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Astron. Astrophys. 328, 253-260 (1997)

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2. The model

In a stellar convection zone or a fully convective star the turbulent convective motions cause an additional stress on the mean (global) motion known as Reynolds stress. While this stress can be described as an additional viscosity in case of a non-rotating convection zone this is no longer correct as soon as the rotation period becomes comparable with or smaller than the convective turnover time. In that case a non-viscous contribution, the [FORMULA] -effect, arises that forces differential rotation.

In Küker et al. (1993), Reynolds stress was considered as the only transporter of angular momentum in the solar convection zone. This model yields a rotation pattern that agrees almost perfectly with the observations from helioseismology and thus confirms the theory of the Reynolds stress as well as the assumption, that the meridional flow is of minor importance for the problem of solar differential rotation. The latter assumption can, however, not be correct for stars in general since a model based on Reynolds stress alone always yields a normalized differential rotation that increases with increasing rotation rate, in contradiction to the observations mentioned above.

We therefore present a model of a rapidly rotating fully convective pre-main sequence star in which Reynolds stress and meridional circulation are treated consistently, i.e. we solve the full Reynolds equation,

[EQUATION]

rather than only its azimuthal component. In (2), [FORMULA] is the correlation tensor of the fluctuating part [FORMULA] of the velocity field while [FORMULA] denotes the mean velocity. The molecular stress tensor [FORMULA] can be neglected since it is many orders of magnitude smaller than the Reynolds stress.

We only treat the axisymmetric case. The velocity field can then be separated into a global rotation and the meridional flow:

[EQUATION]

where [FORMULA] is the unit vector in the azimuthal direction. The azimuthal component of the Reynolds equation reads,

[EQUATION]

where

[EQUATION]

The meridional circulation can be expressed by a stream function A,

[EQUATION]

An equation for the stream function is obtained by taking the azimuthal component of the curl of (2):

[EQUATION]

where [FORMULA] is the curl of the meridional flow velocity and [FORMULA] is the gradient along the axis of rotation. In (7), we have omitted all nonlinear terms except the one including [FORMULA]. Since the code described below allows the inclusion of nonlinear terms we made some test calculations including the full advective term, but the results did not change significantly. We use the anelastic approximation,

[EQUATION]

i.e. the density is constant with time but varies with depth. The stream function and the vorticity [FORMULA] are related via the equation

[EQUATION]

where [FORMULA] Eq. (9) can be reduced to a set of ordinary differential equations by an expansion in terms of Legendre functions [FORMULA], which are eigenfuntions of the latitudinal part of D.

In the correlation tensor Q, a viscous and a non-viscous part can be distinguished:

[EQUATION]

where [FORMULA]. The viscous part,

[EQUATION]

is given by

[EQUATION]

(Kitchatinov et al. 1994). The viscosity coefficients,

[EQUATION]

depend on the angular velocity as well as on the convective turnover time, [FORMULA], via the Coriolis number, [FORMULA]. In the limiting case of very slow rotation, [FORMULA], the viscous stress becomes isotropic and reduces to the well known expression,

[EQUATION]

The second contribution, the [FORMULA] -effect, is the source of differential rotation. In spherical polar coordinates, it is only present in the components [FORMULA] and [FORMULA]:

[EQUATION]

The functions [FORMULA] have been derived in Kitchatinov et al. (1994) and those for [FORMULA], [FORMULA], and [FORMULA] can be found in Küker et al. (1993).

We take the stratifications of density, temperature and luminosity from a model of a fully convective PMS star by Palla & Stahler (1993) and calculate the rms value of the turbulent convective velocity from mixing-length theory. The star has 1.5 solar masses and is at the beginning of his contraction phase, with a radius of 4.6 solar radii. Both density and pressure are assumed to be functions of the fractional stellar radius only and their gradients are thus aligned. As a consequence, the last term in (7) vanishes and the rotation pattern is determined by the balance between the meridional flow and the Reynolds stress.

Standard mixing-length theory does not include the rotational influence on the turbulent heat transport. We therefore use the eddy heat flux derived by Kitchatinov et al. (1994),

[EQUATION]

where

[EQUATION]

is the turbulent heat conductivity tensor, [FORMULA] the deviation from adiabatic stratification, and [FORMULA] denotes the Coriolis number. For slow rotation, i.e.  [FORMULA], the heat conductivity reduces to the scalar quantity

[EQUATION]

For arbitrary rotation rates, the functions [FORMULA] and [FORMULA] give the rotational influence on the isotropic and anisotropic part of the heat conductivity, respectively. [FORMULA] is the convective turnover time. We are only interested in the magnitude of the heat transport and do not regard any anisotropy. We further replace [FORMULA] with its limit for large Coriolis numbers,

[EQUATION]

With these approximations the radial convective heat flux is

[EQUATION]

For fully ionized matter and the composition of a standard solar model the specific heat capacity [FORMULA] is approximately given by

[EQUATION]

The superadiabatic temperature gradient [FORMULA] is eliminated using the standard mixing-length relation

[EQUATION]

We equate the convective heat flux and the stellar luminosity,

[EQUATION]

yielding

[EQUATION]

We choose [FORMULA] for a stratification very close to the adiabatic case, and use the relation

[EQUATION]

for the convective turnover time.

To solve Eqs. (4) and (7), we use a time-dependent code based on a finite difference method in both space and time. Every time step includes the solution of a second order partial differential equation to compute the stream function A from [FORMULA]. The domain of the computation should principally be the whole star, but we must exclude a small volume around the center for numerical reasons. We take a value of 0.1 for the fractional radius of this artificial inner boundary. As boundary conditions, we require that both the radial component of the meridional motion and [FORMULA] vanish, i.e. neither matter nor angular momentum may be transported across the boundaries.

The numerical scheme used to solve Eqs. (4) and (7) is an explicit finite difference scheme with a staggered grid. The main advantage of this scheme is that it is relatively easy to implement a complicated tensor structure for Q and to ensure the conservation of angular momentum. The ordinary differential equations resulting from the expansion of Eq. (9) in terms of Legendre functions are solved by applying a standard second order finite difference scheme and solving the resulting tridiagonal linear equations. The results for the T Tauri model were tested for sensitivity to changes in the resolution of the grid. The necessary number of grid points turned out to be determined by the requirement of stability rather than accuracy, especially in case of large Taylor numbers. Once the resolution is sufficient for stability, further increase does not change the result significantly. We use a non-uniform distribution of the radial grid points with enhanced resolution close to the boundaries and a uniform distribution of latitudinal grid points. A grid of ([FORMULA]) points is appropriate for [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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