Solar Atmospheric Magnetic Energy Coupling: Broad Plasma Conditions and Spectrum Regimes

5.7 MB·

by  N. Brice Orange

Draft version September 21, 2016
PreprinttypesetusingLATEXstyleemulateapjv.12/16/11
SOLAR ATMOSPHERIC MAGNETIC ENERGY COUPLING: BROAD PLASMA CONDITIONS AND
SPECTRUM REGIMES
N. Brice Orange1,2, David L. Chesny1,3, Bruce Gendre2,4,5, David C. Morris2,4, and Hakeem M. Oluseyi3
1OrangeWaveInnovativeScience,LLC,MoncksCorner,SC29461
2EtelmanObservatory,St. Thomas,UnitedStatesVirginIslands00802
3DepartmentofPhysics&SpaceSciences,FloridaInstituteofTechnology,Melbourne,FL32901
4CollegeofScienceandMath,UniversityofVirginIslands,St. Thomas,UnitedStatesVirginIslands00802and
5LaboratoireARTEMIS,Universit´eCoˆted’Azur,Observatoire´eCoˆted’Azur,CNRS,
boulevarddel’Observatoire,BP34229,F-06304NiceCedex04,France
Draft version September 21, 2016
6
1
ABSTRACT
0
2 Solar variability investigations that include magnetic energy coupling are paramount to solving many
key solar/stellar physics problems, particularly for understanding the temporal variability of mag-
p
netic energy redistribution and heating processes. Using three years of observations from the Solar
e
Dynamics Observatory’s Atmospheric Imaging Assembly and Heliosemic Magnetic Imager; radiative
S
and magnetic fluxes were measured from gross features and at full-disk scales, respectively. Magnetic

0 energycouplinganalysessupportradiativefluxdescriptionsviaaplasmaheatingconnectivityofdom-
2 inant (magnetic) and diffuse components, specifically of the predominantly closed field corona. Our
workshowsthatthisrelationshipfavorsanenergeticredistributionefficiencyacrosslargetemperature

] gradients, and potentially sheds light on the long withstanding issue of diffuse unresolved low corona
R
emission. Theintimacyofmagneticenergyredistributionandplasmaconditionsrevealedbythiswork
S holdssignificantinsightforthefieldofstellarphysics,aswehaveprovidedpossiblemeansforprobing
. distant sources in currently limited and/or undetectable radiation distributions.
h
p
-
o 1. INTRODUCTION been implicated as the source, and origin of the solar
r wind, respectively (e.g., Cranmer 2012; McIntosh et al.
t TheSun’satmosphereexistsintwophases; onethatis
s magneticallyconfinednearthesolarsurfaceandonethat 2013), which emanates from open field magnetic struc-
a tures (e.g., Li et al. 2012). Though investigations have
consists of the extended atmosphere that interfaces with
[ sought the existence of a self-similar magnetically open
andcomprisesthesolarwind. Thesolaratmosphere,ob-
and closed field heating mechanism (e.g., Lee & Magara
6 served on the disk and above the limb, can be divided
v into three distinct regions: active regions (ARs); regions 2014; Che & Goldstein 2014), little support exists for
6 of “quiet” Sun (QS); and coronal holes (CH), i.e., gross such (Klimchuk 2014).
8 feature classes. It has been established that the redis- Key in pinning down a single dominant solar/stellar
9 tribution of magnetic energy appears to dominate the atmospheric heating mechanism of closed magnetic field
2 structures, is the linear relationship of coronal X-ray lu-
heating of the corona (e.g., Klimchuk 2015), but the
0 minositytounsignedmagneticflux(Pevtsovetal.2003).
mechanisms responsible for and heights at which plasma
. Theseresultsaresupportedbyevidenceofself-organized
1 heating occurs remain outstanding puzzles.
criticality (SOC; Bak et al. 1987), where heating events
0 Solar atmospheric heating of plasmas to coronal tem-
resultfromnon-linearprocessesoverbroadspatialscales
6 peratures (logT ≥6.0) is believed to result from the dis-
(e.g., Lu & Hamilton 1991; Oluseyi et al. 1999b). More-
1 sipationofmagneticfree(i.e.,viareconnectionevents)or
: wave energy; i.e., such energy conversion events lead to over, Alvarado-G´omez et al. (2016) have recently pre-
v sented evidence from numerical simulations of the Sun
bundles of nanoflare heated loop strands (Parker 1963).
Xi However, emerging evidence is challenging the standard and other cool main sequence stars of an extension of
the X-ray luminosity to unsigned magnetic flux relation-
r coronal heating model, e.g., fast transition region (TR;
a 4.9≤logT ≤6.0) upflows (Tripathi et al. 2012; Orange shiptotheextremeultra-violet(EUV).Therefore,itisof
distinct interest to investigate if observational evidence
etal.2013),andstronglypeakedactiveregioncoreemis-
supports an extension of linear radiative to magnetic
sion measure distributions (Warren et al. 2012).
coupling descriptions across previously unexplored elec-
Throughout the last few decades, extensive work has
tromagnetic spectrum regimes (i.e., visible, ultra-violet
been carried out on magnetically confined structures
(UV), far UV (FUV), EUV, etc.), as well as tempera-
(e.g.,Aschwanden&Schrijver2002; Spadaroetal.2006;
ture regimes (i.e., photospheric through coronal), mul-
Mackay et al. 2010 Orange et al. 2013; Chesny et al.
tiple epochs of solar activity, and comparisons between
2013). These works, mainly in relation to the corona,
largescaleopenandclosedmagneticfieldstructures(i.e.,
have greatly influenced and enhanced our understand-
CH versus QS, etc.) remains unexplored.
ing of solar atmospheric heating (e.g., Aschwanden &
Constraints on plausible heating mechanism(s) (e.g.,
Nightingale 2005), and revealed that both steady-state
Mandrini et al. 2000) can be ascertained from energetic
(e.g., Winebarger et al. 2011) and impulsive heating
coupling investigations of radiative and magnetic flux
contribute to their generation (e.g., Viall & Klimchuk
(e.g., Fludra & Ireland 2003). That is, observed intensi-
2012). Basal heating of cooler atmospheric layers has
tiesaredependentonthermodynamicdistributions,sub-
2 Orange et al.
sequently governed by heating rates (e.g., Fludra & Ire- photospheric to coronal temperatures at a pixel size of
land 2003; Warren & Winebarger 2006). Importantly, ≈0.6arcsecpixel−1. TheHMIdataareimagesofthefull
the established magnetic field strength’s role in heating disk line-of-sight (LOS) magnetic field with a cadence of
modelsindicatesthatmuchstandstobelearnedofheat- 45 s, and spatial resolution of ≈0.5 arcsec pixel−1. AIA
ing processes, and possibly variations thereof, via gross andHMIpassbandimageswerepre-processedusingstan-
feature class comparison studies, considering that large dard Solar SoftWare (SSW), corrected for solar rotation
thermodynamicgradients(e.g.,O’Dwyeretal.2010)and effects, and co-aligned using the techniques of Orange
starkly differing magnetic field geometries (e.g., Orange et al. (2014a). Note that rotation effects between
et al. 2015) should prevail between these features. passbands were negligible through using observa-
ARs are composed of the hottest and densest plas- tionaltimedifferencesbelowAIA’sthermaljitter
mas (e.g., Del Zanna et al. 2015) across large temper- motion (≈0.(cid:48)(cid:48)3; Aschwanden et al. 2011; Lemen
aturegradients,andhence,arethemostluminousinthe et al. 2012; Orange et al. 2014a), and that the
FUV, EUV, and soft X-ray. Of interest to ARs is that applied alignment technique centered on utilizing
themosthighlyenergetictransientphenomenaintheso- the 1700˚A observations as the fiducial passband
lar atmosphere, e.g., flares (FL), predominantly occur in to which all others were co-registered.
their cores, i.e., “inter moss” regions, where plasma of AIA 193˚A images, per observational date, were used
logT >6.3 resides (Warren et al. 2010; Del Zanna et al.
to select two CH, QS, ARs and ARCs (e.g., Figure 1).
2015) and densities exceed ≈1010 cm−3 (O’Dwyer et al. For each selected feature all AIA passband and HMI
2010, 2011). The heating of ARs and their core (ARC) LOS magnetogram data were aggregated, and the typ-
loops is a matter of much debate. Observations of the ical radiative and unsigned magnetic fluxes measured,
loops favor both low-frequency, i.e., stable high temper- respectively. Errors were propagated using a summa-
ature emission from plasma heating rates much larger tion of photon counting statistics, and the standard er-
than cooling time scales, as well as high-frequency, i.e., ror on the mean. We note here; all investigations of
shallow temperature gradients from heating rates less solar atmospheric thermal to magnetic energy coupling
than cooling time scales (e.g., Tripathi et al. 2011; in this work are carried out via the common approxi-
Winebarger et al. 2013; see references therein). As other mation that energy flux is proportional to “data num-
gross features and cooler atmospheric layers commonly bers” (DNs; e.g., Wolfson et al. 2000; Benevolenskaya
indicate magnetically confined structures far from equi- et al. 2002), i.e., AIA data are not calibrated to physical
librium,i.e.,characterizedbynarrowtemperaturedistri- units. Wealsorecognizethatnoobjectivemethodexists
butions (e.g., Aschwanden & Nightingale 2005; Warren inrelationtoidentifyinggrossfeaturessuchasCHs, QS,
et al. 2008; Hansteen et al. 2014), it is apparent that ARs, and ARCs. As such, spurious feature signal over-
their comparison to ARCs, across large electromagnetic lap should be expected within analyzed radiative distri-
spectrum regimes, are useful for deciphering the nature butions. However, as observed in Figure 2, particularly
of plasma heating and its rates. in the 193˚A panel (i.e., passband used to identify fea-
In relation to the above presentation, and specifically
tures), we find confidence in the implemented selection
toourgoalofseekingapossibleextensionoftheX-rayra-
methodology. Specifically and importantly, in being one
diativetomagneticcouplingdescriptionofPevtsovetal.
which provided radiatively differing gross feature sam-
(2003) across broad electromagnetic spectrum regimes
pleswhosedistributionsalignwithexpectations, i.e., see
in the presence of large open and closed magnetic field
Figure 1 of Pevtsov et al. (2003).
structures,theremainderofthispaperisorganizedasfol-
For each observational date the typical solar disk ra-
lows. Observationaldataprocessingandanalysisofgross
diative and unsigned magnetic fluxes were also charac-
solar atmospheric feature classes (i.e., CHs, QS, ARs,
terized, againwitherrorspropagatedasdescribedprevi-
and ARCs), as well as at full-disk (FD) scales are pre-
ously. Note, solar disk radiative and magnetic flux mea-
sented in Section 2. Within Section 3 we present radia-
surements were derived from a region comprising ≈95%
tive versus magnetic energy measurements (Section 3.1)
ofthevisibledisk(i.e.,seeFigure1),andishereafterare
andtheirlinearenergeticcouplingdescriptions,withand
referredtoasourFDfeature. Additionallywepointout,
withoutfeaturedependence(Section3.2and 3.3,respec-
prior to FD magnetic field characterizations, sunspot re-
tively). Section 3.4 presents a general coronal heating gionsweremasked(i.e.,onlyfluxes(cid:46)|103|Gwereconsid-
theory based on the compilation of our magnetic energy
ered)tominimizedownwardbiasingoferroneousfeature
coupling analyses, and our conclusions are provided in
results in our FD typical magnetic field strengths (e.g.,
Section 4, respectively.
see Warren & Winebarger 2006).
3. ANALYSIS&RESULTS
2. OBSERVATIONS
3.1. Radiative Versus Magnetic Energy
Observational data was obtained from SDO’s Atmo-
sphericImagingAssembly(AIA;Lemenetal. 2012)and In Figure 2 we provide plots of radiative (covering all
Heliosemic Magnetic Imager (HMI; Schou et al. 2012) AIA passbands, with exception of 4500˚A) versus mag-
at approximately 3 – 5 day intervals from May 2010 netic fluxes (from HMI observations) with respect to
through July 2013. AIA data consisted of the follow- our feature set. Hereafter, we note, the terminologies
ing ten passbands: 94˚A, 131˚A, 171˚A, 193˚A, 211˚A, of chromospheric, TR, and corona are used interchange-
304˚A, 335˚A, 1600˚A, 1700˚A, and 4500˚A, which image ablyfor304˚A;131˚Aand171˚A;and193˚A,211˚A,335˚A,
the Sun’s full disk approximately every 12 s, with the and 94˚A passbands, respectively. Additionally, in terms
exception of 4500˚A which observes at a typical cadence of 1600˚A, 1700˚A, and 4500˚A observations these are
of ≈30 min. These bands observe solar plasma from used interchangeably for cooler atmospheric layers (i.e.,
Magnetic Energy Coupling 3
Figure 1. Fromlefttorightandtoptobottom,respectively,HMILOSmagnetogram,andAIA1600˚A,304˚A,131˚A,171˚A,193˚A,211˚A,
335˚A, and 94˚A radiative images, respectively, observed 30 June 2010. Note, on HMI the circle (blue) indicates the region representing
95%ofthesolardiskutilizedtostudythefulldiskfeaturereportedonherein,whileoneachAIAradiativeimageexamplesofeachofthe
othergrossfeatureclassesanalyzedhereinhavebeenidentified.
logT (cid:46)4.8; Lemen et al. 2012). band observations can be assumed to originate from the
It is recognized that AIA passbands are multithermal, expected dominant emission lines.
and arguments exist for cool and hot emission contami- Though not shown, 4500˚A radiative to magnetic field
nation of a number of passbands (e.g., Del Zanna et al. comparisons provide no evidence of a thermal to mag-
2011; Schmelz et al. 2013; Boerner et al. 2014), likely netic coupling: that is, little to negligible variations in
as a function of gross solar atmospheric feature classes its radiative energy occurs for increasing magnetic field
(e.g., O’Dwyer et al. 2010). Important here though, is strengths, independentoffeature. Theseresultsarecon-
thatpredominantlyAIAobservedradiativefluxesderive sistentwiththeexpectedhighβ (i.e.,ratioofgastomag-
from the expected emission line (i.e., Lemen et al. 2012; netic pressure) conditions that should dominate here.
Boerner et al. 2014). However, more careful considera- The results shown in Figure 2 reveal as a function of
tion must be taken for 94˚A and 131˚A results, as spec- analyzedfeaturesandforsolaratmospherictemperatures
tralmodelsinthe50–150˚Arangestillrevealsignificant oflogT (cid:46)4.8(i.e.,1700˚Aand1600˚Aplotstherein),min-
amounts of unaccounted emission, and remain an active imal radiative energy distinctions exist. However, there
area of investigation (Boerner et al. 2014). Outside of is a slight “knee,” at approximately logB∼1.0 where a
flaring conditions, cool emission, originating around 1 blending of the radiative energies observed in CH, QS,
MK, is likely contaminating 94˚A observed fluxes (e.g., AR, ARC (to a lesser degree), and FD occurs. We point
O’Dwyeretal.2010;Boerneretal.2014),whilefor131˚A outthatsuchresultsareexpected, againconsideringthe
hotemissioncontamination(i.e.,Fexxiflarelineformed β(cid:38)1 conditions that should prevail here (Abbett 2007).
In contrast, the ARC results of these regimes are trend-
around logT ≈7.0) dominates in flaring conditions (e.g.,
ing towards a possible linear thermal to magnetic en-
Del Zanna et al. 2011; Del Zanna 2013; Boerner et al.
ergy relationship (e.g., Pevtsov et al. 2003). Thus, rela-
2014). In that respect, to first order approximations,
tivetoothergrossfeatures, enhancedARCphotospheric
that beyond the 94˚A and 131˚A observations, AIA pass-
magnetic field strengths could be leading to frozen-in-
4 Orange et al.
Figure 2. Radiativefluxes(arbitraryunits)versusunsignedmagneticflux(arbitraryunits)forthe1700˚A,1600˚A,304˚A,131˚A,171˚A,
193˚A,211˚A,335˚A,and94˚Apassbands,fromlefttorightandtoptobottom,respectively. OneachplotCH,QS,AR,ARC,andFDregions
are denoted by squares (purple), asterisks (red), x’s (black), pluses (blue), and triangles (orange), respectively. Though small spurious
feature signal overlap exists, these results emphasize the confidence in our selection methodology of generating statistically significant
radiativelydifferinggrossfeaturesamples,whilealigningwithradiativedistributionexpectations(e.g.,Pevtsovetal.2003).
flux conditions (i.e., β<1) at cooler atmospheric lay- vationisconsistentwiththeworkofPevtsovetal.(2003)
ers/heights. for regions dominated by single polarity magnetic fluxes
Inthechromosphereresultsareconsistentwiththeex- (i.e., see their Figure 1). Note, 131˚A and 171˚A pro-
pectationsofalinearradiativetomagneticenergytrend, vide evidence for similarly distributed CH and QS ra-
which scales across the gross feature classes. These re- diative flux distributions, relative to their respective un-
sultsrevealanemergingdistinctionbetweenobservedra- derlying magnetic field strengths and other studied fea-
diances of CH and QS conditions; correlating with simi- tures. Thereby, elevating arguments adopted here that
lar strengths in their underlying magnetic field energies. 131˚AemissionpredominantlyreflectsupperTRregimes
Notethatthekneeidentifiedincoolerpassbandsremains in nonflaring conditions, as expected (e.g., Lemen et al.
distinctly discernable in chromospheric emission. For 2012). The upper TR knee is emerging where a portion
ARs and ARCs, similar observations to the cooler at- ofARCobservationshave“migrated”tohigherenergies,
mospheric layers prevail, i.e., radiative to magnetic flux compared to their AR counterparts. Here we point out,
distributions provide evidence to a linear linking. asobservedinFigure2,131˚Ato171˚A,comparisonspar-
InpassbandsdominatedbyemissionfromTRtemper-
ticularlyofARandARCdistributions, revealsubtledif-
atures, with the possibility of lower and/or upper coro-
ferences that support hot emission contamination of the
nal contributions (i.e., 131˚A; O’Dwyer et al. 2010; Del 131˚A passband. Specifically, 131˚A AR and ARC radia-
Zanna et al. 2011; Schmelz et al. 2013; Boerner et al. tive distributions are more reminiscent to the 211˚A and
2014), the knee structure of cooler atmospheric layers 335˚A passbands.
has “smoothed” out. However TR radiative versus mag-
In the warm corona, described here by AIA’s 193˚A
netic energy distributions give rise to signatures of an
and 211˚A passbands (Lemen et al. 2012; Boerner et al.
ankle and knee. The ankle corresponds to CH condi-
2014)), results are generally similar to those of the TR.
tions, which is distributed downward to lower radiative
TheonlydistinctionofcoronaltoTRobservationsexists
energies than other studied feature classes. This obser-
Magnetic Energy Coupling 5
inacomparisonoftheirCHandQSradiativeenergydis- magnetic free energy (e.g., Parker 1963), and therefore,
tributions. Botharecharacterizedbydecreasedradiative per x, i.e., x ∈ {CH,QS,FD,AR,ARC}, and AIA pass-
energy distributions relative to other analyzed features. band, λ, the following linear equation
The “upper TR – coronal” ARC knee suggest the pos-
sibility of heating not directly attributable to the mag- Fx,λ ∝Bpx,λ, (1)
neticfield. Thisideaalignswithrecentchallengestothe
was fitted to our data to obtain the energetic magnetic
standard coronal heating model interpretation (Parker
to radiative coupling descriptions, i.e., p , in a simi-
1983),particularlyobservationsofinverselyproportional x,λ
lar fashion to the previous works of Golub et al. (1980);
emission measure (EM) to underlying field strengths of
Hara (1996); Fisher et al. (1998); Roald et al. (2000);
ARCs (Warren et al. 2012), as well as the absence of
Wolfson et al. (2000); Schrijver (2001); Benevolenskaya
non-thermal velocity to temperature trends (Brooks &
et al. (2002); Pevtsov et al. (2003); Saar (1996); Vidotto
Warren 2016). Moreover, they provide observational ev-
etal.(2014);Alvarado-G´omezetal.(2016). Featurecoef-
idence in favor of the presence of unresolved emission
ficients(i.e.,p )werederivedfromlinearleast-squarere-
(e.g., Del Zanna & Mason 2003; Viall & Klimchuk 2012; x
gressionfits(MPFIT;Markwardt2009)toEquation1for
Subramanian et al. 2014).
varying physical constraints (detailed below) as a func-
Athottercoronaltemperatures,i.e.,thoseof335˚Aand
tion of λ (i.e., passband; Figure 3), and then, comput-
94˚A observations, with the possibility of cooler emission
ing(cid:104)p (cid:105)(Figure3)usingthebootstrapprocedure(Press
contributions (i.e., 94˚A; O’Dwyer et al. 2010; Boerner x
etal.2002). Therefore,(cid:104)p (cid:105)coefficientswerederivedsim-
x
et al. 2014), similar characteristics prevail to that of the
ilarly to the methodologies of Pevtsov et al. (2003), as
middle corona, and to a lesser degree the TR. In opposi-
they reflect defining power law indices via least-squares
tiontocoolerregions,however,atuppercoronalregimes
and bootstrap procedures, with steps implemented to
theARCkneeappearssmoother. Assuchamoredistinct
provide more realistic coefficients and errors. In relation
linear relation of QS, FD, AR, and ARC features, listed
to latter, this is, they include weighting from statisti-
inaccordancewithincreasingmagneticfieldstrengthsof
calandnonstatisticaluncertaintiesfromvaryingphysical
the respective distributions, is witnessed. We again em-
constraint applications.
phasize;theseresults,andthepreviouslydescribedupper
The differing physical constraints from which least-
TR – coronal ARC trend, further elevate arguments for
squares regression fits were applied, included the follow-
the existence of unresolved coronal emission (e.g., Del
ingdataperturbations: radiativeandmagneticfluxmod-
Zanna & Mason 2003; Viall & Klimchuk 2012; Subra-
ulations (±≤20%); truncation to upper and lower fea-
manian et al. 2014). Recognizing the likelihood of
ture energetic distributions (≤10% of observed data);
cool emission contamination of 94˚A observations (e.g., and random observational data subset sampling. Note
Boerner et al. 2014), the following artifacts are high- that our (cid:104)p (cid:105)’s reflect radiative and magnetic energy
x
lighted. ItsCH,QS,andFDradiativedistributionsqual- truncations, perturbations which aided in simulating
itatively favor such, based on feature radiative flux dis- more fully sampled distributions (Figure 2), their de-
tributionconsistencieswithcoolerpassbands(Figure2). rived power-law slopes include weighting from energetic
In contrast, however, if such notions held for ARs and boundaries expected in the solar atmosphere. Passband
ARCs, their respective distributions would be expected magnetic coupling uncertainties per feature (i.e., p )
x,λ
to exhibit similarities of cooler atmospheric layers, i.e., were defined as the summation of their fit 1σ coefficient
the upper TR – coronal ARC knee, which is clearly not deviations and resultant fit errors (Figure 3). In terms
asdistinctasforexamplein171˚Aor193˚Aobservations. of (cid:104)p (cid:105) uncertainties, they were propagated during pass-
x
A direct comparison of Figure 2’s 94˚A results to Fig- band smoothing via summing the afore described fit en-
ure5bofBenevolenskayaetal.(2002)revealsdistinctive sembledeviationswithsaidsubsetsstandarderroronthe
similarities, particularly our results align with the sug- mean (Figure 3).
gestions of Benevolenskaya et al. (2002) and Fludra & To summarize, our reported coefficients and uncer-
Ireland (2003) for two differing dependencies of radia- taintiesincludeweightingfromnonstatisticalerrorssuch
tive energy versus that of the underlying magnetic field. as instrument sensitivity, varying physical plasma con-
As one progresses to cooler atmospheric layers, i.e., the ditions, and selection bias. We also emphasize that
TR to the chromosphere (171˚A and 304˚A, respectively) previous works (e.g., Pevtsov et al. 2003; Warren &
though,ourresultsarereminiscentofthenotionofalin- Winebarger 2006) have highlighted the importance of
earlinking ofthecoronato magneticfields(i.e., Pevtsov two-side significance from zero measurements as a de-
et al. 2003). In summary, radiative versus magnetic flux termination of the fit qualities over that of the χ2 distri-
presentations(Figure2),withcoverageofthesolaratmo- bution, in radiative to magnetic energy coupling investi-
sphere’s gross features and FD scales across broad spec- gations. Deviationsfromzeroofourmeasuredtwo-sided
trum regimes, qualitatively favor an extension of linear significance between radiative and magnetic fluxes from
magnetic coupling, although with the possibility of ad- Spearmansrankcorrelationcoefficientsforallourpower-
ditional non magnetic heating as revealed by low corona law indices were statistically significant (i.e., s≈0).
observations. Table1givesourcoefficients,aswellasliteraturecoun-
terparts,pl,andrange(i.e.,pl /pl ). Significant
x x,min x,max
3.2. Magnetic Energy Redistribution variations of (cid:104)p (cid:105) exist between the feature coefficients,
x
In this section we investigate the magnetic to radia- as expected, i.e., similar trends prevail for pl subsets. In
tiveenergycouplingofourgrossfeatureclassesandAIA contrast to literature, this work embodies (cid:104)p (cid:105) results
x
passband observations. We employ the assumption that derived from broad electromagnetic spectrum regimes of
observed radiative energy results from the dissipation of the solar atmosphere (Figures 2 and 3). Note, reported
6 Orange et al.
Figure 3. Histogram distributions of Equation 2 derived power-law indices for sample features and passbands. Passband coefficient
ensembles(toprow)reflectvaryingphysicalconstraintapplications(see§3.2),whilefeaturecouplingcoefficients(bottomrow)reflectall
analyzedpassbandsandsubsetsampling. Onthelatterpanels,thesolid(green)verticallinesindicatethefeatures’derivedmagneticenergy
couplingcoefficient(Table1),withuncertaintiesdenotedbyshaded(green)regions.
coefficientanduncertainties,thelatterofwhichissignif- bles pl , considered here to stem from the fact
AR
icantlylargerthantypicalworks(e.g.,Fisheretal.1998; this passband most closely resembles SXT obser-
Wolfsonetal.2000;Benevolenskayaetal.2002;Fludra& vations. The 94˚A and 335˚A ARC coefficients
Ireland 2003; Warren & Winebarger 2006; Warren et al. align well with the 1.6 value reported by War-
2012), are remarkably consistent with the preal data set ren & Winebarger (2006) in a soft X-ray AR
of Pevtsov et al. (2003). We emphasize here that this study, where it was noted that their integrated
datasetwasestimatedviaobjectaveragingofsoftX-ray radiative fluxes were dominated by the bright-
power-law indices under varying physical constraints, in est AR regions. Given this ARC over AR coeffi-
order to provide more realistic errors. cient to pl consistency, we speculate this results
AR
We have included 94˚A and 335˚A results in Ta- from the fact that the bulk of previous works uti-
ble 1, as these passbands’ electromagnetic spec- lizedfluxintegrations,whichpossiblybiasedtheir
trum coverage most closely resemble that de- analysestowardstheAR’smostluminouscompo-
tailed by existing works (i.e., X-ray). First, note nent – the core. Note that we therefore consider
that a comparison of their results to literature our analytic approach of investigating the typical 94˚A
reveal a feature independent alignment, particu- in ARs and ARCs predominantly reflects the passbands
larlyasa“splitting”oftheliteratureminedcoun- expecteddominancebythehotFexxandxxiiiemission
terparts. Thereby, largely, energetic coupling co- lines (i.e., logT ≈7.0; Boerner et al. 2012).
efficient similarities of this work to existing stud- For(cid:104)p (cid:105)results(noteincludingerrors)noreportedval-
x
iesareconfinedtoself-similaranalyzedspectrums uesaregrosslydisproportionatetoexpectations,particu-
(Figure 3). These results importantly elevate support larly, in consideration of the wide literature ranges (e.g.,
of our previous speculations (i.e., § 3.1) for the possi- column 5 of Table 1). Akin to said wide literature vari-
bility of wavelength dependence with magnetic energy ances, similar artifacts prevail in our feature coefficient
coupling and/or plasma heating beyond that available fitensembles, mainly largeuncertaintiesand widecoeffi-
from the photospheric magnetic field. cientensembledistributionsinthepresenceofbroadelec-
From a qualitative standpoint, we previously high- tromagneticspectrumregimes(i.e.,Figure3). Therefore,
lighted that 94˚A AR and ARC radiative distributions wefindfurtherevidencesupportingspeculationsthatra-
favored the expected upper coronal origin (i.e., ≈3 MK; diativetomagneticenergycouplingdescriptionspossibly
Boerner et al. 2012), notions self-consistent with their exhibit a wavelength dependence, particularly as such
derived magnetic coupling coefficients (Table 1). Specif- aligns with the recent numerical simulations of Equa-
ically, its AR result is remarkably similar to the 1.19 tion 1 to cool main sequence stars by Alvarado-G´omez
of Fisher et al. (1998), derived from average Soft X- et al. (2016), for EUV, soft X-ray, and X-ray portions
Ray Telescope (SXT), on Yohkoh Tsuneta et al. (1991), of the spectrum. However, as additionally highlighted
radiative fluxes of 333 ARs. Note that the 335˚A above, it can not be ruled out such observations could
AR power-law index more appropriately resem- be suggestive of the presence of an additional plasma
Magnetic Energy Coupling 7
Table 1
Perfeature(x)studiedinthiswork,wepresentpower-lawindicesderivedfrom: allAIApassbands,(cid:104)px(cid:105);AIA’s94˚Aand335˚A
passbands,(cid:104)p94˚A,x(cid:105)and(cid:104)p335˚A,x(cid:105),respectively;literaturemining,plx;andthesubsequentliteraturereportedrange(i.e.,Min/Max).
Note,reportedpower-lawindiceshavebeenderivedfromfitstoEquation1,andareimmediatelyfollowedbytheir1σ deviations.
Feature(x) (cid:104)px(cid:105) (cid:104)p94˚A,x(cid:105) (cid:104)p335˚A,x(cid:105) plx pl –Min/MaxRange
CH 2.38±0.23 2.01±0.12 2.26±0.13 2.06±0.07†1
QS 1.73±0.73 1.69±0.06 1.91±0.12 1.74±0.21†2 0.93/2.03
AR 1.65±0.30 1.15±0.06 1.47±0.06 1.43±0.40†3 0.98/2.30
ARC 1.81±0.22 1.45±0.09 1.65±0.10 — —
FD 2.05±0.35 1.52±0.05 1.86±0.08 1.73±0.19†4 1.47/2.10
†1Pevtsovetal.(2003)
†2Roaldetal.(2000);Benevolenskayaetal.(2002);Pevtsovetal.(2003)
†3Fisheretal.(1998);Pevtsovetal.(2003);Fludra&Ireland(2003);Warren&Winebarger(2006);Warrenetal.(2012)
†4Wolfsonetal.(2000);Pevtsovetal.(2003)
heating component, possibly married to the dominant Thereby,Equation2describestheobservedradiativeen-
mechanism (e.g., Tan 2014; Uritsky & Davila 2014). ergy via the standard assumption of its linear linkage to
Similarspeculationsonthepresenceofplasmaheating thedissipationoffreemagneticenergy(i.e.,Bpλ)coupled
beyond the standard flare model in magnetic coupling withanadditionalnon-magneticplasmaheatingcompo-
descriptions were presented by Pevtsov et al. (2003) for nentwithwavelengthdependence(i.e.,a ). Equation2’s
λ
their FD indices. They attributed a FD knee and higher lower magnetic energy cutoff (i.e., |B|>10 G) reduces
coupling coefficient (i.e., ≈2.06) to CH open field con- thisanalysestoradiativedistributionsabovetheCHand
tributions, a value emphasized as remarkably consistent QS knee (§ 3.1), a feature particularly prevalent in hot-
with our (cid:104)p (cid:105)≈2.05. Then, of interest is that our FD ter atmospheric layers (Figure 2), previously indicated
FD
radiative fluxes were derived in a manner that sought as possible evidence for a differing dependency of radia-
to minimize sunspots, and thus large open field contri- tiveversusmagneticenergy(Benevolenskayaetal.2002;
butions (e.g., Warren & Winebarger 2006). The 94˚A Fludra & Ireland 2003; Pevtsov et al. 2003 and § 3.1).
FD coefficient of Table 1 aligns with that reported by As observed in Figure 2, those radiative distributions
Pevtsov et al. (2003), when they accounted for large- abovesaidmagneticenergycutofffavorapredominantly
scaleopencontributions,i.e.,1.61. Couplingtheseobser- self-similar radiative to magnetic energy coupling (e.g.,
vations with emerging evidence that cooler atmospheric Benevolenskaya et al. 2002; Pevtsov et al. 2003), and
heights are the origin of large scale open field structures therefore, derivation of feature dependent magnetic en-
(Cranmer2012),aswellasthat94˚Aobservationsshould ergy coupling coefficients are dropped. Below, further
predominantly reflect cool emission contamination (i.e., reasons are provided for invoking a lower magnetic en-
CHsandQS;O’Dwyeretal.2010),weemphasizethefol- ergy cutoff.
lowing. Though our radiative and magnetic energy cou- First, the 10G magnetic energy cutoff reduces unde-
pling study supports an extension of the common solar sired mathematical applications to our data, i.e., largely
atmospheric magnetic coupling description (i.e., Equa- avoidsdiagnosticsonclustereddatasubsets. Itaidsinre-
tion 1; Golub et al. 1980; Hara 1996; Fisher et al. 1998; ducingnonstatisticaleffectsfrominstrumentalnoise,i.e.,
Roald et al. 2000; Wolfson et al. 2000; Schrijver 2001; avoidsinfluencesfromLOSmagneticfluxestypicalofthe
Benevolenskaya et al. 2002; Pevtsov et al. 2003; Saar noiselevel(≈10G;A.Sterling2015;privatecommunica-
1996; Vidotto et al. 2014; Alvarado-G´omez et al. 2016) tion). Note,asobservedinFigure2belowthismagnetic,
across broad electromagnetic spectrum regimes, it has and subsequent radiative boundary our data is a trun-
elevated the necessity of such an investigation seeking cation of a hypothetically complete sample. Therefore,
to elucidate the possibility of additional plasma heating theoreticallyenforcingthislowerlimitcriteriaactstore-
components and/or a potential spectrum dependence. duceourobservationstoamorehypotheticallycomplete
sample. An upper limit energetic stipulation has been
3.3. Magnetic Energy Redistribution – Revisited avoided, given previous observationally and numerically
centered works (e.g., Schrijver et al. 1989; Pevtsov et al.
Here we investigate an extension of the “universal” X-
2003; Vidotto et al. 2014; Alvarado-G´omez et al. 2016)
ray luminosity to unsigned magnetic flux description re-
have established the validity of linear magnetic energy
ported by Pevtsov et al. (2003) (i.e., see their Figure 1),
couplingtootherdistantstellarsources,i.e.,moreradia-
in light of our previous results which suggested the pres-
tively and magnetically energetic.
ence of an additional plasma heating component and/or
Equation 2 coefficients were derived similarly to the
a spectrum dependence (i.e., § 3.1 and 3.2). In that re-
prescriptions presented in § 3.2. First, passband coeffi-
spect, we have modified Equation 1 as follows,
cient ensembles were obtained from application of least-
Fλ ∝aλ+Bpλ, |B|>10 G, (2) square fits across the same varying physical constraints,
perfeature(i.e., FD,AR,andARC).Fromtheseensem-
to include an additional free parameter, a , which ex-
λ bles (cid:104)a (cid:105) and (cid:104)p (cid:105) were then defined by averaging across
hibits no dependence on the photospheric field strength. λ λ
8 Orange et al.
soft X-ray simulations of cool stars (i.e., 2 – 30˚A); and
the universal description of Pevtsov et al. (2003), i.e.,
1.13. Again, consistent with 94˚A results presented in
§’s3.1and3.2theseobservationsareconsideredsupport
of previous arguments that the radiative distributions
investigated here (i.e., those accompanied by underlying
field strengths > 10 G) predominantly reflect emission
from hot coronal temperatures (i.e. ≈3 MK). For AIA
passbands typical of cooler atmospheric layers (i.e., 304,
131, 171, 193, and 211˚A), we determined 0.91±0.16,
a result agreing with Pevtsov et al. (2003)’s reports for
XBPs,QS(noaveraging),anddwarfstars,aswellasthe
X-ray and EUV simulations of Alvarado-G´omez et al.
(2016), i.e., 5 – 100˚A and 100 – 920˚A, respectively. As
highlighted in § 3.1, 131˚A observations, mainly in rela-
tion to the feature radiative distributions focused on in
this analysis (i.e., FD, AR, and ARC), favored hot emis-
sion contamination (e.g., O’Dwyer et al. 2010; Boerner
et al. 2014). However, marginal power-law index vari-
ations in the cooler atmospheric group result if 131˚A
is avoided, while weighting the hotter passbands by its
coefficients, leads to (cid:104)p(cid:105)≈1.15. Thereby, as previously
speculated,131˚Aobservationsfavormulti-thermalemis-
sion,particularlyinrelationtoradiativefluxesassociated
with |B|>10 G. In terms of AIA passbands reflective of
TR and cooler atmospheric regimes, with avoidance of
131˚A, (cid:104)p(cid:105)≈0.65. This aligns quite well with the EUV
coolstarsimulationofAlvarado-G´omezetal.(2016). In-
dependentofpassband,andavoiding1600˚Aand1700˚A,
a power-law index of 1.0±0.2 is measured. This result
is emphasized as support for self-similar plasma heating
Figure 4. Resultant fit parameters (cid:104)p (cid:105) and (cid:104)a (cid:105), top and bot- of the predominantly closed field corona (e.g., Pevtsov
λ λ
tompanels,respectively,derivedfromapplicationofEquation2as et al. 2003), while elevating evidence for an extension to
discussed in the text (see § 3.2), both plotted as a function tem-
cooler atmospheric layers.
perature for the expected passband dominant emission line (e.g.,
Boerneretal.2014). Note,inrelationtosuch,passbandtempera- Figure 4 presents Equation 2 derived coefficients as a
ture uncertainties have been assumed at their expected moderate function of passband, where passbands have been plot-
resolution of 0.3 in logT (Guennou et al. 2012). The exception tedinaccordancewiththeirexpecteddominantemission
is 94˚A and 131˚A, which both have temperature errors appropri- lines formation temperature (e.g., Boerner et al. 2014),
ate to their expected cool and hot emission contamination (e.g.,
and given enhanced uncertainties reflective of their ex-
O’Dwyer et al. 2010; Boerner et al. 2014), discussed at length in
§ 3.3. The shaded gray region on the (cid:104)p (cid:105) plot, corresponds to pectedmoderatetemperatureresolution,i.e.,0.3inlogT
λ
the“universal”power-lawindexreportedbyPevtsovetal. (2003), (Guennou et al. 2012). Furthermore, in line with our
i.e.,p=1.13±0.05. Note,wehavepropagatedthisuniversaltrend abovediscussion,aswellaswiththeresultsof§’s3.1and
across the entire plotted temperature space, but it was derived
from only X-ray observations. To that effect, it potentially high- 3.2, within Figure 4 both 94˚A and 131˚A temperature
lightsasourceofpreviouslyreportedvariabilityinlinearmagnetic uncertainties have been modified to reflect hot and cool
toradiativecouplingstudiesasspectrumdependence. emission contamination. Specifically, we have enhanced
their temperature errors as the difference described by
various feature subsets (i.e., bootstrapping). Again we
O’Dwyeretal.(2010,2012)tothosereportedbyBoerner
point out, as such they reflect results derived from the
et al. (2012). Distinctly interesting to our (cid:104)p (cid:105) results,
same object computed under various constraints, and λ
when observed across large solar atmospheric spectrum
thusprovidemorerealisticassessmentsastheyhavebeen
and thus, temperature regimes (Figure 4), is that it fa-
weighted by statistical and nonstatistical effects. Uncer-
vors a linear correlation, i.e.,
tainties in (cid:104)a (cid:105) and (cid:104)p (cid:105) were propagated as the sum of
λ λ
fit ensemble deviations with the standard error on the (cid:104)p(cid:105)∝Tγ, (3)
mean during subset averaging, and deviations from zero
of two-sided significance for all passband to feature sub- where γ would be a proxy for the efficiency of mag-
samples were statistically significant (i.e., s≈0). netic energy redistribution with temperature. It is em-
For a direct literature comparison, (cid:104)pλ(cid:105) results were phasized that such a functional dependence of the ef-
smoothed across various passband subsets, and are sum- ficiency of magnetic energy deposition with thermody-
marized as follows. For 94˚A and 335˚A passbands, a namic conditions (Figure 4) are results previously spec-
magnetic energy coupling coefficient of (cid:104)p(cid:105)=1.21±0.17 ulated on, e.g., Longcope 1998; Longcope & Kankelborg
was found, a result consistent with: the 1.19 reported 1999. Then, along with the fact that large statistical
by Fisher et al. (1998) in an X-ray study of 333 ARs; samples were utilized in deriving said energetic coupling
the 1.29 of Alvarado-G´omez et al. (2016) obtained from descriptions(e.g.,Rosneretal.1978;Dere1982),wefind
Magnetic Energy Coupling 9
Here we speculate on the source of an additional
Table 2
plasma heating component, as possibly revealed by the
Spearmanrankcorrelationcoefficients(rs)andtwo-sided
significanceofitsdeviationfromzero(s)tabulatedbetweena function form of (cid:104)a (cid:105). That is, it exhibits: a high in-
λ λ
andtypicalAR,QS,andFD(definedhereasaverageofQSand dex (“energy”) like tail, correlating with cooler atmo-
AR).
spheric layers; a TR upturn; an approximate upper TR
to lower corona peak; and thereafter, decreases for in-
Feature rs s creasingly hotter temperatures (Figure 4). Interesting
to this a description is its resemblance to a “typical”
λ
AR 0.69 0.06 solar atmospheric differential emission measure (DEM)
QS 0.75 0.05
distribution, e.g.,
AR/QSAverage 0.85 0.01
dh
DEM(n ,T)=n2 , (4)
e edT
confidence in Equation 3’s validity. Even in the pres-
ence of the moderate temperature resolution provided with h the LOS coordinate and n the electron density
e
bythebulkofAIAchannels,andthelikelihoodofsignif- (e.g., see O’Dwyer et al. 2010), which provide signifi-
icantcool/hotemissioncontaminationfromthe94˚Aand cant insight regarding solar atmospheric thermal struc-
131˚A passbands. Further contributing to these specula- turing. Table 2 presents the Spearmans rank correlation
tions is the upper TR – low coronal dip, “ankle,” wit- coefficients, rs, and two-sided significance of its devia-
nessed in Figure 4’s (cid:104)pλ(cid:105) panel. Specifically, said fea- tion from zero, between aλ results to solar atmospheric
ture is possibly reminiscent to the: expected upper TR QS and AR DEMs obtained from the CHIANTI atomic
peak in current dissipation per particle (Hansteen et al. database(e.g., Dere etal.1997), aswellasanaverageof
2010; Bingert & Peter 2011); switch in EM to underly- those two features. Though Table 2’s correlation coeffi-
ing magnetic field strength dependence (Warren et al. cientsrepresentcrudeapproximations,inrelationtothis
2012); and/or enhanced coronal abundances resultant study they elevate the possible connectivity of aλ to the
from atmospheric energy redistribution processes (e.g., solar atmosphere’s thermodynamic conditions, based on
Bradshaw2003;Laming2004). Possiblymoreimportant their strong correlations. This investigation, therefore,
to the ankle is its remarkable alignment with the atmo- possibly highlights an entanglement of thermodynamic
spheric temperature regime where significant unresolved and magnetic energy contributions in previous energetic
emission is expected to reside (e.g., Del Zanna & Ma- coupling studies (Equation 1), particularly for descrip-
son 2003; Viall & Klimchuk 2012; Subramanian et al. tions of broad plasma conditions (i.e., AR vs QS, etc.)
2014); discussions deferred to the proceeding section. and spectrum regimes (i.e., soft X-ray through UV), as
As the overarching goal of this work remains an investi- our results (Figure 4) present reasonably well the ex-
gation of magnetic energy coupling across broad electro- pected manifestation of diffuse unorganized emission at
magneticspectrumregimes,weappliednomethodologies coronal temperatures (Subramanian et al. 2014).
seeking explicit estimates of temperature to derived en- In Figure 5 the fits of Equation 2 to our data, i.e.,
ergy coupling coefficients. We emphasize such analyses (cid:104)aλ(cid:105) and (cid:104)pλ(cid:105) per AIA passband, are presented. As
would introduce further nonstatistical biases given their observed the fits are consistent with expectations, no-
reliance on systematic calibration assumptions for nar- tions largely confined to X-ray portions of the spectrum
rowband observations. Additionally in support of these (to the best of our knowledge). That is, across broad
arguments,spectralmodelwavelengthrangescorrelating solar atmospheric electromagnetic spectrum regimes ra-
with AIA channels where the bulk of our multi-thermal diative fluxes approximately linearly scale with those of
emission contamination is expected to reside, i.e., 50 – the underlying magnetic field; albeit with varying mag-
170˚A (CHIANTI; Dere et al. 1997), remain deficient netic energy coupling descriptions as a function of the
(Boerner et al. 2014). electromagnetic spectrum (i.e., Figure 4). These results
In terms of the derived (cid:104)a (cid:105) coefficients (Figure 4), we are akin to the numerically derived results of Alvarado-
λ
first note its hot corona results (i.e., 94˚A and 335˚A) G´omez et al. (2016), who reported a wavelength depen-
denceinthemagneticenergyredistributionofcoolmain
possibly explains well that this work’s Equation 1 de-
sequence stars, and whose work further elevates argu-
rivedcoefficientsimilaritieswithliteraturewerepredom-
ments presented here for Equation 3 (Figure 4). It is
inantly constrained to self-consistent spectrum regimes,
recognized, energetic scatter about Equation 2 descrip-
i.e., Figure 4 favors a less significant role of a in these
λ tions remain in Figure 5, predominantly correlated with
passbands. For cooler atmospheric layers, mainly where
ARCs. An artifact of interest as it aligns well with spec-
β(cid:38)1conditionsshouldprevail(i.e.,1600˚Aand1700˚A),
trumregimesfavoredbynumerousARCworksasthelo-
(cid:104)a (cid:105) coefficients support expectations of plasma heat-
λ cationofsignificantunresolvedemission(e.g.,DelZanna
ing not dominated by the magnetic field, i.e., consis-
& Mason 2003; Viall & Klimchuk 2012; Subramanian
tent with their marginal magnetic energy coupling co-
et al. 2014; Alvarado-G´omez et al. 2016).
efficients compared to other AIA channels. At tempera-
ture regimes intermediate to the afore described results,
3.4. Coronal Heating
correlating with the highlighted (cid:104)p(cid:105) ankle, an increased
contribution to observed radiative fluxes from (cid:104)a (cid:105) oc- First,recallan(cid:104)p (cid:105)anklewashighlightedin§3.3that
λ λ
curs. Therefore, indicating as hypothesized in §’s 3.1 correlated well with spectral and temperature regimes
and 3.2, a possible rise in plasma heating contributions favored as the location of significant unresolved coro-
not directly attributable to the freely available magnetic nal emission (e.g., logT ≈6.0 – 6.5; Del Zanna & Ma-
energy, however, likely intimately linked to it. son 2003; Viall & Klimchuk 2012; Subramanian et al.
10 Orange et al.
Figure 5. Same as Figure 2, except observational data <10G have been avoided (i.e., § 3.3. Note, the shaded region represents the fit
ofEquation2toourresults,whichisextensivelydiscussedinthetext,anditsrespectiveparameterspacesarepresentedinFigure4.
2014), and aligned with a “bump” in our (cid:104)a (cid:105) distri- ofwhichexperienceslocalheatingviathefreelyavailable
λ
bution (Figure 4). Hereafter the feature is referred to magnetic energy (i.e., H , H , and H , respec-
cool warm hot
as logT , and we hypothesize that as Equation 3 favors tively; Equation 3). Within the cool layer, atmo-
w
that across the bulk of solar atmospheric temperature spheric heating, via chromospheric evaporation,
spacetheefficiencyofmagneticenergyredistributionap- E (e.g.,Fisheretal.1985; Craig&McClymont1986;
cool
proximatelylinearlyscales. Asthecompilationofresults Hansteen et al. 2010), would contribute to warm en-
presented in this work support, logT is considered to hanced plasma emission. Additionally, under the stan-
w
correlate with upper TR – low corona temperatures. To dard coronal heating picture (e.g., Oluseyi et al. 1999a;
supportsuchassumptions,wepointoutourEquation3’s Oluseyietal.1999b),downwardconductedhotlayerheat
alignment with previously established linear correlations flux (C ) would provide a source of radiatively bright
hot
of temperature to EM distributions (e.g., Warren et al. warm emission. Assuming for simplicity that heated
2012; Subramanian et al. 2014; Del Zanna et al. evaporating plasma (E) and conduction (C) processes
2015), and pressure and loop length (e.g., Rosner et al. represent a portion (δ) of local layer heating, we arrive
1978; Kano & Tsuneta 1995). Thereby, it is necessary at a total warm heating (Ht ) given by
warm
here to explain the “obscured” logT radiative observa-
tionsinourmagneticcouplingdescriwptions,whereobser- Hwtarm ≈Hwarm+δHcool+δHhot. (5)
vationalevidenceindicatesthat“cool”plasmaexcessex-
Using similar arguments the total cool and hot heating
ists (i.e., Figure 4). Below we present a generalized
would be described as
coronal heating theory, that centers on the domi-
nant energy sources, i.e., magnetic, enthalpy, and Ht ≈H +δH , (6)
cool cool warm
thermal conduction (Bradshaw & Cargill 2010a).
Consider a generalized and hypothetical solar at- and
mosphere segmented into cool (logT (cid:46)logTw), warm Hhtot ≈Hhot+δHwarm, (7)
(logT ), and hot (logT (cid:38)logT ) layers (Figure 6), each
w w respectively(Figure6). Inotherwords,warmconduction