11institutetext:  Institut de Recherche en Astrophysique et Planétologie,
Université de Toulouse, CNRS, UPS, CNES 14 avenue Edouard Belin, 31400
Toulouse, France 22institutetext: Charles University, Astronomical Institute,
V Holešovičkách 2, CZ-18000, Prague 8, Czech Republic 33institutetext:
Astronomical Institute of the Czech Academy of Sciences, Fričova 298, CZ-25165
Ondřejov, Czech Republic 44institutetext: Observatoire de Paris, LESIA, 5
place Janssen, 92195 Meudon, France, PSL Research University, CNRS, Sorbonne
Universités, Sorbonne Paris Cité 55institutetext: Lockheed Martin Solar and
Astrophysics Laboratory, Palo Alto, 3251 Hanover Street, CA 94303, USA

# Photospheric downflows observed with SDO/HMI,
HINODE, and an MHD simulation

T. Roudier 11 M. Švanda 2233 J.M. Malherbe 44 J. Ballot 11 D. Korda 22 Z.
Frank 55<EMAIL_ADDRESS>
(Received date / Accepted date )

Downflows on the solar surface are suspected to play a major role in the
dynamics of the convection zone, at least in its outer part. We investigate
the existence of the long-lasting downflows whose effects influence the
interior of the Sun and the outer layers. We study the sets of Dopplergrams
and magnetograms observed with Solar Dynamics Observatory and Hinode
spacecrafts and an magnetohydrodynamic (MHD) simulation. All of the aligned
sequences, which were corrected from the satellite motions and tracked with
the differential rotation, were used to detect the long-lasting downflows in
the quiet-Sun at the disc centre. To learn about the structure of the flows
below the solar surface, the time-distance local helioseismology was used. The
inspection of the 3D data cube $\left(x,y,t\right)$ of the 24-hour Doppler
sequence allowed us to detect 13 persistent downflows. Their lifetimes lie in
the range between 3.5 and 20 hours with sizes between $2\arcsec{}$ and
$3\arcsec{}$ and speeds between $-0.25$ and $-0.72$ km s-1. These persistent
downflows are always filled with the magnetic field with an amplitude of up to
600 Gauss. The helioseismic inversion allow us to describe the persistent
downflows and compare them to the other (non-persistent) downflows in the
field of view. The persistent downflows seem to penetrate much deeper and, in
the case of a well-formed vortex, the vorticity keeps its integrity to the
depth of about 5 Mm. In the MHD simulation, only sub-arcsecond downflows are
detected with no evidence of a vortex comparable in size to observations at
the surface of the Sun. The long temporal sequences from the space-borne
allows us to show the existence of long-persistent downflows together with the
magnetic field. They penetrate inside the Sun but are also connected with the
anchoring of coronal loops in the photosphere, indicating a link between
downflows and the coronal activity. A link suggests that EUV cyclones over the
quiet Sun could be an effective way to heat the corona.

###### Key Words.:

Sun: Granulation, Sun: Photosphere, Sun: Atmosphere

††offprints: Th. Roudier,

##  1 Introduction

Except magnetic structures, the whole solar surface is almost completely
renewed every 10–15 minutes because of the convective motions carrying the
energy in the solar envelope. At the solar surface convective cells of hot
rising buoyant plasma in which energy is released by radiation exchange are
observed. Around these cells, the cold plasma falls down. The cooler plasma
flow is more dense than its surroundings, which triggers the formation of
turbulent plumes (Stein R. F. and Nordlund Å. 1989; Rieutord & Zahn 1995) and
drives the dynamics of the flow. Downdrafts are sinks where the cold plasma
goes back into the Sun. These plumes undergo secondary instabilities along
their descending trajectories producing a turbulent mixture of vorticity
filaments (Rast 1999; Belkacem et al. 2006; Stein et al. 2009; Rincon &
Rieutord 2018). The vertical downflows are located in the intergranular
regions and occasionally become supersonic (Stein & Nordlund 1998). Their
works generally indicate the existence of strong long-lasting downdrafts
penetrating into the convection zone. The models of Rast (2003) predict that
supergranular downflows are concentrated near the vertices of supergranular
cells and merge at deeper layers. This results in large, long-lived downflow
regions and, in consequence, the formation of a supergranular flow system. The
kinetic energy in the convection zone is tightly connected to the evolution of
the innumerable minuscule diving plumes that formed at intergranular lanes
(Hanasoge & Sreenivasan 2014).

Various studies revealed the existence of vortex flows (Bonet et al. 2008;
Attie et al. 2009) having a wide range of spatial extents from 1 to 20 Mm with
lifetimes between 5 min and 2 h. The scale of the vortex flows seem to be
comparable to scales of supergranules and mesogranules. Some of the long-
lasting vortex flows were located at supergranular junction vertices (Attie et
al. 2009, 2016; Requerey et al. 2018). The related downflows also represent
transient processes for magnetic field intensification associated with the
formation of bright points in the continuum (Berger et al. 2004; Bello
González et al. 2009; Narayan 2011).

Figure 1: Region of interest on the 29 November 2018 datacube. The region was
localised at the disc centre; the field of view has dimensions of
$353\arcsec{}\times 353\arcsec{}$. Left: 5-hour mean of the HMI Dopplergrams,
where black areas are considered as persistent downflows. Middle: 5-hour mean
of the line-of-sight magnetic induction $\left\lvert
B_{\parallel}\right\rvert{}$. Right: 5-hour mean of AIA 193 Å filtergrams.

Recently, the Lagrangian coherent structures (LCS; see, e.g. Chian et al.
2019, 2020) detection method was applied to locate repulsion and attraction
regions as well as the shear and swirling of particle motions on the solar
surface. With such a contribution, they highlight that supergranular cells are
interconnected by ridges of repelling LCSs that facilitate the formation of
vortices and magnetic concentration in the valley of the repelling LCS. The
attractive LCS reveals the location of sinks of photospheric flows at
supergranular junctions associated with persistent vortices and intense
magnetic flux.

The downflows are important ingredients of the convective motions. Until now,
they had only been studied only in small fields of view and with a short
cadence. The only exception known to us is the study by Duvall & Birch (2010),
who studied the vertical component of supergranulation. The downflows at
mesogranular downdraft boundaries act as ‘collapsars’. Small granules vanish
and excite the upward-propagating waves (Rast 1995, 1999; Skartlien & Rast
2000; Skartlien et al. 2000).

The role of surface layers in the global convective simulations was recently
discussed by many studies. For instance Nelson et al. (2018) show that the
introduction of the near-surface small-scale downflows into the global 3D
simulation changes the convective driving motions throughout the convection
zone. In particular, the coalescence of downflow plumes into giant cells at
larger depths linked to the self-organisation of the near-surface plumes
provides a new approach towards the convection conundrum.

Today, the main difficulty seems to be the overestimation of the convective
velocity in the convection zone by forward models when forward and inverse
models are compared. On the other hand, Hotta et al. (2019) show that the
surface region has an unexpectedly weak influence on the deep convection zone
and does not resolve the problem of the high convective velocity in the deep
solar interior of the state-of-the-art forward models. Nevertheless, Hotta et
al. (2019) indicate the possibility of an unknown influence of the unresolved
small-scale turbulence.

In this paper, we investigate the properties of near-surface downflows through
the observations of the Solar Dynamics Observatory (SDO; Pesnell et al. 2012)
and Hinode (Kosugi et al. 2007) satellites and finally a 3D MHD simulation. In
Section 2, we describe the data selection and reduction. The analysis of the
persistent downflows is presented in Section 3. Section 4 is devoted to the
potential link between vortex and the persistent downflows. The non-detection
of a vortex comparable in size to observations at the surface of the Sun in
the simulation is described in Section 5. The discussion and conclusion are
given in Section 6.

Figure 2: Details of Fig. 1 around one of the persistent downflows, with a
field of view of $59.5\arcsec\times 73.5\arcsec$. Left: Mean Doppler velocity.
Middle: Mean $\left\lvert B_{\parallel}\right\rvert{}$. Right: Mean AIA 193 Å
filtergram.

Figure 3: Coalignment of panels from Fig. 2. Background: Mean AIA 193 Å
filtergram. Blue contours: Mean $\left\lvert B_{\parallel}\right\rvert{}$
level=120 Gauss. Red contours: Mean Doppler velocity level=-0.2 km s-1 Red and
blue contours largely overlap.

##  2 Observations and simulation

To study the properties of the medium-scale surface downflows, we used two
24-h SDO Doppler observations, covering 29 November 2018, and also 2–3
November 2010. Hereafter, we refer to the latter series as the ’Requerey’s
vortex’, where a strong vortical downflow was already studied by Requerey et
al. (2018). The regions of interest were around the disc centre in the quiet-
Sun regions, with a small inclination of the solar rotation axis with respect
to the observer (the heliographic latitudes of the disc centre were
$B_{0}=1.1$∘ in the first case and $B_{0}=4.25$∘ in the second case). The
original pixel size of images were 0.5040″ and 0.5042″, respectively, and a
time step of 45 seconds. Mainly, we used the observations from the
Helioseismic and Magnetic Imager (HMI; Schou et al. 2012). The 24-h sequences
were aligned, corrected from the satellite motions and the limbshift, and
finally tracked with the differential rotation. From the full-disc
Dopplergrams we extracted a region around the disc centre having $701\times
701$ pixels on a side $\left(353\arcsec{}\times 353\arcsec{}\right)$. The
tracked Doppler datacube was filtered in the $k\text{--}\omega$ domain with a
threshold phase velocity of 6 km s-1. Line-of-sight HMI magnetograms capturing
the longitudinal magnetic induction $B_{\parallel}$ were used for both
sequences and processed similarly to the Dopplergrams, except for the
application of the velocity filter. For the 2018 series, we also used
ultraviolet filtergrams from the Atmospheric Imaging Assembly (AIA; Lemen et
al. 2012) at a wavelength of 193 Å. The AIA sequence recorded at the original
pixel size of 0.6″ was mapped to the HMI frame and tracked to coaling with HMI
sequences.

We also used datasets from the Solar Optical Telescope (SOT; Tsuneta et al.
2008) on-board the Hinode mission spacecraft. The observations were recorded
continuously on 4 September 2009, from 7:34:34 to 10:16:34 UT. For our study,
we used observations of Fe I at $\lambda=5250$ Å from the Hinode/SOT-NFI
(Narrow band Filter Imager), where the spectral line was scanned at five
wavelength positions along the line profile. The Doppler shift at the disc
centre gives us the vertical (radial) flow component $v_{z}$, the Stokes $V$
and hence the vertical component of the magnetic induction $B_{z}$. The pixel
size of images is 0.16″ and the time step is 60 seconds. After the flat field
and dark corrections, the images were coaligned and filtered for $p$-modes in
the $k\text{--}\omega$ space with a threshold phase velocity of 6 km s-1.We
note that arcsecond-kilometre conversion, for each dataset considered, takes
the distance between the corresponding instrument and the Sun into account.

In order to learn about the structure of the flows also below the solar
surface, we utilised time-distance local helioseismology (Duvall et al. 1993).
This method comprises tools that measure and analyse the travel times of the
waves propagating throughout solar interior.

Travel times of the waves may be measured via cross-correlations of the
signals at spatially different points on the solar surface. Doppler shifts of
the photospheric absorption lines contain clear indications about solar
oscillations. We therefore utilised HMI Dopplergrams and measured travel times
using a set of filters, averaging geometries and distances. Since we are
dealing with the quiet-Sun region, we measured the travel times using
linearised Gizon & Birch (2004) method.

These travel times were then inverted for flows using multi-channel
subtractive optimally localised averaging method (MC-SOLA; see Jackiewicz et
al. 2012) when involving the cross-talk minimisation (Švanda et al. 2011;
Korda & Švanda 2019). When discussing the flows, thanks to the cross-talk
minimisation our methodology allows one to not only infer the horizontal
velocities ($v_{x}$ in the zonal direction and $v_{y}$ in the meridional
direction), but also the vertical component $v_{z}$. The inferred velocity
vector $\@vec{v}$ is a function of the horizontal position $\@vec{r}$ and
height $z$. Horizontal velocities are strong perturbers that may be
successfully retrieved to the depth, whereas the vertical velocity is a weak
perturber and only inversions in the first 1 Mm of depth are possible with the
signal-to-noise ratio larger than one when averaging the data over 24 hours or
so.

Therefore, except for the surface, the vertical velocity was reconstructed
from the horizontal components by integrating the equation of continuity:

$v_{z}\left(\@vec{r}_{0},z_{0}\right)=\frac{-1}{\rho\left(z_{0}\right)}\int\limits_{0}^{z_{0}}{\rm
d}z\,\@vec{\nabla}_{h}\cdot\left[\rho\left(z\right)\,\@vec{v}_{h}\left(\@vec{r}_{0},z\right)\right],$
(1)

where $\@vec{\nabla}_{h}\cdot\@vec{v}_{h}$ indicates the horizontal part of
the velocity divergence. The integration does not yield the surface
$v_{z}\left(z=0\right)$, where we directly use the inverted values by
helioseismology. The surface inversions for plasma flows were validated
against the granule-tracking inferences by Švanda et al. (2013).

For the deeper layers, we performed a set of inversions targeted such that the
vertical sampling was 2 Mm, starting at the depth of 1 Mm and ending at 25 Mm.
We basically followed the vertical sampling given by Greer et al. (2016). With
the depth, the extents of the averaging kernels defining the effective
resolution increased monotonically, reaching 10 Mm at 1 Mm depth, 34 Mm at 13
Mm depth and finally 58 Mm at 25 Mm depth. The vertical extent defining the
vertical resolution increased as well, having 1 Mm at the depth of 1 Mm, 2.9
Mm at the depth of 13 Mm and finally 8.7 Mm at the depth of 25 Mm.

Figure 4:  Top left: 24-hour divergence field measured with the LCT applied to
Dopplergrams. The supergranule boundaries detected by the watershed method are
shown as white network. The circles represent the location of the 13 studied a
persistent downflows. These persistent downflows are located in convergent
flows and more particularly where several supergranules meet. Top right:
24-hour mean Doppler map overlapped with the 13 studied persistent downflows
(circles) and supergranule boundaries. Downflows are visible as dark
structures in each circle. Far from the disc centre, in the lower right part
of the figure, some large darker regions are visible. These are caused by the
horizontal velocities which become a dominant contribution due to the
projection effects. Bottom left: Corks position (grey points) after 24-hour
diffusion by horizontal flows. Their locations are, in accordance with
supergranule boundaries, found by the watershed method (superimposed white
network). Bottom right: 24-hour mean magnetic field and the location of the 13
studied persistent downflows (circles).

Figure 5: Doppler (top) and magnetic (bottom) temporal cuts centred on the 13
selected persistent downflows. The time (vertical axis) stands for 24 hours,
the spatial axis (y cut, horizontal) is 15″ for each column and the numbers
indicate the labels of the selected downflows corresponding to the bottom
right of Figure 4. The persistent downflows move across the surface but are
visible in the cuts at least from 3.5 h to 20 h.

Figure 6: The scatter plot of the dependence of Doppler velocity and
$\left\lvert B_{\parallel}\right\rvert{}$ obtained for the SDO data on 29
November 2018.

##  3 Analysis of persistent downflows with HMI, Hinode, and simulation data

The first detection of persistent (long-lived) downflows surviving for at
least 4 hours is described in Duvall & Birch (2010) (their Figs. 1a, 1c). The
results were obtained by the analysis of the Dopplergrams and magnetograms
from SOHO/MDI (Scherrer et al. 1995) with a resolution of 1.2″. Duvall & Birch
(2010) selected a 4-hour temporal window to average Dopplergrams at the disc
centre to follow the evolution of supergranular cells. They tested a possible
contamination of the Doppler signal by magnetic fields but concluded that
there was a rather small contribution from the magnetic field. Their
simultaneous 4-hour average of Dopplergrams and magnetograms showed a
correlation between the persistent downflows and the magnetic induction. We
note that the persistent downflows were called apparent downflows in their
paper.

For our study, we selected a 24-hour sequence of disc centre quiet-Sun
Dopplergrams and magnetograms observed on 29 November 2018 with the SDO/HMI.
The spatial resolution of these observations was 1.0″. To mimic the downflow
detection like Duvall & Birch (2010), we used the 5-hour average Dopplergrams.
Figure 1 (left) convincingly shows persistent downflows as localised black
areas. The large amplitude of the negative vertical velocity indicates that
the downflow survived 24-h averaging. These downflows with sizes around 2–3″
and velocity amplitudes between $-0.25$ and $-0.72$ km s-1 spatially correlate
with the magnetic field in location (see the middle panel of Fig. 1) and also
with the regions of larger activity in the corona (images from AIA 193 Å).
Figure 2 shows a zoom-out of the larger downflows from Figure 1. Figure 3
displays the persistent downflows (red) and $\left\lvert
B_{\parallel}\right\rvert{}$ (blue) at the exact locations relative to the
coronal loop. In this example the persistent downflows appear to correlate
with the coronal-loop anchor in the photosphere, indicating a link between the
persistent downflows and the coronal activity.

The inspection of the 3D $\left(x,y,t\right)$ Dopplergram datacube from 29
November 2018 allowed us to detect 13 persistent downflows with a minimum
duration of 3 hours (see below). This gives a rate of occurrence of $2\times
10^{-4}$ cases per Mm2 and 24 h. This rate is lower than those found by
Requerey et al. (2017) during sequences of 32.0 and 22.7 minutes of $6.7\times
10^{-2}$ per Mm2. These two rates of occurrence are not directly comparable
because the first one deals with very long-lasting downflows (a few hours)
although the second rate is related to shorter-lasting downflows (30 minutes).

To detect the supergranule boundaries, horizontal velocities were computed
with the local correlation tracking technique (LCT; November L. 1986) applied
to the sequence of Dopplergrams. From the derived horizontal velocities, we
computed the 24-h average divergence field and the supergranule boundaries
using the watershed method (Roudier et al. 2020). These boundaries plotted
over the divergence field are shown as a white network in Figure 4 (top left).
In Fig. 4 (top right) the 24-hour mean Doppler map is overlapped by locations
of the 13 studied persistent downflows (circles) and supergranular boundaries.
Downflows are visible as dark structures in each circle. The persistent
downflows are identified in the convergent flows and more particularly at
positions, where several supergranules meet. This is in agreement with the
conclusion that vortex flows are usually located at supergranular junction
vertices found by Attie et al. (2009, 2016). Using the 24-hour horizontal-
velocity fields, we computed the corks diffusion during that period of time.
Fig. 4 (bottom left) confirms that the locations of the persistent downflows
(circles) at the supergranule boundaries are situated where the corks are
accumulate.

Temporal cuts of the Doppler velocities centred on the 13 persistent downflows
are shown in Fig. 5 (top). The downflows are visible as dark structures in the
cuts and they last for a long time from 3.5 h to 20 h in our examples.
However, the downflows move across the surface during their lifetimes which
makes it difficult to catch them on single positional cuts. Nevertheless, the
Doppler temporal cuts clearly exhibit their long duration. Some of them are
not fully continuous but are visible at the same position. The sizes of these
downflows are around 2–3″.

Fig. 5 (bottom) shows magnetic field temporal cuts centred on the 13
persistent downflows. The correlation between the persistent downflows and the
magnetic field is well observed and conforms to the one found by Duvall &
Birch (2010). The magnetic field is always observed at the location of the
persistent downflows. From our set, only one persistent downflow (number 6)
seems to be without a strong magnetic counterpart (see bottom right panel of
Fig. 4).

To quantify the velocity-magnetic field correlation, we plotted the mean
Doppler velocity versus $\left\lvert B_{\parallel}\right\rvert{}$ from the
24-hour sequence in Fig. 6. In the regions where $\left\lvert
B_{\parallel}\right\rvert{}>60$ G, a linear relationship between the magnetic
field and the observed Doppler velocity is most clear as described by Duvall &
Birch (2010). The strongest magnetic concentrations are found in the strongest
downflows. As we observe with high-resolution HMI data (0.5″ per pixel), the
positive correlation probably indicates a suppression of the granular flows
due to the magnetic field. Our observations thus confirm the hypothesis by
Duvall & Birch (2010).

###  3.1 Depth structure of the persistent downflows

The persistent downflows identified in the HMI frame were also localised in
the helioseismic datacubes. Unlike full-resolution Dopplergrams or velocity
fields obtained by LCT, the helioseismic inferences have a much coarser
spatial resolution, 10 Mm effective resolution at the surface at best.
Therefore these spatially confined persistent downflows cannot be directly
identified in the helioseismic frames. We verified that no clear signal of
these confined downflows is present in the travel-time maps. This is due to
the fact that the typical wavelength of the $p$-modes and the surface gravity
$f$-modes is a few megametres, which is much larger than the horizontal extent
of the identified persistent downflows.

Hence, the downflows were localised by their coordinates. In Fig. 7 we see
their locations on the vertical-flow maps derived at the surface (a
helioseismic inversion) and at the depth of 25 Mm (estimate obtained from
integrating Eq. (1)). At the surface frame the locations of the persistent
downflows do not seem to be special except that they are mostly located at the
edges of supergranules. On the other hand, at a depth of 25 Mm most of them
seem to be present at locations of the negative vertical velocity or very
close to it.

Figure 7: Locations of the 13 selected downflows (open circles) on vertical
velocity map at the surface (top) and at the depth of 25 Mm (bottom). The
flows were averaged over 24 hours on the day of 29 November 2018.

Vertical cuts through vertical velocity generally show a complicated structure
(see an example in Fig. 8). In most cases, it is not possible to track the
downflows or upflows through the datacube in a depth strictly vertically. In
some cases the vertical velocities create compact regions that are shifted
laterally in between the consecutive depths. Merging of the downflows and
separation of the upflows are visible in the cuts.

Figure 8: Example vertical cut through the 3D vertical velocity at
$y=210\arcsec{}$.

In this study, we did not focus on a general analysis of downflows, we rather
studied the 13 representatives that prevailed for a long time and were
associated with the magnetic elements. The vertical cuts through these
downflows are seen in Fig. 9. As one can see, in most cases the downflow seems
to be present deep, in eight out of 13 cases all the way to the bottom of the
domain of interest.

Figure 9: Vertical cuts through the vertical velocity around the 13 selected
downflows from 29 November 2018. Around each downflow, a vicinity of 20″ was
segmented and the cut was plotted in both zonal and meridional directions.

These 13 selected persistent downflows are not the only downflows present in
the field of view. By doing an automatic search we identified 183 downflows in
the region of interest as the local minima of the vertical velocity which, at
the same time, exhibited the negative value. From those downflows we selected
the 13 strongest to represent a comparative set to the 13 persistent
downflows. For each of the three sets, we computed the average vertical flow
profile in the centre of the downflow. These plots are given in Fig. 10. We
note that our results are only weakly influenced by the downflow changing the
position in the coordinate frame, as these positional changes are rather small
(about 5″). These positional changes are significantly smaller than the
effective resolution of the helioseismic flow maps (about 13″ at the surface,
the number increases with depth).

There is an obvious difference between the common and the persistent
downflows. The persistent downflows, on average, may have a smaller amplitude
both at the surface and in the near-surface layers, but on average the flows
remain negative all the way to the bottom of our inversion domain. Compared to
that the average over all common downflows tends to be zero at the depth of
about 8 Mm and remains negligible further down. The comparative set of the
strongest downflows has a much larger magnitude peaking at the depth of about
3 Mm on average, but it reaches zeros at the depth of about 13 Mm on average
and deeper down it even turns positive.

Figure 10: Average depth profiles of the vertical flows in downflows. The
dashed line represents the average over all 183 downflows located in the field
of view. The dot-dashed line is an average over the 13 strongest (at the
surface) downflows in the region. The solid line then shows the plot for the
13 persistent downflows.

###  3.2 Persistent downflows in 2h30min Hinode observations

Figure 11: Two persisting downflows during 2h30min of Hinode observations.
Top: Averaged Dopplergrams and $\left\lvert B_{\parallel}\right\rvert{}$
around two different persistent downflows in a field of view of
$4.4\arcsec{}\times 4.4\arcsec{}$. Middle: Cuts around the persistent
downflows in the $\left(y,t\right)$ plane of the Dopplergram and $\left\lvert
B_{\parallel}\right\rvert{}$ datacubes. The dimensions of the arrays are
($4.4\arcsec{}\times$ 2h30min). Bottom: The same in the $\left(x,t\right)$
plane.

Figure 12: Correlation between $\left\lvert B_{\parallel}\right\rvert{}$ and
the Doppler velocity of the mean data 2h30min of the mean Hinode data.

As a complementary series, we analysed the 2h 30min series of Dopplergrams
(based on Fe I at $\lambda=5250$ Å line) and the longitudinal magnetic field.
SOT/NFI on-board Hinode has a spatial resolution of 0.35″. Two examples of
persistent downflows found in the datacube sequences are displayed in Figure
11. There, in the top row, we see an average of the Doppler and $\left\lvert
B_{\parallel}\right\rvert{}$ over 2h30min. In the bottom row, the cuts in the
$\left(x,t\right)$ and $\left(y,t\right)$ planes through Dopplergram and
$\left\lvert B_{\parallel}\right\rvert{}$ datacubes are plotted at the centres
of the two persistent downflows. Here, the persistent downflows are clearly
associated with the presence of the magnetic field. The relationship between
the magnetic field and the downflows (Fig. 12) is identical to that obtained
with the SDO data. It is important to note that the amplitude of the magnetic
field is lower in the Hinode data, which is due to the smaller field of view
and a quieter region.

##  4 Link between persistent downflows and potential vortex

Figure 13: $\Gamma_{1}$ computed from 24-h mean velocity overlapped by
horizontal velocity field. Blue and red represent counter clockwise and
clockwise velocities respectively. The locations of the 13 studied persistent
downflows are indicated by circles.

Figure 14:  Top: Mean Doppler downflow number 5 from time 11.4 h to time 15.2
h (left) and from time 15.2 h to 19.7 h (right) relative to the beginning of
the sequence, overlapped with the mean horizontal velocities. Bottom:
Vorticity characterised by the $\left\lvert\Gamma_{1}\right\rvert>0.4$
parameter (downflow number 5) from time 11.4 h to 15.2 h (left) and from time
15.2 h to 19.7 h (right), overlapped with the mean horizontal velocities.
White and black represent counter-clockwise $\Gamma_{1}$ and clockwise
velocities, respectively.

Figure 15:  Left: Mean Doppler downflow number 8 between time 5 h and 10.7 h
overlapped with the mean horizontal velocities. Right: Vorticity characterised
by the $\left\lvert\Gamma_{1}\right\rvert>0.5$ parameter (downflow 8). White
and black represent counter-clockwise $\Gamma_{1}$ and clockwise velocities,
respectively.

Requerey et al. (2018) analysed a long-lived and large vortex (having a size
of about 5 Mm) using high spatial (0.2″) observations from the Hinode
satellite. In the literature, the terms ’vortex’ and ’swirl’ are both employed
to identify rotating structures in the flows. A large review of the vortex (or
swirls) detection is given in Murabito et al. (2020). Photospheric vortex
flows are usually correlated with a network of magnetic elements at
supergranular vertices (Requerey et al. 2018).

###  4.1 Vortex detection

Here, medium spatial resolution observations are used over a larger field of
view to detect a potential vortex (or swirls). From the horizontal velocity
field, we applied the vortex detection to identify ’swirls’ as described in
Giagkiozis et al. (2017); de Souze e Almeida Silva et al. (2018); Liu et al.
(2019), with the two dimensionless parameters $\Gamma_{1}\left(P\right)$ and
$\Gamma_{2}\left(P\right)$ at the target point $P$:

$\displaystyle\Gamma_{1}\left(\mathrm{P}\right)$
$\displaystyle=\frac{1}{N}\sum\limits_{M\in
S}\frac{\@vec{n}_{\mathrm{PM}}\times\@vec{v}_{\mathrm{M}}}{\left\lVert\@vec{v}_{\mathrm{M}}\right\rVert},$
(2) $\displaystyle\Gamma_{2}\left(\mathrm{P}\right)$
$\displaystyle=\frac{1}{N}\sum\limits_{M\in
S}\frac{\@vec{n}_{\mathrm{PM}}\times\left(\@vec{v}_{\mathrm{M}}-\@vec{v}_{\mathrm{P}}\right)}{\left\lVert\@vec{v}_{\mathrm{M}}-\@vec{v}_{\mathrm{P}}\right\rVert},$
(3)

where $\@vec{v}_{\mathrm{M}}$ and $\@vec{v}_{\mathrm{P}}$ are the velocity
vectors at the points $\mathrm{M}$ and $\mathrm{P}$, $S$ is the two
dimensional region with the size $N$ pixels surrounding the target point
$\mathrm{P}$, $\mathrm{M}$ is the point within the region $S$,
$n_{\mathrm{PM}}$ is the normal vector pointing from $\mathrm{P}$ to
$\mathrm{M}$, and $\times$ stands for the vector product (Liu et al. 2019).

Here for convenience, we use only the term vortex and the parameter
$\Gamma_{1}$ to detect the core of a vortex. More sophisticated vortex
detection have been recently developed (LCS approach; see de Souze e Almeida
Silva et al. 2018 or Chian et al. 2019) but our small sample allows us to
control the vortex detection by eye. For an ideal vortex,
$\Gamma_{1}\left(P\right)$ is maximum (value is equal to 1) at the centre of
the vortex and decreases towards zero outside of it. Hence, it allows us to
determine both the vortex location and the radius. Figure 13 shows the 13
persistent downflows relative to the measured $\Gamma_{1}$ parameter computed
with the 24h mean velocity field. In that figure only a few of the 13
downflows correspond to the detected vortex (here via $\Gamma_{1}>0.5$).
However, a detailed inspection reveals a vorticity, at least, in six of them
(46%) (downflows 1, 2, 4, 5, 8 and 13) during a portion of their lifetimes,
including 9.9 h–17.3 h, 0.9 h–7.1 h, 10.9 h–18.1 h, 10.7 h–19.7 h, and 10.9
h–16.3 h, respectively. Examples of downflows number 5 and 8 are given in the
figure.

Figure 14 shows the mean Doppler and $\left\lvert\Gamma_{1}\right\rvert>0.4$
parameter for the two periods of time relative to the beginning of the
sequence: 11.4 h–15.2 h and 15.4 h–19.7 h of downflow number 5. The temporal
evolution of the flows indicates, very close to the downdraft, a direction
change in the vortex rotation. Therefore, the vortex direction seems to be
very sensitive to the evolution of local large-amplitude flows and can be
reversed as in our example. Another vortex is presented in Figure 15, where we
observe a clear link between $\left\lvert\Gamma_{1}\right\rvert>0.23$ and
persistent downflow number 8. One important point is that we never detect a
vortex before the appearance of the magnetic field in the downflow, when it is
possible to observe that in a few cases. We do not know if this is due to the
1″ of spatial resolution and 45s temporal step.

###  4.2 Requerey’s vortex and long-lived downflows

Figure 16: $\Gamma_{1}$ ’vortex’ (in circles) overlapped by horizontal
velocity field.

Figure 17: Mean Dopplergram overlapped by horizontal velocity field and the
location of the $\Gamma_{1}$ ’vortex’ (in circles).

Figure 18: Requerey’s vortex in a large field of view of the module of the
mean horizontal velocities (24-h mean)

Figure 19: Persisting downflows during 24 h. $\left(y,t\right)$ cuts through
the Dopplergram (left) and $B_{\parallel}$ (right). Requerey’s vortex
corresponds to the long-lasting downflow and $B_{\parallel}$ is located around
$y\approx 120\arcsec{}$.

To validate our vortex detection via the parameters $\Gamma_{1}$ and
$\Gamma_{2}$, we used the SDO observation used in the analysis of Requerey et
al. (2018); de Souze e Almeida Silva et al. (2018); Chian et al. (2019).
Luckily, SDO observations started just before Requerey’s observations were
made with the Hinode satellite. We remind the readers that Requerey et al.
(2018); de Souze e Almeida Silva et al. (2018); Chian et al. (2019) observed
and analysed a nice long-lasting vortex. The advantage of the SDO observations
is the access to various kinds of context observations, on the same day, to a
larger field of view. Despite the spatial resolution of SDO being lower than
that of Hinode, we detected the vortex described in Requerey et al. (2018)
without any problem. The locations of $\Gamma_{1}$, vortex detection
overlapped by a horizontal velocity field are plotted in Fig. 16. The
Requerey’s vortex is well detected by the gamma-method and is clearly visible
in the plotted horizontal velocities (at the coordinates
$\left(62\arcsec\times 49\arcsec\right)$. Figure 17 shows the mean Doppler
overlapped by horizontal velocity field and the location of the $\Gamma_{1}$
corresponding to downflows around the supergranule.

In our observations, five larger-amplitude regions in the $\Gamma_{1}$
parameter are seen in total (see Fig. 16 and 17 ) and only one of them
corresponds to the vortex studied by Requerey. The others have vorticity but
also a non-negligible shear component which contributes to the final amplitude
of $\Gamma_{1}$ and hence does not allow for one to call them a vortex.
However, Chian et al. (2019, 2020) with higher spatial resolution, reported on
the same field of view, more persistent objective vortices with shorter
lifetimes corresponding to the gap regions of ’Lagrangian chaotic saddles’.

In the larger field of view, the horizontal velocity module (Figure 18) shows,
in the central part, the roundish supergranule corresponding to the
supergranule shown in Figure 16, where Requerey’s vortex is observed. This
supergranule is a particularly symmetrical, well-formed supergranule with
large amplitude horizontal outflows. The large horizontal velocity amplitude
of that supergranule and the combination of the velocities from the large
supergranules on its right location on the eastern edge, produce the large
Requerey’s vortex. The temporal and spatial coincidence is probably the reason
for producing this large and long-lasting vortex. The other large amplitude
downflows around this supergranule do not show identical properties.

Figure 20: Depth profile of the 24h averaged vertical velocity in the middle
of Requerey’s vortex.

Figure 21: Depth profile of the vertical component of the flow field vorticity
in the middle of Requerey’s vortex.

Helioseismic inversion allows us to study the general properties of the large
Requerey’s vortex in depth. In the map of the inverted component, nothing
extraordinary is seen at the location of this vortex, likely because the
spatial resolution of the flow maps is again coarser than the extent of the
vortex. The plot of the vertical profile of the vertical velocity, on the
other hand, clearly shows that the downflow extends deep into depth of 20 Mm
and more (see Fig. 20). The plotted profile is similar to the profile of the
strongest downflows recorded in the studied field of view on 29 November 2018
(compare to the dash-dotted line in Fig. 10).

In the maps of the vertical vorticity computed from the horizontal flow
components as

$\Omega_{z}=\frac{\partial v_{x}}{\partial y}-\frac{\partial v_{y}}{\partial
x},$ (4)

Requerey’s vortex is clearly visible. The depth profile of $\Omega_{z}$ shows
that the vorticity keeps its integrity to the depth of about 5 Mm (see Fig.
21), where it seems to start to oscillate around zero and vanishes at the
depth of about 11 Mm. We do not think that the peak at the depth of 9 Mm that
is visible in both vertical velocity and vertical vorticity profiles is
particularly real. The formal uncertainty of the vertical velocity at this
depth is about 10 m s-1 and that of the vertical velocity vorticity about
$10^{-5}$ s-1, which corresponds to the magnitude of these ’peaks’. The
comparison of Requerey’s vortex and the downflows in the field of view on 29
November 2018 indicates that the long-living vortex is linked to strong
downflows.

##  5 Simulation link between magnetic field and persistent downflows

We used two different datasets based on the 3D MHD numerical simulation. These
runs apply to the purely quiet Sun (Stein & Nordlund 1998).

The first dataset provides the velocity vector $\@vec{v}\left(x,y,t\right)$
and the magnetic field vector $\@vec{B}\left(x,y,t\right)$ over the solar
surface $\left(z=0\right)$ for 27 hours (spatial resolution: 96 km/pixel;
temporal resolution: 60 s; field of view (FOV): $96\leavevmode\nobreak\
\mathrm{Mm}\times 96\leavevmode\nobreak\ \mathrm{Mm}$). The corresponding
resolution is 0.13″ on the Sun, as the computational pixel size is 48 km,
which is much better than observations. We used the $v_{x}$, $v_{y}$, $v_{z}$
as well as $B_{z}$ quantities. The $z$ axis is positive above the surface. The
second dataset, which is much shorter, gives only the vertical velocity
$v_{z}\left(x,y,z,t\right)$ during 4 hours from the surface $\left(z=0\right)$
to $z=-20$ Mm, with the same horizontal FOV and time step as the first
dataset.

In order to eliminate 5-minute oscillations, the vertical velocity field
$v_{z}$ was filtered in the $k\text{--}\omega$ diagram low frequencies such
that $\omega$ ¡ $c$ $k$ were kept and other frequencies were eliminated ($c=6$
km s-1). Then, we averaged velocities and magnetic fields over time intervals
12 and 24 hours for the first sequence close to the supergranulation lifetime
and 4 hours for the second sequence in order to keep only long-lasting
features.

Figure 22: $B_{z}$, $\@vec{\nabla}_{h}\cdot\@vec{v}_{h}$, $\@vec{\nabla}v_{z}$
at the surface (FOV $96\times 96$ Mm2). Long-lasting downward velocities are
displayed as red points using the threshold of $-0.25$ km s-1. Left: $B_{z}$
(grey levels), averaged over 12 hours. Middle:
$\@vec{\nabla}_{h}\cdot\@vec{v}_{h}$ averaged over 12 hours. Bright and dark
correspond to diverging and converging flows, respectively. Right:
$\mathrm{d}v_{z}/\mathrm{d}z$ averaged over 4 hours. Bright and dark
correspond to upward and downward vertical gradient, respectively.

Figure 23: Cross sections of the vertical velocity $v_{z}$ (in $x\times z$ and
$y\times z$ planes) at pixel $\left(x=171\leavevmode\nobreak\
\mathrm{Mm},y=452\leavevmode\nobreak\ \mathrm{Mm}\right)$ indicated by the
vertical lines, where a typical downflow occurs. The surface is at $z=0$ Mm.
Bright and dark correspond to upflows and downflows, respectively.

Figure 22 displays 12-hour averages (first dataset) of the vertical magnetic
field $B_{z}$ and the divergence of the horizontal velocities
$v_{h}=\left(v_{x},v_{y}\right)$. The long-duration downflows (red points) are
spatially much more concentrated and also much more numerous than in
observations. They occur almost everywhere, even in non-magnetic areas. Such
points are systematically characterised by
$\@vec{\nabla}_{h}\cdot\@vec{v}_{h}<0$ (converging flows) with the mean value
of $-1.8\times 10^{-3}$ s-1. The second dataset shows that these points are
also characterised by $\mathrm{d}v_{z}/\mathrm{d}z>0$ at z = 0, which means
that the amplitude of $v_{z}$ decreases with height in these points at the
surface. The mean value is $1.9\times 10^{-3}$ s-1, so that we are close to
$\@vec{\nabla}\cdot\@vec{v}=0$ at the surface. Therefore, the fluid exhibits
an uncompressible behaviour in this layer.

Figure 23 shows two typical cross sections of the vertical flows based on the
second dataset, averaged over 4 hours. Some downflows extend from the surface
down to $z=-13$ Mm. However, most downflows are not deeper than $z=-4$ Mm. We
also point out that the continuous downflow regions do not extend purely
vertically to a certain depth, but they depict a lateral displacement between
the depths. This behaviour is very similar to what we found in the
helioseismic inversions (cf. Fig. 8).

We conclude that the numerical simulation predicts the existence of spatially
concentrated lasting downflows in the quiet Sun with a typical size of about
0.25″. However, the spatial resolution of observations used in this paper does
not allow us to confirm simulation results. In the simulations, we did not
find evidence of a vortex comparable in size to the observed vortex on the
surface of the Sun.

##  6 Results and conclusion

The properties of solar plasma below the surface is mostly inferred from the
observations of solar surface oscillations. The formation of vigorous
downflowing plumes and the collective interaction of these (R03a; Cossette &
Rast 2016) also provides the opportunity, combined with local helioseismology,
to probe the first megameters of the solar interior. Lot of observations of
vortices (vortex) associated with downflows are described in the literature
(Attie et al. 2009; Duvall & Birch 2010; Bonet et al. 2010; Vargas Domínguez
et al. 2011, 2015; Attie et al. 2016; Requerey et al. 2017, 2018), generally
studied on short sequences. Three of them are observed on long temporal
sequences from 4 h to 24 h (Attie et al. 2009; Duvall & Birch 2010; Attie et
al. 2016; Requerey et al. 2018). Taking advantage of the long temporal and
homogeneous SDO/HMI observations (24 h here) of the Dopplergrams and
magnetograms, we investigate the properties of the downflows, particularly the
long-lasting one called persistent downflows here, and their links to a
concentrated magnetic field and finally the link with the corona observations
(AIA 193 Å).

In the 3D data cube $\left(x,y,t\right)$ of the 24-hour Doppler sequence (29
November 2018), at the disc centre, we detect 13 persistent downflows giving a
rate of occurrence of $2\times 10^{-4}$ cases per Mm2 and 24 h. This rate is
lower than those found by Requerey et al. (2017) during sequences of 32.0 and
22.7 minutes of $6.7\times 10^{-2}$ per Mm2 which was found with higher
spatial resolution but on shorter time sequences. The lifetimes of our 13
persistent downflows were found between 3.5 h to and 20 h with a sizes between
2 and 3″ and speeds between $-0.25$ and $-0.72$ km s-1. They are well located
at the junction of several supergranules as observed by Attie et al. (2009,
2016); Chian et al. (2019, 2020). In our examples, only 46% of the persistent
downflows were connected with an observed vortex, but not all the time and not
for all the studied downflows. At higher spatial resolution (Chian et al.
2019, 2020), more persistent objective vortices were found with shorter
lifetimes corresponding to the gap regions of ’Lagrangian chaotic saddles’.

All the studied persistent downflows are always associated with a magnetic
field (up to 600 Gauss). No significant downflows were observed before the
appearance of the magnetic field as described in Vargas Domínguez et al.
(2015). The advantage of their higher spatial resolution (0.15″) probably
allows them to detect downflows before the magnetic field appearance and
follow the enhancement of this magnetic field by the strong downflows. The
detection of the vortex described by Requerey et al. (2018) with our SDO/HMI
data gives confidence in our method of vortex detection. This very beautiful
vortex, with our large field of view, seems to be the result of the
composition of the larger-scale horizontal flows where the combination of the
flow phases, temporal and spatial, creates it.

We observe that this downflow is always associated with the presence of the
magnetic field. The helioseismic inversion allows us to describe the
persistent downflow properties in depth below the solar surface and, for
instance, to compare the properties of the persistent downflows to the other
(non-persistent) in the field of view. Persistent downflows seem to penetrate
deeper, whereas the common downflows seem to reach zero vertical velocity
already at the depth of around 7 Mm. The persistent downflows remain slightly
negative on average all the way to the bottom of our region of interest, which
is 25 Mm.

In the high spatial resolution MHD simulation (0.13 ″), long-duration
downflows (4 h and 12 h) are spatially much smaller and numerous than in
observations where the resolution lies around 2 to 3″. We did not find
evidence of a vortex comparable in size to observations. The spatial
resolution of observations does not allow us to see small scale downflows, so
that observations with more precise satellites (as the PHI on-board Solar
Orbiter) or ground-based large telescopes are required.

From the space-borne SDO instruments, downflow evolution is studied at
different wavelengths. We observe a persistent downflow to be correlated with
the coronal-loop anchor in the photosphere, indicating a link between
downflows and the coronal activity. This observation is to be compared with
the observed EUV cyclones over the quiet Sun, anchored in the rotating
magnetic field network magnetic fields, suggesting an effective way to heat
the corona Zhang & Liu (2011). The link between the flow evolution at the
junctions of supergranular cells, giving mini-coronal-mass ejection and X-ray
network flares, is described in Innes et al. (2009); Attie et al. (2016) .
Hence, the persistent downflows must be investigated with a greater number of
time sequences.In the same way, more studies have to be performed on the link
between persistent downflows and the coronal phenomena to confirm the link
between them and describe the temporal dynamic behaviour of such structures.

The new non-linear methods (Lagrangian Coherent Structure and Lagrangian
Chaotic Saddles) developed recently by Chian et al. (2019, 2020), applied to
the full Sun SDO/HMI observations, is probably the best way for future
research in this domain to access to large-events statistics.

###### Acknowledgements.

This work was granted access to the HPC resources of CALMIP under the
allocation 2011-[P1115]. Thanks to SDO/HMI, IRIS, and Hinode/SOT teams. MŠ and
DK were supported by the Czech Science Foundation under the grant project
18-06319S.

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