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Lecture 9
Disc Driven Winds and Magnetic Towers

Overview

This lecture continues the wind theme from the solar-wind lecture, but changes the engine. There, coronal pressure drove a transonic outflow to supersonic speed. Here, large-scale magnetic flux anchored in a differentially rotating disc lets matter be flung outward centrifugally and then further accelerated electromagnetically. The Blandford–Payne disc wind and Lynden-Bell magnetic tower are two limits of that same rotating, flux-frozen geometry.

The useful division is this. Launch and acceleration are along-field problems controlled by Bernoulli balance and the critical surfaces. Collimation is a cross-field problem controlled by the transfield force balance. In a matter-loaded wind, the rotating field extracts angular momentum from the disc and carries it away partly as matter and partly as electromagnetic stress, converting Poynting flux into poloidal kinetic energy. In a magnetically dominated tower, the same differential rotation keeps injecting toroidal flux, which is then confined by an external pressure and grows vertically.

Historical Perspective

The bridge from the solar-wind lecture runs through Parker and Weber–Davis. Parker showed that a hot atmosphere can be driven by pressure through a sonic point into a supersonic wind, while Weber and Davis showed that once the wind is magnetized and rotating it also carries angular momentum along an Alfvén lever arm Parker [1958], Weber and Davis [1967]. Blandford and Payne’s 1982 paper then made the magnetocentrifugal launching picture unforgettable by reducing it to a bead sliding on a rotating wire Blandford and Payne [1982]. The later literature filled in the full steady ideal-MHD structure: mass loading, critical surfaces, angular-momentum extraction, and asymptotic collimation Pelletier and Pudritz [1992], Heyvaerts and Norman [1989], Spruit [1996], Ostriker [1997], Vlahakis et al. [2000]. Lynden-Bell’s magnetic-tower papers emphasized a complementary limit in which toroidal field accumulates faster than matter escapes, so the outflow behaves more like a pressure-confined magnetic column than a classical cold wind Lynden-Bell [1996, 2003]. Historically, this is a very MHD story: once flux freezing and differential rotation are taken seriously, acceleration, torque, collimation, and magnetic inflation all become parts of one calculation.

Caution

Three cautions are worth stating at the outset.

First, the famous \(30^\circ \) launching criterion is only a launch criterion. It does not by itself guarantee a global transcritical solution: solar winds must pass a sonic point, while disc winds must pass the slow, Alfvén, and fast critical surfaces.

Second, the cartoon of “hoop stress collimates the jet” is incomplete. The real statement is a transfield one involving magnetic tension, magnetic pressure, gas pressure, rotation, and current closure.

Third, a magnetic tower is not a different branch of electrodynamics. It is the magnetically dominated, laterally confined limit of the same anchored, twisted magnetic geometry.

9.1 Steady axisymmetric ideal-MHD structure

We begin from the ideal-MHD subset of the basic equations already introduced in Eqs. (1.7), (1.8), and (4.8), now specialized to a steady, axisymmetric flow:

\[\begin{aligned}\divergence (\rho \uvec ) &= 0, \\ \rho \,\uvec \cdot \grad \uvec &= -\grad p - \rho \grad \Phi _g + \J \times \B , \\ \E + \uvec \times \B &= 0, \\ \divergence \B &= 0, \qquad \muo \J = \curl \B .\end{aligned} \tag{9.1}\]

Throughout this lecture we assume axisymmetry, \[ \pp {}{\phi }=0, \] and cylindrical coordinates \((R,\phi ,Z)\).

Represent the poloidal field by a flux function. As in the Grad–Shafranov lecture, write

\[\B = \grad \psi \times \grad \phi + I(R,Z)\,\grad \phi , \tag{9.5}\]
where \(\psi (R,Z)\) is the poloidal flux function and \(I=RB_\phi \). The poloidal field is therefore
\[\B _p = \grad \psi \times \grad \phi = -\frac {1}{R}\pp {\psi }{Z}\eR + \frac {1}{R}\pp {\psi }{R}\eZ . \tag{9.6}\]
Magnetic surfaces are given by \(\psi ={\rm constant}\).

The first key fact: the poloidal flow is parallel to the poloidal field. Because the electric field is electrostatic in steady state, its toroidal component must vanish: \[ E_\phi = 0. \] Using Eq. (9.3),

\[0 = -(\uvec \times \B )_\phi = -(u_R B_Z-u_Z B_R). \tag{9.7}\]
Hence
\[\uvec _p \parallel \B _p. \tag{9.8}\]
This is the steady axisymmetric version of the frozen-in constraint: matter does not cut across the poloidal field in ideal MHD.

Tutorial

Write the poloidal velocity as

\[\uvec _p = \alpha (R,Z)\,\B _p. \tag{9.9}\]
Insert this into continuity,
\[\divergence (\rho \uvec ) = \divergence (\rho \alpha \B _p + \rho u_\phi \ephi )=0.\]
Axisymmetry removes the toroidal divergence, and \(\divergence \B _p=0\), so
\[\B _p\cdot \grad (\rho \alpha )=0. \tag{9.11}\]
Since \(\B _p\cdot \grad \psi =0\), the quantity \(\rho \alpha \) must be constant on a magnetic surface. Define
\[\boxed { k(\psi ) \equiv \rho \alpha = \frac {\rho u_p}{B_p}. } \tag{9.12}\]
This is the mass loading: how much matter is carried per unit poloidal magnetic flux.

Now examine the poloidal part of the ideal Ohm law. Because \(\uvec _p=(k/\rho )\B _p\),

\[\begin{aligned}\E _p &= -(u_\phi \ephi \times \B _p + \uvec _p\times B_\phi \ephi ) \\ &= -\left (u_\phi - \frac {k B_\phi }{\rho }\right )\ephi \times \B _p.\end{aligned}\]

Using \(\ephi \times \B _p = \grad \psi /R\), we get

\[\E _p = -\frac {1}{R}\left (u_\phi -\frac {kB_\phi }{\rho }\right )\grad \psi . \tag{9.15}\]
But steady state implies \(\curl \E =0\), so the coefficient of \(\grad \psi \) must be a function of \(\psi \) alone. Define the field-line angular velocity
\[\boxed { \Omega _F(\psi ) = \frac {1}{R}\left (u_\phi -\frac {kB_\phi }{\rho }\right ). } \tag{9.16}\]
Then
\[\E _p = -\Omega _F(\psi )\,\grad \psi . \tag{9.17}\]
So a magnetic surface rotates as though it were a rigid wire with angular velocity \(\Omega _F\), even though different surfaces may rotate at different rates.

Two more invariants: angular momentum and Bernoulli constant. The azimuthal component of the momentum equation and the projection of the momentum equation along the flow give two additional surface invariants, usually written as

\[\boxed { L(\psi ) = R\left (u_\phi -\frac {B_\phi }{\muo k}\right ), } \tag{9.18}\]
for the total specific angular momentum, and
\[\boxed { E(\psi ) = \frac {u^2}{2}+h+\Phi _g-\frac {\Omega _F R B_\phi }{\muo k}, } \tag{9.19}\]
for the Bernoulli constant. The enthalpy per unit mass is \(h=\int dp/\rho \) for a barotropic gas.

The physics is already visible. Equation (9.19) says that the energy transported along a field line is partly mechanical and partly electromagnetic. Equation (9.18) says that the field can carry angular momentum as well as matter.

9.2 Keplerian footpoints and magnetocentrifugal launch

The crucial assumption for a disc wind is that a field line is anchored at a disc radius \(R_0\) whose angular velocity is approximately Keplerian,

\[\Omega _K(R_0)=\sqrt {\frac {GM}{R_0^3}}, \qquad \pp {\Omega _K}{R_0}=-\frac {3}{2}\frac {\Omega _K}{R_0}<0. \tag{9.20}\]
Thus neighboring magnetic surfaces inherit different rotation rates because they are frozen into different annuli of the disc. To lowest order,
\[\Omega _F(\psi ) \simeq \Omega _K(R_0). \tag{9.21}\]
This differential rotation is the source of the wound-up toroidal field and therefore of the extracted angular momentum.

A useful rearrangement for \(B_\phi \). Equation (9.16) may be solved for the toroidal field:

\[B_\phi = \frac {\rho }{k}\left (u_\phi -\Omega _F R\right ) = \frac {u_\phi -\Omega _F R}{u_p}\,B_p. \tag{9.22}\]
Beyond the Alfvén surface, one typically has \(u_\phi \ll \Omega _F R\), so for a prograde outflow \(B_\phi <0\). The field therefore becomes a trailing helix, just as in the Parker spiral derived later for the solar wind.

Tutorial

In the cold “bead-on-a-wire” limit, a parcel constrained to slide along a rigidly rotating field line feels the effective potential

\[\Phi _{\rm eff}(R,Z) = -\frac {GM}{\sqrt {R^2+Z^2}} - \frac {1}{2}\Omega _F^2 R^2. \tag{9.23}\]
Let the line emerge from the disc at \((R_0,0)\) and make an angle \(\theta \) with the vertical. Introduce distance \(s\) along the field near the footpoint:
\[R = R_0 + s\sin \theta , \qquad Z = s\cos \theta . \tag{9.24}\]
The derivative along the field is
\[\dd {}{s} = \sin \theta \,\pp {}{R}+\cos \theta \,\pp {}{Z}.\]
At the disc surface, the first derivative vanishes because the footpoint corotates with the Keplerian disc:
\[\left .\dd {\Phi _{\rm eff}}{s}\right |_{s=0}=0.\]
The second derivative is
\[\left .\dd {^2\Phi _{\rm eff}}{s^2}\right |_{0} = \sin ^2\theta \,\left .\pp {^2\Phi _{\rm eff}}{R^2}\right |_0 + 2\sin \theta \cos \theta \,\left .\pp {^2\Phi _{\rm eff}}{R\partial Z}\right |_0 + \cos ^2\theta \,\left .\pp {^2\Phi _{\rm eff}}{Z^2}\right |_0.\]
At \(Z=0\), the mixed derivative vanishes. Using \(\Omega _F=\Omega _K(R_0)\), one finds
\[\left .\pp {^2\Phi _{\rm eff}}{R^2}\right |_0 = -3\Omega _K^2, \qquad \left .\pp {^2\Phi _{\rm eff}}{Z^2}\right |_0 = +\Omega _K^2.\]
Therefore
\[\left .\dd {^2\Phi _{\rm eff}}{s^2}\right |_{0} = \Omega _K^2\bigl (1-4\sin ^2\theta \bigr ). \tag{9.29}\]
Outward motion is unstable, and therefore launch is possible, when this second derivative is negative:
\[\sin \theta > \frac {1}{2} \qquad \Longleftrightarrow \qquad \theta > 30^\circ\]
from the vertical, or equivalently when the field line makes an angle less than \(60^\circ \) with the disc surface.

This is the Blandford–Payne criterion. It says that a sufficiently inclined open field line lets centrifugal acceleration win over gravity along the wire.

What the launch criterion does and does not say. The \(30^\circ \) condition tells us that the disc can centrifugally feed matter onto the field line. It does not tell us the asymptotic speed, whether the flow crosses the critical surfaces smoothly, or whether the outflow eventually collimates. Those are separate pieces of the steady MHD problem.

The jet analogue of Parker’s supersonic wind. In the solar-wind lecture the key question was whether the outflow could pass smoothly through the sonic point and become supersonic. For a disc wind, especially in the cold limit, the more informative language is magnetosonic rather than purely sonic: the relevant global solution must become superslow, then super-Alfvénic, and ideally super-fast-magnetosonic. That is the correct MHD version of saying that the jet is a genuine high-Mach-number outflow rather than a gentle surface breeze. In that sense the Blandford–Payne jet is the disc-wind cousin of the Parker and Weber–Davis winds: local launch is easy to sketch, but the global solution is selected by its critical points Weber and Davis [1967], Vlahakis et al. [2000].

9.3 Angular-momentum extraction and the Alfvén lever arm

A useful quantity is the poloidal Alfvén Mach number,

\[M_A^2 \equiv \frac {\muo \rho u_p^2}{B_p^2} = \frac {\muo k^2}{\rho }. \tag{9.31}\]
At the Alfvén surface, \(M_A=1\).

Solve for \(u_\phi \) and \(B_\phi \). Equations (9.16) and (9.18) are two linear equations for \(u_\phi \) and \(B_\phi \). It is convenient to write them as

\[\begin{aligned}\Omega _F R &= u_\phi - M_A^2\frac {B_\phi }{\muo k}, \\ \frac {L}{R} &= u_\phi - \frac {B_\phi }{\muo k}.\end{aligned}\]

Subtract the second from the first:

\[\Omega _F R - \frac {L}{R} = \left (1-M_A^2\right )\frac {B_\phi }{\muo k},\]
so that
\[\boxed { \frac {B_\phi }{\muo k} = \frac {\Omega _F R^2-L}{R(1-M_A^2)}. } \tag{9.35}\]
Then
\[\boxed { u_\phi = \frac {\Omega _F R^2 - L M_A^2}{R(1-M_A^2)}. } \tag{9.36}\]
These expressions make the Alfvén point visible: both denominators vanish as \(M_A^2\to 1\). A smooth solution is therefore possible only if the numerator vanishes there as well. This gives the regularity condition
\[\boxed {L = \Omega _F R_A^2.} \tag{9.37}\]
The quantity \(R_A\) is the Alfvén lever arm.

Why the disc loses angular momentum. If \(R_A>R_0\), then the wind carries specific angular momentum \(L=\Omega _F R_A^2\), which is larger than the matter possessed at its footpoint, \(\Omega _K R_0^2\). The magnetic field therefore exerts a torque on the disc, allowing accretion and ejection to occur together. This is one of the conceptual triumphs of the theory: the same magnetic stress that launches the wind also solves the angular-momentum problem of accretion discs Pelletier and Pudritz [1992], Ostriker [1997].

Mechanical and electromagnetic angular-momentum flux. Multiply Eq. (9.18) by the poloidal mass flux \(\rho u_p\) and use \(k=\rho u_p/B_p\). Then

\[\boxed { \rho u_p L = \rho u_p R u_\phi - \frac {R B_\phi B_p}{\muo }. } \tag{9.38}\]
The first term is the mechanical transport of angular momentum by the matter. The second is the magnetic stress, equivalently the angular-momentum flux associated with the torsion of the field and the accompanying Poynting flux. Close to the disc the magnetic term usually dominates; farther out, part of that flux is converted into matter angular momentum and poloidal kinetic energy. This is why disc winds are so important for accretion: they provide a vertical escape valve for the excess angular momentum of the inflowing gas, instead of forcing the disc to move it all radially Pelletier and Pudritz [1992], Ostriker [1997].

Preview of the MRI lecture. Later, in the MRI lecture, we will see that once differential rotation becomes unstable, the distinction between “disc transport” and “wind transport” becomes dynamical rather than absolute. MRI-driven stresses move angular momentum radially within the disc, while large-scale magnetic fields can remove it vertically in an outflow; in strongly magnetized states the instability and the wind can be part of the same story Lesur et al. [2013].

Where the energy is stored. The Bernoulli invariant, Eq. (9.19), can be read as a bookkeeping statement. The poloidal electromagnetic energy flux is

\[\vect {S}_p = \frac {1}{\muo }\E \times \B \Big |_p = -\frac {\Omega _F R B_\phi B_p}{\muo }\,\frac {\B _p}{B_p}. \tag{9.39}\]
Thus a rotating disc exports energy electromagnetically, and part of that Poynting flux is converted downstream into poloidal kinetic energy. In a cold wind this conversion is dramatic; in a magnetically dominated jet it may remain incomplete over large distances.

The full steady problem includes critical surfaces. The launch condition loads matter onto the field line, but a global steady solution must still pass smoothly through the slow, Alfvén, and fast magnetosonic points. In modern formulations the acceptable branch is selected by precisely these regularity conditions Vlahakis et al. [2000]. Only then does one obtain the stronger MHD statement behind the casual phrase “supersonic jet”: a super-fast-magnetosonic outflow whose distant shocks and termination region cannot communicate back to the disc. The clean conceptual split is therefore: \[ \text {launch near the disc} \quad \Longrightarrow \quad \text {critical-point selection} \quad \Longrightarrow \quad \text {asymptotic acceleration and collimation}. \]

9.4 Cross-field balance, hoop stress, and magnetic towers

Launch and acceleration are along-field processes. Collimation is not. It is controlled by the force balance perpendicular to the magnetic surfaces, that is, by the steady-flow generalization of the Grad–Shafranov problem.

A cylindrical-force-balance cartoon that is actually useful. Far from the disc, where the jet is approximately cylindrical, a schematic radial balance is

\[0 \simeq -\dd {}{R}\left (p+\frac {B_p^2+B_\phi ^2}{2\muo }\right ) -\frac {B_\phi ^2}{\muo R} +\frac {\rho u_\phi ^2}{R}. \tag{9.40}\]
The second term is the familiar inward hoop stress associated with toroidal magnetic tension. The first term contains both gas-pressure and magnetic-pressure gradients, which are typically outward. The final term is the outward centrifugal force.

Equation (9.40) is not the full transfield equation, but it already explains why collimation is a more delicate question than launch. One needs the toroidal field to be large enough to bend the flow toward the axis, yet not so large that magnetic pressure simply inflates the configuration laterally.

Why current closure matters. The phrase “hoop stress collimates” is only part of the story because \(B_\phi \) cannot be prescribed independently of the current system that generates it. The asymptotic structure depends on where the return current closes, how the magnetic surfaces open, and whether an external medium provides lateral confinement Heyvaerts and Norman [1989], Ostriker [1997]. The local inward tension is real, but the global current circuit decides whether it can actually produce a narrow jet.

Observed jets really do mark a preferred axis. Collimation is not just a theorist’s cartoon. Radio jets in active galaxies can propagate far beyond the optical body of the host, even to megaparsec scales, while pulsar systems show persistent jet–torus morphologies aligned with a well-defined axis Smith [2012], Reynolds et al. [2017]. The observed longevity of those structures is a strong clue that the central engine plus magnetic field configuration selects and maintains a preferred rotation/collimation axis. The open question is not whether narrow jets exist, but how the current system, external pressure, and magnetic stresses cooperate to keep them narrow.

Tutorial

The magnetic-tower limit is obtained when toroidal field accumulates faster than matter escapes. A simple scaling argument shows how the tower grows.

Suppose a bundle of anchored poloidal flux \(\psi _p\) is threaded by field lines whose footpoints rotate with a differential rate \(\Delta \Omega \). Each turn winds poloidal flux into toroidal flux, so the toroidal-flux injection rate scales as

\[\dd {\Phi _\phi }{t} \sim \Delta \Omega \,\psi _p. \tag{9.41}\]
After a time \(t\), the accumulated toroidal flux is therefore
\[\Phi _\phi (t) \sim \Delta \Omega \,t\,\psi _p. \tag{9.42}\]
Now suppose the tower has radius \(a\), height \(H\), and is laterally confined by an external pressure \(p_{\rm ext}\). Its toroidal flux scales as
\[\Phi _\phi \sim B_\phi \, a H. \tag{9.43}\]
Pressure balance at the tower edge gives
\[p_{\rm ext} \sim \frac {B_\phi ^2}{2\muo }, \qquad B_\phi \sim \sqrt {2\muo p_{\rm ext}}. \tag{9.44}\]
Equating Eqs. (9.42) and (9.43) yields the tower height
\[\boxed { H(t) \sim \frac {\Delta \Omega \,\psi _p}{a\sqrt {2\muo p_{\rm ext}}}\,t. } \tag{9.45}\]
So the tower grows approximately linearly in time as successive turns are stacked on top of one another. This is the essence of the Lynden-Bell picture: differential rotation injects toroidal flux, and external pressure prevents that flux from simply expanding sideways.

Disc wind versus tower. The classical Blandford–Payne solution is matter loaded and transcritical: the rotating field accelerates plasma from subslow conditions near the disc to a super-Alfvénic and, in physically complete versions, super-fast-magnetosonic jet. The tower limit is more magnetically dominated: the toroidal field itself becomes the main dynamical agent and the structure grows as a pressure-confined magnetic column Lynden-Bell [1996, 2003]. These are neighboring limits of the same anchored, twisted field geometry, not contradictory pictures.

9.5 Connections to Parker, Weber–Davis, and rotating mirrors

Pressure-driven wind versus centrifugally driven wind. The solar-wind lecture emphasized Parker’s central point: a hot corona naturally drives a transonic outflow, so pressure alone can produce a supersonic wind. The Blandford–Payne problem keeps the same global critical-point logic but changes the driver. Here the energy reservoir is the gravitational binding energy released by accretion, tapped through a rotating field line that first centrifugally lifts matter from the disc and then transfers electromagnetic energy into poloidal motion. Both are transcritical winds; they differ mainly in what launches them.

The Weber–Davis bridge. Equation (9.22) is the jet analogue of the Parker-spiral relation. The clean bridge between the two lectures is the rotating, magnetized wind of Weber and Davis Weber and Davis [1967]. Once rotation and magnetic flux freezing are included, the same invariants \(\Omega _F(\psi )\), \(L(\psi )\), and \(E(\psi )\) appear in both the stellar-wind and disc-wind problems, and the outflow removes angular momentum through an Alfvén lever arm. For the Sun the thermal pressure gradient remains the primary launch agent, but rotation still imprints the Parker spiral and produces magnetic braking. For sufficiently fast magnetic rotators, centrifugal acceleration can become dynamically important as well Johnstone [2017]. In that sense the Blandford–Payne wind is not alien to the solar-wind story; it is the disc version of the same rotating-wind machinery in a regime where centrifugal forcing is front and center.

A laboratory Parker-spiral cousin. The laboratory Parker-spiral experiments make this bridge even more explicit. They are not pure thermal Parker winds: imposed rotation helps launch the plasma, centrifugal flinging participates alongside pressure, and the field is then wound into the same trailing spiral geometry by frozen-in advection Peterson et al. [2019]. That makes them a useful halfway house between the pressure-driven solar wind and the magnetocentrifugal disc wind.

The rotating-mirror cousin. There is also a formal similarity to rotating mirrors. In both problems one thinks in terms of force balance along a rotating magnetic field line. But the physical question is different. In a disc wind the field line is open and the effective potential is arranged to let matter escape. In a rotating mirror the field line closes or returns, and the effective potential is used to confine particles rather than launch them. The mathematics of the along-field balance is related; the topology and the interpretation are not.

Takeaways
  • In steady axisymmetric ideal MHD, each magnetic surface carries line invariants such as the mass loading \(k(\psi )\), the field-line angular velocity \(\Omega _F(\psi )\), the total angular momentum \(L(\psi )\), and the Bernoulli constant \(E(\psi )\).
  • The Blandford–Payne \(30^\circ \) criterion is a local launch criterion obtained from the effective potential along a rotating field line; a global jet still has to pass the slow, Alfvén, and fast critical surfaces.
  • The Alfvén regularity condition \(L=\Omega _F R_A^2\) explains why the wind extracts more angular momentum than matter possessed at the footpoint, carrying that torque in both matter and field.
  • The disc-wind problem is the rotating-disc cousin of the Parker/Weber–Davis wind problem: pressure is central in the solar wind, centrifugal forcing is central here, and both are transcritical MHD outflows.
  • Collimation is a transfield problem. Hoop stress matters, but only together with magnetic pressure, gas pressure, rotation, external confinement, and current closure; observed jets from AGN to pulsars show that such preferred axes are not a theoretical fiction.
  • A magnetic tower is the magnetically dominated, pressure-confined limit of the same differentially rotating, anchored-flux geometry.

Bibliography

    E N Parker. Dynamics of the interplanetary gas and magnetic fields. The Astrophysical Journal, 128:664, 1958. doi:10.1086/146579.

    Edmund J. Weber and Leverett Davis, Jr. The angular momentum of the solar wind. The Astrophysical Journal, 148:217–227, 1967. doi:10.1086/149138.

    R. D. Blandford and D. G. Payne. Hydromagnetic flows from accretion discs and the production of radio jets. Monthly Notices of the Royal Astronomical Society, 199(4):883–903, 1982. doi:10.1093/mnras/199.4.883.

    G. Pelletier and R. E. Pudritz. Hydromagnetic disk winds in young stellar objects and active galactic nuclei. The Astrophysical Journal, 394:117–138, 1992. doi:10.1086/171565.

    J. Heyvaerts and C. A. Norman. The collimation of magnetized winds. The Astrophysical Journal, 347:1055–1081, 1989. doi:10.1086/168195.

    H. C. Spruit. Magnetohydrodynamic winds and jets from accretion disks. arXiv e-prints, 1996.

    E. C. Ostriker. Self-similar magnetocentrifugal disk winds with cylindrical asymptotics. The Astrophysical Journal, 486:291–306, 1997. doi:10.1086/304513.

    N. Vlahakis, K. Tsinganos, C. Sauty, and E. Trussoni. A disc-wind model with correct crossing of all magnetohydrodynamic critical surfaces. Monthly Notices of the Royal Astronomical Society, 318(2):417–428, 2000. doi:10.1046/j.1365-8711.2000.03703.x.

    D. Lynden-Bell. Magnetic collimation by accretion discs of quasars and stars. Monthly Notices of the Royal Astronomical Society, 279(2):389–401, 1996. doi:10.1093/mnras/279.2.389.

    D. Lynden-Bell. On why discs generate magnetic towers and collimate jets. Monthly Notices of the Royal Astronomical Society, 341(4):1360–1372, 2003. doi:10.1046/j.1365-8711.2003.06506.x.

    Geoffroy Lesur, Jonathan Ferreira, and Gordon I. Ogilvie. The magnetorotational instability as a jet launching mechanism. Astronomy & Astrophysics, 550:A61, 2013. doi:10.1051/0004-6361/201220395.

    Michael D. Smith. Observations of extragalactic jets. In Astrophysical Jets and Beams, pages 50–69. Cambridge University Press, 2012. doi:10.1017/CBO9780511994562.005.

    Stephen P. Reynolds, George G. Pavlov, Oleg Kargaltsev, Noel Klingler, Matthieu Renaud, and Sandro Mereghetti. Pulsar-wind nebulae and magnetar outflows: Observations at radio, x-ray, and gamma-ray wavelengths. Space Science Reviews, 207(1–4):175–234, 2017. doi:10.1007/s11214-017-0356-6.

    C. P. Johnstone. On the fast magnetic rotator regime of stellar winds. Astronomy & Astrophysics, 598:A24, 2017. doi:10.1051/0004-6361/201629609.

    Ethan E. Peterson, Douglass A. Endrizzi, Matthew Beidler, Kyle J. Bunkers, Michael Clark, Jan Egedal, Ken Flanagan, Karsten J. McCollam, Jason Milhone, Joseph Olson, Carl R. Sovinec, Roger Waleffe, John Wallace, and Cary B. Forest. A laboratory model for the parker spiral and magnetized stellar winds. Nature Physics, 15(10):1095–1100, 2019. doi:10.1038/s41567-019-0592-7.

Problems

Problem 9.1. Surface invariants in steady axisymmetric ideal MHD

Part A. Starting from Eq. (9.7), show explicitly that \(\uvec _p\parallel \B _p\).

Part B. Insert Eq. (9.9) into continuity and derive Eq. (9.12) for the mass loading.

Part C. Starting from the poloidal ideal-Ohm-law calculation in Eqs. (9.15)–(9.17), derive the field-line angular velocity \(\Omega _F(\psi )\).

Problem 9.2. The Blandford–Payne launch criterion

Part A. Starting from Eq. (9.23), compute the first derivative of the effective potential along the field line and show that it vanishes at the disc surface when \(\Omega _F=\Omega _K(R_0)\).

Part B. Work through the second-derivative calculation in full and recover Eq. (9.29).

Part C. Explain in words why the criterion \(\theta >30^\circ \) from the vertical is a launch criterion but not yet a proof of a global steady jet solution.

Problem 9.3. The Alfvén lever arm and electromagnetic torque

Part A. Starting from Eqs. (9.16) and (9.18), solve for \(u_\phi \) and \(B_\phi \) and recover Eqs. (9.35) and (9.36).

Part B. Show that smooth passage through \(M_A=1\) requires the regularity condition \(L=\Omega _F R_A^2\).

Part C. Use Eq. (9.39) to explain why the disc loses rotational energy when \(B_\phi <0\) for an outward-pointing poloidal field.

Problem 9.4. A scaling model of a magnetic tower

Part A. Reproduce the argument leading from Eqs. (9.41)–(9.45).

Part B. Suppose the external pressure decreases with height as \(p_{\rm ext}(Z)\propto Z^{-\alpha }\). How would this modify the tower-growth estimate?

Part C. Explain physically why a lightly loaded outflow is more likely to behave like a magnetic tower than a cold matter-dominated wind.