Keith B. MacGregor

Solar Interface Dynamos. II. Linear, Kinematic Models in Spherical Geometry

Numerical models of interface dynamos are constructed, and their properties discussed in some detail. These models are extensions in spherical geometry of the Cartesian interface models considered by Parker and in the first paper of this series. The models are cast in the framework of classical mean-field electrodynamics and make use of a realistic solar-like internal differential rotation profile. The magnetic diffusivity is assumed to vary discontinously by orders of magnitude across the core-envelope interface. This allows the buildup of very strong toroidal magnetic fields below the interface, as apparently required by recent models of erupting bipolar magnetic regions. Distinct dynamo modes powered either by the latitudinal or radial shear can coexist and, under certain conditions, interfere destructively with one another. Hybrid modes, relying on the latitudinal shear both in the envelope and below it, are most easily excited in some portions of parameter space, and represent a class of dynamo solutions distinct from the true interface modes previously investigated in Cartesian geometry. Which mode is preferentially excited depends primarily on the assumed ratio of magnetic diffusivities on either side of the core-envelope interface. For an α-effect having a simple cos θ latitudinal dependency, the interface mode associated with the radial shear below the polar regions of the interface is easier to excite than its equatorial counterpart. In analogy with more conventional dynamo models, interface modes propagate equatorward if the product of the radial shear (∂Ω/∂r) and α-effect coefficient (Cα) is negative, and poleward if that product is positive. Interface dynamo modes powered by the positive radial shear localized below the core-envelope interface in the equatorial regions can be produced by artificially restricting the α-effect to low latitudes. For negative dynamo number, those modes are globally dipolar, propagate toward the equator, and are characterized by a phase relationship between poloidal and toroidal magnetic field components that is in agreement with observations. While the models discussed in this paper are linear and kinematic, and consequently rather limited in their predictive power, results obtained so far certainly suggest that interface dynamos represent a very attractive alternative to conventional solar mean-field dynamo models.

Angular Momentum Transport in Magnetized Stellar Radiative Zones. IV. Ferraro’s Theorem and the Solar Tachocline

We consider the circumstances under which the latitudinal differential rotation of the solar convective envelope can (or cannot) be imprinted on the underlying radiative core through the agency of a hypothetical weak, large-scale poloidal magnetic field threading the solar radiative interior. We do so by constructing steady, two-dimensional axisymmetric solutions to the coupled momentum and induction equations under the assumption of a purely zonal flow and time-independent poloidal magnetic field. Our results show that the structure of the interior solutions is entirely determined by the boundary conditions imposed at the core-envelope interface. Specifically, in the high Reynolds number regime a poloidal field having a nonzero component normal to the core-envelope interface can lead to the transmission of significant differential rotation into the radiative interior. In contrast, for a poloidal field that is contained entirely within the radiative core, any differential rotation is confined to a thin magnetoviscous boundary layer located immediately beneath the interface, as well as along the rotation/magnetic axis. We argue that a magnetically decoupled configuration is more likely to be realized in the solar interior. Consequently, the helioseismically inferred lack of differential rotation in the radiative core does not necessarily preclude the existence of a weak, large-scale poloidal field therein. We suggest that such a field may well be dynamically significant in determining the structure of the solar tachocline.

MAGNETIC BRAKING FORMULATION FOR SUN-LIKE STARS: DEPENDENCE ON DIPOLE FIELD STRENGTH AND ROTATION RATE

We use two-dimensional axisymmetric magnetohydrodynamic simulations to compute steady-state solutions for solar-like stellar winds from rotating stars with dipolar magnetic fields. Our parameter study includes 50 simulations covering a wide range of relative magnetic field strengths and rotation rates, extending from the slow- and approaching the fast-magnetic-rotator regimes. Using the simulations to compute the angular momentum loss, we derive a semi-analytic formulation for the external torque on the star that fits all of the simulations to a precision of a few percent. This formula provides a simple method for computing the magnetic braking of Sun-like stars due to magnetized stellar winds, which properly includes the dependence on the strength of the magnetic field, mass loss rate, stellar radius, surface gravity, and spin rate, and which is valid for both slow and fast rotators.

The evolution of angular momentum among zero-age main-sequence solar-type stars

We consider a survey of rotation among F, G, and K dwarfs of the Pleiades in the context of other young clusters (Alpha Persei and the Hyades) and pre-main-sequence (PMS) stars (in Taurus-Auriga and Orion) in order to examine how the angular momentum of a star like the sun evolves during its early life on the main sequence. The rotation of PMS stars can be evolved into distributions like those seen in the young clusters if there is only modest, rotation-independent angular momentum loss prior to the ZAMS. Even then, the ultrafast rotators (UFRs, or ZAMS G and K dwarfs with v sin i equal to or greater than 30 km/s) must owe their extra angular momentum to their conditions of formation and to different angular momentum loss rates above a threshold velocity, for it is unlikely that these stars had angular momentum added as they neared the ZAMS, nor can a spread in ages within a cluster account for the range of rotation seen. Only a fraction of solar-type stars are thus capable of becoming UFRs, and it is not a phase that all stars experience. Simple scaling relations (like the Skumanich relation) applied to the observed surface rotation rates of young solar-type stars cannot reproduce the way in which the Pleiades evolve into the Hyades. We argue that invoking internal differential rotation in these ZAMS stars can explain several aspects of the observations and thus can provide a consistent picture of ZAMS angular momentum evolution.

Magnetic Fields in Massive Stars. II. The Buoyant Rise of Magnetic Flux Tubes through the Radiative Interior

We present results from an investigation of the dynamical behavior of buoyant magnetic flux rings in the radiative interior of a uniformly rotating, early-type star. Our physical model describes a thin, axisymmetric, toroidal flux tube that is released from the outer boundary of the convective core and is acted on by buoyant, centrifugal, Coriolis, magnetic tension, and aerodynamic drag forces. We find that rings emitted in the equatorial plane can attain a stationary equilibrium state that is stable with respect to small displacements in radius, but is unstable when perturbed in the meridional direction. Rings emitted at other latitudes travel toward the surface along trajectories that largely parallel the rotation axis of the star. Over much of the ascent, the instantaneous rise speed is determined by the rate of heating by the absorption of radiation that diffuses into the tube from the external medium. Since the timescale for this heating varies like the square of the tube cross-sectional radius, for the same field strength, thin rings rise more rapidly than do thick rings. For a reasonable range of assumed ring sizes and field strengths, our results suggest that buoyancy is a viable mechanism for bringing magnetic flux from the core to the surface, being capable of accomplishing this transport in a time that is generally much less than the stellar main-sequence lifetime.

Solar Interface Dynamos. I. Linear, Kinematic Models in Cartesian Geometry

We describe a simple, kinematic model for a dynamo operating in the vicinity of the interface between the convective and radiative portions of the solar interior. The model dynamo resides within a Cartesian domain, partioned into an upper, convective half and lower, radiative half, with the magnetic diffusivity η of the former region (η2) assumed to exceed that of the latter (η1). The fluid motions that constitute the α-effect are confined to a thin, horizontal layer located entirely within the convective half of the domain; the vertical shear is nonzero only within a second, nonoverlapping layer contained inside the radiative half of the domain. We derive and solve a dispersion relation that describes horizontally propagating dynamo waves. For sufficiently large values of a parameter analogous to the dynamo number of conventional models, growing modes can be found for any ratio of the upper and lower magnetic diffusivities. However, unlike kinematic models in which the shear and α-effect are uniformly distributed throughout the same volume, the present model has wavelike solutions that grow in time only for a finite range of horizontal wavenumbers. An additional consequence of the assumed dynamo spatial structure is that the strength of the azimuthal magnetic field at the location of the α-effect layer is reduced relative to the azimuthal field strength at the shear layer. When the jump in η occurs close to the α-effect layer, it is found that over one period of the dynamos operation, the ratio of the maximum strengths of the azimuthal fields at these two positions can vary as the ratio (η1/η2) of the magnetic diffusivities.

On the Structure and Properties of Differentially Rotating, Main-Sequence Stars in the 1-2 M☉ Range

We present models for chemically homogeneous, differentially rotating, main-sequence stars with masses in the range 1-2 M☉. The models were constructed using a code based on a reformulation of the self-consistent field method of computing the equilibrium stellar structure for a specified conservative internal rotation law. Relative to nonrotating stars of the same mass, these models all have reduced luminosities and effective temperatures, and flattened photospheric shapes (i.e., decreased polar radii) with equatorial radii that can be larger or smaller, depending on the degree of differential rotation. For a fixed ratio of the axial rotation rate to the surface equatorial rotation rate, increasingly rapid rotation generally deepens convective envelopes, shrinks convective cores, and can lead to the presence of a convective core (envelope) in a 1 (2) M☉ model, a feature that is absent in a nonrotating star of the same mass. The positions of differentially rotating models for a given mass in the H-R diagram can be shifted in such a way as to approximate the nonrotating ZAMS for lower mass stars. Implications of these results include (1) possible ambiguities arising from similarities between the properties of rotating and nonrotating models of different masses, (2) a reduced radiative luminosity for a young, rapidly rotating Sun, (3) modified rates of lithium destruction by nuclear processes in the layers beneath an outer convective envelope, and (4) the excitation of solar-like oscillations and the operation of a solar-like hydromagnetic dynamo in some 1.5-2 M☉ stars.

Pulsation modes in rapidly rotating stellar models based on the self-consistent field method

Context. New observational means such as the space missions CoRoT and Kepler and ground-based networks are and will be collecting stellar pulsation data with unprecedented accuracy. A significant fraction of the stars in which pulsations are observed are rotating rapidly. Aims. Our aim is to characterise pulsation modes in rapidly rotating stellar models so as to be able to interpret asteroseismic data from such stars. Methods. A new pulsation code is applied to stellar models based on the self-consistent field (SCF) method. Results. Pulsation modes in SCF models follow a similar behaviour to those in uniformly rotating polytropic models, provided that the rotation profile is not too differential. Pulsation modes fall into different categories, the three main ones being island, chaotic, and whispering gallery modes, which are rotating counterparts to modes with low, medium, and high

Models for the Rapidly Rotating Be Star Achernar

\l-|m|

Angular momentum transport by internal gravity waves

values, respectively. The frequencies of the island modes follow an asymptotic pattern quite similar to what was found for polytropic models. Extending this asymptotic formula to higher azimuthal orders reveals more subtle behaviour as a function of m and provides a first estimate of the average advection of pulsation modes by rotation. Further calculations based on a variational principle confirm this estimate and provide rotation kernels that could be used in inversion methods. When the rotation profile becomes highly differential, it becomes more and more difficult to find island and whispering gallery modes at low azimuthal orders. At high azimuthal orders, whispering gallery modes, and in some cases island modes, reappear. Conclusions. The asymptotic formula found for frequencies of island modes can potentially serve as the basis of a mode identification scheme in rapidly rotating stars when the rotation profile is not too differential.