Coupling the solar surface and the corona: coronal rotation, Alfvén wave-driven polar plumes

2.8 MB·English

by  R. F. Pinto

Coupling the solar surface and the corona: coronal rotation,
Alfvén wave-driven polar plumes
R. F. Pinto∗, R. Grappin†, M. Velli∗∗ and A. Verdini‡
∗LaboratoireAIM,CEASaclay,DSM/Irfu/SAp,U.Paris-Diderot,Gif-sur-Yvette,France
†LUTh,ObservatoiredeParis,Meudon,France,andLPP,ÉcolePolytechnique,Palaiseau,France
∗∗Dip.diFisica,UnversitádiFirenze,Italy,andJPL,CaliforniaInstituteofTechnology,Pasadena,CA,USA
3 ‡Solar-TerrestrialCentreofExcellence-SIDC,RoyalObservatoryofBelgium,Brussels,Belgium
1
0
2 Abstract. Thedynamicalresponseofthesolarcoronatosurfaceandsub-surfaceperturbationsdependsonthechromospheric
stratification,andspecificallyonhowefficientlytheselayersreflectortransmitincomingAlfvénwaves.Whileitwouldbe
n
desirabletoincludethechromosphericlayersinthenumericalsimulationsusedtostudysuchphenomena,thatismostoften
a
notfeasible.Wedefinedandtestedasimpleapproximationallowingthestudyofcoronalphenomenawhiletakingintoaccount
J
aparametrisedchromosphericreflectivity.Weaddressedtheproblemsofthetransmissionofthesurfacerotationtothecorona

2 andthatofthegenerationofpolarplumesbyAlfvénwaves[1,2].Wefoundthatahigh(yetpartial)effectivechromospheric
reflectivityisrequiredtoproperlydescribetheangularmomentumbalanceinthecoronaandthewaythesurfacedifferential

] rotationistransmittedupwards.Alfvénwave-drivenpolarplumesmaintaintheirpropertiesforawiderangeofvaluesforthe
R
reflectivity,buttheybecomebursty(andeventuallydisrupt)whenthelimitoftotalreflectionisattained.
S
Keywords: Sun,MHD,Solarwind,Polarplumes,Solarcycle
.
h PACS: 96.60.-j,96.60.P-,96.60.pc,96.60.Vg
p
-
o 1. INTRODUCTION whichactasresonantcavities.Finite-frequencyfootpoint
r motionsinducelong-lastinglargeamplitudeoscillations
t
s The dynamics of the outer layers of the Sun strongly [3] and null frequency motions shear the loops by arbi-
a
constraint the global topology of the solar corona and trarily high amounts. Ofman [4] pointed out that foot-
[
areattheoriginofmanytransienteventsobserved.The point leakage of coronal loop oscillations could be im-

1 time-varying magnetic field generated by the solar dy- portantinsomecases.Grappinetal.[5]haveshownthat
v
namo connects the sub-surface layers of Sun up to the the line-tying approximation is inappropriate to charac-
1
corona and heliosphere, and effectively couples the dy- terisethecoronalloop’sdynamicalresponsetofootpoint
4
3 namics of those regions. The photospheric movements motions even for very high effective chromospheric re-
0 are expected to shear the coronal magnetic arcades at flectivities.Analternativeapproximationwhichdiamet-
. their footpoints. This shearing may be transmitted up- rically opposes the line-tied condition relies on the so-
1
0 wards by the MHD wave-modes and so affect the dy- called fully transparent boundaries, as used by Pinto
3 namics of the solar wind and of the corona. The range et al. [1, 2]. In these works, the authors neglected all
1 of characteristic time and spatial scales for these sur- chromosphericreflectionandpurposelyconsidered“pes-
: face motions is very broad. Slow (null frequency) mo- simistic” configurations in what concerns the destabil-
v
i tionsarewidelybelievedtocontributetothequasi-steady isation of coronal structures and searched for mecha-
X
accumulation of magnetic energy in the corona, as re- nisms which nonetheless are able to produce coronal
r quirede.g.,bythecurrentstorage-and-releasetypeerup- events such as polar plumes, jets and inflows. Both ap-
a
tion models. Finite frequency movements, on the other proximations are extreme ones, the physical reality lay-
hand,injectwavesintothesamemagneticstructuresand ing necessarily in between. The line-tied case is prone
provideforthevastwealthofoscillatoryphenomenade- to severely overestimate the energy input from the pho-
tectedinthecorona.Thestandardapproachtothemod- tosphere into the corona within magnetically connected
ellingoftheeffectsofsurfacemovementsonthecorona regions,andthefullytransparent caseunderestimatesit
is that of the line-tied forcing of the magnetic loops by is the absence of the denser chromospheric (and photo-
theirfootpoints,whicharerigidlyattachedtothephoto- spheric)layers.Ideally,onewouldincludetheselayersin
sphere.Animportantconsequenceofthisapproximation the numerical domain. But that demands prohibitive in-
isthatthelowercoronalboundaryactsasperfectreflec- creasesinCPUtimeandmodelcomplexity,speciallyin
tor to MHD waves instead of letting the chromospheric multi-dimensional configurations. Here, we test a semi-
layers define the amount of reflection occurring there. reflective lower boundary condition applied at the base
Total reflection occurs at the foot-points of the loops, ofthecoronawhichrepresentstheeffectiveresponseof
thechromospheretoimpingingAlfvénwaves.Thiscon- by the ratio ε of Alfvén wave speeds above and below.
dition is comparable to that used by Verdini et al. [6] Thechromosphericreflectivityaisthendefinedas
but with a different implementation, more appropriated
toourMHDsetup. ε−1 Cphotosph
a= , withε = A , (1)
ε+1 Ccorona
A
2. NUMERICALMODEL C representing the Alfvén speed B/(µ ρ)1/2. The ap-
A 0
proximationisvalidforperturbationswhosecharacteris-
We used the DIP numerical code, which is a 2.5 D ax-
tic wavelength is much larger than the thickness of the
isymmetric model of the solar corona solving the fol-
chromosphere. We defined two types of semi-reflective
lowingsystemofMHDequationsdescribingaone-fluid,
conditions,onefortheAlfvéncharacteristicandanother
isothermal,fullyionisedandcompressibleplasma:
oneforpurelyazimuthalmotions(notethat,ononehand,
2 perturbations with purely azimuthal velocity are a mix-
∂ρ + ∇·ρu=0; P= ρk T
t m B ture of MHD wave modes, and on the other hand that
H
pure Alfvén modes are circularly polarised). The first
∇P J×B
∂tu + (u·∇)u=− + −g+ν∇2u kindtranslatessimplyin
ρ µ ρ
0
∂tB = ∇×(u×B)+η∇2B LA+=−aLa−+(1+a)δLA+(t) , (2)
The magnetic field B decomposes into a time- where, L+ is the amplitude of the upward-propagating
independent external component B0 and an induced A
Alfvén characteristic (i.e, propagating into the domain)
one b. The solar wind develops into a stable transsonic and L− is the amplitude of its downward-propagating
solution where the magnetic tension cannot counteract A
counterpart. The second kind assumes the following
its dynamical pressure, hence forming the open-field
threeconditions
regions of the corona. The upper boundary (placed at
r = 15) is transparent. The lower boundary (placed ∂u+= f(θ,t); ∂u+=0; ∂u+=0, (3)
t φ t r t θ
at r = 1.01 R ) is semi-reflective with respect to the
(cid:12)
Alfvén mode, but transparent with respect to all others where f(θ,t)describesaperturbationexertedatthesur-
(cf.Sect.2.1).Werecurtothedescriptionofthesystem face.Thisresultsinacombinationofperturbationsofthe
of equations above in terms of the MHD characteristics slow-mode,thefast-modeandAlfvénmodecharacteris-
atthedomain’sboundariesinordertoachievetotaltrans- tics, all defined as a function of f(θ,t). The two kinds
parency or to control explicitly its reflectivity. Alfvén of semi-reflective boundaries are used here with differ-
waves are injected at the lower boundary by perturbing entpurposes.WeusethefirstkindinSect.4,whichad-
the corresponding characteristic there. The diffusive dresses a problem where the injection and reflection of
termsareadaptedsothatgridscale(∆l)fluctuationsare pureandfinite-frequencyAlfvénwavesisofimportance.
correctly damped. The kinematic viscosity is defined as Wefavourthesecondkindofsemi-reflectiveboundaryin
ν =ν0(∆l/∆l0)2,typicallywithν0=2×1014 cm2·s−1 Sect.3,onwhichwestudyquasi-steadysolarwindsolu-
and0.01(cid:46)(∆l/∆l )2(cid:46)10.Thegridisnon-uniformand tionsforwhichabettercontrolofthesurfaceazimuthal
0
∆lisminimalclosetothelowerboundary.Themagnetic velocity is required (and the effects of perturbing non-
diffusivity η is scaled similarly. Both coefficients were Alfvénicmodesarenotimportant).
kept as low as possible in all the runs performed here
(yetinevitablylargerthanthoseintherealSun).Grappin
etal.[7]giveamorethoroughdescriptionofthespatial 3. CORONALROTATION
andtemporalschemesused.
Pintoetal.[2]studiedtheresponseofthesolarwindand
coronatotheslow(quasi-steady)variationsoftheglobal
2.1. Semi-reflectivelowerboundary magneticfieldduringasolarcycle.Theyproduceda11
yrtime-seriesofmagneticfieldsgeneratedbya2.5Dax-
As stated in Sect. 2, our lower and upper boundary isymmetrickinematicdynamocode[STELEM,8]which
conditions consist of imposing the amplitudes of the was used to constraint the magnetic coronal topology
MHDcharacteristicspropagatingintothenumericaldo- in the coronal MHD numerical code DIP. They quanti-
main (the outgoing ones being already completely de- fiedthetemporalevolutionofthepoloidaldistributionof
termined by the dynamics of the system). We treat the the wind velocity, mass and angular momentum fluxes,
chromosphere as a discontinuity, hence the effective re- which were found to depend strongly on the magnetic
flectivity(oftheAlfvénmode)iscompletelydetermined topology(henceonthemomentofthesolarcycle).The
FIGURE 1. Coronal magnetic field, wind velocity and rotation rate at two illustrative moments of the solar cycle (t =0 and
t=3.8yrinFig.3of[2]).Thebluelinesaremagneticfield-lines.Theblue-yellowcolour-scalerepresentsthewindspeedinMach
number(blue:vr/cs<1;yellow:vr/cs>1).Theorangecolour-scalerepresentstherotationrateΩ.Theplotsontherightside
representtherotationperiodasafunctionoflatitudeatdifferentradii(r=1, 2.5, 7, 15R(cid:12)).
determination of the angular momentum flux relied on latitudinalrange,andthereforeawiderangeoffootpoint
the assumption of a solid-body and time-invariant so- rotation rates. As a result, the surface differential rota-
lar rotation. We revisit here this work and apply semi- tion pattern reflects more visibly in such streamers than
reflectiveconditionstothelowerboundaryofthenumer- onthesmallerones.Ontheoverall,thestrongestlatitudi-
ical domain. We selected two illustrative instants of the nal shears appear at the streamer/coronal hole frontiers.
Pinto et al. [2] simulations. These correspond to t =0 Thefinitemagneticresistivityaffectsthewidthofthese
(activity minimum) andt =3.8 yr (activity maximum), regions, but has a negligible effect on the overall rota-
andarerepresentedinthefirstandfourthpanelsoftheir tionrates.Figure1representsthecoronalmagneticfield
Fig. 3. We started off from a non-rotating state and ap- geometry,thewindspeed(inunitsofMachnumber)and
plied a torque at the surface which accelerated it to the thecoronalrotationratefort=0andt=3.8yr.Theplots
followingrotationrateprofile totherightofthefiguresshowstherotationperiodasa
function of latitude at different radii (r=1, 2.5, 7 and
Ω(θ)=Ω +Ω sin2θ+Ω sin4θ, (4)
a b c 15R ).Wenotethatinordertoachievelong-lasting(i.e,
(cid:12)
with θ being the latitude, Ωa = 14.713 ◦/day, Ωb = stable)coronalrotationprofileswehadtoresorttopartic-
−2.396◦/dayandΩc=−1.787◦/day.Equation(4)fits ularly low values for ε. The calculations presented here
thetime-averageddifferentialrotationpatternatthesur- were made with ε =10−3 rather than a more solar-like
face of theSun [9]. The durationof the initial accelera- ε =10−2.Thereasonforthisisthataftertheinitialand
tion was ∼1/4 of the average (final) rotation period at short-lived forcing, the surface was left to evolve under
the surface. The initial transient propagates upwards as the influence of the braking torque exerted by the wind
anAlfvénwavefront,acceleratingtheopenfieldplasma flow alone without any counter-balancing torque due to
in the azimuthal direction and exciting a few global os- the underlying convective dynamics. This issue will be
cillationsintheclosedfieldregions.AfterafewAlfvén addressedinaforthcomingpaper.
transit times, and for a small enough ε, the corona and
windsettledownintoaquasi-steadystate.Forε∼1,the
highlytransparentsurfacespinsdownquickly.Notethat 4. POLARPLUMEGENERATION
therotatingwindflowcarriesanetoutwardangularmo-
mentum flux, and that a magnetic braking torque is ap- Pintoetal.[1]studiedthepropertiesofpolarplumesgen-
pliedatthesurfaceinresult.Forε∼0,thesurfacemain- eratedbyAlfvénictorsionalmotionsofthefootpointsof
tainstheimposedrotation,smallerclosedloopskeepos- a bipolar magnetic structure in a coronal hole using a
cillatingresonantlyforalongtimeandthelargeronesare fully transparent lower boundary. Such magnetic struc-
shearedindefinitely.Intermediatesolar-likevaluesallow turesappearnaturallyinunipolarregionssuchasthepo-
forthesurfacerotationtobemaintainedintheopen-flux lar coronal holes. They are also visible in simulations
regionswhiletheminimalrequiredamountoffootpoint suchasthatinFig.1.Weextendedthisworkbyaccount-
leakageisallowedfortheclosed-fluxregionstostabilise. ing for the wave reflection occurring at the dense chro-
The streamers gradually assume a nearly rigid rotation, mospheric layers by means of the reflectivity parameter
whiletheopenfieldregions(coronalholes)shapeupas ε.Asbefore,Alfvénwavesarecontinuouslyinjectedat
aParker’sspiral.Therotationrateoftheformerisdeter- the coronal base and drive the formation of a dense jet
mined by that of its magnetic footpoints. Large stream- (plume) along the magnetic axis of the structure. The
erslikethatonthefirstpanelofFig.1encompassawide
axis switches to a regime where it is dominated by the
slow-mode wave-fronts (the “blobs”) rather than by the
smoothoverdenseupflow.Forε≈0,theplumesareeas-
ilydisrupted.
5. CONCLUSIONS
We studied two different problems concerning the cou-
pling between surface movements and the solar corona.
The first one consists of the transmission of the sur-
face rotation to the corona and the second one of the
generation of polar plumes by Alfvén waves. For this
FIGURE2. Leftpanel:thestandardcase,withε=1.Right purpose, we implemented and tested a semi-reflective
panel:ε=0.01.Thecolour-scalerepresentsthedensity.White
lower boundary condition for our coronal model which
linesaremagneticfield-lines.
accounts for the Alfvén wave partial reflection taking
placeinthechromosphere.Theeffectivechromospheric
reflectivityisparametrisedbytheratioofAlfvénspeeds
acrossthechromosphereandtransitionregion.Wefound
that a high (yet partial) effective reflectivity is required
forsustainingthecoronalrotationagainstthesolarwind
magnetic breaking torque, while still allowing for the
necessaryamountoffootpointleakage.Futureworkwill
consider the underlying time-dependent convective sur-
face dynamics. We verified that the generation of polar
plumesbyAlfvénwavesworkswellbeyondtheinitially
proposedlimitoffulltransparency[1].Theplumeprop-
FIGURE3. Densitynasafunctionoftimeatthebottomof erties are maintained for a wide range of values for the
thejet(r=0.2∼1.4 R(cid:12)).Eachcurvecorrespondstoagivenε. reflectivity, but become bursty (and eventually disrupt)
Notetheregimechangehappeningforε=0.01.Theordinates when the limit of total reflection is reached. Additional
areinlog-scale.
workisinprogresstoevaluatemorefinelytheparameter
range0<ε ≤10−2.
density and velocity contrast between the axis of the
plumeandthebackgroundwindflowdecayswithheight,
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