Energy Input Flux in the Global Quiet-Sun Corona
C. Mac Cormack, A. M. Vásquez, M. López Fuentes, F. A. Nuevo, E. Landi, R. A. Frazin
EEnergy Input Flux in the Global Quiet-Sun Corona
Cecilia Mac Cormack , Alberto M. V´asquez , , Marcelo L´opez Fuentes and Federico A.Nuevo Instituto de Astronom´ıa y F´ısica del Espacio (IAFE), CONICET-UBA,CC 67 - Suc 28, (C1428ZAA) Ciudad Aut´onoma de Buenos Aires, Argentina.
Enrico Landi and Richard A. Frazin
Department of Climate and Space Sciences and Engineering (CLaSP),University of Michigan, 2455 Hayward Street, Ann Arbor, MI 48109-2143, US.
ABSTRACT
We present first results of a novel technique that provides, for the first time, con-straints on the energy input flux at the coronal base ( r ∼ .
025 R (cid:12) ) of the quiet-Sunat a global scale. By combining differential emission measure tomography (DEMT) ofEUV images, with global models of the coronal magnetic field, we estimate the energyinput flux at the coronal base that is required to maintain thermodynamically stablestructures. The technique is described in detail and first applied to data provided by the
Extreme Ultraviolet Imager (EUVI) instrument, on board the
Solar TErrestrial REla-tions Observatory (STEREO) mission, and the
Atmospheric Imaging Assembly (AIA)instrument, on board the
Solar Dynamics Observatory (SDO) mission, for two solarrotations with different levels of activity. Our analysis indicates that the typical en-ergy input flux at the coronal base of magnetic loops in the quiet-Sun is in the range ∼ . − . × (erg sec − cm − ), depending on the structure size and level of activ-ity. A large fraction of this energy input, or even its totality, could be accounted forby Alfv´en waves, as shown by recent independent observational estimates derived fromdeterminations of the non-thermal broadening of spectral lines in the coronal base ofquiet-Sun regions. This new tomography product will be useful for validation of coronalheating models in magnetohydrodinamic simulations of the global corona. Subject headings:
Sun: corona — Sun: magnetic fields — Sun: UV radiation — Sun: funda-mental parameters Departamento de F´ısica, Facultad de Ciencias Ex-actas y Naturales (FCEN), Universidad de BuenosAires (UBA),Pabell´on I, Ciudad Universitaria, (C1428ZAA) CiudadAut´onoma de Buenos Aires, Argentina. Departamento de Ciencia y Tecnolog´ıa, Universi- dad Nacional de Tres de Febrero (UNTREF),Valent´ın G´omez 4752, (B1678ABH) Caseros, Provinciade Buenos Aires, Argentina. a r X i v : . [ a s t r o - ph . S R ] J un . Introduction The heating of the solar corona remains amain topic of current research in solar physics.While there is a wide consensus in the solarphysics community on the magnetic nature ofthe phenomena responsible for the heating ofthe million-degree corona, the precise mecha-nisms by which this occurs are an open field ofresearch and active debate.Coronal heating theories are traditionallyclassified into two broad groups: those basedon the dissipation of magnetic stress, infor-mally called DC-heating, and those basedon the dissipation of magnetohydrodinamic(MHD) waves, also known as AC-heating. Thefirst group includes models in which the stressproduced in the coronal field by photosphericmotions is released in situ by reconnectionand current sheet formation (see e.g., Parker(1988); Priest (2011)). Other MHD simula-tions (Gudiksen & Nordlund 2005) model thesystem as a numerical enclosure in which theenergy is injected by photospheric motions atthe base and released by Ohmic dissipation inthe corona. Also in the first group are modelsbased on the energy release due to the tur-bulent interplay between photospheric plasmamotions and the magnetic field (Gomez et al.2000). AC heating mechanisms are based onthe propagation of disturbances produced atthe feet of the magnetic structures that dissi-pate at certain atmospheric layers releasing en-ergy that translates into plasma heating (Hey-vaerts & Priest 1983; De Moortel et al. 2000;O’Neill & Li 2005). Recently, van Ballegooi-jen et al. (2011) proposed a mixed mechanismby which the diffusion of MHD waves at thechromosphere and transition region (TR) in-terface produces a turbulent regime that heatsthe plasma.The heating of the plasma in differentstructures (coronal holes, bright loops in ac-tive regions, quiet-Sun corona, etc.) is prob- ably dominated by different physical mecha-nisms. To advance our understanding of thephysics underlying this complex phenomenon,advances on observational constraints are key.The majority of the observational literatureon coronal heating focus on active region (AR)structures, where individual bright extreme ul-traviolet (EUV) and X-ray loops provide directdiagnostics of magnetic structures (Reale 2010;Schmelz et al. 2010; Aschwanden & Boerner2011; Klimchuk 2015). Although not evidentfrom EUV/X-ray images due to its diffuse ap-pearance, the quiet-Sun corona is of course alsofully threaded by magnetic fields along whichenergy transport and deposition takes place.The observational study of the heating in thequiet-Sun diffuse corona has comparatively re-ceived less attention (Benz & Krucker 2002;Wilhelm et al. 2004; Hahn & Savin 2014).In general, the aforementioned works pro-vide insight on the heating phenomenon at alocal scale, for the specific structures selectedfor observation, and are affected by line-of-sight projection effects.
Differential emissionmeasure tomography (DEMT) provides a pow-erful tool to study the quiet-Sun corona at aglobal scale and in three dimensions (Frazin,V´asquez & Kamalabadi 2009; V´asquez 2016).Based on full solar rotation time series ofEUV images taken in channels sensitive to dif-ferent temperatures, DEMT provides three-dimensional (3D) maps of the electron densityand temperature of the lower corona, in the he-liocentric height range 1.02 to 1.225 R (cid:12) . Thecoronal magnetic field in these regions can beglobally modeled by means of potential fieldsource surface (PFSS), or MHD models. Com-bination of the DEMT and global magneticmodels has provided useful insight in the 3Dthermodynamical structure of the global quiet-Sun corona (Huang et al. 2012; Nuevo et al.2013, 2015), making it an ideal tool to pro-vide constraints on coronal heating for theseregions. However, until now such a DEMT ap-2lication had not been developed.In this work, we develop the first versionof a new DEMT tool capable of providing con-straints on the coronal heating of the quiet-Sunglobal corona. Specifically, it provides spatialtwo-dimensional (2D) maps of the energy in-put flux required at the coronal base to main-tain thermodynamically stable coronal struc-tures under hydrostatic assumption. This newtool is applied to two specific selected rota-tions with different levels of activity, studiedby means of two EUV instruments, namely the
Extreme Ultraviolet Imager (EUVI, Wuelser etal. 2004), on board the
Solar TErrestrial REla-tions Observatory (STEREO) mission, and the
Atmospheric Imaging Assembly (AIA, Lemenet al. 2012) telescope on board the
Solar Dy-namics Observatory (SDO) mission.Section 2 summarizes the techniques, in-struments and data sets used. Section 3 de-tails the energy model and the loop-integratedquantities that are introduced, which form thenew DEMT tool. Section 4 shows the DEMTresults for the selected periods, while Section 5details the new results on energy input flux atthe coronal base. In Section 6 these resultsare compared to a 0D hydrodynamic (HD)model. Section 7 discusses the main conclu-sions and its implications, and anticipates fur-ther planned work.
2. Methodology and Data
This work is based on the study of theglobal corona by means of the DEMT tech-nique to determine its 3D thermodynamicalstructure, the PFSS modeling of its globalmagnetic field, and the combination of both.The technique is applied to two specific Car-rington rotations (CRs): CR-2081 (2009, 09March through 05 April), a deep minimum pe-riod between solar cycles (SCs) 23 and 24 char-acterized by virtually no ARs, and CR-2099(2010, 13 July through 9 August), a rotation during the early rising phase of SC 24.Both periods were tomographically recon-structed from data taken by the EUVI/STEREOinstrument. In the case of CR-2099, the periodwas also reconstructed using data taken by theAIA/SDO instrument. For both rotations, themagnetic field was modeled by means of the
Fi-nite Difference Potential-Field Solver (FDIPS)PFSS model developed by T´oth, van der Holst& Huang (2011), using as boundary condi-tions synoptic magnetograms built from datataken by the
Michelson Doppler Imager (MDI,Scherrer et al. 1995) on board the Solar andHeliospheric Observatory (SOHO) mission.In DEMT the inner corona, in the heightrange 1 . − . R (cid:12) , is discretized in a spheri-cal computational grid. The size of the grid cell(or voxel) is set to 0 . R (cid:12) in the radial direc-tion (or ∼ ◦ in both the latitu-dinal and longitudinal directions (or ∼
27 Mmat an intermediate grid height of r = 1 . R (cid:12) at the equator). With this angular resolutionone image every 6 hours is the cadence neededto fully constrain the inversion problem, for atotal of about 110 images to cover a full solarrotation. This is the standard DEMT resolu-tion used over the past few years in all previ-ously published work based on this technique(V´asquez 2016), providing a good compromisebetween resolution and computational load.Two consecutive procedures are then per-formed. In a first step, time series of EUVimages in different bands, covering a full solarrotation, are used to perform EUV tomogra-phy. The product of the tomographic inversionin each band is the 3D distribution of the filterband emissivity (FBE), defined as the wave-length integral of the coronal EUV spectralemissivity and the telescope’s passband func-tion of each band.In a second step, the FBE values obtainedfor all bands in each tomographic cell (or voxel)are used to constrain the determination of a local differential emission measure (LDEM)3istribution. The LDEM describes the tem-perature distribution of the electron plasmacontained in each individual tomographic gridvoxel. The LDEM is defined so that the elec-tron density N e and electron mean tempera-ture T m (averaged over temperature) of eachvoxel are computed as, N e = (cid:90) dT LDEM( T ) , (1) T m = 1 N e (cid:90) dT T LDEM( T ) . (2)Due to their ill-posed nature DEM in-version problems are difficult to treat, bothin bright loops observed in ARs, as well asin the diffuse quiet Sun corona here ana-lyzed. If based on high resolution spectraldata, DEM analysis is best performed throughthe Monte Carlo Markov Chain (MCMC) ap-proach (Kashyap & Drake 1998), or regular-ized inversion techniques (Hannah & Kontar2012) (also applicable to filter band data sets).In the case of narrowband filter telescopes, theAIA instrument has improved temperature di-agnostic capabilities over previous EUV tele-scopes instruments (such as EUVI). Applica-tion of MCMC methods based on AIA imageshave been explored by several works (Testaet al. 2012; Del Zanna 2013), as discussed inNuevo et al. (2015). Schmelz et al. (2013) haverecently shown advances of MCMC multiter-mal DEM analysis based on AIA data of brightEUV loops in ARs, using all six AIA channelssimultaneously, with updated temperature re-sponse functions based on CHIANTI v7.1. (seealso Schmelz et al. 2016 for an application toan extended study of bright loops).In the case of DEMT, the DEM parametricapproach is suitable due to the limited numberof data points. Nuevo et al. (2015) have devel-oped a detailed study of the capabilities of theparametric technique in DEMT when appliedto both EUVI and AIA data. In the case of EUVI data its three coronal bands are used(171, 195, and 284 ˚A), with maximum sensi-tivity temperatures in the range 1.0-2.15 MK(see Table 1 in Nuevo et al. 2015). In the caseof AIA data, as DEMT targets the diffuse quietSun corona, the filters that are currently usedin DEMT are those of 171, 193, 211, and 335 ˚A,with maximum sensitivity temperatures in therange 0.85-2.5 MK (see Table 1 in Nuevo et al.2015). These four AIA bands cover the maintemperature range characteristic of the quietSun regions targeted by DEMT. Adding the94 and 131 ˚A bands of AIA in DEMT studieshas also been attempted (exploring other para-metric models for the LDEM as well), but ata global scale (as required in tomography) thesignal from these channels in the diffuse quietSun is in general too weak to provide meaning-ful information.In this study the EUVI and AIA temper-ature responses have been computed based onCHIANTI v7.1, as discussed in detail in Nuevoet al. (2015). As shown in that work, when us-ing the aforementioned four coronal bands ofAIA, the tomographic emissivity 3D distribu-tions are best explained by an ubiquitous bi-modal LDEM, described by two Gaussian dis-tributions with characteristic centroids of or-der 1.5 MK and 2.6 MK, dubbed “warm” and“hot”, respectively. In the case of using allthree EUVI channels DEMT detects only thewarm component. It is shown 1) that if onlythe three AIA channels of 171, 193 and 211 ˚Aare used (as in this work) then also only thewarm component is detected, and 2) that inthat case the results are very similar to thosebased in EUVI data, with some small system-atic differences due to the different responsefunctions of the respective trios of filters. TheLDEM obtained are typically broad, with vari-able FWHM depending on the region. Indeed,in that same work, a controlled study invert-ing synthetic data from modeled DEM distri-butions has shown DEMT to be able to detect4early isothermal and multithermal plasmas,reflected in the resulting value of the FWHMof the LDEM.The parametric approach is at the mo-ment the best implementation of DEMT. Evenwith its limitations, the resulting LDEM dis-tributions, as well as the electron density andmean temperature maps they produce, havebeen validated with other types of observa-tional studies. For example, in Nuevo et al.(2015) the LDEM parametric technique hasbeen validated comparing its results when ap-plied to LOS-DEM analysis of image pixelsagainst those obtained by other authors usingMCMC techniques (based on EIS data). Simi-lar results are obtained, in particular in termsof the bimodal DEMs. Applied to many differ-ent rotations, DEMT has so far provided con-sistent results across many studies and coronalregions, which have also been used as a valida-tion tool for MHD simulations of the globalcorona (see review in V´asquez 2016). Futurenew developments could explore the implemen-tation of regularized inversion techniques forthe determination of the LDEM at each voxel,or even MCMC methods as applied to AIA im-ages by Schmelz et al. (2016), who have beenable to exploit those if at least 3 data pointsare available.In this work, all three coronal bands ofEUVI are used to study CR-2081 and CR-2099, and in the latter case an alternate studybased on the AIA trio 171-193-211 ˚A is alsocarried out. When using three bands, as inthis work, the LDEM is modeled by a Gaus-sian function (Nuevo et al. 2015) dependenton three free parameters: centroid tempera-ture, temperature width, and total area. Ineach voxel the values of the free parameters arefound so that the synthetized emissivity val-ues FBE ( k )syn best match the tomographic valuesFBE ( k )tom for all 3 bands k = 1 , ,
3, achieved byminimizing the score, R = 13 (cid:88) k =1 (cid:12)(cid:12)(cid:12) − FBE ( k )syn / FBE ( k )tom (cid:12)(cid:12)(cid:12) . (3)A score of R ∼ N e , electron meantemperature T m , and score R . The readeris referred to Frazin, V´asquez & Kamalabadi(2009) and Nuevo et al. (2015) for a detaileddescription of the DEMT technique, and toV´asquez (2016) for a recent review on all pub-lished work based on it.The DEMT results can be then combinedwith the global magnetic field model, by trac-ing the results of the former along the mag-netic field lines of the latter. This approachwas first used by Huang et al. (2012) to studythe temperature structure of the solar coronaduring the last minimum, and later on appliedby Nuevo et al. (2013) to expand the analysisto rotations with different level of activity. Inthis paper the same approach is used to study,in an original fashion, the energy balance alongindividual magnetic loops, as it is described inSection 3. In doing so, a new set of numeri-cal tools within the suite of DEMT codes wasdeveloped.
3. Loop Model and Energy Balance
A simple hydrostatic model for a coronalmagnetic flux tube, sketched in Figure 1 is con-sidered. The position along the tube is givenby the variable s , with s = 0 and s = L rep-resenting the positions at the coronal base. Atany position s , the (unknown) coronal heat-ing power E h ( s ) is balanced by the two ma-jor coronal losses, namely the radiative power5 r ( s ) and the thermal conduction power E c ( s )(Aschwanden 2004), E h ( s ) = E r ( s ) + E c ( s ) , (4)where the three power quantities are per unitvolume, i.e., have units of [erg sec − cm − ]. Inthe optically thin corona the radiative energyis emitted isotropically, while the heat conduc-tive flux is constrained to flow along magneticfield lines. Considering the coronal magneticflux tube as a whole, the conductive flux atthe coronal base represents a net gain or lossof energy for the system.The thermal conduction power E c equalsthe divergence of the conductive heat flux F c ,which can be expressed as the derivative alongthe flux tube, E c ( s ) = 1 A ( s ) dds [ A ( s ) F c ( s )] . (5)where, for the quiescent solar coronal plasmaregime, the conductive heat is dominated bythe electron thermal conduction described bythe usual Spitzer model (Spitzer 1962), F c ( s ) = − κ T ( s ) / dTds ( s ) , (6)where the Spitzer thermal conductivity is κ ≈ . × − erg s − cm − K − / .For each of the three power quantities perunite volume in Equation (4), a correspondingtotal power γ [erg sec − ] in the coronal part ofthe magnetic flux tube is obtained by integrat-ing them over its whole volume, γ i ≡ (cid:90) L ds A ( s ) E i ( s ) . (7)where i = c, r, h denotes the conductive, ra-diative, and heating terms, respectively. UsingEquation (5), the integrated thermal conduc-tion power can be expressed as, γ c = A L F c,L − A F c, , (8) where A and A L are the values of thetransversal area at s = 0 and s = L , respec-tively, and F c, and F c,L are the respectivevalues of the conductive heat flux.Dividing the three integrated power quan-tities in Equation (7) by the total basal areaof the flux tube A + A L , three associated fluxquantities φ [erg sec − cm − ] can be defined as, φ i ≡ γ i A + A L ; i = h, r, c. (9)Using these last three quantities, integra-tion of Equation (4) over the whole coronalvolume of the magnetic flux tube implies theintegrated energy balance, φ h = φ r + φ c . (10)The energy required to heat the plasmacontained in the magnetic flux tube is ul-timately injected into it through its coronalbase. The quantity φ h represents the total en-ergy input flux due to all mechanisms exceptheat conduction (accounted for by the term φ c ). The quantity φ h will hereafter be referredto as the “energy input flux” at the coronalbase. The quantity φ c is total energy flux en-tering (or leaving) the coronal part of the mag-netic tube due to heat conduction. The quan-tity φ r is the total radiative power emitted bythe plasma contained in the coronal part of themagnetic flux tube, divided by the total areaof its coronal base. Begging proportional tothe squared local electron density, which de-creases strongly with height, most of the theradiative loss occurs in the lower heights, sothat φ r provides a characteristic value of thecoronal radiative flux.The magnetic null divergence condition, in-tegrated along the magnetic flux tube, reads A ( s ) B ( s ) = A B = A L B L = c , where c is aconstant specific to each magnetic flux tube,and B and B L are the values of the magneticfield strength at s = 0 and s = L , respectively.6ig. 1.— A closed coronal magnetic flux tube and the coordinate axis s along it. The radiativepower (per unit volume) E r ( s ) [erg sec − cm − ] emitted at an arbitrary position s along the tubeis indicated, along with the local transverse area A ( s ) of the tube. The conductive heat flux F c atboth footpoints of the magnetic tube are also sketched.Using these relations into Equations (7), (8),and (9), the radiative and conductive terms ofthe RHS of Equation (10) can be expressed as φ r = (cid:18) B B L B + B L (cid:19) (cid:90) L ds E r ( s ) B ( s ) , (11) φ c = B F c,L − B L F c, B + B L . (12)Note that, by introducing these integratedflux quantities, the energy balance equation(10) is freed from transversal area values, hold-ing then for individual magnetic field lines ,rather than magnetic flux tubes. In studiescombining DEMT with magnetic models, in-dividual magnetic field lines from the modelare assigned the DEMT values of N e and T m of the tomographic voxels through which theypass through (as described in Section 4.2 be-low). The DEMT values for density and tem-perature in each computational voxel representan average description of the plasma containedin it. Assigning these values to field lines of themagnetic model provides then a semi-empiricalsteady state model for quiet Sun long-lived coronal structures, aiming at describing theiraverage state over the DEMT temporal resolu-tion ( ∼ / B ( s ) along thefield line and, in particular, its values at thecoronal base B and B L , are one of the prod-ucts directly available from the global coronamagnetic extrapolation. Hence, the terms in-volved in Equations (11)-(12) can be computedfrom the results of DEMT and the magneticextrapolation.In the optically thin corona, the radiative7ower of an isothermal plasma at temperature T is computed as E r = N e Λ( T ), where the ra-diative loss function Λ( T ) is in turn computedby means of a model, such as the CHIANTIatomic database and plasma emission model(Dere et al. 1997), used in this work in its lat-est version. Hence, at each tomographic voxelthe radiative power is computed from the tem-perature distribution LDEM( T ) obtained fromthe DEMT technique as, E r = (cid:90) dT LDEM( T ) Λ( T ) . (13)where, from Equation (1), dT LDEM( T ) is thecontribution to the quadratic electron densityin the voxel of the plasma with temperature inthe range T ± dT . The radiative power E r , initself a novel DEMT product introduced in thiswork, can be numerically traced along the fieldlines of the global magnetic coronal extrapola-tion, as explained in Section 4.2 below. Thisallows computation of the quantity φ r fromEquation (11) for each magnetic field line inthe model.Finally, the quantity φ c given by Equation(12) requires computation of the conductiveheat flux F c at both coronal base points ofthe field line. To this end, in Equation (6)the temperature T and temperature gradient dT /ds are computed from the DEMT resultstraced along the each magnetic field line.The next section details the numerical im-plementation of the tracing of the DEMT re-sults along the magnetic field lines, and thecomputation of the corresponding quantities φ r and φ c . Once these two quantities are com-puted for each field line, Equation (10) allowscomputation of the energy input flux at thecoronal base φ h , the new DEMT product thatconstitutes the main result of this work.
4. DEMT-PFSS Results4.1. 3D Reconstruction of Density andTemperature
This section shows and describes theDEMT results corresponding to the two rota-tions, CR-2081 and CR-2099, based on EUVIdata in both cases, as well as AIA/SDO datain the latter one. In the case of CR-2099, thevery similar EUVI based results are omitted tosave space. CR-2099 has been the subject ofDEMT analysis based on both EUVI and AIAdata in Nuevo et al. (2015), where the resultsare compared in detail and shown to be con-sistent. Density values obtained with AIA are ∼
2% smaller compared those obtained withEUVI, while temperatures are ∼
8% larger.These systematic differences are due to slightdifferences in the temperature response be-tween respective channels of both instrumentalsets.The 3D distribution of the FBE was tomo-graphically reconstructed for the EUVI bandsof 171, 195 and 284 ˚A in both selected ro-tations, as well as for the AIA bands of 171,193 and 211 ˚A in the case of CR-2099. Us-ing the three FBE reconstructions in each ro-tation, the 3D distribution of the LDEM wasfound. Finally, from Equations (1) through(3), the 3D maps of the electron density N e ,electron mean temperature T m , and score R were computed for both rotations.As way of example, Figure 2 shows the re-sults obtained from the DEMT analysis at aselected height of the tomographic computa-tional sphere. The results are shown as Car-rington maps , the distribution of quantities inlatitude and longitude. Similar maps are ob-tained at all heights of the tomographic grid.The left panels correspond to the EUVI databased CR-2081 reconstruction, while the rightpanels do to the AIA data based CR-2099 re-construction.The top panels show, as an example, FBE8ig. 2.— DEMT results at r = 1 .
075 R (cid:12) for CR-2081 based on EUVI data (left panels) and for CR-2099 based on AIA data (right panels). The top panels show Carrington maps of the reconstructedFBE [ph cm − sr − s − ] in the bands EUVI 195 ˚A (left) and AIA 193 ˚A (right). The second to fourthrow panels show, in descending order, Carrington maps of the following DEMT results: R -score(see text), electron density N e [10 cm − ] and electron mean temperature T m [MK]. In all panels,the overplotted thick-black curves indicate the boundary between the open and closed magneticregions, as derived from the PFSS model. 9aps in the bands of EUVI 195 ˚A and AIA193 ˚A. Due to unresolved coronal dynamics,tomographic reconstructions exhibit artifactssuch as smearing and negative values of thereconstructed FBEs, or zero when the solutionis constrained to positive values. These arecalled zero-density artifacts (ZDAs). In theFBE Carrington maps, ZDA voxels are indi-cated as solid-black regions. The overplottedthick-black curves in all panels indicate theboundary between the open and closed mag-netic regions, as derived from the PFSS model.The second row of panels in Figure 2 showsCarrington maps of the R , with ZDA voxels in-dicated as solid-black regions. It can be readilyseen that, in both cases, most of the closed-corona volume is characterized by R < − (dark-green color), meaning that the LDEMpredicts the tomographic FBEs with a preci-sion better than 1%. Note that, in the caseof the AIA analysis, some regions of the opencorona are characterized by somewhat larger R scores, but in any case the focus of this paperis the closed corona.The bottom two rows of panels show theDEMT electron density N e and electron meantemperature T m . In the temperature maps,solid-black regions indicate ZDA voxels, whilewhite solid-white regions indicate voxels forwhich the parametric LDEM has a score R > .
1, i.e. the discrepancy between the tomo-graphic and synthetic FBEs is more than 10%.In these cells, dubbed anomalous emissivityvoxels (AEVs), the Gaussian LDEM modeldoes not accurately reproduce the tomographicresults. In the electron density maps, ZDAsand AEVs are indicated as black regions.It is interesting to note that the open/closedboundaries of the PFSS model quite accuratelymatch contour levels of the tomographic den-sity and temperature. In other words, alongthe open/closed boundary the gradient of thetomographic results is approximately perpen-dicular to it. This is an interesting consistency check between the PFSS and DEMT mod-els, which can also be verified in all previousDEMT works.In the tomographic density maps, the vox-els with the largest density values (yellowishregions) always correspond to observed activeregions (AR). This has been verified by carefulcomparison of the reconstructions to the cata-logue provided by the National Oceanic andAtmospheric Administration (NOAA) SpaceWeather Prediction Center. These regions arecharacterized by threshold values of the tomo-graphic density, as detailed in Nuevo et al.(2015). As tomographic reconstructions arenot suitable to study the fast-evolving ARs,these regions are left out of the analysis of thispaper. Voxels belonging to ZDA and AEV re-gions are also excluded from the analysis.
Figure 3 shows 3D visualizations of thefield lines of the PFSS models computed fromMDI synoptic magnetograms for CR-2081 andCR-2099. For each traced field line the 3Dcoordinates of a starting point must be spec-ified. In order to evenly cover the whole vol-ume spanned by the DEMT reconstructions,one starting point was selected at the center ofeach tomographic cell at 10 uniformly spacedheights, ranging from 1.035 to 1.215 R (cid:12) , andevery 2 ◦ in both latitude and longitude, fora total of 162 ,
000 starting points and tracedfield lines. The geometry of the magnetic fieldline of the PFSS model passing through eachstarting point is then computed, both out-ward and inward, until its footpoint (1.0 R (cid:12) )and/or the source surface (2.5 R (cid:12) ) is reached.To do so, the first order differential equationsd r/B r = r d θ/B θ = r sin( θ ) d φ/B φ are nu-merically integrated using the PFSS Solarsoftpackage. o of latitude and230 o longitude.The next step is to trace the DEMT resultsalong the computed magnetic field lines. Tothat end, once the field line geometry in highresolution is completed, only one sample pointper tomographic cell is kept, the median one.To each sample point, the DEMT products( N e , T m , and E r ) in the tomographic cell whereit is located are assigned to it, namely the elec-tron density N e , the electron mean tempera-ture T m , and the radiative power E r .Figure 4 shows an example along a closedmagnetic loop. Note how the results are sepa-rated into the two legs of the loop, defined asthe two segments that go from the coronal baseup to the apex of the loop. For each leg sepa-rately, exponential least-square fits are appliedto both the density and the radiative powerdata points as a function of height. Giventheir much less strong variation with height,temperature data points are fitted to a linearfunction. In this case the Theil-Sen estimatorwas preferred over the least-square fit, being more robust to outliers. These are commonin solar rotational tomography results, mainlydue to the effect of unresolved coronal dynam-ics on the assumed static solution of the posedglobal optimization problem.At this point, for every traced magneticfield line, its 3D geometry has been deter-mined, the magnetic field strength along it B ( s ) has been computed, and the radiativepower E r ( s ) and electron mean temperature T m ( s ) have been determined from the fits tothe traced DEMT data. As a result, all quan-tities involved in Equations (11) and (12) canbe now numerically computed. Note that, asthe quantity φ c is sensitive to both the basaltemperature and temperature gradient of theDEMT results (Equations (12) and (6)), itscomputation from the fits to the traced DEMTdata mitigates the effect of its stochasticity,mainly due to unresolved coronal dynamics.Once these quantities are known, the energyinput flux φ h is computed from Equation (10).11 oop shape -70 -60 -50 -40 -30Latitude [Deg]1.021.041.061.081.101.12 r [ R s un ] DEMT Temperature T e [ M K ] F = 0.93F = 0.88
DEMT Electron density N e [ c m - ] r = 0.98r = 0.95 Radiative power E r [ - e r g c m - s - ] r = 0.97r = 0.98 Fig. 4.— DEMT results traced along magnetic loops of the PFSS model. The two legs of the loopare colored red and blue, with diamonds indicating data points. Top-left: loop shape projected inthe radial-latitudinal plane. Top-right: electron density and exponential fits (see text). Bottom-left: electron temperature and linear fits (see text). Bottom-right: radiative power and exponentialfits (see text). T he parameters r and F are explained in the text of Section 4.3. (A color versionof this figure is available in the online journal). 12 .3. Selection of Loops The analysis of results is based on a selec-tion of closed loops for which there are enoughDEMT data points, they are evenly distributedover the range of heights spanned by the loop,and they are fairly described by their respec-tive functional fits, as in the case shown in Fig-ure 4.As the electron density data points exhibitsstrong variations with height, the quality oftheir exponential fit is reasonably measured byits coefficient of determination r . In the caseof the linear fit to the temperature, variationswith height are sensibly smaller. Some loopsmay even be quasi-isothermal, and the coeffi-cient of determination can be nearly zero, evenfor excellent fits, when the temperature gradi-ent is low. Measuring the quality of the linearfit to the temperature based on the coefficientof determination, as done in previous works(Huang et al. 2012; Nuevo et al. 2013), wouldonly select strong enough gradients. Interestedin keeping loops with both strong and weaktemperature gradients, for the present studythese criteria have been modified as describedbelow.A recent work by Lloveras et al. (2017)quantifies the impact of the main sources ofsystematic uncertainty of the DEMT techniqueinto its products. In particular, the character-istic value of the temperature uncertainty is oforder ∼ −
10% (depending on the coronalregion). In order for a loop to be selected foranalysis, the linear fit to the temperature isrequired to match the data within that uncer-tainty for a majority of the data points.Based on the previous considerations, thenumerical selection criteria listed below arebased on actual experimentation with thedata. These criteria aim at maximizing the r ≡ − S res /S tot , where S res is the sum of the squaredresiduals and S tot is the sum of data deviations fromthe mean. sample size, while keeping only those loops forwhich the tomographic data can be fairly de-scribed by the exponential and linear fits tothe electron density and temperature, respec-tively. As shown below, the selected samplesize is larger than in previous studies, andevenly sample the coronal volume covered bythe tomographic technique, resulting in a goodrepresentation of the complete tomographic re-sults. Specifically, to be selected for analysis aclosed loop must meet all following conditions:1. Each leg of the loop must go through atleast five tomographic grid cells with us-able data (i.e. not labeled as ZDA orAEV), and there must be at least onedata point in each third of the range ofheights spanned by the loop.2. The quality of the exponential fit to thedensity is r > .
75 in each leg of theloop.3. The linear fit to the temperature matchesthe DEMT values within their estimatederror for at least a fraction
F > .
75 ofthe data points in each leg of the loop.In Sections 5.1 through 5.3 below, the anal-ysis is performed over all loops that meet thelisted criteria.
5. Energy Flux Results
After analyzing all of the 162 ,
000 tracedfield lines in each rotation, about 54% and60% are closed in CR-2081 and CR-2099, re-spectively. Some closed field lines do not haveenough DEMT data points and/or are not welldistributed, as specified by the first selectionrequirement listed above (Section 4.3), andsome belong to ARs (as discussed in Section4.1). Those loops amount to about 17% ofthe closed field lines in CR-2081, and to about53% for CR-2099 based on EUVI data, or 49%13ased on AIA data. On this remaining popula-tion the 2nd and 3rd selection criteria listed atthe end of Section 4.3 are met by about 40% ofloops for CR-2081, 46% of loops for CR-2099based on EUVI, and 50% of loops for CR-2099based on AIA. The resulting number of loops is ∼ ,
000 for CR-2081, and ∼ ,
000 loops forCR-2099 based on EUVI, or ∼ ,
000 basedon AIA.Visual inspection of the temperature mapof CR-2081 in Figure 2 reveals that in theclosed corona the low-latitudes are character-ized by relatively cooler temperatures, whilemid-latitudes are hotter. This is character-istic of the last solar minimum (V´asquez etal. 2010; Nuevo et al. 2015), as well as theprevious period of minimum activity betweenSCs 22 and 23 (Lloveras et al. 2017). A sim-ilar behavior can be verified in the quiescentclosed corona of CR-2099. In the analysis thatfollows these diverse thermodynamical regionsare separated. To that end, magnetic loopswere discriminated into those with both foot-points within a specific low-latitude range de-fined by | latitude | < ◦ , and those within amid-latitude range defined by | latitude | > ◦ .After applying this selection criteria, the re-maining population is ∼ ,
000 for CR-2081,and ∼ ,
000 or ∼ ,
000 for CR-2099 whenbased on EUVI or AIA data, respectively.
Results for CR-2081 are shown in Figure5, where the left and right panels correspondto the low- and mid-latitude regions, respec-tively. The top panels show the spatial lo-cation of the footpoints of the loops (i.e. at r = 1 . (cid:12) ), while the middle ones show theirlocation at an intermediate height of the tomo-graphic computational grid (at r = 1 .
075 R (cid:12) ).Comparison between top and middle panelsgives a feeling of the divergence of the mag-netic field lines. Loops for which the apex iswithin the range of heights covered by DEMT (dubbed as “small” loops hereafter) are indi-cated in violet color in the on-line version ofthe Figure, while those with a higher apex areindicated in red color (dubbed as “large” loopshereafter). For the selected loops, the bottompanels show the corresponding distribution ofvalues of the loop-integrated quantities φ r , φ c ,and φ h . The total number N of analyzed loopsis shown, along with the median m and stan-dard deviation σ values of each distribution.It is readily seen that the integrated radia-tive loss of the loops, measured by the quan-tity φ r , is larger in the low-latitudes. This ismainly due to the fact that, in the range of sen-sitivity of the EUVI instrument, namely 0.5-3.0 MK (Nuevo et al. 2015, see), the radiativeloss function Λ( T ) used in this work has a localmaximum at T ≈ .
17 and1 .
38 MK, respectively (see Table 1), which ex-plains a larger radiative loss at low-latitudes.This happens despite the fact that the averageloop-length for the low- and mid-latitudes re-gions are 0 .
55 and 0 .
74 R (cid:12) , respectively, whichimplies a larger length-integral in the mid lati-tude loops. Still, most of the coronal radiativeloss occurs at lower heights as E r ∝ N e , whichdecays very rapidly with height.Statistical results for CR-2081 are shown inTable 1, discriminating low- and mid-latitudeloops. For both populations, the table showsthe median value (indicated as (cid:104) (cid:105) ) and thestandard deviation ( σ ) of the flux quantities,as well as of the characteristic DEMT electrondensity ¯ N e and temperature ¯ T m of the loops,where the bar indicates the height-averagedvalue for each loop.While the quantity φ r is defined positive,the conductive flux quantity φ c is not. Notethat low-latitudes are dominated by φ c < φ c >
0. Thesign of φ c is closely related to the temperaturegradient with height. From Equation (6) itcan be easily shown that in magnetic loops for14 S Low closed loops -2 -1 0 1 2 3 4 5 φ [10 erg cm -2 sec -1 ]0.000.050.100.150.20 F r e c uen cy H i s t og r a m m σ .-.- φ r φ c -0.13 0.35__ φ h QS Mid closed loops -1 0 1 2 3 φ [10 erg cm -2 sec -1 ]0.000.020.040.060.080.100.12 F r e c uen cy H i s t og r a m m σ .-.- φ r φ c φ h Fig. 5.— Statistical results of energy flux quantities for CR-2081. Left/right panels correspond tothe low/mid-latitude regions. Top: physical location of the footpoints of the loops ( r = 1 . (cid:12) ).Middle: physical location of the loops at a larger height ( r = 1 .
075 R (cid:12) ). In the top and middlepanels, the violet/red colored dots correspond to small/large loops (see text). Bottom: distributionof values of the loop-integrated quantities φ r , φ c , and φ h . The median m and standard deviation σ values of each distribution is tabulated, and the total number N of analyzed loops is indicated.(A color version of this figure is available in the online journal).Instrument Latitude N tot (cid:10) ¯ N e (cid:11) ( σ ) (cid:10) ¯ T m (cid:11) ( σ ) (cid:104) φ r (cid:105) ( σ ) (cid:104) φ c (cid:105) ( σ ) (cid:104) φ h (cid:105) ( σ )[10 cm − ] [MK] [10 erg cm − sec − ]EUVI Low 9645 0.93 (0.18) 1.17 (0.10) 1.25 (0.83) -0.13 (0.35) 1.11 (0.89)Middle 5861 0.99 (0.17) 1.38 (0.11) 0.84 (0.43) 0.16 (0.22) 1.05 (0.45)Table 1: Global statistics for the CR-2081 results, discriminating low and mid latitudes. For bothpopulations the table shows the sample size N tot , and the median value (indicated as (cid:104) (cid:105) ) andstandard deviation ( σ ) of the height-averaged DEMT electron density ¯ N e and temperature ¯ T m , aswell as of the energy flux quantities φ r , φ c , and φ h .15hich φ c < (cid:12) , have been dubbed as “up”/“down”loops by Huang et al. (2012) and Nuevo etal. (2013), who first observed their presenceby means of DEMT. As speculated by the au-thors of those works, loops of type down can beexpected if the heating deposition is stronglyconfined near the coronal base of a magneticloop. Down loops were first predicted by Serioet al. (1981), and later on by Aschwanden &Schrijver (2002). In a recent study by Schiff &Cranmer (2016), down and up loops have beensuccessfully reproduced by a numerical imple-mentation of a 1D steady state model that con-siders time-averaged heating rates. The anal-ysis of these structures in the context of thenew tool here developed is shown in Section5.3 below.The resulting distributions of the energyinput flux at the coronal base φ h have similarmedian values in both regions, with a smallerstandard deviation in the mid latitudes. Thecharacteristic range of values considering bothregions is φ h ∼ . − . × erg cm − sec − . Figure 6 shows the results for CR-2099based on AIA data, where the left and rightpanels correspond to low- and mid-latitude re-gions, respectively. The same analysis was alsoperformed for this rotation based on EUVIdata, leading to results consistent to thosebased in AIA data. Table 2 details the statis-tical results for CR-2099 based on both datasets. The results obtained with both instru-ments are more consistent in mid-latitudes,due to the similar sample size. In the case oflow latitudes, the EUVI sample size is consid-erably smaller (about half) than for AIA. Dif- ferences in sample size are due to the same se-lection criteria being applied to different datasources, but the precise reasons for which theAIA based analysis is able to retrieve a largerdata sample is not clear. In any case, resultsfrom both data sets lead to similar character-istic distributions.Comparing both rotations, in the low-latitudes the results are similar, being themost notable difference that the distributionof values of the quantity φ c for CR-2099 isnot dominated by negative values as it is forCR-2081. This is consistent with the findingby Nuevo et al. (2013) that down loops areprominent during solar minimum. In the mid-latitudes of CR-2099, φ c is virtually positiveeverywhere, which is consistent with the factthat down loops not only diminish in num-ber with increasing activity but also tend tobe found only at low-latitudes as activity in-creases (Nuevo et al. 2013). It is also to benoted an increase of the characteristic valuesof the input energy flux φ h for CR-2099 com-pared to CR-2081, specially at mid-latitudeswhere it shows a ∼
20% larger median value.This is consistent with the relatively highertemperatures in the mid-latitude regions forCR-2099 (see bottom panels in Figure 2).
As discussed in Section 5.1, the sign of theconductive flux quantity φ c is related to that ofthe temperature gradient with height. Down-loops, i.e., those for which the temperature de-creases with height, are characterized by φ c <
0, while the opposite holds for up-loops. Whilein previous DEMT works (Huang et al. 2012;Nuevo et al. 2013) the loop-selection criteriafocused only on down/up loops, our criteria inSection 4.3 allow selection not only of downand up loops, but also of quasi-isothermalstructures.A loop can be regarded as quasi-isothermal16
S Low closed loops -2 -1 0 1 2 3 4 5 φ [10 erg cm -2 sec -1 ]0.000.050.100.150.20 F r e c uen cy H i s t og r a m m σ .-.- φ r φ c -0.13 0.35__ φ h QS Mid closed loops -1 0 1 2 3 φ [10 erg cm -2 sec -1 ]0.000.020.040.060.080.100.12 F r e c uen cy H i s t og r a m m σ .-.- φ r φ c φ h Fig. 6.— Similar to Figure 5, but for CR-2099 based on DEMT reconstructions from AIA data.(A color version of this figure is available in the online journal).Instrument Latitude N tot (cid:10) ¯ N e (cid:11) ( σ ) (cid:10) ¯ T m (cid:11) ( σ ) (cid:104) φ r (cid:105) ( σ ) (cid:104) φ c (cid:105) ( σ ) (cid:104) φ h (cid:105) ( σ )[10 cm − ] [MK] [10 erg cm − sec − ]EUVI Low 3243 0.89 (0.15) 1.40 (0.16) 1.19 (0.81) 0.15 (0.31) 1.35 (0.83)Middle 14724 0.83 (0.12) 1.60 (0.10) 0.83 (0.35) 0.42 (0.20) 1.31 (0.34)AIA Low 6891 0.92 (0.17) 1.47 (0.11) 1.08 (0.84) 0.03 (0.48) 1.12 (0.94)Middle 15820 0.86 (0.14) 1.61 (0.11) 0.77 (0.33) 0.44 (0.28) 1.27 (0.41)Table 2: Similar to Table 1, but for CR-2099 based on DEMT reconstructions from both EUVI andAIA data, alternatively. 17 R-2081 F r equen cy H i s t og r a m m σ CR-2099 F r equen cy H i s t og r a m m σ Fig. 7.— Histograms of length L of the up and down loops for CR-2081 (left panel) and CR-2099(right panel). The vertical lines indicate bin limits, defined for the study as a function of looplength.in the coronal region studied by DEMT ifthe temperature gradient is weak enoughcompared to the range of coronal heights δr spanned by the loop. More specifically, δr = r max − r base , where r max = r appex forsmall loops, and r max = 1 . R (cid:12) (the maxi-mum height of the tomographic computa-tion ball) for large loops. In this study aloop is then classified as quasi-isothermal if | dT /ds | < ∆( T ) / δr , where dT /ds is the tem-perature gradient of the linear fit to the tem-perature data points, and ∆( T ) is the charac-teristic uncertainty in temperature data pointsdue to systematic errors, which is in the range5 −
10% as shown in Lloveras et al. (2017).Down/up loops can then be separated by re-questing | dT /ds | > ∆( T ) / δr , and further dis-criminated in their two populations accordingto the sign of the gradient.This classification criterion was applied toall closed loops analyzed in Sections 5.1 and5.2 to separate the up and down loops. Theloops were then further classified according totheir length L . For both analyzed rotations,Figure 7 shows histograms of the loop length L of all up and down loops. The vertical linesindicate the limits of five loop length bins setto have a similar sample size within each bin.Tables 3 and 4 show the statistical results ofthe whole population as well as for each bin separately.For the four smaller loop length bins in Ta-ble 3 for CR-2081, Figure 8 shows the statis-tical distributions of all energy flux quantities φ r , φ c , and φ h for the up and down loops only,i.e. without the quasi-isothermal loops. Tosave space the same graph for CR-2099 is notincluded.Note that, after having filtered out thequasi-isothermal loops, all four panels of Fig-ure 8 exhibit a distribution of φ c lacking itspopulation with values ∼
0, as expected. Next,note how the number of down loops, measuredby the area with φ c <
0, decreases with in-creasing mean loop length (cid:104) L (cid:105) . This is quan-titatively measured in Table 3 in the column N down /N tot . The last column in that tableshows the ratio between the half length (cid:104) L/ (cid:105) of the loops with their mean scale height (cid:104) λ N (cid:105) .Note that this ratio increases with increasingloop length, being of order ∼ ∼ φ r increases with loop length18 min L max (cid:104) L (cid:105) N down N tot N down N tot (cid:104) λ N (cid:105) (cid:104) L (cid:105) / (cid:104) λ N (cid:105) L min , L max ] ofthe loop length L . For each range of lengths, the median value of loop length (cid:104) L (cid:105) is tabulated, thenumber of down loops N down , the total number of loops N tot (up and down), the median value ofdensity scale height (cid:104) λ N (cid:105) of loops, and two ratios of discussed in the text. L min L max (cid:104) L (cid:105) N down N tot N down N tot (cid:104) λ N (cid:105) (cid:104) L (cid:105) / (cid:104) λ N (cid:105) : (0.50 - 0.70) Rsun -1 0 1 2 3 φ [10 erg cm -2 sec -1 ]0.000.050.100.15 F r equen cy H i s t og r a m m σ .-.- φ r φ c φ h L: (0.36 - 0.50) Rsun -1 0 1 2 3 φ [10 erg cm -2 sec -1 ]0.000.050.100.15 F r equen cy H i s t og r a m m σ .-.- φ r φ c φ h L: (0.70 - 0.98) Rsun -1 0 1 2 3 φ [10 erg cm -2 sec -1 ]0.000.050.100.150.200.25 F r equen cy H i s t og r a m m σ .-.- φ r φ c φ h Fig. 8.— Statistical distribution of the energy flux quantities φ r , φ c , and φ h of the up and downloops for CR-2081, discriminated in the four different loop length bins defined in Figure 7. (A colorversion of this figure is available in the online journal). L . In this analysis each bin contains a mixof all types of temperature structures (up anddown) and latitudes (low and mid), so thestructures grouped in each panel of the Figure(corresponding to each bin in Table 3) havea similar average temperature. Consequently,the mean radiative loss is the same for all bins,and hence the characteristic value of the radia-tive loss quantity φ r increases with loop length L due to an increasing integral length. Consis-tently, the input energy flux φ h also increaseswith loop length. In the case of CR-2099 downloops are much less prominent, and their ten-dency of a decreasing population with increas-ing loop length is much subtler, though stillverified.For a very marginal population of loops,it is found that φ h <
0, which is an unphysi-cal result. This affects only the smallest loops,as revealed by the analysis of structures as afunction of their size (Figure 8). The radiativeloss term is calculated based on plasma emis- sion detected by the 3 coronal bands of EUVIand the 3 used of AIA. Though this shouldaccount for most of the coronal plasma, theresurely is additional emission that is out of therange of sensitivity of the used instruments.Thus, the radiative loss term φ r is most proba-bly under-estimated which, along with the factthat this is an additive positive term in Equa-tion (9), helps to explain slightly negative val-ues of φ h . Also, the systematic errors of theDEMT technique derived by Lloveras et al.(2017), once propagated into the energy fluxquantities (which will be informed in a futurework) can easily explain the marginal popula-tion of loops with negative values of φ h .Finally, statistical results of all the fluxquantities φ r , φ c , and φ h for CR-2081, dis-criminating up and down loops, are shown inTable 5. Note that (cid:104) φ r (cid:105) is larger for downloops, consistently with their location at lowlatitudes where temperatures are lower. Downloops are also characterized by a negative con-20uctive flux term (cid:104) φ c (cid:105) , which contributes thento the input of energy in the balance Equation(4). Thus, a smaller value of the input energyflux (cid:104) φ h (cid:105) is needed for down loops (comparedto up loops) to compensate for the radiativeflux and keep them stable. As a closing com-ment, note that as this study deals with thecoronal section of magnetic loops, the charac-teristic values of the conductive flux at theirbase are smaller than those of the radiativelosses, because the temperature gradient thereis not as large as in the layers underlying theCorona. Still, the conductive flux plays a partin the balance Equation (10), as shown by thehistograms in Figure 8.
6. Statistical Comparison with a Hy-drodynamic Loop Model
In this section, a first comparison betweenthe results from Section 5 and a theoreticalmodel is carried out, specifically using the 0Dhydrodynamic model
Enthalpy-Based ThermalEvolution of Loops (EBTEL Klimchuk et al.(2008)). Here we highlight the key aspects ofthe comparison, and the reader is referred toMac Cormack et al. (2017) for full details.EBTEL considers the time dependentequation of energy conservation, assuming aconstant area along the magnetic loop, and apiece-wise continuous radiative loss functionΛ E ( T E ) given by Klimchuk et al. (2008). Themodel separates the loop in two segments, onecorresponding to the corona and the other oneto the transition region (TR). By integrating (cid:104) φ r (cid:105) ( σ ) (cid:104) φ c (cid:105) ( σ ) (cid:104) φ h (cid:105) ( σ )up 0.98 (0.64) 0.20 (0.10) 1.21 (0.61)down 1.27 (0.83) -0.29 (0.31) 0.94 (0.93)Table 5: Statistical results of the quanti-ties φ r , φ c , φ h [10 erg cm − sec − ] for CR-2081,discriminating up and down loops. the energy balance equation in each segment,the following relationship between the radia-tion loss in the TR and the downwards con-ductive flux in the corona is obtained, H ≈ − F − φ r,T R (14)where H and F < φ r,T R > • If | F | > φ r,T R the excess conductive fluximplies a positive enthalpy flux, so thatmatter evaporates to the corona, increas-ing its density. • If | F | < φ r,T R there is a deficit of con-ductive flux which results in a negativeenthalpy flux (promoting radiation in theTR), so that matter condensates fromthe corona, decreasing its density.Treating the loop length and its mean coro-nal heating rate as free parameters, EBTELcombines the equations of the corona and theTR to predict the temporal evolution of thepressure ¯ P E , electron temperature ¯ T E and elec-tron density ¯ N E height-averaged along theloop (Klimchuk et al. 2008). For the purposeof the comparison, the values of the two freeparameters are drawn from the DEMT+PFSSresults of the previous section. The averageheating rate is then set equal to the input en-ergy flux divided the loop length, φ h /L , foreach magnetic field line in the model.The comparative analysis was performedfor every loop of the sample corresponding toeach latitudinal region of the two analyzedrotations in the previous section. For eachloop, the ratio between the loop-height aver-age of the DEMT electron density ¯ N e and theEBTEL average density ¯ N E is computed, aswell as the ratio between the two average tem-perature, i.e. ¯ T m / ¯ T E . Table 6 summarizes the21edian value and standard deviation of thesetwo ratios for each of the two latitudinal re-gions in both analyzed rotations.While the DEMT and EBTEL electrontemperatures are similar for all analyzed pop-ulations, the electron densities differ by an av-erage factor of ∼ .
2. This systematic discrep-ancy can be traced down to assumptions inthe EBTEL model. Most importantly, specificphysical scale laws that are used in the model,as well as the characteristic size assumed forthe modeled loops (of the order of their ther-mal scale height), are consistent with observa-tions of AR loops rather than with quiet-Suncoronal structures. Furthermore, the radiationloss function used by EBTEL and the one usedin Section 5 differ in the temperature range ofinterest, which has an impact on the deriveddensities. All these factors build up to explainthe observed systematic difference in densities(Mac Cormack et al. 2017).
7. Discussion and Conclusions
A new DEMT tool was developed that al-lows calculation of the energy input flux φ h required at the coronal base ( r ∼ .
025 R (cid:12) )of magnetic loops of the quiet-Sun corona tosustain hydrostatic thermodynamically stablestructures. First results of applying the toolto two solar rotations (CR-2081 and CR-2099)with different level of activity are shown. Thecharacteristic values obtained are in the range φ h ∼ . − . × (erg sec − cm − ), depend-ing on the particular coronal structure and thelevel of activity of the corona.For CR-2081, a solar minimum rotation,the mid-latitude hotter regions of the streamerbelt and the cooler low-latitude regions exhibitsimilar median values in their distribution ofenergy input flux, with mid-latitudes charac-terized by a considerably smaller standard de-viation. The same characteristics are observedfor CR-2099. For CR-2099, during the early rising phaseof SC 24, the analysis was performed with boththe EUVI and AIA instruments. Results ob-tained with both instruments are highly con-sistent in the mid-latitude region, where thesample size is similar, with a mean value ofthe energy input flux in this region being about ∼
20% larger than during solar minimum. Inthe low latitudes, the results with both instru-ments are somewhat less consistent (thoughstill comparable), being the case that the AIAdata produced a considerably larger popula-tion size. The low-latitude results of CR-2099,based in AIA, show virtually the same energyinput fluxes as the the low latitudes of CR-2081, based on EUVI.The characteristic values of energy inputflux in different sub-regions of the equatorialstreamer belt are related to the presence ofdifferent types of thermodynamic structures,namely the up and down loops first discoveredby Huang et al. (2012) and further studiedby Nuevo et al. (2013). The study here pre-sented added new insight on the characteristicsof down loops, showing that they are charac-terized by smaller values of energy input fluxdue to the extra energy source of heat con-duction, and that their population is larger forsmaller scales (see Table 3).The characteristic values obtained for theenergy input flux φ h in the quiet-Sun coronaare consistent with observational estimatesfor the quiescent corona reported in previ-ous works by Withbroe & Noyes (1977), As-chwanden (2004), and more recently by Hahn& Savin (2014). Based on spectroscopic dataof quiet sun regions taken by the ExtremeUltraviolet Imaging Spectrometer (EIS) at aheight range 1 . − .
20 R (cid:12) , on board the
Hin-ode mission, Hahn & Savin (2014) estimate thenon-thermal component in the observed broad-ening of spectral lines. Assigning the non-thermal line broadening to Alfv´en waves, theyderive the corresponding wave energy flux as a22atitude (cid:10) ¯ N e / ¯ N E (cid:11) ( σ ) (cid:10) ¯ T m / ¯ T E (cid:11) ( σ )CR-2081 Low 2.0 (1.5) 1.1 (0.4)(EUVI) Middle 2.3 (1.3) 1.1 (0.2)CR-2099 Low 2.2 (1.4) 1.1 (0.3)(AIA) Middle 2.4 (1.1) 1.0 (0.2)Table 6: Statistics of the ratio between, a) the loop-height averaged DEMT electron density ¯ N e and the EBTEL average electron density ¯ N E , and b) the loop-height averaged DEMT electrontemperature ¯ T m and the mean EBTEL electron temperature ¯ T E . Values corresponding to low andmid-latitudes for CR-2081 and CR-2099 are discriminated.function of height, estimating the local plasmadensity based on DEM analysis, and the localAlfv´en speed relying on a magnetic potentialextrapolation to estimate the magnetic fieldstrength. Their study includes observationsboth around the equator and mid-latitudes,for which the authors find their respective dis-tributions of wave energy flux at the coronalbase to be in the range ∼ (2 . ± .
4) and ∼ (1 . ± . × (erg sec − cm − ), respec-tively, which compare well to the characteris-tic distributions found in this work (Figures5 and 6). Based on those estimates, a largefraction of the coronal base energy input flux φ h estimated in this work, or even its totality,could be accounted for by Alfv´en waves.Under the assumption that all the energyinput flux is in the form of Alfv´en waves,their quadratic velocity amplitude (cid:10) δv (cid:11) is re-lated to the input energy flux through φ h = ρ (cid:10) δv (cid:11) V A (Moran 2001; Hahn & Savin 2014),where ρ and V A = B/ √ π ρ are the localplasma mass density and Alfv´en speed. Ac-counting for a ∼
8% helium abundance in thecorona, the mass density can be estimated as ρ ≈ . m p N e in terms of the electron den-sity N e and the proton mass m p . Applyingthis relationship to each magnetic field line,using the PFSS and DEMT models to esti-mate B and N e , respectively, at the coronalbase, the range obtained for the energy in-put flux translates into a characteristic Alf´en wave velocity amplitude range (cid:112) (cid:104) δv (cid:105) ∼ −
40 (km / seg). The upper limit corresponds tothe low-latitudes in the streamer belt of CR-2081, and the lower limit to mid-latitudes.This range of Alfv´en wave amplitudes, consis-tent with the quiet-Sun coronal estimates byHahn & Savin (2014), are also in agreementwith characteristic ranges reported in coronalhole studies, see for example Banerjee et al.(2011) and references therein.Based on a 0D HD physical model for coro-nal loops (Klimchuk et al. 2008), we confirmedthat the characteristic values obtained for thecoronal base energy input flux φ h are consis-tent with the height-averaged values of elec-tron temperature and density obtained fromthe DEMT analysis. In a future effort, resultswill be compared to 1D HD models.In a recent work, Nuevo et al. (2015) ex-tended the DEMT technique to also use the335 ˚A coronal band of the AIA instrument,which, combined with the other 3 bands usedin this paper, expands the sensitivity rangeto 0.5-4.0 MK. In that work, the authors finda ubiquitous bimodal coronal LDEM distri-bution, with two distinct characteristic tem-peratures. Their result is interpreted as re-vealing the ubiquitous presence of “warm”( T ∼ . T ∼ . φ h at the coronal base layerof the quiet-Sun closed corona. This will beused as a validation tool at the coronal baselayer of steady-state 3D MHD coronal simula-tions of the Space Weather Modeling Frame-work (SWMF), developed by van der Holst etal. (2014). This will be the subject of a futureeffort.DEMT studies provide a time-averaged de-scription of the state of the corona during theobserving time of each structure ( ∼ / REFERENCES
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