Mausumi Dikpati

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

Flux-Transport Dynamos with α-Effect from Global Instability of Tachocline Differential Rotation: A Solution for Magnetic Parity Selection in the Sun

We propose an αΩ flux-transport dynamo for the Sun that is driven by a tachocline α-effect. This α-effect comes from the global hydrodynamic instability of latitudinal differential rotation in the tachocline, as calculated using a shallow-water model. Growing, unstable shallow-water modes propagating longitudinally in the tachocline create vortices that correlate with radial motion in the layer to produce a longitude-averaged net kinetic helicity and, hence, an α-effect. We show that such a dynamo is equally successful as a Babcock-Leighton-type flux-transport dynamo in reproducing many large-scale solar cycle features. The success of both dynamo types depends on the inclusion of meridional circulation of a sign and magnitude similar to that seen on the Sun. Both α-effects (the Babcock-Leighton-type and tachocline α-effect) are likely to exist in the Sun, but it is hard to estimate their relative magnitudes. By extending the simulation to a full spherical shell, we show that the flux-transport dynamo driven by the tachocline α-effect selects a toroidal field that is antisymmetric about the equator, while the Babcock-Leighton flux-transport dynamo selects a symmetric toroidal field. Since our present Sun selects antisymmetric fields, we argue that the tachocline α-effect must be more important than the Babcock-Leighton α-effect.

Diagnostics of Polar Field Reversal in Solar Cycle 23 Using a Flux Transport Dynamo Model

Motivated by observed anomalous features in cycle 23, as inferred from records of photospheric magnetic flux, we develop a flux transport dynamo-based scheme in order to investigate the physical cause of such anomalies. In this first study we focus on understanding anomalies occurring in the polar field evolutionary pattern in cycle 23, namely, why the polar reversal in cycle 23 was slow, why after reversal the buildup of the polar field was slow, and why the south pole reversed approximately a year after the north pole did. We construct a calibrated flux transport dynamo model that operates with dynamo ingredients such as differential rotation, meridional circulation, and large-scale poloidal field source derived from observations. A few other dynamo ingredients, such as diffusivity and quenching pattern, for which direct observations are not possible, are fixed by using theoretical guidance. By showing that this calibrated model can reproduce major longitude-averaged solar cycle features, we initialize the model at the beginning of cycle 22 and operate by incorporating the observed variations in meridional circulation and large-scale surface magnetic field sources to simulate the polar field evolution in cycle 23. We show that a 10%-20% weakening in photospheric magnetic flux in cycle 23 with respect to that in cycle 22 is the primary reason for a ~1 yr slowdown in polar reversal in cycle 23. Weakening in this flux is also the reason for slow buildup of polar field after reversal, whereas the observed north-south asymmetry in meridional circulation in the form of a larger decrease in flow speed in the northern hemisphere than that in the southern hemisphere during 1996-2002 and the appearance of a reverse, high-latitude flow cell in the northern hemisphere during 1998-2001 caused the north polar field to reverse before the south polar field.

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

A solar mean field dynamo benchmark

Context. The solar magnetic activity and cycle are linked to an internal dynamo. Numerical simulations are an e cient and accurate tool to investigate such intricate dynamical processes. Aims. We present the results of an international numerical benchmark study based on two-dimensional axisymmetric mean field solar dynamo models in spherical geometry. The purpose of this work is to provide the scientific community with reference cases that can be analyzed in detail and that can help in further development and validation of numerical codes that solve such problems. Methods. The results of eight numerical codes solving the induction equation in the framework of mean field theory are compared for three increasingly computationally intensive models of the solar dynamo: an dynamo with constant magnetic di usivity, an dynamo with magnetic di usivity sharply varying with depth and an example of a flux-transport Babcock-Leighton dynamo which includes a non-local source term and one large single cell of meridional circulation per hemisphere. All cases include a realistic profile of di erential rotation and thus a sharp tachocline. Results. The most important finding of this study is that all codes agree quantitatively to within less than a percent for the dynamo cases and within a few percents for the flux-transport case. Both the critical dynamo numbers for the onset of dynamo action and the corresponding cycle periods are reasonably well recovered by all codes. Detailed comparisons of butterfly diagrams and specific cuts of both toroidal and poloidal fields at given latitude and radius confirm the good quantitative agreement. Conclusions. We believe that such a benchmark study will be a very useful tool for the scientific community since it provides detailed standard cases for comparison and reference.

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.

Joint Instability of Latitudinal Differential Rotation and Concentrated Toroidal Fields below the Solar Convection Zone

Motivated by observations of sunspot and active-region latitudes that suggest that the subsurface toroidal field in the Sun occurs in narrow latitude belts, we analyze the joint instability of solar latitudinal differential rotation and the concentrated toroidal field below the base of the convection zone, extending the work of Gilman & Fox (hereafter GF). We represent the profile of the toroidal field by Gaussian functions whose width is a variable parameter and solve the two-dimensional perturbation equations of GF by relaxation methods. We reproduce the results of GF for broad profiles, and we find instability for a wide range of amplitudes of differential rotation and toroidal fields (103-106 G fields at the base of the solar convection zone), as well as a wide range of toroidal-field bandwidths. We show that the combination of concentrated toroidal fields and solar-type latitudinal differential rotation is again unstable, not only to longitudinal wavenumber m=1 as in GF, but also to m>1 for sufficiently narrow toroidal-field profiles. For a fixed peak field strength, the growth rate first increases as the toroidal-field band is narrowed, reaching a peak for bandwidths between 10° and 20° in latitude, depending on the peak field strength, and then decreases to a cut-off in the instability for toroidal field bands of 3°-4°. Irrespective of bandwidth, the differential rotation is the primary energy source for the instability for weak fields, and the toroidal field is the primary source for strong fields. The weaker (stronger) the peak toroidal field is, the narrower (broader) is the bandwidth for which the toroidal field becomes the primary energy source. The Reynolds, Maxwell, and mixed stresses required to extract energy from the differential rotation and toroidal field are most active in the neighborhood of the singular or turning points of the perturbation equations. This first study focuses on toroidal fields that peak near 45° latitude, as in GF; later papers will place the toroidal-field peak at a wide variety of latitudes, as we might expect to occur at different phases of a sunspot cycle.

FLUX TRANSPORT SOLAR DYNAMOS WITH NEAR-SURFACE RADIAL SHEAR

Corbard & Thompson analyzed quantitatively the strong radial differential rotation that exists in a thin layer near the solar surface. We investigate the role of this radial shear in driving a flux transport dynamo operating with such a rotation profile. We show that despite being strong, near-surface radial shear effectively contributes only ~1 kG (~30% of the total) to the toroidal fields produced there unless an abnormally high, surface α-effect is included. While 3 kG spot formation from ~1-2 kG toroidal fields by convective collapse cannot be ruled out, the evolutionary pattern of these model fields indicates that the polarities of spots formed from the near-surface toroidal field would violate the observed polarity relationship with polar fields. This supports previous results that large-scale solar dynamos generate intense toroidal fields in the tachocline, from which buoyant magnetic loops rise to the photosphere to produce spots. Polar fields generated in flux transport models are commonly much higher than observed. We show here that by adding enhanced diffusion in the supergranulation layer (originally proposed by Leighton), near-surface toroidal fields undergo large diffusive decay preventing spot formation from them, as well as reducing polar fields closer to the observed values. However, the weaker polar fields lead to the regeneration of a toroidal field of less than ~10 kG at the convection zone base, too weak to produce spots that emerge in low latitudes, unless an additional poloidal field is produced at the tachocline. This is achieved by a tachocline α-effect, previously shown to be necessary for coupling the north and south hemispheres to ensure toroidal and poloidal fields that are antisymmetric about the equator.

Joint Instability of Latitudinal Differential Rotation and Concentrated Toroidal Fields below the Solar Convection Zone. II. Instability of Narrow Bands at All Latitudes

Assuming that concentrated toroidal field bands occur in the solar tachocline at different latitudes as the solar cycle progresses, we examine the joint instability of latitudinal differential rotation and coexisting narrow bands placed at a wide range of latitudes. Following the basic formalism developed by Gilman & Fox and employing the numerical technique of Dikpati & Gilman, we show that the instability exists for almost all phases of the solar cycle, i.e., for a wide range of latitudinal positions of the bands of field strengths between 500 G and 200 kG. Modes with longitudinal wavenumber m = 1 up to m = 7, depending on parameter values, are unstable for both kinds of symmetries. Mid-latitude bands are the most unstable; the instability disappears if the band is at very high or low latitudes. High-latitude bands are highly unstable, with e-folding growth times of a few months, even when the field strength is low (about a few hundred gauss); low-latitude bands are unstable with longer growth times (≥1 yr), but for high field strengths (104-2 × 105 G). We argue that, because of the instability, the high-latitude bands would undergo turbulent mixing in the latitudinal direction on a timescale that is short compared to their build-up time from shearing of poloidal field by the differential rotation, and thus they may not be buoyant enough to appear as active regions at the surface, but the low-latitude bands can build up simultaneously with this instability and can eventually manifest as active regions. Instability for modes with m ≥ 1 could help determine the longitude distribution of active regions; nonlinear changes in the toroidal field due to the instability may contribute to their decay.

Energy Transport to the Solar Corona by Magnetic Kink Waves

We show that the magnetic kink waves generated by the motions of photospheric footpoints of the coronal flux tubes can supply adequate energy for heating the quiet corona, provided there are occasional rapid motions of these footpoints as found in recent observations. Choudhuri et al. (1992) modeled the solar corona as isothermal atmosphere and showed that these rapid motions are much more efficient for transporting energy compared to the slow footpoint motions taking place most of the time. We extend these calculations for a two-layer atmosphere, with the lower layer having chromospheric thickness and temperature, and the upper layer having coronal temperature. Even in the presence of such a temperature jump we find that the rapid footpoint motions are still much more efficient for transporting energy to the corona and the estimated energy flux is sufficient for quiet coronal heating. We discuss the general problem of the propagation of kink pulses in a two-layer atmosphere for different possible values of the basic parameters. We find a fairly complicated behavior which could not be anticipated from the analysis of a pure Fourier mode.