Dense Gas Formation via Collision-induced Magnetic Reconnection in a Disk Galaxy with a BiSymmetric Spiral Magnetic Field

Hereafter, we name the fiducial model as MW0. The third column of Table 1 lists the key parameters of the model. Figure 6 shows the collisions between the four pairs of clouds from t=0.02 (0.4 Myr) to t=0.05 (1 Myr). Each panel shows a pair of colliding clouds at a time step. The coordinates are changing due to the disk rotation. The collisions happen at t=0.02 and create compression layers between cloud pairs.

As introduced in §1, the key feature of CMR is a dense filament at the center of the collision midplane. The filament is wrapped by helical fields. Since our simulation is 2D, we expect to see a density peak at the center of the collision midplane with enclosing field loops. At t$\gtrsim$0.03, we do see field loops in the compression layer. However, the loops remain elongated, failing to pull gas in the layer and create a centrally peaked structure. For t$\gtrsim$0.04 in Figure 6, the compression layer does have a higher density at the center. But the density contrast is not as high as those in K21. Nor do we see magnetic islands (plasmoids).

To see the reconnected field loops and confirm CMR, we zoom into the central 4 pc region of the dense midplane. This field of view has the same size as the computation domain in K21. Figure 7 shows the zoom-in view at t=0.6 Myr. Here, the field streamline separation in both directions is 0.03 pc, i.e., about half of the cell size. For comparison, the streamline separation in Figure 6 is 0.25 pc, which is a factor of 4 larger than the cell size.

In Figure 7, we see field loops forming in all four collision sites. The magnetic tension forces at the top and bottom sides of the loops point toward the loop center, trying to pull gas to the center. The loops and the tension forces confirm the triggering of CMR. This result shows that CMR indeed happens. While we only model four pairs of colliding clouds, there are likely many more colliding clouds along field-reversal interfaces (see Figure 2). Therefore, we should expect more CMR events. Since our modified-BSS field is based on studies of our Milky Way, we can address the question from §1 by speculating that CMR should be more common along the Milky Way field-reversal interfaces, which coincide with the spiral arms (HQ94). Note that the BSS field is the large-scale component of the overall magnetic field. There is also small-scale stochastic field which may promote or prevent CMR, partially depending on the net field orientation (K21).

As shown in Figures 6 and 7, although CMR is triggered at the collision sites, the field loops fail to produce very high density gas. On the contrary, the CMR filament in K21 had a factor of $>$100 density increase. They compared different forces in MHD and showed that the magnetic tension force dominated the other forces. In particular, self-gravity was negligible during the initial filament formation phase. Below, we evaluate contributions from different forces in our model.

The MHD momentum equation (6) can be rewritten as (ignoring the disk potential)

It includes three force terms, i.e., the thermal pressure term $-\nabla P$, the magnetic pressure term $-\nabla(B^{2}/8\pi)$, and the magnetic tension term $(\mathbf{B}\cdot\nabla)\mathbf{B}/4\pi$. Here, $\rho$ is the gas density, $\mathbf{v}$ is the velocity, $P$ is the thermal pressure, and $\mathbf{B}$ is the magnetic field.

As shown in K21, magnetic tension was the dominating force that created the central dense gas. It pulled material from two sides of the elongated field loop to the loop center, overcoming the counteracting thermal pressure force. In our case, however, the magnetic tension is not strong enough. Figure 8 shows the relative importance of the forces. Each row shows the three force terms for one CMR site. The color scale indicates the force magnitude. All panels have the same color scale for comparison. The arrows indicate the direction of the force. Their lengths are all the same (not proportional to the magnitude).

Most obviously, the thermal pressure force dominates the other two forces for all four CMR sites. For CMR1, the median value of the ratio between thermal pressure force and magnetic pressure force is $3.8\times 10^{2}$; the median value of the ratio between thermal pressure force and magnetic tension force is $2.6\times 10^{3}$. For the other sites, the ratios are higher due to their weaker fields. As shown in Figure 6, the dense midplane expands at later times due to the overwhelming thermal pressure. While adding self-gravity may help confining the dense gas, the expansion shows that CMR needs to be stronger to make the compressed gas even denser in an environment like the WNM.

The magnetic pressure force within the compression layer is typically a factor of a few stronger than the tension force. The pressure force is pushing the gas from both left and right. Together, the magnetic forces try to hold the gas in the collision midplane, which shows that CMR still helps maintaining the dense gas.

Since the magnetic forces in MW0 were dominated by the thermal pressure, a natural idea is to test if stronger fields would enhance the CMR effect. In light of this, we run another simulation with a factor of 3 stronger magnetic field ($B_{0}=4.2\mu$G). All other parameters remain the same. Hereafter the new model is named as MW1. Figure 9 shows the results for MW1. The field strength for CMR1 is $B_{1}\approx$5.1 $\mu$G, for CMR2 is $B_{2}\approx$2.3 $\mu$G, for CMR3 is $B_{3}\approx$0.84 $\mu$G, for CMR4 is $B_{4}\approx$0.33 $\mu$G.

Immediately from Figure 9 we can see a density enhancement inner layer in the CMR1 collision midplane. From $t=0.02$ to $t=0.03$, all four sites show the compression layer due to the collision. Their densities reach about 5. Starting from $t=0.04$, there is a denser inner layer in the CMR1 compression layer. At $t=0.05$ (1 Myr), the dense layer becomes more prominent. Inside the layer, multiple dense clumps form. Their density reaches an even higher value of $\sim$8 ($n_{H}=136$ cm^-3), while the collision compression reaches about 5.7. The formation of the clumps indicates the triggering of the plasmoid instability. The clumps are plasmoids in the current sheet with a density enhancement.

Meanwhile, the dense clump formation is not seen in the other three CMR sites. Perhaps CMR2 shows a dense inner layer at $t=0.05$ but its density enhancement is not very prominent. Note that the three sites still show field loops indicating magnetic reconnection. The fact that they fail to form dense clumps is probably due to their weaker field strengths. Again, this strength difference is because of our modified-BSS field. Based on the aforementioned field strengths, we anticipate that the critical value for clump formation is $\sim$5 $\mu$G.

As discussed in §4.1, CMR can be common along field-reversal interfaces if more collision events happen there. The production of dense gas in MW1 indicates that CMR can be a significant source of dense gas under appropriate conditions. Moreover, the colliding clouds do not have to be molecular since we are just modeling the MHD fluid. As long as they are protruding-shaped structures colliding at the field reversal, we should expect CMR. The structure may be a cloud of ionized or atomic gas. Due to the accumulated column density after CMR, the dense gas may become molecular and form stars. Currently, we only aim to demonstrate the dense gas production by CMR in the disk. We have not included self-gravity. Nor have we run the simulations long enough to see the follow-up gas evolution. Future work will address these aspects step by step.

The K21 fiducial model was able to form a filament with a $>100$ density increase compared to their colliding clouds. However, in our fiducial model MW0, although magnetic reconnection happens, no dense structure clearly stands out within the compression midplane. Understanding this difference between the two models is crucial to evaluating the importance of CMR in the Galactic disk.

First, it is necessary to distinguish between magnetic reconnection in a current sheet and fast reconnection due to the plasmoid instability. Generally speaking, there are three phases of magnetic reconnection (see, e.g., Pucci & Velli, 2014; Comisso et al., 2016; Ji et al., 2022). In phase-1, antiparallel magnetic fields are brought together to form a current sheet at the field-reversing interface. Here, magnetic reconnection should happen if diffusion becomes important compared to convection (see Equation 7). Accordingly, the magnetic Reynolds number at the scale of the thickness (or width, $\delta_{s}$) of the sheet $R_{\rm m,s}\equiv\delta_{s}v_{s}/\eta$ should be $\lesssim$1. Here $v_{s}$ is the characteristic velocity at the same scale. In this phase, the reconnection rate is relatively slow while the thinning of the current sheet continues (the aspect ratio increases).

In phase-2, small plasmoids emerge and enter the linear growth phase $\propto e^{\gamma t}$ as the current sheet thins, where $\gamma$ is the growth rate that is a function of the Lundquist number $S_{L}\equiv Lv_{A}/\eta$ ($L$ being the sheet half-length and $v_{A}\equiv B/\sqrt{4\pi\rho}$ the Alfvén velocity). For a steady-state Sweet-Parker current sheet (Parker, 1957; Sweet, 1958), its aspect ratio ($a/L$ where $a$ is the half-width), growth rate, and mode wave number ($k$) scale with $S_{L}$ as (Tajima & Shibatu, 1997; Loureiro et al., 2007)

where $\tau_{A}\equiv L/v_{A}$ is the Alfvén timescale as the outward velocity is typically the Alfvén velocity. For large $S_{L}$, these scaling relations break and the growth rate saturates at the maximum rate (Bhattacharjee et al., 2009). The saturation is because the fastest growing mode amplitude exceeds the inner resistive layer with the thinning of the current sheet, when the plasmoid instatibility enters the next phase (a nonlinear growth phase, Comisso et al., 2016, 2017). Before that, the plasmoid growth rate not only depends on the Lundquist number but also the mode wave number and the current sheet width. The latter is a function of time because of the sheet thinning (Furth et al., 1963; Coppi et al., 1976). Comisso et al. (2016) showed that the mode that takes the least time to surpass the linear phase dominates the plasmoid instability.

In phase-3, the plasmoid instability enters a nonlinear growth phase in which fast magnetic reconnection begins. The magnetic reconnection becomes impulsive and bursty, breaking the original current sheet into multiple shorter sheets and secondary plasmoid instabilities occur. If no continuous driving, the system enters a low-energy state by quickly converting magnetic energy to other forms (gas internal energy or energetic particles). Eventually, the reconnection stops. Note, we are not discussing the turbulent reconnection (Lazarian & Vishniac, 1999). Nor do we include turbulence in our simulations. It is a separate topic for future explorations.

However, the CMR mechanism includes more than just the reconnection process itself. Simply speaking, it works through a series of reconnection events initiated by an appropriate geometry. As shown in K21 (see their figure 24), the first two reconnection events happen at the two ends of the compression pancake, with each end ejecting a gas parcel away from the pancake. The ongoing collision (orthogonal to the midplane) creates a field X-point between the pancake end and the parcel. Each X-point thus triggers magnetic reconnection, resulting in two outward reconnected fields. The outward field that faces the compression pancake joins its counterpart from the other end and forms a big field loop enclosing the pancake (also see Figure 1). At this point, two additional processes happen simultaneously. First, the field loop pinches the compression pancake, trying to squeeze it to the center (its symmetry axis). Second, the ongoing cloud-cloud collision continues to compress the pancake. The pancake is essentially a big current sheet and the continuous compression is essentially the sheet thinning process.

Now, two scenarios for each of the two processes shall follow. First, the field loop can either overcome the thermal pressure in the compression pancake or fail to do so. The outcome is either a filament along the symmetry axis or the largely unchanged pancake. Note that the pancake still has a wrapping field which should show up in observations as a field reversal. Second, the plasmoid instability may or may not develop timely in the compression pancake. Following the above description of magnetic reconnection, the development depends on the Lundquist number, the mode wave number, and the thinning of the pancake. Regardless of the plasmoid instability, the enclosing field loops should bind everything into a single structure if magnetic tension dominates. For example, the high-resolution runs in K21 (see their figure 25) generated many plasmoids along the midplane. They merged as they moved to the center due the enclosing field loops.

Given the above discussion, we need to consider three separate concepts when comparing our models with the K21 model. The first is whether the two reconnections at the two ends of the compression pancake are triggered and a field loop enclosing the pancake follows. Hereafter we call this the seeding-phase of CMR. The second is whether plasmoids develop along the collision midplane. Hereafter we call this the plasmoid-phase of CMR. The third is whether the field loops are able to squeeze the pancake and form a filament. Hereafter we call this the pinching-phase of CMR. With these definitions, we are in a better position to compare our models with the K21 model.

Because we see field loops in MW0 (§4.1 and Figure 6) and MW1 (§4.3 and Figure 9), we believe that the seeding-phase of CMR happens in the two models, which is what we conclude in §4. In the same section, we also notice that the loops fail to squeeze the compression layer into a central dense structure, indicating that the pinching-phase of CMR does not happen. In §4.2, we compare the different force terms and find that the critical tension force is not strong enough to dominate the thermal pressure, which is why the pinching-phase fails. This failure is also evidenced by the large plasma $\beta$ in our models compared to that in K21. Table 1 lists key initial conditions for the K21 fiducial model, our fiducial model MW0 (for CMR1), and our strong field model MW1 (for CMR1). One can see that our models have a plasma $\beta$ that is far greater than that in K21 (the sound speed is superAlfvénic).

There is a geometrical artifact that could impact the seeding-phase and the pinching-phase. Although we include AMR, the mesh refinement is performed after the cloud initialization on the coarser base grid. Consequently, the clouds have a boxy shape. At the beginning of the collision, the contacting geometry is a flat interface instead of a point. Therefore, the first loop, albeit still being created by the seeding-phase, has to confine a larger pancake than that in the K21 model. It is likely taking more time and energy. To make things worse, the follow-up cloud boundaries are like step-functions. So the seeding-phase is like an episodic process. However, the seeding-phase in K21 was more like a continuous process because of the circular geometry of the cloud boundary. So we suspect that the continuous seeding-phase in K21 was able to create a large number of uninterrupted field loops that kept pinching the gradually-growing pancake. Moreover, since we do not include self-gravity, the colliding clouds quickly expand along the field lines and become sparser. They are less capable of overcoming the magnetic pressure and transporting materials to the collision midplane. So it becomes harder to accumulate dense gas.

The plasmoid-phase, however, does not happen in our models, except for CMR1 in MW1. In K21, the plasmoid-phase happened in all of their models (e.g., see their figures 21(a) and 25). Based on the above discussion, we know that our models have experienced the seeding-phase. Therefore, the triggering of magnetic reconnection is present in our models. It should provide an initial perturbation. Below, we provide some qualitative discussion about why we do not see the plasmoid-phase in most of our models.

First, if we focus on K21 and MW0, we see that MW0 has a factor of 8 lower resolution. It is possible that coarser grid cells cannot capture plasmoid modes with large wave numbers. If we consider the steady-state Sweet-Parker context (Equation 11), we see that the wave number $k$ is positively related to the Lundquist number $S_{L}$. MW0 has a factor of 30 higher Lundquist number than K21, which means a plasmoid mode of smaller size scales. So we probably need a factor of 2 higher resolution than K21, meaning another 4 levels of AMR in our simulations. Future work with the desired resolving power that also correctly initializes the spherical cloud morphology should be able to address this issue.

MW0 has a factor of 700 higher plasma $\beta$ than K21 (see Table 1). The role of $\beta$ in the onset of plasmoid instability is not fully explored. A recent work by Ni et al. (2012) investigated the onset of the instability as a function of plasma $\beta$ and Lundquist number $S_{L}$. They found that the higher the $\beta$, the smaller the critical Lundquist number for the instability. In their case with $\beta=50$, they found a critical $S_{L}$ of $\sim 10^{3}$. However, in our case, even with $\beta=23$ and $S_{L}\sim 10^{4}$ in MW0, we do not see the onset of the instability. But we notice that Ni et al. (2012) assumed an adiabatic equation of state while we assumed an isothermal equation of state. So the thermal energy converted from the magnetic energy is instantly lost. Nonetheless, if we still follow the $\beta-S_{L}$ trend found by Ni et al. (2012), the fact that the K21 model with $\beta=0.033$ and $S_{L}\sim 10^{3}$ exhibited the plasmoid instability indicates that MW0 should also show the instability. Then, the fact that we do not see the instability implies that $\beta$ is not the cause.

The discussion above also indicates that $S_{L}$ is probably not the reason MW0 does not show the plasmoid-phase of CMR. However, as we increase $S_{L}$ by another factor of 3 in MW1, we start to see plasmoids along the collision interface in CMR1 which has the highest $S_{L}$ among the four collision sites. The plasmoid development indicates an elevated critical $S_{L}$ for the instability. The elevation is probably not due to the plasma $\beta$, as discussed above.

A major difference between our model and K21 is that the colliding clouds follow the rotation of the disk. In our fiducial model, the CMR sites are in the flat rotation part of the disk. The rotation velocity is $\sim$200 km s^-1. Also, gas at different radii feels a differential rotation which tries to tear up the cloud, making it harder for the magnetic tension to pull the dense gas. This effect directly impacts the pinching-phase of CMR. For the plasmoid-phase, it is unclear how the differential rotation promotes or prevents the development of plasmoids.

We run another test simulation (hereafter MW2) with a factor of 4 slower rotation. All other conditions are the same as MW0. Figure 10 shows the field strength $\lvert{\bf B}\rvert$ for CMR1-CMR4 sites at $t=0.02$ (0.4 Myr) and $t=0.03$ (0.6 Myr). From the figure, we clearly see plasmoids in all CMR sites. CMR1 has more plasmoids than other sites. But they are smaller than those in other sites. The CMR4 plasmoid is the largest at $t=0.03$. For comparison, Figure 11 shows the same quantity for MW0. No plasmoids develop in the compression layer. This comparison shows that reducing the disk rotation indeed helps the onset of the plasmoid instability. The pinching-phase of CMR is still absent because of the high plasma $\beta$.

If we re-examine the setup in K21, their initial $\textbf{v}\times\textbf{B}$ were actually curl-free. So the convective term in the induction Equation (7) vanished in K21. So in their case, the field diffusion dominated. However, in our case, the convective term dominates the induction equation, which is also manifested by the large magnetic Reynolds number $R_{m}$ in our models. In Table 1, we compute $R_{m}$ using the collision velocity. $R_{m}$ becomes a factor of 20 larger if we use the disk rotation velocity. Therefore, it is possible that the disk rotation limits the initial growth of the reconnection (phase-1, see §5.1). It is also possible that the plasmoid instability is suppressed if the convection rate exceeds the plasmoid growth rate. More explorations are needed in the future.

In fact, the nonequilibrium model in Comisso et al. (2016) indicated that magnetic reconnection depends on the actual thinning process of the current sheet (also see Furth et al., 1963; Coppi et al., 1976). For our models, the thinning process is largely determined by the cloud-cloud collision and the magnetic field configuration. One of the key features of CMR is that it is a path of current sheet creation in the ISM. Therefore, the inflow velocity to a current sheet originates from the colliding velocity in CMR. On the other hand, the modified-BSS field in our model has an increasing width between reversed fields as a function of increasing radii (see Figure 2). So it might be harder for CMR to produce dense gas in the outer disk. An example in Comisso et al. (2016) showed that the fastest mode property depends on the sheet thinning time scale, the Lundquist number, and is a logarithmic function of the timescale, the Lundquist number, the initial sheet width, and the initial perturbation amplitude (see their equation 19). So it is a very complex process. On top of that, one should also consider the differential rotation in a disk environment. However, if the colliding clouds are self-gravitated entities and immune to the differential rotation, CMR may be more prominent.