Table Of ContentMem.S.A.It.Vol.,1
(cid:13)c SAIt 2008 Memoriedella
Chromospheric heating and structure as
determined from high resolution 3D simulations
M.Carlsson1,2,V.H.Hansteen1,2,andB.V.Gudiksen1,2
0
1
0 1 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-
2 0315Oslo,Norway
n 2 Center of Mathematics for Applications, University of Oslo, P.O. Box 1052 Blindern,
a N-0316Oslo,Norway
J
0
1 Abstract. Wehaveperformed3DradiationMHDsimulationsextendingfromtheconvec-
tionzonetothecoronacoveringabox16Mm3at32kmspatialresolution.Thesimulations
] showveryfinestructureinthechromospherewithacousticshocksinteractingwiththemag-
R
neticfield.Magneticfluxconcentrationshaveatemperaturelowerthanthesurroundingsin
S thephotosphere buthigherinthelowchromosphere. Theheatingistheremostlythrough
. ohmicdissipationpreferentiallyattheedgesofthefluxconcentrations.Themagneticfield
h
isoftenwounduparoundthefluxconcentrations.Whenacousticwavestravelupalongthe
p
fieldthistopologyleadstoswirlingmotionsseeninchromosphericdiagnosticlinessuchas
-
o thecalciuminfraredtriplet.
r
st Key words. Methods: numerical – hydrodynamics – MHD – radiative transfer – Stars:
a atmospheres
[
1
v 1. Introduction the chromosphere can sustain a multitude of
6 wavemodes that become degenerate and cou-
4 Inferringphysicalconditionsinthesolarchro- ple when the sound speed equals the Alfve´n
5 mosphereisdifficult,becausethefewspectral speed.Major progressin ourunderstandingis
1 lines that carry information from this region
dependentoncomparisonbetweendetailedob-
. (e.g. hydrogenBalmer-α, CaII H and K lines
1 servationsand detailed modelling.Such mod-
0 and infrared triplet and He 1083 nm line) are ellingneedstogoallthewayfromtheconvec-
0 formed outside local thermodynamic equilib- tion zone (where waves are excited and mag-
1 rium (LTE). Thus they do not directly couple
netic field foot-point braiding takes place) to
: to the local conditions. In addition, the chro-
v the corona (to enable magnetic field connec-
mosphere is very dynamic with large varia-
i tivity) and include a reasonable treatment of
X tionsintemperatureanddensityandaconcept
theradiativetransfer.Thefirstsuchmodelling
r like “formationheight”isnotveryuseful.We was reported by Hansteen (2004). These sim-
a
also go from plasma domination in the pho-
ulations have been used to study flux emer-
tosphere to a magnetically dominated regime
gencethroughthephotosphereintothecorona
in the upper chromosphere. This means that
(Mart´ınez-Sykoraetal.2008,2009a)anddriv-
ingofspicules(Mart´ınez-Sykoraetal.2009b).
Sendoffprintrequeststo:M.Carlsson
2 Carlsson:Chromospheric3Dsimulations
We here analyze a high resolution simulation ionized calcium. These latter losses dominate
focusingonthechromosphericstructure,heat- the radiative losses from the chromosphere
ing and dynamicsclose to magnetic flux con- (Vernazzaetal. 1981). We include these
centrations. The layout of this contribution is radiative losses without assuming LTE by
asfollows:inSection2weoutlinethemethods usinganexpression
used,inSection3wedescribehowthesimula-
tionanalyzedwassetup,inSection4wegive
theresultsandwefinishinSection5withcon- Φ=θ(T)e−τf(τ)
clusions.
whereΦgivestheradiativelossesperelectron
2. Methods peratomfroma givenelement(hydrogenand
singly ionized calcium, respectively), θ(T) is
Earlier simulations were carried out with the thesumofallcollisionalexcitations(obtained
“Oslo Stagger Code”, a code where the par- fromatomicdata),T istemperature,τisanop-
allelization was performed assuming shared tical depth and e−τf(τ) is an effective escape
memory and thereforerestricted to rather few probability. The optical depth τ is set propor-
CPUs, see Hansteen,Carlsson,&Gudiksen tionaltotheverticalcolumnmasswiththepro-
(2007) for a description. We have now com- portionalityconstantobtainedfromthe1Dde-
pletedafullrewriteusingtheMessagePassing tailed non-LTEradiationhydrodynamicsimu-
Interface (MPI) parallelization library and a lationsofCarlsson&Stein(1992,1995,1997,
completelynewradiationtransfermoduleem- 2002)Thefunction f(τ) isobtainedfromafit
ploying domain decomposition (Hayeketal. usingthesame1Dsimulations.
2010). This new code, named BIFROST, is
outlinedbelow.Adetailedwriteupisinprepa-
ration(Gudiksenetal.).
We solve the equations of radiation- 3. Simulationsetup
magnetohydrodynamics using a compact,
high-order, finite difference scheme on a The simulation analyzed in this contribution
staggered mesh (6th order derivatives, 5th coversaregion16.6×16.6×15.5Mm3withthe
orderinterpolation).Theequationsarestepped bottom boundary in the convection zone 1.4
forward in time using the explicit 3rd order Mm below the heightwhere the mean optical
predictor-corrector procedure of Hyman depthat500nmisunity(inthefollowingthis
(1979), modified for variable time steps. height defines our z=0) and the top boundary
High-order artificial diffusion is added both inthecorona14.1Mmabovethisheight.Both
in the forms of a viscosity and in the form thebottomandtopboundariesaretransparent.
of a magnetic diffusivity. The radiative trans- The simulationdomainis periodicin boththe
fer equation is solved along 24 rays using horizontal directions with a fixed spacing of
a domain-decomposed short-characteristic 32.5 km while the vertical grid size is 24-28
method.Theopacityfrommillionsofspectral km up to 4 Mm height, increasing gradually
linesisincludedusingmulti-groupopacitiesin abovethisheighttoafinalmaximumvalueof
fourbins(Nordlund1982)modifiedtoinclude 213kmatthetopofthecomputationaldomain.
the effects of coherent scattering (Skartlien The final simulation has 512×512×325 grid-
2000). Conduction along the magnetic field- points.Theinitialrelaxationoftheconvection
lines is solved implicitly using a multigrid wasperformedatlowerresolution.
method. In addition to the radiative energy The mean, unsigned field strength at z=0
exchange treated with the detailed solution is initially 40 G but falls to about10 G at the
of the radiative transfer (including scattering beginningofthehigh-resolutionrunduetothe
effects) we include optically thin radiative factthat no field is injected into the computa-
losses in the the corona and radiative losses tional domain and the existing, tangled field,
from strong lines of hydrogen and singly decaysslowly.
Carlsson:Chromospheric3Dsimulations 3
4. Results temperature structure of granules and inter-
granularlanesisshowninFig.2.Theaverage
After the increase of the resolution, there is a
temperature for granules has been taken over
slow increasein the spatialpowerat highfre-
all columnswith a verticalupflow largerthan
quencies.Aftersome400stheevolutionstarts 100 ms−1 at z=0. The intergranular lane av-
todivergefromthecontinuationofthelowres-
erage has been taken over all columns with a
olution run — granules fragment more easily vertical downflow larger than 100 ms−1 and
and the velocity power at small spatial scales
the average for magnetic flux concentrations
increases. The mean unsigned field strength
has been taken over all columns where the
alsostartstoincrease,presumablyduetolocal
unsigned magnetic field in the photosphere is
dynamoactionmadepossibleatthehigherres-
largerthan300G.Intergranularlanesarehot-
olution, and the mean unsigned field strength
ter than granules above the surface (inverse
ends up at 15 G around t=1500s and stays
granulation).Magneticfluxconcentrationsare
at this level until the end of the simulation at
cooler than intergranularlanes below the sur-
t=1670s. The vertical magnetic field strength
faceatagivengeometricheightbuthotterthan
atz=0,t=1500sisshowninFig.1.Ascanbe
bothgranulesandintergranularlanesintheup-
expected from the low average unsigned field
perphotosphereandlowerchromosphere.
of15Gtherearelargeregionswithonlyweak
field.Thereareseveralregionswithonepolar-
ity dominating but in general, both polarities
co-exist. This is even more evident if the im-
agescalingisadjustedtoemphasizetheweak
field.Themagneticfluxconcentrationshavea
fieldstrengthcloseto1kG.
Fig.2.Meantemperatureasfunctionofheightfor
granules(dashed),intergranularlanes(dash-dotted)
andmagneticfluxconcentrations(solid).
This increased temperature in the upper
photosphere-lowerchromosphereof magnetic
flux concentrationsis clearlyshown in Fig. 3.
The region shown with a square in Fig. 1
is shown in more detail at a height of 0.29
Fig.1. Vertical magnetic field strength at z=0, Mmwithtemperature,magneticfieldstrength,
t=1500s.Thecolourscalehasbeenclippedat500 Jouleheatingandviscousheatingdisplayed.It
G to emphasize weaker field. The maximum field
isclearthatmostmagneticfluxconcentrations
strengthis±1kG.Thesquaremarkstheregionof
have a higher temperature than the surround-
interestshowninmoredetailinFig.3.
ingsatthisheightandthatJouleheatingtakes
place at the locations of high magnetic field
The temperature structure of these mag- strength. These locations evolve only slowly
netic flux concentrations compared with the in time while the viscous heating is concen-
4 Carlsson:Chromospheric3Dsimulations
Fig.3.Temperature(upperleftpanel),magneticfieldstrength(upperright),Jouleheating(lowerleft)and
viscousheating(lowerright)ataheightof0.29Mmatt=1500s.Atthisheight,magneticfluxconcentrations
arehotterthanthesurroundingsduetoJouleheating
trated at the edges of granules evolving on a force-free state. This interplay between
muchshortertimescales. field-braiding and expansion into a force-free
Due to foot-point motions, the magnetic state often leads to spiral topologies around
fieldtopologyinthechromospherechangesall small magnetic flux concentrations. One
the time. In the upper photosphere and lower examplefromthesimulationisshowninFig.4
chromosphere the plasma β (ratio between where the field-topology is shown around
gas pressure and magnetic pressure) is larger a magnetic flux concentration at t=1010s.
than unity and the magnetic field is far from Field-lines from the negative polarity in the
force-free. In the upper chromosphere, β flux concentration connect to various positive
drops below unity and the field approaches polarity patches with a clear spiral pattern as
Carlsson:Chromospheric3Dsimulations 5
Fig.4. Magnetic field above a magnetic field concentration as seen from above (left panel) and in per-
spective fromthelower leftcorner (rightpanel). Thevertical component of thefieldat z=0isshown in
greyscale.ThesignofBzisshownincolourwithredfieldlinescorrespondingtonegativeBz(darkinthe
planeatz=0).Thehighestfield-linesshownextendtoaheightof5.8Mm.Notethespiraltopologyasseen
fromaboveandtheabundanceoflow-lyingalmosthorizontalfield-lines.
Fig.5.Synthesizednarrowbandimageatthecoreofthechromosphericλ854.2nmlinefromsinglyionized
calciumattwotimesshowingtheapparentswirlingmotion.
seen from above (left panel in Fig 4) and a In synthesized, narrowband,imagesfrom the
largeamountofalmosthorizontal(butcurved) core of the λ854.2 nm line of singly ionized
field (right panel in Fig 4). At this instant in calcium (Fig.5) this is visible as a swirling
timeinthesimulationthereisanacousticwave motion around the flux-concentration very
that turn into a slow-mode wave in the low-β similar to observations recently reported by
region travelling along the curved field-lines.
6 Carlsson:Chromospheric3Dsimulations
Wedemeyer-Bo¨hm&RouppevanderVoort & R. J. Rutten (ed.) The Physics of
(2009). Chromospheric Plasmas, vol. 368 of
Astronomical Society of the Pacific
ConferenceSeries,107–114
5. Discussionandconclusions
Hayek W., Asplund M., Carlsson M., et al.,
2010,A&A,(submitted)
A multitude of phenomena are seen in these
Hyman J.M., 1979, In: Advances in com-
high resolution radiation-MHD simulations
puter methods for partial differential
that extend from the convection zone to the
equations - III; Proceedings of the Third
corona and we have here only focused on
International Symposium, Bethlehem, Pa.,
the heating and dynamics around magnetic
June 20-22,1979.(A80-2666309-64)New
flux concentrations. Although these simu-
Brunswick, N.J., International Association
lations are the most ambitious performed
for Mathematics and Computers in
to date, there are still important ingredients
Simulation,1979,p.313-321.,313–321
missing. The current simulations do not
include the effects of out-of-equilibrium Leenaarts J., Wedemeyer-Bo¨hm S., 2006,
A&A,460,301
hydrogen ionization even though it is known
that such effects are very important for the LeenaartsJ.,CarlssonM.,HansteenV.,Rutten
R.J.,2007,A&A,473,625
energy balance of especially the middle-
Mart´ınez-SykoraJ.,HansteenV.,CarlssonM.,
higher chromosphere (Carlsson&Stein
2008,ApJ,679,871
2002; Leenaarts&Wedemeyer-Bo¨hm 2006;
Leenaartsetal. 2007). Including such effects Mart´ınez-SykoraJ.,HansteenV.,CarlssonM.,
2009a,ApJ,702,129
wouldlikely increasethe amplitudeof shocks
Mart´ınez-Sykora J., Hansteen V., DePontieu
in the middle chromosphere. Another limita-
B.,CarlssonM.,2009b,ApJ,701,1569
tion is the neglect of the back-radiation from
NordlundA.,1982,A&A,107,1
thecoronaontotheupperchromosphere.This
SkartlienR.,2000,ApJ,536,465
radiation would heat the chromosphereabove
Vernazza J.E., Avrett E.H., Loeser R., 1981,
about 1 Mm height (Carlsson&Stein 2002)
but have little effect on the heights discussed ApJS,45,635
Wedemeyer-Bo¨hm S., Rouppe van der Voort
inthiscontribution.
L.,2009,A&A,507,L9
Acknowledgements. This research was supported
by the Research Council of Norway through the
grant “Solar Atmospheric Modelling” and through
grants of computing timefrom theProgramme for
Supercomputing.
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