Magnetohydrodynamics of Cloud Collisions in a Multi-phase Interstellar Medium

We extend previous studies of the physics of interstellar cloud collisions by beginning investigation of the role of magnetic fields through 2D magnetohydrodynamic (MHD) numerical simulations. We study head-on collisions between equal mass, mildly supersonic diffuse clouds. We include a moderate magnetic field and two limiting field geometries, with the field lines parallel (aligned) and perpendicular (transverse) to the colliding cloud motion. We explore both adiabatic and radiative cases, as well as symmetric and asymmetric ones. We also compute collisions between clouds evolved through prior motion in the intercloud medium and compare with unevolved cases. We find that: In the (i) aligned case, adiabatic collisions, like their HD counterparts, are very disruptive, independent of the cloud symmetry. However, when radiative processes are taken into account, partial coalescence takes place even in the asymmetric case, unlike the HD calculations. In the (ii) transverse case, collisions between initially adjacent unevolved clouds are almost unaffected by magnetic fields. However, the interaction with the magnetized intercloud gas during the pre-collision evolution produces a region of very high magnetic energy in front of the cloud. In collisions between evolved clouds with transverse field geometry, this region acts like a ``bumper'', preventing direct contact between the clouds, and eventually reverses their motion. The ``elasticity'', defined as the ratio of the final to the initial kinetic energy of each cloud, is about 0.5-0.6 in the cases we considered. This behavior is found both in adiabatic and radiative cases.


Introduction
Our understanding of the physical processes of the interstellar medium (ISM) of the Galaxy (and of external ones) has progressed tremendously observationally and theoretically in the last decade. Part of the required effort has been stimulated by possible implications for galaxy formation in the early epochs of the universe, but it is clear that many aspects of the subject pose specific physics questions that are still unsolved and, therefore, interesting to study in their own right.
That the ISM of our galaxy should present a multi-phase structure has been put forward for more than three decades, in its various versions including a two-, three-(and even four-) phase medium. The concept of a number of thermal phases coexisting in pressure equilibrium has now been developed further by Norman & Ferrara (1996) who found that, once turbulence is taken into account, a generalization to a continuum of phases is required.
One of the aspects that has been recognized by essentially all the authors of the above mentioned studies as a crucial physical phenomenon in such a multi-phase environment is represented by "cloud collisions" (CCs). Of course, the term "cloud" might literally be appropriate only for an approach that is based on a somewhat simplified thermal characterization of the ISM, whereas there is growing evidence that the dynamics of the gas, either in ordered or random/turbulent form, could govern its large-scale distribution of the gas. Nevertheless, collisions among fluid elements, in general, should be quite frequent and common in the ISM, and without sticking to any particular global model, the first aim of the present series of papers (Ricotti, Ferrara & Miniati 1997, RFM;Vietri, Ferrara & Miniati 1997;Miniati et al. 1997, Paper I) is to clarify the physics of such events, with special emphasis on the fate of "clouds" (i.e., , gas clumps) and the dissipation of their kinetic energy.
So far only minor attention has been devoted to magnetized CCs . Some pioneering analytical MHD work can be found in the literature (Clifford & Elmegreen 1983) but numerical studies have been overwhelmingly limited to hydrodynamical calculations. This is surprising, since by now magnetic fields have been detected throughout our galaxy by a number of dedicated experiments. Observations (Spitzer 1978, Zeldovich, Ruzmaikin, & Sokoloff 1983 and references therein) suggest that its orientation is mainly parallel to the galactic plane and, according to some authors, it becomes toroidal at high latitudes (Gomez De Castro, Pudritz, & Bastien 1997). The magnetic field is further believed to consist of a mean systematic component and of a random one. The strength of both is found to be approximately a few µG. The evidence is provided by Faraday rotation, synchrotron radiation emitted by energetic electrons (cosmic rays), starlight polarization and Zeeman effect measurements (see e.g., Zeldovich, Ruzmaikin, & Sokoloff 1983, for more details).
Direct information about the magnetic field along the line of sight (B ) in galactic diffuse clouds, is obtained by observing the Zeeman splitting of the 21 cm radio line. With this technique the magnetic field strength is found to range on average between 3 and 12 µG both in H I and CO diffuse clouds . Also Heiles (1989) finds B ∼ 6.4 µG observing "morphologically distinct H I shells". Recent measurements carried out by Myers & Khersonsky (1995) have substantially improved our knowledge of the properties of magnetic fields in interstellar clouds. For the H I diffuse clouds in their sample, log x e is found to range between -2.7 and -4.9, x e being the electron fraction. The kinetic Reynolds number, Re (=vr/ν), and even the magnetic Reynolds number, Re M (=vℓ/ν M ) turn out to be large for v ∼ a few km sec −1 and r ∼ ℓ ∼ 1 pc. Those results validate our use of an ideal MHD code (see §3.1) for these simulations.
Large Reynolds numbers are very familiar in astrophysics and characterize non-viscous flows. In addition when Re M ≫ 1, the field lines are well coupled with the neutrals and the magnetic flux is frozen into the fluid. Finally the ambipolar diffusion time is given by (1-1) and therefore the field should not decay through this process, in the timescales relevant for the clouds under study. In particular this ambipolar diffusion should affect the clouds neither during their propagation through the ISM prior to, nor during the collisions.
Real structures in the ISM will have complex geometries, so any attempt to model specific interactions in detail will require 3D simulations. Yet, ours is the first explicit MHD study of CC, where the physical effects due to the presence of a magnetic field are being investigated. Because any 3D studies will certainly require interpretation of very complex patterns and behaviors, we anticipate those works with an explorative 2D study that should contain many of the same physical behaviors, and, being far simpler to understand, offers a practical basis for comparison. Several important and fundamentally different aspects of the physics of CCs are expected to come out of 3D calculations. One important example is the appearance of Kelvin-Helmholtz (KHI) and Rayleigh-Taylor (RTI) instabilities of the cloud surface along the cylinder axial direction, which are suppressed in 2D calculations.
This might be particularly relevant when the magnetic field is transverse to the cloud motion, because in this case instabilities on the plane perpendicular to the cylinder axis can be suppressed by the development of an intense magnetic field at the cloud nose . This limits the time over which the 2D flows are really representative. At the same time, an initial look to our preliminary result of 3D single cloud calculations (Gregori et al. 1998) reveal that part of the MHD structure relevant to CCs developed by three-dimensional clouds is qualitatively similar to that seen in 2D clouds. Even though other quantitative differences must occur (see §5), that result certainly supports the validity of our approach consisting of an initial explorative study of this yet uninvestigated problem.
Since our objective is the examination of explicit MHD effects in 2D cloud collisions, we try to follow as closely as possible the analogous HD simulations presented in Paper I. The plan of the paper is as follows. In Sec 2 we give some general considerations and introduce the relevant physical quantities of the problem; in Sec. 3 we briefly describe the numerical code and the experimental setup. Sec. 4 is devoted to the results, which are discussed and summarized in Sec 5.

Gas Dynamics
In this section we review some basic aspects of purely hydrodynamical CC (HD CC), a problem already studied in great detail by previous authors (e.g., Paper I, Klein et al. 1995 and references therein). The natural timescale for CC is given by which is approximately the time required for the shock generated by the collision to propagate across the cloud radius, R c . We suppose that clouds initially have a circular cross section. The main parameter determining the character of non self-gravitating HD CCs is (Klein et al. 1995) where N rad = n c v c τ rad is the radiative cooling column density through one cloud, n c is the cloud number density, and τ rad is the cooling time (Spitzer 1978). Combining Eqs. 2-1 and 2-2 we have η = τ rad /τ coll . Accordingly, if η ≤ 1 significant radiative cooling takes place during the collision; when η ≫ 1 emission processes become unimportant and the flow behaves adiabatically. In Paper I we concluded, after several low resolution tests, that the latter condition can be well represented by the weaker relation η > 1. Once the two phase model for the ISM is assumed, for a given cloud velocity these conditions imply that impacts between larger clouds are more influenced by radiative cooling (RFM, Paper I).
Since much of the basic physics of CCs has been explored using head-on events, which can provide a standard for comparison, in this paper we focus entirely on head-on CCs . In general, head-on symmetric HD CCs evolve through four main phases (Stone 1970a(Stone , 1970b; namely, compression, re-expansion, collapse and under some circumstances dispersal (Paper I). The occurrence of these four phases was first pointed out by Stone in his pioneering work (1970a, 1970b). It was also subsequently confirmed by high resolution hydrodynamic calculations (Klein, McKee & Woods 1995, Paper I), which allows for the highest resolution.
Similar results were also found through SPH calculations (e.g., Lattanzio et al. 1985, Lattanzio & Henriksen 1988. However, the limitations of these calculations, both in resolution and in the SPH method of simulation, have led to some misinterpretations of the physics of CC, like the hypothesis of "isothermality", as discussed in Paper I. To a certain extent the same depiction of four phases can be drawn for magnetohydrodynamical CCs (MHD CCs) as well, although as we shall see, there are some important differences.
However, we adopt this general terminology as a useful tool to refer to the various stages of the CCs .

Cloud Propagation through a Magnetized ISM
The HD of a dense cloud moving into a low density medium has been central to the study of several authors (Jones et al. 1994Schiano, Christiansen & Knerr 1995;Murray et al. 1995;Vietri et al. 1997;Malagoli, Bodo & Rosner 1996) who generally concentrated on the growth of KHI and RTI in such conditions. We refer the interested reader to these works for an exhaustive description of this topic. When considering the motion of a cloud through a magnetized medium, new parameters, in addition to those introduced so far ( §2.1), must be considered. In particular, the initial magnetic field is completely defined by its strength and orientation. The former parameter is usually expressed in terms of where p g and p B = B 2 /8π are the gas and magnetic pressure, and M = v c /c s and are the sonic and Alfvénic Mach number respectively. The magnetic field orientation is, in general, determined by two angles; in 2D simulations, with the field lying in the computational plane, they reduce to θ, the angle between the cloud velocity and the field lines. As long as the unperturbed magnetic field is dynamically unimportant (β 0 ≫ 1) the initial evolution of a MHD cloud is similar to a HD one. This is the case we consider below, where we adopt β = 4. So, in the presence of a weak field, a stationary bow shock develops on a timescale τ bs ∼ 2τ coll . Further, a "crushing" shock is generated and propagates through the cloud with relative speed v cs ≃ v c /χ 1/2 , on a timescale (crushing time) where χ = ρ c /ρ i is the ratio of the cloud and intercloud medium densities. Finally a low pressure region (wake) forms at the rear of the cloud and, interacting with the converging flow reflected off the symmetry axis (X-axis), generates a relatively strong tail shock.
However, over time, new features develop in response to the magnetic field. , in a 2D study of individual MHD supersonic clouds, identified several of these for an adiabatic, high Mach number cloud with modest density contrast, χ = 10. Mac Low et al. (1994) also studied the MHD evolution of 2D individual, shocked clouds, which behave in a qualitatively similar manner. Utilizing these works, in the next paragraphs we give a brief review of the main features that result from inclusion of a magnetic field during the interaction with the intercloud medium.
First, the orientation of the magnetic field with respect to the direction of the cloud motion is particularly important. So far only two extreme cases have been published; namely, cloud motion parallel (aligned) or perpendicular (transverse) to the initially uniform field. We present elsewhere calculations with oblique magnetic fields . In the aligned case, the field lines, following the flow, are swept over the cloud.
As a result, those lines anchored at the cloud nose, are pulled, stretched and folded around the cloud. Eventually, these lines experience magnetic reconnection, forming new flux tubes passing around the cloud, somewhat like streamlines in a smooth flow. In this region the magnetic field never becomes dynamically dominant, although its realignment around the cloud contributes to smoothing and, therefore, stabilizing the flow. On the other hand, field lines are drawn into the cloud wake. Flow in the wake stretches lines anchored in the cloud material, causing the intensity of the magnetic field to increase. This feature, referred to as the post-cloud "flux rope" (Mac Low et al. 1994, is the only one that becomes magnetically dominated (β ∼ > 0.1) for an aligned geometry. A similar wake region forms also in the transverse field case, where the field lines drape over the cloud, converging in its wake. In this case, however, those field lines are anti-parallel across the symmetry axis.
Above and below the symmetry axis the field structures in the wake initially are relatively uniform with a very sharp transition between them corresponding to a thin current sheet.
Classically such a thin current sheet is unstable to the resistive, "tearing-mode" instability (e.g., Biskamp 1993, p. 73), in which the current sheet breaks into line currents and the magnetic field reconnects across the sheet. The instability condition is that the thickness of the sheet is much smaller than its width (e.g., Biskamp 1993, p. 152 for details). Indeed, we see these sheet transitions break up into a series of closed field loops that are the signature of this instability (e.g., Melrose 1986, p 151). This rapid modification of the magnetic field topology is often called "tearing-mode reconnection" (e.g., Melrose 1986) and typifies the reconnection that occurs in our simulations. A very clear illustration of the evolution of one such example is shown in Miniati et al. (1998, Figure 3). Because of this behavior the magnetic field intensities in the transverse case wakes are lower than in the aligned case. On the other hand, the field lines in front of the cloud are compressed and, more importantly, stretched around the cloud nose. In this region, unlike the previous aligned field case, reconnection does not occur for these field lines. Therefore, the magnetic field becomes very intense (10 −2 ≤ β ≤ 10 −1 ) on the cloud nose, forming a magnetic shield. As pointed out by Jones et al. (1996), the main reason of magnetic energy enhancement is the stretching of the lines as these are swept up by the cloud. The timescale for the growth of the magnetic energy is given in eq. 9 of Jones et al. (1996), in the spirit of a first order approximation quantity, as τ ∼ v c R −1 c = τ coll . Soon however, nonlinear effects become important and a more realistic timescale, as long as 2D approximation is valid, is provided by Further details on the development of the magnetic shield and on its dependence on cloud characteristics can be found in . It is important to notice that the magnetic shield acts to prevent the growth of KHIs and RTIs on the cloud surface.
When radiative losses are included, as in the HD calculations, the thermal energy of the compressed gas is lost, reducing its pressure and allowing the cloud material to be compressed to very large densities. Particularly in the transverse field case, the cloud aspect ratio (length, x, to height, y) is highly increased by this effect.

The Code
Our CCs simulations have been performed using an ideal MHD code, based on a second order accurate, conservative, explicit TVD method. Details of the MHD code are described in  and ; a brief description of some aspects concerning the inclusion of cooling and of a mass tracer is given in Appendix A. The ∇ · B = 0 condition is maintained during the simulations by a scheme similar to the Constrained Transport (CT) scheme (Evans & Hawley 1988;Dai & Woodward 1997), which is reported in Ryu et al. (1998). We have used the 2D, Cartesian version of the code. The computational domain is on the xy plane, and the Z-components of velocity and magnetic field have been set to zero.

Grid, Boundary Conditions and Tests
In each CC simulation only the plane y ≥ 0 is included in the computational box, and reflection symmetry is assumed across the X-axis. The length scale is chosen for each case so that R c = 1.0 and the computational domain is adjusted to minimize boundary influences, as listed in Table 1 Table 1) and is reflective (as the bottom boundary) otherwise, when there is a mirror symmetry to the collision. With this latter choice we are allowed to use only half of the grid and reduce the computational time of the calculation. Only the highest resolution calculations, characterized by 50 zones across the initial cloud radius, are presented here. Lower resolution (25 zones per initial cloud radius) tests were also performed in order to check for consistent behavior. It turns out that beside inevitable quantitative differences, for each case we studied there is absolute consistency in the CC outcome. In particular it is apparent the qualitative agreement between low and high resolution calculations, in the density distribution and magnetic field structures that form out of the CC.

Initial Conditions
In this section we will discuss the initial conditions for our CCs. It is worth pointing out from the very beginning that, following the results of Paper I, most of the simulated CCs involved evolved clouds, i.e., clouds that have propagated through the ISM for about τ cr before colliding. Initially, individual clouds have a circular cross section and uniform density and are in pressure equilibrium with a uniform background medium; the magnetic field is also assumed uniform throughout the domain. The relevant parameters, whose numerical values are given below, are the density contrast χ, the Mach number M and the cloud radius R c . The exact thermodynamic quantities characterizing the initial equilibrium state are not particularly important as their memory is lost soon after the beginning of the cloud evolution: for a supersonic motion, the thermal pressure of the shocked gas and the ram pressure of the flow are dynamically far more important than the initial pressure balance. In addition, as already pointed out in §2.2, the cloud motion through the intercloud medium produces a variety of features which strongly alter the initial configuration. As a result, despite the simplicity of the conditions at the onset of the cloud motion, before the collision takes place both the gas and the magnetic field are characterized by a rich structure. A comparison with the collision of two unevolved clouds is provided in the results section ( §4.2.1), to show the importance of considering prior cloud evolution in this study.
The initial values of the above parameters are the same as in Paper I and are listed below. These values are inspired by the most recent observational studies and thought to be representative of the magnetized ISM. We assume a specific heat ratio γ = 5/3 throughout. The inter-cloud medium has a density n i = 0.22 cm −3 and temperature T i = 7400 K. Clouds are characterized by a density contrast χ = n c /n i =100, so that the cloud density and temperature are n c = 22 cm −3 and T c = 74 K respectively and τ rad ≈ 3.7 × 10 4 yr. The sound speed in the inter-cloud medium turns out c si ≈ 10 km s −1 and the equilibrium thermal pressure for the ISM p eq /k B = 1628 K cm −3 . When initially set in motion parallel to the X-axis, each cloud has a Mach number M = v c /c si = 1.5, and therefore v c ≈ 15 km s −1 . Setting R c = 0.4 pc, yields τ coll = R c /v c ≈ 2.6 × 10 4 yr. Since this implyes η = τ rad /τ coll ≈ 1.4 > 1, in accord with §2.1 these collisions behave adiabatically and therefore the cooling can be turned off. In the following collisions involving these clouds are referred to as adiabatic cases. On the other hand, setting R c = 1.5 pc we have: τ coll ≈ 9.7 × 10 4 yr and therefore η ≈ 0.38. These are the radiative cases. Table 2 summarizes these cloud characteristics.
With a magnetic field included, three new parameters with respect to the HD case have to be determined: the field orientation and strength. The orientation is determined in 2D by a single parameter, namely the angle between the initial cloud velocity and magnetic field. We explore two cases: magnetic field parallel (aligned case) and perpendicular (transverse case) to the cloud velocity. For the initial strength of the magnetic field, conveniently expressed by the parameter β 0 (see Eq. 2-3), we assume β 0 = 4, corresponding to B = 1.2 µG. It could be argued that this value is somewhat smaller than what is usually observed. However, during the following cloud evolution, the field is stretched and amplified and in several regions becomes energetically dominant (β ≪ 1 and B > 1µG, §2.2). In the resulting configuration, therefore, the magnetic field influence is not highly sensitive to this choice.
Initially, the Jeans length of our typical diffuse cloud is λ j ≈ 29 pc≫ R c . Even though large density enhancements are produced during the compression phase in symmetric radiative collisions, λ j never becomes smaller than the vertical size of the clouds. For this reason we have neglected self-gravity throughout our calculations (see also Klein, McKee, & Woods 1995). Since we have concentrated on diffuse clouds, as opposed to molecular complexes, this approximation is justified.
Similarly to Paper I, we consider collisions between both radiative and adiabatic clouds. However, for brevity, we discuss here only the most significant new results for the two different cases. Table 1 summarizes the parameters of the collisions discussed below.
For Cases 1 and 2 in Table 1, the cloud begins its independent evolution at t = − 3 / 4 τ cr , where t = 0 corresponds to the instant when the bow shock of the two clouds first touch.
Analogously, for Cases 3 and 4 the two clouds are placed on the grid at t = 0 with their bow shock next to each other, after evolving for 1 / 2 τ cr and τ cr respectively. In these four cases the initial magnetic field was aligned with the cloud motion, whereas for Cases 5-7 it was transverse. Case 5 is the only non-evolved calculation we present: by that we mean a uniform cloud of circular cross section placed on the grid in such a way that its boundary is only 2 zones from the (reflecting) Y-axis at t = 0.0. Finally in Cases 6 and 7 the cloud starts its evolution at t = − 3 / 4 τ cr and then is treated as in Cases 1 and 2 respectively.
Animations of each simulation have been posted on our World Wide Web site at the University of Minnesota.

Aligned Field (BX, Cases 1-4)
In the aligned field case MHD CC show many similarities with HD CC. These are illustrated in Figure 1, where the evolution of various MHD cloud integral properties are plotted as a function of time. The curves of kinetic and thermal energy as well as Y cm closely resemble those for HD clouds in Figures 8 and 9 of Paper I and the occurrence of all the phases characteristic of a HD CC (see §2.1 and Paper I) is clearly seen. In general a stronger compression is generated when the clouds have mirror symmetry across the impact plane and only a weak reexpansion takes place in radiative cases, because most of the thermal energy is radiated away. The top right panel of Figure 1 displays the total magnetic energy. Since its variations are related to compression and/or stretching of the field lines, this quantity gives an approximate measure of the overall interaction between the magnetic field and the gas. In the adiabatic cases (solid and dotted lines) the large expansion undergone by the cloud gas produces both significant stretching and compression of the field lines. As a result, during the re-expansion phase the total magnetic energy increases of about 30%. On the other hand, in all radiative cases the total magnetic energy suffers only slight variations.
We now begin specific comparison, considering the adiabatic collision of two evolved identical clouds with aligned fields.

Symmetric Cases
The evolution of Case 1 is reported in Figs During the reexpansion phase a thin shell of dense (4ρ i ≤ ρ ≤ 10ρ i ) cloud gas forms behind the reverse shock of the expanding material. A long finger appears on the X-axis, due to the fact that the reexpansion finds an easy way through the flux rope, where the density and the pressure are quite low (Mac Low et al. 1994. Also, in contrast to the HD case, the shell boundaries in Figure 2a (bottom left) are quite sharp.
This difference has an important physical base. In fact, although the initial magnetic field has not sufficient strength to inhibit the onset of KHI (Chandrasekhar 1961), nevertheless it is able to reduce and eventually stop its growth. In fact, as the magnetic field lines, frozen in the gas, get stretched in the turbulent flow, their strength is increased until during eventual reconnection they redesign the flow pattern to a more stable configuration. The criterion for this field dominance is that the local alfvenic Mach number falls to order unity or less (Chandrasekhar 1961, Frank et al. 1996. This is evinced by the presence of several field line loops on the external side of the gas shell (lower left panel in Figure 2a). Inside the shell the magnetic field intensity has severely dropped. According to Figure 5 the magnetic energy density (dotted line) has been reduced with respect to its initial value (solid line) by a factor ranging from 10 2 to 10 4 . This cannot be accounted for by expansion alone; indeed complex reconnection processes at the beginning of the reexpansion phase are responsible as well.
The collapse phase (at t ∼ 18.75τ coll ) is chaotic and turbulent: in and around the cloud gas the density distribution is rather clumpy and the magnetic field has a tangled structure (top panels of Figure 2b and dash lines in Figs. 3 and 5). This situation will persist until the end of the simulation (dot-dash lines in Figs. 3 and 5). It is during this phase that the magnetic field produces a qualitative change in the evolution of the CC. In Paper I we showed that during the HD collapse phase the reverse shock propagates toward the inner region of the expanded cloud gas, whereas the external layer is shredded by KHIs and RTIs.
At the end numerous filaments fill a region of about the same size and shape of the shell at its maximum extent. In the present case, on the other hand, the magnetic field lines which have been stretched by the vertical expansion of the gas shell, begin to relax, accelerating the gas at the top of the shell (near the Y-axis) toward the X-axis (top panels of Figure   2b). At the end of the collision (t = 75τ coll ) the initial cloud material is confined in a layer beneath the relaxed magnetic field lines, with a mean density ρ ∼ 4ρ i (bottom panels of Figure 2b) and is still laterally expanding. Within this new structure, with a thickness of several initial cloud radii, we find a clumpy density distribution (dot-dash lines in Figure 3) and a weak, irregular magnetic field (dot-dash lines in Figure 5). Outside it, on the other hand, the magnetic field has the same initial configuration but greater strength, often twice as much as its initial value (dot-dash line in bottom panel of Figure 5).
The analogous radiative case (Case 2) is illustrated in Figure 6 and its quantitative properties plotted in Figure 7. Again, an aligned magnetic field does not seem to influence the collision. As in the HD case efficient radiative cooling allows a much higher gas compression (solid line in top panel of Figure 7) and inhibits the strong reexpansion that would be driven by the high pressure of the shocked gas (solid line in bottom panel of Figure 7). During these phases the clouds coalesce, generating a well defined high density round structure about twice as large as the initial single cloud (this can be inferred from the position of the sudden drop in the dot line in top panel of Figure 7). No collapse phase ever happens. In addition a narrow jet of gas, characteristic also of radiative HD CC, is formed and extends along the Y-axis. As it propagates transverse to magnetic field lines a high density spot is created at the leading edge. However those features are usually unimportant because they only involve a negligible fraction of the total mass. Of more interest is instead the final fate of the mentioned structure. At the end of our simulation (t = 30τ coll ) its edge near the X-axis is still expanding along the X-axis with a speed v ∼ 0.1c si . According to the three density cuts in the top panel of Figure 7 the cloud edge (located at the sharp drop in each line) has been expanding at roughly the same speed (∼ 0.1c si ) throughout the evolution. Based on this velocity we can, therefore, estimate the time τ α for the density of the new structure to drop by a factor α. Assuming that only one dimension of the cloud volume increases (at the speed of 0.1 c si ) as a result of its expansion, we have τ α ≃ αR c /(0.1c si ) = α10Mτ coll = α 1.5 × 10 6 yr (we used τ coll ∼ 10 5 yr from §3.3). Since inside the new structure ρ ∼ 40 − 80, α can be as large as 5-10, and τ α comparable with the time between two cloud collisions.

Asymmetric Cases
As in Paper I we produce a simple asymmetric CC by colliding clouds with the same initial mass, velocity and radius, but evolved individually for a different time interval before they collide. By contrasting with mirror symmetric cases we can begin to see properties that are symmetry dependent. In the following cases (Case 3 and 4) the two clouds have been evolved for about 1 / 2 τ cr and for τ cr respectively ( Table 1).
The evolution of the adiabatic asymmetric CC resembles the analogous symmetric Case 1. Figure 8 shows the field lines (top panel) and the density distribution (bottom panel) at the end of the re-expansion phase (t = 22.5τ coll ). We can identify a relatively dense shell (ρ ∼ 5ρ i ) with a large clump of gas on top of it as well as loops of magnetic field lines generated by reconnection events. Some new features appear, however, as a result of the broken symmetry. An example is the long tail of cloud gas on the right hand side of Figure 8, due to the unbalanced momentum distributions in the two clouds along the X-axis. Nevertheless they are not so significant as to alter the general character of this CC with respect to Case 1. Therefore we expect that the collapse phase and the remaining following evolution of this case will not differ qualitatively from that of Case 1. We also point out that as long as the asymmetry does not prevent the development of a reexpansion phase, Case 1 can be considered as qualitatively well representative of adiabatic MHD CC with magnetic field aligned to the initial cloud motion.
The radiative case (Case 4), shown in Figs. 9a, 9b, presents new and interesting insights. The most crucial part of the evolution is represented by the cloud interaction during the compression phase. During that phase the older and more compact cloud (C2), moving from the right, attempts to plow through the other cloud (C1) (Figure 9a t = 6.75τ coll ). Some of the features that are visible in Case 2, such as the vertical jet of gas, can also be recognize here if one carefully accounts for the distortions due to the asymmetry.
However, we give particular attention to the new feature on the X-axis toward the left of Figure 9a (X≃ −5R c ). This is a compact clump with density about 5 × 10 2 ρ i and velocity along the X-axis, v x ∼ − 1 / 2 c si . At the end of the simulation (t = 22.5τ coll ) it has expanded and is still moving toward the left along the X-axis (Figure 9b). At this time the "mass tracer" variables allow us to conclude that, despite the large prevalence of gas from the more compact cloud (C2), the new clump is the result of a partial coalescence of the two initial clouds. Its mass is about 10% larger than the initial mass of either cloud. Its density varies between 15ρ i and 100ρ i and the velocity pattern suggests that it is expanding along the X-axis with v ∼ c si /6. By the same argument at the end of §4.1.1 we can, therefore, conclude that for this case τ α ≃ α 9 × 10 5 yr. Again, before the cloud disperses in the background medium, the newly-formed clump is likely to undergo another CC. The remainder of C1's gas (feature to the right), moving transversely to the magnetic field, has formed a long filamentary structure and has created a sharp cusp in the field lines ( Figure   9b).
This result strongly differs from the analogous purely HD calculation, where we found that the collision produced a large, low-density-contrast filamentary structure eventually fading into the background gas.

Transverse Field (BY, Cases 5-8)
We start this section by presenting the case of a CC between two unevolved clouds (Case 5). This calculation is mostly intended to provide a reference case when studying evolved CCs and, thus, to emphasize the importance of initial conditions in calculations of this type.

Unevolved Adiabatic Collision
The results of this calculation (Case 5) are shown in Figure 10, where two density images are superposed on field lines. The left panel captures the reexpansion phase at t = 7.5τ coll ; it closely resembles the analogous HD Case 1 of Paper I and no significant difference from Case 1 (of this paper), where the magnetic field was aligned to the motion, can be pointed out. This all means that no major role is played by the magnetic field, as it has been the case so far for all adiabatic CCs. The shell density has typical values (∼ 10ρ i ) and the thermal pressure drives the reexpansion. RTIs develop on the shell surface near the Y-axis. However, at the end of the collision a thick layer, qualitatively similar to that formed in Case 1 and 3 forms along the Y-axis. We also point out that the evolution of the total magnetic energy (dash line in bottom left panel) is reliable only up to t ∼ 9τ coll , which marks the exit of the forward blast shock from the right boundary of the grid, along with consequent outflow of magnetic energy. The right panel image is from t = 30τ coll . RTIs, aided by the formation of a slow, switch-off shock, which aligns the field with the expansion of the shell, have formed a large finger expanding almost parallel to the Y-axis.

Evolved Symmetric Cases
As already mentioned in §2.2 and discussed by Jones et al. (1996), an individual cloud moving transverse to a magnetized intercloud medium develops a region of strong magnetic field known as the magnetic shield. That feature dominates the collisions of such evolved clouds from the start of their encounter. Consequently, magnetic field effects dominate the evolution in Cases 6 and 7, in striking contrast to Case 5. Consequently, it is clear that meaningful simulations of CCs may depend on understanding the field geometry in their surroundings and on allowing self-consistent magnetic structures to evolve before collisions take place.
Case 6 is shown in Figs. 12a and 12b, both including field line geometry and density images superposed on the velocity field. In addition, the solid lines in Figure 11 show the usual time evolution of integral quantities. Peaks in the thermal energy and corresponding valleys in the Y cm curve, are signatures of the bow shock precompression and the "collision" undergone by the cloud . Note the simultaneous kinetic energy decrease and the magnetic energy increase, primarily due to cloud interaction with the magnetic field  at the beginning of the simulation and to the "collision" event later on.
As the cloud approaches the collision plane (Y-axis) in Figure 12a (bottom panel), the field lines are highly compressed (top panel) generating a strong repulsive force, opposite to the cloud motion. Eventually all the cloud kinetic energy is converted into magnetic form and stored as magnetic pressure. Then the cloud stops and, as the field lines reexpand, is accelerated backward and its motion reversed (Figure 12b). Therefore we can state that the magnetic shield acts almost like an elastic magnetic bumper. At the "apex" of this reversal phase both thermal and magnetic energy peak, whereas the kinetic energy obviously is about null (Figure 11). The latter, however, at the end of the simulation returns to about 60% its initial value ( Figure 11) and the cloud velocity is v x ∼ c si . Also thermal and magnetic energy are back to their initial values. Therefore the only effect of the collision is to dissipate part of the cloud kinetic energy with no other major consequences.
The same qualitative result is also obtained in the radiative Case 7, shown in Figs. 13a and 13b and in Figure 11 (dot line). Whereas the kinetic and magnetic energy evolve as in Case 6, the thermal energy and Y cm parameter have now similar qualitative behavior but much lower values (the scale for the dotted lines in the two bottom panels of Figure 11, is 10 times smaller than for the solid ones); thus, the evolved cloud is mostly supported by magnetic pressure. Figs. 13a and 13b show the magnetic bumper effect for the radiative case as well. The final kinetic energy is about the same as in the previous case (50% of the initial value). However, the cloud gas has extensively spread out, especially in the X-direction, creating a dense elongated structure. Because of the high density reached by the cloud during the reversal phase, the resulting cloud shape is probably limited by the diffusivity of the code. Nevertheless, the final outcome is different from the analogous HD and previous aligned field cases and is more similar to Case 6. The cloud neither disperses nor coalesces; its structure is, however, strongly distorted and its kinetic energy partly conserved.

Summary & Discussion
We have investigated the role of the magnetic field in CCs, through high resolution, fully MHD 2D numerical simulations. This paper is an extension of the gasdynamical study presented in Paper I. Our aim is to provide a first step toward the understanding of the physical role of magnetic fields in interstellar diffuse CCs . In particular, we have studied magnetic influences on: (i) the final fate of the cloud after the collision (i.e., dispersal, coagulation, shattering, filamentation); (ii) the evolution of cloud kinetic energy and (iii) the effects of CCs on the magnetic field structure in and around the clouds. These simulations -25represent only an initial attempt to study a complex problem. To identify the most obvious and simplest behaviors we have restricted the geometrical freedom of the flow to 2D. For all behaviors, but particularly for (iii), our results need to be confirmed by more thorough and extended 3D calculations. The main results can be summarized as follows: • Adiabatic, aligned field CCs are disruptive (as in the HD case), both for symmetric and asymmetric events. The remnant consists of an elongated structure of low magnetic energy in which cloud and background gas are mixed together.
• Addition of an aligned field to radiative, symmetric CCs does not change the fact that they dissipate most of the cloud kinetic energy, which leads to almost complete coalescence of the two clouds. During asymmetric collisions of this type coagulation takes place as well, but the final structure has a mass only slightly (∼ 10% for Case 4) larger than either cloud initial mass; little alteration of the magnetic field line pattern is seen. This result is important, since purely HD asymmetric collisions of diffuse clouds have been shown to be highly disruptive (Paper I, Klein et al. 1995 and references therein).
• In 2D motion of clouds moving transverse to the magnetic field leads to the formation of a magnetic shield in front of each cloud. When two evolved clouds of that kind run into each other, a magnetic shield may prevent direct collision from taking place. In our simulations the clouds remain separated by a magnetic barrier and bounce back with a fraction ǫ of the initial kinetic energy. According to our results ǫ ∼ 0.5 − 0.6 for both the adiabatic and radiative cases. This is probably an upper limit in more realistic situations, including 3D, and especially off-axis collisions, since the magnetic bumper may be less developed and other degrees of freedom (e.g., rotation) are available.In addition, if a third dimension were included, after the collision of the clouds the compressed magnetic shield would partially reexpand perpendicular to the direction of the initial motion, thus reducing ǫ value further.
Despite the caveats mentioned, much of the character represented in the last bullet may be independent of the symmetry of the collision. This was tested in part through a low resolution 2D numerical experiment of an asymmetric head-on, transverse-field collision.
Results were consistent with those cited. In general, the magnetic shield is expected to work at some level for off-axis and largely asymmetric cases; transfer of momentum and angle scattering would occur in a similar way as in head-on collisions. When the magnetic field is aligned to the cloud motion, some of the arguments discussed in Paper I for off-axis HD CC should apply here as well. In particular off-axis CCs with an impact parameter b ≪ R c should be well represented by our asymmetric cases. Also for adiabatic cases, we expect that even for b ∼ > R c , CCs should produce a reexpansion that is strong enough to disperse the clouds. The asymmetric radiative case results, however, caution from extending the HD results to radiative MHD CC when b is comparable to R c . Those cases must be investigated in the future. Finally for b ∼ 2R c the collision should produce only minor perturbation to the clouds.
A striking difference exists between the outcome of CCs when aligned and transverse field geometry are considered. Therefore, in order to model correctly CCs in the ISM it is important to understand the conditions for the formation of the magnetic bumper. Two points are crucially important in this regard: (i) the assumed cloud shape and (ii) the initial configuration.
As for the former, no 3D MHD individual supersonic cloud numerical simulation has been published so far. Especially near the nose of the clouds we should expect to see some kind if magnetic shield develop. On the other hand divergence of the flow away from the nose and transverse to the prevailing field orientation should advect field lines away from the nose, thus limiting its development and extent. The importance of that effect will depend on the geometry of the cloud. A "pointed cloud" will have only a very limited shield. For example, Koide et al. (1996), studying the propagation of extragalactic jets through a medium with an oblique magnetic field, find that the field lines distort to let the jet pass through. This effect reduces the strength of the magnetic shield and could be important for spherical clouds. However, for cylindrical or filamentary clouds, with axes transverse to the motion, field lines may be trapped at the front of the cloud long enough to play a major dynamical role. We point out that elongated clouds are not simply convenient to our 2D approximation. Rather, elongated shapes are expected for clouds that form in a magnetized environment, where the support provided by the magnetic pressure is anisotropic (e.g., Spitzer 1978). As already pointed out by Jones et al. (1996), realistic clouds show strong shape irregularities where the field lines can penetrate, be captured and, therefore, stretched to form some sort of magnetic bumper. But even if not so, when a cylindrical cloud moves in a transverse magnetic field, the motion of the gas along the cloud major axis, away from the stagnation point, is certainly slower at the center of the cloud where the stagnation point is located, than near the sides. Since the field lines are frozen into the gas, the central region of the cylindrical cloud is where the field lines are held longest. Therefore, an uneven magnetic tension is applied on the cloud and a bending of the latter is produced in the central region. As a result it becomes more difficult for the field lines to slip by the cloud enhancing the deformation of the cloud and increasing trapping of the field lines.
Finally we have just begun a set of preliminary, low resolution numerical calculations to be presented in a subsequent paper (Gregori et al. 1998). From those calculations we can anticipate that a magnetic shield always forms and in a fashion qualitatively similar to that seen in 2D cases. Moreover, as expected, initially elongated clouds develop a stronger magnetic shield than "spherical" clouds. However, in the latter case the tension of the magnetic field lines wrapped around the cloud produces a strong deformation of the cloud shape, which grows strongly elongated transverse to the plane containing the field and the motion thus facilitating the formation of the magnetic shield.
On the other hand the effect of the initial configuration on individual 2D cloud evolution has been investigated by Miniati et al. (1998) who study the influence of the initial field orientation with respect to the cloud motion (θ), of the cloud density contrast (χ) and velocity (M) on the magnetic bumper formation. They find that, as long as 2D approximation is valid and θ ∼ > 30 • , the timescale for the formation of the magnetic bumper is of the order of τ ∼ (βχ) 2/3 M 4/3 τ coll . Since clouds are slowed down by the ram pressure of the impinging flux on a timescale τ de ∼ χτ coll , they also concluded that magnetic bumpers are more likely to develop around high density contrast, low Mach number clouds.
Another important 3D issue is the reexpansion of cloud gas in the direction perpendicular to the computational plane (along the Z-axis). This effect is important because it could in principle modify our previous conclusions for the non disruptive cases.
However, as it turns out this only sets a limit on the length of the cloud major axis for the adiabatic Case 6. In radiative cases lateral reexpansion involves only a small fraction of cloud mass, independent of the Y or Z direction (see §4.1.1). On the other hand, in the adiabatic case, the rarefaction wave generating such reexpansion propagates at the sound speed of the postshock gas, whose temperature has been enhanced by a factor ∼ M 2 . Therefore, v exp ∼ Mc si . If the length of the cylinder is ℓ z = µR c , the reexpansion occurs on a timescale τ exp ≃ ℓ z /v exp ∼ µR c /(Mc si ) = µτ coll . Consequently, since τ rad /τ exp = τ rad /µτ coll = η/µ, then if µ ≫ 1 (ℓ z ≫ R c for cylindrical clouds), the cloud behaves as it did in 2D radiative cases discussed before.
The aforementioned reasons justify 2D simulations as a valuable starting point for the more complex 3D MHD CCs. As already mentioned along the paper, however, in the latter we expect to observe new behaviors not seen in 2D calculations. We expect those to -29be mostly related to differences in the evolution of the magnetic field. Since the latter is dynamically dominant in transverse field cases, new behaviors will be more apparent there.
For example, even when the magnetic shield forms in 3D as well, if its strength is much less than in 2D, then we may expect different results from those reported in this paper (more precisely in §4.2). Nevertheless, instabilities and, in general, flows along the third direction, as well as quantitative differences in cloud features developed in 2D and 3D, will certainly affect the dynamics of the collision. We have begun a series of 3D MHD cloud simulations to address these complex issues more fully.

Appendix A
We discuss here some details of the treatment of the radiative losses and of the mass tracing routine. Radiative losses have been taken into account using the same approach as in Paper I, to which we refer for a detailed description. The radiative correction that we have applied is quasi-second order accurate (see, e.g., LeVeque 1997 for a general where D t and D s are the temporal operator and the spatial plus source operator respectively and q is the set of variables describing our system. A second order accurate splitting from time step n to n + 2 for the operator D s would be (Strang 1968): where L x and L y are the differential operator with respect to the X-coordinate and Y-coordinate respectively and S is the operator representing general source terms. The subscript 1 2 means that the operator is applied for only half of a time step. Instead, we have used which assumes that S 1 2 S 1 2 ≡ S. Hence the "quasi-second" order description. In addition we have suppressed cooling inside the shock thickness. Indeed, in the physical shock layer the flow should be non-radiative, because the crossing time of the real shock thickness (artificially spread out by the code) is much shorter than the cooling time. Also, since density and pressure are not accurate inside the shock, but only adjacent to it, radiative cooling could become artificially large, reducing the performance of the code. This turns out of particular importance for the MHD calculation. The radiative cooling function we have used is identical to that in Paper I and is fully described in Ferrara & Field (1994).
It includes cooling due to free-free emission, recombination lines, and collisional excitation lines as well as heating terms provided by collisional ionization and ionization by cosmic rays. For the cosmic ray ionization rate we adopt the value 2 × 10 −17 s −1 as determined from observations by van Dishoeck & Black (1986). This and other rates can be found in Ferrara & Field (1994) and references therein. We neglect dust, and particularly PAHs, photoelectric heating. This is certainly a rough approximation as long as an accurate model of the multiphase ISM is concerned (e.g., Wolfire et al. 1995). However, our aim here is to build a simple, albeit reasonable, model for the two-phase ISM and concentrate on the properties of collisions that do not depend drastically on the details of the multiphase structure. When pressure equilibrium is imposed, a two-phase (cloud + intercloud) ISM structure results.
Finally a routine, based on van Leer's second-order advection scheme (van Leer 1976), has been included to track the fraction of cloud material inside each grid cell. This quantity, referred to as "mass tracer" or "mass fraction" (Xu & Stone, 1995), is initially set to unity inside the cloud and zero elsewhere. The mass fraction allows us to discriminate between the different components in our simulations, which are the two clouds and the intercloud medium. In this way we can calculate various quantities of interest such as each cloud's kinetic and thermal energy as well as Y cm , the Y-coordinate of the center of mass of each cloud . These are used in the analysis of our results. a All models use β = 4, γ = 5/3, χ = ρ c /ρ i = 100, equilibrium pressure p eq /k B = 1628 K cm −3 . Also, at equilibrium, we have T i = 7400 K and n i = 0.22 cm −3 for the intercloud medium and T c = 74 K and n c = 22 cm −3 inside the clouds. b η = τ rad /τ coll c This is the relative Mach number for the cloud pair when the clouds are first set into motion. It is referred to the intercloud sound speed, c si .
d The grid size is expressed in units of cloud radius. One cloud radius R c =50 zones.
e The end time is expressed in terms of collision time τ coll , and represents the total time from the beginning of the collision.  Table 1.
c The actual value of the cooling time for the smaller cloud is also τ rad = 3.7 × 10 4 yr. However, as explained in the text, since η > 1 the cooling does not affect the collision of these clouds and therefore it has been turned off during the simulations. For this reason we have set τ rad = ∞ in the table. to Case 4. The kinetic, thermal total magnetic energy in these plots have been normalized to the sum of the cloud energy (kinetic + thermal + magnetic) and the background magnetic energy at the beginning of the simulation. Therefore, with respect to Paper I the vertical scale of the kinetic and thermal energy of the cloud is reduced by a factor 1.18.           Figure 1 but for transverse field cases. Dash lines refer to Case 5, solid and dot lines to Cases 6 and 7 respectively. The latter two lines have been multiplied by a factor 10 and 100 respectively, in order to make them readable on the same vertical scale as the dashed line.
-40 - Fig. 12.-Field line geometry and density distribution for Case 6 at t = 12τ coll (a) and t = 30τ coll (b) respectively. Note that, since the magnetic field is quite stronger than in previous cases, contours of the magnetic flux (for the field lines) correspond to change in the latter by a factor 14 (instead of 7 as before). Density images, on the other hand, are as in