We have performed the first three-dimensional non-linear simulation of the turbulent convective envelope of a rotating 0.8 M⊙ RGB star using the ASH code. Adopting a global typical rotation rate of a tenth of the solar rate, we have analyzed the dynamical properties of the convection and the transport of angular momentum within the inner 50% in radius of the convective envelope. The convective patterns consist of a small number of large cell, associated with fast flows (∼3000m/s) and large temperature fluctuations (∼300K) in order to carry outward the large luminosity (L ∼ 400L⊙) of the star. The interactions between convection and rotation give rise to a large radial differential rotation and a meridional circulation possessing one cell per hemisphere, the flow being poleward in both hemisphere. By analysing the redistribution of angular momentum, we find that the meridional circulation transports the angular momentum outward in the radial direction, and poleward in the latitudinal direction, and that the transport by Reynolds stresses acts in the opposite direction. From this 3-D simulation, we have derived an average radial rotation profile, that we will ultimately introduce back into 1-D stellar evolution code.

We study convective overshooting by means of local 3D convection calculations. Using a mixing length model of the solar convection zone (CZ) as a guide, we determine the Coriolis number (Co), which is the inverse of the Rossby number, to be of the order of ten or larger at the base of the solar CZ. Therefore we perform convection calculations in the range Co = 0. . .10 and interpret the value of Co realised in the calculation to represent a depth in the solar CZ. In order to study the dependence on rotation, we compute the mixing length parameters αT and αu relating the temperature and velocity fluctuations, respectively, to the mean thermal stratification. We find that the mixing length parameters for the rapid rotation case, corresponding to the base of the solar CZ, are 3-5 times smaller than in the nonrotating case. Introducing such depth-dependent α into a solar structure model employing a non-local mixing length formalism results in overshooting which is approximately proportional to α at the base of the CZ. Although overshooting is reduced due to the reduced α, a discrepancy with helioseismology remains due to the steep transition to the radiative temperature gradient.

In comparison to the mixing length models the transition at the base of the CZ is much gentler in the 3D models. It was suggested recently (Rempel 2004) that this discrepancy is due to the significantly larger (up to seven orders of magnitude) input energy flux in the 3D models in comparison to the Sun and solar models, and that the 3D calculations should be able to approach the mixing length regime if the input energy flux is decreased by a moderate amount. We present results from local convection calculations which support this conjecture.

The Session F round table. and audience, considered the following questions:

1. (i) what determines the differential rotation in stellar convective zones?

2. (ii) If there really is a rapid inward increase in angular velocity in giants will it have significant effects in the late stages of stellar evolution?

3. (iii) How effective is penetration above a convective core?

4. (iv) Why is the solar core nearly uniformly rotating?

The Rossby number Ro is an important quantity related to the well-known magnetic activity-rotation correlation for main sequence, solar-type stars. For such stars, Ro can be obtained by a semi-empirical relationship between the convective turnover time τc and the B-V colour index, but an equivalent activity-rotation correlation seems not to exist for pre-main sequence stars. In this work we report theoretical estimates of τc for low-mass, rotating pre-main sequence stars under either the Full Spectrum of Turbulence (FST) or the classical Mixing Length Theory (MLT) convection models. The results for the MLT models show that the lower the convection efficiency the higher τc, while the FST models give τc lower than those for the MLT. The presence of a parametric magnetic field lowers the convection efficiency, resulting in smaller τc values.

We study the redistribution of the mechanisms of the energy transport with rotation in the interiors of young stars of the Main sequence. In that purpose, we did numerical modelling of the structure of zero age stars of the Main sequence for 26 stellar masses in the interval 0.8–120 Solar masses (M⊙), with constant chemical composition X = 0.75, Y = 0.23, Z = 0.02 (N = 0.3 Z). Rotation of a solid body type with angular velocity varied from zero (spherically symmetrical models) till critical one (models at the border of dynamical stability) for each stellar mass is considered. For numerical integration of the differential equations we used Henyey's method for spherically symmetrical stars, modified by the technique of Kippenhahn and Thomas for rotating stars, with the algorithm used by J. Petrovic. Additional algorithms determine the mass and radius of convective core.

Analysis of the results shows that the radius rf of the convective core decreases with increase of the angular velocity and has the lowest value for critical rotation. The radius of convective core has the lowest value for 1.2 M⊙ for spherically symmetric models, and in the interval 1–1.3 M⊙ for rotating stars at the border of dynamical stability. Absolute values (rf are in a range from 0.03–0.08, and increase with increase of the stellar mass for m >1.2 M⊙, while the relative changes are in the interval 15–22%. The lowest value of the mass qf of the convective core coincides with the lowest values of the radius. The change (qf >0, and increases with increase of the stellar mass for m >1.4 M⊙, while the greatest relative change with rotation is 7%. We show the dependence of qf from the structural parameters in the centre and on the surface of the star.

We emphasize a “critical” mass of the star, or a characteristic interval of stellar masses (1.2–1.4 M⊙), in which a qualitative change in the local structure and the distribution of mechanisms of energy transport of both, spherically symmetric and rotating stars occurs. It is connected with the change of the heat regime of the star in hydrostatic equilibrium. Rotation additionally changes the structure and redistributes the regions in radiative and convective equilibrium.

Differential rotation not only occurs in astrophysical plasmas like accretion disks, it is also measured in laboratory plasmas as manifested in the toroidal rotation of tokamak plasmas. A re-examination of the Lagrangian of the system shows that the inclusion of the angular momentum's radial variation in the derivation of the equations of motion produces a force term that couples the angular velocity gradient with the angular momentum. This force term is a property of the angular velocity field, so that the results are valid wherever differential rotation is present.

The outer surface layers of the sun show a clear deceleration at low latitudes. This is generally thought to be the result of a strong dominance of vertical turbulent motions associated with strong downdrafts. This strong negative radial shear should not only contribute to amplifying the toroidal field locally and to expelling magnetic helicity, but it may also be responsible for producing a strong prograde pattern speed in the supergranulation layer. Using simulations of rotating stratified convection in cartesian boxes located at low latitudes around the equator it is shown that in the surface layers patterns move in the prograde direction on top of a retrograde mean background flow. These patterns may also be associated with magnetic tracers and even sunspot proper motions that are known to be prograde relative to the much slower surface plasma.

Possibilities and difficulties of applying the theory of magnetic field generation by convection flows in rotating spherical fluid shells to the Giant Planets are outlined. Recent progress in the understanding of the distribution of electrical conductivity in the Giant Planets suggests that the dynamo process occurs predominantly in regions of semiconductivity. In contrast to the geodynamo the magnetic field generation in the Giant Planets is thus characterized by strong radial conductivity variations. The importance of the constraint on the Ohmic dissipation provided by the planetary luminosity is emphasized. Planetary dynamos are likely to be of an oscillatory type, although these oscillations may not be evident from the exterior of the planets.

Magnetic fields, convective motions and radiative energy transport all strongly influence each other in the solar photosphere. Attempts to understand, at a quantitative level, the diverse phenomona seen in the photosphere have relied on increasingly powerful computers since the 1980's. This approach has been remarkably successful and some recent examples, particularly cases where all three ingredients are important, will be discussed.

The convection-driven MHD dynamo in a rotating spherical shell is simulated numerically. Convection cells are regarded as a connecting link between the global and local electromagnetic processes. Local (in many cases, bipolar) magnetic structures are regularly produced by convection cells. Dynamo regimes in “thick” and “thin” shells are discussed. In the first case, the “general” magnetic field maintained by the dynamo has a sign-alternating dipolar component, which varies cyclically, although not periodically. The local structures, as they disintegrate, change into background fields, which drift toward the poles. From time to time, reversals of the magnetic fields in the polar regions occur, as “new” background fields expel the “old” fields. In the second case, the system settles down to a nearly stationary regime without polarity reversals.

Three-dimensional global modelling of turbulent convection coupled to rotation and magnetism within the Sun are revealing processes relevant to many stars. We study spherical shells of compressible convection spanning many density scale heights using the MHD version of the anelastic spherical harmonic (ASH) code on massively parallel supercomputers. The simulations reveal that strong magnetic fields can be realized in the bulk of the solar convection zone while still attaining differential rotation profiles that make good contact with helioseismic findings. We find that the Maxwell and Reynolds stresses present in such a turbulent layer play an important role in redistributing angular momentum, with the latter maintaining the differential rotation, aided by baroclinic forcing at the base of the convection zone which is consistent with a tachocline there. The dynamo processes generate strong non-axisymmetric and intermittent fields and weak mean (axisymmetric) fields, but do not possess a regular cyclic magnetism. The explicit inclusion of penetrative convection into the tachocline below is modifying such behavior, serving to build strong toroidal magnetic fields there that may yield more prominent mean fields that have the potential of erupting upward.

Vigorous discussion ensued about conditions under which both small-scale and global-scale dynamo action would be realized within real stars where the flow fields are expected to be highly turbulent and the magnetic Prandtl numbers small. Our nearest star reminds us that intricate boundary-layer phenomena may have to also be considered, such as the presence of a tachocline of rotational shear at the base of the solar convection zone revealed by helioseismology, which suggests that an interface dynamo may be at work to produce the observed 22-year cycles of large-scale magnetic activity.

Full-vector magnetograms and Dopplergrams of selected areas of the solar photosphere are used to compute the vertical component of the right-hand side of the vector induction equation. We attempt to find a criterion for the action of the convective mechanism of amplification and structuring of magnetic fields using the distributions of this quantity and of the vertical magnetic-field component.

The results of applying two algorithmic techniques to time-averaged images of the solar granulation are presented. Some quasi-regular structures, which are hidden in the intense noise due to the presence of light blotches in the images, can be detected in this way.

This investigation is devoted to study the turbulent convective transport of mean (large-scale) magnetic field in the solar convection zone (SCZ). For the SCZ model by Stix (1989) the reconstruction of the toroidal field was calculated as a result of the balance of mean-field magnetic buoyancy and two “negative magnetic buoyancy” effects: i) macroscopic turbulent diamagnetism and ii) the ∇ρ effect. It is shown that at high latitudes negative buoyancy effects block the magnetic fields, about 3000 – 4000 G, near the bottom of the SCZ. This may be the most plausible reason why a deep-seated field here could not become as apparent at the solar surface as sunspots. However, at the near equatorial domain in the deep layers the ∇ρ effect, with allowance for rotation, causes the upward magnetic transport, that facilitates penetration of strong fields to the surface where they emerge as sunspot patters in the “royal zone”.

We have found photometric indications that Interacting Eclipsing Binaries of early to mid F spectral type (and possibly A) have strong magnetic activity which would arise from convective atmospheres. Light curve solutions and periodicity studies revealing spots, magnetic breaking and magnetic cycles are presented in XZ CMi, V965 Cyg and V963 Cyg.

The origin, structure and evolution of sunspots are investigated using a numerical model. The compressible MHD equations are solved with physical parameter values that approximate the top layer of the solar convection zone. A three dimensional (3D) numerical code is used to solve the set of equations in cylindrical geometry, with the numerical domain in the form of a wedge. The linear evolution of the 3D solution is studied by perturbing an axisymmetric solution in the azimuthal direction. Steady and oscillating linear modes are obtained.

We present results from a global 3-D nonlinear simulation of magnetic dynamo action achieved by solar convection in a penetrative geometry. We include within the spherical computational domain both the bulk of the convection zone and a portion of the underlying stable layer. A tachocline of rotational shear is realized below the convection zone, where we have imposed both a hydrodynamic drag term and small thermal perturbations consistent with thermal wind balance. Thus we are capturing many of the dynamical elements thought to be essential in the operation of the global solar dynamo, including differential rotation arising from convection, magnetic pumping, and the stretching and amplification of toroidal fields within the tachocline. In the stable region, the simulation reveals that strong axisymmetric toroidal magnetic fields (about 3000 G in strength) are realized, in contrast to the mostly fluctuating fields that predominate in the convection zone. The toroidal fields in the stable region exhibit a striking antisymmetric parity akin to that observed in sunspots, with fields in the northern hemisphere largely of the opposite sign to those in the southern hemisphere. These deep toroidal fields are accompanied by mostly dipolar mean poloidal fields, whose polarity has retained the same sense over multiple years of simulated evolution.

Spectral properties of convective magnetohydrodynamic (MHD) turbulence in two and three dimensions are studied by means of direct numerical simulations (Skandera D. & Müller W.-C. 2006). The investigated system is set up with a mean horizontal temperature gradient in order to avoid a development of elevator instabilities in a fully periodic box. All simulations are performed without mean magnetic field. The applied resolution is 5123 and 20482. The MHD equation are solved by a numerical code (Müller & Biskamp 2000) that uses a standard pseudospectral scheme. For removing of aliasing errors a spherical truncation method is employed. Obtained results are compared with predictions of various existing phenomenological theories for magnetohydrodynamic and convective turbulence (Müller & Biskamp 2000). While the three-dimensional system is found to operate in a Kolmogorov-like regime where buoyant forces have a negligible impact on the turbulence dynamics (relatively low Rayleigh number achieved in the simulation; Ra ∼106), the two-dimensional system exhibits interesting irregular quasi-oscillations between a buoyancy dominated Bolgiano-Obukhov-like regime of turbulence and a standard Iroshnikov-Kraichnan-like regime of turbulence (Müller & Biskamp 2000). The most important parameter determining the turbulent regime of 2D magnetoconvection, apart from a high Rayleigh number, seems to be the mutual alignment of velocity and magnetic fields. The non-linear dynamics and the interplay between individual fields are examined with different transfer functions that confirm basic assumptions about directions of energy transfer in spectral space. Kinetic, magnetic and temperature energy are transported by a turbulent cascade from large to smaller scales. The local/nonlocal character of the transport is tested for several individual terms in the governing equations. Moreover, other statistical quantities, e.g. probability density functions, are computed as well. A passive character of the temperature field in the investigated three-dimensional magnetoconvection is supported by computations of intermittency using extended self-similarity. The intermittency of the Elsasser field z+ is in agreement with results from numerical simulations of isotropic MHD turbulence (Müller & Biskamp 2000). The intermittency of the temperature field is found to approximately agree with results of passive scalar measurements in hydrodynamic turbulence (Ruiz-Chavarria, Baudet & Ciliberto 1996).

Observations of magnetic fields in the quiet Sun indicate that kilogauss-strength fields can be found in the intergranular lanes. Since the magnetic energy of these localised features greatly exceeds estimates of the kinetic energy of the surrounding granular convection, it is difficult to see how these features could be formed simply by convective flux concentration. Idealised, high-resolution simulations of three-dimensional compressible magnetoconvection are used to investigate the formation of these features numerically. Initially we take a fully developed non-magnetic convective state into which we insert a weak, uniform, vertical magnetic field. Magnetic flux is rapidly swept into the convective downflows, where it is concentrated into localised regions. As the field strength within these regions becomes dynamically significant, the high magnetic pressure leads to partial evacuation (via the convective downflows). Provided that the magnetic Reynolds number is large enough, the strength of the resulting magnetic fields significantly exceeds the (so called) “equipartition” value, with the dynamical effects of the surrounding convection playing an important role in confining these magnetic features to localised regions. These results can be related to the well-known convective collapse instability, although there are some important differences between the two models.