Figure 1 shows the altitude, local time (LT) coverage and time lag between GF1 (black) and GF2 (red) when they fly over the geographic equator. The altitude of the two spacecraft is about 490 ~ 520 km, with GF2 following directly behind GF1. The thicker and thinner lines represent the ascending and descending orbits, respectively. The LT coverage of both GF1 and GF2 orbits slowly progresses, and considering both the ascending and descending orbits it needs about 161 days to cover the 24 local time hours, which is comparable with the LT precession of the GRACE mission (e.g., Xiong et al. 2010). During the period from 1 June 2018 to 31 October 2020, the time lag between GF1 and GF2 varies between 22 and 29 s. Assuming a velocity of 7.5 km/s for GRACE-FO, such a time lag corresponds to a distance of 165–220 km between GF1 and GF2.
The top panel of Fig. 2 shows the time series of FACs derived from GF1 (black) and GF2 (red) for an event on 31 October 2019 when they crossed the southern auroral latitudes. The upper panel shows that the FACs derived from the two GRACE-FO satellites have very similar variations along their orbits, but with a time delay of about 24 s. The middle panel shows the original 1-Hz FAC densities plotted over MLAT. Here MLAT refers to the Apex latitude calculated by the model of Emmert et al. (2010). In this example the FAC signatures compare well to each other both in latitude and amplitude, with enhanced FAC activity observed between − 64° and − 83° as well as − 68° and − 84° MLAT on the dusk and dawn sides, respectively. Outside the auroral latitudes, the FACs from both satellites show amplitudes less than 0.5 μA/m2. This can be regarded as the noise floor. The bottom panel shows the large-scale FACs structures, calculated by applying a low-pass filter with cutoff period of 20 s (corresponding to about 150 km along orbit). Comparing the middle and lower panel, it shows that remaining differences in peak amplitude between GF1 and GF2 FACs are larger for small-scale structures (less than 150 km, middle panel), but the peaks of large-scale structures (lower panel) follow each other well.
To get a more quantitative picture of the temporal variability of small-scale FACs, we performed cross-correlation analyses of the FACs time series for each high-latitude orbital crossing (|MLAT|> 50°) from two spacecraft. The derived maximum correlation coefficients (Rmax) are recorded for further statistics. Note that lower Rmax means the FACs contain dynamic small-scale structures with scale lengths shorter than the distance of the two spacecraft. However, sometimes the GF1 and GF2 satellites did not observe clear FAC currents at high latitudes. Possible reasons could be that either the GRACE-FO satellites did not reach the auroral latitudes, or the FAC signatures are so weak that the GRACE-FO onboard magnetometers are not sensitive enough to reflect the weak currents. Two such examples are shown in Fig. 3a, b. In these two events, the Rmax between GF1 and GF2 are quite low (less than 0.2), but it does not mean that the small-scale FAC structures are dynamic, as argued above. Therefore, we excluded such events from the statistical analysis.
To identify events with no or very weak FAC signatures, the maxima of the absolute FAC intensity from GF1 and GF2 are identified for each high-latitude orbital crossing, and then their mean value, \(\overline{{\left| {\text{FAC}} \right|}}\), is recorded. Figure 4 shows the distribution of Rmax over \(\overline{{\left| {\text{FAC}} \right|}}\) for the two hemispheres. We find the majority of Rmax being low (< 0.5) for low amplitude of \(\overline{{\left| {\text{FAC}} \right|}}\), e.g., 1 μA/m2. Therefore, these events are disregarded from the statistics. The excluded events correspond to 3.9 and 16.1% of the total number of events in the northern and southern hemispheres, respectively.
Figure 5 presents the seasonal distribution of Rmax as expressed by the day of year (DOY) in the two hemispheres. Rmax derived from each high-latitude orbital crossing is presented as grey dots, and the median values in each DOY bin (width of 1 day) are shown as blue circles, that vary between 0.6 and 0.7. In the northern hemisphere the median values of Rmax are slightly larger during June solstice months, while an opposite dependence is found in the southern hemisphere, though not as prominent as that in the northern hemisphere. In other words, the Rmax is larger during local summer for both hemispheres.
Figure 6 further presents the MLT distribution of Rmax in the two hemispheres, and the median values of Rmax in each MLT bin (width of 1 h) are shown similarly as blue circles. A clear feature seen here is that Rmax show larger values around dawn and dusk hours, slightly lower values around midnight, and the lowest value is found around noon. This feature applies to both hemispheres. For a more specified impression of the small-scale FAC structures at different MLT, Fig. 7 presents four individual examples of FACs profiles of GF1 and GF2 for noon, midnight, dusk and dawn hours. The correlations at noon and midnight are low, thus small-scale FAC variations, shorter than spacecraft separation distance, are important. The correlations at dawn and dusk are higher, thus FACs appear persistent at scale lengths larger than the spacecraft separation distance.
Figure 8 shows the dependence of Rmax over magnetic activity, separately for different MLT sectors. Here we used the solar wind merging electric field, Em, to represent the magnetic activity. Xiong et al. (2014) found that the location of the auroral oval equatorward boundary shows nearly linear dependence on the intensity of Em, being an expression of geomagnetic activity. For more details about how to derive Em from the solar wind and IMF parameters, the readers are referred to Newell et al. (2007) and Xiong et al. (2014). Less events of low Rmax appear for larger Em; however, the median values (blue circles) do not show prominent dependence on Em, and this feature applies for both hemispheres (the results for the southern hemisphere are not shown). This result suggests that the small-scale FAC structures do not strongly depend on magnetic activity.
The last question we want to address is the persistence of FAC structures depending on their scale lengths. Figure 9 (top) presents one example of the original FACs data at 1-Hz resolution, and the panels below reflect low-pass filtered time series at different cutoff periods for the same event. The original 1-Hz data show rather strong FACs reaching current densities beyond 9 μA/m2, and the smallest scale-size being resolved is about 15 km (considering the Nyquist sampling theorem and the spacecraft velocity of 7.5 km/s). Here, Rmax is 0.42. The amplitudes decrease for increasing scale lengths and their correlation between the two spacecraft increases. After applying a filter with cutoff periods of 10 s, corresponding to scale-lengths larger than 75 km, FACs seem persistent with a correlation of Rmax = 0.96. At such scale-length, FACs start to reflect the basic R1/R2 pattern as reported by Iijima and Potemra (1976a).
The example shown in Fig. 9 represents the midnight case. To get a more quantitative picture of the temporal variability of small-scale FAC structures at different MLT hours, Fig. 10 shows the distribution of Rmax as a function of MLT and filter lengths, separately for three different seasons and two hemispheres. The three seasons are defined as: December solstice (November to February), equinoxes (March, April, September, October), and June solstice (May to August). For each subpanel, the first vertical column represents correlation results from the original 1-Hz FACs time series, and all the other values are from low-pass filtered data with cutoff periods as listed in the abscissae. The original 1-Hz recordings show correlations below 0.6 at noon and slightly higher values (about 0.75) are observed at dawn and dusk. After applying filters with cutoff period larger than 10 s, correlations are significantly improved (Rmax > 0.9) at all local times, implying small-scale FAC structures have been smoothed out.
The correlations between two spacecraft are generally higher in the northern hemisphere than that in the southern hemisphere, which can also be seen from Figs. 5 and 6. For the seasonal difference, an interesting feature shown in Fig. 10 is that the correlations during local summer are higher than that during local winter, most prominent for the noon hours.