# Importance of the H α Visibility and Projection Effects for the Interpretation of Prominence Fine-structure Observations

Stanislav Gunár, Jaroslav Dudík, Guillaume Aulanier, Brigitte Schmieder, Petr Heinzel

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Stanislav Gunár, Jaroslav Dudík, Guillaume Aulanier, Brigitte Schmieder, Petr Heinzel. Importance of the H α Visibility and Projection Effects for the Interpretation of Prominence Fine-structure Observations. The Astrophysical Journal, 2018, 867 (2), pp.115. 10.3847/1538-4357/aae4e1. hal-02341528

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# Importance of the Hα Visibility and Projection Effects for the Interpretation of Prominence Fine-structure Observations

Stanislav Gunár<fnref n="1" />,<fnref n="2" />✉, Jaroslav Dudík<fnref n="1" />✉, Guillaume Aulanier<fnref n="2" />✉, Brigitte Schmieder<fnref n="2" />✉, and Petr Heinzel<fnref n="1" />

## Abstract

We construct a new 3D Whole-prominence Fine-structure (WPFS) model based on a prominence magnetic field configuration designed to qualitatively approximate the morphology of a quiescent prominence observed on 2010 June 22. The model represents an entire prominence with its numerous fine structures formed by a prominence plasma located in dips in the prominence magnetic field. We use the constructed 3D model and employ a radiative-transfer-based Hα visualization method to analyze the Hα visibility of prominence fine structures and its effect on the perceived morphology of observed and modeled prominences. We qualitatively compare three techniques used for visualization of modeled prominences—visualizations drawing magnetic dips up to a height of 1 pressure scale height, drawing the full extent of magnetic dips, and the synthetic Hα visualization—and discuss their suitability for direct comparison between models and observations of prominences and filaments. We also discuss the role of visibility of the prominence fine structures in the estimation of the total height of prominences, which may indicate the height of pre-erupting flux ropes. This parameter is critical for the observational determination of the flux-rope stability. In addition, we employ the WPFS model to assess the effects caused by a projection of the naturally three-dimensional and heterogeneous prominences onto a two-dimensional plane of the sky. We discuss here how the morphological structures of prominences differ when observed in projections from different viewing angles. We also discuss the shapes of the dipped magnetic field lines and the perceived projection of motions of prominence fine structures along such field lines.

**Key words:** magnetic fields – methods: numerical – Sun: filaments, prominences

## 1. Introduction

Quiescent prominences are very heterogeneous and, on small scales, highly dynamical solar phenomena. This fact is documented by numerous prominence observations summarized, for example, in the review by Parenti (2014). However, prominence observations yield only two-dimensional projections of their naturally three-dimensional reality. Such projections inevitably create effects that may significantly influence the appearance of the prominence features. This is, of course, a well-known fact that can be easily demonstrated by the case of a prominence observed at the solar limb and against the solar disk (a filament). The observed filaments exhibit very different morphological features, shapes, and configurations of fine structures and often exhibit significantly different dynamics from prominences (see, e.g., Labrosse et al. 2010). Similar differences may also be expected in prominences that are observed on the limb, but from different viewing angles. This is because individual small-scale prominence features are often semi-transparent in many wavelengths/spectral lines used for observations, including the Hα line. Therefore, the distribution of these small-scale structures in the 3D space, the actual viewing angle, and the radiative transfer effects in the prominence plasma together play an important role in the way we perceive the morphology and fine structures of observed prominences. Even more susceptible to these effects is the appearance of the prominence dynamics, especially on small scales. Some projection effects inherent to the prominence observations can be mitigated by the use of 3D prominence models, which can be studied from any angle. However, the visibility of the prominence features in the models strongly depends on the choice of the visualization

technique—a choice that is further complicated by the fact that prominence plasma radiates very differently at different spectral lines.

In this paper, we construct a new 3D Whole-prominence Fine-structure (WPFS) model, which represents an entire prominence with its numerous fine structures formed by prominence plasma located in dips in the prominence magnetic field. The model is based on a prominence magnetic field configuration designed to qualitatively approximate the morphology of a quiescent prominence observed on 2010 June 22 (Gunár et al. 2014) and employs the fast approximate radiative transfer method for the Hα visualization developed by Heinzel et al. (2015). We use this model to discuss how the morphological structures of prominences differ when observed from different viewing angles and how the projection effects can influence our understanding of motions of prominence fine structures along field lines. To investigate the role of the Hα visibility of prominence structures, we compare three visualization techniques and suggest their best use in prominence modeling. We consider visualization techniques that draw magnetic dips up to a height of 1 pressure scale height or the full extent of magnetic dips and the synthetic Hα visualization technique of Heinzel et al. (2015). We also investigate the role of visibility of prominence fine structures in the estimation of the total height of prominences, which may indicate the height of pre-erupting flux ropes—a parameter critical for the observational determination of the flux-rope stability. In addition, we employ the 3D WPFS model to assess the effects caused by a projection of the naturally three-dimensional and heterogeneous prominences onto a two-dimensional plane of the sky. We discuss here how the morphological structures of prominences differ when observed in projections from different

<footnote n="1">¹Astronomical Institute, The Czech Academy of Sciences, Czech Republic, 25165 Ondřejov, Czech Republic</footnote>

<footnote n="2">²LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université Paris Diderot, 5 place J. Janssen, F-92190 Meudon, France
Received 2018 August 21; revised 2018 September 25; accepted 2018 September 25; published 2018 November 6</footnote>
viewing angles. We also discuss the shapes of the dipped magnetic field lines and the perceived projection of motions of prominence fine structures along such field lines. Moreover, we study the role of the projection effects on the appearance of the prominence morphological structures, apparent shapes of the dipped magnetic field lines, and the perceived projections of prominence fine-structure motions. This allows us to demonstrate that many aspects of prominences that are presently puzzling could be explained when the combined roles of the visibility and projection effects are taken into account.

In the following paragraphs, we summarize the background of prominence physics with a direct relation to the work presented here. A more detailed description of the nature of prominences can be found in Labrosse et al. (2010), Mackay et al. (2010), Schmieder et al. (2014b), or Vial & Engvold (2015).

This work focuses on quiescent solar prominences, which have mostly stable large-scale structures and lifetimes of up to a few months. However, their fine structures exhibit significant dynamics with velocities on the order of $10 \text{ km s}^{-1}$ and lifetimes as short as 5–10 minutes (see, e.g., Parenti 2014). The typically observed dynamics of quiescent prominences range from mass flows in opposite directions within a limited area (counter-streaming, see, e.g., Schmieder et al. 1991; Lin et al. 2003, 2005; Chae et al. 2005); localized prominence oscillations (see, e.g., Okamoto et al. 2007; Berger et al. 2008; Lin et al. 2009; Zapiór et al. 2015); oscillations of the entire bulk of prominence material (reviewed by Tripathi et al. 2009 and Luna et al. 2018); to movements caused by the evolution of entire prominences due to changes in the underlying photospheric flux (Feynman & Martin 1995) and due to external disturbances (see, e.g., Jing et al. 2003, 2006; Luna & Karpen 2012). These types of prominence dynamics are generally consistent with the usually accepted picture of a configuration of prominence magnetic fields. In this picture, the magnetic field is central to the existence of prominences. First, it allows the prominence plasma to cool radiatively by insulating it from the surrounding hot corona. Second, it supports the dense prominence plasma against gravity. The prominence plasma is located in dips of the mostly horizontal field. This picture is supported by prominence magnetic field measurements, which infer the properties of the magnetic field from spectro-polarimetric observations. Examples of measurements of predominantly horizontal magnetic fields in prominences can be found in, e.g., Leroy et al. (1983), Casini et al. (2003, 2009), Orozco Suárez et al. (2014), or Levens et al. (2016a, 2016b).

However, with increasing resolution of the available prominence observations, quiescent prominences reveal small-scale structures and dynamics that are seemingly inconsistent with the existence of the horizontal magnetic field. For example, significant parts of prominences consist of apparently vertical fine-structure threads (e.g., Chae 2010; Schmieder et al. 2010), whose existence in the horizontal magnetic field was discussed by Leroy (1989). Such vertical prominence fine structures can form as vertical pile-ups of dips in the horizontal magnetic field (see, e.g., Dudík et al. 2008). These structures were modeled by, e.g., Low (1982), Heinzel & Anzer (2001), Gunár et al. (2008), and Hillier & van Ballegooijen (2013). The relationship between the vertical fine structures observed in prominences and the horizontal fine structures observed in filaments (prominences observed in

projection against the solar disk) was discussed by, e.g., Heinzel & Anzer (2006). The prominence fine structures may also exhibit dynamics in the vertical direction in the plane of the sky. This is documented, for example, by Chae et al. (2008) and Chae (2010). Moreover, Berger et al. (2008, 2010, 2011) presented observations of plumes that emanate from bubbles located below the prominence body and rise upward, creating localized vortical flows. Another type of observed prominence dynamics that is seemingly inconsistent with the presence of horizontal magnetic fields is exhibited the apparently rotating prominences. These prominences, sometimes called solar tornadoes, were studied by, e.g., Li et al. (2012), Su et al. (2012), and Levens et al. (2016a, 2016b). However, as Levens et al. (2016a, 2016b) demonstrate, even in such prominence structures, the magnetic field can be preferentially horizontal.

The difficulties in interpretation of the sometimes conflicting prominence observations highlight the need for comprehensive prominence models. To construct a detailed 3D model of a prominence we need to start from a 3D model of its magnetic field configuration. Such models were constructed as static, for example, by Mackay et al. (1997, 1999), Aulanier & Démoulin (1998), Aulanier et al. (1999, 2000), Régner & Amari (2004), van Ballegooijen (2004), Dudík et al. (2008, 2012), and Gunár et al. (2014), and as dynamic, in which the magnetic field evolves from the initial configuration, e.g., by van Ballegooijen & Martens (1989), DeVore & Antiochos (2000), Mackay & van Ballegooijen (2009), and Xia et al. (2012). Prominence modeling was reviewed by Gunár (2014).

The dynamic nonlinear force-free field simulations of Mackay & van Ballegooijen (2009) were used by Gunár & Mackay (2015a) to construct the first 3D WPFS model. This model represents an entire prominence with its numerous fine structures. The prominence plasma in the model is located in magnetic dips that occur naturally in the predominantly horizontal prominence magnetic field. In the present paper, we use the linear force-free field (LFFF) modeling approach designed by Aulanier & Démoulin (1998), which has already led to construction of models of on-disk filaments (Aulanier & Démoulin 1998; Schmieder et al. 2006) and off-limb prominences (Dudík et al. 2012; Gunár et al. 2014). This method was extended to calculate linear magneto-hydrostatic extrapolations from magnetograms to model filaments by Aulanier et al. (1999, 2000), Aulanier & Schmieder (2002), and Dudík et al. (2008). To construct the new 3D WPFS model, we use here the LFFF that was built by Dudík et al. (2012) and used by Gunár et al. (2014) to qualitatively approximate the general morphology of the quiescent prominence observed on 2010 June 22. This is the first step toward applying the 3D WPFS models to the existing series of linear magneto-hydrostatic extrapolations of the observed prominence magnetic fields (Aulanier et al. 1999, 2000; Aulanier & Schmieder 2002 and Dudík et al. 2008).

The paper is organized as follows. In Section 2, we briefly summarize the observations of the 2010 June 22 prominence, which provides the basis for the 3D WPFS model developed here. The magnetic field configuration used is described in Section 3 and the details of the new 3D WPFS model are given in Section 4. In Section 5, we use the 3D WPFS model to analyze the effects of the visibility of prominence and filament fine structures on the perceived morphology of observed and modeled prominences. In Section 6, we employ the model to assess the role of projection effects in our understanding of the
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Top row: Hinode/SOT observations obtained on 2010 June 22 at 15:02 UT in the Ca II H line (left) and the Hα line 0.208 Å from the line center (right). Images are aligned to show the same field of view of 103" × 99". Bottom row: SDO/AIA 171, 193, and 304 Å channel observations obtained on 2010 June 22 at 15:37:50 UT. The x and y axes give the solar X and Y coordinates in arcsec, respectively. This figure is adapted from Figures 2 and 3 of Gunár et al. (2014).</figcaption></figure>

prominence composition, the nature of the prominence fine structures, and prominence dynamics. In Section 7, we offer our conclusions.

## 2. Observations that Inform and Constrain the Model

In this section, we briefly summarize the observations of the quiescent polar-crown prominence of 2010 June 22, which serves as a basis for the 3D prominence model constructed in this work. Details of these observations can be found in Gunár et al. (2014). In Figure 1, we show observations obtained in the Hα and Ca II H lines by the Solar Optical Telescope (SOT, Tsuneta et al. 2008) onboard Hinode and in the 171, 193, and 304 Å channels of the Atmospheric Imaging Assembly (AIA, Lemen et al. 2012) onboard the Solar Dynamics Observatory (SDO).

In the top row of Figure 1, we display the prominence in Ca II H (3968.5 Å) in the left panel and the Hα line (0.208 Å from the line center at 6562.8 Å) in the right panel. These observations show that the central part of the prominence is dominated by a bright, vertical column surrounded by numerous fine structures. On the left side, fine structures are quasi-vertical while on the right they appear more tangled. Two voids (called bubbles) are located underneath the fine structures

on both sides of the central column. One is located immediately to the right from the central column and the other is further to the left of it. These voids appear to give rise to frequent dynamical plumes (Berger et al. 2011) that propagate upwards through the prominence body. The bubbles are better visible in the Hα line, where they appear as dark empty regions. In the Ca II H line, we can see some emission at the location of the bubbles, which implies the presence of cool prominence material somewhere along the lines of sight traversing them—either in the front of, behind, or inside the actual bubbles. This different appearance of the bubbles in the Hα and Ca II H lines can be explained by the different optical thickness in the near wing of the Hα line (below unity) and in the Ca II H line (on the order of 10). This means that a significantly lower amount of cool prominence plasma is needed to produce an intensity detectable in the Ca II H line than in the Hα line. For more details, see Gunár et al. (2014).

In the bottom row of Figure 1, we show the observations by SDO/AIA in 171, 193, and 304 Å channels. The SDO/AIA 171 and 193 Å channels show a column-like absorption structure located in the center of the prominence, oriented nearly vertically from the local solar limb. This absorption structure corresponds to the similarly shaped brightest region in
the *Hinode*/SOT Ca II H and Hα images (top row of Figure 1). The central absorption structure is in the 171 Å channel surrounded by a diffuse emission that outlines the general shape of the prominence observed in the Hα and Ca II H lines. This emission is produced by the Prominence-Corona Transition Region (PCTR) plasma that contributes to the 171 Å channel at temperatures of log T [K] ≳ 5.6 (Parenti et al. 2012). In the SDO/AIA 193 Å channel, the prominence is visible purely in absorption caused by the hydrogen and helium continuum opacity together with the effects of emissivity blocking (Anzer & Heinzel 2005). In the SDO/AIA 304 Å channel, the prominence is observed in emission. Typically, prominences observed in this channel look significantly different from those observed in 171 and 193 Å channels but also in the Hα or Ca II H lines. In our case, in the 304 Å channel, we cannot identify the central absorption region visible in the SDO/AIA 171 and 193 Å channels. Due to the high optical thickness of the He II 303.8 Å line, which dominates the 304 Å channel, the radiation in this channel originates in the front parts of the prominence. This masks the central column-like structure and also the bubbles, which are clearly visible (as bright) in the 193 Å channel. The extent of the prominence in the 304 Å channel is similar to the outline of the diffuse emission visible in the 171 Å channel.

### 3. Prominence Magnetic Field Configuration

The prominence model constructed in this work aims to mimic three well-defined morphological features of the 2010 June 22 prominence. These features are the central vertical column seen in absorption in SDO/AIA 171 and 193 Å channels and in emission in the *Hinode*/SOT Ca II H and Hα observations; the regions with numerous quasi-vertical fine structures surrounding the central column; and the voids (bubbles) lying underneath them. Another constraining factor derived from the 2010 June 22 prominence is the height of the central vertical column above the solar surface, estimated to be around 40,000 km.

#### 3.1. *The Original Magnetic Field Model*

The original model of the 3D prominence magnetic field configuration used in the present work was constructed by Gunár et al. (2014). The model was designed to qualitatively approximate the shape and the morphological features of the 2010 June 22 prominence described above.

This type of modeling uses the well-established linear force-free setups developed by Aulanier & Démoulin (1998) and Aulanier et al. (1998, 1999) for the approximation of the observed Hα filament configurations containing lateral feet (barbs). Dudík et al. (2012) adapted these setups for quiescent, cusp-shaped polar-crown prominences. In this adapted configuration, insertion of a parasitic bipole led to perturbation of an otherwise translationally invariant magnetic flux rope by an arch-like magnetic field of the bipole. Two lateral filament feet surround this arch-like magnetic field, each at a given side of the polarity inversion line (PIL). The filament feet are separated from the arches of the bipole by separatrix layers related to two magnetic null-points.

The model of Gunár et al. (2014) utilized this configuration and introduced an asymmetric bipole, which produced a magnetic field configuration with asymmetric lateral feet. The distribution of magnetic dips in this configuration mimics the morphological features of the 2010 June 22 prominence when

<figure><img src="image_4.png" /><figcaption>**Figure 2.** 3D visualization of the magnetic field model of Gunár et al. (2014) projected off-limb at 30° of solar latitude. Magnetic dips are plotted from their bottom up to a height of 300 km, shown in red. Dimensions are given in units of 1000 km. However, note the different geometrical scales along the x and y axes. This figure is adapted from Figure 6 of Gunár et al. (2014).</figcaption></figure>

viewed in a projection shown in Figure 2. In this figure, we can clearly see the quasi-vertical central column consisting of densely packed magnetic dips, the regions with multiple fine structures represented by lateral feet (barbs) extending on both sides of the central column or the void (bubble) located underneath the right part of the prominence. The visualization technique used in Figure 2 draws a portion of the dipped magnetic field lines from the bottom of each dip up to the height of 300 km. This height corresponds to 1 pressure scale height (PSH) when an isothermal plasma with a temperature of 10,000 K is assumed (Aulanier et al. 1998). This technique was designed to approximate the extent of individual dips filled with cool material and thus to visualize the shape of modeled filaments and prominences (see, e.g., Aulanier et al. 1999; Aulanier & Schmieder 2002; Dudík et al. 2008).

#### 3.2. *Extraction of Magnetic Dips*

To construct a new 3D WPFS model, we scaled the magnetic field configuration of Gunár et al. (2014) to obtain its uppermost magnetic dips at an altitude corresponding to the height of the 2010 June 22 prominence, which is about 40,000 km.

To map the magnetic field, we traced a large number of field lines starting with a dense grid of points covering the volume of the modeled prominence with a spacing of 200 km in x-y plane and 100 km along the z axis (prominence height). In each grid point, we traced a field line and determined whether it is dipped. Only dipped field lines were kept. In such a way, we obtained over 40,000 dipped field lines, each traced from the center of the dip all the way toward either their photospheric footpoints or locations where the field line exits the computational box.

On the extracted field lines we identified magnetic dips, excluding those that are extremely shallow dips with the total depth of less than 20 km and properly accounting for the double dips (for more details, see Section 2.2 of Gunár & Mackay 2015a). To select those field lines that accommodate individual prominence fine structures and are independent of all
others, we used the multiple-randomization method described in Section 3.1 of Gunár & Mackay (2015a). After this random selection, we obtained 1186 field lines that cover the entire volume of the modeled prominence with independent fine structures with the cross-section width of 1000 km (see Section 4.1).

## 4. 3D WPFS model

In this section, we describe how the 3D WPFS model is filled with prominence plasma and how it is visualized. We use here the same techniques used by Gunár & Mackay (2015a) to construct the first 3D WPFS model.

### 4.1. Prominence Fine-structure Plasma

To obtain the 3D distribution of the plasma of the modeled prominence, the magnetic dips extracted from the magnetic field configuration of Gunár et al. (2014) are filled with prominence plasma using the method of Gunár et al. (2013). This method fills force-free magnetic dips with plasma in hydrostatic equilibrium. The method generates individual prominence fine structures by filling magnetic dips iteratively from their bottom until either the lower shoulder of the dip is encountered or the prescribed maximum column mass along the field line is reached. The loaded mass does not modify the shape of the magnetic dips or the structure of the prominence magnetic field configuration (we assume a force-free field). The filling method of Gunár et al. (2013) produces individual prominence fine structures, which are not identical despite the fact that we use a set of global input parameters that are the same for all modeled fine structures. The non-uniformity of the obtained fine structures is caused by the variation of the pressure and temperature inside individual fine structures, which depends on the depth and the geometrical shape of each loaded magnetic dip, both of which are unique.

In the present 3D WPFS model, we use the same set of global input parameters as in Gunár & Mackay (2015a, see Table 1 therein). These parameters prescribe the minimum temperature in the dip center (7000 K), the maximum temperature at the dip boundary (100,000 K), the pressure at the dip boundary (0.015 dyn cm⁻²), and the maximum column mass along the magnetic field (1 × 10⁻⁴ g cm⁻²). Another prescribed global parameter is the width of the cross section of the prominence fine structures. This is set to 1000 km. The cross section is circular and radially symmetric. A sketch of the geometry of a magnetic dip with highlighted circular cross sections is shown in Figure 3. The mass-loading within the cross section decreases with the decreasing column mass, as described by Equation (7) of Gunár & Mackay (2015a).

The temperature variation both along the field lines and within the cross section is prescribed semi-empirically and assumes the presence of the PCTR surrounding the cool cores of the prominence fine structures. The PCTR was shown to be necessary to reproduce the observed prominence spectra by, e.g., Anzer & Heinzel (1999), Heinzel & Anzer (2001), Labrosse & Gouttebroze (2004), and Gunár et al. (2007b, 2008, 2010). The temperature within the PCTR rises from the minimum in the center of each fine structure to its boundaries. This rise is gradual in the direction along the field lines (see Equation (2) of Gunár & Mackay 2015a) and very steep across the field (within the cross section, see Equation (6) of Gunár & Mackay 2015a). The steep temperature gradient corresponds to

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Sketch of a magnetic dip in the 3D geometry used by the WPFS model. Circular cross sections of a loaded prominence fine structure are drawn. Red arrows denote the viewing angles used for the $H\alpha$ visualization. This figure is adapted from Figure 4 of Gunár & Mackay (2015a).</figcaption></figure>

the fact that the cross-field thermal conductivity is significantly inhibited.

### 4.2. $H\alpha$ Visualization

To obtain the synthetic $H\alpha$ images of the modeled prominence we use the fast approximate $H\alpha$ visualization method of Heinzel et al. (2015). This method allows us to compute the synthetic $H\alpha$ intensity along a given LOS. The obtained synthetic $H\alpha$ intensities were shown to be reasonably accurate if the optical thickness of individual prominence fine structures is below unity (see Heinzel et al. 2015). The prominence fine structures constructed in the present work have, in general, an optical thickness below this limit. We note, however, that in dense clusters of fine structures the accuracy of the used $H\alpha$ synthesis may decrease. Therefore, the obtained synthetic $H\alpha$ intensities might not be useful for a direct comparison with calibrated $H\alpha$ observations. However, the possible inaccuracies in the range of tens of percents of the absolute intensity in the brightest regions will not have a significant effect on the appearance of the visualized WPFS model, nor will it affect the conclusions of this work.

To obtain the $H\alpha$ intensity along an LOS with a known distribution of the temperature and pressure, we assume a uniform $H\alpha$ line source function. This allows us to derive the $H\alpha$ intensity from only the optical thickness along the given LOS together with the value of the source function (see Equation (8) of Gunár & Mackay 2015a). Values of the source function have been derived for different altitudes above the solar surface by Heinzel et al. (2015) under the non-LTE (i.e., departures from local thermodynamic equilibrium) conditions. The optical thickness can be derived when the hydrogen second-level population is known along the LOS (see Equation (10) of Gunár & Mackay 2015a). Locally, the population of the second level of hydrogen can be estimated from the electron density, which depends on the local temperature, pressure, and ionization
degree (see Equations (11) and (12) of Gunár & Mackay 2015a). The ionization degree and the factor determining the population of the second level of hydrogen have also been derived in non-LTE by Heinzel et al. (2015) for different values of the altitude, temperature, and pressure (see Table 1 therein).

The constructed 3D WPFS model represents an entire prominence with the total extent of the dipped magnetic field region of several 100,000 km. However, the detailed description of the prominence fine-structure plasma properties needed, for example, to resolve the steep gradient of the cross-field PCTR, requires a resolution as fine as 10 km. This introduces challenging numerical complexities that do not allow us to, at the moment, visualize the modeled prominence with an arbitrary viewing angle. Currently, we can produce synthetic $H\alpha$ images of the model only with viewing angles parallel to the y axis and z axis, as in the case of the WPFS model of Gunár & Mackay (2015a). The viewing angle parallel to the y axis (projection onto the x-z plane) visualizes the model as a prominence. In that case, the synthetic $H\alpha$ is in emission (see Figure 4) and no background intensity is assumed. On the other hand, the viewing angle parallel to the z axis (x-y plane) shows the model as a filament. The synthetic $H\alpha$ is then in absorption (dark) against the bright solar disk (Figure 4). The background intensity of the solar disk is obtained from David (1961). Both synthetic $H\alpha$ images are obtained at the $H\alpha$ line center wavelength and have a resolution of 150 × 150 km. This synthetic resolution is comparable with the *Hinode/SOT* observations shown in Figure 1. The same wavelength and resolution were used by Gunár & Mackay (2015a, 2015b). We note that the synthetic $H\alpha$ image of the modeled prominence in Figure 4 has significant differences from the observed $H\alpha$ prominence in Figure 1. This is due to the fact that the 3D WPFS model constructed here aims to qualitatively approximate the general morphology of the 2010 June 22 prominence but does not intend to match it precisely. Even more importantly, the projection onto the x-z plane displayed in Figure 4 shows the modeled prominence from a significantly different angle compared to the observed prominence (Figure 1). As discussed in Section 6.1, the projection angle is very important for the appearance of the prominence structures in $H\alpha$ observations and thus may account for most of the morphological differences between prominences in Figures 1 and 4.

Besides the synthetic $H\alpha$ images, Figure 4 also shows three orthogonal views of 1186 plotted independent field lines. Entire dipped parts of the field lines, i.e., from the bottom of the dip up to its maximum extent, are plotted in the top view (x-y plane) and both side views (x-z and y-z planes).

## 5. Visibility of Prominence/Filament Fine Structures

Models of entire prominences with their numerous fine structures, such as the 3D WPFS model constructed here, allow us to consistently study morphological details of prominences and filaments. However, individual morphological features in both models and observations strongly depend on the visibility of the prominence/filament fine structures. This visibility may be, however, significantly affected by the choice of the visualization technique or the wavelength/spectral line used for the visualization. In the present work, we focus on the role of the visibility of the prominence fine structures due to the choice of visualization techniques.

### 5.1. Visualization Techniques

The position of the prominence fine structures in a static 3D WPFS model is determined by the location of dips in the prominence magnetic field. Therefore, the natural way to visualize the structure of modeled prominences is to draw these dips. Drawing entire dipped parts of magnetic field lines will visualize the full extent of the volume of the field that is dipped and therefore can accommodate static prominence plasma. This technique was used, for example, by Gunár & Mackay (2015a, 2015b, 2016) to visualize the structure and properties of the magnetic field configuration of their 3D WPFS model. We also employ this technique in Figure 4 to show the structure of the magnetic field configuration of the WPFS model constructed in the present work. However, the technique of plotting the full extent of individual magnetic dips always overestimates the true extent of the prominence and filament fine structures. This is because the dense prominence plasma, which is the dominant contributor to the intensity of observed prominences in many spectral lines, accumulates only in the bottom parts of magnetic dips.

If we assume hydrostatic plasma—as in the filling method of Gunár et al. (2013)—it can be argued that the most important part of the dips that needs to be visualized is their lowermost part. More specifically, the most important part is, in this case, located between the bottom of dips and the height corresponding to for example 1 PSH. Typically, the height corresponding to 1 PSH is in prominences estimated to be around 300 km (see, e.g., Aulanier et al. 1998). The disadvantage of simply drawing dips up to a height of 1 PSH is that such a visualization—hereafter called the PSH-dips visualization technique—assigns the same visual weight (emphasis) to all dips regardless of their total depth (height). This means that shallow dips, even those that do not reach a height of 300 km, are drawn with the same emphasis as the deepest dips. The depth of the dips is, however, the dominant factor in the increase of the hydrostatic pressure inside the dips. Therefore, the pressure inside shallow dips may be several orders of magnitude lower than the pressure inside the deepest dips, as the hydrostatic pressure increases exponentially with dip depth. Such a difference in pressure will result in a significantly different radiation emerging from fine structures located in shallow and deep dips. Therefore, the PSH-dips visualization tends to overestimate the visibility of the shallow dips while it underemphasizes the role of the deepest dips. We note that while these arguments are specific to the hydrostatic plasma filling of the dips used in the WPFS models, they will be valid also for other filling methods of force-free dips where the pressure at the dip boundaries is assumed to be coronal.

A more advanced type of a visualization technique is the synthetic $H\alpha$ visualization described in Section 4.2. This spectrum synthesis technique developed by Heinzel et al. (2015) simulates processes that produce the radiation emerging from actual observed prominences and filaments. Therefore, it offers a visualization tool that is close to actual observations. However, the $H\alpha$ synthesis requires the magnetic field configuration to be filled with realistic prominence plasma. This introduces significant numerical challenges and causes resulting synthetic $H\alpha$ images to be dependent on the choice of input parameters of the loaded plasma, even though this dependence is not very significant. Therefore, the use of the synthetic $H\alpha$ visualization may not be practical in many circumstances, especially where complex 3D prominence
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Entire dipped parts of 1186 field lines of the 3D WPFS model are plotted in three orthogonal projections—onto the *x*–*y*, *x*–*z*, and *y*–*z* planes. Included are also two synthetic Hα line center images of the model. One shows the model as a prominence (*x*–*z* plane) and the other depicts the model as a filament (*x*–*y*) plane. Dimensions are in units of 1000 km.</figcaption></figure>

magnetic field configurations need to be visualized in a fast and
efficient way.

In this work, we use the newly constructed 3D WPFS model
to qualitatively compare these three visualization techniques for
the first time. To do so, we display the 3D model as a filament

(Figure 5) and a prominence (Figure 6). In both figures, the top panel shows the synthetic Hα visualization, the middle panel shows the PSH-dips visualization, and the bottom panel shows the visualization where the entire extent of the magnetic dips is plotted.
<figure><img src="image_7.png" /><figcaption>**Figure 5.** 3D WPFS model visualized as a filament with three different visualization techniques. The top panel shows the synthetic $H\alpha$ line center intensity. In the middle panel, we plot magnetic dips to a height of 300 km—the PSH-dips visualization. In the bottom panel, we plot the full extent of the same dips. In the middle and top panels, we plot the same dipped field lines as in Figure 4. Dimensions are in units of 1000 km.</figcaption></figure>

## 5.2. Comparison of Visualization Techniques

In this section, we analyze the similarities and differences between the visualization techniques described above.

In the case of the filament (Figure 5), both the synthetic $H\alpha$ and the PSH-dips visualization produce the same main morphological features. The spine ($S_1$ and $S_2$) and barbs ($B_1$ and $B_2$) can be easily discerned in both images. The barbs, which correspond to the prominence feet visible in Figure 2 and discussed in Section 3, are clearly separated in both cases. However, the extent of individual filament fine structures is clearly shorter in the synthetic $H\alpha$ image than in the PSH-dips image. The spine ($S_1$ and $S_2$) is also much less visible in the synthetic $H\alpha$ image. Moreover, small-scale features, such as an opening (e) inside the barb $B_1$ are difficult to identify in the PSH-dips image. In addition, the synthetic $H\alpha$ image shows a plethora of fine details in the composition and intensity of the

filament fine structures that are impossible to distinguish in the PSH-dips image. This is caused by the fact that the PSH-dips visualization does not distinguish between deep (and thus more visible) and shallow (and thus often nearly invisible) dips. All dips are drawn with the same emphasis. However, the fact that the main morphological features of the filament are similar in both visualizations means that the PSH-dips visualization is well suited for an efficient imaging of morphology of simulated and observed filaments.

In contrast, when the full extent of magnetic dips is drawn (bottom panel of Figure 5), even some of the larger scale features differ from the synthetic $H\alpha$ image. For example, the gap between the clearly separated barbs $B_1$ and $B_2$ visible in $H\alpha$ is difficult to identify; the spine ($S_1$ and $S_2$) is significantly more pronounced; the apparent extent of individual fine structures has increased; and we can identify several cusp-like features (f, g, h) that do not exist in either the synthetic $H\alpha$ or
<figure><img src="image_8.png" /><figcaption>**Figure 6.** 3D WPFS model visualized as a prominence with three different visualization techniques. The top panel shows the synthetic $H\alpha$ line center intensity. In the middle panel, we plot magnetic dips to a height of 300 km—the PSH-dips visualization. In the bottom panel, we plot the full extent of the same dips. In the middle and top panels, we plot the same dipped field lines as in Figure 4. Dimensions are in units of 1000 km.</figcaption></figure>

the PSH-dips images. These differences mean that drawing the full extent of the magnetic dips may not be the most accurate technique for comparison between models and observations, for example, in the $H\alpha$ line. However, as we show in Section 6, information about entire traces of the dipped field lines in the 3D space is important for the understanding of the true extent of the magnetic dips and for disentangling between true and apparent motions of prominence fine-structure motions along the dipped field lines.

In the case of the prominence (Figure 6), the comparison between the synthetic $H\alpha$ and the PSH-dips visualizations is even worse. The volume of the prominence is significantly more extended if the PSH-dips visualization is used, compared to the synthetic $H\alpha$ intensity. This difference is again caused by the fact that the PSH-dips visualization draws all dips (both deep and shallow) with the same visual weights. However, the shallow dips may be practically invisible in $H\alpha$ while the deep dips are dominantly bright. In the synthetic $H\alpha$ image, we can identify well-defined vertical prominence features that correspond to the barbs $B_1$ and $B_2$. In fact, these vertical features are composed of deeper dips that happen to be aligned quasi vertically in this projection onto the x-z plane. However, it is more difficult to identify these vertical features in the PSH-dips image, specifically in $B_1$. We note that such a quasi-vertical alignment of dips is caused by the three-dimensional structure of the magnetic field and not by any selection effect.

The shallow dips that are visible in the PSH-dips image but mostly invisible in the synthetic $H\alpha$, correspond to the spine ($S_1$ and $S_2$). These dips are filled with cool prominence plasma but due to their shallow depth, the pressure inside them is low, resulting in a very low emerging intensity. For more details of this mechanism, see discussion in Gunár & Mackay (2015a).

Smaller scale features, such as the V-shape of the prominence part $B_1$ or the opening in it (which may correspond to the opening (e) in Figure 5), are difficult to identify in the PSH-dips image. Even finer details are impossible to discern in the PSH-dips image, as is also true in the case of the filament. This comparison shows that the PSH-dips visualization may mask even larger scale prominence features or also display features that would not be visible in $H\alpha$ at all. The PSH-dips visualization also generally overestimates the entire extent (volume) of the prominence compared to the synthetic $H\alpha$ visualization, although this may be different in other spectral lines.

When the full extent of magnetic dips is plotted (bottom panel of Figure 6), the apparent prominence volume is even larger and the prominence features are even more difficult to recognize than in the case of the PSH-dips visualization. This is apparent also from Gunár & Mackay (2015a, 2015b), where the extent of the visualized magnetic dips is significantly larger than the extent of the synthetic $H\alpha$. Therefore, even though the PSH-dips visualization is still efficient for visualization of magnetic field configurations in a prominence view, it should be approached with much more caution than in the case of filaments.

### 5.3. Prominence Height

The choice of visualization technique does not only affect the visibility of larger and smaller scale prominence structures, but it also has an influence on the determination of other fundamental properties such as prominence height. From Figure 6, it is apparent that the modeled prominence appears considerably taller in images where magnetic dips are plotted than in the synthetic $H\alpha$ image. The altitude of the uppermost
dips in the used magnetic field configuration is, by design, 40,000 km. However, the altitude of the uppermost prominence fine structures visible in the synthetic $H\alpha$ image (the prominence apex) is at most 37,000 km—a difference of almost 10%. This difference is caused by the fact, that the uppermost magnetic dips in the modeled magnetic field configuration are very shallow. Such shallow dips can only accommodate plasma with a very low pressure and are thus practically invisible in $H\alpha$. To obtain a precise measure of this difference we would need to study in detail the contrast of the observations, such as those by Hinode/SOT, and compare them with the contrast of synthetic $H\alpha$ images. However, such a precise measure of the difference in prominence height obtained from different visualization techniques would not vary significantly from our simple estimate. This estimate shows that the apparent altitude of the top of the modeled prominence visualized by $H\alpha$ synthesis is approximately 10% lower than the altitude of the uppermost dips in the modeled magnetic field configuration. Similar differences can be also found in the work of Gunár & Mackay (2015a, 2015b) and might also be expected in actual prominences.

This result is important for the estimation of the maximum height that prominences can reach before they become unstable and erupt as coronal mass ejections (CMEs). To illustrate this stability criterion, let us consider the flux-rope model for prominences, in which the flux rope is embedded in a coronal arcade magnetic field. In such a model, a CME is always triggered when the axis of the flux rope reaches a critical height $z_c$. This height is characterized by the drop-off of the coronal arcade field with the altitude $z$. This becomes faster than $z^{-a}$ for $z \gg z_c$, where lower (or, respectively, higher) values of the exponent $a$ correspond to lower (resp. higher) values of the critical height $z_c$. This is the essence of the so-called torus instability. For the value of exponent $a$, idealized analytical calculations predicted a range of $1-1.5$, depending on the shape and width of the flux-rope axis (van Tend & Kuperus 1978; Kliem & Török 2006; Démoulin & Aulanier 2010). On the other hand, 3D numerical simulations systematically obtained higher critical heights $z_c$, with $a = 1.4-1.5$ (Aulanier et al. 2010; Amari et al. 2014; Inoue et al. 2015; Zuccarello et al. 2015), and even $a = 1.5-1.9$ (Fan 2010; Kliem et al. 2013).

These theoretical predictions led to several observational studies. Thus far all of them used the altitude of prominence apex as a proxy for the axis of the flux rope. Most of the studies obtained lower critical heights $z_c$ than in numerical models, mostly with $a = 1-1.1$ (Filippov & Den 2001; Filippov & Zagnetko 2008; Schmieder et al. 2014a; Zuccarello et al. 2014; Su et al. 2015; Roudier et al. 2018). Some others obtained wider ranges, $a = 0.5-2$, but always peaked at the same low value, $a = 1$ (McCauley et al. 2015; Aggarwal et al. 2018). To reconcile the mean observed low critical heights (with $a \simeq 1$) with the mean modeled critical heights (with $a \simeq 1.5$), Zuccarello et al. (2016) proposed an idea that relies on the following geometrical argument. In an arc-shaped flux rope, the topmost plasma-supporting dips are always located below the axis of the rope. The reason is that the twist, which is normally responsible for the existence of dips, can be balanced by the curvature of the rope axis. Using numerical simulations, Zuccarello et al. (2016) were able to recover an apparent range of $a = 1.-1.2$, using the altitude of topmost dips to measure the altitude of the prominence apex (like in observations),

instead of using the altitude of the flux-rope axis. But this geometrical property is obviously not sufficient to explain the lower critical heights associated with the values $a \le 1$, as reported in the most recent statistical studies (McCauley et al. 2015; Aggarwal et al. 2018).

Our present result provides an answer to this puzzle. Indeed, we found that the topmost dips are hardly visible in $H\alpha$. So it should be considered that the apex of a prominence is not only lower than the altitude of the flux-rope axis but is also lower than the altitude of the topmost dips. So any estimation of the exponent $a$, as based on the measurement of the altitude of observed prominences, ought to be even lower than the values given by the topmost dips. This has two consequences. First on the theory side, it shows that even observed values of $a \le 1$ do not necessarily jeopardize the validity of the torus instability for triggering of CMEs. Second, on the space weather side, it implies that the onset times of eruptions must be advanced in time when they are estimated from the altitude of slowly growing prominences in the framework of the torus instability.

# 6. The Role of Projection Effects

The way we see and interpret the composition of prominences and the shapes and dynamics of their fine structures is not affected only by their visibility. As we discussed in the previous section, visibility depends on factors such as the mass-loading (pressure) of individual prominence fine structures or the choice of spectral line used for visualization or observations. In the case of prominence modeling, the visibility of various prominence features also depends on the choice of visualization technique. However, even if we assume that all visibility factors are fixed, the appearance of prominences would vary greatly simply due to projection effects when different viewing angles are assumed for visualization.

## 6.1. Prominence Morphology

It is very difficult to observe the same prominence simultaneously from different angles. On the other hand, the 3D nature of the WPFS models gives us the opportunity to study projection effects in a well-controlled environment. As mentioned previously, we can currently use the synthetic $H\alpha$ visualization only for two specific viewing angles—the projection onto the x-y and x-z planes (see Figures 4–6). This limitation is caused by the numerical challenges in dealing with a very large computational box containing an entire prominence but with very fine spacing needed to properly represent the distribution of the prominence fine-structure plasma. However, by plotting the magnetic dips, we can visualize the modeled prominence with any viewing angle.

In the case of the prominence studied here, projection effects are most pronounced when the viewing angle is nearly parallel with the direction of the magnetic field inside the prominence fine structures. To visualize the modeled prominence with such a viewing angle, we use here the projection onto the y-z plane (see Figure 4). A close-up view of this projection is shown in Figure 7. In the top panel of this figure, we plot magnetic dips drawn up to a height of 300 km. In the middle panel, we plot the entire dipped portions of the same field lines. As we have shown in Section 5.2, there is a large difference between these two visualizations. This difference is even more apparent in the projection onto the y-z plane, which corresponds to a situation
<figure><img src="image_9.png" /><figcaption>**Figure 7.** 3D WPFS model visualized as a prominence in projection onto the y-z plane. In the top panel, we plot magnetic dips to a height of 1 PSH. In the middle panel, we plot the full extent of the same dips. The bottom panel shows a randomly selected subset of the same dipped field lines as in the middle panel. Specific populations of field lines are highlighted with various colors. Dimensions are in units of 1,000 km.</figcaption></figure>

in which the modeled prominence is observed along its axis. This view is not far from the view shown in Figure 2, which simulates the orientation of the 2010 June 22 prominence. In the projection onto the y-z plane, many of the dips plotted up to a height of 300 km appear to be very short. This effect is most pronounced in two regions located above voids $V_1$ and $V_2$, which are devoid of magnetic dips (see the top panel of Figure 7). These very short dips appear short only because the used viewing angle is nearly parallel to the direction of their magnetic field. To be more precise, the magnetic field in these dips is parallel to the used viewing angle only in their lower parts. When the full extent of these field lines is plotted (middle panel of Figure 7) it is clear that what appear to be extremely short dips are in fact very large and deep dips. Moreover, these dips are significantly elongated in the direction of the field. It is

interesting to note that such seemingly very short but in reality long and deep dips would appear as bright, localized plasma blobs if visualized using the $H\alpha$ synthesis in the projection onto the y-z plane. Because some of these apparently short dips seem to be aligned in quasi-vertical stacks, such bright blobs might form quasi-vertical fine structures. In contrast, fine structures located in nearby dips which are more extended would have horizontally elongated shapes in the synthetic $H\alpha$ images.

Other interesting features that are clearly visible in this projection in the PSH-dips image are the central column-like pile up of dips rising vertically from the bottom to the top of the prominence; the vertical gap separating the central column from the left barb (feet); and the left barb and the right barb, which both appear to be detached from the central column.
These morphological features, together with two voids ($V_1$ and $V_2$) located underneath, resemble the general morphology of the 2010 June 22 prominence (Figure 1). However, this resemblance is achieved only with a viewing angle that is close to the projection onto the $y-z$ plane. For example, the central column-like structure clearly visible in the $H\alpha$, $Ca II H$, and SDO/AIA 171, 193 Å channel observations can be identified as such only in the projection onto the $y-z$ plane (Figure 7). In the projection onto the $x-z$ plane (Figure 6), it is difficult to find its counterpart. In fact, this central column is formed by dips belonging to the spine ($S_1$ and $S_2$ in the middle panel of Figure 5). This example demonstrates that prominences, even when represented by rather simplified models, are inherently complex 3D structures in which projection effects play a very important role.

## 6.2. Field Line Traces

In Section 5.1 we argued that bottom parts of dips plotted in PSH-dips visualization may serve as a good representation of the position and extent of the prominence fine structures located in deeper dips. We have also noted that PSH-dips visualization overestimates the importance of shallow dips which are plotted with the same emphasis as the deep dips. However, from Section 6.1 it is clear that even in the case of the deep dips, plotting only their bottom 300 km may not reveal all what we can learn from modeled prominence magnetic field configurations. As we argue in this section, the traces of the dipped magnetic field lines projected with a given viewing angle may also provide important insights into the nature of prominences.

For example, at the top of the modeled prominence shown in the middle panel of Figure 7 one can see a large number of dips that appear to form spiral structures. However, no such structure can be recognized in the PSH-dips image in the top panel of Figure 7. Moreover, in both barbs we can find populations of very deep dips with field lines traces that often appear vertical. Many of these field lines correspond to the very short dips visible above voids $V_1$ and $V_2$ in the PSH-dips image. In the bottom panel of Figure 7 we have removed nearly 75% of the field lines plotted in the middle panel. This was done to allow a clearer view of the remaining dips. However, we stress here that we have not attempted to select only those dips that are in some way interesting. The plotted dips were selected randomly. From the plotted field lines we have selected three populations of dips which we highlight using different colors. The population at the top of the prominence (blue, yellow and light green field lines) appears to trace significant parts of or even complete ellipses. In contrast, populations from the left barb (cyan and magenta) and from the right barb (red and purple) indicate very deep dips that have at least one half strictly vertical. Two dark green field lines (one on the left and one on the right) highlight similar field lines completing a full, vertically highly stretched ellipses. These indicated field lines have traces which seem to be in contradiction with the usual picture of the horizontal prominence magnetic field. However, as we argue in the following section, these traces bear many similarities with trajectories of fine-structure motions that are occasionally observed in prominences.

## 6.3. Apparent Motions

In the previous section, we showed that the traces of dipped field lines of the modeled magnetic field configuration can often be rather surprising. In this section, we try to imagine how the movement of prominence fine structures along field lines such as those highlighted in Figure 7 (bottom panel) would appear to an observer. In doing so, we assume a motion of compact fine structures along fixed field lines. However, even in such a simplified case, the resulting projection of the prominence fine-structure motion onto the $y-z$ plane is rather unusual.

For example, prominence fine structures moving along the blue, yellow, and light green field lines (population at the top of the prominence) would appear to move along circular (elliptical) paths in the plane of the sky (in the projection onto the $y-z$ plane). Such a movement, if coherent, would in many respects resemble observed tornado-like prominence-top features, which seem to be formed by coherently rotating large volumes of prominence material (see, e.g., Li et al. 2012; Su et al. 2012). However, such seemingly rotating prominence structures can be present even in the predominantly horizontal magnetic field, as shown by Levens et al. (2016a, 2016b). An explanation for this phenomenon can also be offered by the 3D WPFS model constructed here. To do so, we plot in Figure 8 the same selected field lines as in the bottom panel of Figure 7 in projections onto the $x-y$, $x-z$, and $y-z$ planes. In Figure 8, we use the same colors as in the bottom panel of Figure 7 to highlight the same field lines in all three projections. From Figure 8 it is clear that all highlighted field lines are in fact predominantly horizontal in the 3D space. This is also the case for the blue, yellow, and light green field lines located at the top of the prominence. These field lines appear to form a spiral shape only in the projection onto the $y-z$ plane. When seen from the top ($x-y$ plane), the blue, yellow, and light green field lines clearly belong to the spine and are only slightly S-shaped. The same field lines projected onto the $x-z$ plane form rather shallow, very elongated dips in the upper part of the prominence. A movement along these field lines would appear as a circular motion only in the projection onto the $y-z$ plane. In the projection onto the $x-z$ plane, the movement would be practically horizontal. Such a scenario is consistent with the findings of Schmieder et al. (2017). These authors reconstructed a movement of observed prominence fine-structure blobs that appeared to move along circular trajectories in the plane of the sky. However, the reconstructed 3D trajectories show that the true movement is dominated by velocity vectors that are predominantly horizontal, either in a direction towards or away from the observed.

Another example of an atypical motion would be a movement of prominence fine structures along the field lines highlighted in the left and right barbs (cyan and magenta, red and purple, and dark green field lines). Such a movement would appear as a more or less vertical motion of prominence fine structures that would rise and fall intermittently. Such a motion would create different, seemingly incoherent patterns due to significant differences in the geometrical shape of magnetic dips in different places in the prominence. Observations of such incoherent more-less vertical dynamics would not be dissimilar from occasionally observed prominence fine-structure motions sometimes attributed to magneto-hydrodynamic instabilities (see, e.g., Berger et al. 2008, 2010, 2011). It is interesting to note that populations of the seemingly vertical field lines are located
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Entire dipped parts of the field lines of the 3D WPFS model are plotted in three orthogonal projections—onto the *x*–*y*, *x*–*z*, and *y*–*z* planes. The same field lines as in Figure 4 are plotted. Specific populations of field lines are highlighted with various colors. Dimensions are in units of 1000 km.</figcaption></figure>

exclusively above regions devoid of magnetic dips ($V_1$ and $V_2$ in Figure 7) that would appear as dark voids (bubbles) in $H\alpha$ observations. In several prominence observations, such voids seem to be the source of plumes, which contribute to the intermittent vertical motions of the prominence fine structures above the voids. However, the cyan and magenta, red and purple, and dark green field lines exhibit dominant vertical orientation only in the projection onto the *y*–*z* plane. From Figure 8 it is clear that these field lines form deep dips in the *x*–*z* plane but are still predominantly horizontal. A movement of prominence fine structures along such dipped field lines would appear to be vertical only in the projection onto the *y*–*z* plane. If observed in the *x*–*z* plane, such motions would appear to be quasi-horizontal.

These two examples clearly demonstrate the significance of the role of projection effects in observing and understanding of prominence fine structures and their dynamics.

## 7. Conclusions

In the present paper, we construct a new 3D WPFS model of an entire prominence with its numerous fine structures. Though this model is based on an idealized magnetic field configuration, it demonstrates that many aspects of prominences that are still puzzling could be explained by currently established views about their structures and magnetic field configuration, if the combined roles of visibility and projection effects are taken into account.

One important aspect of the visibility of prominence structures is the apparent prominence height, which is a significant factor in prominence stability. As we discuss in Section 5.3, the prominence height derived using the synthetic $H\alpha$ visualization is about 10% lower than the actual altitude of the uppermost dips in the model. This is due to the fact that the dips at the top of the modeled prominence are very shallow and thus practically invisible in $H\alpha$. The difference in the derived heights provides an answer to the discrepancy between the height of prominences derived from observations and from numerical simulations. The observed prominence heights are

systematically lower than those obtained from simulations (see Section 5.3). This discrepancy is likely caused by the same mechanism, whereby the topmost dips are shallow and cannot accommodate high-density prominence plasma and are thus invisible in the $H\alpha$.

Because the visibility of prominence features (both small-scale and large-scale) in models is significantly affected by the choice of the used visualization technique, we compare three such techniques (see Section 5.1). The considered techniques are the PSH-dips visualization in which magnetic dips are plotted up to a height of 1 pressure scale height (PSH); visualization drawing the full extent of the magnetic dips; and the synthetic $H\alpha$ visualization technique of Heinzel et al. (2015).

The synthetic $H\alpha$ visualization simulates processes that produce radiation emerging from actual observed prominences. Therefore, it offers a visualization tool that can provide synthetic images of modeled prominences and filaments that are close to actual observations. However, the $H\alpha$ spectral synthesis introduces significant numerical challenges (see Section 4.2) that make its use computationally expensive. It thus becomes impractical in many circumstances, especially where complex 3D prominence magnetic field configurations need to be visualized.

The PSH-dips visualization is fast (portions of dipped field lines are drawn without a need to fill prominence plasma on them) and well suited for the comparison of prominence magnetic field configurations with filament observations. However, it has drawbacks when used for comparison with prominence observations. This is because, in the case of filaments, the PSH-dips visualization shows the same general morphological features as the synthetic $H\alpha$ visualization (Figure 5). However, a prominence visualized by the PSH-dips visualization is much wider than when visualized by the $H\alpha$ synthesis (see Figure 6). Even large-scale prominence morphological structures are difficult to identify properly in the PSH-dips images. Moreover, some features, which appear prominently in the PSH-dips visualization, are practically invisible in the synthetic $H\alpha$ images—an effect which could be
misleading when the PSH-visualization is compared with
observations. We note that the small-scale features and
intensity details visible in synthetic Hα images cannot be
identified in the PSH-dips images in both cases. Therefore, if
such details are a topic of a study, the synthetic Hα
visualization needs to be used.

The visualization technique where the full extent of magnetic dips is plotted is not suitable for determination of the morphology of modeled prominences or filaments. This is because the drawing of entire dips masks even the large-scale structures in filaments and prominences (see Figures 5 and 6). However, the advantage of drawing the full extent of the dips is that it allows the identification of the true depth of individual dips and thus their true importance for the visibility of the prominence/filament structures.

The projection effects are caused by the fact that the complex and naturally three-dimensional prominences are observed in two-dimensional projections. Therefore, observed prominences may appear dramatically different from different viewing angles. In Section 6.1, we show how the modeled prominence differs when projected onto the x-z plane (Figure 6) and the y-z plane (Figure 7). For example, in the projection onto the y-z plane, the prominence has a cusp-like shape. In such a projection we can identify the central vertical column with two lateral feet (barbs) and two voids located underneath the barbs (Figure 7). However, the same model projected onto the x-z plane appears as a significantly different prominence with two separate quasi-vertical features and diffuse areas surrounding them (Figure 6). It would be difficult to argue that these prominences are in fact the same object or even that they have a similar magnetic field configuration.

Even more stunning examples of consequences of projection effects are the apparent traces of the dipped magnetic field lines. In Section 6.2, we demonstrate that in certain projections, the magnetic dip traces may exhibit shapes that dramatically differ from a simple, predominantly horizontal field. Some of these traces outline a magnetic field that appears vertical while others draw elliptical or spiral shapes (see Figure 7). Such shapes are in apparent contradiction with the usual notion that prominence magnetic fields are predominantly horizontal (see, e.g., Leroy et al. 1983; Casini et al. 2003, 2009; Orozco Suárez et al. 2014). On the other hand, the existence of circular or vertical field lines would support the observations of apparently rotational (Li et al. 2012; Su et al. 2012; Levens et al. 2016a, 2016b) and intermittent vertical motions (Chae et al. 2008; Chae 2010; Berger et al. 2011), respectively. However, as we demonstrate in Figure 8 and further argue in Section 6.3, there is no need for a contradiction between the existence of such observed shapes and motions and the existence of the predominantly horizontal magnetic field. Figure 8 clearly shows that even though the highlighted field lines exhibit unconventional shapes in the projection onto the y-z plane, their traces in the 3D space are dominated by the horizontal component of the magnetic field. This means that any motion of the prominence fine structures along these field lines would happen within magnetic dips with the predominantly horizontal field, even though they would appear as circular (as shown by Schmieder et al. 2017) or vertical motions in certain projections.

S.G., J.D., and P.H. acknowledge the support from grant 16-17586S of the Czech Science Foundation (GAČR). S.G. is

thankful for financial support from the Observatoire de Paris
through its “Postes Temporairement Vacants” programme
during his visit at LESIA. S.G., J.D., and P.H. acknowledge
the support from project RVO:67985815 of the Astronomical
Institute of the Czech Academy of Sciences. G.A. and B.S. are
thankful for the financial support and the warm welcome from
the Solar Department during their visits at the Astronomical
Institute of the Czech Academy of Sciences in Ondřejov. B.S.
acknowledges financial support from grant 16-17586S of the
Czech Science Foundation (GAČR) during her visit at the
Astronomical Institute of the Czech Academy of Sciences.
S.G., B.S., and P.H. acknowledge support from the Interna-
tional Space Science Institute, Bern, Switzerland to Interna-
tional Team 374.

ORCID iDs

Stanislav Gunár https://orcid.org/0000-0003-3889-2609

Jaroslav Dudík https://orcid.org/0000-0003-1308-7427

Guillaume Aulanier https://orcid.org/0000-0001-5810-1566

Brigitte Schmieder https://orcid.org/0000-0003-3364-9183

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