Energy Input Flux in the Global Quiet-Sun Corona

The heating of the solar corona remains a main topic of current research in solar physics. Although there is a wide consensus in the solar physics community on the magnetic nature of the phenomena responsible for the heating of the million-degree corona, the precise mechanisms by which this occurs are an open field of research and active debate.

Coronal heating theories are traditionally classified into two broad groups: those based on the dissipation of magnetic stress, informally called DC-heating, and those based on the dissipation of magnetohydrodinamic (MHD) waves, also known as AC-heating. The first group includes models in which the stress produced in the coronal field by photospheric motions is released in situ by reconnection and current sheet formation (see e.g., Parker 1988; Priest 2011). Other MHD simulations (Gudiksen & Nordlund 2005) model the system as a numerical enclosure in which the energy is injected by photospheric motions at the base and released by Ohmic dissipation in the corona. Also in the first group are models based on the energy release due to the turbulent interplay between photospheric plasma motions and the magnetic field (Gomez et al. 2000). AC-heating mechanisms are based on the propagation of disturbances produced at the feet of the magnetic structures that dissipate at certain atmospheric layers, releasing energy that translates into plasma heating (Heyvaerts & Priest 1983; De Moortel et al. 2000; O'Neill & Li 2005). Recently, van Ballegooijen et al. (2011) proposed a mixed mechanism by which the diffusion of MHD waves at the chromosphere and transition region (TR) interface produces a turbulent regime that heats the plasma.

The heating of the plasma in different structures (coronal holes, bright loops in active regions [ARs], quiet-Sun corona, etc.) is probably dominated by different physical mechanisms. To advance our understanding of the physics underlying this complex phenomenon, advances on observational constraints are key. The majority of the observational literature on coronal heating focus on AR structures, where individual bright extreme ultraviolet (EUV) and X-ray loops provide direct diagnostics of magnetic structures (Reale 2010; Schmelz et al. 2010; Aschwanden & Boerner 2011; Klimchuk 2015). Although not evident from EUV/X-ray images due to its diffuse appearance, the quiet-Sun corona is of course also fully threaded by magnetic fields along which energy transport and deposition takes place. The observational study of the heating in the quiet-Sun diffuse corona has received comparatively less attention (Benz & Krucker 2002; Wilhelm et al. 2004; Hahn & Savin 2014).

In general, the aforementioned works provide insight on the heating phenomenon at a local scale, for the specific structures selected for observation, and are affected by line-of-sight projection effects. Differential emission measure tomography (DEMT) provides a powerful tool to study the quiet-Sun corona at a global scale and in three dimensions (Frazin et al. 2009; Vásquez 2016). Based on full solar rotation time series of EUV images taken in channels sensitive to different temperatures, DEMT provides three-dimensional (3D) maps of the electron density and temperature of the lower corona, in the heliocentric height range 1.02–1.225 R_⊙. The coronal magnetic field in these regions can be globally modeled by means of potential-field source surface (PFSS) or MHD models. A combination of the DEMT and global magnetic models has provided useful insight into the 3D thermodynamical structure of the global quiet-Sun corona (Huang et al. 2012; Nuevo et al. 2013, 2015), making it an ideal tool to provide constraints on the coronal heating for these regions. However, until now, such a DEMT application had not been developed.

In this work, we develop the first version of a new DEMT tool capable of providing constraints on the coronal heating of the quiet-Sun global corona. Specifically, it provides spatial two-dimensional (2D) maps of the energy input flux required at the coronal base to maintain thermodynamically stable coronal structures under hydrostatic assumption. This new tool is applied to two specific selected rotations with different levels of activity, studied by means of two EUV instruments, namely the Extreme Ultraviolet Imager (EUVI; Wuelser et al. 2004) on board the Solar TErrestrial RElations Observatory (STEREO) mission, and the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) telescope on board the Solar Dynamics Observatory (SDO) mission.

Section 2 summarizes the techniques, instruments, and data sets used. Section 3 details the energy model and the loop-integrated quantities that are introduced, which form the new DEMT tool. Section 4 shows the DEMT results for the selected periods, while Section 5 details the new results on energy input flux at the coronal base. In Section 6, these results are compared to a 0D hydrodynamic (HD) model. Section 7 discusses the main conclusions and its implications, and anticipates further planned work.

This work is based on the study of the global corona by means of the DEMT technique to determine its 3D thermodynamical structure, the PFSS modeling of its global magnetic field, and the combination of both. The technique is applied to two specific Carrington rotations (CRs): CR 2081 (2009 March 9 through April 5), a deep minimum period between solar cycles (SCs) 23 and 24 characterized by virtually no ARs, and CR 2099 (2010 July 13 through August 9), a rotation during the early rising phase of SC 24.

Both periods were tomographically reconstructed from data taken by the EUVI/STEREO instrument. In the case of CR 2099, the period was also reconstructed using data taken by the AIA/SDO instrument. For both rotations, the magnetic field was modeled by means of the Finite Difference Potential-field Solver PFSS model developed by Tóth et al. (2011), using as boundary conditions synoptic magnetograms built from data taken by the Michelson Doppler Imager (MDI; Scherrer et al. 1995) on board the Solar and Heliospheric Observatory mission.

In DEMT, the inner corona, in the height range 1.0–1.25 R_⊙, is discretized in a spherical computational grid. The size of the grid cell (or voxel) is set to 0.01 R_⊙ in the radial direction (or ∼7 Mm), and to 2° in both the latitudinal and longitudinal directions (or ∼27 Mm at an intermediate grid height of r = 1.1 R_⊙ at the equator). With this angular resolution, one image every 6 hr is the cadence needed to fully constrain the inversion problem, for a total of about 110 images to cover a full solar rotation. This is the standard DEMT resolution used over the past few years in all previously published work based on this technique (Vásquez 2016), providing a good compromise between resolution and computational load.

Two consecutive procedures are then performed. In a first step, time series of EUV images in different bands, covering a full solar rotation, are used to perform EUV tomography. The product of the tomographic inversion in each band is the 3D distribution of the filter-band emissivity (FBE), defined as the wavelength integral of the coronal EUV spectral emissivity and the telescope's passband function of each band.

In a second step, the FBE values obtained for all bands in each tomographic cell (or voxel) are used to constrain the determination of a local differential emission measure (LDEM) distribution. The LDEM describes the temperature distribution of the electron plasma contained in each individual tomographic grid voxel. The LDEM is defined so that the electron density N_e and electron mean temperature T_m (averaged over the temperature) of each voxel are computed as

$$\begin{eqnarray}&&{N}_{{\rm{e}}}^{2}=\int \,{dT}\,\mathrm{LDEM}(T),\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{T}_{{\rm{m}}}=\displaystyle \frac{1}{{N}_{{\rm{e}}}^{2}}\,\int \,{dT}\,T\,\mathrm{LDEM}(T).\end{eqnarray} \tag{ 2 }$$

Due to their ill-posed nature, DEM inversion problems are difficult to treat, both in bright loops observed in ARs, as well as in the diffuse quiet-Sun corona analyzed here. If based on high-resolution spectral data, DEM analysis is best performed through either the Monte Carlo Markov Chain (MCMC) approach (Kashyap & Drake 1998) or regularized inversion techniques (Hannah & Kontar 2012; also applicable to filter-band data sets). In the case of narrowband filter telescopes, the AIA instrument has improved temperature diagnostic capabilities over previous EUV telescopes instruments (such as EUVI). Application of MCMC methods based on AIA images has been explored in several works (Testa et al. 2012; Del Zanna 2013), as discussed in Nuevo et al. (2015). Schmelz et al. (2013) have recently shown advances of MCMC multithermal DEM analysis based on AIA data of bright EUV loops in ARs, using all six AIA channels simultaneously, with updated temperature response functions based on CHIANTI v7.1. (see also Schmelz et al. 2016 for an application to an extended study of bright loops).

In the case of DEMT, the DEM parametric approach is suitable due to the limited number of data points. Nuevo et al. (2015) have developed a detailed study of the capabilities of the parametric technique in DEMT when applied to both EUVI and AIA data. In the case of EUVI data, its three coronal bands are used (171, 195, and 284 Å), with maximum sensitivity temperatures in the range 1.0–2.15 MK (see Table 1 in Nuevo et al. 2015). In the case of AIA data, as DEMT targets the diffuse quiet-Sun corona, the filters that are currently used in DEMT are those of 171, 193, 211, and 335 Å, with maximum sensitivity temperatures in the range 0.85–2.5 MK (see Table 1 in Nuevo et al. 2015). These four AIA bands cover the main temperature range characteristic of the quiet-Sun regions targeted by DEMT. Adding the 94 and 131 Å bands of AIA to DEMT studies has also been attempted (as well as exploring other parametric models for the LDEM), but at a global scale (as required in tomography) the signal from these channels in the diffuse quiet Sun is in general too weak to provide meaningful information.

In this study, the EUVI and AIA temperature responses have been computed based on CHIANTI v7.1, as discussed in detail in Nuevo et al. (2015). As shown in that work, when using the aforementioned four coronal bands of AIA, the tomographic emissivity 3D distributions are best explained by an ubiquitous bimodal LDEM, described by two Gaussian distributions with characteristic centroids of order 1.5 MK and 2.6 MK, dubbed "warm" and "hot," respectively. In the case when all three EUVI channels are used, DEMT detects only the warm component. It is shown (1) that if only the three AIA channels of 171, 193 and 211 Å are used (as in this work), then only the warm component is also detected, and (2) that in that case, the results are very similar to those based on EUVI data, with some small systematic differences due to the different response functions of the respective trios of filters. The LDEM obtained are typically broad, with variable FWHM depending on the region. Indeed, in that same work, a controlled study inverting synthetic data from modeled DEM distributions has shown DEMT to be able to detect nearly isothermal and multithermal plasmas, reflected in the resulting value of the FWHM of the LDEM.

The parametric approach is at the moment the best implementation of DEMT. Even with its limitations, the resulting LDEM distributions, as well as the electron density and mean temperature maps they produce, have been validated with other types of observational studies. For example, in Nuevo et al. (2015) the LDEM parametric technique has been validated by comparing its results when applied to LOS–DEM analysis of image pixels against those obtained by other authors using MCMC techniques (based on Extreme Ultraviolet Imaging Spectrometer [EIS] data). Similar results are obtained, in particular in terms of the bimodal DEMs. Applied to many different rotations, DEMT has so far provided consistent results across many studies and coronal regions, which have also been used as a validation tool for MHD simulations of the global corona (see review in Vásquez 2016). Future new developments could explore the implementation of regularized inversion techniques for the determination of the LDEM at each voxel, or even MCMC methods as applied to AIA images by Schmelz et al. (2016), who have been able to exploit those if at least three data points are available.

In this work, all three coronal bands of EUVI are used to study CR 2081 and CR 2099, and in the latter case, an alternate study based on the AIA trio 171–193–211 Å is also carried out. When using three bands, as in this work, the LDEM is modeled by a Gaussian function (Nuevo et al. 2015) dependent on three free parameters: centroid temperature, temperature width, and total area. In each voxel, the values of the free parameters are found so that the synthesized emissivity values ${\mathrm{FBE}}_{\mathrm{syn}}^{(k)}$ best match the tomographic values ${\mathrm{FBE}}_{\mathrm{tom}}^{(k)}$ for all three bands k = 1, 2, 3, achieved by minimizing the score

$$\begin{eqnarray}&&R=\displaystyle \frac{1}{3}\displaystyle \sum _{k=1}^{3}| \,1-{\mathrm{FBE}}_{\mathrm{syn}}^{(k)}/{\mathrm{FBE}}_{\mathrm{tom}}^{(k)}\,| .\end{eqnarray} \tag{ 3 }$$

A score of R ∼ 0 means the LDEM accurately predicts the three tomographic FBEs at a given voxel, while higher scores mean a less accurate prediction.

Once the LDEM is obtained in each voxel of the tomographic computational grid, Equations (1) through (3) allow the computation of 3D maps of the electron density N_e, electron mean temperature T_m, and score R. The reader is referred to Frazin et al. (2009) and Nuevo et al. (2015) for a detailed description of the DEMT technique, and to Vásquez (2016) for a recent review of all published work based on this technique.

The DEMT results can then be combined with the global magnetic field model by tracing the results of the former along the magnetic field lines of the latter. This approach was first used by Huang et al. (2012) to study the temperature structure of the solar corona during the last minimum, and later on applied by Nuevo et al. (2013) to expand the analysis to rotations with different level of activity. In this paper, the same approach is used to study, in an original fashion, the energy balance along individual magnetic loops, as described in Section 3. In doing so, a new set of numerical tools within the suite of DEMT codes was developed.

A simple hydrostatic model for a coronal magnetic flux tube, sketched in Figure 1, is considered. The position along the tube is given by the variable s, with s = 0 and s = L representing the positions at the coronal base. At any position s, the (unknown) coronal heating power E_h(s) is balanced by the two major coronal losses, namely, the radiative power E_r(s) and the thermal conduction power E_c(s) (Aschwanden 2004),

$$\begin{eqnarray}&&{E}_{{\rm{h}}}(s)={E}_{{\rm{r}}}(s)+{E}_{{\rm{c}}}(s),\end{eqnarray} \tag{ 4 }$$

where the three power quantities are per unit volume, i.e., have units of (erg s⁻¹ cm⁻³). In the optically thin corona, the radiative energy is emitted isotropically, while the heat conductive flux is constrained to flow along magnetic field lines. Considering the coronal magnetic flux tube as a whole, the conductive flux at the coronal base represents a net gain or loss of energy for the system.

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The thermal conduction power E_c equals the divergence of the conductive heat flux F_c, which can be expressed as the derivative along the flux tube,

$$\begin{eqnarray}&&{E}_{{\rm{c}}}(s)=\displaystyle \frac{1}{A(s)}\,\displaystyle \frac{d}{{ds}}[A(s){F}_{{\rm{c}}}(s)],\end{eqnarray} \tag{ 5 }$$

where, for the quiescent solar coronal plasma regime, the conductive heat is dominated by the electron thermal conduction described by the usual Spitzer model (Spitzer 1962),

$$\begin{eqnarray}&&{F}_{{\rm{c}}}{(s)=-{\kappa }_{0}T(s)}^{5/2}\,\displaystyle \frac{{dT}}{{ds}}(s),\end{eqnarray} \tag{ 6 }$$

where the Spitzer thermal conductivity is κ₀ ≈ 9.2 × 10⁻⁷ erg s⁻¹ cm⁻¹ K^−7/2.

For each of the three power quantities per unit volume in Equation (4), a corresponding total power γ (erg s⁻¹) in the coronal part of the magnetic flux tube is obtained by integrating them over its whole volume,

$$\begin{eqnarray}&&{\gamma }_{i}\equiv {\int }_{0}^{L}{ds}\,A(s){E}_{i}(s),\end{eqnarray} \tag{ 7 }$$

where i = c, r, h denotes the conductive, radiative, and heating terms, respectively. Using Equation (5), the integrated thermal conduction power can be expressed as

$$\begin{eqnarray}&&{\gamma }_{{\rm{c}}}={A}_{L}\,{F}_{{\rm{c}},L}-{A}_{0}\,{F}_{{\rm{c}},0},\end{eqnarray} \tag{ 8 }$$

where A₀ and A_L are the values of the transversal area at s = 0 and s = L, respectively, and F_c,0 and ${F}_{{\rm{c}},L}$ are the respective values of the conductive heat flux.

Dividing the three integrated power quantities in Equation (7) by the total basal area of the flux tube ${A}_{0}+{A}_{L}$, the three associated flux quantities ϕ (erg s⁻¹ cm⁻²) can be defined as

$$\begin{eqnarray}&&{\phi }_{i}\equiv \displaystyle \frac{{\gamma }_{i}}{{A}_{0}+{A}_{L}};\,i={\rm{h}},{\rm{r}},{\rm{c}}.\end{eqnarray} \tag{ 9 }$$

Using these last three quantities, integration of Equation (4) over the whole coronal volume of the magnetic flux tube implies the integrated energy balance

$$\begin{eqnarray}&&{\phi }_{{\rm{h}}}={\phi }_{{\rm{r}}}+{\phi }_{{\rm{c}}}.\end{eqnarray} \tag{ 10 }$$

The energy required to heat the plasma contained in the magnetic flux tube is ultimately injected into it through its coronal base. The quantity ϕ_h represents the total energy input flux due to all mechanisms except heat conduction (accounted for by the term ϕ_c). The quantity ϕ_h will hereafter be referred to as the "energy input flux" at the coronal base. The quantity ϕ_c is the total energy flux entering (or leaving) the coronal part of the magnetic tube due to heat conduction. The quantity ϕ_r is the total radiative power emitted by the plasma contained in the coronal part of the magnetic flux tube, divided by the total area of its coronal base. Being proportional to the squared local electron density, which decreases strongly with height, most of the the radiative loss occurs in the lower heights, so that ϕ_r provides a characteristic value of the coronal radiative flux.

The magnetic null divergence condition, integrated along the magnetic flux tube, reads $A(s)B(s)={A}_{0}\,{B}_{0}={A}_{L}\,{B}_{L}=c$, where c is a constant specific to each magnetic flux tube, and B₀ and B_L are the values of the magnetic field strength at s = 0 and s = L, respectively. Using these relations in Equations (7)–(9), the radiative and conductive terms of the r.h.s. of Equation (10) can be expressed as

$$\begin{eqnarray}&&{\phi }_{{\rm{r}}}=\left(\displaystyle \frac{{B}_{0}\,{B}_{L}}{{B}_{0}+{B}_{L}}\right){\int }_{0}^{L}{ds}\,\displaystyle \frac{{E}_{{\rm{r}}}(s)}{B(s)},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{\phi }_{{\rm{c}}}=\displaystyle \frac{{B}_{0}\,{F}_{{\rm{c}},L}-{B}_{L}\,{F}_{{\rm{c}},0}}{{B}_{0}+{B}_{L}}.\end{eqnarray} \tag{ 12 }$$

Note that by introducing these integrated flux quantities, the energy balance (Equation (10)) is freed from transversal area values, holding then for individual magnetic field lines, rather than magnetic flux tubes. In studies combining DEMT with magnetic models, individual magnetic field lines from the model are assigned the DEMT values of the N_e and T_m of the tomographic voxels through which they pass through (as described in Section 4.2 below). The DEMT values for density and temperature in each computational voxel represent an average description of the plasma contained in it. Assigning these values to field lines of the magnetic model thus provides a semi-empirical steady-state model for quiet-Sun long-lived coronal structures, aiming at describing their average state over the DEMT temporal resolution (∼1/2 solar rotation). This approach has been previously used by Huang et al. (2012) and Nuevo et al. (2013) to study thermodynamical properties of the large-scale quiet-Sun corona, and also served as a validation tool for steady-state 3D MHD models of the global corona (Evans et al. 2012; Jin et al. 2012; Oran et al. 2015). It is out of the capabilities of DEMT to analyze the individual bright loops seen in ARs or the fast dynamics of any structure. This study, and in particular the balance Equation (10), is to be understood also as an average description of the energy balance in each loop.

The magnetic field strength B(s) along the field line and, in particular, its values at the coronal base B₀ and B_L are some of the products directly available from the global corona magnetic extrapolation. Hence, the terms involved in Equations (11) and (12) can be computed from the results of DEMT and the magnetic extrapolation.

In the optically thin corona, the radiative power of an isothermal plasma at temperature T is computed as ${E}_{{\rm{r}}}={N}_{{\rm{e}}}^{2}\,{\rm{\Lambda }}(T)$, where the radiative loss function Λ(T) is in turn computed by means of a model, such as the CHIANTI atomic database and plasma emission model (Dere et al. 1997), used in this work in its latest version. Hence, at each tomographic voxel, the radiative power is computed from the temperature distribution LDEM(T) obtained from the DEMT technique as

$$\begin{eqnarray}&&{E}_{{\rm{r}}}=\int \,{dT}\,\mathrm{LDEM}(T){\rm{\Lambda }}(T),\end{eqnarray} \tag{ 13 }$$

where, from Equation (1), dT LDEM(T) is the contribution to the quadratic electron density in the voxel of the plasma with temperature in the range T ± dT. The radiative power E_r, in itself a novel DEMT product introduced in this work, can be numerically traced along the field lines of the global magnetic coronal extrapolation, as explained in Section 4.2 below. This allows computation of the quantity ϕ_r from Equation (11) for each magnetic field line in the model.

Finally, the quantity ϕ_c given by Equation (12) requires computation of the conductive heat flux F_c at both coronal base points of the field line. To this end, in Equation (6) the temperature T and temperature gradient dT/ds are computed from the DEMT results traced along each magnetic field line.

The next section details the numerical implementation of the tracing of the DEMT results along the magnetic field lines and the computation of the corresponding quantities ϕ_r and ϕ_c. Once these two quantities are computed for each field line, Equation (10) allows the computation of the energy input flux at the coronal base ϕ_h, the new DEMT product that constitutes the main result of this work.

4.1. 3D Reconstruction of Density and Temperature

This section shows and describes the DEMT results corresponding to the two rotations, CR 2081 and CR 2099, based on EUVI data in both cases, as well as AIA/SDO data in the latter one. In the case of CR 2099, the very similar EUVI-based results are omitted to save space. CR 2099 has been the subject of DEMT analysis based on both EUVI and AIA data in Nuevo et al. (2015), where the results are compared in detail and shown to be consistent. Density values obtained with AIA are ∼2% smaller compared to those obtained with EUVI, while temperatures are ∼8% larger. These systematic differences are due to slight differences in the temperature response between the respective channels of both instrumental sets.

The 3D distribution of the FBE was tomographically reconstructed for the EUVI bands of 171, 195, and 284 Å in both selected rotations, as well as for the AIA bands of 171, 193, and 211 Å in the case of CR 2099. Using the three FBE reconstructions in each rotation, the 3D distribution of the LDEM was found. Finally, from Equations (1) through (3), the 3D maps of the electron density N_e, electron mean temperature T_m, and score R were computed for both rotations.

As an example, Figure 2 shows the results obtained from the DEMT analysis at a selected height of the tomographic computational sphere. The results are shown as Carrington maps, the distribution of quantities in latitude and longitude. Similar maps are obtained at all heights of the tomographic grid. The left panels correspond to the EUVI-data-based CR 2081 reconstruction, while the right panels correspond to the AIA-data-based CR 2099 reconstruction.

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The top panels show, as an example, FBE maps in the bands of EUVI 195 Å and AIA 193 Å. Due to unresolved coronal dynamics, tomographic reconstructions exhibit artifacts such as smearing and negative values of the reconstructed FBEs, or zero when the solution is constrained to positive values. These are called zero-density artifacts (ZDAs). In the FBE Carrington maps, ZDA voxels are indicated by solid black regions. The overplotted thick black curves in all panels indicate the boundary between the open and closed magnetic regions, as derived from the PFSS model.

The second row of panels in Figure 2 shows the Carrington maps of R, with ZDA voxels indicated by the solid black regions. It can be readily seen that in both cases, most of the closed-corona volume is characterized by R < 10⁻² (dark green color), meaning that the LDEM predicts the tomographic FBEs with a precision better than 1%. Note that, in the case of the AIA analysis, some regions of the open corona are characterized by somewhat larger R scores, but in any case, the focus of this paper is the closed corona.

The bottom two rows of panels show the DEMT electron density N_e and electron mean temperature T_m. In the temperature maps, the solid black regions indicate ZDA voxels, while the solid white regions indicate voxels for which the parametric LDEM has a score R > 0.1, i.e., the discrepancy between the tomographic and synthetic FBEs is more than 10%. In these cells, dubbed anomalous emissivity voxels (AEVs), the Gaussian LDEM model does not accurately reproduce the tomographic results. In the electron density maps, ZDAs and AEVs are indicated as black regions.

It is interesting to note that the open/closed boundaries of the PFSS model quite accurately match the contour levels of the tomographic density and temperature. In other words, along the open/closed boundary, the gradient of the tomographic results is approximately perpendicular to it. This is an interesting consistency check between the PFSS and DEMT models, which can also be verified in all previous DEMT works.

In the tomographic density maps, the voxels with the largest density values (yellowish regions) always correspond to observed ARs. This has been verified by careful comparison of the reconstructions to the catalog provided by the National Oceanic and Atmospheric Administration Space Weather Prediction Center.⁵ These regions are characterized by threshold values of the tomographic density, as detailed in Nuevo et al. (2015). As tomographic reconstructions are not suitable for studying fast-evolving ARs, these regions are left out of the analysis in this paper. Voxels belonging to ZDA and AEV regions are also excluded from the analysis.

4.2. Tracing of Results along Magnetic Field Lines

Figure 3 shows 3D visualizations of the field lines of the PFSS models computed from MDI synoptic magnetograms for CR 2081 and CR 2099. For each traced field line, the 3D coordinates of a starting point must be specified. In order to evenly cover the whole volume spanned by the DEMT reconstructions, one starting point was selected at the center of each tomographic cell at 10 uniformly spaced heights, ranging from 1.035 to 1.215 R_⊙, and every 2° in both latitude and longitude, for a total of 162,000 starting points and traced field lines. The geometry of the magnetic field line of the PFSS model passing through each starting point is then computed, both outward and inward, until its footpoint (1.0 R_⊙) and/or the source surface (2.5 R_⊙) is reached. To do so, the first-order differential equations ${dr}/{B}_{{\rm{r}}}={rd}\theta /{B}_{\theta }=r\,\sin (\theta )d\phi /{B}_{\phi }$ are numerically integrated using the PFSS Solarsoft package.

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The next step is to trace the DEMT results along the computed magnetic field lines. To that end, once the field line geometry in high resolution is completed, only one sample point per tomographic cell is kept, the median one. For each sample point, the DEMT products (namely, the electron density N_e, the electron mean temperature T_m, and the radiative power E_r.) in the tomographic cell where it is located are assigned to it.

Figure 4 shows an example along a closed magnetic loop. Note how the results are separated into the two legs of the loop, defined as the two segments that go from the coronal base up to the apex of the loop. For each leg, exponential least-squares fits are applied to both the density and the radiative power data points as a function of height. Given their much weaker variation with height, temperature data points are fitted to a linear function. In this case, the Theil–Sen estimator was preferred over the least-squares fit, as it is more robust to outliers. These are common in solar rotational tomography results, mainly due to the effect of unresolved coronal dynamics on the assumed static solution of the posed global optimization problem.

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At this point, for every traced magnetic field line, its 3D geometry has been determined, the magnetic field strength B(s) along it has been computed, and the radiative power E_r(s) and electron mean temperature T_m(s) have been determined from the fits to the traced DEMT data. As a result, all quantities involved in Equations (11) and (12) can be now numerically computed. Note that as the quantity ϕ_c is sensitive to both the basal temperature and temperature gradient of the DEMT results (Equations (12) and (6)), its computation from the fits to the traced DEMT data mitigates the effect of its stochasticity, mainly due to unresolved coronal dynamics. Once these quantities are known, the energy input flux ϕ_h is computed from Equation (10).

4.3. Selection of Loops

After analyzing all of the 162,000 traced field lines in each rotation, about 54% and 60% are closed in CR 2081 and CR 2099, respectively. Some closed field lines do not have enough DEMT data points and/or are not well distributed, as specified by the first selection requirement listed above (Section 4.3), and some belong to ARs (as discussed in Section 4.1). Those loops amount to about 17% of the closed field lines in CR 2081, and to about 53% for CR 2099 based on EUVI data, or 49% based on AIA data. In this remaining population, the second and third selection criteria listed at the end of Section 4.3 are met by about 40% of the loops for CR 2081, 46% of the loops for CR 2099 based on EUVI, and 50% of the loops for CR2099 based on AIA. The resulting number of loops is ∼29,000 for CR 2081 and ∼21,000 loops for CR 2099 based on EUVI, or ∼25,000 based on AIA.

Visual inspection of the temperature map of CR 2081 in Figure 2 reveals that in the closed corona, the low latitudes are characterized by relatively cooler temperatures, while midlatitudes are hotter. This is characteristic of the last solar minimum (Vásquez et al. 2010; Nuevo et al. 2015), as well as the previous period of minimum activity between SCs 22 and 23 (Lloveras et al. 2017). A similar behavior can be verified in the quiescent closed corona of CR 2099. In the analysis that follows, these diverse thermodynamical regions are separated. To that end, magnetic loops were discriminated into those with both footpoints within a specific low-latitude range defined by $| \mathrm{latitude}| \lt 30^\circ$, and those within a midlatitude range defined by $| \mathrm{latitude}| \gt 30^\circ$. After applying this selection criteria, the remaining population is ∼16,000 for CR- 2081, and ∼18,000 or ∼23,000 for CR 2099 when based on either EUVI or AIA data, respectively.

5.1. Results for CR 2081

Results for CR 2081 are shown in Figure 5, where the left and right panels correspond to the low and midlatitude regions, respectively. The top panels show the spatial location of the footpoints of the loops (i.e., at r = 1.0 R_⊙), while the middle ones show their location at an intermediate height of the tomographic computational grid (at r = 1.075 R_⊙). A comparison between the top and middle panels gives a sense of the divergence of the magnetic field lines. Loops for which the apex is within the range of heights covered by DEMT (dubbed as "small" loops hereafter) are indicated in violet, while those with a higher apex are indicated in red (dubbed as "large" loops hereafter). For the selected loops, the bottom panels show the corresponding distribution of values of the loop-integrated quantities ϕ_r, ϕ_c, and ϕ_h. The total number N of analyzed loops is shown, along with the median m and standard deviation σ values of each distribution.

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It is readily seen that the integrated radiative loss of the loops, measured by the quantity ϕ_r, is larger in the low latitudes. This is mainly due to the fact that, in the range of sensitivity of the EUVI instrument, namely 0.5–3.0 MK (Nuevo et al. 2015), the radiative loss function Λ(T) used in this work has a local maximum at T ≈ 1 MK. The average temperature of the low and midlatitudes is 1.17 and 1.38 MK, respectively (see Table 1), which explains the larger radiative loss at low latitudes. This happens despite the fact that the average loop length for the low- and midlatitude regions are 0.55 and 0.74 R_⊙, respectively, which implies a larger length integral in the midlatitude loops. Still, most of the coronal radiative loss occurs at lower heights as ${E}_{{\rm{r}}}\propto {N}_{{\rm{e}}}^{2}$, which decays very rapidly with height.

Table 1.  Global Statistics for the CR 2081 Results, Discriminating Low and Midlatitudes

Instrument	Latitude	Ntot	$\langle {\bar{N}}_{{\rm{e}}}\rangle (\sigma )$	$\langle {\bar{T}}_{{\rm{m}}}\rangle (\sigma )$	$\langle {\phi }_{{\rm{r}}}\rangle (\sigma )$	$\langle {\phi }_{{\rm{c}}}\rangle (\sigma )$	$\langle {\phi }_{{\rm{h}}}\rangle (\sigma )$
			(108 cm−3)	(MK)	(105 erg cm−2 s−1)	(105 erg cm−2 s−1)	(105 erg cm−2 s−1)
EUVI	Low	9645	0.93 (0.18)	1.17 (0.10)	1.25 (0.83)	−0.13 (0.35)	1.11 (0.89)
	Middle	5861	0.99 (0.17)	1.38 (0.11)	0.84 (0.43)	0.16 (0.22)	1.05 (0.45)

Note. For both populations, the table shows the sample size N_tot, and the median value (indicated as $\langle \,\rangle$) and standard deviation (σ) of the height-averaged DEMT electron density ${\bar{N}}_{{\rm{e}}}$ and temperature ${\bar{T}}_{{\rm{m}}}$, as well as of the energy flux quantities ϕ_r, ϕ_c, and ϕ_h.

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Statistical results for CR 2081 are shown in Table 1, discriminating low- and midlatitude loops. For both populations, the table shows the median value (indicated as $\langle \,\rangle$) and the standard deviation (σ) of the flux quantities, as well as of the characteristic DEMT electron density ${\bar{N}}_{{\rm{e}}}$ and temperature ${\bar{T}}_{{\rm{m}}}$ of the loops, where the bar indicates the height-averaged value for each loop.

Although the quantity ϕ_r is defined positive, the conductive flux quantity ϕ_c is not. Note that low latitudes are dominated by ϕ_c < 0 values, while midlatitudes are dominated by ϕ_c > 0. The sign of ϕ_c is closely related to the temperature gradient with height. From Equation (6), it can be easily shown that in magnetic loops for which ϕ_c < 0, the temperature decreases with height, while the opposite holds when the temperature increases with height.

Coronal magnetic structures for which the temperature increases/decreases with height in the range of heights covered by DEMT, 1.02–1.22 R_⊙, have been dubbed as "up"/"down" loops by Huang et al. (2012) and Nuevo et al. (2013), who first observed their presence by means of DEMT. As speculated by the authors of those works, loops of type down can be expected if the heating deposition is strongly confined near the coronal base of a magnetic loop. Down loops were first predicted by Serio et al. (1981), and later on by Aschwanden & Schrijver (2002). In a recent study by Schiff & Cranmer (2016), down and up loops have been successfully reproduced by a numerical implementation of a 1D steady-state model that considers time-averaged heating rates. The analysis of these structures in the context of the new tool here developed is shown in Section 5.3 below.

The resulting distributions of the energy input flux at the coronal base ϕ_h have similar median values in both regions, with a smaller standard deviation in the midlatitudes. The characteristic range of values considering both regions is ϕ_h ∼ 0.5–1.5 × 10⁵ erg cm⁻² s⁻¹.

5.2. Results for CR 2099

Figure 6 shows the results for CR 2099 based on AIA data, where the left and right panels correspond to low- and midlatitude regions, respectively. The same analysis was also performed for this rotation based on EUVI data, leading to results consistent with those based in AIA data. Table 2 details the statistical results for CR 2099 based on both data sets. The results obtained with both instruments are more consistent in midlatitudes, due to the similar sample size. In the case of low latitudes, the EUVI sample size is considerably smaller (about half) than that for AIA. Differences in sample size are due to the same selection criteria being applied to different data sources, but the precise reasons why the AIA-based analysis is able to retrieve a larger data sample are not clear. In any case, results from both data sets lead to similar characteristic distributions.

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Table 2.  Similar to Table 1, but for CR 2099 Based on DEMT Reconstructions from Both EUVI and AIA Data

Instrument	Latitude	Ntot	$\langle {\bar{N}}_{{\rm{e}}}\rangle (\sigma )$	$\langle {\bar{T}}_{{\rm{m}}}\rangle (\sigma )$	$\langle {\phi }_{{\rm{r}}}\rangle (\sigma )$	$\langle {\phi }_{{\rm{c}}}\rangle (\sigma )$	$\langle {\phi }_{{\rm{h}}}\rangle (\sigma )$
			(108 cm−3)	(MK)	(105 erg cm−2 s−1)	(105 erg cm−2 s−1)	(105 erg cm−2 s−1)
EUVI	Low	3243	0.89 (0.15)	1.40 (0.16)	1.19 (0.81)	0.15 (0.31)	1.35 (0.83)
	Middle	14724	0.83 (0.12)	1.60 (0.10)	0.83 (0.35)	0.42 (0.20)	1.31 (0.34)
AIA	Low	6891	0.92 (0.17)	1.47 (0.11)	1.08 (0.84)	0.03 (0.48)	1.12 (0.94)
	Middle	15820	0.86 (0.14)	1.61 (0.11)	0.77 (0.33)	0.44 (0.28)	1.27 (0.41)

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Comparing both rotations, in low latitudes the results are similar, with the most notable difference being the distribution of values of the quantity ϕ_c for CR 2099 not dominated by negative values as it is for CR 2081. This is consistent with the finding by Nuevo et al. (2013) that down loops are prominent during solar minimum. In the midlatitudes of CR 2099, ϕ_c is virtually positive everywhere, which is consistent with the fact that down loops not only diminish in number with increasing activity but also tend to be found only at low latitudes as activity increases (Nuevo et al. 2013). An increase of the characteristic values of the input energy flux ϕ_h for CR 2099 compared to CR 2081, especially at midlatitudes where it shows a ∼20% larger median value, is also noted. This is consistent with the relatively higher temperatures in the midlatitude regions for CR 2099 (see bottom panels in Figure 2).

5.3. Analysis of Temperature Structures

As discussed in Section 5.1, the sign of the conductive flux quantity ϕ_c is related to that of the temperature gradient with height. Down loops, i.e., those for which the temperature decreases with height, are characterized by ϕ_c < 0, while the opposite holds for up loops. While in previous DEMT works (Huang et al. 2012; Nuevo et al. 2013) the loop-selection criteria focused only on down/up loops, our criteria in Section 4.3 allow the selection not only of down and up loops, but also of quasi-isothermal structures.

A loop can be regarded as quasi-isothermal in the coronal region studied by DEMT if the temperature gradient is weak enough compared to the range of coronal heights δr spanned by the loop. More specifically, $\delta r={r}_{\max }-{r}_{\mathrm{base}}$, where r_max = r_appex for small loops and r_max = 1.25 R_⊙ (the maximum height of the tomographic computation ball) for large loops. In this study, a loop is classified as quasi-isothermal if $| {dT}/{ds}| \lt {\rm{\Delta }}(T)/\,\delta r$, where dT/ds is the temperature gradient of the linear fit to the temperature data points and Δ(T) is the characteristic uncertainty in temperature data points due to systematic errors, which is in the range 5%–10% as shown in Lloveras et al. (2017). Down/up loops can then be separated by requiring $| {dT}/{ds}| \gt {\rm{\Delta }}(T)/\,\delta r$, and further discriminated in their two populations according to the sign of the gradient.

This classification criterion was applied to all closed loops analyzed in Sections 5.1 and 5.2 to separate the up and down loops. The loops were then further classified according to their length L. For both analyzed rotations, Figure 7 shows the histograms of the loop length L of all up and down loops. The vertical lines indicate the limits of the five loop length bins set to have a similar sample size within each bin. Tables 3 and 4 show the statistical results for the whole population as well as for each bin separately.

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Table 3.  Statistics of Down and Up Loops of CR 2081, Discriminating among Different Ranges $[{L}_{\min },{L}_{\max }]$ of Loop Length L

Lmin	Lmax	$\langle L\rangle$	Ndown	Ntot	$\tfrac{{N}_{\mathrm{down}}}{{N}_{\mathrm{tot}}}$	$\langle {\lambda }_{N}\rangle$	$\tfrac{\langle L\rangle /2}{\langle {\lambda }_{N}\rangle }$
0.10	2.00	0.46	6600	15625	0.42	0.082	2.82
0.10	0.36	0.28	2144	3202	0.67	0.081	1.75
0.36	0.50	0.42	1521	3185	0.48	0.080	2.63
0.50	0.70	0.58	1200	3141	0.38	0.079	3.67
0.70	0.98	0.85	802	3156	0.25	0.085	5.00
0.98	2.00	1.43	933	2941	0.32	0.089	8.01

Note. For each range of lengths, the median value of the loop length $\langle L\rangle$, the number of down loops N_down, the total number of loops N_tot (both up and down), the median value of density scale height $\langle {\lambda }_{N}\rangle$ of the loops, and the two ratios discussed in the text are tabulated.

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Table 4.  Same as Table 3 but for CR 2099

Lmin	Lmax	$\langle L\rangle$	Ndown	Ntot	$\tfrac{{N}_{\mathrm{down}}}{{N}_{\mathrm{tot}}}$	$\langle {\lambda }_{N}\rangle$	$\tfrac{\langle L\rangle /2}{\langle {\lambda }_{N}\rangle }$
0.10	2.00	0.55	1657	18,133	0.09	0.081	3.40
0.10	0.44	0.35	494	3658	0.14	0.071	2.44
0.44	0.56	0.50	351	3753	0.09	0.078	3.24
0.56	0.68	0.60	286	3412	0.08	0.085	3.55
0.68	0.87	0.74	227	3871	0.06	0.097	3.83
0.87	2.00	1.01	299	3439	0.09	0.087	5.81

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For the four smaller loop length bins in Table 3 for CR 2081, Figure 8 shows the statistical distributions of all energy flux quantities ϕ_r, ϕ_c, and ϕ_h for the up and down loops only, i.e., without the quasi-isothermal loops. To save space, the same graph for CR 2099 is not included.

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Note that after having filtered out the quasi-isothermal loops, all four panels of Figure 8 exhibit a distribution of ϕ_c lacking its population with values ∼0, as expected. Next, note how the number of down loops, measured by the area with ϕ_c < 0, decreases with increasing mean loop length $\langle L\rangle$. This is quantitatively measured in Table 3 in the column N_down/N_tot. The last column in that table shows the ratio between the half length $\langle L/2\rangle$ of the loops with their mean scale height $\langle {\lambda }_{N}\rangle$. Note that this ratio increases with increasing loop length, being of order ∼3 in between the second and third bins, where down loops start to become less prominent (less than 50% of the population). This is consistent with the stability criterion of down loops theoretically derived by Serio et al. (1981) that predicts down loops to be unstable for structures with values of this ratio greater than ∼3.

Figure 8 shows that the integrated radiative loss quantity ϕ_r increases with loop length L. In this analysis, each bin contains a mix of all types of temperature structures (up and down) and latitudes (low and mid), so the structures grouped in each panel of the figure (corresponding to each bin in Table 3) have a similar average temperature. Consequently, the mean radiative loss is the same for all bins, and hence the characteristic value of the radiative loss quantity ϕ_r increases with loop length L due to an increasing integral length. Consistently, the input energy flux ϕ_h also increases with loop length. In the case of CR 2099, down loops are much less prominent, and their tendency toward decreasing population with increasing loop length is much subtler, although still verified.

For a very marginal population of loops, it is found that ϕ_h < 0, which is an unphysical result. This affects only the smallest loops, as revealed by the analysis of structures as a function of their size (Figure 8). The radiative loss term is calculated based on plasma emission detected by the three coronal bands of EUVI and the three bands of AIA used. Though this should account for most of the coronal plasma, there surely is additional emission that is out of the range of sensitivity of the instruments used. Thus, the radiative loss term ϕ_r is most probably underestimated, which, along with the fact that this is an additive positive term in Equation (9), helps to explain the slightly negative values of ϕ_h. Also, the systematic errors of the DEMT technique derived by Lloveras et al. (2017), once propagated into the energy flux quantities (which will be investigated in a future work), can easily explain the marginal population of loops with negative values of ϕ_h.

Finally, the statistical results for all the flux quantities ϕ_r, ϕ_c, and ϕ_h for CR 2081, discriminating the up and down loops, are shown in Table 5. Note that $\langle {\phi }_{{\rm{r}}}\rangle$ is larger for down loops, consistent with their location at low latitudes where temperatures are lower. Down loops are also characterized by a negative conductive flux term $\langle {\phi }_{{\rm{c}}}\rangle$, which then contributes to the input of energy in the balance Equation (4). Thus, a smaller value of the input energy flux $\langle {\phi }_{{\rm{h}}}\rangle$ is needed for down loops (compared to up loops) to compensate for the radiative flux and keep them stable. As a closing comment, note that as this study deals with the coronal section of magnetic loops, the characteristic values of the conductive flux at their base are smaller than those of the radiative losses because the temperature gradient there is not as large as in the underlying layers of the corona. Still, the conductive flux plays a part in the balance Equation (10), as shown by the histograms in Figure 8.

Table 5.  Statistical Results of the Quantities ϕ_r, ϕ_c, ϕ_h (10⁵ erg cm⁻² s⁻¹) for CR 2081, Discriminating the Up and Down Loops

	$\langle {\phi }_{{\rm{r}}}\rangle (\sigma )$	$\langle {\phi }_{{\rm{c}}}\rangle (\sigma )$	$\langle {\phi }_{{\rm{h}}}\rangle (\sigma )$
Up	0.98 (0.64)	0.20 (0.10)	1.21 (0.61)
Down	1.27 (0.83)	−0.29 (0.31)	0.94 (0.93)

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In this section, a first comparison between the results from Section 5 and a theoretical model is carried out, specifically using the 0D hydrodynamic model Enthalpy-based Thermal Evolution of Loops (EBTEL; Klimchuk et al. 2008). Here, we highlight the key aspects of the comparison, and the reader is referred to Mac Cormack et al. (2017) for full details.

EBTEL considers the time-dependent equation of energy conservation, assuming a constant area along the magnetic loop, and a piecewise continuous radiative loss function Λ_E(T_E) given by Klimchuk et al. (2008). The model separates the loop into two segments, one corresponding to the corona and the other one to the TR. By integrating the energy balance equation in each segment, the following relationship between the radiation loss in the TR and the downwards conductive flux in the corona is obtained,

$$\begin{eqnarray}&&{H}_{0}\approx -{F}_{0}-{\phi }_{{\rm{r}},\mathrm{TR}},\end{eqnarray} \tag{ 14 }$$

where H₀ and F₀ < 0 are the enthalpy and conductive flux at the coronal base, respectively, and ${\phi }_{{\rm{r}},\mathrm{TR}}\gt 0$ is the radiative loss flux in the TR. Two qualitatively diverse behaviors are predicted:

• 1.  
  If $| {F}_{0}| \gt {\phi }_{{\rm{r}},\mathrm{TR}}$, the excess conductive flux implies a positive enthalpy flux, so that matter evaporates to the corona, increasing its density.
• 2.  
  If $| {F}_{0}| \lt {\phi }_{{\rm{r}},\mathrm{TR}}$, there is a deficit of conductive flux, which results in a negative enthalpy flux (promoting radiation in the TR), so that matter condensates from the corona, decreasing its density.

Treating the loop length and its mean coronal heating rate as free parameters, EBTEL combines the equations of the corona and the TR to predict the temporal evolution of the pressure ${\bar{P}}_{{\rm{E}}}$, electron temperature ${\bar{T}}_{{\rm{E}}}$, and electron density ${\bar{N}}_{{\rm{E}}}$ height-averaged along the loop (Klimchuk et al. 2008). For the purpose of comparison, the values of the two free parameters are drawn from the DEMT+PFSS results of the previous section. The average heating rate is then set equal to the input energy flux divided the loop length, ϕ_h/L, for each magnetic field line in the model.

The comparative analysis was performed for every loop of the sample corresponding to each latitudinal region of the two analyzed rotations in the previous section. For each loop, the ratio between the loop-height average of the DEMT electron density ${\bar{N}}_{{\rm{e}}}$ and the EBTEL average density ${\bar{N}}_{{\rm{E}}}$ is computed, as well as the ratio between the two average temperature, i.e., ${\bar{T}}_{{\rm{m}}}/{\bar{T}}_{{\rm{E}}}$. Table 6 summarizes the median value and standard deviation of these two ratios for each of the two latitudinal regions in both analyzed rotations.

Table 6.  Statistics of the Ratio between (a) the Loop-height-averaged DEMT Electron Density ${\bar{N}}_{{\rm{e}}}$ and the EBTEL Average Electron Density ${\bar{N}}_{{\rm{E}}}$, and (b) the Loop-height-averaged DEMT Electron Temperature ${\bar{T}}_{{\rm{m}}}$ and the Mean EBTEL Electron Temperature ${\bar{T}}_{{\rm{E}}}$

	Latitude	$\langle {\bar{N}}_{{\rm{e}}}/{\bar{N}}_{{\rm{E}}}\rangle (\sigma )$	$\langle {\bar{T}}_{{\rm{m}}}/{\bar{T}}_{{\rm{E}}}\rangle (\sigma )$
CR 2081	Low	2.0 (1.5)	1.1 (0.4)
(EUVI)	Middle	2.3 (1.3)	1.1 (0.2)
CR 2099	Low	2.2 (1.4)	1.1 (0.3)
(AIA)	Middle	2.4 (1.1)	1.0 (0.2)

Note. Values corresponding to the low and midlatitudes for CR 2081 and CR 2099 are discriminated.

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Although the DEMT and EBTEL electron temperatures are similar for all analyzed populations, the electron densities differ by an average factor of ∼2.2. This systematic discrepancy can be traced down to assumptions in the EBTEL model. Most importantly, specific physical scale laws that are used in the model, as well as the characteristic size assumed for the modeled loops (of the order of their thermal scale height), are consistent with observations of AR loops rather than with quiet-Sun coronal structures. Furthermore, the radiation loss function used by EBTEL and the one used in Section 5 differ in the temperature range of interest, which has an impact on the derived densities. All of these factors contribute to explaining the observed systematic difference in densities (Mac Cormack et al. 2017).

A new DEMT tool was developed that allows the calculation of the energy input flux ϕ_h required at the coronal base (r ∼ 1.025 R_⊙) of magnetic loops of the quiet-Sun corona to sustain hydrostatic thermodynamically stable structures. The first results of applying the tool to two solar rotations (CR 2081 and CR 2099) with different levels of activity are shown. The characteristic values obtained are in the range ϕ_h ∼ 0.5–2.0 × 10⁵ (erg s⁻¹ cm⁻²), depending on the particular coronal structure and the level of activity of the corona.

For CR 2081, a solar minimum rotation, the hotter midlatitude regions of the streamer belt and the cooler low-latitude regions exhibit similar median values in their distribution of energy input flux, with the midlatitudes characterized by a considerably smaller standard deviation. The same characteristics are observed for CR 2099.

For CR 2099, during the early rising phase of SC 24, the analysis was performed with both the EUVI and AIA instruments. Results obtained with both instruments are highly consistent in the midlatitude region, where the sample sizes are similar, with a mean value of the energy input flux in this region being about ∼20% larger than that during solar minimum. In the low latitudes, the results with both instruments are somewhat less consistent (though still comparable), since the AIA data produced a considerably larger population size. The low-latitude results of CR 2099, based on AIA, show virtually the same energy input fluxes as the the low latitudes of CR 2081, based on EUVI.

The characteristic values of energy input flux in different subregions of the equatorial streamer belt are related to the presence of different types of thermodynamic structures, namely, the up and down loops first discovered by Huang et al. (2012) and further studied by Nuevo et al. (2013). The study presented here added new insight into the characteristics of down loops, showing that they are characterized by smaller values of energy input flux due to the extra energy source of heat conduction, and that their population is larger for smaller scales (see Table 3).

The characteristic values obtained for the energy input flux ϕ_h in the quiet-Sun corona are consistent with observational estimates for the quiescent corona reported in previous works by Withbroe & Noyes (1977), Aschwanden (2004), and more recently Hahn & Savin (2014). Based on spectroscopic data of quiet-Sun regions taken by the EIS at a height range 1.05–1.20 R_⊙, on board the Hinode mission, Hahn & Savin (2014) estimate the non-thermal component in the observed broadening of spectral lines. Assigning the non-thermal line broadening to Alfvén waves, they derive the corresponding wave energy flux as a function of height, estimating the local plasma density based on DEM analysis, and the local Alfvén speed by relying on a magnetic potential extrapolation to estimate the magnetic field strength. Their study includes observations both around the equator and midlatitudes, for which the authors find their respective distributions of the wave energy flux at the coronal base to be in the range ∼(2.0 ± 0.4) and ∼(1.8 ± 0.5) × 10⁵ (erg s⁻¹ cm⁻²), respectively, which compare well to the characteristic distributions found in this work (Figures 5 and 6). Based on those estimates, a large fraction of the coronal base energy input flux ϕ_h estimated in this work, or even its totality, could be accounted for by Alfvén waves.

Under the assumption that all the energy input flux is in the form of Alfvén waves, their quadratic velocity amplitude $\langle \delta {v}^{2}\rangle$ is related to the input energy flux through ${\phi }_{{\rm{h}}}=\rho \langle \delta {v}^{2}\rangle \,{V}_{{\rm{A}}}$ (Moran 2001; Hahn & Savin 2014), where ρ and ${V}_{{\rm{A}}}=B/\sqrt{4\pi \,\rho }$ are the local plasma mass density and Alfvén speed. Accounting for an ∼8% helium abundance in the corona, the mass density can be estimated as ρ ≈ 1.14 m_p N_e in terms of the electron density N_e and the proton mass m_p. Applying this relationship to each magnetic field line, using the PFSS and DEMT models to estimate B and N_e, respectively, at the coronal base, the range obtained for the energy input flux translates into a characteristic Alfén wave velocity amplitude range $\sqrt{\langle \delta {v}^{2}\rangle }\sim 25\mbox{--}40\,(\mathrm{km}\,{\mathrm{seg}}^{-1})$. The upper limit corresponds to the low latitudes in the streamer belt of CR 2081, and the lower limit to midlatitudes. This range of Alfvén wave amplitudes, consistent with the quiet-Sun coronal estimates by Hahn & Savin (2014), is also in agreement with the characteristic ranges reported in coronal hole studies; see, for example, Banerjee et al. (2011) and references therein.

Based on a 0D HD physical model for coronal loops (Klimchuk et al. 2008), we confirmed that the characteristic values obtained for the coronal base energy input flux ϕ_h are consistent with the height-averaged values of electron temperature and density obtained from the DEMT analysis. In a future effort, results will be compared to 1D HD models.

In a recent work, Nuevo et al. (2015) extended the DEMT technique to also use the 335 Å coronal band of the AIA instrument, which, combined with the three other bands used in this paper, expands the sensitivity range to 0.5–4.0 MK. In that work, the authors find a ubiquitous bimodal coronal LDEM distribution, with two distinct characteristic temperatures. Their result is interpreted as revealing the ubiquitous presence of "warm" (T ∼ 1.5 MK) and "hot" (T ∼ 2.6 MK) loops throughout the quiet-Sun closed corona. The loops analyzed in this paper correspond to the warm class. In a future work, this new tool will be applied to also study the hot loops.

The new tomographic tool developed in this paper provides a semi-empirical constraint on the global coronal heating models. Its main new product is in the form of 2D maps of the energy input flux ϕ_h at the coronal base layer of the quiet-Sun closed corona. This will be used as a validation tool at the coronal base layer of steady-state 3D MHD coronal simulations of the Space Weather Modeling Framework, developed by van der Holst et al. (2014). This will be the subject of a future effort.

DEMT studies provide a time-averaged description of the state of the corona during the observing time of each structure (∼1/2 solar rotation). The new results of this study are to be interpreted in this context and can be compared with steady-state models of large-scale coronal structures, such as those recently developed by Schiff & Cranmer (2016), who successfully reproduced the up/down loops reported by Huang et al. (2012) and Nuevo et al. (2013), and studied here in Section 5.3.

The authors wish to thank the anonymous referee for a careful reading of the manuscript and insightful review, which helped to improve the content and clarity of the manuscript. C.M.C. acknowledges the CONICET doctoral fellowship (Res. No. 4870) to IAFE that supported her participation in this research. C.M.C., A.M.V., F.A.N., and M.L.F. acknowledge ANPCyT grant PICT # 2012/0973 and CONICET grant PIP # 11220120100403 to IAFE that partially supported their participation in this research.