Thermodynamic Structure of the Solar Corona: Tomographic Reconstructions and MHD Modeling
D. G. Lloveras, A. M. Vásquez, F. A. Nuevo, C. Mac Cormack, N. Sachdeva, W. Manchester IV, B. Van der Holst, R. A. Frazin
SSolar PhysicsDOI: 10.1007/ ••••• - ••• - ••• - •••• - • Thermodynamic Structure of the Solar Corona:Tomographic Reconstructions and MHD Modeling
Diego G. Lloveras · Alberto M. V´asquez · Federico A. Nuevo · Cecilia MacCormack · Nishtha Sachdeva · Ward Manchester IV · Bartholomeus Vander Holst · Richard A. Frazin c (cid:13) Springer ••••
Abstract
We carry out a study of the global three-dimensional (3D) structureof the electron density and temperature of the quiescent inner solar corona( r < .
25 R (cid:12) ) by means of tomographic reconstructions and magnetohydrody- (cid:66)
D.G. Lloveras [email protected]
A.M. V´asquez [email protected]
F.A. Nuevo [email protected]
C. Mac Cormack [email protected]
N. Sachdeva [email protected]
W. Manchester IV [email protected]
B. Van der Holst [email protected]
R.A. Frazin [email protected] Instituto de Astronom´ıa y F´ısica del Espacio (IAFE), CONICET-UBA, CC 67 - Suc28, (C1428ZAA) Ciudad Aut´onoma de Buenos Aires, Argentina Universidad Nacional de Tres de Febrero (UNTREF). Departamento de Ciencia yTecnolog´ıa, S´aenz Pe˜na, Argentina. Ciclo B´asico Com´un (CBC), Universidad de Buenos Aires (UBA), Buenos Aires,Argentina Department of Climate and Space Sciences and Engineering (CLaSP), University ofMichigan, 2455 Hayward Street, Ann Arbor, MI 48109-2143, USA
SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 1 a r X i v : . [ a s t r o - ph . S R ] A p r .G. Lloveras et al. namic simulations. We use differential emission measure tomography (DEMT)and the Alfv´en Wave Solar Model (AWSoM), in their latest versions. Two targetrotations were selected from the solar minimum between solar cycles (SCs) 23and 24 and the declining phase of SC 24. We report in quantitative detail on the3D 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/uploops), as reported first in previous DEMT studies. We also estimate the heat-ing energy flux required at the coronal base to keep these structures stable,found to be or order 10 erg cm − s − , consistently with previous DEMT andspectroscopic studies. We discuss how these findings are consistent with coronaldissipation of Alfv´en waves. We compare the 3D results of DEMT and AWSoMin distinct magnetic structures. We show that the agreement between the prod-ucts of both techniques is the best so far, with an overall agreement (cid:46) Keywords:
Solar Cycle, Observations; Corona, E; Corona, Structures; Corona,Models; Magnetohydrodynamics
1. Introduction
Being the region where the solar atmosphere plasma is heated to million degreetemperatures, the solar wind accelerated, and where impulsive events such assolar flares and coronal mass ejections are energized, observing and modelingof the solar corona are of great relevance to improving our understanding ofthe Sun-Earth environment. To advance our knowledge of the physics of thesolar corona, as well as to enhance and validate three-dimensional (3D) models,information derived from observational data plays a key role. Solar rotationaltomography (SRT) is currently the sole observational technique able to providea quantitative empirical description of the 3D distribution of some fundamentalplasma parameters of the solar corona at a global scale.To study the 3D structure of the quiet-Sun global corona, SRT has provento be a powerful tool. In SRT, solar rotation is taken advantage of, so thatinstruments gather time series of images covering all viewing angles of the solarcorona. This allows posing an inversion problem to solve for the unknown 3Ddistribution of specific quantities of the solar corona. Based on extreme ultravi-olet (EUV) images, taken in several channels sensitive to different temperatures,differential emission measure tomography (DEMT) allows reconstruction of the3D distribution of the differential emission measure (DEM). The final product ofDEMT is in the form of 3D maps of electron density and temperature, coveringthe range of heliocentric heights (cid:46) .
25 R (cid:12) . The technique was first developed
SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 2 omparative Coronal Study of CR-2082 and CR-2208 by Frazin et al. (2009), and first applied to the observational study of coronalstructures by V´asquez et al. (2009). A recent review on DEMT was publishedby V´asquez (2016). The technique is summarized in Section 2.1.Non-tomographic studies of localized regions of the quiet-Sun corona havebeen carried out by means of DEM analysis. Mackovjak et al. (2014) used regular-ized inversion techniques to study characteristic temperatures in the quiet-Sun.L´opez et al. (2019) used a parametric method to study EUV dimmings aftercoronal mass ejections (CMEs) to estimate the coronal mass evacuated by theevents. At a global scale, DEM analysis has been used by Morgan and Taroyan(2017) to characterize the evolution of the temperature of the quiet-Sun coronaduring most of solar cycle (SC) 24.The combination of DEMT with global magnetic models provides insightinto the 3D thermodynamical structure of the global corona. DEMT was firstcombined with a potential field source surface (PFSS) model by Huang et al.(2012) and Nuevo et al. (2013). More recently, Lloveras et al. (2017) combinedDEMT with PFSS models to study the thermodynamics of the global solarcorona in specific magnetic structures for two target rotations selected from thelast two solar minimum epochs. Also combining DEMT with a PFSS model,Mac Cormack et al. (2017) developed a new DEMT product that estimatesthe energy input flux required at the coronal base to maintain stable coronalloops. In this article, DEMT is first combined with the magnetic field of amagnetohydrodynamic (MHD) model.The Alfv´en Wave Solar atmosphere Model (AWSoM) within the Space Wea-ther Modeling Framework (SWMF) is a three-dimensional (3D) physics-based,data-driven MHD model extending from the upper chromosphere, to the uppercorona and to 1 AU and beyond (van der Holst et al., 2010, 2014). The only datainput of the model is a magnetogram of the global corona, used as boundarycondition for the simulation. As new improvements are implemented, the modelis continuously being validated with observations. DEMT results were used byJin et al. (2012) and Oran et al. (2015) to validate AWSoM results finding anagreement within 50% in density and electron temperature in the low corona.More recently, Sachdeva et al. (2019) compared the results of the latest versionof AWSoM model with DEMT products in a global fashion.In this work, the AWSoM model is used with two purposes. Firstly, to providean MHD model of the coronal magnetic field to be used to study the DEMTresults along magnetic field lines. Secondly, to provide thermodynamic resultsto be compared with those reconstructed by DEMT.Combining the DEMT and AWSoM models, we carry out a detailed quantita-tive analysis of two target rotations. We selected Carrington rotation (CR)-2082(2009, 05 April through 03 May) during the SC 23/24 minimum, and CR-2208(2018, 02 September through 29 September) during the end of the decliningphase of SC 24. In the case of CR-2082, the DEMT analysis is based on datataken by the
Extreme UltraViolet Imager Behind (EUVI-B: Wuelser et al. 2004)on board the
Solar TErrestrial RElations Observatory (STEREO), while theAWSoM model uses the synoptic magnetogram provided by the Global Oscil-lation Network Group (GONG: Hill et al. 1994). In the case of CR-2208, theDEMT analysis is based on data taken by the
Atmospheric Imaging Assembly
SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 3 .G. Lloveras et al. instrument (AIA: Lemen et al. 2012) on board the
Solar Dynamics Observatory (SDO), while the AWSoM model uses the magnetogram provide by the Air ForceData Assimilation Photospheric flux Transport (ADAPT)-GONG model.Section 2.1 and 2.2 summarize the DEMT technique and the AWSoM model,respectively. Section 2.3 details the method used to trace the DEMT resultsalong the field lines of the magnetic model. Section 2.4 details the method thatallows determination of the energy input flux at the coronal base. In Section 3.1the quantitative detailed DEMT analysis of both target rotations is shown, andin Section 3.2 the AWSoM and DEMT results are compared. Finally, Section 4summarizes and discusses the main conclusions of this analysis, and anticipatesfurther planned work.
2. Methodology
As detailed in Section 1, rotations CR-2082 and CR-2208 were selected to carryout DEMT reconstructions, based on data taken by the STEREO/EUVI-B andSDO/AIA instruments, respectively. The EUVI and AIA data were preparedusing the latest processing tools and calibration corrections provided by theirteams through the SolarSoft package. In the case of EUVI data, stray-lightcontamination is removed by deconvolution of the point spread function (PSF),carefully determined for each detector by Shearer et al. (2012). In the case ofAIA data, we have not yet implemented such a procedure as we were not awareof reliable determinations of their PSF, and we also understand that stray lightcontamination is expected to be less important for this instrument. A recentstudy by Saqri et al. (2020) indicates that the effect is noticeable in DEM analysisof 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 spatialbinning of the images used in tomography the effect of stray-light removal inDEMT products turns out to be mild, being (cid:46)
10% for density products andnegligible for temperature products. In the future, we will explore the effect ofstray-light removal in AIA images on DEMT tomography, which we expect to besmaller than for EUVI images. For this work, we introduced two improvementsin the implementation of the DEMT technique, as described next.While in all previous DEMT studies full-disk data was used to perform tomog-raphy, in this work we opt to only use off-limb data. In this way, the smallestscale and brighter coronal features seen on disk (most typically in the 171˚Aband) are not included. This has two implications. Firstly, the fast dynamicsthat typically characterizes those structures is absent from the data. Secondly,only half synodic rotation worth of data is needed to constrain the inversionproblem for the whole coronal volume. As a result, artifacts induced by coronaldynamics are reduced compared to previous DEMT reconstructions.The solution of the tomographic problem involves inversion of a very largesparse matrix. Such inversion problems are characterised by spurious high-fre-quency artifacts in the solution, which can be mitigated through regularization
SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 4 omparative Coronal Study of CR-2082 and CR-2208 techniques (Frazin, 2000). In the case of DEMT, all previous efforts used the2D scheme implemented by Frazin et al. (2009), using a finite difference matrixoperator to approximate angular derivatives in both latitude and longitude. Alsonew to the present work, is the implementation of an expanded 3D regularizationscheme, which adds to the previous scheme a finite difference matrix operator toapproximate radial derivatives. In this way, the tomographic inversion problemis performed penalizing nonphysical high-frequency artifacts in all three spatialdirections. As a result, tomographic reconstructions behave more smoothly closeto the radial boundaries of the computational grid when compared to previousreconstructions.In DEMT, the inner corona in the range of heliocentric heights (cid:46) .
25 R (cid:12) isdiscretized in a spherical computational grid. The size of the tomographic gridcell (or voxel) is typically set to 0 .
01 R (cid:12) in the radial direction and 2 ◦ in boththe latitudinal and longitudinal directions. The cadence of the data time-seriesis set to 6 hr. The main product of the technique is the local DEM (LDEM) ateach voxel, a measure of the temperature distribution of the plasma containedin it. We summarize next the main aspects of DEMT required for the analysisof this work.In a first step, the time series of EUV images is used to solve a solar SRTproblem, for each EUV band independently. As a result, the 3D distribution ofthe so called filter band emissivity (FBE) is determined for each band separately.The FBE, an emissivity-type quantity, is defined as the wavelength integral ofthe coronal EUV spectral emissivity and the telescope’s passband function ofeach EUV channel. Line-of-sight (LOS) integration of the FBE provides syntheticimages that can be quantitatively compared to the real data in the time series. Tofind the FBE, the tomographic problem is posed as a global optimization problemin which the quadratic norm of the difference between all pairs of synthetic andreal images is minimized.Due to unresolved coronal dynamics, tomographic reconstructions exhibitnegative values of the reconstructed FBE, or zero when the solution is con-strained to positive values (Frazin, 2000; Frazin et al., 2009). These non-reconstructedvoxels are indicated in black color in the latitude-longitude (Carrington) mapsof DEMT results in Section 3.In a second step, the FBE values obtained for all bands in each voxel ofthe tomographic grid are used to constrain the determination of the local-DEM(LDEM) which, as described in Section 1, describes the temperature distributionof 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 toFBE ( k ) i = (cid:90) d T LDEM i ( T ) TRF ( k ) ( T ) , k = 1 , ..., K , (1)where K is the number EUV bands, and TRF ( k ) is the temperature responsefunction of the k -th detector. In this work, the TRFs are computed based onthe (known) channel’s passband times the coronal emissivity at that temperature(normalized by the squared electron density). The emissivity model used here SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 5 .G. Lloveras et al. is provided by the latest version of the CHIANTI atomic database and plasmaemission model (Del Zanna et al., 2015; Landi et al., 2013).In this work, data from three EUV bands was used: 171, 193 and 211 ˚A inthe case of AIA, and 171, 195 and 284 ˚A bands in the case of EUVI. Whenusing data from three bands, a Gaussian model for the LDEM is able to accu-rately predict the FBEs (Nuevo et al., 2015). In each tomographic voxel, theproblem is then reduced to finding the values of the three free parameters of theGaussian (centroid, standard deviation, and area) that best reproduce the threetomographically reconstructed values of FBE in that voxel.As the LDEM describes the temperature distribution of the plasma in aspecific voxel, it does not deal with different large scale structures, as it maybe 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 works by Aschwanden and Boerner (2011); Plowman et al. (2013);Del Zanna (2013). Other methods applied to DEM analysis of narrow bandimages include Monte Carlo Markov Chain (MCMC) techniques (Schmelz et al.,2016), regularized inversion techniques (Hannah and Kontar, 2012), and iterativesolvers (Pickering and Morgan, 2019; Morgan and Pickering, 2019) that use theknown TRFs of all filters as a functional base. The latter work in particular,introduced a fast iterative solver named SITES, which can easily be adapted forits use in DEMT. We will explore this in the future and compare results withthose provided by our parametric technique.Once the LDEM is determined at each voxel, the LDEM-averaged squaredelectron density N and electron temperature T m in the voxel can be computedby taking its zeroth and first moments over temperature. More specifically, atthe i -th voxel, N ,i = (cid:104) N (cid:105) i = (cid:90) d T LDEM i ( T ) , (2) T m ,i = (cid:104) T e (cid:105) i = 1 (cid:104) N (cid:105) i (cid:90) d T T
LDEM i ( T ) . (3)Next, we define a measure of the accuracy of the LDEM model to predict thetomographic FBEs in each voxel, as R i ≡ (1 /K ) K (cid:88) k =1 (cid:12)(cid:12)(cid:12) − FBE ( k ) i, syn / FBE ( k ) i, tom (cid:12)(cid:12)(cid:12) , (4)being the average absolute relative difference between the tomographic and thesynthetic FBEs. The final product of DEMT is in the form of 3D maps of theLDEM-averaged quantities (cid:112) N and T m , as well as of the measure R . For a fulldescription of the DEMT technique we refer the reader to Frazin et al. (2009). SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 6 omparative Coronal Study of CR-2082 and CR-2208
AWSoM is a three-temperature, MHD model of the solar corona and innerheliosphere which provides the 3D distribution of density and temperatures aswell as the 3D magnetic structure and velocity of the solar wind. In this work,we use AWSoM model simulated results below 1.25 R (cid:12) to correspond to theDEMT analysis region.Heating of the solar corona is addressed by including the non-linear interac-tion of forward and counter-propagating (reflected) Alfv´en waves which resultsin a turbulent cascade. This dissipated turbulent energy is distributed overanisotropic (parallel and perpendicular) proton temperatures and isotropic elec-tron temperature using theories of linear wave damping and stochastic heating.The model accounts for both collisional and collision-less electron heat con-duction and does not use ad-hoc heating functions. The extended MHD equa-tions including radiative cooling, heat conduction and wave turbulence withinAWSoM (van der Holst et al., 2014) are solved using the Block-Adaptive-Tree-Solarwind-Roe-Upwind Scheme (BATS-R-US, Powell et al., 1999; T´oth et al.,2012) numerical scheme.In a previous version of the model, the cascade time of the major wave wasused to determine the wave damping rate (Chandran et al., 2011; van der Holstet al., 2014). In its present version, the energy partitioning is improved by usingthe Alfv´en wave number associated with the damping rate as determined bythe critical balance condition, which uses the cascade time of the minor wave(Lithwick et al., 2007). This leads to more electron heating and less solar windacceleration (van der Holst, 2019).The inner boundary of AWSoM is located at the base of the transition region(at ≈ (cid:12) ). In reality, the thin transition region (TR) has steep gradients intemperature and density as a result of the balance between coronal heating,heat conduction and the radiative losses. To resolve these gradients in a globalmodel would require excessive numerical resources. As described in Lionello et al.(2009) and Sokolov et al. (2013), the TR is artificially broadened to be resolvedwith a finest grid resolution of 0.001 R (cid:12) . To ensure that the base of the TRis not affected by chromospheric evaporation we overestimate the density atthe inner boundary, N e = N i = N (cid:12) = 2 × m − corresponding to theisotropic temperature values, T e = T i = T i (cid:107) = T (cid:12) = 50 ,
000 K, where thesubscripts represents electrons and ions. The upper chromosphere is required toextend radially for the density to fall rapidly to correct (lower) values (Lionelloet al., 2009). At this level, the radiative losses are sufficiently low so that thetemperature can increase monotonically with height and form the transitionregion. Since the broadening of the transition region pushes the corona outwards,the AWSoM model achieves coronal conditions at height ≈ .
05 R (cid:12) , below whichresults can not then be compared to coronal tomographic reconstructions.To drive the AWSoM model, estimates of the photospheric magnetic field ofthe Sun are the main input. Synoptic magnetograms are used to specify the initialand the boundary conditions of the magnetic field. We use the PFSS model to ex-trapolate the 3D magnetic field (from the 2D photospheric magnetic field maps)using spherical harmonics. The source surface is taken to be at 2.5 R (cid:12) . GONG
SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 7 .G. Lloveras et al. provides synoptic full-disk surface maps of the radial magnetic field component ofthe Sun. However, since the polar regions are not well observed from the ecliptic,GONG estimates the polar fields by fitting a polynomial to neighboring observedlatitudes, which might lead to inaccuracies. An improvement over these mapsis provided by the Air Force Data Assimilation Photospheric Flux Transport(ADAPT) model (Worden and Harvey, 2000), which creates synchronic-synopticmaps by incorporating supergranulation, meridional circulation, and differentialrotation. These maps provide a physics-based description of the unobservedpolar magnetic fields (Arge et al., 2010; Henney et al., 2012). In this work, weuse the GONG synoptic map as input for CR-2082 (ADAPT-GONG maps areunavailable for CR-2082) and the ADAPT-GONG global magnetic field map forCR-2208. Based on results from previous efforts, for CR-2082 the magnetic fieldfrom the GONG map is scaled up by a factor of 1.85 for weak fields ( B r < S A ) of the outgoing wave,( S A /B ) (cid:12) = 1 . × W m − T − and 1 . × W m − T − for CR-2082 andCR-2208, respectively, with B (cid:12) being the field strength at the inner boundary.The correlation length of the Alfv´en waves is set to, L ⊥ √ B = 1 . × m √ T ,where L ⊥ is transverse to the magnetic field direction.The computational domain of the solar corona extends from 1 to 24 R (cid:12) . Thespherical grid has an adaptive grid that has fine resolution near the Sun, andincreases outward with the z-axis aligned with the rotation axis in HeliographicRotation coordinates. The Adaptive Mesh refinement (AMR) resolves the an-gular cell size to 1 . ◦ between 1 − . (cid:12) and to 2 . ◦ outside this radius range.The solar corona component uses about 3 million cells on 6 × × × × To determine the electron density and temperature along individual magneticfield lines, first both the thermodynamic results and the magnetic field obtainedwith the AWSoM model were interpolated into the DEMT grid. Then, thegeometry of the field lines is determined by numerical integration of the firstorder differential equations d r/B r = r d θ/B θ = r sin( θ ) d φ/B φ , both inwardsand outwards, from the specified 3D coordinates of a starting point. In orderto evenly sample the whole volume spanned by the DEMT reconstructions, onestarting point is selected at the center of each tomographic cell at 6 uniformlyspaced heights, ranging from 1 .
025 to 1 .
225 R (cid:12) , and every 2 ◦ in both latitudeand longitude, for a total of 97 ,
200 starting points.
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For analysis purposes, the traced magnetic field lines are classified as open orclosed according to their full geometry. Each closed field line is further classifiedas “small” or “large”, according to its coronal length L being respectively smalleror larger than the median value of the whole population, which is Md( L ) ≈ . (cid:12) for both rotations. Finally, each closed magnetic field line is separatedin its two “legs”, defined as the two segments that go from each of its twofootpoints ( i.e. their location at r = 1 R (cid:12) ) to its apex ( i.e. , the location ofmaximum height).At this stage, DEMT and AWSoM products are traced along open and closedmagnetic field lines. Once the field line geometry is computed in high spatialresolution, 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 struc-tures in the solar minimum corona and by Nuevo et al. (2013) to expand thatanalysis to rotations with different level of activity.For each open field line and for each closed field leg, an exponential fit wasapplied to the electron density data points and a linear fit applied to the electrontemperature data points. For DEMT the data points used are (cid:112) N ( r ) and T m ( r ), and in case of the AWSoM models the data points used are N e ( r ) and T e ( r ). The exponential and linear fit equations are described by (cid:112) N = N exp [ − ( h/l ) / ( r/ R (cid:12) ) ] , (5) T m = T + a h , (6)where h ≡ r − (cid:12) is the coronal height measured from the photosphere. In theelectron density fit, l [R (cid:12) ] is the density scale height and N [cm − ] is the electrondensity at the footpoint ( h = 0) of the loop. In the electron temperature fit, a [MK / R (cid:12) ] is the slope and T [MK] is the electron temperature at the footpointof the loop. The slope a estimates the radial gradient of the electron temperaturealong the loop, which we denote as a = (cid:79) r T m hereafter, with (cid:79) r ≡ e r · ∇ beingthe radial derivative operator and e r the heliocentric radial unit vector.In the case of the electron density, the fitted function corresponds to theisothermal hydrostatic equilibrium solution, allowing for variation of the solargravitational acceleration with height. This choice of function provides a straight-forward means to directly compare the observed coronal thermodynamical statewith the hydrostatic solution.Coronal magnetic structures for which temperature increases/decreases withheight (in the inner corona) were dubbed as “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 those works, down loops can be expected if theheating 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 Aschwandenand Schrijver (2002). In a recent study, Schiff and Cranmer (2016) reproducedboth down and up loops by means of numerical simulations, using a 1D steady-state model and considering time-averaged heating rates. SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 9 .G. Lloveras et al.
To determine if the leg of a traced field line is of type up or down, we firstdetermine the Pearson correlation coefficient ρ ( T, r ) between the DEMT tem-perature T m and the heliocentric height r . We then select field lines for which thetemperature is significantly correlated with height by requiring | ρ ( T, r ) | > . (cid:79) r T m > (cid:79) r T m <
0, respectively. The linear fit allowscharacterization of the variation of T m with height by means of a characteristictemperature gradient (cid:79) r T m [MK / R (cid:12) ] along each leg. The chi-squared test toevaluate the quality of the fit considers the uncertainty level in the DEMTproducts due to systematic sources (radiometric calibration and tomographicregularization), that Lloveras et al. (2017) estimated to be ∆ T m ≈
10% and∆ (cid:112) N ≈ | ρ ( T, r ) | > . • Type 0: closed-small-down with footpoints in the range | latitude | < ◦ . • Type I: closed-small-up with footpoints in the range | latitude | < ◦ . • Type II: closed-large-up with footpoints in the range | latitude | > ◦ . • Type III: open with footpoint in the range | latitude | > ◦ .In the case of closed-small field lines, the population of down and up legs be-comes comparable for CR-2082, so we classify them into the two complementaryclasses of legs of type 0 (down) and legs of type I (up). In the case of closed-largefield lines, down legs are (cid:46)
15% of the population for both rotations. In the caseof open field lines, down legs are (cid:46)
10% of the population for both rotations.Hence, the requirement of being up for legs of type II and III is included toselect the vastly dominating population in each case. On the other hand, theinclusion of latitude limits for the footpoints in the classification of legs fromtype 0 through III is purposely set to study the streamer belt in progressivelyouter layers, as well as to separate the field lines of the high latitude CHs (legsof type III). In Section 3, the results of both the DEMT and AWSoM models inthe four classes of legs are statistically analysed.
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The high temperature of the corona requires heating mechanisms to compensatefor the energy losses. While the vast majority of the existing literature on coro-nal heating focuses on active regions (ARs), some studies have been dedicatedto the heating of quiet-Sun regions. In particular, Mac Cormack et al. (2017)developed a new application of the DEMT technique to estimate the energyinput 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 globalcoronal magnetic model, as described in Section 2.3.Consider a static energy balance for each magnetic flux tube, in which thedominating 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 the power quantities are in unitsof [erg sec − cm − ].The thermal conduction power E c equals the divergence of the conductive heatflux F c , i.e. E c ( s ) = [1 /A ( s )] d[ A ( s ) F c ( s )] / d s , where A ( s ) is the cross-sectionalarea of the magnetic flux tube at position s . Under a quiescent solar coronaplasma regime, the conductive flux is assumed to be dominated by the electronthermal conduction, described by the usual Spitzer model (Spitzer, 1962) F c ( s ) = − κ T ( s ) / dTds ( s ) , (8)where κ = 9 . × − erg sec − K − / is the Spitzer thermal conductivity.In the corona, EUV emission is dominated by collisions of the emitting ionswith free electrons, so that the radiative power scales as N e . The radiative powerof an isothermal plasma at temperature T is then computed as E r = N e Λ( T ),where the radiative loss function Λ( T ) is calculated by means of an emissionmodel. In this work we used the latest version of the atomic database and plasmaemission model CHIANTI (Del Zanna et al., 2015). The radiative power in termsof the LDEM is then given by: E r = (cid:90) d T LDEM( T ) Λ( T ) . (9)The energy balance given by Equation 7 is then integrated in the volume ofany given coronal magnetic flux tube. Dividing the result by the flux tube areaat the coronal base, and making use of the soleidonal condition of the magneticfield, a field line integrated version of that energy balance is found, φ h = φ r + φ c , (10)where the line-integrated flux quantities φ r,c [erg sec − cm − ] are given by (MacCormack et al., 2017): SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 11 .G. Lloveras et al. φ r = (cid:18) B B L B + B L (cid:19) (cid:90) L ds E r ( s ) B ( s ) , (11) φ c = (cid:18) B F c,L − B L F c, B + B L (cid:19) , (12)where L is the length of the loop, and B and B L are the values of the magneticfield at the footpoint locations of the loop in the coronal base, namely s = 0and s = L . For any given field line, all quantities in these two expressions canbe computed from the DEMT results traced along field lines, and the AWSoMmagnetic field model, through Equations 8-9. Once computed, the quantity φ h can be calculated, which is the energy input flux required at the coronal base ofeach coronal field-line to maintain a stable coronal structure.
3. Results
As described in Section 2.1, we carried out DEMT reconstructions of the coro-nal structure for rotations CR-2082 and CR-2208 using STEREO/EUVI andSDO/AIA data, respectively. Once the LDEM was determined for each rotation,the square root of the mean value of the electron density squared ( (cid:112) N ) and theelectron mean temperature ( T m ) were computed at each voxel of the tomographiccomputational grid by means of Equations 2 and 3, and the measure R wascalculated by means of Equation 4.Figures 1 and 2 show latitude-longitude maps of DEMT results for both rota-tions. Three different heights of interest are selected from the tomographic grid,providing also a detailed 3D view of the tomographic results: the lowest heightof the tomographic grid (1 .
025 R (cid:12) ), the lowest height where the AWSoM resultsare fully consistent with coronal conditions (1 .
065 R (cid:12) ), and a middle height ofthe tomographic grid (1 . (cid:12) ). Black voxels correspond to non-reconstructedvoxels (see Section 2.1). Thick-black curves indicate the open/closed boundariesof the magnetic field of the AWSoM model, detailed in Section 2.2.Both target rotations are highly axisymmetric, i.e. characterised by a highazymuthal symmetry. Rotation CR-2082 has two small ARs, both near latitude+30 ◦ and around longitudes 50 ◦ and 120 ◦ (not identified in the NOAA catalog).Rotation CR-2208 has two ARs, both near latitude +5 ◦ and around longitudes140 ◦ and 300 ◦ (NOAA 12722, 12721).The magnetically open and closed regions of the AWSoM model are associ-ated with CHs and the equatorial streamer belt, respectively. The location ofthe open/closed boundaries derived from the respective AWSoM model quiteaccurately matches the regions of the DEMT maps, which exhibit the strongestlatitudinal gradient of both the electron density and temperature.Figures 1 and 2 show that the DEMT reconstruction of the streamer belt ischaracterised by relatively higher densities and temperatures in comparison to SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 12 omparative Coronal Study of CR-2082 and CR-2208 N m N m N m Figure 1.
Carrington maps of DEMT products (cid:112) N (left panels) and T m (right panels)for CR-2082. Top, middle and bottom panels show the results at three heliocentric heights,1 . .
065 and 1 .
105 R (cid:12) respectively. Black voxels correspond to non-reconstructed regions(see text in Section 3.1) and thick-black curves indicate the open/closed boundaries. the CHs. They also show that the streamer belt region of CR-2082 was denser andcolder than that of CR-2208. In the case of CR-2082, which belongs to the deepminimum epoch between SCs 23 and 24, the low latitudes of the streamer belt arecharacterised by lower electron temperature than in its mid-latitudes. A similarbehavior is seen in CR-2208, belonging to the end of the declining phase of SC24, but having a somewhat less axisymmetric structure this characteristic is notso obvious. This thermodynamic structure of the streamer has been reported forother solar minimum rotations in previous DEMT works (Lloveras et al., 2017;Nuevo et al., 2013; V´asquez et al., 2010).Latitude-longitude maps of the score R defined by Equation 4 show that theagreement between the tomographic and synthetic FBEs is 5% or better in morethan 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 foundin each voxel accurately predicts the reconstructed FBEs.For both rotations, the top panels of Figure 3 show the latitude-longitudelocation (at heliocentric height r = 1 .
105 R (cid:12) ) of all traced field line legs for
SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 13 .G. Lloveras et al. N m N m N m Figure 2.
Same as Figure 1 but for CR-2208. which criterion (i) of Section 2.3 is met. Open legs are indicated in gray color andclosed ones in black color. Considering the DEMT data points and the resultingfits along each leg, the middle panels of Figure 3 show the latitude-longitudelocation of the subset for which also both criteria (ii) and (iii) of Section 2.3are met. Using a four-color code, type 0, I, II and III legs are shown in blue,orange, red, and cyan color, respectively. The bottom panels show histogramsof the latitude distribution of the legs of the middle panel, using the same colorcode. Of the ≈ ≈ SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 14 omparative Coronal Study of CR-2082 and CR-2208
CR-2082 -50 0 50Latitude [deg]0.000.010.020.03 F r eq . H i s t og r a m CR-2208 -50 0 50Latitude [deg]0.000.010.020.030.040.05 F r eq . H i s t og r a m Figure 3.
Top panels: latitude-longitude location at heliocentric height r = 1 .
105 R (cid:12) of allopen (grey color) and closed (black color) traced field line legs for which criterion (i) of Section2.3 is met, for both CR-2082 (left) and CR-2208 (right). Middle panels: latitude-longitudelocation of the subset for which also both criteria (ii) and (iii) of Section 2.3 are met. Thelocation of type 0, I, II and III legs is shown in blue, orange, red, and cyan color, respectively.Bottom panels: frequency histograms of the latitude of the four types of legs of the middlepanel. 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 byNuevo et al. (2013). Type I (small-closed-up) legs are present at all latitudeswithin the streamer belt. Their population peaks in the mid-latitudes of bothhemispheres for CR-2082, and in the mid-latitudes of the northern hemispherefor CR-2208. The latitude distribution of legs of type 0 and I in CR-2082 isconsistent to the distribution of down and up loops of CR-2081 in the analysisby Nuevo et al. (2013), which did not place any limits on the latitude locationof the analysed structures.Type II (large up) legs are mostly very large trans-equatorial field lines form-ing the envelope of the streamer belt (the requirement of footpoints locatedbeyond 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 (cid:79) r T m for legs of type 0, I, II and III.The lack of population around values close to zero is due to the requirement | ρ ( T, r ) | > . SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 15 .G. Lloveras et al.
CR-2082 -5 0 5Temperature gradient [MK/Rsun]0.000.010.020.030.040.05 F r eq . H i s t og r a m CR-2208 -5 0 5Temperature gradient [MK/Rsun]0.000.010.020.030.040.05 F r eq . H i s t og r a m Figure 4.
Frequency histograms of the temperature radial gradient for the four types of legsin Figure 3 (using the same color code) for CR-2082 (left panel) and CR-2208 (right panel). median value of the temperature radial gradient is Md ( (cid:79) r T m ) ≈ − .
2, +2 . . . / R (cid:12) , for legs of type 0, I, II and III, respectively. DEMT - Type 0 CB [10 cm -3 ]0.000.020.040.060.080.10 F r eq . H i s t og r a m m= 1.24m= 1.01 DEMT - Type 0 λ N [10 -2 r sun ]0.000.020.040.060.080.10 F r eq . H i s t og r a m m= 7.14m= 7.35 DEMT - Type 0 m > [MK]0.000.020.040.060.080.100.12 F r eq . H i s t og r a m m= 1.09m= 1.35 DEMT - Type I CB [10 cm -3 ]0.000.020.040.060.080.100.120.14 F r eq . H i s t og r a m m= 1.15m= 1.03 DEMT - Type I λ N [10 -2 r sun ]0.000.020.040.060.080.10 F r eq . H i s t og r a m m= 7.52m= 9.68 DEMT - Type I m > [MK]0.000.020.040.060.080.10 F r eq . H i s t og r a m m= 1.25m= 1.55 DEMT - Type II CB [10 cm -3 ]0.000.020.040.060.08 F r eq . H i s t og r a m m= 0.98m= 0.79 DEMT - Type II λ N [10 -2 r sun ]0.000.010.020.030.040.050.06 F r eq . H i s t og r a m m= 9.91m= 11.8 DEMT - Type II m > [MK]0.000.020.040.060.080.100.120.14 F r eq . H i s t og r a m m= 1.36m= 1.58 DEMT - Type III CB [10 cm -3 ]0.000.020.040.060.080.100.12 F r eq . H i s t og r a m m= 0.66m= 0.52 DEMT - Type III λ N [10 -2 r sun ]0.000.020.040.060.080.100.12 F r eq . H i s t og r a m m= 7.70m= 8.85 DEMT - Type III m > [MK]0.000.020.040.060.080.100.120.14 F r eq . H i s t og r a m m= 0.97m= 1.13 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 CB ≡ (cid:112) N ( r = 1 .
055 R (cid:12) ), electron density scale height λ N , and loop-averaged temperature (cid:104) T m (cid:105) . In each panel the median value m is indicated. SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 16 omparative Coronal Study of CR-2082 and CR-2208
For both rotations, Figure 5 shows, in a statistical fashion, the DEMT resultstraced along field lines discriminated by leg type. From top to bottom results areshown for type 0 to type III legs, respectively. From left to right the panels showthe statistical distribution of the electron density value N CB ≡ (cid:112) N ( r = 1 . λ N , and the height-averaged (along the leg) electron temperature (cid:104) T m (cid:105) , with the median value m indicated in each plot. Table 1.
Median value (indicated as “Md”) of thestatistical distribution of N CB , λ N , and (cid:104) T m (cid:105) foreach coronal type of legs defined in Section 2.3. ForCR-2082 values are expressed in absolute terms, whilefor CR-2208 they are given as a percentual variationrelative to the CR-2082 value in the parentheses.Type Md( N CB ) Md( λ N ) Md( (cid:104) T m (cid:105) )[10 cm − ] [10 − R (cid:12) ] [MK]0 1.24 ( - + + - + + - + + - + + Table 1 summarizes the results of the quantitative comparative analysis be-tween the two target rotations. For CR-2082 quantities are expressed as absolutevalues, while for CR-2208 they are expressed relative to the corresponding resultsfor CR-2082.Throughout the magnetically closed region of both rotations, type 0, I andII legs, associated to increasingly outer layers of the equatorial streamer belt,exhibit progressively decreasing coronal base density, increasing density scaleheight, and increasing electron temperature. In both rotations also, type III legsin the CHs are characterised by sub-MK temperatures, and electron densityvalues of order ≈ / ≈ −
20% lower values of the electron density at the coronal base, ≈ − ≈ −
25% larger values of the electrontemperature.In comparing the DEMT results obtained for the two selected targets, wehighlight they rely on data provided by two different instruments, namely EUVIand AIA for CR-2082 and CR-2208, respectively. To quantify the systematicdifference of the DEMT products due to the different filter sets of both in-struments, Nuevo et al. (2015), who were the first to apply DEMT to AIAdata, analysed a single target using both instruments independently. They foundthat while the density products are essentially equal, the temperature based onAIA data is systematically 8% larger than the one based on EUVI data, i.e. T (AIA)m /T (EUVI)m ≈ .
08. Considering this error, Figure 5 and Table 1 indicatethat CR-2208 was characterised by temperatures ≈ −
15% larger relativeto CR-2082 throughout the streamer belt region. As for the electron density
SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 17 .G. Lloveras et al.
CR-2082 Type 0 -1 0 1 2 3 φ [10 erg cm -2 sec -1 ]0.000.050.100.150.20 F r eq . H i s t og r a m CR-2208 Type 0 -1 0 1 2 3 φ [10 erg cm -2 sec -1 ]0.000.020.040.060.080.100.12 F r eq . H i s t og r a m CR-2082 Type I -1 0 1 2 3 φ [10 erg cm -2 sec -1 ]0.000.050.100.150.200.250.30 F r eq . H i s t og r a m CR-2208 Type I -1 0 1 2 3 φ [10 erg cm -2 sec -1 ]0.000.020.040.060.080.100.120.14 F r eq . H i s t og r a m CR-2082 Type II -1 0 1 2 3 φ [10 erg cm -2 sec -1 ]0.00.10.20.3 F r eq . H i s t og r a m CR-2208 Type II -1 0 1 2 3 φ [10 erg cm -2 sec -1 ]0.000.050.100.150.20 F r eq . H i s t og r a m Figure 6.
Statistical results of the loop-integrated energy flux quantities φ r , φ c , and φ h incolors blue, red and green, respectively for CR-2082 (left) and CR-2208 (right). From top tobottom, panels show the results for loops of type 0, I and II, which are loops for which bothlegs meet the criteria from Section 2.3. products, CR-2208 was found to be ≈ −
20% less dense than CR-2082throughout the streamer belt region. These systematic differences are around orbeyond the uncertainty level in the DEMT products due to systematic sources(radiometric calibration and tomographic regularization), that Lloveras et al.(2017) estimated to be ∆ T m ≈
10% and ∆ (cid:112) N ≈ (cid:79) r T m . In this way, according to the SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 18 omparative Coronal Study of CR-2082 and CR-2208 classification of both its legs, each loop was classified as of type 0 (small downloop), I (small up loop), or II (large up loop). For both target rotations, and forloops of type 0, I and II, Figure 6 shows the frequency histogram of the loop-integrated energy flux quantities φ r , φ c and φ h in blue, green and red color,respectively.For both rotations, the value of the loop-integrated radiative power E r , mea-sured by the quantity φ r , is largest for loops of type 0. This is due to E r ∝ N e Λ( T e ), with both factors contributing to maximize E r for loops of type 0. Asshown in Figure 5 and Table 1, loops of type 0 are characterised by the largestvalues of electron density. Also, in the range of sensitivity of the EUVI and AIAinstruments, namely 0.5–3.0 MK (Nuevo et al., 2015), the radiative loss functionΛ( T ) has a local maximum at T c ≈ T c , for both rotations.The sign of the quantity φ c depends on that of the conductive flux F c . Equa-tions 8 and 12 imply that, by definition, down loops (type 0) and up loops (typeI and II) are characterised by φ c < φ c >
0, respectively, as verified inFigure 6.Adding the radiative and conductive terms, the characteristic energy inputflux at the coronal base is in the range φ h ≈ . − . × erg cm − s − ,depending on the rotation and the type of loop, matching the values reportedby Mac Cormack et al. (2017). For type 0 loops there is a marginal populationcharacterised by the unphysical result φ h <
0. As shown by Mac Cormack et al.(2017), this affects only the smallest sized loops of the type 0, and it is likelydue to emission out of the instrumental sensitivity range. Though accountingfor most of the coronal plasma, there surely is additional emission out of theinstrumental sensitivity range. As a result, the positive term φ r is most likelyunderestimated, leading to values φ h < φ c < Figure 7 shows carrington maps of the radial magnetic field ( B r ) for bothrotations at 1 .
005 R (cid:12) . Both maps clearly show the large-scale dipolar field,characteristic of solar minimum conditions. Differences between both maps areobserved in the sub-polar latitudes, due to the different treatments applied thereby the GONG (CR-2082) and the ADAPT-GONG (CR-2208) maps.As described in Section 2.2, the AWSoM model includes an artificially thickTR, achieving coronal conditions above height ≈ .
06 R (cid:12) . Results for the AW-SoM model are shown here above that height. For both target rotations, Figures8 and 9 show latitude-longitude maps of the AWSoM electron density and tem-perature. Maps are shown at the two largest heights selected for visualization ofthe DEMT results in Figures 1 and 2. Thick-black curves indicate the magneticopen/closed boundaries based on the magnetic field of the AWSoM model. Visualinspection of these maps shows that the AWSoM model for both rotations ishighly axisymetric, as the tomographic results.
SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 19 .G. Lloveras et al.
Figure 7.
Carrington maps of the radial magnetic field B r of the AWSoM model at 1 .
005 R (cid:12) for CR-2082 (left) and CR-2208 (right).
Figure 8.
Carrington maps of density (left panels) and temperature (right panels) obtainedwith AWSoM model at heliocentric heights 1 .
065 (top panels) and 1 .
105 R (cid:12) (bottom panels).
When compared to DEMT results (Figures 1 and 2), the latitude-longitudemaps of the AWSoM model for heights 1 .
065 and 1 .
105 R (cid:12) capture well thedenser and hotter equatorial streamer belt surrounded by the less dense andcolder CHs. Furthermore, for both rotations, the temperature maps show thelow latitudes of the equatorial streamer belt to be characterised by lower tem-peratures 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 unitsand scales, so that a visual comparison reveals similar values of electron densityand temperature in both models.Being highly axisymmtric rotations, the longitude-averaged latitudinal profileof the results of both models is an informative way to compare their large-scalestructure. Such a comparison is shown in Figure 10 at height 1 .
105 R (cid:12) , with thetop panels comparing electron density and mid panels electron temperature. Thelongitude-averaged latitudinal profile of B r is shown in the bottom panels. Inthese longitude-averaged profiles, longitudes containing ARs or low latitude CHs SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 20 omparative Coronal Study of CR-2082 and CR-2208
Figure 9.
Same as Figure 8 for CR-2208. were excluded. In each panel the averaged latitudinal variation for the DEMTmodel is shown in solid-line style, while the result for the AWSoM model is shownin dashed-linesyle. Left panels show the comparison for CR-2082 (in blue) andright panels for CR-2208 (in red). In each panel the vertical black lines denotethe corresponding longitude-averaged latitude of the open-closed boundary inboth hemispheres.Several details from Figure 10 are worth highlighted. Firstly, at most latitudesthe overall agreement of the electron density of both models is within ≈
20% forCR-2082, and ≈
5% for CR-2208. The noticeable exception is to be found nearthe open/closed boundaries of both target rotations, where the disagreementbetween both models can be up to twice as much. In the case of the electrontemperature, for both rotations the models agree within ≈
15% at all latitudes.Secondly, for both rotations, and for both models, these plots clearly show therelatively lower temperatures characterizing the low-latitudes of the equatorialstreamer belt compared to its mid-latitudes. Thirdly, for both rotations, thelatitude of the open/closed magnetic boundary in both hemispheres matchesthe location of the strongest latitudinal gradient of the DEMT electron density.Note this is not the case for the AWSoM model, that shows a minimum densityat the open/closed boundary. Lastly, the DEMT electron density decreases fromthe open/closed boundary towards the poles (in both hemispheres of the tworotations), while the AWSoM model shows the opposite trend. For comparison, B r in the CHs increases from the open/closed boundary towards the poles forCR-2082, while showing local maxima around latitudes − ◦ and +70 ◦ in thecase of CR-2208.To characterize the results of the AWSoM model in distinct magnetic struc-tures, its results for electron density and temperature were traced along itsmagnetic field lines. For each field line leg, the results were then fit to Equations SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 21 .G. Lloveras et al. -50 0 50Latitude [deg]0.30.40.50.60.7 N e [ c m - ] DEMTAWSoM
Lat. Profile CR-2082 at 1.105 Rsun -50 0 50Latitude [deg]0.30.40.50.60.7 N e [ c m - ] DEMTAWSoM
Lat. Profile CR-2208 at 1.105 Rsun -50 0 50Latitude [deg]0.91.01.11.21.31.4 T e [ M K ] DEMTAWSoM
Lat. Profile CR-2082 at 1.105 Rsun -50 0 50Latitude [deg]1.01.21.41.6 T e [ M K ] DEMTAWSoM
Lat. Profile CR-2208 at 1.105 Rsun -50 0 50Latitude [deg]-4-2024 B r [ G ] Lat. Profile CR-2082 at 1.105 Rsun -50 0 50Latitude [deg]-4-2024 B r [ G ] Lat. Profile CR-2208 at 1.105 Rsun
Figure 10.
Longitude-averaged latitudinal variation of the electron density (top panels), elec-tron temperature (middle panels) and radial magnetic field B r (bottom panels), for rotationsCR-2082 (blue color, left panels) and CR-2208 (red color, right panels) at 1 .
105 R (cid:12) . Dashedand solid lines indicate AWSoM and DEMT results, respectively. Vertical black lines indicatethe longitude-averaged latitude of the open/closed magnetic boundary in both hemispheres.
Figure 11.
Same as Figure 3, but using the density and temperature of the AWSoM modelto classify its legs in types I, II and III. The model does not exhibit legs of type 0.
SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 22 omparative Coronal Study of CR-2082 and CR-2208 .
055 R (cid:12) . Wethen classified the traced legs into types I, II and III, according to the criteriadescribed in Section 2.3. Legs of type 0 are not included for AWSoM as itcurrently can not simulate down loops, as discussed in Section 4.
CR-2082 - Type I CB [10 cm -3 ]0.000.020.040.060.080.100.120.14 F r eq . H i s t og r a m m= 1.15m= 0.99 CR-2082 - Type I λ N [10 -2 r sun ]0.000.050.100.15 F r eq . H i s t og r a m m= 7.52m= 8.09 CR-2082 - Type I m > [MK]0.000.020.040.060.080.10 F r eq . H i s t og r a m m= 1.25m= 1.13 CR-2082 - Type II CB [10 cm -3 ]0.000.020.040.060.080.100.12 F r eq . H i s t og r a m m= 0.99m= 0.74 CR-2082 - Type II λ N [10 -2 r sun ]0.000.020.040.060.080.100.12 F r eq . H i s t og r a m m= 9.88m= 9.67 CR-2082 - Type II m > [MK]0.000.020.040.060.080.100.120.14 F r eq . H i s t og r a m m= 1.36m= 1.33 CR-2082 - Type III CB [10 cm -3 ]0.000.050.100.150.200.25 F r eq . H i s t og r a m m= 0.66m= 0.55 CR-2082 - Type III λ N [10 -2 r sun ]0.00.10.20.30.4 F r eq . H i s t og r a m m= 7.70m= 7.00 CR-2082 - Type III m > [MK]0.000.050.100.150.20 F r eq . H i s t og r a m m= 0.97m= 0.88 Figure 12.
Statistical distribution of the results of the DEMT (solid line-style) and AWSoM(dashed-line style) models traced along legs of type I, II and III (from top to bottom), asdefined in Section 2.3. From left to right: electron density at the lowest coronal height of theAWSoM model N e ( r = 1 .
055 R (cid:12) ), electron density scale height λ N , and leg-averaged electrontemperature (cid:104) T m (cid:105) . In each panel the median values m are indicated. For both target rotations, the top panels of Figure 11 show the latitude-longitude location (at heliocentric height 1 .
105 R (cid:12) ) of all traced field line legs forwhich criterion (i) of Section 2.3 is met. That criterion is adapted here, requiringthat at least five voxels of the tomographic grid are threaded by the leg. Openlegs are indicated in gray color and closed ones in black color. For each leg, thefits to tomographic temperature and density were applied, as given by Equations5 and 6. Considering the AWSoM data points and the resulting fits along eachleg, the bottom panels of Figure 11 show the latitude-longitude location of thesubset for which also both criteria (ii) and (iii) of Section 2.3 are met. Usinga three-color code, type I, II and III legs are shown in red, magenta and cyancolor, respectively. This figure can be compared with the corresponding Figure3 for DEMT results. The AWSoM maps are more densely populated than thoseof DEMT. This is due to the 3D MHD model having having spatially smootherdistributions of electron density and temperature than those of DEMT.For rotation CR-2082, Figure 12 shows the statistical distribution of theresults of the DEMT (solid line-style) and AWSoM (dashed line-style) models
SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 23 .G. Lloveras et al.
CR-2208 - Type I CB [10 cm -3 ]0.000.020.040.060.080.100.12 F r eq . H i s t og r a m m= 1.03m= 1.09 CR-2208 - Type I λ N [10 -2 r sun ]0.000.020.040.060.080.100.120.14 F r eq . H i s t og r a m m= 9.69m= 8.86 CR-2208 - Type I m > [MK]0.000.020.040.060.080.10 F r eq . H i s t og r a m m= 1.55m= 1.29 CR-2208 - Type II CB [10 cm -3 ]0.000.020.040.060.080.100.12 F r eq . H i s t og r a m m= 0.79m= 0.86 CR-2208 - Type II λ N [10 -2 r sun ]0.000.020.040.060.080.100.120.14 F r eq . H i s t og r a m m= 11.8m= 10.2 CR-2208 - Type II m > [MK]0.000.020.040.060.080.100.12 F r eq . H i s t og r a m m= 1.58m= 1.45 CR-2208 - Type III CB [10 cm -3 ]0.000.050.100.15 F r eq . H i s t og r a m m= 0.52m= 0.56 CR-2208 - Type III λ N [10 -2 r sun ]0.00.10.20.30.4 F r eq . H i s t og r a m m= 8.85m= 7.21 CR-2208 - Type III m > [MK]0.000.020.040.060.080.100.120.14 F r eq . H i s t og r a m m= 1.13m= 0.93 Figure 13.
Same as Figure 12 for CR-2208. traced along legs of type I, II and III (from top to bottom), as defined in Section2.3. From left to right: electron density at the lowest coronal height of theAWSoM model N CB ≡ N e ( r = 1 .
055 R (cid:12) ), electron density scale height λ N ,and leg-averaged electron temperature (cid:104) T m (cid:105) . In each panel the median values m are indicated. Figure 13 shows the same analysis for rotation CR-2208.For the two target rotations, Table 2 summarizes a quantitative comparativeanalysis between the results of the DEMT and AWSoM models based on theresults shown in Figures 12 and 13. The DEMT results are expressed as absolutevalues, while the ASWSoM results are informed as a percentual variation relativeto the corresponding result for DEMT.For rotation CR-2082, the median value of the electron density N CB of bothmodels agree within ≈ − λ N agree within ≈
10% in all regions. The leg-averagedelectron temperature (cid:104) T m (cid:105) of both models also agree within 10% in all regions.For rotation CR-2208 the agreement of the median value of N CB and λ N of bothmodels is within 10%, while median values of (cid:104) T m (cid:105) agree within 15%. Thesedetailed results, being consistent with the large-scale comparison provided inFigure 10, show in detail how the AWSoM model performs compared to DEMTin different magnetic structures.Finally, to provide a graphical comparison of both models across the full rangeof heliocentric heights covered by the DEMT results, Figure 14 shows the averagefits of N e ( r ) and T e ( r ) for legs of type I (red), II (magenta), and III (cyan) for SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 24 omparative Coronal Study of CR-2082 and CR-2208
Table 2.
Median value (indicated as “Md”) of the statisti-cal distribution of N CB , λ N , and (cid:104) T m (cid:105) for each coronal typeof leg defined in Section 2.3. DEMT values are expressed inabsolute terms, while AWSoM results are expressed relativeto the corresponding DEMT value.Type Md( N CB ) Md( λ N ) Md( (cid:104) T m (cid:105) )[10 cm − ] [10 − R (cid:12) ] [MK]CR-2082I 1.15 ( - + - - - - - - - + - - + - - + - - both target rotations. In each panel the DEMT and AWSoM results are plottedin solid and dashed line styles, respectively. CR-2082 - Radial Profile N e [ c m - ] Solid: DEMTDashed: AWSoM
CR-2082 - Radial Profile T e [ M K ] Solid: DEMTDashed: AWSoM
CR-2208 - Radial Profile N e [ c m - ] Solid: DEMTDashed: AWSoM
CR-2208 - Radial Profile T e [ M K ] Solid: DEMTDashed: AWSoM
Figure 14.
Average fits to N 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. SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 25 .G. Lloveras et al.
As discussed above, Figure 10 shows that the longitude-averaged latitudinalprofile 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 R (cid:12) , 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 thesouthern CH in Figures 10. Similarly, all velocity data points to the north of theHCS position map down to the northern CH in Figures 10. This clearly showsan anti-correlation between the DEMT electron density at low heights and theAWSoM wind speed at larger heights. Lat. Profile CR-2082 at 6 Rsun -50 0 50Latitude [deg]200300400500 V r [ m / s ] -50 0 50Latitude [deg]200250300350400450 V r [ m / s ] Lat. Profile CR-2208 at 6 Rsun
Figure 15.
Longitude-averaged latitudinal dependence of the AWSoM model wind speed V r at 6 . (cid:12) 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 withincreasingly outer layers of the equatorial streamer belt (Figure 3). These mag-netic structures progressively exhibit decreasing coronal base density, increasingdensity scale height, and increasing electron temperature, as informed in 3Dquantitative detail in Figure 5 and Table 1. For both rotations we find thatdown legs populate the low latitudes of the streamer belt, while up legs domi-nate its mid-latitudes. Also, in the case of CR-2082 the fraction of down legs issignificantly larger than for CR-2208. These findings are consistent with previousstudies by Huang et al. (2012) and Nuevo et al. (2013). In the case of the latter,they include in their analysis target CR-2081, which is a rotation almost identicalto our target CR-2082. Our results for target CR-2082 compare very well withthose of Nuevo et al. (2015) and Lloveras et al. (2017) for target CR-2081. Asour study uses the improved version of the DEMT technique, such comparisonprovided a consistency check. For both rotations, type III field lines in the CHsare characterised by sub-MK temperatures, and electron density values of order ≈ / φ h at the coronal base, required to maintain stablecoronal loops, is in the range φ h ≈ . − . × erg cm − s − , dependingon the rotation and the type of loop, matching the values reported by MacCormack et al. (2017). Based on spectroscopic data of the EIS instrument in SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 26 omparative Coronal Study of CR-2082 and CR-2208 quiet-Sun regions Hahn and Savin (2014) showed that, if the observed non-thermal broadening are assigned to Alfv´en waves, their energy flux at the coronalbase is estimated to be in the range ≈ . − . × erg cm − s − . A largefraction of the coronal base energy input flux φ h estimated in this work, or evenits totality, could then be accounted for by Alfv´en waves (see the discussion inMac Cormack et al. 2017).The comparison of the results of the AWSoM model to the DEMT reconstruc-tions can be summarized as follows. For CR-2082, the electron density of bothmodels agree within ≈
20% in most regions, while for CR-2208 the agreementis within ≈ ≈ − ≈
50% compared to the DEMTreconstructions, both in the equatorial streamer and CH regions.The overall better match of the results of the current version of the AWSoMmodel compared to DEMT reconstructions is partly due to the improved energypartitioning scheme of the model, described in Section 2.2. The simulation ofCR-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 moreaccurate match to the DEMT reconstructions in the case of CR-2208.For both rotations, the AWSoM model reproduces the relatively lower tem-peratures found by DEMT to characterize the low-latitudes of the equatorialstreamer belt compared to its mid-latitudes. On the other hand, while thelatitude of the open/closed magnetic boundary in both hemispheres matchesthe 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, that shows a minimum density at theopen/closed boundary. Also, while the DEMT electron density decreases fromthe open/closed boundary towards the poles (in both hemispheres of the tworotations), as expected in transitioning from the source region of the slow to thefast component of the solar wind, the AWSoM model shows the opposite trend.This behavior is notoriously opposite to that reported in the AWSoM modelversion used by Oran et al. (2015), in which the electron density decreases fromthe open/closed boundary towards the poles. These unphysical characteristicsof the results of the AWSoM model in the range of low heights analysed here( r (cid:46) . (cid:12) ), may be attributed to the less reliable values of B r provided byboth the GONG and ADAPT-GONG maps at subpolar latitudes. This will beinvestigated 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 ofcoronal structures. Schiff and Cranmer (2016) numerically simulated stable downloops by means of a 1D steady-state model, requiring that the initial populationof Alfv´en waves is efficiently converted into compressive modes. Mode conversion SOLA: lloveras_diego_2020_arxiv.tex; 16 April 2020; 0:35; p. 27 .G. Lloveras et al. is favored by the β (cid:38) Acknowledgments
D.G.L. and C.M.C. acknowledge CONICET doctoral fellowships (Res.Nr. 4870) to IAFE that supported their participation in this research. D.G.L, A.M.V., F.A.N.and C.M.C. acknowledge ANPCyT grant 2016/0221 to IAFE that partially supported theirparticipation in this research. A.M.V. also acknowledges UBACyT grant 20020160100072BAto DCAO-UBA to FCEyN-UBA and IAFE that partially supported his participation in this re-search. W.M. and B.v.H acknowledge NSF grant 1663800 that partially supported his participa-tion in this research. W.M. also acknowledges NASA grants NNX16AL12G and 80NSSC17K0686.Disclosure of Potential Conflicts of Interest: The authors declare that they have no conflictsof interest.
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