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Extra resources for Accretion discs, jets and high energy phenomena in astrophysics: Les Houches Session LXXVIII, 29 July-23 August, 2002
6) r|∇a| ≤ α The function λ(a, r) approaches zero when r approaches inﬁnity at a given a in the set. This provides the following constraint on z(a) at ﬁxed r: z≥r Aeq a da = r Λ(a, r). 7) Since Λ(a, r) diverges as r → ∞, all magnetic surfaces polewards of those on which ρr2 approaches zero are asymptotically above some paraboloid. 2 Asymptotic transﬁeld equation The Bernoulli equation provides some general constraints on the asymptotic shape of polytropic winds. However, a complete understanding of their asymptotic structure requires an analysis of the transﬁeld equation.
Using this remark, it can be shown that the ﬂaring magnetic surfaces are asymptotically conical when the wind subtends a ﬁnite magnetic ﬂux [14, 15]. 16) implies that at large distances no current ﬂows between ﬂaring magnetic surfaces. For non-zero total current this implies that the current that they enclose ﬂows in some region about the polar axis, where the magnetic surfaces are cylindrical. Paraboloidal magnetic surfaces do not enclose a ﬁnite current asymptotically, unless the magnetic ﬂux subtending the wind source is inﬁnite .
Figure 6 illustrates this. Regions bounded by the polar axis and the polar-most null surface, or regions bounded on both sides by a null surface, are cells in which the current exactly closes. There is pressure balance between both sides of any of these null surface boundary layers. Since, outside of the boundary layers, the toroidal magnetic pressure dominates, the azimuthal magnetic ﬁeld just changes sign at their crossing. Thus I∞ (a) changes sign when passing from one to the neighbouring current cell.