Thermodynamic Structure of the Solar Corona: Tomographic Reconstructions and MHD Modeling

Abstract

We carry out a study of the global three-dimensional (3D) structure of the electron density and temperature of the quiescent inner solar corona ($r<1.25\,\mathrm{R_{\odot }}$) by means of tomographic reconstructions and magnetohydrodynamic simulations. We use differential emission measure tomography (DEMT) and the Alfvén Wave Solar Model (AWSoM), in their latest versions. Two target rotations were selected from the solar minimum between Solar Cycles (SCs) 23 and 24 and the declining phase of SC 24. We report in quantitative detail on the 3D thermodynamic structure of the core and outer layers of the streamer belt, and of the high latitude coronal holes (CH), as revealed by the DEMT analysis. We report on the presence of two types of structures within the streamer belt, loops with temperature decreasing/increasing with height (dubbed down/up loops), as reported first in previous DEMT studies. We also estimate the heating energy flux required at the coronal base to keep these structures stable, found to be of order $10^{5}~\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$, consistently with previous DEMT and spectroscopic studies. We discuss how these findings are consistent with coronal dissipation of Alfvén waves. We compare the 3D results of DEMT and AWSoM in distinct magnetic structures. We show that the agreement between the products of both techniques is the best so far, with an overall agreement $\lesssim 20\%$, depending on the target rotation and the specific coronal region. In its current implementation the ASWsoM model cannot reproduce down loops though. Also, in the source region of the fast and slow components of the solar wind, the electron density of the AWSoM model increases with latitude, opposite to the trend observed in DEMT reconstructions.

1 Introduction

Being the region where the solar atmosphere plasma is heated to million degree temperatures, the solar wind accelerated, and where impulsive events such as solar flares and coronal mass ejections are energized, observing and modeling of the solar corona are of great relevance to improving our understanding of the Sun–Earth environment. To advance our knowledge of the physics of the solar corona, as well as to enhance and validate three-dimensional (3D) models, information derived from observational data plays a key role. Solar rotational tomography (SRT) is currently the sole observational technique able to provide a quantitative empirical description of the 3D distribution of some fundamental plasma parameters of the solar corona at a global scale.

To study the 3D structure of the quiet-Sun global corona, the SRT has proven to be a powerful tool. In SRT, solar rotation is taken advantage of, so that instruments gather time series of images covering all viewing angles of the solar corona. This poses an inversion problem: to solve for the unknown 3D distribution of specific quantities of the solar corona. Based on extreme ultraviolet (EUV) images, taken in several channels sensitive to different temperatures, differential emission measure tomography (DEMT) allows reconstruction of the 3D distribution of the differential emission measure (DEM). The final product of DEMT is in the form of 3D maps of electron density and temperature, covering the range of heliocentric heights $\lesssim 1.25\,\mathrm{R_{\odot }}$. The technique was first developed by Frazin, Vásquez, and Kamalabadi (2009), and first applied to the observational study of coronal structures by Vásquez, Frazin, and Kamalabadi (2009). A recent review on DEMT was published by Vásquez (2016). The technique is summarised in Section 2.1.

Non-tomographic studies of localized regions of the quiet-Sun corona have been carried out by means of DEM analysis. Mackovjak, Dzifčáková, and Dudík (2014) used regularized inversion techniques to study characteristic temperatures in the quiet Sun. López et al. (2019) used a parametric method to study EUV dimmings after coronal mass ejections (CMEs) to estimate the coronal mass evacuated by the events. At a global scale, DEM analysis has been used by Morgan and Taroyan (2017) to characterise the evolution of the temperature of the quiet-Sun corona during most of Solar Cycle 24.

The combination of DEMT with global magnetic models provides insight into the 3D thermodynamical structure of the global corona. DEMT was first combined with a potential field source surface (PFSS) model by Huang et al. (2012) and Nuevo et al. (2013). More recently, Lloveras et al. (2017) combined DEMT with PFSS models to study the thermodynamics of the global solar corona in specific magnetic structures for two target rotations selected from the last two solar minimum epochs. Also combining DEMT with a PFSS model, Mac Cormack et al. (2017) developed a new DEMT product that estimates the energy input flux required at the coronal base to maintain stable coronal loops. In this article, DEMT is first combined with the magnetic field of a magnetohydrodynamic (MHD) model.

The Alfvén Wave Solar Atmosphere Model (AWSoM) within the Space Weather Modeling Framework (SWMF) is a three-dimensional (3D) physics-based, data-driven MHD model extending from the upper chromosphere to the upper corona and reaching 1 AU and beyond (van der Holst et al., 2010, 2014). The only data input of the model is a magnetogram of the global corona, used as boundary condition for the simulation. As new improvements are implemented, the model is continuously being validated with observations. DEMT results were used by Jin et al. (2012) and Oran et al. (2015) to validate AWSoM results finding agreement within 50% in density and electron temperature in the low corona. More recently, Sachdeva et al. (2019) compared the results of the latest version of AWSoM model with DEMT products in a global fashion.

In this work, the AWSoM model is used with two purposes. Firstly, to provide an MHD model of the coronal magnetic field to be used to study the DEMT results along magnetic field lines. Secondly, to provide thermodynamic results to be compared with those reconstructed by DEMT.

Combining the DEMT and AWSoM models, we carry out a detailed quantitative analysis of two target rotations. We selected Carrington rotation (CR)2082 (2009, 05 April through 03 May) during the minimum between SCs 23 and 24 and CR 2208 (2018, 02 September through 29 September) during the end of the declining phase of SC 24. In the case of CR 2082, the DEMT analysis is based on data taken by the Extreme Ultraviolet Imager Behind (EUVI-B: Wuelser et al. 2004) on board the Solar Terrestrial Relations Observatory (STEREO), while AWSoM uses the synoptic magnetogram provided by the Global Oscillation Network Group (GONG: Hill et al. 1994). In the case of CR 2208, the DEMT analysis is based on data taken by the Atmospheric Imaging Assembly instrument (AIA: Lemen et al. 2012) on board the Solar Dynamics Observatory (SDO), while AWSoM uses the magnetogram provide by the Air Force Data Assimilation Photospheric Flux Transport (ADAPT)-GONG model.

Sections 2.1 and 2.2 summarise the DEMT technique and the AWSoM model, respectively. Section 2.3 details the method used to trace the DEMT results along the field lines of the magnetic model. Section 2.4 details the method that allows determination of the energy input flux at the coronal base. In Section 3.1 the quantitative detailed DEMT analysis of both target rotations is shown, and in Section 3.2 the AWSoM and DEMT results are compared. Finally, Section 4 summarises and discusses the main conclusions of this analysis, and anticipates further planned work.

2 Methodology

2.1 DEMT Reconstructions

As detailed in Section 1, rotations CR 2082 and CR 2208 were selected to carry out DEMT reconstructions, based on data taken by the STEREO/EUVI-B and SDO/AIA instruments, respectively. The EUVI and AIA data were prepared using the latest processing tools and calibration corrections provided by their teams through the SolarSoftware package. In the case of EUVI data, stray-light contamination is removed by deconvolution of the point spread function (PSF), carefully determined for each detector by Shearer et al. (2012). In the case of AIA data, we have not yet implemented such a procedure as we were not aware of reliable determinations of their PSF, and we also understand that stray-light contamination is expected to be less important for this instrument. A recent study by Saqri et al. (2020) indicates that the effect is noticeable in DEM analysis of coronal holes (CHs). That is also the case for DEM analysis of EUVI images. Nonetheless, as shown by Lloveras et al. (2017), due to the temporal and spatial binning of the images used in tomography the effect of stray-light removal in DEMT products turns out to be mild, being $\lesssim 10\%$ for density products and negligible for temperature products. In the future, we will explore the effect of stray-light removal in AIA images on DEMT tomography, which we expect to be smaller than for EUVI images. For this work, we introduced two improvements in the implementation of the DEMT technique, as described next.

While in all previous DEMT studies full-disc data were used to perform tomography, in this work we opt to only use off-limb data. In this way, the smallest scale and brighter coronal features seen on disc (most typically in the 171 Å band) are not included. This has two implications. Firstly, the fast dynamics that typically characterises those structures is absent from the data. Secondly, only half synodic rotation worth of data is needed to constrain the inversion problem for the whole coronal volume. As a result, artifacts induced by coronal dynamics are reduced compared to previous DEMT reconstructions.

The solution of the tomographic problem involves inversion of a very large sparse matrix. Such inversion problems are characterised by spurious high-frequency artifacts in the solution, which can be mitigated through regularization techniques (Frazin, 2000). In the case of DEMT, all previous efforts used the 2D scheme implemented by Frazin, Vásquez, and Kamalabadi (2009), using a finite difference matrix operator to approximate angular derivatives in both latitude and longitude. Also, new to the present work is the implementation of an expanded 3D regularization scheme, which adds to the previous scheme a finite difference matrix operator to approximate radial derivatives. In this way, the tomographic inversion problem is performed penalizing nonphysical high-frequency artifacts in all three spatial directions. As a result, tomographic reconstructions behave more smoothly close to the radial boundaries of the computational grid when compared to previous reconstructions.

In DEMT, the inner corona in the range of heliocentric heights $\lesssim 1.25\,\mathrm{R_{\odot }}$ is discretized in a spherical computational grid. The size of the tomographic grid cell (or voxel) is typically set to $0.01\,\mathrm{R_{\odot }}$ in the radial direction and 2^∘ in both the latitudinal and longitudinal directions. The cadence of the data time-series is set to 6 hr. The main product of the technique is the local DEM (LDEM) at each voxel, a measure of the temperature distribution of the plasma contained in it. We summarise next the main aspects of DEMT required for the analysis of this work.

In a first step, the time series of EUV images is used to solve a solar SRT problem, for each EUV band independently. As a result, the 3D distribution of the so called filter band emissivity (FBE) is determined for each band separately. The FBE, an emissivity-type quantity, is defined as the wavelength integral of the coronal EUV spectral emissivity and the telescope passband function of each EUV channel. Line-of-sight (LOS) integration of the FBE provides synthetic images that can be quantitatively compared to the real data in the time series. To find the FBE, the tomographic problem is posed as a global optimisation problem in which the quadratic norm of the difference between all pairs of synthetic and real images is minimised.

Due to unresolved coronal dynamics, tomographic reconstructions exhibit negative values of the reconstructed FBE, or zero when the solution is constrained to positive values (Frazin, 2000; Frazin, Vásquez, and Kamalabadi, 2009). These non-reconstructed voxels are indicated in black colour in the latitude–longitude (Carrington) maps of DEMT results in Section 3.

In a second step, the FBE values obtained for all bands in each voxel of the tomographic grid are used to constrain the determination of the local DEM (LDEM) which, as described in Section 1, describes the temperature distribution of the plasma within the individual voxel. Specifically, at each tomographic voxel $i$, the FBE of the band $k$ is related to the LDEM of the voxel according to

$$ {\mathrm{FBE}}_{i}^{(k)} = \int \,\textrm{d}T\,\mathrm{LDEM}_{i}(T)\,{\mathrm{TRF}}^{(k)}(T), \ \ \ {k=1,\ldots,K} \,, $$

(1)

where $K$ is the number EUV bands, and $\mathrm{TRF}^{(k)}$ is the temperature response function of the $k$th detector. In this work, the TRFs are computed based on the (known) channel passband times the coronal emissivity at that temperature (normalised by the squared electron density). The emissivity model used here is provided by the latest version of the CHIANTI atomic database and plasma emission model (Del Zanna et al., 2015; Landi et al., 2013).

In this work, data from three EUV bands were used: 171, 193 and 211 Å in the case of AIA, and 171, 195 and 284 Å in the case of EUVI. When using data from three bands, a Gaussian model for the LDEM is able to accurately predict the FBEs (Nuevo et al., 2015). In each tomographic voxel, the problem is then reduced to finding the values of the three free parameters of the Gaussian (centroid, standard deviation, and area) that best reproduce the three tomographically reconstructed values of FBE in that voxel.

As the LDEM describes the temperature distribution of the plasma in a specific voxel, it does not deal with different large-scale structures, as it may be the case for the DEM describing the plasma along a full LOS. As a result, LDEMs are usually successfully modeled with simpler profiles (such as Gaussian) than those returned by DEM studies constrained by LOS-integrated intensities. Parametric techniques are also used for DEM analysis of narrowband images, such as in the work by Aschwanden and Boerner (2011), Plowman, Kankelborg, and Martens (2013), Del Zanna (2013). Other methods applied to DEM analysis of narrow band images include Monte Carlo Markov chain (MCMC) techniques (Schmelz, Christian, and Chastain, 2016), regularized inversion techniques (Hannah and Kontar, 2012), and iterative solvers (Pickering and Morgan, 2019; Morgan and Pickering, 2019) that use the known TRFs of all filters as a functional base. The latter work in particular, introduced a fast iterative solver named Solar Iterative Temperature Emission Solver, which can easily be adapted for its use in DEMT. We will explore this in the future and compare results with those provided by our parametric technique.

Once the LDEM is determined at each voxel, the LDEM-averaged squared electron density $N_{\mathrm{m}}^{2}$ and electron temperature $T_{\mathrm{m}}$ in the voxel can be computed by taking its zeroth and first moments over temperature. More specifically, at the $i$th voxel,

$$\begin{aligned} N^{2}_{\mathrm{m},i}= \langle N_{\mathrm{e}}^{2} \rangle _{i} =& \int \, \textrm{d}T\,\mathrm{LDEM}_{i}(T), \end{aligned}$$

(2)

$$\begin{aligned} T_{\mathrm{m},i}= \langle T_{\mathrm{e}}\rangle _{i} =& \frac{1}{\langle N_{\mathrm{e}}^{2} \rangle _{i}}\,\int \,\textrm{d}T\,T\,{ \mathrm{LDEM}}_{i}(T). \end{aligned}$$

(3)

Next, we define a measure of the accuracy of the LDEM model to predict the tomographic FBEs in each voxel, as

$$ R_{i} \, \equiv \ (1/K) \, \sum _{k=1}^{K} \left | \, 1 - \mathrm{FBE}^{(k)}_{i, \mathrm{syn}}\,/\,\mathrm{FBE}^{(k)}_{i,\mathrm{tom}}\, \right | \,, $$

(4)

being the absolute relative difference between the tomographic and the synthetic FBEs. The final product of DEMT is in the form of 3D maps of the LDEM-averaged quantities $\sqrt{N^{2}_{\mathrm{m}}}$ and $T_{\mathrm{m}}$, as well as of the measure $R$. For a full description of the DEMT technique we refer the reader to Frazin, Vásquez, and Kamalabadi (2009).

2.2 AWSoM Simulations

AWSoM is a three-temperature (accounts for anisotropic, parallel and perpendicular, proton temperatures and isotropic electron temperatures), MHD model of the solar corona and inner heliosphere which provides the 3D distribution of density and temperatures, as well as the 3D magnetic structure and velocity of the solar wind. In this work, we use AWSoM simulated results below 1.25 $\mathrm{R_{\odot }}$ to correspond to the DEMT analysis region.

Heating of the solar corona is addressed by including the non-linear interaction of forward and counter-propagating (reflected) Alfvén waves which results in a turbulent cascade. This dissipated turbulent energy is distributed over anisotropic (parallel and perpendicular) proton temperatures and isotropic electron temperature using theories of linear wave damping and stochastic heating.

The model accounts for both collisional and collisionless electron heat conduction and does not use ad-hoc heating functions. The extended MHD equations including radiative cooling, heat conduction and wave turbulence within AWSoM (van der Holst et al., 2014) are solved using the Block-Adaptive Tree Solarwind Roe-type Upwind Scheme (BATS-R-US, Powell et al., 1999; Tóth et al., 2012).

In a previous version of the model, the cascade time of the major wave was used to determine the wave damping rate (Chandran et al., 2011; van der Holst et al., 2014). In its present version, the energy partitioning is improved by using the Alfvén wave number associated with the damping rate as determined by the critical balance condition, which uses the cascade time of the minor wave (Lithwick, Goldreich, and Sridhar, 2007). This leads to more electron heating and less solar wind acceleration (van der Holst et al., 2019).

The inner boundary of AWSoM is located at the base of the transition region (at ≈1.0 $\mathrm{R_{\odot }}$). In reality, the thin transition region (TR) has steep gradients in temperature and density as a result of the balance between coronal heating, heat conduction, and radiative losses. To resolve these gradients in a global model would require excessive numerical resources. As described in Lionello, Linker, and Mikić (2009) and Sokolov et al. (2013), the TR is artificially broadened to be resolved with a finest grid resolution of 0.001 $\mathrm{R_{\odot }}$. To ensure that the base of the TR is not affected by chromospheric evaporation we overestimate the density at the inner boundary, $N_{e}=N_{i}=N_{\odot } =2\times 10^{17}$ m⁻³, corresponding to the isotropic temperature values, $T_{e}=T_{i}=T_{i\parallel }=T_{\odot }=50,000$ K, where the subscripts represent electrons and ions. The upper chromosphere is required to extend radially for the density to fall rapidly to correct (lower) values (Lionello, Linker, and Mikić, 2009). At this level, the radiative losses are sufficiently low so that the temperature can increase monotonically with height and form the transition region. Since the broadening of the transition region pushes the corona outwards, the AWSoM model achieves coronal conditions at a height $\approx 1.05\,\mathrm{R_{\odot }}$, below which results cannot then be compared to coronal tomographic reconstructions.

To drive the AWSoM model, estimates of the photospheric magnetic field of the Sun are the main input. Synoptic magnetograms are used to specify the initial and the boundary conditions of the magnetic field. We use the PFSS model to extrapolate the 3D magnetic field (from the 2D photospheric magnetic field maps) using spherical harmonics. The source surface is taken to be at 2.5 $\mathrm{R_{\odot }}$. GONG provides synoptic full-disc surface maps of the radial magnetic field component of the Sun. However, since the polar regions are not well observed from the ecliptic, GONG estimates the polar fields by fitting a polynomial to neighboring observed latitudes, which might lead to inaccuracies. An improvement over these maps is provided by the ADAPT model (Worden and Harvey, 2000), which creates synchronic-synoptic maps by incorporating supergranulation, meridional circulation, and differential rotation. These maps provide a physics-based description of the unobserved polar magnetic fields (Arge et al., 2010; Henney et al., 2012). In this work, we use the GONG synoptic map as input for CR 2082 (ADAPT-GONG maps are unavailable for CR 2082) and the ADAPT-GONG global magnetic field map for CR 2208. Based on results from previous efforts, for CR 2082 the magnetic field from the GONG map is scaled up by a factor of 1.85 for weak fields ($B_{r}< 5\,\mathrm{{G}}$), while no modification is applied in the case of the ADAPT-GONG map for CR 2208. The AWSoM steady-state simulation set-up and input parameters for both rotations are described below.

To account for the energy partitioning between electrons and protons, based on Chandran et al. (2011), we set the stochastic heating exponent equal to 0.21 and the amplitude equal to 0.18, for both rotations. The boundary condition for Alfvén wave energy is given by the poynting flux $S_{A}$ of the outgoing wave, $({S_{A}/}{B})_{\odot } = 1.1\times 10^{6}~\mbox{W}\,\mbox{m}^{-2}\,\mbox{T}^{-1}$ and $1.0\times 10^{6}~\mbox{W}\,\mbox{m}^{-2}\,\mbox{T}^{-1}$ for CR 2082 and CR 2208, respectively, with $B_{\odot }$ being the field strength at the inner boundary. The correlation length of the Alfvén waves is set to, $L_{\perp } \sqrt{B} = 1.5 \times 10^{5}~\mbox{m}$$\sqrt{T}$, where $L_{\perp }$ is transverse to the magnetic field direction.

The computational domain of the solar corona extends from 1 to $24 \,\mathrm{R_{\odot }}$. The adaptive spherical grid has a finer resolution near the Sun, and increases outward with the z-axis aligned with the rotation axis in heliographic rotation coordinates. The adaptive mesh refinement (AMR) resolves the angular cell size to $1.4^{\circ }$ between 1 – $1.7\,\mathrm{R_{\odot }}$ and to $2.8^{\circ }$ outside this radius range. The solar corona component uses about 3 million cells on $6 \times 8 \times 8$ grid blocks (for CR 2208) and $6 \times 4 \times 4$ grid blocks for CR 2082. Local time stepping is used to speed up the steady-state convergence. The AWSoM simulation of the solar corona for solar minimum conditions represented by CR 2082 and CR 2208 are compared to the DEMT in the results section.

2.3 Tracing Results Along Magnetic Field Lines

To determine the electron density and temperature along individual magnetic field lines, first both the thermodynamic results and the magnetic field obtained with the AWSoM model were interpolated into the DEMT grid. Then, the geometry of the field lines is determined by numerical integration of the first order differential equations $\mathrm{d}r/B_{r} = {r\,\mathrm{d}\theta }/B_{\theta }= r\,\sin (\theta )\,{ \mathrm{d}}\phi /B_{\phi }$, both inwards and outwards, from the specified 3D coordinates of a starting point. In order to evenly sample the whole volume spanned by the DEMT reconstructions, one starting point is selected at the center of each tomographic cell at 6 uniformly spaced heights, ranging from 1.025 to $1.225\,\mathrm{R_{\odot }}$, and every 2^∘ in both latitude and longitude, for a total of $97,200$ starting points.

For analysis purposes, the traced magnetic field lines are classified as open or closed according to their full geometry. Each closed field line is further classified as “small” or “large”, according to its coronal length $L$ being respectively smaller or larger than the median value of the whole population, which is $\mathrm{Md}(L)\approx 0.5\,\mathrm{R_{\odot }}$ for both rotations. Finally, each closed magnetic field line is separated in its two “legs”, defined as the two segments that go from each of its two footpoints (i.e. their location at $r=1\,\mathrm{R_{\odot }}$) to its apex (i.e. the location of maximum height).

At this stage, DEMT and AWSoM products are traced along open and closed magnetic field lines. Once the field line geometry is computed with high spatial resolution, only one sample point per tomographic cell is kept (the median one). As a result, for each field line one data point per tomographic cell is obtained. This approach was first used by Huang et al. (2012) to study temperature structures in the solar minimum corona and by Nuevo et al. (2013) to expand that analysis to rotations with different level of activity.

For each open field line and for each closed field leg, an exponential fit was applied to the electron density data points and a linear fit applied to the electron temperature data points. For DEMT, the data points used are $\sqrt{N^{2}_{\mathrm{m}}}(r)$ and $T_{\mathrm{m}}(r)$, and in the case of the AWSoM models the data points used are $N_{\mathrm{e}}(r)$ and $T_{\mathrm{e}}(r)$. The exponential and linear fit equations are described by

$$\begin{aligned} \sqrt{N^{2}_{\mathrm{m}}} =& N_{0}\,\exp {\left [\,-\,\left (h/\lambda _{ \mathrm{N}}\right )\,/\,\left (r/{\mathrm{R_{\odot }}}\right )\,\right ]} \,, \end{aligned}$$

(5)

$$\begin{aligned} T_{\mathrm{m}} =& T_{0}\,+\,a\,h\,, \end{aligned}$$

(6)

where $h \equiv r - 1\,\mathrm{R_{\odot }}$ is the coronal height measured from the photosphere. In the electron density fit, $\lambda _{\mathrm{N}}\,[{\mathrm{R_{\odot }}}]$ is the density scale height and $N_{0}\,[{\mathrm{cm}}^{-3}]$ is the electron density at the footpoint ($h=0$) of the loop. In the electron temperature fit, $a\,[{\mathrm{MK}}/{\mathrm{R_{\odot }}}]$ is the slope and $T_{0}\,[{\mathrm{MK}}]$ is the electron temperature at the footpoint of the loop. The slope $a$ estimates the radial gradient of the electron temperature along the loop, which we denote as $a = \triangledown _{r}T_{\mathrm{m}}$ hereafter, with $\triangledown _{r}\equiv \mathbf{e}_{r}\cdot \nabla$ being the radial derivative operator and $\mathbf{e}_{r}$ the heliocentric radial unit vector.

In the case of the electron density, the fitted function corresponds to the isothermal hydrostatic equilibrium solution, allowing for variation of the solar gravitational acceleration with height. This choice of function provides a straightforward means to directly compare the observed coronal thermodynamical state with the hydrostatic solution.

Coronal magnetic structures for which temperature increases/decreases with height (in the inner corona) were dubbed up/down loops by Huang et al. (2012) and Nuevo et al. (2013), who first reported the presence of down loops. As speculated by the authors of this work, down loops 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 by Aschwanden and Schrijver (2002). In a recent study, Schiff and Cranmer (2016) reproduced both down and up loops by means of numerical simulations, using a 1D steady-state model and considering time-averaged heating rates.

To determine if the leg of a traced field line is of type up or down, we first determine the Pearson correlation coefficient $\rho (T,r)$ between the DEMT temperature $T_{\mathrm{m}}$ and the heliocentric height $r$. We then select field lines for which the temperature is significantly correlated with height by requiring $|\rho (T,r)| > 0.5$. To test the goodness of the fit we perform a chi-squared test (Press et al., 2002) considering the uncertainty of the DEMT data, selecting legs for which the fit matches the data with a 90% confidence level. In this way, legs for which the DEMT temperature does not show a significant correlation with height, or the linear fit to temperature is weak, are excluded from the analysis. The test is also applied to the density-height data points, to ensure the trend is reasonably represented by the exponential fit. Finally, selected legs are then classified as up or down according to $\triangledown _{r}T_{\mathrm{m}}> 0$ or $\triangledown _{r}T_{\mathrm{m}}< 0$, respectively. The linear fit allows characterization of the variation of $T_{\mathrm{m}}$ with height by means of a characteristic temperature gradient $\triangledown _{r}T_{\mathrm{m}}\,[{\mathrm{MK}}/{\mathrm{R_{\odot }}}]$ along each leg. The chi-squared test to evaluate the quality of the fit considers the uncertainty level in the DEMT products due to systematic sources (radiometric calibration and tomographic regularization), which Lloveras et al. (2017) estimated to be $\Delta T_{\mathrm{m}}\approx 10\%$ and $\Delta \sqrt{N^{2}_{\mathrm{m}}}\approx 5\%$. In summary, a selected leg must meet three conditions:

• i) The leg must go through at least five tomographic grid cells with reconstructed data, and there must be at least one data point in each third of the range of heights spanned by the leg. This requirement is set to ensure a reasonably spread sample of heights along the leg.

• ii) The DEMT temperature and height points must meet $|\rho (T,r)| > 0.5$.

• iii) The confidence level of both the exponential and linear fits must be larger than 90%.

To characterise the global thermodynamic state of the inner solar corona in distinct magnetic structures, the DEMT and AWSoM results were traced along the magnetic field lines of the latter model. Based on the geometry and size of the loops, as well as on their thermodynamical properties, their legs were classified in four different types:

• Type 0: closed-small down with footpoints in the range $|{\mathrm{latitude}}|<50^{\circ }$.

• Type I: closed-small up with footpoints in the range $|{\mathrm{latitude}}|<50^{\circ }$.

• Type II: closed-large up with footpoints in the range $|{\mathrm{latitude}}|>40^{\circ }$.

• Type III: open with footpoint in the range $|{\mathrm{latitude}}|>60^{\circ }$.

In the case of closed-small field lines, the population of down and up legs becomes comparable for CR 2082, so we classify them into the two complementary classes of legs of type 0 (down) and legs of type I (up). In the case of closed-large field lines, down legs are $\lesssim 15\%$ of the population for both rotations. In the case of open field lines, down legs are $\lesssim 10\%$ of the population for both rotations. Hence, the requirement of being up for legs of type II and III is included to select the vastly dominating population in each case. On the other hand, the inclusion of latitude limits for the footpoints in the classification of legs from type 0 through III is purposely set to study the streamer belt in progressively outer layers, as well as to separate the field lines of the high latitude CHs (legs of type III). In Section 3, the results of both the DEMT reconstruction and AWSoM model in the four classes of legs are statistically analysed.

2.4 Energy Input Flux

The high temperature of the corona requires heating mechanisms to compensate for the energy losses. While the vast majority of the existing literature on coronal heating focuses on active regions (ARs), some studies have been dedicated to the heating of quiet-Sun regions. In particular, Mac Cormack et al. (2017) developed a new application of the DEMT technique to estimate the energy input flux required at the base of quiet-Sun coronal loops to maintain stability. The technique is based on tracing the DEMT results along field lines of a global coronal magnetic model, as described in Section 2.3.

Consider a static energy balance for each magnetic flux tube, in which the dominating losses of radiative power $E_{r}$ and thermal conduction power $E_{c}$ are compensated by a coronal heating power $E_{h}$ (Aschwanden, 2004):

$$ E_{h}(s) = E_{r}(s) + E_{c}(s), $$

(7)

where $s$ is the position along the flux tube and these quantities are in units of $[\mbox{erg}\,\mbox{sec}^{-1}\,\mbox{cm}^{-3}]$.

The thermal conduction power $E_{c}$ equals the divergence of the conductive heat flux $F_{c}$, i.e. $E_{c}(s)=[1/A(s)]\,\mathrm{d}[A(s)\,F_{c}(s)]/{\mathrm{d}}s$, where $A(s)$ is the cross-sectional area of the magnetic flux tube at position $s$. Under a quiescent solar corona plasma regime, the conductive flux is assumed to be dominated by the electron thermal conduction, described by the usual Spitzer model (Spitzer, 1962)

$$ F_{c}(s)=-\kappa _{0}\,{T(s)}^{5/2}\,\frac{dT}{ds}(s), $$

(8)

where $\kappa _{0} = 9.2 \times 10^{-7}~{\text{erg}\,\text{sec}^{-1}\,\text{K}^{-7/2}}$ is the Spitzer thermal conductivity.

In the corona, EUV emission is dominated by collisions of the emitting ions with free electrons, so that the radiative power scales as $N_{e}^{2}$. The radiative power of an isothermal plasma at temperature $T$ is then computed as $E_{r}=N_{e}^{2}\,\Lambda (T)$, where the radiative loss function $\Lambda (T)$ is calculated by means of an emission model. In this work we used the latest version of the atomic database and plasma emission model CHIANTI (Del Zanna et al., 2015). The radiative power in terms of the LDEM is then given by

$$ E_{r} = \int \,\textrm{d}T\,\mathrm{LDEM}(T)\,\Lambda (T). $$

(9)

The energy balance given by Equation 7 is then integrated in the volume of any given coronal magnetic flux tube. Dividing the result by the flux tube area at the coronal base, and making use of the solenoidal condition of the magnetic field, a field line integrated version of that energy balance is found,

$$ \phi _{h} = \phi _{r} + \phi _{c}, $$

(10)

where the line integrated flux quantities $\phi _{r,c}\,[\mbox{erg}\,\mbox{sec}^{-1}\,\mbox{cm}^{-2}]$ are given by (Mac Cormack et al., 2017):

$$\begin{aligned} \phi _{r} =& \left ( \frac{B_{0} \ B_{L}}{B_{0} + B_{L}} \right ) \int _{0}^{L} ds \ \frac{E_{r}(s)}{B(s)}, \end{aligned}$$

(11)

$$\begin{aligned} \phi _{c} =& \left ( \frac{B_{0} \ F_{c,L} - B_{L} \ F_{c,0}}{B_{0} + B_{L}} \right ) , \end{aligned}$$

(12)

where $L$ is the length of the loop, and $B_{0}$ and $B_{L}$ are the values of the magnetic field at the footpoint of the loop in the coronal base, namely $s=0$ and $s=L$. For any given field line, all quantities in these two expressions can be computed from the DEMT results traced along field lines, of the AWSoM magnetic field model, through Equations 8–9. Once computed, the quantity $\phi _{h}$ can be calculated, which is the energy input flux required at the coronal base of each coronal field line to maintain a stable coronal structure.

3 Results

3.1 Tomographic Results

As described in Section 2.1, we carried out DEMT reconstructions of the coronal structure for rotations CR 2082 and CR 2208 using STEREO/EUVI and SDO/AIA data, respectively. Once the LDEM was determined for each rotation, the square root of the mean value of the electron density squared $\sqrt{N^{2}_{\mathrm{m}}}$ and the electron mean temperature $T_{\mathrm{m}}$ were computed at each voxel of the tomographic computational grid using Equations 2 and 3, and the measure $R$ was calculated with Equation 4.

Figures 1 and 2 show latitude–longitude maps of DEMT results for both rotations. Three different heights of interest are selected from the tomographic grid, providing also a detailed 3D view of the tomographic results: the lowest height of the tomographic grid ($1.025\,\mathrm{R_{\odot }}$), the lowest height where the AWSoM results are fully consistent with coronal conditions ($1.065\,\mathrm{R_{\odot }}$), and a middle height of the tomographic grid ($1.105{\mathrm{R_{\odot }}}$). Black voxels correspond to non-reconstructed voxels (see Section 2.1). Thick-black curves indicate the open/closed boundaries of the magnetic field of the AWSoM model, detailed in Section 2.2.

Figure 1

Carrington maps of DEMT products $\sqrt{N^{2}_{\mathrm{m}}}$ (left panels) and $T_{\mathrm{m}}$ (right panels) for CR 2082. Top, middle, and bottom panels show the results at three heliocentric heights, 1.025, 1.065, and $1.105\,\mathrm{R_{\odot }}$, respectively. Black voxels correspond to non-reconstructed regions (see text in Section 3.1) and thick-black curves indicate the open/closed boundaries.

Figure 2

Same as Figure 1 but for CR 2208.

Both target rotations are highly axisymmetric, i.e. characterised by a high azimuthal symmetry. Rotation CR 2082 has two small ARs, both near latitude $+30^{\circ }$ and around longitudes $50^{\circ }$ and $120^{\circ }$ (not identified in the NOAA catalog). Rotation CR 2208 has two ARs, both near latitude $+5^{\circ }$ and around longitudes $140^{\circ }$ and $300^{\circ }$ (NOAA 12722, 12721).

The magnetically open and closed regions of the AWSoM model are associated with CHs and the equatorial streamer belt, respectively. The location of the open/closed boundaries derived from the AWSoM model quite accurately matches the regions of the DEMT maps, which exhibit the strongest latitudinal gradient of both the electron density and temperature.

Figures 1 and 2 show that the DEMT reconstruction of the streamer belt is characterised by relatively higher densities and temperatures in comparison to the CHs. They also show that the streamer belt region of CR 2082 was denser and colder than that of CR 2208. In the case of CR 2082, which belongs to the deep minimum epoch between SCs 23 and 24, the low latitudes of the streamer belt are characterised by an electron temperature lower than in its mid-latitudes. A similar behaviour is seen in CR 2208, belonging to the end of the declining phase of SC 24 but, having a somewhat less axisymmetric structure, this characteristic is not so obvious. This thermodynamic structure of the streamer has been reported for other solar minimum rotations in previous DEMT work (Lloveras et al., 2017; Nuevo et al., 2013; Vásquez et al., 2010).

Latitude–longitude maps of the score $R$ defined by Equation 4 show that the agreement between the tomographic and synthetic FBEs is 5% or better in more than 90% of the reconstructed coronal volume (i.e. where FBEs are non-zero), and of order 10% in the rest of the volume. This implies that the LDEM found in each voxel accurately predicts the reconstructed FBEs.

For both rotations, the top panels of Figure 3 show the latitude–longitude location (at heliocentric height $r=1.105\,\mathrm{R_{\odot }}$) of all traced field line legs for which criterion (i) of Section 2.3 is met. Open legs are indicated in grey colour and closed ones in black colour. Considering the DEMT data points and the resulting fits along each leg, the middle panels of Figure 3 show the latitude–longitude location of the subset for which also both criteria (ii) and (iii) of Section 2.3 are met. Using a four-colour code, type 0, I, II, and III legs are shown in blue, orange, red, and cyan colour, respectively. The bottom panels show histograms of the latitude distribution of the legs of the middle panel, using the same colour code. Of the $\approx 44000$ legs selected for CR 2082, 20% are type 0, 31% are type I, 23% are type II, and 26% type III. On the other hand, of the $\approx 50000$ legs selected for CR 2208, 10% are type 0, 38% are type I, 27% are type II and 25% type III.

Figure 3

Top panels: latitude–longitude location at heliocentric height $r=1.105\,\mathrm{R_{\odot }}$ of all open (grey colour) and closed (black colour) traced field line legs for which criterion (i) of Section 2.3 is met, for CR 2082 (left) and CR 2208 (right). Middle panels: latitude–longitude location of the subset for which both criteria (ii) and (iii) of Section 2.3 are met. The location of type 0, I, II, and III legs is shown in blue, orange, red, and cyan colour, respectively. Bottom panels: frequency histograms of the latitude of the four types of legs of the middle panel.

For both rotations, the population of type 0 (small-closed down) legs peaks at the core of the streamer belt, around the equator in the case of CR 2082, and towards the southern hemisphere in the case of CR 2208. This kind of structure was originally found by Huang et al. (2012) and their existence was shown to be anti-correlated with the solar activity level around the solar minimum between SCs 23 and 24 by Nuevo et al. (2013). Later on, Lloveras et al. (2017) showed that equatorial down loops in streamers were also to be found in the deep minimum between SCs 23 and 24. Here, we verify the existence of this type of structure for the two rotations. The relatively smaller population of down legs seen in CR 2208, as compared to CR 2082, is consistent with the aforementioned results by Nuevo et al. (2013). Type I (small-closed-up) legs are present at all latitudes within the streamer belt. Their population peaks in the mid-latitudes of both hemispheres for CR 2082, and in the mid-latitudes of the northern hemisphere for CR 2208. The latitude distribution of legs of type 0 and I in CR 2082 is consistent to the distribution of down and up loops of CR 2081 in the analysis by Nuevo et al. (2013), which did not place any limits on the latitude location of the analysed structures.

Type II (large up) legs are mostly very large transequatorial field lines forming the envelope of the streamer belt (the requirement of footpoints located beyond mid-latitudes was included precisely to select this kind of loop). Finally, type III (open) legs populate the high latitude CHs.

Figure 4 shows frequency histograms of $\triangledown _{r}T_{\mathrm{m}}$ for legs of type 0, I, II, and III. The lack of population around values close to zero is due to the requirement $|\rho (T,r)| > 0.5$ which discards quasi-isothermal legs. For both rotations, the median value of the temperature radial gradient is $\mathrm{Md} \left (\triangledown _{r}T_{\mathrm{m}}\right ) \approx -2.2$, $+2.3$, $+2.4$, and $+2.3\,\mathrm{MK}/{\mathrm{R_{\odot }}}$, for legs of type 0, I, II, and III, respectively.

Figure 4

Frequency histograms of the temperature radial gradient for the four types of legs in Figure 3 (using the same colour code) for CR 2082 (left panel) and CR 2208 (right panel).

For both rotations, Figure 5 shows, in a statistical fashion, the DEMT results traced along field lines discriminated by leg type. From top to bottom results are shown for type 0 to type III legs, respectively. From left to right the panels show the statistical distribution of the electron density value $N_{\mathrm{CB}}\equiv \sqrt{N^{2}_{\mathrm{m}}}(r=1.055)~[{\mathrm{cm}}^{-3}]$, the scale height $\lambda _{N}$, and the height-averaged (along the leg) electron temperature $\left < T_{\mathrm{m}}\right >[{\mathrm{MK}}]$, with the median value $m$ indicated in each plot.

Figure 5

Statistical distribution of DEMT results for rotations CR 2082 (blue) and CR 2208 (red) traced along legs of type 0, I, II, and III (from top to bottom), as defined in Section 2.3. From left to right: electron density $N_{\mathrm{CB}}\equiv \sqrt{N^{2}_{\mathrm{m}}}(r=1.055\,\mathrm{R_{\odot }})$, electron density scale height $\lambda _{N}$, and loop-averaged temperature $\left < T_{\mathrm{m}}\right >$. In each panel the median value $m$ is indicated.

Table 1 summarises the results of the quantitative comparative analysis between the two target rotations. For CR 2082 quantities are expressed as absolute values, while for CR 2208 they are expressed relative to the corresponding results for CR 2082.

Table 1 Median value (indicated as Md) of the statistical distribution of $N_{\mathrm{CB}}$, $\lambda _{N}$, and $\left < T_{\mathrm{m}}\right >$ for each coronal type of legs defined in Section 2.3. For CR 2082, values are expressed in absolute terms, while for CR 2208 they are given as a percentual variation relative to the CR 2082 value in the parentheses.

Throughout the magnetically closed region of both rotations, type 0, I, and II legs, associated to increasingly outer layers of the equatorial streamer belt, exhibit progressively decreasing coronal base density, increasing density scale height, and increasing electron temperature. In both rotations also, type III legs in the CHs are characterised by sub-MK temperatures, and electron density values of order $\approx 1/2$ of those observed for the type 0 and type I legs in the core of the equatorial streamer. A comparison of the results between the two rotations shows that, compared to CR 2082, rotation CR 2208 was characterised by $\approx 10$ – $20\%$ lower values of the electron density at the coronal base, $\approx 5$ – $30\%$ larger values of density scale height, and $\approx 15$ – $25\%$ larger values of the electron temperature.

In comparing the DEMT results obtained for the two selected targets, we highlight they rely on data provided by two different instruments, namely EUVI and AIA for CR 2082 and CR 2208, respectively. To quantify the systematic difference of the DEMT products due to the different filter sets of both instruments, Nuevo et al. (2015), who were the first to apply DEMT to AIA data, analysed a single target using both instruments independently. They found that, while the density products are essentially equal, the temperature based on AIA data is systematically 8% larger than the one based on EUVI data, i.e. $T_{\mathrm{m}}^{\mathrm{(AIA)}}/T_{\mathrm{m}}^{\mathrm{(EUVI)}} \approx 1.08$. Considering this error, Figure 5 and Table 1 indicate that CR 2208 was characterised by temperatures $\approx 10$ – $15\%$ larger relative to CR 2082 throughout the streamer belt region. As for the electron density products, CR 2208 was found to be $\approx 10$ – $20\%$ less dense than CR 2082 throughout the streamer belt region. These systematic differences are around or beyond the uncertainty level in the DEMT products due to systematic sources (radiometric calibration and tomographic regularization), which Lloveras et al. (2017) estimated to be $\Delta T_{\mathrm{m}}\approx 10\%$ and $\Delta \sqrt{N^{2}_{\mathrm{m}}}\approx 5\%$.

To analyse the loop-integrated energy flux quantities introduced in Section 2.3, we selected closed loops for which both legs have the same sign of the radial gradient of the electron temperature $\triangledown _{r}T_{\mathrm{m}}$. In this way, according to the classification of both legs, each loop was classified as of type 0 (small down loop), I (small up loop), or II (large up loop). For both target rotations, and for loops of type 0, I, and II, Figure 6 shows the frequency histogram of the loop-integrated energy flux quantities $\phi _{r}$, $\phi _{c}$, and $\phi _{h}$ in blue, green, and red colour, respectively.

Figure 6

Statistical results of the loop-integrated energy flux quantities $\phi _{r},\phi _{c}$, and $\phi _{h}$ in colours blue, red, and green, respectively for CR 2082 (left) and CR 2208 (right). From top to bottom, panels show the results for loops of type 0, I and II, which are loops for which both legs meet the criteria from Section 2.3.

For both rotations, the value of the loop-integrated radiative power $E_{r}$, measured by the quantity $\phi _{r}$, is the largest for loops of type 0. This is due to $E_{r}\propto N_{e}^{2}\,\Lambda (T_{e})$, with both factors contributing to maximize $E_{r}$ for loops of type 0. As shown in Figure 5 and Table 1, loops of type 0 are characterised by the largest values of electron density. Also, in the range of sensitivity of the EUVI and AIA instruments, namely 0.5 – 3.0 MK (Nuevo et al., 2015), the radiative loss function $\Lambda (T)$ has a local maximum at $T_{\mathrm{c}}\approx 1\,\mathrm{MK}$. According to Figure 5, loops of type 0, I, and II are characterised by temperatures that are progressively larger, and farther from $T_{\mathrm{c}}$, for both rotations.

The sign of the quantity $\phi _{c}$ depends on that of the conductive flux $F_{c}$. Equations 8 and 12 imply that, by definition, down loops (type 0) and up loops (type I and II) are characterised by $\phi _{c}<0$ and $\phi _{c}>0$, respectively, as verified in Figure 6.

Adding the radiative and conductive terms, the characteristic energy input flux at the coronal base is in the range $\phi _{h}\approx 0.5-1.5 \times 10^{5}\,\mathrm{erg}\,\mathrm{cm}^{-2}\,{ \mathrm{s}}^{-1}$, depending on the rotation and the type of loop, matching the values reported by Mac Cormack et al. (2017). For type 0 loops there is a marginal population characterised by the unphysical result $\phi _{h}<0$. As shown by Mac Cormack et al. (2017), this affects only the smallest sized loops of the type 0, and it is likely due to emission out of the instrumental sensitivity range. Though accounting for most of the coronal plasma, there surely is additional emission out of the instrumental sensitivity range. As a result, the positive term $\phi _{r}$ is most likely underestimated, leading to values $\phi _{h}<0$ in loops of type 0, being characterised by $\phi _{c}<0$.

3.2 Comparison of the DEMT and AWSoM Models

Figure 7 shows Carrington maps of the radial magnetic field $B_{r}$ for both rotations at $1.005\, \mathrm{R_{\odot }}$. Both maps clearly show the large-scale dipolar field, characteristic of solar minimum conditions. Differences between both maps are observed in the sub-polar latitudes, due to the different treatments applied there by the GONG (CR 2082) and the ADAPT-GONG (CR 2208) maps.

Figure 7

Carrington maps of the radial magnetic field $B_{r}$ of the AWSoM model at $1.005 \, \mathrm{R_{\odot }}$ for CR 2082 (left) and CR 2208 (right).

As described in Section 2.2, the AWSoM model includes an artificially thick TR, achieving coronal conditions above height $\approx 1.06\,\mathrm{R_{\odot }}$. Results for the AWSoM model are shown here above that height. For both target rotations, Figures 8 and 9 show latitude–longitude maps of the AWSoM electron density and temperature. Maps are shown at the two largest heights selected for visualization of the DEMT results in Figures 1 and 2. Thick-black curves indicate the magnetic open/closed boundaries based on the magnetic field of the AWSoM model. Visual inspection of these maps shows that the AWSoM model for both rotations is highly axisymmetric, as the tomographic results.

Figure 8

Carrington maps of density (left panels) and temperature (right panels) obtained with AWSoM model at heliocentric heights 1.065 (top panels) and $1.105\,\mathrm{R_{\odot }}$ (bottom panels).

Figure 9

Same as Figure 8 for CR 2208.

When compared to DEMT results (Figures 1 and 2), the latitude–longitude maps of the AWSoM model for heights 1.065 and $1.105\,\mathrm{R_{\odot }}$ capture well the denser and hotter equatorial streamer belt surrounded by the less dense and colder CHs. Furthermore, for both rotations, the temperature maps show the low latitudes of the equatorial streamer belt to be characterised by lower temperatures than its mid-latitudes, as also seen in the DEMT results. The latitude–longitude maps of the AWSoM and DEMT results are shown in the same units and scales, so that a visual comparison reveals similar values of electron density and temperature in both models.

Being highly axisymmetric rotations, the longitude-averaged latitudinal profile of the results of both models is an informative way to compare their large-scale structure. Such a comparison is shown in Figure 10 at height $1.105\,\mathrm{R_{\odot }}$, with the top panels comparing electron density and middle panels the electron temperature. The longitude-averaged latitudinal profile of $B_{r}$ is shown in the bottom panels. In these longitude-averaged profiles, longitudes containing ARs or low latitude CHs were excluded. In each panel the averaged latitudinal variation for the DEMT model is shown in solid line style, while the result for the AWSoM model is shown in dashed line style. Left panels show the comparison for CR 2082 (in blue) and right panels for CR 2208 (in red). In each panel the vertical black lines denote the corresponding longitude-averaged latitude of the open-closed boundary in both hemispheres.

Figure 10

Longitude-averaged latitudinal variation of the electron density (top panels), electron temperature (middle panels), and radial magnetic field $B_{r}$ (bottom panels), for rotations CR 2082 (blue colour, left panels) and CR 2208 (red colour, right panels) at $1.105\,\mathrm{R_{\odot }}$. Dashed and solid lines indicate AWSoM and DEMT results, respectively. Vertical black lines indicate the longitude-averaged latitude of the open/closed magnetic boundary in both hemispheres.

It is worth highlighting several details from Figure 10. Firstly, at most latitudes the overall agreement of the electron density of both models is within $\approx 20\%$ for CR 2082, and $\approx 5\%$ for CR 2208. The noticeable exception is to be found near the open/closed boundaries of both target rotations, where the disagreement between both models can be up to twice as much. In the case of the electron temperature, for both rotations the models agree within $\approx 15\%$ at all latitudes. Secondly, for both rotations, and for both models, these plots clearly show the relatively lower temperatures characterizing the low-latitudes of the equatorial streamer belt compared to its mid-latitudes. Thirdly, for both rotations, the latitude of the open/closed magnetic boundary in both hemispheres matches the location of the strongest latitudinal gradient of the DEMT electron density. Note this is not the case for the AWSoM model, which shows a minimum density at the open/closed boundary. Lastly, the DEMT electron density decreases from the open/closed boundary towards the poles (in both hemispheres of the two rotations), while the AWSoM model shows the opposite trend. For comparison, $B_{r}$ in the CHs increases from the open/closed boundary towards the poles for CR 2082, while showing local maxima around latitudes $-75^{\circ }$ and $+70^{\circ }$ in the case of CR 2208.

To characterise the results of the AWSoM model in distinct magnetic structures, its results for electron density and temperature were traced along its magnetic field lines. For each field line leg, the results were then fit to Equations 5 and 6, considering only data points above heliocentric height $1.055\,\mathrm{R_{\odot }}$. We then classified the traced legs into types I, II, and III, according to the criteria described in Section 2.3. Legs of type 0 are not included for AWSoM as it currently cannot simulate down loops, as discussed in Section 4.

For both target rotations, the top panels of Figure 11 show the latitude–longitude location (at heliocentric height $1.105\,\mathrm{R_{\odot }}$) of all traced field line legs for which criterion (i) of Section 2.3 is met. That criterion is adapted here, requiring that at least five voxels of the tomographic grid are threaded by the leg. Open legs are indicated in grey colour and closed ones in black colour. For each leg, the fits to tomographic temperature and density were applied, as given by Equations 5 and 6. Considering the AWSoM data points and the resulting fits along each leg, the bottom panels of Figure 11 show the latitude–longitude location of the subset for which also both criteria (ii) and (iii) of Section 2.3 are met. Using a three-colour code, type I, II, and III legs are shown in red, magenta, and cyan colour, respectively. This figure can be compared with the corresponding Figure 3 for DEMT results. The AWSoM maps are more densely populated than those of the DEMT. This is due to the 3D MHD model having spatially smoother distributions of electron density and temperature than those of the DEMT.

Figure 11

Same as Figure 3, but using the density and temperature of the AWSoM model to classify its legs in types I, II, and III. The model does not exhibit legs of type 0.

For CR 2082, Figure 12 shows the statistical distribution of the results of the DEMT reconstruction (solid line style) and AWSoM (dashed line style) model traced along legs of type I, II, and III (from top to bottom), as defined in Section 2.3. Figure 13 shows the same analysis for CR 2208.

Figure 12

Statistical distribution of the results of the DEMT reconstruction (solid line style) and AWSoM (dashed line style) traced along legs of type I, II, and III (from top to bottom), as defined in Section 2.3. From left to right: electron density at the lowest coronal height of the AWSoM model $N_{\mathrm{e}}(r=1.055\,\mathrm{R_{\odot }})$, electron density scale height $\lambda _{N}$, and leg-averaged electron temperature $\left < T_{\mathrm{m}}\right >$. In each panel the median values $m$ are indicated.

Figure 13

Same as Figure 12 for CR 2208.

For the two target rotations, Table 2 summarises the quantitative comparative analysis between the results of the DEMT reconstruction and AWSoM model based on the results shown in Figures 12 and 13. The DEMT results are expressed as absolute values, while the AWSoM results are informed as a percentual variation relative to the corresponding result for the DEMT.

Table 2 Median value (indicated as Md) of the statistical distribution of $N_{\mathrm{CB}}$, $\lambda _{N}$, and $\left < T_{\mathrm{m}}\right >$ for each coronal type of leg defined in Section 2.3. DEMT results are expressed in absolute values, while AWSoM results are expressed relative to the corresponding DEMT value.

For CR 2082, the median value of the electron density $N_{\mathrm{CB}}$ of both models agree within $\approx 10-25\%$, depending of the type of leg, with the largest discrepancy found for legs of type II (near the open/closed boundary). The median value of the scale height $\lambda _{N}$ agrees within $\approx 10\%$ in all regions. The leg-averaged electron temperature $\left < T_{\mathrm{m}}\right >$ of both models also agree within $10\%$ in all regions. For CR 2208 the agreement of the median value of $N_{\mathrm{CB}}$ and $\lambda _{N}$ of both models is within $10\%$, while median values of $\left < T_{\mathrm{m}}\right >$ agree within $15\%$. These detailed results, being consistent with the large-scale comparison provided in Figure 10, show in detail how the AWSoM model performs compared to the DEMT in different magnetic structures.

Finally, to provide a graphical comparison of both models across the full range of heliocentric heights covered by the DEMT results, Figure 14 shows the average fits of $N_{\mathrm{e}}(r)$ and $T_{e}(r)$ for legs of type I (red), II (magenta), and III (cyan) for both target rotations. In each panel the DEMT and AWSoM results are plotted in solid and dashed line styles, respectively.

Figure 14

Average fits to $N_{\mathrm{e}}(r)$ (left panels) and $T_{e}(r)$ (right panels) for legs of type I (orange), II (red), and III (cyan), for CR 2082 (top panels) and CR 2208 (bottom panels). Solid lines correspond to DEMT results, while dashed lines correspond to AWSoM results.

As discussed above, Figure 10 shows that the longitude-averaged latitudinal profile of the DEMT electron density in the CHs decreases towards the poles. Figure 15 below shows the longitude-averaged AWSoM radial wind speed $V_{r}$ at $6\,\mathrm{R_{\odot }}$, where all field lines are open. The heliocentric current sheet (HCS) location is indicated by the minimum of the speed curve. For each rotation, all velocity data points to the south of the HCS position map down to the southern CH in Figures 10. Similarly, all velocity data points to the north of the HCS position map down to the northern CH in Figures 10. This clearly shows an anticorrelation between the DEMT electron density at low heights and the AWSoM wind speed at larger heights.

Figure 15

Longitude-averaged latitudinal dependence of the AWSoM model wind speed $V_{r}$ at $6.0\,\mathrm{R_{\odot }}$ for CR 2082 (left panel) and CR 2208 (right panel).

4 Discussion and Conclusions

Magnetic field lines of type 0, I, and II were selected to be associated with increasingly outer layers of the equatorial streamer belt (Figure 3). These magnetic structures progressively exhibit decreasing coronal base density, increasing density scale height, and increasing electron temperature, as informed in 3D quantitative detail in Figure 5 and Table 1. For both rotations we find that down legs populate the low latitudes of the streamer belt, while up legs dominate its mid-latitudes. Also, in the case of CR 2082 the fraction of down legs is significantly larger than for CR 2208. These findings are consistent with previous studies by Huang et al. (2012) and Nuevo et al. (2013). In the case of the latter, they include in their analysis CR 2081, which is a rotation almost identical to our target CR 2082. Our results for CR 2082 compare very well with those of Nuevo et al. (2015) and Lloveras et al. (2017) for CR 2081. As our study uses the improved version of the DEMT technique, such comparison provided a consistency check. For both rotations, type III field lines in the CHs are characterised by sub-MK temperatures, and electron density values of order $\approx 1/2$ of those observed for the type 0 and type I lines in the core of the equatorial streamer.

The energy input flux $\phi _{h}$ at the coronal base, required to maintain stable coronal loops, is in the range $\phi _{h}\approx 0.5$ – $1.5 \times 10^{5}\,\mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$, depending on the rotation and the type of loop, matching the values reported by Mac Cormack et al. (2017). Based on spectroscopic data of the EIS instrument in quiet-Sun regions Hahn and Savin (2014) showed that, if the observed non-thermal broadenings are assigned to Alfvén waves, their energy flux at the coronal base is estimated to be in the range $\approx 1.5$ – $2.5\times 10^{5}\,\mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$. A large fraction of the coronal base energy input flux $\phi _{h}$ estimated in this work, or even its totality, could then be accounted for by Alfvén waves (see the discussion in Mac Cormack et al. 2017).

The comparison of the results of the AWSoM model to the DEMT reconstructions can be summarised as follows. For CR 2082, the electron density of both models agree within $\approx 20\%$ in most regions, while for CR 2208 the agreement is within $\approx 5\%$. The noticeable exception is to be found near the open/closed boundaries of both target rotations, where the disagreement between both models can be up to twice as much. In the case of the electron temperature, both models agree within $\approx 10$ – $15\%$. This level of agreement between both models (within or slightly beyond the uncertainty level of the DEMT results) is considerably better than that reported in previous works. Jin et al. (2012) and Oran et al. (2015), who used previous versions of the AWSoM model, reported electron density values differing by $\approx 50\%$ compared to the DEMT reconstructions, both in the equatorial streamer and CH regions.

The overall better match of the results of the current version of the AWSoM model compared to DEMT reconstructions is partly due to the improved energy partitioning scheme of the model, described in Section 2.2. The simulation of CR 2082 used GONG maps as boundary condition, while the simulation of CR 2208 used the improved ADAPT-GONG maps. This is likely the cause of a more accurate match to the DEMT reconstructions in the case of CR 2208.

For both rotations, the AWSoM model reproduces the relatively lower temperatures found by the DEMT to characterise the low-latitudes of the equatorial streamer belt compared to its mid-latitudes. On the other hand, while the latitude of the open/closed magnetic boundary in both hemispheres matches the location of the strongest latitudinal gradient of the DEMT electron density (physically expected in transitioning from magnetically closed to open regions), this is not the case for the AWSoM model, which shows a minimum density at the open/closed boundary. Also, while the DEMT electron density decreases from the open/closed boundary towards the poles (in both hemispheres of the two rotations), as expected in transitioning from the source region of the slow to the fast component of the solar wind, the AWSoM model shows the opposite trend. This behaviour is notoriously opposite to that reported in the AWSoM model version used by Oran et al. (2015), in which the electron density decreases from the open/closed boundary towards the poles. These unphysical characteristics of the results of the AWSoM model in the range of low heights analysed here ($r\lesssim 1.2\,\mathrm{R_{\odot }}$), may be attributed to the less reliable values of $B_{r}$ provided by both the GONG and ADAPT-GONG maps at sub-polar latitudes. This will be investigated in a follow up article focusing on the current deep minimum epoch, during which the large-scale corona shows the simplest possible structure.

Down loops are to be expected if heating is enhanced at the footpoints of coronal structures. Schiff and Cranmer (2016) numerically simulated stable down loops by means of a 1D steady-state model, requiring that the initial population of Alfvén waves is efficiently converted into compressive modes. Mode conversion is favored by the $\beta \gtrsim 1$ condition found to characterise the regions where down loops are observed in DEMT analysis (Nuevo et al., 2013). In the AWSoM model, heating is controlled by two key parameters, namely, the Alfvén wave dissipation length, and its reflection coefficient. So far, attempts to reproduce observed down loops in AWSoM simulations by increasing the reflection coefficient have not been successful.

A detailed empirical description of the 3D thermodynamic structure of the inner corona at a global scale is currently only possible with tomographic techniques, such as the DEMT. Using tomographic results for continuous validation of 3D MHD models is of high relevance for the continued improvement of models. In follow up articles we will carry out the 3D DEMT reconstruction and MHD modeling of new target rotations selected from the current solar minimum epoch between SCs 24 and 25, as well as CR 2219 corresponding to the July 2, 2019 total solar eclipse.