Astronomy and Astrophysics – Astronomy
Scientific paper
Feb 1995
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1995apj...440..789t&link_type=abstract
Astrophysical Journal v.440, p.789
Astronomy and Astrophysics
Astronomy
14
Hydrodynamics, Stars: Early-Type, Stars: Interiors, Stars: Rotation
Scientific paper
An analytical-numerical study is made of the thermally driven meridional currents and concomitant differential rotation in the radiative envelope of a nonmagnetic, early-type star. Since we cannot possibly encompass all the timescales and space scales of motion in a star, we make use of steady state models to represent these large-scale motions, while applying parametric expressions to describe the effects of all smaller scales. Assuming that the departures from spherical symmetry are not too large, we expand the solutions about hydrostatic equilibrium in powers of the rotational parameter ɛ, which is the ratio of the centrifugal force to gravity at the equator. Thence, by making allowance for turbulent friction acting on the mean flow (i.e., the differential rotation and the meridional currents), we obtain second-order solutions in ɛ that satisfy all the basic equations and all the boundary conditions. As a matter of fact, there exist thin thermoviscous boundary layers, so that the currents do not become infinite near the boundaries, the meridional velocities remaining uniformly small throughout the radiative envelope. Simultaneously, the frictional force acting on the differential rotation can be made to balance the transport of angular momentum by the large-scale meridional flow. The proposed solution is a diffusive one, in the sense that the effects of vertical dissipation are assumed to be larger than those caused by the thermally driven currents.
Detailed solutions have been obtained in the case when the basic state of motion is a uniform, or almost uniform, rotation. Neglecting turbulent friction altogether, one always obtains 1/ρ singularities in the second-order corrections for the r- and θ-components of the circulation velocity u. Strict solid-body rotation to order ɛ1/2 is not an admissible solution, therefore, since the second-order terms may dominate over the first-order ones in the outermost surface layers. On the contrary, in the case of almost uniform rotation to order ɛ1/2, one finds 1/ρ singularities to all orders (i.e., ɛ and ɛ2) in the inviscid part of the solution for the meridional flow. Hence, by making allowance for turbulent friction in the surface layers, one can obtain self-consistent solutions in these regions. Owing to the presence of the 1/ρ terms in the inviscid solution, the radial component ur at first increases toward the surface, then drops rapidly to zero at the outer boundary. In agreement with the prediction made by Baker & Kippenhahn, there is indeed an intensification of the meridional flow near the surface. However, it is shown that the frictional force acting on the circulatory currents prevents huge surface velocities. Typically, one finds that |ur| ≲ 1 cm s-1 and |uθ| ≲ 102 cm s-1 near the surface of a 3 Msun star, which is a long remove from the evaluations based on the inadequate formulae ur ∝ ɛ/ρ or ur ∝ ɛ2/ρ.
It is also found that the topology of the circulatory currents in the surface layers is quite dependent on the gradient of angular velocity in these regions. Assuming almost solid-body rotation to order ɛ1/2, we demonstrate that the meridional flow in a very slow rotator may consist either of a single cell or of two distinct cells. For moderate or large rotation rates, it consists of two or even three distinct cells the exact location of their separation in the latter case depending on the magnitude and spatial variations of the basic rotation law. This is completely different from the Gratton-Öpik double-cell pattern, which is not observed in our self-consistent, second-order solutions. As a sideline, we demonstrate that their double-cell pattern is the result of at least one of the following causes: (1) an overabundance of conflicting assumptions in the model, (2) a mathematical divergence of the expansions, (3) an inadequate separation of r and θ in the expansions, or (4) demonstrable algebraic errors.
To sum up, we have shown that Sweet's original solution is remarkably successful in describing the thermally driven currents in the deep interior of a stellar radiative envelope in almost uniform rotation. Near the lower boundary, however, there exists a thin thermoviscous boundary layer that allows the meridional velocity to make a smooth transition to the value prescribed at the convective core. His inviscid solution must be amended in the surface layers also, since one always finds a slight intensification of the currents in these low-density regions. Again, there exists a thin thermoviscous boundary layer that acts to satisfy the surface boundary conditions while preventing huge meridional velocities in the region where a double-cell pattern may occur. Unfortunately, because it is impossible at this time to perform a meaningful evaluation of the eddy viscosities, we cannot calculate the small departures from solid-body rotation with any accuracy.
Tassoul Jean Louis
Tassoul Monique
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