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

**4.1. Global Dynamics of Confined Jets**

The propagation of a laminar jet confined by external
agents in stationary hydrodynamic conditions is governed by the mass and
momentum conservation laws. In the Newtonian, nonrelativistic limit and for
a polytropic gas *P* = *K*
^{},

(12) |

(13) |

where *M*_{j} is the flow Mach number,
the
gravitational potential,
the adiabatic index
of the gas, and
_{j} the mass
discharge. Equations 12 and 13 can be solved in the limit of given jet
power and discharge
_{j}. The
cross-sectional area typically shows a minimum value for
*P*_{min} ~ (1/2)*P*_{0} (*P*_{0}
is the speed at the base of the flow, the stagnation point),
where the jet speed becomes transonic. An adiabatic jet accelerates to
supersonic speed passing through a converging nozzle. In the subsonic
part of the flow, pressure and density are approximately constant, and
*A*
*v*^{-1}
(Blandford & Rees
1974).

For the large-scale hypersonic part of the flow, *M*_{j}
>> 1, Equations 12 and 13 give *M*_{j}
*P*^{(1-)/2} and *A*
*P*^{-1/},
implying *v* = const; for an adiabatic flow
= 5/3, the scaling is
*M*_{j}
*P*^{-4/5} *A*^{-1}, *A*
*P*^{-3/5}. Pressure distribution in galaxies can be
modeled as *P*_{ext}
*r*^{-n} (here *r* is the distance from the core), and
the angle subtended by the jet width as seen from the nucleus decreases
as
~ *A*^{1/2} *r*^{-1}
*r*^{(n-2)/2}. For an adiabatic flow
in a halo with *n* = 2, the opening angle is
~
*r*^{-2/5}.
Therefore steady jets can be collimated even though their area expands. On
the other hand, the estimated equipartition pressure inside a jet scales
as *P*
*A*^{-2/7}, i.e. falls much less rapidly than the
equilibrium solution *P*
*A*^{-5/3}. This means that some form of internal
dissipation must favor collimation.

A more complete analysis of the global dynamics of jets that takes into account confinement in channels, nonthermal momentum deposition or subtraction by external fields (radiation, plasma waves, etc) and in extended gravitational potential wells has been developed in compressible hydrodynamics by Ferrari et al (1985). They discussed quasi-2-D solutions of the problem of relativistic equilibrium flows for given profiles of the propagation channels, obtaining an equation for the flow velocity along the rotation axis:

(14) |

where

(15) |

The first term on the right-hand side is the gravitational term. The
second is very interesting: It easily can be derived that when the channel
*S*(*Z*) expands more (less) rapidly than spherically
(
*Z*^{2}), its walls will deposit (subtract) momentum. The
last term
indicates momentum addition by external forces; for the case of radiation
in the limit of optically thin plasma,

(16) |

where *H, J*, and *k* are the momenta of the radiation field.

Relativistic flows from accretion funnels exhibit many critical points, as compared with the single critical point (de Laval nozzle) of the nonrelativistic hydrodynamic wind solution. In particular, the sudden expansion at the exit of the funnel brings the first critical point close inside the nucleus. In these solutions, the thrust is not constant, 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 is slowed down (accelerated).

Steady Newtonian and relativistic solutions
show that variations in the channel cross section due to the physics of
external confinement act exactly like momentum deposition. Additional
critical points of complex mathematical topologies produce jets with
several transitions from a subsonic to supersonic regime through
shocks. The production of shocks
is very interesting in connection with the observations of extended jets
with bright knots, as they provide suprathermal particle acceleration and
field compression that locally enhance nonthermal emission (see
Section 6).
In addition, the flow pattern depends on the profile of the extended
gravitational potential outside the nucleus. In particular for adiabatic
jets, a mass distribution *M*_{gal}
*r*^{-s}, with *s*
, does not allow the
flow to reach a transonic point: The flow is stopped inside the galaxy
(Ferrari et al 1986).