A Statistical Model of Vorticity in the Polar Ionosphere and Implications for Extreme Values

Measurements of vorticity in the Earth's ionosphere enable the characterization of turbulent structure in the ionospheric plasma flow and how it varies spatially in relation to large‐scale magnetic field‐aligned current (FAC) systems. We have determined the spatial variation of the probability density function (PDF) of ionospheric vorticity measurements made by the Super Dual Auroral Radar Network (SuperDARN) across the northern polar ionosphere for the 6‐year interval 2000–2005 inclusive. These PDFs are highly leptokurtic, with heavy tails, and are well‐modeled by Tsallis q‐exponential probability distributions. The parameters of the model q‐exponential distributions have been determined using maximum likelihood estimation, resulting in a statistical model of ionospheric vorticity that covers the polar ionosphere. The spatial variation of the model parameters is highly variable, with the shape and scale of the model distributions varying systematically in relation to the well‐known FAC regions, showing that FACs have a major influence on the character of ionospheric plasma vorticity. From the model distributions, we estimate the probability of observing extreme vorticity values with the SuperDARN radars (beyond thresholds of 5, 10, 20, and 40 mHz) across the northern polar ionosphere.

2 of 16 turbulence has significant consequences for forecasting the impact of space weather on space-and groundbased technological systems, such as the impact of geomagnetically induced currents (GICs) on power grids (Barbosa et al., 2017;Freeman et al., 2019).
In the aforementioned studies of Abel et al. (2006Abel et al. ( , 2007Abel et al. ( , 2009, it was suggested that the intermittency in the open geomagnetic field line region of the ionosphere is inherited from the turbulent solar wind to which it is magnetically connected. This is evidenced not only by a similar degree of intermittency in the ionosphere and solar wind (Abel et al., 2007), but also the fact that the intermittency disappears when the interplanetary magnetic field (IMF) is northward , and the flow on remnant open field lines is no longer directly driven by the solar wind (Lockwood et al., 1990). Outside the open field line region, Abel and colleagues did not measure intermittency directly but inferred it must be different, both because of the different physics expected in the closed field line auroral zones and because the first-order structure function differs from that in the open field line region and to that of a solar wind coupling function (Abel et al., 2006).
Further evidence for a spatial variation in ionospheric intermittency is provided by the non-Gaussian nature of flow vorticity (Chisham & Freeman, 2010). Measurements of vorticity in the natural environment are very limited due to the difficulties involved in making two-and three-dimensional flow measurements (e.g., Hughes et al., 2010;Sardeshmukh & Sura, 2009). However, in the ionosphere, vorticity can be measured using the Super Dual Auroral Radar Network (SuperDARN) Sofko et al., 1995). Here, vorticity in the plasma flow is associated with the magnetic field-aligned currents (FACs) that flow into and out of the ionosphere (Hasegawa & Sato, 1979;Vasyliunas, 1984), and which transfer energy and momentum between the magnetosphere and ionosphere (Cowley, 2000;Stern, 1983). This relationship has been explicitly demonstrated by the organization of the average ionospheric vorticity measured at fixed geomagnetic coordinates relative to the Earth-Sun line into clearly defined spatial regions ), corresponding to the so-called "region 0" (R0), "region 1" (R1), and "region 2" (R2) large-scale FAC systems (Erlandson et al., 1988;Iijima & Potemra, 1976a, 1976b. The average spatial pictures of ionospheric vorticity  and FACs (Anderson et al., 2008;Edwards et al., 2020;Weimer, 2001Weimer, , 2005 in the polar ionosphere and their large-scale variation with respect to changes in the IMF and season are well known. However, studies of the probability distributions of the fluctuations of these quantities and the physical reasons for the spatial variation of these distributions are in their infancy. By analyzing millions of ionospheric F-region magnetic field-aligned vorticity measurements made by Su-perDARN, Chisham and Freeman (2010) showed that the probability density function (PDF) of ionospheric vorticity is intermittent. Specifically, the PDF is leptokurtic (higher kurtosis than a Gaussian) at all locations throughout the high-latitude polar ionosphere and is most often heavy-tailed (higher kurtosis than an exponential). The shape and scale of measured PDFs can be described by the moments of the distributions (e.g., mean, variance, skewness, and kurtosis, etc.). However, the fitting of model distributions to the observed PDFs can provide more insight into their origins. It also allows the estimation of the probability of the occurrence of extreme values. Chisham and Freeman (2010) successfully modeled ionospheric vorticity PDFs using Tsallis q-exponential distributions (Picoli et al., 2009;Tsallis, 2009). In space plasmas, q-distributions have been used to characterize interstellar turbulence (Esquivel & Lazarian, 2010), turbulence in the solar wind (Burlaga et al., 2007), and geomagnetically induced currents (GICs) (Barbosa et al., 2017).
Here, we build on the study of Chisham and Freeman (2010) by analyzing the spatial variation of the shape and scale parameters of the q-exponential PDFs of vorticity fluctuations across the northern hemisphere polar ionosphere. We further use these model distributions to determine the probability of observing extreme values of vorticity in different regions of the high-latitude ionosphere. Section 2 outlines the methodologies used to determine vorticity from SuperDARN data, to compile the vorticity PDFs, to fit the model distributions to the PDFs, and to determine the probabilities of extreme values. Section 3 shows the results of the modeled PDF parameter variation across the northern polar ionosphere, and the estimates of the probabilities of extreme vorticity. Section 4 discusses the analysis and results. Section 5 summarizes the paper.

SuperDARN Vorticity Measurements
SuperDARN is a network of coherent scatter HF radars designed to measure large-and meso-scale plasma flow in the Northern and Southern polar and mid-latitude ionosphere Greenwald et al., 1995;Nishitani et al., 2019). Each radar in the network measures the line-of-sight component of the plasma velocity in the E-and F-region ionosphere over a 16-to 24-beam scan field of view (with beams typically separated by  E 3.24° azimuth). Hence, the total field of view of each radar covers a spatial area of ∼  6 2 5 10 km E . The duration of a full scan of the field of view is typically 1 or 2 min.
Many of the SuperDARN radars are organized in pairs that have overlapping fields of view. Using the method presented by Chisham et al. (2009), it is possible to estimate the magnetic field-aligned vorticity of the ionospheric plasma flow across these overlapping fields of view. The method makes use of Stokes' theorem; determining localized measurements of vorticity by measuring the ionospheric plasma flow around closed loops defined by the geometry of the overlapping radar beams (see also Sofko et al., 1995).
Following the original study of Chisham et al. (2009), we have used vorticity measurements from three northern hemisphere SuperDARN radar pairs (Prince George/Kodiak, Saskatoon/Kapuskasing, and Goose Bay/Stokkseyri), from the 6-year interval 2000-2005 inclusive (Chisham, 2020). Figure 1 shows a map that indicates the overlapping fields of view of these radars on a geographic coordinate grid (dashed lines), overlaid with Altitude-Adjusted Corrected GeoMagnetic (AACGM) latitude contours at 10° intervals (bold lines). The look directions of the beams of each radar are shown as either blue lines (for Prince George, Kapuskasing, and Goose Bay) or green lines (for Kodiak, Saskatoon, and Stokkseyri), except where the radar fields of view overlap, where the beams are colored red, defining the closed loops discussed above. Hence, the red quadrilaterals in Figure 1 represent the vorticity measurement cells, and the vorticity estimates in these cells are tagged to the coordinates at the center of these cells. The mapping of the fields of view uses the empirical virtual height model of Chisham et al. (2008) that has been shown to map far-range backscatter more accurately than the standard SuperDARN virtual height model (Yeoman As in Chisham et al. (2009), a positive field-aligned vorticity (clockwise rotation when looking in the direction of the magnetic field into the northern hemisphere ionosphere) is equivalent to a negative (upward) field-aligned current, whereas a negative field-aligned vorticity (anticlockwise rotation when looking in the direction of the magnetic field into the northern hemisphere ionosphere) is equivalent to a positive (downward) field-aligned current.
We sorted the measured vorticity values from all the 1-and 2-min SuperDARN common mode radar scans during the 6-year epoch into spatial bins based on where the center of the relevant measurement cell was located at the time of the measurement. These bins were chosen to be 1 hr of MLT by 1° of AACGM latitude as in Chisham et al. (2009). The vorticity values in each spatial bin were then used to compile the vorticity PDFs for the different spatial regions.

Modeling the Measured PDFs
In the preliminary work of Chisham and Freeman (2010), we used maximum likelihood estimation (MLE) to compare how well three different leptokurtic model distributions (exponential, Weibull, and q-exponential) could be fit to the observed PDFs at all locations. We considered the negative and positive vorticity measurements as separate distributions because the observed PDF variations for the different vorticity polarities were distinctly asymmetric and associated with oppositely directed FACs. Hence, for each vorticity distribution we considered two data sets,     . Hence, we also determine  m E across the whole polar region, similar to the determination of the mean value presented by Chisham et al. (2009).
We consider a q-exponential PDF of the form where the function tends to an exponential as  1 , are determined from a numerical solution of the partial differentials of the log likelihood function for the q-exponential (Shalizi, 2007). We require  E ω and  E ω to be comprised of at least 100 elements each for an acceptable MLE output.
In Figure 2 . Here, the mean of the distribution is proportional to  E , changing the scale of the distribution, but the shape of the distribution remains the same. The q-exponential distributions all tend , which is a subset of heavy-tailed distributions commonly referred to as fat-tailed.

Determining the Probability of Extreme Values
Once the model q-exponential parameters for the measured PDFs have been determined, it is possible to use the survival function to estimate the probability of extreme values of vorticity (both negative and positive). The survival function (  ( ) E S ) is the probability of observing a vorticity greater than, or equal to,  E . It is often called the complementary cumulative distribution function and is related to the cumulative distribution function (CDF,  where Ω E is a continuous random variable representing the vorticity measurements. Hence, it is trivial to determine the survival function directly from the model q-exponential PDFs. For the case of "positive" vorticity (    m E ), the survival function can be written Similarly, for the case of "negative" vorticity (    m E ), the survival function can be written

Example PDFs From Region 1 and Region 2
In this paper, the primary aim is to model the PDFs of vorticity measurements at all spatial locations in order to be able to estimate the probability of the occurrence of extreme vorticity and its spatial variation across the polar ionosphere. First, for illustrative purposes, we present example PDFs from the large-scale FAC regions. To select these regions, we look at the average vorticity variation across the polar ionosphere.
In Figure 3a , we reproduce the spatial distribution of mean vorticity measured by the three SuperDARN radar pairs, as originally presented in Figure 3 of Chisham et al. (2009). Here, the data are presented in the AACGM coordinate system, extending from 66° latitude to  E 86° AACGM latitude. The vorticity value in each spatial bin (of size 1° AACGM latitude by 1 hr of MLT) was determined by averaging all the vorticity values measured in that bin location. The mean vorticity values were only presented for spatial bins where there were more than 100 measurements in that bin (see Chisham et al., 2009, for the occurrence distribution of vorticity values across all bins). As described in Chisham et al. (2009), the mean vorticity map shows clear features, which match those of the average picture of the distribution of large-scale R1 and R2 FAC systems observed in average FAC maps (Anderson et al., 2008;Edwards et al., 2020;Weimer, 2001Weimer, , 2005. In order to highlight the main R1 and R2 current regions, in Figure 3b we replot the spatial variation of the mean vorticity, but filter out mean vorticity values that have magnitudes of less than 0.4 mHz (shown as white). This leaves four distinct regions that represent the mean vorticity variation associated with the morning and afternoon R1 and R2 FAC systems.
In Figure 4, we present vorticity PDFs that are typical of the R1 and R2 FAC regions. Figure 4a presents the vorticity PDF (black square symbols and error bars) determined for the red hatched region in the morning sector R1 as illustrated in Figure 3b (74°-78° AACGM latitude and 0700-0800 MLT). Figure 4b presents the vorticity PDF (black square symbols and error bars) determined for the red hatched region in the morning sector R2 as illustrated in Figure 3b (67°-70° AACGM latitude and 0600-0700 MLT). The PDFs are plotted on a log-linear scale, with a vorticity bin size of 1 mHz, to enable visualization of both the variations in the core and the tails of the PDFs.
The vorticity PDFs shown are all leptokurtic. This is the case for all the measured PDFs across the whole of the northern hemisphere polar ionosphere. The R2 PDFs in Figure 4 are more leptokurtic, with the R1 PDFs being closer to exponential (i.e., closer to a straight line on the log-linear plot). The PDFs across the whole polar region are typically asymmetric showing that the character of the two senses of vorticity in any location are typically different. This is particularly clear in the R1 PDF in Figure 4a where the positive vorticity distribution is more leptokurtic than an exponential, whereas the negative vorticity distribution is less leptokurtic than an exponential. This asymmetry in the PDF shape is partially responsible for the variations seen in the spatial pattern of mean vorticity.
The MLE q-exponential fits to the measured PDFs are presented in Figure 4 as solid red lines. Inset in red text in Figure 4 are the MLE values of Ê q and Ê associated with the fitted curves. In both Figures 4a and 4b, the model curves appear to fit well to the observed PDFs over most of the range displayed. There are departures of the fits in the tails of the distributions, where the measurements approach the one count level and the statistics are poor. In Figure 4a, the values of Ê q are close to 1.0 denoting that the R1 PDFs are reasonably close to exponential. In Figure 4b, the values of Ê q are significantly higher than 1.0 denoting that the R2 PDFs  Figure 4c (R1), the mode is very close to zero, but the significant asymmetry in the PDF results in a much larger mean value. In Figure 4d (R2), the mode is further from zero than in the R1 case, but much closer to the mean, as the asymmetry in the PDF is reduced.
As in Figures 4a and 4b, the solid red lines show the MLE q-exponential fits to the PDF. Again, in both Figures 4c and 4d, the model curves provide good fits to the observed PDFs. At the very center of the distributions, close to the mode, the measured PDF is below the fit value. This is potentially the result of smoothing of the measured PDF due to vorticity measurement errors. We discuss the potential sources of measurement error later in the paper. Figure 5 presents the spatial variation of the mode of the vorticity PDFs across the northern hemisphere polar ionosphere, in a similar format to the mean variation presented in Figure 3a. Again, the data are presented in the AACGM co-ordinate system, extending from 66° latitude to  E 86° latitude. The range of the vorticity scale has been reduced (from −0.4 to 0.4 mHz) compared to Figure 3, owing to the mode values being consistently lower than the mean values for all the PDFs.

Spatial Variation of the Mode
As suggested by the example PDFs presented in Figures 4c and 4d, the relationship between the PDF modes and means is typically different in R1 to what it is in R2. In R1, the mode is typically between a magnitude of 0 and  E 0.2 mHz, whereas the mean is typically  E 1.0 mHz. Hence, it is the asymmetry in the PDFs (essentially the differences between the clockwise and counter-clockwise vorticity populations) that are predominantly responsible for determining the mean value. In R2, the mode is typically  E 0.2 to  E 0.4 mHz in magnitude compared to a mean value of  E 0.5 to  E 0.8 mHz. Here, the shift of the mode away from zero plays as big a role in the determination of the mean as any asymmetry in the PDF.

Spatial Variation of the q-Exponential Fit Parameters
The MLE analysis demonstrated in Figure 4 has been used to model the vorticity PDFs across the complete northern polar ionosphere, for the data in each 1° AACGM latitude by 1 hr of MLT bin. Figure 6 shows the spatial variation of the q-exponential MLE parameters ( , E q ) determined , at any location, allow the reconstruction of the model vorticity PDF at that location. Hence, by using the q-exponential fit parameters from Figure 6, the mode from Figure 5, and the survival function Equations 3 and 4, we can determine the probability that the ionospheric vorticity on the spatial measurement scales allowed by the SuperDARN configuration will exceed any particular magnitude.
In Figure 7, we present examples of the spatial variation of this probability for four different thresholds of vorticity magnitude (5, 10, 20, and 40 mHz). Looking at Figures 4a and 4b suggests that 5 mHz is close to the edge of the core of the distribution. By 10 mHz, most PDFs have dropped to less than 1% of the peak value. By 20 mHz, the measured PDFs are typically very noisy due to poorer statistics, and so the q-exponential model fit most probably provides a better estimate of the true PDF at this point. The probability of identifying a vorticity above 40 mHz can only be estimated by an extrapolation of the q-exponential model fit as it is beyond the range of the measured values.
The higher probabilities for each threshold in Figure 7 (black) occur when the variance of the distributions is high. This is a combination of high values of E q (which represent more leptokurtic distributions), and high

Factors Affecting the Measured Vorticity PDFs
The physical processes driving ionospheric vorticity are the primary factors that determine the PDF of vorticity fluctuations. However, there are other factors associated solely with the measurement technique that affect the measured PDFs and consequently, the spatial variation of the PDF characteristics. These factors need to be fully appreciated in order to minimize the misinterpretation of the PDFs and their spatial variation. Here, we discuss the effects of some of these factors.

Mixture Distributions
It is likely that there are multiple processes that cause fluctuations in vorticity. These processes may occur at spatial locations that vary with time. Consequently, the distribution at a given location will be the mixture distribution of all processes that can create vorticity at that location.
The polar ionosphere is continually changing in response to variations in the energy input from the solar wind to the magnetosphere via magnetic reconnection, and the explosive release of energy from the magnetospheric system via substorms. The imbalance of these dayside and nightside reconnection processes results in the continual expansion and contraction of the open magnetic field line region within the polar cap (e.g., Cowley & Lockwood, 1992).
The location of the R1 FACs is intimately associated with the location of the polar cap boundary between the ionospheric footprints of open and closed magnetic field lines Clausen et al., 2012). Hence, the R1 and R2 current systems expand and contract in tandem with the motion of the polar cap boundary (Clausen et al., 2013;Coxon et al., 2014). If we assume that the R1 FAC systems are responsible for one type of vorticity distribution and that the R2 FAC systems are responsible for another type of vorticity distribution, then the motion of these current systems will result in a mixture distribution in the overlap region (where both R1 and R2 vorticities may both exist, at different times). This mixture distribution will be that sum of the R1 and R2 distributions weighted by their probability of occurrence at that location.
Hence, it is likely that some of the spatial variability in the estimated MLE parameters for vorticity distributions near boundaries between different FAC regions is a result of the temporal motion of these regions. Moving equatorward from the R1 peak (where the vorticity distribution may be purely R1), the probability of observing R1 vorticity would be expected to decrease, while that of R2 vorticity would be expected to increase, leading to mixture distributions. Moving further equatorward would result in the vorticity distributions tending toward purely R2 distributions. This might suggest that the "true" R1 and R2 distributions are those seen at  E 74° and  E 67°, respectively.
The adverse effect of the expanding and contracting polar cap on ionospheric climatologies and empirical models is a long standing problem. Combining measurements from the auroral zone and the polar cap has been known to blur sharp boundaries in empirical convection models (as discussed in Chisham, 2017). Similarly, auroral zone and sub-auroral measurements are often averaged together near the equatorward edge of the auroral oval. The observed data at fixed coordinates are generally a combination of measurements from the different spatial regions either side of these boundaries. In recent years, these problems have been addressed by the use of adaptive high-latitude co-ordinate systems (Chisham, 2017;Kilcommons et al., 2017;Redmon et al., 2010;Zhu et al., 2020), where measurements are organized in spatial bins that adapt relative to the boundaries in question. However, this analysis is only possible if boundary location estimates for the measurement epoch are available. This is not the case for the whole interval of data presented here.
Another source of mixture distributions is the ionospheric asymmetries introduced by changes in IMF Y E B .
Variations in IMF Y E B result in variations in the MLT extents of the R1 and R2 FAC and vorticity regions, and hence have similar mixing effects to the expanding-contracting polar cap when combining data on fixed coordinate grids. The spatial variations of average vorticity for different IMF orientations were presented in Chisham et al. (2009). These show that while the patterns of vorticity around dawn and dusk are similar regardless of the IMF orientation, those around noon and midnight vary significantly, where the MLT of the switch between the dominant vorticity directions depends heavily on the IMF orientation. In this study, we have combined all vorticity measurements made at each location, regardless of the IMF orientation. This will result in mixture distributions in the measured PDFs close to noon and midnight.
In future work, we intend to separate the vorticity data by IMF direction (as in Chisham et al., 2009) to see if this addresses this problem. However, the consequent reduced data in the PDFs in many of the spatial bins (using the present data set) would increase the uncertainty in the MLE q-exponential fitting. This would be particularly significant at the high and low latitude extents of the field-of-view, where the overall occurrence of vorticity estimates reduces. The estimated values of E q and  E in this region are already relatively noisy (see Figure 6) without being subdivided by IMF direction. Consequently, to confidently cover a similar field-of-view, this proposed analysis would probably require either an increase in the spatial bin size or an extension of the measurement epoch beyond the 2000-2006 year interval.

Measurement Uncertainties
As with all measured quantities, there is a level of uncertainty in the individual vorticity estimates. Hence, the measured PDFs are a convolution of the real PDF at that location with a function that represents the measurement and algorithmic errors and uncertainties. These errors are most likely to result from missing or poor velocity estimates around the closed loops used to determine the vorticity (see Chisham et al., 2009 for full details of the vorticity determination), although they could also result from inaccurate mapping of the SuperDARN measurements (see e.g., Chisham et al., 2008). In the absence of systematic errors, we would expect this error function to be approximately Gaussian due to these random errors, although its width is difficult to estimate without realistic simulations of the measurement process. This convolution will have the effect of increasing the variance of the measured PDFs, although it is unlikely to significantly change their shape.
There is also an effect on the PDFs owing to the different measurement scales of the closed loops formed by the overlapping radar beams. As shown in Figure 1, these loops are larger at greater distances from the radars due to the divergence of the radar beams. Hence, these larger loops are typically at higher latitudes. Studying the variation of the vorticity PDFs with measurement scale (not shown) shows that the PDFs become more kurtotic and narrower as the measurement scale reduces; as the scales increase (beyond the measurement scales used in this study), the PDFs increasingly tend toward a Gaussian. Hence, the probability of measuring large vorticity values will decrease if the measurement scale increases significantly. This may partially explain the higher values of E q determined in R2 at the lowest latitudes where the measurement scales are the smallest. However, as there is no obvious monotonic variation of E q and  E with latitude, this is only likely to be a minor factor affecting the PDF shape and scale.

Interpretation of the Model Fit Parameters
In this paper, we have presented a statistical representation of ionospheric vorticity on the smallest measurement scales allowed by SuperDARN and how this statistical representation varies spatially across the northern polar ionosphere. These statistical results show that the large-scale averaged picture of ionospheric vorticity presented by Chisham et al. (2009) represents a gross simplification of the true picture, much like the averaged pictures of FAC structure (Anderson et al., 2008;Edwards et al., 2020;Weimer, 2001Weimer, , 2005. The PDFs show that the measured vorticity value at any location can be vastly different from the average value, often in the different sense, and often significantly larger than the average. Hence, a similar picture would be expected for FACs. The occurrence of these larger values will become more probable as the measurement scale decreases. Increased FAC structure has been shown to exist on small scales (Lühr et al., 1994;Neubert & Christiansen, 2003;Rother et al., 2007) with extreme FAC values increasing with decreasing scale size. Similarly, we would expect increased vorticity structure on smaller scales with the magnitude and probability of extreme vorticity values increasing with decreasing scale size. In this study, the measurement scale is restricted by the SuperDARN instrumental setup, and so it is not possible to measure vorticity on the smallest scales, below  E 100 km. However, the statistical analysis presented here has helped to better understand how the large-scale averaged picture of vorticity emerges from the aggregation of smaller scale vorticity measurements. It is likely that FACs can be understood in a similar way.
We have shown that the measured PDFs of ionospheric vorticity fluctuations are well modeled by q-exponential probability distributions at all locations in the polar ionosphere. These distributions have previously been shown to successfully model fluctuations in other space plasma measurements (e.g., Barbosa et al., 2017;Burlaga et al., 2007;Esquivel & Lazarian, 2010). The parameter E q is associated with the level of kurtosis in the distribution, and hence, the amount of turbulent intermittency. Higher values of E q reflect more leptokurtic tails and hence increasing intermittency (see Figure 2). The value of  E is associated with the variance in the distribution, and hence, how much the vorticity varies from the typical magnitude of the rotational plasma flow.
Based on the results discussed in Section 3.3 and shown in Figure 6, we find that vorticities in the average R2 current region below 70° latitude are generally relatively weak (low  E ) but intermittent (high E q ). This also suggests that the current systems are more fragmented and filamentary in R2, resulting in an increase in the small-scale structure, such as seen in parts of the auroral zone. As the R2 FACs connect to the partial ring current, this is likely to be the source of this turbulence.
Weak and intermittent vorticity is also seen in the open field line region poleward of the R1 current region, that is, poleward of 80° on the dayside and poleward of 75° on the nightside. This open field line intermittency is slightly less than that in the R2 region and more spatially structured, suggesting that it has a different physical cause. Abel et al. (2007Abel et al. ( , 2009 showed that in the nightside open field line region moments of velocity gradients in plasma flow (i.e., structure functions) were consistent with the so-called p model of intermittent Kraichnan-Iroshnikov (i.e., 2-D) turbulence. They argued that the intermittency was inherited from that of the solar wind because both ionospheric and solar wind turbulence could be described by a p model with a similar value for the intermittency parameter p, and also because the intermittency disappeared during northward IMF when ionospheric flows would be decoupled from the solar wind. Thus, we propose the same cause for the intermittency seen here in open field line vorticity.
In contrast to the open field line and R2 regions, vorticities are typically stronger but less intermittent in the average R1 current region between 70° and 80° latitude on the dayside and between 70° and 75° on the nightside. However, there are distinct differences between the PDFs on the dayside and nightside.
On the dayside, the R1 vorticities are strongest but the least intermittent, even though we would expect them to be directly driven by the solar wind like in the open field line region. To explain this, we speculate that intermittency may be reduced by the relatively high and homogeneous conductance from solar illumination in the dayside R1 region compared to the open field line region.
On the nightside, the R1 currents connect to the magnetotail current sheet and are driven by substorms and nightside reconnection, which are indirectly driven by the solar wind. Thus, here, a difference in vorticity to that in the directly driven open field line region is understandable. The simplest large-scale picture of substorms depicts FACs at either end of a westward electrojet, forming the substorm current wedge (SCW). However, the real situation is more complex, with significant small-scale structure being observed within the traditional large-scale SCW (Murphy et al., 2013). Forsyth et al. (2014) have shown that the SCW comprises lots of filamentary current sheets and substructure on scales of 100 km, some of which are oppositely directed to the larger scale current. These small-scale filamentary current sheets seen during substorms suggest that the nightside R1 vorticities are typically more intermittent than those observed on the dayside.
The larger values of E q determined for the nightside R1 PDFs support this interpretation.

Regarding the Measurement of the Probability of Extreme FACs
The interpretation of the character of the vorticity PDFs above has focused heavily on the role of FACs. Given the intimate connection between vorticity and FACs, we might also expect distributions of FAC to have similar leptokurtic and heavy-tailed forms. However, although small-scale measurements of extreme FACs have been made, the distribution of all FAC fluctuations and the probability of extreme values in different ionospheric regions has not been studied. A long-tailed distribution of "intense kilometer-scale" FAC densities has been reported (Rother et al., 2007), but this was of peak currents during an "event," and not of all the FAC measurements. No attempt was made to model this distribution or to assess whether it was leptokurtic or heavy-tailed.
To fully understand FACs and the relationship between measurements on different scales, a similar statistical analysis to that of vorticity presented here, is required. Given the close relationship of FACs with vorticity, one question is whether it is possible to estimate the equivalent PDFs of FAC fluctuations directly from the PDFs of vorticity. The connection between FAC and vorticity in the ionosphere can be expressed in terms of the  E E B drift velocity E v Sofko et al., 1995) as where  E    E v is the ionospheric flow vorticity. Hence, in the limit of constant (fixed Σ P E ) and uniform ionospheric conductance (     Σ Σ 0 P H E ), the FAC distributions should match those of vorticity. However, such conditions are generally limited to the sunlit region of the ionosphere.
Equation 5 implies that extreme FACs are not solely associated with regions of extreme vorticity, but also with regions where extreme spatial gradients in the Hall or Pedersen conductance exist, as might occur at the poleward and equatorward boundaries of the auroral zones. These gradients in conductance are difficult to measure over large areas with present instrumentation. Hence, the production of statistical distributions of these gradients is not presently feasible. This may change with the advent of new instrumentation, such as the upcoming EISCAT 3D system (McCrea et al., 2015).
The alternative is to determine PDFs of FACs, and hence the probability of extreme FACs, independent of vorticity measurements, using direct FAC measurements. The only data set large enough to allow the production of reliable PDFs across the whole polar ionosphere would be that from Active Magnetosphere and Planetary Electrodynamics Response Experiment (AMPERE) Coxon et al., 2018). The AMPERE data set consists of FACs at a 10-min cadence, determined by combining magnetic field fluctuations from the constellation of Iridium spacecraft. Many years of AMPERE FAC observations are now available that would enable the direct estimation of PDFs of FAC using a similar analysis to that used to analyze the SuperDARN vorticity data. These would be on a similar measurement scale to the SuperDARN vorticity measurements. However, additional challenges may exist owing to the spatial averaging and limited temporal resolution of the AMPERE measurements.

Summary and Conclusions
The statistical study of ionospheric vorticity presented here helps to understand the connections between flow turbulence, vorticity, and FACs in the polar ionosphere, and the different effects of the drivers of these processes.
From the analysis presented in this study, we can make the following conclusions: 1. The observed systematic spatial variation of the q-exponential fit parameters across the northern hemisphere polar ionosphere shows that the variations in vorticity are organized similar to the different FAC systems. 2. The vorticity PDFs show that at any location a much wider range of values are observed, of both senses of vorticity, that cannot be easily predicted from the local average value. 3. Modeling of the observed vorticity PDFs allows us to understand differences between the R1 and R2 regions. Although the vorticity has greater variance in R1, vorticity is more intermittent in R2, meaning that extreme flows are more likely there. 4. Extreme values of vorticity (and by association, of field-aligned current), are observed with a nonnegligible probability in most regions of the polar ionosphere and hence, likely have a disproportionate effect on M-I coupling processes.