Enthalpy-based Thermal Evolution Of Loops Research Articles

The Effect of Nanoflare Flows on EUV Spectral Lines

The nanoflare model of coronal heating is one of the most successful scenarios to explain, within a single framework, the diverse set of coronal observations available with the present instrument resolutions. The model is based on the idea that the coronal structure is formed by elementary magnetic strands which are tangled and twisted by the displacement of their photospheric footpoints by convective motions. These displacements inject magnetic stress between neighboring strands that promotes current sheet formation, reconnection, plasma heating, and possibly also particle acceleration. Among other features, the model predicts the ubiquitous presence of plasma flows at different temperatures. These flows should, in principle, produce measurable effects on observed spectral lines in the form of Doppler shifts, line asymmetries and nonthermal broadenings. In this work we use the two-dimensional cellular automaton model (2DCAM) developed in previous works, in combination with the enthalpy-based thermal evolution of loops (EBTEL) model, to analyze the effect of nanoflare heating on a set of known EUV spectral lines. We find that the complex combination of the emission from plasmas at different temperatures, densities and velocities, in simultaneously evolving unresolved strands, produces characteristic properties in the constructed synthetic lines, such as Doppler shifts and nonthermal velocities up to tens of km s−1 for the higher analyzed temperatures. Our results might prove useful to guide future modeling and observations, in particular, regarding the new generation of proposed instruments designed to diagnose plasmas in the 5–10 MK temperature range.

Enthalpy-based modeling of tomographically reconstructed quiet-Sun coronal loops

The structure of the solar corona is made of magnetic flux tubes or loops. Due to the lack of contrast with their environment, observing and studying coronal loops in the quiet Sun is extremely difficult. In this work we use a differential emission measure tomographic (DEMT) technique to reconstruct, from a series of EUV images covering an entire solar rotation, the average 3D distribution of the thermal properties of the coronal plasma. By combining the DEMT products with extrapolations of the global coronal magnetic field, we reconstruct coronal loops and obtain the energy input required to keep them at the typical million-degree temperatures of the corona. We statistically study a large number of reconstructed loops for Carrington rotation (CR) 2082 obtaining a series of typical average loops of different lengths. We look for relations between the thermal properties and the lengths of the constructed typical loops and find similar results to those found in a previous work (Mac Cormack et al., 2020).. We also analyze the typical loop properties by comparing them with the zero-dimensional (0D) hydrodynamic model Enthalpy-Based Thermal Evolution of Loops (EBTEL, Klimchuk et al., 2008). We explore two heating scenarios. In the first one, we apply a constant heating rate assuming that typical loops are in quasi-static equilibrium. In the second scenario we heat the plasma in the loops using short impulsive events. We find that the reconstructed typical loops are overdense with respect to quasi-static equilibrium solutions of the hydrodynamic model. Impulsive heating, on the other hand, reproduces better the observed densities and temperatures for the shorter and approximately semicircular loops. The thermal properties of longer loops cannot be correctly reproduced with the EBTEL model. We suggest that to properly assess the physical characteristics of the analyzed loops in future works, it would be necessary to use a more sophisticated 1D model, with which to study the loop temperature and density profiles and test localized heating at different locations along the loops.

Flows in Enthalpy-based Thermal Evolution of Loops

Abstract Plasma-filled loop structures are common in the solar corona. Because detailed modeling of the dynamical evolution of these structures is computationally costly, an efficient method for computing approximate but quick physics-based solutions is to rely on space-integrated 0D simulations. The enthalpy-based thermal evolution of loops (EBTEL) framework is a commonly used method to study the exchange of mass and energy between the corona and transition region. EBTEL solves for density, temperature, and pressure, averaged over the coronal part of the loop, velocity at coronal base, and the instantaneous differential emission measure distribution in the transition region. The current single-fluid version of the code, EBTEL2, assumes that at all stages the flows are subsonic. However, sometimes the solutions show the presence of supersonic flows during the impulsive phase of heat input. It is thus necessary to account for this effect. Here, we upgrade EBTEL2 to EBTEL3 by including the kinetic energy term in the Navier–Stokes equation. We compare the solutions from EBTEL3 with those obtained using EBTEL2, as well as the state-of-the-art field-aligned hydrodynamics code HYDRAD. We find that the match in pressure between EBTEL3 and HYDRAD is better than that between EBTEL2 and HYDRAD. Additionally, the velocities predicted by EBTEL3 are in close agreement with those obtained with HYDRAD when the flows are subsonic. However, EBTEL3 solutions deviate substantially from HYDRAD’s when the latter predicts supersonic flows. Using the mismatches in the solution, we propose a criterion to determine the conditions under which EBTEL can be used to study flows in the system.

Energy partition in a confined flare with an extreme-ultraviolet late phase

Aims. In this paper, we reanalyze the M1.2 confined flare with a large extreme-ultraviolet (EUV) late phase on 2011 September 9, with a focus on its energy partition. Methods. The flare was observed by the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamics Observatory (SDO). The three-dimensional (3D) magnetic fields of the active region 11283 prior to the flare were obtained using nonlinear force free field modeling and the vector magnetograms observed by the Helioseismic and Magnetic Imager (HMI) on board the SDO. Properties of the nonthermal electrons injected into the chromosphere were obtained from the hard X-ray observations of the Ramaty Hight Energy Solar Spectroscopic Imager (RHESSI). Soft X-ray fluxes of the flare were recorded by the GOES spacecraft. Irradiance in 1−70 Å and 70−370 Å were measured by the EUV Variability Experiment (EVE) on board the SDO. We calculated various energy components of the flare. Results. The radiation (∼5.4 × 1030 erg) in 1−70 Å is nearly eleven times larger than the radiation in 70−370 Å, and is nearly 180 times larger than the radiation in 1−8 Å. The peak thermal energy of the post-flare loops is estimated to be (1.7−1.8) × 1030 erg based on a simplified schematic cartoon. Based on previous results of the enthalpy-based thermal evolution of loops (EBTEL) simulation, the energy inputs in the main flaring loops and late-phase loops are (1.5−3.8) × 1029 erg and 7.7 × 1029 erg, respectively. The nonthermal energy ((1.7−2.2) × 1030 erg) of the flare-accelerated electrons is comparable to the peak thermal energy and is sufficient to provide the energy input of the main flaring loops and late-phase loops. The magnetic free energy (9.1 × 1031 erg) before flare is large enough to provide the heating requirement and radiation, indicating that the magnetic free energy is sufficient to power the flare.

Transition Region Contribution to AIA Observations in the Context of Coronal Heating

Abstract We investigate the ratio of coronal and transition region intensity in coronal loops observed by the Atmospheric Imaging Assembly (AIA) on the Solar Dynamics Observatory (SDO). Using Enthalpy-based Thermal Evolution of Loops (EBTEL) hydrodynamic simulations, we model loops with multiple lengths and energy fluxes heated randomly by events drawn from power-law distributions with different slopes and minimum delays between events to investigate how each of these parameters influences observable loop properties. We generate AIA intensities from the corona and transition region for each realization. The variations within and between models generated with these different parameters illustrate the sensitivity of narrowband imaging to the details of coronal heating. We then analyze the transition region and coronal emission from a number of observed active regions and find broad agreement with the trends in the models. In both models and observations, the transition region brightness is significant, often greater than the coronal brightness in all six “coronal” AIA channels. We also identify an inverse relationship, consistent with heating theories, between the slope of the differential emission measure (DEM) coolward of the peak temperature and the observed ratio of coronal to transition region intensity. These results highlight the use of narrowband observations and the importance of properly considering the transition region in investigations of coronal heating.

Three-Dimensional Reconstruction and Thermal Modeling of Observed Loops

Due to their characteristic temperature and density, loop structures in active regions (ARs) can be seen bright in extreme ultraviolet (EUV) and soft X-ray images. The semiempirical determination of the three-dimensional (3D) distribution of basic physical parameters (electronic density and temperature, and magnetic field) is a key constraint for coronal heating models. In this work we develop a technique for the study of EUV bright loops based on differential emission measure (DEM) analysis and we first apply it to AR structures observed by the {Atmospheric Imaging Assembly} (AIA) on board the {Solar Dynamics Observatory} (SDO). The 3D structure and intensity of the magnetic field of the observed EUV loops are modelled using force-free field extrapolations based on magnetograms taken by the {Helioseismic and Magnetic Imager} (HMI) on board SDO. In this work we report the results obtained for several bright loops identified in different ARs. Our analysis indicates that the mean and width of the temperature distributions are nearly invariant along the loop lengths. For a particular loop we study its temporal evolution and find that these characteristics remain approximately constant for most of its life time. The appearance and disappearance of this loop occurs at time-scales much shorter than its life time of $\approx 2.5$ hours. The results of this analysis are compared with numerical simulations using the zero-dimensional (0D) hydrodynamic model, Enthalpy-Based Thermal Evolution of Loops (EBTEL). We study two alternative heating scenarios: first, we apply a constant heating rate assuming loops in quasi-static equilibrium, and second, we heat the loops using impulsive events or nanoflares. We find that all the observed loops are overdense with respect to a quasi-static equilibrium solution and that the nanoflare heating better reproduces the observed densities and temperatures.

Extremely Large Extreme-ultraviolet Late Phase Powered by Intense Early Heating in a Non-eruptive Solar Flare

Abstract We analyzed and modeled an M1.2 non-eruptive solar flare on 2011 September 9. The flare exhibited a strong late-phase peak of extreme-ultraviolet (EUV) warm coronal emissions (∼3 MK), with peak emission over 1.3 times that of the main flare peak. Multiple flare ribbons are observed, whose evolution indicates a two-stage energy release process. A nonlinear force-free field extrapolation reveals the existence of a magnetic null point, a fan-spine structure, and two flux ropes embedded in the fan dome. Magnetic reconnections involved in the flare are driven by the destabilization and rise of one of the flux ropes. In the first stage, the fast ascending flux rope drives reconnections at the null point and the surrounding quasi-separatrix layer (QSL), while in the second stage, reconnection mainly occurs between the two legs of the field lines stretched by the eventually stopped flux rope. The late-phase loops are mainly produced by the first-stage QSL reconnection, while the second-stage reconnection is responsible for the heating of main flaring loops. The first-stage reconnection is believed to be more powerful, leading to an extremely strong EUV late phase. We find that the delayed occurrence of the late-phase peak is mainly due to the long cooling process of the long late-phase loops. Using the model enthalpy-based thermal evolution of loops, we model the EUV emissions from a late-phase loop. The modeling reveals a peak heating rate of 1.1 erg cm−3 s−1 for the late-phase loop, which is obviously higher than previous values.

Probing the Production of Extreme-ultraviolet Late-phase Solar Flares Using the Model Enthalpy-based Thermal Evolution of Loops

Abstract Recent observations in extreme-ultraviolet (EUV) wavelengths reveal an EUV late phase in some solar flares that is characterized by a second peak in warm coronal emissions (∼3 MK) several tens of minutes to a few hours after the soft X-ray (SXR) peak. Using the model enthalpy-based thermal evolution of loops (EBTEL), we numerically probe the production of EUV late-phase solar flares. Starting from two main mechanisms of producing the EUV late phase, i.e., long-lasting cooling and secondary heating, we carry out two groups of numerical experiments to study the effects of these two processes on the emission characteristics in late-phase loops. In either of the two processes an EUV late-phase solar flare that conforms to the observational criteria can be numerically synthesized. However, the underlying hydrodynamic and thermodynamic evolutions in late-phase loops are different between the two synthetic flare cases. The late-phase peak due to a long-lasting cooling process always occurs during the radiative cooling phase, while that powered by a secondary heating is more likely to take place in the conductive cooling phase. We then propose a new method for diagnosing the two mechanisms based on the shape of EUV late-phase light curves. Moreover, from the partition of energy input, we discuss why most solar flares are not EUV late flares. Finally, by addressing some other factors that may potentially affect the loop emissions, we also discuss why the EUV late phase is mainly observed in warm coronal emissions.

Two-phase Heating in Flaring Loops

Abstract We analyze and model a C5.7 two-ribbon solar flare observed by the Solar Dynamics Observatory, Hinode, and GOES on 2011 December 26. The flare is made of many loops formed and heated successively over one and half hours, and their footpoints are brightened in the UV 1600 Å before enhanced soft X-ray and EUV missions are observed in flare loops. Assuming that anchored at each brightened UV pixel is a half flaring loop, we identify more than 6700 half flaring loops, and infer the heating rate of each loop from the UV light curve at the footpoint. In each half loop, the heating rate consists of two phases: intense impulsive heating followed by a low-rate heating that is persistent for more than 20 minutes. Using these heating rates, we simulate the evolution of their coronal temperatures and densities with the model of the “enthalpy-based thermal evolution of loops.” In the model, suppression of thermal conduction is also considered. This model successfully reproduces total soft X-ray and EUV light curves observed in 15 passbands by four instruments GOES, AIA, XRT, and EVE. In this flare, a total energy of 4.9 × 1030 erg is required to heat the corona, around 40% of this energy is in the slow-heating phase. About two-fifths of the total energy used to heat the corona is radiated by the coronal plasmas, and the other three fifth transported to the lower atmosphere by thermal conduction.

Heating and Cooling of Coronal Loops with Turbulent Suppression of Parallel Heat Conduction.

Using the "enthalpy-based thermal evolution of loops" (EBTEL) model, we investigate the hydrodynamics of the plasma in a flaring coronal loop in which heat conduction is limited by turbulent scattering of the electrons that transport the thermal heat flux. The EBTEL equations are solved analytically in each of the two (conduction-dominated and radiation-dominated) cooling phases. Comparison of the results with typical observed cooling times in solar flares shows that the turbulent mean free path λT lies in a range corresponding to a regime in which classical (collision-dominated) conduction plays at most a limited role. We also consider the magnitude and duration of the heat input that is necessary to account for the enhanced values of temperature and density at the beginning of the cooling phase and for the observed cooling times. We find through numerical modeling that in order to produce a peak temperature ≃1.5 × 107 K and a 200 s cooling time consistent with observations, the flare-heating profile must extend over a significant period of time; in particular, its lingering role must be taken into consideration in any description of the cooling phase. Comparison with observationally inferred values of post-flare loop temperatures, densities, and cooling times thus leads to useful constraints on both the magnitude and duration of the magnetic energy release in the loop, as well as on the value of the turbulent mean free path λT .

A time dependent relation between EUV solar flare light-curves from lines with differing formation temperatures

Extreme ultraviolet (EUV) solar flare emissions evolve in time as the emitting plasma heats and then cools. Although accurately modeling this evolution has been historically difficult, especially for empirical relationships, it is important for understanding processes at the Sun, as well as for their influence on planetary atmospheres. With a goal to improve empirical flare models, a new simple empirical expression is derived to predict how cool emissions evolve based on the evolution of a hotter emission. This technique is initially developed by studying 12 flares in detail observed by the EUV variability experiment (EVE) onboard the Solar Dynamics Observatory (SDO). Then, over 1100 flares observed by EVE are analyzed to validate these relationships. The Cargill and Enthalpy Based Thermal Evolution of Loops (EBTEL) flare cooling models are used to show that this empirical relationship implies the energy radiated by a population of hotter formed ions is approximately proportional to the energy exciting a population of cooler formed ions emitting when the peak formation temperatures of the two lines are up to 72% of each other and above 2 MK. These results have practical implications for improving flare irradiance empirical modeling and for identifying key emission lines for future monitoring of flares for space weather operations; and also provide insight into the cooling processes of flare plasma.

TWO-DIMENSIONAL CELLULAR AUTOMATON MODEL FOR THE EVOLUTION OF ACTIVE REGION CORONAL PLASMAS

We study a 2D cellular automaton (CA) model for the evolution of coronal loop plasmas. The model is based on the idea that coronal loops are made of elementary magnetic strands that are tangled and stressed by the displacement of their footpoints by photospheric motions. The magnetic stress accumulated between neighbor strands is released in sudden reconnection events or nanoflares that heat the plasma. We combine the CA model with the Enthalpy Based Thermal Evolution of Loops (EBTEL) model to compute the response of the plasma to the heating events. Using the known response of the XRT telescope on board Hinode we also obtain synthetic data. The model obeys easy to understand scaling laws relating the output (nanoflare energy, temperature, density, intensity) to the input parameters (field strength, strand length, critical misalignment angle). The nanoflares have a power-law distribution with a universal slope of -2.5, independent of the input parameters. The repetition frequency of nanoflares, expressed in terms of the plasma cooling time, increases with strand length. We discuss the implications of our results for the problem of heating and evolution of active region coronal plasmas.

ON THE NATURE OF THE EXTREME-ULTRAVIOLET LATE PHASE OF SOLAR FLARES

The extreme-ultraviolet (EUV) late phase of solar flares is a second peak of warm coronal emissions (e.g., Fe XVI) for many minutes to a few hours after the GOES soft X-ray peak. It was first observed by the EUV Variability Experiment (EVE) on board the Solar Dynamics Observatory (SDO). The late phase emission originates from a second set of longer loops (late phase loops) that are higher than the main flaring loops. It is suggested as being caused by either additional heating or long-lasting cooling. In this paper, we study the role of long-lasting cooling and additional heating in producing the EUV late phase using the "enthalpy-based thermal evolution of loops" (EBTEL) model. We find that a long cooling process in late phase loops can well explain the presence of the EUV late phase emission, but we cannot exclude the possibility of additional heating in the decay phase. Moreover, we provide two preliminary methods based on the UV and EUV emissions from the Atmospheric Imaging Assembly (AIA) on board SDO to determine whether an additional heating plays some role or not in the late phase emission. Using nonlinear force-free field modeling, we study the magnetic configuration of the EUV late phase. It is found that the late phase can be generated either in hot spine field lines associated with a magnetic null point or in large-scale magnetic loops of multipolar magnetic fields. In this paper, we also discuss why the EUV late phase is usually observed in warm coronal emissions and why the majority of flares do not exhibit an EUV late phase.

HEATING AND DYNAMICS OF TWO FLARE LOOP SYSTEMS OBSERVED BY AIA AND EIS

We investigate heating and evolution of flare loops in a C4.7 two-ribbon flare on 2011 February 13. From SDO/AIA imaging observations, we can identify two sets of loops. Hinode/EIS spectroscopic observations reveal blueshifts at the feet of both sets of loops. The evolution and dynamics of the two sets are quite different. The first set of loops exhibits blueshifts for about 25 minutes followed by redshifts, while the second set shows stronger blueshifts, which are maintained for about one hour. The UV 1600 observation by AIA also shows that the feet of the second set of loops brighten twice. These suggest that continuous heating may be present in the second set of loops. We use spatially resolved UV light curves to infer heating rates in the few tens of individual loops comprising the two loop systems. With these heating rates, we then compute plasma evolution in these loops with the "enthalpy-based thermal evolution of loops" (EBTEL) model. The results show that, for the first set of loops, the synthetic EUV light curves from the model compare favorably with the observed light curves in six AIA channels and eight EIS spectral lines, and the computed mean enthalpy flow velocities also agree with the Doppler shift measurements by EIS. For the second set of loops modeled with twice-heating, there are some discrepancies between modeled and observed EUV light curves in low-temperature bands, and the model does not fully produce the prolonged blueshift signatures as observed. We discuss possible causes for the discrepancies.

OBSERVATIONS AND MODELING OF THE EMERGING EXTREME-ULTRAVIOLET LOOPS IN THE QUIET SUN AS SEEN WITH THESOLAR DYNAMICS OBSERVATORY

We used data from the Helioseismic and Magnetic Imager (HMI), and Atmospheric Imaging Assembly (AIA) on the \textit{Solar Dynamics Observatory} (SDO) to study coronal loops at small scales, emerging in the quiet Sun. With HMI line-of-sight magnetograms, we derive the integrated and unsigned photospheric magnetic flux at the loop footpoints in the photosphere. These loops are bright in the EUV channels of AIA. Using the six AIA EUV filters, we construct the differential emission measure (DEM) in the temperature range $5.7 - 6.5$ in log $T$ (K) for several hours of observations. The observed DEMs have a peak distribution around log $T \approx$ 6.3, falling rapidly at higher temperatures. For log $T <$ 6.3, DEMs are comparable to their peak values within an order of magnitude. The emission weighted temperature is calculated, and its time variations are compared with those of magnetic flux. We present two possibilities for explaining the observed DEMs and temperatures variations. (a) Assuming the observed loops are comprised of hundred thin strands with certain radius and length, we tested three time-dependent heating models and compared the resulting DEMs and temperatures with the observed quantities. This modeling used Enthalpy-based Thermal Evolution of Loops (EBTEL), a zero-dimensional (0D) hydrodynamic code. The comparisons suggest that a medium frequency heating model with a population of different heating amplitudes can roughly reproduce the observations. (b) We also consider a loop model with steady heating and non-uniform cross-section of the loop along its length, and find that this model can also reproduce the observed DEMs, provided the loop expansion factor $\gamma \sim$ 5 - 10. More observational constraints are required to better understand the nature of coronal heating in the short emerging loops on the quiet Sun.

ENTHALPY-BASED THERMAL EVOLUTION OF LOOPS. II. IMPROVEMENTS TO THE MODEL

This paper develops the zero-dimensional (0D) hydrodynamic coronal loop model "Enthalpy-based Thermal Evolution of Loops" (EBTEL) proposed by Klimchuk et al (2008), which studies the plasma response to evolving coronal heating, especially impulsive heating events. The basis of EBTEL is the modelling of mass exchange between the corona and transition region and chromosphere in response to heating variations, with the key parameter being the ratio of transition region to coronal radiation. We develop new models for this parameter that now include gravitational stratification and a physically motivated approach to radiative cooling. A number of examples are presented, including nanoflares in short and long loops, and a small flare. The new features in EBTEL are important for accurate tracking of, in particular, the density. The 0D results are compared to a 1D hydro code (Hydrad) with generally good agreement. EBTEL is suitable for general use as a tool for (a) quick-look results of loop evolution in response to a given heating function, (b) extensive parameter surveys and (c) situations where the modelling of hundreds or thousands of elemental loops is needed. A single run takes a few seconds on a contemporary laptop.

Highly Efficient Modeling of Dynamic Coronal Loops

Observational and theoretical evidence suggests that coronal heating is impulsive and occurs on very small cross-field spatial scales. A single coronal loop could contain a hundred or more individual strands that are heated quasi-independently by nanoflares. It is therefore an enormous undertaking to model an entire active region or the global corona. Three-dimensional MHD codes have inadequate spatial resolution, and one-dimensional (1D) hydrodynamic codes are too slow to simulate the many thousands of elemental strands that must be treated in a reasonable representation. Fortunately, thermal conduction and flows tend to smooth out plasma gradients along the magnetic field, so zero-dimensional (0D) models are an acceptable alternative. We have developed a highly efficient model called enthalpy-based thermal evolution of loops (EBTEL), which accurately describes the evolution of the average temperature, pressure, and density along a coronal strand. It improves significantly on earlier models of this type—in accuracy, flexibility, and capability. It treats both slowly varying and highly impulsive coronal heating; it provides the time-dependent differential emission measure distribution, DEM(T), at the transition region footpoints; and there are options for heat flux saturation and nonthermal electron beam heating. EBTEL gives excellent agreement with far more sophisticated 1D hydrodynamic simulations despite using 4 orders of magnitude less computing time. It promises to be a powerful new tool for solar and stellar studies.