Another important clue that magnetic fields are involved with wind production comes from the extensive data on stellar rotation. As one proceeds down the main sequence toward lower surface temperatures, one finds an abrupt decrease in rotational speed at F-type stars. This is also the point at which main-sequence objects develop an outer convection zone. Quite generally, stellar magnetic fields are thought to arise from dynamo action, i. e., the interplay of rotation and turbulence within convectively unstable regions. Suppose now that a late-type star ejects gas that travels out along rotating magnetic field lines anchored to the stellar surface. If the field maintains its rigidity for some distance, then the ejected matter picks up a high angular velocity before it is flung into space. Such a centrifugal wind can therefore provide efficient rotational braking of the star, accounting for the observations. More powerful versions of these winds may be occurring in younger stars. Indeed, the bipolar anisotropy of flows from embedded sources plausibly arises from the reflection symmetry of the underlying field.

Let us now explore the elements of centrifugal wind theory. Our goal, to be achieved in equation (13.36) below, will be to derive a relation governing the velocity analogous to equation (13.13) for pressure-driven flows. Along the way, we shall also gain a better understanding of wind braking. We assume from the outset that both the wind gas and magnetic field are azimuthally symmetric about the z-axis of a cylindrical coordinate system, and that this axis coincides with the direction of stellar rotation. It is then convenient to split both the field B and the velocity u into poloidal and toroidal (i. e., azimuthal) components (recall Figure 10.13). Sup pose that the wind material is sufficiently ionized that the ideal MHD equation (9.45) applies. Assuming the flow to be steady-state, this condition of flux freezing reduces to v x (u x B) =0 . (13.15)

Matter subject to flux freezing can only slide along field lines or rotate with them around the z-axis. To develop this picture mathematically, we note that equation (13.15) implies that uxB is expressible as the gradient of a scalar field. But no quantities, including this scalar, vary azimuthally. Thus u x B must itself be a poloidal vector. If we now write u = up + u$ and B = Bp + B$, we quickly see that the only toroidal contribution to uxB is up xBp. For this to vanish, up must be everywhere parallel to Bp. That is, up = k Bp , (13.16)

where k is a scalar function of position. Within any meridional plane, matter indeed slides along the unchanging field lines.

Having eliminated the toroidal component of uxB, equation (13.15) is now v x (up xB$ + x Bp) = 0 . (13.17)

The velocity may be recast as wQe$, where is the unit toroidal vector. Using equation (13.16) for up, equation (13.18) becomes v x (Bp x awe$) = 0 . (13.18)

Here, a is a new scalar quantity:

This equation and (13.16) together imply that u = k B + aw . (13.20)

To ascertain the physical meaning of a, we expand the triple vector product in equation (13.18).3 In axisymmetry, Maxwell's equation V B = 0 reduces to V Bp = 0, while V-(awe$) also vanishes. We thus find dB

Noting that de|d$ = and (BpV)e$ = 0, this last equation simplifies to aB- Bp-V(aw) = 0 , from which it follows that

3 The relevant identity is Vx (CX D) = (D-V)C - D(V-C) - (C-V)D + C(V-D), for any two vector fields C and D.

Figure 13.17 A stellar magnetic field is dragged in the counterclockwise direction at angular speed a. Depicted are the projection of the field and gas flow onto the equatorial plane. At radial distance w, the material velocity has poloidal and azimuthal components, as indicated. The former makes an angle 0B with the field vector B. The coordinate s measures distance along the field line.

But B^-Va = B^ da/d<p = 0 by axisymmetry, so that finally

The scalar a is thus unchanging in the direction of B. This quantity represents the angular speed at which each field line rotates about the z-axis. Equation (13.20) states that the total azimuthal speed of a fluid element is aw plus kBthe slippage along the field itself. These two contributions are of opposite sign if the field line forms a trailing spiral, as illustrated in Figure 13.17.

13.4.2 Flow Dynamics p{u-V)u = -VP - pV$g + — (Vx B)x B . (13.23)

Within the constraints imposed by flux freezing, the wind is accelerated by the ambient pressure, gravitational and magnetic forces. The governing momentum equation (3.3) becomes, in steady state,

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