
on events occurring just prior to and during the passage of a non-compressive
density enhancement, a type of solar wind transient, on 3 November 1993,
and during a more normal substorm event on 10 November 1993. We present
instantaneous convection and current maps for several times during the
events and 'difference maps', which show how the currents and convection
change over short periods of time. The 'differencing' capability, especially
when combined with other information, such as particle injection data and
magnetometer pulsation data can provide a powerful diagnostic for identifying
substorm onset and investigating substorm and reconnection response.

INTRODUCTION

Instantaneous ionospheric convection and current maps which incorporate
multiple data sources are making significant contributions to the body
of knowledge regarding substorms and reconnection events [Kamide et al.,
1994, Lu et al., 1996, submitted, Siscoe and Lu, 1996, submitted, and Taylor
et al., 1996, submitted]. These maps are generally synthesized from observations
made by ground-based radar, ground magnetometers and low-earth orbiting
satellites. One means of combining such diverse types of observations into
"snapshots" of ionospheric features is the Assimilative Mapping
of Ionospheric Electrodynamics (AMIE) procedure [Richmond and Kamide, 1988].
We have used the AMIE procedure to assimilate observations from 116 high-latitude
northern hemisphere ground magnetometer stations to produce the convection
and current maps in Figures 1,
2 and 3. Satellite particle precipitation
data, as well as ground magnetometer data, were also used to produce distributions
of ionospheric conductance (not shown) which link the current maps to the
electric field maps. Although other data sources are available for the
current and convection mappings, we have chosen in this study to use only
ground magnetometer observations as the data input in order to eliminate
artifacts in the mapping which sometimes appear when data sources, such
as satellites, enter and exit the mapping region .

The AMIE procedure is an optimally constrained, weighted, least-squares
fit of coefficients to observed data. Estimated Pedersen and Hall conductivities
allow the ionospheric electric fields and currents to be related though
Ohm's law. The first data fit within the procedure is one that estimates
the height integrated Pedersen and Hall conductivities by modifying statistical
models. Assuming those patterns and values have been reliably determined,
the procedure goes on to fit ionospheric convection patterns from direct
observations of ion drift observations made by satellites, radars, and
digisondes and from indirect observations associated with the inversion
of magnetic perturbations. These patterns from statistical data bases can
be used as information to insure physical solutions to the fit in data
sparse regions. The procedure also has an objective means of estimating
the uncertainty in the electric field maps. These uncertainties are typically
50% or less [see for example Richmond and Kamide, 1988]. During intervals
of good data coverage much of the field is estimated with an uncertainty
of ~30%.

THE ASSIMILATIVE MAPPING OF IONOSPHERIC ELECTRODYNAMICS PROCEDURE

TECHNIQUE

The procedure determines internally consistent patterns of the electrostatic
potential the electric field
**E**, the height integrated horizontal ionospheric electric current
density **I**, the field-aligned current density J_{||}, and
the magnetic perturbations **B**.
By performing a weighted, least squares fit the procedure insures the best
possible pattern match to the available observations. Constraints are placed
on the fit by *a priori* information, typically statistical patterns,
that force the fit to a physical solution in areas of sparse data. Each
of the quantities is expanded in a series of basis functions in magnetic
latitude and longitude as

**(,)
= **_{i}**a**_{i}** **_{i}**(,)**

**E(,)
= **_{i}**a**_{i}**E**_{i}**(,)**

**I(,)
= **_{i}**a**_{i}**I**_{i}**(,)**

**J _{||}(,)
= **

**B(,)
= **_{i}**a**_{i} **B**_{i}**(***h***,,)**

where *h* is height. For simplicity ,
**E**, **I**, and J_{||} are treated as being mapped to an
ionospheric reference height of 110 km, but the height dependence of **B
**must be kept explicitly, since it does not simply map along geomagnetic
field lines.

We can currently use up to 244 electric potential (_{i})
basis functions for the estimation (although fewer, typically, 79, can
be used for a work-station version of AMIE output, as is the case for this
paper). The functions are generated by combining generalized associated
Legendre functions at high latitude with low latitude functions that satisfy
a zero Laplacian requirement and are capable of representing any general
pattern of electric potential poleward of 40° magnetic latitude. The
244 basis functions have effective wavelengths, down to about 6° in
the north-south direction, signifying effective distance scales down to
6°/(2) or roughly 1°. The effective
distance for fewer coefficients increase as the square root of the ratio
of 244 to the number of new coefficients, so 79 coefficients results in
effective distances scales of about 2° in latitude and 8° in longitude.
In longitude the minimum wavelength is 30°, corresponding to a distance
scale of about 5°. The other basis functions for the electric field
**E**_{i}, the horizontal current density **I**_{i},
and the field-aligned current density J_{||i}, are derived from
_{i} by the following relationships:

**E**_{i}**=-**_{i}

**I**_{i}**=E**_{i}

**J**_{||i}**=I**_{i}

where is the ionospheric
conductance tensor. A basis function _{i}
for the equivalent current function is obtained by solving:

^{2}_{hi}=**r**x **I**_{i}

where h^{2}
is the component of the Laplacian differential operator involving only
horizontal derivatives and **r** is the unit radial vector. This relation
is valid under the simplifying assumption used in AMIE that all electric
currents above the ionosphere are purely radial, thus ignoring the tilt
of the magnetic field lines, and the existence of the ring current, the
magnetopause currents and the magnetotail currents.

Relating the magnetic deviations to the three dimensional current patterns
is conceptually and computationally simplified if the height integrated
horizontal ionospheric current is considered as two parts: a toroidal current
**I**_{T}, which circulates without divergence
in the ionosphere, and a poloidal current **I**_{P},
which is the ionospheric horizontal closure of field-aligned currents generated
in the distant magnetosphere. Both of these components make a contribution
to the height integrated horizontal current **I**

**I = (x)-=I**_{T}**+I**_{P}

The ground-based magnetic variations can be associated with only the
divergence-free (toroidal) component of the height integrated horizontal
current, the so-called 'equivalent current.' The ionospheric horizontal
equivalent sheet current, **I**_{equiv}, is related to the equivalent
current function by

This fictitious current would produce the same ground magnetic effects
as the actual three-dimensional overhead current including the field-aligned
currents. The basis functions for the equivalent current are used to calculate
those for the ground-level magnetic potential V_{i}, by assuming
the sheet current to be at an altitude of 110 km. Such an overhead current
will induce currents in the earth which in turn give rise to ground magnetic
effects. AMIE uses the approximation of a perfectly conducting layer at
a depth of 250 km, overlain by a perfect insulator to estimate the surface
magnetic effects of the induced currents. The basis functions **B**_{i}
for the magnetic perturbations on the ground are then calculated from

The procedure incorporates both horizontal and vertical components of
magnetic perturbations as data input. The most direct use of the vertical
information is in determining the locations and intensity of the auroral
electrojets, although the procedure gains collateral information about
field-aligned currents and electric field distributions from this data.

The effects of this three dimensional (poloidal) current system on a satellite
magnetometer are calculated in AMIE by assuming that the poloidal contribution
to height-integrated horizontal current comes from the closing of the field-aligned
currents across the ionosphere. The local divergence of the height-integrated
current arises from the entry and exit of the vertical field-aligned currents.
Since these closing currents are curl-free they can be expressed in terms
of a current function, . We express
the field-aligned current density as

Information about the field-aligned currents is derived from the divergence of the height integrated current and from the measurements of current flow above the ionosphere (i.e. satellite magnetic perturbations). We assume that these currents produce no magnetic effect at the ground. Satellite magnetic perturbations measured above the ionosphere can be expressed as:

PRIOR INFORMATION

In an ideal situation the observations would provide most if not all of
the information about the coefficients. In some situations or locations,
however, observations may be very limited, unreliable, or simply unavailable.
In data sparse regions the AMIE technique relies partly on statistical
information to reconstruct reasonable electrodynamic patterns. In this
manner we bypass the typical non-uniqueness problems usually associated
with data poor regions. We accept *a priori *information about both
the electric potential distribution and the conductance distribution. A
10-level Hemispheric Power Index (HPI) computed from NOAA and/or DMSP satellite
particle precipitation data, can be used to parameterize the electric potential
[Foster et al., 1986] and conductance [Fuller-Rowell and Evans, 1987] distributions.
These precipitation data are interpolated in time to provide HPI values
throughout the period of interest. Statistical patterns of electric potential
have been derived by Foster et al. [1986] who used seven years of Millstone
Hill radar plasma drift measurements and binned the measurements according
to the HPI of auroral precipitating particles, the latitude and magnetic
local time, and then derived statistical models of the electric potential
from the binned data values. The electric potential distribution can also
be parameterized by IMF orientation [Heppner and Maynard, 1987] if solar
wind data are available.

UNCERTAINTIES

The method of determining the uncertainty in the estimated fields is discussed
by Richmond and Kamide [1988]. This uncertainty arises both from the total
effective errors in the observations (which includes the effects of errors
in the assumptions used by the AMIE procedure, as well as, measurement
error) and from the incomplete coverage of the observations. Data-poor
regions like the Arctic Ocean, for example, have large uncertainties in
the mapped fields. The weight for any given datum is the inverse square
of its expected error. The error includes not only measurement errors,
but also errors due to finite truncation of the series of coefficients,
errors in the assumptions employed in the AMIE procedure and errors introduced
by accepting data from a window of time about the period of interest. In
general the observations windows are plus or minus 15 minutes for satellite
data, and approximately 5 minutes for ground-based data. We expect greater
errors to be associated with longer temporal mismatches. We impose exponentially
decreasing weights on observations temporally distant from the estimation
time. In addition, we now compare the mean square differences between computed
fields and observations with what the error analysis of AMIE predicts for
these differences, and when necessary we adjust the effective data errors
to improve the agreement in this comparison. This adjustment increases
the objectivity of the data error specifications, which are a necessary
input to the AMIE procedure.

CONDUCTANCE

The ability of the AMIE procedure to accurately relate the currents
to the electric field via Ohm's Law requires knowledge of the ionospheric
conductance distribution. The AMIE technique can incorporate direct and
indirect information about the height integrated conductivities. The procedure
has an explicit estimation of the distributions of auroral energy flux
and mean particle energy, based on the direct observations from satellites
and indirect information from ground magnetometers. The energy flux and
mean energy of auroral particles are closely related to the height-integrated
ionospheric Pedersen and Hall conductivities, which are estimated in the
first stage of the AMIE procedure. Our initial estimates of the auroral
energy flux are made by the statistical auroral conductivity model of Fuller-Rowell
and Evans [1987], parameterized by the 10-level Hemispheric Power Index
(HPI). Energy fluxes and average energies of the auroral precipitating
electrons measured by DMSP, and most recently UARS [Lu et al, 1996], are
used to calculate Hall and Pedersen conductance modifiers. The conversion
from flux to conductance for DMSP data are based on empirical formulae
from Robinson et al. [1987] and a UV conductance model is added. The electron
transport code of Fuller-Rowell and Evans [1987] estimates conductivities
from the NOAA satellites which are also used as modifiers for the statistical
patterns. Simultaneous observations from both hemispheres are often incorporated
into the conductance fitting routine, assuming the auroral precipitation
is approximately conjugate. But such observations are typically given only
half the weight of observations from the hemisphere of interest. More indirect
information about the conductance distribution can also be obtained from
empirical relations of conductance and ground magnetic perturbations [Ahn
et al., 1983]. Additional discussion of the technique can be found in Richmond
[1992] and Richmond et al. [1996, submitted].

MAPPING CURRENT AND CONVECTION VARIATIONS ASSOCIATED WITH SUBSTORMS

In the past much of the emphasis in substorm research has focused on the AE index and its components AU and AL. With the maturation of global mapping schemes and the advent of new auroral observing techniques and instruments, it will be possible to be far more quantitative about the spatial and temporal development of substorms. As we show below, it is possible to locate the peak current, and the latitude of substorm onset. As suggested by Cooper et al. [1995], it is also possible to use gridded AMIE output to recalculate the AE index during intense geomagnetic activity when the electrojets have moved beyond the locations of the AE observatories. This further suggests that it will be possible to estimate, from the gridded output, the size of the mid-latitude bays associated with substorms and to compare these estimates with individual and observatory chain estimates. By using the differencing scheme, that we illustrate below, it is possible to study the evolution of the convection current enhancements and the development of the surge or wedge current. It is also possible to estimate the energy dissipated in Joule and particle heating. Finally, when auroral images are available it will be possible to estimate the intensity area moment of the auroral bulge and trace the bulge development along with the development of the current systems.

EXAMPLE 1. MODERATE AND STRONG GEOMAGNETIC ACTIVITY

We illustrate an event which begins with a near classic substorm generated
by a southward turning of the IMF. However, shortly after the recovery
phase began a large density enhancement in the solar wind caused considerable
compression of the magnetosphere. A new substorm developed almost immediately
upon arrival of the density enhancement. We provide a snapshots of both
disturbances in Figure 1.

Fig. 1. AMIE maps viewed from the north geomagnetic pole,
which is at the center of each plot. The outer ring is 50° magnetic.
The time and IMF values are in the upper left corner of each plot. Column
1 contains horizontal current vectors with vector length representing 0.5
A as indicated in the lower right corner. Column 2 contains electric potential
contours at 10 kV intervals, the potential difference between the maximum
value and minimum value is printed in the upper right corner. column 3
contains the field-aligned currents. The contour interval is 0.3 µA/m2,
beginning at ± 0.15 µA/m2. The downward currents have solid
contours. The hemispherically integrated value of the downward field-aligned
current is shown in the upper right corner of the plot.

Each row of Figure 1 contains a horizontal current, electric potential
and field-aligned current map for the times designated in the upper left
corner of each map. Also in the upper left corner of each map are the interplanetary
field components observed by the GEOTAIL spacecraft which was 200 Re downtail
of the earth (the observations have been "backed" to time of
earth passage). Figure 2 shows 'difference plots', which are made by subtracting
the present 5-minute snapshot from that of the previous 5-minute interval.
We use these difference maps to illustrate the changes occurring in convection
and currents on the 5-minute time-scale.

Fig. 2. Difference plots for column 1) horizontal currents, column2) electric potential and column 3) field-aligned current. The differences are created by subtracting the gridded values for the previous 5-minute interval from those at the time shown in the upper left of the map. Note that for clarity the current difference vectors have been rescaled to 0.25 A/m as indicated in the lower right corner. The potential contours are 5 kV and the field-aligned current contours are 0.15 µA/m2, starting at 0.075 µA/m2.

Figures 1a-c show the ionospheric state of horizontal currents, electric
potential and field aligned currents, respectively at 2155 UT on 3 November.
The maps are indicative of quiescent ionospheric conditions which had prevailed
for approximately one hour. The cross polar cap voltage in Figure
1b was 27 kV and the area-integrated downward field-aligned current
(Figure 1c) was 1.0 MA. Within the subsequent 5 minute interval the IMF
turned southward. Figures 1d-f show an immediate dayside response in terms
of both enhanced currents and convection. The cross polar cap voltage rose
by 19 kV and the field aligned current strength increased by 30% in the
5-minute interval. The elements of Figure 2, row 1 shows the 5-minute ìdifference
mapsî produced by numerically subtracting the maps of Figure 1, row
1 from Figure 1, row 2. Consistent with the ideas presented in Lockwood
et al. [1990], these maps indicate enhancements of the dayside horizontal
and field aligned current systems and increasing electric field in the
region whose field lines should map to dayside reconnection sites. Between
2155 UT and 2200 UT all of the new potential was added on the dayside.
The nightside region, not yet affected by the addition of newly reconnected
flux, remained unchanged.

Figures 1g-i show the ionospheric state some 30 minutes later. The convection
system had extended nightward and equatorward as new flux entered the polar
cap in response to the increasingly southward IMF. The cross polar cap
voltage rose to 65 kV. An intense new post-midnight field-aligned current
system (Fig. 1i) and strengthening horizontal currents (Figure 1g) developed
in the post-midnight region as the first substorm occurred. The more tailward
position of the electric potential maximum at 2235 UT (Figure 2h), as well
as the intense midnight field-aligned current system are in agreement with
the ideas of substorm expansion phase presented by Kamide et al. [1994].

In order to more clearly delineate the changes associated with substorm
expansion, we show 5-minute difference plots (2235 UT minus 2230 UT) of
the horizontal currents in Figure 2, row 2. The growing
convection currents are evident near the dawn-dusk line in Figure 2d. Even
more evident is the substorm current surge across the midnight meridian
covering most of the latitude interval between 60 and 70 magnetic. Figure
2e indicates that nightside electric fields were increasing rapidly between
2230 and 2235 UT, while the dayside had reached an equilibrium state. Figure
2f shows not only the enhancement of the global field aligned current system,
but the intensification of the upward and downward currents in the vicinity
of the midnight current wedge.

The last row of Figure 1 shows the ionospheric response
at 2320 UT to the leading edge of a non-compressive density enhancement,
a type of solar wind transient. (N. Crooker, personal communication, 1996).
The passage began at 2310 UT on 3 November and lasted for approximately
45 minutes. During the 45 minute interval the IMP-8 satellite, which was
~ 30 Re downtail, recorded Bz component values as large as -35 nT and solar
wind densities in excess of 50 amu cm^{-3}. The GEOTAIL satellite
was enveloped by the magnetosheath during the transient passage and did
not make IMF observations. Figure 2 row 3 shows the 5-minute difference
plots and illustrates the dayside response to the dayside magnetospheric
compression and the onset of a second substorm.

At 2320 UT, just ten minutes into the passage, the cross-polar-cap-voltage
had risen to 81 kV and intense currents flowed both horizontally and vertically.
The 5 minute potential difference plots (2320 UT minus 2315 UT) shown in
Figure 2 row 3, indicates continuing growth of the
dayside convection with nearly 15 kV of new potential in the dayside negative
cell and 10 kV of new potential in the dayside positive cell (Fig 2h).
The current difference plot (2220 UT minus 2315 UT) shows clearly defined
and separate regions of current enhancement. Dayside current enhancements
were the most extreme, perhaps owing to the extraordinary compression associated
with the high densities. (The magnetopause crossed geosynchronous orbit
at approximately 2325 UT, J. Borovsky, personal communication, 1996). A
simultaneous large current enhancement occurred across the midnight meridian.
Figure 1j shows horizontal currents in excess of 1 A/m occurred across
much of the midnight region while the solar wind density enhancement and
strong southward field influenced the ionospheric behavior. Consistent
with a new expansion phase at 2320 UT, geosynchronous satellites indicated
a strong particle injection event (G. Reeves, personal communication, 1996).
During the 60 minute interval after 2320 UT the entire current system was
so excited that it was not possible to separate the convection current
system from the substorm system.

EXAMPLE 2. A MORE "NORMAL" SUBSTORM

The substorm that we show in Figure 3 is more illustrative of those mapped during typical solar wind conditions. This event took place on 10 November 1993. The IMF was slowly varying in orientation and solar wind flow speed was near 500 km/s.

Fig. 3. Same as for Figure 1 except that row 4 contains difference plots as described in Figure 2 with the electric potential contours at 3 kV.

Figures 3a-c show a well-developed convection and current system representative
of the conditions for the previous hour. The IMF had been steadily southward
with a magnitude of ~3 nT. Between 1225 UT (not shown) and 1230 UT, as
the expansion phase began, a new local nightside minimum in the evening
convection cell developed and the morning convection cell migrated further
across the midnight meridian into the pre-midnight region. The field-aligned
current flow in the pre-midnight region also strengthened. Despite this
rather substantial reconfiguration, the total cross polar cap voltage was
nearly steady. In the plots for 1235 UT (Fig. 3g-i) these features further
intensified and the horizontal current flow across the midnight meridian
increased. All channels of the low energy particle detectors on the midnight
geosynchronous satellite registered a slight increase in particle flux
at 1235 UT, indicative of a small particle injection.

In the bottom row of Figure 3 we show the 'difference plots' for 1235 UT
minus 1230 UT There is evidence of a horizontal current surge in the midnight
region (Figure 3j) , and enhancement of the nightside electric field (Figure
3k) and enhancements of the near midnight field-aligned current system
(Figure 3l). By 1245 UT (not shown) these features were absent.

SUMMARY

We have shown three different substorm conditions, each illustrative
of a different type of solar wind and forcing. In the order of presentation,
these substorm examples generally typify moderately, greatly, and slightly
disturbed geomagnetic conditions. In Figures 1 and
2 we provided hemispheric snapshots and difference
plots that illustrate the electrodynamics associated with the addition
of open flux to the polar cap and the onset of substorm expansion phase.
The difference plots in Figures 2 and 3 provide a method
by which we can identify substorm onset and track the progress of substorm
development. For these events we illustrated here auroral imagery was not
available as a means of identifying substorms, however, the substorm expansion
phases were verified with geosynchronous particle injection data. We also
provided a snapshot of the combined response to a magnetospheric compression
and a simultaneous substorm expansion. The difference map for that event
clearly illustrated the separate dayside and nightside response. A more
detailed treatment of the compression and ensuing storm will be the subject
of a future publication.

We anticipate that a more complete view of substorm dynamics will be possible
with the combination of data assimilation procedures and global auroral
coverage like that soon to be available from the POLAR satellite. In short
order we expect to be able to provide snapshots of the spatial and temporal
development of the peak and total substorm current and the total energy
dissipated by individual substorms. We will also be able to determine the
latitude and longitude of substorm onset, and the size of the mid-latitude
bay. When auroral imagery are available for assimilation we anticipate
making estimates of the temporal and area integral of the auroral bulge
associated with a given substorm. Further we anticipate making useful estimates
of the temporal and spatial variations of the field-aligned current contributions
to substorm dynamics.

ACKNOWLEDGMENTS

Data for this study were made available by M. Nakamura, T. Mukai, S.
Kokubun, T. Terasawa, A. Lazarus, F. Rich, D. Evans, L. Cafarella, B. Clauer,
E. Friis-Christensen, A. Green, J. Hughes, M. Engebretson, L. H. Luehr,
T. Moretto, L. Morris, J. Olson, D. Orr and D. Milling, T. Rosenberg G.
Rostoker and T. Hughes, P. Stauning, C. Szuberla, O. Troshichev, W. Worthington,
K. Yumoto and K. Shiokawa, R. Lepping, R. Coles and D. Boteler. This work
is partially supported by NSF Grant ATM 93-02144 and a grant from the Air
Force Office of Scientific Research.

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