On the Occurrence of Thermal Nonequilibrium in Coronal Loops

We choose to focus on three different loop geometries and to scan various heating configurations for these loops. In addition to the loop geometry that matches the pulsating loop bundle already used in Paper I (noted here as loop B), we use a semicircular geometry as a control sample (loop A), and we picked another loop geometry from the LFFF extrapolation that matches with a nonpulsating loop bundle as observed with AIA (loop C). The field lines corresponding to loop B and loop C, and matching visible loop bundles in the 171 Å AIA image, are indicated in Figure 1. Loop A is an ad hoc loop and is therefore not present in the field of view.

The 1D hydrodynamic simulations are made using the same initial conditions and assumptions as in Paper I (see description in Section 2.2). The only difference here is that instead of using the Spitzer thermal conductivity as in Paper I, we use an option present in the code (Mikić et al. 2013) to artificially broaden the transition region at low temperatures. This modification of ${\kappa }_{\parallel }$ allows us to reduce the steep gradients below a cutoff temperature (chosen as T_c = 250,000 K here) with a minimal effect on the coronal solutions, as described in Lionello et al. (2009) and Mikić et al. (2013). In that way we can afford to use bigger mesh cells and thus fewer mesh points, i.e., 10,000 points, than in Paper I. This technique is particularity suited for this study, since we wish to scan a large area of the space of parameters. Some of the runs presented in this paper were repeated with the classic, unmodified, Spitzer conductivity using more mesh points (typically 100,000 points). It is the case for the particular simulations that we choose to present in detail in Section 4 (see Figure 16) and some examples presented in Froment (2016). The overall pattern is not affected by this technique, but some differences related to the precise nature of condensations can appear between the simulations using the Spitzer thermal conductivity and the ones using modified conductivity.

We choose a simple heating function that can be tuned with three free parameters. This heating function is the same as the one used in Paper I (see Equation (2)):

$$\begin{eqnarray}&&H(s)={H}_{0}+{H}_{1}({e}^{-g(s)/{\lambda }_{1}}+{e}^{-g(L-s)/{\lambda }_{2}}),\end{eqnarray} \tag{ 1 }$$

where $g(s)=\max (s-{\rm{\Delta }},0)$ and Δ = 5 Mm is the thickness of the chromosphere, where the heating is constant.

H(s) is the volumetric heating rate, expressed in W m⁻³. H₀ is the value to which H(s) tends at the apex, and $({H}_{0}+{H}_{1})$ is the value of the heating in the chromosphere. λ₁ and λ₂ are the scale lengths for the energy deposition at the eastern and western leg of the loop, respectively.

For each loop geometry we test several values of H₁, λ₁, and λ₂. We choose to fix the H₀ value for each loop geometry, to a value that allows us to obtain a static loop (we set H₁ = 0 to use a uniform heating) with an apex temperature around 1 MK.

For each loop geometry, we explored values of H₁ in increments of a factor of 2. In that way we can easily compare the simulations. Note that the value of the factor itself is arbitrary. For each value of H₁ explored, we test a large set of combinations of λ₁ and λ₂, specified in percentage of the total length L of each loop. The scan cube is then (H₁, λ₁, λ₂).

For each simulation, we can define the heat flux, i.e., the total heat that the loop receives over its length, normalized to the first loop footpoint cross-sectional area:

$$\begin{eqnarray}&&{Q}_{0}=\displaystyle \frac{1}{{A}_{0}}{\int }_{0}^{L}H(s)\times A(s){ds}\hspace{1cm}[{\rm{W}}\,{{\rm{m}}}^{-2}],\end{eqnarray} \tag{ 2 }$$

with A(s) the cross-sectional area of the loop, which is $\sim 1/B(s)$. As we will see in Section 2.2 (Figures 2 and 3), the magnetic field strength is similar at both ends of each loop geometry chosen. The heat flux normalized at the second footpoint, Q₁, is thus always similar to Q₀. We thus only consider this latest value. The Q₀ heat flux value does not give information about the asymmetry of the heating. However, looking at Equations (1) and (2), we can see that the heating in a leg will be dominant when the scale height of energy deposition is larger, and the more the loop area expansion is important in this leg.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

It is worth noting that the way we choose to explore the parameter space of the heating strength and stratification implies that Q₀ changes between two slices through the scan cube, i.e., between simulations with the same stratification (λ₁, λ₂) but with different values of the heating at the footpoints (H₁). Indeed, Q₀ changes from one simulation to another. We could have chosen a different parameterization, fixing Q₀ instead of H₁ for each exploration of λ₁ and λ₂. However, we confirmed that this different parameterization did not change our conclusions. In addition, whatever the parameterization, since Q₀ and H₁ are linked, the results can be explored at Q₀ constant. We tested as well an alternate way of creating asymmetry in the heating along the loop by applying a different amount of heating at each footpoint. We verified that this did not change the conclusions of this paper either.

In Figure 2, we present the loop geometry of loop A, the test case semicircular geometry we use for the first set of simulations. In Figure 3, we present the loop geometry of loops B and C, which are from two field lines extracted from the LFFF extrapolations of the active regions presented in Figure 1. These loop geometries, even if from single field lines, are selected to model the behavior of the corresponding loop bundles. When we compare the simulations with observations, a simulated loop represents the average behavior of a loop bundle, whose individual threads all have similar properties.

In these figures, we show the quantities that are input to the simulations, i.e., the loop profile (the altitude of each point of the loop), the gravity projected along the loop (see Equation (1) of Paper I), the magnetic field strength along the loop, and the loop area A(s), normalized to the first footpoint. For both loops, s = 0 corresponds to the eastern footpoint.

These three loop geometries have the following characteristics:

Loop A—This loop is a semicircular loop with the same length as loop B, i.e., L = 367 Mm. The magnetic field along the loop is given by

$$\begin{eqnarray}&&B(s)={B}_{0}+{B}_{1}({e}^{-s/l}+{e}^{-(L-s)/l}),\end{eqnarray} \tag{ 3 }$$

with B₀ = 1 G, B₁ = 10 G, and l = 14 Mm (see Equation (4) of Mikić et al. 2013). The loop expansion factor reaches a value of 11 at the loop apex (i.e., at 58 Mm of height).

In input of the simulations we select the mesh spacing to be Δs = 19 km in the chromosphere and transition region, increasing to Δs = 190 km in the corona.

Loop B—This is the same loop that we studied in Paper I. It corresponds to the pulsating loop bundle detected in AIA data. It is quite a large and asymmetric loop with L = 367 Mm. As in Paper I, s = 0 corresponds to the eastern footpoint, while s = L corresponds to the western footpoint. The apex is at an altitude of 87 Mm at s = 212 Mm (i.e., at 0.58 L), i.e., the loop is skewed toward one footpoint. The magnetic field strength is about the same at s = 0 (∼315 G), where ${B}_{z0}\gt 0$, as at s = L (∼275 G), where ${B}_{z0}\lt 0$. The cross-sectional area at each footpoint is thus similar. The loop expansion factor reaches a value of 38 at s = 162 Mm, i.e., to the east of the loop apex. As seen in Paper I, the large, low-lying portion at the eastern footpoint is due to the magnetic topology in this area, i.e., a low-lying null point and many bald patches. As for loop A, the mesh spacing is Δs = 19 km in the chromosphere and transition region, and Δs = 190 km in the corona.

Loop C—This field line corresponds to a nonpulsating loop bundle in the AIA data. It is located in the small active region west of NOAA AR 11499. We did not find any intensity pulsations in any of the loops of this region. The loop chosen is shorter than the previous ones, with L = 139 Mm. As for loop B, s = 0 corresponds to the eastern footpoint. The apex of this loop has an altitude of 42 Mm at s = 77 Mm, i.e., s = 0.55 L. The loop expansion is quite large, with a maximum value of 82 at s = 92 Mm, i.e., to the west of the loop apex. The magnetic field strength is about the same at each footpoint with 1100 and 1200 G, respectively. Using the same number of mesh points as for the two other loops, the mesh spacing is smaller for this loop: Δs = 7 km in the chromosphere and transition region, increasing to Δs = 70 km in the corona.

For each loop geometry, we present hundreds of simulations, which is still only a fraction of the parameter space we explored. We choose here to focus only on the area of the parameter space surrounding the simulations showing TNE cycles.

The results of the exploration for each loop geometry are displayed using grid plots, each individual plot showing the temperature evolution along the loop for 3 days of simulation time. Each figure represents a scan of the λ₁ and λ₂ values for a given value of H₁, i.e., a cut through the scan cube. By this means we can analyze the global behavior of the simulations within the parameter space. The simulations conducted for loop a are shown in Figure 4, the ones for loop b in Figures 5 and 6, and the ones for loop c in Figures 7 to 9.

We choose arbitrarily to show only the temperature, although the density or the velocity would be also suitable to present a different view of the loop behaviors. Nevertheless, the density and the velocity profiles are shown for a few examples analyzed in detail (see Figure 16). Temperature, density, and velocity averaged around the apex are also shown in Section 2.3. Throughout the present paper the apex area is defined, as in Paper I, as the part of the loop above 90% of the apex height. Temperature, density, and velocity are averaged in this area, to avoid quoting values at a single location.

2.3.1. Physical Limitations on the Domains Explored

As mentioned earlier, we explored a very large range of heating for each loop geometry, in particular in terms of scale heights: λ₁ and λ₂. We did not limit the study to specific ranges that would be appropriate to the magnetic field configuration of each loop system. As a consequence, it is likely that the domain of the parameter space presented in this paper is larger than a "realistic" one that would be constrained by the magnetic field strength, for example. This was done deliberately, to separate the heating mechanism from the magnetic field, so as not to make any strong assumptions about the heating. Thus, we are maybe scanning scale heights that are not possible in reality for a given loop geometry.

The choice of the scanned H₁ values is justified by the temperature and density of the simulated loops, i.e., we aimed to produce loops with coronal temperature (close to 1 MK and at most 4 MK for the temperature and ${10}^{14}\mbox{--}{10}^{15}\,{{\rm{m}}}^{-3}$ for the density). We explored as well H₁ producing cooler loops, but for these cuts through the scan cube, not enough density was injected, so no TNE was produced. We tested also higher H₁, but then the loops are extremely hot, which does not correspond to our observations. Note that given the large loop expansion of loops B and C (38 and 82, respectively), some of the values of H₁ give really high Q₀ (up to $20\times {10}^{4}\,{\rm{W}}\,{{\rm{m}}}^{-2}$ for loop C; see Table 1). However, it can be delicate to interpret the absolute values of Q₀. In our simulations the chromosphere and low-transition region are not well modeled. The part of the heating that is radiated away in these layers may be unrealistic, and further studies would be needed to quantify it. Therefore, we are not interpreting the absolute values of the strength of the heating, but rather the evolution of the size of the TNE domain when H₁ is doubled (relative heating strength).

Table 1.  Summary of the Main Characteristics of the TNE Cases from the Heating Parameter Scan for Loops A, B, and C

	L	H1	Q0	$\langle {T}_{e}\rangle$	$\langle {n}_{e}\rangle$	$\langle v\rangle$	Periods	Time Lag Te–ne	% IC
	(Mm)	${10}^{-5}({\rm{W}}\,{{\rm{m}}}^{-3})$	${10}^{4}({\rm{W}}\,{{\rm{m}}}^{-2})$	(MK)	${10}^{14}({{\rm{m}}}^{-3})$	$(\mathrm{km}\,{{\rm{s}}}^{-1})$	(hr)	% Period
Loop A	367	6.4	0.2–0.5	0.7–1.4	0.3–3.5	−22/+24	4.6–15.5	16–60a	18
		12.8	0.3–0.9	0.5–2.1	0.2–6.9	−32/+39	2.5–15.5	9–60a	13
		25.6	0.6–1.9	0.6–2.7	0.4–12.0	−68/+68	2.5–12.4	2–60a	25
Loop B	367	6.4	2.2–7.8	0.8–2.4	0.9–4.3	−8/+6	6.9–12.4	11–33	56
		12.8	4.4–14.0	1.1–2.8	1.4–6.7	−12/+11	5.0–12.4	13–26	45
Loop C	139	4.0	2.6–5.6	0.8–1.2	2.5–4.0	−3/+3	4.0–5.6	27–45	0
		8.0	2.5–10.7	0.7–1.7	2.0–6.1	−4/+5	2.5–5.9	22–52	0
		16.0	3.8–20.3	0.8–2.5	2.4–10.3	−6/+7	2.4–4.0	17–47	1

Note. The different ranges correspond to the ranges covered by each grid plot presented in Figures 4–9. T_e, n_e, and v values are the mean values around the loop apex. The percentage of IC is given compared to the number of TNE cases for each group of simulations.

^aWe saturated the delay exploration at 60% of the period. Further analysis shows that some time lags are close to 100% of the period. For some other cases the time lag is underestimated because the cross-correlation technique captures only the first peak of density. This is happening in the case of very strong condensations formed at the loop apex.

Download table as:  ASCIITypeset image

2.3.2. Criteria to Distinguish between the Different Behaviors

2.3.3. Loop A

For this group of simulations ${H}_{0}=1\times {10}^{-7}\,{\rm{W}}\,{{\rm{m}}}^{-3}$. We scan three values of H₁: $640{H}_{0}$, 1280H₀, and 2560H₀. For each value of H₁, ${\lambda }_{1}$ and λ₂ are scanned between 2% and 11% of L, i.e., we test 10 values between 7.3 and 44.0 Mm. Every combination is tested, so we have eventually 3 × 10 × 10 different heating configurations, i.e., 300 simulations. All these simulations are presented in Figure 4.

Zoom In Zoom Out Reset image size

Some of the simulations show cyclic cases of evaporation and condensation. Around a restricted domain of TNE cycles, the simulations produce either continuous siphon flows or loops reaching a static equilibrium. For each simulation, we indicate the following:

• 1.  
  If it is a TNE case with a white dot, and either CC (for complete condensation) or IC (for incomplete condensation). CC cases can be visually identified by the dark-blue and purple drops in the temperature evolution (temperature ≤0.5 MK);
• 2.  
  SF is stated for continuous siphon flows;
• 3.  
  and SE for static equilibrium.

In order to determine whether a simulation exhibits TNE cycles and what the nature of the condensations are, we use the criteria presented in Section 2.3.2. We see that TNE cycles are encountered only around the diagonal of each of these square grid plots, i.e., for simulations for which a symmetric heating function is applied. The upper limit for theses cycles is λ₁ = λ₂ = 33.0 Mm (i.e., 9% of L), i.e., the solutions with λ₁ or λ₂ larger than this value are stable. We also notice that the more heating is applied (H₁ high, and consequently Q₀ high), the more the TNE domain extends away from the diagonal.

In Figure 10, we show the maximum temperature and density averaged near the loop apex and the averaged velocity over the loop apex. This temperature is most of the time coronal (with very few simulations below ∼0.6 MK). It increases to 4 MK for large values of H₁, λ₁, and λ₂. We see a clear signature of heating for symmetric heating cases (with λ₁ ∼ λ₂). Indeed, comparing with Figure 4, we can identify that within the TNE domain (indicated by the field of white dots in Figure 4), high temperatures are reached more easily. The maximum density plot shows also clearly a condensation pattern for the TNE simulations. We notice that this maximum density is up to $\sim {10}^{15}\,{{\rm{m}}}^{-3}$, which is a reasonable value for a large coronal loop (Reale 2014). However, these values are quite low for CC cases, but we have to bear in mind that we average these quantities around the apex before determining the maximum, and as we will see for other loop geometries, the density peak is not necessarily reached around the apex.

We notice that the velocity around the loop apex is quite high ($\sim 100\,\mathrm{km}\,{{\rm{s}}}^{-1}$) when ${\lambda }_{1}+{\lambda }_{2}\lt 40\,\mathrm{Mm}$, when the heating is very stratified. The velocities are lower close to TNE cases.

We use the velocity maps⁸ to analyze the loops without cycles. For each H₁, in the region where λ₁ > λ₂, we encounter loops whose evolution is dominated by siphon flows to the right footpoint (i.e., the less heated footpoint; see SF+ in Figure 4). We witness the reversed siphon flows when λ₁ < λ₂ (see SF− in Figure 4). We have thus continuous siphon flows to the less heated footpoint when the heating is strongly asymmetric. The last main behavior encountered is static equilibrium (velocity close to zero along the loop; see SE in Figure 4), for the loops along the diagonal and with λ₁, λ₂ >33.0 Mm. Note that the TNE cases show periodic siphon flows (see Figure 16) but that the cases pointed out as SFs here are simulations dominated by continuous siphon flows for several days. For each value of H₁, the TNE, SF +, SF−, and SE domains do not overlap.

To conclude for this semicircular loop geometry, we found that the majority of TNE cases produced CC cycles: between 82% and 89% of the TNE cases are CCs, depending on the H₁ used. This is consistent with the results of Mikić et al. (2013; see, e.g., Case 7). A few ICs are encountered at the boundaries of the TNE domain (i.e., when the heating is asymmetric) when the total heating is increasing (i.e., for higher H₁). The domain within the parameter space in which loops are undergoing TNE cycles is rather restricted to symmetric (or close to) heating cases. We notice that there is more dispersion around the diagonal when the total heating is increased.

2.3.4. Loop B

We use the same H₀, i.e., $1\times {10}^{-7}\,{\rm{W}}\,{{\rm{m}}}^{-3}$, for the simulations with this loop geometry. We scan two values⁹ of H₁: 640 H₀, as presented in Figure 5, and 1280H₀, as presented in Figure 6. Note that the temperature scale between these two figures is different. It will be the same for the plots concerning loop C. For each value of H₁, λ₁ is scanned between 7% and 18% of L, i.e., we test 12 values between 25.7 and 66.1 Mm, and λ₂ is scanned between 2% and 13% of L, i.e., 12 values between 7.3 and 47.7 Mm, for a total of 2 × 12 × 12 = 288 simulations.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The TNE cycles are also located in a restricted domain but are now shifted to the region where λ₁ > λ₂, i.e., asymmetric heating cases when the eastern footpoint (leg) is heated more than the western one. Compared to loop A, the subfield of the parameter space presented here is thus not centered around the symmetric heating cases (indicated by the black dashed line). The scale heights for the energy deposition have to be larger than for loop A to reach TNE conditions.

Only one simulation shows TNE cycles with a symmetric heating function. For this simulation, λ₁ = λ₂ = 33.0 Mm (for ${H}_{1}=1280{H}_{0}$; see Figure 6). However, this simulation is at the edge of the TNE domain. The envelope of the TNE domain is roughly restricted to the following values: 1.0 < λ₁/λ₂ < 3.4, though the exact shape of the TNE domain is more complex. We notice that the range of λ₁ taken by TNE cases (between 7% and 17% of L) is wider than the one of λ₂ (between 2% and 9% of L). This is probably due to the asymmetry of the loop geometry, the field line from the LFFF extrapolations being skewed toward one footpoint.

As for loop A, the more the loop is heated, the wider the TNE domain is. Looking now at the condensations in these TNE cases for the two H₁ values scanned, we notice that 56% and 45% of them, respectively, have cycles with ICs. Moreover, these IC cases tend to be at the edges of the TNE domain.

In the same way as for loop A, the maximum of the averaged apex temperature and density and the mean apex velocity are displayed in Figure 11. The values reached for both temperature and density are similar to the ones for loop A. We also see a larger apex temperature in the TNE domain, compared to the surrounding SF cases, as was the case for loop A. The velocities at the apex are much smaller than for loop A (maximum 15 km s⁻¹). But we observe the same pattern of velocity evolution within the parameter space, i.e., higher velocities when the heating is highly stratified, λ₁ + λ₂ < 60 Mm in that case. Moreover, as for loop A, velocities at the apex are lower in the TNE domain.

At each side of the TNE domain, the simulations are dominated by siphon flows, the direction depending on the asymmetry of the heating. We also find a few SE cases (see location in Figures 5 and 6). Note that the white pixels in Figure 11 do not necessarily indicate SE cases, as TNE cases can show zero velocities at the apex. SE cases point out no flows along the loop for most of the 3 days of the simulation.

Finally, it is worth noting the presence of high-frequency fluctuations (wavy pattern) in these simulations, especially around the eastern footpoint. We surmise that it is probably due to a combination of the thick chromosphere at this footpoint and the numerical treatment of the transition region. Loop B has a portion that is almost tangent to the photosphere at its eastern leg (i.e., with small projected gravity; see Figure 3 and Paper I). Indeed, this sawtooth pattern is not observed for the other loop geometries or for the high-resolution simulations in Figure 16.

2.3.5. Loop C

For this last loop geometry, we parameterize the heating function with ${H}_{0}=2\times {10}^{-6}\,{\rm{W}}\,{{\rm{m}}}^{-3}$. We scan three values of H₁: 20H₀, as presented in Figure 7, 40H₀, as presented in Figure 8, and 80H₀, as presented in Figure 9. For each value of H₁, ${\lambda }_{1}$ and λ₂ are scanned between 4% and 15% of L, i.e., we test 12 values between 5.6 and 20.9 Mm. Every combination is tested, so we have in total 3 × 12 × 12 = 432 simulations.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

With this loop geometry, we notice that TNE cycles appear first for symmetric or slightly asymmetric heating conditions (λ₁ > λ₂) when ${H}_{1}=20{H}_{0}$. Then when we increase H₁, more TNE appears for asymmetric heating conditions, especially for λ₁ > λ₂. Finally, when ${H}_{1}=80{H}_{0}$, we can notice that the TNE domain becomes much larger than for loops A and B. In particular, there does not appear to be a limit to TNE for large scale heights. However, the TNE domain remains limited, as barely any TNE cases appear for a very high stratification of the heating (λ₁ or λ₂ smaller than 8.3 Mm).

We notice also that most of the TNE cases have CC cycles (0% of the TNE events for the first two values of H₁ and 1% for the last one). The siphon flow cases surround the TNE domain, with IC cycles starting to appear on the boundary of this domain.

Zoom In Zoom Out Reset image size

Figure 12 shows the maximum of the averaged apex temperature and density and the mean apex velocity. The temperature and density reached are similar to those of the other loops, but we clearly see the large range covered by the CC events (high temperature and high density). The velocities are close to the ones observed for loop B, but the pattern we observed for loops A and B, i.e., higher velocities for small heating scale heights, is not as clear here.

Zoom In Zoom Out Reset image size

Scanning the parameter space of heating configurations for different loop geometries, we have noticed that the distribution of the occurrence of TNE depends on the loop geometry. However, from this study, it seems that we are able to produce TNE-favorable conditions for any loop geometry. TNE will occur if the heating strength is sufficient to produce a loop dense enough to create a thermal runaway at high altitudes, and if this heating is deposited on specific scale heights.

From the heating parameter-space scan that we conducted with three different loop geometries, we can conclude that a stratified heating is a necessary condition to produce TNE, but it is not sufficient, as already found by many authors (e.g., Müller 2004; Susino et al. 2010; Mikić et al. 2013). For each loop geometry, the system undergoes TNE cycles for specific heating stratifications:

• 1.  
  ${\lambda }_{1}\simeq {\lambda }_{2}$ for loop A;
• 2.  
  λ₁ > λ₂ for loop B;
• 3.  
  for loop C, we observe two behaviors: λ₁ ∼ λ₂ when H₁ is small, λ₁ or λ₂ > 8.3 Mm when H₁ is large.

As stated before, in this paper we present only a subfield of the parameter space that we scanned. For each loop geometry, we also tested smaller and larger values of H₁ than the values presented here. However, when H₁ is too small, we do not reach typical coronal loop temperatures and densities, and a large majority of the loops do not show any TNE cycles. When the H₁ are too large, the temperature of the loops is too high (>4 MK) compared to the warm pulsating loops observed with AIA.

For all the loop geometries, the more heating (H₁) is applied at the footpoints, the less the system requires heating symmetry to achieve TNE cycles. In other words, higher H₁ leads to more TNE within the parameter space. Higher H₁ induces more chromospheric evaporation, which results in denser plasma at coronal altitudes, favoring the thermal runaway. Indeed, the input heating H₁ has to be sufficient to inject the density required for triggering TNE events. The loops that we are modeling in this paper are quite large (367 and 139 Mm), and thus a large chromospheric evaporation is needed to inject enough density, which can explain the large Q₀ values required to have TNE cycles (see values in Table 1).

It is worth remembering that the loop C geometry is extracted from a field line corresponding to a loop bundle that is not undergoing cycles in the observations. Interestingly, it is for this loop geometry that the TNE cycles have the highest probability¹⁰ to occur, according to our simulation results with the highest H₁ tested. If our model (quasi-steady stratified heating) is indeed correct, it would mean that we can constrain the heating for these particular loops. That would mean that loop C, which is not showing any pulsations in the AIA data, is not heated enough at the footpoints to inject the excess of density needed in the loop bundle to trigger the thermal runaway, and/or that the heating is very stratified. In the same way, loop B is showing pulsations in the AIA data, so we can guess that for this loop bundle the heating is asymmetric, stratified, and relatively important.

For each of the geometries tested, not all the stratified heating configurations lead to TNE. The area where TNE occurs is limited to some range in the heating parameter space. This leads to the question as to whether the area explored within the parameter space is realistic. In particular, the heating is somewhat correlated to the magnetic field strength (see, e.g., turbulent models in Rappazzo et al. 2007), and therefore we may have explored heating parameters that are unrealistic.

On the other hand, the strength of the magnetic field along the loops does not take into account the magnetic topology, which necessarily influences the heating as well (formation of separatrices, preferential reconnection sites; e.g., Aly & Amari 1997; Pariat et al. 2009; Parnell et al. 2010; Wyper et al. 2012). The heating parameter-space scan for loop B shows that we can produce TNE cycles for this loop geometry only with asymmetric heating profiles. This heating configuration was validated a posteriori in Paper I by the magnetic topology found around the eastern footpoints of this loop bundle, which can favor continuous reconnection and thus enhanced heating.

From the analysis of the flows, using in particular the averaged apex velocity (see Section 2.3), we noticed that the siphon flows are more intense for short heating scale heights. Moreover, they tend to be weaker close to the TNE conditions (see Figures 13–15).

Zoom In Zoom Out Reset image size

We examine also some characteristics of the cycles of the TNE cases. Figures 13–15 show, for each simulation, the periods of the cycles and time lags between the temperature and density averaged around the apex.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Period of the TNE cycles—We can notice an evident dependence on the loop length. The periods are from 2.5 to 15.5 hr for loop A, and from 5.5 to 15.5 hr loop B, which are both 367 Mm long, and from 2.4 to 5.9 hr for loop C, which is 139 Mm long. This dependence has already been seen in the EIT event statistics of Auchère et al. (2014), the AIA event statistics of Froment (2016), and the three events of Froment et al. (2015). For loop A, the period of the cycles increases along the diagonal of each H₁-constant grid and between two grids, i.e., when Q₀, and thus the maximum T_e at the apex, is increasing. For the other loops, we find the same general dependence, but the detailed behavior is more complex. The period increases with Q₀ and L.

Time lag between T_e and n_e—We compute the time lag between the temperature and density evolution. This delay is also a characteristic of TNE cycles; it is a signature of TNE when combined with the periodicity. It also explains the systematic cooling pattern observed between EUV channels, the intensity peaking first in the hotter channels and then in the cooler ones (e.g., Viall & Klimchuk 2012). In case of TNE events this observed cooling can be explained by a faster rise of the temperature than the temperature fall combined with, or only if, the density is low during the heating phase compared to the cooling phase (see Section 3.2.3 in Paper I for more details).

This time lag is given here by the peak of the cross-correlation between the average temperature and density curves around the loop apex. We choose to display them as a fraction of the period in order to compare the cases more easily. We explore systematically positive time delays between 0% and 60% of the TNE cycle period. Indeed, to our knowledge, no TNE simulations have been reported to show an increasing of the density before the temperature and the signal being periodic we avoid in that way to detected spurious negative time lags between T_e and n_e. For loop A, we notice that there are very long delays when there are very strong CCs. In some cases this delay is close to the period (see the notes in Table 1). Moreover, for some CC cases the shapes of the temperature and density curves are very different, which leads to poor cross-correlation values and underestimated time lags (not catching the strongest density peak). For loops B and C the time lags tend to be maximum close to symmetric heating cases, otherwise becoming quite uniform within the TNE domain, i.e., about 20%–30% of the period, which is what was observed in Froment et al. (2015).

Figure 16 shows three TNE cases for loop B. We display the temperature, density, and velocity evolution along the loop, for 3 days of simulation, i.e., about eight evaporation/condensation cycles (giving thus a period close to the one detected for event 1 in Froment et al. 2015). These simulations are extracted from Figure 6 and thus correspond to ${H}_{1}\,=12.8\times {10}^{-5}\,{\rm{W}}\,{{\rm{m}}}^{-3}$. They are indicated by a red dot in Figure 6. They all have similar heating conditions. However, and as discussed earlier, these simulations are repeated using the unmodified Spitzer conductivity and 100,000 mesh points, as in Paper I.

Zoom In Zoom Out Reset image size

The first simulation is an IC case for which λ₁ = 40.4 Mm and λ₂ = 33.0 Mm. Note that this simulation is not the same as the one presented in Paper I (with λ₁ = 50 Mm and λ₂ = 20 Mm) that has the same loop geometry. The two other simulations present CCs, with different locations of the condensations. For the middle simulation, λ₁ = 44.0 Mm and λ₂ = 29.4 Mm, and the condensations form close to the apex. For the last simulation, λ₁ = 40.4 Mm and λ₂ = 29.4 Mm, and the condensations form closer to the eastern footpoint. At t₁, we display the loop profiles when the temperature reaches a maximum at the apex for one of the cycles, and at t₂ the profiles when the temperature reaches a local minimum. These three loops have a maximum apex temperature around 3 MK. During the cooling phase, when the condensations are established, the temperature drops to ∼1 MK in the eastern leg of the IC simulation, while the density increases by a factor of ∼2. For the CC simulations, the temperature drops locally to 0.01 MK and the density increases by a factor of 10.

We also notice a larger velocity for CC (up to 140 km s⁻¹) compared to the IC case (about 10 km s⁻¹ at the footpoints), probably due to the increase of the density of the condensation that falls compared to the density of the loop itself. The velocity is also higher when the CC starts closer to the apex, due to the longer acceleration time. We notice periodic siphon flows for both the IC and the CC cases, the ones for CCs being stronger.

As already indicated before, around the apex, the amplitudes of the temperature (and even the density) evolution are not dramatically different for the complete and incomplete cases. The CC, which is triggered in the eastern leg, outside of the apex area, has a minimal effect on the temperature evolution around the apex. It is thus not possible to use the evolution of the parameters at the apex to distinguish between complete and incomplete cases.

In Figure 17, we trace the evolution of the EUV intensity as it would have been seen in the 171, 193, and 335 Å channels of AIA (see Section 3.2.1 of Paper I for computation details), for these three loop systems (noted as IC, CC 1, and CC 2). As in Paper I, we used the AIA response functions to isothermal plasma for each channel, calculated with CHIANTI version 8.0 (Del Zanna et al. 2015). For comparison, we add the same plot from the AIA observations. The intensity is given along the loop defined by a smoothed version of the orange contour¹¹ displayed in Figure 1. However, as we already pointed out in detail in Paper I, the synthetic intensity can only be examined in the coronal part of the loop (due to the limitations of the model, we exclude the chromosphere and the low-transition region of the intensity analysis). We will discuss in more detail the intensity variation and values along the observed contour in the next section.

Zoom In Zoom Out Reset image size

The overall pulsating behavior is well reproduced in the three simulations. In both complete and incomplete cases we can find the same global cooling pattern, with the intensity peaking first at 335 Å, then 193 Å, and finally 171 Å, following the order of the peak responses of the channels. Note that we choose these three simulations because they show condensations to the eastern footpoint and thus match the intensity pattern (higher intensities close to the eastern footpoint) along the observed loop bundle. The simulation presented in Paper I showed condensations to the western leg of the loop. This heating case showed the most convincing intensity light curves at the apex (and directly comparable to the observed light curves) among the simulations of the parameter space explored. However, the IC case chosen in the present paper shows a more convincing pattern along the loop for the reason detailed above (asymmetry of the intensity between the two loop legs).

Between the IC and CC simulations, the biggest differences¹² occur at the location of the triggering of the condensations, where we can see another 335 Å peak, corresponding to the 0.2 MK peak of that band.¹³ However, it seems quite challenging to look for this smaller peak in AIA data, as it is probably hidden in the line-of-sight integration to distinguish between CC and IC cases.

The pulsations that we observed in Froment et al. (2015) may also include co-spatial and simultaneous coronal rain events. However, it is quite difficult to distinguish between complete and incomplete cases using only the coronal channel of AIA, as discussed in Froment et al. (2017). It also remains difficult to conclude firmly for on-disk observations. However, the time lag between the 171 Å and 131 Å channels can help to identify the nature of the condensations even if we only have access to the mean behavior of the loop bundles. In Auchère et al. (2018), the authors detect long-period EUV pulsations coincident with coronal rain, in a region observed off-limb. In this study the 131 Å intensity peaks after 171 Å, which was not the case for the events studied on-disk in Froment et al. (2015). We found no time lag between these channels, which indicates that the temperature of the plasma did not decrease on average below the peak response of 171 (around 0.8 MK).

We have previously seen that some of the TNE cases can reproduce very well the average behavior observed with AIA, in the case of long-period intensity pulsations (see also the results of Paper I). Looking beyond the TNE cases, we can ask ourselves whether the non-TNE cases produced in the parameter space are realistic. Only a few simulations are hydrostatic, and most of the non-TNE simulations are dominated by continuous siphon flows lasting for most of the 3 days of the simulations. For these cases the simulated intensity would not show any temporal variations, which is inconsistent with observed EUV loops. In this regard, we have to bear in mind that in this simplified analysis we have modeled the average behavior of loop systems and have not included the variability of the heating that is likely to occur in the corona. Further studies, including an exploration of the temporal variations of the heating and/or loop geometry, are needed to quantify the dynamics of these systems and their stability.

One way to check whether the densities produced by our simulations are consistent with observations is to compare the observed AIA intensities with the simulated ones. In Figure 17, we display the intensity along loop B for three different simulations, as it would be seen with AIA, considering a radius of the cross section of the loop bundle of 100 km at s = 0. The intensity values in DN s⁻¹ are between 0.1 and 10, except during the cooling phases. Considering that the loop intensity is about 10% above the background emission (Del Zanna & Mason 2003; Viall & Klimchuk 2011), this is consistent with the intensity counts derived from AIA observations (between about 1 and 100 DN s⁻¹) displayed in the same figure. The fact that we model only one loop also explains why we can easily identify the condensations in the profiles. In contrast, in the AIA observations the difference of intensity between the heating and cooling phase profiles chosen at t₁ and t₂ is quite small. The signature of the condensations is probably hidden by the background and foreground emission.