tokens: Mathematically, this is a two-point boundary problem and it will assure that we have a well-behaved solution in the whole r − θ space. We have tried to obtain such solutions but failed. Therefore, we only require the solution satisfying the boundary condition at θ = 90°. As we will explain in the next section, we think the solution obtained in this way should still be physically meaningful. 3 RESULTS We solve the above equations ( 34 )–( 40 ) numerically. We integrate these equations from the equatorial plane (θ = 90°) towards the rotational axis (θ = 0°). We adopt the values of β θ0 and β φ0 with reference to the numerical simulation data of Yuan et al. ( 2012b ), but our results are not sensitive to the exact values. As we have explained before, the toroidal component of the magnetic field, which is produced by shear, should be negative. We do find that the toroidal component is negative at the beginning of integration. But at a certain critical value of θ, denoted as θ B , we find that it begins to change its sign. At the region of θ θ B , the toroidal component of the magnetic field becomes positive thus the solution is no longer physical so we stop our integration at θ B . We think our solution in the region of θ B θ 90° is still physical. 4 This is because, on the one hand, the solution satisfies the equations and the boundary conditions at θ = 90°. On the other hand, the values of all physical quantities of the accretion flow at θ = θ B are physical. Therefore, we can reasonably treat these values as boundary conditions at θ B . In fact, as we will describe below, the main properties of the solutions we obtain in this way are in good agreement with those obtained in Yuan et al. ( 2015 ) from numerical simulations. From the bottom-right plot of Fig. 2 , it seems that there is a sonic point for v θ / c s and the integration stops when the sonic point is approached because of the singularity there. This is not true. This is because, the value of v θ must be equal to zero at the pole because of the axisymmetric boundary condition there. Therefore, v θ / c s should not pass through a sonic point. We note that Jiao Wu ( 2011 ) obtained a similar result with us, i.e. their solution also has to stop at a certain θ angle. An interesting explanation for such a ‘truncation’ is given in their work. Figure 2. Angular profile of a variety of variables. The up-left plot shows that the radial velocity changes its sign at θ ∼ 47°. 3.1 Overall inflow-wind structure of the solutions Fig. 1 shows the poloidal magnetic field configuration. We emphasize that while the even symmetry of the field is our assumption, the strength of the field is obtained by solving equations ( 34 )–( 40 ). Above the equatorial plane, the radial component of the magnetic field is positive (pointing to large radii). So the toroidal component of the magnetic field, which is produced by shear, is negative. We find that such kind of magnetic field configuration is similar to the
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