Gaetano Belvedere

CONVECTION IN A ROTATING DEEP COMPRESSIBLE SPHERICAL SHELL: APPLICATION TO THE SUN

In this paper we study the interaction of rotation with convection in a deep compressible spherical shell as the Suns convection zone. We examine how the energy transport and the large scale motions can be affected by rotation. In particular we study how a large scale meridional circulation can give rise to variations of angular velocity with latitude and depth.It is assumed that the energy transport is only due to convection and that the mixing-length theory gives an adequate representation of it. Furthermore we assume that rotation acts as a perturbation of the turbulent convective flux through its transport coefficient.The equations involved in the model are integrated numerically in the limit of large viscosity and slow rotation. After having expanded all physical quantities to the first order in terms of Legendre polynomials, the fitting with the observed solar differential rotation gives the expansion parameter, which represents the coupling constant between rotation and convection.The results show a three-cell circulation extending from the poles to the equator. The first one is located in the lower half of the convection zone with the fluid rising at the equator and sinking at the poles. In the second one the direction of the motion is reversed while the third one, located in a thin upper layer, shows the same characteristics of the first one. The meridional velocities at the surface are directed towards the poles and are about 20 cm s-1. In the other cells the meridional velocities are typically of a few cm s-1 while the radial velocities are of the order of a few tenths of cm s-1.The heat flux relative variation at the surface is about 10-4 (3 × 10-3 at the bottom) with a polar excess. The temperature variation at the surface is of the same order, with an equatorial excess however. The convection seems to be stabilized stronger at the equator. The angular velocity increases inwards and varies about 6% between the surface and the bottom of the convection zone.An attempt is made for explaining the picture which emerges. In particular the negligible flux and temperature variations at the surface are explained in terms of equalization by the particular structure of the latitudinal flow. This configuration of large scale circulation is attributed to the high stratification of the convection zone with depth.

Differential rotation set up by latitude-dependent heat transport

Abstract Models of differentially rotating compressible deep spherical shells are computed according to the method of Belvedere and Paterno (1977): the heat transport is entirely convective, small-scale motions are parametrized by a thermal diffusivity and a kinematic viscosity, and the limit of slow rotation and large viscosity is considered. In order to adapt the resulting differential rotation to the observed equatorial acceleration of the Sun, the heat transport must be more effective in the vicinity of the equator. In all models the latitude dependence of the transport coefficient induces meridional circulation in the form of a large cell, with rising material at high latitudes and sinking material near the equator. On top of this cell, one or two thin countercells develop in a minority of cases. Large pole-equator temperature differences and meridonal velocities at the surface are obtained when the Prandtl number is 1. But values of, say, 1/10 are sufficiently small to allow the models to be applied...

Nonlinear dynamics of a stellar dynamo in a spherical shell

Abstract A simple mean-field model of a nonlinear stellar dynamo is considered, in which dynamo action is supposed to occur in a spherical shell, and where the only nonlinearity retained is the influence of the Lorentz forces on the zonal flow field. The equations are simplified by truncating in the radial direction, while full latitudinal dependence is retained. The resulting nonlinear p.d.e.s in latitude and time are solved numerically, and it is found that while regular dynamo wave type solutions are stable when the dynamo number D is sufficiently close to its critical value, there is a wide variety of stable solutions at larger values of D. Furthermore, two different types of dynamo can coexist at the same parameter values. Implications for fields in late-type stars are discussed.

Large scale circulation in the convection zone and solar differential rotation

AbstractIn this paper we study the dependence on depth and latitude of the solar angular velocity produced by a meridian circulation in the convection zone, assuming that the main mechanism responsible for setting up and driving the circulation is the interaction of rotation with convection. We solve the first order equations (perturbation of the spherically symmetric state) in the Boussinesq approximation and in the steady state for the axissymmetric case. The interaction of convection with rotation is modelled by a convective transport coefficient kc = kco + ℰkc2P2(cos θ) where ℰ is the expansion parameter, P2 is the 2nd Legendre polynomial and kc2 is taken proportional to the local Taylor number and the ratio of the convective to the total fluxes. We obtain the following results for a Rayleigh number 103 and for a Prandtl number 1: (1)A single cell circulation extending from poles to the equator and with circulation directed toward the equator at the surface. Radial velocities are of the order of 10 cm s−1 and meridional ones of the order of 150 cm s−1.(2)A flux difference between pole and equator at the surface of about 5 percent, the poles being hotter.(3)An angular velocity increasing inwards.(4)Angular velocity constant surfaces of spheroidal shape.The model is consistent with the fact that the interaction of convection with rotation sets up a circulation (driven by the temperature gradient) which carries angular momentum toward the equator against the viscous friction. Unfortunately also a large flux variation at the surface is obtained. Nevertheless it seems that the model has the basic requisites for correct dynamo action.

Solar and Stellar Activity: The Theoretical Approach

The unified sight of solar and stellar activity has revealed a worthwhile concept under several aspects, gaming in the last decade the increasing favour of observers and theorists, and the term solar-stellar connection has recently been introduced to point out the complementarity of solar and stellar observations in the background of the basic role played by the magnetic field.

On the sun's pole-equator flux differences

We study the possibility that large flux differences between the poles and the equator at the bottom of the solar convective zone are compatible with the small differences observed at the surface. The consequences of increasing the depth of the convective zone due to overshooting are explored.A Boussinesq model is used for the convective zone and we assume that the interaction of the global convection with rotation is modelled through a convective flux coefficient whose perturbed part is proportional to the local Taylor number. The numerical integration of the equations of motion and energy shows that coexistence between large pole-equator flux differences at the bottom and small ones at the surface is possible if the solar convective zone extends to a depth of 0.4R⊙. The angular velocity distribution inside the convective zone is in agreement with the αω-dynamo theories of the solar cycle.

Long term variation of the solar equatorial velocity and its relation to non-axisymmetric convection

The long term variations of solar equatorial velocity are considered, as determined by spectroscopic observations of several authors since 1900. By eliminating Storeys observations covering the period 1914–1932 which seem to be affected by casual errors, a computer analysis picks out a period of about 34 yr in the velocity variation.An interpretation is given of this period in the framework of the interaction of non-axisymmetric convection with rotation.

On the influence of nonlinearities on the eigenfrequencies of five-minute oscillations of the Sun

Fitting the results of linear normal-mode analysis of the solar five-minute oscillations to the observed k - ω diagram selects a class of models of the Suns envelope. It is a property of all the models in this class that their convection zones are too deep to permit substantial transmission of internal g modes of degree 20 or more. This is in apparent conflict with Hill and Caudells (1979) claim to have detected such modes in the photosphere.A proposal to resolve the conflict was made by Rosenwald and Hill (1980). They pointed out that despite the impressive agreement between linearized theory and observation, nonlinear phenomena in the solar atmosphere might influence the eigenfrequencies considerably. In particular, they suggested that a correct nonlinear analysis could predict a shallow convection zone. This paper is an enquiry into whether their hypothesis is plausible.We construct k - ω diagrams assuming that the modes suffer local nonlinear distortions in the atmosphere that are insensitive to the amplitude of oscillation over the range of amplitudes that are observed. The effect of the nonlinearities on the eigenfrequencies is parameterized in a simple way. Taking a class of simple analytical models of the Suns envelope, we compute the linear eigenfrequencies of one model and show that no other model can be found whose nonlinear eigenfrequencies agree with them. We show also that the nonlinear eigenfrequencies of a particular solar model with a shallow convective zone, computed with more realistic physics, cannot be made to agree with observation. We conclude, therefore, that the hypothesis of Rosenwald and Hill is unlikely to be correct.

The Influence of the Stellar Mass Ratio on Spiral Shocks in Accretion Disks Around Compact Objects

We investigated, in the Smooth Particle Hydrodynamics (SPH) framework, the development of spiral structures and shock fronts in the radial flow of accretion discs in close binary systems. These shock waves take place when the initially radial flow penetrating the disc bulk, reduces substantially its speed becoming suddenly subsonic. To this purpose, keeping constant the mass of the compact primary (M1= 1M⊙), the separation between the two components and the injection speed at the inner lagrangian point L1(close to the local sound speed), we carried out four 2D SPH simulations for four values of the stellar mass ratio M2/M1.

Alpha-Effect, Current and Kinetic Helicities for Magnetically Driven Turbulence, and Solar Dynamo

Recent numerical simulations lead to the result that turbulence is much more magnetically driven than believed. In particular the role of magnetic buoyancy appears quite important for the generation of α-effect and angular momentum transport (Brandenburg & Schmitt 1998). We present results obtained for a turbulence field driven by a (given) Lorentz force in a non-stratified but rotating convection zone. The main result confirms the numerical findings of Brandenburg & Schmitt that in the northern hemisphere the α-effect and the kinetic helicity are positive (and negative in the northern hemisphere), this being just opposite to what occurs for the current helicity which is negative in the northern hemisphere (and positive in the southern hemisphere). There has been an increasing number of papers presenting observations of current helicity at the solar surface, all showing that it is negative in the northern hemisphere and positive in the southern hemisphere (see Rudiger et al. 2000, also for a review). Mass conservation requires that Notice, that density has been assumed as homogeneous and density fluctuations vary in time. We do not adopt the inelastic approximation. For the turbulent energy equation we simply adopt a polytropic relation. In the sense of the ‘ -approximation’, the spectrum of the given magnetic fluctuations field has been approximated by with The turbulence may develop under the influence of a large-scale magnetic field B and the gravity g. For the current helicity we find