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Intuitively, longer loops will lead to TNE cycles with longer periods as the volume of the flux to fill from the chromospheric reservoir at the footpoints increases, and radiative cooling timescales are longer due to the lower densities on average. This is clearly shown in e.g., Froment et al. (2018). Starting form the classical RTV scalling laws (Rosner et al., 1978), it was also shown analytically by Auchère et al. (2014) that the characteristic evolution time of loops should be correlated with their length, regardless of TNE occurrence.

Müller et al. (2003, 2004) were the first to explore the influence of the heating scale height, that we will note here λ, or “damping length” in their papers, on the thermal evolution of a coronal loop. They showed that for the same loop length L, different λ would lead to TNE with different periods and maximum temperatures and densities, including no TNE at all. Before that, Karpen et al. (2001) showed that only large enough structures would lead to prominences for the similar reason that the λ/L should not be too great to foster TNE. The typical parametrization used for describing the loop volumetric heating follows that introduced by Serio et al. (1981): H = H _base   exp(−s/λ), with H _base the volumetric heating at the footpoint of the loops and s the curvilinear abscissa along the loop (see Mikić et al., 2013; Froment et al., 2018; Johnston et al., 2019; Pelouze et al., 2022, for more details).

In numerical simulations exploring condensations phenomena, the loop geometry has for a long time chosen to be exclusively semi-circular (e.g., She et al., 1986). On top of that, the heating geometry, represented by the heating scale height in each loop leg, was symmetric (e.g., Müller et al., 2003; Klimchuk et al., 2010). Even the loop cross-section was chosen to be constant, even though the expected expansion of the magnetic field with height has important consequences in terms of enthalpy and thermal conduction. This was based on the observed moderate to no expansion of coronal loops in EUV (Klimchuk, 2000; Watko and Klimchuk, 2000). Recent studies at high resolution in the EUV further support the circularity of the loop cross-section and the lack of expansion (Klimchuk and DeForest, 2020). Numerical studies have suggested that this may be an apparent effect caused by a combination of temperature, density, cross-section area variation and instrumental factors (e.g., Peter and Bingert, 2012; Malanushenko and Schrijver, 2013).

Mikić et al. (2013) were the first to explore different geometries of loops and heating, especially different degrees of asymmetry. They investigate different configurations that deviate from a semi-circular loop, exploring constant and non-constant loop cross-sections, symmetric and asymmetric heating profiles with different heating strengths. They showed that the previous oversimplifications in loop simulations, namely not taking into account the full range of loop geometrical properties, can lead to unrealistic loop observables (see Klimchuk et al., 2010; Lionello et al., 2013, for more details and Section 3.3). The inclusion of an expanding area cross-section with height greatly increases the thermal conduction flux into the transition region, thereby increasing the radiative losses there and leading to greater chromospheric evaporation and thus greater coronal density. The onset of TNE is therefore more easily achieved for loops expanding with height. Exploring loop and heating geometrical parameters, Mikić et al. (2013) actually showed that asymmetries lead to persistent siphon flows that can provide an important source of energy (in the form of an enthalpy flux) to the cooling coronal plasma, as well as severely reducing the time it stays in the coronal region. Such asymmetries can therefore play a central role in TNE onset, strongly vary the TNE cycle properties, as well as the cooling rate and morphology of the condensations. Through this process, it was observed that the catastrophic cooling of condensations down to chromospheric temperatures could therefore be aborted, leading to “incomplete” condensations for which the temperatures remain at transition region temperatures. For these cases, the apex temperature of the loop was found to cool down to 1 MK only, with a density of at most 5 × 10⁹ cm⁻³.

In Froment et al. (2018), these results were revisited following the discovery of long-period intensity pulsations (section 2.2.1). Using realistic loop geometries from photospheric magnetic field extrapolations (different L, asymmetries and loop expansion), the authors explored the heating parameter space by varying λ, and the volumetric heating rate at the foopoints and at the apex. They found that in principle any loop geometry could lead to TNE but only under very specific geometrical heating conditions varying from a loop to another. The need of a specific match between the loop geometry and the heating geometry in order to produce TNE reveals that for a particular loop geometry, the domain where TNE is possible is rather limited in the heating parameter space as it can be seen on the left of Figure 2. On the right panel of Figure 2, we show two simulations that are extracted from Froment et al. (2018) parameter space exploration. Both simulations use the same heating parameters with a symmetric heating function, but a different loop geometry. While the simulation with Loop A (semicircular loop) produces TNE with coronal rain-like signatures, the simulation with Loop B (asymmetric loop) shows no TNE signatures. This need of a specific match between the loop geometry and the heating geometry is relaxed when the heating strength increases, as we will see below and on Figure 2 as the TNE domain increases for each loop geometry when the heating rate increases.

Pelouze et al. (2022) show that the asymmetric loops are less likely to produce complete condensations. Indeed, it was found that more stringent conditions on the heating geometry are needed for this purpose when geometrical asymmetries are considered. This can be understood from the fact that stronger geometrical asymmetries more readily lead to siphon flows. This condition was quantified through analytical formulae (Klimchuk and Luna, 2019). The authors found that “as a rough rule of thumb,” asymmetries between the two values of the heating parameter λ for a loop should be less than a factor of 3 in order to produce TNE cycles (see Eqs 1, 4).

In the parametrization of the volumetric heating, the parameter H _base has also a significant influence on the generation of TNE cycles and their properties. It was shown by Froment et al. (2018) that the TNE cycles are more common when H _base increases and that these cycles tend to produce more complete condensations rather than incomplete condensations. This is expected since the average coronal density increases with footpoint heating due to stronger chromospheric evaporation (see section 3.1). This increase of coronal density then shortens the radiative cooling timescale and increases the growth of thermal modes leading to complete condensations.

Klimchuk and Luna (2019) draw a similar conclusion analytically. The ratio of the apex to footpoint volumetric heating rate should be less than 0.1 in order to get TNE. The heating at the apex is usually parametrized by a constant: H = H _background + H _base   exp(−g(s)/λ). Here g(s) = max{s − Δ, 0}, with Δ the thickness of the chromosphere (see Mikić et al., 2013; Froment et al., 2017). With such an additional term H _background + H _base becomes the new volumetric heating in the chromosphere and H _background the value to which H tends to at the apex (only if λ ≪ L 2 − Δ ).

This background heating term is almost always considered in 1D hydrodynamical simulations, mostly to achieve hydrostatic equilibrium prior to the start of an experiment. Its value is generally small compared to H _base and thus was rarely considered to play an important role. However, Johnston et al. (2019) showed that this parameter has in reality a great influence on the onset and characteristics of the TNE cycles, in support of Klimchuk and Luna (2019). This is clearly shown in Figure 3, where the TNE period is seen to vary greatly when the background heating is varied in the range of 10^–5 − 10^–4 J m⁻³ s⁻¹, and when it is larger than ≈ 1 0 − 4 J m^{−3 s−1} TNE is entirely suppressed. This is of particular importance for 3D MHD simulations, where large diffusion coefficients are usually used in favour of numerical stability and computational efficiency. Such large diffusivity can act as a background heating term, even though it is artificial and therefore non-physical.

In TNE simulations, the volumetric heating rate is generally constant for the full duration of the simulation. It is important to note that the quasi-steadiness of the heating is a crucial criterion in order to maintain TNE cycles as it was pointed out by Peter et al. (2012). Without a sufficiently high frequency heating there seems to be not enough density supply to maintain the production of condensations. The effect of periodic or randomly occurring heating events on TNE cycles has been explored to some extent (see e.g., Mendoza-Briceño et al., 2002, 2005; Antolin et al., 2008; Susino et al., 2010). Johnston et al. (2019) show that TNE is produced when the time between each heating event t _w is lower than the radiative cooling time of the loop. When these timescales are on the same order, condensations may still occur if the heating is sufficiently strong. However, the observed periodicity in the cycles is then governed by the heating timescale, meaning that TNE stops occurring. This sets in evidence the fact that condensations such as prominences or coronal rain can occur without TNE whenever the heating varies significantly enough to impose its own timescales.

As seen above, TNE crucially depends on the coronal density, and factors such as the heating scale length or background heating naturally have a great impact on the density. On the computational side, a major parameter that also affects coronal density is spatial resolution, as demonstrated by Bradshaw and Cargill (2013). Indeed, coarse spatial resolution affects the amount of energy that is radiated away in the transition region, leading to lower levels of chromospheric evaporation and therefore lower coronal density. Johnston et al. (2019) show that in order to properly capture the radiative losses in the transition region and the resulting enthalpy flux into the corona through the process of chromospheric evaporation a spatial resolution of at least 2 km should be achieved in the transition region. This is of course prohibitive for 3D numerical simulations with uniform grids and challenging even for codes with adaptive grids. Fortunately, numerical methods now exist that circumvent this process by either artificially broadening the transition region (Mikić et al., 2013) or by adapting a jump condition that properly captures the radiation in the unresolved transition region (Johnston et al., 2017; Johnston and Bradshaw, 2019; Johnston et al., 2021; Zhou et al., 2021), leading to correct results even with coarse resolution.

We focused here on 1D studies that are very useful to comprehend the influence of the different geometrical and heating parameters. However, we note that 3D MHD simulations of TNE have been performed (Mok et al., 2016). In these simulations the heating was based on the turbulent Alfvén wave dissipation theory (Rappazzo et al., 2008), with the volumetric heating rate depending on the loop’s length, the magnetic field strength and the plasma density. Such simulations are important for proper comparison with observations since 3D aspects (such as line-of-sight superimposition) are taken into account in the forward modelling. However, these simulations are similar to most of the 1D studies mentioned above in the sense that the magnetic field structure is kept fixed during the simulation. Moschou et al. (2015), Xia et al. (2017) have successfully simulated in 3D MHD a (fixed) loop arcade undergoing 1 cycle of TNE, together with catastrophic cooling and the accompanying coronal rain. More recently, Li et al. (2022) in 2.5D MHD at sufficiently high resolution have managed to produce several TNE cycles with accompanying coronal rain in a fixed loop arcade. However, it remains a challenge for future studies to self-consistently include the magnetic field evolution in multi-dimensional simulations that are long enough to include several EUV pulsations (see Section 2.2.3 and Section 4.1). 

EUV cooling is a well-known feature of the corona. As reported by many observational studies and understood through numerical simulations (e.g., Warren et al., 2002; Winebarger et al., 2003; Winebarger and Warren, 2005; Ugarte-Urra et al., 2006; Mulu-Moore et al., 2011; Warren et al., 2011; Viall and Klimchuk, 2011, 2012, 2017), the corona is observed to be in a constant and widespread state of cooling. Indeed, although heating events have to happen in order to sustain the corona at its characteristic high temperatures, only the cooling events are usually observable (at least with the current instrumental capabilities). Different types of loop models (TNE, nanoflare storms, etc) all show that the heating phases occur at a low plasma density, which means at low emission measure, making its detectability in the EUV challenging. However, during the cooling phases the density is high and the radiative cooling times are longer, making it possible to observe the temperature variations. These are observed gradually in different EUV lines and/or passbands, according to the peak temperature response of these lines/passbands. This state of cooling is thus there, regardless of the heating mechanism and its properties.

As it will be detailed in Section 2.2.3 the first and most well-known manifestations of TNE and/or TI in the solar atmosphere are prominences and coronal rain, that were discovered in chromospheric lines such as Hα. Coronal counterparts of TNE-TI cycles, in EUV lines, have been more discrete and not straightforward to reveal. However sparse, EUV manifestations of TNE were observed quite early following the discoveries of their cool cousins, even though they were not directly reported as such. Foukal (1978) reported dark absorption features in the EUV passbands that were related to coronal rain events. With the Coronal Diagnostic Spectrometer (CDS, Harrison et al., 1995) onboard the Solar and Heliospheric Observatory (SoHO, Domingo et al., 1995), Kjeldseth-Moe and Brekke (1998) observed an important variability, stronger at lower transition region temperatures, in a few loops where they also found downflows in the O V lines that reveal material at chromospheric temperatures. EUV variability characteristic of cooling, in relation with coronal rain, was also found in the Transition Region and Coronal Explorer (TRACE, Handy et al., 1999) observations by Schrijver (2001). In this study, time delays were reported between the light curves of coronal channels and transition region/chromospheric channels. This observation thus provided evidence of cooling from coronal to chromospheric temperatures. Cool material was reported falling toward loop footpoints by De Groof et al. (2004) in the 304 Å   channel of the Extreme Ultraviolet Imaging Telescope (EIT, Delaboudinière et al., 1995) and was later associated to coronal rain in De Groof et al. (2005). However no imprint at higher temperatures were reported. O’Shea et al. (2007) with SoHO/CDS reported a decrease in the intensities of the coronal lines simultaneous with the appearance of plasma condensations.

The rapid cooling trend and increase in variability across transition region lines was further elucidated with Hinode/EIS observations in the Mg VI 269 Å  (0.4 MK) and Si VII 275 Å  (0.63 MK) lines by Ugarte-Urra et al. (2009), seen in Figure 5. An increase in strand-like structure (and respective decrease in diffuse, more extended EUV emission) is clear, associated with the fan-like peripheral cool loops. Similar reports have been made with Hinode/EIS and SDO/AIA, sometimes accompanied by dark or bright downflowing features interpreted as coronal rain (Tripathi et al., 2009; Kamio et al., 2011; Orange et al., 2013).

Major steps, both observational and theoretical, toward bringing together the pieces of the multi-scale observations of TNE-TI puzzle happened in the past few years. Using more than a solar cycle of observations with SoHO/EIT, Auchère et al. (2014) reported the widespread occurrence of long-period intensity pulsations in the corona using the 195 Å passband that had both a sufficient cadence of 12 min and an almost continuous coverage of the full disk corona from January 1997 to July 2010. These quasi-periodic EUV variations have periods ranging from 2 to 16 h (same as the range explored by the authors) and are most remarkably very common in coronal loops (about 25% of the events). They can be observed continuously in the same area for up to 6.5 days, a duration only limited by observational constraints. Using the Atmospheric Imaging Assembly (AIA, Lemen et al., 2012) on board the Solar Dynamics Observatory (SDO, Pesnell et al., 2012), Froment et al. (2015) confirmed the existence of such widespread pulsations, and through thermal analyses connected them to evaporation and condensation cycles in coronal loops. The discovery of long-period EUV pulsations was critically assessed in Auchère et al. (2016a) with Fourier and wavelets analysis. The high confidence in these detections was demonstrated using a proper noise model, global confidence levels and evidencing no source of artefacts such as pre-filtering of the data.

These studies brought evidence, for the first time, of what has been predicted by TNE simulations for decades: TNE cycles are periodic and they have a strong manifestation in the coronal EUV observations through the observation of pulsations. In parallel, using the wide temperature range observation capabilities brought by coordinated Hinode (Kosugi et al., 2007), SDO and the newly Interface Region Imaging Spectrograph (IRIS, De Pontieu et al., 2014) observations at the time, Antolin et al. (2015) uncovered the multi-thermal aspects of coronal rain, spanning chromospheric, transition region and coronal temperatures with a progressive cooling through them (see section 2.2.3). On the modelling side, a series of papers (Lionello et al., 2013; Mikić et al., 2013; Winebarger et al., 2014) demonstrated that TNE, that was previously considered to be a sporadic phenomenon, mostly linked to prominences and coronal rain, could occur in coronal loops with standard properties and explain several observational constraints (see Klimchuk et al., 2010; Lionello et al., 2013, for the key loop observables).

While the widespread occurrence of long-period intensity pulsations was confirmed in Froment (2016), its relationship with TNE was confirmed by Froment et al. (2017) confronting the SDO/AIA observations to 1D hydrodynamic TNE simulations. A comparison of the synthetic intensities derived from such simulations and the observed intensities is presented in Figure 6. To the best of our current knowledge, no other model besides TNE is able to reproduce the key properties of the observed long-period intensity pulsations. The X-ray emission in the model of Imada and Zweibel (2012) shows pulsations whose periods are compatible with those observed by Auchère et al. (2014) and Froment (2016), given the length of the loops studied. However, in these simulations the temperature and the density show a different correlation than that produced from the usual TNE, allowing a differentiation based on observations. Auchère et al. (2016b) demonstrated that properties of the EUV signal itself is consistent with TNE. Indeed the imprint of the long-period intensity pulsations on the Fourier spectra are consistent with a succession of periodic pulses of random amplitudes (rather than oscillations linked to vibration modes). As we have seen in Section 2.1.1, the variation of loop and heating geometries and heating magnitudes will impact the average temperature and density reached in the TNE cycles and thus ultimately the EUV intensity. This means that even without considering the very important line-of-sight effects, TNE cycles will likely not repeat themselves identically and we should thus expect varying amplitudes and possibly periods (see Section 4.1).

The thermal structure of loops exhibiting long-period intensity pulsations was investigated combining Differential Emission Measure (DEM) analysis with time lag analysis. The DEM represents the amount of emitting plasma along the line-of-sight as a function of the temperature. In active regions, the DEM generally follows a power law in the low temperature range (below the DEM peak, e.g. Warren et al., 2011). Using the DEM inversion method of Guennou et al. (2012, 2013), Cheung et al. (2015), Froment et al. (2015), and Froment et al. (2020) showed that long-period intensity pulsations events can show a periodic variation of the DEM slope (i.e., the power law index). Even though this parameter is usually poorly constrained in the DEM inversion process, this behaviour is consistent with synthetic DEMs produced with numerical simulations of loops in TNE (private communication, Cooper Downs). DEM analysis also revealed a time delay between the temperature (via the peak temperature of the DEM) and density (via the total emission measure, proportional to the density squared). This time delay combined with the periodicity of their evolution strongly matches the TNE predictions, and thus constitute strong evidence of TNE.

Using the time-lag method developed and popularized by Viall and Klimchuk (2012), both observational and modelling studies (Froment et al., 2015; Winebarger et al., 2016; Auchère et al., 2018; Froment et al., 2020) showed that long-period intensity pulsations and TNE are consistent with the widespread EUV cooling patterns observed in the solar corona.

The discovery of widespread long-period intensity pulsations in the corona as a manifestation of TNE led to two major puzzles: How these cycles, linked to a particular heating distribution in space and time, can be maintained for several days given the stochasticity of the physical conditions on the Sun? Auchère et al. (2014) estimated that about 50% of the active regions of solar cycle 23 underwent long-period intensity pulsations. Why are some loops entering this particular thermodynamic regime while others do not?

While the first question remains open (see Section 4.1), the second one was tackled by Froment et al. (2018) by exploring the heating parameter space for several loops geometries with 1D hydrodynamical simulations (see Section 2.1.1). The authors found that for each geometry the domain where TNE is found in the heating parameter space is limited and that these geometrical heating conditions are case dependent. This explains why every loop bundle does not show quasi-periodic intensity pulsations. Within the same active region for example, the heating conditions imposed at low altitude in the chromosphere may be similar from a loop bundle to a neighboring one. However, only the loop bundle that has a loop geometry compatible with the heating conditions will undergo TNE. It is also very likely that the line-of-sight integration may in some cases prevent clear detections. As shown in Section 4.1, the current method used to detect these pulsations is very likely leading to non-exhaustive statistics, and more like a tip-of-the-iceberg scenario.

It is important to note that the loops showing long-period intensity pulsations evolve on average in phase across the same bundle (Froment et al., 2015). While these loops may have similar geometries and heating conditions at their footpoints, their lengths are expected to be different (shorter in the inner part of the bundle), and thus their cycle properties (such as the period) are expected to be different (see Section 2.1.1). This remains to be fully explained. However, a promising mechanism behind this behaviour known as “sympathetic cooling,” seen in coronal rain simulations, might be the solution (see section 2.2.3).

Periodic EUV pulsations are an expected signature of TNE. As seen on Figure 6, the temperature and density in a loop undergoing TNE have a periodic evolution. The density evolution is delayed compared to the temperature evolution. The maxima of the temperature and density are even more delayed than the minima. This reveals that the heating phase occurs at low density. As we saw previously, the temperature variations are expected to reflect gradually in the coronal passbands following the ordering of their peak response temperature during the cooling phase, when the density is high enough.

Another prediction from the TNE model is the existence of multi-thermal periodic flows across the coronal, transition region and chromospheric temperatures. These are periodically recurrent upflows of hot plasma (evaporation phase) and downflows of cooling plasma (condensation phase). Pelouze et al. (2020) undertook this investigation, using Hinode/EIS observations of EUV pulsating loops. They demonstrated that detecting periodic flows is very challenging due to the fact that a pulsating loop only accounts for 10–30% of the total emission integrated along the line-of-sight. The resulting Doppler shift is thus at the limit of the current instrumental capabilities (see their Figure 13). Detecting this signal would require a very high signal-to-noise ratio combined with a sufficient cadence and continuous observations of the same region for several hours, depending on the period of the cycles. Despite this major instrumental and observational handicap, they detected some flow signatures compatible with the ones predicted by TNE simulations (in terms of velocity variations).

A major TNE observational signature is the presence of periodic coronal rain showers appearing in phase with EUV pulsations. The periodic cooling phases of TNE produce periodic EUV pulsations as we have seen previously. The decrease of the temperature is seen first in EUV coronal passbands and can then appear as coronal rain if a TI is triggered. This combined TNE-TI manifestation was revealed by detections of long-period pulsations off-limb in SDO/AIA observations by Auchère et al. (2018). Contrary to on-disk observations, the off-limb viewpoint has the advantage of a dark background against which coronal rain can easily be seen. Periodic coronal rain showers were reported in the 304 Å passband in a large trans-equatorial loop bundle exhibiting pulsations with similar cooling signatures as described in Froment et al. (2015). A summary Figure extracted from this study is presented in Figure 7. These observations show undeniably that long-period EUV pulsations and coronal rain are two manifestations of the same phenomenon. The vast temperature and length scale range encompassed by TNE-TI was evidenced in Froment et al. (2020), where a full thermal diagnosis was performed, using the ground-based instruments at the Swedish 1-m Solar Telescope (SST, Scharmer et al., 2003) observing chromospheric lines. A summary of this study is presented in Figure 8. The cooling of plasma all the way down to chromospheric temperatures and the condensation formation process was studied. The rain showed characteristics similar to the “usual” coronal rain observations (e.g., Antolin and Rouppe van der Voort, 2012; Ahn et al., 2014). These observations show the ability of TNE and TI to shape the dynamics of the coronal loops across a very large temperature and density range, and also across multiple spatial scales: from the size of a loop bundle ( ≈ 100 Mm) to the size of a coronal rain clump ( ≈ 200 km or less).

Unlike in Auchère et al. (2018), Froment et al. (2020), the main case of long-period intensity pulsations presented in Froment et al. (2015) showed little to no coronal rain as it was re-examined off-limb (but unpublished) with observations of the extreme Ultraviolet Imager (EUVI; Wuelser et al., 2004) from the Sun Earth Connection Coronal and Heliospheric Investigation instrument suite (SECCHI; Howard et al., 2008) onboard the Solar Terrestrial Relations Observatory (STEREO; Driesman et al., 2008; Kaiser et al., 2008). Such cases may be the manifestation of incomplete condensation events (Mikić et al., 2013), as discussed in Section 2.1.1, but it remains to be properly investigated.

The quiescent kind of coronal rain is linked to active region coronal loops and seems to be the most common kind of rain at times of high solar activity. Multi-wavelength observations combining AIA and another instrument probing the cool temperature range such as IRIS, SST or NST clearly show that coronal rain corresponds to a catastrophic cooling phenomenon in loops (Ahn et al., 2014; Antolin et al., 2015; Kohutova and Verwichte, 2016). Accordingly, as explained in section 2.2.1, a timelag in emission is seen from hotter to cooler passbands (Froment et al., 2020). However, during the formation of coronal rain and throughout its lifetime, emission from EUV, UV and visible wavelengths is common, indicating multi-thermal and strongly inhomogeneous fine-scale structure.

On the low temperature end, estimates have been obtained based on spectral line widths in Hα, Ca II H and Ca II 8542 (Antolin and Rouppe van der Voort, 2012; Antolin et al., 2012), where average temperatures oscillate between 5000 K and 33 000 K, with minima of 2000 K or less (Antolin et al., 2015). Observing with multiple lines simultaneously offers the possibility of more precise temperature measurements, along with non-thermal broadening estimates (see Figure 10), by assuming a common temperature and non-thermal line broadening between lines forming in similar conditions (Ahn et al., 2014; Froment et al., 2020; Kriginsky et al., 2021). The reported non-thermal broadening average values oscillate between 6 and 12 km s⁻¹, with maxima of 20 km s⁻¹. These values are particularly important for coronal heating, since they provide upper estimates for the turbulence in thermally unstable coronal loops. The optical thickness has also been estimated using the cloud model (Beckers, 1964), with values of 0.5–1.5 for Hα and 0.1–0.3 for Ca II 8542 (Ahn et al., 2014). Interestingly, while progressive cooling was found in Antolin et al. (2015), with lower temperatures at lower heights, an opposite trend was found in (Ahn et al., 2014). These trends may be anti-correlated with velocity, with higher final velocity (110 km s⁻¹) for the cooler rain (while 80 km s⁻¹ was reported for the warmer rain), which may reflect an effect from the compression downstream.

Claes and Keppens (2019) indicate growth rates of minutes under coronal conditions, reflecting the exponential growth of the thermal mode in the linear stage. Similarly, when taking the wave-caused perturbation of the heating function into account, Kolotkov et al. (2020) show that the growth (or damping) times of the acoustic and thermal modes in typical coronal conditions can vary from several minutes to several tens of minutes. This suggests that a single rain clump, without any additional effect than its cooling, should exhibit the intermediate TR temperatures of the cooling range for only a very short duration. The fact that we see continuously multi-thermal emission for single clumps at high resolution suggests a more complex structure in which a clump is surrounded by a warm shell at TR temperatures, which itself is in contact with the ambient coronal plasma (Antolin, 2020). This shell corresponds to the CCTR, and is observed to be very thin, under the 0′′.33 resolution of IRIS (Figure 9), in agreement with 1D modelling results Xia et al. (2011). Furthermore, this UV and EUV emitting shell is expected to be more extended than the chromospheric emission in the longitudinal direction with respect to the magnetic field, also supported by observations (Ahn et al., 2014; Antolin et al., 2015; Vashalomidze et al., 2015). This is because the rain produces strong compression downstream, particularly prior to impact with the chromosphere, which then increases both the density and temperature (Müller et al., 2003, 2004). The shell is also expected to extend along the wake of the rain due to flux-freezing conditions. In addition, the inhomogeneous character of the rain means that highly localised high density regions are produced, as shown in multi-dimensional simulations (Fang et al., 2015). As we will see below, this leads to a differential longitudinal velocity with denser clumps falling faster than lighter clumps, leading to V-shape structure in 2D where the head is denser than the tail, thereby further extending the UV and EUV emission along the rain’s path.

Coronal rain densities between 2 × 10¹⁰ and 2.5 × 10¹¹ cm⁻³ have been determined through EUV absorption (Antolin et al., 2015), assuming that the EUV emission from the rain clumps themselves is negligible and assuming ionisation equilibrium. Given the very thin CCTR shell, the former assumption should hold as long as most of the EUV emission observed comes from the region upstream or downstream of the dense part of the rain. Also, these number densities correspond to the neutral H, He and singly ionised He, which are the elements causing the continuum absorption. Electron number densities can be inferred assuming an ionisation fraction for the rain. A word of caution here is that ionisation equilibrium is likely not to hold, given that rain clumps are cooling rapidly, likely on similar timescales as recombination and ionisation, and it is unclear if heating (either through an unknown coronal heating source of through downstream compression or shear flows) prevents further cooling. For instance, temperatures as low as 2000 K have been measured in coronal rain (Antolin et al., 2015). However, the energy released through the recombination process should slow down the cooling rate (Ballester et al., 2018), which has found observational support (Antolin et al., 2015). The very fast cooling timescales combined with heating mechanisms produced by the flows and the short lifetime of the rain clumps makes the ionisation degree of coronal rain uncertain, but very likely much higher than for the prominence case. Further work is necessary to properly determine the ionisation degree in coronal rain, and properly assess the assumption of ionisation equilibrium.

Another, more robust electron number density estimation method for coronal rain is provided by Gouttebroze et al. (1993), and applied in Froment et al. (2020) to find average values of (6.7–8) × 10¹⁰ cm⁻³ with maxima at 10¹² cm⁻³, similar to the EUV absorption values stated above (see Figure 10). This method relies on the good correlation between absolute intensity in Hα and the emission measure E M = ∫ 0 D n e 2 d z , where D denotes the thickness along the LOS of a rain clump and n _e is the electron number density. This relation is based on the dominance of photoionisation in the Lyman continuum and scattering of the incident radiation from the solar disk.

Regarding the dynamics of coronal rain clumps, they are characterised by their average downward velocities. Spectroscopic high-resolution observations have allowed to determine the distribution of total velocities, taking into account the plane-of-the-sky and Doppler components (Antolin and Rouppe van der Voort, 2012; Antolin et al., 2012; Ahn et al., 2014; Froment et al., 2020), leading to average total velocities of (30–50) km s⁻¹ at the loop apex and (80–100) km s⁻¹ towards the footpoints, with a long tail towards higher velocities of 200 km s⁻¹ or so (Kleint et al., 2014; Schad et al., 2016). It has been known for several decades that other forces besides gravity must be present in order to observe the lower than free-fall accelerations. Indeed, early observations with TRACE and EIT have shown that downward accelerations are about a third of the solar gravity value (Schrijver, 2001; De Groof et al., 2004), and about a half of what is expected from the effective gravity along an elliptical loop (Antolin and Verwichte, 2011; Antolin and Rouppe van der Voort, 2012). Gas pressure forces have been shown to play a major role (Mackay and Galsgaard, 2001), leading to deceleration and even reversal of the direction of the rain (Antolin et al., 2010; Kohutova and Verwichte, 2017a). In 1D simulations of fully and partially ionised plasma Oliver et al. (2014), Oliver et al. (2016) have shown that the formation of a condensation produces a restructuring of the gas pressure downstream, which ultimately leads to constant downward velocities that are indeed sometimes observed (Antolin et al., 2010). On a timescale determined by the sound speed, the clump stops accelerating and the final speed is found to be determined by the density with a well-defined power law of ≈ ρ ^0.64. This process has been shown to hold in 2D for magnetic fields above 10 G (Martínez-Gómez et al., 2020), shown in Figure 11. For large magnetic field values, the downstream compression produced by the clumps leads to expansion of the field lines, which lead to an upward magnetic tension that further decelerates the rain (Kohutova and Verwichte, 2017a). A combined gas pressure and magnetic tension force must therefore be at the origin of this striking relation. It is unclear, however, how much does this relation apply to the complex solar atmosphere. The very fast speeds of the rain prior to impact suggest that this physical process may not be dominant. In a large statistical survey of supersonic downflows observed with IRIS, Samanta et al. (2018) find that those with a chromospheric component are 10 km s⁻¹ faster, which may find an explanation in this effect (see Figure 16).

As suggested above, the fact that a variety of dynamics are observed, including very fast speeds close to free-fall indicate that many other factors are at play, among which magnetic forces. Transverse MHD waves are often observed in coronal loops, which can be particularly well traced for coronal loops with rain thanks to the higher resolution available when observing in cool UV or visible lines (Antolin and Verwichte, 2011; Kohutova and Verwichte, 2016). It was therefore speculated that the ponderomotive force could have an effect, acting on rain clumps as beads on an oscillating string (see Figure 19). This was analysed with a one-dimensional analytical model by Verwichte et al. (2017), in which it was shown that the loop behaves like a dynamical system with stable points (locations where rain can be stopped) and unstable points (such as saddle points, that prevent the accumulation of rain). In the analysis of Verwichte et al. (2017), 2 cases were considered depending on the rain’s mass being significant with respect to the loop’s initial mass. In the case of little rain, the analysis applies to individual clumps, and it was shown that the acceleration from the ponderomotive force preventing a clump from falling needs to satisfy: a > a crit = 2 g ◦ R v A n , (16) where n refers to the longitudinal harmonic and needs to be odd for holding a clump at the loop apex, R is the loop radius (assuming a semi-circular loop) and v _A denotes the Alfvén speed. Taking typical coronal values of R = 50 Mm, v _A = 1,000 km s⁻¹, yields a _crit = 0.17/n, corresponding to a displacement of 8/n Mm. Such amplitudes are large for usual transverse oscillations in active regions (Anfinogentov et al., 2015), but are observed for external perturbations such as eruptions. The analysis was applied to a specific database and shown to explain successfully the observed rain kinematics. However, on average, the ponderomotive force from transverse MHD waves was found to be insufficient to fully explain the observed reduced downward velocities.

Another interesting consequence of coronal rain is the excitation of transverse oscillations, particularly targeting the large rain events or “showers” (described further below) for which the mass fraction is large relative to the initial loop mass. Verwichte et al. (2017) show that the transverse displacement is approximately linearly proportional to the ratio m/M, of rain mass m relative to the loop’s mass M, and is expected to be on the order of several hundred km, matching observations. This scenario is further supported by 2.5D MHD simulations by Kohutova and Verwichte (2017b), where the process is found to be characterised by a decrease of the oscillation period as the rain falls down the loop. This mechanism could therefore be partly responsible for the observed decay-less small-amplitude transverse oscillations (Nisticò et al., 2013; Anfinogentov et al., 2015), whose periods are found to be correlated with the loop length (Nechaeva et al., 2019). Although in this scenario the coronal rain excites primarily the fundamental mode, the fact that the period changes with time due to the varying temperature and density, would lead to scatter in the period-length correlation, which may or may not agree with observations. This mechanism is particularly interesting when parameters such as the loop inclination, the clump’s location, speed and displacement can be well determined, since it provides a seismological method. This has been successfully applied in Verwichte and Kohutova (2017), where it is found that the rain comprises two thirds of the loop mass.

Further seismological applications are presented by the longitudinal oscillations of the rain, produced by the combination of gas pressure and magnetic forces (Kohutova and Verwichte, 2017a). When the cool material is far from the footpoint, the dynamics are governed by the gas pressure and gravity, and the system presents a series of stable and unstable points, of particular interest for prominence formation. This has been recently explored by Adrover-González et al. (2021).

Coronal morphology varies greatly according to wavelength and spatial resolution. As explained above, coronal rain is multi-thermal, and the cold and dense core surrounded by thin but elongated warm sheath at transition region temperatures means that the clumpy character is particularly appreciated in chromospheric wavelengths such as Hα or Ca II H. Accordingly, the widths are sharply distributed at these temperatures around a few hundred km on average, with, however, an abrupt drop at the lower end that suggests limitations from spatial resolution (see Figure 10). Observations with the SST with ≈ 100 km resolution lead to averages of 200–300 km, with minima and maxima around 150 and 500 km (Antolin and Rouppe van der Voort, 2012; Froment et al., 2020), while the flare-driven rain observed in Hα at higher resolution with the NST ( ≈ 40 km), exhibits averages of 120 km, with minima/maxima of 70/250 km (Jing et al., 2016). It is therefore possible that the currently resolved structure is a tip-of-the-iceberg distribution, with far more clumps at higher resolution (Scullion et al., 2014). This is supported by 2.5D simulations by Fang et al. (2013), Li et al. (2022), where a sharp increase in numbers for blob sizes of 50–100 km is obtained. Further support can be found in observations, by noting the stark increase in width values with observations in the SJI 2796 passband with IRIS at ≈ 240 km resolution, dominated by the Mg II k line at similar formations temperatures than Hα (Antolin et al., 2015). Indeed, the Mg II k rain clumps appear with widths of 580 km, with minima/maxima of 300/900 km.

The clumpy morphology is accompanied by a peculiar multi-stranded structure that is particularly visible at high resolution in chromospheric lines (Antolin et al., 2015; Froment et al., 2020). While the clumpy morphology is well-reproduced by multi-dimensional simulations, and the shear flows produced by TI naturally lead to a strand morphology, it is unclear whether the observed multi-stranded structure is well reproduced. Indeed, the strands obtained through the shear flow process are 1–2 Mm apart (see Figure 12), while the observed strands are just a few hundred km apart, extending in the transverse direction for up to 3 Mm (Antolin et al., 2015). It was then hypothesised that TI may be generating this structure through a perpendicular thermal conduction effect (van der Linden and Goossens, 1991).

It is good to note, however, that opacity may also greatly influence the morphology. This is supported by observations with the Solar Optical Telescope (SOT, Suematsu et al., 2008) on board Hinode at ≈ 150 km resolution in the Ca II H line, or with observations with CHROMIS (Löfdahl et al., 2021) at the SST (albeit with average seeing conditions) (Antolin et al., 2015). Despite the high spatial resolution of ≈ 150 km, the Ca II H widths are 430 km on average, with minima/maxima of 150/700 km.

In contrast with the widths, the lengths of rain clumps are more sparsely distributed, with averages between 1,000–3,000 km but long tails to even tens of thousands of km (Antolin et al., 2015). This is expected given the various effects leading to clump elongation, such as shear flows (Fang et al., 2015; Li et al., 2022) and density-regulated downward velocities (Oliver et al., 2014; Martínez-Gómez et al., 2020).

A peculiar property of coronal rain is the fact that rain clumps falling along a similar trajectory appear quasi-simultaneously over a relatively wide region (Antolin and Rouppe van der Voort, 2012). This coherency has a transverse length scale of a few Mm (with respect to the flow of the rain, assumed to be the direction of the magnetic field), and is observed in both large loop systems or relatively isolated loops (Şahin and Antolin, 2022, manuscript in preparation). This is about an order of magnitude larger than a cool and dense clump width (see Figure 9). The timescale of this synchronous behaviour seems to be on the order of minutes or less. This rules out a random behaviour or solely TNE in different field lines without transverse communication. Indeed, if the latter, assuming common footpoint heating conditions and therefore approximately similar TNE cycles across neighboring field lines, would yield an apparent expansion effect in which the rain is seen in gradually longer and higher loops, which is not observed for quiescent rain.

Multi-dimensional simulations by Fang et al. (2013, 2015) have provided a clue behind this behaviour (see also Li et al., 2022). Indeed, a region composed of loops exposed to the same heating function and thus differentiating themselves only on the length, exhibit thermal instability leading to coronal rain quasi-simultaneously, a behaviour denoted as “sympathetic cooling” (see Figure 12). The growth of the unstable region in the direction perpendicular to the loop occurs on a timescale of a minute or so. The physical mechanism behind this is the trigger of the thermal instability in neighboring field lines due to the slow and fast mode perturbations produced by the first TI at the centre of the loop system. In a 3D MHD simulation encompassing a large loop system Moschou et al. (2015), Xia et al. (2017) have confirmed this behaviour, adding the fact that the locations of condensations also coincide with locations unstable to the convective continuum instability (CCI), leading to interchange mode or Rayleigh-Taylor type instabilities at the apex of the loop. However, it is important to note that the coronal magnetic field in these simulations is more of a Quiet Sun type (a few Gauss), and it is therefore unclear whether the CCI would also be effective in active regions. More recently, Jenkins and Keppens (2021) have shown that CCI precedes TI in flux rope configurations, and therefore may play a role in the formation of condensations and showers. In any case, these simulations suggest that showers may be a clear signature of TI within TNE cycles in the sense that TI acts as a locking/syncing mechanism across field lines that are only critically stable. In that sense, the observed few Mm transverse coherency length is produced by a combination of similar footpoint heating conditions across the different field lines and the cross-field effects associated with TI.

While 3D MHD simulations can reproduce to some extent relatively large-scale structures such as showers, they cannot reproduce small observed clumps, due to the limited resolution achieved in the corona (a necessary sacrifice to allow high resolution in the lower atmosphere in non-adaptive codes). They neither can reproduce the high magnetic Reynolds of the corona, meaning that important effects are likely missing. However, these simulations can shed light into the formation process of coronal rain in more realistic atmospheres. Recently, Kohutova et al. (2020) reported for the first time coronal rain generated self-consistently in a 3D radiative MHD simulation with the Bifrost code (see Figure 13). The heating in Bifrost is dominated by Ohmic dissipation (Kanella and Gudiksen, 2019). The heating per unit mass exhibits an apex to footpoint ratio of about 0.1. Also, as seen in Figure 13, the loop with coronal rain experiences high frequency heating, with a few large heating events with durations on the order of 100–200 s. This matches the theoretical conditions for TNE.

Because of their larger widths and lengths, showers can often produce clear EUV absorption features (Antolin et al., 2015). An extreme example of a shower is the flare-driven rain, resulting from the simultaneous and similar heating to a group of field lines during the reconnection event in the corona. Accordingly, dark EUV flaring loops, also known as “Dark Post-Flare Loops” (Song et al., 2016), can be explained by the shower mechanism, as shown by Ruan et al. (2021).