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Cross Polar Cap Potential Determination

As mentioned in section 1 we use APL FIT to determine a global solution of $\Phi$ in the high-latitude ionosphere from which $\Phi_\mathsf{PC}$ is easily found. Ruohoniemi and Baker, [1998] give explicit details of this technique and subsequent improvements are explained in the appendix of Shepherd and Ruohoniemi, [2000]. Briefly, the LOS velocity measurements from each SuperDARN HF radar are mapped onto a grid of roughly equal area cells (~110 km $\times$ 110 km) in the region >50$^\circ$ latitude, using the geomagnetic coordinate system described by Baker and Wing, [1989]. Additional data vectors from the statistical model of Ruohoniemi and Greenwald, [1996] are sparsely added to the grid in order to prevent the solution from becoming nonphysical in regions where no data are available. The choice of the particular model data is determined by the magnitude and orientation of the IMF conditions at the magnetopause.

An expression for $\Phi$ is obtained by fitting the LOS and model data to an expansion of spherical harmonic basis functions. The order of the expansion is chosen in such a manner as to represent the global character of the convection while retaining local features observed by the radars. For this study all fittings were performed to order 8.

Figure 3. Solutions of the electrostatic potential $\Phi$ using APL FIT for the example period shown in Figure 1. The lag time of the IMF measured at ACE is calculated using equations (3), (4), and (5) and the fitting is performed to order 8. Colored arrows indicate the position of SuperDARN measurements and denote the direction of the fitted velocity determination at that location. The magnitude of each fitted velocity determination is indicated by the color and length of the arrow. Contours are spaced at 6-kV increments to represent the electrostatic potential $\Phi$.

Figure 3 shows the solution of $\Phi$ obtained from APL FIT for the example period in Figure 1. Each 10-min period is shown on a grid of magnetic local time (MLT) and magnetic latitude $\ge 60^\circ$ [Baker and Wing, 1989]. The locations of SuperDARN measurements are denoted by markers consisting of colored-coded dots and vector tails. The color and length of the tail indicate the magnitude of each velocity determination according to the scales in the upper right corner of Figure 3. The tail points in the direction of the solved velocity at that location.

Contours of $\Phi$ are spaced at 6-kV intervals. The potential extrema are indicated in each cell by a plus sign and negative sign for the dawn and dusk cells, respectively. $\Phi_\mathsf{PC}$ is simply the difference between these two values and is shown in the lower left corner of each plot. In the lower right corner the $(Y$-$Z)_{GSM}$ components of the IMF, measured at ACE and lagged according to equations (3), (4), and (5), are shown.

The fitted solutions of $\Phi$ in Figures 3a-3e show a two-cell convection pattern with antisunward flow over the polar cap and sunward return flow along the dawn and dusk flanks that is typical of IMF $B_Z < 0$. Evidence of the relatively strong ($\sim$10 nT) IMF $B_Y > 0$ can be seen in the dayside ionosphere in the form of flow toward the dawn sector across 1200 MLT between 75$^\circ$ and 80$^\circ$ and the existence of a more crescent-shaped dawn cell and a more circular dusk cell [Heppner, 1972; Crooker, 1979; Heelis, 1984; Reiff and Burch, 1985; Greenwald et al., 1990].

During the example period shown in Figures 3a-3e backscatter from SuperDARN HF radars was observed over a large region of the dayside between $\sim$0600 and 1800 MLT and, in some areas, from <65$^\circ$ to nearly 90$^\circ$ latitude. There is also a large region of the post-midnight sector from which backscatter was observed. During this period, $\Phi$ is much more structured than statistical models would prescribe for the given IMF [e.g., Ruohoniemi and Greenwald, 1996; Weimer, 2001]. While mesoscale structures evolve throughout the 50-min period, the main feature of these patterns is the steady increase in $\Phi_\mathsf{PC}$, from 67 kV to 86 kV, attributed to an expansion of the region containing large (>1 km s$^{-1}$) zonal velocities in the postnoon dayside sector and the increase in large sunward velocities in the dusk sector around 0400 MLT.

Figure 1j shows a time series of $\Phi_\mathsf{PC}$ during this period. The red line represents $\Phi_\mathsf{PC}$ as determined using APL FIT with the standard 2-min resolution SuperDARN data [Greenwald et al., 1995]. The 10-min averaged $\Phi_\mathsf{PC}$ values and the average for the entire 50-min period are shown in blue and green, respectively.

Despite the quasi-stable solar wind and IMF conditions, there is quite large variability in $\Phi_\mathsf{PC}$. The range of the 10-min averaged $\Phi_\mathsf{PC}$ is 67-86 kV and the range of the 2-min $\Phi_\mathsf{PC}$ is 60-87 kV.

For this study a solution of $\Phi$ is determined using APL FIT for each 10-min period that satisfies equation (1). For each of these periods the number of data points (a data point is defined as a grid cell containing LOS data from a single SuperDARN radar) in each MLT sector is extracted and used to select a subset of periods for which the SuperDARN data provide sufficient coverage to adequately define $\Phi_\mathsf{PC}$. While complete coverage of the entire high-latitude ionosphere is ideal for a truly definitive determination of $\Phi_\mathsf{PC}$, this situation never occurs in practice. It is, however, possible to accurately determine $\Phi_\mathsf{PC}$ with significantly less coverage. For instance, a "polar cap" determination of $\Phi_\mathsf{PC}$ is possible by measuring only the flow in the polar cap region between the two potential extrema. Likewise, an "auroral" determination of $\Phi_\mathsf{PC}$ is also possible by measuring only the flow at latitudes below each of the potential extrema. Our usual approach is the `polar cap' solution, which, in practice, can be obtained with as few as two SuperDARN radars, provided the backscatter is sufficient in extent and the radars are making measurements in the proper MLT sector (usually the dayside near 1200 MLT and looking into the convection throat). Periods with much less than total coverage of the high-latitude ionosphere can therefore be suitable for determining $\Phi_\mathsf{PC}$.

Several definitions of adequate coverage are possible, and after trying various formulations involving the number and location of data points we define suitable coverage as those times when >200 data points exist in the dayside (0600-1800 MLT) ionosphere or >400 data points exist anywhere in the high-latitude region. This criteria does not guarantee that SuperDARN measurements are made over the entire region spanning the potential extrema, but it is our experience that this is most often the case. More than 200 data points in the dayside region almost always ensures that the convection throat region is adequately sampled, and more than 400 data points overall includes periods when the nightside convection out of the throat is well defined and periods when the polar cap is contracted and the former criteria is overly restrictive.

One final selection criteria is imposed on the data set. Because there is some uncertainty in the propagation time of the solar wind observations at ACE, the first and last 10-min period of each quasi-stable period $\ge$40 min is dropped from the final data set to allow for $\pm10$ min uncertainty in the propagation time.

To summarize the various restrictions imposed on the data sets and the corresponding decimations to the number of periods included in the study, we begin by selecting quasi-stable periods of the solar wind and IMF conditions. A total of 9464 10-min periods result from searching the ACE level 2 MAG and SWEPAM data for events that satisfy equation (1) for $\ge$40 min. Of these matches, 2721 10-min periods satisfy the condition that either >200 SuperDARN data points are present in the dayside sector or >400 total SuperDARN data points are present in the high-latitude region. Finally, the first and last 10-min periods for each event lasting $\ge$40 min are dropped, reducing the number of periods to 1638. This subset of 10-min periods represents those times when (1) the solar wind driving conditions at the magnetopause are well-known and (2) the convection in the dayside high-latitude ionosphere is well-known. These high-confidence periods form the basis of our statistical study of $\Phi_\mathsf{PC}$ and the solar wind driver.

Figure 4. $\Phi_\mathsf{PC}$ as a function of $E_{KL}$ determined using APL FIT for (a) all 10-min periods satisfying equation (1) and (b) those periods where the SuperDARN data sufficiently determines $\Phi_\mathsf{PC}$ (see section 2.3). Each 10-min period is represented by a dot. A sliding, linear least squares fit to data within a 10 kV $R_E^{-1}$ window, and corresponding 2$\sigma$ deviations, are shown for each unit of $E_{KL}$ up to 40 kV $R_E^{-1}$. Due to the sparsity of data in the range >40 kV $R_E^{-1}$, a single fit was performed on this data. Four larger dots indicate specific periods shown in later figures.
%% \ref{fig:indi_15} and \ref{fig:indi_40}.}

Figure 5. $\Phi_\mathsf{PC}$ computed from the solar wind observations of this study using the model of Boyle et al., [1997] for the periods shown in Figure 4. A sliding, linear least squares fit to the data and 2$\sigma$ deviations are computed and shown in the same format as Figure 4.

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Next: Results Up: Procedure Previous: Lag Time Determination

Simon Shepherd 2002-06-04