Simple Scaling Relationships for Stellar Dynamos

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by  Kyle Augustson

The19thCambridgeWorkshoponCoolStars,StellarSystems,andtheSun
EditedbyG.A.Feiden
Simple Scaling Relationships for Stellar Dynamos
KyleAugustson1,StéphaneMathis1,AllanSachaBrun1
1LaboratoireAIMParis-Saclay,CEA/DRF–CNRS–UniversitéParisDiderot,IRFU/SApCentredeSaclay,F-91191Gif-sur-YvetteCedex,
France
7
Abstract
1
0 Thispaperprovidesabriefoverviewofdynamoscalingrelationshipsforthedegreeofequipartitionbetweenmagneticand
2 kineticenergies. Threebasicapproachesareadoptedtoexplorethesescalingrelationships, withafirstlookattwosimple
models: oneassumingmagnetostrophyandanotherthatincludestheeffectsofinertia. Next, athirdscalingrelationshipis
n
derivedthatutilizestheassumptionsthatthedynamopossessestwointegralspatialscalesandthatitisdrivenbythebalance
a
ofbuoyancyworkandohmicdissipationasstudiedinDavidson(2013). Theresultsofwhicharethencomparedtoasuiteof
J
convectivedynamosimulationsthatpossessafullyconvectivedomainwithaweakdensitystratificationandthatcaptured

0 thebehavioroftheresultingdynamoforarangeofconvectiveRossbynumbers(Augustsonetal.,2016).
1

]
R
1 Introduction for thought experiments. Further, one can attempt to iden-
S
tify a few regimes for which some global-scale aspects of
. Apreciserubricforpredictingthenatureofthesaturated
h those dynamos might be estimated with only a knowledge
stateofturbulentconvectivedynamosremainsaselusiveas
p of the basic parameters of the system. The following ques-
tracing individual convective eddies: they can be identified
- tions are examples of such parametric dependencies: how
o for a time, but they eventually are lost in the tumult. Nev-
the magnetic energy contained in the system may change
r ertheless, one can hope to approximate the shifting nature
t withamodifiedlevelofturbulence(orstrongerdriving),or
s ofthosedynamos. Theeffectsofastrophysicaldynamoscan how does the ratio of the dissipative length scales impact
a bedetectedatthesurfaceandintheenvironmentofagiven
thatenergy,orhowdoesrotationinfluenceit? Establishing
[ magnetically-active object, such as stars (e.g., Christensen
the global-parameter scalings of convective dynamos, par-
1 et al., 2009; Donati & Landstreet, 2009; Donati, 2011; Brun ticularly with stellar mass and rotation rate, is useful given
v et al., 2015). One direct way to approximate the dynamics that they provide an order of magnitude approximation of
occurringwithinsuchanobjectistoconductlaboratoryex-
2 themagneticfieldstrengthsgeneratedwithintheconvection
8 perimentwithfluidsthathavesomeequivalentglobalprop- zonesofstarsastheyevolvefromthepre-main-sequenceto
5 erties,whileobservingtheirresponsetocontrollableparam- aterminalphase. Thisinturnpermitstheplacementofbet-
2 eters, such as the strength of thermal forcing or rotation ter constraints upon transport processes, such as those for
0 rate. In those cases, all the observable scales of the system
elementsandangularmomentum,mostofwhichoccurover
. canbeaccountedfor,fromtheglobalordrivingscaletothe
1 structurally-significantevolutionarytimescales.
dissipation scales. In practice, this has proven to be quite
0
difficultwhentryingtomimicgeophysicalorastrophysical
7 2 FundamentalEquations
dynamos, but they are still fruitful endeavors (e.g., Gailitis
1
: etal.,2001;Laguerreetal.,2008;Spenceetal.,2009). How- In the hunt for a simple set of algebraic equations to de-
v ever, recent experiments with liquid gallium have shown scribethebasicprocessesatwork,itisusefultoconsiderthe
Xi that magnetostrophic states, where the Coriolis force bal- followingsetofMHDequations:
ances the Lorentz force, seem to be optimal for heat trans-
r port (King & Aurnou, 2015), which is interesting given the ∂ρ
a = ∇·(ρv),
strong likelihood that many astrophysical dynamos are in
∂t −
such a state. Another method is to simulate a portion of
∂v ∇P J×B ∇·σ
thoseexperiments,butthesenumericalsimulationsarelim- = (v·∇)v 2Ω×v ∇Φ + + ,
∂t − − − ρ − eff cρ ρ
itedinthescalestheycancapture:eitheranattemptismade
(1)
to resolve a portion of the scales in the inertial range to
down to the physical dissipation scale (e.g., Mininni et al., ∂B
=∇×[v×B ηJ/c],
2009; Mininni & Pouquet, 2009; Brandenburg, 2014), or an ∂t −
attemptismadetoapproximatetheequationsofmotionfor
∇·B=0,
theglobalscaleswhilemodelingtheeffectsoftheunresolved
∂E
dynamicalscales(e.g.,Gilman,1983;Brunetal.,2004;Chris- = ∇·[(E+P σ)v+q]+ρ(cid:15),
tensen&Aubert,2006;Strugareketal.,2016). ∂t − −
Thesevaryingapproachestogatheringdataaboutthein- ∂ρs 1 (cid:20)4πη (cid:21)
ner workings of convective dynamos provide a touchstone = ∇·[ρsv]+ J J+σ : v+ρ(cid:15) ∇·q ,
∂t − T c2 · ∇ −
1
Augustson
where v is the velocity, B is the magnetic field, ρ is the that there are three other forces at work in addition to the
density, P thepressure, andsistheentropyperunitmass. Coriolis, inertial, and Lorentz forces, namely the forces re-
Moreover,thefollowingvariablesarealsodefinedasΦ = sulting from pressure gradients, buoyancy, and viscous dif-
eff
Φ + λ2Ω2/2, Φ is the gravitational potential, Ω is the ro- fusion. Assumingthatsomefraction(1 β,for0 < β < 1)
tation rate of the frame, λ is the distance from the axis of of the inertial force accounts for these−remaining forces, it
rotation, the current is J = c∇×B/4π, with c being the canbeseenthat
speed of light, σ is the viscous stress tensor, q = κ T,
κisthethermaldiffusivityand(cid:15)isaninternalheati−ng∇rate βρv v+2ρΩ ×v 1 (∇×B)×B,
per unit mass that is due to some prescribed exoergic pro- ·∇ 0 ≈ 4π
cess(e.g.,chemical,nuclear,orotherwise). Thetotalenergy β B2
= ρv2 +2ρv Ω ,
isE =ρv2/2+B2/8π+ρΦeff+ρe,whereeistheinternal ⇒ (cid:96) rms rms 0 ≈ 4π(cid:96)
energyperunitmass.
B2 1
Since only the most basic scaling behavior of the stellar = ρv2 (β+2(cid:96)Ω /v),
system is sought, the following assumptions are made: the ⇒ 8π ≈ 2 rms 0
totalenergyofthesystemisconservedandsecondthesys- = ME β+Ro−1, (4)
tem is in a time-steady, nonlinearly-saturated equilibrium. ⇒ KE ≈
Thefirstassumptionleadstotheeliminationofthevolume-
integratedenergyequation. withΩ0 =Ω0eˆztherotationrateofthereferenceframeand
with Ro the convective Rossby number (hereafter denoted
asonlytheRossbynumber).
3 ScalingofMagneticandKineticEnergies
As demonstrated in Augustson et al. (2016), Equation (4)
Onewaytoassesswhatthescalingbehaviorofthemag- may hold for a subset of convective dynamos, wherein the
neticandkineticenergiesinadynamoistofindthebalance ratioofthetotalmagneticenergy(ME)tothekineticenergy
of forces acting in the system when in a quasi-statistically (KE)dependsontheinverseRossbynumberandaconstant
steady, but nonlinear regime. To get a feeling for the ba- offset. The constant is sensitive to details of the dynamics
sicbalances,considerEquation(1)firstforaslowlyrotating and, in some circumstances, it may also be influenced
system,wherethebuoyancy,Coriolis,pressure,andviscous by the Rossby number. In any case, convective dynamos
forces are neglected. Such a simple force balance involves are sensitive to the degree of the rotational constraint on
theLorentzandinertialforceas the convection, as it has a direct impact on the intrinsic
ability of the convection to generate a sustained dynamo.
1
ρv v (∇×B)×B, (2) Yet, even in the absence of rotation, there appears to be
·∇ ≈ 4π dynamo action that gives rise to a minimum magnetic
wherevisthevelocity,Bisthemagneticfield,andρisthe energy state in the case of sufficient levels of turbulence.
density. Supposethattheflowandmagneticfieldvarywith Hence, there is a bridge between two dynamo regimes:
some characteristic length scale (cid:96). Therefore, this balance the equipartition slowly rotating dynamos and the rapidly
yieldsamagneticfieldstrengthinequipartitionwiththeki- rotatingmagnetostrophicregime, whereME/KE Ro−1.
neticenergycontainedintheconvection,suchas For low Rossby numbers, or large rotation rates, i∝t is even
possible that the dynamo can reach superequipartition
Be2q ≈4πρvr2ms. (3) stateswhereME/KE > 1. Indeed,itmaybemuchgreater
thanunity,asisexpectedfortheEarth’sdynamo(seeFigure
Convective flows often possess distributions of length 6ofRoberts&King(2013)).
scales and speeds that are peaked near a single character-
isticvalue. Onesimplemethodtoestimatethesequantities To better characterize the force balance without directly
is to divine that the energy containing flows have roughly resorting to the parameterization above, consider again
the same length scale as the depth of the convection zone Equation (1) but this time taking its curl, wherein one can
and that the speed of the flows is directly related to the seethat
rateofenergyinjection(givenherebythestellarluminosity)
andinverselyproportionaltothedensityofthemediuminto ∂ω (cid:20) 1(cid:18)J×B (cid:19)(cid:21)
=∇× v×ω + +∇·σ ∇P , (5)
whichthatenergyisbeinginjected(Augustsonetal.,2012). ∂t P ρ c −
The latter is encapsulated as v (2L/ρ )1/3, where
rms CZ
ρ istheaveragedensityinthecon∝vectionzone. However, whereω =∇×vandω =2Ω+ω.
CZ P
suchamixing-lengthvelocityprescriptiononlyprovidesan Takingthedotproductofthisequationwithω givesrise
orderofmagnitudeestimateasthepreciselevelofequipar- totheequationfortheevolutionoftheenstrophy. Integrat-
tition depends sensitively upon the dynamics (e.g., Yadav ingthatequationoverthevolumeoftheconvectivedomain
et al., 2016). Since stars are often rotating fairly rapidly, andoverareasonablenumberofdynamicaltimessuchthat
taking for instance young low-mass stars and most inter- thesystemisstatisticallysteadyyields
mediate and high-mass stars, their dynamos may reach a
quasi-magnetostrophic state wherein the Coriolis accelera- (cid:90) (cid:20) 1(cid:18)J×B (cid:19)(cid:21)
dS· v×ω + +∇·σ ∇P ×ω
tion also plays a significant part in balancing the Lorentz P ρ c −
force. Such a balance has been addressed and discussed at (cid:90) (cid:20) 1(cid:18)J×B (cid:19)(cid:21)
length in Christensen (2010) and Brun et al. (2015) for in- + dV (∇×ω)· v×ω + +∇·σ ∇P =0.
stance. To again have a zeroth-order suggestion of the be- P ρ c −
(6)
havior of more rapidly-rotating convective dynamos, note
2 Zenodo,2016
The19thCambridgeWorkshoponCoolStars,StellarSystems,andtheSun
Ifnoenstrophyislostthroughtheboundariesoftheconvec-
tivedomains,thenthesurfaceintegralvanishes,leaving ∂ (cid:18)B2(cid:19) B (cid:104) η (cid:105)
= ·∇× v×B J ,
(cid:90) (cid:20) 1(cid:18)J×B (cid:19)(cid:21) ∂t 8π 4π − c
dV (∇×ω)· v×ω + +∇·σ ∇P =0. (cid:20) (cid:21)
P ρ c − = 1 ∇·(ηJ×B)+ 4πηJ2+v·(J×B) .
(7) −c c
(13)
This assumption effectively means that magnetic stellar
windswillnotbepartofthisscalinganalysis.Fortimescales If this equation is averaged over many dynamical times τ,
whenitisinaquasi-steadystate,andifitisintegratedover
consistent with the dynamical timescales of the dynamos
thevolumeoftheconvectiveregion,ityields
consideredhere,thisisareasonablyvalidassumption.Then,
since ∇×ω is not everywhere zero, the terms in square (cid:90) dt (cid:20)4πη (cid:21) (cid:90) dt
bracketsmustbezero,whichimpliesthat dV J2+v·(J×B) = dS·(ηJ×B). (14)
τ c − τ
ρv×ω + J×B +∇·σ ∇P =0. (8) The Lorentz force (J×B) can vanish at the boundaries of
P c − theconvectiveregionforanappropriatechoiceofboundary
conditions. As an example, if the magnetic field satisfies a
Takingthecurlofthisequationeliminatesthepressurecon- potentialfieldboundarycondition, thenitiszero. Orifthe
tributionandgives field is force-free (e.g., J B), then it is also zero. Sup-
posing that this is the cas∝e, then one has that the average
(cid:20) J×B (cid:21) Lorentz work ((cid:82)dt/τdVv·(J×B)) is equal to the average
∇× ρv×ω+2ρv×Ω+ c +∇·σ =0. (9) Joule heating (cid:15)η = 4π(cid:82)dt/τdVηJ2/c. This is an important
pointthatisoftenbrushedasideinastrophysics,asitshows
that the nature of the convection and magnetic field struc-
This is the primary force balance, being between inertial,
tures are directly impacted by the form of the resistive dis-
Coriolis,Lorentz,andviscousforces. Takingfiducialvalues
sipation. Hence, the use of numerical dissipation schemes
fortheparameters,andscalingthederivativesastheinverse
couldyieldunexpectedresults.
ofacharacteristiclengthscale(cid:96),thescalingrelationshipfor
In a fashion similar to that used for the magnetic energy
theaboveequationyields
evolutionabove,onecanfindthatthekineticenergyevolves
as
ρv2 /(cid:96)2+2ρv Ω /(cid:96)+B2/4π(cid:96)2+ρνv /(cid:96)3 0,
rms rms 0 rms ≈ (10) 1∂ρv2 = ∇·(cid:2)(cid:0)ρv2/2+P σ(cid:1)v(cid:3)+P∇·v σ:∇v
2 ∂t − − −
v
whichwhendividedthroughbyρvr2ms/(cid:96)2gives −ρv·∇Φeff + c·(J×B). (15)
ME/KE 1+Re−1+Ro−1. (11) Toeliminatethepressure,thetotalinternalenergymustalso
∝
beaddedtothesystemas
TheReynoldsnumberusedintheaboveequationistakento
beRe=vrms(cid:96)/ν. Sincethecurlistaken,theapproximation ∂ (cid:2)ρv2/2+B2/8π+ρe(cid:3)=
for the pressure gradient and buoyancy terms employed in ∂t
Equation(4)iseliminatedfromtheforcebalance. However, ∇·(cid:104)(cid:0)ρv2/2+ρe+P σ(cid:1)v ηJ×B(cid:105)
the leading term of this scaling relationship is found to be − − − c
lessthanunity,atleastwhenassessedthroughsimulations. 4πη
ρv·∇Φ J2 σ:∇v+ρ(cid:15) ∇·q. (16)
Hence,itshouldbereplacedwithaparametertoaccountfor − eff − c2 − −
dynamosthataresubequipartition,leavingthefollowing
One can also consider the time-averaged and volume-
integrated evolution equation for this energy equation,
ME/KE β(Ro,Re)+Ro−1, (12)
∝ whichyields
whereβ(Ro,Re)isunknownaprioriasitdependsuponthe (cid:90) dtdV (cid:20)ρ(cid:15) ∇·q ρv·∇Φ 4πηJ2 σ :∇v(cid:21)=
intrinsicabilityofthenon-rotatingsystemtogeneratemag- τ − − eff − c2 −
onfettihcefiseylsdtse,mwshuicchhainstthuernboduenpdenardyscuopnodnittiohnessapnecdifigecodmeteatirlys (cid:90) dtdS·(cid:104)(cid:0)ρv2/2+ρe+P σ(cid:1)v ηJ×B(cid:105). (17)
oftheconvectionzone. τ − − c
The surface integral is zero if there are no outflows or net
4 An Alternative Approach to Scaling Rela- torquefromtheLorentzforceatthedomainboundaries,im-
tionshipsforME/KE plyingthefollowing:
Analternativeapproachtobuildingascalingrelationship (cid:90) dt (cid:20) 4πη (cid:21)
dV ρ(cid:15) ∇·q ρv·∇Φ J2 σ :∇v =0.
fortheratioofmagnetictokineticenergyconsidersthebal- τ − − eff − c2 −
ancesestablishedingeneratingentropy,kineticenergy,and (18)
magnetic energy. To begin, note that the evolution of the
magneticenergyis
Zenodo,2016 3
Augustson
Note that (cid:82)dVρ(cid:15) = L(r), where L is the total luminosity whereg = ∇Φ .
eff eff
of the star at a given radius r for a spherically symmet- This provid−es the basis of a scaling relationship. Follow-
ric heating. Likewise, the radiative luminosity of the star ing Davidson (2013), the Rossby number is assumed to be
is given by (cid:82)dV∇·q = L (r) = 4πr2κ∂T/∂r for the small enough so that the flow becomes roughly columnar
r
spherically-symmetric component of−the temperature field, and moderately aligned with the rotation axis. In such a
whichshouldbedominant. Forthecaseofstarsthatareon case, there are two integral length scales: one parallel to
the main-sequence, there are three configurations of their therotationaxis(cid:96) andanotherperpendiculartoit(cid:96) ,with
(cid:107) ⊥
primary regions of convection: either a convective core for (cid:96) < (cid:96) . Here, unlikeDavidson(2013), thedensitystratifi-
⊥ (cid:107)
highmassstars,aconvectiveenvelopeforlowermassstars, cationisretained.So,densityperturbationsratherthantem-
or both for F-type stars. In all these cases, one can assume peratureperturbationsarecontainedinthebuoyancywork
that the region of integration is over the entire convection integral and the force balance below. The full density can
zone and so the volume-integrated luminosity will be the beretainedintheintegralandinthescalinggiventhatthe
nuclearluminosity,orthecurrenttotalluminosityL . Fur- gradient of the mean density is parallel to g . So, their
∗ eff
thermore,theinwardly-directedradiativeluminositywillbe cross product vanishes, leaving only the product of the ve-
nearly,butnotexactly,equalinmagnitudetoL .Thereason locity and gradients of the density perturbations. Assum-
∗
being that the thermal evolution of the system is a largely ing further that the magnetic energy density per unit mass
passiveresponsetothechangesinthenuclearburningrates. scales only with (cid:96) and W , unit consistency requires that
(cid:107) B
Because the nuclear luminosity is slowly increasing along B2/(4πρ) F (cid:0)(cid:96) ,W (cid:1) (cid:96)2/3W2/3, where W is the
themain-sequence,Lr willalwayslagbehindL∗ duetothe rate of buoy∝ancy w(cid:107)orkBper≈unit(cid:107)massBas above. ThBerefore,
time required for thermal diffusion to modify the thermal
given Equation (22) and assuming that each term is of the
gradient. FromEquation(18),onecanseethat
sameorderofmagnitude,thebasicproportionalityis
(cid:90) dt
Lr+(cid:15)ν +(cid:15)η−L∗ =− τ dVρv·∇Φeff, (19) ρΩ·∇v≈∇ρ×geff ≈∇×(J×B)/c
Ω v g B2
= 0 rms . (23)
vwohluermee(cid:15)-iνntaengdrat(cid:15)eηdadriesstiphaetipoonsirtaitvees-ddeufientiotev,itsicmosei-tayvaernadgered-, ⇒ (cid:96)(cid:107) ≈ (cid:96)⊥ ≈ 4πρ(cid:96)2⊥
sistivity, respectively. Thus, the rate of buoyancy work is Thus, comparing the curl of the Lorentz force to the curl
directly proportional to the mismatch of the two luminosi- oftheCoriolisforce,onehas
ties and the rates of viscous and resistive dissipation, im-
plyingthatthelatterresultfromtheformer. Assumingthat Ω v /(cid:96) B2/4πρ(cid:96)2 (cid:96)2/3W2/3/(cid:96)2, (24)
L L andfollowingBrandenburg(2014),theratioofthe 0 rms (cid:107) ≈ ⊥ ≈ (cid:107) B ⊥
∗ r
diss≈ipationratescanbedescribedas
and moreover it can be shown that an estimate of the rms
velocityis
(cid:15) /(cid:15) =kPmn, (20)
ν η
v (cid:96)5/3W2/3/(cid:96)2Ω . (25)
where Pm = ν/η is the magnetic Prandtl number, and rms ≈ (cid:107) B ⊥ 0
where k and n could be determined from a suite of direct
Withinthecontextofsuchestimates,thebuoyancytermcan
numericalsimulations. Inparticular,whenkinetichelicityis
bewrittenas
injectedatthedrivingscale,Brandenburg(2014)foundthat
k = 7/10 and n = 2/3. Note that their results have also (cid:82) dtdVρv·g
considered rotating driven turbulence and found that this W = τ eff gv , (26)
scaling is effectively independent of the rotation rate, and B (cid:82) dtdVρ ≈ rms
τ
thus the Rossby number. However, other studies indicate
that there may be a stronger rotational influence (Plunian which implies that the estimated magnitude for the curl
&Stepanov,2010). Sincethereisambiguityinthatscaling, of the buoyancy force in Equation (23) is g/(cid:96)
⊥
let k(Ro) be an unknown function of the Rossby number. W /(v (cid:96) ). Then,itiseasilyseenthat ≈
B rms ⊥
Subsequently,thebuoyancyworkW perunitmasscanbe
B
describedas (cid:96) /(cid:96) W /(Ω v2 ). (27)
⊥ (cid:107) ≈ B 0 rms
(cid:82) dtdVρv·∇Φ (cid:16) (cid:17) So,theratioofintegrallengthscalesshouldvaryas
W = τ eff = 1+k(Ro)Pm2/3 (cid:15) /M.
B − (cid:82) dτtdVρ η (21) (cid:96)⊥ WB1/9 =(cid:32) WB (cid:33)1/9. (28)
(cid:96)(cid:107) ≈ Ω10/3(cid:96)2(cid:107)/9 Ω30(cid:96)2(cid:107)
whereM isthemassintheintegratedvolume.
Now, returning to the time-averaged curl of the momen-
Likewisetheratioofmagnetictokineticenergythenscales
tum equation, though neglecting the viscous and inertial
as
terms,onecanfind
(cid:32) (cid:33)−2/9
∇×(cid:20)2ρv×Ω+ 1cJ×B(cid:21)+∇ρ×geff =0, (22) MKEE ≈ 8π(1/B22ρvr2ms) ≈ ΩW30B(cid:96)2(cid:107) . (29)
4 Zenodo,2016
The19thCambridgeWorkshoponCoolStars,StellarSystems,andtheSun
(a) asetofMESAstellarmodelswithasolar-likemetallicityto
4 obtain the density and temperature profiles (Paxton et al.,
− 2011), one can define the average expected molecular mag-
neticPrandtlnumberineithertheconvectivecoreofamas-
6
− sive star or the convective envelope of a lower mass star.
r First, note that the molecular values of the diffusive coeffi-
P
g10 −8 cientsintheNavier-StokesMHDequationsaregivenby
o
l
−10 Envelope ν =3.210−5Te5/2, (32)
ρΛ
12 Core
− T5/2
κ =1.810−6 e , (33)
cond ρΛ
(b) 2 η =1.2106ΛT−3/2, (34)
e
Envelope
0 Core where ν is the kinematic viscosity, η is the magnetic diffu-
sivity,κ istheelectronthermaldiffusivity,κ isthera-
cond rad
m 2 diativethermaldiffusivity,Te istheelectrontemperaturein
P − electronvolts,andΛistheCoulomblogarithm. Thesequan-
g10 titiesarecomputedundertheassumptionsthatchargeneu-
o 4
l − trality holds and that the temperatures are not excessively
high, sothattheCoulomblogarithmiswell-defined. Inthe
6 charge-neutral regime, the magnetic diffusivity happens to
−
be density independent because the electron collision time
8 scales as the inverse power of ion density and the conduc-
−−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 tivityisproportionaltotheelectrondensitytimestheelec-
log M/M
10 (cid:12) tron collision time. Therefore, the thermal Prandtl number
PrandmagneticPrandtlnumberPmscaleas
Figure1:TheaveragethermalPrandtlnumberPr(a)andav-
eragemagneticPrandtlnumberPm(b)forstarswithmasses ν
Pr= , (35)
between0.03and100M(cid:12),withtheaveragebeingtakenover (κcond+κrad)
substantialconvectionzones.Thezonebeingaveragedisin-
T4
dicated by triangles for a convective core and circles for a Pm=2.710−11 e . (36)
convectiveenvelope. ρΛ2
To make a comparison to the earlier results in §3, consider
In most cases, the radiative diffusivity is several orders of
thattheRossbynumberisdefinedas
magnitudelargerthantheelectronconductivityanditthus
dominatesthethermalPrandtlnumber.
(cid:32) (cid:33)4/9
v W TheaboveprescriptionforthePrandtlnumbershasbeen
Ro= rms B , (30)
Ω (cid:96) ≈ Ω3(cid:96)2 applied to MESA models of stars near the zero-age main-
0 (cid:107) 0 (cid:107) sequence in the mass range between 0.03 and 100 M .
(cid:12)
The resulting convective-zone-averaged Prandtl numbers
whichimpliesthat
are shown in Figure (1), where it is clear that all stars pos-
sessconvectiveregionswithalowthermalPrandtlnumber,
ME
Ro−1/2. (31) whereasonecanfindtworegimesofmagneticPrandtlnum-
KE ∝
ber. The existence of these two regimes is directly related
towheretheconvectiveregionislocated. Formassivestars
However, ifonetakesintoaccountthescalingofthebuoy-
withconvectivecores,thetemperatureanddensityaveraged
ancyworkwithRossbynumberandmagneticPrandtlnum-
over the convective volume are quite high when compared
ber, Equation (30) becomes an implicit relationship for the
tolower-massstarswithaconvectiveenvelope. Suchahigh
Rossby number that is indeterminate for large magnetic
temperatureleadstoalargemagneticPrandtlnumber. The
Prandtlnumber.
dichotomy in magnetic Prandtl number implies that there
maybetwofundamentallydifferentkindsofconvectivedy-
5 ComparisonofScalingRelationships
namoactioninlow-massversushigh-massstars.
Since the scaling described in the previous section has a Tocomparethesethreeschemes,considerthedataforthe
magnetic Prandtl number dependence, it is useful to quan- evolution of a set of MHD simulations using the Anelas-
tify the specific regimes in which each of the three differ- tic Spherical Harmonic code presented in Augustson et al.
ent models are most applicable. To do so, consider a fully- (2016). These simulations approximate the convective core
resolved convective dynamo and its associated dynamics, dynamo that likely exists within massive stars. In such
whereinthedissipationoftheenergyinjectedintothesys- 10 M stars, the average Pm is about four throughout the
(cid:12)
temisgovernedbythemolecularvaluesofthediffusivities. core, and it drops to roughly 1/10 close to the stellar sur-
Using Braginskii plasma diffusivities (Braginskii, 1965) and faceintheradiativeexterior. Therefore,theconvectivecore
Zenodo,2016 5
Augustson
(a) (a)102
101
101
E
K
/E ΛD
M 100
100
10−110−2 10−1 100 101 102 10−110−2 10−1 100 101 102
Ro−1 Ro−1
Figure2: Thescalingoftheratioofmagnetictokineticen- Figure3: ThescalingofthedynamicElsassernumber(Λ )
D
ergy (ME/KE) with inverse Rossby number (Ro−1). The withinverseRossbynumber(Ro−1). Theuncertaintyofthe
black curve indicates the scaling defined in Equation (12), measuredRossbynumberanddynamicElsassernumberthat
with β = 0.5. The blue dashed line is for magnetostrophy, arises from temporal variations are indicated by the size of
e.g. β = 0. The green dashed line is that for buoyancy- thecrossforeachdatapoint.
work-limited dynamo scaling, given in Equation (31). The
red dashed line indicates the critical Rossby number of the in Figure (2). This transition to the magnetostrophy is fur-
star, corresponding to its rotational breakup velocity. The ther evidenced in Figure (3), which shows the dynamic El-
uncertaintyofthemeasuredRossbynumberandenergyra- sassernumber
tiothatarisesfromtemporalvariationsareindicatedbythe
sizeofthecrossforeachdatapoint. ΛD =Brms2/(8πρ0Ω0vrms(cid:96)), (37)
where (cid:96) is the typical length scale of the current density J.
can be considered as a large magnetic Prandtl number dy- As a point of reference, when Λ tends toward unity the
namo.WhatmakessuchPmregimesinterestingisthatthey balancebetweentheLorentzandtDheCoriolisforcesalsoap-
are a numerically accessible, but still astrophysically rele- proaches unity, which indicates that the dynamo is nearly
vant, dynamo. Moreover, these convective cores represent magnetostrophic.
an astrophysical dynamo in which large-eddy simulations
can more easily capture the hierarchy of relevant diffusive
6 Conclusions
timescales. Thisisespeciallysogiventhatthedensitycon-
trast across the core is generally small, meaning that one Thisconferenceproceedinghopefullyhasshedsomelight
can simplify the problem to being Boussinesq without the ontheexistenceoftwokindsofastrophysicaldynamos,and
lossoftoomuchphysicalfidelity. Also,theheatgeneration providedscalingrelationshipsforthelevelofthepartition-
fromnuclearburningprocessesdeepwithinthecoreandthe ing of magnetic energy and kinetic energy. Particularly,
transition toradiative cooling nearer theradiative zone are there may be a shift in the kind of dynamo action taking
smoothlydistributedinradius. So,therearenoinherentdif- place within stars that possess a convective core and those
ficultieswithresolvinginternalboundarylayers. that possess an exterior convective envelope. The scaling
WithinthecontextofthesimulationsexhibitedinAugust- relationship between the magnetic and kinetic energies of
son et al. (2016), the rotation rates employed lead to nearly suchconvectivedynamosinturnprovideanestimateofthe
three decades of coverage in Rossby number, as shown in rmsmagneticfieldstrengthintermsofthelocalrmsvelocity
Figure(2). Inthatfigure, theforce-basedscalingderivedin anddensityataparticulardepthinaconvectivezone.
§3andgivenbyEquation12isdepictedbytheblackcurve, Twosuchscalingrelationshipsappeartobethemostap-
whichdoesareasonablejobofdescribingthenatureofthe plicable to the simulations carried out in Augustson et al.
superequipartition state for a given Rossby number. Note, (2016): one for the high magnetic Prandtl number regime
however, that the constant of proportionality has been de- and another for the low magnetic Prandtl number regime.
terminedusingthedataitself. Thisistruealsooftheother Within the context of the large magnetic Prandtl number
twoscalinglawsshowninFigure(2). Surprisingly,thescal- systems,themagneticenergyofthesystemscalesastheki-
ing law derived in §4 does not capture the behavior of this netic energy multiplied by an expression that depends the
set of dynamos very well, in contrast to the many dynamo inverse Rossby number plus an offset, which in turn de-
simulationsanddatashowninChristensenetal.(2009)and pends upon the details of the non-rotating system (e.g., on
Christensen (2010) for which it performs well. This could Reynolds,Rayleigh,andPrandtlnumbers).Forlowmagnetic
be related to the fact that for large Pm the buoyancy work Prandtlnumberandfairlyrapidlyrotatingsystems,suchas
potentiallyhasanadditionalRodependence. the geodynamo and rapidly rotating stars, another scaling
These simulated convective core dynamos appear to en- lawthatreliesuponanenergeticbalanceofbuoyancywork
ter a magnetostrophic regime for the four cases with the andmagneticdissipation(aswellasaforcebalancebetween
lowest average Rossby number. For emphasis, the magne- thebuoyancy,Coriolis,andLorentzforces)maybemoreap-
tostrophicscalingregimeisdenotedbythedashedblueline plicable. When focused on in detail, this yields a magnetic
6 Zenodo,2016
The19thCambridgeWorkshoponCoolStars,StellarSystems,andtheSun
energythatscalesasthekineticenergymultipliedbythein- Spence,E.J.,Reuter,K.,&Forest,C.B.2009,TheAstrophys-
verse square root of the convective Rossby number. Such icalJournal,700,470.
a scaling relationship has been shown to be fairly robust Strugarek, A., Beaudoin, P., Brun, A. S., Charbonneau, P.,
(Davidson,2013). Mathis, S., et al. 2016, Advances in Space Research, 58,
Yet, more work is needed to establish more robust scal- 1538.
ingrelationshipsthatcoveragreaterrangeinbothmagnetic Yadav,R.,Gastine,T.,Christensen,U.,Wolk,S.J.,&Poppen-
PrandtlnumberandRossbynumber.Likewise,numericalex- haeger,K.2016,Proc.Nat.Acad.Sci.,113.
perimentsshouldexplorealargerrangeofReynoldsnumber
andlevelofsupercriticality. Indeed,asinYadavetal.(2016),
someauthorshavealreadyattemptedtoexaminesuchanin-
creasedrangeofparametersforthegeodynamo. Neverthe-
less, to be more broadly applicable in stellar physics, there
is a need to find scaling relationships that can bridge both
thelowandhighmagneticPrandtlnumberregimesthatare
showntoexistwithinmain-sequencestars. Theauthorsare
currently working toward this goal, as will be presented in
anupcomingpaper.
Acknowledgments
K.C.AugustsonandS.Mathisacknowledgesupportfrom
theERCSPIRE647383grant. A.S.Brunacknowledgesfund-
ing by ERC STARS2 207430 grant, INSU/PNST, CNES So-
larOrbiter,PLATOandGOLFgrants,andtheFP7SpaceInn
312844grant.
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