Stability of Toroidal Magnetic Fields in Rotating Stellar Radiation Zones

0.29 MB·English

by  L. L. Kitchatinov

Astronomy&Astrophysicsmanuscriptno.core2 c ESO2008
(cid:13)
February5,2008
Stability of toroidal magnetic fields in rotating stellar radiation
zones
L.L.Kitchatinov1,2 andG.Ru¨diger1
1 AstrophysikalischesInstitutPotsdam,AnderSternwarte16,D-14482,Potsdam,Germany
e-mail:[email protected]; [email protected]
2 InstituteforSolar-TerrestrialPhysics,POBox4026,Irkutsk664033,Russia
7
e-mail:[email protected]
0
0
Received;accepted
2

n ABSTRACT
a
J Aims.Thequestionsofhowstrongmagneticfieldscanbestoredinrotatingstellarradiativezoneswithoutbeingsubjectedtopinch-
0 typeinstabilitiesandhowmuchradialmixingisproducedifthefieldsareunstableareaddressed.
3 Methods.Linearequationsarederivedforweakdisturbancesofmagneticandvelocityfieldswhichareglobalinhorizontaldimen-
sionsbutshort–scaledinradius.Theequationsaresolvedtoevaluatethestabilityoftoroidalfieldpatternswithoneortwolatitudinal
1 beltsundertheinfluenceofarigidbasicrotation.Hydrodynamicstabilityoflatitudinaldifferentialrotationisalsoconsidered.
v Results.Themagneticinstabilityisessentiallythree–dimensional.Itdoesnotexistina2Dformulationwithstrictlyhorizontaldistur-
7 bancesondecoupledsphericalshells.Onlystable(magneticallymodified)r-modesarefoundinthiscase.Theinstabilityrecoversin
4 3D.ThemostrapidlygrowingmodesfortheSunhaveradialscalessmallerthan1Mm.Thefinitethermalconductivitymakesastrong
destabilizingeffect.Themarginalfieldstrengthfortheonsetoftheinstabilityintheupper partofthesolarradiativezoneisabout
8
600G.Thetoroidalfieldcanonlyslightlyexceedthiscriticalvalueforotherwisetheradialmixingproducedbytheinstabilitywould
1
betoostrongtobecompatiblewiththeobservedlithiumabundance.Alsothethresholdforhydrodynamicinstabilityofdifferential
0
rotationwhichexistsin2Disloweredin3D.Whenradialdisplacementsareincluded,thevalueof28%forcriticalshearisreduced
7
to21%.
0
h/ Keywords.instabilities–magnetohydrodynamics(MHD)–stars:interiors–stars:magneticfields–Sun:magneticfield
p
-
o
1. Introduction still excited. How much radial mixing the Tayler (1973) insta-
r
t bilityproducesisnotwell-knownsofar.Astheradialmixingis
s The old problem of hydromagnetic stability of stellar radia-
a relevanttothetransportoflightelements,atheoryofthemixing
tive zones attains a renewed interest in relation with Ap stars
: comparedwiththeobservedabundancescanhelptorestrictthe
v magnetism (Braithwaite & Spruit 2004), the solar tachocline
amplitudesof the internalmagnetic fields (Barnes et al. 1999).
Xi (Gilman 2005) and transport processes in stably stratified stel- InthepresentpapertheverticalmixingproducedbytheTayler
larinteriors(Barnesetal.1999).
instabilityisestimatedandthenusedtoevaluatetheupperlimit
r
a Whichfieldsthestellarradiativecorescanpossessisrather onthemagneticfieldamplitude.
uncertain.Theresistivedecayinradiationzonesissoslowthat
primordialfields can be stored there (Cowling 1945). Whether
In this paper, the equations governing linear stability of
thefieldsof105Gwhichcaninfluenceg-modesofsolaroscilla- toroidalmagneticfieldsin differentiallyrotatingradiationzone
tions(Rashbaetal.2006)orstillstrongerfieldswhichcancause
are derived. They are solved for two latitudinal profiles of the
neutrinooscillations(Burgessetal.2003)canindeedsurvivein-
toroidal field with one and with two belts in latitude but only
sidetheSunmainlydependsontheirstability.
forrigidrotation.Theunstablemodesare expectedto havethe
Among several instabilities to which the fields can be sub- longest possible horizontal scales (Spruit 1999). Accordingly,
jected(Acheson1978),thecurrent-driven(pinch-type)instabil- our equations are global in horizontal dimensions. They are,
ityoftoroidalfields(Tayler1973)isprobablythemostrelevant however,localinradius,i.e.theradialscalesareassumedshort,
one because it proceeds via almost horizontal displacements. kr 1 (k is radial wave number). The computations con-
The radial motions in radiative cores are strongly suppressed firm≫that the most rapidly growing modes have kr 103 but
by buoyancy. Watson (1981) estimated the ratio of radial (ur) theyareglobalinlatitude.Thederivedequationsrepr∼oducethe
tohorizontal(uθ)velocitiesofslow(subsonic)motionsinrotat- 2D approximation as a special limit. It is shown with an ex-
ing stars as ur/uθ Ω2/N2, where Ω is the basic angular ve- actly solvable case that for rigid rotation the Tayler instability
∼
locityandN thebuoyancy(Brunt-Va¨isa¨la¨)frequency.Thisratio ismissinginthe2Dapproximation.Itrecovers,however,in3D
is small in radiative cores (Fig. 1). If the radial velocities are case. Finitediffusionisfoundimportantforthe instability.The
completely neglected, the stability analysis can be done in 2D minimumfield producingthe instability is stronglyreducedby
approximation with decoupled spherical shells (Watson 1981; allowance for finite thermal conductivity. This field amplitude
Gilman & Fox 1997). There is, however, some radial motion is about 600 G for the upper part of the solar radiation zone.
Consideringthe lightelementstransportby the Tayler instabil-
Sendoffprintrequeststo:G.Ru¨diger itywefindthatthefieldstrengthcanbeonlyslightlyabovethe
2 L.L.Kitchatinov&G.Ru¨diger:Magneticinstabilitiesinstellarradiationzones
+ eφ ∂Tv 1 ∂2Pv ,
r ∂θ − sinθ∂r∂φ
!
1 ∂ ∂ 1 ∂2
ˆ = sinθ + (3)
L sinθ∂θ ∂θ sin2θ∂φ2
(cf. Chandrasekhar 1961). Here and in the following distur-
bances are marked by dashes. For zero radial velocity, the
whole class of disturbancescorrespondingto the poloidalflow
vanishes. Then the remaining toroidal flow can produce only
toroidalmagneticdisturbancesso thatin the expressionforthe
magneticfieldperturbations(B),
′
e e 1 ∂T ∂2P
B = r ˆP θ m + m +
′ r2L m− r sinθ ∂φ ∂r∂θ
!
+ eφ ∂Tm 1 ∂2Pm , (4)
Fig.1. The buoyancyfrequency(2) in the upperpart of the ra- r ∂θ − sinθ ∂r∂φ
!
diativecoreoftheSunafterthemodelbyStix&Skaley(1990).
The convectionzoneof the modelincludesthe overshootlayer thepoloidalfieldpotentialbecomeszero,Pm =0.Reducingthe
sothatN/Ω islargeimmediatelybeneaththeconvectionzone. classofdisturbancesmayswitch-offsomeinstabilities.
⊙ It can be seen from Eq. (3) that the horizontal part of the
poloidalflowcanremainunchangedwhentheradialvelocityis
marginalvalue.Otherwise,themixingwouldnotbecompatible reducedand the radial scale of the flow is reducedproportion-
withtheobservedlithiumabundance. ally.Thedisturbancescanthusavoidthestabilizingeffectofthe
stratificationbydecreasingtheirradialscale.
Our stability analysis is local in the radial dimension, i.e.
2. Themodel weuseFouriermodesexp(ikr)withkr 1.Itwillbeconfirmed
≫
thatthemostunstablemodesdoindeedprefershortradialscales.
2.1.Backgroundstateandbasicassumptions
Theanalysisremains,however,globalinhorizontaldimensions.
Thestabilityofrotatingradiationzoneofastarcontainingmag- Instabilitiesoftoroidalfieldsordifferentialrotationproceed
neticfieldisconsidered.Thefieldisassumedaxisymmetricand vianotcompressivedisturbances.Characteristicgrowthratesof
purelytoroidal.Hence,itcanbewrittenas the instabilities are small comparedto p-modesfrequencies.In
theshort-waveapproximationthevelocityfieldcanbeassumed
B=eφrsinθ√µ0ρΩA(r,θ) (1) divergence-free,divu = 0. Note that even a slow motion in a
′
stratified fluid can be divergent if its spatial scale in radial di-
in terms of the Alfve´n angular frequencyΩ . In this equation,
A rectionisnotsmallcomparedtoscaleheight.Intheshort-wave
ρ is the density, r,θ,φ are the usual spherical coordinates and
approximation (in radius) the divergency can, however, be ne-
e it the azimuthal unit vector. Equation (1) automatically en-
φ glected.
suresthatthetoroidalfieldcomponentvanishes–asitmust–at
Ournextassumptionconcernsthepressure.Moreprecisely,
the rotation axis. Centrifugaland magnetic forces are assumed
local thermal disturbances occur at constant pressure so that
smallcomparedtogravity,g/r Ω2 andg/r Ω2.Deviation
of the fluid stratification from s≫pherical symm≫etryAcan thus be ρ′/ρ = −T′/T or s′ = −Cpρ′/ρ.Thisassumptionisagainjusti-
fied bythe incompressiblenatureofthe perturbations.Another
neglected.
interpretation of this assumption is given by Acheson (1978).
The stabilizing effect of a subadiabatic stratification of the
Acheson assumed zero disturbances of total (including mag-
radiativecoreischaracterizedbythebuoyancyfrequency,
netic)pressuretoinvolvemagneticbuoyancyinstability.Inour
g ∂s derivations, assumptionson constant total or only gas pressure
N2 = , (2) areidenticalbecausetheeffectofmagneticbuoyancyappearsin
C ∂r
p higher order in (kr) 1 compared to the terms kept in the equa-
−
where s = C ln(P/ργ)isthespecificentropyofidealgas.The tionsofthenextsection.
v
frequencyNisverylargeintheradiativecoresofnottoorapidly
rotatingstarsliketheSun(N Ω,N Ω ,seeFig.1).
≫ ≫ A 2.2.Equations
The larger is N the more the radial displacements are op-
posedbythebuoyancyforce.Radialvelocitiesshouldtherefore Westartfromthelinearizedequationsforsmallvelocitypertur-
be small. They are often neglected in stability analysis what bations,i.e.
might be dangerous as certain instabilities may even be sup-
∂u 1
pressed by the neglect(we shall see later how this indeed hap- ′ + (u )u + u u+ B B
′ ′ ′
pensfortheTaylerinstability). ∂t ·∇ ·∇ µ0ρ ∇ ·
Theconsequencesoftheneglectoftheradialvelocitypertur- (cid:0) (cid:1) (cid:0) 1(cid:0) ′ (cid:1)
bations,u′r,canbeseenfromtheexpressionfor(divergence-free) − (B·∇)B′− B′·∇ B =− ρ∇P +ν∆u′ (5)
velocity,u,insphericalgeometryintermsofthetwoscalarpo- !
′
(cid:0) (cid:1) (cid:1)
tentialsforthepoloidal,Pv,andtoroidal,Tv,flows magneticfield,
u′ = re2rLˆPv− erθ si1nθ∂∂Tφv + ∂∂2rP∂θv + ∂∂Bt′ =∇× u×B′+u′×B−η∇×B′ , (6)
!
(cid:0) (cid:1)
L.L.Kitchatinov&G.Ru¨diger:Magneticinstabilitiesinstellarradiationzones 3
andentropy, whereµ=cosθ,and
N
∂∂st′ +u·∇s′+u′·∇s= CTpχ∆T′. (7) λˆ = Ω0kr (10)
can be understood as special normalization for radial wave-
The basic flow is a rotation with nonuniformangular velocity,
length.Thefirsttermontherighthandsidedescribesthestabiliz-
Ω = Ω(r,θ), andthemeanmagneticfieldis thetoroidaloneof ingeffectofthestratification.Itvanishesforsmallλˆ.Apartfrom
theform(1).Perturbationsofvelocityandmagneticfieldareex-
thisstabilizingbuoyancyterm,thewavelengthisonlypresentin
pressedintermsofscalarpotentialsafter(3)and(4).Theiden- diffusiveterms.ThesecondtermoftheRHSincludestheaction
tities
offiniteviscosity,
r r B = ˆT , r r B = ˆP , νN2
·∇× ′ L m · ′ L m ǫ = . (11)
∂2 ν Ω3r2
r(cid:0)r u(cid:1) = ˆT , r3(cid:0)r (cid:1) u = ˆ+r2 ˆP 0
·∇× ′ L v ·∇×∇× ′ L ∂r L v Similarly,weusebelow
!
(cid:0) (cid:1) (cid:0) (cid:1)
areusedtoreformulatetheequationsintermsofthepotentials. ηN2 χN2
ǫ = , ǫ = (12)
The radial component of Eq. (6) then gives the equation for η Ω3r2 χ Ω3r2
thepoloidalmagneticfieldandtheradialcomponentsofcurled 0 0
forthediffusiveparametersηandχ.Thesecondandthefollow-
equations (5) and (6) give the equations for toroidal flow and
toroidalmagneticfield,respectively1. inglinesofEq.(9)describetheinfluencesofthebasicrotation
and the toroidalfield. Only latitudinalderivativesof Ω and Ω
The perturbationsare consideredas Fouriermodesin time, A
azimuthandradiusintheformofexp(i( ωt+mφ+kr)).Foran appear.Allradialderivativesareabsorbedbydisturbanceswhich
− varyonmuchshorterradialscalesthanΩorΩ .Thecomplete
instability,theeigenvalueωshouldpossessapositiveimaginary A
system of five equationsalso includesthe equationfortoroidal
part. Only the highest-order terms in kr for the same variable
flow
werekeptinthesameequation.
ǫ
Whenderivingthepoloidalflowequation,thepressureterm ωˆ ˆW = i ν ˆW +mΩˆ ˆW mΩˆ ˆB
wastransformedasfollows L − λˆ2 L L − A L
(cid:16) (cid:17) ∂(cid:16)2 (cid:17) (cid:16) (cid:17) ∂2 (cid:16) (cid:17)
r 1 P ′ = r 1 ( ρ) ( P) ′ = − mW∂µ2 1−µ2 Ωˆ +mB∂µ2 1−µ2 ΩˆA
·∇×∇× ρ∇ − ·∇× ρ2 ∇ × ∇
! ! ∂(cid:16)(cid:16) (cid:17) (cid:17) ∂(cid:16)(cid:16) (cid:17) (cid:17)
r 1( s) ( P) ′ = r g s = g ˆs. + LˆV ∂µ 1−µ2 Ωˆ − LˆA ∂µ 1−µ2 ΩˆA
′ ′
Cp ·∇× ρ ∇ × ∇ ! −Cp ·∇×(cid:0) ×∇ (cid:1) rCpL + (cid:16) ∂ (cid:17) 1 (cid:16)(cid:16)µ2 2 ∂Ωˆ(cid:17) (cid:17) 2(cid:16)1 (cid:17)µ2 Ωˆ(cid:16)(cid:16) ∂V (cid:17) (cid:17)
Inordertousenormalizedvariablesthetimeismeasuredin ∂µ − ∂µ − − ∂µ
miutnieeiastssauorrefeΩdsci−0an1leu(dΩnii0tnsisuonfaiΩtcsho,afaranrcdΩtet0rh,iestthoiecthnaenorgrnmuolaarlmrizvaeeldilzofecrdeitqvyua),erintahcbeylev(seωˆla)orceis- − ∂∂µ (cid:16)1−µ2(cid:17)2 ∂∂ΩˆµA! −2(cid:16) 1−µ(cid:17)2 Ω!ˆA ∂∂Aµ, (13)
0 ! !
(cid:16) (cid:17) (cid:16) (cid:17)
theequationfortoroidalmagneticfield
k 1 k
A = P , B= T , V = P , ǫ
Ω0r2√µ0ρ m Ω0r2√µ0ρ m Ω0r2 v ωˆ LˆB = −iλˆη2 LˆB +mLˆ ΩˆB −mLˆ ΩˆAW
1 ikrg Ω Ω
W = Ω r2 Tv, S = C rN2 s′, Ωˆ = Ω , ΩˆA = ΩA. (8) (cid:16) (cid:17) m2∂Ωˆ(cid:16)A (cid:17) ∂ 1(cid:16) µ2(cid:17)2 ∂Ωˆ ∂(cid:16)A (cid:17)
0 p 0 0
− ∂µ − ∂µ − ∂µ ∂µ
!
tIenntrtoiadluscminagkethsethfeamctoerqsukarlibnytohredneroromfamliazgantiiotundseotfoptohleoitdoarolipdoa-l + m2∂ΩˆAV + ∂ (cid:16) 1 µ(cid:17)2 2 ∂ΩˆA∂V , (14)
∂µ ∂µ − ∂µ ∂µ
potentials,whileinEqs.(3)and(4)itwasPv ≪rTv,Pm ≪rTm. (cid:16) (cid:17) !
Theequationforthepoloidalflowthenreads thepoloidalfieldequation
ǫ
ωˆ LˆV = −λˆ2 LˆS −iλǫˆν2 LˆV ωˆ LˆA =−iλˆη2 LˆA +mΩˆ LˆA −mΩˆA LˆV (15)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) ∂ µΩˆ ∂W ∂Ωˆ an(cid:16)dthe(cid:17)entropy(cid:16)equa(cid:17)tion, (cid:16) (cid:17) (cid:16) (cid:17)
− 2µΩˆ LˆW −2 1−µ2 (cid:16)∂µ (cid:17) ∂µ −2m2∂µW ωˆS =−iλǫˆχ2S +mΩˆS +LˆV. (16)
+ 2µΩˆ (cid:16) ˆB(cid:17) +2(cid:16)1 µ2(cid:17) ∂ µΩˆA ∂B +2m2∂ΩˆAB InthesimplestcaseofΩ = ΩA = 0andvanishingdiffusivi-
A L − (cid:16)∂µ (cid:17)∂µ ∂µ tiestheonlynontrivialsolutionoftheaboveequationsareshort
mΩˆ (cid:16)ˆA (cid:17) 2m(cid:16)∂ µΩˆA(cid:17) A 2m 1 µ2 ∂ΩˆA∂A (inraNdius)gravitywaveswith
− A L − (cid:16)∂µ (cid:17) − − ∂µ ∂µ ω= l(l+1), l=1,2,... (17)
kr
(cid:16) (cid:17) ∂ µΩˆ (cid:16) ∂(cid:17)Ωˆ ∂V The instpabilities we shall find are thus due to either magnetic
+ mΩˆ ˆV +2m V+2m 1 µ2 , (9)
field or nonuniformrotation.It shouldbe keptin mindthatthe
L (cid:16)∂µ (cid:17) − ∂µ ∂µ
above equations are only valid for kr 1. We shall see that
(cid:16) (cid:17) (cid:16) (cid:17)
≫
1 theradialcomponentofEq.(5)curledtwiceprovidestheequation themaximumgrowthratesdoindeedbelongtotheshortradial
forthepoloidalflow scales.
4 L.L.Kitchatinov&G.Ru¨diger:Magneticinstabilitiesinstellarradiationzones
2.3.2Dapproximation(λˆ 1) ItissatisfiedwithconstantΩ form=1andclosetothepoles.
A
≫
From Section 2.3 we know that the instability can appear
Generally,theratioofN/Ω inradiativezonesissolarge(Fig.1)
0 onlyin3Dformulationwhenradialdisplacementsareallowed.
thatλˆ can also be large in spite of kr 1. Equation(9) in the
leading order in λˆ then gives S = 0.≫Then Eqs. (16) and (15) Forthis case the equations(9), (13)–(16) have beensolvednu-
merically.
successively yield V = 0 and A = 0. Diffusive terms can also
beneglected.Theequationsystemreducestotwocoupledequa-
tionsfortoroidalmagneticfieldandtoroidalflow(Gilman&Fox
1997),
ωˆ ˆW = mΩˆ ˆW mΩˆ ˆB
A
L L − L
(cid:16) (cid:17) (cid:16)∂2 (cid:17) (cid:16) (cid:17) ∂2
mW 1 µ2 Ωˆ +mB 1 µ2 Ωˆ ,
− ∂µ2 − ∂µ2 − A
ωˆ ˆB = m ˆ ΩˆB(cid:16)(cid:16) m ˆ (cid:17)Ωˆ (cid:17)W . (cid:16)(cid:16) (cid:17) (cid:17) (18)
A
L L − L
He(cid:16)re th(cid:17)e wave n(cid:16)umb(cid:17)er k dr(cid:16)ops ou(cid:17)t and the equations describe
2Ddisturbanceswithindecoupledsphericalshells.
Theparticularcase whereΩandΩ arebothconstantsim-
A
plifiesequations(18)strongly.Theequationshaveconstantco-
efficients in this case and can be solved analytically. The solu-
tionintermsofLegendrepolynomialsW,B Pm(µ)leadstothe
∼ l
eigenfrequencies
ωˆ 1 2 1
=1 Ωˆ2 1 + (19)
m − l(l+1) ± s A − l(l+1) l2(1+l)2
!
of magnetically modified r-modes (Longuet-Higgins 1964;
Fig.2.ThestabilitymapforconstantΩ andzerodiffusivities.
Papaloizou&Pringle1978)describingstablehorizontalpatterns A
ThereisnoinstabilityforweakfieldswithΩ <Ω.Thethresh-
driftinginlongitude.WeshallseeinSection3.1thatthecaseof A
oldfieldstrengthfortheinstabilityincreaseswithincreasingver-
constantΩ and Ω showsTayler instability with 3D approach.
A ticalwavelengthλˆ.
2D approximation,however,misses the instability. Thoughthe
result is obtained for particular case of constant Ω it is most
A
probablyvalidingeneral.2Dincomressivedistortionsonspher-
Considerfirstidealfluidswithzerodiffusivities,χ=η=ν=
icalsurfacesdonotchangeareaencircledbytoroidalmagnetic
0.Figure 2givestheresultingstabilitymap.Standardnotations,
field lines. The circular lines of background field have mini-
SmandAm,areusedforthesymmetrytypesoftheglobalmodes
mum length for given encircled area. Any distortion increases
withazimuthalwavenumbermsymmetricandantisymmetricto
thelengthofclosedfieldlinesthusincreasingmagneticenergy.
theequator.However,thedisturbancesofdifferentfieldsofthe
Thereisnopossibilitytofeedaninstabilitybymagneticenergy
samemodepossessdifferentsymmetries.Ratherarbitrarily,our
release.Onlywithdifferentialrotation2Dinstabilitiesarepossi-
symmetrynotationscorrespondtothesymmetryofthepotential
ble(Gilman&Fox1997).
W oftoroidalfield.Wedidnotfindinstabilitiesform,1.
Thesmallestfieldstrengthproducinginstabilitycorresponds
3. Resultsanddiscussion toshortestradialscales,λˆ 0.Theradialvelocityisalsozero
→
inthislimit.Notethatputtingu = 0regardlessofwhichvalue
We proceedby discussing numericalsolutions of the perturba- ′r
theverticalwavelength,λ,hasreturnstothe2Dcasewithoutany
tion equations for special profiles of Ω . In the present paper
A instability. The pointhere is that for u = 0 the entire poloidal
theinteractionoftoroidalmagneticfieldsandthebasicrotation ′r
flowvanishes.Thehorizontalpartoftheflowisproportionalto
isformulatedonlyforrigidrotation.Theworkwithdifferential
u r/λ.Ahorizontalpoloidalflowisnecessaryforthepinch-type
rotationis much morecomplicated,butit is in progress.In the ′r
instability and the flow remainsfinite when radialvelocity and
Appendixasafirstannouncementthehydrodynamicstabilityof
radialscalebothapproachzerokeepingtheirratioconstant.The
latitudinaldifferentialrotationin3Disdiscussed.
largestgrowthratescorrespondtothislimit.Inthissense,thein-
Forthetoroidalfieldtwosimplegeometriesareconsidered.
stabilityindeedproceedsvia‘almosthorizontal’displacements.
First,thequantityΩ istakenconstantsothatthetoroidalfield
A Thetendencyfortheinstabilitytopreferindefinitelyshortradial
hasonlyonebeltsymmetricwithrespecttotheequator.Second,
scalesshowsthatfinitediffusivitiesshouldbeincluded.
twomagneticbeltsareconsideredwithequatorialantisymmetry,
Figure2 doesnotshow anyinstability for weak fields with
i.e.withanodeofB attheequator.
φ Ω < Ω. With other words, the basic rotation makes a strong
A
stabilizing effect. This may be a special propertyof our model
3.1.Fieldswithequatorialsymmetry wheretheratioΩA/Ωisuniform(Pitts& Tayler1985).We in-
deed find that the threshold field for Ω cosθ is about ten
Constant Ω and Ω give the simplest realization of the Tayler A ∼
A timessmallercomparedtoconstantΩ .Amuchstrongerdesta-
A
instability.Theinstabilitycriterionfornonaxisymmetricdistur-
bilization,however,isproducedbyfinitediffusion.
bances(Goossensetal.1981)reads
Fromnowonthevalues
∂ 2µ2 m2
∂µln B2φ < µ 1−µ2 , forµ≥0. (20) ǫν =2 10−10, ǫη =4 10−8, ǫχ =10−4, (21)
− · ·
(cid:16) (cid:17)
(cid:0) (cid:1)
L.L.Kitchatinov&G.Ru¨diger:Magneticinstabilitiesinstellarradiationzones 5
whicharecharacteristicfortheupperpartofthesolarradiative thegrowthrate.ThedependenceofFig.4doesindeedapproach
core, are used for the diffusion parameters (11). The relations the σ Ω relation (σ is the growth rate) in the strong-field
A
∼
χ η ν of Eq. (21) are quite typical of stellar radiation limit. The growth rate drops by almost four orders of magni-
zon≫es. ≫ tudewhenΩˆ isreducedbelow1.Intheweak-fieldregimeitis
A
σ Ω2/Ω(Spruit1999).
∼ A
Fig.3.ThesameasinFig.2butwiththefinitediffusivities(21).
The critical field strengths for the onset of the instability are
stronglyreducedcomparedtoidealMHD.
ForidealfluidstheTaylerinstabilityoperateswithextremely
small radial scales. When there is no stabilizing stratification,
Fig.5. Toroidal field lines of the most rapidly growing eigen-
however, finite vertical scales are preferred (Arlt et al. 2007).
modesofFig.4forweakbackgroundtoroidalfield,Ω = 0.1Ω
The thermalconductivitydecreasesthe stabilizing effect of the A
(upper panel), and strong field, Ω = 10Ω (lower panel). The
stratificationandreducesstronglythecriticalfieldstrengthsfor A
horizontalaxisgivesthelongitude.
theinstability.ThecharacteristicminimainFig.3correspondto
small,λˆ <1,butfiniteverticalscales.
The g∼rowth rates for weak fields (Ω < Ω) where the in- The structures of the unstable modes differ between strong
A
stabilityexistsduetofinitediffusion,areverysmall.Forstrong andweakfieldregimes.Fortheweakfieldstheresultingpattern
fields(Ω >Ω)thebasicrotationisnotimportantandΩ scales isdistributedovertheentiresphere.Forstrongfieldsitismuch
A A
more concentratedto the poles (Fig 5) but it remainsglobal in
latitude.
3.2.Fieldswithequatorialantisymmetry
Nowa simpletoroidalfieldwith twobeltsofoppositepolarity,
i.e.
Ωˆ =bcosθ, (22)
A
is considered with the diffusivity set (21). Such a field geome-
try can resultfromthe action ofdifferentialrotationondipolar
poloidalfields.Figure6showsthecorrespondingstabilitymap.
ItissimilartothemapofFig.3.Instabilityisagainfoundonly
form=1.
Inphysicalvaluesfortheupperpartofsolarradiationzone
(withρ 0.2g/cm3)onefinds
≃
B 105b G (23)
φ
≃
and
Fig.4.Thenormalizedgrowthrateσˆ asfunctionofthemagnetic λ 10λˆ Mm, (24)
≃
fieldamplitudefortheS1modeandforλˆ = 0.6(wheretheline
sothatfromFig.6followsacriticalmagneticfieldfortheinsta-
ofFig.3hasaminimum).Thedottedlinesforstrong,Ω > Ω,
A bilityslightlybelow600G. Themodewhichfirstbecomesun-
and weak, Ω < Ω, fields give approximations by the power
A stableifthefieldexceedsthiscriticalvaluehasaverticalwave-
lawsσˆ Ωˆ andσˆ Ωˆ2 respectively.
∼ A ∼ A lengthbetween1and2Mm.Forlargerfieldstrengths,ofcourse,
6 L.L.Kitchatinov&G.Ru¨diger:Magneticinstabilitiesinstellarradiationzones
Fig.8.Thegrowthrateasfunctionofthetoroidalfieldamplitude
forλˆ = 0.1.Thedottedlineshowstheparabolicapproximation
Fig.6.Stabilitymapforthefieldmodel(22)withtwobeltsand σˆ = 0.1b2.Thescaleontherightgivestheestimatedradialdif-
equatorialantisymmetry. fusivityofchemicalspeciesafter(25).
thereisarangeofunstablewavelengths.Themaximumgrowth
ratesappears,however,atwavelengths <1Mm(Fig.7). instabilityintheupperradiationzoneoftheSun.Weshouldnot
∼ forget,however,thatthesuperrotation(∂Ω/∂r>0)atthebottom
of the convectionzone in the equatorial region acts stabilizing
sothatthecriticalfieldamplitutesforTaylerinstabilitymaybe
higherthanthecomputed600G.
4. Summary
Linear stability of toroidalmagnetic field in rotating stellar ra-
diation zones is analyzed assuming that the vertical scale of
the fluctuations is short compared to the local radius (‘short-
wave approximation’). The analysis is global in horizontal di-
mensions.Stability computationsconfirmthatthe mostrapidly
Fig.7.GrowthratesinunitsofΩforb=0.01(left)andb=0.1
growingperturbationshaveshortradialscales:kr 103.
(right).The largestrates exist for λˆ < 0.1 independentof the ∼
∼ We have shown that pinch-type instability of toroidal field
magneticfieldamplitude.AllforS1modes.
requirenonvanishingradialdisplacements.The instabilitydoes
notappearin2Dapproximationwithzeroradialvelocities.The
Theflowfieldofanyinstabilitymixeschemicalspeciesalso maximum amplitude of stable toroidal magnetic fields for the
inradialdirection.Suchaninstabilitycanthusberelevanttothe Sunwhichwehavefoundisabout600G.Thisvalueresultsonly
radial transport of the light elements (Barnes et al. 1999). The forrigidrotation.Itwillmostprobablyincreaseifthestabilizing
effective diffusivity, D uℓ (u and ℓ are rms velocity and influence of the positive radial gradient of Ω in the equatorial
T ′ ′
correlationlength in radia≃l direction)can roughlybe estimated regionofthetachoclineisincludedintothemodel.
from our linear computationsassuming that σ ℓ/u and ℓ The field strength in the upper part of the solar radiative
′
λ/2.WithEq.˜(24)thisyields ≃ ≃ interior can only marginally exceed the resulting critical val-
ues. Otherwise the instability would produce too strong radial
D 7 109σˆ cm2s 1, (25)
T ≃ · − mixing of light elements. After our results all the axisymmet-
where σˆ is the normalizedgrowth rate given in Fig. 8. For the ric hydromagnetic models of the solar tachocline (Ru¨diger &
rangeof0.01 < b < 0.2wheretheplotiscloselyapproximated Kitchatinov1997;Garaud2002;Suleetal.2005;Kitchatinov&
by the parabolic law σˆ 0.1b2, Eq. (25) can be rewritten in Ru¨diger 2006) have stable toroidal fields. On the contrary, the
termsofB (Eq.(23))as≃ strongfields 105Gwhichareabletomodifyg-modesoreven
φ
∼
strongerfieldswhichmayinfluencesolarneutrinosarestrongly
B 2
D 7 104 φ cm2s 1. (26) unstablewithe-foldingtimesshorterthanonerotationperiod.
T −
≃ · 1kG 3D computationsof joint instabilities of toroidalfields and
!
differentialrotation(Gilman&Fox1997;Cally2003,Ru¨digeret
As known, diffusivities exceeding 103cm2s 1 in the upper
− al.2007)canbeaperspectiveforfurtherwork.Anothertempting
radiativecorearenotcompatiblewiththeobservedsolarlithium
extensionistheinclusionofthepoloidalfield.Thefieldcanbe
abundance.Hence,thetoroidalfieldamplitudecanonlyslightly
importantinviewoftheveryshortverticalscalesoftheunstable
exceed the marginal value of about 600 G. In our (simplified)
modes.
formulation,the observed solar lithium abundance seem to ex-
clude2anyconceptofhydromagneticdynamosdrivenbyTayler IntheAppendixwepresentacalculationwiththesameequa-
tionsforthe hydrodynamicinstability oflatitududinaldifferen-
2 orrisenewproblems tialrotation.Thisinstabilitycanalreadybefoundin2Dapprox-
L.L.Kitchatinov&G.Ru¨diger:Magneticinstabilitiesinstellarradiationzones 7
imations(Watson1981)butitissubstantiallymodifiedinthe3D Weseethatshortratherthanlongradialscalesarepreferred.The
theory. minimumaappearsforλˆ 0.6,sothatthecharacteristicwave-
≃
lengthofλ 6MmresultsafterEq.(24)forthesolartachocline.
Acknowledgements. This work was supported by the Deutsche ≃
Forschungsgemeinschaft and by the Russian Foundation for Basic Research
(project05-02-04015).
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AppendixA: TheWatsonproblemin3D
Latitudinal differential rotation can be unstable even without
magnetic field if the shear ∂Ω/∂θ is sufficiently large (Watson
1981; Dziembowski & Kosovichev 1987). The instability may
reducethedifferentialrotationtoitscriticalvalue(Garaud2001)
whichcanberelevanttothetheoryofthesolartachocline.The
critical relative value of 28% for differential rotation found by
Watson resulted from a 2D theory (cf. Section 2.3). The value
hasalsoappearedina3Dnumericalprobeofmarginalstability
ofashellrotatingfastenoughwiththerotationlaw
Ω=Ω 1 acos2θ , (A.1)
0
−
butforn(cid:16)otstratified(cid:17)material(Arlt,Sule& Ru¨diger2007).The
critical shear increases to much higher values, however, if the
real rotation law (including its radial variation) of the solar
tachoclineisadopted.
Equations(9),(13),and(16)oftheSection2.2(intheirhy-
drodynamicalversion)canalsobeappliedtoextendtheWatson
approach by allowance for radial displacements. Note that the
ReynoldsnumberRe=Ωr2/νcanbewrittenas
N2/Ω2
Re= , (A.2)
ǫ
ν
whichwithEq.(21)givesaverylargevalue,O(1015).
For positive and sufficiently large a, the modes A1 and S2
becomeunstable.FigureA.1showsthedependenceofthecrit-
icalvaluesofaonthenormalizedwavelengthλˆ (10).Forlarge
enoughradialwavelengthsthe28%-valueofthe2Dtheoryisre-
produced.Itisreduced,however,toa=0.21in3Dcalculations.