Annu. Rev. Astron. Astrophys. 1988. 36:
539-598
Copyright © 1998 by . All rights reserved |

The phenomenology of AGNs, namely their exceedingly large powers (up to
10^{47} erg s^{-1}) and concentration in small volumes
( 10^{-3} pc), leads
to consideration of models for jet formation based on processes around
supermassive black holes (*M*
10^{7}
*M*_{})
(Rees 1984).
Other suggested origins, namely star clusters, pulsar clusters, and spinars,
appear to be inadequate to explain the total power, the long-term stability
of quasars and radio galaxies, and short-term variability. The launch and
collimation of supersonic (eventually relativistic) outflows from
supermassive
black holes can originate from two distinct mechanisms: (*a*) accretion
of matter onto the black hole, liberating gravitational binding energy that
is transferred to matter flung along the rotational axis; (*b*)
electrodynamic processes, tapping black hole rotational energy and
feeding it into large-scale
magnetic fields. In both cases, the global picture leads to the formation
of twin opposite jets as in the early proposal by
Blandford & Rees
(1974).

**3.1. Collimated Outflows from Accretion Disks**

The structure of gaseous, magnetized or
unmagnetized disks is reviewed by
Papaloizou & Lin
(1995).
Two characteristic configurations can be obtained analytically by
solving the structure equations: (*a*) thin disks and (*b*)
thick disks or tori, depending on whether the energy dissipated into the
plasma by stresses can be rapidly radiated away or if the local pressure
is competitive with gravity instead
(Begelman et al 1984).
In both cases, the activity at the disk surface creates a hot corona
that drives winds along the rotation axis.

HYDRODYNAMIC WINDS Hot matter at the surface of a disk can
be accelerated outwards against the gravitational pull of a compact core
of mass *M*_{h}
by hydrodynamic pressure forces. This idea was first applied to jet
acceleration by
Blandford & Rees
(1974)
in the twin-exhaust model whose physics is derived from the stellar wind
theory. Among the many subsequent papers, we refer to
Fukue & Okada (1990),
who presented a complete axisymmetric solution of the balance equations
along and perpendicular to streamlines
(Bernoulli and Grad-Shafranov equations,
respectively), also including the effect of centrifugal forces generated
by the disk rotation. In cylindrical geometry
(*r*, ,
*z*) with *ds* as a line element along a streamline and
*dn* perpendicular to it, these equations are

(1) |

(2) |

where *L* is the specific angular momentum. Energy is injected
into each streamline at the base on the disk, and a flow pattern is set up
that crosses transonic surfaces to produce a supersonic wind. If the
temperature distribution on the disk is flatter than 1 /
*r*_{e} (*r*_{e}
is the equatorial distance), the gas ejected into streamlines close
to the axis is gravitationally confined and forms a corona, while the gas
on external streamlines can form a wind. The opposite is true for a
temperature distribution steeper than 1 / *r*_{e}.
These two patterns correspond to hollow jets and well-collimated jets,
respectively. The flow goes through multiple critical points, passing from
subsonic to supersonic and vice versa several times with formation of
shocks.
The novelty of these solutions with respect to the original twin-jet model
is the possibility of studying the wind solution stability and especially
of bringing the first critical point very close to the core and immediately
making the jet supersonic, as is observed. The link between jets and disks
in the hydrodynamic models is only through the injection of energy at the
base of streamlines. There is no direct back-reaction from the jets to the
disk.

MAGNETOHYDRODYNAMIC WINDS Magnetohydrodynamic (MHD) winds are an important source of advection losses from disks; they occur in the presence of an initial poloidal magnetic field anchored in the accreting material that is wound up by rotation of the disk and generates a collimating toroidal field. Thus accretion disks can naturally drive winds by centrifugal or magnetocentrifugal mechanisms (Mestel 1961, Weber & Davis 1967, Sakurai 1985). The set of MHD nonlinear partial differential equations is

(3) |

(4) |

(5) |

where *µ* is the specific enthalpy and
the specific heating/cooling rate. Steady-state solutions for the full
two-dimensional (2-D) problem of jet acceleration from accretion disks
exist for the axisymmetric
case only. In this case, the set of MHD equations can be reduced to two
coupled
equations, the Bernoulli and Grad-Shafranov (or transfield) equations, which
fully describe the wind dynamics along and across streamlines, respectively.
However, these equations are formidably difficult, and some simplifying
assumptions
must be adopted. The main problem encountered is the presence of several
critical points where equations become singular. Physically acceptable
solutions
must go through these points smoothly. In one-dimensional geometries, they
correspond to the flow reaching the characteristic propagation speeds in
the fluid (Alfvènic, slow, and fast magnetosonic); in 2-D
geometries, this
is no longer true for the bulk velocity but applies to specific velocity
components
(Tsinganos et al 1996).

*Self-similar solutions: the Blandford & Payne model*
The seminal paper in the context of magnetized disks and jets is by
Blandford & Payne
(1982),
who found self-similar steady-state solutions of the ideal MHD equations
of a cold, axially symmetric magnetospheric flow from a Keplerian disk. They
assumed that the disk is threaded by open poloidal magnetic field lines
corotating with the disk at the Keplerian velocity; in cylindrical
coordinates (*r*,
, *z*),

(6) |

where is the flux
function. Total magnetic field lines are wrapped around
= constant magnetic
surfaces. Matter is centrifugally driven outwards in the corona and is
frozen along field lines **v** =
*k*()
**B**_{p} / 4
+ *r*
()
,
where *k* is a structure constant and
the angular velocity;
*k* and are
related as (**B** ^{.}
)*k* =
(**B** ^{.}
)
.
The second term implies that a toroidal field is generated:

(7) |

which becomes dominant at large radii close to the rotation axis and consequently collimates the flow into a jet. The self-similar solutions are obtained with scaling in terms of the radial distance from the center:

(8) |

For a fixed colatitude
= z/*r* (i.e. same ),
all physical quantities scale with the spherical radius, and the MHD
equations reduce to a second-order differential equation in
for the Alfvènic Mach
number *v* / *v*_{A}, *v*_{A} =
(*B*^{2} / 4
)^{1/2},
and a first-order equation for the field/streamline geometry. Both equations
have singular points and are studied with the above-mentioned technique of
wind solutions. Physical solutions must cross singular points with
regularity;
the study of the topology of these critical points is rather involved and
defines the characteristics of solutions. In fact, the Blandford & Payne
solutions are not imposed to cross the fast magnetosonic point, and this
causes a collapse of the solutions on the symmetry axis at large distances
(eventually at infinity).

Blandford & Payne found two classes of collimated wind solutions
depending on the final flow velocity: (*a*) fast magnetosonic winds
with paraboloidal asymptotic streamlines and (*b*)
trans-fast-magnetosonic winds that focus onto the rotation axis. In the
first
class about one third of the energy is carried as bulk kinetic energy and
two thirds as Poynting flux; the second class is instead dominated by the
kinetic flux and in this sense is very interesting, although it has the
drawback of an excess of pinching force corresponding to a divergence in
the electric current on the axis. Far from the axis, the flow continues
its free expansion and is instrumental in extracting angular momentum
from the system by magnetic
torque. Even a small mass loss can carry a large specific angular momentum
given the large lever arm of the field acting on the matter.

Other radial self-similar solutions have been studied by
Contopoulos &
Lovelace (1994),
Pelletier & Pudritz
(1992),
Rosso & Pelletier
(1994).
In particular Contopoulos & Lovelace derived a relation between the
shape of streamlines and the poloidal current consistent with the
magnetic structure.
Also, the Blandford & Payne model has been extended to the special
relativistic case by
Li et al (1992),
which shows that a kinetic energy flux comparable to the Poynting flux can
be obtained. Contopoulos has proposed a steady solution for jets without
imposing an original poloidal field
(Contopoulos 1995).
A toroidal component originally present in the disk is increased by
differential rotation. The strong pressure gradient between the disk
(large *B*_{}) and the corona above it (*B*_{}
0) ejects plasma perpendicular to the disk. For this to happen, the vertical
velocity at the surface of the disks must be comparable to the Keplerian
velocity *v*_{z0} ~ *v*_{0}. The ejected
plasma convects azimuthal magnetic field and is self-collimated.

A different type of scaling has been proposed by Tsinganos, Trussoni, and Sauty (Sauty & Tsinganos 1994, Trussoni et al 1996, Tsinganos et al 1996), in which latitudinal self-similarity is used with all physical quantities expressed in separable form. In particular, the magnetic potential is written as

(9) |

and the Bernoulli and Grad-Shafranov equations for *f*(*r*)
are obtained. This scaling permits a better representation of the regions
around the rotation axis of the system that are singular in the Blandford
& Payne solution. Solutions correspond to super-Alfvènic winds,
and one class provides self-confined outflows: After an initial quasi
radial expansion, poloidal streamlines undergo some oscillations and
then settle into a cylindrical pattern. Also note that these solutions
do not require the use of a polytropic equation of state; in fact, they
correspond to a given profile of the streamlines
in ,
which fixes the profile of the propagation channel. The heating/cooling
conditions to maintain the outflow (i.e. the local equation of state)
can be derived a posteriori from the solutions, in order to determine
whether they are physically reproducible.

Another class of MHD winds is based on simplifying the Grad-Shafranov equation, with the consequence, however, that solutions are valid only in restricted regions. For instance, close to the equatorial plane, the flow can be modeled in thin cylindrical shells by averaging physical quantities in the direction perpendicular to the axis (Lovelace et al 1991).

*Persistence of the bead-on-wire configuration*
The connection between disks and jets is through
the transfer of angular momentum. This allows the stability of disks and
is the driving element for the spontaneous initiation of outflows. However,
the field structure must be maintained in conditions to have flux lines
inclined at 60° or less to the disk plane. Advection of field lines
by supersonic
mass flow tends to increase the inclination angle; resistive or ambipolar
diffusion has been proposed to balance this effect, but in fact the angle
is likely to move very close to 90°
(Königl 1989,
1994).
Consequently other ways must be found to launch and maintain a wind.

Ferreira & Pelletier (1993a, b, 1995) have shown that, taking into account viscous and magnetic turbulent effects in the disk plasma, the disk magnetic pressure develops a vertical component that pushes matter to the surface, leading to a continuous transition from the resistive plasma disk to the ideal MHD jet.

Shu et al (1994)
have proposed that MHD winds exist only along field lines originating from
the inner edge of the disk, where the central object field lines penetrate
into the disk and almost corotate with it: The corotation radius is
*R*_{cor} = (*GM* /
_{star}^{2})^{1/3}. The disk is
actually truncated inside this radius, as matter diffuses across the
magnetic field (by microscopic mechanisms of ambipolar diffusion and
ohmic dissipation), bends its lines inward, and accretes
onto the central object. Also outside *R*_{cor},
matter diffuses onto field lines but bows them outward, transferring
angular momentum to the disk, which can then reach super-Keplerian
velocities and reaches an ideal configuration to start a funnel
flow. This mechanism is referred to as magnetocentrifugal
acceleration. For large accretion rates, the corotation radius moves
close to the central object, which is forced to rotate at breakup
conditions.

Recently,
Contopoulos (1996)
initiated the study of general solutions for axisymmetric flows without
imposing *E*_{}
= 0, i.e. the poloidal velocity parallel to the poloidal magnetic field.
This condition allows the magnetic field to be advected by the accretion
flow and accumulated onto the axis of symmetry.

*Relativistic flows*
Camenzind (1986)
has pointed out that, when dealing with fast rotators and a strong
gravitational field, use of the full machinery of the general
relativistic MHD theory is unavoidable. Relativistic winds have the same
critical points as the nonrelativistic ones: slow magnetosonic,
Alfvènic, and fast magnetosonic points. A wind equation
can be obtained along the flux tube. Camenzind showed that up to ~
80% of the initial Poynting flux at the base of the flux tube is converted
into kinetic energy beyond the light cylinder and the wind velocity reaches
Lorentz factors up to
_{bulk}
~ 8. A current *I* is carried by the wind and is essentially
determined by the total angular momentum lost through the outflow.

*Radiation pressure acceleration*
Radiation-supported thick accretion disks have been
studied for a way to stabilize them against the Papaloizou & Pringle
instability
(Frank 1979,
Meier 1979).
In addition, deep funnels, replenished by ultraviolet photons from the walls
with a net outward momentum component, produce bulk acceleration of a wind
by radiation pressure.
Ferrari et al (1985)
discussed the quasi-2-D hydrodynamic problem of relativistic equilibrium
flows for given profiles of the propagation channels. Solutions exhibit many
critical points. In particular, the sudden expansion at the exit of the
funnel brings the first critical point close to the nucleus. The jet
thrust is not constant but decreases in the subsonic regime and
increases in the supersonic
regime. In fact, the surface of the jet is not parallel to the flow, and
this corresponds to an external pressure force acting consistently on the
field. Where the channel narrows (widens), the flow slows down
(accelerates).
Nobili (1998) consistently solved the radiation transport equation in the
funnel, extending the results to the optically thick case. In all models
related to acceleration by radiation pressure, the critically limiting
factor
is Compton drag by the same radiation. Particles moving at large velocities
along the funnel overtake the photons emitted by the walls and have strong
Compton losses. In fact, the asymptotic flow velocity cannot exceed a
Lorentz factor
_{bulk}
2. A solution to this
problem has been proposed by
Ghisellini et al (1990),
who considered the possibility that clouds of electrons in the flow can
synchrotron
self-absorb the radiation from the disk or funnel. In this case, absorption
is selective so that red-shifted photons fall below an absorption cutoff
and therefore cannot brake the flow. Lorentz factors up to
_{bulk}
~ 10 are attained.