**4.2. Evolution of multiphase systems**

By averaging the equations of mass and energy conservation over a volume
(*V*) which
contains many clouds, one can derive equations for the global evolution
of the hot intercloud medium in the presence of mass exchange with
embedded clouds
(Begelman and McKee 1990).
The mass of intercloud gas in
is
_{h}
*µ*_{H}
, where
_{h} is the mean
density of the
intercloud gas in .
Choosing the volume
to comove with the intercloud gas implies that
this mass can change only by cloud evaporation, at a rate
_{ev}
*µ*_{H}
. Mass conservation for
the intercloud gas then becomes

(24) |

Integration of equation (11) over then implies

(25) |

where *s* is the surface bounding
. By assuming that the
characteristic dimension of the
averaging volume is large compared to the Field length, we ensure that
the conductive heat
flux term in equation (25) is negligible compared to the heating and
cooling term. The global energy equation then simplifies to

(26) |

where
< >_{}
denotes an average over the volume
. Note that mass
exchange does not enter
this equation: mass exchange alters the density and temperature of the
intercloud gas, but not its pressure.

Additional constraints are required to solve for the evolution of a
specific system.
Since the cloud evaporation rate depends on the typical cloud size, it
is necessary to have
an equation for the evolution of *r*_{c} in time. There are
also likely to be externally imposed constraints on the intercloud medium;
Begelman and McKee (1990)
considered two limiting
cases. In the * isochoric* limit, tile comoving volume
is
held constant as the system evolves.
If one assumes that the clouds are fixed as well, then the mean density
also remains
constant. This condition would apply in a system in which the sound
crossing time *R*/*c*_{s} is
long compared to both the characteristic heating/cooling time and the
evaporation time.
Such a situation might apply locally within a supersonic accretion flow
or wind. Setting
= 0 in
eqs. (24) and (26), we obtain

(27) |

and

(28) |

In the *isobaric* limit, the intercloud medium can exchange mass
with a reservoir in order
to maintain a constant pressure, so we set *p*(*t*) =
constant. Such a situation might apply,
for example, if the system were in contact with an X-ray heated wind
above an accretion disk
(Begelman, McKee and Shields
1983).
We then have

(29) |

The instantaneous state of the intercloud medium can be described by the
location of a point in the *p* - *V* plane (cf.
Fig. 1);
*V*
*n*^{-1}_{h}
is the specific volume of the intercloud
gas. The radiative equilibrium curve
*n*^{2}_{h}
= 0
divides the plane into two regions: above
the curve, radiative cooling exceeds the external heating, whereas below
the curve the converse is true. The net cooling rate
*n*^{2}_{h}
may be
assumed to be a known function of *p*
and *V* everywhere on the plane. However, the evaporation rate
_{ev} also depends
on the distribution of cloud sizes and separations.

The character of a trajectory in the *p* - *V* plane is
determined by the
relative importance of energy exchange and mass exchange, which is expressed
quantitatively by the radiation/evaporation ratio
(eq. [16]). in the
isochoric case, the slope of trajectories in
the *p* - *V* plane is governed by the ratio of equations (27)
and (28):

(30) |

Trajectories are nearly vertical if radiative cooling and heating are
dominant
(|| >> 1),
and nearly horizontal if mass exchange is dominant
(|| << 1). It
is immediately obvious
that trajectories must be locally horizontal
(*dp* / *dV* = 0) where they cross the equilibrium curve
( = 0), provided that
_{ev}
0 at the
point of crossing. Points at which
=
_{ev} = 0
represent stationary states. The temperature evolves as

(31) |

Trajectories with
= - 3/5 are
isothermal. Isobaric trajectories are constrained to be
horizontal in the *p* - *V* plane. The direction (and rate) of
motion is given by equation (29), which may be written in the form

(32) |

Since *T* *pV*
*V* in the
isobaric case, this equation also describes the temperature
evolution of the system. The point
= - 1 represents a
steady state for the hot phase,
although mass continues to be lost (if
< 0) or gained (if
> 0) by clouds in
this state (unlike the "true" steady state
=
_{ev} = 0
in the isochoric case). For a system
dominated by radiative heating or cooling
(|| >> 1),
evolution is leftward above the
thermal equilibrium curve and rightward below the curve. In a system
dominated by mass exchange
(|| << 1),
evolution is leftward for evaporation and rightward for condensation.
A more detailed discussion of *p* - *V* plane trajectories may
be found in
Begelman and McKee (1990),
who considered the specific case of a gas heated by
Compton scattering and
cooled by bremsstrahlung and the inverse Compton effect. They also
discuss the stability properties of evolving systems.