6. Axisymmetric 2D polytropic MHD winds
6.1. Magnetized winds
To obtain an axisymmetric magnetized wind solution, we may simply add a purely radial magnetic field to a 2D HD wind solution and use this configuration as an initial condition for an MHD calculation. Hence, we set and while . Such a monopolar field is rather unrealistic for a real star, but it is the most straightforward way to include magnetic effects. The same type of field was used by Sakurai (1985, 1990), which contained the first 2D generalization of the WD model.
Boundary conditions at equator and pole are imposed by symmetry considerations. At , we extrapolate all quantities linearly. Similarly, the base conditions extrapolate the density profile and all magnetic field components, while the poloidal velocity components are set to ensure a prescribed mass flux. The stellar rotation rate and the coupling between the velocity and the magnetic field enters in the boundary condition at the stellar surface where we demand
Specific attention is paid to ensuring the condition. As explained in Sect. 3, we now switch strategy and use explicit time stepping combined with a projection scheme to obtain the steady state solution.
We calculated the 2D extension of the WD wind corresponding to , , and . The mass flux is set to be . We show in Fig. 4 the streamlines, and the positions of the critical curves where the poloidal Alfvén Mach number and the poloidal slow and fast Mach numbers equal unity. The squared poloidal Alfvén Mach number is given by
with Alfvén speeds . The squared poloidal slow and fast Mach numbers are defined by
At the pole, the fast Mach number coincides with the Alfvén one since vanishes there and the parameters are such that . Away from the pole, the toroidal field component does not vanish, so that Alfvén and fast critical curves separate. Note how the equatorial solution strongly resembles the WD wind solution for the same parameters shown in Fig. 2. The obtained wind solution is mostly thermally driven, like the solar wind. The rotation rate and magnetic field effects are minor and an almost spherically symmetric wind results. Sakurai (1985, 1990) demonstrated that for stronger fields, the magnetic force of the spiraling fieldlines deflect the outflow poleward. This magnetic pinching force can produce a polar collimation of the wind. These effects have also been addressed by analytical studies of self-similar outflows in Trussoni et al. (1997).
For these axisymmetric, steady-state MHD outflows, the solutions can be verified to obey the following conservation laws. Mass conservation is ensured when writing the poloidal momentum vector as , with the stream function . The zero divergence of the magnetic field yields, likewise, , with the flux function. The poloidal part of the induction equation then leads to , provided that the toroidal component of the electric field vanishes. This can easily be checked from , and the solution shown in Fig. 4 satisfies this equality to within 1%. This allows us to write . A fair amount of algebra shows that the toroidal momentum and induction equation introduce two more flux functions, namely the specific angular momentum and a function related to the electric field . The Bernoulli function derivable from the momentum equation can be written as
Note how the constants of motion found in the WD solution immediately generalize in this formalism (mass flux, magnetic flux, corotation as in Eq. (8), specific angular momentum L and Bernoulli function E). The hydrodynamic limit is found for zero magnetic field and vanishing electric field . Across the poloidal streamlines, the momentum balance is governed by the generalized, mixed-type Grad-Shafranov equation. The numerical solutions we obtained indeed have parallel poloidal streamlines and poloidal fieldlines and conserve all these quantities along them.
6.2. Winds containing a `dead' zone
The monopolar field configuration used above is unrealistic. However, it should be clear that our method easily generalizes to bipolar stellar fields by appropriately changing the initial condition on the magnetic field. In fact, a star like our sun has open fieldlines at both poles and closed fieldlines around its equator. To obtain a steady-state stellar wind containing a `wind' zone along the open fieldlines and a `dead' zone about the stellar equator, we can simply initialize the polar regions up to a desired polar angle as above. The equatorial `dead' zone is then initialized as follows: the density and the toroidal momentum component is taken from the 2D HD wind with the same rotational and polytropic parameters while the poloidal momentum components are set to zero. The initial magnetic field configuration in the `dead' zone is set to a dipole field which has
The strength of the dipole is taken to keep the radial field component constant at . The initial component is again zero throughout. In summary, we now have the following set of parameters used in the simulation: the escape speed , the polytropic index , the rotational parameter , the field strength through , and the extent of the dead zone through . In addition, the mass flux is used in the boundary condition of the poloidal momentum components. Boundary conditions at the stellar surface are identical as above, but now the dead zone has a zero mass flux, so that . Note that in a completely analoguous way, we could allow for a latitudinal dependence of the stellar rotation rate , or the magnetic field strength .
This guess for an axisymmetric MHD wind is then time-advanced to a stationary solution. Fig. 5 shows the final stationary state, for the parameter values , , a constant rotation rate corresponding to , , and the mass flux in the wind zone set to the constant , while it is zero in the dead zone. These parameters are as in the WD solution and the Sakurai wind presented earlier. The initial field geometry has evolved to one where the open fieldlines are draped around a distinct bipolar `dead' zone of limited radial extent and the prescribed latitudinal range. The outflow nicely traces the field geometry outside this dead zone. As seen from the figure, we have calculated the full poloidal halfplane and imposed symmetry boundary conditions at north and south pole. We used a polar grid of resolution of radial extent with a radial grid accumulation at the base. The north-south symmetry of the final solution is a firm check of the numerics. The critical surfaces are also indicated in Fig. 5 and they differ significantly from the monopolar field solution shown in Fig. 4. Again, at the polar regions, the Alfvén and fast critical surface coincides. Now, the also vanishes at the equator where conditions are such that the slow and the Alfvén critical surfaces coincide. The component changes sign when going from north to south, as the rigid rotation shears the initial, purely poloidal bipolar magnetic field. This is different from the Sakurai wind presented above, where the boundary condition on was taken symmetric about the equator. Note how the equatorial acceleration to super-Alfvénic velocities occurs very close to the end of the dead zone. The critical surfaces are all displaced inwards as compared to the monopolar case.
Fig. 5 shows that poloidal streamlines and fieldlines are parallel. The is below 3%. In Fig. 6, we show the latitudinal variation of the (scaled) density and the velocity at two fixed radial distances in a polar plot. The spacecraft Ulysses and the on-board SWOOPS experiment provided the solar community with detailed measurements of these quantities for the solar wind (McComas et al. 1998). Qualitatively, the measured poloidal density and velocity variation resembles the curves from Fig. 6: the density is higher about the ecliptic and there is a decrease in wind speed associated with the equatorial `dead' zone. However, our computational domain extended to 50 stellar radii, while Ulysses measurements apply to larger radial distances. Note that we could use observed solar differential rotation profiles, as well as mass fluxes and magnetic field strengths, to obtain a better MHD proxy of solar wind conditions. The extent of the solar coronal active region belt suggests the use of a `dead' zone larger than modeled in Fig. 5.
© European Southern Observatory (ESO) 1999
Online publication: March 1, 1999