Open Access
Issue
A&A
Volume 661, May 2022
Article Number A94
Number of page(s) 12
Section The Sun and the Heliosphere
DOI https://doi.org/10.1051/0004-6361/202243111
Published online 06 May 2022

© Y. Chen et al. 2022

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Open Access funding provided by Max Planck Society.

1. Introduction

The Solar transition region is the interface region between the chromosphere and corona, where the temperature rises from ∼104 to ∼106 K within ∼100 kilometers, at least in a simple 1D picture (e.g., Mariska 1992). The dynamics of the transition region plays a key role in understanding the coronal heating problem and the acceleration of the solar wind.

It has been well known since the 1970 s that the emission lines formed in the lower transition region, such as the C IV 1548 Å line, exhibit prevailing redshifts in the quiet Sun (e.g., Doschek et al. 1976; Dere et al. 1989). Using observations of hundreds of spectral lines with a wide range of formation temperatures taken by Solar Ultraviolet Measurements of Emitted Radiation (SUMER, Wilhelm et al. 1995) onboard the Solar and Heliospheric Observatory (SOHO), variations of the Doppler velocities with formation temperatures can be investigated. It was found that the average redshifts increase with the formation temperatures and peak around 105.2 K, and then the redshifts decrease with the formation temperatures (e.g., Brekke et al. 1997; Chae et al. 1998). In the corona, for example in the Ne VIII 770 Å line, with a formation temperature of ∼105.8 K, blueshifts prevail (e.g., Peter 1999; Peter & Judge 1999; Teriaca et al. 1999; Tian et al. 2010). While blueshifts in the quiet Sun corona can be related to true outflows into the solar wind, for instance from coronal holes (e.g., Hassler et al. 1999; Xia et al. 2003) or the chromospheric network (e.g., Tian et al. 2021), most of the blueshifts in the quiet Sun corona are related to magnetically closed regions – simply because these blueshifts are everywhere (Peter 1999; Tian et al. 2009). Detailed studies also found that the redshifts of lines formed in the lower transition region are larger in the bright network lanes, suggesting a positive correlation between the intensity and Doppler velocity (Brynildsen et al. 1998; Curdt et al. 2008; Tian et al. 2008; Wang et al. 2013).

Many scenarios are proposed to explain the dominant redshifts in the lower transition region, for example siphon flows (e.g., Mariska & Boris 1983), downward propagating waves induced by nanoflares (Hansteen 1993), and transient heating around the loop footpoints (Spadaro et al. 2006). Tu et al. (2005) and He et al. (2008) suggested that the redshifts and blueshifts in the coronal holes are associated with bidirectional flows generated by magnetic reconnection between open coronal funnels and the neighboring closed loops. This is also supported by 2D models (Yang et al. 2013). A similar scenario of continuous reconnection at the boundaries of network lanes may cause the redshifts and blueshifts in the quiet-Sun regions (Aiouaz 2008).

Another popular interpretation for the transition region redshifts is the return of previously heated spicules which was first suggested by Pneuman & Kopp (1978). The transition region images taken by the Interface Region Imaging Spectrograph (IRIS, De Pontieu et al. 2014) exhibit prevalent network jets (Tian et al. 2014), some of which appear to be the heating signatures of chromospheric spicules (Pereira et al. 2014; Rouppe van der Voort et al. 2015). Many studies also suggested that spicules can be heated to coronal temperatures (e.g., De Pontieu et al. 2011; Martínez-Sykora et al. 2017; Samanta et al. 2019). Considering the mass cycle in the solar atmosphere, the transition region spectral line profiles are the superposition of rapid heated upflows generated in the chromosphere, slow cooling downflows of the previously heated plasma, and a steady background (Wang et al. 2013). The varying contributions of these components at different temperatures may cause the change of Doppler velocities for different spectral lines.

To understand the temperature dependence of Doppler shifts, Peter et al. (2004, 2006) synthesized spectral line profiles of several transition region and coronal emission lines from a 3D magnetohydrodynamic (MHD) model of a (scaled-down) active region, in which the redshifts of the transition region lines are essentially caused by asymmetric heating along field lines which results in upflows and downflows occurring at different temperatures and densities. However, their model failed to reproduce blueshifts of the spectral lines formed in the upper transition region and corona, such as the Ne VIII 770 Å and Mg X 625 Å lines. Through the analysis of magnetic field topology in a 3D MHD model, Zacharias et al. (2009) suggested that the redshifts of the transition region lines are caused by cooling plasma draining from the reconnection site. Hansteen et al. (2010) constructed a series of 3D MHD models for the magnetic network extending from the upper convection zone to the corona, and their models present redshifts and blueshifts in the transition region and low coronal lines, respectively. In their models, the rapid and episodic heating events result in a high-pressure plug of plasma at the upper transition region, because there the heating per particle is largest. The expansion of the plasma produces the redshifts and blueshifts in the transition region and corona.

While the above models explained several of the observed properties of the Doppler shifts in the quiet Sun transition region and corona, none of the models are consistent with all aspects (cf. Sect. 2). In contrast to previous studies, we use a 3D MHD model in which the magnetic field in the quiet Sun is produced self-consistently through a small-scale dynamo that leads to a pattern of the surface magnetic field that is similar to the supergranular network. In our study, we synthesize spectral profiles of several emission lines from the MHD model and then investigate the resulting Doppler shifts. Our model can reproduce average redshifts and blueshifts in the transition region and lower corona similar to observations, but, unfortunately, the spatial distribution of the Doppler shifts of the Si IV line in our model still deviates from observations.

2. Quiet Sun observations

2.1. Spectral raster map with IRIS

To put the results of our quiet Sun simulation into the context of observations, we briefly present a quiet Sun raster map of a transition region line. For this we use a large dense raster acquired by IRIS in the Si IV line at 1394 Å which, under ionization equilibrium conditions, forms just below 0.1 MK. The data were acquired during the early phase of the IRIS mission from 13 Oct. 2013 at 23:27 UT to 14 Oct. 2014 at 02:59 UT. Because of telemetry problems during the first hour, the data we show in Fig. 1 start only at 00:21 UT and cover a field of view of ca. 104″ by 174″ centered around solar (X,Y) = (−120″, −48″).

thumbnail Fig. 1.

Observation of the line intensity and Doppler shift of Si IV 1394 Å in the quiet Sun. The data were taken with IRIS with a field of view of 104″ × 174″ centered around solar (X,Y) = ( − 120″, −48″), i.e., close to disk center. The intensity map is displayed on a logarithmic scale. Gray regions in the Doppler map mark areas where the signal is too low to perform a reliable Gaussian fit (below ca. 15% of the mean intensity or 80 DN/line for this data set). See Sect. 2 for more details.

The data were taken with a raster step being roughly equal to the slit width of ca. 0.35″ and a plate scale along the slit of ca. 0.17″ pixel−1. The spectral plate scale is ca. 0.013 Å pixel−1. The spectra were taken with an exposure time of 30 s. The details of the instrument can be found in De Pontieu et al. (2014). Here we use level 2 data1. We fine-tuned the absolute wavelength calibration by assuming zero average Doppler shifts of the chromospheric lines of Fe II at 1392.82 Å and Ni II at 1393.33 Å that are close to the Si IV line (see also Peter et al. 2014).

2.2. Observational properties of quiet Sun Doppler shifts

In this study, we use the Doppler shifts seen in the upper atmosphere as a test for the numerical model. For the comparison with the observations, we performed single-Gaussian fits to the observed profiles of the Si IV. The resulting Doppler shift with respect to the rest wavelength of the line is shown in Fig. 1 together with the intensity of Si IV integrated across the whole line. In regions of low count rates in the internetwork, the Gaussian fits are unreliable, and the Doppler map is very noisy. Therefore we masked out these low-intensity regions in the Doppler map.

On average, the Si IV line is redshifted by 8 km/s. This net average redshift of transition region lines has been known since the work of Doschek et al. (1976). Also, only a small fraction of the area of about 10% shows blueshifts as first reported by Dere et al. (1989). The histogram of the Doppler shifts in the map shown in Fig. 1 is close to a Gaussian with a full width at half maximum of ca. 15 km s−1 (as is discussed later in Sect. 3.3 and Fig. 4a). This is consistent with data from earlier instruments (e.g., Peter 1999; De Pontieu et al. 2015). In addition, the Si IV line profiles often exhibit blue wing enhancement in the network regions, and the blue wing enhancement may be associated with intermittent high-speed upflows or spicules (e.g., De Pontieu et al. 2009; McIntosh & De Pontieu 2009; Tian et al. 2014; Chen et al. 2019).

In terms of the spatial structure, the Doppler map in Fig. 1 seems to show a structure similar to tufts of grass. This is reminiscent of the spatial structure of spicules as already reviewed by Beckers (1968) or more recently detailed by Samanta et al. (2019).

After a first indication for a net Doppler shift of coronal lines toward the blue by Sandlin et al. (1977), this was firmly established through the center to-limb variation of the line shift by Peter (1999). The turn from transition region redshifts to coronal blueshifts happens around 0.5 MK (Peter & Judge 1999; Xia et al. 2004; Tian et al. 2021).

In conclusion, a good model for the upper atmosphere in the quiet Sun would have to reproduce and explain (at least) the following properties of quiet Sun Doppler shifts: transition region lines show an average net redshift; transition region lines show redshifts almost exclusively, and only ca. 10% of the quiet Sun is covered by blueshifts; Doppler maps show patterns reminiscent of nests of spicules; the net Doppler shifts change from red in the transition region to blue in emission lines from coronal temperatures. So far, models have addressed aspects of these properties, but they have not given a complete and comprehensive explanation. This is discussed in more detail in Sect. 4.

3. Numerical quiet Sun model including the corona

3.1. Radiation MHD simulation

A 3D radiation MHD simulation was performed using the coronal extension version of the MURaM code (Vögler et al. 2005; Rempel 2017) following the setup in Chen et al. (2021). The model spans the region from ∼20 Mm below to ∼17.5 Mm above the photosphere with a grid spacing of 25 km in the vertical direction. In other words, our model covers the regions from the upper convection zone to the lower corona. The model contains a region of 50 × 50 Mm2 in the horizontal direction with a grid spacing of ∼48.8 km, allowing several supergranular cells to appear.

To set up for the coronal model, we first ran the simulation with an upper boundary at the temperature minimum (1 Mm). The simulation was run for 53 hours with no magnetic field to allow convection to develop. A small vertical seed field was then added, with zero net flux and a RMS field strength of only 10−3 G. The simulation was then run for another 58 hours to allow the field to saturate. We then extended the computational domain into the corona. A potential field extrapolated from the top of the previous domain was used as the initial condition. We extended the domain in a two-step process (first up to 6 Mm, to allow a transition region to form, and then up to 17.5 Mm above the surface). Because of the faster timescales in the corona (the Alfvén crossing time is only a few minutes), it took only 60 and 20 additional minutes for the two steps until the corona settled. At the end, the magnetic field was evolving self-consistently in interplay with the plasma all the way from 20 Mm below the surface in the convection zone up to 17.5 Mm above the surface in the corona. Once the system had settled, we collected 61 snapshots at a cadence of half a second, that is for a total duration of half a minute, for the analysis.

Radiation transfer in the photosphere and chromosphere of the simulation is gray and treated in local thermodynamic equilibrium (LTE). Above the transition region, optically thin losses were used as described in Rempel (2017). A pretabulated equation of state (EoS) was used, and the equation of state was made by smoothly merging two tables as described in Rempel (2017). The OPAL EoS (Rogers et al. 1996) was used in the convection zone (ρ > 10−6), and an equation of state based on the Uppsala Opacity Package (Gustafsson et al. 1975) was used at the photosphere and in the atmosphere. The EoS was used in LTE. A time-dependent treatment of hydrogen ionization (Leenaarts et al. 2007) and nonequilibrium helium ionization (Golding et al. 2014, 2016) were not included.

The simulation gives a model of the quiet Sun with the magnetic field generated by the small-scale dynamo. The super-granulation self-consistently results from the convective dynamics of the simulation. The size of the resulting super-granular cells is constrained by the limited depth of the simulation and the extent in the (periodic) horizontal directions. The magnetic field was generated from a small seed field by a small-scale dynamo mechanism in the convection zone, and once it reached a steady-state, the field was extrapolated into the corona. The horizontally averaged, unsigned vertical magnetic field is 59.48 G at the photosphere. The small-scale dynamo is not a local mechanism (e.g., Vögler & Schüssler 2007; Nordlund et al. 2009; Abbett & Fisher 2012; Martínez-Sykora et al. 2019), but it acts throughout the convection zone (Rempel 2014; Hotta & Kusano 2021). Rempel (2014) shows that in order to match the inferred photospheric magnetic field strengths (e.g., Trujillo Bueno et al. 2004; Danilovic et al. 2010, 2016; Shchukina & Trujillo Bueno 2011), recirculation deeper in the convection zone is required. We used a boundary condition which allows a horizontal field at roughly equipartition field strengths to emerge into the domain (OSb of Rempel 2014). Due to the limited extent of the simulated convection zone and the lack of global-dynamo generated magnetic fields, the resulting simulation represents a lower limit of solar activity. Consequently, this model is well suited to describe the processes in the quiet Sun.

3.2. Emission and spectra synthesized from the model

In this study, we focus on nine emission lines listed in Table 1. These lines have been abundantly observed by past and present EUV spectrometers including SUMER, the Coronal Diagnostic Spectrometer (CDS, Harrison et al. 1995), the EUV imaging spectrometer (EIS, Culhane et al. 2007), IRIS, and the Spectral Imaging of the Coronal Environment (SPICE, SPICE Consortium 2020). The formation temperatures of these lines range from ∼104.9 to ∼106.2 K, covering the temperatures of the transition region and corona and providing good spacing of the temperature in logarithmic scale.

Table 1.

Emission lines used in this study.

We synthesized these emission lines and their spectral profiles following the procedure described in Peter et al. (2006) based on the CHIANTI atomic database (version 10.0; Dere et al. 1997; Del Zanna et al. 2021). The abundance of Si, Fe, and other elements are taken from Scott et al. (2015b), Scott et al. (2015a), and Asplund et al. (2009), respectively. Furthermore, we chose the line of sight (LOS) as the vertical direction, that is to say the model mimics an observation at the center of the solar disk. At each grid point, the emissivity ε of each line can be written as follows:

(1)

where ne, T, and G(ne, T) are the electron density, temperature, and the contribution function of the line given by CHIANTI, respectively. We assume that the line profile at each grid point is a Gaussian with a width given by the thermal width,

(2)

where m is the mass of the ion. Thus the spectral line profile with the wavelength written in units of Doppler shifts reads as

(3)

where v0 is the component of velocity along the LOS, that is to say in the vertical direction.

Finally we integrated the line profiles along the LOS for each column which provides a spatial map of spectral profiles of the same format as acquired by spectroscopic observations. From this we derived the intensity maps by integrating over the line and the Doppler shifts by taking the first moment of the profile, which is equivalent to the position of a fitted Gaussian profile in the case of a symmetric profile.

3.3. Doppler shift of the Si IV line in the model

Many spectral observations of the Si IV line in the quiet-Sun regions with IRIS have an exposure time of 30 s in order to achieve a sufficient signal-to-noise ratio. Thus we synthesized the Si IV spectra for the whole time sequence (with the model having a write-out cadence of 0.5 s) and then summed them up in time for each pixel to mimic a real observation with an exposure time of half a minute. The original model data and also the synthesized observable have a grid spacing of just below 50 km. To better compare them with the observations, we degraded the intensity and Doppler shift images to the spatial resolution of IRIS and SUMER. Firstly, we convolved the spectral maps at each wavelength position of the Si IV line with a 2D Gaussian function and rebinned the spatial maps to the pixel scale of IRIS and SUMER. As the pixel size of the IRIS observation is 0.33″ and 0.17″ perpendicular and parallel to the slit, respectively, we chose a full width at half maximum (FWHM) of 480 km in the x-direction and 240 km in the y-direction for the kernel. The typical SUMER observations have a pixel size of ∼1″, so we chose an FWHM of 1.45 Mm for the kernel. Secondly, the Si IV resulting line profiles of the spectral maps at reduced resolution were fitted with single Gaussian functions to calculate their Doppler velocities. The intensity maps were derived by integrating in wavelength over the line profile. The results for the original full resolution of the model and for a resolution comparable to IRIS and to SUMER are shown in Figs. 2 and 3 for both one single snapshot of the model and integrated over 30 s.

thumbnail Fig. 2.

Intensity maps of the Si IV line at different spatial and temporal resolution. Intensity images of the Si IV line shown at (a) original resolution of the simulation, (b) IRIS resolution, and (c) SUMER resolution after integration over 30 s. (d)–(f) The same as (a)–(c), but for the first snapshot. The images are shown in a logarithmic scale normalized to the median value. The dynamic ranges of the intensity at full, IRIS, and SUMER resolution are 1000, 500, and 50, respectively. See Sect. 3.3 for more details.

thumbnail Fig. 3.

Doppler maps in Si IV and underlying magnetic field. Panels a to f: are similar to Fig. 2, but for Dopplergrams of the Si IV line saturated at ±50 km s−1. Panel g: vertical magnetic field in the photosphere. See Sect. 3.3 for more details.

It is evident that both the intensity maps and Dopplergrams do not change much after the integration over 30 s. In other words, the single snapshot is already a good representation of the observation with an exposure time of 30 s. Essentially, this shows that the dynamics on timescales of less than 30 s do not play a significant role in the appearance of the intensity and Doppler maps, even though (small) changes can be seen on those short timescales (see Figs. 2 and 3).

The network patterns are present in all the intensity maps of the Si IV line in Fig. 2, irrespective of the resolution, and they are related to magnetic concentrations in the photosphere as shown in Fig. 3g. Furthermore, the Dopplergrams of the Si IV line are dominated by redshifts at different spatial resolutions, in particular at the coarsest resolution of SUMER, which is similar to the observations.

To further investigate the distribution of the Doppler shift of the Si IV line, we derived histograms and cumulative histograms. For this we calculated the frequency and cumulative frequency in bins of equal Doppler shifts. Because of the similarity of the single snapshot and the time-integrated maps, here, we show results only for the time-integrated maps. For these we display the original, IRIS, and SUMER resolutions of our synthesized observable, and the IRIS observation in Fig. 4. In our model, more than half of the Si IV line profiles show redshifts at different resolutions, and the proportion of the redshifts increases when the spatial resolution gets worse. Nevertheless, more than 20% of profiles show blueshifts in our model even at the SUMER resolution, while only ∼10% of profiles show blueshifts in the IRIS observation. The analyses of a single snapshot led to the same results. Thus, our model shows too many blueshifts of the Si IV line compared to the observations, but it still shows an areal fraction of redshifts that is significantly larger than one half.

thumbnail Fig. 4.

Histograms of the Doppler shifts of the Si IV line. The frequency (a) and cumulative frequency (b) of the Doppler shifts of the Si IV line after integration over 30 s as shown in Fig. 3a–c. The solid black, red, and blue lines indicate the distribution of the Si IV line at the original, IRIS, and SUMER resolution, respectively. The red dashed line indicates the distribution of the Si IV line calculated from the observation shown in Fig. 1. The vertical dotted line shows the zero shift. See Sect. 3.3 for more details.

3.4. Doppler shift of other lines in the model

We also calculated the intensity maps and Dopplergrams for other emission lines listed in Table 1. As the integration over 30 s does not significantly change the intensity map and Dopplergram of the Si IV line, we just calculated the intensities and Doppler velocities of a single snapshot for the other emission lines. We present the results of the C IV, O VI, Ne VII, Ne VIII, and Fe XII lines in Fig. 5 because these lines provide good coverage in temperature. Small-scale scattered structures are gradually replaced by large-scale diffuse ones in the intensity maps as the formation temperature of the line increases. There are many small-scale structures in the C IV line intensity image, but the Fe XII line intensity image only reveals large-scale loop-like structures. Furthermore, the proportion of the blueshifts increases with the formation temperature of the line. The Doppler map of the C IV line forming at 0.1 MK is dominated by redshifts. In contrast, for the Fe XII line forming at 1.5 MK, the proportion of the blueshifts is larger than that of the redshifts.

thumbnail Fig. 5.

Maps of line intensity and Doppler shift for emission lines from the transition region into the corona. For each line as listed on the very top, the top panel shows the intensity map and the bottom panel shows the Doppler map. The line formation temperature T is given with each ion name. The intensity images are shown on a logarithmic scale with a dynamic range of 100. The Doppler shift saturates at ±50 km s−1. See Sect. 3.4 for more details.

As a first step to check how the Doppler shifts change with temperature, we averaged the line profiles over the whole domain. The resulting profiles of Si IV, O VI, and Fe XII are shown in Fig. 6. We performed single Gaussian fits to these three average profiles and found the Doppler shifts of 7.2, 5.6, and −1.1 km s−1 for the Si IV, O VI, and Fe XII lines, respectively. In other words, the Doppler shift changes from redshift in the lower transition region to blueshift in the lower corona.

thumbnail Fig. 6.

Average line profiles. The black, red, and blue curves show the profiles of the Si IV, O VI, and Fe XII lines averaged over the whole domain for a single snapshot. The Doppler shift of the profiles are +7.2, +5.6, and −1.1 km s−1, respectively. The vertical black dashed line indicates zero shift. See Sect. 3.4 for more details.

To further investigate the temperature-dependence of the Doppler shifts in our model, we study the distribution of shifts in the Doppler maps. We present this in Fig. 7. The average shift of the Doppler map in the respective line is shown as a diamond with a bar indicating the scatter in the Doppler map (standard deviation). The trend in observations from Peter & Judge (1999) is also added for comparison. The lines formed in the lower transition region around 105.0 K show average redshifts. The redshifts decrease with an increasing formation temperature above 105.2 K and turn to blueshifts around 105.8 K.

thumbnail Fig. 7.

Average Doppler shifts as a function of line-formation temperature. The diamonds show the average shifts of the synthesized Doppler map in the respective line, and the bars indicate the standard derivation of the Doppler shift in each map. The solid black line shows the average vertical velocity in the MHD model as a function of temperature. The black dashed line shows the trend from Peter & Judge (1999). The red dashed line indicates zero shift. See Sect. 3.4 for more details.

To check this trend of Doppler shift with the temperature, we also calculated the average vertical velocities in the MHD model as a function of temperature. For this we binned the computational domain in temperature and calculated the average vertical velocity in each temperature bin. This average vertical velocity as a function of temperature (dashed lines in Fig. 7) matches the average Doppler shifts. This indicates that the Doppler shifts of the emission lines are caused by mass flows in our model.

As a final step, we calculated the joint probability density function (PDF) of the vertical velocity and density at different temperature ranges as shown in Fig. 8. Where the temperature is below 105.7 K, most of the plasma moves downward. The proportions of upward and downward plasma become roughly the same for the temperature range of 105.7–106.0 K. At temperatures above 106.0 K, more than half of the plasma shows upward motions. Thus, the plasma changes from downflow dominance in the transition region to upflow dominance in the corona, and the mass motion causes the change from redshifts to blueshifts as the formation temperatures increase. We discuss the processes that are at the basis of these flow patterns in Sect. 4.2.

thumbnail Fig. 8.

Joint probability density function (PDF) of vertical velocity and density at different temperature ranges. The respective temperature ranges (in logarithmic scale) are given with each plot. The white curves indicate the distribution of density in the respective temperature range. The horizontal white dashed lines indicate zero velocity. See Sect. 3.4 for more details.

4. Discussion

4.1. Effects of spatial resolution and temporal integration

Previous spectral observations have shown that downflows are dominated in the lower transition region. IRIS provides unprecedented subarcsecond high-resolution spectral observations of the lower transition region (De Pontieu et al. 2014), and our analyses suggest that the Si IV line is still dominated by redshifts in IRIS observations. In other words, Doppler shifts in the lower transition region are always dominated by redshifts in observations in spite of the variations of the spatial resolutions. In our model, the Si IV line is also dominated by redshifts at different spatial resolutions.

However, the proportion of blueshifts of the Si IV line is larger than 20% in our model, while it is less than 10% in observations by IRIS (and earlier instruments). Likewise, the Dopplergrams in the earlier models of Peter et al. (2006) and Hansteen et al. (2010) also show much more blueshifts in the lower transition region compared to observations. Our impression is that our model shows more area covered by redshifts than previous models did. Unfortunately the previous authors did not provide the proportion of redshifts in their models, so here we have to rely on our visual judgment.

We first suspected that selecting a single snapshot from our model might be responsible for the differences between the observations and our model. This is because the typical exposure time is around 30 s in high-signal-to-noise-ratio spectral observations of the quiet-Sun regions. However, our results remain almost the same after integrating the line profiles over 30 s in our model. This can be nicely seen by comparing the single snapshot maps (lower row) and the map integrated in time (top row) in Figs. 2 and 3. These show some minor differences, but no major ones. Essentially this is because the fluctuations on short timescales do not impact the intensity or flows significantly, and because the flows in the horizontal direction only lead to minor changes in the intensity patterns and to the Doppler shifts that sample the vertical motions here.

The above discussion is only valid for timescales up to about half a minute, which are typically the longest exposure times that spectrographs such as SUMER or IRIS and now SPICE use for the quiet Sun. Obviously, fluctuations on longer timescales can be expected, and they should be studied in the future. However, these would show up mostly as small-scale fluctuations (on scales of granules), but the overall pattern of the super-granular patterns should be stable for much longer times.

4.2. Possible processes contributing to the redshifts in the transition region in our model

The redshifts of the Si IV line in the bright network lanes should be larger than in the internetwork (Brynildsen et al. 1998; Curdt et al. 2008; Tian et al. 2008). Some observations even hint at small blueshifts in dark internetwork regions (e.g., Peter 2000, his Fig. 3). In the observations we present here in Sect. 2 and Fig. 1, the signal-to-noise ratio is not sufficient to derive Doppler shifts in the internetwork. So one might suspect that our model blueshifts in the internetwork regions might cause the overestimated proportion of blueshifts of the Si IV line, that is to say 20% instead of 10%. However, we found that the proportion of the redshifts and blueshifts in the network lanes is almost the same as the whole region. To further investigate the distribution of Doppler shifts, we calculated the joint PDF of the intensity and Doppler shift of the Si IV and Fe XII lines over the whole domain in Fig. 9. There is no clear correlation between the intensity and Doppler shift of the Si IV line. The results including only the network are similar to Fig. 9. Previous studies have shown a positive correlation between the intensity and Doppler shifts of the Si IV line (Curdt et al. 2008; Tian et al. 2008), and the positive correlation disappears for the network region (Peter 1999). Our model fails to reproduce such complexity in the observations.

thumbnail Fig. 9.

Joint probability density function (PDF) of line intensity and Doppler shift of the Si IV and Fe XII lines. See Sect. 4.2 for more details.

To understand what causes the Doppler shifts in our model, we isolated the conditions along several selected field lines. This leads us to conclude that there is no single process at the heart of the Doppler shifts, but several mechanisms operate under different circumstances. For this, we traced four field lines in the calculation domain and projected these onto the x-y-plane. Along this curve in the x-y-plane, we extracted the parameters along the vertical direction and through this constructed a vertical map. This is not a plane, but generally a type of warped curtain. We thus display various parameters interpolated to this warped curtain in Fig. 10. These are the temperature, electron density, vertical velocity, heating rate (resistive plus viscous), pressure, Si IV emission, and Fe XII emission.

thumbnail Fig. 10.

Vertical maps of various properties in and around selected loops. From top to bottom: we display the temperature, electron number density, vertical velocity, total heating rate, pressure, Si IV emission, and Fe XII emission. The four columns show sample cases for the processes described in Sects. 4.2.1 to 4.2.4. The colored line in each panel shows the respective magnetic field lines. The contours outline regions with enhanced Si IV emission. The vertical maps were constructed by first projecting each field onto the horizontal x-y-plane and then combining the variations along the z-direction into a single map. Essentially, these vertical maps are close to vertical cuts. See Sect. 4.2 for more details.

4.2.1. Si IV brightenings unrelated to coronal emissions

Most scenarios consider the redshifts and blueshifts in the transition region and coronal lines in a single common physical process. The plasma along a given field line is believed to have a wide temperature range, from 104.0 to 106.0 K, which is responsible for both the redshifts and blueshifts. However, the field line shown in the first case in Fig. 10 goes through a region with greatly enhanced Si IV emission, and the temperature along the field line does not reach the typical coronal temperature (106.0 K). There is no Fe XII emission along the field line in this case.

We also examined the magnetic field structures around the bright patch of the Si IV intensity in Fig. 10f1, and two bundles of field lines fork around the brightening, similar to the campfires studied by Chen et al. (2021); readers can refer to their Fig. 4c,d for more information. Probably related to this, the region with enhanced Si IV emission shows an enhanced heating rate and pressure. Thus the enhancement of the Si IV emission may be caused by component reconnection through loop interactions (Chen et al. 2021), and the Si IV brightening might be the remnant of an explosive event, which is a type of small-scale magnetic reconnection event occurring in the transition region (Brueckner & Bartoe 1983; Innes et al. 1997).

In some cases, the regions with enhanced Si IV emission show bi-directional reconnection flows, which is similar to the scenario proposed by Innes et al. (1997). When the heating rate decreases and fails to balance the radiation cooling, the plasma around the event starts to cool down and condense. Then the plasma falls. As a result, downflows are dominated within the brightening in the Si IV intensity image as shown in Fig. 10c1, and the corresponding line profiles exhibit redshifts.

4.2.2. Flows driven by local pressure enhancement

The second case in Fig. 10 shows a scenario similar to the one suggested by Hansteen et al. (2010). In this case, a thin loop structure seen in the Fe XII intensity image is aligned with a field line as shown in Fig. 10g2. Panel f2 reveals that the Si IV emission is enhanced around the right footpoint of the loop within a thin layer. The heating rate around that footpoint is enhanced, and the plasma with strong Fe XII and Si IV emission exhibit an upflow and downflow, respectively. The heating is caused by the dissipation of currents induced by the braiding of the field lines. The small-scale heating event around the footpoint of the coronal loop largely increases the local pressure and naturally generates upflows and downflows in the corona and transition region, respectively.

4.2.3. Possible siphon-type flows

The field line of the third case also has a loop-like counterpart in the Fe XII intensity image, and the Si IV emission is enhanced at the footpoints of the loop (Fig. 10f3, g3). However, there is no strong heating along the field line. Interestingly, the loop shows dominated upflows and downflows in the left and right parts, respectively. Around the left footpoint, pressure in the transition region (outlined by the contours) is much higher, and the pressure imbalance may trigger upflows from the left footpoint. As a result, the plasma moves along the field line and piles up at the part of the loop on the right, causing the density and pressure enhancement there. Such flows in loops have been studied extensively in early 1D models (as reviewed in the book by Mariska 1992). The role of such flows caused by asymmetric heating for the dynamics of the evolution of loops has been highlighted more recently by Mikić et al. (2013). Such flows have also been reported in synthetic observations based on 3D MHD models (Zacharias et al. 2011; Lionello et al. 2013; Peter 2015), albeit only for active regions.

4.2.4. Boundaries between cold and hot plasma

In the fourth case, the field line above the heights of 2 Mm appears to be the separatrix of the hot and cool plasma (Fig. 10a4, b4), and the Si IV emission is enhanced around the top of the field line (g4). There is a region of low density and pressure, a cavity, below the field line (panels b4, e4). The pressure gradient results in downflows dominating around the separatrix, and the corresponding Si IV line profiles show redshifts.

As our model is established under the assumption of thermal equilibrium, the recombination rates are overestimated in the chromosphere. Consequently, the temperature and pressure decrease faster than they probably do on the real Sun. Therefore the pressure collapse in the cool dense plasma might be more pronounced in our model than on the Sun. However, even if we consider the nonequilibrium ionization for hydrogen and helium, which can impact the thermodynamics in the chromosphere and transition region, radiation cooling in the cold dense plasma would still decrease the pressure. So the separatrix would still move downward, only a bit slower.

4.2.5. Relative importance of the scenarios

Here we give a rough first estimate on how important the processes driving the four cases discussed are, that is to say how often they are found with respect to each other. To keep things manageable for a manual inspection, we reduced the resolution synthesized spectral maps to a grid pacing equivalent to on a spatial pixel of SUMER (i.e., 1″). In the vertical column of the 3D data cube corresponding to each pixel of the spectral map, we found the location of the peak emission in Si IV. We then traced the field line from that grid point and constructed the vertical slice following the field line, similar to Fig. 10. Examining all of these almost 5000 slices, we found that almost all fall into one of the four categories shown in Fig. 10. About 5% of the cases were too complex to classify into these categories.

Based on the investigation of these slices, we established the following classification:

  • Case 1: 30% – Transition region brightenings unrelated to coronal emission,

  • Case 2: > 50% – Flows driven by pressure enhancements,

  • Case 3: few cases – Siphon-type flows,

  • Case 4: 14% – Separatrix between hot and cold plasma.

From this we see that in about half of the cases, we find the Doppler shifts to be caused by a (local) pressure enhancement in the transition region, in a way similar as described by Hansteen et al. (2010). In almost one-third of the cases, the Doppler shifts in the transition region are completely unrelated to coronal emission, that is they are driven by the dynamics of the cool plasma alone. In a bit less than one-sixth of the locations, the field line separates cold and hot plasma and it is dragged down by a pressure reduction. And finally, there are only a few instances that siphon-type flows with their asymmetry of redshifts and blueshifts cause the observed Doppler shifts.

4.3. Reasons why the proportion of redshifts in the lower transition region is underestimated in our model

Although the plasma moves up and down in the chromosphere and lower transition region, there are few elongated jet-like chromospheric and transition-region structures in our model. However, transition region network jets and spicules are prevalent around the network lanes in observations (e.g., Beckers & Schultz 1972; Sterling 2000; De Pontieu et al. 2004, 2011; Tian et al. 2014; Pereira et al. 2014; Rouppe van der Voort et al. 2015; Chen et al. 2019; Samanta et al. 2019). Nevertheless, spicules and network jets are crucial in the mass cycle in the solar atmosphere (e.g., De Pontieu et al. 2011; Wang et al. 2013; Martínez-Sykora et al. 2017; Samanta et al. 2019). Furthermore, spicule-like downflows have been reported (e.g., De Pontieu et al. 2012; Samanta et al. 2019; Bose et al. 2021a,b), and they can be related to redshifts of the Si IV line (Bose et al. 2021a).

Thus, the lack of spicules and network jets in our model – as well as in all earlier models investigating the systematic Doppler shifts in the transition region – may cause the underestimation of redshifts of the Si IV line, especially around the network lanes. Missing the network jets and spicules in our 3D model may result from the absence of ambipolar diffusion (see e.g., the 2D model of Martínez-Sykora et al. 2017). Also the limited spatial resolution in the horizontal direction (ca. 50 km in our model) could play a role.

In addition, the spatial resolution in the vertical direction of our model is not sufficient at properly resolving the transition region. As a result, the heat flux jumps from the corona to the chromosphere without heating and impacting the plasma in the transition region, and the determination of the radiation loss in the transition region is not fully accurate (e.g., Bradshaw & Cargill 2013; Rempel 2017). Furthermore, we did not include nonequilibrium ionization for hydrogen and helium, which might play an important role in the dynamics in the transition region (e.g., Nóbrega-Siverio et al. 2020; Martínez-Sykora et al. 2020). Also, we did not consider nonequilibrium ionization for the spectral syntheses. Considering this effect in a 3D MHD model seems to result in more blueshifts of the Si IV line (Olluri et al. 2015). Spadaro et al. (1990) already found in their 1D loop models that nonequilibrium ionization would have the tendency to lead to (small) blueshifts in transition region lines, which is the opposite to what is desired in order to explain the redshifts. Thus, the limited grid spacing and missing physics such as ambipolar diffusion and nonequilibrium ionization for hydrogen and helium in our model might lead to systematic changes in the proportion and magnitude of redshifts of the lower transition region lines, and it remains to be seen which effect goes into which direction.

4.4. Importance of the upper transition region observations

Finally, the intensity image of the Ne VII line reveals many more large-scale diffuse structures compared to the C IV line. Even so, it presents many more small-scale compact structures than to the Fe XII line. Hence, Ne VII would be an ideal line to capture the transition from the cooler transition lines, such as Si IV as imaged with IRIS, to the coronal lines, such as Fe XII as imaged with AIA. In addition, the proportion of the blueshifts in the Dopplergrams starts to significantly increase from the Ne VII line. Thus, although the direct imaging observations of the upper transition region have not been achieved yet, direct imaging and spectroscopic observations of the Ne VII line would be essential to understand the mass and energy cycle of the transition region (Tian 2017).

5. Conclusions

In order to understand the persistent Doppler shifts in the quiet Sun transition region and corona, a model not only has to explain the average line shift, but also the following: the 90% dominance in terms of area coverage of redshifts in the transition region, the change from net redshifts to blueshifts into the corona, and the appearance of the redshifts in the Doppler maps similar to the nest of spicules which needs to be reproduced. We constructed a 3D radiation MHD model using the MURaM code. The calculation domain of our model is significantly larger than previous models (e.g., Hansteen et al. 2010; Abbett & Fisher 2012), which self-consistently maintains network fields and allows a steady corona of 1 MK. A similar simulation of the quiet Sun has already been achieved by Rempel (2017). We ran our model at a significantly higher spatial resolution, which reduced the diffusivity and allowed us to resolve heating events on smaller scales.

In our model we find redshifts in the transition region and blueshifts in the corona. While this has been found before, in contrast to previous models, we also see a significant dominance of redshifts, even though this dominance is not as extreme as in observations. Part of the reason for this might be that in our model, the chromosphere is not captured with a sufficient accuracy, for example, we do not see proper spicules.

Despite this shortcoming, we could isolate at least four processes that cause the systematic Doppler shifts: the redshifts are found in regions of transition region brightenings unrelated to the coronal emission; the redshifts and blueshifts are caused by pressure enhancements in the transition region; the Doppler shifts are found at the separators (or boundaries) along the magnetic field that separate lower-lying cool from hotter plasma above; and, in very few cases, we also found siphon-type flows to cause the net shifts. Thus processes that have been suggested before are found in our model. So there is not only one process that drives the transition region and coronal net Doppler shifts, but, unsurprisingly, we have to deal with a mixture of mechanisms.

Future investigations of 3D models that also can properly account for the formation of spicules will have to show what role the spicules play in this. In particular, one might wonder if spicules are responsible for the majority of the transition region redshifts or if they are merely another piece in the puzzle of the net Doppler shifts.


Acknowledgments

This work is supported by NSFC grants 12073004, 11825301 and 11790304, the Strategic Priority Research Program of CAS (grant no. XDA17040507). Y.C. also acknowledges partial support from the China Scholarship Council and the International Max Planck Research School (IMPRS) for Solar System Science at the University of Göttingen during his stay at MPS. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 695075). IRIS is a NASA small explorer mission developed and operated by LMSAL with mission operations executed at NASA Ames Research center and major contributions to downlink communications funded by ESA and the Norwegian Space Centre. We thank Dr. L. P. Chitta for helpful discussion. We gratefully acknowledge the computational resources provided by the Cobra and Raven supercomputer systems of the Max Planck Computing and Data Facility (MPCDF) in Garching, Germany.

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All Tables

Table 1.

Emission lines used in this study.

All Figures

thumbnail Fig. 1.

Observation of the line intensity and Doppler shift of Si IV 1394 Å in the quiet Sun. The data were taken with IRIS with a field of view of 104″ × 174″ centered around solar (X,Y) = ( − 120″, −48″), i.e., close to disk center. The intensity map is displayed on a logarithmic scale. Gray regions in the Doppler map mark areas where the signal is too low to perform a reliable Gaussian fit (below ca. 15% of the mean intensity or 80 DN/line for this data set). See Sect. 2 for more details.

In the text
thumbnail Fig. 2.

Intensity maps of the Si IV line at different spatial and temporal resolution. Intensity images of the Si IV line shown at (a) original resolution of the simulation, (b) IRIS resolution, and (c) SUMER resolution after integration over 30 s. (d)–(f) The same as (a)–(c), but for the first snapshot. The images are shown in a logarithmic scale normalized to the median value. The dynamic ranges of the intensity at full, IRIS, and SUMER resolution are 1000, 500, and 50, respectively. See Sect. 3.3 for more details.

In the text
thumbnail Fig. 3.

Doppler maps in Si IV and underlying magnetic field. Panels a to f: are similar to Fig. 2, but for Dopplergrams of the Si IV line saturated at ±50 km s−1. Panel g: vertical magnetic field in the photosphere. See Sect. 3.3 for more details.

In the text
thumbnail Fig. 4.

Histograms of the Doppler shifts of the Si IV line. The frequency (a) and cumulative frequency (b) of the Doppler shifts of the Si IV line after integration over 30 s as shown in Fig. 3a–c. The solid black, red, and blue lines indicate the distribution of the Si IV line at the original, IRIS, and SUMER resolution, respectively. The red dashed line indicates the distribution of the Si IV line calculated from the observation shown in Fig. 1. The vertical dotted line shows the zero shift. See Sect. 3.3 for more details.

In the text
thumbnail Fig. 5.

Maps of line intensity and Doppler shift for emission lines from the transition region into the corona. For each line as listed on the very top, the top panel shows the intensity map and the bottom panel shows the Doppler map. The line formation temperature T is given with each ion name. The intensity images are shown on a logarithmic scale with a dynamic range of 100. The Doppler shift saturates at ±50 km s−1. See Sect. 3.4 for more details.

In the text
thumbnail Fig. 6.

Average line profiles. The black, red, and blue curves show the profiles of the Si IV, O VI, and Fe XII lines averaged over the whole domain for a single snapshot. The Doppler shift of the profiles are +7.2, +5.6, and −1.1 km s−1, respectively. The vertical black dashed line indicates zero shift. See Sect. 3.4 for more details.

In the text
thumbnail Fig. 7.

Average Doppler shifts as a function of line-formation temperature. The diamonds show the average shifts of the synthesized Doppler map in the respective line, and the bars indicate the standard derivation of the Doppler shift in each map. The solid black line shows the average vertical velocity in the MHD model as a function of temperature. The black dashed line shows the trend from Peter & Judge (1999). The red dashed line indicates zero shift. See Sect. 3.4 for more details.

In the text
thumbnail Fig. 8.

Joint probability density function (PDF) of vertical velocity and density at different temperature ranges. The respective temperature ranges (in logarithmic scale) are given with each plot. The white curves indicate the distribution of density in the respective temperature range. The horizontal white dashed lines indicate zero velocity. See Sect. 3.4 for more details.

In the text
thumbnail Fig. 9.

Joint probability density function (PDF) of line intensity and Doppler shift of the Si IV and Fe XII lines. See Sect. 4.2 for more details.

In the text
thumbnail Fig. 10.

Vertical maps of various properties in and around selected loops. From top to bottom: we display the temperature, electron number density, vertical velocity, total heating rate, pressure, Si IV emission, and Fe XII emission. The four columns show sample cases for the processes described in Sects. 4.2.1 to 4.2.4. The colored line in each panel shows the respective magnetic field lines. The contours outline regions with enhanced Si IV emission. The vertical maps were constructed by first projecting each field onto the horizontal x-y-plane and then combining the variations along the z-direction into a single map. Essentially, these vertical maps are close to vertical cuts. See Sect. 4.2 for more details.

In the text

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