Next Contents Previous

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 curlyV is nbarh µH curlyV, where nbarh is the mean density of the intercloud gas in curlyV. Choosing the volume curlyV to comove with the intercloud gas implies that this mass can change only by cloud evaporation, at a rate ndotev µH curlyV. Mass conservation for the intercloud gas then becomes

Equation 24 (24)

Integration of equation (11) over curlyV then implies

Equation 25 (25)

where s is the surface bounding curlyV. 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

Equation 26 (26)

where < >curlyV denotes an average over the volume curlyV. 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 rc 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 curlyV is held constant as the system evolves. If one assumes that the clouds are fixed as well, then the mean density n bar also remains constant. This condition would apply in a system in which the sound crossing time R/cs 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 dot curly V = 0 in eqs. (24) and (26), we obtain

Equation 27 (27)


Equation 28 (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

Equation 29 (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 ident n-1h is the specific volume of the intercloud gas. The radiative equilibrium curve n2h curly L = 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 n2h curly L may be assumed to be a known function of p and V everywhere on the plane. However, the evaporation rate ndotev 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 curly R (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):

Equation 30 (30)

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

Equation 31 (31)

Trajectories with curly R = - 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

Equation 32 (32)

Since T propto pV propto V in the isobaric case, this equation also describes the temperature evolution of the system. The point curly R = - 1 represents a steady state for the hot phase, although mass continues to be lost (if curly L < 0) or gained (if curly L > 0) by clouds in this state (unlike the "true" steady state curly L = ndotev = 0 in the isochoric case). For a system dominated by radiative heating or cooling (|curly R| >> 1), evolution is leftward above the thermal equilibrium curve and rightward below the curve. In a system dominated by mass exchange (|curly R| << 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.

Next Contents Previous