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,  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.,  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., 
an empirical relationship for
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.,  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., , 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.,  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.,  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,  and Shepherd and Ruohoniemi, , 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.,  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.,  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., , 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.,  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.