THE EFFECTS OF FLOW-INHOMOGENEITIES ON MOLECULAR CLOUD FORMATION: LOCAL VERSUS GLOBAL COLLAPSE

In the Smooth model (left column), material enters the collision region on the equilibrium curve at 1.0 cm⁻³ which coincides with the peak in the equilibrium cooling curve. As the material collides with the oppositely directed material it is initially compressed adiabatically up to the flow ram pressure at 1.04 × 10⁴ cm⁻³. It then cools and compresses onto the thermal equilibrium curve. With time, more material piles up at higher densities. Eventually, self-gravity takes over at the highest densities and compresses this material further, above the ram pressure provided by the flow. At this point, gas collapses and forms a core, or is being accreted by an existing core. The core formation or accretion explains the lack of material at densities above ≈10^4.6 cm⁻³.

The Clumpy flow, on the other hand (right column), has material entering the collision region at both 0.25 cm⁻³ and 15.2 cm⁻³. Additionally, some mixing occurs between the clumps and the background flow which causes the thick band of material below the thermal equilibrium curve. At these densities, the thermal time scales are longer than the dynamical time (see Figure 3 of Heitsch et al. 2008a)—so this material does not equilibrate before colliding with the oppositely directed flow. The low-density background appears to also compress adiabatically though not to as high pressures as the Smooth run. At the interface between the two flows there are three possible types of interactions due to the two densities present in the flow. (1) For background–background collisions, the ram pressure will be one-fourth of that for the Smooth flow. (2) Collisions between background material from one side and clumps from the other will produce bow shocks equivalent to background material running over a stationary clump with velocity 2v₀ = 16.5 km s⁻¹, resulting in a ram pressure equal to that in the Smooth model, at 1.04 × 10⁴ K cm⁻³. (3) Finally, head-on clump–clump collisions, though rare, can produce pressures 15.2 times the ram pressure in the Smooth model. Yet this material will cool fairly quickly due to the high densities.

Figures 2 and 3 show column densities taken along the flow axis (left) and normal to the flow axis (right) for the Smooth and Clumpy runs, respectively. Also shown are the boundaries of the cylindrical region used for the following analysis. The Smooth run exhibits the usual filamentary structure due primarily to the non-linear thin shell instability (NTSI; Vishniac 1994; Hueckstaedt 2003; Heitsch et al. 2005; Vázquez-Semadeni et al. 2006), and at later times gravity. Also visible is material which has been "splashed" radially outward from the collision region due to the high ram pressures. The NTSI focuses material into various nodes and by 10.1 Myr the first core (solid black square) has formed in one of these nodes (see also Heitsch et al. 2008b). By 20.1 Myr, nine cores have formed throughout the complex and by 27.2 Myr, 27 cores have begun to arrange themselves into clusters.

The morphology of the Clumpy run (Figure 3) is quite different from the Smooth run. The view along the flow axis shows that the clumps are confined to be within the stream by a few parsecs and they have a tendency to cluster around the perimeter. This is just an artifact of the clump placing algorithm and can be safely ignored. There also tends to be much less radial splashing than in the Smooth run because the uniform background component prone to being splashed out is only one-fourth as dense. The interaction region is much more extended along the flow axis than in the Smooth run. Early on, dense clumps run into the lighter background and travel a distance before being destroyed. The timescale for the clump destruction will be of order the clump crushing time t_cc = χr_c/v_w (Klein et al. 1994) where v_w is the "wind velocity" as seen by the clump and χ is the density contrast. Since the clump is itself traveling into an oppositely directed flow, v_w = 2v₀ and the distance the clump will travel will be of the order of

$$\begin{equation} \mathcal {D}= t_{{\rm cc}} v_0 = \frac{\chi r_c}{2} = 4.3{\rm pc}. \end{equation} \tag{ 2 }$$

If the clump survives for a few clump crushing times, it will travel distances of ≈10 pc before being destroyed. This explains the more extended interaction region in the upper right panel of Figure 3. Later in time, the clumps pass through a denser wall of material that has built up and they are also pulled back by gravity—so the extent of the collision region shrinks over time. While the Smooth run has formed nine isolated cores by 20.1 Myr distributed throughout the collision region, the Clumpy run has only just formed a group of three cores near the center of the global potential well. By 27 Myr, the entire region is undergoing rapid global collapse and a dense group of 20 cores has formed again near the center of the potential well. The Smooth run has formed 27 cores by the same time.

In order to construct power spectra, typically one uses a fast Fourier transform (FFT). However, the FFT requires a uniform grid of points. The process of mapping AMR data onto a fixed grid often leads to spurious signals at wavelengths corresponding to coarser cell sizes depending on the method used to interpolate the data. In AMR simulations, coarse data is often interpolated onto a finer mesh through a process called prolongation. Typically a single coarse cell will be divided into R^D finer cells, where R refers to the refinement ratio—or the ratio between cell width—and D is the dimension of the grid. For constant prolongation, these finer cells just inherit the values for density, momentum, etc.. from the coarser parent cell. However, this results in smooth gradients in coarse cells turn into stair steps on finer cells. This stair stepping artificially increases power at the wavelength of the coarser cell. One can do linear interpolation, or higher order interpolation to reduce the effect, but here we prefer to do interpolation in Fourier space—so that the process of interpolating does not introduce any new power in the resulting spectra. The procedure involves first coarsening (averaging) the data everywhere to a uniform coarse grid and then prolongating it to a finer uniform grid using Fourier prolongation described in the Appendix. The finer uniform grid is then updated with existing finer data (where available) and then the process is repeated until we have a uniform grid of points on the finest level of the AMR hierarchy. For details on the prolongation see the Appendix.

The FFT also expects the data to be periodic, however, the physical boundaries are not periodic, and treating them as such introduces a step function at the x-boundaries where the inflowing material changes sign. To obtain the power spectrum for a non-periodic velocity field, we introduced a window of d = 40.625 pc in diameter centered on the collision region. The velocity field at a distance r from the center of the collision region is first multiplied by this windowing function before calculating the power spectra:

$$\begin{eqnarray*} w(r)=\left\lbrace \begin{array}{@{}ll}\cos (\pi r/d) & : r\,{<}\,{=}\, d/2 \\ 0 & : r \,{>}\, d/2. \end{array} \right. \end{eqnarray*}$$

Figure 4 shows spectra of the kinetic and gravitational energy densities, E_k and E_g. The contribution of cores (i.e., sink particles) to the energy densities are not included. Snapshots have been taken at three times, early in the cloud evolution (corresponding to a diffuse CO cloud) at t = 10.1 Myr after simulation start, an intermediate stage (t = 20.1 Myr), and a late stage (t = 27.2 Myr). In the following discussion, we will distinguish between "large" and "small" scales by considering scales larger or smaller than the clump crushing distance $\mathcal {D}$ (Equation (2)).

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At early times (t = 10.1 Myr, light gray lines), the Clumpy run shows a kinetic and gravitational energy excess on small scales, with E_k ≫ E_g. While both runs have the same influx of kinetic energy driven by the flows through the boundaries, the Clumpy run contains most of the kinetic energy in dense clumps on small scales. When these flows collide, the clumps are able to maintain coherence until they have travelled a clump shredding distance (Equation (2)) at which point their kinetic energy is able to dissipate. This clump shredding distance can be thought of as a "driving scale" for the small scale "turbulence" visible in the excess, and it roughly corresponds to the break in the power spectrum, justifying our choice of this scale to separate between "large" and "small" scales.

By 20.1 Myr, the Clumpy run (dark gray, dashed lines) has gained large-scale kinetic energy due to the onset of global collapse mirrored in an increase of the gravitational energy on large scales, with E_g ≈ E_k. By 27.2 Myr the Clumpy run has gained kinetic and gravitational energy on all scales due to both the continued global collapse as well as the onset of local collapse. Note that the overall apparent drop in gravitational energy in the Clumpy run between 20.1 and 27.2 Myr is due to the very rapid accretion onto sink particles, following the global collapse (as stated before, the spectra only refer to the energy in the gas phase, not in the sink particles).

The Smooth run (solid lines) shows only a modest increase in the small-scale kinetic and gravitational energy, with E_g ≈ E_k except at very early times on large scales, when E_k ≫ E_g. With advancing time, E_g > E_k for small scales, whereas E_g < E_k on large scales.

The time evolution of the energy spectra highlights a fundamental ambiguity in the interpretation of the kinetic energy. On the one hand, for situations with E_k ≈ E_g, the models show gravitational collapse on the respective scales, suggesting that the kinetic energy actually follows the gravitational one, i.e., that the increase in kinetic energy is due to gravitational collapse (see Vázquez-Semadeni et al. 2007; Heitsch et al. 2008b). On the other hand, for situations E_k ≫ E_g (as for small scales in the Clumpy run), the kinetic energy seems to take on an active role preventing gravitational collapse.

We attempt to throw some more light on this ambiguity by following the time evolution of the mean energy densities $\bar{E}_g$ and $\bar{E}_k$ for large and small scales (Figure 5), where "large" and "small" is defined in Section 3.3. The mean energy densities are derived by averaging over scales ${<}\mathcal {D}$ (small scales) and ${>}\mathcal {D}$ (large scales) in the energy spectra.

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For the Smooth run (right panel of Figure 5) we note that $\bar{E}_k\gg \bar{E}_g$ on large scales. This is consistent with the lack of global collapse observed for the Smooth run, and also with the absence of large-scale modes in the energy spectra (Figure 4). However, the small scale gravitational energy becomes comparable to the small scale kinetic energy after 10 Myr which is when we begin to see the formation of isolated cores, indicating local collapse. The Clumpy run on the other hand shows the opposite—the energy is dominated by small scale kinetic energy ($\bar{E}_k\gg \bar{E}_g$) at times for which no local collapse occurs. Only after 20 Myr, when on large scales $\bar{E}_k\approx \bar{E}_g$, do we begin to see global collapse. By 23 Myr, much of the gas has been accreted into several large cores near the center of the potential well, which explains the drop in energies.

While both runs begin with the same total kinetic energy, and while they have the same influx of kinetic energy, the Smooth run is much more efficient at dissipating this energy in large coherent shocks resulting in a smaller overall kinetic energy within the collision zone. In the Clumpy run, the density contrast between the clumps and the opposing ambient material leads to a less efficient dissipation of kinetic energy. This excess kinetic energy on small scales suppresses local collapse (remember that the clumps themselves are gravitationally stable) but cannot prevent global collapse—while in the Smooth run, the larger amount of kinetic energy on large scales resists global collapse but not local collapse. Another way to see this is that the shocks in the Smooth run will fragment quickly due to the thermal instability. Yet, the velocity dispersion between the fragments will be small, at least smaller than for the Clumpy run (see Figure 6, right), thus forming structures that are more or less coherent in velocity space. Thus, local collapse is seeded. For the Clumpy run, thermal instability does not play much of a role, and while the clumps can accrete their own mass within a few Myr, they are still (thermally) Jeans-stable. Thus, local collapse is suppressed, while global collapse sets in once enough mass in clumps has been assembled. Note that the excess of small-scale kinetic energy in the clumpy run results from the relative motions of the clumps, not from turbulence within the clumps. The power spectrum shows this as an excess energy content at small scales.

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Summarizing, we point out that in our interpretation the fact that $\bar{E}_k\approx \bar{E}_g$ is not a sign of virialization, and thus balance between gravitational contraction and turbulent support, but actually a natural consequence of gravitational collapse (Ballesteros-Paredes et al. 2011). Only for $\bar{E}_k\gg \bar{E}_g$ there is enough kinetic energy in the system to prevent gravitational collapse of the highly structured, irregular, and non-linear cloud.

The story of local versus global collapse also can be gleaned from the mass histories (left panel of Figure 6). The mass histories are restricted to the analysis region (see beginning of Section 3). The dotted line in Figure 6 shows the theoretical upper bound using the mass influx $\dot{M}$ through the boundaries. Both runs collect mass at the inflow rate $\dot{M}$ for t < 2 Myr, after which the growth rate drops. For the Smooth run, material is being splashed radially outwards, and at later times, some of the NTSI fingers develop past the analysis region (see Figure 2). Eventually, after 15 Myr, material is falling back in from the edges of the analysis region, increasing the mass collection rate again, i.e., steepening the slope. In the Clumpy run, some of the clumps exit the analysis region on the far side after 2 Myr. The overall mass collection rate slowly increases after that, with material falling back in, and eventually collapsing globally.

The Smooth run begins forming cores at 10 Myr, and by 25 Myr, the rate of total mass growth and core mass growth have become equal. This implies that material is being accreted by the cores at the same rate it is entering the collision region. The Clumpy run (Figure 3) does not begin to form cores until 20 Myr, but then quickly accretes gas at a rate higher than the mass flux into the region. This suggests a degree of global collapse not present in the Smooth run.

To better understand the degree of gravitational instability in both models, we calculate the Jeans mass, averaged over the analysis region. Given the geometry, we compare the Jeans mass for a thin, infinite sheet at a given sound speed c_s and column density Σ,

$$\begin{equation} M_{J0} = 1.17\frac{c_s^4}{G^2 \Sigma } \end{equation} \tag{ 3 }$$

(Larson 1985), and the critical mass for an isothermal, infinite sheet,

$$\begin{equation} M_c = 4.67\frac{c_s^4}{G^2\Sigma } \end{equation} \tag{ 4 }$$

(Larson 1985). Both expressions are obviously an approximation, since our model clouds is neither a perfect sheet, nor infinite. The mass column density Σ is averaged over the analysis region, as is the sound speed. The latter is calculated as

$$\begin{equation} c_s^2 = \frac{5}{3} \frac{k_B}{\chi } \frac{\int \rho T dV + 10 {\rm K} \sum {m_n}}{\int \rho dV + \sum {m_n}}, \end{equation} \tag{ 5 }$$

where the integrals extend over the analysis region and m_n refers to the mass of the nth core. We also assume that the cores (i.e., the sinks) are at a temperature of 10 K. These critical masses are plotted in the left panel of Figure 6 as well as the locations at which the actual mass becomes theoretically gravitational unstable (large squares).

Since the gas is compressed and thus cools, the Jeans mass will drop with time, as shown in Figure 6 (left panel). It levels out once the minimum temperature of ≈10 K is reached (this is only obvious in the Clumpy run, dashed medium or thick lines, for t > 25 Myr). The Jeans mass for the Clumpy run is smaller by at least an order of magnitude, because of the clumps at higher densities and lower temperatures. Yet, since these clumps do not form a coherent region with M > M_J, local gravitational collapse is suppressed until ≈20 Myr, and sinks form only once global collapse sets in, indicated by the increasing slope of the total mass, black dashed line. The onset of global collapse in the Clumpy run is also visible in Figure 5 (left), and in the velocity dispersions shown in the right panel of Figure 6.

The Smooth run has a substantially larger Jeans mass that does not level out at a minimum within the model run time. Yet, because of the rapid local fragmentation into structures larger than a local Jeans mass, local collapse (and sink formation) sets in at ≈10 Myr. If the steepening of the total mass curve with time at ∼15 Myr is interpreted as a sign of fall-back, and thus global collapse, then the Smooth run shows a substantially weaker signature of global collapse compared to the Clumpy run, consistent with the evolution of the energy budgets and the velocity dispersions.

Note that Equations (3) and (4) refer to the global Jeans mass. This may explain the somewhat counter-intuitive notion of the Smooth run having a larger Jeans mass than the Clumpy run, while it develops gravitational collapse at an earlier time. Since the flow compression in the Clumpy run is less important than in the Smooth run, the Jeans mass of the Clumpy run is largely determined by the cold clumps. For the Smooth run, the flow compression drives the density up and the temperature down, leading to a locally much smaller Jeans mass, hence rapid local fragmentation and collapse. Yet, comparing Figures 2 and 3 at 10.1 Myr shows that in fact the filamentary structures of the Smooth run are less volume-filling than the clumps of the Clumpy run. Thus, the global Jeans mass is a better predictor for the behavior of the Clumpy run than that of the Smooth run.

Figure 7 shows the distribution of core masses for both runs at 27.2 Myr. Most of the cores in the Smooth run have masses <100 M_☉, while there is a substantial fraction of cores in the Clumpy run with masses >100 M_☉. This is consistent with the findings of Vázquez-Semadeni et al. (2009), who compared "low-mass regions" (LMR) and "high-mass regions" (HMR) in their simulations of flow-driven cloud formation. They found that cores in LMRs tend to have lower masses, while those in the HMR formed due to global collapse of the whole cloud tend to have higher masses. Unlike Vázquez-Semadeni et al. (2009), we do not observe LMRs and HMRs in one and the same simulation. In fact, in our models, the total mass in cores for the Clumpy and Smooth run differ only by 50%, while the LMR and HMR of Vázquez-Semadeni et al. (2009) seem to have a mass ratio of nearly 10. Nevertheless, the cores in the Clumpy run form at the center of the potential well (Figure 3) late in the simulation, i.e., at a time by which global collapse has set in, while the cores in the Smooth run start to form much earlier (Figure 6, light gray lines), at a shallower gravitational potential. That the Smooth core mass distribution is slightly wider than the Clumpy one, would also be consistent with this scenario: with a deepening gravitational potential, more massive cores are formed. Yet, for more definitive statements, we would need better statistics.

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It also should be pointed out that we measure the mass in cores at equal times (27.2 Myr), while Vázquez-Semadeni et al. (2009) derive the statistics for their LMRs and HMRs within one and the same simulation at different times. Due to the global collapse and the more massive "seed"-cores, one might expect the mass difference between our Clumpy and Smooth cores to increase further with time.

In both runs, material injected from the left and right side was marked with a tracer (ρ_L and ρ_R) proportional to the density so that the amount of mixing could be investigated. We then define a local mixing ratio

$$\begin{equation} {\rm MR}=\frac{2 \min {\left(\rho _L, \rho _R \right)}}{\max {\left(\rho, \rho _L+\rho _R \right)}}. \end{equation} \tag{ 6 }$$

Thus, MR = 0 indicates the presence of only one tracer at a location (or none—as is the case in the ambient medium outside of the flow), and MR = 1 indicates equal amounts of both tracers with no ambient material mixed in. Since we are confining our analysis to the colliding flow region, there should be no ambient material present so ρ = ρ_L + ρ_R—and the definition is equivalent to

$$\begin{equation} {\rm MR}=\frac{2 \min {\left(\rho _L, \rho _R \right)}}{\rho _L+\rho _R}. \end{equation} \tag{ 7 }$$

The Smooth run shows a higher mass-fraction of well-mixed material (Figure 8, left panel). The mass in the gas phase (i.e., not including the cores/sink particles) is comparable for both runs (see left panel of Figure 6), at around 4 × 10³ M_☉. The Clumpy run tends to have a more spread out distribution of mixing ratios than the Smooth run. As clumps drive through the opposing stream—they will shed some of their material and provide varying amounts of mixing. In the Smooth case, the flows interact along a thin interface and it is difficult to get unequal amounts of material from either side in the same region.

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While the gas in the Smooth run shows a higher mixing ratio (i.e., more uniform composition) than in the Clumpy run, the cores in the Smooth run (Figure 8, right panel) have lower mixing ratios. This can be understood in two ways.

• 1.  
  In the Smooth run, the NTSI creates nodes that act to funnel material streaming into the "trough" from only one side, while diverting material from the other side. If so, then cores that formed to the left of the collisional mid-plane should be biased towards the right tracer and vice versa. To test this, we define a mixing bias as

  $$\begin{equation} {\rm MB}=\frac{\rho _R-\rho _L}{\rho _R+\rho _L} \end{equation} \tag{ 8 }$$

  and plot it against the core's offset from the y–z mid-plane shown in the right panel of (Figure 7). Indeed, cores with negative offsets (left of mid-plane) tend to have a higher value for the mixing bias so they have more right tracer and are comprised of material primarily from the right side, and vice versa. Note that this bias is absent in the cores formed in the Clumpy run, whose cores are clustered around 0.
• 2.  
  In the Clumpy run, instantaneous mixing in the gas is suppressed in the absence of a strong NTSI: the clumps just plow through the oncoming gas. Thus, mixing in the gas phase will be generally low. Yet, as Banerjee et al. (2009) discussed, the boundaries of the clumps are not rigid. Due to the thermal bistability, material streaming onto the clump cools rapidly and thus is accreted quickly. If a clump of radius R_c and density n_c traveled through the opposing flow of density n₀ at a velocity v_c, then it would accrete its own mass (assuming a cross section of $\pi R_c^2$ and a constant relative velocity) after

  $$\begin{equation} t_{{\rm mix}} = \frac{3R_cn_c}{4v_cn_0} = 79 \frac{n_c}{n_0}\left(\frac{R_c}{{\rm pc}}\right) \left(\frac{v_c}{{\rm km\, s}^{-1}}\right)^{-1}{\rm Myr}, \end{equation} \tag{ 9 }$$

  or, for our parameters, after t_mix ≈ 5 Myr. If the clump has accreted its own mass (of the opposing flow's metallicity), the mixing ratio (Equation (7)) will be one. The bulk of the mass is residing in the clumps, thus, once global collapse sets in, the cores are (for the most part) being assembled from the well-mixed clumps.