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

Data and Discussion

Figure 2. a) $\Phi_\mathsf{PC}$ determined using APL Fit ( $\Phi_\mathsf{PC}^\mathsf{SD}$) and the best-fit Hill model ( $\Phi_\mathsf{PC}^\mathsf{Hill}$, shown in Figure 4b) plotted against $E_\mathsf{KL}$. A linear fit to a 5-kV/R$_\mathsf{E}$ window is shown for these data and for several other comparison models: the Boyle model, $\Phi_\mathsf{PC}^\mathsf{Hill}$ with a low (2 S) and high (44 S) $\Sigma$, and $\Phi_\mathsf{PC}^\mathsf{Hill}$ for the best-fit in Figure 4a. b) Distribution of events in $E_\mathsf{KL}$.
\begin{figure}\figbox*{\hsize}{}{\epsfig{file=figs/sd_hill_boyle_data.eps}}
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Figure 2a shows $\Phi_\mathsf{PC}^\mathsf{SD}$ plotted against $E_\mathsf{KL}$ in red and $\Phi_\mathsf{PC}^\mathsf{Hill}$($\Sigma$=23 S) + 17 kV in green (a best-fit solution to $\Phi_\mathsf{PC}^\mathsf{SD}$ discussed shortly). For each model a linear fit to $\Phi_\mathsf{PC}$ in a 10 kV/R$_\mathsf{E}$ window of $E_\mathsf{KL}$ is shown for each unit value of $E_\mathsf{KL}$ in kV/R$_\mathsf{E}$. We also plot just the linear fits of several other models for comparison: $\Phi_\mathsf{PC}^\mathsf{Hill}$($\Sigma$=2 S) and $\Phi_\mathsf{PC}^\mathsf{Hill}$($\Sigma$=44 S) in grey, and $\Phi_\mathsf{PC}$ for the Boyle model in black. Figure 2b shows the distribution of $E_\mathsf{KL}$ for the dataset used in this study with a peak in the distribution near 13 kV/R$_\mathsf{E}$.

There is a great deal of variability in $\Phi_\mathsf{PC}$ (particularly in $\Phi_\mathsf{PC}^\mathsf{SD}$) for nearly all ranges of $E_\mathsf{KL}$. We, therefore, show the mean and standard deviation of the windowed, linear fits to $\Phi_\mathsf{PC}$($E_\mathsf{KL}$) in Figure 3a.

Figure 3. a) $\Phi_\mathsf{PC}$ (mean and standard deviation) and b) $\Delta$ $\Phi_\mathsf{PC}$/$\Delta$$E_\mathsf{KL}$ of $E_\mathsf{KL}$-windowed, linear fits to the data shown in Figure 2a.
\begin{figure}\figbox*{\hsize}{}{\epsfig{file=figs/sd_hill_boyle_fits.eps}}
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The slopes ($\Delta$ $\Phi_\mathsf{PC}$/$\Delta$$E_\mathsf{KL}$) of these fits are shown in Figure 3b.

The two Hill models ($\Phi_0$ = 0 and $\Sigma$ = 2 S and 44 S) are the minimum and maximum values of $\Sigma$ used by Siscoe et al., [2002] and illustrate the extremes of this model. The other $\Phi_\mathsf{PC}^\mathsf{Hill}$ shown in Figures 2a and 3 corresponds to the best-fit solution of $\Phi_\mathsf{PC}^\mathsf{Hill}$($\Sigma$) + $\Phi_0$ to $\Phi_\mathsf{PC}^\mathsf{SD}$. One expects a non-zero minimum potential ($\Phi_0$) due to viscous magnetospheric convection or other such processes so we have added a constant $\Phi_0$to $\Phi_\mathsf{PC}^\mathsf{Hill}$ which we attribute to the effects of these processes.

In order to determine the best-fit $\Phi_\mathsf{PC}^\mathsf{Hill}$($\Sigma$) + $\Phi_0$ to $\Phi_\mathsf{PC}^\mathsf{SD}$, total root-mean-square (RMS) differences between the two datasets were calculated in two ways.

Figure 4. Unit contours of RMS differences between $\Phi_\mathsf{PC}^\mathsf{Hill}$($E_\mathsf{KL}$) and $\Phi_\mathsf{PC}^\mathsf{SD}$($E_\mathsf{KL}$), in $\Sigma$,$\Phi_0$ space, for a) all data points (Figure 2) and b) $E_\mathsf{KL}$-windowed, linear fits to the data (Figure 3). The minimum total RMS values are indicated by the as the intersection of the dotted lines.
\begin{figure}\figbox*{\hsize}{}{\epsfig{file=figs/hill_search.eps}}
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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 $\Sigma = 27$ S and $\Phi_0$ = 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 $E_\mathsf{KL}$-windowed, linear fits shown in Figure 2b, and has a well-defined minimum. We therefore consider this, $\Phi_\mathsf{PC}^\mathsf{Hill}$($\Sigma$=23 S) + 17 kV, the best-fit solution to $\Phi_\mathsf{PC}^\mathsf{SD}$ for these data, and note there is a possible range of solutions as shown in Figure 4a.

The value of $\Phi_0$ = 17 kV is reasonable for the minimum potential. The lowest value of $\Phi_\mathsf{PC}$ for the Boyle model is $\sim$20 kV for this dataset and the lowest $\Phi_\mathsf{PC}^\mathsf{SD}$ is $\sim$18 kV. Some studies report $\Phi_0$ = 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 $\Phi_0$ is in good agreement with other studies, the value of ionospheric conductance, $\Sigma$ = 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 $\Sigma$ for the family of solutions in Figure 4, the minimum $\Sigma$ for these solutions is $>$10 S, which is still quite high.

The ram pressure dependence of $\Phi_\mathsf{PC}^\mathsf{Hill}$ can also be compared to that of $\Phi_\mathsf{PC}^\mathsf{SD}$. Figure 5

Figure 5. Ram pressure ($p$) dependence of $\Phi_\mathsf{PC}$ for four of the models in Figure 2. The data have been binned in 5 kV/R$_\mathsf{E}$ intervals of $E_\mathsf{KL}$.
\begin{figure}\figbox*{\hsize}{}{\epsfig{file=figs/sd_hill_means.eps}}
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shows $\Phi_\mathsf{PC}^\mathsf{SD}$($p$) in red, the best-fit $\Phi_\mathsf{PC}^\mathsf{Hill}$($p$) in green, and the two extreme-value Hill models, $\Sigma$ = 2 S in black, and $\Sigma$ = 44 S in grey. The data have been binned in 5 kV/R$_\mathsf{E}$ intervals of $E_\mathsf{KL}$ and plotted in Figures 5a-f along with 1-$\sigma$ error bars. The horizontal dotted lines represent the average $\Phi_\mathsf{PC}^\mathsf{SD}$ for each range of $E_\mathsf{KL}$. The general trend of increasing $\Phi_\mathsf{PC}^\mathsf{SD}$ with increasing $E_\mathsf{KL}$ is obvious from the panels in Figure 5. Also obvious is the lack of any $\Phi_\mathsf{PC}^\mathsf{SD}$ dependence on $p$. Unlike $\Phi_\mathsf{PC}^\mathsf{SD}$, $\Phi_\mathsf{PC}^\mathsf{Hill}$ for the best-fit solution shows a very clear $p$ dependence and very narrow distributions, i.e., small $\sigma$s. On the other hand, $\Phi_\mathsf{PC}^\mathsf{Hill}$($\Sigma$=2 S) shows no $p$ dependence and the $\sigma$s are comparable to those of $\Phi_\mathsf{PC}^\mathsf{SD}$, suggestive of a lower value of $\Sigma$. Of course, the $E_\mathsf{KL}$ dependence of $\Phi_\mathsf{PC}^\mathsf{Hill}$($\Sigma$=2 S) is inconsistent with $\Phi_\mathsf{PC}^\mathsf{SD}$, i.e., the value of $\Phi_\mathsf{PC}$($E_\mathsf{KL}$) is too high.

A final observation is the large amount of variability present in $\Phi_\mathsf{PC}^\mathsf{SD}$($E_\mathsf{KL}$), seen in Figure 2a. Although the events were selected to minimize uncertainties in determining $\Phi_\mathsf{PC}$, 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 $\Phi_\mathsf{PC}$ 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 $p$ dependence of the data also suggest a lower $\Sigma$ but there remain some inconsistencies with the SuperDARN results.


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


Simon Shepherd 2002-06-04