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Astron. Astrophys. 359, 1124-1138 (2000)

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4. Numerical simulations

4.1. The simulation

The formalism and the results presented in the previous section allow us to qualitatively understand the behaviour and the effects of the magnetic field in a converging and very compressible flow. Except for the solutions of Appendix B, they are only valid for loss functions proportional to temperature ([FORMULA]) and this is an important restriction. In particular, these functions present no thermal bistability. In order to avoid this restriction and to understand more quantitatively the thermal condensation in a bistable flow like the ISM, we now present a numerical simulation of the system of Eq. (1-6). As in Paper I, the cooling function is a fit (Lioure 1991) of the function calculated by Dalgarno & McCray (1972) and is equal to:

[EQUATION]

The heating rate per atom is taken as a constant equal to [FORMULA] ([FORMULA]), and the ionization fraction is equal to [FORMULA]. The resulting loss function is bistable and comparable to more recent calculations (Wolfire et al. 1995).

The main difficulty of the present problem is due to the presence of several very different time and space scales. The smallest space scale in the simulation is the Field length in the CNM ([FORMULA]), and the largest is the scale of the initial flow. This difficulty is overcome by the use of an adaptive spatial grid with adaptive time steps. A description of the code can be found in Paper I. The condition for adapting the resolution is that the spatial gradients of density and temperature must be smaller than 1%. Tests of the robustness of the results with respect to this criterion have been made for a simulation (not very close to the condensation threshold) with a tolerance of respectively 0.5% and 2%. The differences between these and the original results never exceed 1 and 2% respectively for any of the [FORMULA] and [FORMULA] fields. The total energy (thermal, kinetic and magnetic), the mass and the transverse magnetic flux (which is conserved in slab geometry) are conserved with an accuracy better than [FORMULA]. The comparison between numerical and approximate theoretical values of the fronts velocity gives agreement of [FORMULA].

As in Paper I, the simulations start from WNM with initially uniform density and temperature. The converging velocity field has no initial transverse component (see Fig. 4). The gas is initially linearly thermally stable. The initial magnetic field is also uniform and is defined by its intensity and the angle [FORMULA].

[FIGURE] Fig. 4. The six fields considered for [FORMULA] and initial magnetic intensity equal to [FORMULA]. Time 0 (full line), 42.40 (dotted line) and [FORMULA] s (dashed line). After a short contraction, the gas re-expands and no thermal condensation occurs. However, at time 42.40 and at the center point, [FORMULA] and [FORMULA] are comparable, whereas at time 72.43, [FORMULA] is less than 1 and [FORMULA] is greater than 2. Magnetic field and density decorrelated rapidly.

4.2. Cloud formation with magnetic field

We present two simulations that differ only in the initial values of [FORMULA] and choose initial conditions (amplitude and typical spatial scale of the velocity field, thermal pressure) that would lead to thermal condensation in the absence of a magnetic field. The initial value of [FORMULA] is equal to [FORMULA] ([FORMULA]), which is comparable to the measurements of Troland & Heiles (1986) and Myers et al. (1995). The results can be seen in Fig. 4 and Fig. 5.

[FIGURE] Fig. 5. The six fields considered for [FORMULA] and [FORMULA]. Time 0 (full line), 39.62 (dotted line) and [FORMULA] (dashed line). At time 73.83, very stiff gradients appear, due to the condensation of WNM into CNM. At time 39.62, the value of [FORMULA] at the center is about 3 times the initial value, which is also the ratio between density at origin and initial density. At time 73.83, [FORMULA] is about 3 times less than the initial value whereas the ratio between density at the center and initial density is greater than 150 (see Fig. 7).

For [FORMULA], the thermal condensation is not obtained, magnetic pressure prevents it. At time [FORMULA] s [FORMULA] and [FORMULA] are correlated, whereas at time [FORMULA] s the central density peak corresponds to a minimum of [FORMULA].

For [FORMULA] part of the gas, initially WNM, condenses into CNM, forming a cloud. As in the lower panel of Fig. 2, the magnetic tension is strong enough to force the magnetic pressure to decrease after an increase phase before the flow stops. Hence, the thermal condensation is possible. This highly compressible MHD process is more clearly illustrated in Fig. 6. The magnetic field lines and the velocity field of Fig. 5 are displayed in the x-y plane for three different times. The magnetic field lines are first bended forward by the longitudinal flow and the gas is put in forward motion ahead of the compression, first towards positive y, then towards negative y, when the field tension reverses the flow (upper panel). This side flow advects the field lines and inverts their curvature in the neighbourhood of the convergence focus (middle panel), where the flow nearly stops. But the density in the center has been sufficiently increased in the meanwhile, for the thermal pressure drop to start pumping the neighbouring gas towards the center along the field lines, as clearly appears in the lower panel. The lines slowly unbend at subsequent times.

[FIGURE] Fig. 6. Two dimensional display of Fig. 5 (half part only). Magnetic field lines and velocity field are presented for three different times. The initial conditions are superimposed for easier reference (thin lines).

Simulations with a twice stronger field show a similar initial field bending, but the field relaxes much faster to its original direction, aligning the whole flow parallel to itself.

Fig. 7 is a spatio-temporal zoom of Fig. 5. The density reached in the center is more than a hundred times the density of the WNM, whereas the value of the transverse magnetic field is comparable to the initial value. For this last example, we can conclude that the presence of a magnetic field does not prevent strong condensations.

[FIGURE] Fig. 7. Spatio-temporal zoom around the center of the simulation of Fig. 5 at time 72.15 (full line), 73.21 (dotted line) and [FORMULA] (dashed line), the spatial scale considered is a thousand times smaller than the scale of Fig. 5. The density increases up to [FORMULA] ([FORMULA])

The self-similar behaviour of the fields is apparent at times 72.15 and [FORMULA] (Fig. 7). The time evolution can be approximatively described by a spatial dilatation and renormalization of the different fields. It progressively disappears when part of the gas reaches thermal equilibrium because at this point the thermal loss function much deviates strongly from [FORMULA].

4.3. Cloud evolution with magnetic field

Let us further investigate the consequences of the magnetic field during the condensation process. Fig. 8 shows the evolution of the cloud column density (only pixels with density 10 times higher than the initial density are taken into account) for four values of [FORMULA]. The condensation occurs later when [FORMULA] increases, and is less efficient. However, the growth curves are similar for all angles, starting with a fast dynamical growth followed by a slow conductive growth (not well described because in our simulation [FORMULA] does not depend on [FORMULA]). To allow for the first parcel of gas (WNM) to reach the second branch of equilibrium (CNM), the magnetic pressure has to be small. According to the mechanism described in Sect. 2 and 3 it is more difficult and slower to decrease the magnetic pressure when the value of [FORMULA] increases. But, once the condensation process starts, the effect of the magnetic field on the growth of the cloud becomes weak.

[FIGURE] Fig. 8. Evolution of the column density of the cloud. All corresponding initial parameters are defined as in Fig. 4 except [FORMULA]. Top curve is [FORMULA], second one [FORMULA], third and fourth one, [FORMULA] and [FORMULA] respectively. For [FORMULA] no thermal condensation occurs. As expected, the condensation starts later and is less efficient when [FORMULA] increases.

The evolution of the cloud formed in Fig. 7, is shown in Fig. 9 at subsequent times. Most of the gas in the cloud reaches thermal equilibrium, two fronts separate the two phases. Pressure and density decrease slowly and the cloud relaxes until it reaches pressure equilibrium with the surrounding medium. The fronts propagate slowly in a quasi-isobaric regime (see Ferrara & Shchekinov 1993 and Paper I for a quantitative comparison in the non-magnetized case). Due to a small transverse velocity gradient (induced by the magnetic tension), the transverse magnetic field, that decreased during the condensation process, rises back to the mean large scale value. The field lines slowly unbend until they become straight. At the end of the process, thermal and magnetic pressure are uniform.

[FIGURE] Fig. 9. Long time evolution of the cloud formed in Fig. 5 for time 84.9 (full line), 96.2 (dotted line) and [FORMULA] s (dashed line). The density decreases and the temperature is constant so that thermal pressure decreases until the gas reaches pressure equilibrium with the surrounding medium. The transverse magnetic field re-increases until it reaches the average large scale value.

4.4. Threshold dependence on magnetic and velocity fields

We now investigate the condensation dependence on magnetic intensity and velocity amplitude. In Paper I, we already studied how the nucleation threshold varies with the initial pressure, amplitude and size of the convergent flow. We shall not consider it further in this paper, although an extended study including these parameters and the magnetic field has to be done at some stage. The initial pressure [FORMULA] ([FORMULA]) and the initial size of the convergent flow is [FORMULA] [FORMULA] (twice the distance between the center and the peaks of the initial velocity field). These two values are typical of the WNM. For several values of [FORMULA] and for the two values [FORMULA] and [FORMULA], we look for the largest value of [FORMULA], [FORMULA] at which thermal condensation occurs. The method used is dichotomy and the accuracy is one degree. The results are presented on Fig. 10. In both cases two branches are found, between which no condensation occurs.

[FIGURE] Fig. 10. Dependence of [FORMULA] on initial magnetic intensity [FORMULA] for two initial values of [FORMULA]. In both cases two branches can be seen: the condensation does not occur at intermediate values of B. When the initial velocity increases, the minimum value of [FORMULA] increases also, the condensation is easier.

Before making further comments on Fig. 10, we use Eq. (13) to derive a rough expression of the magnetic field that allows us to understand qualitatively some features of Fig. 10 and summarizes the conclusions i) , ii) and iii) of Sect. 3. The l.h.s. of Eq. (13) is comparable to [FORMULA] and the first term of the r.h.s. to [FORMULA]. The order of magnitude of the second term of l.h.s. is given by [FORMULA] where [FORMULA] is the dynamical time. The matter conservation leads to [FORMULA]. With these relations, we find

[EQUATION]

If [FORMULA], clearly Eq. (38) is not valid anymore.

With Eq. (38) three regimes are expected:

  • weak magnetic fields, [FORMULA]: [FORMULA] is independent of [FORMULA] and almost proportional to density; [FORMULA] rapidly decreases with [FORMULA] because of the magnetic pressure.

  • strong magnetic fields, [FORMULA]: [FORMULA] is no longer proportional to [FORMULA], its evolution depends on [FORMULA], and [FORMULA] increases with [FORMULA]. This result is in good qualitative agreement with the analytical result (Fig. 3). The poor quantitative agreement is due to the significant differences between the thermal processes considered in the two approaches.

  • The minimum [FORMULA] is found for an intermediate value of the magnetic field: [FORMULA]. There is a rough equipartition between magnetic and other energies (thermal and kinetic). When the kinetic energy increases, so does the value of [FORMULA] corresponding to the minimum [FORMULA].

The typical values of [FORMULA] range from 20 to [FORMULA].

The distribution of [FORMULA] results from the development of the MHD turbulence and is not straightforwardly quantified. The assumption of a random distribution likely yields an upper limit on the efficiency loss ([FORMULA]) with respect to the unmagnetized cases. We have:

[EQUATION]

[FORMULA] gives [FORMULA] and [FORMULA], [FORMULA]. As expected the magnetic field stabilizes partially the flow against thermal condensation, reducing the number of condensations that would occur in a non-magnetized gas by a factor 5 to 20. However, the condensations are still possible, the rate of cloud formation is reduced but is not equal to zero.

For larger velocities, several structures appear and the problem becomes more complex because these structures collide. Such cloud collisions, well described in the literature (Mac Low et al. 1994, Ricotti et al. 1997, Miniati et al. 1999), lead to strong density contrasts in the CNM and strong enhancement of the local magnetic field.

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© European Southern Observatory (ESO) 2000

Online publication: July 13, 2000
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