Magnetohydrodynamic wave propagation from the subphotosphere to the corona in an arcade

13 Pages·2015·5.93 MB·English

A&A577,A70(2015) Astronomy
DOI:10.1051/0004-6361/201424701 &
(cid:2)c ESO2015 Astrophysics
Magnetohydrodynamic wave propagation
from the subphotosphere to the corona
in an arcade-shaped magnetic field with a null point(cid:2)
I.C.Santamaria1,2,E.Khomenko1,2,3,andM.Collados1,2
1 InstitutodeAstrofísicadeCanarias,38205LaLaguna,Tenerife,Spain
e-mail:[email protected]
2 DepartamentodeAstrofísica,UniversidaddeLaLaguna,38205LaLaguna,Tenerife,Spain
3 MainAstronomicalObservatory,NAS,03680Kyiv,Ukraine
Received29July2014/Accepted26February2015
ABSTRACT
Aims.TheaimofthisworkistostudytheenergytransportbymeansofMagnetohydrodynamic(MHD)wavespropagatinginquiet-
Sunmagnetictopologyfromlayersbelowthesurfacetothecorona.Upwardlypropagatingwavesfindobstacles,suchastheequipar-
titionlayer withplasmaβ = 1, thetransitionregion, andnull points, and theyget transmitted, converted, reflected, andrefracted.
UnderstandingthemechanismsbywhichMHDwavescanreachthecoronacangiveusinformationaboutthesolaratmosphereand
themagneticstructures.
Methods.Wecarriedouttwo-dimensionalnumericalsimulationsofwavepropagationinamagneticfieldstructurethatconsistsof
twoverticalfluxtubeswiththesamepolarityseparatedbyanarcade-shapedmagneticfield.Thisconfigurationcontainsanullpoint
inthecorona,whichsignificantlymodifiesthebehaviorofthewavesastheypassnearit.
Results.Wedescribeindetailthewavepropagationthroughtheatmosphereunderdifferentdrivingconditions.Wealsopresentthe
spatialdistributionofthemeanacousticandmagneticenergyfluxesforthecaseswherethesecalculationsarepossible,aswellasthe
spatialdistributionofthedominantfrequenciesinthewholedomain.
Conclusions.We conclude that the energy reaches the corona preferably along almost vertical magnetic fields, that is, inside the
vertical flux tubes. This energy is acoustic in nature. Most of the magnetic energy stays concentrated below the transition region
owingtotherefractionofthemagneticwavesandthecontinuousconversionofacoustic-likewavesintofastmagneticwavesinthe
equipartition layer locatedinthe photosphere whereplasma β = 1.However, part of themagnetic energy reaches thelow corona
whenpropagatingintheregionwherethearcadesarelocated,butwavesaresentbackdownwardintotheloweratmosphereatthe
null-pointsurroundings.Thisphenomenon,togetherwiththereflectionandrefractionofwavesintheTRandthelowerturningpoint,
actasare-feedingoftheatmosphere,whichkeepsoscillatingduringallthesimulationtimeevenifadriverwithasinglepulsewas
usedasinitialperturbation.Inthefrequencydistribution,wefindthathighfrequencywavescanreachthecoronaoutsidethevertical
fluxtubes.
Keywords.magnetohydrodynamics(MHD)–waves–magneticfields–Sun:photosphere–Sun:chromosphere–Sun:corona
1. Introduction concentrationsembeddedin the intergranularlanes suffer foot-
pointmotions,andthisdriveswavesinthesestructuresthrough
Quiet-Sun magnetic structures are much less organized and
the solar atmosphere (Hasan et al. 2000; Kato et al. 2011).
have a less intense magnetic field compared to sunspots Dependingon the foot-pointmotion,differentwaves are going
(SánchezAlmeida & Martínez González 2011). At the same
to be excited. There is a large number of works on numerical
time, the quiet Sun covers at least about 90% of the solar sur- simulations that reproduce these different scenaria: horizon-
face (Sheeley 1967; Lites et al. 1996; Lin & Rimmele 1999;
tal motions producingtransverse kink waves; pressure fluctua-
Khomenkoetal.2003;TrujilloBuenoetal.2004)independently
tionsproducinglongitudinalwaves;twistingmotionsgenerating
of the activity level, which makes quiet-Sun magnetic struc-
torsional Alfvén waves; observationally-driven or impulsive
tures become important contributors to the solar magnetism.
motions (Ulmschneider et al. 1991; Choudhuri et al. 1993;
Theseweakstructuresareverydynamicandareconstantlyper-
Rosenthal et al. 2002; Bogdan et al. 2003; Hasan et al. 2003,
turbed by convection and propagatingwaves (see Muller et al.
2005; DePontieu et al. 2004, 2005; Khomenko et al. 2008;
1994; Berger & Title 1996; Krijger et al. 2001; DePontieu
Vigeesh et al. 2009;Hegglandet al. 2009;Khomenko& Cally
et al. 2003; Vecchio et al. 2007, 2009; Centeno et al. 2009;
2011; Fedun et al. 2011b,a; Nutto et al. 2012). With some ex-
Fujimura & Tsuneta 2009; Kontogiannis et al. 2010a,b, 2011;
ceptions(Choudhurietal. 1993;De Pontieuetal.2005;Fedun
MartínezGonzálezetal.2011;Chittaetal.2012,tonameafew).
et al. 2011a), these works model the wave propagation in the
Waves are stochastically excited by turbulent convection
photosphericandchromosphericlayers.
at the base of the photosphere (Goldreich & Keeley 1977;
The dominant oscillations observed in the quiet-Sun pho-
Balmforth 1992; Nordlund & Stein 2001). The magnetic field
tosphere are acoustic-gravity waves with periods of five min-
(cid:3) MoviesrelatedtoFigs.3,7,and11areavailableinelectronicform utes (Leightonet al. 1962;Ulrich 1970).As these waves prop-
athttp://www.aanda.org agate upward, the frequency distribution changes. Essentially
ArticlepublishedbyEDPSciences A70,page1of13
A&A577,A70(2015)
three-minute period waves are observed in the quiet chromo- the frequencydistribution of waves with height and horizontal
sphereinsidesupergranularlanes(Deubner&Fleck1990;Lou distance.
1995; Hoekzema & Rutten 1998; Rutten & Uitenbroek 1991;
Litesetal.1993),andfive-minuteoscillationsaredetectedinand
aroundmagnetic elements in facular regions (DePontieu et al. 2. Numericalmethod
2004,2005;Hansteenetal.2006;Jefferiesetal.2006;McIntosh
& Jefferies 2006; Centeno et al. 2006; Kostik & Khomenko Wesolvethe2Didealmagnetohydrodynamic(MHD)equations
ofconservationofmass, momentum,energy,andthe induction
2013) and network regions (Lites et al. 1993; Krijger et al.
equationforthemagneticfield:
2001;DePontieuetal.2003;Bloomfieldetal.2006;Tritschler
et al. 2007; Vecchio et al. 2007), with some differences in be- (cid:2) (cid:3)
∂ρ ∂ρ
havior (see recent review by Khomenko & Calvo Santamaria +∇(ρu)= , (1)
∂t ∂t
2013). Both three- and five-minute oscillations appear in the (cid:2) (cid:4) diff (cid:5) (cid:3) (cid:2) (cid:3)
corona (De Moortel & Nakariakov 2012). The distribution of ∂(ρu) B2 BB ∂(ρu)
the dominant frequency of oscillations along the atmosphere +∇ ρuu+ p+ I− =ρg+ , (2)
∂t 2μ μ ∂t
appears to be different inside the quiet network cells, network (cid:2) (cid:3)0 0 diff
borders, and facular regions, which contain stronger magnetic ∂p +u∇p+γp∇u= ∂p , (3)
fields. There are observations of the five-minute waves chan- ∂t ∂t
(cid:2) (cid:3)diff
nelled along the inclined magnetic field lines to the chromo-
∂B ∂B
sphere in facular regions (via the so-called “ramp-effect” or =∇×(u×B)+ , (4)
∂t ∂t
“magneto-acoustic-portals”),whichhavebeenexplainedbythe diff
reductionoftheacousticcut-offfrequencyininclinedmagnetic
where all the notations are standard. We use an ideal equation
fields (Michalitsanos 1973; Suematsu 1990; De Pontieu et al. of state and γ = 5/3. The terms with subscript “diff” are the
2005;Jefferiesetal.2006;Hegglandetal.2007,2011).Centeno artificialhyper-diffusivetermsrequiredforthecode’snumerical
et al. (2009) and Khomenko et al. (2008) have proposed that
stability (see Felipe et al. 2010a). After removing the equilib-
radiativelossesin thinfluxtubescanalsohelppropagatelong-
riumcondition(seebelow),wesolvethesenonlinearequations
period oscillations vertically upward (see Roberts 1983). This by meansof the codemancha, whichis describedin detail in
mechanism,however,wasnotfoundtobedominantinthesim-
Khomenko&Collados(2006)andFelipeetal.(2010a).Thenu-
ulationsbyHegglandetal.(2011),wheretheauthorsuseacom-
mericalcodesolvestheequationsforperturbations.Throughout
plextreatmentofradiativetransfer.
the present study, we keep the amplitude of the perturbations
Asthewavespropagateupward,theyfindobstacles,suchas smallenoughtoconcentrateonthelinearregime.Thenonlinear
theequipartitionlayerwhereplasmaβ = 1,thelayerwherethe effectswillbereportedinaforthcomingstudy.
localacousticcut-offfrequencyisequaltothewavefrequency,
the steep temperature gradient at the transition region, etc. In
2.1.MHSmodel
these regions waves suffer transformations, refraction, and re-
flection. Since the atmospheric properties change as we move Forsimplicity,wehavechosenapotentialmagneticfieldstruc-
fromonelayertoanother,andevenfromonecolumntoanother, ture.Suchamagneticfielddoesnotinteractwiththeequilibrium
these effects can appear at different horizontaland vertical po- atmosphere.Thisallows usto computethe hydrostaticequilib-
sitions. For instance, while the gas pressure dominates in the riummodelseparatelyfromthemagneto-staticequilibrium.
photosphere,themagneticpressurebecomesdominantfromthe
middlechromosphereupward.
2.1.1. Hydrostaticmodel
Particularlyinterestingarethequestionsofhow,andinwhat
amount, the energy of waves excited in subphotospheric lay- To obtain the hydrostatic equilibrium, we start by defining the
temperature structure as a function of the vertical coordinate
ers can reach the chromosphere and corona and whether the
wave modes reaching there can be dissipated to convert their z = {−5,10} Mm. For that, we join three models. The sub-
energyintoheat,i.e.,howthesolaratmosphericlayersaremag- photosphericlayersaredescribedbytheconvectivelystabilized
modelbyParchevsky&Kosovichev(2007).Themainreasonfor
neticallyandenergeticallyconnectedbymeansofwaves.Only
a few numerical studies simulate waves in the whole domain choosingthismodelisthatnoconvectiveinstabilityisdeveloped
fromthephotospheretothechromosphere,transitionregion,and inresponsetotheperturbation,allowingwavestobestudiedsep-
aratelyfromconvection(seeKhomenkoetal.2009).Thephoto-
corona.In their 2D simulationsof high-frequencywave propa-
gation,Fedunetal.(2011a)findthatmagneto-acousticwavesin sphereandchromospherearedescribedbytheVALCmodelfor
awiderangeoffrequenciescanleakenergyeffectivelyintothe quietSun regions(Vernazzaet al. 1981).Finally we choosean
isothermalcoronawithatemperatureofonemillionKelvin.The
corona.Theseauthorsconsideredanisolatedmagneticfluxtube
thatbecomesverticalinthechromosphere.Inthepresentwork, complete temperaturedistribution is shown in Fig. 1. Once the
westudyhowtheenergyreachesthechromosphereandcorona temperature stratification is established, we calculate the scale
byexcitingwavesbelowthephotospherewithdifferentperturba- height H, and then integrate the equation for pressure and re-
tions.Wechoseacomplexmagneticfieldthatconsistsofvertical coverthedensitydistributionsas
fluxtubeswiththesamepolarityseparatedbyanarcade-shaped
R T
magneticfield,whichresemblesanetworkandinternetworkre- H = gas 0 (5)
gions,withwaveperiodicitiesinthe3−5minregime.Wediscuss gμ
indetailthebehaviorofdifferentwavemodesinsuchacomplex dp (z) p
0 + 0 =0 (6)
situation. The length of our simulated time series allows us to dz H
reachthestationaryregimeintwoofthethreeconsideredcases p
ρ = 0 (7)
and to quantify the energyflux reaching the corona,as well as 0 gH
A70,page2of13
I.C.Santamariaetal.:Wavepropagationfromthesubphotospheretothecorona
Pressure
Density
Temperarure
Atmospheric Stratification
10 −4
10 1.5×10
)
K
(
e
r 8
u 10
at
er 2.3×10−7
p
em 106 3m]
T c
d g/
an 104 3.5×10−10 y [
2m) sit
n
c e
n/ 102 D
y
d
e ( 5.4×10−13
sur 100 Fig.1. Temperature (green line), density
s
e (blue line), and pressure (red line) distri-
Pr butions. The left vertical axis gives us the
−2 −16
10 8.3×10 values for temperature and pressure, and
−4 −2 0 2 4 6 8 10
theverticalaxisontherightisthedensity
Height [Mm] scale.
usingtheidealgasequation.Withthis,weensurethattheatmo-
sphereisin hydrostaticequilibrium.Theresultingdistributions
ofpressureanddensityaregiveninFig. 1.Aplane-parallelat-
mosphereis built by replicatingthe obtainedverticalstratifica-
tions. In the above equations, we denote the equilibrium vari-
ablesbyasubscript“0”;R isthegasconstantandμthemean
gas
atomic weight. We set μ = 1 at the lower layers of the atmo-
sphere where we can consider that the gas is made of neutral
hydrogen,andμ=1/2athigherlayerswheretheplasmaisfully
ionized.
2.1.2. Magnetostaticmodel
Themagnetostaticmodelisgivenbyanarcade-shapedmagnetic
fieldplusaconstantcomponent:
B = B exp(−kz)sin(kx) (8)
0x 00
B = B +B exp(−kz)cos(kx) (9)
0z u 00
where B and B arethemagneticfieldstrengthsinthephoto-
00 u
sphereandcorona,respectively,kdefinesthespatial(horizontal Fig.2.Magneticfieldstrengthdistributionfrombelowthephotosphere
andvertical)scales,andxandzarethehorizontalandverticaldi- (−5Mm)tothecorona(10Mm).Theverticalaxisgivestheheightin
rections,respectively.WechooseB =100GandB =10G,so Mm, and the horizontal axis gives the horizontal size in Mm. Height
00 u z = 0 represents the base of the photosphere. The red lines are the
weareworkingwithaweakmagneticfieldregion.Equations(8)
magneticfieldlines,andtheyellowdashedlinesaretheβ=1contours
defines a magnetic field configuration composed by two verti-
(oneonthephotosphereandtheotheronearoundthenullpoint).
calflux tubesseparatedby an arcade-shapedmagneticfield, as
drawninFig.2.Choosinghighervaluesforthephotosphericand
coronalmagneticfield has the disadvantageof significantlyre- Table1. Alfvénandsoundspeedsinthedifferentlayersoftheequi-
ducingthecomputationaltimestepsduetothehighvaluesofthe libriumatmosphere.
Alfvénspeedinthecomputationaldomain.HighervaluesofB
00
and B wouldlowertheeventualpositionoftheβ = 1contour, Photosphere Chromosphere Corona
u
butwouldnotchangetheoverallpictureofthewavepropagation c 8kms−1 10kms−1 100kms−1
s0
and conversion.Table 1 givessoundandAlfvén speedsfor the v 2kms−1 100kms−1 1000kms−1
differentlayersintheequilibriumatmosphere. a0
Ourfieldconfigurationincludesanullpointinthecorona.In
thispointthemagneticfieldstrengthismathematicallyzeroand
theplasmaβgoestoinfinity.Thisnullpointchangesthebehav- the magnetic field strength is zero, the Alfvén speed is null,
iorofthe wavesdrasticallywhile theypropagatenearit. When thereforeonlytheacousticwavescanphysicallypassthroughit.
A70,page3of13
A&A577,A70(2015)
Table2. Summaryofthesimulationruns.
Section Driving Amplitude Period dx/dz/duration Figures
3.1 vertical,harmonic 10−2ms−1 200s 75km/50km/3000s 3–6
3.2 horizontal,harmonic 0.2ms−1 300s 75km/50km/3000s 7–10
3.3 pressurepulse 10−5p − 75km/50km/3000s 11–13
0
2.2.Simulationsetup both definitions is a factor γ/2, which in our case is given by
5/6.Thisvalueisclosetounityand,therefore,almostimpercep-
We analyzed three simulation runs, performed with the same tibleinthefigureorinwaveeffects.Weusethelatterdefinition
modelfortheequilibriumatmosphere,butwithadifferentdriv- throughout the paper. Transformation of different wave modes
ing of the waves. The cases include:(1) verticaldriving,when canoccurinthelayerwithβ≈1(equipartitionlayer).Sinceour
theanalyticalsolutionforanacousticwavewithaperiodof200s
simulations are 2D, the only possible wave modes are fast and
(fast mode) is applied at the lower boundary of the simulation
slow magneto-acoustic-gravity modes, so we exclude the con-
domain;(2)horizontaldriving,whentheanalyticalsolutionfor
versiontoAlfvénmodefromourcurrentanalysis.
theslowmagneticwavewithaperiodof300sisappliedatthe
The propagation of the fast and slow magneto-acoustic-
lowerboundary;(3)instantaneouspressurepulsebelowthepho-
gravitymodesisingeneralrathercomplex,exceptincaseswith
tosphere.Table2givesasummaryfortheseruns,includingthe
verysimplemagnetictopologies.Underamathematicalpointof
sectionandthefigureswheretheresultsarepresented.Wechose
view, the analysis of the behavior of waves comes in a natural
theperiodicitiesof200and300stostudythebehaviorofwaves
way if done accordingto the fast or slow nature of their prop-
with frequenciesbelowand abovethe cut-off frequency,which
agation speed, regardless of their acoustic or magnetic proper-
istypicalofthesolarconditions.Thecut-offfrequencylayerfor
ties.However,fortheinterpretationofobservations,itisingen-
300s periodwavesis located belowthe photosphere,thosefor
eralmoreconvenienttoconsiderthephysicalnatureofthemain
200swavesisatthetemperatureminimum.
restoringforce.Itisunfortunatethatthetermswavetransmission
and conversion have different meanings in these two descrip-
tions. Since this paper focuses on interpreting numerical sim-
2.3.Boundaryconditions
ulations, we preferred to follow the physical convention. This
To avoid spurious reflections of waves at the upper bound- way,weconsiderthatawaveis transmittedwhenthenatureof
aries of the domain, we add a perfectly matched layers (PML itsmainrestoringforcedoesnotchangewhencrossingagiven
Berenger 1994) of 20 grid points; see Felipe et al. (2010b)for layer,keepingitsmainlyacousticormagneticnature(indepen-
details. The PML is especially designed to absorb the pertur- dentlyofwhetherinthemathematicaldescriptionitmaychange
bations that reach the borders of the domain and was shown fromfasttosloworviceversa).Similarly,weusethetermwave
to successfully performin the simulationsof MHD waves (see conversionwhenthenatureofthemainrestoringforcechanges
Parchevsky&Kosovichev2008;Khomenkoetal.2008;Felipe fromacousticormagnetictomagneticoracoustic(keepingthe
et al. 2010b; Hanasoge et al. 2010). In the simulations with a fastorslowcharacterorthewave).
periodicdriver,thelowerboundaryisreplacedbytheanalytical Dependingonthemagneticfieldinclination,theconversion
solutionforagiventypeofwave.Inthesimulationswithanin- andtransmissionareusuallyonlypartial(Zhugzhda&Dzhalilov
stantaneous pulse, the lower boundary contains the PML layer 1982;Cally2005,2006;Schunker&Cally2006;Khomenko&
aswell.Boundaryconditionsforperiodicvariationsareusedin Collados 2006; Khomenko & Cally 2012), so that part of the
thehorizontaldirection. wavekeepsitsacousticormagneticnatureandpartofitchanges.
Thetransmissionefficiencydependsontheattackangle,i.e.,the
anglebetweenthemagneticfieldlinesandthewavevectorinthe
3. Drivingwaveswithlinearperturbations layerofβ=1(Cally2006):
(cid:2) (cid:3)
Thissectionpresentstheresultsofthedifferentsimulationruns. Kπsin2α
Eventhoughthecodecanhandlenonlinearities,theamplitudeof T =exp −|(d/ds)(c2/v2)| , (10)
the perturbation is sufficiently small and the wave propagation s a cs=va
is kept in the linear regime in all the cases, so no shocks are whereαistheattackangleandsthedistancealongthedirection
formed.Thisisdoneonpurposebecauseourintentionistostudy of kandK =|k|.c /v istheratioofthespeedofsoundandthe
s a
the wave propagation, conversion, refraction, and reflection in Alfvénspeed.Thetransmissioniscompletewhenthemagnetic
thelinearregime.Forabetterunderstandingoftheresultsofthe fieldandthedirectionofthewavepropagationarealigned,α =
simulations, we briefly summarize the nomenclature and some 0.Inthiscase, theconversioncoefficient,1−T, iszero.When
basicsofthemodeconversiontheory. α(cid:2)0,thenthetransmissiondecreaseswithincreasingk,thatis,
withincreasingwavefrequency.
3.1.Wavetransformation
3.2.Verticalperiodicdriverof200s
Figure2showstheβ=1contoursinourequilibriummodel,one
inthephotosphereandanotheraroundthenullpoint.Theplasma Inthisrunwedrivewavesbyapplyingaverticalperiodicpertur-
βgivesustheinformationofwhethertheatmosphereismagneti- bationinvelocityandthermodynamicvariablesatthebottomof
callydominatedornot.Strictlyspeaking,theplasmaβisdefined thedomainandlettheperturbationpropagateforatotalduration
asβ = P /P ,butfromthepointofviewofthewaveprop- of3000s. Thisis longenoughforthe wavesto fillthe fulldo-
gas mag
agation, it is more convenientto define it as a squared ratio of mainandbecomestationary.Thedirectionofpropagationofthe
thecharacteristicspeeds;i.e.,β=c2/v2.Thedifferencebetween perturbationin this case is nearlyparallelto the magnetic field
s a
A70,page4of13
I.C.Santamariaetal.:Wavepropagationfromthesubphotospheretothecorona
lines, so acoustic-like fast-mode waves are mostly generated.
vlong vtrans t=210.
Weuseananalyticalsolutionforanacoustic-gravitywaveinan
isothermalatmosphere(seeMihalas&Mihalas1984).Theper-
turbationsaregivenby
TR TR
(cid:6) (cid:7)
z
δv = V exp +k z sin(ωt−k z) (11)
z 0 2H zi zr cut-off cut-off
(cid:6) (cid:7)
δp z
= V |P|exp +k z sin(ωt−k z+φ ) (12)
p 0 2H zi zr P
0 (cid:6) (cid:7)
δρ z
= V |R|exp +k z sin(ωt−k z+φ ) (13) t=360.
ρ 0 2H zi zr R
0
where V0 = 10−2 ms−1 is the amplitude of the velocity (see TR TR
Table2),andHisthepressurescaleheight.Therearenopertur-
bationsinthehorizontalvelocityandthemagneticfield,andkzr cut-off cut-off
andk aretherealandcomplexpartsoftheverticalwavenum-
zi
ber,respectively.Giventhewavefrequencyω,thisisfoundfrom
thedispersionrelation:
(cid:8)
t=450.
ω2−ω2
k =k +ik = c (14)
z zr zi
c
s
TR TR
whereω =γg/2c istheisothermalacousticcut-offfrequency.
c s
Therelativeamplitudesandphaseshiftsbetweenthepertur- cut-off cut-off
bationsaregivenby
(cid:9)
(cid:4) (cid:5)
γ 1 (γ−2) 2
|P| = k2 + k + (15)
ω zr zi 2H γ t=750.
(cid:9)
(cid:4) (cid:5)
1 1 2
|R| = ω kz2r+ kzi− 2H (16) TR TR
(cid:4) (cid:5)
φ = arctan kzi + 1 (γ−2) (17) cut-off cut-off
P
k 2Hk γ
(cid:4) zr zr(cid:5)
k 1
φ = arctan zi − · (18)
R k 2Hk
zr zr t=900.
Theperturbationisimposedintheintervalz={−5,−4.65}Mm,
meaningthatitoccupiessevengridpoints.
TR TR
As is frequentlydone in MHD wave studies, to distinguish
betweenthemodesinthesimulations,weprojectthevelocities
cut-off cut-off
into the directions paralleland perpendicularto the local mag-
neticfield,definedas
e = sinθe +cosθe (19)
long x z
t=1350.
e = −cosθe +sinθe (20)
trans x z
whereθistheinclinationofthemagneticfield.Thisway,inthe
β(cid:5) 1atmosphere,theacousticslowmodeisvisibleinthelon- TR TR
gitudinal velocity (propagation parallel to the magnetic field);
cut-off cut-off
andthefastmagneticmodeisessentiallytransversetothemag-
netic field. The projections do not help to distinguish between
themodesintheβ(cid:6)1region.
3.2.1. Wavepropagation
Figure 3 shows the longitudinal (left panels) and trans√verse
(righ√tpanels)velocityprojectionsscaledwiththefactors ρ0cs √Fig.3.Longi√tudinal(left)andtransverse(right)velocitiesmultipliedby
and ρ0va,respectively.Thisscalinggivesanapproximationfor ρ0cs0 and ρ0vA0,respectively,forthesimulationrunwithavertical
theenergiescontainedinthedifferentwavecomponents.Some periodic driver. The black solid line is the location of the cut-off fre-
importantlayersare also indicated in each panelof this figure: quencyforthe200speriodwaves,thedashedblacklinesaretheβ=1
β = 1 contours, the transition region, and the location of the contours, and the red line shows the transition region. The magnetic
cut-off layer with a period of 200 s, the same period as that of field lines are inclined black lines. Positive velocity (red color) is an
upflow.Themovieofthewavepropagationisavailableonline.
thedriver.Therearehorizontalvariationsinthelongitudinaland
A70,page5of13
A&A577,A70(2015)
transversevelocitiesinthebottomlayerswherethedriverisim-
posed,eventhoughthevelocityperturbationiszerointhehori-
zontaldirection(seeEq.(11)).Thesechangesarecausedbythe
projectionofthevelocityonthedirectionsparallelandperpen-
diculartothemagneticfield,whichhasavaryinginclination.
Themagneto-acousticfastandslowwavesstartpropagating
upwardthroughthelayersbelowthephotosphere,whereplasma
β(cid:6)1(seetherowst=210sandt=360s).Becausethedriver
isparalleltothemagneticfieldatthefootpoints,theamplitude
ofthelongitudinalvelocityisgreaterthanthetransverseone.
Asthewavesreachtheequipartitionlayeratt = 450s,fast
acoustic-like waves are partially converted into fast magnetic-
like waves and partially transmitted into longitudinal slow
acousticwaves.Thesecanbeseencomparingthesecond(360s)
andthird(450s)rowsofFig.3:inthesecondrowthetransverse
componentcrossestheequipartitionlayerwithaverysmallam-
plitude, but once the longitudinalcomponentreaches the layer
where β = 1 at 450 s, the amplitude of the transverse compo-
nentgrowsconsiderablywhiletheamplitudeofthelongitudinal
wavesdecreases.
These waves continue propagatingupward until they reach
the transition region (before 600 s). At this point, part of the
waveenergyisreflectedandpartistransmittedintothecorona.
Thetransversevelocitycomponentsufferssignificantreflection
fromthe transition region(rightpanelsfrom 750s to the end).
Theenergyofthetransversecomponentismostlyconcentrated
betweentheequipartitionlayerandthetransitionregion.
Someoftheenergyofthelongitudinalwavesistransmitted
tothecoronathroughthetransitionregion.Thebehaviorofthese
waves,oncetheypassthetransitionregion,dependsonwhether √ √
Fig.4.Time-heightdiagramof ρ cv (left)and ρ v v (right)
theypropagatenearthenullpointorfarfromit(seeleftpanels 0 s long 0 a trans
forthreeselectedhorizontalpositionsintheverticalperiodicdriversim-
ofFig.3fromt=750stotheend).Thewavespropagatingnear ulations. Upper row: x = 150 km, where themagnetic fieldisnearly
thelateralbordersofthedomain,insidethealmostverticalflux vertical.Middlerow:x=3750km,closetothecenteroftheleftarcade
tubes, are longitudinaland continue propagatingupward along withanearlyhorizontalmagneticfield.Bottomrow:x=7500kmclose
thefieldlinesuptotheuppercorona.Wavespropagatingoutside tothenull-pointlocation.ThecolorscaleisthesameasinFig.3.
theverticalfluxtubesreachthenullpoint,andthischangesdras-
ticallytheirbehavior.Thesewavessufferpartialconversionfrom
magnetictoacousticandviceversa,andtransmissionfromfast showshowthepictureofwaveinterferencechangesatdifferent
toslowandviceversa,attheequipartitionlayeraroundthenull magneticfieldinclinations.
point.TheAlfvénspeediszeroatthenull,soonlyacousticfast
Thelefthandpanelsdemonstratethatwavespropagatingpar-
waves (inside the circle marked by the coronalβ = 1 contour) allelto the nearlyverticalmagneticfield at x = 150kmandat
are able to cross it. The movie of the simulations makes it ap- x = 7500 km travelalmost undisturbedup to the transition re-
parenthowthenullpoint“absorbs”thewavesandthen“ejects”
gion, where reflection is produced and an interference pattern
themawayinalldirections.Thiseffectismainlycausedbythe
is formed between the upwardly and downwardly propagating
refraction of the fast magnetic waves due to the large gradient waves. Since the attack angle is small at the β = 1 contour
of the Alfvén speed. Part of these fast waves propagate down-
at these locations, the transmission from fast acoustic to slow
wardagainthroughthetransitionregionandequipartitionlayer
longitudinal acoustic waves is almost total. Once the waves at
to the lower atmosphere.The null pointacts as a re-feedingof x=150kmreachthetransitionregionatabout600softhesim-
theatmosphere.
ulationtime,theycontinuepropagatingupwardwithsignificant
A few minutes after the waves reach the upper boundary, amplitude (top left panel of Fig. 4). Unlike that, waves prop-
thesimulationentersthestationarystage.Thispropertywillbe agating at x = 7500 km are affected by the null point. In the
usedfocalculatingthemeanwaveenergyfluxes,asdoneinthe bottomleftpanel,onemayappreciatehowthewavespropagate
sectionbelow. downward,owingto the refractionsufferednearthe nullpoint,
Time-heightdiagramsallowthebehaviorofwavesdescribed thenmeetthewavesgoingupward,andfinallytheiramplitudes
abovetobeapp√reciatedmoreclearly√(seeFig.4).Thereweplot cancelout.
the quantities ρ c v (left) and ρ v v (right) at three Thebehaviorofwavespropagatingat x = 3750km(middle
0 s long 0 a trans
differenthorizontallocationsinthedomain.Thecutsarelocated panels of Fig. 4), where the magnetic field is nearly horizontal
at x = 150kminsidethe fluxtube,wherethemagneticfield is belowthetransitionregion,isdifferentfromabove.Onecanob-
nearly vertical at x = 3750 km in the center of the left arcade serveseveralsmall-scalewavecomponents,withtheirenergylo-
and where the magnetic field is nearly horizontal, and at x = catedessentiallybetweentheβ=1contourandthetransitionre-
7500 km in the middle of the domain where the null point is gion(middleleftpanel).Thesesmall-scalewavesarearesultof
located. The inclination of ridges on this diagram reflects the themodeconversionintoslowmodespropagatingalongthein-
wave propagationspeed (more vertical means faster). Figure 4 clinedfield.(Theirpropagationspeedisapparentlylowbecause
A70,page6of13
I.C.Santamariaetal.:Wavepropagationfromthesubphotospheretothecorona
Vertical Mean Acoustic Flux
1×106
x=150 Km
] ]
m m x=3750 Km
M M
[ [ 5×105 x=7500 Km
z z
2m]
c
x [Mm] x [Mm] g/s/ 0
er
[
−5×105
] ]
m m
M M
[ [

z z
−1×106
−4 −2 0 2 4 6 8
Height [Mm]
x [Mm] x [Mm] Vertical Mean Magnetic Flux
1×106
Fig.5. Mean energy fluxes for the simulation with the vertical peri-
odic driver. Top: horizontal and vertical mean acoustic energy fluxes.
x=150 Km
Bottom:horizontalandverticalmagneticmeanenergyfluxes.Thelower x=3750 Km
bthlaecdkaslihneedilsintheseaarceouthseticβc=ut1-ocffonfrteoquur,etnhceyrleadyelirnfeoirstthhee2tr0a0n-ssitpioenriored-, 5×105 x=7500 Km
gion.Themagneticfieldlinesareinclinedblacklines.Theunitsofthe
barare106[erg/s/cm2]. 2m]
c
s/ 0
g/
ofthe longerdistancetheyhavetocrossin theinclinedfields.) er
Fastmagneticmodeshavemuchhigherpropagationspeedsand [
canbeseeninthemiddlerighthandpanelasnearlyverticallines
−5×105
with the maximum power concentrated again in the chromo-
sphere below the transition region. The reflection of longitudi-
nalslowwavesfromthenullpointisalsoapparentinthemiddle
lefthandpanelfromtheoppositeinclinationoftheridgesabove −1×106
thetransitionregionandbelowthenullpoint. −4 −2 0 2 4 6 8
Height [Mm]
Fig.6.Verticalcutoftheverticalcomponent oftheacoustic(top)and
3.2.2. Energyfluxes magnetic(bottom)meanfluxesforthreedifferenthorizontalpositions,
asindicatedineachpanel.Theverticalblacklinesarethelocationsof
We calculate the mean acoustic and magnetic energy fluxes as thecut-offfrequencyfor200-speriodwaves,β =1contours(dashed),
follows(Bray&Loughhead1974): transitionregion,andthenullpoint(fromlefttoright).
F =(cid:8)δpδu(cid:9), (21)
ac
F =(cid:8)δB×(δu×B )(cid:9)/μ . (22)
mag 0 0
To obtainthe meaningfulaveragesofthe flux,oneneedsto in-
x=3750km;andatx=7500kmwherethenullpointislocated.
clude at least several periods of the stationary regime of the
The bottompanel of Fig. 6 clearly shows that all the magnetic
simulations.Forthesimulationspresentedabove,weconsidered
energyisconcentratedbetweenthephotosphere(z = 0km)and
thatthestationaryregimeismaintainedfrom1000to3000sof
the transition region,as a consequenceof the mode conversion
thesimulationtime.
in the inclined magnetic field of the arcades (green line). The
Figure 5 shows the spatial distribution in the domain of
sign of the energy flux is negative, meaning that waves essen-
themagneticandacoustic,vertical,andhorizontalmeanenergy
tiallypropagatedown,whichisexplainedbytheinfluenceofthe
fluxes.Itcanbeseenthatalmostallthemagneticenergy(lower
panels)isconcentratedbetweenthephotosphere(z=0km)and nullpoint.Themagneticenergyfluxinourexperimentisunable
transitionregion(z = 2100km),whiletheupwardacousticen- tocrossthetransitionregion.
ergyfluxissignificantinthecoronainsidetheverticalfluxtubes. TheupperpanelofFig.6showsthattheacousticenergyin
Theonlystrongupwardenergyfluxin thecoronaisdueto up- the photospherepropagatesupward in the nearly vertical mag-
wardlypropagating,longitudinalslowacousticwaves.Nowave netic concentrations(blue and red lines), and downward at the
energy flux is present around the null point. This happens be- locations of the arcade. This acoustic energy flux is signifi-
causewavespropagateinalldirectionsaroundthenullpointand cantly larger in the photosphere and in the coronal part, com-
canceloutwhenwaveswithoppositedirectionsmeet. pared to the magnetic flux. (The latter is almost zero in these
In Fig. 6 three different cuts of the mean vertical energy regions.) The largest upward acoustic energy flux reaches the
fluxesareshowninmoredetail.Asbefore,wemadecutsatthree corona in the nearly vertical flux tubes at the sides of the do-
horizontallocations:insidethefluxtubesat x = 150km;atthe main and reachesless than 10–20%of the originalenergyflux
middle of the arcadeswhere the magnetic field is horizontalat ofacousticwavesbelowthephotosphere.
A70,page7of13
A&A577,A70(2015)
3.3.Horizontalperiodicdriverof300s
vlong vtrans t=210.
Inthisrun,a horizontalperiodicmotionwith aperiodof 300s
is applied at the lower boundary of the simulation domain at
heightsz={−5,−4.65}Mm,sameasfortheverticaldriver: TR TR
δv = V sin(ωt) (23) cut-off cut-off
x 0
δB k
x = zV sin(ωt), (24)
B ω 0
0
t=300.
wherethehorizontalwavelengthisinfinite,sok = 0,andk =
x z
ω/va. The perturbation in the rest of the(cid:10)variables is null. The TR TR
valueofthemagneticfieldstrengthB = B2 +B2 istakenat
0 0x 0z
theboundary.Thehorizontalvelocityamplitudeischosentobe cut-off cut-off
V = 0.2ms−1,seeTable2.Suchdrivinggeneratesessentially
0
noacoustic wavesin verticalmagneticfieldswherethe motion
ofthedriveriscompletelytransverse.Thiscanbeseeninsome
t=450.
pointsoftheedgesofthedomainandinthemiddlebetweenthe
twoarcades.However,acoustic-likewavesappearinalmostall
therestofthedomain,wherethemagneticfieldismoreinclined TR TR
(seeHasanetal.2005;Hasan&vanBallegooijen2008).
cut-off cut-off
3.3.1. Wavepropagation
Figure7showsthepropagationofwavesgeneratedbythehor-
t=900.
izontalperiodicdriver.Thedriverexcitesmagneticslow waves
seenasasmallwavelengthpatternatthebottomofthedomain
at all times. The pattern due to these waves disappears after TR TR
some height around −3 Mm since these short-wavelength per-
turbations are affected by the numerical diffusion effects and
cut-off cut-off
are not fully resolved. Fast acoustic waves are also excited at
the locations with inclined field. These can be appreciated on
the two upperpanels and at later times, as a large-scaledistur-
bancepropagatingfromthebottomboundaryupward.Thesefast t=1650.
wavesreachtheequipartitionlayeratt=300s,andtheygetpar-
tiallyconvertedandtransmitted.Theperturbationseeninlongi-
TR TR
tudinal velocity (left panels) remains almost completely below
the equipartition layer at all posterior time moments. The only
cut-off cut-off
exception is the vertical flux tubes on the sides of the domain
where again some perturbationpropagatesupward through the
transitionregiontothecorona,similartotheverticaldriving.
Perturbations seen in the transverse velocity (right panels) t=2100.
above the equipartition layer and below the transition region
at later times (t > 450 s) correspond to fast magnetic waves.
TR TR
Neither these fast magnetic waves nor slow acoustic waves
(small-scale perturbations between the layer where β = 1 and
the transition region in the left panels) pass throughthe transi- cut-off cut-off
tion regionwith largeamplitudes,andtheir energyremainses-
sentiallybelowit.Theamplitudesofwavesreachingthecorona
aresmallerthaninthecaseoftheverticaldriver.
Unlikethelongitudinalslowwavesinverticalfluxtubes,fast
magnetic transverse waves reach the corona in the whole do-
main. When passing through the null point, they are partially
converted into acoustic-like waves in the equipartition layer Fig.7.SameasFig.3butforthesimulationrunwithahorizontalperi-
around it. The simulation reaches a stationary state after about odicdrivingof300s.Theblacksolidlineisthelayerwherethecut-off
t=1200softhesimulation. √ periodis300s.Themovieisavailableonline.
F√igure8showsthetime-heightdiagramsof ρ0csvlong(left)
and ρ v v (right) at three different horizontalpositions of
0 a trans
the domain.The topand bottomrowsofthis figure(horizontal ofthecut-offlayerisnotveryclearinthesediagrams.Thetrans-
positions x = 150kmand x = 7500km)showwavespropagat- versevelocitypropagationatthesetwopositionsofthedomain
inginthenearlyverticalmagneticfield.Onecanappreciatehow isverysimilar.Themodeconversionresultsinanincreaseinthe
the behavior of these waves changes in the equipartition layer transversewavecomponentasthewavescrosstheequipartition
(dashed-dottedline) and in the transition region.The influence layerinthedownwarddirection.Theridgesinthediagramshow
A70,page8of13
I.C.Santamariaetal.:Wavepropagationfromthesubphotospheretothecorona
] ]
m m
M M
[ [

z z
x [Mm] x [Mm]
] ]
m m
M M
[ [

z z
x [Mm] x [Mm]
Fig.9.SameasFig.5butforthesimulationrunwithahorizontalperi-
odicdrivingof300s.Theunitsofthebarare[erg/s/cm2].
In the case of the horizontal driver, the energy fluxes are
very small (see Fig. 9). This figure shows the spatial distribu-
tionofthemeanenergyfluxes.Theupwardacousticenergyflux
is essentially concentrated below the arcades at heights below
the transitionregion(upperrightpanel)and doesnotreach the
corona.Curiously,theaverageupwardacousticfluxisnegative
insidetheverticaltubesonthesides,unliketheverticaldriving.
Thismightberelatedtothefactthatinthesimulationswiththe
Fig.8.SameasFig.4butforthesimulationrunwithahorizontalperi-
horizontaldriver,thesewavesareexcitedbyhorizontallyprop-
odicdrivingof300s.
agatingdisturbancesinthecoronaafterthewavesarerefracted
downward aroundthe null point. The vertical magnetic energy
flux (bottomrightpanel)is mostlypositiveand ismuchhigher
theoppositeinclination,indicatingdownwardpropagationpro-
abovethephotospherethanbelow.
ducedbythereflectionatthetransitionregion.Inthecorona,the
Figure10showstheverticalmagneticandacousticfluxesat
ridgesarevertical,indicatingthatwavespropagatewithAlfvén
three different horizontal locations. This figure reveals that the
speed,sotheirnatureismagnetic.Asthesefastwavescrossthe
largestacousticenergyfluxinthissimulationrunisconcentrated
null point, they are converted at the equipartition layer around
belowthearcades(greenlineintheupperpanel).Themagnetic
it (bottom right panel of Fig. 8). The propagationspeed of the
fluxisverysmallexceptatheightsaroundtheequipartitionre-
waves is altered, which results in a shift in the position of the
gion.Thevaluesofthefluxaremuchlowerthaninthevertical
ridgesatheightsaroundthenullpoint.
Thelongitudinalvelocitybehavioratx=150kmandatx= drivingcase,despiteitslargerdrivingamplitude.
7500kmisquitesimilar,exceptfortheamplitudeofthewaves.
Someweaklongitudinalslowwavesappearabovethetransition 3.4.Instantaneouspressurepulse
region at x = 150 km, generated by horizontally propagating
perturbationsinthecorona. Inthisrunweexertedalocalizedinstantaneousforceatthebase
ThemiddlepanelofFig.8showsthewavepropagationatthe of the arcade in the middle of the domain, which consists of a
locationoftheinclinedfieldofthearcade.There,thetransverse pressurepulsewithaGaussianshape,asfollows:
(cid:2) (cid:4) (cid:5)(cid:3)
waves propagate undisturbed up to the transition region where
δp (x−x )2 (z−z )2
theyare reflected.Conversionfromfastacousticintofastmag- 1 = Aγexp − 0 + 0 (25)
neticwavesoccursattheequipartitionlevelascanbeseenfrom p0 (cid:2) (cid:4) 2σ2x 2σ2z (cid:5)(cid:3)
the smooth transition in the inclination of the ridgesat that lo- δρ (x−x )2 (z−z )2
cation.Someweakacousticslowwavesarealsogeneratedthere 1 = Aexp − 0 + 0 (26)
ρ 2σ2 2σ2
(middleleft panel).These wavespropagateup to the transition 0 x z
region and are reflected there and form an interference pattern where A = 10−5 is the relative amplitude given to the pulse,
around2Mmheight. [x ,z ] = [7500,−3500] km are the coordinates at which the
0 0
pulseislocated,andσ = 1500kmandσ = 1000kmarethe
x z
widths of the two-dimensional Gaussian profile (see Table 2).
3.3.2. Energyfluxes
Suchapulsegeneratesasuperpositionofwavesinabroadrange
As for the simulations with a vertical driver,we calculated the offrequencies.
mean acoustic and magnetic fluxes (see Eqs. (21)), averaging Unlikethesimulationswiththeverticalandhorizontaldriv-
over the stationary state of the simulations from t = 1200 s to ing, we locate a PML boundary condition at both the top and
t=3000s(6periodsof300swaves). bottomboundariesofoursimulationdomain.
A70,page9of13
A&A577,A70(2015)
Vertical Mean Acoustic Flux
1.5×104 vlong vtrans t=30.
1.0×104 TR TR
5.0×103 cut-off cut-off
2m]
c
s/ 0
erg/ t=210.
[
−5.0×103 x=150 Km
[ TR TR
x=3750 Km
[
x=7500 Km
−1.0×104
cut-off cut-off
−1.5×104
−4 −2 0 2 4 6 8
Height [Mm]
t=300.
Vertical Mean Magnetic Flux
1.5×104 TR TR
1.0×104 cut-off cut-off
5.0×103
2m] t=450.
c
s/ 0
g/
er TR TR
[
−5.0×103 x=150 Km
x=3750 Km cut-off cut-off
x=7500 Km
−1.0×104
t=750.
−1.5×104
−4 −2 0 2 4 6 8
Height [Mm]
TR TR
Fig.10. Same as Fig. 6 but for the simulation run with a horizontal
periodicdrivingof300s.
cut-off cut-off
3.4.1. Wavepropagation
t=900.
Thetimeevolutionofthescaledlongitudinalandtransverseve-
locitycomponentsof the pulsesimulationare givenin Fig.11.
Thepulsestartsexpandingin alldirections(twoupperpanels). TR TR
Asintheotherruns,thewavesgetpartiallyconvertedandtrans-
mitted whenreachingthe equipartitionlayer aroundt = 300s. cut-off cut-off
The acoustic fast componentof the pulse is partiallyconverted
and transmitted into magnetic fast and acoustic slow waves,
respectively.
The acoustic slow waves continue propagatingupward and
reachthetransitionregionbeforet = 450s.Partofthesewaves
is transmitted into the corona, and they reach the null point,
wheretheybecometrappedandareagainsentinalldirections,
Fig.11. Same as Fig. 3 but for the simulations with an instantaneous
except those waves inside the vertical flux tubes that continue
pressurepulse.Themovieofthewavepropagationisavailableonline.
propagatingto upperlayers along the field lines (leftpanels of
Fig.11).
Acoustic waves reflected from the transition region reach
equipartition layer again on their downward propagation, and up around −4 Mm at the lower turning point. Therefore, these
theysufferanotherpartialtransmission.Fastacousticwavespro- wavesstartpropagatingupwardagain(see leftandrightpanels
ducedafterthissecondarytransmissionpropagatebackdownto of Fig. 11 at t = 750 s at the lower part of the domain), suf-
thelowerboundaryofthedomain,wheretheyarerefractedow- fering the same phenomena as before. Thus, even though our
ingtothegradientsoftheacousticspeedandarefinallyreflected driverisnotperiodic,theatmospherekeepsoscillating,reaching
A70,page10of13