Anomalous compressible mode generation by global frame projections of pure Alfven mode

In this section we review the underlying assumptions of the mode decomposition method as introduced by CL03 and the development since then. In CL03 they consider a volume $d\Omega$ in which the perturbation of magnetic field is small compared to the mean field $\delta B(d\Omega)<\langle B\rangle$, so does the density fluctuations $\delta\rho/\langle\rho\rangle<1$. Fig.1 shows how the volume $d\Omega$ is selected. Readers should be careful that once the volume is selected the mean magnetic field direction $\hat{\lambda}$ is also defined respectively. In this scenario, the small perturbation in the presence of a strong mean magnetic field will provide a linearized set of MHD equations in which three non-trivial eigenvectors would be found. In this localized box, the Alfven, slow and fast mode eigenvectors are¹¹1However, recent publication from Gan et al. (2022) suggests that a significant portion of the projected spectral powers are not in the form of propagating waves, but fluctuations with miniature frequencies. The nature of the non-wave fluctuations as dubbed in Gan et al. (2022) requires further clarifications from the theory of MHD turbulence. See Beresnyak & Lazarian (2019); Fu et al. (2022); Schekochihin (2022).:

where $\alpha=\beta\gamma/2$, $D=(1+\alpha)^{2}-4\alpha\cos^{2}\theta_{\lambda}$, $\cos\theta_{\lambda}=\hat{\bf k}\cdot\hat{\bf\lambda}$, plasma $\beta\equiv P_{gas}/P_{mag}$ measures the compressibility and $\gamma=\partial P/\partial\rho$ is the polytropic index of the adiabatic equation of state. The presence of $\hat{\bf k}$ suggests that the direction of the three mode vectors are changing as ${\bf k}$ changes. In this scenario, the perturbed quantities, say for the velocity fluctuations ${\bf v}_{1}={\bf v}-\langle{\bf v}\rangle$ can be written as:

for some power spectrum $E_{v}(k)=F^{2}_{0}=k^{-n/2}$ (Yuen et al., 2022), some anisotropy weighting function $F_{1}$ and the mode constants $C_{X}$ denoting the relative weight of the three modes. Notice that the decomposition (Eq.(1)) is a global decomposition method since the magnetic field fluctuations within the volume $d\Omega$ is not considered when computing the eigenvectors of the three modes. One of the most important consequences of performing global decomposition is the loss of the GS95 scaling for small $k$. In fact, Cho & Lazarian (2002) (see also Lazarian & Pogosyan (2012)) pointed out that in the global system of reference the anisotropic scaling is scale independent, meaning that the elongation of turbulence eddies is fixed and does not change as the eddies cascade. Another important consequence of the concept of $d\Omega$ is that, when one changes the sampling volume, e.g. from the volume in blue to that of yellow or orange in Fig.1, the weighting of the three modes will also change due to the change of the mean direction of magnetic field from volume to volume.

Notice that the selection of the volume $d\Omega$ has to fulfill the conditions as assumed in CL03: Both the sonic and Alfven Mach number within the volume must be smaller than unity. Notice that if the volume is smaller than the volume defined by the correlation length of the turbulence, the dispersion of the turbulence observables will be scaled as a function of distance according to their respective turbulence statistics, meaning that if $\rho,v$ follows GS95,

To address the issue of the locality, the community has explored a number of ways to include the local fluctuations of magnetic field during the calculation of statistics of MHD modes. For instance, one of the most notable ways of obtaining the statistics of MHD modes is to compute the local structure functions Beresnyak et al. (2005). The mathematical expression of the 3D structure function of the turbulence variable $v$ in the local frame of reference is given by:

where

The anisotropy computed throughout this manner is scale dependent and exhibit the GS95 scaling $r_{\parallel}\propto r_{\perp}^{2/3}$. The use of the local structure function correctly recovers the GS95 statistics, yet it is not possible to obtain the realization of the modes in this manner, meaning that further study of the features of the modes, e.g. how does the mode look like when projected, are prohibited when using the structure functions.

Another important way of improving the CL03 is to rewrite the turbulence variables into the linear sum of small localized patches through the wavelet transform (Kowal & Lazarian, 2010). Physically, they are looking for specific functional forms obeying the orthogonality requirement and represent the volumes as shown in Fig.1.By considering the set of orthogonal wavelets $\phi$, one can write the velocity field ${\bf v}({\bf r})$ as:

where $\phi$ is the set of wavelet functions. For (Kowal & Lazarian, 2010) they select the 12-tap Daubechies wavelet. To perform mode decomposition, they first convert the velocity field into the linear combinations of wavelets, and then proceed with the procedures of CL03 for the wavelet transformed variable. The contributions for all wavelets for a specific mode are added before the inverse Fourier transform takes place. Notice that the wavelet functions is simply a mathematical construction that may contain spatially dispatched regions (e.g. D4 and D12 of the Daubechies wavelet), for which taking the statistical calculations within the wavelet might not be physically justifiable. Nevertheless, the improved mode decomposition method proposed by Kowal & Lazarian (2010) still retrieve seemingly the correct GS95 statistics from the simulations.

In realistic MHD simulations the magnetic field lines are fluctuating within any selected volume, which is named wandering effect as in CL03. However the mode decomposition algorithm available in the community had not considered any of these wandering effect, which makes the estimation of modes be rather unrealistic for larger $M_{A}$. To model the additional effect when decomposing the modes in the global frame of reference, Lazarian & Pogosyan (2012) model the magnetic field correlation function as the linear combination of the isotropic tensor $\hat{T}_{E}$ and the axis-symmetric tensor $\hat{T}_{F}$ (See Yan & Lazarian 2004 also our Appendix for a detailed mathematical construction). In general the direct tensor of the magnetic field in the Fourier space at a given wavevector ${\bf k}$ in the local frame of reference can be written as:

The transformation from the local to global frame as modelled by LP12 is:

where $W_{I,L}$ are two modelling constants that are functions of $M_{A}$ and also ${\bf k}$.

Since $\hat{T}_{E}=\hat{T}_{C}+\hat{T}_{A}$ and $\hat{T}_{F}=\hat{T}_{C}$ (See the Appendix A.2) one can actually write the magnetic field in the Fourier space in the global frame of reference using vector notations due to the orthogonality of the base vectors:

where

(See Appendix). The modelling of LP12 simply means that, assuming if in the local frame of reference the magnetic field only has the Alfven component $\tilde{H}({\bf k})=\sqrt{E({\bf k})}C_{0}\zeta_{A}$, then the transformation from local to global reference frame simply means a vector projection:

where $C_{C}=C_{0}\sin(\theta_{f})$, $C_{A}=C_{0}\cos(\theta_{f})$ for some $\theta_{f}$ that we will explore in the coming subsection. For Alfven waves, $E({\bf k)}=-F({\bf k})$, and the relation between $C_{C,A}$ and $W_{I,L}$ can be easily derived:

Notice that when $M_{A}\ll 1$, there is no magnetic field wandering effect. In this limit, $C_{A}(M_{A}\rightarrow 0)=1$, $C_{C}(M_{A}\rightarrow 0)=0$.

The vector notation (cf. Eq.(9)) allows us to think of this problem geometrically and relates to a very important physical effect that mentioned in both Yuen et al. (2018) and more recently Yuen & Lazarian (2020): The Alfven leakage effect. Alfven leakage describes the effect that the locally Alfven component of the turbulent variables are projected as linear combinations of Alfven and non-Alfven components in the presence of a curved magnetic field averaged over selected volume (local mean magnetic field). In Yuen & Lazarian (2020) we consider the effect that the gravitational forces creates extra compression to the Alfven waves and thus some of the Alfven waves from the self-gravitating systems are transferred into compressible components. In fact, in the presence of non-trivial magnetic field structures, there exist non-zero (Yuen & Lazarian, 2020) field lines in any volume due to magnetic field wandering. Similar effect has been considered in Maron & Goldreich (2001) but being corrected by Cho & Vishniac (2000) as the ”rotation effect” by recognizing that the global frame anisotropy axis ratio is a function of $M_{A}$. However the actual relation between the local and global frame of reference has yet to be explored.

Fig.2 gives a pictorial illustration on how the Alfven leakage happens during mode decomposition. For a given magnetic field $H_{i}({\bf r})$, its Fourier transform $\tilde{H}_{i}({\bf k})$ can be written as the linear sum of Alfven and compressible components. To simplify our argument, we consider a pure Alfven wave B-field in the local frame of reference, which $\tilde{H}_{i}$ is simply proportional to the local Alfven vector $\zeta_{A,local}$, which is represented by the green vector. However if the volume $d\Omega$ is selected (the orange dash circle), the mean field is defined in $d\Omega$. In this case, if we consider a pure Alfven mode magnetic field in the local frame of reference as we show in Fig.2, the projection of the Alfven mode in the global frame of reference will be an linear combination of Alfven and compressible modes as written in Fig.2. By selecting a k-vector that points inside the plane, the local Alfven wave unit vector (green) , which is defined by the cross product between the k-vector and the local magnetic field direction, makes an angle to that of the Alfven wave unit vector defined by $\hat{k}\times\hat{\lambda}$ which we will name that angle $\delta\theta_{d\Omega}$. The projection effect causes artificial compressible modes to be detected within $d\Omega$. It is very apparent that if the magnetic field line is aligned with the mean field vector $\hat{\lambda}$ in $d\Omega$, then there is no artificial compressible modes due to the projection effect. We call this effect Alfven Leakage. This effect is artificial and does not involve the change of the cascade laws or anisotropies.

We can further model the leakage phenomenon through the use of $\delta\theta_{d\Omega}$. From Fig.2 we see that $\delta\theta_{d\Omega}$ is a measure of the average magnetic field angle dispersion in this selected volume. We can postulate that the dispersion of angles are related to the Alfven Mach number measured within $d\Omega$:

Notice that the dispersion of magnetic field angle can scale up to $M_{A}\approx 2$ as shown in the appendix of Lazarian et al. (2022b). Notice that according to the definition of the Alfven Mach number and the self-similarity of turbulence cascade, any localized calculation of statistical observables in a turbulence system can be approximated by the scale relation through the observable’s structure function. For the case of Alfven Mach number, the corresponding observable is $\delta B/B$, then

where the last equality comes from the fact that the magnetic field fluctuation is small and Kolmogorov: $\langle\delta B^{2}\rangle\sim r^{2/3}$. We can then relate the $C_{C,A}$ with Eq.(13) and (14).

We can perform a very simple numerical test to illustrate the behavior of the Alfven leakage, which can be done by the following steps:

1. 1.

   We select the local frame vectors $\zeta_{A,C}$, where they can be approximated by selecting a very small volume in the numerical simulations and use Eq.9. We shall call this volume 1.

2. 2.

   We then perform Alfven wave decomposition using $\zeta_{A}$ from simulations so that we can put $C_{A,local}=1$ in all our case (See Fig.1).

3. 3.

   We then select a larger volume with size $\sigma$. According to §2, if there is indeed the effect of Alfven leakage happening, then $C_{C}=\sin M_{A}\sim\sin(C\sigma^{1/3})$, where $C=M_{A,global}L^{-1/3}$. In this new volume, there will be another pair of $\zeta_{A,C}$ being defined. We shall call this volume 2.

4. 4.

   We plot the quantity $C_{C}^{2}=1-(\langle\hat{B}_{Alf,1}\cdot\hat{B}_{Alf,2})^{2}\rangle$. Notice we only compare the directions of $B$ within volume 1, since this calculation only makes sense there.

5. 5.

   There could be three outcome from this test

   1. (a)

      If there is no such leakage effect, $C_{C}$ should be a constant zero as we already removed all non-compressible components in the previous step.

   2. (b)

      If there is indeed $C_{C}$ but our model in the previous section is incorrect, then there should not be a dependence of $C_{C}\sim\sin(C\sigma^{1/3})$.

   3. (c)

      If the Alfven leakage effect indeed exists, we expect $C_{C}\sim\sin(C\sigma^{1/3})$.

Fig.3 shows how the $C_{C}$ behave as a function of the size of the volume $\sigma$, where we plot the regime when $\sigma$ is not in the dissipation range. Notice that here we intentionally pick simulation cubes from various numerical codes and with different conditions to show that the leakage effect is universal and rather independent to what choices of MHD code one works with. For each figure, we draw the predicted proportionality $C_{C}\sim\sin^{2}(C\sigma^{1/3})$ as the red dash curve. As we can see from these three subplots, our prediction $C_{C}\sim\sin^{2}(C\sigma^{1/3})$, which came analytically from previous sections and not from a fitting algorithm, follows the trend reasonably well.

This indicates that (1) the Alfven leakage effect actually exists even in incompressible mode turbulence (2) the leakage is smaller when one goes to smaller scales (3) our postulate that $\delta\theta_{d\Omega}\sim M_{A,local}$ is a good approximation. These three consequences indicate that the mode decomposition as proposed by CL03 requires an additional update in addressing the contributions of large scale magnetic field wandering to the relative composition of modes within the volume.

As a direct consequence of this section, the $W_{I,L}$ constants originated from Lazarian & Pogosyan (2012) as a function of $M_{A,local}$ ($\ll 1$) are given by Eq.(8) are :

In Zhang et al. (2020) the authors discussed a novel implementation called SPA in obtaining the modes from synchrotron emission maps. Their argue that, since the tensor structures for Alfven or compressible waves being different at the synthesis of the Stokes parameter, the signature of the dominance of the modes are left in the ”signature parameter”. Let us first recap the formulation of Zhang et al. (2020) and their main results (See Method section of Zhang et al. 2020). To start with, the authors consider the emissivity of the synchrotron emissions under a locally defined reference frame:

where our symbols follow the Zhang et al. (2020) notations and we employ Einstein notation of summation. Very importantly, the angle $\phi_{s}$ is the angle between the polarization vector to the currently defined x-axis of the local magnetic reference frame. In Zhang et al. (2020) they select a small area and compute the change of $\epsilon_{xx}$ as the reference frame rotates. They recognize that the variance of $\epsilon_{xx}$ contains factors that could reflect the relative contributions of MHD modes.

The variance of the emissivity contains the linear term (the first term) and the quadratic term depending on the power of the tensor $\hat{T}$:

Zhang et al. (2020) pointed out that the linear term, namely the signature parameter:

can be expressed in the following format with some constants $a_{xx},b_{xx}$ defined according to MHD theory (See Zhang et al. 2020 for details):

where the classification parameter $r_{xx}$ is defined as

In practice, we need to compute the parameter

This term contains both the ”linear” term (Eq.19) and the ”quadratic” term as defined in Zhang et al. (2020).

As discussed in LP12, two-point turbulence statistics contains three types of contributions: The spectrum which measures the cascade as a function of scales, the anisotropy which records whether there is a preferred direction for the cascade to happen, and the tensor structures which records the projection effect of the mode components. From the discussion above, the SPA technique actually did not consider two-point statistics. In particular, Eq.(21) is a one-point statistics, in which the anisotropy of the observable does not play a role in the value of the output. Since all scales are summed up when computing the mean and variances, the spectrum also do not play an important role for Eq.(21). As a result, a natural guess on how the SPA technique works is the projection effect from the tensor structures. In this scenario using the vector formulation (See Appendix) we can understand quantitatively better how the SPA technique works. Notice that, in the case of one-point statistics, the 2D (i.e. the average operator above) and 3D statistics (which we will consider later below) should be the same.

Let us consider a 3D magnetic field line written as sum of the mean and perturbed contribution in a selected volume $d\Omega$ with $M_{A,d\Omega}\ll 1$:

From now on we are going to choose the frame to be the PCA frame (See Appendix A.2, Fig.11), and assuming the line of sight direction is at the z-axis and the magnetic field in the plane of sky defines the x-axis. The x-component magnetic field dispersion, which is just the mean value of the emissivity subtracted by a constant (cf.Eq.(16)), is given by:

Notice that the only difference between the compressible and the Alfven component can be observed when we expand the dot product for the above equation:

We can model Eq.(23) via the frame definition of $\phi_{s}$ in Eq.(16), where the frame angle $\phi_{s}=0$ when the projection of magnetic field is along the x-axis:

where $\cos\gamma=\hat{\lambda}\cdot\hat{z}$ is the line of sight angle, and $A_{xx},B_{xx}$ are:

The factors within the bracket of each equation above are the geometric factors as discussed in LP12. Here we consider the general case of the leakage, which applies to both Alfven and compressible modes (See §3), i.e. the observed amplitudes of Alfven and compressible modes $C_{A,obs},C_{C,obs}$ undergo an orthogonal rotation of angle $M_{A}<1$ (See Eq.13):

The expressions inside the brackets of $A_{xx},B_{xx}$ are the geometric factors that considered in both LP12 and Zhang et al. (2020). We can see from Eq.(25) that the contributions of Alfven and compressible modes are separated when one considers the frame rotation even for $\langle\epsilon\rangle$. It is not necessary to compute Eq.(21) in extracting the contributions of the modes. Moreover, we can now quantify the contributions of modes via Eq.(26) by using the modes for $C_{A,C}$ by simply comparing the values of $A_{xx}$ and $B_{xx}$ while analyzing the observed synchrotron emission. In particular, if $M_{A}$ is small and there is no compressible mode, then $B_{xx}=0$. i.e. that contribution of the Alfven mode to $\langle\epsilon\rangle$ is frame independent (i.e. rotating the x-y plane does not alter the result) since $(\hat{\lambda}\cdot\hat{z})$ cannot be changed due to frame rotation, while that for compressible mode is a frame dependent quantity since $\hat{\lambda}\cdot\hat{x}$ is a function of the reference frame.

In Zhang et al. (2020) they study the effects of Faraday rotation to the SPA technique. Pictorially the Faraday depolarization effects shields information up to a certain distance along the line of sight. This distance has been adequately discussed in Lazarian & Pogosyan (2012) in the presence of galactic MHD turbulence and is called the Faraday screening effect (Lazarian & Yuen, 2018). Qualitatively, the SPA technique can only determine the mode fraction before the Faraday screen. However, we would like to perform the analysis based on the formalism of Lazarian & Yuen (2018).

In general, the synchrotron emission depends both on the distribution of relativistic electrons

with intensity of the synchrotron emission being

where ${\bf X}=(x,y)$ is the 2D position of sky (POS) vector and $B_{\perp}=\sqrt{B_{x}^{2}+B_{y}^{2}}$ being the magnitude of the magnetic field perpendicular to the LOS $z$. In general, $\gamma=0.5(\alpha+1)$ is a fractional power, which was a serious problem that was successfully addressed in LP12. LP12 proves that the statistics of $I(\alpha)$ is similar to that of $I(\alpha=3)$. Therefore it suffices to discuss the statistical properties of the case $\alpha=3$.

Per Lazarian & Pogosyan (2012), Synchrotron complex polarization function with Faraday rotation is given by:

where $\epsilon_{synch}$ is the emissivity of synchrotron radiation,

is the Faraday Rotation Measure ²²2It is usually more convenient to use $H_{z}=B_{z}/\sqrt{4\pi}$ for analysis. . Notice that $\rho_{rel}$ is the relativistic electron density, while $\rho_{thermal}$ is the thermal electron density. The C-factor $\approx 0.81$ (Lee et.al 2016). The projected magnetic field orientation is then given by:

where $\tan^{-1}_{2}$ is the 2-argument arc-tangent function.

For frequencies lower than $O(1GHz)$, the amplitude of the Faraday Rotation measure will exceed $2\pi$. The physical picture of the synchrotron polarization with Faraday rotation measure can be understood as: photons that passes through a section of ISM has to experience a certain amount of phase shift. If this phase shift exceeds $2\pi$, all information coming from the source is completely lost. Therefore an important concept called the Faraday screening emerges, which indicates the maximal line of sight distance that the observed synchrotron emissions can measure in the presence of line of sight magnetic field. In the case of sub-Alfvenic turbulence, the source term $P_{i}=\rho_{rel}\exp(2i\theta({\bf R},z))$ is dominated by the mean field rather than the fluctuating one. The two regimes: (1) strong and (2) weak Faraday Rotation depend on whether the ratio $L_{eff}/L$, is smaller (strong) or larger (weak) than unity:

where $\phi=\max(\sqrt{2}\sigma_{\phi},\bar{\Phi})$ with $\sigma_{\phi}$ is the dispersion of random magnetic field. The difference between the two regimes are, the Faraday rotation and the emission happens together in the former regime ($\phi=\sqrt{2}\sigma_{\phi}$), while the latter has the Faraday rotation happens after the emission of the polarization. We shall name the two regimes ”Variance-driven Faraday Rotation” (VFR) and ”Mean-field Faraday Rotation” (MFR), respectively. Notice that both regimes have been considered in Zhang et al. (2020).

Fig.5 shows a plot on how VFR and MFR could change the value of $r_{xx}$. For this current plot we intentionally plot $r_{xx}$ with values that are not typically considered in previous literature (See, e.g. Zhang et al. 2020, $r_{xx}\in[-1,1]$). This allows us to better characterize whether the value of $r_{xx}$ came from the effect of compressibility or from Faraday rotations. We can observe from Fig.5 that there are two new regimes of $\lambda$ that could make $r_{xx}$ fluctuates well beyond the values previously considered in Zhang et al. (2020). From Fig.5 we classify the ranges of values of $\lambda$ via the fluctuations of $r_{xx}$ into three regimes: he ”weak” regime correspond to the case where $r_{xx}$ is small ($\in[-1,1]$ as in Zhang et al. 2020). The intermediate regime correspond to the case where $r_{xx}$ start to grow exponentially, and the strong regime correspond to the case where the $r_{xx}$ basically loses traces on the compressibility. We can see that obviously the technique of SPA does not work when we are in the strong regime. However an interesting question is whether the SPA technique actually works in the intermediate regime which will be the subject for future studies.

The essence of on why $\gamma$ is encoded in $I+Q$ and $I-Q$ is based on the fact that tensor formulation (c.f. Eq.2) contains different expressions for observables that $\parallel B$ and $\perp B$. In Fig. 6, we present a set of figures showing the anisotropy of $I+Q$ and $I-Q$ for both A and F type contributions (c.f. Fig.11). We present two extreme cases for $\gamma$ in Fig.6 ³³3Notice that the projection of pure Alfven wave fluctuations when $\gamma$ is exactly $90^{o}$ will vanish, see Lazarian et al. (2022b) for the analysis. that is sufficient to illustrate the differences of behaviors for the anisotropy of A and F type fluctuations. The left group of figures in Fig.6 shows the case when $\gamma=89^{o}$, while the group of figures on the right shows the case when $\gamma=9^{o}$. We can observe from Fig.6 a few interesting phenomena which is not covered in previous anisotropy studies:

1. 1.

   The anisotropies of A and F type tensor do not necessarily align with the mean magnetic field direction. We discussed this effect already from Fig.4. The reason behind is that both the anisotropy and tensor contribution are anisotropic (c.f. §4.2). However, the direction of anisotropy for the tensor contribution (with the Alfven leakage in effect) does not necessary be parallel to B-field and is a function of $\gamma$. Notice that the change of anisotropy is highly tied with the $\gamma$ value (See Fig.7)

2. 2.

   For the case of pure Alfven fluctuations, the anisotropy is more or less parallel to magnetic field for $I+Q$, while $\perp B$ for $I-Q$. Yet, the compressible mode does not carry the same trend as its Alfven counterpart: When $\gamma\approx 9^{o}$, the F-type anisotropy for $I+Q$ is actually $\perp B$, while that for $I-Q$ is $\parallel B$. In contrast, when $\gamma\approx 89^{o}$ the F-type anisotropy varies very similarly to that of the Alfven counterpart.

3. 3.

   The measurement of relative anisotropies between $I+Q$ and $I-Q$ allows us to characterize the $\gamma$ value. From Fig.6 we can see that if we consider the anisotropies of $I+Q$ and $I-Q$ at $\gamma\approx 89^{o}$, $I+Q$ tends to be parallel to magnetic field, while that for $I-Q$ tends to be perpendicular to magnetic field. We utilize the formulation in Appendix B that the minor-to-major axis ratio $l_{\perp}/l_{\parallel}=\sqrt{1-\epsilon^{2}}$, which the eccentricity $\epsilon$ is related to the quadropole-to-monopole ratio $|D_{2}/D_{0}|$ via Eq.60. The quadropole-to-monopole ratio is the key parameter in parametrizing the anisotropy in previous literature (Lazarian & Pogosyan, 2012, 2016; Kandel et al., 2016, 2017; Lazarian et al., 2022b).

We will start from the parameters $I+Q$ and $I-Q$ in which we will assume the projected mean field is right now along the $x$ direction ⁴⁴4For a general magnetic field configuration, one could always consider the combination $I+(Q\cos(2\phi_{pol})-U\sin(2\phi_{pol}))$, where we perform an inverse orthogonal transform with twice of the polarization angle $2\phi_{pol}=\tan^{-1}_{2}{U/Q}$ for this analysis.. For the case of $I+Q$, we adopt the structure function expression from Eq.(E20) of Lazarian et al. (2022b):

where those factors are simply the expressions of $\zeta_{A}\zeta_{A}$ and $\zeta_{F}\zeta_{F}$ in the global frame of reference (i.e. after leakage). The main takeaway here is, This D factor depends on the following form

Similarly, the structure function for $I-Q$ can be also modelled similarly as:

Based on Fig.6 we can see that the construction:

contains the information on $\gamma$. Here we take the convention that $\text{Anisotropy}(D)>1$ when the anisotropy of structure function is parallel to the global magnetic field direction, and vice versa. In particular, from Fig.6 we expect that $\bar{y}_{A}>1$ for all $\gamma$, while that for F-type contribution changes from smaller than 1 to greater than 1. Detecting the value of $\bar{y}$ for compressible modes (in global frame of reference) detected in observation is the key to extract the value of $\gamma$.

The key reason why we consider the ratio of structure functions instead of individual quantity is because, from our expressions in the global frame of reference, the structure function of some observables carries factors on spectrum, anisotropy and tensors. For the case of structure functions of $I+Q$ and $I-Q$, their only difference is coming from the tensor factor as spectrum and anisotropy factors are fixed once the turbulence is set-up.

To proceed with our analysis, we consider the multipole expansion up to quadrupole (See Appendix §B for the condition for the expansion. In particularly, the expansion is valid only for $M_{A}\sim 0.5-1.0$.). Formally we can express the anisotropy function that we defined above with the monopole and quadrupole coefficients $D_{0},|D_{2}|$:

Recall from the previous discussion that the factors $d_{0,2}$ can be literally written as the spectrum, anisotropy and the tensor contribution, and the first two contributions are cancelling out under our treatment, we can formally write $y$, which is the quadrupole approximation of $\bar{y}$ to be (c.f. Eq.(E30) of Lazarian et al. 2022b):

where we notice that under our current configurations, $I+Q\sim{\cal L}_{z}b_{x}^{2}$ and $I-Q\sim{\cal L}_{z}b_{y}^{2}$ where $L_{z}$ is the length of the integration. Keeping only the tensor term, we will have an expression that is purely based on $W_{I,L}$ in Eq.15, and also functions of $\gamma$ (See Eqs.35 and 36).

Fig.8 shows how numerically the factor $y$ depends on the line of sight angle $\gamma$ for Alfven mode (black) and the compressible mode (green). We notice that the qualitative phenomenon happened in Fig.6 is exactly described by $y$ for compressible modes: $y<1$ for $\gamma\rightarrow 90^{o}$, while $y>1$ for $\gamma\rightarrow 0^{o}$. We recognize that there are fluctuations in terms of the variation of $y$ relative to $\gamma$ for the compressible case. Surprisingly, the Alfven mode $y$ also exhibits some interesting properties that we can exploit in obtaining $\gamma$ in observation. Notice that $y$ for Alfven mode stays $<1$ from what we observe in Fig.6, we see that the Alfven mode’s $y$ has very similar trend when $\gamma\gtrapprox 55^{o}$ , but when $\gamma\lessapprox 55^{o}$, the Alfven mode $y$-value went exactly opposite to that of compressible mode. Moreover, we observe that the change of values of $y$ as a function of $\gamma$ is more or less monotonic if we consider $\gamma\lessapprox 55^{o}$ and $\gamma\gtrapprox 55^{o}$. Notice that the modes that we are talking about here are all in the global frame of reference.

To see whether the trend that we observed in Fig.8 is robust, we select some of the numerical cubes from Tab.1 and to plot $y$ as a function of $\gamma$ for both $A$ and $F$ type contribution and plot it as Fig.9. The selected numerical cubes cover a wide range of sonic and Alfvenic Mach numbers. We can see from Fig.9 that the trends of the two curves are very similar to that of Fig.8. Furthermore, the exact values of $y$ are also very similar across different turbulent conditions. Originally, the formalism of $A$ and $F$ type tensor applies only for $M_{s,A}<1$. However, we perform the calculation of $y$ also for supersonic sub-Alfvenic simulations, which is closer to the environment of molecular clouds (See, e.g. Draine 2011) and still observe the same trend. We therefore conclude that the $\bar{y}$ parameter tracers $\gamma$. In fact, we observe from Fig.9 that when the plasma $\beta\propto M_{A}^{2}/M_{s}^{2}$ is smaller, it is easier to recover the trend that we see in Fig.8.

At last, we provide the empirical formula (units in degrees) for the case of low $\beta$ ($\beta<1$). For $\gamma<40^{o}$

for $\gamma>40$ degrees

The full study on how the y-parameter can be applied to situation with different mixture of driving will be discussed in Malik et al. (in prep).

The analysis of turbulence properties generally from observations requires the consideration of the local-to-global frame problem, which is modelled as the ”magnetic field wandering problem”. While the theory of MHD turbulence is well-established, how the local scaling laws are projected globally is still mysterious, despite models have been proposed from both Lazarian & Pogosyan (2012) and Lazarian et al. (2020). Here, we propose the first physical model in explaining how the wandering of magnetic field happens when projected along the line of sight, and how we could utilize the magnetic field wandering in deducing a number of important physical quantities such as the line of sight angle and also the mode fractions.

The problem of the local-to-global frame transition in theoretical MHD turbulence studies have puzzled the community for a while. While the anisotropic scaling $k_{\parallel}\sim k_{\perp}^{2/3}$ is well motivated from the simple constant energy cascade and critical balance condition (GS95), we cannot retrieve the local scaling from the global frame of reference. In fact, the global correlation function usually gives a constant scaling rather than a geometrically-driven, size-dependent scaling as predicted by GS95. Before the availability of MHD simulations (e.g. Cho & Lazarian 2003; Beresnyak et al. 2005), it is not yet possible to validate the GS95 relation even from numerical simulations.

The more puzzling effect comes when $M_{A}$ is very large. Traditionally the numerical test on GS95 are done in small $M_{A}$ systems and in small scales. However as we see from the previous sections, in moderate and small $k$ the Alfven mode acquired from the Cho & Lazarian (2003) decomposition method contains non-negligible contributions along the $\hat{\zeta}_{C}$ vector, indicating the presence of anomalous compressive wavevector even after Alfven mode decomposition. The only plausible reason why this happens is because the mode decomposition method from Cho & Lazarian (2003) is done on a global frame of reference. As a result when we are looking at small scales, in average the mean field is not very different from its local field. Yet for larger scales the mean field is very different from the local field, so that there exists anomalous compressible terms even the data are supposed to be ”Alfven modes” according to Cho & Lazarian (2003). We named this effect ”Alfven leakage” in our previous section since this effect happens even for Alfven waves as long as the Alfvenic Mach number is not zero.

In this paper, we further show that the Alfven leakage effect is a global function of $M_{A}$. In fact, the presence of the leakage effect suggests that the mode decomposition method by Cho & Lazarian (2003) should subject to the a correction term for moderate and small $k$. However since most of the calculation from Cho & Lazarian (2003) are done in small scales, i.e. large $k$, the results of their work are not affected.

The novel invention of the SPA technique (Zhang et al. (2020)) utilizes the fact that the tensor projections have different contributions for Alfven and compressible modes to identify them in observations. This work further strengthens their argument through the use of Alfven leakage picture and suggests a few important improvements to their method. For instance, it is not necessary to compute the parameter $s_{xx}$ as in Eq.21 to distinguish the modes. The tensor properties are encoded in the Stokes parameters and thus ignoring the tensor contribution would make dramatically different predictions in astrophysical applications.

One very important factor that is accounted by Zhang et al. (2020) is the use of one point statistics under Stokes frame transformation. The traditional turbulence statistical studies usually utilize multi-point statistics since they are either directly related to the spectra (e.g. two-point) or is used to validate scaling relations for higher order structure functions (e.g. Kolmogorov 4/5 law). The reason of why single point statistics was not useful before is because the spectrum and anisotropy are the main characteristics of turbulence studies for the past 60 years. However, how the tensor projection affects the geometry of the structures for each of the turbulence variable is not really explored. Tensor forms of turbulence modes were not much explored beyond the physics of cosmic rays(Schlickeiser, 2002; Yan & Lazarian, 2002, 2004). In fact, the previous anisotropy analysis also did not consider what is the statistics of a single component of a 3D turbulence, i.e.tensor projection, after projection along the line of sight. While the series of papers by Lazarian & Pogosyan started to consider how the single component statistics works, not until recently did both numerically (e.g. Lazarian et al. 2018) and observationally (Zhang et al., 2020) found the effect of tensors to be that important during single component projection. The anisotropy of projected fast modes with the direction opposite to the Alfvenic anisotropy was shown in Lazarian & Pogosyan (2012). In fact, one of the most common belief that is circulating in the earlier studies of MHD turbulence theory (e.g. the discussion section of Lazarian et al. (2017)) is the presumption that the projection of the observables (e.g. velocities, magnetic field) from fast modes will be isotropic since the fast modes in 3D are. This is empirically proven wrong by Lazarian et al. (2018) through the development of velocity gradient and also utilized through the development of SPA in Zhang et al. (2020).

In fact, in a number of astrophysical applications, the observables are constructed through not all three directions of velocities or magnetic field, but just some of them. For instance, the Davis-Chandrasekhar-Fermi (DCF) technique utilizes both the line of sight velocity dispersion and the plane of sky polarization angle dispersion to estimate the magnetic field strength through the use of Alfven relation (See also Cho & Yoo 2016). However, as found in Lazarian et al. (2022b), the direction of the velocity and magnetic field fluctuations as collected in DCF technique are exactly perpendicular to each other. Moreover, in this work we also show both analytically and numerically that the tensor term contains anisotropy and can be dominant as long as the $\gamma$ fulfills some conditions. As a result, one should not ignore the contributions of the tensor term in studying the properties of MHD turbulence.

In MHD turbulence studies, there are a few length scales that determine whether the underlying turbulence is hydrodynamic or GS95-like. It is a general phenomenon that for 3D saturated turbulence the small scale fluctuations are GS95 like. However for both sub-Alfvenic and super-Alfvenic there exist a transition scale that the turbulence becomes non-GS95 like. For instance, for the sub-Alfvenic case there exist the transition from weak to strong turbulence (Cho & Lazarian 2003; Makwana & Yan 2020) at the length scale $LM_{A}^{2}$, while for the super-Alfvenic case above the scale $LM_{A}^{-3}$ the turbulence is hydrodynamic. This might suggest that the removal of large scale fluctuations could allow observers to obtain the desired GS95 statistics with the use of high pass filters.

However upon projection the high pass filter in 2D acts a little bit differently compared to 3D. Fundamentally the high pass filter (HPF) in 3D serves as the high frequency extractor. As noticed in Lazarian et al. (2020) , HPF in 2D acts as a lower bound of the HPF in 3D, i.e. if we explicitly want $K=\sqrt{k_{x}^{2}+k_{y}^{2}}>K_{0}$, this will automatically apply to $k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}\geq K>K_{0}$. However since the sampling of turbulence statistics upon projection is not statistically complete, meaning that the wavevectors with $k>K_{0}$ but $K<K_{0}$ is not sampled, it is hard to determine whether we will obtain back the same turbulence spectrum anisotropy just by inspection here since we did have additional knowledge on how the LOS direction is related to the inclination angle. More importantly, if we are considering the case when $M_{A}$ is not small, the randomness of the magnetic field fluctuation will make the filtering in 2D in Stokes parameters being completely different from that of the 3D. Fig.10 shows an example on how different the Stokes Q look like. On the left of Fig.10 we perform filtering after projection (i.e. 2D), while on the right it is the projection after 3D filtering. We can see that, while the statistical anisotropies for the two maps are roughly the same, the differences of the features are prominent.