Determining Electric Fields in Thunderclouds With the Radiotelescope LOFAR

Abstract An analysis is presented of electric fields in thunderclouds using a recently proposed method based on measuring radio emission from extensive air shower events during thunderstorm conditions. This method can be regarded as a tomography of thunderclouds using cosmic rays as probes. The data cover the period from December 2011 till August 2014. We have developed an improved fitting procedure to be able to analyze the data. Our measurements show evidence for the main negative‐charge layer near the −10° isotherm. This we have seen for a winter as well as for a summer cloud where multiple events pass through the same cloud and also the vertical component of the electric field could be reconstructed. On the day of measurement of some cosmic‐ray events showing evidence for strong fields, no lightning activity was detected within 100 km distance. For the winter events, the top heights were between 5 and 6 km, while in the summer, typical top heights of 9 km were seen. Large horizontal components in excess of 70 kV/m of the electric fields are observed in the middle and top layers.

TRINH ET AL.: ATMOSPHERIC ELECTRIC FIELDS BY LOFAR X -3 For all events the same procedure was followed where the parameters defining the structure of the atmospheric electric fields were fitted, using MGMR3D, in order to obtain the best agreement with the measured radio profile. In the fitting we optimized a chi-square criterium using the semi-analytic MGMR3D code for the calculation. This calculation was done for three different values for X max since it was observed that there is a strong dependency in the parameters and thus it was not possible to keep X max as an independent parameter. Once an atmospheric electric field configuration was obtained this was used as an input to a fully microscopic calculation, using CoREAS, to verify the results.
In many cases the results of MGMR3D and CoREAS are very similar, but there were also some cases where they differ significantly. These cases were excluded from further analysis. Of the three calculation a preferred one was selected based on the agreement of the CoREAS calculation with the data and the value of the normalization constant, as explained in the main text. The preferred result is indicated with an asterisk attached to the simulation number in the table giving the fit parameters. For the preferred case a figure is shown giving the comparison of the results of the CoREAS calculation with the LOFAR data for the stokes parameters.
The determination of the position of the core of the air-shower is important for the analysis of the radio footprint. As a guidance the position as estimated from the particledetector array (called LORA) is taken, in most cases this position was part of the chisquare search. The approximate shift in the core position is also noted.
February 19, 2020, 11:15am X -4 TRINH ET AL.: ATMOSPHERIC ELECTRIC FIELDS BY LOFAR 1.1. General summary All events show that over a wide range of X max values, a reasonable agreement between MGMR3D and the data can be obtained. The extracted field configuration, in particular the boundaries of the different layers, is rather in-sensitive to the value of X max .
In events #1, #2, #4, #7, #8, and #10, the intensity pattern has a ring-like structure as can be seen from Fig. S1 which displays the intensity footprint of event #8. In the plots of the Stokes parameters (see the middle panels of Fig. S4, Fig. S5, Fig. S8, Fig. S15, Fig. S21), the ring-like structure is seen as a peak in the Stokes I at a distance between 100 m to 250 m from the shower axis. The ring-like structure in intensity is due to a destructive interference between the radio emission from the upper and lower layers for these events since the field points almost in opposite directions in these layers. In addition, the radius of the ring in the intensity is strongly correlated to the height where the field is inverted. For smaller radii, the heights are smaller. For example, in event #2, the radius of the ring is 100 m and thus the field is inverted at 2.1 km (see Table S2) while in event #10, the radius of the ring is about 250 m and thus the field is inverted at 5.0 km (see Table S10).
In event #2, the amount of circular polarization (Stokes V ) is very small (see the middle panel of Fig. S5). Thus, this event can be fitted by a two-layered electric field where the fields in the two layers are almost opposite to each other (see Table S2). In contrast, events #1, #7, #8, and #10 which also show the ring-like structure in the intensity have a large amount of circular polarization. Therefore, these events cannot be reconstructed well by a two-layered electric field structure. The electric field needs to have at least three layers (see Table S1, Table S7, Table S8, Table S10). A third layer is needed to February 19, 2020, 11:15am TRINH ET AL.: ATMOSPHERIC ELECTRIC FIELDS BY LOFAR X -5 introduce the change in the orientation of the electric fields and thus the rotation of the transverse current which results in a large amount of circular polarization. In addition, a third layer also gives rise to the change in the linear polarization which causes a 'wavy' pattern. Fig. S2, as an example, shows the 'wavy' pattern of event #7 where the linear polarization rotates about 90 • from small distances near the shower axis to large distances beyond 100 m from the shower axis.
Event #4 is an odd one since there is a large amount of circular polarization near the shower axis but the linear polarization is the same all antennas (see the middle panel of Fig. S8). Therefore, as shown in Table S4, the electric fields in the bottom and the middle layers are not fully opposite but they have an angle of about 150 • .
In contrast to the events just discussed above, the intensity patterns in events #3, #6, #9, and #11 are similar to those in fair-weather events (see Fig. S6, Fig. S13, Fig. S18,   Fig. S24). However, unlike for fair-weather events, with the exception of event #11, the signals are not polarized along the v×B-direction because Q/I is not equal to 1 (see Fig. S6, Fig. S13, Fig. S18). In events #3 as well as #6, since U/I is about -1, the linear polarization makes an angle of about -45 • with the v×B-direction. For this reason, the electric fields in the layers where the current is large, i.e. the middle layer of event #3 and the top layer of event #6 (see Table S3 and Table S6), make an angle of about -55 • with respect to the v×B-direction. There is some amount of circular polarization in these two events but it is small. In event #9 which has been discussed in detail in (Trinh et al., 2017), the polarization footprint shows a 'wavy' pattern and there is a large amount of circular polarization, varying as a function of distance from the shower axis. Therefore, February 19, 2020, 11:15am X -6 TRINH ET AL.: ATMOSPHERIC ELECTRIC FIELDS BY LOFAR the electric field in the middle and the bottom layer rotates 90 • giving rise to the rotation of the linear polarization as well as the amount of circular polarization (see Table S9). Event #11 is an odd case because not only the intensity but also the linear polarization looks like that of a fair-weather events at shorter distances to the shower axis, while at large distances Q/I is much smaller than 1 (see the middle panel of Fig. S24). The main difference from a fair-weather event is, however, that the circular polarization is large and changes its handedness with distances which is caused by the rotation of the electric field orientation with height. Near the shower axis, the signal at the bottom layer arrives earlier than the signal from the other layers because the showers propagates with the speed of light while the signal moves at a reduced speed due to the finite refractivity of air. This gives rise to a large amount of circular polarization at small distances, V /I = 0.4. At 150 m from the shower axis, the signal from the middle layer arrives earlier than the signal from the bottom layer, so V /I = -0.4. Similarly, beyond 150 m, the signal from the top layer arrives before that in from the other layers, so the circular polarization continues to decrease at large distances.

December 14 th , 2011
For the three events detected on December 14 th , 2011, there was no lightning activity detected in the vicinity of the Superterp. The nearest lightning activity was detected at a distance of 200 km and we have thus not included a lightning map. Radar-reflectivity measurements, Fig. S3, show that at the time of the air-shower detections an active cell of a cloud was passing over the Superterp. This event turned out to be the most difficult one to understand from our whole collection. Partly because of the lower intensity and thus relatively large error bars, partly because of the structure of the polarization. In addition the core location for this event is difficult to determine because of the low intensity of the radio signal and because the core appears to lie at the rim of the Superterp and is thus not surrounded by antennas.
For a best fit in MGMR3D the core moved by about 70 m. In MGMR3D a good fit could be obtained for the three different values of X max that does not differ much in structure.
However, for each of these cases the results of the CoREAS calculation were showing considerable differences. This can be seen from the values of chi-square given in Table S1 by comparing χ 2 3D (for MGMR3D) with χ 2 C (for CoREAS). To give an example the results of the CoREAS calculation is compared with the data in Fig. S4. We do not understand the reasons for the major discrepancies between the semi-analytic and microscopic calculations for this event but it is probably due to strong destructive interference.
Because of the significant discrepancy between the data and the microscopic calculation we cannot be sure about the structure of the atmospheric fields and thus have decided to drop this event from further analysis.

Event #2
The peak in the radio intensity for this event, see Fig. S5, is reached at distances of 100 m, indicative of a strong interference of the radiation from different layers. The radio intensity reaches a local maximum at the core telling that the radiation from the bottom layer is relatively large. Since X max in Sim. I is small, the height where the number of particles reaches a maximum is in the top layer and thus the particle density becomes small in the bottom layer. For this reason, to have a large current at the bottom layer, the February 19, 2020, 11:15am X -8 TRINH ET AL.: ATMOSPHERIC ELECTRIC FIELDS BY LOFAR electric field in this layer needs to be strong, as shown in Table S2. With increasing X max the air shower penetrates deeper into the atmosphere and thus a smaller value for the electric field results in a similar emission strength for the bottom layer. The core position in the fit is shifted by mere 3 m from the position determined by the LORA data.
For this event the results of the MGMR3D calculation agree rather well with those from CoREAS for all three values of X max .

Event #3
The determined field configurations for the different values for X max do not differ greatly.
In addition the results of the MGMR3D and the CoREAS calculation are close, although a discrepancy is seen for the intensity near the shower axis. We have selected Sim. III as the preferred one because of the slightly better, event hough the ratio f r is large.
February 19, 2020, 11:15am At the time of event #4 there was lightning activity detected at a distance of about 100 km however none within the vicinity of the LOFAR core. The radar-reflectivity data, Fig. S7, show that the shower passed through the edge of a rather extended cloud system with a very active core at about 20 km from the Superterp.
For this event the first two cases yielded a comparable agreement between CoREAS and MGMR3D as well as the structure of the electric fields, the values for the reduced χ 2 .
The values for the norm factor f r as shown in Table S4 differ considerably. Initially we restricted the fields not to exceed the echo-top heights by much, however we noted that the chi-square improved by more than 1 unit when allowing for an electric field above the cloud. The maximum height of this layer is set at 15 km, any larger height would produce very similar results. To allow for this we have increased the number of layers to four for this event. We have selected calculation I as the preferred one. Since for this case the values for the electric fields are not very large one could have increased these and found a solution with a smaller value for f r .
The observed ring-like structure in the radio intensity indicates that there is a considerable amount of destructive interference between the emissions from different layers. The rather large value of the circular polarization near the core is a sign that the fields must make a finite angle and are not completely pointing in opposite directions.
Particular for this event is that we needed to introduce a moderate electric field that extents well above the cloud height. We have tries to reproduce the structure without this field at large altitudes, but this resulted in a considerably worse value for the chi-square in the MGMR3D fits.
February 19, 2020, 11:15am For event #5 the only lightning activity is detected more than 12 hours after detecting the event. The radar-reflectivity data of Fig. S9 show that at the time of this event an extensive cloud was overhead that was not moving much. The shower passed through the edge of the cloud with an active core in close vicinity.
The radio intensity, Fig. S10, shows a clear maximum at the core position and one thus can deduce that there is little destructive interference in the radio footprint and the electric fields all point in basically the same direction, as is seen indeed from Table S5.
The circular polarization shows a pattern that is similar to a fair-weather event. The direction of the linear polarization is however orthogonal to what one would expect for fair weather. All three fits show a very similar structure for the electric field configuration and similar values for the reduced χ 2 . Also for all three there is a good agreement between the CoREAS and the MGMR3D results. Since the norm factor for simulation I is smallest, combined with a good value for the reduced χ 2 we have chosen this as the preferred result.

August 26 th , 2012
During the time of of detecting events #6, #7, and #8 there was some lightning activity observed by the Météorage lightning-detection network in the close vicinity of the The three fits given in Table S6 show a three-layer field structure with similar strengths and orientations of the fields. The intensity and circular polarization patterns, see Fig. S13, are very reminiscent of those for fair weather circumstances however the linear polarization is deleted: orthogonal new: at 45 degree. The dip in the intensity near the core is an indication of some destructive interference between the emissions from different layers and thus a similar orientation of the electric field that is at a large angle to the of the geomagnetic force. Since for calculation I the agreement between the results of the CoREAS and the MGMR3D calculation is best we have selected this one as the preferred one. In addition the norm factor f r is best for this case.

Event #7
The results shown in Table S7 shows that the extracted field configurations for the electric fields do not differ much for the three fits. In addition the CoREAS and the MGMR3D results agree well for all three cases. The radio-intensity pattern, see Fig. S14, shows a clear ring-structure with a rather large diameter. One thus obtains a field configuration where there is a strong destructive interference between the top layer and the lower two. The strong circular polarization is evidence for fields that are at a finite angle with respect to each other. We have selected simulation II as the preferred fit.

Event #8
Also for event #8 one observes that the three different fits converge to very similar electric field configurations, see Table S8. Also for this event the results of the CoREAS and the MGMR3D calculations agree reasonably well for all three cases. For this event the For this event one observes, see Fig. S18, an intensity pattern that closely resembles that of a fair weather event, however the polarization data differ completely. This results in rather stable orientations of the fields for the three fits, see Table S9, however the extracted strengths differ. The reason for this is that the height of X max is right around h 3 and thus changes in X max require a sizable compensation in the electric field to keep similar currents. Since this case could also be fitted well with a simple 2 layer structure, this is given preference. Simulation II is preferred based on the reduced χ 2 as well as the ratio f r . The core position was moved by 19 m. Even though the χ 2 C is worse than χ 2 3D one can see that the CoREAS fit is actually quite good since all the features in the data are reproduced. The χ 2 C is relatively poor because the data for this event has very small errors.
1.7. July 26 th , 2013; Event #10 Considerable lightning activity was observed in the vicinity of the core at the time of detection of event #10, see Fig. S19. The radar-reflectivity data shown in Fig. S20 February 19, 2020, 11:15am  Also for this event one observes in    Table S1. The values of the fit parameters describing the structure of the atmospheric electric field as obtained from a chi-square fit using MGMR3D. Also given are the values for the chi-square for MGMR3D (χ 2 3D ) and that for CoREAS (χ 2 C ) using the same field and almost the same X max . The normalization factor for the intensity of the radio signal is given by f r . This calculation is performed for event #1.  Figure S4.  February 19, 2020, 11:15am   Figure S6. Same as Fig. S4 but for event #3 where simulation III is selected.
new: Since at large distances the footprint deviates most strongly from the fair-weather case one expects strong fields in the high layers.