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# Data and Discussion

Figure 2a shows plotted against in red and (=23 S) + 17 kV in green (a best-fit solution to discussed shortly). For each model a linear fit to in a 10 kV/R window of is shown for each unit value of in kV/R. We also plot just the linear fits of several other models for comparison: (=2 S) and (=44 S) in grey, and for the Boyle model in black. Figure 2b shows the distribution of for the dataset used in this study with a peak in the distribution near 13 kV/R.

There is a great deal of variability in (particularly in ) for nearly all ranges of . We, therefore, show the mean and standard deviation of the windowed, linear fits to () in Figure 3a.

The slopes ( /) of these fits are shown in Figure 3b.

The two Hill models ( = 0 and = 2 S and 44 S) are the minimum and maximum values of used by Siscoe et al., [2002] and illustrate the extremes of this model. The other shown in Figures 2a and 3 corresponds to the best-fit solution of () + to . One expects a non-zero minimum potential () due to viscous magnetospheric convection or other such processes so we have added a constant to which we attribute to the effects of these processes.

In order to determine the best-fit () + to , total root-mean-square (RMS) differences between the two datasets were calculated in two ways.

Figure 4a shows unit contours of the total RMS difference for the 1317 periods shown in Figure 2a, a 'fit' to all the data points. A minimum occurs when S and = 22 kV, but it is clear that there is a family of solutions for which the RMS difference is within a few kV of this minimum. This fact is partly due to the distribution of data points shown in Figure 2b. Figure 4b, however, shows unit contours of the total RMS difference between the -windowed, linear fits shown in Figure 2b, and has a well-defined minimum. We therefore consider this, (=23 S) + 17 kV, the best-fit solution to for these data, and note there is a possible range of solutions as shown in Figure 4a.

The value of = 17 kV is reasonable for the minimum potential. The lowest value of for the Boyle model is 20 kV for this dataset and the lowest is 18 kV. Some studies report = 22-39 kV [e.g., Reiff et al., 1981] but these values were obtained by linear fitting of data and lower values were clearly present.

While is in good agreement with other studies, the value of ionospheric conductance, = 23 S, is quite high. A typical value of uniform ionospheric conductance used in MHD models is 5 S, and a few S is not unusual [e.g., Ridley, 2001]. Even if one considers the range of for the family of solutions in Figure 4, the minimum for these solutions is 10 S, which is still quite high.

The ram pressure dependence of can also be compared to that of . Figure 5

shows () in red, the best-fit () in green, and the two extreme-value Hill models, = 2 S in black, and = 44 S in grey. The data have been binned in 5 kV/R intervals of and plotted in Figures 5a-f along with 1- error bars. The horizontal dotted lines represent the average for each range of . The general trend of increasing with increasing is obvious from the panels in Figure 5. Also obvious is the lack of any dependence on . Unlike , for the best-fit solution shows a very clear dependence and very narrow distributions, i.e., small s. On the other hand, (=2 S) shows no dependence and the s are comparable to those of , suggestive of a lower value of . Of course, the dependence of (=2 S) is inconsistent with , i.e., the value of () is too high.

A final observation is the large amount of variability present in (), seen in Figure 2a. Although the events were selected to minimize uncertainties in determining , some part of this variability may be due to variation in the data coverage or variability in the solar wind. However, it is also possible that variability of this magnitude can only be explained by internal effects and that a model of the ionospheric potential requires detailed knowledge of the coupling between the magnetosphere and ionosphere and its time history.

The Hill model is certainly an advance in the sophistication of representing in terms of measurable quantities. Saturation of the transpolar potential is a salient feature of this model missing in many others. It appears, however, that the Hill model, as formulated by Siscoe et al. [2002], needs some modification to be more consistent with the SuperDARN results. One would expect a lower uniform ionospheric conductance than the 23 S of the best-fit solution. The dependence of the data also suggest a lower but there remain some inconsistencies with the SuperDARN results.

Next: Conclusions Up: Testing the Hill model Previous: Procedure

Simon Shepherd 2002-06-04