Astron. Nachr. / AN 332, No. 9/10, 983 – 987 (2011) / DOI 10.1002/asna.201111610

Numerical simulations of magnetic relaxation in rotating stellar radia-tion zones

V. Duez�

Argelander Institut fur Astronomie, Universitat Bonn, Auf dem Hugel 71, D-53121 Bonn, Germany

Received 2011 Oct 1, accepted 2011 Oct 28Published online 2011 Dec 12

Key words magnetohydrodynamics (MHD) – methods: numerical – stars: magnetic fields – stars: rotation

I detail the results of simulations of magnetic relaxation, as is assumed to happen in stellar radiation zones: startingfrom an initially stochastically organized magnetic field, this one self-organizes towards a large-scale, mainly dipolarconfiguration with both poloidal and toroidal components. While previous studies focused on the description of staticequilibria, I include here the rotation in the numerical setup which substantially modifies the outcome of the simulations.In particular, magnetic equilibria are found, whose configuration remain stable over diffusive timescales (in general of theorder of the main-sequence lifetime) rather than dynamic ones, as would be the case in presence of MHD instability. Theirprivileged axis are inclined with respect to the rotation axis, as observed in early-type, magnetic main sequence stars. Thestationary internal flows and the dependence of the final equilibrium state on the rotation rate are described. Implicationson stellar evolution are discussed.

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1 Introduction

Spectropolarimetric observations routinely reveal the pres-ence of magnetic fields at the surface of a substantial pro-portion of early type stars, in particular of the chemicallypeculiar ones. It is thus clear that magnetism is somehowlinked with their evolution as it potentially modifies mi-croscopic transport processes, as well as macroscopic onesby interfering with rotation and meridional flows. Besides,current stellar evolution models including all known hydro-dynamic transport processes still present difficulties in ex-plaining many observed features, among which internal so-lar rotation profile recovered from helioseismology, chemi-cal abundances anomalies in intermediate as well as massivestars (nitrogen enrichments in slowly rotating massive stars,hydrogen in type IIn supernovae progenitors, etc.). How-ever, the details of the interplay between magnetic and hy-drodynamic processes remain uncertain.It is generally admitted that these magnetic fields of-ten display large-scale organized features, since observa-tions can in general be interpreted in terms of an inclined(the “oblique rotator”) dipole model (Bagnulo et al. 2002;Babcock 1949; Donati & Landstreet 2009; Stibbs 1950)with lower contributions of the higher multipoles, althoughsmaller scale features are mostly unresolved. But the waysuch a large-scale magnetic equilibrium is reached, startingfrom a field certainly organized on much smaller scales, stilllacks of a coherent theoretical frame.I here present the results of numerical simulations re-producing the relaxation process occurring from an initialstochastic magnetic field (presumably of primordial origin

� Corresponding author: vduez@astro.uni-bonn.de

or being the result of a dynamo having operated earlier, dur-ing the pre-main-sequence phase) in order to decipher thephysical processes at work during the formation of a large-scale magnetic field in equilibrium in a stratified region. Inboth cases, it is taken for granted that the initial field is orga-nized on much smaller scales than the final equilibrium; thislatter one appears to be a minimum-energy state in agree-ment with the hydrodynamical constraints on the system.For the first time, the effects of rotation are included in thenumerical set-up.

2 Numerical set-up

2.1 Implementation

Use is made of the STAGGER code (Gudiksen & Nord-lund 2005; Nordlund & Galsgaard 1995), a high-orderfinite-difference Cartesian MHD code which uses a “hyper-diffusion” scheme, a system whereby diffusivities are scaledwith the length scales so that badly resolved structures nearthe Nyquist spatial frequency are damped, while preservingwell-resolved structure on longer length scales. This, andthe high-order spatial interpolation and derivatives (sixth or-der) and time-stepping (third order) increase efficiency bygiving a low effective diffusivity even at modest resolution.Since one does not need to resolve turbulent features, a res-olution of 1283 is here adopted. The code includes Ohmicas well as thermal and kinetic diffusion. Using Cartesian co-ordinates avoids problems with singularities and simplifiesthe boundary conditions: periodic boundaries are used here.The rotation is implemented through its Coriolis acceler-ation only, while the centrifugal acceleration is neglected

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984 V. Duez: Magnetic relaxation in rotating stars

to avoid undesired effects arising from the enhanced ro-tation rate used in the simulation. However, the hierarchy(detailed infra) between sound crossing time (τs), Alfvencrossing time (τA), rotation period (T ) and Ohmic diffusivetimescale (τη) is well-preserved: τs � τA, T � τη .

2.2 Initial conditions

The star is modeled as a self-gravitating ball of ideal gas(γ = 5/3) with radial density and pressure profiles initiallyobeying the polytropic relation P ∝ ρ1+(1/n), with indexn = 3. This approximates a stably stratified radiation zoneof an early-type, main sequence star. The influence of aconvective core is neglected: owing the high turbulent dif-fusivity the large scale field penetrates this region easily – itis thus assumed to be unaffected.I start from an initial normal distribution for the magneticenergy distribution: |B| = B0 exp(−r2/2B2

w) with thescale parameter Bw = 0.35 R∗; |B|2,B0, R∗ and r beingrespectively the magnetic energy, the central magnetic fieldintensity, the stellar radius and the radius. This distributionand the initial seed for the random distribution in the direc-tion of the stochastic field are chosen so that the resultingfield in the non-rotating case is a simple nested torus, todeal with the simplest case. As was shown by Braithwaite(2008), different choices for the initial field distribution canresult in non-axisymmetric equilibria. The influence of ro-tation on the development of these non-axisymmetric con-figurations will be explored in a forthcoming work.The magnetic energy is kept fixed throughout the differentruns; in contrast the rotation rate changes from one run toanother, in order to cover a wide range of values spanningfrom T � τA (this is the case e.g. for a typical Ap star,where τA � 10 yrs and T � 100 days), to T � τA (whichcan easily be reached for slowest rotating stars, if we as-sume that the inner field strength is two orders of magnitudehigher than that at the surface).

3 Results

3.1 Settling of the equilibrium

The evolution covering approximately 8 Alfven crossingtimes (up to t = 600 in arbitrary code units) happens inthree consecutive phases: (i) the relaxation itself for 0 ≤t ≤ 120 (∼ 1.5 τA); (ii) the stationary evolution once thefield has reached the relaxed state for 120 ≤ t ≤ 450(corresponding to ∼ 4.5 τA); (iii) the diffusive limit for450 ≤ t ≤ 600 (2 τA) .

Relaxation phase In this first phase, the system evolvesfrom small scale to large scale coherence. The ideal MHDinvariants are no longer conserved. During this phase themagnetic diffusivity η plays a key role in helping the systemto organize into a larger scale equilibrium, by smoothing out

larger angular modes structures: a spherical harmonics de-composition of the spectrum readily shows that δB/δt �−η m2 B in the limit of high azimuthal wavenumbers m.As the magnetic energy is kept constant throughout the runhowever, it seems hard to follow the actual MHD invariantsconserved during this phase. However, one has to keep inmind that this energy is injected continuously on the threecomponents of the magnetic field, without giving privilegeto a single mode; hence the final equilibrium doesn’t differqualitatively from the one obtained without this numericalartifact. It has been checked by running simulations deacti-vating the upscaling routine that it does not affect qualita-tively the outcome of the situation.

Stationary evolution During the second phase, theemergence of a large-scale equilibrium is observed, and cor-responds to a relaxed state: internal flows reach a stationarystate, as well as the overall magnetic configuration. In thefaster rotating case investigated, the simulation covers about40 rotation periods. The well-identified relaxed phase cov-ers more than 20 rotation periods, over which the internalflows are observed to be stationary (cf. following section).Furthermore, although the diffusivity is always the same,some ideal MHD invariants are indeed conserved. Amongthem the magnetic helicity, and higher order invariants areconserved (cf. Fig. 1): in particular, the mass enclosed inmagnetic flux surfaces which was shown by Chandrasekhar& Prendergast (1958) to be conserved in the ideal MHD,axisymmetric case. It was highlighted by Duez & Mathis(2010) that taking into account this invariant in the varia-tional method used to find the lower energy state in a self-gravitating system was necessary to take into account thenon-force-free character of the global magnetic field. Hencethis equilibrium is of non-force-free type. Furthermore, onecan draw the characteristic magnetic scale: in the static, ax-isymmetric case, the magnetic field can be expressed as afunction of a poloidal flux function Ψ(r, θ) and a toroidalpotential F (r, θ) according toB(r, θ) = (1/r sin θ) (∇Ψ× eϕ + F eϕ) . (1)When the configuration is axisymmetric, eϕ is colinear with(eT, the toroidal unit vector. In the non-axisymmetric case,if the magnetic field present a main dipolar configurationone can simply identify the toroidally averaged stream func-tions Ψ(r, δ) and F (r, δ) according toB(r, δ) = (1/r sin δ) (∇Ψ × eT + F eT) , (2)where δ = β−θ, β being the angle between the rotation andthe magnetic dipole axis. The simple linear non-force-freemodel predicts F (r, θ) = F (Ψ), where F (Ψ) = λ Ψ/R∗,the proportionality factor being the magnetic characteristiclength. Figure 1 shows that this factor is constant in the re-laxed phase, even in the non-axisymmetric case.

diffusive limit The third and last phase of the simula-tion, corresponds to the diffusive limit: the neutral line mi-grates towards the surface; the high poloidal field compo-nent reaches the surface and the flux is lost from the star. In

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Astron. Nachr. / AN (2011) 985

this limit, the invariants are no longer conserved, as can beseen on Fig. 2.

Fig. 1 (online colour at: www.an-journal.org) Three-dimensional representation of magnetic field lines for theperpendicular rotator.

3.2 Final state: generalities

Including rotation, I obtained a set of stationary MHDequilibria similar to the one obtained in the non-rotatingcase. Figure 2 shows the representation of the perpendic-ular dipole resulting from the relaxation in the slowly rotat-ing regime. The field lines ( parallel to B ) are drawn. Thefinal magnetic configuration is a twisted torus, with a strongtoroidal component inside the star while the magnetic fieldlines in the vicinity of the stellar surface (not shown here)match with a quasi dipolar, potential field at the surface andare thus constrained to be poloidal in all cases.

3.3 Final state: magnetic dipole inclination

It is interesting to evaluate the dependency of the inclinationangle β between the rotation axis and the magnetic privi-leged axis (torus revolution axis) B =

∫V

r × B dV , assome tendencies have been reported following recent obser-vations on Ap stars Bagnulo et al. (2002); Mathys (2008).Large inclinations (β → 90◦) seem to arise in fast rotators(T < 100 d); slow rotators (100 d < T < 1000 d) display atendency to axisymmetry (β → 0◦) while for very slow ro-tators (T > 1000 d) β increases again. Figure 3 shows theevolution of this angle β as a function of the rotation rateimprinted to the model for seven simulations covering sevendifferent rotation rates spanning approximately 8 τA. Forhigh rotation rates, the inclination angle goes to 0◦, while

Fig. 2 (online colour at: www.an-journal.org) Temporal evolu-tion of the invariants in the slowly rotating case, showing thethree phases: (i) for 0 ≤ t ≤ 120 the relaxation phase; (ii) for120 ≤ t ≤ 450 the relaxed, stationary phase; (iii) for 450 ≤ t ≤

600, the diffusive limit. Top: mass∫V

ρ dV (solid blie line), idealMHD invariant

∫V

ρ Ψ dV (mass included in polodal magneticflux surfaces, dotted green line), and

∫V

ρ Ψ2 dV (dash-dotted red

line). Bottom: toroidal flux∫V

F dV (solid blue line), ideal MHDinvariant

∫V

F Ψ dV (magnetic helicity, dotted green line), and∫V

F Ψ2 dV (dash-dotted red line).

for low rotation rates, the final state tends to a perpendicularrotator (β → 90◦). On Fig. 3 is plotted the final inclina-tion angle as a function of the rotation rate. Two distinctsbehaviours emerge. One below Ω = 0.75, where the finalinclination angle decreases linearly from 90◦ to 0◦ i.e. fromthe perpendicular dipole to the aligned one, and above thisvalue a saturated regime with β in the vicinity of 0◦. As thetransition between the two regimes occurs for Ω � ωA, it islikely that some rotational smoothing process is responsiblefor the observed behaviour. This will need to be confirmedby other similar simulations starting with different initialconditions.

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986 V. Duez: Magnetic relaxation in rotating stars

Fig. 3 (online colour at: www.an-journal.org) Temporal evolu-tion of the angle β (modulo 180

◦; actual values given in legend)between the rotation and the magnetic privileged axis.

Fig. 4 (online colour at: www.an-journal.org) Temporal evolu-tion of the final inclination angle β (and modulo 180

◦) as a func-tion of the rotation rate.

3.4 The slow rotator case: inner flows

The models investigated differ not only because of their in-clination angle, but also due to the way they develop dif-ferent stationary internal flows. I detail here the slowly ro-tating case. Figure 5 shows cross sections of the magneticfunctions and the angular frequency Ω = vϕ/(r sin θ),toroidally averaged (in frame associated with the magneticfield, here with a quasi-perpendicular inclination). One canreadily see two main properties: on one hand a dependencyof the type F ∝ [Ψ/(r sin δ)]α, with α of the order unity,in agreement to what was assumed in earlier works, wherein the axisymmetric case one had F ∝ Ψ/(r sin θ) (Duez& Mathis 2010); on the other hand a departure to the Fer-raro state (Ferraro 1954), with two counter rotating toroidalflows. Therefore, though on azimuthal average responsiblefor a solid-body rotation as was predicted by Moss (1989),these flows mainly concentrated deep inside the star may be

Fig. 5 (online colour at: www.an-journal.org) Top: cross-sectionin the magnetic reference frame (toroidal average) of the toroidalflux function F amplitude (color scale); overimposed are the iso-contours of the flux function Ψ (top); Bottom: same with the am-plitude of the toroidal velocity vT (color scale) and the flux func-tion Ψ r sin δ (isocontours). All quantities are normalized to theirmaximum.

responsible for an extra-mixing mechanism associated withthe presence of a large-scale magnetic field. This is to beobserved in parallel to the recent results obtained from the

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Astron. Nachr. / AN (2011) 987

VLT-Flames Survey, showing evidences of extra-mixing inintrinsically slowly rotating massive stars through surfaceenrichments in Nitrogen (Brott et al. 2011).

4 Discussion

The question of a dynamo operating directly inside the radi-ation zone (might it be on a transient timescale), or ratherhaving operated earlier during the PMS is not addressedhere. Recently, Arlt & Rudiger (2011) explored the pos-sibility that the observed fields are remnants of the Taylerinstability of toroidal fields in the stellar interiors. It is in-teresting to compare the gradual evolution into stable fieldsobtained in the present set-up with an eruptive emergenceof such fields (which will later evolve on diffusive time-scale), as Arlt & Rudiger’s scenario suggests. Their studyindicates that this timescale will be way shorter than evo-lutionary timescales: the longest emergence delays seen intheir simulations are about 8 years, with a strongly stablestratification. This is obtained in the case of a fast rotation,but since rotation present a stabilizing effect, the emergencetime is even shorter for slow rotators. As a consequence,if magnetic fields would appear on Ap stars at later timesduring their evolution as suggested by Hubrig et al. (2000),it should be by other factors influencing the stability limitsof toroidal fields potentially stored below the surface. In oursimulations, in the slow rotation regime the relaxation phaseoccurs on the order of one Alfven timescale, which is alsoof the order of 10 years in the case of a typical Ap star witha 1kG surface field, even shorter if we assume the internalmagnetic field strength is orders of magnitude higher. Fromthis time on, the poloidal magnetic field lines are organizedover large scales at the surface and match with the exte-rior potential field so should also be observable after such ashort timescale, provided the surface field strength is strongenough. Therefore though the method and the approach arequite different, the present simulations lead to a similaritieswith the one obtained by Arlt et al.

5 Conclusion and perspectives

It has been shown how the numerical relaxation methodcan be used as a test-case tool to understand magnetic con-figurations evolution in early-type, main sequence stars. Itprovides precious informations about the equilibrium statebetween differential rotation and meridional flows, whichwill be important when implementing MHD transport pro-cesses in upcoming stellar evolution codes including mag-netic fields. In the cases explored here, magnetic dipoles,presenting a toroidal component inside the star result fromthe relaxation process. Their inclination with respect to themagnetic field depends of the rotation rate: for the simula-tions performed we found that slower rotators give perpen-dicular dipoles while faster rotators give aligned dipoles.Some ideal MHD invariants are shown to be conserved

during the relaxed phase, although we are still in a resis-tive regime. These invariants are described and allows toidentify the type of equilibrium settling: it is similar insome extent to the static equilibria described in recent lit-erature. However, toroidal flows are observed, especially inthe slowly rotating case, although the star remains on aver-age in a solid-body rotation regime. These flows might beresponsible for an extra source of mixing.

Acknowledgements. The author would like to thank J. Braithwaiteand N. Langer for stimulating discussions on the subject.

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