EMERGENCE OF GRANULAR-SIZED MAGNETIC BUBBLES THROUGH THE SOLAR ATMOSPHERE. II. NON-LTE CHROMOSPHERIC DIAGNOSTICS AND INVERSIONS

We analyze the same data set that was used in Paper I. Our observations of AR11024 were acquired on 2009 July 05 starting at 09:48 UT at coordinates S27${}^{^\circ }$, W12${}^{^\circ }$. The data were taken with the Swedish 1 m Solar Telescope (SST, Scharmer et al. 2013) and the CRisp Imaging Spectro-Polarimeter (CRISP, Scharmer et al. 2008) in the Ca ii $\lambda 8542$ Å line (hereafter $\lambda 8542$). We sequentially acquired narrow band images sampling the $\lambda 8542$ profile at 17 line positions in steps of 100 mÅ between ${\rm{\Delta }}\lambda =\pm 800$ mÅ from line center. An additional point was measured at ${\rm{\Delta }}\lambda =2400$ mÅ.

The data were reduced using the CRISPRED pipeline (de la Cruz Rodríguez et al. 2015). The images were restored with multi-frame–multi-object-blind-deconvolution techniques (MOMFBD, van Noort et al. 2005), and residual seeing distortions within the line scan have been compensated using methods described in Henriques (2012).

It is not possible to reach the continuum of the $\lambda 8542$ line with CRISP because the prefilter is not wide enough. Therefore, to scale our observations relative to the continuum of the quiet-Sun spatial average at disk-center, we use a 3D HD simulation (de la Cruz Rodríguez et al. 2011). The idea is to compute a simulated spatio-temporal average of quiet-Sun spectra at $\mu =0.87$ (where our observations were acquired). Then we normalize the average intensity profile by the continuum intensity at $\mu =1$. Finally, we scaled our observations in such a way that the average quiet-Sun spectrum at ${\rm{\Delta }}\lambda =2.4$ Å has the same intensity as the normalized synthetic average profile. The same technique was used in Paper I and by Watanabe et al. (2012).

The absolute wavelength calibration is performed using a spatio-temporal average from a patch of quiet Sun, such that the line center is at rest. It is hard to find velocity references in the solar atmosphere, particularly in the chromosphere. The wavelength coverage of our observations do not allow to compute a photospheric convective blushift that could be used as a reference for Doppler velocities (de la Cruz Rodríguez et al. 2011).

In Paper I we described the peculiar intensity profile of the $\lambda 8542$ line during the flux emergence event. Figure 1 illustrates a time series of the profile within the region of interest, indicated in the corresponding images with a white square. The profile has three distinctive features compared with a spatial quiet-Sun average.

• 1.  
  The line center is clearly blueshifted.
• 2.  
  Close to the line center, the line profile is strongly asymmetric, with a very sharp gradient on the red side and a much more extended blue side.
• 3.  
  At approximately $+250$ mÅ there is an emission feature that is not similarly visible on the blue wing.

The degree of asymmetry and the exact strength of the three features enumerated above suffer slight changes over time, but the overall shape of the profile is persistently similar. In our analysis, we will try to understand the formation of the $\lambda 8542$ line and the atmospheric parameters that are responsible for these features.

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We use the non-LTE inversion code Nicole to compute inversions in the $\lambda 8542$ line (Socas-Navarro et al. 2015). The code iteratively modifies the physical parameters of a guessed model atmosphere to reproduce observed spectra (Socas-Navarro et al. 2000). Nicole solves the non-LTE problem as described in Socas-Navarro & Trujillo Bueno (1997) assuming statistical equilibrium, which is reasonable for Ca ii lines (Wedemeyer-Böhm & Carlsson 2011). Our atom model consists of five bound levels plus an ionization continuum, similar to that used by Leenaarts et al. (2009). The atomic data used in this study has been extracted from the VALD database (Piskunov et al. 1995).

Nicole assumes Zeeman-induced polarization to compute the full-Stokes vector, neglecting scattering polarization and the Hanle effect. This assumption works fine in regions with relatively strong magnetic fields, but in the quiet Sun all these effects must be considered to model Stokes Q and U observations (Manso Sainz & Trujillo Bueno 2010; Carlin et al. 2012; Carlin & Asensio Ramos 2015).

The radiative transfer equation is solved using a piece-wise cubic Bezier spline interpolant both to calculate the non-LTE populations (assuming unpolarized light) and to compute the final polarized formal solution (Auer 2003; de la Cruz Rodríguez & Piskunov 2013).

The spatially averaged quiet-Sun spectrum of the $\lambda 8542$ line shows a reversed C-shape bisector (Uitenbroek 2006b; Chae et al. 2013; Pietarila & Harvey 2013). Only recently, Leenaarts et al. (2014) realized that the asymmetry in solar observations is produced by the presence of different Ca ii isotopes, apparently a well-known phenomenon in chemically peculiar stars (Castelli & Hubrig 2004; Cowley & Hubrig 2005; Cowley et al. 2007; Ryabchikova et al. 2008).

We have included the effect of isotopic splitting in our inversions using a simple approximation. First, the non-LTE problem is solved with a regular ⁴⁰Ca atom. Once the atom population densities are known, we compute a final formal solution of the radiative transfer equation, assuming that the absorption coefficient is a weighted sum of Voigt profiles, each of them centered at the rest-wavelength of the line for each isotope. In our case, the weight is proportional to the relative abundance of each isotope (Carlsson 1986).

Our observations are under-sampled by almost a factor $\times 2$ in the spectral dimension, a necessary trade-off to keep the cadence sufficiently high while observing three spectral lines. This situation is less than ideal for inversions, where critically sampled observations constrain the solution much better. Therefore we have run a set of tests to assess the optimal combination of nodes that allows us to reproduce the observed profiles with the lowest number of degrees of freedom that is possible. The inversions are performed in several cycles to improve the convergence (Ruiz Cobo & del Toro Iniesta 1992): during the first cycle, a reduced number of nodes is considered. Once the solution cannot be improved further, a second cycle is started from that point with more nodes. Table 1 summarizes the number of nodes used in each of the cycles.

Table 1.  Number of Nodes Used in Each Cycle of Our Inversions for Temperature, Line-of-sight (LOS) Velocity, and Longitudinal Magnetic Field (B_Long)

	Cycle 1	Cycle 2	Cycle 3
Temp	4	5	5
${v}_{\mathrm{LOS}}$	2	4	4
Blong	0	0	1

Download table as:  ASCIITypeset image

In our first tests with the number of nodes, we realized that placing the nodes equidistantly along the depth-scale is not an optimal choice. After some trial and error, we found that we obtain much better fits, at least whenever there is a node placed at $\mathrm{log}{\tau }_{500}\sim -4.5$ both in temperature and in velocity.

We have modified Nicole to compute the response functions (see definitions in Section 3.2) using centered numerical derivatives instead of the default derivatives based on one-side perturbations of the model parameters. This makes the code roughly 33%–50% slower, but it greatly improves the convergence of the inversion that now reaches more consistently similar values of ${\chi }^{2}$ from different randomizations of the initial model parameters. Additionally, inspired by van Noort (2012), we have removed some inversion noise⁴ by inverting the data once, horizontally smoothing the parameters of the model at the locations of the nodes, and then restarting the inversion again. This trick only seems to help in pixels where the inversion clearly got stuck in a local minimum.

To study the formation of the Ca ii $\lambda 8542$ line we use a 3D magnetohydrodynamical simulation performed with the Bifrost code (Gudiksen et al. 2011), which has also been used in Paper I. Synthetic full-Stokes $\lambda 8542$ observations have been calculated with Nicole, as described in Section 2.2.

The simulation includes the upper convection-zone, photosphere, chromosphere, and corona covering a physical domain that extends vertically from ${z}_{0}=-2.6$ Mm to ${z}_{1}=14$ Mm from the surface of the photosphere. Horizontally, the simulation covers approximately 24 × 24 Mm. This domain is discretized in a grid of $504\times 504\times 496$ points, with a horizontal step of 47.6 km and a vertical step of 20 km in the radiative zone (−400 ≤ z ≤ 5000 km). In the convection zone and corona the vertical distance between grid points becomes larger with increasing scale height.

The calculations are performed with periodic boundary conditions in the horizontal plane, and open boundary conditions at the top and at the bottom. At the beginning of the simulation, the magnetic flux is injected at the bottom boundary. The injected field is a flux sheet with no twist oriented along the y-axis from ${x}_{0}=4$ Mm to ${x}_{1}=16$ Mm with strength B = 3300 G. This injection continues for 1 hr 45 minutes before being turned off. The field rises steadily through the convection zone to the photosphere before stalling there after about an hour. Eventually the field breaks through the photosphere, in certain locations, and into the chromosphere, forming expanding bubbles of magnetic field as described in greater detail in Paper I. Later stages of this simulation were used to study small flares by Archontis & Hansteen (2014).

We note that compared to the latest runs performed with Bifrost, this simulation does not include the effect of collisions between neutrals and ions that can influence the temperature stratification in the chromosphere (Khomenko & Collados 2012; Martínez-Sykora et al. 2012; Martinez-Sykora et al. 2015) or the effect of hydrogen non-equilibrium ionization that can increase the amount of free electrons in the chromosphere, affecting the opacity of chromospheric lines (Leenaarts et al. 2007).

In this section we discuss the results of the non-LTE inversions of the observed Ca ii $\lambda 8542$ Stokes profiles. We will focus on the inferred temperature and line-of-sight velocity maps.

Figure 2 illustrates the inferred temperature as a function of time at $\mathrm{log}{\tau }_{500}=-4.4,-3.0,-1.9,-1.0,0.0$. We note that the panels at $\mathrm{log}{\tau }_{500}=0.0$ may be an extrapolation of the model in the upper layers because our observations do not reach the continuum. Even so, those deep photospheric layers show typical granulation patterns with hot granules, colder intergranules, and hot bright-points (though geometrically the latter are probably located in deeper layers than the former, see Steiner et al. 2001).

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At $\mathrm{log}{\tau }_{500}=-1.0$ the granulation pattern is reversed showing hotter intergranular lanes than granules (Cheung et al. 2007). Here the footpoints of the bubble are clearly visible, but otherwise, there are no signatures of the bubble in the temperature map. Only at $\mathrm{log}{\tau }_{500}\sim -1.9$, in the center of the bubble above the granule, does the temperature show a significant drop compared to the surroundings, especially between ${\rm{\Delta }}t=1$ minute and ${\rm{\Delta }}t=4$ minute. Later on, the temperature rises again in the area where the bubble appeared.

At $\mathrm{log}{\tau }_{500}=-3.0$ there is a dramatic change in the structuring of the temperature, and features that resemble fibrils start to be visible. Therefore we associate this panel with the lower chromosphere. The magnetic bubble reaches these layers with a temporal delay of roughly 2 minutes compared to $\mathrm{log}{\tau }_{500}=-1.9$, similarly to what was reported in Paper I from the analysis of images at different line positions.

Above the lower footpoint (in the image) there are signatures of heating, quite localized until ${\rm{\Delta }}t=4$ minute when they extend around the bubble. We speculate that this heating can be produced by three different processes:

• 1.  
  by magnetic interaction of the footpoints with the existing magnetic canopy of fibrils.
• 2.  
  by wave dissipation, assuming that waves are channeled through the footpoints and persistently release energy into the chromosphere.
• 3.  
  by Joule heating at the boundary of the magnetic bubble, where steep gradients in the magnetic field drive electrical currents (Solanki et al. 2003).

Unfortunately, we would need very accurate estimates of the magnetic field at these locations to study whether the brightenings are due to reconnection and/or to estimate the location and effect of electrical currents in our maps. Therefore, this aspect must be clarified in future studies, hopefully with higher sensitivity data.

At $\mathrm{log}{\tau }_{500}=-4.4$ our inversions show a more homogeneous temperature structuring, with no signature of the emerging field. This event is right at the end of the time series, so we cannot confirm whether the bubble manages to reach the middle/upper chromosphere: in our data set it does not. The canopy may hinder the ascent of the bubble and we find evidences of magnetic interaction between the emerging flux and the existing field especially at the boundary of the bubble. Interestingly, the entire region in the temperature panels at $\mathrm{log}{\tau }_{500}=-4.4$ seems to get slightly warmer by the end of the time series.

It is also clear that the apex of the bubble must be located between $\mathrm{log}{\tau }_{500}=-3.0$ and $\mathrm{log}{\tau }_{500}=-4.4$ in our observations. We note that the panels in Figure 2 may show reduced temperature contrast due to 3D non-LTE effect (see de la Cruz Rodríguez et al. 2012) or to stray light in the observations.

Similarly, 2D maps of inferred velocities are presented in Figure 3. Our results in the deepest layers of the inverted model seem to be an extrapolation of the velocity field  around $\mathrm{log}{\tau }_{500}\approx -1.5$. We are not surprised because the Lorentzian (photospheric) wings are likely to be much more sensitive to changes in temperature than to changes in the velocity field, since the intensity gradient is not very steep. Close to the line center, the profiles carry more complete information of the velocity field.

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Our inversion results at $\mathrm{log}{\tau }_{500}=-1.9$ are very similar to those inferred using Fe i lines in Paper I. Gas flows up in the center of the bubble and drains down at the sides. These downflows coincide with the location of intergranular lanes of photospheric granulation. At $\mathrm{log}{\tau }_{500}=-3.0$, the imprint of these downflows in the surroundings has almost vanished, but in some locations they are visible at the end of the series, when the bubble has reached high enough in the atmosphere. Finally, at $\mathrm{log}{\tau }_{500}=-4.4$ the images are dominated by the canopy of fibrils, which appear up-flowing along our line of sight. The entire region seems to be moving upward with speeds of $-6$ km s⁻¹.

We now discuss the inferred longitudinal component of the magnetic field, shown in Figure 4. The maps at ${\rm{\Delta }}t=0,1,2$ minutes do not show any features at the location of the bubble. However, from ${\rm{\Delta }}t=3$ minutes to ${\rm{\Delta }}t=5$ minutes two patches of opposite polarities appear and slowly separate during the ascent of the bubble. From ${\rm{\Delta }}t=6$ minutes, one of the footpoints is still visible at the bottom of the image. The inferred longitudinal field has amplitudes peaking ±140 G, which seems to lay just above our sensitivity threshold. This inversion has been performed with a constant magnetic field since the polarization signals are too small to attempt to infer gradients.

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Initially we were surprised to see these opposite polarity patches. In Paper I, we used the far wings and the inner core of the 8542 line to estimate the photospheric and chromospheric fields, respectively. The results show half-moon shaped footpoints in the photosphere (with opposite polarities), whereas we only detected signals in a very small patch above the footpoints in the chromosphere, but unfortunately not in the interior of the bubble. We suspect this is due to the too restrictive wavelength ranges we used to compute the weak-field approximation both at the core and at the wings, skipping those parts of the line profile where the bulk of the signals are contained.

Now we discuss the quality of the fits to the observed profiles that are reasonably good, but not perfect. Figure 5 shows an example from the upper opposite polarity patch, at ${\rm{\Delta }}t=3$ minutes. The general aspect of the Stokes I profile is reproduced by the inversion code, but the sharpest features in the observations appear smoothed in the inversion, an expected result from the discretization of the nodes in the inversion (de la Cruz Rodríguez et al. 2012): the location and separation of the nodes are not ideal to capture sharp gradients in the atmosphere. The Stokes V profiles are very noisy, but the code seems to find a solution that is compatible with the observations assuming a constant magnetic field.

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We focus now on the strongly tilted blue wing of the profile (see Figure 1). As we described in Paper I, the imprint of the bubble is more prominent in the blue wing of the line. This effect is particularly visible from ${\rm{\Delta }}\lambda =-800$ mÅ to ${\rm{\Delta }}\lambda =-300$ mÅ. As we move to the center of the line, the monochromatic images are dominated by fibrils. Therefore, the profile is sampling to different regimes in the atmosphere, the bubble in the outer photospheric wings, which causes a strong asymmetry, and a canopy of fibrils close to line center. The exact steepness of the velocity gradient and the projection of the canopy of fibrils along the line of sight will define how strong these effects are and how the line center is shifted relative to the wings.

The advantage of the MHD simulation is that we have all of the thermodynamical properties in hand, making it possible to study the formation of the line. We have been puzzled by the peculiar shape of the line profiles inside the bubble. To find an explanation for such profiles, our first attempt was to inspect the properties of the 3D simulation.

Figure 6 shows synthetic images in the Ca ii $\lambda 8542$ line at different wavelengths, for one time-step of the simulation. In the left-most panel we illustrate the line-core intensity for each pixel, and in the consecutive panels, monochromatic images at ${\rm{\Delta }}\lambda =-100,-75,0,75$ mÅ relative to rest wavelength. In the first panel, we have effectively removed intensity fluctuations due to Doppler motions, and therefore, the dark circular shape of the flux emerging region becomes clearer. In the monochromatic panels, the intensity also changes due to Doppler motions, but it is clear that spectra within the emerging region are blueshifted, and therefore, that it is darker at blue wavelengths. This is similar to what is observed, as described in Paper I. However, the synthetic profiles are significantly narrower than those from real observations, with a much steeper Gaussian core than in the observations. Therefore, the imprint of Doppler shifts is quite strong in the simulation, and in the red wing there is an intensity enhancement from Doppler motions that is not obviously visible in the observations.

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In Figure 7, we show vertical slices of the corresponding snapshot from the MHD simulation: in the top row, temperature and line-of-sight velocity, in the middle row, density⁵ and magnetic field strength, indicated in black for vertical fields and in red for horizontal fields. For completeness, the velocity field has been over-plotted using arrows in all panels, except in the middle-right panel, where the arrows indicate the direction of the magnetic field in the plane of our slices ($Y-Z$). For context, we indicate the location of these slices in Figure 6 using a dotted white line.

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In the simulation, the emergence of magnetic flux produces a cold up-flowing bubble, where the plasma flow is mostly vertical in the center and turns almost horizontal on the sides. At the boundary with the environment, there seems to be a draining effect, and the plasma flows down. We note that density is higher within the emerging region than in the surroundings because the gas is compressed at the upper boundary of the bubble.

We have computed the contribution function to the intensity and the response function to temperature to assess whether the emission feature observed in the Ca ii 8542 intensity profile can be related to temperature enhancements at some height in the atmosphere.

The contribution function (${C}_{\lambda }$) is computed using the monochromatic source function (${S}_{\lambda }$), and therefore, it encodes information from many thermodynamical variables that implicitly affect the source function:

$$\begin{eqnarray}&&{C}_{\lambda }(z)={S}_{\lambda }(z)\cdot {e}^{-{\tau }_{\lambda }(z)}\cdot \displaystyle \frac{d{\tau }_{\lambda }(z)}{{dz}},\end{eqnarray} \tag{ 1 }$$

where ${\tau }_{\lambda }(z)$ is the monochromatic optical depth at height z.

The response function (${R}_{\lambda }$) is computed by directly perturbing the physical quantities in the model, at each layer, and directly quantifying the impact of such a perturbation on the synthetic profile (see, e.g., Landi Degl'Innocenti & Landi Degl'Innocenti 1977; Ruiz Cobo & del Toro Iniesta 1992; Socas-Navarro & Uitenbroek 2004; Orozco Suárez & Del Toro Iniesta 2007):

$$\begin{eqnarray}&&{R}_{\lambda }(z,T)=\displaystyle \frac{\delta {I}_{\lambda }(z)}{\delta T(z)\delta z},\end{eqnarray} \tag{ 2 }$$

where ${I}_{\lambda }$ is the emerging intensity and T is the temperature at height z in the model. Fossum & Carlsson (2005) proposed an equivalent method to compute the response function, which is numerically more stable, with a slightly different form than Equation (2), that has been used in our study. We applied a perturbation of $\delta T(z)=\pm 0.01T(z)$.

In the bottom row of Figure 7 we display the $\lambda 8542$ line, computed through the column at the center of the bubble, indicated with a dashed vertical line in the panels above. The blue curve corresponds to the temperature stratification (in the left panel) and to the velocity stratification (in the right panel). The line profile seems to have an emission peak in the red wing; similar to what is observed, but contrary to intuition, the temperature stratification does not exhibit clear enhancements that could be associated with such a peak. Additionally, we present the response function to the temperature and the contribution function for each wavelength (Uitenbroek 2006a). These two functions quantify how sensitive the spectral line is to different layers of the model; however, the contribution function does not account for radiative coupling of the source function to non-local properties of the atmosphere.

In our calculations, both methods seem to provide similar results: far in the wings, the line is mostly influenced by the photosphere, between 0 and 500 km. Close to the line center, the line is sensitive to higher layers. The blue curve represents the temperature stratification with height (left panel) and the line-of-sight velocity profile (right panel). Their corresponding scales are indicated in the upper axis of each of the panels. Both plots indicate that the core of the line is formed within the rising magnetic region, which is denser than the surroundings in the simulation.

The contribution function indicates that the emission peak in the red wing originates at around 450 km above the surface, coinciding with the base of the bubble. The velocity profile also changes there, with a steeper velocity gradient inside the bubble. The response function shows a complicated sensitivity to changes in temperature within a large range of heights, making it very hard to identify a specific regime responsible for the emission.

Therefore, we attacked the problem with a slightly different approach. To test whether the emission is an effect of Doppler velocities, we ran an experiment where the profile from this column is calculated using the original velocity stratification, with the velocities set to zero, and with the velocities decreased by a factor $\times 2$. The result is illustrated in Figure 8 (top panel). The red curve is computed neglecting the velocity field, and therefore it is symmetric. The emission feature is then visible in both line wings, although there are no temperature enhancements along the line of sight. The plot also demonstrates that the emission continues to be visible when the amplitude of the velocities is half of that in the simulations, but the presence of strong Doppler velocities enhance the emission in the red wing. The latter is because the gradient of velocities within the bubble (to the blue, as corresponds to up-flowing material) reduces the line opacity in the red wing (see discussion by Scharmer 1984). Therefore, it is the bubble, and not the downflowing material above it, that enhances the emission in the red wing and attenuates it in the blue wing.

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To understand the origin of the emission, we inspected the shape of the monochromatic source function and analyzed its dependence on the local temperature profile. In Figure 8 (middle panel) we illustrate the temperature profile of the column and the total source function at Δλ = 60 mÅ, which has been converted to radiation temperature. The source function decouples from the temperature profile at around 350 km from the surface and peaks at 500 km. Then it decreases again monotonically. At that wavelength, the ${\tau }_{\nu }=1$ layer coincides with the bump in the source function and therefore the emission feature is formed. However, this behavior is not due to an increase in the temperature (unlike de la Cruz Rodríguez et al. 2013) or in the density profiles, rather the opposite. In the lower panel of Figure 8, we show the departure coefficients for the upper and lower levels of the 8542 line as a function of height. In this case, the departure coefficient for an atomic level is defined using Zwaan's notation (Wijbenga & Zwaan 1972):

$$\begin{eqnarray}&&b=\displaystyle \frac{n}{{n}_{\mathrm{LTE}}},\end{eqnarray} \tag{ 3 }$$

where n and n_LTE are the actual and LTE populations of that level. At z = 500 km, the upper level departs from LTE more vigorously than the lower level, explaining the peak of the source function at that height.

Uitenbroek (1989) reports that such a behavior can occur because the Ca ii K and the 8542 lines share the upper level, while their lower levels are collisionally coupled. Therefore, the source function of the 8542 line is set by the K line and displays the same temperature sensitivity as 3933 Å. We emphasize that the source function in Figure 8 decouples from the temperature profile exactly at the temperature drop produced by the magnetic bubble. Given the behavior observed in the departure coefficients and the shape of the source function, we have no reason to think that this is not the same phenomenon.

We now focus on the polarization signals in our synthetic observations. Figure 9 illustrates some non-consecutive snapshots from the simulation. The left column displays the minimum intensity of the profile at each pixel. In the middle column we made a color composite: in orange we represent $\mathrm{max}(\sqrt{{Q}_{\lambda }^{2}+{U}_{\lambda }^{2})}$ and in blue $\mathrm{max}(| {V}_{\lambda }| )$. In this way we can visualize where the field is predominantly inclined. The column on the right illustrates Stokes V in the photospheric wing, and provides information about the footpoints of our emerging region. In the beginning of the time series, linear polarization is dominant in the picture, suggesting that the $\lambda 8542$ line is sampling the top of the bubble where the field is mostly inclined. As time goes on and the bubble reaches higher layers in the atmosphere, the imprint of vertical fields becomes stronger on the sides, where the magnetic field connects with the footpoints. This is also consistent with the results from LTE inversions (using the Fe i $\lambda 6301/6302$ lines) presented in Paper I. Since we are tracking the maximum signal of the Stokes profiles, which can change wavelength due to Doppler motions or a change in the formation regime, the observables depicted in Figure 9 may be sensitive to slightly different heights in the model.

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In our observations we did not detect linear polarization within the magnetic bubble and the circular polarization levels were just above the detection limit, probably because the magnetic field is highly inclined and the sensitivity of Stokes Q and U to the field strength is much lower than for Stokes V. Interestingly, we only detected Stokes V signals when the synthetic Stokes I profile showed emission around ${\rm{\Delta }}\lambda =60$ mÅ (e.g., panel 1 of Figure 10 in Paper I). Therefore, we inspected the polarization levels in the Stokes profiles computed from the MHD simulation. We note that the longitudinal component of the magnetic field inferred from the observations ranges between 100–200 G, and these values are quite similar to those present in the 3D MHD simulation.

The synthetic Stokes V profiles show polarization levels peaking at $0.2\%$ relative to the continuum intensity for the linear and circular components. We convolved the spectra with a theoretical CRISP transmission profile at $\lambda 8542$ and most of the signal is washed away, showing peak values one order of magnitude lower than the unconvolved ones. However, the profiles computed from our simulation are significantly narrower than the observed profiles (see discussion by Leenaarts et al. 2009; de la Cruz Rodríguez et al. 2012; Rubio da Costa et al. 2015). Therefore, our test with the CRISP transmission profile is likely to be significantly more pessimistic than reality, and it should only be taken as a lower limit. Regions with stronger magnetic fields could produce polarization levels of $\approx 0.1\%$, which is our detection limit. Figure 10 illustrates this experiment. Incidentally, we note that the model shows Stokes Q, U, and V signals only at the position of the emission feature, i.e., the emission brings out polarization signals that would otherwise be hidden. This is also what happens in our observations.

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Summarizing, in this section we used a 3D MHD simulation to study the formation of the $\lambda 8542$ within a magnetic bubble emerging into the solar chromosphere. Our analysis reveals that the emission component in the red wing of the line is consistent with an increase of the source function peaking at z = 500 km. The presence of a strong upflow in the bubble makes the profile highly asymmetric, hides the emission feature in the blue wing, and makes the emission in the red wing stronger.