7. Conclusions and outlook
We obtained polytropic stellar winds as steady-state transonic outflows calculated with the Versatile Advection Code. We could relax an isothermal, spherically symmetric Parker wind, to a polytropic wind model. Subsequently, we included stellar rotation and a magnetic field, to arrive at the well-known Weber-Davis solution. We used fully implicit time stepping to converge to the steady-state solutions. The correctness of these 1D wind solutions can be checked precisely .
We generalized to 2D axisymmetric, unmagnetized and magnetized winds. Noteworthy is our prescription of the stellar boundary conditions in terms of the prescribed mass flux and the way in which the parallelism of the flow and the fieldlines in the poloidal plane is achieved. In Bogovalov (1996), the stellar boundary specified the normal magnetic field component and the density at the surface, while keeping the velocity of the plasma on the stellar surface in the rotating frame constant. Our approach differs markedly, since we impose the mass flux and ensure the correct rotational coupling of velocity and magnetic field. We refrain from fixing the density, as the analytical treatment shows that the algebraic Bernoulli equation together with the cross-field momentum balance really determines the density profile and the magnetic flux function concurrently, and should not be specified a priori. In fact, we let the density and all magnetic field components adjust freely at the base. This allows for the simultaneous and self-consistent modeling of both open and closed fieldline regions, which is not immediately possible when using the method of Sakurai (1985). By an appropriate initialization of the time-marching procedure used to get the steady-state solutions, we can find magnetized winds containing both a `wind' and a `dead' zone.
The method lends itself to investigate thermally and/or magneto-centrifugally driven polytropic wind solutions. One could derive angular momentum loss rates used in studies of stellar rotational evolution (Keppens et al. 1995, Keppens 1997). However, our immediate interest is in the relaxation of the assumptions inherent in our approach.
In this paper, we assume a polytropic equation of state throughout. All solutions are smooth and demonstrate a continuous acceleration from subslow outflow at the stellar surface to superfast outflow at large radial distances. Our polytropic assumption has to be relaxed to investigate the combined coronal heating/solar wind problem within an MHD context. This involves adding the energy equation. We plan to study possible discontinuous transonic solutions containing shocks. We can then address the puzzling paradox recently raised by analytic investigations of translational symmetric and axisymmetric transonic MHD flows (Goedbloed & Lifschitz 1997, Lifschitz & Goedbloed 1997, Goedbloed et al. 1998). The generalized Grad-Shafranov equation describing the cross-fieldline force balance has to be solved concurrently with the algebraic condition expressed by the Bernoulli equation. Rigorous analysis of the generalized mixed-type Grad-Shafranov partial differential equation, in combination with the algebraic Bernoulli equation, shows that only shocked solutions can be realized whenever a limiting line appears within the domain of hyperbolicity. Moreover, in Goedbloed & Lifschitz (1997) and Lifschitz & Goedbloed (1997), it was pointed out that there are forbidden flow regimes for certain translationally symmetric, self-similar solutions of the MHD equations. The Alfvén critical point is in those solutions situated within a forbidden flow regime, which can only be crossed by shocks. It is of vital importance to understand what ramifications this has on analytic and numerical studies of stellar winds, or on accretion-type flows where shocked solutions are rule rather than exception. Since the schemes used in VAC are shock-capturing, we have all ingredients needed to clarify this paradox. Numerical studies of self-similar solutions as those discussed in Trussoni et al. (1996) and Tsinganos et al. (1996) are called for. Combined analytic and numerical studies of such axisymmetric steady-state flows have been initiated in Goedbloed et al. (1998) and in Ustyugova et al. (1998).
After those paradoxes are resolved, we will be in a position to relax the conditions of axisymmetry and stationarity. While several authors have already initiated this daunting task (Gibson & Low 1998, Guo & Wu 1998, Wu & Dryer 1997, Usmanov & Dryer 1995), we believe that an in-depth study of the subtleties involved with the various restrictions mentioned is still warranted.
© European Southern Observatory (ESO) 1999
Online publication: March 1, 1999