Dynamics of the solar tachocline – I. An incompressible study

Gough & McIntyre have suggested that the dynamics of the solar tachocline are dominated by the advection–diffusion balance between the differential rotation, a large-scale primordial field and baroclinicly driven meridional motions. This paper presents the first part of a study of the tachocline, in which a model of the rotation profile below the convection zone is constructed along the lines suggested by Gough & McIntyre and solved numerically. In this first part, a reduced model of the tachocline is derived in which the effects of compressibility and energy transport on the system are neglected; the meridional motions are driven instead by Ekman–Hartmann pumping. Through this simplification, the interaction of the fluid flow and the magnetic field can be isolated and is studied through non-linear numerical analysis for various field strengths and diffusivities. It is shown that there exists only a narrow range of magnetic field strengths for which the system can achieve a nearly uniform rotation. The results are discussed with respect to observations and to the limitations of this initial approach. A following paper combines the effects of realistic baroclinic driving and stratification with a model that closely follows the lines of work of Gough & McIntyre.     

Analytical theory of tachocline transport at mid latitudes

We provide a theory of magnetic diffusion, momentum transport, and mixing in the solar tachocline by considering magnetohydrodynamics (MHD) turbulence on a β plane subject to a large scale shear (provided by the latitudinal differential rotation). In the strong magnetic field regime, we find that the turbulent viscosity and diffusivity are reduced by magnetic fields only, similarly to the two-dimensional MHD case (without Rossby waves). In the weak magnetic field regime, we find a crossover scale (L_R) from a Alfvén dominated regime (on small scales) to a Rossby dominated regime (on large scales). For parameter values typical of the tachocline, L_R is larger than the solar radius so that Rossby waves are unlikely to play an important role in the transport of magnetic field and angular momentum. This is mainly due to the enhancement of magnetic back-reaction by shearing which efficiently generates small scales, thus strong currents. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rotation of the magnetic elements in polar regions on the Sun

Using the Michelson Doppler Imager (MDI) data from Solar and Heliospheric Observatory (SOHO), the rotation rate of the unipolar magnetic regions in North high-latitude regions of the Sun is estimated by tracking individual magnetic elements. The analysis reveals a strong spin down near the pole, which is greater than the Doppler and magnetic rotation rates estimated by Snodgrass & Ulrich (1990), and rotation rate inferred from helioseismology (Birch & Kosovichev 1998), and is probably related to variation of velocity gradient in the subsurface shear layer. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solar internal rotation rate and the latitudinal variation of the tachocline

A new set of accurately measured frequencies of solar oscillations is used to infer the rotation rate inside the Sun, as a function of radial distance as well as latitude. We have adopted a regularized least-squares technique with iterative refinement for both 1.5D inversion, using the splitting coefficients, and 2D inversion using individual m splittings. The inferred rotation rate agrees well with earlier estimates showing a shear layer just below the surface and another one around the base of the convection zone. The tachocline or the transition layer where the rotation rate changes from differential rotation in the convection zone to an almost latitudinally independent rotation rate in the radiative interior is studied in detail. No compelling evidence for any latitudinal variation in the position and width of the tachocline is found, although it appears that the tachocline probably shifts to a slightly larger radial distance at higher latitudes and possibly also becomes thicker. However, these variations are within the estimated errors and more accurate data would be needed to make a definitive statement about latitudinal variations.     

Latitudinal shear instability in the solar tachocline

In an attempt to explain the observed rotation profile in the solar radiative zone and the tachocline, Spiegel & Zahn proposed a model based on anisotropic turbulent angular momentum transport. Although very successful in reproducing some of the features of the solar tachocline, their model assumes without verification that the origin of the turbulence could be caused by latitudinal shear instability. This paper studies the weakly non-linear evolution of two-dimensional shear instability, in which the interaction between the global rotation profile and the Reynolds stresses can be described self-consistently. Provided that the initial rotation profile is sufficiently close to marginal stability (which is the case of the solar tachocline), the instability is shown to saturate and to relax to a marginally stable state, which differs very little from the observed rotation profile. It is therefore likely that the tachocline is in a state of marginal stability with respect to latitudinal shear instability, and shows that angular momentum transport in the tachocline is unlikely to be caused by shear-induced turbulence.     

The angular momentum transport in rotating turbulent convection

The angular momentum transport in rotating turbulent convection is simulated with the NIRVANA code for Taylor numbers up to 10⁶. The box consists of an unstable layer embedded in two stable overshoot layers. We find the expected anisotropies in the rotating anisotropic turbulence field: 〈u^′2_r〉 exceeds 〈u^′2_ϕ〉, and 〈u^′2_ϕ〉 exceeds 〈u^′2_θ〉. The resulting radial angular momentum transport is directed inwards and peaks in the middle of the convective layer. The resulting latitudinal angular momentum transport is equatorwards, peaks at the surface and is highly concentrated to the equatorial region. Apart from a factor of 2–3 the total amplitudes of the cross‐correlations are of the same order of magnitude. In the lower overshoot region (‘tachocline’) the cross‐correlations are negative. It is argued that the concentration of the latitudinal angular momentum transport towards the surface and towards the equator does not too strongly reduce its potential to produce rotation laws with accelerated equators. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A study of possible temporal and latitudinal variations in the properties of the solar tachocline

Temporal variations of the structure and the rotation rate of the solar tachocline region are studied using helioseismic data from the Global Oscillation Network Group (GONG) and the Michelson Doppler Imager (MDI) obtained during the period 1995–2000. We do not find any significant temporal variation in the depth of the convection zone, the position of the tachocline or the extent of overshoot below the convection zone. No systematic variation in any other properties of the tachocline, like width, etc., is found either. The possibility of periodic variations in these properties is also investigated. Time-averaged results show that the tachocline is prolate with a variation of about 0.02 R_⊙ in its position. Neither the depth of the convection zone nor the extent of overshoot shows any significant variation with latitude.     

Subdiffusive transport in intergranular lanes on the Sun. The Leighton model revisited

In this paper we consider a random motion of magnetic bright points (MBP) associated with magnetic fields at the solar photosphere. The MBP transport in the short time range [0–20 minutes] has a subdiffusive character as the magnetic flux tends to accumulate at sinks of the flow field. Such a behavior can be rigorously described in the framework of a continuous time random walk leading to the fractional Fokker-Planck dynamics. This formalism, applied for the analysis of the solar subdiffusion of magnetic fields, generalizes the Leighton’s model.      

Dynamo action in simulations of penetrative solar convection with an imposed tachocline

We summarize new and continuing three-dimensional spherical shell simulations of dynamo action by convection allowed to penetrate downward into a tachocline of rotational shear. The inclusion of an imposed tachocline allows us to examine several processes believed to be essential in the operation of the global solar dynamo, including differential rotation, magnetic pumping, and the stretching and organization of fields within the tachocline. In the stably stratified core, our simulations reveal that strong axisymmetric magnetic fields (of ∼ 3000 G strength) can be built, and that those fields generally exhibit a striking antisymmetric parity, with fields in the northern hemisphere largely of opposite polarity to those in the southern hemisphere. In the convection zone above, fluctuating fields dominate over weaker mean fields. New calculations indicate that the tendency toward toroidal fields of antisymmetric parity is relatively insensitive to initial magnetic field configurations; they also reveal that on decade-long timescales, the magnetic fields can briefly enter (and subsequently emerge from) states of symmetric parity.We have not yet observed any overall reversals of the field polarity, nor systematic latitudinal propagation. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of the solar tachocline with magnetic fields

The stability of magnetic fields in the solar tachocline is investigated. We present stability limits for higher azimuthal wave numbers and results on the dependence of the stability on the location of toroidal magnetic fields in latitude. While the dependence of the wave number with the largest growth rate on the magnetic field strength and the magnetic Prandtl number is small, the dependence on the magnetic Reynolds number Rm indicates that lowest azimuthal modes are excited for very high Rm. Upon varying the latitudinal position of the magnetic field belts, we find slightly lower stability limits for high latitudes, and very large stability limits at latitudes below 10°, with little dependence on latitude in between. An increase of the maximum possible field was achieved by adding a poloidal field. The upper limit for the toroidal field which can be stored in the radiative tachocline is then 1000 G, compared to about 100 G for a purely toroidal field as was found in an earlier work. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Perspectives on the interior of the Sun

The interior of the Sun is not directly accessible to observations. Nonetheless, it is possible to infer the physical conditions inside the Sun with the help of structure equations governing its equilibrium and with the powerful observational tools provided by the neutrino fluxes and oscillation frequencies. The helioseismic data show that the internal constitution of the Sun can be adequately represented by a standard solar model. It turns out that a cooler solar core is not a viable solution for the measured deficit of neutrino fluxes, and the resolution of the solar neutrino puzzle should be sought in the realm of particle physics.     

Nonlinear simulations of magnetic instabilities in stellar radiation zones: The role of rotation and shear

Using the 3-dimensional ASH code, we have studied numerically the instabilities that occur in stellar radiation zones in presence of large-scale magnetic fields, rotation and large-scale shear. We confirm that some configurations are linearly unstable, as predicted by Tayler and collaborators, and we determine the saturation level of the instability. We find that rotation modifies the peak of the most unstable wave number of the poloidal instability but not its growth rate as much as in the case of the m = 1 toroidal instability for which it is changed to σ = /Ω. Further in the case with rotation and shear, we found no sign of the dynamo mechanism suggested recently by Spruit even though we possess the essential ingredients (Tayler's m = 1 instability and a large scale shear) supposedly at work. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Axial rotation, orbital revolution and solar spin–orbit coupling

The orbital motion of the Sun has been linked with solar variability, but the underlying physics remains unknown. A coupling of the solar axial rotation and the barycentric orbital revolution might account for the relationships found. Some recent published studies addressing the physics of this problem have made use of equations from rotational physics in order to model particle motions. However, our standard equations for rotational velocity do not accurately describe particle motions due to orbital revolution. The Sun's orbital motion is a state of free fall; in consequence, aside from very small tidal motions, the associated particle velocities do not vary as a function of position on or within the body of the Sun. In this note, I describe and illustrate the fundamental difference between particle motions in rotation and revolution, in order to dispel some part of the confusion that has arisen in the past and that which may yet arise in the future. This discussion highlights the principal physical difficulty that must be addressed and overcome by future dynamical spin–orbit coupling hypotheses.     

Stability of toroidal magnetic fields in the solar tachocline and beneath

Stability of toroidal magnetic field in a stellar radiation zone is considered for the cases of uniform and differential rotation. In the rigidly rotating radiative core shortly below the tachocline, the critical magnetic field for instability is about 600 G. The unstable disturbances for slightly supercritical fields have short radial scales ∼1 Mm. Radial mixing produced by the instability is estimated to conclude that the internal field of the sun can exceed the critical value of 600 G only marginally. Otherwise, the mixing is too strong and not compatible with the observed lithium abundance. Analysis of joint instability of differential rotation and toroidal field leads to the conclusion that axisymmetric models of the laminar solar tachocline are stable to nonaxisymmetric disturbances. The question of whether sun-like stars can posses tachoclines is addressed with positive answer for stars with rotation periods shorter than about two months. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling helioseismic modes in the magnetic atmosphere of the Sun

The pulsation of the solar surface is caused by acoustic waves traveling in the solar interior. Thorough analyses of observational data indicate that these f and p helioseismic oscillation modes are not bounced back completely at the surface but they partially penetrate into the atmosphere. Atmospheric effects and their possible observational application are investigated in one‐dimensional magnetohydrodynamic models. It is found that f and p mode frequencies are shifted of the order of μHz due to the presence of an atmospheric magnetic field. This shift varies with the direction of the wave propagation.Resonant coupling of global helioseismic modes to local Alfvén and slow waves reduce the life time of the global modes. The resulting line width of the frequency line is of the order of nHz, and it also varies with propagation angle. These features enable us to use helioseismic observations in magnetic diagnostics of the lower atmosphere. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)