**4.1. Mass exchange vs. radiative heating/cooling**

When the bulk of the energy content and the bulk of the mass content reside in different phases, a relatively small amount of mass or energy transfer between phases can have a large effect on the structure of a multiphase medium. Such a situation is believed to exist in the three-phase model of the ISM (McKee and Ostriker 1977), where the hot phase occupies most of the volume while most of the mass is in cold clouds. Physically, the effect of mass exchange (either by conduction or ablation followed by effective mixing) is to cool the hot phase, since a fixed amount of energy is being distributed among a larger number of particles. Since radiative heating and cooling depend on both density and temperature, mass exchange can affect the radiative evolution of the medium as well. We can illustrate the global consequences of mass and energy exchange between phases by considering a medium with uniform pressure and subsonic motions, in which mass exchange is driven by thermal conduction. The approximate time-dependent equations governing the medium are then

(10) | |

(11) |

where represents the conductive heat flux. Note that these equations are appropriate in the single-fluid limit, corresponding to classical or mildly saturated conduction (Cowie and McKee 1977): in the highly saturated suprathermal limit (Balbus and McKee 1982), both the single-fluid approximation and the assumption of pressure balance break down. The evolution of the medium is driven by the terms on the right-hand-side of eq. (11), and may be dominated either by the effects of conduction or by the effects of radiative heating and cooling.

There is a crucial distinction between the heat flux term and the
radiative loss term in
the energy equation: the energy entering or leaving a volume of radius
*r* due to conduction is proportional to *r*^{2}
*q*( *r*
for classical conduction) whereas that due to heating or cooling
is proportional to *r*^{3}. Thus, there is a critical
length scale which enters the problem, which
turns out to be the Field length,
_{F}
(eq. [8]), when mass exchange is driven by classical conduction.

The temperature structure of the intercloud medium in a system of clouds
extending
over a region of size *R* depends on the ratio of the Field length to
*R*.
Balbus (1985)
developed an elegant electrostatic analogy for an ensemble of clouds in
a hot intercloud
medium under conditions in which radiative heating and cooling are
negligible and the
temperature is specified on the boundary of the ensemble. This
corresponds to the case *R* <<
_{F}. In a
steady state, the
evaporation rate and the temperature structure in the intercloud
medium are then determined by a solution of Laplace's equation with
Dirichlet boundary
conditions. In the complementary case, *R* >>
_{F}, global heat
flows are insignificant and
the temperature structure of the intercloud gas is determined by a
competition between
cloud evaporation on the one hand and heating and cooling on the
other. Numerically, the Field length is

(12) |

where *T*_{6}
*T*/10^{6}
K and _{M-23}
_{M} /
(10^{-23} erg cm^{3} s^{-1}) is normalized to a
characteristic value of the radiative cooling rate for astrophysical
plasmas.

To quantify the competition between mass exchange and radiative
heating/cooling,
consider the spatially averaged effect of cloud evaporation on the hot
phase. This is
meaningful only when the characteristic temperature of the hot phase
changes over a length
scale which exceeds the mean intercloud separation
*r*_{0}. This condition is guaranteed to be
satisfied when *r*_{0} is smaller than
_{F}. When
*r*_{0} exceeds
_{F}, then
mass exchange cannot compete with radiative heating and cooling anyway,
so the point is moot. If there are
*n*_{cl}(
3/4
*r*^{3}_{0}) clouds per unit volume
evaporating at a mean rate
_{ev} per
cloud, then the density of the intercloud medium changes at a rate

(13) |

where _{ev} is
negative for condensation. We may then use
_{ev} to define an
*effective evaporative cooling rate*,

(14) |

(Begelman and McKee 1990).
The evaporative cooling coefficient
_{ev} is
analogous to the radiative cooling coefficient
in that both
reduce the specific entropy *s*, but there are
crucial differences between the two:
reduces the total
entropy of a given volume of intercloud gas, whereas
_{ev}
increases it;
reduces the energy density of the intercloud gas, whereas
_{ev}
leaves it unchanged. Because of these distinctions,
_{ev}
should not be included in the net radiative cooling function
. However, the relative
impact of mass exchange and
radiative heating/cooling on the thermal state of the intercloud medium
can be expressed by the *radiation/evaporation ratio*
(Begelman and McKee 1990),

(15) |

When _{M}
_{M} /
_{ev} is
>> 1( << 1), then radiative heating and cooling (mass exchange)
determines the thermal state of the intercloud medium.

One can express in
terms of quantities which characterize the structure of the
two-phase medium. Writing the evaporation rate in the form
_{ev} =
4 *r*^{2}_{c}
_{h}
*c*_{s}*f*, where *c*_{s}
is the isothermal sound speed in the hot phase, we have

(16) |

*r*_{0} may be eliminated in favor of the cloud filling
factor *f* (assumed to be << 1) by substituting
*r*_{c}/*f* for
*r*_{0}^{3} / *r*_{c}^{2}. In
the limit of saturated evaporation *F* ~ a few
(Cowie and McKee 1977),
while
*F* ~ / 4 in the
Nulsen (1982)
model of ablation. In the classical conduction
limit, *F* is twice the "saturation parameter"
'_{0} derived by
Cowie and McKee (1977),
with the result that

(17) |

Cloud evaporation thus determines the intercloud temperature for
*r*_{c} > *r*^{3}_{0} /
_{F}^{2}.
The corresponding condition on the filling factor is

(18) |

or *f* > (*r*_{0} /
_{F})^{6};
for *f* ~ 0.03, as in the ISM, this will
be true if _{F}
2*r*_{0}. In terms of the
sound-crossing time across *r*_{c} (measured in the hot phase),
*t*_{s} ~ *r*_{c} / 2*c*_{s}, and
the radiative cooling time in the hot phase,
*t*_{c} ~ 5*kT*_{h} / 2*n*_{h}
(*T*_{h})
, we may express the condition for mass
exchange to dominate globally in the form

(19) |

Writing (18) in the form
*r*_{0} < *r*^{1/3}_{c}
_{F}^{2/3},
we see that the intercloud spacing in a
conduction-dominated medium must also be smaller than the Field
length. This has
important consequences for the thermal stability of the hot phase in a
conduction-dominated system. Since
_{F} is
roughly the minimum wavelength which is thermally unstable
(Field 1965),
potentially unstable regions must contain many clouds. The
conduction-modified
condition for thermal instability is obtained simply by including
_{ev}
in Balbus's (1986*a*)
criterion (eq. [4]):

(20) |

(Begelman and McKee 1990).
*n*_{h}
_{ev} /
*T*_{h} is
generally an increasing function of *T*_{h}: for
isobaric perturbations *n*_{h}
_{ev} /
*T*_{h} ~ *q*
*T*_{h}^{7/2}(*T*_{h}^{1/2})
for classical (saturated) conduction.
Therefore, evaporation has a * stabilizing* influence on the hot
phase, and in a
conduction-dominated medium thermal instability will be inhibited by the
presence of evaporating clouds
(Begelman and McKee 1990).
This will be true even if the radiative processes place
the hot phase in a thermally unstable regime. However, it should be
noted that the cooling
time scale in the hot phase of a conduction-dominated medium is shorter
than the radiative
cooling time scale. Therefore, conduction cannot stabilize a hot phase
over a time scale
which is longer than the time scale for radiative thermal
instability. However, it can lead
to the hot phase cooling down somewhat before the onset of thermal
instability. Since
thermal conduction generally becomes less important at low temperatures,
such a system
may evolve to a state in which evaporative cooling no longer dominates,
whereupon thermal
instability may occur. Since the Field length is generally a strongly
increasing function of
temperature, the operation of thermal instability may lead to the
production of smaller
and more closely spaced clouds than would have formed in the hotter medium.

It is instructive to apply the ideas discussed above to the three-phase
model of the ISM
(McKee and Ostriker 1977).
The three-phase ISM consists of cold HI clouds
surrounded by warm HI and HII envelopes, all embedded in a pervasive hot
ionized medium
(HIM). The physical conditions in the HIM are governed by mass exchange
with the clouds
and energy injection by supernovae. The model is intrinsically
time-dependent: A given
element of gas is compressed and heated by SNRs at intervals of about
5 × 10^{5} yr, and
this makes its evolution difficult to analyze using concepts developed
to treat steady-state
or slowly evolving systems. Nonetheless, it is of interest to evaluate
the Field length and
the radiation/evaporation ratio. Using the fit to the cooling function
of Raymond, Cox and Smith
(1976)
for cosmic abundances,
(*T*)
1.6 ×
10^{-19}*T*^{-1/2} erg cm^{3}
s^{-1} (10^{5} K
< *T* < 4 × 10^{7} K), multiplied by an
enhancement factor
10_{1}
10 to take account of
nonequilibrium ionization and density inhomogeneity near conduction fronts
(McKee and Ostriker 1977),
we have
_{F} =
44(_{c}^{1/2}
*T*^{3}_{6} /
_{4}
_{1}^{1/2}) pc, where
_{4}
*p* /
10^{4}*k* and
*T*_{6}
*T*_{h}/10^{6} K. We can write condition (18) in
the form

(21) |

It is evident that the relative importance of conductive and radiative
energy exchange
is sensitive to conditions in the hot phase, particularly to
*T*_{h}, as well as to the filling
factor and typical size of clouds. Under the conditions deduced by McKee
and Ostriker (*T*_{6} = 0.45,
_{4} = 0.36,
_{1}
= 1, *r*_{c} = 2.1
pc, and *f* = 0.23), condition (21) is marginally
satisfied. Equivalently, the radiation/evaporation ratio is given by
*R* = 0.38, which implies
that evaporative cooling dominates radiative cooling and that the HIM is
thermally stable.
(The argument that evaporation can stabilize the HIM is originally due
to McCray [1986].)
However, even a small amount of cloud ablation by hydrodynamic processes
(Section 3.2) or a
slight increase in the HIM temperature would lead to a drastic increase
in the energetic
importance of evaporation. If the typical clouds were sufficiently small
that conduction
were saturated, i.e., for

(22) |

(Cowie and McKee 1977; Balbus and McKee 1982), the appropriate version of condition (19) would be

(23) |

In the McKee-Ostriker picture, each region of HIM is overrun by another
supernova remnant before it has time relax to a stationary state with
1. Stochastic heating
by SNR shocks thus leads to discontinuous trajectories in the *p* -
*V* plane, and heating
is balanced by radiative cooling. In the galactic fountain model
(Shapiro and Field 1976;
Wang and Cowie 1988),
the heat is advected into the galactic halo. If clouds are effectively
ablated then the accelerated lowering of *T*_{h} by
evaporation may allow the large tracts of
the ISM to cool to a homogeneous state at
*T*
10^{4} K. The intercloud medium remains
thermally stable until radiative cooling begins to dominate over
evaporative cooling, at *T*_{h}
10^{5}
K. Since the Field length is a strongly increasing
function of *T*_{h}, the onset of
instability at a reduced temperature may lead to the formation of
sub-parsec size clouds.