3D WHOLE-PROMINENCE FINE STRUCTURE MODELING

Any simulated prominence magnetic field configuration intended for use in WPFS modeling has to have sufficiently high resolution (Figure 1(a)). This is needed to allow the WPFS model to naturally resolve the internal structure of the modeled prominence without creating any numerical artifacts, such as grid-dependent features. Such artifacts could be layers of dips at the same height with gaps in between them. This can be caused by tracing field lines only from sparsely distributed grid points. The level of the required resolution (density of points from which field lines are traced) is related to the dimensions of the assumed cross-section of the individual prominence plasma fine structures. For example, the size of the gaps of unresolved magnetic field lines between layers of (resolved) dips must be smaller than the cross-section of the plasma fine structures. We note that such a relation is strongly dependent on the nature of the prominence magnetic field configuration used and should be considered individually for various WPFS models.

Special attention needs to be paid to the identification of the dipped portions of magnetic field lines (Figure 1(b)). An automatic procedure identifying all of the dips within a magnetic field configuration may recognize series of very small dips located along a single field line inside a much large dip but not take into account the presence of the large dip itself. Because the large and deep dips have a much greater impact on the internal structure of the prominence than smaller shallow ones (they can hold significantly more dense mass and thus be much more visible), they need to be all properly identified. In the case where small dips are identified within a much larger dip, a small hump of the magnetic field should be removed (see Figure 1(b)).

High resolution magnetic field configurations result in a large number of often densely packed field lines with their respective minimum distances less then the assumed cross-field dimensions of the prominence plasma fine structures. As any overlap between prominence plasma fine structures needs to be avoided (for numerical reasons), only field lines that are at a given distance from all other selected field lines are to be chosen (Figure 1(c)). The given distance is based upon the chosen cross-section of the plasma fine structures. In this work we call such field lines independent. To ensure that no arbitrary selection criteria will influence the appearance of the modeled prominence, a random selection process can be used to find all independent field lines.

In the case where the configuration of the magnetic field is not varying significantly on the spatial scales comparable with the assumed cross-section width of the prominence plasma fine structures, one can use a selected field line as a representation of the shape of the magnetic field inside the whole cross-section. The dipped portion of such a field line can be filled with prominence plasma, for example, by a plasma in hydrostatic equilibrium using the method developed by Gunár et al. (2013a). This provides the variation of the plasma parameters along the field line. Next, a realistic plasma distribution within the cross-section can be assumed to derive a full 3D description of the plasma forming individual prominence fine structures. In this way all independent field lines can be filled with prominence plasma. Each of the resulting prominence fine structures will depend on the shape of the dipped portion of its respective guiding field line and thus will be composed of plasma influenced by the local configuration of the whole-prominence magnetic field.

We use the results of the 3D NLFF simulations of Mackay & van Ballegooijen (2009) as a realistic prominence magnetic field configuration. However, it is not our intention to validate these simulations by comparing them with other models or observations. We use them as the starting point for the development of the fine structure models of entire prominences. In these simulations the initial distribution of the magnetic flux in the photosphere was chosen to be that of a magnetic arcade, but with flux concentrations that are displaced from one-another. One unit in the dimensionless length-scale is chosen to be 60,000 km, with the arcades separated by two length-scale units, i.e., 120,000 km. These choices are arbitrary but are taken such that the arcades lie four typical supergranular cells apart. The initial flux distribution in the photosphere is shown in Figure 2(a). The solid/dashed contours denote positive/negative flux. From this photospheric distribution we construct a 3D linear force-free field with the value of α fixed at $\alpha =-1.477\times {{10}^{-8}}$ m⁻¹. This is sufficiently large to produce a dipped magnetic field configuration representing the basic structure of a prominence. This linear force free field is used as the initial condition. We note that the chosen value of α is well below the critical value, which is −2.14 × 10⁻⁸ m⁻¹.

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To detect the presence of magnetic dips in the resulting configuration, each grid point within the 3D computational domain is tested. Dips occur when the vertical component of the field satisfies

$$\begin{eqnarray}&&{{B}_{z}}=0\qquad {\rm and}\qquad {\boldsymbol{B}} {\boldsymbol{.}} \nabla {{B}_{z}}\gt 0.\end{eqnarray} \tag{ 1 }$$

All grid points satisfying this criterion are used as the starting points for plotting magnetic field lines.

In Figure 2 portions of dips are drawn up to the depth of one pressure scale-height (PSH) or until the height of the lower peak is reached. For the case of an isothermal plasma and assuming a temperature of 10,000 K, the dip depth equal to one PSH is approximately 300 km. We note that this technique, commonly employed in simulations of the prominence magnetic field, is not used elsewhere in this work, apart from Figure 2.

After the initial magnetic flux distribution is established a bipole is inserted into the initial configuration through the technique described in Section 2.2.3 of Mackay & van Ballegooijen (2009). The bipole is placed in the positive polarity side of the polarity-inversion line (PIL) and is aligned parallel to the overlying arcade as can be seen in Figure 2(b). The ratio of bipole flux (${{F}_{{\rm b}}}$) to arcade flux (${{F}_{{\rm a}}}$) is ${{F}_{{\rm b}}}/{{F}_{{\rm a}}}=0.0084$, while the ratio of bipole to filament flux (${{F}_{{\rm f}}}$) is ${{F}_{{\rm b}}}/{{F}_{{\rm f}}}=2.7$. With this orientation the minority polarity for this side of the PIL lies closest to the dips and is then advected toward the main body of the filament by a boundary flow. As the minority polarity is advected toward the main body of the filament, the coronal field is perturbed and evolves through a series of quasi-static NLFF states as described by the magneto-frictional technique of van Ballegooijen et al. (2000) and Mackay & van Ballegooijen (2006). In Figure 2(b) we show an enlarged view of the filament after 3000, 60 s time-steps. Due to the presence of the bipole the main body of the filament separates into two parts. The gap between them is caused by the absence of dips, not an absence of plotted field lines. Full details of the 3D NLFF prominence magnetic field simulations can be found in Mackay & van Ballegooijen (2009).

In this paper we use the simulation snapshot after 2000, 60 s time steps (similar to Figure 2(b)) as the 3D magnetic field configuration of the WPFS model. To obtain a magnetic field resolution that is sufficiently high to resolve prominence plasma fine structures with cross-section width of 1000 km, we increased the original resolution of the grid used in the NLFF simulations from 256³ to 1280³ grid points. In such a high-resolution configuration we identify nearly 40,000 dipped field lines. For selection of those field lines that can accommodate non-overlapping plasma fine structures we use multiple randomizations that help us to overcome any arbitrary selection effects. We randomly select the first field line to be taken as independent and then we scan through all other dipped field lines in a random order. The minimum distance of the dipped portion of each scanned field line is evaluated against all field lines already found to be independent. If the minimum distance is above the assumed cross-section width of the prominence plasma fine structures (in this case 1000 km), the scanned field line is also taken to be independent. Then the next field line (in random order) is considered, until all dipped field lines in the magnetic field configuration are scanned. The total number and composition of a set of independent field lines depends on the random choice of the starting field line and on the random order of the subsequently scanned field lines. In Figure 3 we show the configuration of 835 independent field lines (out of about 40,000) in the top view (x–y plane) and side views (x–z and y–z planes). Only dipped portions that will be filled with prominence plasma (see next section) are plotted.

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To obtain the plasma distribution within the entire prominence we fill the dipped portions of all (or a randomly selected part of) the independent field lines with plasma in hydrostatic equilibrium using the method developed by Gunár et al. (2013a). This was shown to produce individual prominence fine structures with a realistic distribution of the gas pressure and temperature including the PCTR.

In this work we use a special 3D grid on which we define all physical quantities of all fine structures. The choice of this grid is based on the fact that all dips in the magnetic field configuration used are preferably oriented along the x-axis of our computational domain. This allows us to use less dense spacing along the x-axis (along the length of the dips) and a denser spacing only in the y–z plane (see Figure 4). We further assume that cross-sections of all prominence fine structures are parallel to the y–z plane instead of being strictly perpendicular to the local magnetic field vector. This allows us to significantly simplify the development of this first 3D WPFS model and ease the computational demands. However, a general WPFS model will require dense spacing in all three-dimensions. In this work we use a grid spacing of 10 km in the y–z plane and 150 km along the x-axis for the plasma distribution.

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Each magnetic dip is filled iteratively from its bottom until it either reaches the required column mass, or until the height of the lower shoulder of the dip is reached. The temperature variation of the plasma inside the dip is prescribed semi-empirically and considers two distinct shapes of the PCTR. These represent a steep gradient of the temperature in the direction perpendicular to the magnetic field and a more gradual temperature increase in the direction along the field. The temperature variation along the field is defined for the left and right part of the dip individually as

$$\begin{eqnarray}&&T(\xi )={{T}_{0}}+({{T}_{{\rm tr}}}-{{T}_{0}}){{\left| \frac{\xi }{L} \right|}^{{{\gamma }_{{\rm al}}}}}.\end{eqnarray} \tag{ 2 }$$

Here ξ represents the geometrical coordinate parallel to x, but defined from the center of the local dip toward either its left or right edge. T₀ is the central minimum temperature and ${{T}_{{\rm tr}}}$ represents the transition region temperature. The temperature gradient along the magnetic field lines is described by the exponent ${{\gamma }_{{\rm al}}}$. This formula is based on the temperature variation used by Heinzel & Anzer (2001). The exponent ${{\gamma }_{{\rm al}}}$ considers the effect of the thermal conductivity along the magnetic field lines resulting in a gradual rise of the temperature from the cool central part toward the PCTR. In this work, we use values of T₀ = 7,000 K, T_tr = 100,000 K, and ${{\gamma }_{{\rm al}}}=2$ for all prominence plasma fine structures (see Table 1).

The hydrogen ionization degree i is estimated using the equation

$$\begin{eqnarray}&&i(\xi )=1-(1-{{i}_{{\rm c}}}){{\left[ \frac{{{T}_{{\rm tr}}}-T(\xi )}{{{T}_{{\rm tr}}}-{{T}_{0}}} \right]}^{2}},\end{eqnarray} \tag{ 3 }$$

adapted by Gunár et al. (2013a) from Heinzel & Anzer (2001). The parameter ${{i}_{{\rm c}}}=0.3$ represents the estimated degree of ionization at the dip center, as suggested by Anzer & Heinzel (1999). The mean molecular mass μ of a hydrogen–helium plasma is given by

$$\begin{eqnarray}&&\mu (\xi )=\frac{1+4\ {{A}_{{\rm He}}}}{1+{{A}_{{\rm He}}}+i(\xi )},\end{eqnarray} \tag{ 4 }$$

where ${{A}_{{\rm He}}}=0.1$ is the helium-to-hydrogen abundance ratio.

To obtain the gas pressure variation along the central field line we define the transition region pressure ${{p}_{{\rm tr}}}$ at both edges of the local dip and then evaluate an integral of the form

$$\begin{eqnarray}&&p(z)={{p}_{{\rm tr}}}{\rm exp} \left[ \int _{0}^{Z}\frac{\mu (z)\ {{m}_{{\rm H}}}\ g}{k\ T(z)}dz \right]\end{eqnarray} \tag{ 5 }$$

over the depth z-coordinate from both edges toward the bottom of the dip over the local total depth Z. Here, ${{m}_{{\rm H}}}$ is the mass of the hydrogen atom and g is the gravitational acceleration at the solar surface. For all prominence fine structures we assume that the transition region gas pressure ${{p}_{{\rm tr}}}$ is equal to 0.015 dyn cm⁻². This corresponds to the prominence boundary values listed e.g., by Engvold et al. (1990; see also Labrosse et al. 2010), and is consistent with our previous work. To obtain the column mass we integrate the density in the local dip along the ξ-coordinate.

After we specify the plasma distribution along the guiding field line we next need to define the variation of all physical quantities within the fine structure cross-section. We assume that the magnetic field does not change considerably on the scale of the cross-section width and thus we use a uniform dipped magnetic field within an individual prominence plasma fine structure. For simplicity we set the geometrical cross-section planes to be parallel to the y–z plane. Furthermore we do not assume any physical mechanism influencing the plasma distribution within the fine structure cross-section, but we use an angularly symmetric distribution, resulting in the de facto circular cross-section. We describe it in Cartesian coordinates within the y–z plane on a grid of 101 × 101 equidistantly spaced grid points. This gives us a resolution of 10 km when we assume the cross-section width of 1000 km. Such fine spacing is necessary to resolve the steep temperature gradient in the direction perpendicular to the magnetic field lines. Such a grid is defined locally at each x position along each field line and is always centered at the guiding field line. The cross-section width of 1000 km assumed here is consistent with the 2D models of Heinzel & Anzer (2001). However, we note that observations of filaments in the hydrogen Hα line, such as those of Lin et al. (2005), indicate that the fine structure widths in some filaments may be as small as 100 km.

The temperature variation within the fine structure cross-section in the center of the dip is defined as

$$\begin{eqnarray}&&{{T}_{{\rm cen}}}(\zeta )={{T}_{0}}+({{T}_{{\rm tr}}}-{{T}_{0}}){{\left| \frac{\zeta }{\delta } \right|}^{{{\gamma }_{{\rm ac}}}}},\end{eqnarray} \tag{ 6 }$$

where $\delta =500$ km (total cross-section dimension is 2 $\delta =1000$ km). Here ζ represents the geometrical coordinate parallel to y, but defined from the center of the fine structure cross-section. The exponent ${{\gamma }_{{\rm ac}}}$ describes the steep temperature gradient across the field. Here we use ${{\gamma }_{{\rm ac}}}=60$. Such a steep gradient reflects the effect of the inhibited thermal conductivity in the direction across the magnetic field.

To obtain the gas pressure and density variation across the field lines, we specify the variation of the column mass in the center of the dip in the cross-field direction as

$$\begin{eqnarray}&&{{M}_{{\rm cen}}}(\zeta )={{M}_{0}}\left( 1-{{\left| \frac{\zeta }{\delta } \right|}^{2}} \right).\end{eqnarray} \tag{ 7 }$$

Thus at the given y-position ${{M}_{{\rm cen}}}(\zeta )$ is used as the required column mass and ${{T}_{{\rm cen}}}(\zeta )$ is used as the minimum temperature at the bottom of the dip.

The values of the parameters that are fixed for every fine structure in the WPFS model are listed in Table 1. The current values are consistent with the work of Gunár et al. (2013a). We discuss the possible effects of varying these parameters in Section 5.

To obtain the Hα intensity along a given LOS we assume a uniform Hα line source function S in the whole medium (see Section 2.1 of Heinzel et al. 2015). This reduces the radiative transfer equation to the well-known form

$$\begin{eqnarray}&&I(\nu )=\int _{0}^{{{\tau }_{\nu }}}S({{t}_{\nu }}){{e}^{-{{t}_{\nu }}}}d{{t}_{\nu }}=S[1-{{e}^{-{{\tau }_{\nu }}}}],\end{eqnarray} \tag{ 8 }$$

where

$$\begin{eqnarray}&&d{{t}_{\nu }}=\kappa (\nu ,l)\ dl.\end{eqnarray} \tag{ 9 }$$

The term d ${{t}_{\nu }}$ is the optical-depth increment corresponding to the geometrical path increment dl along the given LOS and $\kappa (\nu ,l)$ is the absorption coefficient for the Hα line. ${{\tau }_{\nu }}$ is the total optical thickness along the LOS, integrated through all structures crossed by the given LOS. The absorption coefficient $\kappa (\nu ,l)$ has the standard form (Heinzel et al. 2015)

$$\begin{eqnarray}&&\kappa (\nu ,l)=\frac{\pi {{e}^{2}}}{{{m}_{e}}c}{{f}_{23}}{{n}_{2}}\phi (\nu )=1.7\times {{10}^{-2}}{{n}_{2}}\phi (\nu ),\end{eqnarray} \tag{ 10 }$$

where f₂₃ is the Hα line oscillator strength, n₂ the hydrogen second-level population, and $\phi (\nu )$ is the normalized line profile. Since we can assume that the individual fine structures have a low optical thickness, the normalized line profile is a Gaussian with dependence on the thermal and micro-turbulent broadening. In this work we use a uniform micro-turbulent velocity of 5 km s⁻¹ (see e.g., Engvold et al. 1990; Labrosse et al. 2010). To estimate n₂ we use the relation between the electron density ${{n}_{{\rm e}}}$ and n₂

$$\begin{eqnarray}&&n_{{\rm e}}^{2}=f(T,p)\ {{n}_{2}}\end{eqnarray} \tag{ 11 }$$

discussed in Heinzel et al. (2015); see also Heinzel et al. (1994). The electron density can be estimated from the local values of gas pressure and temperature provided by the WPFS model using the well known formula (Heinzel et al. 2015)

$$\begin{eqnarray}&&{{p}_{{\rm g}}}=(1+\frac{1.1}{i})\ {{n}_{{\rm e}}}kT,\end{eqnarray} \tag{ 12 }$$

and assuming 10% abundance of helium and no helium ionization. Values for the factor f and the hydrogen ionization degree i, together with the source function, are tabulated in Heinzel et al. (2015) for various values of temperature, gas pressure, and altitude above the solar surface.

To visualize our WPFS model as a prominence, i.e., in emission above the solar limb without any background radiation, we use the LOS oriented perpendicular to the x–z plane (parallel to the y-axis). This is consistent with the x–z plane view in Figure 3. To obtain the synthetic images we construct a grid with 150 × 150 km spacing covering the x–z plane. This results in a synthetic resolution comparable with the Hinode/SOT Hα observations.

We evaluate Equation (8) at each grid point along the LOS parallel to the y-axis taking into account the local distribution of temperature and gas pressure in any prominence plasma fine structure intersected by the given LOS. For simplicity and to save computational resources we assume an altitude of 10,000 km above the solar surface in the whole model. We also assume representative values of f and i instead of interpolating these values from the tables in Heinzel et al. (2015). While such simplification introduces a certain inaccuracy, it is sufficiently precise for the purpose of this paper. For an altitude of 10,000 km we use tabulated values for T = 8000 K and ${{p}_{{\rm g}}}=0.1$ dyn cm⁻² giving $f=4.8\times {{10}^{16}}$ and i = 0.44. We also at this stage evaluate Equation (8) only at the frequency of the Hα line center.

In Section 3.1 we identified a set of independent field lines that produce non-overlapping prominence fine structures. However, not all of these field lines have to necessarily contain prominence plasma. To illustrate the effect of the so-called filling factor—ratio of the volume filled by prominence mass to the total volume taken up by a prominence—we use three different scenarios. We either fill all independent field lines, or we randomly select 25% or 50% of the field lines to be filled. In Figure 5 we plot for each of these cases the synthetic Hα line center image in emission (in color) and also as a negative image to ensure a better visibility of the plotted prominence fine structures. The body of the modeled prominence is broken-up into two parts, similar to the results of Mackay & van Ballegooijen (2009); see Figure 2. The left detached part of the prominence is considerably fainter than the right part and the gap between them appears much larger than that seen in the plot of the dipped portions of the magnetic field lines in Figure 3. From the image it is clear that there is a significant amount of fine structures produced by the dips in the model. These fine structures show a predominantly horizontal pattern. However, as the number of filled field lines increases, some thin vertical structures also become visible. We note that most of the bright fine structures are located between roughly 120,000 and 140,000 km marks on the x-axis. This is where the majority of the deep dips are present. For an interested reader we recommend enlarging Figure 3 in the electronic format to distinguish these deep dips.

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To obtain the synthetic Hα image of our WPFS model as a (dark) filament seen in absorption against the bright solar disk, we use the LOS oriented downward in a direction perpendicular to the x–y plane (parallel to the z-axis). This is consistent with the x–y plane view in Figure 3. We again construct a grid with 150 × 150 km spacing, now covering the x–y plane. In filament visualization we have to take into account the background intensity from the solar disk that is absorbed by the prominence plasma. The total intensity emerging at the top of the prominence/filament structure is then given by the formula

$$\begin{eqnarray}&&I(\nu )=S[1-{{e}^{-{{\tau }_{\nu }}}}]+{{I}_{{\rm bgr}}}\ {{e}^{-{{\tau }_{\nu }}}}.\end{eqnarray} \tag{ 13 }$$

The background intensity ${{I}_{{\rm bgr}}}$ in the Hα line center can be obtained, e.g., from David (1961).

In Figure 6 we show the synthetic Hα line center images of the WPFS model, seen as a filament (top panels of each pair) and a prominence (bottom panels), where 25%, 50%, and 100% of independent field lines are filled. Top panels show the filament synthetic image in absorption (not as a negative). Bottom panels show the prominence as a negative image. From this figure it can be seen that the bright prominence fine structures pile-up one above each other with seemingly random horizontal shifts. In contrast, when viewed as a filament they appear aligned and resemble ensembles of fibrils forming the structure of observed filaments. Note that the filament fine structures in the synthetic Hα images of our WPFS model are wider than those seen in observations. In observations they appear to have widths as small as 100 km (Lin et al. 2005).

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For the visualization of the WPFS model and for a casual comparison with the Hα prominence/filament observations, it is sufficient to produce synthetic images in the Hα line center, such as those in Figures 5 and 6. However, for a qualitative comparison of a prominence model derived to reproduce observations of a specific prominence, one would need to pay attention to the consistency of the calibrations of the synthetic and observed images. First, case-specific values of the background Hα intensity would be needed and the uniformity of the source function should be properly justified. Values of the source function, together with the f and i values, could also be interpolated in altitude using the tables from Heinzel et al. (2015). Second, the resulting Hα intensity in each synthetic pixel should be an integral over a part of the synthetic Hα line profile corresponding to the bandwidth of the filter used for observations. Third, the sensitivity of the detector and the contrast of the observed images has to be considered and the same image properties applied to the synthetic filtergrams.