The temporal and spatial development of dB/dt for substorms

Ground induced currents (GICs) due to space weather are a threat to high voltage power transmission systems. However, knowledge of ground conductivity is the largest source of errors in the determination of GICs. A good proxy for GICs is dB/dt obtained from the Bx and By components of the magnetic field fluctuations. It is known that dB/dt values associated with magnetic storms can reach dangerous levels for power transmission systems. On the other hand, it is not uncommon for dB/dt values associated with substorms to exceed prior Pulkkinen and Molinski critical thresholds of 1.5 nT/s and 5 nT/s, respectively, and the temporal and spatial changes of the dB/dt associated with substorms, unlike storms, are not well understood. Using two dimensional maps of dB/dt over North America and Greenland derived from the spherical elementary currents, we investigate the temporal and spatial change of dB/dt for both a single substorm event and a two dimensional superposed epoch analysis of many substorms. Both the single event and the statistical analysis shows a sudden increase of dB/dt at substorm onset followed by an expansion poleward, westward, and eastward after the onset during the expansion phase. The area of dB/dt values exceeding the two critical thresholds from the initial onset dB/dt values showed little to no expansion equatorward. The temporal and spatial development of the dB/dt resembles the temporal and spatial change of the auroral emissions. Substorm values of dB/dt peak shortly after the auroral onset time and in at least one event exceeded 35 nT/s for a non-storm time substorm. In many of our 81 cases the area that exceeds the threshold of 1.5 nT/s is over several million square kilometers and after about 30 minutes the dB/dt values fall below the threshold level. These results address one of goals of the Space Weather Action Plan, which are to establish benchmarks for space weather events and improve modeling and prediction of their impacts on infrastructure. Plain language: The change in the ground magnetic field with respect to time (dB/dt) associated with magnetic storms (a large disturbance of the magnetic field of the earth)


Introduction
Geoelectric fields due to geomagnetically induced currents (GICs) [1] associated with space weather phenomena are a threat to high voltage power transmission systems and oil pipe lines [2]. However, the largest source of uncertainty in determining GICs is our limited knowledge of the ground conductivity [3,4]. Therefore we must look at other measurements to quantify threats to power transmission systems. The dB/dts associated with GICs are an excellent proxy for GICs and can be quickly and easily calculated from ground magnetometers [5]. It has been recently shown there is a good correlation between the GIC magnitude in New Zealand and the dB/dt determined from the horizontal component (i.e., the combined X and Y components) measured by ground magnetometers [6]. It is known that dB/dt values associated with magnetic storms can reach dangerous levels for power transmission systems for short periods [7,8]. However, it is not uncommon for dB/dt values associated with non-storm time substorms to exceed critical thresholds of 1.5 nT/s [9,10], and 5 nT/s [11,12], which are dB/dt values associated with problems in the electrical grid. What is not yet well understood is the temporal and the two dimensional spatial development of dB/dt associated with substorms during non-storm time conditions. The dB/dt and GICs associated with geomagnetic storms have been studied in details for decades [6,[9][10][11][12][13][14]. Geomagnetic storms occur when the interplanetary magnetic field turns southward and remains southward for a prolonged period of time. These magnetic storms typically last several days. While the number of storms per year varies with the solar cycle, there are on average around 5 storms per year where the strength of the storm (Dst) is greater than 100 nT [15]. It is well known that during magnetic storms ground magnetic field perturbations, which can be observed in ground magnetometer data and in both the Dst and the AL indices, can induce potentially large electric currents in high voltage power transmission lines and other electrically conducting infrastructure, such as pipelines and communications systems etc, stretched over long distances. Some of the most widely studied storms over the last few decades are the March 1989 storm, the Halloween storm of 2003, and more recently the March 2015 St Patrick's day storms and most of these storms are known to have induced large currents in power grids. The March 1989 storm is well known for the costly Hydro Quebec power grid failure and the Sydkraft Group in Sweden, which is a large power utility, experienced transformer problems during the Halloween 2003 storm that led to a system failure and a subsequent power outage [16]. However, despite the large AL index drops no power outages are known to have occurred for the March 2015 storm. Table 1 provides the geomagnetic conditions, dB/dt values, Voltages per km of power lines, and GIC current amplitudes available in the literature for the three well studied magnetic storms. Note that the dB/dt value for the March 2015 storm (~13.5 nT/s) has been published by [17], but using the method discussed in this study we obtain a value of about 16.5 nT/s, which has been previously presented and discussed at the International Community Coordinated Modeling Center-Living With a Star working meeting in Cape Canaveral Florida in April 2017. Recently a value of ~16.7 nT/s has been published for the Scandinavia region [5] for the September 2017 storm.
Prior work has been done to determine the general dB/dt thresholds at which power grid operators need to be concerned. [12] developed a geomagnetic hazard map of North America for dB/dt = 5 nT/s, which he felt represented dB/dt's during large storm time intervals, working directly with Geomagnetic Laboratory of the Geological Survey of Canada in Ottawa and the Electric Power Research Institute. Their map shows that the highest probability (0.2% in any given year) of exceeding this threshold occurs in the nominal location of the auroral oval (i.e., between about 52° to 56° MLat). Furthermore, the threshold of 5 nT/s is within a factor of three of the storms known to cause power grid failures reported in Table 1. On the other hand, a much lower level of 1.5 nT/s was found in another study [18] by examining conditional probability distributions developed using a local electrojet index that used 10 s resolution IMAGE ground magnetometer data in the local time sector of 18 to 24 LT during a large ground induced current event associated with a ΔB of ~600 nT. However, later studies [1,2] consider thresholds of 0.3, 0.7, 1.1, 1.5 nT/s in studies that compare ground magnetometer results for a number of large storm events with Community Coordinate Modeling Center model results. These threshold are considerably lower than the previously reported threshold of 5 nT/s [11,12], but other studies reported that some power grids have had problems at dB/dt's of 1.7 nT/s [8]. For our study we will consider the upper threshold of 1.5 nT/s [9,10,18], which is associated with dangerous ground induced currents reported in Scandinavia, and the threshold of 5 nT/s [12] that is associated with dangerous ground induced currents levels reported in Canada. Compared to geomagnetic storms, substorms occur much more frequently. In fact, it has been shown using particle injection data measured by geosynchronus spacecraft that substorms occur approximately every 2.75 hr [23]. A similar frequency of 2.75 hr has been reported elsewhere [24] and it is well known that substorms continue to occur during magnetic storms. A typical substorm consists of three phases: onset, expansion, and recovery [25]. During the onset there is a sudden localized brightening of the aurora at the equatorward edge of the auroral oval somewhere between 18 and 3 MLT and 55° and 74° MLat [26]. At about the same time and geographic location the H component of the magnetic field suddenly decreases, which is associated with an enhancement of the westward electrojet and a sharp drop in the AL index. This sharp drop can be as much as 1000 nT or more. Following the auroral onset the aurora begins to expand poleward, westward, and eastward and the field aligned currents increase in intensity over a period of about 15-20 min. The eastward electrojet also intensifies during the expansion phase although not as much as the westward electrojet. During the recovery phase the auroral luminosity, magnetic field, and current systems return to their nominal configuration prior to the substorm onset. This brief description of the substorms implies that large values of dB/dt can occur during substorms. It has been shown that these changes can be as large as 10.7 nT/s typically occurring within the first 10-20 min of the substorm onset, but can occur at any point during the substorm [27]. More recently, dB/dt changes as large as ~16.7 nT/s during a storm time substorm have been measured [5]. A lot of work has been done to examine the dB/dt and the ground induced currents associated with storm time periods. However, little work has been done to examine the dB/dt and the ground induced currents during substorms and no studies have examined the simultaneous temporal and spatial development of values of dB/dt associated with substorms (both storm time and non-storm time substorms). In this study we will examine that temporal and spatial development for a single non-storm time substorms on 4 April 2010. In addition, we perform a 2 dimensional superposed epoch analysis on a set of 81 substorms, most of which are non-storm time substorms, to understand when and where the peak dB/dt occur with respect to the auroral onset time and location.

Data
The data for this study come from two distinct sources: the THEMIS all sky image (ASI) array and a large number of ground magnetometers across North America and Greenland.
Magnetometer data comes from eleven different ground magnetometer arrays. The magnetometer data from these arrays are used to produce a two dimensional map of ionospheric currents over North America and Greenland using the spherical elementary current system (SECS) technique. More details on the description of the SECS technique over Greenland and North America and the calculation of the spherical elementary currents (SECs) [28][29][30][31]. The number of available magnetometer stations for each two dimensional map of ionospheric currents typically changes from day to day due to data gaps, changes in baseline, and measurement errors. Information related to the ionospheric currents can be found at [29,32].
THEMIS ASIs are used to identify substorm auroral onset times and locations. White light ASIs with the temporal resolution of 3 s are obtained from an array of Ground-Based Observatories (GBOs) spread over Alaska, Canada and Greenland. More details on the imagers and their geographic positions can be found in [33] and [34].
The list of substorms used in this study has been published elsewhere [35] and uses the midlatitude positive bay index (MPB). The compiled list of substorms consists of several thousand midlatitude positive bay substorm onsets and was developed from an inversion technique to calculate parameters determining the onset time, intensity, and geometry of the substorm current wedge system using magnetic field data from 20 midlatitude ground magnetometers [35]. See [24,36] for more details. We have taken a subset of the midlatitude bay substorms from the substorm list between 2008 and 2012 and determined an auroral onset time and location within the ASIs when clear auroral images are available. From the original substorm list [35] consisting of about 1600 midlatitude bay substorms between 2008 and 2012 we have identified the auroral onset time and location for 81 substorms between 22 and 23 MLT. See Supplementary materials. Six of these substorms are storm time substorms with a Dst between −50 nT and −100 nT in the time range of 10 minutes before the auroral onset time to 60 min after the auroral onset time. The remaining 75 substorms are non-storm time substorms.

Procedure
The dB/dt values to be calculated in this study will be determined by using the Biot-Savart law applied to both the equivalent ionospheric currents and the spherical elementary current amplitudes. Specifically, where dBx and dBy represent the two components of the fluctuations in the magnetic field horizontal to the Earth's surface determined from the spherical elementary current (SEC) contributions and dt is the time resolution of the SEC data, which is 10 s in our study. This determination of dB/dt is employed to be consistent with prior methods [9,10].
We are applying the Biot-Savart law to the ionospheric currents of obtained ΔB = (dBx 2 + dBy 2 ) 1/2 values because the interpolation of dB/dt across North America and Greenland directly from the ground magnetometer data using the ground magnetometers results in either data gaps, unrealistic data (i.e., data spikes), or inaccurate data. To check the accuracy of our SEC ΔB values we calculated the SECs using all the available magnetometer stations. From these SECs we calculate the ΔB values using the Biot-Savart law. We then remove one station in the central part of the magnetometer array and recalculate the SECs and new ΔB values. The difference in the ΔB values before and after the station is removed are compared at the location of the station. Our results suggest that the differences are on the order of 6% or less for stations located within the center of North America between Hudson bay and the west coast and surrounded by additional magnetometers. Comparisons of the interpolated ΔB sometimes show large differences on the order of more than 10% for stations located in or near Hudson Bay compared to the ΔB measure at the individual magnetometer sites. At the edges of the magnetometer array these difference can grow to tens of percentage point because removing a magnetometer at the edge of the array can significantly change the SEC pattern in that region. However, similar poor results are obtained we linear interpolation is used at the edges of the array. Fortunately, none of our auroral substom onset occur near the edge of the magnetometer array.

Observations
We now examine the dB/dt associated with a specific substorm on 4 April 2010 at 0654 UT and demonstrate that the dB/dt exceeds the critical threshold of 1.5 nT/s [9,10] and 5 nT/s [11,12]. Just prior to this stubstorm the solar wind speed is about 500 km/s and the interplanetary magnetic field is about (Bx = 4, By = 0, Bz = −2) nT GSM. The solar wind number density is just over 2 #/cm 3 Figure 1) and one time step is given in each row. Each panel is in geographic coordinates and the black north-south line on the right side of the map indicates local midnight. The keys for the equivalent ionospheric currents and the spherical elementary current amplitudes are located in the lower right corner. The color bar for the dB/dt values is below each panel. The stars in each plot indicate the ground magnetometers with data used to derive the SECs. In the first column are the equivalent ionospheric currents. The dot indicates the location at which the currents were derived and the bar indicates the magnitude and direction of the equivalent current. The middle column shows the current amplitudes, which are a proxy for the field aligned currents. In each panel the blue squares extending from northeastern Alaska across Canada to northern Quebec display where the region-1 current flows into the ionosphere and the red crosses equatorward of the blue indicate where the region-2 current flows out of the ionosphere. The Harang current system is located on the eastern side of Alaska at the Canadian border.
In the top row at 0646 UT just prior to the substorm a westward electrojet is visible in the equivalent currents from northern Quebec and Newfoundland over Hudson bay and across to Alaska. Region 1, Region 2, and the Harang current system are present in the current amplitudes, and two small regions of dB/dt > 1.5 nT/s are outline with blue contours in the upper right panel. In the second row at 0654 UT the westward electrojet has begun to strengthen just to the left of local midnight, the Region 1 and Harang current systems have also begun to strengthen, and the region of dB/dt > 1.5 nT/s has grown to 2.4 Mkm 2 . For comparison, Alberta, Canada covers 0.66 Mkm 2 . Two small regions of dB/dt > 5 nT/s outlined with mauve have also appeared.
At 0701 UT about 7 minutes after the substorm onset in the third row the both the westward electrojet and eastward electrojet, which is below the westward electrojet in Alaska, have significantly enhanced. The current amplitudes associated with the substorm have also strengthened. The dB/dt values > 1.5 nT/s cover large parts of Canada including areas in eastern Canada far from the substorm onset location. The dB/dt > 5 nT/s has peaked at 14.5 nT/s over an area of in northwestern Canada at the Alaskan border. By 0708 UT the equivalent ionospheric currents and current amplitudes are still quite strong, but the dB/dt has significantly decreased. Only a few small areas of dB/dt > 5 nT/s are still present, but a large area of dB/dt > 1.5 nT/s on the order of 2.4 Mkm 2 is still present in northwestern Canada. By 0724 UT the AL index has returned to pre-substorm levels (~200 nT) and both the equivalent currents and the current amplitudes have decreased in magnitude. Furthermore, no more dB/dt > 5nT/s is present, but an area where dB/dt > 1.5 nT/s is still apparent in northwestern Canada at the Alaskan border.   [9,10] suddenly increases to between 2 and 3 Mkm 2 and both thresholds appear to peak about 7 min after onset and the critical threshold covers ~2 Mkm 2 for about 16 min. Furthermore, the Molinski threshold [11] reaches just under 1 Mkm 2 in area at 6 min after the onset and then nearly 2 Mkm 2 about 26 minutes after the onset albeit for just 1 min. This large area appears to be related to a spike in the Inuvik ground magnetometer data. Similar spikes have been observed before [37,38]. We note here that the location of the region that exceeds 5 nT/s is generally in the northwestern portion of Canada, but not consistently in the exact same location. A similar statement can be made for the region that exceeds 1.5 nT/s, but the area exceeding the Pulkkinen threshold [9,10] tends to be at the same location more frequently.

Discussion
In the Observations section we demonstrated for the large 4 April 2010 substorm at 0654 UT that critical dB/dt thresholds were exceeded and covered large areas. What is unclear is how frequently do substorms exceed the critical thresholds of 1.5 nT/s and 5 nT/s, and if large dB/dts are common in substorms, then what is the temporal and spatial development of values of dB/dt associated with substorms?
We examined 60 minutes after substorm onset for 81 substorms between 22 and 23 MLT with AL minimums from −149 to −1408 nT and found in all 81 substorm at some time within the 60 min dB/dt exceeded the Pulkkinen threshold [9,10] of 1.5 nT/s. The median number of minutes above the threshold is 32 min, although this time is not necessarily consecutive. We note that the dB/dt values are at 1 min temporal resolution and we have not used the full 10 s resolution data. Furthermore, in 63 of 81 substorms the dB/dt exceeded the Molinski threshold [11] of 5 nT/s. The median number of minutes above the 5 nT/s threshold is 4 min. We note one substorm event on 9 October 2012 during the latter half of a storm dB/dt exceeded 5 nT/s for approximately 40 min.
The top of Figure 4 displays the minimum AL index value between the substorm onset and 60 min after the substorm onset versus the maximum dB/dt in the nightside in the same period for the 81 different substorm events. Note that we have not differentiated between storm time substorms and non-storm time substorms because of the limited number of storm time substorms. The black line is a linear fit to the events where the slope is 0.014 s −1 and the intercept is 3.0 nT/s. However, there is a great deal of scatter in the data points and the correlation coefficient is 0.54. These results are not as well correlated with a similar study using the IMAGE ground magnetometer array study [27], but that study is systematically different from our study. That study compared maximum dH/dt to the maximum value of the H component, which points toward magnetic north, for a specific station for both non-storm time substorms and storm time substorms and obtain correlation coefficients of 0.75 and 0.66, respectively [27]. In comparison, our dB/dt values are the combination of both the X and Y components, which point toward geographic north and east, and we have compared to the minimum in AL. Furthermore, we examined the maximum in dB/dt wherever it occur in the nightside over North America and not at a specific station location. That being said, it is interesting to note that the maximum dH/dt value reached in the study using the IMAGE array was just under 11 nT/s [27] whereas our study found a maximum dB/dt of just under 40 nT/s.
The bottom of Figure 4 shows the minimum AL index value between the substorm onset and 60 min after the substorm onset versus number of minutes above the critical threshold. The black point are associated with the Pulkkinen threshold [9,10] and the blue points are associated with the Molinski threshold [11]. The black line is a linear fit to the events associated with the Pulkkinen threshold [9,10] where the slope is 0.42 s −1 and the intercept is 11.7 nT/s and the blue line is a linear fit to the events associated with the Molinski threshold [11] where the slope is 0.019 s −1 and the intercept is −3.7 nT/s. The correlation coefficients for the datasets are 0.73 and 0.73, respectively. These plots indicate that the time associated with each threshold seems to be well correlated with the minimum in the AL index associated with each substorm.
In statistical studies it is common to combine many events with a one dimensional superposed epoch analysis to show a systematic temporal changes in the parameter of interest. In this study we  [26]. Figure  9 in [39] helps illustrate the rotation and expansion/contraction of the currents to the epoch zero location. That is to say, we treat the location 66° MLat and 22.5 MLT as an epoch origin location in the same way the auroral onset time as an epoch zero time. For current maps before and after the substorm onset time the same expansion (or contraction) of the currents and rotation of the currents is done. The purpose is to align the current systems in such a way to produce a statistical two dimensional median of the dB/dts and currents and indicate how these systematically develop throughout the substorm growth, expansion, and recovery phases. Once the current maps have been aligned, the data are binned to 3 in Mlat by 0.5 hr in MLT to determine the median values of the dB/dts and the SECs.  However, it is difficult to determine the peak dB/dt value within the two dimensional superposed epoch maps and when that peak value occurs. For Figure 6 we have extracted the peak dB/dt value from the two dimensional superposed epoch dB/dt maps within the vicinity of the substorm per epoch time. The curves shown in Figure 6 are the peak (solid curve) dB/dt values of the superposed epoch maps and the upper and lower quartiles (dashed curves). To obtain the upper and lower quartiles we had to construct additional two dimensional superposed epoch maps of the upper and lower quartiles and extract the dB/dt peaks for each epoch time. The median curve demonstrates that the dB/dt begins to increase just prior to the substorm onset time at epoch time 0 min and peaks at about epoch time 5 min. This time of the peak is within the range of occurrence time of the max (|dH/dt|) after the substorm onset at each available magnetometer site within the IMAGE array study [27] for both storm time substorms and non-storm time substorms. The range shown in Figure 3 of the IMAGE array study [27] is 4 to 6 minutes after susbtorm onset. Our Figure 6 also shows that the upper quartile extends above the critical threshold of 1.5 nT/s, which indicates that a number of large substorms are present in our 81 events. Shortly, we will examine these larger substorms in more detail.  In Figures 5 and 6 we showed the temporal and spatial development of all 81 substorms between 22 and 23 MLT. These figure shows the sudden increase of dB/dt and the expansion of the dB/dt area within the first 10 minutes of the susbtorm in the two dimensional maps. However, Figures 2, 4, and 7 indicate that a number of larger substorms are present in this study. In Figure 8 we show the temporal and spatial development of dB/dt at the 95% (the top 5 largest susbtorm events). The format of the dB/dt polar plots is the same as those in Figure 5 and we show the epoch times −5, 0, 5, 20, 40, 60 min. This series of dB/dt maps shows the sudden increase of dB/dt at onset above the critical threshold of 1.5 nT/s in the pre-midnight sector. See the black contour in the pre-midnight sector. The area above the 1.5 nT/s threshold continues to expand westward, eastward, and poleward during the expansion phase in much the same manner as the auroral emissions evolve. The area above the 1.5 nT/s threshold expands because the field aligned currents and ionospheric Hall currents associated with the substorm current wedge strengthen and widen during the onset and expansion phase of the substorm as shown in Figure 5a,b. Throughout the recovery phase (epoch time > 20 min) the dB/dt remains above 1.5 nT/s and the peak dB/dt shifts poleward. The purpose of this sequence of dB/dt maps is to show during large substorms dB/dt exceeds the critical threshold of 1.5 nT/s for periods on the order of tens of minutes over large areas, which increases the potential for dangerous geoelectric fields and GICs.

Summary and conclusions
We have examined both a single dB/dt enhancement during a substorm on 4 April 2010 and have also provided a statistical analysis of 81 substorm events. We summarize the important findings here. 5. The area that exceed the threshold of 1.5 nT/s shows a systematic pattern similar to the auroral onset of the substorm. That is to say, the dB/dt suddenly increases at the auroral onset time and expands westward, eastward, and poleward throughout the substorm expansion phase. No, significant equatorward dB/dt expansion was observed, contrary to was is seen for substorm associated with intense storms. During the recovery phase the dB/dt gradually returns to pre-substorm levels and the peak shifts westward and poleward. We believe that the latter two results of the evolution of the dB/dt area have not yet been published. The results of our study may be of interest to the Space Weather Action Plan (SWAP). Two of the goals of the SWAP are improving the timeliness and accuracy of space weather forecasts and enhancing the protection of national security assets and critical infrastructure. We have demonstrated for a limited number of substorms that the dB/dt exceeds critical thresholds associated with problems in the electrical grid over significant areas. Furthermore the temporal and spatial development of the dB/dt follows a systematic pattern that could be used to improve the accuracy and timeliness of space weather forecasting. A larger scale statistical study is required to confirm these statements.