Energy Budget of Plasma Motions, Heating, and Electron Acceleration in a Three-loop Solar Flare

The flare is observed with a unique combination of space- and ground-based instruments throughout the entire electromagnetic spectrum from radio waves to HXRs; see Figure 1. Nevertheless, there are substantial gaps in these data, which we attempt to fill with 3D modeling.

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In the HXR domain the event is observed by a few instruments: RHESSI (Lin et al. 2002), Konus-Wind (Aptekar et al. 1995; Pal'shin et al. 2014), and X-Ray Telescope (XRT; Golub et al. 2007) on board the Hinode mission (Kosugi et al. 2007). RHESSI missed the impulsive phase owing to the orbital night and recorded only the thermal response phase (see Figure 1(a)). Konus-Wind was not in triggered mode, so only the low-resolution G1 (21–80 keV) light curve is available (dark-blue line, Figure 1(a)).

IRIS (De Pontieu et al. 2014) carried out a 400-step raster scan on 2014 February 16 from 06:16 to 07:19 UT. Each raster step took 9.5 s, consisting of an 8 s exposure time plus overhead, and encompassed an area of 033 × 174''. This gives a total field of view (FOV) of 141'' × 174'', with the caveat that the different slit positions are not obtained simultaneously. Fortunately, the IRIS slit was just above the region of activity at the time of the flare. For each slit position, IRIS recorded near-UV (NUV) and far-UV (FUV) spectra in several spectral lines that are described below. Figure 2 shows a context image and the IRIS slit positions at different time steps.

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Soft X-rays are available from the Geostationary Operational Environmental Satellite (GOES; White et al. 2005; see Figure 1(b)), while optical and EUV data are available from Solar Dynamics Observatory/Helioseismic and Magnetic Imager (SDO/HMI; Scherrer et al. 2012) and SDO/Atmospheric Imaging Assembly (SDO/AIA; Lemen et al. 2012) respectively. The EUV light curves integrated over the region of interest (ROI; see Figure 3) are shown in Figure 1(c).

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In the microwave domain the flare occurred in the time range covered by the Siberian Solar Radio Telescope (SSRT; Grechnev et al. 2003) and Nobeyama instruments. Nobeyama Radioheliograph (NoRH; Nakajima et al. 1994) finished observations a few minutes before the event. SSRT was in a transition mode to a new instrument: the data were taken, but no calibration has been available at the time of the flare to produce images. The Solar Radio Spectropolarimeters (SRS; Muratov 2011) 2–24 GHz data were lost because of disk failure (A. T. Altyntsev 2021, private communication). The available microwave data set includes Nobeyama Radio Polarimeter (NoRP; Torii et al. 1979) data at a few frequencies, Radio Solar Telescope Network (RSTN; Guidice et al. 1981), and the Badary Broadband Microwave Spectropolarimeters (BBMS; Zhdanov & Zandanov 2015) data at 4–8 GHz; the combined dynamic spectrum is shown in Figure 1(d).

HXR observations of the flare impulsive phase are only available in the Konus-Wind wide G1 channel covering the 21–80 keV range. The light curve with the time cadence of 2.994 s was recorded in the waiting mode, which we combine with the microwave light curves to evaluate the nonthermal electron escape time from the radio source.

RHESSI HXR observations with high temporal (2 s) and energy (1 keV) resolution provide information on electrons in the range from ∼3 keV up to ∼30–50 keV, and maps of X-ray sources with a spatial resolution of ∼7'' are only available in the decay phase of the flare. RHESSI data available after the terminator ≈06:45 UT are used to produce images in the flare decay phase at a few spectral intervals between 3 and 25 keV using the CLEAN algorithm (Hurford et al. 2002) with clean beam width factor (CBWF) = 1.8; see Figure 3.

A loop-like structure connecting footpoints is observed during the impulsive peak with the Hinode/XRT using various filters. The RHESSI images were rotated by the roll angle rhessi_roll_angle = −0.2 using rhessi_roll_center = (0, 0) to co-align with these Hinode/XRT images, which are properly co-aligned with the SDO/AIA data.

GOES soft X-ray data are shown in Figure 1(b). There is a preflare enhancement well seen in both low- and high-energy channels at ∼06:43 UT, which are roughly co-temporal with IRIS enhancement from box 1, shown in Figure 4, in 1400 and 1330 Å (see below for IRIS data) and with SDO/AIA enhancement, seen in time profiles in Figure 1(c) (see next section for SDO/AIA data analysis).

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The standard EUV data set of the full solar disk is available from six SDO/AIA (94, 131, 171, 193, 211, 335 Å) coronal passbands. The AIA images with ∼12 spatial resolution have been taken with 12 s cadence, calibrated using the aia_prep.pro routine and normalized by the exposure time. We focus on the EUV emission from the ROI shown in Figure 3 to quantify the thermal energy and its evolution in the coronal part of the flare. We employ the differential emission measure (DEM) analysis technique applied to the entire ROI, as well as using the DEM maps with the methodology developed and applied by Motorina et al. (2020, hereafter Paper I) to a nonthermally dominated SOL2013-11-05T035054 solar flare. Some of the EUV images contain saturation artifacts. Unsaturated images with shorter exposure times are taken for the quantitative analysis of thermal energy of the flare.

The IRIS data show activity (impulsive enhancements in a localized area) temporally coinciding with impulsive emission from Konus-Wind and microwave. This is visible in the slit-jaw images with passbands around 1330 and 1400 Å, which were obtained at a cadence of about 20 s. Figure 4 shows that the enhancements of the more southern loop coincided temporally with the X-ray and microwave emission, while the northern loop brightened a few minutes earlier (while RHESSI was still in orbital night). The average intensity inside the two marked boxes is drawn in the right panels, for the IRIS 1330 and 1400 Å passbands.

The spectra allow us to probe the bulk velocities of the chromospheric and transition region plasma. We analyzed the spectral lines Si iv λ1394, Si iv λ1403, O iv λ1399, O iv λ1401, O iv λ1405, and Fe xxi λ1354, which form at different temperatures typical for the transition region or corona.

We verified that these spectral lines do not show irregularities (double peaks as, for example, observed in strong flares or supersonic flows) in our FOV and fitted a Gaussian function, with a central wavelength λ₀ and FWHM Δλ, to each pixel for each spectral line to determine their Doppler velocities and Doppler widths and thus to quantify the kinetic energy of the plasma flows and turbulent motions.

As mentioned, there are no microwave imaging data for this event. Only total power (spatially integrated) spectroscopic data are available from several instruments, including NoRP (with a significant flux at 3.75 and 9.4 GHz and a weak signal at 17 GHz), RSTN (at 2.8, 5, 8.8, and 15.4 GHz), and BBMS data at 4–8 GHz. These data were combined in a single synthetic dynamic spectrum (see Figure 1) as described in Lysenko et al. (2018). The dynamic spectrum and single-frequency light curves are employed in conjunction with the Konus-Wind light curve and the 3D model of the flare to constrain the nonthermal energy deposition in this flare.

Given that RHESSI data are only available after the Konus-Wind impulsive peak is over, our expectation is to employ RHESSI spectroscopy to quantify the hottest thermal component of the flaring plasma using OSPEX. ⁸ The spectral fits were applied to the background-subtracted data (detectors 1, 4, 5) every 8 s from 06:45:06 to 06:46:18 UT, every 12 s for the interval 06:46:18–06:47:30 UT, and every 20 s for the 06:47:30–06:53:10 UT time interval for better statistics (the intervals are respectively highlighted by light-, dark-, and darkest-gray areas in Figure 1(a)).

We attempted a two-temperature fit in the 3–17 keV range, which returned a cool (T ∼ 10 MK) and a hotter (T ∼ 25 MK) component. However, uncertainties of the cool component parameters are very large, likely because RHESSI calibration is unreliable below ∼6 keV. In addition, the χ² metrics were unacceptably high, due mainly to high residuals in the 3–7 keV range. Thus, we only employ the hot component parameters (T2_RHESSI and EM2_RHESSI) of this two-temperature fit, which are shown in dark green in Figure 6.

To cross-check these results, we attempted an isothermal plus thick-target spectral model fit in the 7–25 keV range, which resulted in a better χ² metric (χ² < 3). Two examples of the RHESSI fits, shown in Figure 5, indicate that the nonthermal component is either very weak or nonexistent. The time evolution of the fit parameters is shown in Figure 6. The thermal part of the fit resulted in the well-constrained temperature (green histogram in panel (a)) and emission measure (EM; green histogram in panel (b)) consistent with those derived from the two-temperature fit (dark-green lines).

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The nonthermal component, even though it appeared to be needed to return acceptably good fits, has large uncertainties of the total integrated electron flux F₀ (panel (c)), the low-energy cutoff E_c (panel (d)), and the spectral index δ of the electron distribution function above E_c (panel (e)). These figures confirm that there is a marginal, if any, nonthermal component in the RHESSI data.

The RHESSI spectral model fits are used to quantify the thermal energy of the hottest component of the flaring plasma similarly to the estimate reported in Paper I. The only distinction is that now we cannot estimate the hot loop volume from data directly. Instead, we rely on the loop volume (Loop II, V_II = 7.75 × 10²⁶ cm⁻³; see Table 1) determined from the 3D model devised in Section 5.

Table 1. Summary of the 3D Model

Parameter	Symbol, Units	Loop I	Loop II	Loop III
Geometry:
Length of the central field line	l, cm	1.448 × 109	6.345 × 109	2.535 × 109
Reference radius of the flux tube at loop top	r, cm	1.9 × 108	1.52 × 108	1.52 × 108
Model volume, ${\left[\int {n}_{0}{dV}\right]}^{2}/\int {n}_{0}^{2}{dV}$	V, cm3	7.27 × 1025	7.75 × 1026	4.03 × 1025
Thermal plasma:
Emission measure, $\int {n}_{0}^{2}{dV}$	EM, cm−3	8.91 × 1046	1.57 × 1046	2.58 × 1046
Mean number density, $\int {n}_{0}^{2}{dV}/\int {n}_{0}{dV}$	nth , cm−3	3.5 × 1010	0.45 × 1010	2.53 × 1010
Temperature	T, MK	9	25	10
Instant total thermal energy	Wth , erg	0.95 × 1028	3.61 × 1028	4.23 × 1027
Nonthermal electrons:
Total electron number	Nb , cm−3	⋯	2.41 × 1035	⋯
Low/high-energy cutoff	E0, MeV	⋯	0.01/2	⋯
Spectral index	δ	⋯	3.9	⋯
Instant total nonthermal energy	Wnth , erg	⋯	5.87 × 1027	⋯

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SDO/AIA observations are used to characterize plasma at temperatures 0.5–25 MK. To infer EM and temperature, we first calculate the DEM (cm⁻⁵ K⁻¹) using a regularization technique (e.g., Tikhonov 1963). The total EM, EM_AIA (cm⁻³), and mean temperature, 〈T_AIA〉, obtained from DEMs ⁹ calculated from the ROI (see Equations (1)–(2) in Paper I) are shown in Figure 6. The saturation effects, including a secondary saturation (blooming), are within the errors. Figure 6 shows the evolution of EM_AIA and 〈T_AIA〉 with the minimum preflare EM subtracted. After the HXR impulsive peak, shown by the Konus-Wind light curve, both the EM and temperature increase. Similar to the 2013 November 5 solar flare studied in Paper I, the values obtained from RHESSI and SDO/AIA are different from each other, which could mean that the two instruments see different sources with different temperatures.

Applying the regularized inversion code to the SDO/AIA data (Hannah & Kontar 2012, 2013), the DEM maps have been created using the methodology described in Section 2.3.2 of Paper I. The DEM maps are then used to calculate the EM maps ${\mathrm{EM}}_{{ij}}^{\mathrm{AIA}}$ (cm⁻³), the mean temperature maps $\langle {T}_{{ij}}^{\mathrm{AIA}}\rangle$ (K), and the thermal energy density maps ${w}_{{ij}}^{\mathrm{AIA}}$ (erg cm⁻³) for all (i, j) pixels.

The 3D modeling of the flare described in Section 5 shows that the essential contribution to the EUV images comes from relatively cold and dense Loop I (see Figure 11 Table 1). Thus, to estimate the thermal energy density ${w}_{{ij}}^{\mathrm{AIA}}$, the adopted length along the line of sight (LOS) has been taken as a characteristic diameter of the modeled Loop I (see Table 1), which is l_depth = 2r ≈ 3.8 × 10⁸ (cm). The values ${{EM}}_{{ij}}^{\mathrm{AIA}}$, $\langle {T}_{{ij}}^{\mathrm{AIA}}\rangle$, and ${w}_{{ij}}^{\mathrm{AIA}}$ are calculated using Equations (3)–(5) (Paper I) and are demonstrated along with the chi-squared (χ² < 2; bottom left panel) in the animated version of Figure 7. Figure 7 shows the 29th time interval (06:46:37–06:46:49 UT) of the animation. The mean temperature is close to 12–14 MK during the flare, while the EM and the thermal energy density evolve dramatically. The saturation effects become visible after 06:45:01 UT, just after the nonthermal HXR peak time. A distinct boundary between two temperatures (∼8 and ∼12 MK) is seen in the $\langle {T}_{{ij}}^{\mathrm{AIA}}\rangle$ maps. We interpret this as a projection effect when, due to the complexity of the flare loop structure, hot Loop II (reddish in Figure 7, top left panel) projects on cooler Loop I (greenish). Note that the (pre)heating takes place in both loops (I and II) before the impulsive energy release, seen in the microwave and Konus-Wind data.

The coronal Fe xxi line is sensitive to plasma temperatures around 10⁷ K. Signatures of Fe xxi are seen in flare loops, indicating the hot plasma. However, these signatures are weak (maximum of 1–2 DN s^–1, compared to >10 DN s^–1 for other spectral lines), which precludes a reliable Gaussian fitting and thus a more quantitative analysis. This also precludes the use of this spectral line to estimate a coronal portion of the kinetic energy.

Gaussian fitting allows us to determine the displacement of the Gaussian peak value λ₀ from the expected rest-frame wavelength λ_ji of the transition between the levels i and j, which yields the bulk velocity due to the Doppler effect:

$$\begin{eqnarray}&&\displaystyle \frac{{v}_{\mathrm{bulk}}}{c}=\displaystyle \frac{{\lambda }_{0}-{\lambda }_{{ji}}}{{\lambda }_{{ji}}},\end{eqnarray} \tag{ 1 }$$

where c denotes the speed of light.

The Doppler fitting of Si iv λ1394 is very reliable, with few exclusions necessary: about 5% of the pixels were below our selected cutoff for the noise (1.25 DN s^–1) and were therefore excluded. Furthermore, we excluded pixels with cosmic rays, spikes, or defects (based on the fitted FWHM Δλ), amounting to 2% of the FOV.

Figure 8 shows enhanced Doppler widths at the location of the RHESSI source in Si iv λ1394 and also in the more northern loop (0.5 Å vs. 0.2–0.3 Å in the quiet Sun). Nearly identical values were obtained from Si iv λ1403 (therefore not shown). The Si iv Doppler velocities were not very noteworthy and ranged between ±20 km s⁻¹, which is a common range also in the quiet Sun as shown in the left panel. Excluded pixels are shown in black for the Doppler widths and with a value of zero (gray) for the Doppler velocity.

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The total line broadening depends on three quantities: the thermal line width $\tfrac{2{k}_{B}{T}_{\mathrm{ion}}}{{M}_{\mathrm{ion}}}$ (km s⁻¹), the instrumental broadening σ_I (km s⁻¹), and the nonthermal line width v_turb (km s⁻¹). Thus, the FWHM Δλ of a spectral line in units of Å can be written as

$$\begin{eqnarray}&&{\rm{\Delta }}\lambda =\displaystyle \frac{{\lambda }_{0}}{c}\sqrt{4\mathrm{ln}2\left(\displaystyle \frac{2{k}_{B}{T}_{\mathrm{ion}}}{{M}_{\mathrm{ion}}}+{v}_{\mathrm{turb}}^{2}+{\sigma }_{I}^{2}\right)}.\end{eqnarray} \tag{ 2 }$$

The IRIS instrumental broadening was determined both by lab measurements (Tian et al. 2014) and by measuring the O i λ1355.598 line (De Pontieu et al. 2014) and yielded for the spectral range around the Si iv line about 30 mÅ (6.4 km s⁻¹). The thermal broadening for Si iv is 50 mÅ (≈11 km s⁻¹), and for O iv it is 90 mÅ (≈19 km s⁻¹). ¹⁰

Because of the better signal-to-noise ratio, we use the Si iv line to calculate the nonthermal line width based on its Gaussian fits, which allowed us to determine the total FWHM and solve Equation (2) for v_turb. We caution that this is only an approximation because we derive the FWHM by fitting a single Gaussian, which may not be entirely accurate in some flare pixels (Jeffrey et al. 2016). A visual inspection of the spectra shows that this seems reasonable in most pixels, but there are pixels with broad asymmetric spectra, whose origin probably is a superposition of multiple atmospheric components, for example, multiple downflows of different velocities within one pixel. Such pixels lead to an overestimation of the turbulent velocities, and therefore our number given here is rather an upper limit.

Several O iv lines are observed in IRIS's spectrum. They are generally weak, but they are visible in some locations, particularly around our flare loops. We carried out a similar Doppler fitting of the O iv λ1401.156 line, and in this case 73% of all pixels had to be excluded owing to being too noisy, and an additional 5% because of cosmic rays or the fitted FWHM being below our cutoff (50 mÅ). Nevertheless, the good pixels appeared to have very similar properties to Si iv λ1394. We therefore later use the Si iv λ1394 Doppler velocity as a proxy for the O iv λ1401.156 Doppler velocity.

Line intensity ratios of several O iv lines around 1400 Å can be used as diagnostics for electron densities in the range of 10⁹−10¹¹ cm⁻³ and weakly up to 10¹² cm⁻³ (e.g., Flower & Nussbaumer 1975; Doschek et al. 2016; Polito et al. 2016; Young et al. 2018). Our goal is to estimate the electron density near the RHESSI source to be able to calculate the kinetic energy density. We use the ratio of the O iv λ1399.8/λ1401.2 lines because it is least blended as shown in the example spectra in Figure 9. A drawback is the low intensity of the λ1399.8 line, which is below the noise level for many pixels. We integrate over a range of 0.8 Å (indicated with blue lines in Figure 9) and apply selection criteria on the integrated intensity and the ratios to exclude noise, cosmic rays, or unusual spectra. We visually verified that the remaining spectra show regular line profiles and produce ratios in the expected range (0.18–0.42). We assumed thermal and ionization equilibrium to apply this density diagnostic. Although it is difficult to firmly justify, considering the weakness of the flare, plus that the slit crossed it after its impulsive phase, we do not expect big errors with this assumption. Figure 10 shows our resulting ratios and densities. The ratios (middle panel) can only be determined near the flare site. Black indicates a ratio near zero, meaning that the λ1399 line is below the noise level, while white pixels indicate that one of the exclusion criteria was met, which includes too small integrated intensities and abnormal ratios, e.g., due to cosmic rays or large Doppler velocities and thus blending of the Si iv line. Even though the accuracy of this method can be debated, mostly because of the low O iv signals in the present observation, the figure shows that the densities of our ROI are around 10¹¹–10¹² cm⁻³ with few (measurable) densities below those values.

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In this study we employed the automated model production pipeline (G. M. Nita et al. 2021, in preparation) based on the NLFFF extrapolation code (Fleishman et al. 2017a) initiated with an SDO/HMI vector magnetogram taken at 06:34:12 UT. Visual inspection of automatically downloaded base maps does not reveal any apparent inversion or π-disambiguation artifact, so the model did not require any manual correction (see Anfinogentov et al. 2019).

The NLFFF 3D magnetic data cube is imported in the modeling tool GX Simulator (Nita et al. 2015, 2018), where the flaring flux tubes are interactively created. The GX Simulator functionality permits computation (and visualization) of selected magnetic field lines such as to match available flare images. In our case, we used only images of thermal emission obtained from RHESSI data at various energy ranges and also EM maps obtained from the DEM/AIA maps. This yielded two distinct flux tubes: Loops I and II; see Figure 11 and Table 1.

Guided by the thermal plasma parameters derived from the OSPEX fit to the RHESSI data and EM maps obtained from AIA data, we succeeded in reproducing most of the imaging data, except for the eastern source in the RHESSI 3–6 keV image. To reproduce this source, we created one more flux tube (Loop III) filled with a relatively cool plasma (10 MK). The magnetic field profiles along the central (reference) field lines in all three of these loops are shown in Figure 12. With these three loops all thermal images are closely reproduced in the model; see Figure 13. Loop III projects on the IRIS Fe xxi source, indicative of the 10 MK plasma there; see Figure 13. Figure 13 also illustrates the effect of the RHESSI point-spread function (PSF), which smears contributions from various thermal flux tubes out, producing a (misleading) visual impression that only a single loop is present.

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The next step of flare model creation is populating the flux tubes with nonthermal electron components, which are ideally guided by images of nonthermal emission. However, such images are not available in our case; thus, we are forced to employ the only available microwave spectral data. To be specific, we focus on the microwave emission peak time at 06:44:41 UT.

There could be ambiguity about how exactly to distribute nonthermal electrons between three available thermal loops of our model given the absence of the relevant images. Moreover, the bulk of the flaring microwave emission from a flare can come from entirely different flux tubes, not visible as a thermal source (Fleishman et al. 2017b; Kuroda et al. 2018).

Luckily, for the given flare this ambiguity can be largely removed owing to strong sensitivity of the gyrosynchrotron spectrum to the magnetic field in the emission source (Fleishman et al. 2020, Supplementary Materials including movie S2): the stronger the magnetic field, the larger the spectral peak frequency. Combining this property of the gyrosynchrotron spectrum with the dependence of its high-frequency slope on the nonthermal electron energy spectrum and of its low-frequency slope on the source geometry (Fleishman et al. 2020), we found that populating either Loop I or Loop III with a noticeable amount of nonthermal electrons results in overestimating the microwave emission at high frequencies and, simultaneously, underestimating it at the low frequencies.

This conclusion appears model independent and related only to the fact that the magnetic field is too high in those two loops: if we match the spectral peak frequency, then the model flux level is way too low; if we match the flux level (at a high frequency), then the spectral peak frequency is much higher than observed. After careful investigation, we concluded that an energetically dominant fraction of the nonthermal electrons must be located in Loop II; ¹¹ see Table 1, with which we can nicely reproduce the synthetic microwave spectrum as shown in Figure 14. The model outlined in Table 1 is consistent with all available observational constraints and thus validated by the data.

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With this model we can also check whether the presence of a modest nonthermal component derived from the RHESSI fit in Section 3.1 is consistent with the data. To this end, we populated flux tube II with nonthermal electrons consistent with the RHESSI fit and computed the microwave emission. We found that the microwave emission produced by this nonthermal electron population is below the 1 sfu level at all frequencies and thus not observable. Therefore, we cannot confidently exclude the presence of a nonthermal electron population as derived from the RHESSI fit even though the uncertainties of the associated parameters are rather large.

The total AIA-derived thermal energy of the flare in the FOV is obtained from the spatial distribution of the thermal energy density ${w}_{{ij}}^{\mathrm{AIA}}$, computed from the regularized DEM maps in Section 3.2, by adding up the contributions from all pixels in the FOV:

$$\begin{eqnarray}&&{W}_{\mathrm{therm}}^{\mathrm{AIA}}(t)={S}_{\mathrm{px}}\ {l}_{\mathrm{depth}}\sum _{i=1}^{{N}_{\mathrm{px}}}\sum _{j=1}^{{N}_{\mathrm{px}}}{w}_{{ij}}^{\mathrm{AIA}}(t)\,[\mathrm{erg}].\end{eqnarray} \tag{ 3 }$$

The animated Figure 7 shows that in addition to the evolution of the main flaring source, there is some dynamics in the top and bottom boxes (outlined in the figure by gray and light-gray contours). We cannot reliably conclude whether this dynamics is related to the flare or independent. To estimate contributions from those boxes to the thermal energy, we show them along with the one from the middle (black) box in Figure 15(a). The middle box uses the LOS depth l_depth = 3.8 × 10⁸ cm, which is comparable to the width of the loops, while two other boxes employ l_depth = 7.6 × 10⁷ cm, because the dynamic features are only about 1'' wide. The figure shows that the contributions from the top and bottom boxes are small compared with the main one; however, they display a behavior correlated in time with the main flaring box. This could indicate that the flaring process spreads out over a region larger than the main flaring loops. In what follows we, however, focus on the main flare region—the middle box.

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Figure 15(b) displays further details of the thermal energy distribution and evolution within the middle (black) box in Figure 7. Here the green/magenta lines show the thermal energy computed for two different green/magenta ROIs in Figure 7. These two ROIs are selected such as to inscribe the two different flaring loops as closely as possible. Note that this loop separation cannot be perfectly done because one of the loops projects onto the other. The magenta and green symbols show the thermal energies computed by the volume integration of the thermal energy densities in flux tubes I and II, respectively. Although they agree with the data within a factor of two, they do not match each other perfectly. There are two possible causes of these mismatches: (i) the already-mentioned projection effect, and (ii) the ambiguity in determining the depth of the source, for which only a rough estimate is available from the data. Potentially, the values derived based on the 3D model are more precise, as they are free from the LOS ambiguity. The dashed green-magenta line shows the sum of those two contributions, while the black line shows the thermal energy from the entire middle (black) box. This suggests that there is some thermal flare energy outside the most distinct loops, even though the loops give dominant contributions to the thermal energy budget.

In all cases, a minimal preflare energy from the corresponding box or ROI was subtracted; for the middle (black) box this minimal preflare value is ${W}_{\mathrm{therm},\ \min }^{\mathrm{AIA}}=2.54\times {10}^{28}$ (erg), which comes from the nonflaring pixels in the FOV. Evolution of this energy (from the black box) is shown in Figure 16 in black. Its peak value is about 7 × 10²⁸ (erg).

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To calculate the thermal energy detected by RHESSI, we use the EM and temperature obtained from the RHESSI fit (to be specific, we only employ here the single-temperature+thick-target fit; see Section 3.1 and green lines in Figure 6; the alternative, two-temperature fit, yields very similar results):

$$\begin{eqnarray}&&{W}_{\mathrm{therm}}^{\mathrm{RHESSI}}=3{k}_{B}{T}_{\mathrm{RHESSI}}\sqrt{{{EM}}_{\mathrm{RHESSI}}\times V}\,[\mathrm{erg}],\end{eqnarray} \tag{ 4 }$$

where V is the volume of the corresponding thermal source. Here we use the model volume of Loop II from Table 1. The evolution of this energy is shown in Figure 16 in green; the peak value is about 5 × 10²⁸ (erg).

We note that RHESSI is mostly sensitive to the hottest plasma within the FOV, namely, for emission from flaring Loop II in our case, which is also quantified by the AIA data in the previous section; see the green line in Figure 15(b). It is instructive to compare these two green curves. The RHESSI-derived thermal energy has a peak around 06:47 UT, where the temperature is the largest (about 30 MK), while the AIA-derived thermal density of this loop has a peak 1 minute later, when the plasma cooled down to ∼20 MK or less. The two energies agree well after that. The reason of the mismatch between these energies at the early decay phase is the well-known fact that AIA is not sensitive to plasma above ∼20 MK; thus, AIA-based thermal diagnostics underestimates the thermal energy of that hot plasma, to which RHESSI is the most sensitive.

Knowing the densities and the velocities, we can calculate the energy density ${w}_{\mathrm{bulk}}^{\mathrm{IRIS}}$ as

$$\begin{eqnarray}&&{w}_{\mathrm{bulk}}^{\mathrm{IRIS}}=1.2\cdot {n}_{e}\cdot {m}_{p}\cdot \displaystyle \frac{{v}_{\mathrm{bulk}}^{2}}{2}[\mathrm{erg}\,{\mathrm{cm}}^{-3}],\end{eqnarray} \tag{ 5 }$$

where n_e is the electron density, the factor 1.2 appears because of ions, and m_p is the proton mass (1.67 × 10⁻²⁴ g). We use the electron density determined from the O iv line ratios and the Si iv Doppler velocities because of the better Gaussian fits due to the higher signal and because Si iv velocities were found to be very similar to the O iv velocities.

We determined which IRIS pixels lie inside the 50% RHESSI 6–9 keV contour and calculated the energy density for all of them (where possible, unless the densities were unavailable). We then obtained an average energy density inside this area (0.12 erg cm⁻³) by averaging over the valid pixels. To estimate the total bulk kinetic energy within the flare footpoints ${W}_{\mathrm{bulk}}^{\mathrm{IRIS}}=\int {w}_{\mathrm{bulk}}^{\mathrm{IRIS}}{dV}\sim {w}_{\mathrm{bulk}}^{\mathrm{IRIS}}V$, we estimate the volume V as a product of the number of pixels (1711) inside the contour and the pixel volume, assuming a height of 1 Mm. IRIS observed with 0166 pixel⁻¹ = 120 km pixel⁻¹, giving a pixel area of 1.44 × 10¹⁴ cm² and thus a volume of 1.44 × 10²² cm³. This gives a total bulk kinetic energy for this volume of ${W}_{\mathrm{bulk}}^{\mathrm{IRIS}}\sim 3\cdot {10}^{24}$ erg.

We determined the turbulent kinetic energy density ${w}_{\mathrm{turb}}^{\mathrm{IRIS}}$ similarly to ${w}_{\mathrm{bulk}}^{\mathrm{IRIS}}$ in Section 6.3 by replacing v_bulk with v_turb in Equation (5) and considering the same valid pixels inside the 50% RHESSI contour. The average turbulent energy density in this area is 2.7 erg cm⁻³. Similarly to Section 6.3, we obtain the total energy by multiplying this energy density by the same volume as in Section 6.3. The total turbulent kinetic energy for this volume is then ${W}_{\mathrm{turb}}^{\mathrm{IRIS}}\leqslant 7\cdot {10}^{25}$ erg, which is less than 0.1% of the flare thermal energy and about 0.15% of the nonthermal energy deposition. Neither ${W}_{\mathrm{bulk}}^{\mathrm{IRIS}}$ nor ${W}_{\mathrm{turb}}^{\mathrm{IRIS}}$ is shown in Figure 16, as they are indistinguishable from the bottom axis.

We compare this turbulent energy detected in the flare footpoints with the turbulent energy in a coronal flare volume. Kontar et al. (2017) employed the Hinode/EIS data to find the turbulent flare energy in a cusp region, which appeared to be less than 1% of the released energy at any given time instance. The kinetic turbulent energy at the flare footpoints detected here, presumably driven by precipitating particles, is one order of magnitude lower than that in the corona reported by Kontar et al. (2017).

As previously mentioned, RHESSI missed the impulsive phase of the flare, while Konus-Wind recorded the impulsive phase in the G1 channel only in the waiting mode. Thus, nonthermal energy and the rate of the nonthermal energy deposition can only be estimated indirectly using the microwave spectrum and available microwave and Konus-Wind light curves. In Section 5.3 we determined the population of nonthermal electrons needed to reproduce the observed peak microwave spectrum that occurred at 06:44:41 UT. The corresponding instant nonthermal energy is W_nth = 5.87 × 10²⁷ erg at that time.

Now, to estimate the nonthermal energy deposition rate, we need to estimate the escape time τ_esc of the nonthermal electrons from Loop II. To do so, we consider lag-correlation between the HXR G1 Konus-Wind light curve and the microwave light curves at 3.75 and 9.4 GHz obtained with NoRP. This lag-correlation analysis shows that the 9.4 GHz light curve is delayed by ∼0.5 s relative to the Konus-Wind light curve, while the 3.75 GHz one is delayed by ∼2 s. Thus, for our order-of-magnitude estimate, we select a characteristic value τ_esc = 1 s, which yields ${\dot{W}}_{{nth}}\approx 5.87\times {10}^{27}$ erg s⁻¹ at the flare peak time. To obtain the total deposition of the nonthermal energy, we multiply this peak value by the HXR light-curve duration, 8 s, at the level of half-maximum: W_{nth tot} ≈ 4.7 × 10²⁸ erg.

The nonthermal energy ${W}_{\mathrm{nonth}}^{\mathrm{RHESSI}}$ potentially captured by RHESSI after the impulsive phase was computed as a cumulative sum using the parameters from the RHESSI fits:

$$\begin{eqnarray}&&{W}_{\mathrm{nonth}}^{\mathrm{RHESSI}}={\int }_{{t}_{0}}^{t}{F}_{0}{E}_{c}\displaystyle \frac{\delta -1}{\delta -2}{dt}\,[\mathrm{erg}],\end{eqnarray} \tag{ 6 }$$

where t₀ is the end of the RHESSI night, while the F₀, E_c , and δ are the thick-target parameters (see Section 3.1) from the fitting of the RHESSI data displayed in Figures 6(c)–(e); ${W}_{\mathrm{nonth}}^{\mathrm{RHESSI}}$ is shown in dashed blue in Figure 16. We note that these thick-target fit parameters, especially F₀, come with very large uncertainties and might overestimate the nonthermal energy deposition. However, as has been shown, the presence of such a nonthermal population does not contradict any available data. For example, the microwave emission produced by such a population does not overestimate the observed microwave emission. The presence of this nonthermal component is also consistent with the behavior of the thermal energy at the hot loop, which reaches the peak exactly when this nonthermal component is over.

Here we briefly summarize the evolution of the energy partitions at the course of the three phases of the flare: the pre-impulsive, the impulsive, and the main phase. We only consider Loops I and II, as we do not have enough constraints to quantify evolution of Loop III. At the pre-impulsive phase, we observe a modest heating of Loops I (T ∼ 9–10 MK) and II (T ∼ 13–14 MK). No signature of any nonthermal component is present at this phase. This phase is followed by a very short (20 s) impulsive phase, when the release of the nonthermal energy observed in the microwave and HXR ranges is immediately followed by a thermal response in Loops I and II. The heating continues for a few more minutes into the main phase of the flare. Both pre-impulsive and main flare phases require an extra heating in addition to the heating due to nonthermal electron loss.

The above-described components of the flare energy must have come from the free magnetic energy available in the AR (Fleishman et al. 2020). To check whether the needed magnetic energy existed in the AR before the flare and to nail down its changes associated with the flare, we resort to 3D magnetic reconstructions. To do so we fixed the Carrington coordinates of our model FOV and created sequences of potential and NLFFF reconstructions computed for seven consecutive time frames between 6:00 and 7:12 UT roughly centered at the flare time. This approach preserves the reconstruction volume for different time frames and minimizes possible uncertainties due to solar rotation. We employed two different disambiguation algorithms for the transverse component of the photospheric magnetic field (Metcalf 1994; Rudenko & Anfinogentov 2014) and found only very minor differences between solutions, which cross-validates them.

The total magnetic energy in each data cube is straightforwardly computed as W_B = ∫B² dV/(8π), which fluctuates less than 1% around ≈1.2 × 10³³ erg for all our potential extrapolations and around ≈1.1 × 10³³ erg for the NLFFF extrapolations. Therefore, we cannot derive the free magnetic energy because the magnetic energy in all NLFFF cubes appeared slightly smaller, by less than 10% in all cases, than the corresponding potential magnetic energy. This is a known problem of reconstruction algorithms associated with numerical residuals of ∇ · B , which are theoretically equivalent to zero, but not in the numerical solutions (De Rosa et al. 2015).

We estimated uncertainties of the total magnetic energy as follows: (i) by finding differences of the total magnetic energy between two successive (in time) data cubes and (ii) by finding differences between two simultaneous models extrapolated from the bottom boundary condition obtained with two different π-disambiguation methods (Metcalf 1994; Rudenko & Anfinogentov 2014). The magnetic energy uncertainties estimated this way are above ∼3 × 10³⁰ erg. This is more than one order of magnitude above the required flare energy, ∼10²⁹ erg; thus, the decrease of the magnetic energy due to the flare cannot be detected using the NLFFF extrapolation method. The total energy released in the flare composes less than 0.01% of the total magnetic energy. We conclude that the presence of the required free magnetic energy in this AR is highly likely.