Relations between dynamo-region geometry and the magnetic behavior of stars and planets
aa r X i v : . [ a s t r o - ph . E P ] J a n epl draft Relations Between Dynamo-Region Geometry and the MagneticBehavior of Stars and Planets
Laure Goudard and
Emmanuel Dormy
MAG (IPGP & ENS), CNRS UMR7154, LRA D´epartement de Physique, 24 rue Lhomond, 75005 Paris France.
PACS – Origins and models of the magnetic field; dynamo theories
PACS – Magnetohydrodynamics
Abstract. - The geo and solar magnetic fields have long been thought to be very different objectsboth in terms of spatial structure and temporal behavior. The recently discovered field structure ofa fully convective star is more reminiscent of planetary magnetic fields than the Sun’s magnetic field[1], despite the fact that the physical and chemical properties of these objects clearly differ. Thisobservation suggests that a simple controlling parameter could be responsible for these differentbehaviors. We report here the results of three-dimensional simulations which show that varyingthe aspect ratio of the active dynamo region can yield sharp transition from Earth-like steadydynamos to Sun-like dynamo waves.
Introduction. – Observations of the magnetic fields dueto dynamo activity appear to fall into two categories:fields dominated by large-scale dipoles (such as the Earthand a fully convective star), and fields whith smaller-scaleand non-axisymmetric structures (such as the Sun).Moreover two kinds of different temporal behaviour havebeen identified so far: very irregular polarity reversals (asin the Earth), and quasi-periodic reversals (as in the Sun).Since the Earth and the Sun provide the largest databaseof magnetic field observations, these objects have beenwell studied and described in terms of alternative physicalmechanisms: the geodynamo involves a steady branch ofthe dynamo equations, perturbed by strong fluctuationsthat can trigger polarity reversals, whereas the solardynamo takes the form of a propagating dynamo wave.The signature of this wave at the Sun’s surface yields thewell-known butterfly-diagram (Sunspots preferentiallyemerge at a latitude that is decreasing with time duringthe solar cycle).
Modelling. – Because of their very different natures (liq-uid metal in one case, plasma in the other), planetary andstellar magnetic fields are studied by different communi-ties. Non-dimensional numbers controling the dynamics ofthe Earth and the Sun, for example, do significantly differ(see [2, 3]). As a practical matter however, the techniquesas well as the typical parameters used in numerical stud-ies of these two systems are surprisingly similar. To someextent this is due to the restricted parameter space avail- able to present day computations. The parameter regimenumerically accessible is rather remote from the actual ob-jects. For planetary dynamos the main discrepancy reliesin the rapid rotation in the momentum equation (charac-terized by the
Ekman number ), whilst for stellar dynamosit relies in solving the induction equation with weak resis-tive effects (characterized by high values of the magneticReynolds number ). Yet within this restricted domain, thesharply different key characters to both geo [4] and so-lar [5, 6] magnetic fields have been reproduced. This leadsus to argue that the important parameter controlling themagnetic field behaviour is the aspect ratio of the dynamoregion (i.e. the radius ratio of the inner bounding sphereto the outer bounding sphere). Indeed, in the Earth, theinert solid inner core extends to less than 40% of the coreradius, whereas in the Sun, the radiative zone fills 70%of the solar radius. One expects the convective zones ofstars and planets to have all possible intermediate aspectratios, even extending to fully convective spheres.In order to isolate and understand this purely geometri-cal effect, we have carried out three-dimensional numericalsimulations of self-excited convective dynamos in whichthe domain aspect ratio was slowly varied, with all otherparameters held constant. The governing equations as wellas parameter regimes used here were originally introducedfor a geodynamo reference calculation [7]. The only dis-tinction being the use of stress-free boundary conditionson the outer sphere of the domain, while imposing no-slipboundary conditions at the bottom of the convective re-p-1aure Goudard & Emmanuel Dormy
Fig. 1: Time evolution of the radial magnetic field averaged in longitude (for an aspect ratio of 0 . gion. This choice was made in order to create a strongshear at the base of the model, and thus try to mimic thesolar tachocline [8]. The inner sphere is here assumed tobe insulating, and we use differential heating. The gov-erning equations are in non-dimensional form:E [ ∂ t u + ( u · ∇ ) u ] = − ∇ π + E ∆ u − e z × u + f Ra r θ + Pm − ( ∇ × B ) × B , (1) ∂ t B = ∇ × ( u × B ) + Pm − ∆ B , (2) ∂ t θ + ( u · ∇ )( θ + T s ) = Pr − ∆ θ , (3) ∇ · u = ∇ · B = 0 , (4)whereE = ν Ω D , f Ra = αg ∆ T Dν Ω , Pr = νκ ,
Pm = νη . (5)All simulations reported here were performed keeping thefollowing parameters constant E = 10 − , f Ra = 100, Pr =1, Pm = 5 . The above system is integrated in three–dimensions of space (3D) using the Parody code [9].When the inner (non dynamo generating) body oc-cupies less than about 60% of the convective body inradius, the flow generates a dipolar field, very similar tothat of the Earth. It features patches of intense flux athigh latitudes and some reversed patches at low latitude,similar to the ones revealed by a downward continuationof the Earth’s field to the Core-Mantle boundary [10].This strongly dipolar solution becomes unstable with afurther increase of the aspect ratio. For an aspect ratio of0 .
65 –close to that of the Sun– the strong dipole is firstmaintained and then strongly weakens, but dynamo ac-tion continues in a different form: that of a wavy solutionwith quasi-periodic reversals (Fig. 1), reminiscent of someaspects of the solar magnetic field behavior. Driftingfeatures can be observed both on the radial field at thesurface of the model (Fig. 1 & 2b) and on the azimuthal(east-west) field below the surface of the model (Fig. 2c).Due to the complex nature of these fully tri-dimensionalsimulations, many waves can co-exist. Some of thedominant structures appear to propagate toward the equator; others propagate poleward. Reversed waves arealso observed at the surface of the Sun at higher lati-tudes [11]. Let us stress however that the model cannotbe expected to capture all the features either of the geoor solar magnetic fields. In particular due to the param-eters regime and the lack of stratification in our modelling.
Physical interpretation. – In order to investigate thephysical mechanisms associated with these waves, we haveperformed some kinematic simulations. During the courseof the simulation the Lorentz force was suppressed. Thewavy nature of the dynamo field was unaltered by thismodification. This rules out the possibility of an inter-pretation in terms of pure Alfv`en waves or Alfv`en wavesmodified by rotation (so called MC or MAC waves), whichboth require the back-reaction of the Lorentz force. Ofcourse, suppressing the Lorentz force is not without con-sequences: the flow slowly evolves to a different purelyhydrodynamical state, and the magnetic field now growsexponentially, but both of these effects are sufficiently slowfor the wavelike character to persist over many wave peri-ods.Two other interpretations for the nature of these wavesremain possible: either hydrodynamic fluctuations (e.g.inertial waves or Rossby waves) or dynamo waves, as ex-pected on the Sun. These possibilities were tested by com-paring oscillations in the velocity field and in the magneticfield in the kinematic simulations. We found that a highfrequency signal is present both in the flow and in themagnetic field. This demonstrates the presence of hydro-dynamic waves, which induce magnetic fluctuations. Thelower frequency signal is however absent in the flow. Thisprovides a proof of their “dynamo wave” nature.We have numerically observed such dynamo waves foraspects ratio up to 0 .
8. For the parameters investigatedhere, the transition from a dynamo dominated by afluctuating dipole to a dynamo wave occurs for an aspectratio close to 0 .
65. This transition exhibits hysteresis:once a dynamo-wave solution is present, the aspect ratiocan be reduced again down to 0 .
6, while maintaining thisdynamo mode.p-2elations Between Dynamo-Region Geometry and the Magnetic Behavior of Stars and Planets
Fig. 2: Radial magnetic field at the surface of the outer sphere,for aspect ratios of 0 .
45 (a) and 0 .
65 (b). Azimuthal magneticfield below the surface of the 0 .
65 aspect ratio model (c).
Connections with parameterized models. – But-terfly diagrams indicative of the solar cycle are usuallyproduced using simplified parameterized models or “meanfield” models. These models require a prescription of theturbulent induction, the so-called “ α –effect” (which canalso be introduced in terms of deviation from axisymme-try [12]). We should stress that this is a valid approxi-mation only if certain conditions are satisfied (e.g. [13]).Such butterfly–like diagrams are generally not producedby direct three-dimensional modelling, with the notableexception (only in the reverse direction) of the pioneeringwork of Gilman and Glatzmaier [5, 14].Because of the strong symmetry of the convective flowsinfluenced by the rapid rotation of the planet or the star,it is well known that two independent families of solu-tions exist, namely with dipole symmetry (antisymmetricwith respect to the equator) and quadrupole symmetry(symmetric with respect to the equator). Both familiesof solutions are often described in reduced parameterisedmodels [15,16], and we have observed these two families inour fully 3D simulations (Fig. 3). Both branches are stablein our simulations for long periods of time, but can also bedestabilised to yield a change of symmetry. In fact, despitethe relative complexity of our model, the temporal behav-ior of both symmetries is clearly reminiscent of kinematicstudies of earlier reduced models (Fig. 4c,d and [15]).The simpler meanfield equations for the axisymmetricfield are obtained by writing the flow and field as u = s ω e φ , B = B p + B e φ = ∇ × ( A e φ )+ B e φ , (6)i.e. assuming a mean flow in the form of a zonal shearonly. In the isotropic case, the axisymmetric part of (2)yields (e.g. [3]) ∂A∂t = αB + Rm − D A , (7) ∂B∂t = s B p · ∇ ω + ( ∇ × α B p ) · e φ + Rm − D B , (8)where s denotes the cylindrical radius and D = ∆ − /s (note that Pm in (2) is here changed to Rm as the flow isnow assumed to be given). Fig. 3: Time evolution of the zonal average of the azimuthalmagnetic field below the surface of the model, for an aspect ra-tio of 0 .
65: the antisymmetric (a) and symmetric (b) solutions.
For an instability of (7-8) to exist, these equations mustnot decouple (this is the essence of Cowling’s anti-dynamotheorem [17]). Equation (8) involves A through two terms.Reduced models have been classified in two categories de-pending on the dominant term. The first term on the RHSof (8) involves the zonal shear and is referred to as the Ω–effect. The second term in the RHS of (8), as well as thefirst term on the RHS of (7), involve mean induction fromnon-axisymmetric features in the flow and are referred toas the α –effect.Dropping the α –effect term in (8) and writing the re-sulting equations in a simplified cartesian geometry yields ∂A∂t = αB + Rm − ∆ A , ∂B∂t = G ∂A∂x + Rm − ∆ B , (9)where G = d u y / d z . Parker [18] was the first to identifytravelling waves solutions (dynamo waves) of the abovesystem. These oscillatory dynamos, named Parker waves,were obtained by Roberts [15] for nearly axisymmetric dy-namos in spherical geometries (following the formalism ofBraginsky [12]). It was found that while the α Ω–dynamostended to be oscillatory (complex growth rate), for α –dynamos the simplest dipole solutions tended to be sta-tionary (real growth rate). A similar behavior can eas-ily be traced in the simpler cartesian example above (seealso [19,20] for a discussion of the generic behavior of suchnearly axisymmetric mean field dynamos).We can perform further comparisons with reducedmodels by studying only the axisymmetric componentof the simulated field. Figure 4 shows the azimuthallyaveraged field for some of our fully 3D simulations.p-3aure Goudard & Emmanuel Dormy Fig. 4: The zonal average of the magnetic field in our 3D simulations. Contours of the toroidal (east-west) part of the field areplotted in the left hemisphere and lines of force of the meridional (poloidal) part of the field plotted in the right hemisphere.The aspect ratio is increased from 0 .
45 (a) to 0 . .
65 (c-d). The sequence of dynamo waves is represented for theantisymmetric mode (c) and symmetric mode (d). It is similar in nature to that produced by parameterized models [15]. p-4elations Between Dynamo-Region Geometry and the Magnetic Behavior of Stars and PlanetsThe Earth-like mode is represented for aspect ratiosof 0 .
45 and 0 . α type to a dynamo of the α Ωtype as the aspect ratio is increased. Indeed, Earth-likethree-dimensional models have been interpreted in termsof regeneration by convective vortices only, and thuscloser to the α formalism [23] (sometimes referred toas “giant α –effect”), whereas the α Ω formalism providesthe classical framework to model solar dynamo-waves, asguided by the strong shear at the base of the convectionzone [15, 16]. Such nearly axisymmetric dynamos [15]produce cyclic magnetic behaviour very similar to thecycles examplified on Figure 3.
Conclusions. – By varying the aspect ratio, we have ob-served a sharp transition from a dipole dominated largescale-magnetic field to a cyclic dynamo with a weakerdipole. This indicates that the geometry of the dynamoregion severly constrains the existence of the dipole dom-inated solution. We should however stress that other pa-rameters, involving ratio of typical forces, could affect theprecise value of the critical aspect ratio for transition. Thevalues of these parameters in our simulations (as in all nu-merical models to date) are indeed very remote from theactual relevant values for the Sun, or for the Earth. Thepotentially strong effect of this parameter change on thedynamo solution should not be under-estimated. It is in-deed quite striking, that despite these shortcomings, nu-merical models can capture a good part of the qualitativefeathures of the solar and geo-magnetic fields.Recent observations of stellar magnetism appear to cor-roborate this mechanism. Donati et al. [1] reported ob-servations of a strongly dipolar field in a fully convectivestar (V374 Peg). More recently, Donati et al. [24] reportmagnetic observations of τ –Bootis, a rapidly rotating Fstar, i.e. one with a relatively shallow outer convectionzone. Not only did they observe a rather complex mag-netic field structure, but they also report that the overallpolarity of the magnetic field has reversed after one year of observation. They interpreted this observation as anindication that the large aspect ratio τ –Bootis star is un-dergoing magnetic cycles, similar to those of the Sun.Futher observations of planets and stars are needed, butclearly the observations available so far seem to confirmthe important role of the aspect ratio in controling thetransition from steady to cyclic dynamo modes. ∗ ∗ ∗ The computing resources for this work were provided bythe CNRS-IDRIS, the ENS-CEMAG and the IPGP-SCPcomputing centers. We are grateful to Pr. N.O.Weiss fordiscussions on a preliminary version of this work.
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