11institutetext: Key Laboratory of Dark Matter and Space Science, Purple
Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China
11email<EMAIL_ADDRESS>22institutetext: School of Astronomy and Space
Science, University of Science and Technology of China, Hefei 230026, China

# Energy estimation of small-scale jets from the quiet-Sun region

Fanpeng Shi 1122 Dong Li 1122 Zongjun Ning 1122 Jun Xu 1122 Yuxiang Song 1122
Yuzhi Yang 1122

###### Abstract

Context. Solar jets play a role in the coronal heating and the supply of solar
wind.

Aims. This study calculated the energies of 23 small-scale jets emerging from
a quiet-Sun region to investigate their contributions for coronal heating.

Methods. We used the data from the the High Resolution Imager (HRI) of the
Extreme Ultraviolet Imager (EUI) on board the Solar Orbiter (SolO). Small-
scale jets were observed by the HRIEUV $174$ Å passband in the high cadence of
$6$ s. These events were identified by the time-distance stacks along the
trajectories of jets. Using the simultaneous observation from the Atmospheric
Imaging Assembly (AIA) on board the Solar Dynamics Observatory (SDO), we also
performed the differential emission measure (DEM) analysis on these small-
scale jets to obtain the physical parameters of plasma, which enabled the
estimation of the kinetic and thermal energies of jets.

Results. We found that most of jets exhibited the common unidirectional or
bidirectional motions, while some showed the more complex behaviors, namely
the mixture of unidirection and bidirection. A majority of jets also presented
the repeated eruption blobs (plasmoids), which may be the signatures of the
quasi-periodic magnetic reconnection that have been observed in the solar
flares. The inverted Y-shaped structure can be recognized in several jets.
These small-scale jets typically have the width of $\sim$$0.3$ Mm, the
temperature of $\sim$$1.7$ MK, the electron number density of
$\gtrsim$$10^{9}\ \mathrm{cm^{-3}}$, with speeds in a wide range of
$\sim$$20$–$170$ $\mathrm{km\ s^{-1}}$. Most of these jets have the energy of
$10^{23}$–$10^{24}\ \mathrm{erg}$, which is marginally smaller than the energy
of typical nanoflares. The thermal energy fluxes of 23 jets are estimated to
be (0.74–2.96) $\times 10^{5}\ \mathrm{erg\ cm^{-2}\ s^{-1}}$, which is almost
on the same order of magnitude as the energy flow required for heating the
quiet-Sun corona, albeit the kinetic energy fluxes vary over a large range
because of their strong dependence on velocity. Furthermore, the frequency
distribution of thermal energy and kinetic energy both follow the power-law
distribution $N(E)\propto$ $E^{-\alpha}$.

Conclusions. Our observations suggest that although these jets cannot provide
sufficient energy for the heating of the whole quiet-Sun coronal region, they
are likely to account for a significant portion of the energy demand in the
local regions where the jets occur.

###### Key Words.:

Sun: corona – Sun: UV radiation – Sun: magnetic fields

##  1 Introduction

Solar jets are collimated beam-like plasma ejections in the solar atmosphere.
They are prevalent and can occur at different scales, and are considered as
the important source of coronal heating and the solar wind. (Raouafi et al.,
2016; Shen, 2021). Solar jets also represent the characteristics of magnetic
reconnection, with bidirectional jets strongly supporting this explanation due
to their double outflows (Innes et al., 1997; Zheng et al., 2018; Shen et al.,
2019). Over the decades, thanks to the improvement of the spatiotemporal
resolution of observation, small-scale jets in different layers of the solar
atmosphere have been studied in detail. Shibata et al. (2007) investigated the
inverted Y-shaped chromospheric anemone jets in active regions, suggesting
that the heating of the chromosphere and corona may be related to small-scale
ubiquitous reconnection. Ji et al. (2012) reported the ultrafine channels for
coronal heating, in which continuous small-scale upward energy flows originate
from photosphere and subsequently light up the corona. Tian et al. (2014)
observed the prevalent small-scale jets from the networks of the solar
transition region and chromosphere, with speeds on the order of $\sim$$100\
\mathrm{km\ s^{-1}}$ and temperatures of $>$$10^{5}$ K, which are likely the
source for the solar wind. There are also some other detailed studies, such as
the chromospheric small-scale jets (Wang et al., 2021; Hong et al., 2022),
chromospheric spicules (Samanta et al., 2019), recurrent jets from sunspot
light bridges (Tian et al., 2018), small-scale IRIS jets (Li et al., 2018),
and coronal nanojets (Antolin et al., 2021).

The high spatiotemporal resolution coronal extreme ultraviolet (EUV) images
observed by the Hi-C 2.1, provided new insights into the study of small-scale
jets owing to the discoveries of dot-like, loop-like, and surge/jet-like EUV
brightenings at small spatial scales (Tiwari et al., 2019; Panesar et al.,
2019). Recently, the observations by the Extreme Ultraviolet Imager (EUI;
Rochus et al. 2020) on board the Solar Orbiter (SolO; Müller et al. 2020),
have revealed the transient small-scale brightenings in the quiet solar corona
termed campfires (Berghmans et al., 2021; Chen et al., 2021; Panesar et al.,
2021), the moving structures in ultraviolet bright points (Li, 2022), the
persistent null-point reconnection (Cheng et al., 2023), as well as the small-
scale plasma flow and jets (Hou et al., 2021; Chitta et al., 2021b; Mandal et
al., 2022; Chitta et al., 2023).

The energy of a small-scale jet usually has the order of magnitude of
nanoflare-like ($\sim$$10^{24}\ \mathrm{erg}$, Parker 1988). Hou et al. (2021)
estimated the energy of a typical coronal microjets, with thermal energy of
$\sim$$3.9\times\ 10^{24}\ \mathrm{erg}$ and kinetic energy of
$\sim$$2.9\times\ 10^{23}\ \mathrm{erg}$. Antolin et al. (2021) estimated the
total energy (”thermal + kinetic”) released by a coronal nanojet to be
$(7.8$–$17.3)\times 10^{24}\ \mathrm{erg}$, and they expected that this range
of energies corresponds to the high end of the nanojet distribution, most of
which is likely to be unresolved by observations. Additionally, the small-
scale EUV brightening or bursts also have the energy of $10^{24}\
\mathrm{erg}$ per event (Chitta et al., 2021a). Li et al. (2018) estimated
that an upper limit of the thermal energy of the coronal EUV brightening
caused by the IRIS jets is around $7\times 10^{24}\ \mathrm{erg}$. Using the
observation of EUV brightening, Purkhart & Veronig (2022) studied the
nanoflare energy distributions in quiet-Sun regions, and found that the mean
energy flux of $(3.7\pm$$1.6)\times 10^{4}\ \mathrm{erg\ cm^{-2}\ s^{-1}}$ is
one order of magnitude smaller than the coronal heating requirement. They also
found that clusters of high energy flux (up to $3\times 10^{5}\ \mathrm{erg\
cm^{-2}\ s^{-1}}$) are surrounded by extended regions with lower activity.

In this study, we investigated 23 small-scale jets emerging from a quiet-Sun
region and estimated their energies. Several events showed the ambiguous
inverted Y-shaped structures resembling the standard or blowout jets as
described by Moore et al. (2010) and Sterling et al. (2015). Most of them
exhibited the nearly collimated simple structures similar to the coronal
microjets (Hou et al., 2021) and picoflare jets (Chitta et al., 2023). This
paper is organized as follows: Section 2 introduces the observations and data
analysis. Section 3 analyzes three examples of small-scale jets. Section 4 is
the discussion and conclusions.

##  2 Observations and Data Analysis

On 2023 March 30, the observation of the 174 Å EUV High Resolution Imager
(HRIEUV) of SolO/EUI started at 15:00:15 UT and ended at 16:07:09 UT (Earth
time) in a cadence of 6 s (a total of 670 frames). The spacecraft was at a
distance of $\sim$0.379 AU from the Sun, and its position respect to solar
longitude was $\sim$$6.2^{\circ}$ west from Sun-Earth line. The HRIEUV 174 Å
plate scale is 0.492 arcsec per pixel, corresponding to a linear scale of
$\sim$134 km per pixel at this distance. This passband is sensitive to
emissions from plasma at temperatures of roughly 1 MK. The level-2 calibrated
data (Kraaikamp et al.,
2023)111https://www.sidc.be/EUI/data/releases/202301_release_6.0/release_notes.html
were used and the remaining jitter in image sequence was removed using the
cross-correlation method.

The HRIEUV 174 Å had a field-of-view (FOV) at a quiet-Sun region on the north
side of the NOAA active region (AR) 13262, as outlined by the white box in
Fig. 1 (left panel). HRIEUV 174 Å observation is shown in the movie
(movie_hri174.mp4). During this observation period, there were plenty of jet
eruptions in this area. A total of twenty two regions were detected and marked
by R1–R22 in Fig. 1 (right panel). Some jets ejected matter over long
distances, while others ejected short distances. We used the white boxes with
different scales to cover the jets.

Six EUV channels of the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012)
and the line-of-sight (LOS) magnetograms of Helioseismic and Magnetic Imager
(HMI; Scherrer et al. 2012) on board the Solar Dynamics Observatory (SDO;
Pesnell et al. 2012) were also used in this study. AIA 94 Å ($T$$\sim$$6$ MK),
131 Å ($T$$\sim$$10$ MK), 171 Å ($T$$\sim$$0.7$ MK), 193 Å ($T$$\sim$$1.6$
MK), 211 Å ($T$$\sim$$2$ MK), and 335 Å ($T$$\sim$$2.5$ MK), mainly reflect
the emissions from the corona. The response of AIA 171 Å passband is similar
to the HRIEUV 174 Å. These EUV passbands have a cadence of 12 s and a plate
scale of 0.6 arcsec ($\sim$432 km). The HMI LOS magnetograms have a cadence of
45 s and a plate scale of 0.5 arcsec ($\sim$360 km). All of the above data
have been removed solar rotation by derotating to 15:00:09 UT as seen in Fig.
1. The differential emission measure (DEM) analysis (Cheung et al., 2015; Su
et al., 2018) was performed on co-aligned AIA images to extract the physical
parameters of plasma. In this procedure, we averaged the AIA intensities over
two frames, i.e., the reconstructed AIA image sequences have a cadence of 24
s. The temperature range of inversion is set as $5.5\leq\log_{10}T\leq 7.6$,
following with previous suggestions (Hannah & Kontar, 2012; Cheung et al.,
2015; Su et al., 2018).

Figure 1: AIA $171$ Å image of the solar disk (left), and the white box
represents the selected HRIEUV $174$ Å FOV ($\sim$$136\times 136\
\mathrm{Mm^{2}}$). Twenty two regions (R1–R22) analyzed in this study are
labeled in the HRIEUV $174$ Å image (right).

The DEM function is an instrument-independent function that characterizes the
electron and temperature of optically thin plasma in thermal equilibrium. The
DEM function (in units of $\mathrm{cm^{-5}\ K^{-1}}$) is defined as the
squared electron density $n_{\mathrm{e}}^{2}$ integrated along the LOS depth
$z$,

$\mathrm{DEM}(T)=n_{\mathrm{e}}^{2}\frac{dz}{dT},$ (1)

and the emission measure (EM)-weighted temperature is,

$\bar{T}=\frac{\int\mathrm{DEM}(T)\times
TdT}{\int\mathrm{DEM}(T)dT}=\frac{\int\mathrm{DEM}(T)\times
TdT}{\mathrm{EM}},$ (2)

the $\mathrm{DEM}(T)$ also allow us to estimate the unknown electron density
$n_{\mathrm{e}}$ via the measurement of $z$,

$\begin{array}[]{l}\mathrm{EM}=\int\mathrm{DEM}(T)dT=\int
n_{\mathrm{e}}^{2}dz=n_{\mathrm{e}}^{2}z,\end{array}$ (3)

where $n_{\mathrm{e}}$ defines a mean electron density that is averaged over
the LOS depth $z$ with a filling factor of unity (Aschwanden et al., 2000,
2015).

Figure 2:  (a)–(c) AIA 171 Å map, the EM map, and the EM-weighted temperature
map, respectively. The FOV of map covers the R1–R22 regions.

Fig. 2(a) shows the AIA 171 Å map taken at 15:45:21 UT, from which we can see
the similar structure as the HRIEUV $174$ Å in Fig. 1. Based on the DEM
analysis in this region, Fig. 2(b)–(c) give the corresponding EM and EM-
weighted temperature maps. Twenty two regions are marked with small boxes.

We describe here the assumptions and formulae used to calculate the jet
energies. If we assume that these small-scale jets are elongated cylinders and
our LOS is perpendicular to the direction of jet motion, then the depth $z$ is
equal to the jet width ($w$). Assuming that the structure of jet is isothermal
and homogeneous (with a constant electron density), the total thermal energy
can be expressed as (e.g., Aschwanden et al. 2015),

$E_{\mathrm{th}}=3n_{\mathrm{e}}k_{\mathrm{B}}T_{\mathrm{e}}V_{\mathrm{{jet}}},$
(4)

where $n_{\mathrm{e}}=\sqrt{\dfrac{\Delta\mathrm{EM}}{w}}$ is the jet density
after considering the variation in EM relative to the pre-event value
($\Delta\mathrm{EM}$ is the increase in EM, e.g., Purkhart & Veronig 2022;
Long et al. 2023), $k_{\mathrm{B}}=1.38\times 10^{-16}\ \mathrm{erg\ K^{-1}}$
is Boltzmann constant, $T_{\mathrm{e}}$ is temperature around the time when
the EUV emission peaked, and $V_{\mathrm{{jet}}}=\pi(\dfrac{w}{2})^{2}l$ is
the volume of jet. The jet width $w$ and length $l$ can be measured from
HRIEUV $174$ Å image.

We can also calculate the mean thermal energy flux, which presumes that the
total thermal energy is dissipated through the surface area of the jet
throughout its entire lifetime,

$F_{\mathrm{th}}=\frac{3n_{\mathrm{e}}k_{\mathrm{B}}T_{\mathrm{e}}V_{\mathrm{{jet}}}}{\tau_{\mathrm{{tot}}}S_{\mathrm{{jet}}}},$
(5)

where $\tau_{\mathrm{{tot}}}$ is the total lifetime,
$S_{\mathrm{jet}}\approx\pi wl\ (\pi wl\gg\pi(\dfrac{w}{2})^{2})$ is the
surface area of jet, then the expression of $F_{\mathrm{th}}$ can be reduced
as,

$F_{\mathrm{th}}=\frac{3n_{\mathrm{e}}k_{\mathrm{B}}T_{\mathrm{e}}w}{4\tau_{\mathrm{{tot}}}},$
(6)

which is independent of the jet length.

The kinetic energy flux carried by a parcel of fluid moving with velocity $v$
is (e.g., Chitta et al. 2023)

$F_{\mathrm{k}}=\frac{1}{2}\rho v^{3},$ (7)

where $\rho\approx n_{\mathrm{e}}m_{\mathrm{p}}$ is the plasma mass density,
and $m_{\mathrm{p}}=1.67\times 10^{-26}\ \mathrm{g}$ is the proton mass. The
$F_{\mathrm{k}}$ is the energy per unit area per unit time on a plane
perpendicular to motion. Subsequently, the total kinetic energy can be
derived,

$E_{\mathrm{k}}=\sum_{i}F_{\mathrm{k}i}\tau_{i}S_{\mathrm{bot}},$ (8)

where subscript $i$ denotes the $i$th blob of jet, $\tau_{i}$ is its lifetime,
and $S_{\mathrm{bot}}=\pi(\dfrac{w}{2})^{2}$ is bottom area of jet.

##  3 Results

###  3.1 Unidirectional jets

Figure 3: (a) HRIEUV $174$ Å image of the R12 jet. (b) The estimation of jet
width. (c) Time-distance diagram in the position of blue slit. (d) Smooth-
subtracted image of panel (c). (e) AIA 171 Å image of the R12 jet. Contour
level represents the 90% of the peak intensity of AIA 171 Å. (f) The EM map.
(g) The EM-weighted temperature map. The mean EM and temperature are averaged
over the 90% area. The $\Delta\mathrm{EM}$ is the net increase in EM relative
to pre-jet value.

Each blob (plasmoid) of small-scale jets recognized and analyzed in this study
lasts for at least three frames (i.e., 18 s), which enabled the motion
identification from HRIEUV $174$ Å images. Fig. 3(a) shows the R12 jet in
HRIEUV $174$ Å image, which presents a simple linear motion along a straight
line. The blue slit is outlined to obtain the time-distance stacks, and the
”end” represents the end of slit. The black dashed line is the position used
for calculating the jet width. Fig. 3(b) shows the estimation for jet width.
The black profile is the reconstructed intensity along the black slit in Fig.
3(a), and the blue profile represents the singe Gaussian fitting with a linear
background. The FWHM of fitting result is considered as the width, which
corresponds to a linear scale of $\sim$0.32 Mm. Fig. 3(c) shows the time-
distance diagram derived from the blue slit. To make it easier to show the
variability, we also give the corresponding smooth-subtracted images for all
time-distance diagrams in this study. Fig. 3(d) is the smooth-subtracted image
of panel (e), i.e., after subtracting the slowly-varying component, which is a
4-frame (24 s) running average of original intensity. The smoothing window is
chosen individually for each event, and its size can be seen through the blank
frames on both sides of the smooth-subtracted image. In Fig. 3(d), there is a
slanted bright stripe spanning five frames, which outlines the blob motion
with velocity of $\sim$$\mathrm{98\ km\ s^{-1}}$. The jet lifetime is defined
as the time interval between the beginning and end of the moving blob, i.e.,
30 s.

Fig. 3(e)–(f) is the zoom-in of the R12 region as labeled in Fig. 2. It is
obvious that the moving blob is only several bright pixels in the FOV of AIA
171 Å. We choose the 90% of the peak intensity as the contour of jet, which is
a good representation due to its elongated structure. Therefore, the mean EM
and EM-weighted temperature can be obtained within the 90% area, their values
are 4.25 $\times\ 10^{26}\ \mathrm{cm^{-5}}$ and 1.78 MK, respectively. As
noted in Section 2, the variation in EM relative to pre-jet value should be
considered to estimate the jet density. The $\Delta\mathrm{EM}$ below Fig.
3(f) is the net increase that is calculated over the same area as the
$\mathrm{EM}$. These EMs and EM-weighted temperatures are computed over the
temperature range of $5.6\leq\log_{10}T\leq 6.5$, where the DEM solution is
well constrained. This range is almost the same as previous literature about
coronal microjets (Hou et al., 2021). All other jets also exhibit emissions
originating from a similar temperature range and lack higher temperature
components, and thus we can apply the same range to EM analysis of all jets.

Using the $\Delta\mathrm{EM}=0.36\times 10^{26}\ \mathrm{cm^{-5}}$ and
$w=0.32$ Mm in Equation 3, then the $n_{\mathrm{e}}$ is $1.06\times 10^{9}\
\mathrm{cm^{-3}}$, which is a reasonable value for the coronal condition. The
jet length $l\sim$1.5 Mm can be roughly measured from the HRIEUV $174$ Å
image. Substituting these known quantities into the Equation 4, the estimated
total thermal energy is $9.43\times 10^{22}\ \mathrm{erg}$, which is one order
of magnitude smaller than the energy of typical nanoflares ($\sim$$10^{24}\
\mathrm{erg}$, Parker 1988; Chitta et al. 2021a). Thus, the thermal energy
flux of R12 jet is $2.08\times 10^{5}\ \mathrm{erg\ cm^{-2}\ s^{-1}}$,
contributing $2/3$ of the canonical amount of energy required to heat the
quiet-Sun corona ($3\times 10^{5}\ \mathrm{erg\ cm^{-2}\ s^{-1}}$, Withbroe &
Noyes 1977).

Using the Equation 7 and 8, the kinetic energy flux of R12 jet is estimated to
be $8.34\times 10^{5}\ \mathrm{erg\ cm^{-2}\ s^{-1}}$, and the kinetic energy
is $2.01\times 10^{22}\ \mathrm{erg}$. Note that the velocity calculated above
is the projected velocity and may depend on the viewing angle. The kinetic
energy estimation can be heavily affected by the projection effect and may not
be accurate, and based on the calculation, the kinetic energy is very small
compared to the thermal energy. It may not be accurate to represent the total
energy of the jet simply as ”$E_{\mathrm{k}}+E_{\mathrm{th}}$” and hence we do
not present the total energy in this study.

###  3.2 Bidirectional jets

Figure 4:  (a)–(g) Similar to Fig. 3, the results for R13 jets.

Fig. 4 gives an example of bidirectional jets in R13. Similar to the analysis
for R12 jet, Fig. 4(a)–(d) show the HRIEUV $174$ Å image, the estimation of
jet width, the time-distance diagram, and the smooth-subtracted image,
respectively. In this event, two blobs have different velocities ($\sim$$80\
\mathrm{km\ s^{-1}}$ and $\sim$$38\ \mathrm{km\ s^{-1}}$) , but their
lifetimes are the same as the total lifetime, i.e., 30 s. The jet length has a
rough value of $\sim$3 Mm. Fig. 4(e) shows the AIA 171 Å image of the R13
jets, which just exhibits a few of bright pixels. The 90% contour level is
also marked. The EM and EM-weighted temperature maps are shown in Fig.
4(f)–(g).

According to the above procedures, the electron density, thermal energy flux,
and total thermal energy, are estimated to be $1.25\times 10^{9}\
\mathrm{cm^{-3}}$, $2.56\times 10^{5}\ \mathrm{erg\ cm^{-2}\ s^{-1}}$, and
$2.46\times 10^{23}\ \mathrm{erg}$, respectively. The derived thermal energy
flux almost satisfies the energy demand for coronal heating in quiet Sun
region. The total thermal energy is in the range of $10^{23}$–$10^{24}\
\mathrm{erg}$. As for the kinetic energy flux, two blobs should be calculated
individually, they are $5.34\times 10^{5}$ and $0.57\times 10^{5}\
\mathrm{erg\ cm^{-2}\ s^{-1}}$. Therefore, the total kinetic energy is
$1.61\times 10^{22}\ \mathrm{erg}$, which is one order of magnitude smaller
than the total thermal energy.

###  3.3 Hybrid jets

Besides the unidirectional and bidirectional jets, the third type of jets was
also detected. In this study, we refer to it as hybrid jets, which are
mixtures of former two types. Fig. 5 and 6 give an example of the third type
that is taken in R15. Fig. 5(a)–(f) present the temporal evolution of jets.
The first eruption is bidirectional, followed by a unidirectional propagation,
as indicated by the black arrows. Fig. 5(g) gives the HMI LOS magnetogram, and
the red and blue contours represent magnetic field strength with the levels of
$\pm$10 G, respectively. Fig. 5(h) shows AIA 171 Å map with overplotted HMI
contours. From this we can see that the jet-base is locates in a weak
opposite-polarity region, which is surrounded by several patches with magnetic
field strength greater than 10 G. The magnetic flux cancellation in opposite-
polarity regions is usually the trigger of quiet-region coronal jet eruptions
(Panesar et al., 2016). The phenomenon of hybrid jets may be caused by the
changes of magnetic field configuration during the first reconnection, which
result in the subsequent unidirectional jet.

The total lifetime of hybrid jets is defined as the time interval between the
initiation of first blob and the termination of last blob, it is 96 s in this
event. The jet length is around 2 Mm based on Fig. 6(a). Using the Equation 3,
4, and 6, the electron density, thermal energy flux, and total thermal energy,
are $1.81\times 10^{9}\ \mathrm{cm^{-3}}$, $1.06\times 10^{5}\ \mathrm{erg\
cm^{-2}\ s^{-1}}$, and $1.98\times 10^{23}\ \mathrm{erg}$, respectively. The
thermal energy flux accounts for $1/3$ of the energy required to heat the
quiet-Sun corona. The total thermal energy is also in the range of
$10^{23}$–$10^{24}\ \mathrm{erg}$.

The respective lifetime of three blobs is 78, 60, and 48 s, with speed of
$\sim$$19$, $\sim$$21$, and $\sim$$21\ \mathrm{km\ s^{-1}}$. Using the
Equation 7 and 8, the kinetic energy fluxes are $0.10\times 10^{5}$,
$0.14\times 10^{5}$, and $0.14\times 10^{5}\ \mathrm{erg\ cm^{-2}\ s^{-1}}$,
as well as the total kinetic energy is $1.76\times 10^{21}\ \mathrm{erg}$.

Using the same method, the jets in other 19 regions are analyzed and shown in
Appendix A. In order to display the jet contour appropriately, the contour
level is chosen individually for every jet within the intensity range of
50–90%. Their parameters and energies are listed in Table 1.

Figure 5:  (a)–(f) The temporal evolution of R15 jets. The black arrows
indicate the propagation directions of jets. (g) HMI LOS magnetogram. The red
and blue contours represent magnetic field strength with the levels of $\pm$10
G, respectively. (h) AIA 171 Å map with overplotted HMI contours. Figure 6:
(a)–(g) Similar to Fig. 3, the results for R15 jets. Table 1: Parameters and
the calculated energies of the small-scale jets.

Label |  $w$aa$a$Width. |  $T$bb$b$Mean EM-weighted temperature. |  $\tau_{\mathrm{{tot}}}$cc$c$The total lifetime ($\tau_{\mathrm{{tot}}}$) was uesd to calculate the thermal energy flux ($F_{\mathrm{th}}$).($\tau$dd$d$Each blob lifetime ($\tau$) was used to calculate the total kinetic energy ($E_{\mathrm{k}}$).) |  $v$ee$e$Velocity of each blob. |  $n_{\mathrm{e}}$ff$f$Mean electron density. |  $F_{\mathrm{k}}$gg$g$Kinetic energy flux. |  $E_{\mathrm{k}}$hh$h$Total kinetic energy. |  $F_{\mathrm{{th}}}$ii$i$Thermal energy flux. |  $E_{\mathrm{{th}}}$jj$j$Total thermal energy. | Typekk$k$Type1: Unidirectional, 2: Bidirectional, 3: Hybrid.
---|---|---|---|---|---|---|---|---|---|---
| (Mm) | (MK) | (s) | ($\mathrm{km\ s^{-1}}$) | ($\mathrm{cm^{-3}}$) | ($\mathrm{erg\ cm^{-2}\ s^{-1}}$) | (erg) | ($\mathrm{erg\ cm^{-2}\ s^{-1}}$) | (erg) |
R1 | 0.39 | 1.63 | 36 | 120 | 1.62 $\times\ 10^{9}$ | 23.33 $\times\ 10^{5}$ | 1.00 $\times\ 10^{23}$ | 2.96 $\times\ 10^{5}$ | 3.91 $\times\ 10^{23}$ | $\mathit{1}$
R2 | 0.31 | 1.39 | 108 |  | 2.04 $\times\ 10^{9}$ | (1.55–12.84) $\times\ 10^{5}$ | 6.08 $\times\ 10^{22}$ | 0.84 $\times\ 10^{5}$ | 2.22 $\times\ 10^{23}$ | $\mathit{2}$
|  |  | (30, 30, | 48, 45, |  |  |  |  |  |
|  |  | 24, 24, | 91, 69, |  |  |  |  |  |
|  |  | 24, 24) | 72, 64 |  |  |  |  |  |
R3 | 0.34 | 1.67 | 42 | 64 | 1.16 $\times\ 10^{9}$ | 2.55 $\times\ 10^{5}$ | 9.71 $\times\ 10^{21}$ | 1.63 $\times\ 10^{5}$ | 1.46 $\times\ 10^{23}$ | $\mathit{1}$
R4_a | 0.43 | 1.61 | 66 |  | 2.19 $\times\ 10^{9}$ | 0.32 $\times\ 10^{5}$ | 5.60 $\times\ 10^{21}$ | 2.38 $\times\ 10^{5}$ | 5.30 $\times\ 10^{23}$ | $\mathit{2}$
|  |  | (66, 54) | 26, 26 |  |  |  |  |  |
R4_b | 0.29 | 1.73 | 30 |  | 1.41 $\times\ 10^{9}$ | (2.68–4.79) $\times\ 10^{5}$ | 9.94 $\times\ 10^{21}$ | 2.45 $\times\ 10^{5}$ | 1.00 $\times\ 10^{23}$ | $\mathit{1}$
|  |  | (18, 24) | 74, 61 |  |  |  |  |  |
R5 | 0.43 | 1.58 | 144 |  | 4.97 $\times\ 10^{9}$ | (77.2–189.7) $\times\ 10^{5}$ | 3.83 $\times\ 10^{24}$ | 2.42 $\times\ 10^{5}$ | 4.24 $\times\ 10^{24}$ | $\mathit{1}$
|  |  | (48, 66, | 164, 166, |  |  |  |  |  |
|  |  | 36, 24) | 129, 123 |  |  |  |  |  |
R6 | 0.36 | 1.73 | 48 | 65 | 1.14 $\times\ 10^{9}$ | 2.62 $\times\ 10^{5}$ | 1.28 $\times\ 10^{22}$ | 1.53 $\times\ 10^{5}$ | 1.67 $\times\ 10^{23}$ | $\mathit{1}$
R7 | 0.57 | 1.65 | 90 |  | 0.98 $\times\ 10^{9}$ | (19.3–31.7) $\times\ 10^{5}$ | 8.24 $\times\ 10^{23}$ | 1.06 $\times\ 10^{5}$ | 1.03 $\times\ 10^{24}$ | $\mathit{1}$
|  |  | (42, 48, | 157, 145, |  |  |  |  |  |
|  |  | 36) | 133 |  |  |  |  |  |
R8 | 0.49 | 1.70 | 108 |  | 1.55 $\times\ 10^{9}$ | (4.64–26.0) $\times\ 10^{5}$ | 5.62 $\times\ 10^{23}$ | 1.24 $\times\ 10^{5}$ | 1.65 $\times\ 10^{24}$ | $\mathit{3}$
|  |  | (42, 54, | 126, 106, |  |  |  |  |  |
|  |  | 48, 24) | 115, 71 |  |  |  |  |  |
R9 | 0.34 | 1.67 | 36 |  | 1.58 $\times\ 10^{9}$ | (0.57–0.62) $\times\ 10^{5}$ | 2.94 $\times\ 10^{21}$ | 2.58 $\times\ 10^{5}$ | 2.48 $\times\ 10^{23}$ | $\mathit{1}$
|  |  | (18, 36) | 35, 36 |  |  |  |  |  |
R10 | 0.35 | 1.61 | 72 |  | 2.39 $\times\ 10^{9}$ | (25.1–32.0) $\times\ 10^{5}$ | 1.87 $\times\ 10^{23}$ | 1.94 $\times\ 10^{5}$ | 6.13 $\times\ 10^{23}$ | $\mathit{1}$
|  |  | (24, 42) | 108, 117 |  |  |  |  |  |
R11 | 0.32 | 1.74 | 78 |  | 1.19 $\times\ 10^{9}$ | (0.03–0.12) $\times\ 10^{5}$ | 6.22 $\times\ 10^{20}$ | 0.88 $\times\ 10^{5}$ | 1.72 $\times\ 10^{23}$ | $\mathit{3}$
|  |  | (30, 54, | 23, 18, |  |  |  |  |  |
|  |  | 30) | 15 |  |  |  |  |  |
R12 | 0.32 | 1.78 | 30 | 98 | 1.06 $\times\ 10^{9}$ | 8.34 $\times\ 10^{5}$ | 2.01 $\times\ 10^{22}$ | 2.08 $\times\ 10^{5}$ | 9.43 $\times\ 10^{22}$ | $\mathit{1}$
R13 | 0.34 | 1.75 | 30 |  | 1.25 $\times\ 10^{9}$ | (0.57–5.34) $\times\ 10^{5}$ | 1.61 $\times\ 10^{22}$ | 2.56 $\times\ 10^{5}$ | 2.46 $\times\ 10^{23}$ | $\mathit{2}$
|  |  | (30, 30) | 80, 38 |  |  |  |  |  |
R14 | 0.26 | 1.76 | 66 |  | 4.67 $\times\ 10^{9}$ | (0.47–1.08) $\times\ 10^{5}$ | 6.10 $\times\ 10^{21}$ | 1.34 $\times\ 10^{5}$ | 1.81 $\times\ 10^{23}$ | $\mathit{2}$
|  |  | (42, 42, | 34, 31, |  |  |  |  |  |
|  |  | 36, 36) | 41, 38 |  |  |  |  |  |
R15 | 0.31 | 1.75 | 96 |  | 1.81 $\times\ 10^{9}$ | (0.10–0.14) $\times\ 10^{5}$ | 1.76 $\times\ 10^{21}$ | 1.06 $\times\ 10^{5}$ | 1.98 $\times\ 10^{23}$ | $\mathit{3}$
|  |  | (78, 60, | 19, 21, |  |  |  |  |  |
|  |  | 48) | 21 |  |  |  |  |  |
R16 | 0.33 | 1.63 | 60 |  | 2.22 $\times\ 10^{9}$ | (0.41–1.02) $\times\ 10^{5}$ | 6.79 $\times\ 10^{21}$ | 2.06 $\times\ 10^{5}$ | 4.49 $\times\ 10^{23}$ | $\mathit{2}$
|  |  | (54, 60) | 38, 28 |  |  |  |  |  |
R17 | 0.48 | 1.68 | 120 |  | 1.63 $\times\ 10^{9}$ | (0.24–23.5) $\times\ 10^{5}$ | 6.03 $\times\ 10^{23}$ | 1.13 $\times\ 10^{5}$ | 1.43 $\times\ 10^{24}$ | $\mathit{2}$
|  |  | (42, 36, | 116, 62, |  |  |  |  |  |
|  |  | 66, 84, | 120, 26, |  |  |  |  |  |
|  |  | 24, 24, | 110, 58 |  |  |  |  |  |
|  |  | 30, 30) | 82, 43 |  |  |  |  |  |
R18 | 0.36 | 1.65 | 66 |  | 1.50 $\times\ 10^{9}$ | (0.13–3.44) $\times\ 10^{5}$ | 1.58 $\times\ 10^{22}$ | 1.40 $\times\ 10^{5}$ | 2.61 $\times\ 10^{23}$ | $\mathit{2}$
|  |  | (24, 30, | 65, 45, |  |  |  |  |  |
|  |  | 24, 24) | 49, 22 |  |  |  |  |  |
R19 | 0.33 | 1.77 | 42 |  | 1.54 $\times\ 10^{9}$ | (0.46–6.82) $\times\ 10^{5}$ | 3.23 $\times\ 10^{22}$ | 2.21 $\times\ 10^{5}$ | 2.41 $\times\ 10^{23}$ | $\mathit{2}$
|  |  | (24, 24, | 81, 61, |  |  |  |  |  |
|  |  | 24, 30) | 75, 33 |  |  |  |  |  |
R20 | 0.38 | 1.90 | 102 |  | 2.61 $\times\ 10^{9}$ | (32.3–69.3) $\times\ 10^{5}$ | 4.84 $\times\ 10^{23}$ | 1.91 $\times\ 10^{5}$ | 1.16 $\times\ 10^{24}$ | $\mathit{1}$
|  |  | (42, 42) | 147, 114 |  |  |  |  |  |
R21 | 0.40 | 1.78 | 54 | 37 | 1.62 $\times\ 10^{9}$ | 0.69 $\times\ 10^{5}$ | 4.65 $\times\ 10^{21}$ | 2.21 $\times\ 10^{5}$ | 3.75 $\times\ 10^{23}$ | $\mathit{1}$
R22 | 0.27 | 1.70 | 156 |  | 2.42 $\times\ 10^{9}$ | (0.55–9.96) $\times\ 10^{5}$ | 6.15 $\times\ 10^{22}$ | 0.74 $\times\ 10^{5}$ | 5.85 $\times\ 10^{24}$ | $\mathit{3}$
|  |  | (36, 30, | 75, 58, |  |  |  |  |  |
|  |  | 18, 54 | 30, 79, |  |  |  |  |  |
|  |  | 48) | 47 |  |  |  |  |  |

222

##  4 Discussion and Conclusions

Using the high spatiotemporal resolution observation of HRIEUV $174$ Å of
SolO/EUI, 23 small-scale jets were identified in a quiet-Sun region that was
observed at 15:00:15–16:07:09 UT on 2023 March 30. Three types can be
classified based on their motion behaviors, i.e., unidirectional,
bidirectional, and hybrid jets. The third type may be caused by the changes of
magnetic field configuration during the first reconnection, which affects the
subsequent eruption. Most of these jets also showed the repeated blobs, which
could arise from the quasi-periodic reconnection that have been observed in
some solar flares (Shi et al., 2022a, b; Li et al., 2022). Combining the
simultaneous observation from SDO/AIA, their physical parameters and energies
were measured. According to the Table 1, Fig. 7(a)–(c) plot the histograms of
width, temperature, and velocity, with average value of $0.37$ Mm, $1.69$ MK,
and $70\ \mathrm{km\ s^{-1}}$, respectively. These jets typically have the
width of $\sim$$0.3$ Mm, the temperature of $\sim$$1.7$ MK, with velocity in a
wide range of $\sim$$20$–$170\ \mathrm{km\ s^{-1}}$.

Figure 7:  (a)–(c) The histograms of width, temperature, and velocity,
respectively. The average values are given. (d) Jet frequency distribution and
the fitting results in a normalized scale in units of $10^{-50}$ jets per unit
time ($\mathrm{s^{-1}}$), unit area ($\mathrm{cm^{-2}}$), and unit energy
($\mathrm{erg^{-1}}$). The power-law distributions $N(E)\propto$ $E^{-\alpha}$
(red for thermal energy, blue for kinetic energy) are also given.

Most of these jets have the energy of $10^{23}$–$10^{24}\ \mathrm{erg}$, which
is marginally smaller than the energy of typical nanoflares ($\sim$$10^{24}\
\mathrm{erg}$, e.g., Parker 1988; Chitta et al. 2021a). We also found that the
kinetic energy of small-scale jets usually one to two orders of magnitude
lower than the thermal energy, except for several relative intense jets, such
as R5, R7, and R8, which showed the ambiguous inverted Y-shaped structures
similar to standard or blowouts jets (Moore et al., 2010; Sterling et al.,
2015). As shown in Table 1, the thermal energy fluxes of 23 jets are estimated
to be (0.74–2.96) $\times 10^{5}\ \mathrm{erg\ cm^{-2}\ s^{-1}}$, which is
almost on the same order as the energy flow required to heat the quiet-Sun
corona ($3\times 10^{5}\ \mathrm{erg\ cm^{-2}\ s^{-1}}$, Withbroe & Noyes
1977), although the kinetic energy fluxes are changing over a wide range due
to their strong dependence on velocity. Additionally, based on the current 23
samples, we did not find significant discrepancies in energetics between the
different jet types.

Fig. 7(d) plots the jet frequency distribution and the fitting results in a
normalized scale in units of $10^{-50}$ jets per unit time
($\mathrm{s^{-1}}$), unit area ($\mathrm{cm^{-2}}$), and unit energy
($\mathrm{erg^{-1}}$). The thermal energy (red) and kinetic energy (blue) both
follow the power-law distribution $N(E)\propto$ $E^{-\alpha}$, with slope of
$-1.57$ and $-1.20$, respectively. Note that the distribution shown here comes
from relatively small sample size of 23 jets, thus the distribution only shows
a rough approximation. Our results are roughly consistent with the composite
flare frequency distribution as shown by Aschwanden et al. (2000), who
integrated multiple studies over the energy domain of $10^{24}$–$10^{32}$ erg
(Crosby et al., 1993; Shimizu & Tsuneta, 1997; Krucker & Benz, 1998; Parnell &
Jupp, 2000).

As shown by Withbroe & Noyes (1977), the coronal energy losses in quiet-Sun
region consist of three components: conduction flux ($2\times 10^{5}\
\mathrm{erg\ cm^{-2}\ s^{-1}}$), radiative flux ($10^{5}\ \mathrm{erg\
cm^{-2}\ s^{-1}}$), and solar wind flux ($\lesssim$$5\times 10^{4}\
\mathrm{erg\ cm^{-2}\ s^{-1}}$). The area of quiet-Sun region investigated in
this study is $136\times 136\ \mathrm{Mm^{2}}$ as seen in Fig. 1, and thus the
total energy required to sustain the heating of whole region during
15:00:15–16:07:09 UT is around $2\times 10^{29}\ \mathrm{erg}$. Summing the
total energy of the 23 events in Table 1, we can obtain a value of about
$3\times 10^{25}\ \mathrm{erg}$, which means that the 23 events only account
for a very small portion of the total energy demand during this observation.
Considering that the 23 events are only a part of all the jets, there are
still substantial jets in the image sequence of HRIEUV $174$ Å
(movie_hri174.mp4) that are difficult to be recognized because of relative
strong background emissions or transient properties, or several jets occurring
in the same region (such as R4_a and R4_b), but are not listed. It is
reasonable to infer that those jets have the similar thermal energy fluxes as
calculated here (comparable to $3\times 10^{5}\ \mathrm{erg\ cm^{-2}\
s^{-1}}$), contributing the heating of local corona. Thus, actually the
contribution of small-scale jets for coronal heating will become greater.
Besides the energy input into corona, the dissipation mechanisms of energy is
also important. The relevant dissipation mechanisms need to be investigated in
future studies. In our observations, although these jets cannot provide
sufficient energy to heat the whole quiet-Sun coronal region, they are likely
to account for a significant portion of the energy demand in the local regions
where the jets occur.

###### Acknowledgements.

We thank the referee for valuable comments. This work is supported by the
Strategic Priority Research Program of the Chinese Academy of Sciences, Grant
No. XDB0560000. This work is also funded by the National Key R&D Program of
China2022YFF0503002 (2022YFF0503000), NSFC under grants 12073081. Solar
Orbiter is a space mission of international collaboration between ESA and
NASA, operated by ESA. We appreciate the teams of SDO for their open data-use
policy.

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##  Appendix A The remaining small-scale jets

Figure 8: Similar to Figure 3, the results for R1, R2, and R3.

Figure 9: Similar to Figure 3, the results for R4_a, R4_b, and R5.

Figure 10: Similar to Figure 3, the results for R6, R7, and R8.

Figure 11: Similar to Figure 3, the results for R9, R10, and R11.

Figure 12: Similar to Figure 3, the results for R14, R16, and R17.

Figure 13: Similar to Figure 3, the results for R18, R19, and R20.

Figure 14: Similar to Figure 3, the results for R21 and R22.