Spectral Characteristics of Solar Flares in Different Chromospheric Lines and Their Implications

Using the spectral data of representative solar flares observed with the infrared detector system of the solar spectrograph at Purple Mountain Observatory, we study the spectroscopic characteristics of solar flares in the Hα, the Ca i i 8?542 Å, and the He i 10?830 Å lines in different phases and various locations of flares and discuss their possible implications coupled with space observations. Our results show that in the initial phase of a flare the Hα line displays a red shift only with no wide wing. Large broadenings of the Hα line are observed a few minutes after the flare onset within small regions of 3?–?5^′′ in both disk and limb flares with and without nonthermal processes. Far wings similar to those of damping broadening appear not only in the Hα line but in the He i 10?830 Å line as well in flares with nonthermal processes. Sometimes we even detect weak far-wing emission in the Ca i i 8?542 Å line in disk flares. Such large broadenings are observed in both the footpoints and the flare loop-top regions and possibly result from strong turbulence and/or macroscopic motions. Therefore, the so-called nonthermal wing of the Hα line profile is not a sufficient condition to distinguish whether nonthermal electrons are accelerated or not in a flare. The Ca i i 8?542 Å line shows lower intensity in the loop-top regions and higher intensity in the parts close to the solar surface. Emissions larger than nearby continuum in the He i 10?830 Å line are detected only in small regions with strong X-ray emissions and avoid sunspot umbrae.     

The Origin of Sequential Chromospheric Brightenings

Sequential chromospheric brightenings (SCBs) are often observed in the immediate vicinity of erupting flares and are associated with coronal mass ejections. Since their initial discovery in 2005, there have been several subsequent investigations of SCBs. These studies have used differing detection and analysis techniques, making it difficult to compare results between studies. This work employs the automated detection algorithm of Kirk et al. (Solar Phys. 283, 97, 2013) to extract the physical characteristics of SCBs in 11 flares of varying size and intensity. We demonstrate that the magnetic substructure within the SCB appears to have a significantly smaller area than the corresponding \(\mbox{H}\upalpha\) emission. We conclude that SCBs originate in the lower corona around \(0.1~R_{\odot}\) above the photosphere, propagate away from the flare center at speeds of \(35\,\mbox{--}\,85~\mbox{km}\,\mbox{s}^{-1}\), and have peak photosphere magnetic intensities of \(148\pm2.9~\mbox{G}\). In light of these measurements, we infer SCBs to be distinctive chromospheric signatures of erupting coronal mass ejections.     

IRIS Observations of Spicules and Structures Near the Solar Limb

We have analyzed Interface Region Imaging Spectrograph (IRIS) spectral and slit-jaw observations of a quiet region near the South Pole. In this article we present an overview of the observations, the corrections, and the absolute calibration of the intensity. We focus on the average profiles of strong (Mg?ii h and k, C?ii and Si?iv), as well as of weak spectral lines in the near ultraviolet (NUV) and the far ultraviolet (FUV), including the Mg?ii triplet, thus probing the solar atmosphere from the low chromosphere to the transition region. We give the radial variation of bulk spectral parameters as well as line ratios and turbulent velocities. We present measurements of the formation height in lines and in the NUV continuum from which we find a linear relationship between the position of the limb and the intensity scale height. We also find that low forming lines, such as the Mg?ii triplet, show no temporal variations above the limb associated with spicules, suggesting that such lines are formed in a homogeneous atmospheric layer and, possibly, that spicules are formed above the height of \(2''\). We discuss the spatio-temporal structure of the atmosphere near the limb from images of intensity as a function of position and time. In these images, we identify p-mode oscillations in the cores of lines formed at low heights above the photosphere, slow-moving bright features in O?i and fast-moving bright features in C?ii. Finally, we compare the Mg?ii k and h line profiles, together with intensity values of the Balmer lines from the literature, with computations from the PROM57Mg non-LTE model, developed at the Institut d’ Astrophysique Spatiale, and estimated values of the physical parameters. We obtain electron temperatures in the range of \({\sim}\, 8000~\mbox{K}\) at small heights to \({\sim}\, 20\,000~\mbox{K}\) at large heights, electron densities from \(1.1\times 10^{11}\) to \(4\times 10^{10}~\mbox{cm}^{-3}\) and a turbulent velocity of \({\sim}\, 24~\mbox{km}\,\mbox{s}^{-1}\).     

A New Solar Imaging System for Observing High-Speed Eruptions: <Emphasis Type="Italic">Solar Dynamics Doppler Imager</Emphasis> (SDDI)

A new solar imaging system was installed at Hida Observatory to observe the dynamics of flares and filament eruptions. The system (Solar Dynamics Doppler Imager; SDDI) takes full-disk solar images with a field of view of \(2520~\mbox{arcsec} \times 2520~\mbox{arcsec}\) at multiple wavelengths around the \(\mathrm{H}\alpha\) line at 6562 Å. Regular operation was started in May 2016, in which images at 73 wavelength positions spanning from \(\mathrm{H}\alpha -9~\mathring{\mathrm{A}}\) to \(\mathrm{H}\alpha +9~\mathring{\mathrm{A}}\) are obtained every 15 seconds. The large dynamic range of the line-of-sight velocity measurements (\({\pm}\,400~\mbox{km}\,\mbox{s}^{-1}\)) allows us to determine the real motions of erupting filaments in 3D space. It is expected that SDDI provides unprecedented datasets to study the relation between the kinematics of filament eruptions and coronal mass ejections (CME), and to contribute to the real-time prediction of the occurrence of CMEs that cause a significant impact on the space environment of the Earth.     

Probing the Quiet Solar Atmosphere from the Photosphere to the Corona

We investigate the morphology and temporal variability of a quiet-Sun network region in different solar layers. The emission in several extreme ultraviolet (EUV) spectral lines through both raster and slot time-series, recorded by the EUV Imaging Spectrometer (EIS) on board the Hinode spacecraft is studied along with \(\mbox{H}\upalpha\) observations and high-resolution spectropolarimetric observations of the photospheric magnetic field. The photospheric magnetic field is extrapolated up to the corona, showing a multitude of large- and small-scale structures. We show for the first time that the smallest magnetic structures at both the network and internetwork contribute significantly to the emission in EUV lines, with temperatures ranging from \(8\times 10^{4}~\mbox{K}\) to \(6\times 10^{5}~\mbox{K}\). Two components of transition region emission are present, one associated with small-scale loops that do not reach coronal temperatures, and another component that acts as an interface between coronal and chromospheric plasma. Both components are associated with persistent chromospheric structures. The temporal variability of the EUV intensity at the network region is also associated with chromospheric motions, pointing to a connection between transition region and chromospheric features. Intensity enhancements in the EUV transition region lines are preferentially produced by \(\mbox{H}\upalpha\) upflows. Examination of two individual chromospheric jets shows that their evolution is associated with intensity variations in transition region and coronal temperatures.     

Multi-wavelength correlation studies of diffuse emission from the Aquila Rift

We present a multi-wavelength correlation study of diffuse ultraviolet radiation using GALEX observations towards the Aquila Rift. Apart from airglow and zodiacal emissions, we find a diffuse background of \(1300\mbox{--} 3700~\mbox{ph}\,\mbox{cm}^{-2}\,\mbox{s}^{-1}\,\mbox{sr}^{-1}\,\mathring{\mathrm{A}}^{-1}\) in the far-ultraviolet (FUV, 1350–1750 Å) band and \(1300\mbox{--}2800~\mbox{ph}\,\mbox{cm}^{-2}\,\mbox{s}^{-1}\,\mbox{sr}^{-1}\,\mathring{\mathrm{A}}^{-1}\) in the near-ultraviolet (NUV, 1750–2850 Å) band. The observed diffuse UV emissions are saturated with total as well as neutral hydrogen column density in the region due to high optical depth in UV (\(\tau \), 0.91–23.38). Higher values of FUV/NUV ratio in the region, greater than the threshold value of 0.6, along with the positive correlation between the ratio and FUV intensity are due to excess emission in the FUV band which is absent in the NUV band. We estimated the excess emission to be in the range \(\sim 400\mbox{--} 2700~\mbox{ph}\,\mbox{cm}^{-2}\,\mbox{s}^{-1}\,\mbox{sr}^{-1}\,\mathring{\mathrm{A}}^{-1}\), plausibly due to H₂ fluorescence, ion line emissions and two-photon continuum emissions from the region in the FUV band, which also shows saturation in optically thick regions with N(H₂) as well as \(\mbox{H}\alpha \) emissions. Since N(H₂) and \(\mbox{H}\alpha \) emissions spread all over the region, the excess emission from the field is composite in nature and a detailed spectroscopic analysis is needed to disentangle the contribution from individual components.     

Turbulence and Heating in the Flank and Wake Regions of a Coronal Mass Ejection

As a coronal mass ejection (CME) passes, the flank and wake regions are typically strongly disturbed. Various instruments, including the Large Angle and Spectroscopic Coronagraph (LASCO), the Atmospheric Imaging Assembly (AIA), and the Coronal Multi-channel Polarimeter (CoMP), observed a CME close to the east limb on 26 October 2013. A hot (\({\approx}\,10~\mbox{MK}\)) rising blob was detected on the east limb, with an initial ejection flow speed of \({\approx}\, 330~\mbox{km}\,\mbox{s}^{-1}\). The magnetic structures on both sides and in the wake of the CME were strongly distorted, showing initiation of turbulent motions with Doppler-shift oscillations enhanced from \({\approx}\, \pm 3~\mbox{km}\,\mbox{s}^{-1}\) to \({\approx}\, \pm 15~\mbox{km}\,\mbox{s}^{-1}\) and effective thermal velocities from \({\approx}\,30~\mbox{km}\,\mbox{s}^{-1}\) to \({\approx}\,60~\mbox{km}\,\mbox{s}^{-1}\), according to the CoMP observations at the Fe?xiii line. The CoMP Doppler-shift maps suggest that the turbulence behaved differently at various heights; it showed clear wave-like torsional oscillations at lower altitudes, which are interpreted as the antiphase oscillation of an alternating red/blue Doppler shift across the strands at the flank. The turbulence seems to appear differently in the channels of different temperatures. Its turnover time was \({\approx}\,1000\) seconds for the Fe 171 Å channel, while it was \({\approx}\,500\) seconds for the Fe 193 Å channel. Mainly horizontal swaying rotations were observed in the Fe 171 Å channel, while more vertical vortices were seen in the Fe 193 Å channel. The differential-emission-measure profiles in the flank and wake regions have two components that evolve differently: the cool component decreased over time, evidently indicating a drop-out of cool materials due to ejection, while the hot component increased dramatically, probably because of the heating process, which is suspected to be a result of magnetic reconnection and turbulence dissipation. These results suggest a new turbulence-heating scenario of the solar corona and solar wind.     

A model of solar flares and faculae

Theories of solar flares based on the storage of energy (usually as magnetic energy) in the solar atmosphere are shown to be incompatible with observational data.The sunspot energy deficit and the photospheric faculae both involve energy fluxes comparable with the flare requirement ( 3 × 10²⁹ erg s^–1). Both also require a subsurface system of waves or oscillations, perhaps those discussed by Danielson and Savage and by Wilson. The flare model proposed is based on a temporary diversion of this energy carried by Alfvén waves through spots and magnetic elements or micro-pores; the calculated plasma perturbation velocity in the umbra is about 6 km s^–1 for a major flare.In the atmosphere the wave energy divides into two parts to produce the cool, stationary optical flare and the particle flare. The first part is dissipated around flux tubes which are mainly horizontal in the chromosphere and which tend to concentrate along the magnetic neutral line (B = 0). Each tube vibrates individually as a taut wire in a viscous fluid, to excite the fluid just outside the tube. The second part of the energy emerges along tubes mainly vertical in the chromosphere and is converted to shock waves in the corona and then to particle energy for the radio and X-ray flare and the blast wave.The model includes white-light faculae, quasi-permanent X-ray and fast-particle emissions, sympathetic flares and surges. An unambiguous test would be provided by observations of plasma motions of a few kilometres per second in spots and micro-pores.     

Solar flare activity and the structure of the coronal neutral line

In this study we analyse the positions of major flares from 1978 and 1979, with respect to the magnetic structure of the solar corona, as described by a potential field model. We find that major flares exhibit no strong association with the neutral line at the chromospheric level. However, when we calculate the neutral line's position at higher and higher altitudes in the corona, we find that major flares show an increasing tendency to be found close to these high-altitude coronal neutral lines. The correlation between flares and higher-altitude coronal neutral lines reaches a maximum at an altitude of 0.35R , and thereafter decreases as the neutral line is moved out to the source surface at an altitude of 1.50R . This indicates that major flares are strongly associated with coronal structure at the 0.35R level ( 250 000 km) - an altitude surprisingly high in the corona. This reinforces the idea that flares are associated with large-scale coronal magnetic fields and also indicates that the region of coronal magnetic topology important to solar flare processes may be larger than previously thought.     

Extreme Ultraviolet Late-Phase Flares: Before and During the Solar Dynamics Observatory Mission

The solar extreme-ultraviolet (EUV) observations from the Solar Dynamics Observatory (SDO) have revealed interesting characteristics of warm coronal emissions, such as Fe xvi 335 Å emission, which peak soon after the hot coronal X-ray emissions peak during a flare and then sometimes peak for a second time hours after the X-ray flare peak. This flare type, with two warm coronal emission peaks but only one X-ray peak, has been named the EUV late phase (Woods et al., Astrophys. J. 739, 59, 2011). These flares have the distinct properties of i) having a complex magnetic-field structure with two initial sets of coronal loops, with one upper set overlaying a lower set, ii) having an eruptive flare initiated in the lower set and disturbing both loop sets, iii) having the hot coronal emissions emitted only from the lower set in conjunction with the X-ray peak, and iv) having the first peak of the warm coronal emissions associated with the lower set and its second peak emitted from the upper set many minutes to hours after the first peak and without a second X-ray enhancement. The disturbance of the coronal loops by the eruption is at about the same time, but the relaxation and cooling down of the heated coronal loops during the post-flare reconnections have different time scales with the longer, upper loops being significantly delayed from the lower loops. The difference in these cooling time scales is related to the difference between the two peak times of the warm coronal emission and is also apparent in the decay profile of the X-ray emissions having two distinct decays, with the first decay slope being steeper (faster) and the delayed decay slope being smaller (slower) during the time of the warm-coronal-emission second peak. The frequency and relationship of the EUV late-phase decay times between the Fe xvi 335 Å two flare peaks and X-ray decay slopes are examined using three years of SDO/EUV Variability Experiment (EVE) data, and the X-ray dual-decay character is then exploited to estimate the frequency of EUV late-phase flares during the past four solar cycles. This study indicates that the frequency of EUV late-phase flares peaks before and after each solar-cycle minimum.     

Analysis of a Failed Eclipse Plasma Ejection Using EUV Observations

The photometry of eclipse white-light (W-L) images showing a moving blob is interpreted for the first time together with observations from space with the PRoject for On Board Autonomy (PROBA-2) mission (ESA). An off-limb event seen with great details in W-L was analyzed with the SWAP imager (Sun Watcher using Active pixel system detector and image Processing) working in the EUV near 174 Å. It is an elongated plasma blob structure of 25 Mm diameter moving above the east limb with coronal loops under. Summed and co-aligned SWAP images are evaluated using a 20-h sequence, in addition to the 11 July, 2010 eclipse W-L images taken from several sites. The Atmospheric Imaging Assembly (AIA) instrument on board the Solar Dynamics Observatory (SDO) recorded the event suggesting a magnetic reconnection near a high neutral point; accordingly, we also call it a magnetic plasmoid. The measured proper motion of the blob shows a velocity up to \(12~\mbox{km}\,\mbox{s}^{-1}\). Electron densities of the isolated condensation (cloud or blob or plasmoid) are photometrically evaluated. The typical value is \(10^{8}~\mbox{cm}^{-3}\) at \(r=1.7~\mathrm{R}_{\odot}\), superposed on a background corona of \(10^{7}~\mbox{cm}^{-3}\) density. The mass of the cloud near its maximum brightness is found to be \(1.6\times10^{13}\) g, which is typically \(0.6\times10^{-4}\) of the overall mass of the corona. From the extrapolated magnetic field the cloud evolves inside a rather broad open region but decelerates, after reaching its maximum brightness. The influence of such small events for supplying material to the ubiquitous slow wind is noticed. A precise evaluation of the EUV photometric data, after accurately removing the stray light, suggests an interpretation of the weak 174 Å radiation of the cloud as due to resonance scattering in the Fe IX/X lines.     

耀斑日冕硬X射线源观测研究

对于足点被日面边缘遮挡住的耀斑的观测研究是诊断日冕硬X射线辐射的一个重要方法.通过统计分析RHESSI (Reuven Ramaty High-Energy Solar Spectroscopic Imager)卫星观测到的71个此类耀斑硬X射线源发现,前人提出的两类源,即日冕X射线辐射中热辐射与非热辐射源区空间分离较小的源和分离较大的源,在能谱、成像、光变曲线以及GOES持续时间等方面都没有显著的区别,其中辐射区的面积、耀斑总热能以及GOES持续时间与分离距离之间有很好的相关性.这些结果支持近年来提出的一些耀斑统一模型.同时也表明Masuda耀斑只是一类非常特殊的事件,不具有日冕硬X射线辐射的一般特征.     

Investigation on M-class Flare-Associated Coronal Mass Ejections with and Without DH Type II Radio Bursts

We perform a statistical analysis on 157 M-class soft X-ray flares observed during 1997?–?2014 with and without deca-hectometric (DH) type II radio bursts aiming at the reasons for the non-occurrence of DH type II bursts in certain events. All the selected events are associated with halo Coronal Mass Ejections (CMEs) detected by the Solar and Heliospheric Observatory (SOHO) / Large Angle Spectrometric and COronograph (LASCO). Out of 157 events, 96 (61%; “Group I”) events are associated with a DH type II burst observed by the Radio and Plasma Wave (WAVES) experiment onboard the Wind spacecraft and 61 (39%; “Group II”) events occur without a DH type II burst. The mean CME speed of Group I is \(1022~\mbox{km}/\mbox{s}\) and that of Group II is \(647~\mbox{km}/\mbox{s}\). It is also found that the properties of the selected M-class flares such as flare intensity, rise time, duration and decay time are greater for the DH associated flares than the non-DH flares. Group I has a slightly larger number (56%) of western events than eastern events (44%), whereas Group II has a larger number of eastern events (62%) than western events (38%). We also compare this analysis with the previous study by Lawrance, Shanmugaraju, and Vr?nak (Solar Phys. 290, 3365L, 2015) concerning X-class flares and confirm that high-intensity flares (X-class and M-class) have the same trend in the CME and flare properties. Additionally we consider aspects like acceleration and the possibility of CME-streamer interaction. The average deceleration of CMEs with DH type II bursts is weaker (\(a = - 4.39\mbox{ m}/\mbox{s}^{2}\)) than that of CMEs without a type II burst (\(a = -12.21\mbox{ m}/\mbox{s}^{2}\)). We analyze the CME-streamer interactions for Group I events using the model proposed by Mancuso and Raymond (Astron. Astrophys. 413, 363, 2004) and find that the interaction regions are the most probable source regions for DH type II radio bursts.     

Characteristics of gamma-ray line flares as observed in hard X-ray emissions and other phenomena

Observations of gamma-ray lines from solar flares by SMM demonstrated that energetic protons and heavy ions are accelerated during the impulsive phase. In order to understand the acceleration mechanism for gamma-ray producing protons and heavy ions, we have studied the characteristics of the flares from which gamma-ray lines were observed by SMM In order to identify the characteristics unique to the gamma-ray line flares, we have also studied intense hard X-ray flares with no gamma-ray line emissions. We have found the following characteristics: 1) Most of the gamma-ray line flares produced intense radio bursts of types II and IV. 2) For most of the gamma-ray line flares, the time profiles of high-energy (? 300 keV) hard X-rays are delayed by order of several seconds with respect to those of low-energy hard X-rays. The delay times seem to be correlated with the spatial sizes of the flares. 3) In Hα importance, the gamma-ray line flares range from sub-flares to importance-3 flares. 4) The hard X-ray spectra of the gamma-ray line flares are generally flatter (harder) than those of flares with no gamma-ray line emission. From these characteristics, we conclude that the first-order Fermi acceleration operating in a flare loop is likely to be the acceleration mechanism for energetic protons and heavy ions as well as relativistic electrons.     

Observations and Modelling of the Pre-flare Period of the 29 March 2014 X1 Flare

On 29 March 2014, NOAA Active Region (AR) 12017 produced an X1 flare that was simultaneously observed by an unprecedented number of observatories. We have investigated the pre-flare period of this flare from 14:00 UT until 19:00 UT using joint observations made by the Interface Region Imaging Spectrometer (IRIS) and the Hinode Extreme Ultraviolet Imaging Spectrometer (EIS). Spectral lines providing coverage of the solar atmosphere from the chromosphere to the corona were analysed to investigate pre-flare activity within the AR. The results of the investigation have revealed evidence of strongly blue-shifted plasma flows, with velocities up to \(200~\mbox{km}\,\mbox{s}^{-1}\), being observed 40 minutes prior to flaring. These flows are located along the filament present in the active region and are both spatially discrete and transient. In order to constrain the possible explanations for this activity, we undertake non-potential magnetic field modelling of the active region. This modelling indicates the existence of a weakly twisted flux rope along the polarity inversion line in the region where a filament and the strong pre-flare flows are observed. We then discuss how these observations relate to the current models of flare triggering. We conclude that the most likely drivers of the observed activity are internal reconnection in the flux rope, early onset of the flare reconnection, or tether-cutting reconnection along the filament.     

Addendum to: Strength of the Solar Coronal Magnetic Field – A Comparison of Independent Estimates Using Contemporaneous Radio and White-Light Observations

This addendum uses an alternate fit for the electron density distribution \(N(r)\) (see Figure 1) and estimates the coronal magnetic field using the new model. We find that the estimates of the magnetic field are in close agreement using both the models.

We have fit the \(N(r)\) distribution obtained from STEREO-A/COR1 and SOHO/LASCO-C2 using a fifth-order polynomial (see Figure 1). The expression can be written as

$$\begin{aligned} N_{\text{cor}}(r) &= 1.43 \times 10^{9} r^{-5} - 1.91 \times 10^{9} r^{-4} + 1.07 \times 10^{9} r^{-3} - 2.87 \times 10^{8} r^{-2} \\ &\quad {} + 3.76 \times 10^{7} r^{-1} - 1.91 \times 10^{6} , \end{aligned}$$

(1)

where \(N_{\text{cor}}(r)\) is in units of cm^?3 and \(r\) is in units of \(\mathrm{R}_{\odot}\). The background coronal electron density is enhanced by a factor of 5.5 at 2.63 \(\mathrm{R}_{\odot}\) during the coronal mass ejection (CME). The estimated coronal magnetic field strength (\(B\)) using radio data indicates that \(B(r) \approx(0.51\text{\,--\,}0.48) \pm 0.02\ \mathrm{G}\) in the range \(r \approx2.65\text{\, --\,}2.82\ \mathrm{R}_{\odot}\). The field strengths for STEREO-A/COR1 and SOHO/LASCO-C2 are ≈?0.32 G at \(r \approx 3.11\ \mathrm{R}_{\odot}\) and ≈?0.12 G at \(r \approx 4.40\ \mathrm{R}_{\odot}\), respectively.

     

Flare morphologies and coronal field configurations

Chromospheric flares are the footpoints of closed coronal field lines. In this paper we present different flare morphologies from observations and examine the implied coronal field configurations above the flaring region. Flares are grouped according to the number of ribbons, from unresolved compact point-like flare to four-ribbon flares. Quiet region flares having characteristics all their own are also presented here.We find that compact, unresolved point-like flares have two distinct footpoints when viewed in offband H. The footpoints of some of the compact flares also show increased separation as a function of time.Unlike large two-ribbon flares, the ribbons of many small and/or short-lived two-ribbon flares usually have no measurable separation of ribbons.Multiple-ribbon (three or more ribbon) flares consist of two or more pairs of two-ribbons, or two or more sets of field lines. Parity of the ribbons in multiple-ribbon flares, or the lack of it, depends on the magnetic makeup of the locale of the ribbons.Flares in old quiet regions resulting from sudden filament eruptions show discrete small patches of emissions reflecting the spottiness of decayed and dispersed field of quiet region.     

The Most Intense Electron-Scale Current Sheets in the Solar Wind

Previous analysis of magnetohydrodynamic-scale currents in high-speed solar wind near 1 AU suggests that the most intense current-carrying structures occur at electron scales and are characterized by average current densities on the order of \(1~\mbox{pA}/\mbox{cm}^{2}\). Here, this prediction is verified by examining the effects of the measurement bandwidth and/or measurement resolution on the analysis of synthetic solar wind signals. Assuming Taylor’s hypothesis holds for the energetically dominant fluctuations at kinetic scales, the results show that when \(\nu_{c}\gg \nu_{b}\), where \(\nu_{c}\) is the measurement bandwidth and \(\nu_{b} \approx 1/3~\mbox{Hz}\) is the break frequency, the average scale of the most intense fluctuations in the current density proxy is approximately \(1/\nu_{c}\), and the average peak current density is a weakly increasing function that scales approximately like \(\nu_{c}^{0.1}\).