Analytic characterization of sub-Alfv\'enic turbulence energetics

Magnetohydrodynamic (MHD) turbulence is a cross-field process relevant to many systems. A prerequisite for understanding these systems is to constrain the role of MHD turbulence, and in particular the energy exchange between kinetic and magnetic forms. The energetics of strongly magnetized and compressible turbulence has so far resisted attempts to understand them. Numerical simulations reveal that kinetic energy can be orders of magnitude larger than fluctuating magnetic energy. We solve this lack-of-balance puzzle by calculating the energetics of compressible and sub-Alfv\'enic turbulence based on the dynamics of coherent cylindrical fluid parcels. Using the MHD Lagrangian, we prove analytically that the bulk of the magnetic energy transferred to kinetic is the energy stored in the coupling between the ordered and fluctuating magnetic field. The analytical relations are in striking agreement with numerical data, up to second order terms.

A widely employed approximation is the incompressibility of the gas (Sridhar & Goldreich 1994;Goldreich & Sridhar 1995), although this is only applicable to a limited number of systems. Compressible MHD turbulence is more complex, and additional energy terms contribute to the energy cascade. One main difference in the energy cascade rate of incompressible and compressible turbulence is that in the latter, the background magnetic field (B 0 ) appears with leading-order terms (Banerjee & Galtier 2013;Andrés & Sahraoui 2017). In contrast, the incompressible turbulence energy cascade is dominated by the increments of the magnetic and velocity fluctuations, and B 0 only appears in higher-order statistics (Wan et al. 2012). This result motivated the hypothesis that B 0 might also appear in the total (kinetic and magnetic) fluctuating energy of compressible MHD turbulence (Andrés & Sahraoui 2017), whereas in incompressible turbulence, the total fluctuating energy is dominated by the fluctuating (second-order) kinetic and magnetic energy.
In incompressible and sub-Alfvénic turbulence, the fluctuating magnetic energy is completely transferred to kinetic energy, and the volume-averaged quantities are in equilibrium, ρ u 2 /2 ∼ δB 2 /8π, when turbulence is maintained in a steady state. In contrast, direct numerical simulations of sub-Alfvénic and compressible turbulence show that the volume-averaged kinetic energy is much higher than the second-order fluctuating skalidis@caltech.edu magnetic energy, ρ u 2 /2 δB 2 /8π (Heitsch et al. 2001;Li et al. 2012a,b), and their relative ratio depends on the amplitude of B 0 (Andrés et al. 2018;Lim et al. 2020;Beattie et al. 2022b). The excess of the kinetic energy suggests that B 0 might provide additional energy to the fluid.
The role of B 0 in the energetics can be intuitively understood when we decompose the total magnetic field into a background and a fluctuating component. In incompressible turbulence, the fluctuating magnetic energy comes only from the perturbations of the magnetic field, which are of second order. However, in compressible turbulence, the background field appears in the the total fluctuating magnetic energy due to the coupling between the background field and magnetic perturbations (δB). The magnetic coupling, expressed as B 0 ·δB, can only be realized in compressible turbulence (Montgomery et al. 1987;Bhattacharjee & Hameiri 1988;Bhattacharjee et al. 1998;Fujimura & Tsuneta 2009) and is the dominant (first-order) term of the fluctuating magnetic energy.
In sub-Alfvénic and compressible turbulence, numerical data show that B 0 · δB stores most of the magnetic energy, and that the kinetic energy approximately reaches equipartition with the fluctuations of the coupling term Beattie et al. 2022b,a). Thus, the magnetic coupling holds the key for understanding the energetics of strongly magnetized and compressible turbulence. However, there is still a lack of first-principle understanding of the role of B 0 · δB in MHD turbulence dynamics and how it contributes to the averaged energetics.
We present an analytical theory of the role of the coupling potential in the energy exchange of sub-Alfvénic and compressible turbulence, which is encountered in systems such as tokamaks (Strauss 1976(Strauss , 1977Zocco & Schekochihin 2011), in the interstellar medium (Mouschovias et al. 2006;Panopoulou et al. 2015Panopoulou et al. , 2016Planck Collaboration et al. 2016;Skalidis et al. 2022), and the Sun (Verdini & Velli 2007;Tenerani & Velli 2017;Kasper et al. 2021;Zank et al. 2022). We write the Lagrangian of coherent flux structures (Crowley et al. 2022), which allows us to approximate turbulence properties in a deterministic manner, and calculate analytically the energy exchange between kinetic Article number, page 1 of 7 arXiv:2209.14143v5 [physics.flu-dyn] 7 Apr 2023 A&A proofs: manuscript no. aanda_corr and magnetic forms as a function of the Alfvénic Mach number (M A ). We find remarkable agreement between the analytically calculated energetics and numerical data. We conclude that the majority of the fluctuating magnetic energy transferred to kinetic energy is provided by the coupling between the background and the fluctuating magnetic field.
Magnetized fluid consisting of multiple coherent cylindrical fluid parcels. Red arrows show the initial magnetic field morphology. Untwisted fluid parcels are elongated, ⊥ , and their motion is longitudinal along or perpendicular to B 0 . In sub-Alfvénic turbulence, the motion of these fluid parcels can be decomposed into two independent velocity components, parallel (black arrows) and perpendicular (orange arrows) to B 0 .

Model
We considered a turbulent fluid characterized by the commonly employed properties: 1) spatial homogeneity, 2) infinite magnetic and kinetic Reynolds number, and 3) time stationarity. We considered that the fluid consists of coherent flux tubes (e.g., Fig. 1 in Banerjee & Galtier 2013) (or fluid parcels) with coordinates (r(t), φ(t), z(t)), as shown in Fig. 1. Cylindrical coordinates are motivated by studies showing that the properties of strongly magnetized turbulence are axially symmetric, with B 0 being the axis of symmetry (Goldreich & Sridhar 1995;Maron & Goldreich 2001). We assumed the following initial conditions: 1) uniform temperature, 2) uniform density, 3) no bulk velocity, 4) uniform static magnetic field (B 0 = B 0ẑ ), and 5) no self-gravity. We henceforth adopt the following notation: z = and r = ⊥ , where and ⊥ denote parcel sizes parallel and perpendicular to B 0 , respectively.
We perturbed the magnetic field of a coherent fluid structure with a length scale = 2 + 2 ⊥ by δB such that |B 0 | |δB |, which applies to sub-Alfvénic turbulence. Magnetic perturbations tend to redistribute the magnetic flux within a fluid. For ideal-MHD (flux-freezing) conditions, the magnetic flux is preserved. Thus, the surface of the perturbed fluid parcel (S ) follows the magnetic field lines. The motion of the field lines, and hence of S , can be either parallel or perpendicular to B 0 ( Fig. 1): 1) Squeezing and stretching of S along B 0 leads to parallel motions,˙ 0. 2) Fluctuations of ⊥ lead to perpendicular motions, ⊥ 0. Finally, 3) twisting leads to rotational motions,φ 0.
This naturally defines and ⊥ as the coherence lengths of the perturbed volume parallel and perpendicular to B 0 , respectively. We focused on large scales since coherent structures are prominent there (De Giorgio et al. 2017). We invoke as a boundary condition a local environment beyond (pressure wall).
The flux freezing theorem is The cross section of the coherent volume perpendicular and parallel to B 0 is S ⊥, = 2π ⊥ r, and S , = π 2 ⊥ẑ , respectively. The cross section related to the rotational motion is S φ, = ⊥φ . The total magnetic field in cylindrical coordinates can be expressed as B = δB ⊥r, r + δB ⊥φ, φ + B 0 + δB , ẑ. From Eq. 1, we obtain that when |B 0 | |δB|, magnetic perturbations along S , are associated with a longitudinal motion such that where we have considered that the initial dimension of the perturbed volume ⊥,0 is much larger than its perturbations. Along S ⊥, , we find that while the azimuthal velocity along S φ, is As a result of assuming |B 0 | |δB|, we have obtained that parallel and perpendicular motions are decoupled. The coupling of parallel and perpendicular motions becomes inevitable when |B 0 | ∼ |δB| (Eq. 3).
The difference in the scaling is due to the Lorenz force by B 0 , which affects perpendicular motions, while it has no effect on parallel motions.

MHD Lagrangian of coherent structures
We write the Lagrangian for the perturbed volume. We place the reference frame at the center of mass of the target volume, hence there is no bulk velocity term in the Lagrangian. Therefore, all the velocity components are due to internal motions induced by magnetic perturbations. We focus on low plasma-beta fluids 1 , which for sub-Alfvénic turbulence corresponds to high sonic Mach numbers (M s ). The perturbed Lagrangian (Newcomb 1962;Andreussi et al. 2016;Kulsrud 2005) of the coherent cylindrical fluid parcel, with surface S , can be split into a parallel and a perpendicular term (Appendix A), Due to Eqs. 2 and 3, δB , , and δB ⊥, are generalized coordinates of δL and (t) = C/δB ⊥, (t) (Eq. 6), where C is a constant determined from the initial conditions. With this expression, we eliminate from the Lagrangian, which up to second-order terms is separable into a parallel and a perpendicular part, and is analytically solvable, We solve the Euler-Lagrange equations for δL (Appendix B) and δL ⊥ (Appendix C) and derive the analytical solutions of the velocity (u , (t), u ⊥, (t) ) and magnetic fluctuations (δB , (t), δB ⊥, (t)) of S . We find that δB , ∼ t 2 , u ⊥, ∼ t, and δB ⊥, ∼ t −1 , while u , is set by the initial conditions (free streaming of the gas). We used these analytical solutions in order to calculate the averaged energetics of a strongly magnetized and compressible fluid.

Energetics
The total energy of fully developed turbulence is stationary because energy diffusion is balanced by injection. Time stationarity enables us to approximate turbulence energetics with the leading-order solutions that we obtained because our approximations preserve time symmetry, and thus energy is conserved. The statistical properties of large-scale coherent structures accurately approximate the volume-averaged turbulent statistical properties. Thus, for an ergodic fluid (Monin & I'Aglom 1971;Galanti & Tsinober 2004), averaging the turbulent statistical properties over the volume of the fluid at a given time step f V = V f is approximately equivalent to averaging over multiple realizations of a typical large-scale coherent structure f T = T f , hence f V ∼ f T , where f denotes an energy term, and T corresponds to the coherent structure crossing time. We next analytically compute the f T energy contribution of each Lagrangian term (Eq. 7) and their relative ratios. Since coherent cylindrical parcels are characterized by two different coherence lengths and ⊥ , they also have two different crossing times: T and T ⊥ , respectively. We compare the f T analytical energy ratios with the f V numerical values. The numerical results correspond to simulations of ideal isothermal MHD turbulence without selfgravity, and turbulence is maintained in a quasi-static state by injecting energy with an external forcing mechanism (Beattie et al. 2022b). These simulation are forced with a mixture of compressible and incompressible modes, but the driving modes do not affect the energetics of sub-Alfvénic and compressible turbulence .

Kinetic energy
The total averaged kinetic energy (E kinetic ) of the coherent fluid parcel with scale is The kinetic energy is dominated to first order by u ⊥, . Thus, the average Alfvénic Mach number to first order is

Harmonic potential
From Eqs. B.3 and C.2, we find that δB 2 , T ⊥ = 7δB 2 ,max /15, and δB 2 ⊥, T = δB 2 ⊥,max /2. The total time-averaged harmonic potential energy (E harmonic ) density is equal to where ζ = δB ⊥,max /δB ,max . Sub-Alfvénic turbulence is anisotropic (Shebalin et al. 1983;Higdon 1984;Oughton et al. 1994;Goldreich & Sridhar 1995), with the anisotropy between δB ⊥ and δB depending on M A (Beattie et al. 2020). To account for this property, we assumed that ζ is a function of M A . When M A → 0, B 0 suppresses any bending of the magnetic field lines with the amplitude of δB being larger than that of δB ⊥ (Beattie et al. 2020), hence ζ → 0; this is also a consequence of ∇ · B = 0 for anisotropic fluid parcels with ⊥ . For M A → 1, fluctuations tend to become more isotropic, and hence ζ → √ 2. These limiting behaviors are consistent with numerical simulations (Beattie et al. 2020(Beattie et al. , 2022b.

Coupling potential
According to Eq. 10, B 0 · δB contributes to E kinetic since to first order. This equation demonstrates that the energy stored in the coupling potential is in equipartition with the averaged kinetic energy when turbulence is sub-Alfvénic.

Energetics ratios
The E kinetic /E coupling ratio is Article number, page 3 of 7 A&A proofs: manuscript no. aanda_corr For M A → 0, E coupling ≈ E kinetic , while for M A → 1, E kinetic E coupling . E kinetic becomes higher than E coupling because u , contributes more to E kinetic as M A increases. When M A → 1, ζ ≈ √ 2 , so that the E kinetic /E coupling ratio in trans-Alfvénic turbulence scales as Regarding the E harmonic /E coupling ratio, we find that which for the two limiting cases of ζ (M A ) becomes

Comparison between analytical and numerical results
In Fig. 2 we compare the analytically calculated energy ratios with numerical results from the literature (Beattie et al. 2022b). The lines correspond to the analytical relations for E harmonic /E coupling (Eq. 17) and E kinetic /E harmonic (Eq. 15), while the colored points correspond to the numerical values. The numerical data behave as predicted by the analytical relations. The scatter of triangles increases at higher M A because thermal pressure starts becoming important there, and hence the contribution of thermal motions to the kinetic energy increases. In the limit of M A 1, thermal pressure is subdominant and β → 0. For M A = 1, we obtain that β → 0 when M s 1, while when M s 1, β → 1. Thus, for trans-Alvfénic turbulence, thermal pressure becomes important only for low M s , while at high M s , it has a minor contribution to the energetics. In our calculations, we neglected thermal pressure, and for this reason, at M A = 1, triangles are consistent with the analytical ratio of E kinetic /E coupling (blue line) when M s ≥ 2, while at lower M s , the deviation between numerical and analytical results increases because β, hence the relative contribution of thermal pressure, increases. For M A < 1, β 1, and for this reason, the numerical data agree perfectly with the analytical ratio (blue line). Finally, when we account for the contribution from both B 0 · δB and δB 2 , the total energy stored in magnetic fluctuations E m,total = E coupling + E harmonic is very close to equipartition with kinetic energy, as shown by the red boxes.

Discussion and conclusions
Analytical calculations of strongly magnetized and compressible (isothermal) turbulence show that B 0 appears in the energy cascade with leading-order terms (Banerjee & Galtier 2013;Andrés et al. 2018). This is in striking contrast to incompressible turbulence, where B 0 appears in higher-order terms (Wan et al. 2012). In the formalism presented here, the incompressible limit is approximated when B 0 · δB = 0. In this case, B 0 does not appear in the dominant Lagrangian terms, hence the averaged kinetic energy would scale linearly with the fluctuating magnetic energy, δu ∼ δB (or equivalently, M A ∼ δB ). However, in agreement with previous works (Wan et al. 2012), our formalism shows that B 0 appears in the energetics of incompressible turbulence because of the coupling between δB , and δB ⊥, (Eqs. 3) when higher-order terms are considered. 2), respectively. Numerical data are shown with colored dots. The blue line corresponds to the analytically obtained E kinetic /E coupling ratio, while colored triangles show the same quantities calculated from numerical data. The red boxes correspond to E kinetic /E m,total . The thin green line shows the energy terms in equipartition. The color bar shows the sonic Mach number (M s ) of the simulations.
For sub-Alfvénic and compressible turbulence, we find that B 0 · δB is the leading term in the dynamics, and as a result, the scaling between velocity and magnetic fluctuations becomes δu ∼ √ B 0 δB , or equivalently, M A ∼ δu /V A ∼ √ δB /B 0 , which is supported by numerical data (Beattie et al. 2020). In compressible and strongly magnetized turbulence, compression and dilatation of the gas locally changes the energy cascade rate (Banerjee & Galtier 2013). These local energy fluctuations can only be realized in compressible turbulence and might be related to the fluctuations of the B 0 · δB potential. Our analytical results prove that the total averaged magnetic energy transferred to kinetic is equal to 2B 0 δB 2 + δB 2 /8π.
The consistency between our analytical relations and numerical data is remarkable. It is not the first time that simple analytical arguments agree quantitatively with numerical simulations of nonlinear problems (e.g., Mouschovias et al. 2011). However, an analytical theory is always advantageous because it allows us to achieve a deeper understanding of complex problems. For this reason, the formalism we presented might offer new insights into the energetics of strongly magnetized and compressible turbulence. We hope that it motivates future works about the role of magnetic couplings in the energy cascade.