The Drag Instability in a 2D Isothermal C-shock

Since there is no background structure parallel to the shock front in a 2D plane-parallel, perpendicular shock, no fluid motion is normal to the shock flow. Resultantly, the background states of a 2D perpendicular shock are the same as those in a 1D perpendicular shock. Therefore, a detailed description of the background states can be found in GC20. For clarity, here we simply summarize the shock model to smoothly connect to the linear analysis to be presented later in the paper. We consider the gas flow toward the $+x$-direction across the shock with the magnetic field in the y-direction. The equilibrium equations for a plane-parallel shock (i.e., $\partial /\partial t=\partial /\partial y=0$) are derived from Equations (1)–(5), which are given by Equations (6)–(10) in GC20 to provide the background state of our linear analysis in the shock frame. In these equilibrium equations, GC20 also employed a strong coupling approximation under which the ion-neutral drag is balanced by the magnetic pressure gradient of the ions, i.e., $\gamma {\rho }_{n}{L}_{B}{V}_{d}={V}_{A,i}^{2}$, where ${L}_{B}\equiv {(-d\mathrm{ln}B/{dx})}^{-1}$ and ${V}_{A,i}$ is the Alfvén speed of the ions. Additionally, the equilibrium between cosmic ionization and recombination is assumed; namely, $\beta {\rho }_{i}^{2}={\xi }_{\mathrm{CR}}{\rho }_{n}$ (see Chen & Ostriker 2012 for justifications of such a choice).

Applying the zero-gradient boundary conditions ($d/{dx}=0$) far upstream and downstream (i.e., no structures in the steady pre- and postshock regions), Chen & Ostriker (2012) derived the 1D structure equation of a C-shock. Together with the preshock conditions described by n₀ (neutral number density), ${v}_{0}={V}_{i,0}={V}_{n,0}$ (shock velocity), B₀, and ${\chi }_{i0}$ (here and throughout this paper, we use the subscript zero to denote a physical quantity in the preshock region), the field compression ratio ${r}_{B}\equiv B/{B}_{0}={V}_{i,0}/{V}_{i}$, as well as the neutral compression ratio ${r}_{n}\equiv {\rho }_{n}/{\rho }_{n,0}={V}_{n,0}/{V}_{n}$, can be solved, and ${\rho }_{i}$, ${\rho }_{n}$, B, V_i , and V_n can be subsequently obtained throughout the shock width. Following GC20, we place the shock front at x = 0 pc for convenience to plot and refer to shock properties as a function of x. In this setup of the problem, the background drift velocity ${V}_{d}={V}_{i}-{V}_{n}\lt 0$ inside the C-shock (i.e., within the smooth shock transition). As in GC20, we adopt the 1D steady C-shock model shown in Figure 3 of Chen & Ostriker (2012) as the fiducial model for the background state in the shock frame, with the preshock parameters ${n}_{0}=500$ cm⁻³, ${v}_{0}=5$ km s⁻¹, ${B}_{0}=5\mu$ G, and ${\chi }_{i0}=10$. The left panel of Figure 1 shows ${r}_{n}/{r}_{B}$ in the fiducial model, which is the same as Figure 1 in GC20.

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We now consider the perturbation eigenvector given by $U{(\omega ,{k}_{x},{k}_{y})\equiv (\delta {\rho }_{i},\delta {v}_{x,i},\delta {v}_{y,i},\delta {B}_{x},\delta {B}_{y},\delta {\rho }_{n},\delta {v}_{x,n},\delta {v}_{y,n})}^{T}$ multiplied by $\exp [i({k}_{x}x+{k}_{y}y+\omega t)]$ under the WKBJ approximation, where k_x is the longitudinal wavenumber, k_y is the transverse wavenumber, and ωおめが is the eigenvalue evaluated in the shock frame. By substituting these perturbations and the background states into Equations (1)–(5), the following linearized equations are obtained (see Gu et al. 2004 and GC20 for the 1D case):

$$\begin{eqnarray}&&{PU}=i\omega U,\end{eqnarray} \tag{ 6 }$$

where

$$\begin{eqnarray}P=\left[\begin{array}{cccccccc}-{{ik}}_{x}{V}_{i}-2\beta {\rho }_{i} & -{{ik}}_{x}{\rho }_{i} & -{{ik}}_{y}{\rho }_{i} & 0 & 0 & {\xi }_{\mathrm{CR}} & 0 & 0\\ -{{ik}}_{x}\displaystyle \frac{{c}_{s}^{2}}{{\rho }_{i}} & -{{ik}}_{x}{V}_{i}-\gamma {\rho }_{n} & 0 & {{ik}}_{y}\displaystyle \frac{{V}_{A,i}^{2}}{{B}_{y}} & -{{ik}}_{x}\displaystyle \frac{{V}_{A,i}^{2}}{{B}_{y}} & -\gamma {V}_{d} & \gamma {\rho }_{n} & 0\\ -{{ik}}_{y}\displaystyle \frac{{c}_{s}^{2}}{{\rho }_{i}} & 0 & -{{ik}}_{x}{V}_{i}-\gamma {\rho }_{n} & 0 & 0 & 0 & 0 & \gamma {\rho }_{n}\\ 0 & {{ik}}_{y}{B}_{y} & 0 & -{{ik}}_{x}{V}_{i} & 0 & 0 & 0 & 0\\ 0 & -{{ik}}_{x}{B}_{y} & 0 & 0 & -{{ik}}_{x}{V}_{i} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -{{ik}}_{x}{V}_{n} & -{{ik}}_{x}{\rho }_{n} & -{{ik}}_{y}{\rho }_{n}\\ \gamma {V}_{d} & \gamma {\rho }_{i} & 0 & 0 & 0 & -{{ik}}_{x}\displaystyle \frac{{c}_{s}^{2}}{{\rho }_{n}} & -{{ik}}_{x}{V}_{n}-\gamma {\rho }_{i} & 0\\ 0 & 0 & \gamma {\rho }_{i} & 0 & 0 & -{{ik}}_{y}\displaystyle \frac{{c}_{s}^{2}}{{\rho }_{n}} & 0 & -{{ik}}_{x}{V}_{n}-\gamma {\rho }_{i}\end{array}\right].\end{eqnarray} \tag{ 7 }$$

Without the loss of generality, we adopt a constant wavenumber k_x of $1/0.015$ pc ${}^{-1}(\equiv {k}_{\mathrm{fid}}$), as used in GC20. The choice of k_x is made to satisfy the WKBJ approximation, as demonstrated in the right panel of Figure 1, where ${k}_{x}{L}_{p}$ and ${k}_{x}{L}_{B}$ are shown to be much larger than unity (also refer to Figure 1 in GC20).

Figure 2 shows the growth rate ${{\rm{\Gamma }}}_{\mathrm{grow}}$ (=Im[$\omega ]\lt 0$; left panel) and its corresponding wave frequency ${\omega }_{\mathrm{wave}}$ (=Re[$\omega ]$; right panel) of the unstable wave mode as a function of ${k}_{y}/{k}_{x}$ in the shock frame, at the location of $x\approx 0.2$ pc in the middle of the C-shock. It is the only unstable mode among all eigenmodes computed from Equation (6). When the longitudinal wavelengths are not exceptionally short (i.e., ${k}_{x}={k}_{\mathrm{fid}}$ and 5k_fid in the figure), the growth rate decreases approximately with increasing ${k}_{y}/{k}_{x}$. However, when k_x is as large as 20k_fid (green curve) and 50k_fid (red curve), the growth rate decreases and even drops to zero, except when ${k}_{y}/{k}_{x}$ is sufficiently large in this particular example, which corresponds to a jump of the wave frequency around ${k}_{y}/{k}_{x}\gtrsim 12.7$, as shown in the right panel of Figure 2. We explore the underlying physics for the mode properties with simplified dispersion relations in the following subsection.

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The eigenvalue problem for Equation (6) amounts to a complicated equation containing a polynomial of ωおめが to the eighth power. To make the physical analysis of the unstable mode extractable, we attempt to numerically solve Equation (6) by removing as many terms as possible in the matrix P but still obtain the same growth rate and wave frequency as those shown in Figure 2. After this exercise, we realize that Equation (6) can be further reduced to

$$\begin{eqnarray}&&({{\rm{\Gamma }}}_{i}+2\beta {\rho }_{i})\displaystyle \frac{\delta {\rho }_{i}}{{\rho }_{i}}+{{ik}}_{y}\delta {v}_{i,y}-{\xi }_{{CR}}\displaystyle \frac{{\rho }_{n}}{{\rho }_{i}}\displaystyle \frac{\delta {\rho }_{n}}{{\rho }_{n}}=0,\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&({{\rm{\Gamma }}}_{i}+\gamma {\rho }_{n})\delta {v}_{i,y}-\gamma {\rho }_{n}\delta {v}_{n,y}=0,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{n}\displaystyle \frac{\delta {\rho }_{n}}{{\rho }_{n}}+{{ik}}_{x}\delta {v}_{n,x}+{{ik}}_{y}\delta {v}_{n,y}=0,\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&-\gamma {\rho }_{i}{V}_{d}\displaystyle \frac{\delta {\rho }_{i}}{{\rho }_{i}}+{{ik}}_{x}{c}_{s}^{2}\displaystyle \frac{\delta {\rho }_{n}}{{\rho }_{n}}+({{\rm{\Gamma }}}_{n}+\gamma {\rho }_{i})\delta {v}_{n,x}=0,\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&-\gamma {\rho }_{i}\delta {v}_{i,y}+{{ik}}_{y}{c}_{s}^{2}\displaystyle \frac{\delta {\rho }_{n}}{{\rho }_{n}}+({{\rm{\Gamma }}}_{n}+\gamma {\rho }_{i})\delta {v}_{n,y}=0,\end{eqnarray} \tag{ 12 }$$

where ${{\rm{\Gamma }}}_{i}=i(\omega +{k}_{x}{V}_{i})$ is the eigenvalue in the comoving frame of the ions and ${{\rm{\Gamma }}}_{n}=i(\omega +{k}_{x}{V}_{n})$ is the eigenvalue in the comoving frame of the neutrals. Since the collisional rate for an ion with the ambient neutrals $\gamma {\rho }_{n}$ is tremendously large in a weakly ionized cloud, Equation (9) implies that $\delta {v}_{i,y}\sim \delta {v}_{n,y}$, which allows Equation (12) to be further simplified to

$$\begin{eqnarray}&&{{ik}}_{y}{c}_{s}^{2}\displaystyle \frac{\delta {\rho }_{n}}{{\rho }_{n}}+{{\rm{\Gamma }}}_{n}\delta {v}_{n,y}\approx 0,\end{eqnarray} \tag{ 13 }$$

and consequently, Equation (8) can be approximated to

$$\begin{eqnarray}&&({{\rm{\Gamma }}}_{i}+2\beta {\rho }_{i})\displaystyle \frac{\delta {\rho }_{i}}{{\rho }_{i}}+\displaystyle \frac{{k}_{y}^{2}{c}_{s}^{2}}{{{\rm{\Gamma }}}_{n}}\displaystyle \frac{\delta {\rho }_{n}}{{\rho }_{n}}-{\xi }_{{CR}}\displaystyle \frac{{\rho }_{n}}{{\rho }_{i}}\displaystyle \frac{\delta {\rho }_{n}}{{\rho }_{n}}\approx 0.\end{eqnarray} \tag{ 14 }$$

The absence of the induction equations for $\delta B$ and the momentum equation for $\delta {v}_{i,x}$ in the above reduced set of linear Equations (8)–(12) indicates that the disturbance of the ion flow along the shock direction and that of magnetic fields plays a minor role in the dynamical evolution of the 2D drag instability.

In a 2D perpendicular shock, the linearized equations should approach the 1D result when ${k}_{y}\ll {k}_{x}$ (i.e., a transversely large-scale mode). The drag instability can occur in a 1D steady C-shock (GC20). Specifically, for the definite occurrence of the instability in 1D, the rate of the mode ${{\rm{\Gamma }}}_{n}$ observed in the comoving frame of the neutrals is considerably smaller than both the recombination rate $2\beta {\rho }_{i}$ and the ion-neutral drift rate across a distance of one wavelength (i.e., ${k}_{x}| {V}_{d}|$), whereas it is considerably larger than the neutral collision rate with the ions (i.e., $\gamma {\rho }_{i}$) and the sound-crossing rate over one wavelength (i.e., ${k}_{x}{c}_{s}$). When the aforementioned conditions are satisfied, the 2D linearized equations (i.e., Equations (10), (11), and (14)) are reduced to those for a 1D perpendicular shock, which yield the simplified dispersion relation (Gu et al. 2004; GC20)

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{n} & \approx & \pm \displaystyle \frac{(1+i)}{2}\sqrt{{k}_{x}{V}_{d}\gamma {\rho }_{i}}\\ & = & \pm \displaystyle \frac{\left(1+i\right)}{2}\sqrt{\tfrac{{k}_{x}}{{L}_{B}}{V}_{A,n}},\end{array}\end{eqnarray} \tag{ 15 }$$

where the positive sign corresponds to a growing wave. As pointed out by GC20, the wave frequency in the shock frame ${\omega }_{\mathrm{wave}}$ is dominated by ${k}_{x}{V}_{n}$ instead of the imaginary part of ${{\rm{\Gamma }}}_{n}$ due to the fast background flow across the shock width downstream.

On the other hand, in the regime where ${k}_{y}\gg {k}_{x}$ (i.e., a transversely small-scale mode) such that ${k}_{y}{c}_{s}$ is much larger than $2\beta {\rho }_{i}$, ${k}_{x}{V}_{i}$, and ${k}_{x}{V}_{n}$, Equation (14) suggests that the terms associated with ionization, recombination, and the background shock flows are less significant, revealing a mode with the wave frequency Re[ωおめが]$\approx {k}_{y}{c}_{s}$, much larger than Im[$\omega ]$. Therefore, Equation (14) suggests the relation $\delta {\rho }_{i}/{\rho }_{i}\approx \delta {\rho }_{n}/{\rho }_{n}$. In this regime, the in-phase relation between $\delta {\rho }_{i}$ and $\delta {\rho }_{n}$ is no longer maintained by the ionization–recombination equilibrium in the regime of a small ${k}_{y}/{k}_{x}$ but is a consequence of the fast acoustic wave along the background magnetic fields, ¹ i.e., the so-called slow mode, because ${c}_{s}\lt {V}_{A,n}$ in our fiducial model (e.g., Shu 1992). This result, along with Equations (10), (11), and (13), yields the following simplified dispersion relation:

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{n}\approx \pm {\left(-{{ik}}_{x}| {V}_{d}| \gamma {\rho }_{i}-{k}^{2}{c}_{s}^{2}\right)}^{1/2}\approx \pm \left(\displaystyle \frac{1}{2}\displaystyle \frac{{k}_{x}| {V}_{d}| \gamma {\rho }_{i}}{{k}_{y}^{2}{c}_{s}^{2}}+i\right){k}_{y}{c}_{s},\end{eqnarray} \tag{ 16 }$$

where ${k}^{2}\equiv {k}_{x}^{2}+{k}_{y}^{2}$; moreover, we have used ${{\rm{\Gamma }}}_{n}^{2}\gamma {\rho }_{i}\approx {k}_{y}^{2}{c}_{s}^{2}\gamma {\rho }_{i}$ from Equation (14) and ${k}_{y}{c}_{s}\gg \sqrt{{k}_{x}| {V}_{d}| \gamma {\rho }_{i}}$ in deriving the above equation. The inequality ${k}_{y}{c}_{s}\gg \sqrt{{k}_{x}| {V}_{d}| \gamma {\rho }_{i}}$ holds in the regime where ${k}_{y}{c}_{s}\gg 2\beta {\rho }_{i}$ is being considered here, along with $2\beta {\rho }_{i}\gg \sqrt{{k}_{x}| {V}_{d}| \gamma {\rho }_{i}}$ in the fiducial model (see the left panel of Figure 3 in GC20). ²

The positive sign of Equation (16) corresponds to a growing wave with the growth rate related to the ion-neutral drag and the wave frequency ${k}_{y}{c}_{s}$ associated with the slow mode. More specifically, in the comoving frame of the neutrals, the phase velocity of the unstable wave is about $-{k}_{y}{c}_{s}/{k}_{x}\hat{x}-{c}_{s}\hat{y}$, and the group velocity is approximately $-({k}_{x}/{k}_{y}){c}_{s}\hat{x}-{c}_{s}\hat{y}$. Hence, in this regime, while both the phase and signal of the unstable wave propagate mainly along the background field lines at the sound speed, the signal also propagates upstream slowly at a speed much smaller than the sound speed. In the shock frame, the phase velocity of the unstable wave is about $[-{k}_{y}{c}_{s}/{k}_{x}\,+{V}_{n}]\hat{x}+[-{c}_{s}+({k}_{x}/{k}_{y}){V}_{n}]\hat{y}\approx -{k}_{y}{c}_{s}/{k}_{x}\hat{x}-{c}_{s}\hat{y}$.

Based on the above simplified dispersion relations for unstable modes on different transverse scales, we may be able to interpret the results illustrated in Figure 2. When ${k}_{y}/{k}_{x}\to 0$, both the growth rate and wave frequency converge to the 1D result for the drag instability (GC20), as illustrated by the flat curves for ${k}_{y}/{k}_{x}\lt 1$ in Figure 2. However, as ${k}_{y}/{k}_{x}$ increases, a new property of the growing mode associated with 2D emerges when the acoustic wave along the background field line, i.e., the slow mode, becomes important. This happens when ${c}_{s}{k}_{y}\sim {k}_{x}{V}_{n}$, as suggested by the transition of the dispersion relation from Equation (15) to Equation (16). Physically, it occurs when the slow-mode rate ${c}_{s}{k}_{y}$ introduced by the additional dimension along the background field in the y-direction becomes comparable to the shock-crossing rate ${k}_{x}{V}_{n}$ in the x-axis through the C-shock. Owing to this transition at the large k_y , the jump of the wave frequency from a negative to a positive value occurs at ${k}_{y}/{k}_{x}\sim {V}_{n}/{c}_{s}\equiv {\hat{k}}_{\mathrm{jump}}\approx 16.84$ at $x\approx 0.2$ pc, which agrees with the right panel of Figure 2. While we show the value of ${\hat{k}}_{\mathrm{jump}}$ at one particular x, the range of ${\hat{k}}_{\mathrm{jump}}$ within a C-shock can be estimated by the preshock conditions. The V_n decreases from v₀ at the beginning of a shock to ${v}_{0}/{r}_{f}$ at the end of a shock, where r_f is the final compression ratio of the neutral density in the postshock region given approximately by $\sqrt{2}{v}_{0}/{v}_{A,n,0}$ (Chen & Ostriker 2012). Hence, ${\hat{k}}_{\mathrm{jump}}$ decreases from ${v}_{0}/{c}_{s}$ to $({v}_{0}/{r}_{f})/{c}_{s}\,\approx {B}_{0}/(\sqrt{4\pi {\rho }_{n,0}}{c}_{s})$. In our fiducial model, ${\hat{k}}_{\mathrm{jump}}$ changes from about 25 to 3 across the shock width. In the typical environments of star-forming clouds, ${v}_{0}\,\sim$ 1–6 km s⁻¹, ${n}_{0}\sim$ 100–1000 cm⁻³, ${B}_{0}\sim 5\mbox{--}10$ μみゅーG, and the temperature is about 10 K (e.g., see Table 1 in Chen & Ostriker 2012). Therefore, ${\hat{k}}_{\mathrm{jump}}$ can decrease from ∼10–30 at the beginning of a C-shock to ∼1.5–5 at the end of a C-shock.

Table 1. MTG${}_{{k}_{y}}$ for Various k_y and ${\theta }_{0}$ in the Fiducial Model

${k}_{y}/{k}_{\mathrm{fid}}$	${\theta }_{0}$	Lshock (pc)	${\omega }_{\mathrm{wave}}$ (1/s)	${k}_{y}/{k}_{x}$	MTG${}_{{k}_{y}}$
0.1	70°	0.48	−3e−11	0.035–0.005	10.0
0.1	45°	0.40	−2e−11	0.052–0.007	10.3
0.1	20°	0.24	−2e−11	0.052–0.005	10.9
1	70°	0.48	−3e−11	0.35–0.05	9.57
1	45°	0.40	−3e−11	0.35–0.05	9.49
1	20°	0.24	−3e−11	0.35–0.05	9.55
10	70°	0.48	−5e−11	1.97–0.32	4.4
10	45°	0.40	−5e−11	2.4–0.32	3.8
10	20°	0.24	−5e−11	2.4–0.08	3.4

Note. The shock width L_shock is estimated using Equation (A19) in Chen & Ostriker (2012). Given a uniform value of k_y (in units of k_fid), the mode frequency ${\omega }_{{\rm{wave}}}$ and the corresponding range of k_x (x) (presented in terms of ${k}_{y}/{k}_{x}$) associated with the MTG at a constant k_y (i.e., MTG${}_{{k}_{y}}$) of the drag instability are listed for the fiducial model of a steady C-shock with various oblique angles.

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As ${k}_{y}/{k}_{x}$ further increases to an even larger value, the behavior of the growing mode follows the dispersion relation described by Equation (16). This is demonstrated in Figure 3, which zooms in to the region where ${k}_{y}/{k}_{x}\gt 20$ in Figure 2. The dashed lines present the growth rate Re[${{\rm{\Gamma }}}_{n}]$ (left panel) and the wave frequency Im[${{\rm{\Gamma }}}_{n}-{k}_{x}{V}_{n}]$ (right panel) calculated from Equation (16). As ${k}_{y}/{k}_{x}$ increases, the exact growth rates of the unstable mode for various k_x computed from Equation (6) converge to Re[${{\rm{\Gamma }}}_{n}]$, while the exact wave frequencies ${\omega }_{\mathrm{wave}}$ overlap with Im[${{\rm{\Gamma }}}_{n}-{{kV}}_{n}]$ in the entire zoomed-in region. The slow-mode frequency ${k}_{y}{c}_{s}$, as illustrated by the dashed–dotted lines, is also plotted for each value of k_x . As ${k}_{y}/{k}_{x}$ increases, the wave frequency ${\omega }_{\mathrm{wave}}$ in all cases converges toward ${k}_{y}{c}_{s}$, confirming that the slow mode emerges and predominates on small transverse scales for the drag instability. Note that the positive wave frequency for ${k}_{y}/{k}_{x}\gt {\hat{k}}_{\mathrm{jump}}$ indicates a growing wave propagating upstream within the 2D perpendicular C-shock, in stark contrast to the growing mode for ${k}_{y}/{k}_{x}\lt {\hat{k}}_{\mathrm{jump}}$, which is more dynamically dominated by the wave convected with the downstream flow across the shock width, similar to the result for a 1D C-shock.

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In the case of a moderate value of k_x (i.e., ${k}_{x}={k}_{\mathrm{fid}}$ and 5k_fid), the growth rate decreases almost monotonically with increasing k_y when ${k}_{y}/{k}_{x}\gtrsim {\hat{k}}_{\mathrm{jump}}$, in approximate agreement with the trend described by Equation (16). On the other hand, in the regime where ${k}_{y}/{k}_{x}\lt 1$, the growth rate is small or nearly zero when k_x is large enough (i.e., k_x  = 50 and $100{k}_{\mathrm{fid}}$ in Figure 2) for the small-scale gas pressure perturbation along the shock direction to suppress the growth (Gu et al. 2004; GC20). However, a growing mode exists when ${k}_{y}/{k}_{x}\gtrsim 12.7$. Its growth rate rises significantly during the wave frequency transition and then declines, according to Equation (16), as ${k}_{y}/{k}_{x}$ increases to a large value. We are unable to derive a simple dispersion relation to describe the rise of the growth rate during the wave frequency transition. It is because more terms in Equations (8)–(12) become comparably important and thus cannot be neglected as a proper approximation for deriving Equation (16). Nevertheless, this complexity of the distinct behaviors of the growth rate at the wave frequency transition can be unraveled by studying the phase differences between perturbed quantities, which will be presented in the next subsection.

We illustrate the phase differences in Figure 4 corresponding to the growth rate and wave frequency shown in Figure 2. Figure 4 includes the phase difference between the density perturbations, as well as those between the neutral velocity and density perturbations. The flat part of the curves in the range of ${k}_{y}/{k}_{x}\lesssim 1$ corresponds to the flat part of the curves in Figure 2, resembling the 1D drag instability for the 2D unstable modes. In contrast to the cases for the small k_x (i.e., blue and orange curves), the phase difference between $\delta {\rho }_{i}$ and $\delta {\rho }_{n}$ in the cases for the large k_x (green and red curves in panel (a) of Figure 4) exhibit a dramatic change when ${k}_{y}/{k}_{x}\approx {\hat{k}}_{\mathrm{jump}}=16.8$, corresponding to the jump of the wave frequency. Since the wave frequency of the mode ${\omega }_{\mathrm{wave}}$ is small around this jump transition, the Doppler-shift frequency ${k}_{x}{V}_{i}$ with a large k_x becomes nonnegligible compared to the ionization rate in the continuity equation of the ions. Consequently, the phase difference between the ion and neutral density perturbations does not stay small by the ionization–recombination equilibrium but becomes large. The phase difference decreases steeply with increasing ${k}_{y}/{k}_{x}$ because the rate of the mode is predominated quickly by the slow mode $\approx {k}_{y}{c}_{s}$, resulting in the small phase difference between the ion and neutral density perturbations for a large ${k}_{y}/{k}_{x}$. It is in accordance with the result that $\delta {\rho }_{i}/{\rho }_{i}\approx \delta {\rho }_{n}/{\rho }_{n}$ when deriving Equation (16).

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The drastic change in ${\phi }_{\delta {\rho }_{i}}-{\phi }_{\delta {\rho }_{n}}$ also causes the steep change in ${\phi }_{\delta {v}_{n,x}}-{\phi }_{\delta {\rho }_{n}}$ from πぱい to $\approx 0.8\pi$ around the frequency transition (see the green and red curves in panel (b) of Figure 4) through the linearized continuity and momentum equations, leading to a bump in the growth rate of the drag instability near the frequency transition, as shown in the left panel of Figure 2. ³ Since an acoustic mode can be characterized by the phase difference πぱい between density and velocity perturbations, the nearly out-of-phase difference between $\delta {v}_{n,y}$ and $\delta {\rho }_{n}$ shown in panel (c) of Figure 4 for ${k}_{y}/{k}_{x}\gtrsim 16.8$ reconfirms the right panel of Figure 3, i.e., the emergence of the slow mode when ${k}_{y}/{k}_{x}\gtrsim {\hat{k}}_{\mathrm{jump}}$. It also agrees with Equation (13) on the relation ${{\rm{\Gamma }}}_{n}\sim {{ik}}_{y}{c}_{s}$ for ${k}_{y}/{k}_{x}\gt {\hat{k}}_{\mathrm{jump}}$, which leads to $\delta {\rho }_{n}/{\rho }_{n}\,\sim -\delta {v}_{n,y}/{c}_{s},$ i.e., out of phase between $\delta {\rho }_{n}$ and $\delta {v}_{n,y}$ for a slow mode.

It was shown by GC20 that the growth of the drag instability in a 1D perpendicular shock is limited by the short time span for an unstable wave to stay within a C-shock as the wave is convected downstream by a fast flow across the shock; specifically, the local growth rate ${{\rm{\Gamma }}}_{\mathrm{grow}}$ is much smaller than the wave frequency in the shock frame ${\omega }_{\mathrm{wave}}$. Consequently, a stronger shock with a larger shock width favors a more appreciable growth of the 1D drag instability in the typical environments of star-forming clouds investigated by the authors. However, for a 2D perpendicular shock, as has been studied in the preceding subsection, the transition of the wave frequency from a negative to a positive value allows for the presence of a slowly traveling mode with an arbitrarily small wave frequency in the shock frame when ${k}_{y}/{k}_{x}\sim {\hat{k}}_{\mathrm{jump}}$, thereby allowing a tremendous amount of time for the growth of the drag instability within a shock width in this transition regime.

We investigate this expectation by simply considering a mode with a small wave frequency ${\omega }_{\mathrm{wave}}=-{10}^{-14}$ 1/s ≪ the flow crossing time over one longitudinal wavelength ${k}_{x}{V}_{n}$. According to the eigenvalue problem, a family of unstable modes should exist with the proper combination of the wavenumbers k_x and k_y as a function of x. For the purpose of computational convenience without loss of generality, we first consider the uniform k_x given by the fiducial wavenumber ${k}_{x}={k}_{\mathrm{fid}}=$ 1/(0.015 pc) while allowing the k_y of the mode to vary with x. The results are shown in Figure 5. The resulting growth rate ${{\rm{\Gamma }}}_{\mathrm{grow}}$ is indeed larger than ${\omega }_{\mathrm{wave}}$ in most of the region within the C-shock, as illustrated in the left panel of Figure 5. The corresponding profile of ${k}_{y}/{k}_{x}$ decreases from 25 to 3 with increasing x, in agreement with the transition wavenumber ratio ${k}_{y}/{k}_{x}\approx {\hat{k}}_{\mathrm{jump}}={V}_{n}/{c}_{s}$, where V_n decreases across the shock width due to density compression, as has been explained in Section 2.2. Next, we consider the uniform k_y given by 15k_fid (i.e., a transversely small-scale mode) while allowing the k_x of the mode to vary with x. The results of the growth rate and wave frequency are considerably similar to those shown in Figure 5, affirming the slow propagation of the unstable modes with ${k}_{y}/{k}_{x}\approx {\hat{k}}_{\mathrm{jump}}$ in a 2D perpendicular C-shock. We also study the "counterpart" mode with the small positive wave frequency ${\omega }_{\mathrm{wave}}={10}^{-14}$ 1/s for the same case of either the uniform k_x or uniform k_y . We find that the profiles of the growth rate and ${k}_{y}/{k}_{x}$ are almost identical to those presented in Figure 5, except that the unstable wave slowly propagates upstream rather than downstream due to the sign change of ${\omega }_{\mathrm{wave}}$.

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As the unstable wave propagates slowly at the phase velocity ${v}_{{ph},x}$, the unstable mode is nearly comoving with the shock during the cloud lifetime, which is typically about tens of millions of years (Engargiola et al. 2003; Blitz et al. 2007; Kawamura et al. 2009; Murray 2011; Miura et al. 2012; Meidt et al. 2015; Jeffreson & Kruijssen 2018). Consequently, the MTG is no longer a proper measure for the growth of the slowly propagating modes in the shock frame. For instance, it takes about 82 million yr for the unstable mode that we have considered with ${\omega }_{\mathrm{wave}}=-{10}^{-14}$ 1/s and uniform ${k}_{x}={k}_{\mathrm{fid}}$ to travel across the entire shock width, ⁴ which is ≳the cloud lifetime. It is evident from Figure 2 that there are modes that are almost/exactly stationary in the shock frame (i.e., ${\omega }_{\mathrm{wave}}\approx 0$) and have ${{\rm{\Gamma }}}_{\mathrm{grow}}\sim {10}^{-13}$ 1/s $\sim 1/0.3$ 1/Myr. Hence, the growth of these modes is approximately given by $\exp ({{\rm{\Gamma }}}_{\mathrm{grow}}t)$, which can be substantially larger than unity if t is some fraction of the cloud lifetime. In contrast, the maximum growth of the unstable mode in the 1D case is limited by the shock width and thus given by MTG, which is merely about 9.9 in the fiducial model (GC20). The 2D mode ought to grow to a nonlinear phase according to the expectation from the linear theory.

Figure 6 shows the amplitude of perturbations for the unstable mode presented in Figure 5. The perturbed quantities are normalized by the y-component of the magnetic field perturbation $| \delta {B}_{y}| /B$. It is evident from the figure that within the shock width where the unstable mode exists, the density perturbations $\delta {\rho }_{n}/{\rho }_{n}$ and $\delta {\rho }_{i}/{\rho }_{i}$ dominate over the others, implying that the growth of density perturbations plays a decisive role in the dynamical evolution of the instability. Although this outcome of promoting the local density growth of a wave turns out to be the same as that in the 1D case (GC20), the drag instability is coupled with the slow mode in the regime ${k}_{y}/{k}_{x}\gt {\hat{k}}_{\mathrm{jump}}$ for a 2D perpendicular shock, as described in the previous section. This can also be realized from Figure 6, which shows $| \delta {v}_{n,y}| \gg | \delta {v}_{n,x}|$ and $| \delta {v}_{i,y}| \gg | \delta {v}_{i,x}|$ as a result of the presence of the slow mode propagating along the background magnetic fields in the y-direction.

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It is worth noting that $\delta {B}_{x}$ and $\delta {B}_{y}$ are generated from B due to the gradients of the longitudinal motion ${\partial }_{y}(\delta {v}_{i,x})$ (i.e., magnetic wiggling) and ${\partial }_{x}(\delta {v}_{i,x})$ (i.e., magnetic compression), respectively, through the induction equation such that $| \delta {B}_{x}| /| \delta {B}_{y}| ={k}_{y}/{k}_{x}$, in agreement with the ratios $| \delta {B}_{x}| /| \delta {B}_{y}|$ in Figure 6 and ${k}_{y}/{k}_{x}$ in the right panel of Figure 5. However, given their small amplitudes, magnetic field perturbations simply grow passively through the induction equation and do not play any significant role in the dynamics of the drag instability, confirming that the absence of the induction equation in the reduced set of linearized equations from Equations (8)–(12) still leads to the same growth rate and wave frequency of the drag instability. Finally, as a reminder, in the transversely large-scale regime where ${k}_{y}/{k}_{x}\ll {V}_{n}/{c}_{s}={\hat{k}}_{\mathrm{jump}}$ , the property of the drag instability in 2D resembles that in 1D, according to the analysis in the previous subsection. Therefore, the density growth also governs the dynamics of the drag instability for the transversely large-scale mode (GC20).

When ${k}_{y}/{k}_{x}\lesssim {V}_{n}/{c}_{s}\equiv {\hat{k}}_{\mathrm{jump}}$ (i.e., transversely large-scale modes), the drag instability behaves similarly between 1D and 2D perpendicular C-shocks. When ${k}_{y}/{k}_{x}\gtrsim {\hat{k}}_{\mathrm{jump}}$ (i.e., transversely small-scale modes), a new property of the growing mode emerges for 2D perpendicular C-shocks even for a large k_x , resulting from the ion-neutral drag coupled with the slow mode along the background magnetic field. When ${k}_{y}/{k}_{x}\sim {\hat{k}}_{\mathrm{jump}}$, the sign transition of wave frequency occurs. This leads to an unstable mode of a small wave frequency and thus enables a slowly propagating mode to grow substantially within a C-shock, unlimited by the shock width. The linear result suggests that the density growth dominates the evolution of the perturbation growth driven by the drag instability for the modes on both transversely large and small scales.

In this section, we study the local stability of a 2D steady oblique shock in which magnetic fields are not normal to the shock flow. We adopt the background states constructed by Chen & Ostriker (2012). That is, the preshock flow is still along the x-direction, the shock front in the y–z plane, and the preshock magnetic field in the x–y plane (${B}_{0}={B}_{x,0}\hat{x}+{B}_{y,0}\hat{y}$), at an initial oblique angle ${\theta }_{0}$ to the inflow (${B}_{y,0}/{B}_{x,0}=\tan {\theta }_{0}$). For a steady, parallel-plane oblique shock (i.e., ${\partial }_{t}\,={\partial }_{y}={\partial }_{z}=0$), Chen & Ostriker (2012) derived the equilibrium equations for ${r}_{n}\equiv {\rho }_{n}/{\rho }_{n,0}={v}_{0}/{V}_{n,x}$, ${r}_{B}\equiv {B}_{y}/{B}_{y,0}$ (B_x is constant because ${\rm{\nabla }}\cdot B=0$), and ${r}_{{ix}}\equiv {v}_{0}/{V}_{i,x}$ from Equations (1)–(5) based on the strong coupling approximation and ionization–recombination equilibrium. Note that r_B is not equal to r_ix for an oblique shock. By solving the equilibrium equations with a set of preshock conditions, the background states including ${\rho }_{i}$, ${\rho }_{n}$, ${V}_{i,x}$, ${V}_{n,x}$, ${V}_{i,y}$, ${V}_{n,y}$, and B_y can be subsequently obtained as a function of x within the shock and postshock regions. Since the field line is not perpendicular to the incoming shock flow, both the field line and gas inflow will continuously change their directions relative to their initial direction as they move across the shock width until they reach the postshock region (see Figure 7 in the case of the preshock conditions given by the fiducial model). As expected, ${\boldsymbol{B}}$ and ${{\boldsymbol{V}}}_{i}$ are first tilted toward the shock front (i.e., a large oblique angle shown in Figure 7) due to the earlier compression of ${\rho }_{i}$, followed by the subsequent tilt of ${{\boldsymbol{V}}}_{n}$ toward the shock front due to the later compression of ${\rho }_{n}$ within the shock. Finally, ${V}_{i}={V}_{n}$ in the postshock region as in the preshock region but with a final nonzero oblique angle. The shock width is given by the region between the pre- and postshock. It is evident from Figure 7 that as ${\theta }_{0}$ decreases, the compression ratio increases and shock width decreases. Besides, ${\boldsymbol{B}}$ in the three cases of ${\theta }_{0}$ shown in Figure 7 quickly becomes more or less normal to the shock flow by shock compression. All of these results are anticipated from the analysis in Chen & Ostriker (2012).

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With the background states, the linearized equations are given by

$$\begin{eqnarray}&&{OU}=i\omega U,\end{eqnarray} \tag{ 17 }$$

where

$$\begin{eqnarray}O=\left[\begin{array}{cccc}-{{ik}}_{x}{V}_{i,x}-{{ik}}_{y}{V}_{i,y}-2\beta {\rho }_{i} & -{{ik}}_{x}{\rho }_{i} & -{{ik}}_{y}{\rho }_{i} & 0\\ -{{ik}}_{x}\displaystyle \frac{{c}_{s}^{2}}{{\rho }_{i}} & -{{ik}}_{x}{V}_{i,x}-{{ik}}_{y}{V}_{i,y}-\gamma {\rho }_{n} & 0 & {{ik}}_{y}\displaystyle \frac{{V}_{A,i}^{2}}{{B}_{y}}\\ -{{ik}}_{y}\displaystyle \frac{{c}_{s}^{2}}{{\rho }_{i}} & 0 & -{{ik}}_{x}{V}_{i,x}-{{ik}}_{y}{V}_{i,y}-\gamma {\rho }_{n} & -{{ik}}_{y}\displaystyle \frac{{B}_{x}}{{B}_{y}}\displaystyle \frac{{V}_{A,i}^{2}}{{B}_{y}}\\ 0 & {{ik}}_{y}{B}_{y} & -{{ik}}_{y}{B}_{x} & -{{ik}}_{x}{V}_{i,x}-{{ik}}_{y}{V}_{i,y}\\ 0 & -{{ik}}_{x}{B}_{y} & {{ik}}_{x}{B}_{x} & 0\\ 0 & 0 & 0 & 0\\ \gamma {V}_{d,x} & \gamma {\rho }_{i} & 0 & 0\\ \gamma {V}_{d,y} & 0 & \gamma {\rho }_{i} & 0\\ 0 & {\xi }_{\mathrm{CR}} & 0 & 0\\ -{{ik}}_{x}\displaystyle \frac{{V}_{A,i}^{2}}{{B}_{y}} & -\gamma {V}_{d,x} & \gamma {\rho }_{n} & 0\\ {{ik}}_{x}\displaystyle \frac{{B}_{x}}{{B}_{y}}\displaystyle \frac{{V}_{A,i}^{2}}{{B}_{y}} & -\gamma {V}_{d,y} & 0 & \gamma {\rho }_{n}\\ 0 & 0 & 0 & 0\\ -{{ik}}_{x}{V}_{i,x}-{{ik}}_{y}{V}_{i,y} & 0 & 0 & 0\\ 0 & -{{ik}}_{x}{V}_{n,x}-{{ik}}_{y}{V}_{n,y} & -{{ik}}_{x}{\rho }_{n} & -{{ik}}_{y}{\rho }_{n}\\ 0 & -{{ik}}_{x}\displaystyle \frac{{c}_{s}^{2}}{{\rho }_{n}} & -{{ik}}_{x}{V}_{n,x}-{{ik}}_{y}{V}_{n,y}-\gamma {\rho }_{i} & 0\\ 0 & -{{ik}}_{y}\displaystyle \frac{{c}_{s}^{2}}{{\rho }_{n}} & 0 & -{{ik}}_{x}{V}_{n,x}-{{ik}}_{y}{V}_{n,y}-\gamma {\rho }_{i}\end{array}\right],\end{eqnarray} \tag{ 18 }$$

where ${V}_{A,i}$ is still defined as ${B}_{y}^{2}/4\pi {\rho }_{i}$ in terms of the field component parallel to the shock front, which follows the same definition in the case of the 2D perpendicular shock. When the x-component of the preshock B field is zero, B_x  = 0, ${V}_{i,y}=0$, and ${V}_{n,y}=0$ (thus ${V}_{d,y}=0$) throughout the shock. Thus, the matrix O in the above equation for an oblique shock is reduced to the matrix P in Equation (6) for a perpendicular shock.

We solve the above eigenvalue problem for the fiducial model of the C-shock with a preshock oblique angle ${\theta }_{0}\lt 90^\circ$. As in the perpendicular shock, we find that there is one unstable mode among the eight eigenmodes within a C-shock. Figure 8 shows the growth rate ${{\rm{\Gamma }}}_{\mathrm{grow}}$ and wave frequency in the shock frame ${\omega }_{\mathrm{wave}}$ of the unstable mode as a function of ${k}_{y}/{k}_{x}$ in the case of ${\theta }_{0}=45^\circ$ with four values of k_x at $x\approx 0.2$ pc, approximately in the middle of the oblique C-shock. The figure exhibits two clear general features. First of all, the growth rates and wave frequencies do not vary significantly but have values similar to the case of k_y  = 0 in the range of ${k}_{y}/{k}_{x}\lt 1$, as shown by the flat part of the curves. The second feature is that regardless of the difference in k_x , the growth rates of the unstable mode all drop to zero when ${k}_{y}/{k}_{x}\approx 9.3$, and the corresponding wave frequencies exhibit a discontinuity at the same wavenumber ratio, which turns out to be exactly equal to $| {V}_{d,x}| /{V}_{d,y}$ at $x\approx 0.2$ pc in the fiducial model. We investigate the reason in the next subsections.

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Since ${V}_{n,y}$ and ${V}_{i,y}$ other than ${V}_{n,x}$ and ${V}_{i,x}$ are present in an oblique shock, the Doppler-shift rates in the comoving frame of the ions ${{\rm{\Gamma }}}_{i}$ and neutrals ${{\rm{\Gamma }}}_{n}$ are expressed by $i(\omega +{k}_{x}{V}_{i,x}+{k}_{y}{V}_{i,y})$ and $i(\omega +{k}_{x}{V}_{n,x}+{k}_{y}{V}_{n,y})$, respectively. Following the same approach in Section 2.2 for perpendicular shocks, we focus on the particular mode with ${{\rm{\Gamma }}}_{n}$ smaller than $2\beta {\rho }_{i}$ and kV_d but larger than $\gamma {\rho }_{i}$. The linear equations expressed in Equation (17) may be reduced to the same set of Equations (10), (11), (13), and (14), except that the additional drag term $\gamma {V}_{d,y}\delta {\rho }_{i}$ appears in the y-component of the momentum equation for the neutrals, namely (see Equation (13)),

$$\begin{eqnarray}&&-\gamma {\rho }_{i}{V}_{d,y}\displaystyle \frac{\delta {\rho }_{i}}{{\rho }_{i}}+{{ik}}_{y}{c}_{s}^{2}\displaystyle \frac{\delta {\rho }_{n}}{{\rho }_{n}}+{{\rm{\Gamma }}}_{n}\delta {v}_{n,y}\approx 0.\end{eqnarray} \tag{ 19 }$$

Moreover, as the recombination rate $2\beta {\rho }_{i}\gt {{\rm{\Gamma }}}_{n}$ for this mode, Equation (14) is reduced to $\delta {\rho }_{i}/{\rho }_{i}\approx (1/2)\delta {\rho }_{n}/{\rho }_{n}$ due to the ionization equilibrium. The resulting dispersion reads

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{n}^{2}=-i\displaystyle \frac{\gamma {\rho }_{i}}{4}({k}_{x}{V}_{d,x}+{k}_{y}{V}_{d,y})-{k}^{2}{c}_{s}^{2}.\end{eqnarray} \tag{ 20 }$$

In the regime where ${k}_{y}/{k}_{x}\ll 1$ and ${{\rm{\Gamma }}}_{n}\gt {k}_{x}{c}_{s}$, the above equation is reduced to Equation (15), the typical growth rate and wave frequency for the 1D drag instability (Gu et al. 2004; GC20), but with a background state for an oblique shock. As in the case of a 2D perpendicular shock discussed in the preceding section, this mode behavior of an oblique shock is consistent with the flat part of the curves for both growth rate and wave frequency in Figure 8. As ${k}_{y}/{k}_{x}$ increases to the moderate value $| {V}_{d,x}| /{V}_{d,y}\equiv {\hat{k}}_{\mathrm{discon}}=9.3$, Figure 8 shows that the growth rates decline and become zero around ${\hat{k}}_{\mathrm{discon}}$. Besides, the corresponding wave frequencies decrease gradually with increasing ${k}_{y}/{k}_{x}$ and then abruptly drop around ${\hat{k}}_{\mathrm{discon}}$. This behavior of the growth rate and wave frequency can be understood in terms of Equation (20), which admits a special solution Re$[{{\rm{\Gamma }}}_{n}]=0$ when ${k}_{x}{V}_{d,x}+{k}_{y}{V}_{d,y}=0$. The dynamics attributed to the absence of the growing mode at ${k}_{y}/{k}_{x}={\hat{k}}_{\mathrm{discon}}$ is that the drag force in the x-direction (i.e., $\gamma {V}_{d,x}\delta {\rho }_{i}$) acts out of phase with the drag force in the y-direction for the neutrals (i.e., $\gamma {V}_{d,y}\delta {\rho }_{i}$), therefore suppressing the density enhancement in the neutral continuity equation and thus quenching the drag instability.

To explain the property of the wave frequency shown in the right panel of Figure 8, we investigate the behavior of Equation (20) near ${k}_{y}/{k}_{x}={\hat{k}}_{\mathrm{discon}}$. Because ${{kc}}_{x}\gg {k}_{x}{V}_{d,x}+{k}_{y}{V}_{d,y}\approx 0$ around this wavenumber ratio, the dispersion relation of the unstable mode described by Equation (20) is simplified to

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{n}=i\omega +{{ik}}_{x}{V}_{n,x}+{{ik}}_{y}{V}_{n,y}\approx \displaystyle \frac{\gamma {\rho }_{i}}{4}\displaystyle \frac{| {k}_{x}{V}_{d,x}+{k}_{y}{V}_{d,y}| }{{{kc}}_{s}}\pm {{ikc}}_{s},\end{eqnarray} \tag{ 21 }$$

where the plus (minus) sign of the imaginary part of the above expression (i.e., the wave frequency in the comoving frame of the neutrals) is taken when ${k}_{x}{V}_{d,x}+{k}_{y}{V}_{d,y}\lt 0$ ($\gt 0$). Due to this sign change, the wave frequency in the shock frame ${\omega }_{\mathrm{wave}}$ (=Re[ωおめが]) at ${k}_{y}/{k}_{x}={k}_{\mathrm{discon}}$ exhibits a discontinuity, with the difference given by $\approx 2{{kc}}_{s}$. As ${k}_{y}/{k}_{x}$ increases beyond ${\hat{k}}_{\mathrm{discon}}$ for a given k_x , Re[ωおめが] ($=-{k}_{x}{V}_{n,x}-{k}_{y}{V}_{n,y}-{{kc}}_{s}$) increases more negatively; approximately speaking, Re[ωおめが]$\approx -{k}_{y}({V}_{n,y}+{c}_{s})\propto -{k}_{y}$, in close agreement with the right panel of Figure 8.

Although we attempt to obtain the simplified dispersion relations for guiding us in understanding the key features of the growth rate and wave frequency of an unstable wave, not all of the features can be explained. Figure 8 shows that the unstable mode is suppressed when ${k}_{y}/{k}_{x}$ is sufficiently large, which is unable to be described by Equation (21), because some neglected terms in deriving the simplified dispersion relations can become comparably important for a large ${k}_{y}/{k}_{x}$. To gain more comprehensive insight into the 2D drag instability, we study the phase differences between perturbation quantities of the unstable mode as in the preceding section to complement the limited analysis from the dispersion relations. The results as a function of ${k}_{y}/{k}_{x}$ are illustrated in Figure 9.

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As in the case of perpendicular shocks, the flat part of the curves in panels (a) and (b) of Figure 9 correspond to that in Figure 8 for ${k}_{y}/{k}_{x}\lesssim 1$, arising from the fact that 2D oblique shocks resemble 1D shocks (i.e., k_y  = 0) in terms of the dynamics in the x-direction. In the y-direction, however, the phase difference between $\delta {\rho }_{n}$ and $\delta {v}_{n,y}$ changes gradually from $\approx 1.75\pi$ (equivalent to $-\pi /4$) to πぱい as ${k}_{y}/{k}_{x}$ increases from $\approx 0.01$ to ${\hat{k}}_{\mathrm{discon}}$ ($\approx 9.3$ at x = 0.2 pc). It arises because when ${k}_{y}/{k}_{x}\ll 1$, the ionization equilibrium (i.e., $\delta {\rho }_{i}/{\rho }_{i}=(1/2)\delta {\rho }_{n}/{\rho }_{n}$) and the ion-neutral drag dictate the dynamics of Equation (19), yielding the phase difference $-\pi /4$ in the y-direction as well. As ${k}_{y}/{k}_{x}$ increases for a given k_x , Equation (21) implies that the wave frequency is gradually dominated by ${{kc}}_{s}$ rather than the Doppler-shift frequency ${k}_{x}{V}_{n,x}+{k}_{y}{V}_{n,y}$. Additionally, the drag term becomes subdominant to the pressure term in Equation (19). Thus, Equation (19) leads to the relation $\delta {\rho }_{n}/{\rho }_{n}\approx -\delta {v}_{n,y}/{c}_{s};$ i.e., $\delta {\rho }_{n}$ and $\delta {v}_{n,y}$ gradually become out of phase as ${k}_{y}/{k}_{x}$ increases to ${\hat{k}}_{\mathrm{discon}}$, as illustrated in panel (c) of Figure 9. Once ${k}_{y}/{k}_{x}$ increases to ${\hat{k}}_{\mathrm{discon}}$, the phase difference between $\delta {\rho }_{n}$ and $\delta {v}_{n,y}$ jumps from πぱい to zero because of the sign change of the wave frequency. Furthermore, panel (c) of Figure 9 depicts that the phase difference between $\delta {\rho }_{n}$ and $\delta {v}_{n,y}$ is almost zero when ${k}_{y}/{k}_{x}\gtrsim {\hat{k}}_{\mathrm{discon}}$. It can be realized from Equation (19) with ${{\rm{\Gamma }}}_{n}\approx -{{ik}}_{y}{c}_{s}$, as suggested by Equation (21) for ${k}_{y}/{k}_{x}\gt {\hat{k}}_{\mathrm{discon}}$, resulting in the relation that $\delta {\rho }_{n}/{\rho }_{n}\approx \delta {v}_{n,y}/{c}_{s}$; i.e., $\delta {\rho }_{n}$ and $\delta {v}_{n,y}$ are almost in phase and are associated with an acoustic wave propagating in the y-direction. The resulting ${v}_{{ph},y}$ of the acoustic wave points to the positive y-direction in the frame comoving with the neutrals.

Panel (a) of Figure 9 also shows that when ${k}_{y}/{k}_{x}\gt {\hat{k}}_{\mathrm{discon}}$, the phase difference between $\delta {\rho }_{i}$ and $\delta {\rho }_{n}$ increases with ${k}_{y}/{k}_{x}$, which can be realized from Equation (14). Since ${{\rm{\Gamma }}}_{i}={{\rm{\Gamma }}}_{n}+i({k}_{x}{V}_{d,x}+{k}_{y}{V}_{d,y})$, the term ${k}_{x}{V}_{d,x}+{k}_{y}{V}_{d,y}$ is nearly zero at ${k}_{y}/{k}_{x}={k}_{\mathrm{discon}}$ and thus is negligible compared to the recombination rate $2\beta {\rho }_{i}$, resulting in the ionization–recombination equilibrium. However, as ${k}_{y}/{k}_{x}$ increases from ${\hat{k}}_{\mathrm{discon}}$, the term ${k}_{x}{V}_{d,x}+{k}_{y}{V}_{d,y}$, which is neglected for deriving Equation (20), increases and thus becomes comparable to and even larger than $2\beta {\rho }_{i}$ at a large value of k_y in the term ${{\rm{\Gamma }}}_{i}+2\beta {\rho }_{i}$ of Equation (14) for the ion continuity equation. Consequently, the ionization–recombination equilibrium is poorly attained, and the phase difference between $\delta {\rho }_{i}$ and $\delta {\rho }_{n}$ shifts significantly away from zero at a large value of ${k}_{y}/{k}_{x}$.

As a result of the aforementioned phase shift between $\delta {\rho }_{i}$ and $\delta {\rho }_{n}$, the x-component of the linearized momentum equation for the neutrals, i.e., Equation (11), implies that the phase difference between $\delta {\rho }_{n}$ and $\delta {v}_{n,x}$ shifts accordingly away from $\sim -\pi /4$ (i.e., a typical phase shift for the 1D drag instability; see GC20) to a large negative value as ${k}_{y}/{k}_{x}$ increases from ${\hat{k}}_{\mathrm{discon}}$. Together with the continuity equation for the neutrals described by Equation (10), the phase shift leads to the change of the growth rate with ${k}_{y}/{k}_{x}$ and yields no growth at a certain large value of ${k}_{y}/{k}_{x}$ ($\sim {10}^{2}$) where $\delta {\rho }_{n}$ and $\delta {v}_{n,x}$ are almost out of the phase, as shown in panel (b) of Figure 9. At this point, the wave is no longer unstable but is transformed into an acoustic wave, with ${v}_{{ph},x}$ pointing to the negative x-direction in the comoving frame of the neutrals. The ${v}_{{ph},y}$ of the acoustic wave points to the positive y-direction in the comoving frame with the neutrals, as was explained earlier in this subsection.

The distinct behaviors of wave frequency for a large value of ${k}_{y}/{k}_{x}$—frequency jump versus discontinuity—between Figure 2 (perpendicular shock with ${\theta }_{0}=90^\circ$) and Figure 8 (oblique shock with ${\theta }_{0}=45^\circ$) suggests that a shock with a proper range of the oblique angle between 45° and 90° can allow for the unstable modes with both behaviors of wave frequency. Figure 10 shows the results for the same shock model with ${\theta }_{0}=80^\circ$. Indeed, the right panel of the figure shows that a frequency jump from negative to positive values happens at ${k}_{y}/{k}_{x}\approx {V}_{n,x}/({c}_{s}-{V}_{n,y})\equiv {\hat{k}}_{\mathrm{jump}}\approx 27$, and a frequency discontinuity occurs at an even smaller scale where ${k}_{y}/{k}_{x}=| {V}_{d,x}| /{V}_{d,y}\equiv {\hat{k}}_{\mathrm{discon}}\approx 37$. Consequently, the growth rate of the modes with large k_x (i.e., 20k_fid and 50k_fid) rises around ${\hat{k}}_{\mathrm{jump}}$ and the growth rates all go to zero at ${\hat{k}}_{\mathrm{discon}}$, as shown and expected in the left panel of Figure 10. Note that ${\hat{k}}_{\mathrm{jump}}$ involves ${V}_{n,y}$ for an oblique shock because the Doppler-shift frequency is ${k}_{x}{V}_{n,x}+{k}_{y}{V}_{n,y}$. What happens is that as the oblique angle ${\theta }_{0}$ decreases from 90°, ${V}_{n,y}$ and thus ${V}_{d,y}$ start deviating from zero such that ${\hat{k}}_{\mathrm{jump}}\lt {\hat{k}}_{\mathrm{discon}}\lt \infty$. Consequently, the modes with behaviors of the frequency jump and discontinuity both appear in the case for ${\theta }_{0}=80^\circ$. As ${\theta }_{0}$ continues to decrease (i.e., the shock is more oblique) until ${\hat{k}}_{\mathrm{discon}}$ becomes smaller than ${\hat{k}}_{\mathrm{jump}}$, the mode with the behavior of a frequency jump disappears at ${k}_{y}/{k}_{x}={\hat{k}}_{\mathrm{jump}}$, leaving the sole phenomenon of the frequency discontinuity at ${k}_{y}/{k}_{x}={\hat{k}}_{\mathrm{discon}}$, such as the case for ${\theta }_{0}=45^\circ$. At the location of $x\approx 0.2$ pc in the fiducial model, ${\hat{k}}_{\mathrm{jump}}={\hat{k}}_{\mathrm{discon}}$ occurs between ${\theta }_{0}=78^\circ$ and $79^\circ ;$ i.e., it occurs when the shock is mildly oblique. Analogous to perpendicular shocks, the presence of ${\hat{k}}_{\mathrm{jump}}$ enables substantial growth of the slowly propagating unstable mode (i.e., unlimited by the shock width) in a mildly oblique shock. Consequently, the range of ${\hat{k}}_{\mathrm{jump}}$ is similar to that for the perpendicular shocks in the typical environments of star-forming clouds.

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Because the slowly traveling mode with substantial growth appears only in a background environment close to a perpendicular shock, the growth of the drag instability in a moderately or exceedingly oblique shock is still limited by the short time span for the unstable mode to remain within a shock. We explore this issue by comparing the C-shock model described by the fiducial model and model V06 from Table 2 in GC20. The preshock conditions of model V06 are given by ${n}_{0}=200$ cm⁻³, ${v}_{0}=6$ km s⁻¹, ${B}_{0}=10\mu$ G, and ${\chi }_{i0}=5$. Model V06 is discussed in GC20 because it exhibits a larger MTG than that of the fiducial model in a 1D shock due to its broader shock width. The MTG is easily computed for a 1D shock, whereas the same calculation involves more elaborate work for a 2D shock. In general, the growth rate ${{\rm{\Gamma }}}_{\mathrm{grow}}$ of a mode with a particular wave frequency ${\omega }_{\mathrm{wave}}$ is a function of both k_x and k_y , which vary with x in a 2D shock. In this work, we do not intend to perform an accurate analysis to identify the combination of k_x (x) and k_y (x) that provides the MTG of the unstable mode. Rather, we keep k_y uniform but allow k_x to change with x across a shock for a given wave frequency of an unstable mode. We then estimate the MTG of an oblique shock by varying the wave frequency for a few constant k_y , which we refer to as MTG${}_{{k}_{y}}$. Comparing MTG${}_{{k}_{y}}$ for different k_y and ${\theta }_{0}$ would provide guidance on how the MTG varies with the initial oblique angle ${\theta }_{0}$. The results are enumerated in Table 1 for the fiducial model and Table 2 for model V06. Given the k_y of an unstable mode and ${\theta }_{0}$ for each C-shock model, the tables display MTG${}_{{k}_{y}}$ with the corresponding mode frequency ${\omega }_{\mathrm{wave}}$, as well as the value of k_x (in terms of ${k}_{y}/{k}_{x}$) ranging from the beginning to the end of the C-shock width.

Table 2. Same as Table 1 but for the C-shock Model Given by Model V06

${k}_{y}/{k}_{\mathrm{fid}}$	${\theta }_{0}$	Lshock (pc)	${\omega }_{\mathrm{wave}}$ (1 s−1)	${k}_{y}/{k}_{x}$	MTG${}_{{k}_{y}}$
0.1	70°	2.08	−2e−11	0.062–0.011	36
0.1	45°	1.70	−2e−11	0.062–0.009	40
0.1	20°	1.0	−2e−11	0.063–0.007	47
1	70°	2.08	−3e−11	0.42–0.07	31
1	45°	1.70	−3e−11	0.42–0.07	30
1	20°	1.0	−3e−11	0.45–0.05	31
10	70°	2.08	−7e−11	1.72–0.33	7.3
10	45°	1.70	−8e−11	1.75–0.28	5.9
10	20°	1.0	−8e−11	1.73–0.24	5.7

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Table 1 shows that in the fiducial shock model, MTG${}_{{k}_{y}}$ does not vary significantly with ${\theta }_{0}$. The smaller the ${\theta }_{0}$, the larger the compression of the density and magnetic field (see Figure 7), leading to a slightly larger growth rate. However, a smaller ${\theta }_{0}$ results in a slightly narrower shock width (see Figure 7 and the parameter L_shock in Table 1). As a result of the competition between these two moderate effects, the oblique angle does not noticeably affect the overall growth of the drag instability in the fiducial model. Moreover, when ${k}_{y}/{k}_{\mathrm{fid}}\lesssim 1$, MTG${}_{{k}_{y}}$ is comparable to the MTG for the 1D shock ($=9.9$; refer to GC20) and smaller by more than a factor of 2 for ${k}_{y}/{k}_{{kid}}=10$. It can be expected from the range of the resulting ${k}_{y}/{k}_{x}$ shown in Table 1. When ${k}_{y}/{k}_{x}\ll 1$, the instability behaves as a 1D mode for which k_y  = 0. On the other hand, when ${k}_{y}/{k}_{x}\gtrsim 1$, the overall growth rate decreases within the shock, and thus MTG${}_{{k}_{y}}$ is small.

Similar to the fiducial model, Table 2 shows that MTG${}_{{k}_{y}}$ in model V06 does not vary significantly with ${\theta }_{0}$ as well. Though in the case of ${k}_{y}/{k}_{\mathrm{fid}}=0.1$, MTG${}_{{k}_{y}}$ can increase moderately from 36 for ${\theta }_{0}=70^\circ$ to 47 for ${\theta }_{0}=20^\circ$. This trend also happens in the fiducial model with ${k}_{y}/{k}_{\mathrm{fid}}=0.1$ but is less prominent (see Table 1). For a smaller ${\theta }_{0}$, the effect of the slightly larger growth rate predominates more than the effect of the slightly narrower width, resulting in a larger MTG${}_{{k}_{y}}$. Evidently, a wider shock in model V06 (see the parameter L_shock in Tables 1 and 2) allows an unstable mode to grow more and therefore makes this trend clearer. In contrast, the two competing effects are comparable for ${k}_{y}/{k}_{\mathrm{fid}}=1$ and 10; hence, MTG${}_{{k}_{y}}$ changes less noticeably with ${\theta }_{0}$. Similar to the fiducial model, the overall trend of the decrease in MTG${}_{{k}_{y}}$ with increasing k_y also happens in model V06, for the same reason. Tables 1 and 2 also show that MTG${}_{{k}_{y}}$ in model V06 is in general larger than that in the fiducial model, which is expected because the shock in model V06 has a broader shock width for the instability to acquire more time to grow. In summary, for a C-shock with the oblique angle ≲70°, we expect that the MTG of the drag instability in the fiducial model is about 10 and that in model V06 is about 30–47. Therefore, we expect that the unstable mode responsible for the MTG has ${k}_{y}/{k}_{x}\lesssim 1$.

Figure 11 shows the magnitude of the perturbations across the shock width for the unstable mode corresponding to ${k}_{y}/{k}_{\mathrm{fid}}=0.1$ and ${\theta }_{0}=45^\circ$ in Tables 1 (left panel) and 2 (right panel). It is evident from the figure that the magnitude of the density perturbation is always larger than that of the velocity and magnetic field perturbations of the unstable mode. In fact, it is true for all cases listed in Tables 1 and 2. The density enhancement is expected to play a critical role in the dynamics of the drag instability in an oblique shock as well.

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When ${\theta }_{0}\lesssim 20^\circ$ in our fiducial model, B_y and ${\rho }_{n}$ are compressed extremely quickly near the beginning and end of the C-shock width, respectively. Therefore, the WKBJ approximation with ${k}_{x}={k}_{\mathrm{fid}}$ becomes invalid in these shock regions. Except for these regions, the basic behavior of the drag instability for ${\theta }_{0}\lesssim 20^\circ$ is similar to that for ${\theta }_{0}=45^\circ$, shown in Figure 8 at $x\approx 0.2$ pc. When ${\theta }_{0}$ is smaller than the critical angle ${\theta }_{{\rm{crit}}}\approx 6^\circ$, the background state admits two additional solutions with field reversal, referred to as intermediate shocks (e.g., Wardle 1998; Chen & Ostriker 2014). In this study, we restrict ourselves to the oblique shocks without field reversal and leave the instability analysis for intermediate shocks to a future work.