# Hydrodynamics of small transient brightenings in Solar corona

Abhishek Rajhans Durgesh Tripathi Inter-University Centre for Astronomy and
Astrophysics, Post Bag - 4, Ganeshkhind, Pune 411007, India  Vinay L. Kashyap
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden St. Cambridge MA
02138, USA

(Received 27th December 2020; Revised 13th May 2021; Accepted 18th May 2021)

###### Abstract

Small scale transients occur in the Solar corona at much higher frequencies
than flares and play a significant role in coronal dynamics. Here we study
three well-identified transients discovered by Hi-C and also detected by the
EUV channels of Atmospheric Imaging Assembly (AIA) on board Solar Dynamics
Observatory (SDO). We use 0-D enthalpy-based hydrodynamical simulations and
produce synthetic light curves to compare with AIA observations. We have
modeled these transients as loops of $\sim$ 1.0 Mm length depositing energies
$\sim$ $10^{23}$ ergs in $\sim$ 50 seconds. The simulated synthetic light
curves show reasonable agreement with the observed light curves. During the
initial phase, conduction flux from the corona dominates over the radiation,
like impulsive flaring events. Our results further show that the time-
integrated net enthalpy flux is positive, hence into the corona. The fact that
we can model the observed light curves of these transients reasonably well by
using the same physics as those for nanoflares, microflares, and large flares,
suggests that these transients may have a common origin.

Solar Physics, Solar atmosphere, Solar corona, Solar coronal loops, Solar
coronal transients

††journal: ApJ††software: IDL (https://www.l3harrisgeospatial.com/Software-
Technology/IDL), SolarSoft (Freeland & Handy, 2012), PINTofALE (Kashyap &
Drake, 2000) and EBTEL (Klimchuk et al., 2008; Cargill et al., 2012)

##  1 Introduction

The presence of high-temperature ($>$1 MK) plasma in the solar corona was
discovered in the 1940s. How this plasma, above the much cooler photosphere,
is heated to such high temperatures has been one of the most challenging
questions in astrophysics. Though our understanding of the energy dissipation
in the corona has improved substantially, the full solution to the problem of
coronal heating remains elusive and the transfer of mass and energy between
layers of the solar atmosphere is not completely understood (see e.g.,
Klimchuk, 2006; Reale, 2014, for a review). Multi-wavelength observations of
the Sun show that different layers couple through magnetic fields. By and
large, theories related to omnipresent coronal heating fall into two groups:
AC heating and DC heating (see e.g., Klimchuk, 2015; Walsh & Ireland, 2003).
Depending on the frequency of occurrence, heating events can be classified
into high and low frequency heating, with the former may contribute to steady
and latter to transient events (see e.g., Tripathi et al., 2011; Winebarger,
2012).

Solar flares provide the best observed examples of impulsive events taking
place in the Solar atmosphere. Hudson (1991) conjectured that for maintaining
the corona at a temperature greater than 1 MK with the help of impulsive
events, there must be a large frequency of such events with smaller energy,
and found that the relationship between number of events and their energies
obeys a power law distribution $\frac{dN}{dE}\propto E^{-\alpha}$, where $dN$
is the rate of occurrence of events having energy in the range [$E,E+dE$] and
$\alpha$ is a positive number (see e.g., Hannah et al., 2011, for a review).
The power law distribution of flares of different energies is a sign of
underlying phenomenon of self organized criticality (Lu & Hamilton, 1991). For
the heating to be dominated by nanoflares, the the power law index $(\alpha)$
should be greater than 2. This has led to many observational studies (see
e.g., Shimizu, 1995; Berghmans et al., 1998, 2001; Krucker & Benz, 1998, 2000;
Parnell & Jupp, 2000; Aschwanden et al., 2000a, b; Christe et al., 2008;
Hannah et al., 2008) based on the counting of different types of transient
events which result in varied negative slopes of power law ranging from
$1.6\lesssim\alpha\lesssim 2.2$. However, there are limitations to such
studies due to constraints on the cadence, passbands, and resolutions of
instruments. Also there remains a chance that flares of different energy,
particularly at lower energies, are under counted (see e.g., Pauluhn &
Solanki, 2007; Upendran & Tripathi, 2021). Additionally, it is possible that
different events may be generated due to different mechanisms and hence they
would not necessarily follow the same power law distribution.

Habbal & Withbroe (1981) studied coronal bright points and used Ly-$\alpha$
emission as a proxy for conduction losses from the corona into the
chromosphere. Preś & Phillips (1999) were then able to establish that
conduction losses are at least an order of magnitude larger than radiation
losses, implying that radiation loss from the corona is a small fraction of
total energy dissipated. The smallest brightenings detected thus far (Régnier
et al., 2014; Subramanian et al., 2018) are due to the observations recorded
by Hi-C (Kobayashi et al., 2014). Subramanian et al. (2018) identified 27 such
events in Hi-C images and performed a detailed study to understand their
energetics using simultaneous observations obtained with Atmospheric Imaging
Assembly (AIA; Lemen et al., 2012) on board the Solar Dynamics Observatory
(SDO). The study found conduction to be the dominant cooling mechanism in
corona. This is a feature shared by impulsive events like flares, microflares,
and nanoflares, suggesting that the same physical mechanism is shared by these
small transient brightenings. We note, however, that a number of simplifying
assumptions, such as detailed thermal balance and stationary loop structures,
were made in this study.

In this work we carry out hydrodynamic simulations to gain a theoretical
understanding of the energetics of small brightenings identified by (Régnier
et al., 2014; Subramanian et al., 2018). We have selected three of the
brightenings (BR-00, 07, 26) from Subramanian et al. (2018) for a detailed
study, as they show the simplest profiles, with a single peak, and a clearly
visible decay phase (see §5.1, §5.2 and §5.3). We have used a 0-D numerical
code called Enthalpy Based Thermal Evolution of Loops (EBTEL; Klimchuk et al.,
2008; Cargill et al., 2012). The results obtained from EBTEL simulations were
used along with AIA response functions to produce synthetic lightcurves,
mimicking observations (Subramanian et al., 2018). The rest of the paper is
arranged as follows. In §2 we outline the observational results of Subramanian
et al. (2018) including the assumptions made. In §3 we briefly discuss EBTEL,
the relevant input parameters, and the simulation set up. The analysis is
described in §4 and results are discussed in §5. We summarise our results in
§6.

##  2  Data

Figure 1: The site of the bright points considered here. (Left) The full disk
AIA/193 Å filter image of the Sun, with an inset box showing the Hi-C field of
view (large square box), as well as the location of the fan-loop structure
where bright points were detected (small square box). (Right) The Hi-C image
corresponding to the small square box showing the location of the fan loop
region where the transient brightenings were observed. The figure is taken
from Subramanian et al. (2018).

Hi-C (High-Resolution Coronal Imager) is a sounding rocket mission that
observed the Sun in the 193 Å passband filter with a pixel size of 0.1
$\arcsec$ (Kobayashi et al., 2014). It was launched on July 11, 2012 and
recorded observations of active region AR 11520 for $\approx$5 minutes. One of
several interesting phenomena observed in
detail111https://hic.msfc.nasa.gov/publications.html#hic1_pubs were multiple
tiny brightenings within a system of fan loops rooted in the active region
(Régnier et al., 2014; Subramanian et al., 2018).

Subramanian et al. (2018) identified 27 such point-like brightenings using the
automatic detection algorithm of Subramanian et al. (2010), and performed a
detailed study of energetics involved in these events. For this purpose, the
Hi-C observations were cross calibrated with simultaneously obtained SDO/AIA
(Solar Dynamics Observatory/Atmospheric Imaging Assembly; Lemen et al. (2012);
O’Dwyer et al. (2010)) data.

AIA provides full disk images of the solar atmosphere in 7 separate EUV
passbands with a cadence of 12 s and pixel size of 0.6″. Fig. 1 displays the
AIA full disk image recorded in the 193 Å bandpass filter, the Hi-C field of
view (larger inset white box) and the region where brightenings were found
(smaller inset white box). The latter is zoomed-in in the right panel of Fig.
1.

In order to determine the thermal structure of the brightenings Subramanian et
al. (2018) obtained the Differential Emission Measure (DEM) using six
optically thin filters of AIA. For this purpose the PINTofALE package (Kashyap
& Drake, 1998) was used. The DEMs were used for computing emission measure
weighted temperatures and electron densities. It was found that all these
events have temperatures $\approx 10^{6.2-6.6}$ K, electron number densities
of the order of 109 cm-3 and estimated radiative energy losses of 1024-25
ergs.

Assuming that a magnetic flux density of $\sim$50 G is involved in each of
these brightenings, Subramanian et al. (2018) estimated the total magnetic
energy to be of the order of $10^{26}$ ergs and hence should be sufficient to
power these brightenings with radiative energies of order of 1024 ergs. On the
basis of cooling timescales in static equilibrium, they suggested that thermal
conduction was the dominant cooling mechanism in corona for these
brightenings.

It is important to note that the estimates of Subramanian et al. (2018) are
crude, as they rely on stationary equilibrium estimates of conduction and
radiative cooling timescales of corona and are integrated estimates over the
lifetimes of the brightenings. Furthermore these neglect flows and density
variations. Hence it is necessary to, first, validate the results using more
realistic dynamical loop constructs, and second, to identify the region of the
parameter space, where loops that can mimic the observed light curves exist.

##  3 Hydrodynamic modelling

In this work, we have performed 0-D simulations using single fluid EBTEL
(Klimchuk et al., 2008; Cargill et al., 2012). The idea behind 0-D description
is to study the time evolution of length averaged quantities. It is based on
1-D field aligned simulations showing that temperature, density and pressure
are within factors of a few between the base and apex of coronal part of loop
in 1-D simulations. As a result, the corresponding length averaged quantities
are characteristic of coronal values. The reduced computational cost and
improved speed of computing 0-D solutions makes studying the temporal
evolution of length-averaged physical quantities a fruitful alternative to
carrying out detailed numerical hydrodynamical simulations. It was
demonstrated that the results obtained with the EBTEL differ from 1-D
hydrodynamic simulations by at most 15-20% despite being much faster (few
seconds on a contemporary laptop).

###  3.1 Enthalpy Based Thermal Evolution of Loops (EBTEL)

EBTEL computes length-averaged quantities over time as they respond to varying
heat input. It relies on a simplified form of the Navier-Stokes equation where
flows are assumed to be subsonic and hence the kinetic energy term is
neglected. Heating is uniform both along and across the loop. The loop is
assumed to have a uniform cross section (Klimchuk et al., 1992), and symmetric
about the apex. Due to this symmetry, vector quantities like conduction flux
and velocity vanish at the loop top. The base of corona is defined such that
conduction is a cooling term in corona and heating term in transition region.
Across the base of transition region, any conduction or enthalpy flux is
assumed to be negligible. Fig. 2 depicts the sketch of loop setup along with
the different regions and properties of the loop.

###  3.2 Input parameters

In order to run the EBTEL simulations, we have to specify some fundamental
input parameters. EBTEL requires two inputs: the half length of the loop and
the heating function. The heating function is defined as the rate at which the
heat is deposited in the loop per unit volume (in units of ergs cm-3 s-1). An
illustrative example of how EBTEL is used, is shown in Fig. 3. The upper panel
shows the heating function, applied to a loop of half length 1.0 Mm. The
energy deposited begins to rise linearly (starting at 60 s) from an ambient
value to a peak (at 105 s) and then declines linearly back to the ambient
value (at 150 s). The baseline ambient level is necessary to establish the
presence of a corona, in this case at a temperature of 0.92 MK and a density
of 7.8$\times 10^{9}$ cm-3. This triangular heating pulse leads to an increase
in the temperature of the plasma in the loop (middle panel), which reaches a
maximum at time close to the heating. In contrast, the plasma density (bottom
panel) rises more shallowly, reaches a peak after the plasma fills the loop in
response to the enthalpy flux from the transition region before it declines.
Notice that the loop temperature drops below the ambient value after the
heating pulse is ended. This is due to the heated plasma becoming over dense
and cooling rapidly (due to $T\propto n^{2}$ scaling law ) until the excess
plasma is drained.

Figure 2: Sketch of the loop set-up. (Left) A flux tube filled with multiple
strands, rising up from the chromosphere at the bottom through a loop
footpoint, and submerging at another footpoint. The thick horizontal line
represents the base of the corona, and the loop is assumed to be symmetrical
around the apex. (Right) The representation of half of the loop in EBTEL. The
plasma aligned along the field lines are shown straightened out, with the
various regions corresponding to the physical system labeled.  Figure 3:
Response of a loop of half length of 1.0 Mm to a triangular heating function
computed using EBTEL. (a) The heat given per unit volume to the loop in units
of ergs cm-3 s-1. (b) Time evolution of temperature in MK (c) Time evolution
of number density in 109 cm-3

###  3.3 Synthetic lightcurves

In order to compare observations with theoretical predictions, we need to
compute synthetic intensities using simulations. For a given instrument’s
filter response $R(T)$ as a function of temperature $T$, the expected
intensity,

$I=\int DEM(T)\cdot R(T)~{}dT~{}~{}{\rm[DN~{}s^{-1}~{}pix^{-1}]}\,,$ (1)

where $DEM(T)$ is the differential emission measure222The units of $R(T)$ and
$DEM(T)$ in the solar context are typically [DN cm5 s-1 pix-1] and [cm-5 K-1]
respectively. over the temperature range $[T,T+dT]$. In EBTEL, it includes the
contributions of both the corona and the transition region.

###  3.4 Estimation of Conduction and Radiation Losses and Enthalpy

For studying the various cooling processes and relative importance of
different energy terms, we need to compute the energy fluxes associated with
conduction, radiation and enthalpy from outputs provided by EBTEL.

The conduction flux (Klimchuk et al., 2008)

$F_{c}=\frac{F_{sp}F_{sat}}{\sqrt{F_{sp}^{2}+F_{sat}^{2}}}\,,$ (2)

with $F_{sp}$ and $F_{sat}$ being the Spitzer and saturation fluxes
respectively and

$F_{sp}=-\frac{2}{7}\kappa_{0}\frac{T_{a}^{7/2}}{L}\\\ \hskip 8.5359pt\&\hskip
8.5359ptF_{sat}=-0.25\sqrt{\frac{k_{b}^{3}}{m_{e}}}\bar{n}\bar{T}^{1.5}\,,$
(3)

where $T_{a}$ is the temperature at the apex of loop, $\kappa_{0}=8.12\times
10^{-7}$ [cgs], $k_{b}$ is Boltzmann’s constant, $m_{e}$ is the electron mass,
and $\bar{n}$ and $\bar{T}$ are length-averaged electron number density and
temperature. The radiation flux,

$F_{R}=(1+c_{1})\bar{n}^{2}L\Lambda(\bar{T})\,,$ (4)

where $c_{1}$ is the ratio of the total radiation losses from the transition
region and the corona as computed within EBTEL, and $\Lambda(\bar{T})$ is the
optically thin power loss function of temperature, and is computed using the
routine rad_loss.pro from Solarsoft software, using the Chianti database (Dere
et al., 1997; Landi et al., 2013) and coronal abundances from Grevesse et al.
(2007).

The enthalpy flux through the base of the loop footpoint ($z=0$ in Fig. 2),

$F_{P}=\frac{5}{2}\bar{P}v_{0}\,,$ (5)

where $\bar{P}$ is the average pressure in the coronal part of the loop and
$v_{0}$ is the velocity at the base of the loop.

##  4 Generation of Synthetic Light Curves

We probe the range of input parameters for generating synthetic light curves
consistent with observations. For this purpose, we take the observations
recorded in the AIA 193 Å filter as reference. This is primarily because the
lifetimes of observed brightenings were obtained using this filter by
Subramanian et al. (2018). In order to determine the input parameters that
best describe the transients, we employ two methods: in the first, we assume
that the transients are part of the same dynamical system as the ambient
corona, and use the pre-transient intensity in AIA 193 to constrain the inputs
(§4.1); in the second, we assume that the transients and the ambient corona
are dynamically distinct, we match the characteristics of the background-
subtracted light curves (§4.2).

###  4.1 Method 1: Modeling the transients with background

Here we seek to find a suitable combination of heat input and loop half length
that replicates the observed background intensities, rise time of the
transient, and the average background subtracted intensities of the transient
as observed in AIA 193 Å.

####  4.1.1 Range of parameters used in simulations

Subramanian et al. (2018) estimated the spatial extent of the bright points to
be $\approx$ 2$\times$2 AIA pixels. Assuming that the loop would be
semicircular, the expected half length is $\approx$ 0.65 Mm. In order to
account for uncertainties, we simulate loops with half lengths ranging between
0.1–1.5 Mm. The heating function provided in EBTEL consists of two parts, viz.
steady heating that creates the background and time dependent heating that
causes the transient. For each loop in this range, we set up a background
heating ($Q_{bkg}$) in accordance with the scaling law used in EBTEL,

$Q_{bkg}~{}~{}{\rm[ergs~{}cm^{-3}~{}s^{-1}]}~{}~{}\approx~{}~{}\frac{2}{7}\kappa_{0}\frac{T_{a}^{\frac{7}{2}}}{L^{2}}=\frac{2}{7}\left(\frac{10}{9}\right)^{\frac{7}{2}}\kappa_{0}\frac{\bar{T}^{\frac{7}{2}}}{L^{2}}$
(6)

where $\kappa_{0}=8.12\times 10^{-7}$ in cgs units, $T_{a}$ is the temperature
at loop summit, $\bar{T}$ is average temperature of the coronal part of loop
and $\bar{T}\approx{0.9}T_{a}$ (Cargill et al., 2012). We seek solutions where
$\bar{T}\approx 1$ MK.

To simulate the transient, we use a triangular heating profile characterized
by a maximum heating rate $H_{m}$ ergs s-1 cm-3, and a total duration
$t_{dur}$ seconds (see e.g., Cargill et al., 2012; Klimchuk et al., 2008).
Then the total energy $E$ dissipated in the semi-loop,

$E=\frac{1}{2}~{}A_{\rm foot}~{}L~{}t_{dur}~{}H_{m}~{}~{}{\rm[ergs]}\,.$ (7)

For the exemplar case shown in Fig. 3, we have $H_{m}=0.2$ ergs s-1 cm-3 and
$t_{dur}=90$ s with cross-sectional area $A_{\rm foot}=1.76\times 10^{15}$
cm2, corresponding to one AIA pixel. Note that to fix the time dependent
heating function, we vary the total energy budget instead of volumetric
heating rate. We run the simulations by varying the amount of total energy in
the range $10^{22}$ to $10^{25}$ ergs, in steps of 0.1 dex. This range was
chosen as it brackets the radiative losses for these transients as determined
by Subramanian et al. (2018). The typical life time of these transients of the
order $\approx 100$ s and therefore we consider $t_{dur}$ ranging from 10 to
200 s in steps of 10 s.

####  4.1.2 Parameters for specific brightening

Our goal is to identify the set of parameter values (viz., loop half length,
heating duration, and heating rate, $[L,t_{dur},E]$) which mimic the
characteristics of the observed brightenings. To find the best parameter set,
we seek to compare and match the following characteristics of the observed and
simulated light curves:

  1. 1.

The average background level of observed light curves and the background
subtracted intensities averaged over the lifetime of the events as observed in
AIA 193 Å images

  2. 2.

The rise times of the events in the observed light curves obtained from AIA
193 Å filters. We require to match the rise times instead of total duration
because the decay times are subject to large systematic errors due to the
difficulty of identifying precisely when the model intensities become
indistinguishable from the background.

We find that for all the three brightenings considered in this study, there
exist physically plausible loop lengths, heating rates, and duration that
capture the behavior of the observed AIA light curves. We emphasize, however,
that the best parameters sets were not obtained by performing fits to the
data; such a process would be unrealistic given the simplicity of the models
we consider. To identify the input parameters for EBTEL, we follow the
following procedure:

  1. 1.

First, we identify the most suitable loop half length. We note from Eq. 6 that
for a given temperature, the background heating is a function of loop length.
Therefore, we compare the background intensities in the light curves of AIA
193 Å with those obtained using Eq. 6 for all values of $L$ within the range
0.1–1.5 Mm (see §4.1.1), while fixing the temperature at 1 MK. We select that
value of $L$ that provides the background intensity closest to the observed
values in the 193 Å light curves.

  2. 2.

Second, we identify the total duration of the heating events. For this, we
deposit an heating event with total energy $E=10^{22}$ ergs by varying the
time duration $t_{dur}$ within the range of 10 to 200 s (see §4.1.1), for the
values of $L$ previously obtained. Given $(L,E)$, the rise times are primarily
dependent on $t_{dur}$. The values of $t_{dur}$, which give the closest
agreement with the rise times of the observed events are selected.

  3. 3.

Finally, for the given $L$ and $t_{dur}$, the average intensity of the
brightening in the 193 Å filter is estimated for values of $E$ ranging from
$10^{22}-10^{25}$ ergs. The value of $E$ which generates an intensity closest
to that observed is then selected.

###  4.2 Method 2: Modeling the transients without background

In contrast to the method outlined above, here we detail an alternate method
where background levels are ignored and only the rise time and average
intensities of the transients are matched with model predictions to obtain a
suitable set of EBTEL parameters. This effectively treats the transients as
dynamically distinct events in relation to the ambient corona.

####  4.2.1 Parameter range used in simulations

Using the assumption of semicircular loops confined within an area equivalent
to 2$\times$2 AIA pixels, we fix the loop half length to be 0.65 Mm. The
background heating rate is generally taken to be two to three orders of
magnitude smaller than the main heating event (Klimchuk et al., 2008; Cargill
et al., 2012). For a loop length of order of 1 Mm with cross-sectional area of
1 AIA pixel, we require a volumetric heating rate of the order of 0.01 to 0.1
ergs cm-3 s-1, such that $\sim 10^{23}$ ergs is deposited in $\sim 50$ s.
Hence we have set a uniform background heating rate of $10^{-4}$ ergs cm-3
s-1, such that the background temperature and densities are an order of
magnitude lower than the peak values.

The remaining task is to find the heating duration and total energy budget for
the event. They were determined in the same way as mentioned in §4.1.1 once
loop length and background heating rates are set After fixing the half length
of the loop and background heating rate to 0.65 Mm and $10^{-4}$ ergs cm-3
s-1, respectively, we follow steps 2 & 3 enumerated in §4.2.1 to obtain the
doublet $[t_{dur},E]$ (see Table 1).

##  5 Results

Using the two methods described above, we identify the triplet $[L,t_{dur},E]$
of inputs required for EBTEL that best describes the transient under under
study or under investigation (see Table 1). We plot the observed and simulated
light curves for AIA 193 Å (panels a & d), simulated plasma temperature
(panels b & e) and density (panels c & f) obtained from both methods;
explicitly modeling background levels (left column: method 1; §4.1) and
excluding background from the modeling (right column: method 2; §4.2) for
BR00, BR07 and BR26 in Figs. 4, 7 and 10, respectively. We have investigated
the robustness of both models by bracketing the nominal duration by half and
twice the selected $t_{dur}$. Moreover, we have also obtained the synthetic
light curves using both methods for other AIA filters, viz., 94, 131, 335,
211, and 171, and compared them with the observed light curves in Figs. 5, 8,
& 11.

The average model intensities of the events in each AIA filter, corresponding
to input parameters selected using both methods are reported in Table 2, and
are compared with the measured intensities from Subramanian et al. (2018). We
also show the range in the calculated intensities that arises due to possible
systematic uncertainty in the event duration. This is done by computing
average intensities when the duration of heating is $\frac{1}{2}\times$ and
2$\times$ the most suitable value. These results demonstrate that the observed
intensities in AIA 193 Å filter are robustly modeled, and where they differ
for other filters, point to limitations in the plasma temperature
reconstructions with EBTEL.

We note that EBTEL also allows the inclusion of non-thermal particle flux
within the simulation. However, we find that even requiring as much as half of
the total energy budget be dissipated by non-thermal particles (for loop
parameters relevant to our study, as in §4), the light curves change by $<$5%.
Additionally the presence of non-thermal particles worsens the agreement
between simulated and observed EM weighted temperatures. Therefore, we have
switched off this option in the code.

In order to study the energetics of brightenings of this class, we now look
into the EBTEL simulations to understand how the energy transfer processes
operate. We show in Figs. 6, 9, & 12, the conduction loss (blue), radiation
loss (green), enthalpy (red), and the heating rate (black) for each
brightening simulated using input parameters obtained from method 1 (panels
[a-b]) and method 2 (panels [c-d]). We note that for the sake of visibility,
loss curves are shown only for the most suitable $t_{dur}$. In order to
demonstrate both the absolute and relative magnitudes of the energy losses,
the contribution of background heating is included in the upper panels of
these figures, and are excluded in the lower panels. Note that the values are
as computed for the full loop, not just the semi-loop as computed by EBTEL. We
next discuss the results for each event in sequence.

Table 1: Input parameters Index | Energy (E [log(ergs)]) | Half Length (L [108 $cm$]) | Duration of heating (tdur [$s$])
---|---|---|---
| Method 1 | Method 2 | Method 1 | Method 2 | Method 1 | Method 2
| (bkg. modelled) | (bkg. not modelled) | (bkg. modelled) | (bkg. not modelled) | (bkg. modelled) | (bkg. not modelled)
00 | 23.0 | 23.2 | 1.00 | 0.65 | 60 | 40
07 | 23.1 | 23.3 | 0.90 | 0.65 | 70 | 60
26 | 22.8 | 23.1 | 1.1 | 0.65 | 60 | 70

###  5.1 Modelling the Dynamics of BR-00

####  5.1.1 Method 1: Background modelled

We find that a loop of half length 1 Mm, with 1023 ergs deposited within 60 s
mimics the observed rise times, peak intensity, and approximate duration. The
pre-event ambient background heating, designed to match the base intensity
level in AIA 193, maintains plasma at temperature of 0.92 MK and density
7.8$\times 10^{9}$ cm-3, which is similar to the cooler loops found in active
regions (Ghosh et al., 2017). The EBTEL simulation is run for 60 s before a
triangular heat pulse is applied for a duration $t_{dur}$=60 s. We follow the
evolution of the plasma during and after this pulse and use the DEMs obtained
from the simulation to predict the intensity light curves for all AIA
channels.

Figure 4: Data and EBTEL simulations of brightening BR00 using input
parameters set using method 1, which uses transient + background for
determining input parameters (left column) and method 2 which uses only
transient for determining input parameters (right column). Panels a & d show
observed and simulated light curves as labelled. Panels (b, e) and (c, f) show
the simulated plasma temperature and density respectively. The solid curves
represent model curves obtained using the parameters in Table 1, the pink
dashed and brown dot-dashed curves represent model curves made for heating
durations $\frac{1}{2}\times$ and $2\times$ the nominal, and the green triple-
dot-dashed curves represent observed light curves.

Figure 5: Observed and simulated light curves using input parameters set using
method 1 (which models the background and the transient together; left column)
and method 2 (which considers the background separately from the transient;
right column) for BR-00 for AIA channels 94 (panel a[f]), 131 (panel b[g]),
335 (panel c[h]), 211 (panel d[i]) and 171 Å (panel e[j]) as labelled. The
colors and line styles correspond to those in Fig. 4. The simulated light
curves obtained using method 1, where input parameters have been selected
using transient + background (left column) and method 2, where input
parameters have been selected using only transient (right column) have been
increased or decreased by a constant offset in each filter to match the
observed background level. Note that the best durations chosen for the two
methods are different, and therefore the 1/2x and 2x durations also differ.

Figure 6: Energy loss and transfer components from EBTEL simulations designed
to match BR-00. The panels in the left column correspond to method 1
(background included in the modeling), and those in the right column to method
2 (backgound treated as separate from the transient). The heat input (solid
curves), conduction (blue dashed curves), radiation loss (green dot-dashed
curves), and enthalpy (red triple-dot-dashed curves) are shown. The top panels
show curves that include the background while bottom panels are shown with
background values subtracted. Note that the radiation loss term includes
contributions from both the corona and the transition region.

Simulation light curves of AIA 193 intensity, plasma temperature, and plasma
density are shown in the left column of Fig. 4, for the nominal heating
duration of $t_{dur}$=60 s (black curve), as well as for 30 s (red dashed
curves) and 120 s (red dot-dashed curves). As expected, the simulated
intensity light curves match the achieved intensity and duration of the
observed light curve. Note that the peak intensity, temperature, and density
are all correlated with the heating duration; this is due to more impulsive
events heating the plasma on a smaller timescales, leading to a sharper rise
in temperature and consequently a higher density due to stronger evaporation.

We also show the simulated intensity light curves for the other AIA filters in
the left column of Fig. 5. In these plots the simulated light curves have been
offset by a case specific value to match the observed pre-event background
intensity level. The background level has been offset from that predicted
based on matching to AIA 193 (see Table 3) to match the model light curve
intensities to the pre-event observed intensities for each filter. As seen
above for the 193 Å filter, the anti-correlation between peak intensity and
heating duration persists for all filters. It is seen that observed
intensities in all AIA filters except 94 Å peak before the synthetic light
curves for the most suitable heating duration $t_{dur}=60$ s. The agreement is
best for 211 Å filter and worst for AIA 171 and 131 Å filters.

Next, we compute the average intensities of the simulated light curves in each
of the AIA filters. The light curves are integrated over a lifetime as
determined from when the AIA 193 Å light curve drops to 5% of the peak
intensity, and divided by the lifetime to obtain a comparison with the
observed average intensities (see columns 2 and 3 of Table 2). The simulated
intensities match observations well for AIA 193, though differ by factors
ranging from $0.8\times$ (AIA 335) to $3.1\times$ (AIA 171) in other filters.
This is a remarkable correspondence with observations considering the
limitations of the 0-D monolithic loop system we consider.

The energy loss and transfer terms (conduction and radiation losses, enthalpy,
and heat input) for the simulation with the nominal heating duration is shown
in Fig. 6. The upper panel includes the contribution from the ambient
background, and the lower panel isolates the effects due only to the
brightening event. Note that enthalpy (red triple-dot-dashed curves; Eq. 5)
can be positive or negative depending whether plasma is flowing into or out of
the corona. As expected, conduction losses (blue dashed curve; Eq. 2) dominate
at the beginning (until $\approx$70 s) of the event, and radiation losses
(green dot-dashed curve; Eq. 4) dominate at later times. The enthalpy into the
corona keeps pace with conduction loss in the early phase (for $\approx$40 s),
and drops off $\approx$10 s before conduction does. The enthalpy reverses sign
at approximately the same time that radiation becomes the dominant loss
mechanism. The total time integrated conduction loss from the coronal loop
(which is eventually radiated) is $3.0\times 10^{23}$ ergs. The net enthalpy
is positive (into corona) and is equal to $4.5\times 10^{22}$ ergs. This is an
order of magnitude less than the heating function. Notice that the radiation
loss drops below the ambient background level at the beginning of the event,
as does conduction loss about 120 s after the heat pulse. These represent
small perturbations in the ambient coronal structure, and do not have any
effect on the energetics.

####  5.1.2 Method 2: Background not modelled

An impulse with a triangular profile dissipating a total of $10^{23.2}$ ergs
in 40 s is best suited for a loop of half length of 0.65 Mm subjected to
uniform background heating of $10^{-4}$ ergs $cm^{-3}$ s-1 (see 4.2.1). The
plots in the right column of Fig. 4 display the observed and simulated light
curves for 193 Å (panel d), plasma temperature (panel e), and density (panel
f). Different colors belong to $t_{dur}$ (black), $\frac{1}{2}\times t_{dur}$
(purple) and $2\times t_{dur}$ (brown) as labelled. The simulated background
intensity is almost three orders of magnitude smaller than the peak simulated
values and hence negligible. Note that to bring the background level of the
synthetic light curves to that of observed, we have added a case specific
offset (see 3). As expected, the peak values of temperature, density, and
intensities are correlated with the duration of heating. Even though the
initial density and temperature in this case are an order of magnitude lower
than the values obtained by simulations using method 1, the peak values of
temperature match in both cases and peak density produced by method 2 is less
than that produced by method 1 by a factor $\lesssim 2$.

We have also obtained the synthetic light curves for the other channels of AIA
and plotted them after adding case specific background levels, in the right
column of Fig. 5. The colors have the same meaning as those in the right
column of Fig. 4. For comparison, we have also over-plotted the observed light
curves for the corresponding AIA channels.

We plot the various energy loss and transfer terms, both including and
excluding the background contribution in the right column of Fig. 6 (see
panels c & d). The qualitative feature of an initial conduction-dominated loss
is the same as that obtained using method 1 and is shown in panels a & b of
Fig. 6. Moreover, the time at which enthalpy changes its sign is also
approximately the same as where radiation loss starts dominating conduction
loss, similar to method 1. However, we note that enthalpy plays a more
important role in method 2 than in method 1. The time integrated intensities
obtained from synthetic light curves in all filters show a better
correspondence with those obtained from the observed light curve, in
particular for 171 Å, which shows the highest discrepancy.

Figure 7: Same as in Fig. 4 but for BR-07.

Figure 8: Same as Fig. 5 but for BR-07.

###  5.2 Modelling the Dynamics of BR-07

####  5.2.1 Method 1: Background modelled

Figure 9: Same as in Fig. 6 but for BR-07.

We follow the same procedure to analyze BR-07 here as we did for BR-00 in
§5.1.1. A loop of half-length of 0.9 Mm, with $10^{23.1}$ ergs deposited in
$t_{dur}=$ 70 s mimics the observed rise times, peak intensity and approximate
duration. The background heating used for matching the base intensity levels
in AIA 193 maintains plasma at a temperature and density of 0.92 MK and
$8.7\times 10^{9}$ cm-3. The simulation is run for 60 s before a triangular
heat pulse is applied for a duration of $t_{dur}$=70 s. The evolution of
plasma obtained from simulations were used to predict light curves in all
channels. Simulation light curves of AIA 193 intensity, temperature and
density of plasma for $t_{dur}$, $2\times t_{dur}$ and $\frac{1}{2}\times
t_{dur}$ along with observed AIA 193 light curve are shown in Fig. 7.
Simulated light curves in remaining filters (after increasing or decreasing by
case specific offsets) are shown in Fig. 8. The energy loss and transfer terms
for simulations with nominal heating duration i.e. 70 s in this case, is shown
in Fig. 9. The scheme used in the figures are same as that used for BR-00.

The peak intensity in all filters, temperature and density are largest for
most impulsive heating. The simulated intensities match observations well for
AIA 193. It differs in other filters by factors between 1.4$\times$ (AIA 94)
and 5.5$\times$ (AIA 171). The event has an initial conduction dominated
cooling phase lasting for $\approx$ 70 seconds followed by radiation dominated
cooling phase at later times. Enthalpy starts dropping $\approx$ 10 s before
conduction. The total time integrated conduction loss from the coronal loop is
$3.7\times 10^{23}$ ergs. The net enthalpy is positive (into corona) and is
equal to $5.1\times 10^{22}$ ergs.

####  5.2.2 Method 2: Background not modelled

We follow the same procedure to analyze BR-07 as in §5.1.2. A heating event
having a triangular profile, which dissipates $10^{23.3}$ $ergs$ in 60 $s$ is
best suited for a loop of half length of 0.65 Mm subjected to uniform
background heating of $10^{-4}$ $ergs~{}cm^{-3}~{}s^{-1}$. A case specific
offset has been added to all the simulated light curves to make the observed
and simulated pre-event intensities equal (see Fig. 7 and 8). All features
including an initial conduction dominated cooling of corona, enthalpy changing
sign approximately when radiation starts dominating, are qualitatively the
same as that of BR-00 (see Fig. 9). The average intensity in AIA 171 filter
agrees better with the observed value.

Figure 10: Same as Fig. 4 but for BR-26.

Figure 11: Same as Fig. 5 but for BR-26.

Figure 12: Same as Fig. 6 but for BR-26.

###  5.3 Modelling the Dynamics of BR-26

####  5.3.1 Method 1: Background modelled

We follow the same procedure to analyze BR-26 here as for BR-00 in §5.1.1. A
loop of half-length of 1.1 Mm, with $10^{22.8}$ ergs deposited in $t_{dur}=$
60 s mimics the observed light curves. The background heating used for
matching the base intensity levels in AIA 193 Å maintains plasma at a
temperature and density of 0.92 MK and $7.1\times 10^{9}$ cm-3. Simulated
light curves of AIA 193 intensity, temperature and density of plasma for
$t_{dur}$, $2\times t_{dur}$ and $\frac{1}{2}\times t_{dur}$ along with
observed AIA 193 light curve are shown in Fig. 10. Simulated light curves in
remaining filters (after increasing or decreasing by case specific offsets)
are shown in Fig. 11. The average intensities in filters apart from AIA-193
differ by factors ranging between 1.2$\times$ (AIA 94) and 3.6$\times$ (AIA
171). The energy loss and transfer terms for simulations with nominal heating
duration i.e. 60 s in this case, is shown in Fig. 12. The scheme used in the
plots are same as that used for BR-00. Energy loss and transfer evolution
trends are similar to those for BR-00 and BR-07. The total time integrated
conduction loss from the coronal loop is $1.9\times 10^{23}$ ergs. The net
enthalpy is positive (into corona) and is equal to $3.5\times 10^{22}$ ergs.

####  5.3.2 Method 2: Background not modelled

We follow the same procedure to analyze BR-26 as detailed in § 5.1.2. A
heating event having a triangular profile, which dissipates $10^{23.1}$ ergs
in 70 $s$ is best suited for a loop of half length of 0.65 Mm subjected to
uniform background heating of $10^{-4}$ ergs cm-3 s-1. A case specific offset
has been added to all the simulated light curves to make observed and
simulated background intensities equal (see Figs. 10 and 11). Once again we
see an initial conduction dominated cooling of corona, enthalpy changing sign
approximately when radiation starts dominating, are qualitatively the same as
that of BR-00 (see Fig. 12). The average intensity in AIA-171 filter agrees
better with the observed value.

Table 2: Observed and simulated average Intensities in DN pix-1 s-1 over AIA-193 lifetime of brightenings. The values in the parenthesis cover the range of average intensities if the duration in which energy is deposited in half and double the time duration selected as the most suitable input parameter. AIA | BR-00 (DN pix-1 s-1) | BR-07 (DN pix-1 s-1) | BR-26 (DN pix-1 s-1)
---|---|---|---
Filter | Obs. | Sim. | Obs. | Sim. | Obs. | Sim.
|  | Method 1 | Method 2 |  | Method 1 | Method 2 |  | Method 1 | Method 2
|  | (bkg. modelled) | (bkg. not modelled) |  | (bkg. modelled) | (bkg. not modelled) |  | (bkg. modelled) | (bkg. not modelled)
AIA-094 | 0.50 | 0.92 | 0.31 | 0.89 | 1.2 | 0.43 | 0.40 | 0.48 | 0.17
|  | [0.78-1.0] | [0.25-0.34] |  | [0.96-1.3] | [0.31-0.49] |  | [0.41-0.61] | [0.12-0.21]
AIA-131 | 7.3 | 24.0 | 19.7 | 9.1 | 31.7 | 25.9 | 4.6 | 13.9 | 12.2
|  | [20.9-26.2] | [16.9-21.45] |  | [25.6-35.0] | [20.4-29.3] |  | [12.0-16.9] | [9.3-14.0]
AIA-171 | 104.3 | 376.5 | 146.9 | 88.6 | 489.6 | 207.2 | 60.1 | 214.0 | 84.2
|  | [325.9-410.3] | [124.2-155.4] |  | [401.3-537.1] | [153.5-228.3] |  | [183.3-260.6] | [53.8-97.9]
AIA-193 | 83.8 | 85.6 | 67.3 | 101.4 | 112.2 | 83.0 | 39.7 | 41.5 | 44.0
|  | [70.1-95.1] | [56.8-73.9] |  | [87.8-127.7] | [64.6-95.4] |  | [35.3-55.6] | [33.1-51.0]
AIA-211 | 12.0 | 15.2 | 16.4 | 8.7 | 19.9 | 20.0 | 5.2 | 7.8 | 11.0
|  | [12.7-16.8] | [14.0-17.9] |  | [15.8-22.4] | [15.7-22.7] |  | [6.7-10.1] | [8.3-12.7]
AIA-335 | 2.8 | 2.0 | 2.3 | 1.1 | 2.6 | 2.9 | 0.64 | 1.1 | 1.6
|  | [1.7-2.2] | [2.0-2.5] |  | [2.1-2.9] | [2.3-3.2] |  | [0.94-1.4] | [1.2-1.8]
Table 3: Background levels used to compare observed and model light curves in various filter in Figs. 4,5,7,8,10,11. The values listed for Method 1 are offsets relative to the simulated value derived by comparison with AIA 193 light curves (see text). AIA | BR-00 (DN pix-1 s-1) | BR-07 (DN pix-1 s-1) | BR-26 (DN pix-1 s-1)
---|---|---|---
Filter | Method 1 | Method 2 | Method 1 | Method 2 | Method 1 | Method 2
| (bkg. modelled) | (bkg. not modelled) | (bkg. modelled) | (bkg. not modelled) | (bkg. modelled) | (bkg. not modelled)
AIA-094 | -2.1 | 4.8 | -0.95 | 5.3 | -2.8 | 3.5
AIA-131 | -176.0 | 43.0 | -116.9 | 81.7 | -163.4 | 35.2
AIA-171 | -1908.6 | 1359.8 | -1056.2 | 1908.0 | -1809.8 | 1154.5
AIA-193 | 0 | 679.0 | 0 | 790.7 | 0 | 620.6
AIA-211 | +131.4 | 272.6 | +179.5 | 307.7 | +131.3 | 259.5
AIA-335 | +2.9 | 22.3 | +7.9 | 25.6 | +0.18 | 17.8

###  5.4 Comparison of conductive flux to the radiative flux

Figure 13: Time evolution of the ratio of absolute values of conduction and
radiation loss rates for BR-00 (panel [a]), BR-07 (panel [b]) and BR-26 (panel
[c]). The solid black curves are for the optimized duration while the dashed-
blue and dashed-dotted red corresponds to the result obtained for half and
twice the duration of the heating event as labelled.

To quantitatively assess the relative importance of conduction loss over
radiation loss during different phases of the brightening, in Fig. 13 we plot
the logarithm of the ratio of absolute values of background subtracted
conduction and radiation loss rates obtained from method 1 (left column) and
method 2 (right column); the upper panels correspond to BR-00, the middle
panels to BR-07, and the bottom panels to BR-26. The solid black curves
correspond to the nominal heating durations, while the dashed-blue and dashed-
dotted red curves correspond to heating durations set to $\frac{1}{2}\times$
and $2\times$ the nominal duration. The cusps in the curves (e.g., those
present between 80 and 100 s in the left panels and between 60 and 80 s in the
right panels) are artefacts of correcting for the background, arising from the
radiative loss dropping below and rising above the ambient value (see Fig. 6,
9, 12 and discussion in Section 5.1). The occurrence of these spikes at
different locations for the same brightenings in the two methods is due to
using different input parameters in each case. For all the three events,
irrespective of how the input parameters are selected, we find an initial
phase where conduction is the dominant cooling mechanism in the corona. This
phase ends at a simulation time step of $\approx$140 s i.e., 80 s after the
heating starts.

##  6  Discussion and Summary

Hi-C has provided observations of the solar corona in 193 Å at the highest
spatial resolution so far. It has presented us with a number of intriguing
observations, including those of the faintest transient brightenings observed
(Régnier et al., 2014; Subramanian et al., 2018). These faint brightenings
when studied by multi-wavelength images of AIA, were deemed to be dominated by
conduction loss. Here, we have explored coronal energetics using a simplified
0-D description (EBTEL) to simulate the evolution of coronal plasma in loops
subject to heat inputs.

We identify loop sizes and heat inputs that mimic the intensities and
durations of the observed brightenings using two methods. In first method we
use background intensities for constraining the input parameters and in second
method we don’t use this information. We use the simulations to study the
transfer of energy into and out of the corona. We focus on three of the
simplest brightenings identified by (Subramanian et al., 2018), with single
unambiguous intensity peaks. We adopt triangular heating profiles of duration
30-140 s, and find that loop half lengths of $\approx$108 cm and energy
deposition of $\approx$1023 ergs can generate dynamical intensity profiles
that mimic the observed brightenings.

We find that the average brightness of the simulated loops in the AIA 193 Å
filter matches the observations well, and are within a factor of 2–5.5 in the
other AIA filters if input parameters are selected by method 1 (i.e. both
transient and background are modelled) and within a factor of 2–3 if input
input parameters are selected by method 2 (background is not modelled). The
obtainment of better tally by latter method, can be suggestive of the
transient events and background being dynamically distinct. In either method
the largest discrepancies between the simulated and observed intensities,
irrespective of how input parameters have been selected, arises in the AIA
171Å and AIA 131Å filters.

For all three events studied here, we find that conduction is the dominant
cooling mechanism in corona in the early phase of the transients. About 80 s
after the heat pulse, radiation looses begin to dominate. We observe that
conduction dominated cooling in corona during the early phase of the evolution
has also been reported for flares (see e.g., Cargill et al., 1995),
microflares (see e.g., Gupta et al., 2018), and are also expected for
nanoflares (see e.g., Cargill, 1994). Our results show that transient events
such as the ones observed in Hi-C are similar in character to microflares and
nanoflares, and are likely produced through the same underlying physical
processes. We also note that our simulated plasma temperatures are lower than
the observed values. And since conduction loss increases as $T^{3.5}$ while
radiative loss decreases in this temperature regime, our assessment of the
relative magnitude of conduction loss in the energetics of coronal plasma is
an underestimate.

If the background temperature were raised in the model, it necessarily
requires an increase in the loop length in order to maintain consistency with
observed intensities (see Equation 6), resulting in loops of length $\gtrsim$8
Mm and life times $\gtrsim$6$\times$ the observed values. If the loop lengths
were fixed at $\approx$1 Mm, the heating rate increases and causes the
predicted model intensities to increase non-linearly, worsening the agreement
between the simulated light curves and the observations.

The results obtained here provide further insights into the energetics and
dynamics of transient events occurring in the solar corona. Further work is
required, possibly with multi-stranded simulations, with more high resolution
observations to study their roles in the coronal heating. One possible
direction is to study the ”campfire” events detected by the Extreme
Ultraviolet Imager (EUI; Rochus et al., 2020) on board the Solar Orbiter
mission (Müller et al., 2020).

We thank the referee for the useful comments and suggestions. This work is
partly supported by the Max-Planck Partner Group on ”Coupling and Dynamics of
the Solar Atmosphere” of MPS at IUCAA. AR acknowledges financial support from
University Grants Commission in form of SRF. VLK acknowledges support from
NASA Contract NAS8-03060 to the Chandra X-ray Center, and the hospitality of
IUCAA during several visits. We thank Srividya Subramanian for assistance with
data extraction and comparison. We thank Prof. Gulab Dewangan for useful
discussions. AIA and Hi-C teams are gratefully acknowledged for providing the
data.

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