Formation and Characteristics of Filament Threads in Double-dipped Magnetic Flux Tubes

Previous simulations have demonstrated that the localized heating is a crucial component of the thread formation process in singly dipped flux tubes (Antiochos et al. 1999; Karpen et al. 2006; Karpen & Antiochos 2008; Xia et al. 2011). Here we determine whether the localized heating can also produce two threads. In the following simulations, we keep the geometry of the flux tube unchanged, namely, D = 9 Mm and l = 180 Mm, n = 1.0, and adjust the background heating, E₀, the localized heating duration, τ_l , the localized heating scale length, λ, and the ratio of the localized heating rate between two footpoints, f. In order to make comparisons, we set a benchmark case RF1 with τ_l = 115 min, λ = 10 Mm, f = 1.0 (symmetric heating), and E₀ = 2.0 × 10⁻⁴ erg cm⁻³ s⁻¹. Figure 4(a) depicts the temporal evolution of the temperature distribution along the flux tube in case RF1. Evidently, two threads are formed and remain in the two magnetic dips after t = 3 hr. The pressure, number density, and temperature distributions along the flux tube in case RF1 before and after the coronal condensation are revealed in Figures 5(a)–(c), respectively. At t = 3.067 hr, the temperature and pressure at s = 60 Mm and s = 175 Mm (away but not far from the dip centers) decreases drastically, sucking the surrounding coronal plasma toward these cooling regions, which leads to a catastrophic rise in density. Soon the two cold filament threads shift toward each other and are trapped in dips, and a low-pressure region is formed between the two threads (Figure 5(a)).

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To understand the contributions of different energy sources to the coronal structure with filament threads, Figure 6 shows the absolute value distributions of the energy terms, including heating (H = H₀ + H_l ), radiative losses (n_e n_H Λ(T)), thermal conduction ($\tfrac{\partial }{\partial s}(\kappa \tfrac{\partial T}{\partial s})$), and enthalpy convection ($\tfrac{\partial }{\partial s}(\tfrac{\gamma {pv}}{\gamma -1})$). Figure 6(a) illustrates that the catastrophic coolings occur where the radiative losses exceed the sum of the heating, conductive flux, and enthalpy flux, which is consistent with the thermal nonequilibrium (TNE) theory (Antiochos et al. 1999; Klimchuk 2019; Antolin 2020). In addition, we find that the radiative cooling is dominant in the thread portion, and thermal conduction is important only at the condensation boundaries.

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Table 1 lists the input parameters and the simulation results for all cases in our parameter study of heating characteristics in our helical flux tube, where the simulation results include the number of threads, onset time of the catastrophic cooling, and the thread length. We perform five simulations with localized heating duration τ_l = 143, 172, 201, 229, and 258 minutes, as listed in Table 1. When τ_l > τ_form(τ_form is the onset time of the catastrophic cooling), we find that only one thread is generated even in a double-dipped flux tube. One example is case RF2, whose temperature evolution is illustrated in Figure 4(b), and the pressure, number density, and temperature distributions along the flux tube are shown in Figures 5(d)–(f). In case RF1, the condensations are formed on the outer downslope of the dips (i.e., regions between points A and B in Figure 7) after the entire coronal portion of the flux tube cools, and then persist in dips. In case RF2, the strong localized footpoint heating continues past the thread formation, which suppresses the cooling of the entire flux tube in the coronal portion. Thus, one condensation forms at the midpoint of the flux tube and then falls into one dip. This thermodynamic process is similar to single-dipped cases with symmetric heating (Karpen et al. 2001; Xia et al. 2011). The independently trapped thread in case RF2 is denser (Figure 5(e)) and has larger peak radiative losses than the magnetically connected threads in case RF1 (Figure 6(d)).

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Second, we investigate the effect of the localized heating scale length λ. Previous studies of single-dipped field lines have demonstrated that, if the localized heating scale length λ is small enough, two condensations will form on the two shoulders of the flux tube. Depending on whether the condensations form inside or outside the dip, the condensations can either approach and coalesce or fall to the nearest footpoint (Karpen et al. 2006; Xia et al. 2011). Accordingly, we expect that decreasing λ to some critical value would generate two threads. We take case RF2 as a benchmark rather than case RF1 because only one thread forms in case RF2; hence, the effect of varying λ can be easily recognized. Keeping other parameters the same as in case RF2, we perform five simulations with λ = 3, 4, 5, 7, and 9 Mm. It is found that two threads can exist only when λ = 4 Mm (Figure 4(c)). In case SL1, two condensations form and then fall into the chromosphere. In other cases (cases SL3–SL5), although coronal condensations start to form in two dips, the inward pressure-gradient force drives the two segments to approach each other and coalesce at the midpoint of the flux tube. The coalesced thread finally falls to one dip, as in case SL4 (Figure 4(d)). Thus, localized heating with scales below 10 Mm can indeed drive catastrophic cooling to form two condensations, but only a subset of λ values (λ/L ∼ 1/59) yields two stable threads that persist. In our results, the threshold of the localized heating scale ratio from one condensation (large λ/L) to two condensations (small λ/L) is 1/23, which is slightly larger than 1/28 (Xia et al. 2011) and is between 1/20 and 1/33 (Müller et al. 2004).

The observed magnetic-field distribution in the lower atmosphere is extremely nonuniform, particularly in the vicinity of filament channels, which implies that the heating characteristics of two footpoints might be asymmetric (Karpen et al. 2005; Mok et al. 2005, 2008; Chen et al. 2014). To study the effects of such an asymmetry, we perform simulations with different localized heating ratios f = 1.1, 1.4, and 1.7, and different durations τ_l = 115 and 258 minutes, as listed in Table 1. Comparing case AH3 (Figure 4(e)) and case RF1 (Figure 4(a)), we infer that when the localized heating duration is short the heating asymmetry has little effect on the number of threads. In cases AH1–AH3, the catastrophic cooling takes place after the localized heating is turned off, so the asymmetry of the localized heating cannot regulate the dynamics of threads. However, when the localized heating endures past the commencement of coronal condensation, the condensation tends to form and remain in the dip near the less heated footpoint rather than at the loop center (Figure 4(f)), consistent with the single-dip results of Karpen et al. (2003) and Xia et al. (2011). Basically, the asymmetric localized heating produces a pressure imbalance along the flux tube, placing the stagnation point of the evaporation flows closer to the more weakly heated footpoint and driving the thread toward that footpoint.

The above results indicate that, even when two condensations are formed, they might approach each other and finally merge if the gas pressure between the two segments were low enough. Thus, we conjecture that the amplitude of the background heating E₀ also affects the subsequent evolution of threads. Keeping other parameters the same as in case RF1, we simulate different cases with E₀ = 1.0, 3.0, and 4.0 × 10⁻⁴ erg cm⁻³ s⁻¹, as listed in Table 1. In case BQ1 (Figure 4(g)), coronal condensations take place in both magnetic dips and finally coexist stably. However, in case BQ2, although condensations form in both dips initially, the two approach each other against the force of gravity and coalesce near the flux-tube midpoint under the inward pressure-gradient force; the resulting single condensation passes through the right-hand dip and finally falls to the chromosphere (Figure 4(h)). Thus, we suggest that the strong background heating is not conducive to the preservation of two threads. In particular, coronal condensations cannot be initiated when the background heating exceeds the radiative losses (e.g., case BQ3), as shown in Figures 6(e) and (f). In addition, the background heating dominates over the other energy terms in this case. Recently, Terradas et al. (2021) found that no solutions with cold threads exist when the background heating is larger than radiative losses in a flux tube, which is consistent with our simulations. The scale length of the background heating is assumed to be much larger than that of the localized heating, indicating that the background heating decays more slowly with distance from the footpoints. Therefore, strong background heating may well dominate the fast cooling due to the radiation losses in the locations where condensations would form, so that thermal runaway cannot occur.

Previous simulations of single-dipped flux tubes have demonstrated that the filament formation and dynamic processes depend significantly on the geometry of the flux tube (Karpen et al. 2003). Consequently, it is essential to understand the influences of the magnetic configuration on the formation of two threads in a double-dipped flux tube.

The gravity distribution along the helical flux tube is fully controlled by D, l, and n. Hence, in the following simulations, we investigate the effects of these geometric parameters. To aid in interpreting the next set of simulations, we define three dimensionless parameters to describe the ability of a magnetic dip to host a thread. According to the field-aligned gravity along the flux tube, g_∥, the dip center is determined by the zero-point in the monotonically decreasing interval of g_∥ along the helical portion, and the dip shoulders are determined by two associated zero-points in the increasing interval, as depicted in Figure 7 for case RF1. Points A, C, and B represent two dip shoulders, and the dip center, respectively, S_l (S_r ) is the length of the left (right) part of the dip, S_d = S_l + S_r is the total length, and g_max (g_min) is the maximum (minimum) field-aligned gravity inside the dip. Accordingly, the effective gravity of a dip g_eff is quantified by the absolute value of the field-aligned gravity integrated over the whole dip:

$$\begin{eqnarray}&&{g}_{\mathrm{eff}}=\displaystyle \frac{{\int }_{0}^{{S}_{d}}| {g}_{\parallel }(s)| {ds}}{{S}_{d}{g}_{\odot }}.\end{eqnarray} \tag{ 8 }$$

The gravity asymmetry A_g between the two sides of the dip is

$$\begin{eqnarray}&&{A}_{g}=\displaystyle \frac{{\rm{\min }}(| {g}_{{\rm{\max }}}| ,| {g}_{{\rm{\min }}}| )}{{\rm{\max }}(| {g}_{{\rm{\max }}}| ,| {g}_{{\rm{\min }}}| )},\end{eqnarray} \tag{ 9 }$$

and the geometry asymmetry A_s between two sides of the dip is determined by Equation (10).

$$\begin{eqnarray}&&{A}_{s}=\displaystyle \frac{\min ({S}_{l},{S}_{r})}{\max ({S}_{l},{S}_{r})},\end{eqnarray} \tag{ 10 }$$

Below we use these three dimensionless parameters to describe the ability of the dip to trap a filament thread, i.e., the trapping ability increases as the g_eff becomes larger, and A_g and A_s approach 1. Unlike the heating parameters explored in Section 3.1.1, the characteristics of the flux tube geometry are easier to deduce from observations.

To investigate the effects of varying geometrical factors on thread formation in our helical flux tube, we perform parameter studies of the dimensional flux-tube properties D, L, and n. Table 2 lists the parameters and results, using the same notations as those used for the results in Table 1. The gravity distribution along a flux tube is highly associated with the depth of its magnetic dips. In this group of simulations (cases DD1–DD4), we select four values of D = 3, 5, 7, and 11 Mm, with the localized heating parameters the same as those in case RF1, as listed in Table 2. Note that case RF1, with D = 9 Mm, fits between cases DD3 and DD4. We find that two threads can only exist stably on deeply dipped (D = 9 and 11 Mm) magnetic field lines. When D < 9 Mm (g_eff < 0.19), the gravity cannot overcome the gas pressure-gradient force, so the two threads finally approach each other and merge into one segment, e.g., case DD3, as shown in Figure 8(a). On the contrary, when D ≥ 9 Mm (g_eff > 0.19), two condensations can form and persist in the two dips (Figure 8(b)). In case DD3, the thread temperature remains unchanged basically after the coalescence, whereas the thread density and length increase, as illustrated by Figures 5(h) and (i). The power density distribution in case DD3 (Figure 6(g)) is similar to that in case RF1 (Figure 6(a)) at the onset time of the catastrophic cooling, indicating that the dip geometry is also important for the number of stationary threads trapped in dips.

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To investigate the effects of the flux-tube length, we perform a series of simulations with l ranging from 120–300 Mm, with other parameters being the same as in case RF2. This range is within the observed scope of filament lengths (Tandberg-Hanssen 1995). The reason for selecting the case RF2 as the benchmark is that two threads cannot form in this case, which can highlight the effects of the flux-tube length. Figures 8(c) and (d) depict the results in cases DL1 and DL6, respectively. As displayed in Table 2, only one condensation takes place when l < 240 Mm, while condensations form in both dips when l > 240 Mm, for the following reasons. First, according to Equation (5), increasing l has a similar effect as decreasing λ. Second, we find that the onset time τ_f increases with the flux-tube length, so that the τ_f > τ_l condition can be met more easily, favoring the formation of two threads according to Section 3.1.1.

In the above two parameter surveys, the two magnetic dips are set to be identical widths, i.e., n = 1 in Equation (6). However, on the Sun, the two dips along a single field line are probably different. To obtain different dip widths in the same flux tube, as in Zhou et al. (2017), we adjust n in Equation (6). If n > 1, then w₂ > w₁, i.e., the second (wider) dip is more extended. We take n = 1.1, 1.5, and 1.9, with other parameters being the same as in case RF1. Only one segment of thread is formed in the second dip when n > 1.1 (case AG3, Figure 8(e)), for the following reasons. First, the second dip is longer and can accumulate much more material. Second, the second dip center is further from the footpoints than the first dip. For example, in case AG3, the distances between each dip center and nearest footpoint are about 61 and 100 Mm, respectively. Thus, the heating in the second dip is weaker, which facilitates the occurrence of thermal runaway. According to the above results for cases DD1–DD4, deep magnetic dips favor the formation of two threads. Therefore, in cases AG4–AG6, we fix n = 1.9 and take D = 11, 13, and 15 Mm. As expected, two condensations can also form if both dips are deep enough (Figure 8(f)).

In this section, we consider the condensation process in flux tubes extracted from a TDm force-free magnetic flux rope (see Section 2). To investigate numerically the effects of the flux rope geometry, we select two different field lines hosting two dips, labeled MFL1 and MFL2, whose twists are 2.64 and 1.88 turns, and lengths are 411.4 and 342.6 Mm, respectively. We consider two localized heating scales: more concentrated (λ = 10 Mm), and less concentrated localized heating (λ = 20 Mm). In the simulations in this subsection, we take τ_l = 573 minutes to evaporate enough chromospheric material for cold condensations to form on these long magnetic field lines. The input parameters and the simulation results are listed in Table 3. Figures 9(a)–(d) show the evolution of the temperature distributions along the flux tubes in cases MF1, MF2, MF3, and MF4, respectively. In Figures 9(e)–(h), the final number density distributions along the 3D field lines are rendered in color.

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We found that the localized heating scale is crucial to the occurrence of two cold condensations. When λ = 10 Mm, two condensations form near two dips (Figures 9(a) and (e)). By contrast, only one condensation forms at the loop center when λ = 20 Mm (Figures 9(b) and (f)). In addition, the catastrophic coolings have taken place before the localized heating ended in all four cases. Therefore, we suggest that the localized heating scale is more significant than the heating duration. As long as the localized heating is concentrated enough, two condensations can be formed even under long-duration localized heating.

However, whether threads can survive also depends on the geometry of magnetic dips. For example, condensations can stably persist in magnetic dips in case MF1 (Figures 9(a) and (e)). On the contrary, one thread quickly drains into the chromosphere in cases MF3 and MF4 (Figures 9(c), (d), (g), and (h)). In order to explain this phenomenon, we calculate g_eff, A_g and A_s (Equations (8)–(10)) to describe the magnetic dip characteristics, whose results are listed in Table 3. We find that the second dip in MFL2 is highly asymmetric (A_g = 0.01, A_s = 0.12) so that the thread cannot persist in this dip. Thus, these three dimensionless parameters can well reflect the potential that the magnetic dip traps the filament thread, consistent with the effect of the slope of the dip (Karpen et al. 2003).

As illustrated in Figure 2, the TDm flux-rope core is weakly twisted while the outer shell is highly twisted. This magnetic model includes non-dipped, single-dipped, and multiply-dipped field lines. Thus, the filament thread formation process and characteristics in different flux tubes would be significantly different even in a single flux rope. In this section, we explore the thread characteristics in a magnetic flux-rope system with a pseudo-3D simulation, similar to Luna et al. (2012): many independent 1D simulations, in which the heating scale and duration are the same as in case MF1. To do so, we first select 40 flux tubes extracted from the TDm model, which are almost uniformly distributed inside the flux rope. Then, we omit four flux tubes with more than two dips, and perform 1D simulations on the remaining 36 flux tubes.

Table 4 lists the topological characteristics of the selected magnetic field lines and the simulation results. The selected flux tubes can be classified into three groups: non-dipped cases (55%), single-dipped cases (28%), and double-dipped cases (17%). Note that these aforementioned fractions correspond to the magnetic model presented in this paper. Accordingly, there are three kinds of threads in a filament system: dynamic threads, independently trapped threads, and magnetically connected threads. Regarding the dynamic threads, where the magnetic dips are insignificant, threads flow rapidly in flux tubes (Karpen et al. 2001). Figure 10 depicts the temporal evolution of the temperature distribution along the flux tube in six examples: three rows from the top to bottom show two dynamic threads, an independently trapped thread, and magnetically connected thread cases, respectively.

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Figure 11(a) shows the relationship between the lengths of the selected field lines and their twists. We find that the twist of a field line increases with its length with a correlation of 0.98, and the double-dipped field lines are commonly long and twisted. The effective gravity g_eff is also proportional to the field-line length with a correlation of 0.97 (Figure 11(b)), and increases more slowly in double-dipped cases. In Figures 11(c) and (d), we find that the thread length is roughly inversely proportional to the field-line length and the mean effective gravity of dips on one field line, with correlations of −0.91 and −0.93, respectively. Moreover, the linear functions fitted from the double-dipped cases have smaller slopes and correlation coefficients than those of single-dipped cases. In particular, the threads on double-dipped field lines are usually shorter than those on single-dipped field lines. It is noted that these results are applicable to the TDm model; different flux-rope models might yield different correlations.

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Figure 12 shows the side and top views of the filament thread distribution in 3D. We find that filament threads persist in the magnetic dips below the flux-rope axis as shown in Figure 12(a). Figure 12(b) shows that this filament resembles an arcade with a bulge in the middle, consistent with many observed prominences. The filament is clearly composed of threads with different lengths. Figure 12(c) shows an end-on view of this prominence composed of many vertically distributed horizontal threads. Figure 12(d) clearly displays that the filament threads are right bearing, implying that the filament axial magnetic field is dextral in this flux rope and has negative helicity. Therefore, the observed thread orientation can be used to determine the helicity of the underlying flux rope. Moreover, the angle between the filament axis and the flux-rope axis is around 13°, which is similar to the 3D simulation of Xia et al. (2014a). Finally, we find that the pitch angles (between threads and the filament axis) vary from 25°–35°, within the range of the measured coronal magnetic-field pitch angle in a prominence (López Ariste & Casini 2003).

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