In the following sections I will describe only those models of disk-halo interactions which are based on energy input from areas of high-level SF in galaxy disks, because there is currently no evidence that energy sources outside the galaxies contribute measurably to the total energy balance of the ISM, and thereby to disk-halo interactions. The influence of galaxy interactions on the level of SF will be discussed in Section 8.6.
8.1
First generation (static) models
In the late 1960's, when Field et al. (1969) developed their two-phase model of the ISM, only two phases - the CNM and the WNM - were known. The model was based on the assumption that these two gas phases are in thermodynamic equilibrium, which basically represents a quiescent ISM without disturbances (as observed in galaxies like e.g., NGC 4244).
However, observations with the Copernicus satellite, especially of the O VI line with an ionization energy of 138 eV, indicated the presence of another, hot component of the ISM (Jenkins & Meloy 1974). This detection was consistent with the presence of a diffuse soft X-ray background (Cox & Smith 1974). The two-phase model of the ISM could also not explain the observed scale heights and velocity dispersions of both gas components, which were higher than predicted. Accordingly, a continuously heating energy source, namely massive young (OB) stars, with their winds (Weaver et al. 1977; Oey & Massey 1994) and SNe (Smith 1977; Tomisaka & Ikeuchi 1986) had to be included in the calculations in order to account for the observations. This was done by Cox & Smith (1974) and McKee & Ostriker (1977) in the three-phase model of the Galactic ISM. A three-phase ISM, with a HIM that has a filling factor close to unity, represents a medium in which vigorous SF dominates the energy balance.
8.2
More refined (dynamic) models
Still, the early ISM models had several shortcomings. They did, e.g.,, not take into account that a model of a constantly heated ISM must include streaming motions of hot gas expanding within the disk, preferentially along the direction of the steepest density gradient, i.e., perpendicular to the disk. It is nowadays, 20 years later, common knowledge that the ISM in the disks of spiral galaxies is a very complex mixture of many different constituents/gas phases (Section 2). Theoretical models must also take into account that these phases of the ISM are not static, but dynamic, which means that the different phases/constituents of the ISM can interact with each other and that phase transitions can occur. The multi-phase ISM exists under the external influences of both the gravitational potential of the galaxy and the energy input of the heating sources. The gravitational potential determines the spatial distribution of the gas (especially its confinement into a thin disk), if it were undisturbed. Energy input, e.g., ionizing UV radiation from hot massive stars, stellar winds, and SN shock waves, disturbs this equilibrium. Thus, in the presence of heating sources the ISM in galaxies is locally never in thermodynamic equilibirium.
The distribution of SF in galaxy disks is not stochastic, but spatially and temporally coordinated, leading to locally very high SF and subsequent high SN rates, creating so-called superbubbles. Since expansion of hot gas occurs preferentially in the direction perpendicular to the disk, these superbubbles connect to the halo, creating local outflows (e.g., Habe & Ikeuchi 1980). For more information see also e.g., Tomisaka & Ikeuchi (1986), Mac Low and McCray (1988), and Tomisaka (1990). Oey & Clark (1997) recently studied the size distribution of superbubbles and their influence on, e.g.,, the porosity factor of the ISM. The numerical and analytical treatment of supershell expansion has been reviewed by Tenorio-Tagle & Bodenheimer (1988) and Mac Low (1996) and calculations on non-coeval SF were performed by Shull & Saken (1995). Thus, modern theoretical models of the ISM not only take into account dynamical processes in galaxy disks, but also include disk-halo interactions. A few of the most widely accepted models of the ISM are listed in the following section. These models implicitely contain the assumption that SF-related processes dominate the energy input into the ISM.
8.3
Current models
8.3.1 Galactic fountain model
The ``galactic fountain'' model (Shapiro & Field 1976) was the first model to include disk-halo interactions. It describes outflows as a widespread phenomenon and the outflow velocities are expected to be smaller than the escape velocity of gas (v < vesc). Thus, this model implies that all material will eventually fall back onto the galactic disk, leading to a circulation of matter, especially of the metals produced by the high-mass stars initiating the disk-halo interactions. This model is consistent with most observations of the Galactic ISM and the HVCs (e.g., Pietz et al. 1996).
8.3.2 Chimney model
Based on calculations of SN expansions and on the assumption of a higher spatial and temporal concentration of SF than adopted in the fountain model, Ikeuchi (1988) and NI89 developed their ``chimney'' model. This model also takes into account the total energy requirements for creating and maintaining a gaseous halo, based on the existing observational evidence.
Superbubbles created by tens to hundreds of SNe from massive OB associations or superclusters can lead to more powerful localized outflows, the so-called chimneys. In this scenario stellar winds are thought to blow free cavities in the ISM into which SNe later expand. Since the Strömgren spheres of giant H II regions can be as large as the thickness of the thin gaseous disk of a spiral galaxy (of order 100 pc), the spheres can form vertical channels through which SN-driven shocks can propagate faster.
The basic difference between chimneys and fountains is that chimneys
are more localized and not as pervasive as fountain outflows.
Accordingly, the predicted volume filling factor of the HIM is
smaller than for a galactic fountain. In addition, chimneys can
reach v 
vesc and thus are powerful enough to drive
outflows of matter into intergalactic space.
Chimneys are envisaged as outflows of overpressured hot gas and CRs from superbubbles into the halo, dragging along part of the ambient colder gas and partly heating it up, thereby creating an onion-skin structure. This scenario is consistent with the observed multi-phase nature of the halo ISM (Section 6). Dust within supernova remnants, e.g.,, is known to be destroyed by the strong shocks permeating the medium. This implies that halo dust is material dragged along by outflowing hot gas. The fact that the abundances of halo clouds are closer to Solar than those of diffuse clouds in the Galactic disk also indicates that the halo material consists of processed matter (Sembach & Savage 1996), corroborating the above hypothesis.
The chimney model is roughly consistent with the current observational evidence. Although details of the ionization structure of the DIG are not yet completely resolved and the role of CRs has not been elaborated, the chimney model provides the closest approximation to our data of galaxies with high SFRs. On an energy scale, the chimney mode of the ISM is an intermediate level between a quiescent (two-phase) ISM and a three-phase ISM, in which the HIM dominates. The prototype chimney mode galaxy is probably NGC 891, with widespread high-level SF over most of its disk and an extended gaseous halo. The most extreme case of SF-related energy input into the ISM, a starburst, is a realization of the upper end of the energy scale, where the chimney mode comes close to and merges with the classical three-phase picture of the ISM.
Both the fountain and the chimney model do imply that the halo ISM should have as complex a structure as the disk ISM, and as actually observed (Section 6). This complexity of the ISM and the ever increasing number of gas ``phases'' detected prompted Norman & Ferrara (1996) to formulate a new description of its structure, introducing the idea of a phase continuum of the ISM, based on the size of the turbulence scales.
8.3.3 Worms
The ``worm'' structures found by Heiles (1979, 1984) in H I gas are manifestations of extraplanar gas. However, this is not an independent model of the ISM, but rather a morphological detail which is entirely consistent with both the fountain and the chimney models. Worms can be interpreted as compressed walls of (adjacent) expanding superbubbles (NI89). It is noteworthy, however, that worms can exist in the plane due to radial sweeping of ISM, with no bulk upward motion, as envisaged in the fountain and chimney models. So worms can exist in an ISM without disk-halo interactions.
8.4
Heating processes and total energy balance
Excluding phenomena relating to nuclear activity in galaxies, we restrict our further discussion to SF-related energy sources heating large-scale gaseous halos. As such we have identified OB stars and their winds and subsequent SNe.
The above models take into account the following heating mechanisms: photo-ionization, shock heating, and collisional excitation. These are the non-relativistic sources of ISM energy input (Sections 8.4.1, 8.4.2, 8.4.3). In some cases, neither of these energy sources (or any combination of them) can satisfactorily explain all observations. This led to searches for additional heating sources; other processes considered possibly important are: CR heating, magnetic reconnection, and photo-levitation. These heating sources lead also to new, previously uninvestigated transport mechanisms of gas from galaxy disks into the halos (Sections 8.4.4 and 8.4.5).
The problem is that the sources of energy are far away from the halo gas and it must be determined how mass, energy, and momentum reach the halo. UV, optical, and near-infrared spectroscopy, in particular emission line strengths and ratios, contain crucial information on the gas composition (element abundancies), excitation conditions, density, and other parameters. If these are tightly constrained, one can distinguish between different physical processes that might dominate the heating and thereby the total energy budget of the ISM.
8.4.1 Photo-ionization by UV and optical continuum of massive young stars
The most commonly invoked heating mechanism, which in most cases is consistent with optical line ratios in H II regions in galaxies, is photo-ionization by hot young (OB) stars (e.g., Mathis 1986; Domgörgen & Mathis 1994), i.e., energy sources located in the galaxy disks.
Leitherer et al. (1995) showed that very little ionizing UV radiation escapes from the disks of starburst galaxies. Heckman et al. (1993) also noted that at least 90% of the UV photons are trapped, being absorbed by (gas and) dust and re-radiated in the FIR (Soifer et al. 1987a, b). This provides a natural explanation for why most of the energy leaving galaxies is emitted in the FIR (because of the different optical depths in the optical/UV vs. FIR regime).
The situation is somewhat different in the case of the DIG outside H II regions in galaxy disks. Some observed optical emission line ratios cannot be explained by pure photo-ionization from OB stars (e.g., Dettmar 1992). Only in the particular case of the irregular galaxy NGC 2188 has photo-ionization been shown to explain all data, including optical emission line ratios (Domgörgen & Dettmar 1997). Several attempts have been made to alleviate the problem of line ratios being inconsistent with photo-ionization, e.g., by invoking a hardening of the optical continuum impinging on the halo gas via differential absorption and/or scattering of radiation due to dust (Sokolowski 1994; Ferrara et al. 1996). In some cases this still leaves open the possibility that photo-ionization is the only heating source of halo gas. Domgörgen & Dettmar (1997), on the other hand, argue that the influence of dust scattering should in general be negligible. Nevertheless, in at least one case, NGC 4666, clear evidence exists that the halo gas is not purely photo-ionized (Section 8.4.3).
As a side effect, the radiation pressure of a strong stellar UV continuum might ``levitate'' dust and H I gas into the disk-halo interface (Franco et al. 1991). This is a mechanical force that was previously not considered in calculations of outflows.
8.4.2 EUV emission lines from cooling hot gas - ``turbulent mixing layers''
Slavin et al. (1993) have proposed another way to explain the observed optical line ratios in halos, when pure stellar photo-ionization is insufficient: self-photo-ionization through EUV line emission of collisionally excited gas in mixing layers (with a temperature T ~ 105.0 - 105.5 K) between hot (~ 106 K) and cold (~ 100 K) gas can reproduce many optical/UV/EUV emission line ratios as observed in the DIG of several galaxies. It is not yet known how important this mechanism might be.
8.4.3Shock heating
Chevalier & Clegg (1985) showed for the first time that shocks driven into the ISM by a supersonic wind emanating from a starburst (or - more generally - high-level SF regions) can contribute significantly to the energy balance of the ISM. Such galactic winds are driven by the winds and subsequent SNe of massive stars. Winds of massive stars blow cavities into the ambient ISM on timescales of a few times 106 years; later, these winds are overtaken by the remnants of supernovae of the most massive progenitor stars. Both expansions occur at velocities above the local speed of sound, vs, in the ISM, thus causing strong shocks.
A significant contribution of shock heating to the total energy balance of DIG can also alleviate earlier problems explaining observed optical emission line ratios (Shull & Draine 1987; Binette et al. 1985). Forbes & Ward (1993) argued that the correlation of radio synchrotron and Fe II 1.64 µm line emission they found in starbursts is a sign for a common origin of both kinds of radiation, namely from fast shocks associated with SNRs. This would imply that shock heating dominates in areas where these emission processes are prominent.
The most massive concentration of stellar clusters is found in circumnuclear starbursts. Super star clusters were detected e.g., by O'Connell et al. (1994, 1995) in NGC 1569, NGC 1705, and M82, and by Watson et al. (1996) in NGC 253. Rieke et al. (1980), Scalo (1990), and Rieke et al. (1993) found evidence for top-heavy IMFs in M82 and NGC 253, i.e., a larger number of high-mass stars than expected from a ``standard'' Salpeter IMF (Salpeter 1955). This implies that in starbursts the formation of high-mass stars might be favored over low-mass star production, due to higher masses of the collapsing cores of GMCs.
HAM90 found that shock models are in agreement with all their observations of superwinds, i.e., galactic-scale winds driven by starbursts - both kinematic data as well as optical line ratios. HAM90 derive several basic properties of galactic superwinds, e.g., the rate of energy input into the medium and the mass transport rate. They also identify potential sources of the observed thermal soft X-ray and optical line emission. More details on soft X-ray emission of galactic superwinds were determined by Suchkov et al. (1994), who show that both outflowing hot gas and gas heated by the shocks propagating through the ionized plasma can contribute significantly to the observed X-ray emission. It is particularly noteworthy that starbursts are capable of providing the total energy needed to drive such superwinds. This means that, although other processes might contribute to the total energy balance of the outflows, they are not necessarily required.
Such superwinds are a mechanism capable of transporting metals into intergalactic space and thereby heating and enriching the intergalactic medium. They are observed in all nearby edge-on starburst galaxies, including the most recently detected superwind in NGC 4666, where clear evidence was found for the dominance of shock heating of halo gas (Dahlem et al. 1997; their Fig. 9).
8.4.4 CR heating
Since shock heating is only important if the kinetic energy is efficiently thermalized, both photo-ionization and shocks can be considered as ``thermal'' heating sources of the ISM. An important additional, nonthermal source of energy input comes from the CRs created in SNe. The energy of these highly relativistic particles can heat the ISM through the resonant generation of magneto-hydrodynamic fluctuations (``waves'') and subsequent damping by ion-neutral collisions in the disk (Kulsrud & Pearce 1969) or non-linear Landau damping in the halo (Zirakashvili et al. 1996).
Through different processes (synchrotron and inverse Compton radiation, adiabatic expansion, diffusion, and - at high z - also convection), CRs can lose energy. The total energy released by CRs can be as high as 10% of the entire hydrodynamic energy released by SNRs (Breitschwerdt et al. 1991). Models describing how this energy can be converted into heat and thereby transfered to other phases of the ISM have been developed e.g., by Lerche & Schlickeiser (1982); Pohl & Schlickeiser (1990a, b), Hartquist & Morfill (1986), Parker (1992), Breitschwerdt et al. (1991, 1993). Brandenburg et al. (1993) investigated the role of turbulent dia-magnetism in such CR-driven winds in order to present a model for the vertical magnetic field component.
CR-driven winds can explain several observational results, e.g., the steepening and subsequent relative flatness of the nonthermal radio spectral index distribution away from the planes of galaxies like NGC 891 and NGC 4631, as well as the breaks in the total synchrotron power spectra (e.g., Hummel et al. 1991c).
8.4.5 Magnetic reconnection
Models of the Galactic halo and large-scale magnetic field are based on models originally developed for the Sun (e.g., Parker 1979). Scaling the heating rate and pressure from Solar values to those appropriate for the Galactic halo, Raymond (1992) found that magnetic reconnection, i.e., the annihilation of two magnetic field lines pointing in opposite directions which are compressed in the turbulent ISM, could contribute significantly to the heating of the halo. In particular, magnetic reconnection can explain the production of diffuse soft X-ray radiation, the strength of the line emission of several highly ionized species, and diffuse EUV line emission.
8.4.6 The role of turbulence
Although optical line ratios observed in H II regions are consistent with photo-ionization as the primary heating source, a significant contribution of another heating process can be deduced from the line widths, which imply that the material is supersonic. This indicates that turbulence and kinetic heating (via shocks) must play an important role also in H II regions. The importance of turbulence has been realized by Norman & Ferrara (1996), who base their new description of the general structure of the ISM on a turbulence scale-dependent phase continuum. They show that the turbulent pressure of interstellar gas can significantly exceed its thermal pressure.
8.4.7 Nature is not always simple
Currently, it is not entirely clear which of the above heating
mechanisms dominates under which conditions. As stated by
Sembach & Savage
(1992),
none of the above heating mechanisms alone can
be dominating the energy balance of interstellar matter. Rather,
it is much more likely that several different processes contribute
to the total energy budget of the ISM, but with varying importance
depending on the circumstances. Within H II regions, photo-ionization
models provide good results on the line ratios, but turbulence must
also be important. Outside H II regions, on the other hand, increased
[N II]/H
and [O III]/H
line ratios suggest that
the relative
importance of other heating processes, like e.g., turbulent mixing and/or
shocks, is greater compared to stellar photo-ionization. In addition to
shocks (HAM90),
CR heating and magnetic reconnection in turbulent plasmas
may play an important role in the heating of the HIM in galaxy halos
(Hartquist & Morfill
1986;
Parker 1992).
Galaxies like the Milky Way, with moderate star formation rates and therefore relatively little gas at temperatures above ca. 3 x 106 K (which is required to reach escape velocity), might have predominantly CR-heated halos. Starburst galaxies, on the other hand, produce enough thermal energy to create a sufficiently hot plasma and thereby have predominantly thermally driven winds. Whatever the driving mechanism - the disk ISM is clearly dynamical, undergoing changes which are coupled to variations of the underlying stellar population and the total energy input. The natural extension of the disk ISM into the halo via SF-driven outflows implies that also the halo ISM is dynamical (e.g., NI89; Breitschwerdt et al. 1993).
8.5
Energy scale and size scales
Both starburst and normal spiral galaxies can have gaseous halos. While all nearby edge-on starburst galaxies have gaseous halos (Dahlem et al. 1998), halos are in general a rare phenomenon among normal spirals (Hummel et al. 1991b). However, this might be a bias caused by insufficient sensitivity of the observations.
Table 1 lists all spiral galaxies (and one irregular) with known gaseous halos. This list still contains so few objects that each one of them can be presented here individually. The information in the previous chapters shows that the individuality of the galaxies also imposes a major problem on the interpretation of the results - it is still difficult to establish general rules or trends for gaseous halos.
Numerical simulations by
Rosen & Bregman
(1995)
corroborate
observations by DLG95,
Rand (1996), and
Meurer et al. (1997):
the critical parameter determining the nature of the ISM appears to be
the local energy injection rate (Edot) per unit surface area,
Edot / A. Low energy injection rates
(Edot / A 
0.1 x
10-3 ergs s-1 cm-2) lead
to a quiescent ISM, which is well-represented by the two-phase
model. Intermediate Edot/A (0.1 x 10-3 ergs
s-1 cm-2
Edot / A 0.5
x 10-3 ergs s-1
cm-2) best represents the Galactic ISM, which is well
approximated by the fountain model. The high Edot / A case
(Edot / A 
0.5 x 10-3 ergs s-1 cm-2)
describes an ISM as observed in galaxies like NGC 891, which
is most likely in the chimney mode. The highest Edot / A,
especially when released over a small surface area, lead to starbursts,
which can be approximated by a three-phase model of the ISM
(DLG95).
In this context see also Meurer et al.
(1995;
1997).
The nature of outflows from galaxy disks depends directly on the local level of SF in the area of the disk driving the outflow. In the fountain mode, disk-halo interactions take place, but v < vesc and the energy input over the disk is distributed over large areas. In the chimney mode, the energy injection into the ISM is more localized and the outflow velocity can surpass vesc, leading - in the most extreme case of starbursts - to superwinds with sizes of order 10 kpc. The difference between the fountain and chimney mode does not lie in the total amount of energy deposited into the ISM, but in the value of Edot / A, which is higher in the case of chimneys. The importance of shocks increases with the level of energy input. The highest spatial density of SF regions is found in starbursts and accordingly the influence of shocks appears to be strongest in superwinds (HAM90).
8.6
Environmental influences
Both internal and external processes can influence the SF history, including the distribution and the current level of SF, of a galaxy. Internal processes are, e.g.,, density waves, shocks, stochastic self-propagation of SF, and non-axisymmetric perturbations (by bars) inducing radial streaming motions.
External processes are related to interactions of galaxies with
their environment, i.e., either with other galaxies or possibly an
intergalactic medium. The nature and the amplitude of processes
triggered by such encounters depend on the impact parameter of
the interaction. Many starburst galaxies are interacting and it
has been proven statistically that starburst activity is enhanced
by galaxy interactions
(Hummel et al. 1990;
Lutz 1992,
and references
therein). Normal starbursts are mostly products of weak interactions,
while ultraluminous IR galaxies are most likely merging spirals.
In the event of distant passages (e.g., NGC 1808;
Dahlem et al. 1990),
the effects of tidal forces on the galaxy disks are relatively small,
leading to non-axisymmetric perturbations of the gas motions. These
perturbations in turn lead to radial gas flows and to
subsequent accretion of gas near the inner Lindblad resonance
(Combes 1987)
or near the galactic turnover of rotation (which
are located at almost the same galactocentric radius;
Lesch et al. 1990),
creating rings of cold gas and thus preferred locations
for (enhanced) SF (see also
Lesch & Harnett 1993).
In case of strong interactions, the disks of the galaxies undergoing the
encounter may be stripped of their gas either by tidal forces
or by ram pressure - if in a retrograde orbit - suppressing
SF, or - if on a prograde or polar orbit - accreting gas and
thereby enhancing SF (e.g.,
Sofue 1994).
Lehnert & Heckman
(1996b)
find that the H half-light
radius is correlated with the
location of the galactic turnover of rotation as well. The SFR per
unit surface area is enhanced within the turnover radius.
Meurer et al. (1997)
show that SF in starburst galaxies is probably
caused by cloud instability in the solid-body part of the disks.
They found that there is a global maximum of the surface brightness
of starbursts on a wide range of physical sizes. Due to this
constancy of the surface brightness, the maximum total luminosity
of a starburst depends on its physical size.
In order to minimize the number of free parameters, investigations of disk-halo interactions are best conducted in isolated field galaxies. In the case of interacting galaxies, it is difficult, if possible at all, to distinguish whether matter was expelled from a galaxy disk or dragged out by tidal forces. The H I gas observed outside NGC 3628 (Haynes et al. 1978; Wilding et al. 1993) or NGC 4631 (Weliachew et al. 1978; Combes 1978), for example, is probably dragged out of the disks. No relation has been found between the locations of the gaseous spurs and the local level of SF in those parts of the disks in which the filaments root. NGC 5775 is also interacting and at least the H I gas might be affected as well by tidal forces (Irwin 1994, 1995). The H I of M82 is also affected by its interaction with M81 (Yun et al. 1994). Only in cases of distant passages (see above; NGC 1808) can one assume with reasonable certainty that SF processes dominate the total energy balance of the ISM. A review on the environment of galaxies and its influence on these systems has been presented by Irwin (1995).
8.7
Temporal variations
Neither the total SFR, nor the local level of SF in different areas of a galaxy disk, nor the distribution of SF regions across the disk are constant over time. Galaxies may undergo cycles of SF (e.g., Richter & Rosa 1988), implying that galaxies like NGC 891 might currently be in a ``high'' state of their SF activity, while systems like NGC 4244 are considered to be in a (temporarily) quiescent phase. Accordingly, the existence and excitation state of gas in galaxy halos might also vary considerably over time. However, at present, the sample of galaxies with firmly established disk-halo interactions and the database on star formation cycles of galaxies are too small to draw any firm conclusions on this matter.