Rotational shear near the solar surface as a probe for subphotospheric magnetic fields

0.14 MB·

by  Leonid Kitchatinov

AstronomyLetters,2016,Vol. 42,No. 5
Rotational shear near the solar surface as a probe
for subphotospheric magnetic fields
L. L. Kitchatinov1,2*
Abstract
Helioseismology revealed an increase in the rotation rate with depth just beneath the solar surface. The rel-
6
ative magnitude of the radial shear is almost constant with latitude. This rotational state can be interpreted
1
as aconsequenceoftwoconditions characteristicofthenear-surfaceconvection: thesmallnessofconvective
0
2 turnover timein comparisonwith therotation periodand absenceof ahorizontalpreferreddirectionofconvec-
tion anisotropy. Thelatter condition is violated in the presenceof a magnetic field. This raises the question of
n
whether the subphotospheric fields can be probed with measurements of near-surface rotational shear. The
a
J shearisshowntobeweaklysensitivetomagneticfieldsbutcanserveasaprobeforsufficientlystrongfieldsof

9 theorderofonekilogauss. Itissuggestedthattheradial differentialrotationinextendedconvectiveenvelopes
1 ofredgiantsisofthesameoriginasthenear-surfacerotationalshearoftheSun.

Keywords
]
R Sun: rotation-Sun: magneticfields-stars: rotation-turbulence-magnetohydrodynamics(MHD)
S
1InstituteforSolar-TerrestrialPhysics,LermontovStr.126A,Irkutsk,664033,Russia
.
h 2PulkovoAstronomicalObservatory,PulkovskoeSh.65,St.Petersburg,196140,Russia
p *E-mail:[email protected]
-
o
r
t
s 1. Introduction than to its gradient. This phenomenonis now called the Λ-
a
effect(Ru¨diger1989).Therearealsodiffusivefluxesofangu-
[
Helioseismology revealed large inhomogeneity of solar ro-
larmomentum,whichareproportionaltothegradientofthe

1 tation near both boundaries of the convection zone. There
angularvelocity,duetotheturbulentviscosity. Thestandard
v is a shallow tachocline just beneath the base of the convec-
boundaryconditions,consideredinthenextSection,demand
5 tion zone where the latitudinal differential rotation rapidly
5 thetotalfluxofangularmomentumtobezeronearthestellar
decreaseswithdepthtoconvergetoauniformrotationofthe
8 surface. In other words, the viscosity and the Λ-effect bal-
radiationcore(Antiaetal. 1998;Charbonneauetal. 1999).
4 anceeachotheratthesurface. Inthecaseofsmallturnover
0 Another relatively thin (∼ 30Mm) layer with rotation rate time τ Ω−1 andradialanisotropyof theturbulence, this
. sharplyincreasingwithdepthliesrightbeneaththesolarsur- ≪
1 balance demands the relative radial shear to be constant ir-
0 face(Thompsonetal. 1996).Thispaperconcernsthesurface respective of how complex the dependencies on latitude in
6 shearlayer.
angularvelocityanditsgradientare.
1
Itisremarkablethattherelativerotationalshear,
The above explanation is however valid for weak mag-
:
v ∂log(Ω)/∂ln(r) = 1, in the surface layer is almost con- neticfieldsonly. Asufficientlystrongfieldprovidesanaddi-
i stant from the equato−r to 60◦ latitude (Barekat et al. 2014)
X tionalanisotropythatinvolvesadependenceonlatitudeinthe
though the angular velocity Ω and its gradientin (heliocen- relativerotationalshear. Thisraisesthequestionofwhether
r
a tric) radius r are latitude-dependent. More specifically, the helioseismologicaldatasimilartothatobtainedbyBarekatet
relative shear averaged over 10Mm depth below the photo-
al. (2014)canprovideinformationonsubsurfacesolarmag-
sphereisuniformwithlatitude.
netic fields? The question seems to be topical because the
Weshallseethattheuniformsurfaceshearcaneasilybe near-surfacelayerisconsideredasapossiblesiteofthesolar
explainedintheframeworkofthedifferentialrotationtheory. dynamo because of its large rotational shear (Brandenburg
Itisaconsequenceofthetwoconditionscharacteristicofthe 2005;Pipin&Kosovichev2011).Inviewofthelackofmea-
near-surfaceregion: 1)thecharacteristictimeτ ofturbulent surementsofinternalsolarmagneticfields,anymeansforthe
convection is small compared to the rotation period, and 2) fieldsdetectionaresignificant.
the only preferred direction of the turbulence anisotropy is Inthispaper,theΛ-effectisderivedandtherelativesur-
theradialone. facerotationalshearisestimatedwithallowanceforthemag-
Thesuggestedexplanationisasfollows.Ashasbeenfirst neticfield. Weshallseethattheshearisweaklysensitiveto
shownbyLebedinskii(1941),anisotropicturbulenceproduc- the magnetic field. The shear can, nevertheless, serve as a
estheso-called‘non-diffusive’fluxesofangularmomentum, probeforamagneticfieldwhosetoroidalcomponentreaches
which are proportional to the angular velocity itself rather thevalueoftheorderofonethousandGauss.
Rotationalshearnearthesolarsurfaceasaprobeforsubphotosphericmagneticfields —2/5
2. Continuity equation for the angular
momentum and boundary conditions
Rotationalshearnearthesurfaceoftheconvectiveenvelope
ofastarcanbeinferredfromtheboundaryconditionswhich
in turn follow from the continuity equation for angular mo-
mentum.
The angular momentum equation can be derived by av-
eraging the equation for the fluid velocity v. The velocity
v = V +u of a fluid with convective turbulence includes
themeanlarge-scalepartV andthesmall-scalefluctuations
u. Theaveraging(overanensembleofturbulentflows)sep-
arates the scales: v = V, u = 0, where the angular
h i h i
bracketssignifytheaveraging. Similarly,themagneticfield
isasuperpositionofthemeanlarge-scalefield,B, andfluc-
tuatingsmall-scale field, b. In whatfollows, sphericalcoor- Figure1. DependenceoftheCoriolisnumber(4)onthe
depthDbeneaththesolarsurface.
dinates(r,θ,φ)withtheaxisofrotationasthepolaraxisare
used and axial symmetryof the averagedstellar parameters
isassumed. Thecontinuityequationfortheangularmomen-
while the viscousstress TD constrainsthe magnitudeof ro-
tum, ij
tation inhomogeneity by the action of turbulent viscosities
∂Ω
ρr2sin2θ = div rsinθ(ρ u u b b /4π (Kitchatinov2005). Both the Λ-effectandthe turbulentvis-
φ φ
∂t − h i−h i cosities depend on rotation rate in a complicated way. The
(cid:0)
+ρrsinθΩV BφB/4π) , (1) keyparameterforthedependenceistheCoriolisnumber
−
(cid:1)
resultsfromtheaveragingoftheazimuthalcomponentofthe
∗
Ω =2τΩ (4)
motion equation (Ru¨diger 1989). The vector under the di-
vergence sign on the right-hand side of this equation is the
(Ω∗measurestheintensityofinteractionbetweenconvection
angularmomentumflux.
androtation). ThedependenceofΩ∗ ondepthnearthesolar
ThestandardboundaryconditionforEq.(1)istheconti-
surface is shown in Fig.1. This dependence follows from
nuity of radial flux of angular momentumon the surface of
the solar structure model (Stix 1989) and the value of Ω =
a star. The conditioncan be obtained by integratingEq.(1)
2.87 10−6s−1 fortheangularvelocity. TheFigureshows
over a small volume comprising a part of the surface. On thatΩ×∗ issmallnearthesurface.
taking into accountthat the mass flux ρV on the boundary
r Inthecase ofsmallΩ∗ andwithoutmagneticfields, rel-
issmall(smallmasslossforthestellarwind),thatthelarge-
atively simple expressionsholdfor the stress tensor compo-
scale magnetic field is continuous, and that convective tur-
nentspresentinEq.(3)(Ru¨diger1989),
bulenceispresentinsidethestaronly,wefindtheboundary
condition
TΛ = Sρν Ωsinθ,
rφ − T
T = ρ u u + b b /4π =0. (2) ∂Ω
rφ − h r φi h r φi TD = ρν rsinθ , (5)
rφ T ∂r
This condition for the off-diagonalcomponentof the stress
tensor,T = ρu u +(b b δ b2/2)/4π ,meansthat where ν is the turbulent viscosity and S is the convection
ij h− i j i j − ij i T
the surface density of azimuthal forces is zero and the dif- anisotropyparameter,whichwillbedefinedlatter. Equation
ferential rotation is controlled by internal processes in the (5) omits the contributions of the order of Ω∗2 the relative
convection zone, not by an external impact. The condition magnitudeofwhichdoesnotexceed1%(Fig.1).Thebound-
(2)isformulatedforthesurfacebutholdsvalidityfordepths ary condition (2) with account for Eq.(5) leads to the con-
thataresmallcomparedtothescaleofvariationofthestress stantrelativeradialshear,
tensor.
The stress-tensor for rotating fluids with anisotropic tur- ∂ln(Ω)
=S, (6)
bulenceincludestwodistinctcomponents: non-diffusivestr- ∂ln(r)
essesTΛandthecontributionofturbulentviscositiesTD,
ij ij nearthesolarsurface(Kitchatinov2013).Thisfindingagrees
∂V withthehelioseismologicaldataofBarekatetal. (2014)and
T =TΛ+TD, TD =ρ k, (3)
ij ij ij ij Nijkl ∂r theirvalueoftherotationalshearrestrictstheanisotropypa-
l
rameterS. However,Eq.(6)ismodifiedbyamagneticfield.
where istheeddyviscositytensor.Non-diffusivestress
ijkl
N
isthemainsourceforthedifferentialrotation(theΛ-effect),
Rotationalshearnearthesolarsurfaceasaprobeforsubphotosphericmagneticfields —3/5
3. Lambda-effect in presence of a suchaturbulencearedefinedbythespectraltensor
magnetic field
E(k,ω)
Qˆ (k,ω)= (1+S)(k2δ k k )
3.1 Quasi-linearapproximation ij 16πk4 (cid:20) ij − i j
Turbulentstresseswillbederivedinthequasi-linearapproxi-
S (rˆ k)2δ +k2rˆrˆ (rˆ k)(rˆk +rˆ k ) (10)
mationalsoknownasthefirst-ordersmoothing(Moffatt1978) − · ij i j − · i j j i (cid:21)
(cid:0) (cid:1)
orthesecond-ordercorrelationapproximation(Krause&Ra¨-
ofthevelocitycorrelation
dler1980). Inthisapproximation,thetermsthatarenonlin-
earinturbulentfluctuationsareneglectedintheequationsfor uˆ0(k,ω)uˆ0(k′,ω′) =Qˆ (k,ω)δ(k+k′)δ(ω+ω′). (11)
thefluctuatingvelocityormagneticfield.Theapproximation h i j i ij
allowsthecorrelationsofvelocityormagneticfields,similar Inthisequation,E(k,ω)isthefluctuationspectrum,
to that present in (2), to be derived as functions of rotation
∞ ∞
rate and/or mean magnetic field. The quasi-linear approx-
imation is justified for cases when the Strouhal number or 3 (u0)2 = E(k,ω)dkdω, (12)
h r i Z Z
the Reynolds number is small. Solar convectiveturbulence
0 0
does not belong to either of these cases. Nevertheless, the
presenceofspecialcasesforwhichthequasi-linearapproxi- andS istheanisotropyparameter,
mationisvalidensuresthisapproximationfromnonphysical
(u0)2
results. Numerical experiments of Ka¨pyla¨ & Brandenburg S = h i 3, (13)
(u0)2 −
(2008)showthequasi-lineartheorytokeepthe‘orderofmag- h r i
nitudevalidity’alsobeyondthesespecialcases.
which has been used already in equations (5) and (6). The
The quasi-linear technique for deriving the Λ-effecthas
spectral tensor should be positive definite. This condition
beendiscussedindetailelsewhere(cf.,e.g.,Kitchatinovetal.
imposesrestrictionsontheanisotropyparameter:
1994a)andisdescribedhereonlybriefly. Itisconvenientto
applytheFouriertransformoffluctuatingvelocity, S 1. (14)
≥−
u(r,t)= uˆ(k,ω)ei(r·k−ωt)dkdω, (7) Accordingto (14), the turbulentflow cannotconsistof only
Z radialmotions(itwouldbeS = 2inthiscase). Themean
−
energydensityofhorizontalmotionscannotbesmallerthan
and the same for magnetic fluctuations b. For the uniform
thatofradialmotions.
angularvelocityΩandthemeanfieldB,thisresultsinalge-
TheΛ-effectwithallowanceforthemagneticfieldcanbe
braicequationsfortheFourier-amplitudesofthefluctuating
derivedfromequations(8)to(11)byperformingtheinverse
fields,whichleadtothefollowingrelations
Fouriertransform.
uˆ (k,ω) = D (k,ω,Ω,V )uˆ0(k,ω),
i ij A j 3.2 TheΛ-effect
i(k B)
bˆ(k,ω) = · uˆ(k,ω), (8) The near-surface convection is weakly affected by rotation
ηk2 iω (Fig.1). It, therefore, suffices to derive linear terms in the
−
angularvelocityonlyintheΛ-effect.Thisyields
where V = B/√4πρ is the Alfven velocity, repetition of
A
subscripts meanssummation, and uˆ0 correspondsto the so-
TΛ =Sρν J (rˆε +rˆ ε )rˆ Ω
called‘originalturbulence’whichisnotperturbedbyrotation ij T(cid:20) 1 i jmn j imn m n
or magnetic field. The influence of rotation and magnetic
J (Ω bˆ)(rˆε +rˆ ε )rˆ ˆb
fieldisinvolvedviathetensorDij, − 2 · i jmn j imn m n
+J (rˆ bˆ)(Ω ε +Ω ε )rˆ ˆb
3 i jmn j imn m n
Nδ +Ωˆε k/k ·
Dij(k,ω,Ω,VA)= ijN2+Ωiˆj2l l , +J3(cid:16)ˆbiεjmn+ˆbjεimn(cid:17)(cid:16)(Ω·rˆ)rˆmˆbn+(rˆ·bˆ)rˆmΩn(cid:17)
N =1+ (k·VA)2 , Ωˆ = 2(k·Ω) , (9) J4(Ω bˆ)(rˆ bˆ) ˆbiεjmn+ˆbjεimn rˆmˆbn , (15)
(ηk2 iω)(νk2+iω) k(νk2 iω) − · · (cid:16) (cid:17) (cid:21)
− −
whereνistheviscosityandηisthemagneticdiffusivity.The where bˆ = B/B is the unit vector along the mean field B
original turbulence is assumed to be given. The turbulent and the J -coefficients (n = 1,...,4) depend on the field
n
stressesTΛarederivedfromthegivenpropertiesoftheorig- strengthandonthefluidparameters.Thesedependenciesare
ij
inalturbulencebyusingequations(8)and(9)toaccountfor expressed in terms of integrals of the turbulence spectrum
theinfluenceofrotationandmagneticfield. Theoriginaltur- with complicated weight-functions. Further simplifications
bulenceisassumedtobeanisotropicwiththeradialpreferred are required for quantitative estimations. Sufficient simpli-
directionsignifiedbytheunitvectorrˆ = r/r. Propertiesof fications are provided by the mixing-length approximation
Rotationalshearnearthesolarsurfaceasaprobeforsubphotosphericmagneticfields —4/5
Figure2. DependenceoftheJn-coefficientsofequation Figure3. DependenceoftheequipartitionfieldBeqon
(15)ontherelativestrengthβ (18)ofthelarge-scale depthDbeneaththesolarsurface.
magneticfield.
therefore,performedfortoroidalfieldB =e B,thatlargely
φ
(knownastheτ-approximationalso). Inthisapproximation, simplifiestheestimations.
thespectrumfunctioncontainstheonlyspatialscaleℓ = τu Equation (15) leads to the following expression for the
(u= (u0)2 1/2 istheRMSvelocity): non-dissipativepartTΛ (3)oftheradialfluxofangularmo-
h i rφ
mentum,
E(k,ω)=2u2δ(k ℓ−1)δ(ω). (16)
− TΛ = Sρν ΩJ (β)sinθ, (19)
Viscosityandmagneticdiffusivityarereplacedbytheiresti- rφ − T 1
matedeffectivevalues, which generalises the Eq.(5)1 with allowance for the mag-
neticfield.
ν =η =ℓ2/τ. (17) Estimatingtherotationinhomogeneityrequiresaknowl-
edgeoftheviscousstresseswhichbalancetheΛ-effect. The
TheJn-coefficientsofEq.(15)thendependonlyontheratio viscositytensorofequation(3),derivedwithinthesameap-
proximationsastheΛ-effect(15),reads
β =B/B , B = 4πρu, (18)
eq eq
p =ν ψ (β)(δ δ +δ δ )
of the mean field B to the so-called equipartition field B Nijkl T 1 ik jl jk il
eq +ψ (β)(δ (cid:2)ˆb ˆb +δ ˆb ˆb +δ ˆb ˆb +δ ˆb ˆb )+... , (20)
(forB =B ,themagneticenergydensityequalsthedensity 2 il j k jl i k ik j l jk i l
eq
of turbulent kinetic energy). For small β, only J1 tends to wheredotssignifythetermsirrelevanttothispaper(th(cid:3)efunc-
a finite value different from zero: J1 ≃ 1−6β2/7, J2 ≃ tionsψ1(β)andψ2(β)aregivenintheAppendixofthepaper
12β2/7, J3 2β2/7 and J4 = O(β4) (for β 1). In of Kitchatinov et al. (1994b) and in its figure 4). The con-
≃ ≪
the opposite case of a very strong field, all Jn-coefficients tribution of the eddy viscosity to the radial flux of angular
are inverselyproportionalto the field strength: J1 J2 momentumthenreads
≃ ≃
J 45π/(256β)andJ 15π/(256β)(forβ 1). The
4 ≃ 3 ≃ ≫ ∂Ω
J -functionsforintermediateβ-valuesareshowninFig.2. TD =ρν (ψ (β)+ψ (β))rsinθ . (21)
n rφ T 1 2 ∂r
Substitution of the stresses (19) and (21) into the boundary
4. Near-surface rotational shear
condition(2)finallygives
Weturnnowtotheestimationofthenear-surfaceradialinho-
mogeneityofrotationwithallowanceforthemagneticfield. ∂ln(Ω)
=Sφ(β), (22)
The field can change considerably the angular momentum ∂ln(r)
fluxes only if it is not too small compared with B (B >
eq whereφ(β)=J (β)/(ψ (β)+ψ (β))isthefunctionofthe
0.1B ,Fig.2).Figure3showsthedependenceoftheequipar- 1 1 2
eq normalisedstrengthofthetoroidalmagneticfield(18). This
tition field B (18) on depth beneath the solar surface for
eq functionisshowninFig.4.
the same solar structure model as Fig.1. It follows from
Fig.3 that only the fields of at least several hundred Gauss
5. Conclusion
can influence the near-surface rotational shear. The large-
scale poloidal field is much weaker, Bpol 1Gs (Stenflo Withoutmagneticfields, theshear (22)is constantwith lati-
∼
1988; Obridko et al. 2006). The toroidal field of the solar tude.TheshearvaluefoundbyBarekatetal. (2014)isrepro-
αΩ-dynamo can be much stronger. Further estimations are, ducedwiththemaximumpossibleradialanisotropyS = 1
−
Rotationalshearnearthesolarsurfaceasaprobeforsubphotosphericmagneticfields —5/5
rotation in extended convective envelopesof red giants can
beofthesamenatureasthenear-surfacerotationalshearof
theSun.
Acknowledgments
ThisworkwassupportedbytheRussianFoundationforBa-
sicResearch(project16–02–00090).
References
Antia,H.M.,Basu,S.,&Chitre,S.M.1998,MNRAS298,
543
Barekat,A.,Schou,J.,&Gizon,L.2014,A&A570,L12
Figure4. Thefunctionφ(β)ofthedependenceofthe Beck,P.G.,Montalban,J.,Kallinger,T.,DeRidder,J.,
rotationalshearonthetoroidalfieldstrengthnearthe Aerts,C.,Garc´ıa,R.A.,Hekker,S.,Dupret,M.-A.,et
surfaceofastellarconvectionzone. al. 2012,Nature481,55
Brandenburg,A.2005,ApJ625,539
Charbonneau, P., Christensen-Dalsgaard, J., Henning, R.,
Larsen,R.M.,Shou,J.,Thompson,M.J.,&Tomczyk,S.
(14), i.e, with u2 = 2 u2 = 2 u2 . Numerical experi-
h ri h θi h φi 1999,ApJ527,445
mentsofKa¨pyla¨etal.(2011)showthattheanisotropyofnon-
Deheuvels, S., Garc´ıa, R.A., Chaplin, W.J., Basu, S., An-
rotating convection is indeed close to this maximum value
tia,H.M.,Appourchaux,T.,Benomar,O.,Davies,G.R.,
(seefig.5ofKa¨pyla¨etal.).
etal. 2012,ApJ756,19
For S = 1, the normalizedrotationalshear equalsthe
− Ka¨pyla¨,P.J.,&Brandenburg,A.2008,A&A488,9
φ(β)-function of Fig.4: ∂ln(Ω)/∂ln(r) = φ(β). The
− Ka¨pyla¨,P.J., Mantere,M.J., &Brandenburg,A.2011,As-
shearincreaseswiththefieldstrength.Theincreaseiscaused
tron.Nachr.332,883
bythestrongersuppressionofturbulentviscositybythemag-
Kitchatinov,L.L.2005,PhysicsUspekhi48,449
netic field compared with the Λ-effect. The magnetic field
Kitchatinov, L.L. 2013, in: Solar and Astrophysical Dy-
decreases the turbulence intensity. As a consequence, both
namos and Magnetic Activity, IAU Symp. 294 (Eds.
the Λ-effectand the eddyviscositybalancingthiseffectare
A.G. Kosovichev, E. de Gouveia Dal Pino, Y. Yan,
decreased.However,theviscosityismorestronglyquenched
CambridgeUniv.Press),p.399
thatresultsinanincreaseintherotationalshearwiththemag-
Kitchatinov,L.L., Ru¨diger,G., & Ku¨ker,M. 1994a, A&A
neticfield(Fig.4).Asthetoroidalfielddependsonlatitude,a
292,125
similardependencecanemergeinthenear-surfacerotational
Kitchatinov, L.L., Pipin, V.V., & Ru¨diger, G. 1994b, As-
shear. Figure2ofBarekatetal. (2014)doesindeedshowa
tron.Nachr.315,157
small increase in the rotational shear around the latitude of
Krause,F., & Ra¨dler,K.-H.1980,Mean-FieldMagnetohy-
20◦ (seealsotable1intheirpaper). However,therelatively
drodynamicsandDynamoTheory(Oxford:Pergamon
smallvalueofthisincreasedoesnotallowitsconfidentinter-
Press)
pretation as a manifestation of a subphotospheric magnetic
Lebedinskii,A.I.1941,Astron. Zh.18,10
field. A comparison of Figures 3 and 4 shows that the sur-
Moffatt, H.K. 1978, Magnetic Field Generationin Electri-
facerotationalshearcanserveasanindicatorofsufficiently
callyConductingFluids(CambridgeUniv.Press)
strong subphotospheric magnetic fields of the order of one
Obridko, V.N., Sokoloff, D.D., Kuzanyan, K.M., Shelti-
thousandGauss.
ng,B.D.,&Zakharov,V.G.2006,MNRAS365,827
Thetheoryofthesolarsurfaceshearlayercanberelevant Pipin,V.V.,&Kosovichev,A.G.2011,ApJ727,L45
to the rotationalstates of red giants. Asteroseismology has Ru¨diger, G. 1989, Differential rotation and stellar convec-
revealed a large inhomogeneityof rotationin extendedcon- tion: Sun and solar-type stars (New York: Gordon &
vectiveenvelopesofthesestars(Becketal. 2012;Deheuvels Breach)
etal.2012).Therelativelyfastrotationoftheirdeepinteriors Stenflo,J.O.1988,Astrophys.SpaceSci. 144,321
wasinterpretedasaconsequenceofangularmomentumcon- Stix,M.1989,TheSun:Anintroduction(Berlin:Springer–
servation in the course of contraction of the central region Verlag)
and expansion of the envelope of a star. The characteristic Thompson,M.J.,Toomre,J.,Anderson,E.R.,Antia,H.M.,
time of turbulent diffusion ( 100 yers) in the convection Berthomieu, G., Burtonclay, D., Chitre, S.M., Chris-
∼
zone of a solar-mass giant is however short compared with tensen-Dalsgaard,J.,etal. 1996,Science272,1300
its evolutionarytime. This circumstancetogetherwith long
rotation periods suggests that the quasi-steady non-uniform