Paul Charbonneau

A Babcock-Leighton flux transport dynamo with solar-like differential rotation

We investigate the properties of a kinematic —ux transport solar dynamo model. The model is charac- terized by a solar-like internal diUerential rotation pro—le, a single-cell meridional —ow in the convective envelope that is directed poleward at the surface, and a magnetic diUusivity that is constant within the envelope but decreases sharply at the core-envelope interface. As in earlier —ux transport models of the Babcock-Leighton type, we assume that the poloidal —eld is regenerated as a consequence of the emer- gence at the surface, and subsequent decay, of bipolar active regions exhibiting a systematic tilt with respect to the east-west direction. Inspired by recent simulations of the rise of toroidal magnetic —ux ropes across the solar convective envelope, we model this poloidal —eld regeneration mechanism as a nonlocal source term formulated in such a way as to account for some of the properties of rising —ux ropes revealed by the simulations. For a broad range of parameter values the model leads to solar cycle¨ like oscillatory solutions. Because of the solar-like internal diUerential rotation pro—le used in the model, solutions tend to be characterized by time-latitude (butter—y) diagrams that exhibit both poleward- and equatorward-propagating branches. We demonstrate that the latitudinal shear in the envelope, often omitted in other —ux transport models previously published in the literature, actually has a dominant eUect on the global morphology and period of the solutions, while the radial shear near the core- envelope interface leads to further intensi—cation of the toroidal —eld. On the basis of an extensive parameter space study, we establish a scaling law between the time period of the cycle and the primary parameters of the model, namely the meridional —ow speed, source coefficient, and turbulent diUusion coefficient. In the parameter regime expected to characterize the Sun, we show that the time period of the cycle is most signi—cantly in—uenced by the circulation —ow speed and, unlike for conventional mean —eld a) dynamos, is little aUected by the magnitude of the source coefficient. Finally, we present one speci—c solution that exhibits features that compare advantageously with the observed properties of the solar cycle. Subject headings: diUusionSun: interiorSun: magnetic —eldsSun: rotation

Helioseismic Constraints on the Structure of the Solar Tachocline

This paper presents a series of helioseismic inversions aimed at determining with the highest possible confidence and accuracy the structure of the rotational shear layer (the tachocline) located beneath the base of the solar convective envelope. We are particularly interested in identifying features of the inversions that are robust properties of the data, in the sense of not being overly influenced by the choice of analysis methods. Toward this aim we carry out two types of two-dimensional linear inversions, namely Regularized Least-Squares (RLS) and Subtractive Optimally Localized Averages (SOLA), the latter formulated in terms of either the rotation rate or its radial gradient. We also perform nonlinear parametric least-squares fits using a genetic algorithm-based forward modeling technique. The sensitivity of each method is thoroughly tested on synthetic data. The three methods are then used on the LOWL 2 yr frequency-splitting data set. The tachocline is found to have an equatorial thickness of w/R☉ = 0.039 ± 0.013 and equatorial central radius rc/R☉ = 0.693 ± 0.002. All three techniques also indicate that the tachocline is prolate, with a difference in central radius Δrc/R☉ 0.024 ± 0.004 between latitude 60° and the equator. Assuming uncorrelated and normally distributed errors, a strictly spherical tachocline can be rejected at the 99% confidence level. No statistically significant variation in tachocline thickness with latitude is found. Implications of these results for hydrodynamical and magnetohydrodynamical models of the solar tachocline are discussed.

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.

Stochastic Fluctuations in a Babcock-Leighton Model of the Solar Cycle

We investigate the eUect of stochastic —uctuations on a —ux transport model of the solar cycle based on the Babcock-Leighton mechanism. Speci—cally, we make use of our recent —ux transport model (Dikpati & Charbonneau) to investigate the consequences of introducing large-amplitude stochastic —uc- tuations in either or both the meridional —ow and poloidal source term in the model. Solar cyclelike oscillatory behavior persists even for —uctuation amplitudes as high as 300%, thus demonstrating the inherent robustness of this class of solar cycle models. We also —nd that high-amplitude —uctuations lead to a spread of cycle amplitude and duration showing a statistically signi—cant anticorrelation, compara- ble to that observed in sunspot data. This is a feature of the solar cycle that is notoriously difficult to reproduce with dynamo models based on mean —eld electrodynamics and relying only on nonlinearities associated with the back-reaction of the Lorentz force to produce amplitude modulation. Another note- worthy aspect of our —ux transport model is the fact that meridional circulation in the convective envelope acts as a ii clock ˇˇ regulating the tempo of the solar cycle; shorter-than-average cycles are typi- cally soon followed by longer-than-average cycles. In other words, the oscillation exhibits good phase locking, a property that also characterizes the solar activity cycle. This shows up quite clearly in our model, but we argue that it is in fact a generic property of —ux transport models based on the Babcock- Leighton mechanism, and relies on meridional circulation as the primary magnetic —eld transport agent. Subject headings: Sun: activitySun: interiorSun: magnetic —elds

Avalanche models for solar flares (Invited Review)

This paper is a pedagogical introduction to avalanche models of solar flares, including a comprehensive review of recent modeling efforts and directions. This class of flare model is built on a recent paradigm in statistical physics, known as self-organized criticality. The basic idea is that flares are the result of an ‘avalanche’ of small-scale magnetic reconnection events cascading through a highly stressed coronal magnetic structure, driven to a critical state by random photospheric motions of its magnetic footpoints. Such models thus provide a natural and convenient computational framework to examine Parkers hypothesis of coronal heating by nanoflares.

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.

Stellar structure modeling using a parallel genetic algorithm for objective global optimization

Genetic algorithms are a class of heuristic search techniques that apply basic evolutionary operators in a computational setting. We have designed a fully parallel and distributed hardware/software implementation of the generalized optimization subroutine PIKAIA, which utilizes a genetic algorithm to provide an objective determination of the globally optimal parameters for a given model against an observational data set. We have used this modeling tool in the context of white dwarf asteroseismology, i.e., the art and science of extracting physical and structural information about these stars from observations of their oscillation frequencies: The efficient, parallel exploration of parameter-space made possible by genetic-algorithm-based numerical optimization led us to a number of interesting physical results: (1) resolution of a hitherto puzzling discrepancy between stellar evolution models and prior asteroseismic inferences of the surface helium layer mass for a DBV white dwarf; (2) precise determination of the central oxygen mass fraction in a white dwarf star; and (3) a preliminary estimate of the astrophysically important but experimentally uncertain rate for the 12C(α, γ)16O nuclear reaction. These successes suggest that a broad class of computationally intensive modeling applications could also benefit from this approach.

Stability of the Solar Latitudinal Differential Rotation Inferred from Helioseismic Data

We revisit the hydrodynamical stability problem posed by the observed solar latitudinal differential rotation. Specifically, we carry out stability analyses on a spherical shell for solar-like two-dimensional inviscid shear flow profiles of the form ν = s0 - s2μ2 - s4μ4, where μ is the sine of latitude. We find that stability is remarkably sensitive to the magnitude of the μ4 term. This allows us to reconcile apparently conflicting results found in the published literature. We then use latitudinal differential rotation profiles extracted from various helioseismic inversions of the solar internal rotation and investigate their stability as a function of depth from the base of the tachocline to the top of the convective envelope. In all cases considered, we find that the latitudinal differential rotation in the tachocline is stable while that in the bulk of the convective envelope is unstable. Under the assumption that the instability is not impeded by finite Reynolds number or three-dimensional effects not accounted for in our analysis, we speculate on possible observable consequences of the occurrence of the instability in the top half of the convective envelope.

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.

The Rotation of the Solar Core Inferred by Genetic Forward Modeling

Genetic forward modeling is a genetic algorithm-based modeling technique that can be used to perform helioseismic inversions of the Suns internal angular velocity profile. The method can easily accommodate constraints such as positivity and monotonicity and readily lends itself to the use of robust statistical goodness-of-fit estimators. After briefly describing the technique, we ascertain its performance by carrying out a series of inversions for artificial splitting data generated from a set of synthetic internal rotation profiles characterized by various small inward increases in angular velocity in the deep solar core (r/R☉ ≤ 0.5). These experiments indicate that the technique is accurate down to r/R☉ 0.2, and retains useful sensitivity down to r/R☉ 0.1. We then use genetic forward modeling in conjunction with the LOW degree L (LOWL) 2 year frequency-splitting data set to determine the rotation rate in the deep solar core. We perform a large set of one-dimensional and 1.5-dimensional inversions using regularized least-squares minimization, conventional least-squares minimization with a monotonicity constraint (∂Ω/∂r ≤ 0), and inversions using robust statistical estimators. These calculations indicate that the solar core rotates very nearly rigidly down to r/R☉ ~ 0.1. More specifically, on spatial scales 0.04 R☉ we can rule out inward increases by more than 50% down to r/R☉ = 0.2, and by more than a factor of 2 down to r/R☉ = 0.1. Thorough testing of various possible sources of bias associated with our technique indicates that these results are robust with respect to intrinsic modeling assumptions. Consequences of our results for models of the rotational evolution of the Sun and solar-type stars are discussed.