Magnetic, Kinetic, and Transition regime: Spatially-segregated structure of compressive MHD turbulence
Guang-Xing Li Mengke Zhao



0000-0003-3144-1952]Guang-Xing Li
South-Western Institute for Astronomy Research, Yunnan University, Chenggong District, Kunming 650500, China

0000-0003-0596-6608]Mengke Zhao
School of Astronomy and Space Science, Nanjing University, 163 Xianlin Avenue, Nanjing 210023, People’s Republic of China

§ ABSTRACT 


Turbulence is a complex physical process that emerges in multiple areas of
modern physics, and in ionized environments such as interstellar gas, the
magnetic field can be dynamically important. However, the exact function of the magnetic field in
the ionized gas remains unclear. 
We use the M_ A = √(E_ k/E_B) to describe the importance of the magnetic field measured to the
turbulent motion, and reveal diverse ways of mutual interaction.
At low M_ A (magnetic regime), the magnetic field
is well-described as force-free. Despite the strong magnetic field,
the motion of gas does not stay aligned with the magnetic field. At the regime of
intermediate M_ A (magnetic-kinetic transition regime), the
velocity field and the magnetic field exhibit the highest degree of alignment, which is likely the result of a rapid relaxation.
At high M_ A (kinetic regime), both the magnetic field and the
velocity field are irregular, with no alignment. We find observational counterparts to these regimes in observations of interstellar gas. The results highlight the diverse behavior of gas in MHD turbulence and guide future interpretations of the role of the magnetic field in astrophysical observations. 










§ INTRODUCTION



Turbulence is a complex process that has puzzled scientists since the age of Leonardo da Vinci, and the capability of turbulence in controlling the evolution of interstellar gas has been known for decades 
<cit.>.
Understanding the role of the magnetic field in compressible turbulence can be crucial for our understanding of turbulence and for interpreting astrophysical observations. Turbulence is a complex, multi-scale process  <cit.> best described using scaling relations <cit.>. Past studies of compressible magnetohydrodynamics (MHD) turbulence have followed this tradition where describing the statistic properties of the region has become a priority <cit.>. Others have astrophysical applications in mind and have focused on the global quantities extracted from the simulation box <cit.>. 










The alignment between vector quantities such as the magnetic field B⃗, the velocity v⃗, and the current J offers insight into the behavior of the magnetized fluids. In astrophysical research, the alignment between B⃗ and v⃗ is assumed to be an indicator of the magnetic field's ability to affect the motion of the gas. The alignment between B⃗ and v⃗ in the strongly magnetized
 regime is a fact often taken for granted, and this picture is the foundation for understanding phenomena such as wind from disk-star systems <cit.>, where the picture of beads on a wire have been widely accepted. In this picture, field lines of a magnetic field behave like rigid wires, which guide the motion of the gas. 
 However, it is unclear to what extent can we trust this picture. On the other hand, <cit.> have shown that the rapid alignment between v⃗ and B⃗ is the result of a rapid relaxation process, where the magnetic field does not need to dominant.



The alignment between J⃗ and B⃗ is also critical. 
In magnetized fluids, an interesting phenomenon is the emergence of force-free fields, where the magnetic pressure much exceeds the plasma pressure, such that the Lorentz force must vanish to ensure a global balance.  This force-free field is thus a direct indication of the dominance of the magnetic energy over other energetic terms. 
 We note that the Lorentz for F⃗_⃗l⃗ is proportional to J⃗×B⃗, and vanishes when J⃗ is parallel to B⃗. The alignment between J⃗ and B⃗ is thus a clear indication of the force-free field.








We study the alignment between B⃗, v⃗ and J⃗ at regions of different degrees of magnetization.
To quantify the importance of the magnetic field, we use the Alfven Mach number M_ A = √(E_ k/E_B), where E_ k is the kinetic energy density and E_B is the magnetic energy density. We study the importance of the magnetic field under different conditions as characterized by  M_ A. By analyzing the alignment between the magnetic field B⃗, velocity v⃗ and current J⃗, we reveal different behaviors of the system under different M_ A. 
































§ DATA AND METHOD

We use numerical simulations of MHD equations performed using the Enzo code <cit.> with the constrained transport turned on.
The simulation conducted in this study analyzed the impact of self-gravity and magnetic fields on supersonic turbulence in isothermal molecular clouds, using high-resolution simulations and adaptive mesh refinement techniques <cit.>. The simulation we use has initial β≈ E_ k / E_ B =0.2. However, as the simulation proceeds, different subregions have different β and M_ A.

























The Alfven Mach number M_ A is the indicator of magnetic field in MHD numerical simulation:

    M_ A = √(E_k/E_B) = √(ρσ_v^2/2/B^2/8π) = √(4πρ)σ_v/B,

which is the square root of the energy ratio between kinetic energy density E_ k and magnetic energy density E_B. Based on the M_ A, we divide the MHD turbulence into three regimes: magnetic regime, B-k transition, and kinetic regime, and study the relation between the magnetic field and the gas motion. From the magnetic field B⃗, the  current J⃗ can be evaluated as

    J = 1/μ_0∇×B⃗ ,

and we study the alignment between the magnetic field B⃗ and the current J⃗, and the alignment between the magnetic field B⃗ and the velocity v⃗ at different M_ A. 








































§ THREE REGIMES OF MHD TURBULENCE



Based on the Alfven Mach number M_ A, we divide the simulation box into three regimes: magnetic regime (M_ A < 0.3), B-k transition (0.3< M_ A<3), and kinetic regime (M_ A>3), and study the alignment between the magnetic field B⃗, velocity v⃗, and current J⃗ at different M_ A. The relationship between the magnetic field-velocity angle θ_B⃗-v⃗, the magnetic field-current angle θ_B⃗-J⃗ and the Alfven Mach number is shown in Fig. <ref>. When plotting these distributions, we subtract the distribution expected if the vectors are randomly oriented, and focus on the additional alignment caused by physics (see Sec. <ref>). 

With increasing M_ A, the alignment between the magnetic field B⃗ and the current J⃗ changes from aligned to not aligned, and the alignment between the magnetic field B⃗ and the velocity v⃗ change from almost no alignment (θ_B⃗-v⃗≈ 30^∘), to a weak alignment (θ_B⃗-v⃗≈ 20^∘), back to no alignment (θ_B⃗-v⃗≈ 30^∘). 



We note that the magnetic force  is

    f⃗_ L = J⃗×B⃗ .

When J⃗ is parallel to B⃗, the magnetic force vanishes, and this field configuration is called the force-free configuration. The magnetic force is activated if the angle between J⃗ and B⃗ is large.


The monotonic decrease of alignment between B⃗ and J⃗  at increasing   M_ A  is related to the decrease in the importance of the magnetic field, leading to the system moving away from the  force-free regime. Based on the Alfven Mach number and the behavior of the system, we divide the simulation into three regimes: 
    
  * The magnetic regime (M_ A<0.3): the magnetic energy is far above the local kinetic energy, where the B and v do not stay aligned, yet the B and J aligned.
    
  * The transition regime (B-k transition, 0.3< M_ A<3): the magnetic energy and kinetic energy have similar densities.  the B and v stay aligned, and the B and J evolve from aligned to not aligned as  M_ A increases.
    
  * The kinetic regime: the kinetic energy is far above the local magnetic energy, the B and v are not aligned, and  B and J are not aligned.

which are plotted in Figs. <ref>, <ref> and <ref>, and are summarized in Fig. <ref>. Some addition slices to the simulation box can be found in Sec. <ref>


















 §.§ Magnetic Regime

The first regime we discovered is the magnetic regime, where the magnetic energy is far above the local kinetic energy (M_ A< 0.3). This regime has two properties, the alignment between  B and J which points to the formation of a force-free field and the lack of alignment between B and v. 

The strong alignment between B and J indicates that the field configuration is force-free, with other forces being dynamically unimportant. We also observe a lack of alignment between B and v, which challenges the common understanding of the magnetic field being “wires” that guide the motion of the gas.  In contrast, the motion of gas does not appear to stay aligned with the orientation of the magnetic field line. A strong magnetic field does not necessarily lead to motions that follow the field lines. 




From astronomy observations, we identify the magnetic regime from low-density regions around some existing molecular clouds. One such example is the existence of striations located at the outer part of the Taurus molecular cloud <cit.>.



































 §.§ B-k Transition

At 0.3<M_ A <3, where the magnetic energy and the kinetic energy are similar, we find a transition regime characterized by a breakdown of the alignment between B⃗ and J, and a strong alignment between  B⃗ and v⃗. The breakdown of the B⃗-J alignment results from a decrease in the importance of the magnetic field. However, the alignment between  B⃗ and v⃗  deserves further discussions. 





We find the  B⃗ and v⃗ only occur in the B-k transition regime, where the magnetic and kinetic energy have similar densities. This finding challenges the common understanding that the alignment between B⃗ and v⃗ indicates the dominance of the magnetic field. This alignment has been found by <cit.>,  which results from a “rapid and robust relaxation process in turbulent flows”. Our findings support their conclusion. However, different from the claim that “ the alignment of the velocity and magnetic field fluctuations occurs rapidly in magnetohydrodynamics for a variety of parameters”, in our case, this alignment only occurs at the transition regime with moderate M_ A. 
In the B-k transition regime, the vecB and J⃗ are still moderately aligned with θ≈ 15^∘. 


















From the reported values of the Alfven Mach numbers (or the mass-to-flux ratio) <cit.>, we believe that a large number of observations of the observations are probing gas located in this transition regime. Examples include the envelope of 
massive star formation Orion A <cit.>, and other star formation regions such as Taurus, L1551 and so on <cit.>.




















 §.§ Kinetic Regime


At the kinetic regime with high kinetic energy (high M_ A), the B-v not aligned, and the B-J not aligned. The lack of this alignment is the result of the weak magnetic field.


In astronomical observations, this corresponds to dense, collapsing regions with strong turbulence, such as the Cyg N44 in Dr21 <cit.>. 



















§ CONCLUSIONS

We investigate the effect of a magnetic field on supersonic turbulence. By evaluating the Alfven Mach number M_ A at different locations in a turbulence box and studying the alignment between the magnetic field B⃗, the velocity v⃗ and the current J⃗, we reveal the different behavior of the system under various conditions. These regimes include



 
  * The magnetic regime (M_ A<0.3): the magnetic energy is far above the local kinetic energy, where the B-v do not stay aligned, yet the B-J aligned. The magnetic field is force-free.

 
  * The transition regime (B-k transition, 0.3< M_ A<3): the magnetic energy and kinetic energy have similar densities.  B and v are. aligned, and the B and J evolve from aligned to not aligned as M_ A increases. We also observe alignments between the magnetic field and the gas motion. However, this alignment between B⃗ and v⃗ does not imp a strong magnetic field but a rapid relaxation process in turbulent flows. 

 
  * The kinetic regime: the kinetic energy is far above the local magnetic energy, the B-v not aligned, and the B-J not aligned.



They are summarized in <ref>. Since there is a correlation between the Alfven Mach number and the gas density (Fig. <ref>), the magnetic regime exists in the lower-density part and the kinetic regime in the higher-density part. The transition regime is an intermediate state between the two. Using observational data, we find cases that support the existence of these regimes.

The results guide the interpretation of new observations. It breaks down the common understanding of the magnetic field as a rigid wire that guides gas motion and replaces it with the complex behavior of the gas under different conditions. The alignment between B⃗ and J⃗ points to the dominance of the magnetic field, and the alignment between B⃗ and v⃗ is likely the result of a rapid self-organization process in turbulent flows. Some supporting cases are identified from observations of the interstellar medium. 

We reveal various regimes where the fluid behaves differently under different conditions. 
To our knowledge, this is the first time these different regimes are clearly outlined. The alignment between these quantities has been studied in a recent paper <cit.> where the authors reported strong alignments in the strongly magnetized regime and some scale-dependence alignment behavior.
We are revealing a much clearer picture with the Alfven Mach number as the only parameter dictating the behavior of the fluid and the alignments. 





















































§ REMOVING THE PROJECTION EFFECT IN ANGLE DISTRIBUTIONS
 









The probability density function, p(θ), of the angle between two vectors of random distribution is distributed in an N-dimensional space as:

    p(θ) = Γ(n/2)/Γ(n-1/2) sin^n-2(θ)/√(π)

where n is numbers of dimensions, and θ is the angle between two vectors of random distribution. 
In our alignment analysis, we study the angle between qualities measured in 3D. 
When n=3, we have

    p(θ) = 1/2 sinθ ,

which two randomly-selected vectors have a alignment angle clustered at around θ=45^∘, caused by the projection effect.  
To remove the projection effect, we use the probability density function p(θ) to weight the angle distribution: 

    I( θ_ corrected) = I(θ_ original) / p(θ)

where I(θ_ original) represents the original angle distribution and I(θ_ corrected) represents the corrected angle distribution, with the projected effects removed.





§ 2D SLICE OF MAGNETIC FIELD, VELOCITY  AND CURRENT


Fig. <ref> shows the distribution of density ρ, velocity field v, magnetic field B, and current field J in X-Y plane, which is a slice of 3D space.





