EARLY-TYPE STARS 1. INTRODUCTION: MAGNETIC FIELDS IN

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1 THE ASTROPHYSICAL JOURNAL, 559:1094È1107, 2001 October 1 ( The American Astronomical Society. All rights reserved. Printed in U.S.A. MAGNETIC FIELDS IN MASSIVE STARS. I. DYNAMO MODELS PAUL CHARBONNEAU AND KEITH B. MACGREGOR High Altitude Observatory, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO Received 2001 February 26; accepted 2001 June 1 ABSTRACT Motivated by mounting evidence for the presence of magnetic Ðelds in the atmospheres of normal ÏÏ early-type main-sequence stars, we investigate the various possible modes of dynamo action in their convective core. Working within the framework of mean Ðeld electrodynamics, we compute a2 and a2) dynamo models and demonstrate that the transition from the former class to the latter occurs smoothly as internal di erential rotation is increased. Our models also include a magnetic di usivity contrast between the core and radiative envelope. The primary challenge facing such models is to somehow bring the magnetic Ðeld generated in the deep interior to the stellar surface. We investigate the degree to which thermally driven meridional circulation can act as a suitable transport agent. In all models with strong core-to-envelope magnetic di usivity contrastèpresumably closest to realityè whenever circulation is strong enough to carry a signiðcant magnetic Ñux, it is also strong enough to prevent dynamo action. Estimates of typical meridional circulation speeds indicate that this regime is likely not attained in the interior of early-type main-sequence stars. Dynamo action then remains highly probable, but an alternate mechanism must be sought to carry the magnetic Ðeld to the surface. Subject headings: MHD È stars: magnetic Ðelds 1. INTRODUCTION: MAGNETIC FIELDS IN EARLY-TYPE STARS Magnetic Ðelds have been detected in stars located throughout the Hertzsprung-Russell diagram (see, e.g., Landstreet 1992). For example, there presently exists a signiðcant body of observational evidence indicating that magnetic Ðelds and the activity related to them are ubiquitous among late-type stars with masses [1 M. The origin _ of these Ðelds is believed to be hydromagnetic dynamo activity, occurring somewhere within the outer, convective portion of the stellar interior. In stars of solar and slightly later spectral type, the detection of long-term, cyclic variability in the level of chromospheric emission (see, e.g., Radick et al. 1998) suggests that a Sun-like dynamo produces time-dependent Ðelds that give rise to associated magnetic activity cycles. In stars with masses low enough (i.e., [0.4 M ) that their interiors are essentially fully con- _ vective, the dynamo may be less like that of the Sun and more turbulent in nature (Durney, De Young, & Roxburgh 1993). Unlike the situation for cool stars, the question of how prevalent magnetic Ðelds are among massive, early-type stars still lacks a deðnitive answer. Although direct evidence for hot star magnetism exists in the form of numerous observations of the Zeeman e ect in objects with masses Z1.5 M, the vast majority of these detections have been obtained _ for stars with anomalous surface abundances, such as the slowly rotating, chemically peculiar A and B stars (Landstreet 1992; Donati 1998). As a group, objects of this kind represent about 10% of all early-type mainsequence stars. The Ðelds of these stars are found to be relatively strong (D103 G), with an inferred large-scale structure that is approximately dipolar, the symmetry axis of the magnetic Ðeld being generally inclined to the starïs rotation axis. Such Ðelds are believed to be of fossil origin, with further secular hydromagnetic evolution likely taking place in the course of main-sequence evolution (see, e.g., Moss 1989) Alternatively, information regarding the occurrence of magnetic Ðelds in upperèmain-sequence stars with normal photospheric abundances has been more difficult to acquire, with most attempts at measurement yielding only bounds on the strengths of any Ðelds that might be present in the target stars (Landstreet 1982). In recent years, increasingly stringent upper limits have been established for a number of normal OB stars (e.g., the O7 V star h1 Orionis C; Donati & Wade 1999), restricting the magnitudes of the Ðelds in these objects to values less than or equal to a few hundred gauss. The sole exception to these results is the B1 IIIe star b Cephei, for which spectropolarimetric observations provide conclusive proof of the existence of a magnetic Ðeld with a longitudinal component of approximate strength 100 G (Henrichs et al. 2001). At the present time, the bulk of the evidence for magnetism in chemically normal hot stars is circumstantial in nature, deriving from observations of phenomena that are most readily explained as consequences of the presence of a stellar magnetic Ðeld. Prominent among such indirect indicators of possible magnetic e ects are the so-called discrete absorption components (DACs; Kaper & Henrichs 1994; Kaper et al. 1996, 1999), which are narrow, recurrent features that migrate over time from low to high velocity in the proðles of strong, ultraviolet resonance lines. DACs are observed in the blueshifted absorption troughs of unsaturated P Cygni line proðles that are formed in the radiationdriven winds of O stars. Their time-dependent behavior has been interpreted as arising from the interaction between high- and low-speed streams within the outñow emitted by a rotating, early-type star (Mullan 1984; Cranmer & Owocki 1996). It has been suggested that such a large-scale dynamical structure might be imposed on the Ñow by a magnetic Ðeld at the stellar surface, similar to the way in which corotating interaction regions in the solar wind are formed by high- and low-speed streams that emanate from di erent locations within the magnetically structured corona. Magnetic deðnition of the Ñow geometry close to

2 MASSIVE STAR DYNAMOS 1095 the star can inñuence the initial acceleration of a line-driven wind and lead to substantial modiðcations in asymptotic Ñow speed (see, e.g., MacGregor 1988). The case for magnetic Ðelds of moderate strength in normal upperèmain-sequence stars is also supported by results from a variety of other studies. Among these, the results from an e ort to model the main-sequence rotational evolution of OB stars (MacGregor, Friend, & Gilliland 1992) as well as from an examination of the dynamics of magnetized, line-driven winds (Maheswaran & Cassinelli 1992) both suggest that normal early-type stars should possess modest photospheric Ðelds, with strengths [100 G. In addition, recent Chandra observations of strong X-ray emission lines from the star f Orionis A (O9.7 Ib) have been analyzed in order to estimate the density of the emitting region within the stellar atmosphere (Waldron & Cassinelli 2001). The derived densities are found to be signiðcantly larger than the values expected on the basis of the standard model for emission from outñowing gas that is heated by radiation-driven shocks. This Ðnding has led to the suggestion that the observed radiation originates in magnetically conðned structures at the base of the wind, similar to the loops that are seen in X-ray images of the solar corona. In view of the evidence, both direct and circumstantial, for the occurrence of magnetic Ðelds in hot stars, it is reasonable to inquire as to the means by which such magnetization might be acquired. In contrast to the prevailing interpretation of Ðelds in low-mass stars as being dynamogenerated, the origin of magnetism in massive stars, whether chemically normal or peculiar, is presently not well understood. It is unclear whether the inferred Ðelds are fossil in nature or are instead the product of dynamo activity in the stellar interior (see, e.g., Parker 1979; Moss 1989). The latter explanation, if appropriate, raises additional interesting questions concerning the site of magnetic Ðeld generation inside hot stars. For stars having spectral types O and B, convection occurs in the innermost portion of the core and not in the outer envelope as in late-type stars. Because convection is thought to be necessary in order that Ðeld regeneration via the so-called a-e ect take place, it follows that a hot star dynamo should be located deep within the stellar interior. The possibility of magnetic Ðeld generation taking place within the convective core of an early-type star appears to have received scant attention in the literature, with the investigations by Levy & Rose (1974) and Schu ssler & Pa hler (1978) among the few attempts to ascertain the feasibility of core dynamo activity (see also Moss 1989). Given that stellar magnetic Ðeld measurements are becoming increasingly precise and that magnetic Ðelds have recently been implicated in the interpretation of a diversity of hot star observations, a reexamination of how a dynamo might operate in the deep interior of an early-type star seems both timely and appropriate. Toward this end, we herein present results from an extensive series of numerical calculations, performed in an e ort to elucidate the functioning of a convective core dynamo.1 As described in 2, the investigation 1 We see these models as being applicable primarily to the magnetic Ðelds inferred or detected in upper main-sequence stars of normal chemical abundance and rotation, as opposed to the strong oblique magnetic Ðelds believed to be present in the slowly rotating Ap and Bp stars. In the latter objects, the fossil Ðeld hypothesis appears quite tenable (Moss 1989), although dynamo-based explanations have also been proposed (see, e.g., Ru diger 2001) and certainly cannot be ruled out. was carried out using a kinematic model that was derived within the framework of mean Ðeld electrodynamics. A previously calculated model for the interior of a chemically homogeneous, 9 M, main-sequence star was adopted as the computational domain, _ with the convective and radiative portions of the stellar interior characterized by di erent values of the magnetic di usivity. Solutions to the governing equations were obtained by means of the methodology utilized by Charbonneau & MacGregor (1997) to study interface models for the solar dynamo. The properties of these solutions are discussed in 3 and 4, wherein the behavior of both fully turbulent models (i.e., a2 dynamos) and models including a prescribed di erential rotation proðle (i.e., a2) dynamos) are examined. For reasonable parameter choices, the results indicate that dynamo activity can indeed be sustained within a convective core but that the high electrical conductivity and large spatial extent of the radiative envelope make it difficult to transport the Ðelds so generated to the stellar surface. Advection of Ðelds by rotationally induced, internal circulatory Ñows represents a possible means of circumventing this obstacle; the efficacy of Eddington-Sweet meridional circulation as a transport mechanism and its e ect on the operation of the dynamo are considered in 5. The paper concludes with a brief examination of the inñuence of the nonlinear e ect of a-quenching on the solutions ( 6) and a discussion of the merits and deðciencies of the core dynamo as a possible source of magnetic Ðelds in hot stars ( 7). 2. MATHEMATICAL FORMULATION 2.1. T he Dynamo Equations Numerous textbooks, monographs, and review articles discuss the application of mean Ðeld theory to the MHD induction equation. (e.g., Mo att 1978; Parker 1979; Krause & Ra dler 1980; Hoyng 2001). We should stress that at this writing, there exist no truly satisfactory mean Ðeld models of the solar cycle, and other competing ÏÏ dynamo mechanisms continue to attract considerable attention (see, e.g., Schmitt, Schu ssler, & Ferriz-Mas 1996; Dikpati & Charbonneau 1999 and references therein). Nonetheless, given the current embryonic state of dynamo modeling in massive stars, a reasonably thorough exploration of mean ÐeldÈbased models of magnetic Ðeld generation in these objects is justiðed. Mean Ðeld electrodynamics yields a time evolution equation for the mean magnetic Ðeld B: LB Lt \ $ Â [UÂB] ab [ (g ] b)$ ÂB], (1) e which we refer to below as the dynamo equation. Magnetic Ðeld ampliðcation is evidently possible via the ab term, which is called the a-e ect and represents the inductive contribution of the small-scale Ñow and magnetic Ðeld. The small-scale Ñow also leads to enhanced resistive decay via the b-term, which is akin to a turbulent magnetic di usivity. In practice these coefficients are e ectively speciðed in some physically plausible manner, either directly or in terms of some simpliðed turbulence model (see Kitchatinov & Ru diger 1995 for an example of this latter approach). A particular, but very astrophysically relevant situation, is that in which both the mean magnetic Ðeld and any mean Ñow exhibit axisymmetry about the rotation axis. The Sun, on its largest spatial scale, is the prototypical example.

3 1096 CHARBONNEAU & MACGREGOR Vol. 559 Under such circumstances, a natural representation for B is B(r, h, t) \ $ Â [A(r, h, t)eü Õ ] ] B Õ (r, h, t)eü Õ, (2) where A deðnes the poloidal Ðeld (i.e., the component of the Ðeld contained in meridian planes) and B is the toroidal Ðeld. For now, we just separate the mean Ñow into a meridional circulation u \ u eü and a zonal Ñow corre- p r r ] u h eü sponding to (di erential) rotation, h U(r, h) \ u (r, h) ] -)(r, h)eü p Õ, (3) with - 4 r sin h. It is readily shown that under these assumptions the dynamo equation can be separated into poloidal and toroidal components: LA ga Lt \ +2[ -2B 1 R A [ m u Æ $(-A) ] C ab, - p a LB Lt \ ga $2[ 1-2B B [ ($g) Â ($ Â Beü Õ ) [ R m -$ Æ AḆ u p B ] C) -($ ] A) Æ ($)) ]C + ] [a$ Â (Aeü )]. (5) a Õ The above equations have been cast in nondimensional form by expressing all lengths in units of the stellar radius R and time in units of the magnetic di usion time q \ R2/g c based on the (turbulent) magnetic di usivity in the convective core (g 4 g ] b ^ b, dimension in units of cm2 s~1). c e This has led to the appearance of three-dimensionless numbers: C \ a 0 R, (6) a g c (4) C \ *) 0 R2, (7) ) g c R \ u 0 R, (8) m g c where a (dimension in units of cm s~1), u (dimension in 0 0 units of cm s~1), and *) (dimension in units of rad s~1) are 0 reference values for the a-e ect, di usivity, meridional Ñow, and zonal shear, respectively. The quantities C and C are dynamo numbers measuring the relative importance a ) of inductive versus di usive e ects on the right-hand side of equations (4) and (5). The third dimensionless number R is a (magnetic) Reynolds number, which measures the relative m importance of advection by meridional circulation versus di usion by Ohmic dissipation in the transport of A and B in meridional planes. The formulation of a (kinematic) dynamo model requires that one supply functional dependencies for g, a, u, and ) as well as assign numerical values for C, C, and R p. Even in the case of the Sun, where )(r, h) and, a to ) a lesser m degree, u can be inferred via helioseismology (see, e.g., Brown et al. 1989; p Tomczyk, Schou, & Thompson 1995; Giles et al. 1997; Schou & Bogart 1998; Braun & Fan 1998), there exist quite a few mean Ðeld models that end up producing dynamo solutions that are reasonably solar-like yet rely on markedly di erent choices for the aforementioned functionals. As a reference point, it is still useful to estimate values for the dynamo numbers, although we are venturing on slippery ground given our very severe lack of quantitative knowledge of the properties of turbulence in a rotating stellar convective core. On dimensional grounds, one expects a to be (very roughly) on the same order of magni- tude as the 0 turbulent velocity u and the turbulent di usivity to be on the order of u l, where t l is the mixing length. For our advocated illustrative t purposes, we simply set a \ 104 cm s~1 and g \ 1013 cm2 s~1. For the shear, we suppose 0 that *) D ) c, where ) is the starïs surface angular velocity, for which 0 * we adopt * a value of 100 km s~1 R~1. This yields C ^ 2000 and C ^ 5 ] 105. Note that the ratio C /C (\250 a here) is independent ) of the assumed turbulent di usivity ) a and that our choice of g leads to a di usion time q ^ 200 yr, which is much smaller c than the main-sequence lifetime. Estimates for R are deferred until 5. m 2.2. Solution Procedure In the kinematic limit, i.e., with a, b, u, and ) given, the axisymmetric dynamo equations (4) p and (5) are two coupled, linear homogeneous equations in A and B. This suggests that one can look for eigensolutions of the form [A(r, h, t), B(r, h, t)] \ [a(r, h), b(r, h)]e(p`iu)t, (9) where p ] iu is the eigenvalue. Such linear solutions leave undetermined the absolute strength of the dynamogenerated Ðeld and merely provide an indication of which types of Ðelds are likely to grow. It is reasonable to suppose that the fastest growing linear eigenmode is likely to still dominate the solution in the nonlinear regime, and, in fact, this is often the case. In what immediately follows, we simply proceed with the linear model and solve the dynamo equations numerically using the Ðnite-elementÈbased inverse iteration method described in the appendix of Charbonneau & MacGregor (1997).2 Eigensolutions are sought in a meridional quadrant of a sphere of radius R and are matched to a potential Ðeld in the exterior (r/r [ 1). Under the assumption of axisymmetry, one must impose A \ 0 and B \ 0 on the axis. In practice, it is often useful to solve explicitly for modes having either odd or even parity with respect to the equatorial plane. To select ÏÏ only antisymmetric (i.e., dipole-like) modes, one simply selects the appropriate boundary condition on the equatorial plane: LA(r, n/2) \ 0, B(r, n/2) \ 0 (antisymmetric), (10) Lh LB(r, n/2) A(r, n/2) \ 0, \ 0 (symmetric). (11) Lh In most of the models discussed below, the fastest growing symmetric and antisymmetric modes have comparable linear growth rates. In seeking eigensolutions of the form given by equation (9), one typically Ðnds that there exists a lower threshold on the dynamo numbers (C, C ), below which only decaying solutions are found (p\0). a ) Dynamo action, which in the context of linear kinematic models simply means solutions having positive growth rates, is possible only above that threshold. This is, of course, to be expected on physical 2 Note that the (valid) criticism of some of the Charbonneau & Mac- Gregor (1997) interface dynamo solutions recently raised by Markiel & Thomas (1999) only applies to situations in which the magnetic di usivity varies discontinuously with depth and so has no bearing on the models considered here.

4 No. 2, 2001 MASSIVE STAR DYNAMOS 1097 grounds since the dynamo numbers measure the relative importance of induction (producing magnetic Ðeld) over dissipation (destroying magnetic Ðelds). Linear solutions having p \ 0 are said to be critical, and supercriticality refers to growing solutions, i.e., p[0. From a mathematical point of view, dynamo action in the linear regime is akin to a form of MHD instability. While the absolute scale for the strength of the magnetic Ðeld is left undetermined by linear eigenvalue calculations, the ratio of poloidal to toroidal Ðeld strength extracted from the model is a relevant quantity. It is convenient to deðne this ratio in terms of the associated magnetic energy of the toroidal and poloidal component; i.e., we deðne the ratio # as # \ / B2dV / ($ ] A)2dV. (12) Another important quantity accessible from linear model is the ratio of the surface Ðeld strength to the Ðeld strength in the dynamo region, here the convective core. In what follows, we use toward this purpose the ratio (&) of the rms surface poloidal Ðeld to the rms poloidal Ðeld at the coreenvelope interface r : c & \ AR2/ o$ ] A o 2 sin h r/r. (13) r2 /o$] A o2 sin h dhb1@2 c r/rc In practice, the Ðnite numerical accuracy at which the eigenfunctions are computed leads to a lower bound on meaningful values of &, here at about 10~ A Model for Dynamo Action in Massive Stars In all calculations discussed below, we make use of a 9 M zero-age main-sequence (ZAMS) stellar model recent- _ ly computed by S. Jackson (2000, private communication). This B2 spectral type model has a luminosity L \ 3767 L, _ e ective temperature T \ 23,600 K, and radius R \ eff R. From the dynamo point of view, an important quantity _ is the radius of the convective core (r ), which in our model c is r \ R. Within the core, thermally driven turbulent c Ñuid motions give rise to an a-e ect and turbulent di usivity, which both vanish for r Z r (under the assumption c that the radiative envelope is turbulence-free). Accordingly, we write C a(r, h) \ 1 Ar [ r 1 ] erf c BD A2r erf 2 w wb cos h, (14) C g(r) \ g ] g c [ g Ar [ r e 1 [ erf c BD, (15) e 2 w where erf (x) is the error function. The resulting radial pro- Ðles are plotted on Figure 1a as solid (a; plotted assuming cos h \ 1) and dashed (g) lines. Equation (14) represent minimal ÏÏ assumptions of the spatial dependency of the a-e ect: it changes sign across the equator (h \ n/2), vanishes at r \ 0 since u presumably vanishes there, rises to a maximum value within t the convective core, and falls again to zero for r Z r, with the transition occurring across a spherical layer of c thickness D2w. While we restrict ourselves to proðles of the form given by equation (14), we consider models with both positive and negative a-e ects. The cos h dependency in equation (14) is the latitudinal variation of the Coriolis force felt by a radially rising/ sinking Ñuid element and, in the context of mean Ðeld FIG. 1.ÈMinimal model for mean Ðeld dynamos in a massive star. (a) Assumed variations with depths of the a-e ect (solid line) and net magnetic di usivity (dashed line), as given by eqs. (14) and (15). The underlying stellar model is that of a 9 M ZAMS star. In this model the convective core extends from the center out _ to a depth r \ R. We assume that the transition between the core and envelope c occurs smoothly over a spherical layer of width 2w centered on r. Above the transition layer, the a-e ect vanishes, and the magnetic di usivity c levels o at a constant value g, much smaller than in the core. We treat w and the di usivity ratio g /g as e parameters of the model. The dash-dotted line shows the density proðle e c in the lower envelope, normalized to the density at the core-envelope interface. The dashètriple-dotted line is a modiðed meridional circulation solution and is discussed in 5. (b) Ratio of a(r, 0)/g(r) corresponding to the proðles plotted in (a) for varying magnetic di usivity ratios g /g. Note how in cases with g /g > 1, a/g remains equal to unity for a e signiðcant c distance beyond the nominal e c core-envelope interface depth r. c theory, represents the minimal ÏÏ h-dependency for the a- e ect. It does not appear to hold in the Sun, where it leads to magnetic activity at high heliospheric latitude instead of the equatorial regions. For the sake of simplicity, we nonetheless retain this minimal dependency. One important consequence of a magnetic di usivity contrast between core and envelope is illustrated in Figure 1b, showing the ratio a(r, 0)/g(r) for four di erent values of the envelope-to-core di usivity ratio. While a presumably falls to zero outside the convective core, the di usivity g drops to a small but Ðnite value g. As a consequence, the a/g ratio does fall to zero outside e the core but does so at larger r for a larger di usivity ratio. The quantity a/g being a measure of local dynamo action, the inñuence of the adopted ratio g /g will have some signiðcant consequence for dynamo action e c in the solutions discussed below. While physically motivated and perhaps not obviously unreasonable, the above prescriptions are still made in an ad hoc manner. Since this situation, in one form or another, characterizes all mean Ðeld dynamo models constructed for the Sun, solar-type stars, and planetary cores, we shall proceed undaunted.

5 1098 CHARBONNEAU & MACGREGOR Vol RESULTS FOR THE a2 DYNAMO MODEL We Ðrst consider solutions in which magnetic Ðeld generation occurs exclusively through the agency of the a-e ect; i.e., both $) and u are assumed to vanish. Dynamos of this type are known as p a2 dynamos in the literature and, starting with the seminal work of Steenbeck & Krause (1969), have been studied extensively in the context of planetary magnetism and magnetic activity in fully convective late-type stars (see, e.g., Krause & Ra dler 1980; Ra dler et al. 1990). However, they have as yet received far less attention in the present context of dynamo action in the cores of massive stars (but see Schu ssler & Pa hler 1978) a2 Solution with Constant Magnetic Di usivity We begin with dynamo solutions computed for depthindependent magnetic di usivity (i.e., g \ g in eq. [15]). Figure 2a shows a typical a2 solution, plotted c e in a meridional quadrant with the symmetry axis vertically oriented and coinciding with the left-hand quadrant boundary. Poloidal Ðeld lines are plotted over a gray-scale representation for the toroidal magnetic Ðeld. The dashed line is the coreenvelope interface r \ r, and the two dotted lines corre- spond to depths r ^ w c (w/r \ 0.1 here). The solution transits from decaying c (p\0) to growing (p[0) at C ^ [32.8, and the growth rate keeps increasing as o C a o is further increased. The solution plotted on Figure 2a is computed for C \[34.5 and is supercritical (p \ 10.82q~1). A a solution with a C \ 34.5 has an identical growth rate and eigenfunction but a shows an opposite relative polarity between the poloidal and toroidal components. Linear mean Ðeld dynamos of the a2 type with a timeindependent scalar functional a(r) always produce steady magnetic Ðelds, i.e., the solution eigenvalue is purely real (u \ 0 in eq. [9]). The solution plotted in Figure 2a is dipole-like (i.e., antisymmetric) and is the fastest growing solution for our model with constant g at the adopted value for C. The next fastest growing mode is symmetric with a FIG. 2.ÈFour antisymmetric steady a2 dynamo solutions, computed using varying magnetic di usivity ratios between the core and envelope. The solutions are plotted in a meridional quadrant, with the symmetry axis coinciding with the left-hand quadrant boundary. Poloidal Ðeld lines are plotted superimposed on a gray-scale representation for the toroidal Ðeld (light to dark is weaker to stronger Ðeld). The dashed line marks the core-envelope interface depth r, and the two dotted lines indicates the depths r ^ w corresponding to the width of the transition layer between core and envelope. Note how the solutions c with g /g [ 10~2 have their toroidal Ðeld peaking c across the core-envelope interface. This behavior is generic and materializes for smaller values of w and r and for e symmetric c (i.e., quadrupolar-like) solutions. Parameters for these solutions are listed in Table 1. c

6 No. 2, 2001 MASSIVE STAR DYNAMOS 1099 respect to the equatorial plane and has a growth rate only slightly smaller, p \ 10.79q~1. This situation is typical of a2 dynamo solutions using a scalar a-e ect. Indeed, the dynamo equations admit a large set of eigensolutions with progressively smaller growth ratesèsome negative, i.e., decaying solutions (of no interest in the dynamo context). It should be stressed that p \ 10 in dimensionless units amounts to an e-folding time of about 20 yr in dimensional units, leaving no doubt that ample time is available to amplify a weak seed magnetic Ðeld in the core of a massive star. The a2 form of the dynamo equations also admits growing solutions that are nonaxisymmetric even though the a-e ect proðle exhibits axisymmetry with respect to the rotation axis. Growth rates for nonaxisymmetric modes are often comparable to those of their axisymmetric counterparts (see Ra dler & Wiedemann 1989 for a simple example). Various nonlinear calculations (see, e.g., Krause & Meinel 1988; Brandenburg et al. 1989) suggest that even when the growth rates of various a2 modes are comparable, the fastest growing linear dynamo mode is usually (but not always) the one that dominates in the nonlinear regime. For simplicity, we restrict ourselves here to axisymmetric modes. We note nonetheless that, motivated largely by the challenge posed by planetary magnetic Ðelds, a2 models can and have been constructed in which nonaxisymmetric modes are the fastest growing and dominate in the moderately supercritical nonlinear regime (see, e.g., Ra dler et al. 1990) a2 Solutions with Magnetic Di usivity V arying with Depth As argued earlier, in the context of dynamo action in massive stars, a more realistic model should have the magnetic di usivity decrease signiðcantly outside the convective core. Figures 2bÈ2d show three a2 solutions computed for decreasing envelope-to-core di usivity ratios but identical in all other respects to the constant g solution of Figure 2a. In particular, the dynamo number C was adjusted to yield a growth rates similar to that of the solution on Figure 2a so that the eigenfunctions are, in some sense, comparable. It turns out that for g /g [ 0.1, the symmetric modes have e c slightly larger growth rate (p \ 11.0q~1, 11.8q~1, and 12.5q~1 for g /g \ 0.1, 0.01, and 0.001, respectively). None- theless, to facilitate e c comparison with the constant di usivity solution of Figure 2a, the antisymmetric modes are plotted in Figures 2bÈ2d. DeÐning parameters for all solutions plotted on Figure 2 are listed in the top part of Table 1. The most signiðcant consequence of g /g being smaller than 1 is perhaps the trapping ÏÏ of the magnetic e c Ðeld in the lower part of the radiative envelope, a direct consequence of the difficulty of an external magnetic Ðeld to di usively penetrate a good electrical conductor. This is clearly evident from Table 1 in the rapid decrease of the surface-to-core Ðeld ratio & (see eq. [13]) with decreasing di usivity ratio g /g. This is a long-recognized property of stellar core dynamos e c (see, e.g., Schu ssler & Pa hler 1978) and represents a rather formidable obstacle to be bypassed if the magnetic Ðelds generated by dynamo action in the convective core are to become observable at the stellar surface. As discussed in Schu ssler & Pa hler, the situation is even worse than Table 1 may suggest. In a time-dependent situation, the time needed for the magnetic Ðeld to resistively di use to the surface can become larger than the starïs main-sequence lifetime for masses in excess of about 5 M. _ Less striking but equally important in what follows is the fact that in solutions with g /g \ 1, the locus of peak e c dynamo actionèas measured by the peak in toroidal Ðeld strengthèmoves out to the core-envelope boundary. Note on Figure 2 how, for g /g [ 0.01, toroidal Ðelds are present e c out to r ^ r ] w. This is a direct consequence of the a/g c ratio remaining equal to unity over a signiðcant radial distance outside of the core (see Fig. 1b). As g /g decreases, the e c magnetic Ðeld is increasingly trapped in the interior, yet it is increasingly concentrated near the core-envelope interface. This has far-reaching consequences for the viability of various mechanisms that might be invoked to bring the dynamo-generated magnetic Ðeld to the stellar surface, an issue explored in 5 below. This behavior also materializes in solutions computed using smaller values w/r \ 0.05 and 0.025; in decreasing w/r, one needs to slightly increase the dynamo number to maintain growth rates comparable to those of the solutions shown in Figure 2, but the morphology of the eigenfunctions is not extremely sensitive to the adopted value of C. a Mean Ðeld dynamo models of the a2 variety typically TABLE 1 PARAMETERS AND EIGENVALUES FOR VARIOUS a2 AND a2) SOLUTIONS Figure Type Parity C a C ) R m g e / g c w / R p u #a &b 2a... a2 A [ ] 10~2 2b... a2 A [ ] 10~4 2c... a2 A [ \ 10~8 2d... a2 A [ \10~8 a2) A [ \10~ a2) S [ \10~8 a2) A \10~8 a2) S \10~8 a2) S [ \10~8 a2) S [ \10~8 6b... a2 A [ ] 10~4 a2 A [ ] 10~4 a2 A [ ] 10~4 6c... a2 A [ ] 10~2 a In a2) models, average value over an oscillation period. b Bounded below at & D 10~8 because of the numerical accuracy of eigenfunction computation (see text).

7 1100 CHARBONNEAU & MACGREGOR Vol. 559 generate magnetic Ðelds that have poloidal and toroidal components of comparable strengths. Indeed, we Ðnd here that the toroidal-to-poloidal ratio deðned in equation (12) is on the order of unity and varies very slightly with g /g (see Table 1). e c 4. ENTER DIFFERENTIAL ROTATION: a2) AND a) DYNAMOS The energy source tapped into by the growing magnetic Ðeld of a2 solutions ultimately originates from the thermonuclear fusion reactions that power the turbulent motions in the convective core. In a rotating star, there exists another energy reservoir available to the dynamo process: rotational kinetic energy. This is most readily tapped into via di erential rotation, speciðcally the shearing of a poloidal Ðeld into a toroidal component. This is an essential component of dynamo action in the Sun as well as, presumably, other solar-type stars. Mean Ðeld dynamos including di erential rotation are usually labeled a2) or a), the latter referring the specialèbut probably commonèsituation in which the production of the toroidal Ðeld is dominated by the shearing of the poloidal component by di erential rotation. The core dynamo models constructed by Levy & Rose (1974) are of this type Di erential Rotation in the Interior of Massive Stars Evolutionary models of early-type stars including rotation usually produce main-sequence models characterized by internal di erential rotation and, in particular, with radial di erential rotation between the convective core and radiative envelope (see, e.g., Endal & SoÐa 1978). While the magnitude of this di erential rotation is rather sensitive to various modeling assumptions, its presence at some level appears to be a robust feature. Recall that the core-envelope interface is where the dynamo eigenfunctions have been shown to reside in models having g /g \ 1, so the e ects of e c di erential rotation must be investigated. In view of the aforementioned results by Endal & SoÐa, we restrict ourselves here to purely radial di erential rotation proðles concentrated at the core-envelope interface. SpeciÐcally, we consider the case of a convective core and radiative envelope both rotating rigidly but at di erent rates () and ) ) joined smoothly across a thin spherical c e shear layer coinciding with the core-envelope interface at r \ r : c C )(r, h) \ ) ] ) e [ ) Ar [ r c 1 ] erf c BD. (16) c 2 w The rotation increases inward, i.e., ) [), leading to a negative radial shear in the vicinity of c the e core-envelope interface. Negative radial shear proðles are the only ones considered here since steep positive radial shears are, in all likelihood, hydrodynamically unstable (see, e.g., Tassoul 1978). Note also that the parameter w used to specify the thickness of the shear layer is the same as that used to specify the width of the transition region for the turbulent di usivity and a-e ect in 2.3. We are now solving the dynamo equations (4) and (5) with R \ 0 but with all other terms present. m 4.2. T he Basic a2) Model The inclusion of di erential rotation means that there is now a second source term on the right-hand side of the toroidal component of the dynamo equation (5) in addition to the a-e ect term considered earlier. This opens a wide range of novel dynamo possibilities. Perhaps the most signiðcant di erence between a2) solutions and the a2 solutions considered previously is the fact that while the latter are spatially steady (in the sense that u \ 0), the former usually yield oscillatory solutions, with solution eigenvalues occurring in complex conjugate pairs p ^ iu. Figure 3 illustrates a half-cycle of a representative a2) solution. This symmetric solution has C \[21, C \ 2000, w/r \ 0.1, and g /g \ 10~2 and is characterized a ) by a growth rate p \ 21.8q~1 e c and frequency u \ 186 q~1. For g \ 1013 cm2 s~1, this corresponds to a dynamo period of about c 7 yr, which is quite short compared to any other relevant timescales. The magnetic Ðeld distribution is shown at Ðve distinct phases, at constant intervals of *r \ n/4, in a format identical to that of Figure 2 for each panel (note in particular that the eigenmodes are again plotted only in the inner half of the star). At a given phase the solutions bear some resemblance to the a2 solutions of Figure 2c in that the magnetic Ðeld is again trapped in the interior. As before, the toroidal Ðeld is concentrated near the core-envelope interface and, in fact, here peaks slightly outside r \ r (dashed circular arc). c As with the a2 solutions considered previously, the growth rate of the a2) solution increases with increasing values of either or both the dynamo numbers C and C. a ) The dynamo frequency u also increases with C and C. In a ) the a) limit, in which the a-e ect makes a vanishing contribution to the right-hand side of equation (5), the eigenvalue is completely determined by the value of the product C C, a ) but this property does not hold, in general, for a2) models. Examination of Figure 3 soon reveals that the magnetic Ðeld distribution migrates steadily poleward in the course of the half-cycle shown in Figure 3, with the solutions at r/n \ 1 being a mirror image of that at r/n \ 0; i.e., the magnetic polarity has undergone a polarity reversal after half an oscillation cycle. This behavior is similar to that seen in mean Ðeld dynamo models of the solar cycle. In this context, it is a well-known result that in a model with purely radial shear, the direction of latitudinal propagation of the dynamo wave ÏÏ is controlled by the sign of the product of the a-e ect and radial di erential rotation: the propagation is equatorward if that product is negative in the northern hemisphere (which is the case argued for the Sun) and poleward if that product is positive, as in the model considered here (see, e.g., Parker 1955; Steenbeck & Krause 1969; Stix 1976).3 Note that the toroidal Ðeld gains in strength as the dynamo wave proceeds from low- to midlatitudes, peaking at about 60 and falling thereafter as the wave experiences enhanced dissipation on converging toward the symmetry axis. A solution with C \]21 but otherwise identical to that shown in Figure 3 has a a growth rate and frequency that are comparable, but not identical, to the C \[21 solution (see Table 1). The di erence is due to spherical a geometry; a poleward-propagating dynamo wave su ers greater di usive decay as it converges toward the symmetry axis than an 3 ParkerÏs original dynamo wave solutions were obtained in Cartesian geometry and in the so-called a) limit, in which the a-e ect is omitted on the right-hand side of the toroidal component of the dynamo equation. Similar dynamo wave solutions are also readily found in the more general a2) case; see, e.g., Choudhuri (1990).

8 No. 2, 2001 MASSIVE STAR DYNAMOS 1101 FIG. 3.ÈRepresentative a2) solution. Since this is an oscillatory solution, the eigenfunction is plotted at Ðve equally spaced phase intervals (*r \ n/4) covering half an oscillation cycle. The format in each panel is similar to Fig. 2. White (black) lines indicate Ðeld lines oriented in a clockwise (counterclockwise) direction. Note the wavelike propagation of the magnetic Ðeld from low to high latitudes. Parameter values are listed in Table 1. equatorward propagating wave does converging toward the equatorial plane, where the symmetry imposed via the boundary condition also a ects the dissipation. Solutions with thinner transition layers require a larger value of o C o a to maintain comparable growth rates and are thus characterized by higher oscillation frequencies. Table 1 lists solution parameters and characteristics for a few representative such solutions. Not surprisingly, in a2) models the availability of an additional energy source in the toroidal component of the dynamo equations leads to solutions in which the toroidal Ðeld strength, in general, exceeds that of the poloidal Ðeld. For the solution plotted on Figure 3, the toroidal-to-poloidal Ðeld ratio (see eq. [12]) reaches a value # ^ 3. Further increases of C lead to increasing # (e.g., # ^ 3.4 and 4.3 at C \ 5000 and ) 104, respectively) until in the a) limit, # scales ) roughly as C /C. For a given di usivity ratio g /g, oscillatory a2) solutions ) a have a smaller surface-to-core e c Ðeld strength ratio & than a2 models, a direct consequence of the oscillatory nature of the Ðeld, which restricts the radial extent of the eigenfunction above the core-envelope interface to a distance comparable to the electromagnetic skin depth, which is very much smaller than the stellar radius for g /g > 1. e c 4.3. a2 versus a2) Solutions The markedly di erent spatial distributions and temporal behavior of a2 and a2) eigenmodes naturally lead one to suspect that both dynamo modes should have some difficulty operating simultaneously. That this is indeed the case can be seen in Figure 4, showing isocontours of the linear growth rate p in the [C, C ] plane for antisymmetric nega- ) a tive C solutions. Dynamo solutions (p[0) are located a below the thick contour, and the thick dashed line delineates the regions in which steady (u \ 0, a2-like) and oscillatory (u D 0, a2)-like) solutions are found. At a Ðxed value of C, introducing di erential rotation Ðrst leads to a a decrease of the growth rate, reñecting the perturbative inñuence of di erential rotation on the basic a2 mode. Once C exceeds a certain (C -dependent) threshold at about C ^ ) 300, the dynamo becomes a a2)-like (u D 0). However, ) growth rates comparable to that of the pure a2 mode (C \ 0) materialize only for much larger values of C. Much ) the same behavior is seen in symmetric solutions ) and/or for positive C solutions. Nonetheless, the transition from the a2 to the a2) a dynamo regime occurs smoothly as di erential rotation is increased. For our adopted value g \ 1013 cm2 s~1, C \ 300 amounts to *) /) ^ 10~3, c i.e., very weak di erential ) rotation. The extant 0 * observations and inferences of magnetic Ðelds in upper main-sequence stars reviewed in 1 currently have little to say about the steady/oscillatory character of the underlying Ðeld. Even if it were oscillating with a regular period on the order of a few years, as do the a2) solutions discussed here, it is not at all clear that the mechanism(s) responsible for bringing the Ðeld to the surface may not introduce additional temporal variabilities that would mask the underlying cycle period. If, on the

9 1102 CHARBONNEAU & MACGREGOR Vol. 559 locations. The poleward return Ñow needed to satisfy mass conservation is then concentrated within the boundary layer adjacent to the core-envelope interface (see, e.g., their Figs. 3 and 4), which is where the dynamo eigenfunctions described above have been shown to reside. The more spatially restricted the return Ñow region, the larger the Ñow velocities therein and the higher the local Reynolds number. Since some of those Ñow streamlines eventually come close to the surface, it is possible, in principle, for meridional circulation to carry the dynamo-generated magnetic Ðeld to the surface. In addition, the presence of a vigorous circulation Ñow near the core-envelope interface can be expected to a ect dynamo action itself. Exploring these possibilities is the purpose of this section. FIG. 4.ÈIsocontours of the a2) linear growth rate in the [C, C ]- ) a plane. The thicker contours corresponds to p \ 0 and solid contours to p[0. All solutions are of antisymmetric parity and have w/r \ 0.1, g /g \ 0.01, and R \ 0. Solutions left of the thick dashed line are steady e c m (u \ 0, a2-like), and oscillatory solutions are to its right. At Ðxed C, the growth rate is a nonmonotonic function of internal di erential rotation, a as measured by C. Qualitatively similar diagrams are obtained for symmetric modes and ) other values of w/r and/or solutions with positive C. a other hand, the magnetic Ðelds are shown to be strictly steady, one would then be forced to conclude that the same magnetic Ðelds have obliterated any angular velocity di erence between the core and envelope, something that they can, in fact, achieve quite efficiently in the absence of internal or external forcing. 5. ENTER MERIDIONAL CIRCULATION 5.1. Meridional Circulation in the Radiative Interior of Massive Stars In the radiative interior of an early-type rotating star, rotationally induced departures from sphericity lead to pole-equator temperature di erences that cannot be equilibrated under the assumption of simultaneous radiative and hydrostatic equilibrium. The star reacts by generating a large-scale meridional circulation [u (r, h), known as the p Eddington-Sweet circulation] in order to balance the nondivergence of the radiative Ñux F (see Tassoul 1978, 188 ; Kippenhahn & Weigert 1990 and references therein). Outside of nuclear burning regions, this Ñow must then satisfy $ Æ F \[ c P ot ($ [ + )u É +p (17) p ad p (see 42 of Kippenhahn & Weigert 1990). Note that the presence of the $ [ + term formally implies divergence of u at the boundary between ad the radiative core and convec- tive p envelope. Likewise, the density dependence leads to a unbounded increase of the meridional Ñow near the stellar surface. The Ñow deðned by equation (17) is equatorward in the bulk of the radiative envelope, but evidently mass conservation also requires a poleward return Ñow somewhere in the interior. In the Tassoul & Tassoul (1982) meridional circulation solutions, divergence of u at r/r \ r and at the surface is p c avoided by introducing viscous boundary layers at those 5.2. A SimpliÐed Model for Meridional Circulation We retain the dynamo model used in the preceding two sections but restore a meridional Ñow u to the dynamo equations. Rather than computing a formal p solution for the meridional Ñow in our stellar model, we simply rescale the Ñow solution u(r) listed in Table 8 of Tassoul & Tassoul (1982), artiðcially stretching the core-envelope boundary layer to a thickness w/r ^ 0.025, i.e., roughly of the same order as the half-thickness of our transition layer for the magnetic di usivity, a-e ect, and di erential rotation. The corresponding modiðed function u(r) is plotted as a dashè triple-dotted line in Figure 1a. The marked increase in u immediately above the core-envelope interface can be traced to the term $ [ + ] 0 as r ] r, (see eq. [17]). ad c Given the other uncertainties in the model, especially in specifying the a-e ect and di erential rotation, this rescaling procedure for the meridional Ñow is sufficient for the purposes of the foregoing analysis. Streamlines of the meridional Ñow are plotted in Figure 5, which should be compared to Figure 3 in Tassoul & Tassoul (1982). The Ñow is quadrupolar (i.e., one Ñow cell per meridional quadrant), contained between the spherical shells r \ r and r \ R, and equatorward-directed in most c of the envelope. The return Ñow is concentrated in a thin boundary layer adjacent to the core-envelope interface at r/r \ Note that the meridional Ñow does not penetrate deeper into the convective core than the middle of our transition layer (see Fig. 1). This is in conceptual agreement with hydrodynamical models of meridional circulation in early-type stars, which indicate that the gradient in mean molecular weight building up at the core-envelope interface as a result of nuclear burning in the core is very efficient at deñecting the circulatory Ñow, e ectively shielding the core (see Mestel 1953; Tassoul & Tassoul 1984). Note also that most streamlines close on themselves rather deep down in the radiative envelope, with only the streamlines Ñowing immediately adjacent to the core-envelope interface reaching into the outer half of the envelope. One should keep in mind that a steady, single-cell circulation pattern of the kind shown in Figure 5 is expected to hold under three conditions: (1) di erential rotation is either absent or very weak in the bulk of the radiative envelope, (2) magnetic Ðelds, if present in the envelope, make a negligible contribution to the meridional force balance, and (3) the ratio of centrifugal to gravitational forces is much smaller than unity. For meridional circulation models relaxing the second and third of these assumptions, see Tassoul & Tassoul (1986, 1995, respectively). We deðne the meridional Ñow components u, u in the r h

10 No. 2, 2001 MASSIVE STAR DYNAMOS 1103 eter that is hard to estimate even in an order of magnitude sense, the solutions described here are sought over a comfortably broad range of numerical values for R m :0¹ R m ¹ 103. FIG. 5.ÈMeridional circulation streamlines computed by rescaling the solution listed in Table 8 of Tassoul & Tassoul (1982). In keeping with our working assumption of a Ðnite thickness transition layer between the convective envelope and radiative core, the core-envelope boundary layer in the Tassoul & Tassoul (1982) solutions is stretched ÏÏ to a thickness D0.025 R, i.e., comparable to the transition layer width w/r used for other quantities entering the dynamo model. The corresponding u(r) function is plotted as a dashètriple-dotted line in Fig. 1a. Notice how only streamlines running immediately adjacent to the core-envelope boundary (r /R \ 0.232) reach the upper half of the envelope, with all others closing c in the lower envelope, forming a thin, closed circulation cell pressed against the interface. The Ñow is equatorward everywhere except in this thin layer, and its speed is measured by the Reynolds number R. m form of a two-dimensional stream function ((r, h) as u (r, h) \ u C1 L((r, h) L((r, h) 0 eü [ eü p or sin h r Lh r Lr hd. (18) Note the appearance of the long-lost density (o), the presence of which is required to ensure $ Æ (ou ) \ 0. The p density proðle in the lower envelope is plotted as a dashdotted line in Figure 1a. The stream function itself is constructed from our modiðed Ñow solution u(r) via ((r, h) \ 1eor2u(r)sin2 h cos h, (19) 2 with ( \ 0 for r ¹ r and where e is a rotational parameter deðned as c e \ )2R R1 \ 5.24 ] 10~6 v2, (20) GM M1 e where in the leftmost expression v is the starïs equatorial surface rotational velocity in kilometers e per second and R1 and M1 are expressed in solar units (for more details, see 6 of Tassoul & Tassoul 1982). In the dynamo equations (4) and (5), the relative importance of meridional Ñows is governed by the magnitude of the Reynolds number R, which, as per our nondimensionalization of the dynamo equations, m becomes directly proportional to e. Setting v \ 200 km s~1 then leads to R \ 10~1 in our model for e our choice of m g \ 1013 cm2 s~1. Given that g is e ectively a free param- c c 5.3. a2 Models with Meridional Circulation Even with the Ñow restricted primarily to the radiative envelope (r [ r ), meridional circulation inñuences dynamo action in our model c because the eigenfunctions have signiðcant amplitude outside of the nominal core-envelope boundary (see Figs. 2 and 3), which is itself a consequence of the rapid outward decrease in magnetic di usivity across the core-envelope interface ( 3.2 and 4.2). Figure 6a shows the variation of the surface-to-core rms Ðeld strength ratio & (see eq. [13]) and linear growth rate p for a sequence of antisymmetric a2 solutions with g /g \ 0.1, C \[24, and w/r \ Figures 6b and 6c show e c the eigenfunctions a for the R \ 0 and 500 members of the sequence, respectively, in m the same format as in Figure 2 (full solution parameters and properties are listed at the bottom of Table 1). The circulation streamlines are superimposed as thin black lines, with the circulation in the clockwise direction. It is clear from Figure 6a that meridional circulation does succeed in bringing a signiðcant magnetic Ðeld to the surface, yet strong circulation (i.e., large R ) also leads to a m decrease of the dynamo growth rate, indicating that dynamo action is impeded. This can be traced to the advective action of the circulation immediately outside the coreenvelope interface. The dynamo eigenfunctions are not signiðcantly a ected until R Z 1, beyond which the defor- m mation produced by the circulation becomes increasingly apparent. Interestingly, the growth rates for solutions in the range 1 [ R [ 100 are signiðcantly higher than for the m circulation-free solution of Figure 6b; this is a consequence of the inductive action of the (sheared) meridional Ñow. With the density varying slowly with radius deep in the stellar interior, the meridional Ñow u is nearly incompress- p ible so that, to a good approximation, one can express its inductive contribution as $ Â (u ÂB) ^ [(u Æ $)B ] (B Æ $)u ; (21) p p p the Ðrst term on the right-hand side is a simple advection term describing the transport of the Ðeld by the Ñow. The second term indicates that the poloidal Ðeld is also ampli- Ðed (or destroyed) in proportion to the shear in the meridional Ñow (under the assumption that $ Æ u \ 0, the toroidal Ðeld is not subject to this ampliðcation p mechanism). Evidence that ampliðcation of the poloidal Ðeld by the shear in the circulation is evident in the toroidal-to-poloidal Ðeld ratio #, which is found to decrease with increasing R (see Table 1). The results plotted on m Figure 6 might appear encouraging, but recall that our earlier order-of-magnitude estimate for R suggested R D 10~1, which is orders of magnitude smaller m than the m values required here to produce signiðcant Ðeld dredge up. Moreover, models with g /g ¹ 0.01 do not manage to produce signiðcant surface e Ðelds c (& Z 10~6, say) before circulation shuts o the dynamo a2) Models with Meridional Circulation The inñuence of meridional circulation on the (oscillatory) a2) modes is more complex and hinges on