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Astron. Astrophys. 343, 251-260 (1999)

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1. Introduction

Observational and theoretical research on stellar winds and astrophysical jets has evolved rapidly. For our own sun and its associated solar wind, the current understanding necessitates the combined study of solar wind acceleration and coronal heating in time-dependent modeling (Holzer & Leer 1997). At the same time, Holzer and Leer rightfully stressed that it remains useful to emphasize on early studies of wind acceleration. This, we find, is especially true for numerical modeling of stellar winds. With the ultimate goal of time-dependent heating/wind modeling in mind, we here address the simpler question on how to accurately model 1D and 2D steady-state winds by the numerical solution of the polytropic magnetohydrodynamic (MHD) equations.

Since much of the 1D solutions we obtain is known from the outset, we can verify our results precisely . Indeed, 1D Parker (1958) and Weber-Davis (Weber & Davis 1967) wind solutions can be checked to agree with the analytic description. In their axisymmetric 2D extensions, various physical quantities must be conserved along poloidal streamlines. Sakurai (1985, 1990) presented such 2D generalizations of the magnetized Weber-Davis wind, using a method designed from these conservation laws. With modern numerical schemes, we can recover and extend his solutions to allow for the self-consistent modeling of `dead' and `wind' zones, as in the solar wind. Our steady-state solutions can be checked to conserve quantities along streamlines a posteriori .

Of crucial importance is the choice of boundary conditions used in the simulations. Since the governing equations for steady-state, transonic MHD flows are of mixed-type, their character can change from elliptic to hyperbolic at a priori undetermined internal critical surfaces. Causality arguments have been used to discuss which and how many boundary conditions must be prescribed (Bogovalov 1997). Our choice of boundary conditions used at the stellar surface is therefore discussed in detail.

All solutions presented are obtained with a single software package, the Versatile Advection Code (VAC, see Tóth 1996, 1997 and also http://www.phys.uu.nl/toth). Although we only present steady-state transonic outflows in spherical and axisymmetry, VAC is developed for handling hydrodynamic (HD) and magnetohydrodynamic (MHD) one-, two-, or three-dimensional, steady-state or time-dependent problems in astrophysics. It is therefore capable of achieving our ultimate goal of time-dependent 3D wind modeling. The insight gained in this study of steady-state polytropic flows will then be very useful.

In Sect. 2, we list the equations and discuss the Versatile Advection Code for solving them in Sect. 3. Our calculations are presented in Sects. 4, 5, and 6. Conclusions are given in Sect. 7. The approach taken is a gradual one, where for instance our 1D solutions are used to construct initial conditions for their 2D extension. We will therefore model, in increasing order of complexity: (i) isothermal, spherically symmetric Parker winds; (ii) polytropic, spherically symmetric Parker winds; (iii) polytropic, rotating Parker winds for the equatorial plane; (iv) Weber-Davis magnetized, polytropic, rotating winds for the equatorial plane; and finally axisymmetric, polytropic, rotating 2D winds, both (v) unmagnetized and magnetized, without (vi) and with (vii) a `dead' zone.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999