Superdiffusion of Cosmic Rays in Compressible Magnetized Turbulence

To understand the process of magnetic line separation in a turbulent medium, i.e., wandering of magnetic fields, it is necessary to consider the anisotropy of MHD turbulence. The scaling for incompressible turbulence was proposed in GS95 for the setting of turbulence having velocity $v_{\rm inj}$ equal to the Alfvén velocity $v_{A}$ at the injection scale. This setting corresponds to the Alfvén Mach number $M_{A}=v_{\rm inj}/v_{A}$ equal to unity. The anisotropy of turbulence in GS95 was given by the relation

where $k_{\parallel}$ and $k_{\bot}$ are wavevectors parallel and perpendicular to the magnetic field. Originally, in GS95 these vectors were defined with respect to the mean magnetic field, i.e. in the global system of reference, which is still a frequent confusion among CRs researchers who try to use the GS95 relation. In fact, the theory of turbulent reconnection in LV99 presents a different outlook on the nature of MHD motions. It suggests that the magnetic field presents a collection of eddies that mix up magnetic fields perpendicular to the direction of their local ambient magnetic field. In this setting, the mean global magnetic field gets irrelevant, while the "critical balance" condition introduced by GS95 has a very simple physical interpretation (i.e. the eddy turnover time $l_{\bot}/v_{l}$ equals the wave period $l_{\parallel}/v_{A}$) that is induced by the eddy motion. Above, $v_{l}$ is turbulent velocity at scale $l$ and $v_{A}$ is Alfvén speed. This type of dynamics is possible as the turbulent reconnection reconnects the magnetic field of the eddy within one eddy turnover time and thus it is easier to mix magnetic field lines in the direction perpendicular to the magnetic field rather than bend the field lines. In other words, in strong MHD turbulence, the energy is channeled along the path of the minimal resistance along the perpendicular direction. Naturally, in this case, the spectrum of hydro motions is Kolmogorov, i.e., $v_{l}\propto l_{\bot}^{1/3}$, where $l_{\bot}$ is the perpendicular scale of the turbulent eddy. Incidentally, this scaling of the turbulence anisotropy is not observable in the global reference frame defined by mean magnetic fields, due to the averaging effect (Cho & Vishniac, 2000). At the same time, the CRs sample the magnetic field locally and experience the aforementioned anisotropy.

The importance of the local system of reference was demonstrated numerically in a seminal paper by (Cho & Vishniac, 2000). The turbulence scaling for sub-Alfvénic turbulence was introduced in Lazarian & Vishniac (1999):

where $L_{\rm inj}$ is the turbulence injection scale.

This anisotropy (see Eq. 2) is crucial as it gives an insight into how the separation of magnetic field lines grows when moving along the magnetic field lines.

Following Lazarian & Yan (2014), we firstly consider the case that the parallel mean free path $\langle\lambda_{\parallel}\rangle$ of CRs is much larger than the injection scale of turbulence. Consequently, the scattering of CRs is negligible, and their diffusion perpendicular to the mean magnetic field is determined by the divergence of magnetic field lines.

Supposing that when CRs moves a distance $x$ in the direction parallel to local magnetic field lines, the field lines spread out in the perpendicular direction with distance $l_{\bot}$. Consequently, one can express the rate of field line diffusion as:

where $\sqrt{\langle z^{2}\rangle}$ is the ensemble averaged diffusion perpendicular to magnetic field lines. Considering the anisotropy relation given by Eq. 2, we get:

The solution of Eq. 4 is

which indicates that the diffusion $\sqrt{\langle z^{2}\rangle}$ of magnetic field lines increases as $x^{3/2}$. Owing to the fact the scattering is negligible here, CRs are freely streaming along magnetic field lines with the velocity $c\mu$, where $c$ is the light speed and $\mu$ is the pitch angle cosine so that $x\sim c\mu t$ (see Fig. 1), here $t$ is time. Consequently, we have the superdiffusion in the direction perpendicular to magnetic field:

Note, this is valid only for strong sub-Alfvénic turbulence, i.e., for perpendicular scales that are smaller than $l_{\rm tr}=L_{\rm inj}M_{A}^{2}$. Magnetic field wandering is the source of perpendicular superdiffusion of CRs as presented in Eq. 6. As was established in LV99 the perpendicular squared displacement of the magnetic field is proportional to the cube of the path of the displacement along the magnetic field, i.e. $x^{3}$, which resembles the relation of the squared separation of particles with cubic of time, i.e. $t^{3}$ present in the hydrodynamic turbulence. The latter is known as the Richardson diffusion (Richardson, 1926). The close relation between the LV99 dispersion of magnetic field lines and the Richardson diffusion in a magnetized fluid is established in Eyink et al. (2013). Physically, the analogy between the hydrodynamic behavior and that of MHD arises from the fast magnetic reconnection in turbulent fluids that allows free eddy motions in the direction perpendicular to the local direction of the magnetic field (Lazarian et al., 2020). The superdiffusion of CRs in MHD turbulence arises from the Richardson diffusion of magnetic field lines in terms of magnetic field wandering (Lazarian & Yan, 2014).

Weak sub-Alfvénic turbulence (i.e., $l_{\bot}>l_{\rm tr}$) is wave-like and requires a different treatment (Lazarian & Vishniac, 1999):

which reflects the usual diffision behavior. The special feature of this diffusion is that it takes place on scales $x>L_{\rm inj}$, but the step of the diffusion is $M_{A}L_{\rm inj}<L_{\rm inj}$. This reflects the anisotropy of turbulence on the injection scale predicted in (Lazarian & Vishniac, 1999).

When CRs move along magnetic field lines in a non-ballistic way and have mean free path $\langle\lambda_{\parallel}\rangle$ less than $L_{\rm inj}$, the propagation along magnetic field lines is diffusive. This diffusion is imprinted, as discussed in Lazarian & Yan (2014), in the CRs’ diffusion both in the directions parallel and perpendicular to the local mean magnetic field. However, the mean angle of the magnetic field lines is changing with $x$ and therefore we have parallel diffusion $\langle x^{2}\rangle$ and perpendicular diffusion $\langle z^{2}\rangle$:

where $D_{\|}$ and $D_{\bot}$ are parallel and perpendicular coefficients, respectively (see Fig. 1). Considering that the projection of the mean free path $\langle\lambda_{\parallel}\rangle$to the perpendicular direction is approximately $\langle\lambda_{\parallel}\rangle l_{\bot}/x$, one can get $D_{\bot}\approx D_{\|}l_{\bot}/x$. Consequently, we have:

where $l_{\bot}/x\propto t^{1/2}$ came from magnetic field wondering (see Eq. 6¹¹1Eq. 6 and Eq. 9 can expressed in terms of the distance parallel to local magnetic field lines: $\displaystyle\langle z^{2}\rangle$ $\displaystyle\propto M_{A}^{4}x^{3},~{}~{}{M_{A}\leq 1}$ (10) $\displaystyle\langle z^{2}\rangle$ $\displaystyle\propto D_{\|}x^{5/2},~{}~{}{M_{A}>1}$ ). As demonstrated in Lazarian & Yan (2014) , the diffusion of CRs in respect to the local and global system of reference can differ. This was confirmed by a recent numerical study in Maiti et al. (2021). Anticipating the approximate nature of the scaling employed, we do not attempt to present the exact numerical factor for the dependence.

In super-Alfvénic condition, turbulence is hydro-like in the range of scales [$l_{A}$, $L_{\rm inj}$] because the dynamics of magnetic fields is dominated by turbulent motions for which the magnetic back reaction is negligible. Above, $l_{A}=L_{\rm inj}M_{A}^{-3}$, is the scale at which the turbulent velocity equals the Alfvén speed.

The role of magnetic field becomes more important at scales smaller than $l_{A}$ but larger than the dissipation scale (Lazarian, 2006). In this scale range, turbulent eddy gets elongated in the direction of the local magnetic field in a way similar to the sub-Alfvénic turbulence and $l_{A}$ plays the role of injection scale: $v_{l}\propto v_{A}(l_{\bot}/l_{A})^{1/3}$. Consequently, the anisotropy relation in “critical balance" is:

and we have:

The above relations can be obtained formally considering that in the case of super-Alfvénic turbulence the injection happens scale is $l_{A}$ rather than $L_{\rm inj}$. Therefore, on the scales from the dissipation scale to $l_{A}$ one can use the relations and all what we discussed above in terms of CR propagation for sub-Alfvénic turbulence is applicable with this change.

It is important for the CR propagation that the perturbations of magnetic fields in a turbulent medium can scatter and isotropize cosmic rays. This is the source of the generally accepted parallel to magnetic field diffusion. This is not the only process that governs CR parallel diffusion, however. A new process of mirror diffusion was introduced in Lazarian & Xu (2021). Unlike the scattering, this diffusion does not cause the diffusion in CR pitch angles²²2Lazarian & Xu (2021) pointed out that the dynamics of the pitch angle similar to the pitch angle dynamics for the Transient Time Damping (TTD) acceleration discussed e.g. in Shalchi & Schlickeiser (2006).. Instead CRs bounce from the magnetic mirrors created by magnetic compression. While the earlier studies (see Kulsrud & Pearce 1969) assumed that the magnetic mirrors will induce trapping of CRs, Lazarian & Xu (2021) demonstrated that due to magnetic field wondering that we discussed above, the CRs were not trapped, but every time bounce from a new mirror and, as a result, diffuse parallel to magnetic field. This new type of diffusion, i.e. the "mirror diffusion" is particularly important for the CRs propagation new CR sources (Xu, 2021). However, for the sake of simplicity, within the present study, we do not distinguish between the two types of parallel to magnetic field diffusion.

Compressible turbulence is a non-linear phenomenon, which nevertheless allows an approximate representation as a composition of three cascades (see Beresnyak & Lazarian 2019 and ref. therein). The Alfvén cascade imposes its properties on slow mode turbulence cascade, while the fast mode cascade evolves mostly on its own (Goldreich & Sridhar, 1995; Lithwick & Goldreich, 2001; Cho et al., 2002; Cho & Lazarian, 2003). The exchange of energy between different modes was proven numerically to be insignificant (Cho & Lazarian, 2002) and this substantially simplifies the practical treatment of MHD turbulence. Lazarian & Vishniac (1999) showed that Alfvénic turbulence induces magnetic field wandering. Therefore, the CRs’ superdiffusion is dominated by Alfvénic modes of turbulence. Fig. 2 presents an example of CRs’ superdiffusion using the method explained in § 4. Initially the spatial separation between CRs is one pixel in the direction perpendicular to the mean magnetic field field. The separation, however, explosively grows up when CRs travel along the magnetic field. However, fast modes dominate the scattering, which can affect the scaling of superdiffusion (see Eq. 6). Therefore we will investigate superdiffusion in compressible MHD turbulence.

To trace the trajectories of CRs, we use the method developed in Beresnyak et al. (2011) and Xu & Yan (2013). Magnetic fields are extracted directly from MHD turbulence simulations described above. Considering the fact that the relativistic particles have speed much higher than the plasma’s Alfvén speed, we treat the magnetic field as stationary and neglect the effect of electric field. At each position of a test particle, we compute the Lorentz force on each particle:

where $\boldsymbol{v}$ is the particle’s velocity, $q$ is particle’s charge, $m$ stands for the relativistic mass, $c$ is the light speed, and $\boldsymbol{B}$ is the local magnetic field. The trajectory is then obtained by integrating the Lorentz force.

The integration employs the Bulirsch-Stoer method with adaptive time step (Press et al., 1986). In the case of that the position of a test particle does not locate at integer grid, the corresponding magnetic fields are interpolated using a cubic spline routine. We also adopt periodic boundary conditions.

To investigate the effect of different MHD turbulence modes, we employ the mode decomposition method proposed in Cho & Lazarian (2003). The decomposition is performed in Fourier space by projecting the velocity’s Fourier components on the direction of the displacement vectors $\hat{\boldsymbol{\zeta_{A}}}$, $\hat{\boldsymbol{\zeta_{f}}}$, and $\hat{\boldsymbol{\zeta_{s}}}$ for Alfvén, fast, and slow modes, respectively. The displacement vectors are defined as:

where wavevectors $\boldsymbol{k}_{\parallel}$ and $\boldsymbol{k}_{\bot}$ are the parallel and perpendicular to the mean magnetic field $\boldsymbol{B}_{0}$, respectively. $d=(1+\beta/2)^{2}-2\beta\cos^{2}\vartheta$, $\beta=2(M_{A}/M_{S})^{2}$, and $\cos\vartheta=\hat{\boldsymbol{k}}_{\parallel}\cdot\hat{B}_{0}$.

Fig. 3 presents the kinetic energy spectrum. The velocity is decomposed into fast, slow, and Alfvénic modes using the decomposition method given in § 4. Because magnetic field fluctuation is proportional to velocity fluctuation: $|b_{k}|=(B_{0}v_{k}/c_{p})|\boldsymbol{B_{0}}\times\hat{\boldsymbol{\zeta}}|$, the spectrum also reflects magnetic field’s properties. Here $b_{k}$ and $v_{k}$ are the magnetic field fluctuation and velocity fluctuation in Fourier space, respectively. $c_{p}$ denotes the propagation speed of the slow, fast, Alfvénic mode and $\hat{\boldsymbol{\zeta}}$ is the corresponding displacement vector. The slope of Alfvénic mode’s spectrum is close to -5/3 as a result of the Kolmogorov scaling. The slope is insensitive to the value of $M_{A}$ and $M_{S}$, although the spectrum’s amplitude increases for large $M_{S}$ cases due to more injected kinetic energy. As for fast mode, the slope gets steeper than $-5/2$ being close to the value of $-2$. $-2$ usually corresponds to shocked gas in supersonic condition and Cho & Lazarian (2003) found a slope of $-3/2$ for subsonic turbulence. The steeper slope indicates fast mode dissipates its energy quickly. However, a discrepancy of the slope has been reported by Kowal & Lazarian (2010) and Kempski & Quataert (2021), who also found $k=-2$ for subsonic turbulence. Our results agree with the latter studies. In terms of the slow mode, it behaves in a way similar to Alfvén mode. Its slope is, therefore, still -5/3.

Additionally, to investigate the anisotropy of each mode, we employ the second-order structure function $SF(R,z)$ defined in the local reference frame, which is widely used to statistically describe a turbulent field;

where $\boldsymbol{v}(\boldsymbol{r_{1}})$ and $\boldsymbol{v}(\boldsymbol{r_{2}})$ give the velocity at positions $\boldsymbol{r_{1}}$ and $\boldsymbol{r_{2}}$. We define the local magnetic field following Cho et al. (2002):

where $\boldsymbol{\tilde{B}}$ is the local magnetic field defined by a cylindrical coordinate system $R$ and $Z$. In this system, axis are defined as $R=|\hat{\boldsymbol{Z}}\times(\boldsymbol{r_{1}}-\boldsymbol{r_{2}})|$ and $Z=\hat{\boldsymbol{Z}}\cdot(\boldsymbol{r_{1}}-\boldsymbol{r_{2}})$ with $\hat{\boldsymbol{Z}}=\boldsymbol{\tilde{B}}/|\boldsymbol{\tilde{B}}|$.

As an example, the structure functions for simulation A1 ($M_{S}=0.62,M_{A}=0.56$) and C2 ($M_{S}=6.47,M_{A}=0.61$) are presented in Fig. 4. The calculation randomly uses half million data points. We can see that in both subsonic and supersonic conditions, the iso-contours of Alfvénic mode and slow mode are elliptical. The semi-major axis is along the $Z$ direction, which is parallel to the local magnetic field. Therefore, the Alfvénic and slow mode are anisotropic. However, for fast mode, its iso-contours are more close to isotropic, as suggested by Cho & Lazarian (2003). In Appendix A, we plot the structure function of velocity fluctuations exactly parallel and perpendicular to local magnetic fields, i.e., $SF(0,Z)$ and $SF(R,0)$, respectively. As expected for slow and Alfvén mode, the fluctuation is more significant in the direction perpendicular to the local magnetic fields. This minimum fluctuation, therefore, appears to the local magnetic field direction (Hu et al., 2021). As for fast mode, the perpendicular fluctuation is slightly higher than the parallel fluctuation. The difference, however, is insignificant and the fast mode is isotropic.

Also, we examine the relative significance of fast, slow, and Alfvénic modes in various astrophysical conditions, including subsonic warm gas and supersonic cold gas. We calculate the kinetic energy for each mode over the entire box as:

where the subscripts $A,s,f$ denote for Alfvénic, slow, and fast modes, respectively. The energy fraction for each mode is defined as its fraction in the total energy $E_{tot}=E_{A}+E_{f}+E_{s}$. The results are presented in Fig. 5. Note that we only decompose sub-Alfvénic turbulence here, as super-Alfvénic case appears no mean magnetic field so that the decomposition method is not appropriated. We find that the fast mode always has a smaller fraction of energy than the other two modes. In terms of a fixed $M_{A}\approx 0.5$, the fraction of fast mode decreases until $M_{S}\approx 2$ but get saturated at the value of $\approx 10\%$ when $M_{S}>2$. This trend exactly agrees with the change of mean free path (see Fig. 6), as fast mode is the major agency of pitch angle scattering. Our results suggest that fast mode’s fraction, as well as the mean free path, is insensitive to large $M_{S}$. In addition, the Alfvénic mode usually dominates over the other two modes in terms of energy fractions, occupying a fraction of $\approx 50\%$ for supersonic turbulence. The trend we observe is consistent with the result from Federrath et al. (2011), who found that the fraction of solenoidal turbulence (i.e., the Alfvénic mode) is a nearly a constant irrespective of $M_{S}$ using a solenoidal driving.

Another important factor, which can introduce more fraction of the fast mode, is $M_{A}$. Therefore, we fix $M_{S}\approx 6$ and investigate the fraction of fast mode as a function of $M_{A}$ in Fig. 5. We see that the fraction of fast mode increases when $M_{A}$ is large, while the fraction of Afvénic mode decreases. However, the fraction of slow mode stays at 40% around, showing no apparent relation with respect to $M_{A}$.

To determine the mean free path of particles, we inject 1000 test particles, which are sufficient for the statistics (Xu & Yan, 2013), at random initial positions and initial pitch angles 0 degrees at a snapshot of the MHD simulation after the turbulent cascade is fully developed. The Larmor radius $r_{L}=u/\Omega$ is set as 1/50 of cube size $L$, where $\Omega=(qB)/(\gamma mc)$ is the frequency of a particle’s gyromotion ($\gamma$ is the particle’s Lorentz factor). We trace the change of pitch angle until it achieves 90 degrees. The corresponding averaged distance traveled along the local magnetic field is taken as the parallel mean free path.

In Fig. 6, we present the mean free path $\langle\lambda_{\parallel}\rangle$ as functions of $M_{S}$ and $M_{A}$. We firstly fix $M_{A}\approx 0.5$ and find $\langle\lambda_{\parallel}\rangle$ increases until $M_{S}\approx 2$. The increment of $\langle\lambda_{\parallel}\rangle$ corresponds to weaker scattering, which indicates that the fast mode’s fraction becomes less significant. However, when $M_{S}>2$, $\langle\lambda_{\parallel}\rangle$ stops increasing but being saturated at the value of 4 or 5 around and fast mode’s fraction has little variation. In supersonic condition $M_{S}\approx 6$, $\langle\lambda_{\parallel}\rangle$ decreases in a power-law manner as $M_{A}^{-2}$. As fast mode is the primary agent for scattering, a large $M_{A}$, therefore, corresponds to a small mean free path. It suggests that a observed small mean free path in supersonic condition is mainly contributed by relatively weak magnetic fluctuation, i.e., large $M_{A}$.

Fig. 7 shows the evolution of $\langle\mu\rangle$, i.e., the cosine of pitch angle as a function of time, averaged over 1000 particles. Obviously, the particles are scattered by magnetic fluctuations so that the averaged $\langle\mu\rangle$ monotonically decreases. We can also see that the decrease of $\langle\mu\rangle$ is slower for low $M_{A}$ cases, showing that the increased $M_{A}$ leads to enhanced efficiency in particle scattering. It can be understood as that the particles’ gyroresonant scattering at a smaller $M_{A}$ is less efficient so that the corresponding mean free path is larger. Indeed, we find the mean free path parallel to the local magnetic field $\langle\lambda_{\parallel}\rangle$ are $12.24L_{\rm inj}$, $3.14L_{\rm inj}$, $1.28L_{\rm inj}$, and $0.92L_{\rm inj}$ for the cases of $M_{A}=0.34$, $0.56$, $0.78$, and $0.87$, respectively. Here $L_{\rm inj}=0.5L$ is the injection scale.

Perpendicular propagation of CRs perpendicular magnetic fields has been a long-standing problem. While the cross-field diffusion due to resonance scattering is extremely slow, Jokipii & Parker (1969) identified the magnetic field wandering as the dominant cause of CR displacement perpendicular to the galactic magnetic field.

The quantitative study of the perpendicular propagation of CR became possible after the laws of magnetic wandering were identified in Lazarian & Vishniac (1999). This stimulated further quantitative research in the field. In the works that followed Yan & Lazarian (2008); Lazarian & Yan (2014), perpendicular diffusion was divided into superdiffusion into scales less than the turbulence injection scale³³3For super-Alfvénic turbulence one should use $l_{A}$ instead of the actual injection scale (Lazarian & Yan, 2014). and normal diffusion on scales larger than turbulence injection scale. The former can be subdivided into two sub-cases, depending on whether the mean free path is larger or smaller than the turbulence injection scale. It was also predicted in Lazarian & Yan (2014) that, depending on the relation the laws of the perpendicular propagation should change. As the mean free path $\langle\lambda_{\|}\rangle$ changes, the relation between the perpendicular displacements changes from $\sqrt{\langle z^{2}\rangle}\propto t^{3/2}$ to $\sqrt{\langle z^{2}\rangle}\propto t^{3/4}$. Both laws are superdiffusive, even though the rates of CRs dispersion growth are different.

The pioneering studies numerically of CRs’ (super)perpendicular and parallel diffusion include Zimbardo et al. (2006); Pommois et al. (2007); Xu & Yan (2013). Xu & Yan (2013) focused on the study of superdiffusion in subsonic and sub-Alfv’enic turbulence with post-processed resonant slab fluctuations introduced in some cases. The study confirmed that both diffusive and superdiffusive propagation of CRs scale with $M_{A}^{4}$, which is different from the naive estimates in some accepted studies. However, Cohet & Marcowith (2016) reported the absence of superdiffusion in sub-Alfvénic case $M_{A}<0.7$, which maybe caused by limited range of strong MHD turbulence. A more recent numerical study in Maiti et al. (2021) is complimentary to our study of superdiffusion in the current paper. In particular, Maiti et al. (2021) perform the decomposition of MHD turbulence with various Alfvén Mach numbers and compare propagation both in the global and local reference frame. It provides a study of both CRs diffusion and superdiffusion for a fixed sonic Mach number. The study showed that the superdiffusion is better observed in the local magnetic reference frame for Alfvén mode in agreement with (Lazarian & Yan, 2014).

In comparison with Maiti et al. (2021), our paper is focused on numerical testing of the laws of superdiffusion for a variety of sonic Mach numbers $M_{S}$. We explore the interplay of $M_{S}$ and $M_{A}$ for the CR propagation in the variety of turbulent regimes relevant to the multiphase ISM. To get better insight into CR propagation, we employ MHD simulations with higher numerical resolution. In particular, we investigate the significance of slow, fast, and Alfvén mode. We show that for a fixed $M_{A}$, the energy fraction of fast mode has little variation with the sonic Mach number for $M_{S}\geq 2$, leading to the stabilization of the parallel mean free path of CRs.

The superdiffusion of CRs at scales less than $L_{\rm inj}$ is an important regime since this condition holds in many astrophysical environments. For example, $L_{\rm inj}\approx 100$ pc in ISMs (Elmegreen & Scalo, 2004) and $L_{\rm inj}\approx 50$ pc in the galaxy M51 (Fletcher et al., 2011). For instance, the CRs’ acceleration processes in shocks happen on scales comparable to or smaller (e.g, $<0.1$ pc in the shock precursor; Beresnyak et al. 2009; Xu & Lazarian 2017) than $L_{\rm inj}$. Therefore, the superdiffusion tested in this work can significantly affect some of the popular ideas about CR acceleration in shocks (Lazarian & Yan, 2014; Zimbardo & Perri, 2020). Naturally, the correct description of the CR propagation and related effects requires accounting for the superdiffusion.

In addition, the effects of superdiffusion induce new elements of CR dynamics. For instance, (i) because spontaneous stochasticity of magnetic field lines, the effect retracing of CR is suppressed in turbulent environments (Yan & Lazarian, 2008). (ii) Lazarian & Xu (2021) find a new effect of CR parallel mirror diffusion in magnetic mirrors instead of CR trapping.

This work numerically confirms the perpendicular supperdiffusion in various physical conditions, including sub-Alfvénic, super-Alfvńic, subsonic, and supersonic. We perform numerical study about three MHD modes, i.e., Alfvénic, fast, and slow mode. We find that for incompressible driving adopted, the energy fraction of fast mode, which is the main agent for scattering stop increasing for $M_{S}\geq 2$. As a result, the mean free path or scattering is saturated at large $M_{S}$ (with a fixed $M_{A}$). Consequently, the scattering is inefficient, and the diverging magnetic field lines dominate the supperdiffusion at scales less than the injection scale $L_{\rm inj}$ in supersonic and sub-Alfvénic conditions. Nevertheless, we find the energy fraction of fast mode increases for super-Alfvńic driving, i.e., weak magnetic field, so that the scattering becomes significant. This finding is important in understanding the suppressed diffusion in supersonic molecular clouds (Semenov et al., 2021), as suggested by recent gamma-ray observations. As the energy fraction of fast mode is insensitive to supersonic turbulence, it is likely that the suppressed diffusion is attributed to trans-Alfvénic or super-Alfvénic turbulence in molecular clouds, which is revealed in observation (Federrath et al., 2016; Hu et al., 2019; Liu et al., 2022)