For comparison purposes, is calculated using APL FIT for all 9464 10-min periods satisfying the quasi-stability condition imposed on the solar wind and IMF in equation (1) in addition to the subset of 1638 high-confidence periods described in section 2.3. Figure 4 shows the resulting values of versus for both sets of 10-min periods. A histogram on the right of each plot shows the distribution of values. For each whole number of up to 40 kV , a sliding, linear least squares fit was performed to the data within a 10 kV window centered on that value. The resulting fit and corresponding 2- standard deviations are shown as dark line segments bounded by lighter line segments. For the data in the range 40 kV a single fit was performed due to the sparsity of data at high values of . Four specific 10-min periods are shown by larger dots and marked by the numbers 0-3. The APL FIT solutions for these four periods are shown in later figures.

Several noteworthy features of the data are illustrated by Figure 4. Of particular note is the similarity between the entire set of 9464 10-min periods (Figure 4a) and the subset of 1638 high-confidence periods (Figure 4b). Except for very large values of (60 kV ) the data distributions have much the same character for both sets of periods. For kV the fitted line segments for both data sets have similar values, slopes, and standard deviations (above ~30 kV , low statistics begin to affect the slope determinations). Because the set of all 10-min periods is determined without regard to the degree of data coverage from the SuperDARN radars, it includes periods when the SuperDARN data are insufficient to fully define , and is consequently determined to a large degree by the statistical model. The similarity between the two data sets for kV therefore implies that of the statistical model patterns used in APL FIT are accurate in the statistical sense with those values calculated from the high-confidence periods, i.e., when the SuperDARN data adequately constrain the solution of . Of course, the inherent nature of statistical quantities ensures that the convection patterns derived by Ruohoniemi:96 appear smoothed or averaged when compared to any particular solution of ; however, it seems that is well-defined statistically by these patterns for 40 kV .

The trends for kV are somewhat different between the
two data sets. In Figure 4a the best-fit line segment to the
data from the entire set of 10-min periods is roughly flat in this range,
but Figure 4b shows a definite increase in the mean
as increases. Part of the reason for this difference is due to
the statistical models used in APL FIT.
Values of larger than 40 kV correspond to IMF
with a magnitude 12 nT. The largest IMF magnitude bin of
*Ruohoniemi and Greenwald*, [1996]
is 6-12 nT, where the mean value of the IMF for
the data used to construct these patterns was ~7 nT. Consequently, for
some of the periods shown in Figure 4a, where
kV and the data coverage is below our threshold,
is determined
to a large extent by the statistical
models, which most likely underestimate
for the largest values of
. The full range of
is, therefore, not represented in the
determination of the mean for kV in Figure
4a. Hence the mean is lower than it is for the
high-confidence periods in Figure 4b for which the statistical
models have much less impact.

Another obvious feature in Figure 4 is the significantly
nonlinear relationship between
and . The slope of each
line segment fit to the data in Figure 4 steadily decreases as
increases; that is, there is no evident range of where
is truly linear. In contrast to these results are the linear relations
of
determined in other studies.
*Burke et al.*, [1999]
use the same data
from DE 2 and the same technique used by
*Weimer* [1995,
1996] to show
that
is linear to very good agreement with for values 30
kV (Figure 3a
[*Burke et al.*, 1999]).
However, it should also be noted
that in the same study, and using a limited range of S3-2 data, this linear
relationship appears much less convincing, and much more scatter is evident
in the data (Figure 3d
[*Burke et al.*, 1999]).

In another study that uses low-altitude, high-latitude spacecraft measurements
of drifting ionospheric plasma to estimate
,
*Boyle et al.*, [1997]
determine
an empirical relationship for
given by

Figures 4 and 5 illustrate the two differing views
of the relationship between
and the merging electric field. The APL
FIT data suggest that
is nonlinearly related to the merging electric
field and saturates at large values of , while the
*Boyle et al.*, [1997]
model suggests that
continues to increase without limit. While the
lower limit of
is ~20 kV for both data sets, the APL FIT data
show a deviation from linearity for values of even below ~20 kV . To better show the different behavior of the two data sets, Figure
6

It has long been theorized that
saturates during extremely
strong IMF conditions
[*Hill et al.*, 1976].
Supporting this idea, some
earlier studies using low-altitude spacecraft found that
rarely
exceeded 160 kV
[*Reiff et al.*, 1981;
*Reiff and Luhmann*, 1986.]
There are reports of
reaching values of 230 kV during storm periods
[e.g., *Sojka et al.*, 1994] and
*Boyle et al.*, [1997],
using a larger data set of low-altitude spacecraft that
included DMSP, found that there is no evidence of saturation of
.
It should, however, be noted that because the more
desirable dawn-dusk DMSP passes normally used to determine
were
limited in number for large IMF,
*Boyle et al.*, [1997]
used a fitting technique to estimate
for DMSP passes in all MLT sectors. It should also be noted
that in their study the observed total potential variation was rarely
observed to exceed 150 kV. For the largest values of (100 kV )
in our study the model given by equation
(6) predicts values of
that exceed 450 kV, which to our knowledge, have not been observed. More
recently,
*Siscoe et al.*, [2002]
show evidence during storm periods that
does indeed saturate for large values of the solar wind electric field.

The question of whether the ionosphere can support such large values of
or whether saturation occurs is an important aspect of M-I coupling.
How the ionospheric convection electric field and the magnetospheric and
ionospheric currents systems interact in a self-consistent manner is still
an unresolved issue. The evidence we show in favor of saturation is that
is nonlinear
throughout the range of shown here and that
has an upper limit of
~150 kV. Figure 6 shows the trend of
/
steadily decreases with increasing . In addition,
for no period in the entire study does
exceed 130 kV, even for very
large values of . In fact, it
is rare for
to exceed ~140 kV using the APL FIT technique as
described by
*Ruohoniemi and Baker*, [1998] and
*Shepherd and Ruohoniemi*, [2000],
even at 2-min resolution
[e.g., *Shepherd et al.*, 2000].

It should be noted, however, that while the data from this study suggests
that saturation of
occurs, difficulties arise in using the APL
FIT technique for large values of IMF and . The problem
occurs when the coupling between the solar wind and magnetosphere is
exceptionally favorable for extended periods of time, and the rapidly
reconnecting magnetic flux at the dayside magnetopause causes the lower
latitude boundary of convection to expand to magnetic latitudes equatorward
of ~55. The SuperDARN radars in the northern hemisphere are
located between 56 and 65 magnetic latitude. Because of the
propagation conditions necessary to achieve perpendicularity
to the magnetic field at ionospheric altitudes and detect backscatter, the
effective lowest magnetic latitude for observing backscatter tends to range
from 58 to 63, depending on the radar. That being said,
because the convection region is constrained to relatively higher magnetic
latitudes on the dayside
[e.g., *Heppner and Maynard*, 1987],
significant coverage
of the dayside region and therefore determination of
can be
achieved even when the convection region is expanded to below 50
on the nightside.

In order to determine better whether the statistical results of Figure 4 actually confirm that saturates at high values of , we look at several individual periods from the study in more detail. Figures 7a, 7b, 8a, and 8b show the solutions of APL FIT for the four periods labeled 0-3, respectively, in Figure 4b. These periods are chosen to illustrate relatively high and low values of for two values of , ~15 kV and ~35 kV .

The APL FIT solutions for the periods 0514-0524 UT on 19 March 2000 and 1748-1758 UT on 30 March 2000 are shown in Figure 7. For these periods kV and 13.7 kV , respectively. Despite roughly equal values of , lower latitude limits of convection (~65), and the amount of SuperDARN data coverage, the resulting values of (95 kV and 37 kV) are dramatically different. For both periods the SuperDARN data coverage is sufficiently extended and suitably located to adequately define the solution of . The difference between these two periods is that the observed convection on 19 March 2000 is dominated by a large region of flow 1 km s in the dayside convection throat region, while on 30 March 2000 the convection is observed over most of the high-latitude dayside to be exclusively 1 km s. The character of the convection and hence is dramatically different for these two periods.

Figure 8 shows the APL FIT solutions for the periods
1622-1632 UT on 26 September 1999 and 2252-2302 UT on 22 January 2000.
For these periods kV and 35.0 kV while
=
98 kV and 78 kV, respectively. Despite the lower latitude convection boundary
extending below 60, in both cases there is good coverage from the
SuperDARN radars. The convection on 26 September 1999 shows two regions of
flow 1 km s
in the prenoon dayside and dusk sectors, as would be expected
for higher values of and more effective penetration of the solar
wind electric field. On 22 January 2000 the convection is observed from
1100-0100 UT to be exclusively 1 km s.
For both of these cases the true
is most likely somewhat higher than the computed values given the
expanded nature of the convection region; however, the 22 January 2000 period
clearly indicates that
is much less than the ~188-kV potential
predicted by the
*Boyle et al.*, [1997]
model given by equation (6).

These four periods reinforce the nonlinear trend of
shown in Figure
4b and the low values of
like that in Figure
8b, and together with a maximum value of
~125 kV for
this study these periods
strongly suggests that
does indeed saturate at high values
of . Because of the difficulty previously mentioned in achieving
backscatter during times when the convection region is expanded to
midlatitudes, the saturation value is most likely above the 125-kV maximum
observed. It should also be emphasized that these results are for
10-min-averaged periods during which the solar wind and IMF conditions are
quasi-stable for 40 min. A different conclusion is possible for periods
of non-steady solar wind and IMF conditions; however, since it has recently
been demonstrated that ionospheric convection responds rapidly (2 min)
to changes in the IMF
[*Ruohoniemi et al.*, 2001,
and references therein], these
results are likely to also apply during more dynamic conditions.

Another important aspect shown by the data in Figure 4 and emphasized in Figures 7 and 8 is the amount of variability in for all values of . Where the statistics are greatest (5 20) the standard deviations of the line segment fittings are 9-12 kV. Similar values are found for the other ranges of , but the statistics are lower. These rather large variations are surprising given the stability of the solar wind and IMF during these periods. The red line in Figure 1j shows that determined using APL FIT with the standard 2-min resolution SuperDARN data is even more variable than the 10-min-averaged data.

It is possible that the solar wind and IMF change enough during the transit
from ACE through the solar wind and the magnetosheath to account for the
observed variability in
; however, several studies suggest that the
solar wind remains relatively unchanged over this distance
[e.g., *Prikryl et al.*, 1998].
Maynard et al., [2001]
claim that even small-scale structure in
measured 200 upstream in the solar wind remains coherent to a remarkable
degree into the dayside ionospheric cusp.

Since
is a global parameter and the ionosphere requires a finite
amount of time to reconfigure to changes at the magnetopause
[Ruohoniemi et al., [2001],
small-scale fluctuations in most likely have
little affect on
. It is more likely that some internal processes such
as variable ionospheric conductivity due to particle precipitation or variable
reconnection rates in the magnetotail are responsible for the large variability
in
. Theories have long suggested that the ionosphere is capable of
regulating magnetospheric convection
[*Coroniti and Kennel*, 1973].
It is apparent that
a more complicated expression that includes the contribution of magnetic
field line merging in the magnetotail is needed to fully describe the dynamics
of
and its relationship to other geophysical parameters. It is
undoubtedly the case that reconnection in the magnetotail, possibly during
substorms, will contribute to
and it is possible that some models of
ionospheric flow [e.g.,
*Siscoe and Huang*, 1985]
would account for the observed
variability in
during quasi-stable stable solar wind conditions.
*Siscoe et al.*, [2002]
attempt to provide a more comprehensive description of
the behavior of
by proposing a model based on the work of
Hill:76. In their study an expression for
is given that
includes a contribution from the Region 1 current system in terms of the
solar wind parameters. Their model saturates for large values of ;
however, a further study is necessary to confirm whether the model matches
the data presented in our study.