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MNRAS 000, 1–22 (2020) Preprint 16 February 2021 Compiled using MNRAS LATEX style file v3.0

Diffusion of large-scale magnetic fields by reconnection in MHD

turbulence

R. Santos-Lima,1⋆ G. Guerrero,2† E. M. de Gouveia Dal Pino,1‡ A. Lazarian3,4§1Instituto de Astronomia, Geofısica e Ciencias Atmosfericas, Universidade de Sao Paulo, R. do Matao, 1226, Sao Paulo, SP 05508-090, Brazil2Physics Department, Universidade Federal de Minas Gerais, Av. Antonio Carlos, 6627, Belo Horizonte, MG, Brazil, 31270-9013Department of Astronomy, University of Wisconsin, 475 North Charter Street, Madison, WI 53706, USA4Center for Computation Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010

Last updated 2015 May 22; in original form 2013 September 5

ABSTRACT

The rate of magnetic field diffusion plays an essential role in several astrophysicalplasma processes. It has been demonstrated that the omnipresent turbulence in as-trophysical media induces fast magnetic reconnection, which consequently leads tolarge-scale magnetic flux diffusion at a rate independent of the plasma microphysics.This process is called “reconnection diffusion” (RD) and allows for the diffusion offields which are dynamically important. The current theory describing RD is basedon incompressible magnetohydrodynamic (MHD) turbulence. In this work, we havetested quantitatively the predictions of the RD theory when magnetic forces are dom-inant in the turbulence dynamics (Alfvenic Mach number MA < 1). We employedthe Pencil Code to perform numerical simulations of forced MHD turbulence, ex-tracting the values of the diffusion coefficient ηRD using the Test-Field method. Ourresults are consistent with the RD theory (ηRD ∼ M3

Afor MA < 1) when turbulence

approaches the incompressible limit (sonic Mach number MS . 0.02), while for largerMS the diffusion is faster (ηRD ∼ M2

A). This work shows for the first time simulations

of compressible MHD turbulence with the suppression of the cascade in the directionparallel to the mean magnetic field, which is consistent with incompressible weak tur-bulence theory. We also verified that in our simulations the energy cascading time doesnot follow the scaling with MA predicted for the weak regime, in contradiction withthe RD theory assumption. Our results generally support and expand the RD theorypredictions.

Key words: magnetic fields – magnetic reconnection – (magnetohydrodynamics) MHD– turbulence – methods: numerical — stars: formation

1 INTRODUCTION

One of the most employed and well known concepts in mag-netohydrodynamic (MHD) theory is the magnetic “frozen-in” condition introduced by Alfven. When the time scalesfor Ohmic dissipation of the magnetic fields are much largerthan the typical dynamical time scales of the flow (thedimensionless parameter characterizing the ratio betweenthese two time scales being given by the magnetic Reynoldsnumber RM = UL/η, with U and L the characteristic ve-locity and scale of the flow, and η the magnetic diffusivityprovided by the Ohmic dissipation), one can adopt the idealMHD approximation. It consists in neglecting the resistive

⋆ Contact e-mail: [email protected]† Contact e-mail: [email protected]‡ Contact e-mail: [email protected]§ Contact e-mail: [email protected]

term in the magnetic induction equation. In this limit, it canbe demonstrated that the magnetic flux across a Lagrangianfluid element is conserved, that is, the magnetic field is per-fectly advected by the fluid motions in the direction normalto the field lines.

The ideal MHD description (and consequently, thefrozen-in condition) is usually thought to be a good ap-proximation for most astrophysical plasmas, which havein general huge values of RM . Nevertheless, the frozen incondition when applied, for instance, to star formationregions gives rise to several problems due to observationaland theoretical requirements for diffusive magnetic fluxtransport through the plasma (e.g. Santos-Lima, et al.2010; Santos-Lima, de Gouveia Dal Pino & Lazarian2012, 2013; Leao, et al. 2013;Gonzalez-Casanova, Lazarian & Santos-Lima 2016). Am-bipolar diffusion is usually invoked for breaking the frozen-in

© 2020 The Authors

http://arxiv.org/abs/2005.07775v2

mailto:[email protected]




2 Santos-Lima, Guerrero, de Gouveia Dal Pino & Lazarian

condition and solving the problem of the magnetic fluxtransport during star formation (see for example Shu 1983;Nishi, Nakano, & Umebayashi 1991; Ciolek & Mouschovias1993; Shu et al. 1994; Tassis & Mouschovias 2005). How-ever, several studies revealed weaknesses in this solu-tion (Shu, et al. 2006; Crutcher, Hakobian & Troland2009; Krasnopolsky, Li & Shang 2010, 2011;Li, Krasnopolsky & Shang 2011). A potential solution forthis problem in the framework of protoplanetary disk forma-tion has been proposed by Machida, Inutsuka & Matsumoto(2007, 2009) based on laminar MHD simulations combinedwith local Ohmic resistivity (see also recent studies onthe role of the ambipolar diffusion during protostellardisk formation Wurster & Li 2018; Guillet et al. 2020;Marchand et al. 2020; Zhao et al. 2020).

We believe that the limitations of the approachthat is described above is that the effects of ubiquitousastrophysical turbulence are disregarded in the afore-mentioned studies. The diffuse interstellar medium andmolecular clouds are turbulent. There are overwhelm-ing observational evidences that support this claimthrough the measurements of power spectrum of densi-ties in diffuse ISM (see Armstrong, Rickett & Spangler1995; Chepurnov & Lazarian 2010), broadening ofthe molecular lines (see Larson 1981), statistics ofvelocities (see Lazarian & Pogosyan 2000; Lazarian2009 and references therein, Padoan, et al. 2009;Chepurnov, et al. 2010, 2015; Kandel, Lazarian & Pogosyan2017; Utomo, Blitz & Falgarone 2019; Wolleben, et al. 2019;Xu 2020; Yuen, et al. 2019), variations of the Faraday ro-tation (Haverkorn, et al. 2008; Xu & Zhang 2016), andpower spectrum of synchrotron fluctuations (Chepurnov1998; Cho & Lazarian 2002, 2010). The reviews describ-ing the molecular cloud turbulence are presented inMcKee & Ostriker (2007); Mac Low & Klessen (2004, seealso Elmegreen & Scalo 2004).

The field of MHD turbu-lence (see Montgomery & Turner 1981;Matthaeus, Montgomery & Goldstein 1983;Shebalin, Matthaeus & Montgomery 1983; Higdon 1984)has seen a rapid progress due to both, to Solar windmeasurements (see Tu & Marsch 1995; Goldstein, et al.1995; Bruno & Carbone 2013 for a review), theoretical(see Goldreich & Sridhar 1995; Lazarian & Vishniac1999; Lithwick & Goldreich 2001; Cho & Lazarian2002; Eyink, Lazarian & Vishniac 2011) and numericalprogress (Cho & Vishniac 2000; Maron & Goldreich 2001;Cho, Lazarian & Vishniac 2002; Cho & Lazarian 2003;Kowal & Lazarian 2010; Federrath, et al. 2010; Beresnyak2014; see also a recent book by Beresnyak & Lazarian2019).

As we discuss later, the subject of MHD turbulenceis closely related to the processes of magnetic reconnec-tion in turbulent fluid. The model of turbulent recon-nection in Lazarian & Vishniac (1999) predicts the failureof the traditional flux freezing in highly conducting tur-bulent fluids. This process does not depend on the rateof ambipolar diffusion and motivated a number of ourearlier studies (Lazarian 2005, 2011; Santos-Lima, et al.2010; Santos-Lima, de Gouveia Dal Pino & Lazarian 2012,2013; Gonzalez-Casanova, Lazarian & Santos-Lima 2016)

that consider the processes of magnetic flux transport inturbulent fluids.

The process by which the topology of the magneticfield changes depends on whether the fluid is in a lami-nar or turbulent state. In the presence of turbulence, themotions of the ionized gas produce tangling and wander-ing of the magnetic field lines which give origin to sev-eral micro-sites of magnetic reconnection. This process isindependent on how small is the Ohmic resistivity whichis always present in any real plasma. These reconnectionmicro-sites are continuously formed and spread all overthe turbulent plasma volume. As a consequence, the fieldlines topology can be modified, and large-scale magneticflux can be transported through the gas, implying that theflux freezing concept is seriously altered (Lazarian 2005;Eyink, Lazarian & Vishniac 2011). The speed at which themagnetic flux is transported in such conditions is indepen-dent of the electric resistivity of the plasma, or the degreeof its ionization but is regulated by the turbulence param-eters, as predicted in the theory of fast magnetic reconnec-tion introduced by Lazarian & Vishniac (1999). This theorywas tested numerically in Kowal, et al. (2009, 2017, 2020). Aconvincing quantitative numerical study proving that turbu-lent reconnection violates flux freezing in MHD turbulence ispresented in Eyink, et al. (2013). An extensive body of evi-dence in favor of turbulence reconnection has been collectedby now and we refer the reader to Lazarian, et al. (2020)where the modern state of the turbulent reconnection the-ory and a description of its tests with solar wind as well asnumerical testing with different codes is reviewed.

The concept of magnetic diffusion via turbulent recon-nection —Reconnection Diffusion (henceforth RD)— is dis-tinct from the concept of standard turbulent mixing. Thelatter is based on the idea that the field lines are mixed pas-sively by the turbulent eddies, without taking into accountthe effects of the magnetic field on the turbulent cascade. RDon the other hand, covers the interesting situation in whichthe magnetic forces are dynamically important (e.g. in thelate stages of star formation), and relies on the fact that thefast reconnection induced by the MHD turbulence is inde-pendent of the value of the electric resistivity of the plasma.For many of the astrophysical applications it is importantthat the RD is not altered by the effects of ambipolar drifton the scales where turbulence exists.

This RD theory predicts that the diffusion coefficientηRD for large-scale magnetic fluxes (i.e., scales larger thanthe injection or forcing scale of the turbulence) depends onthe turbulence parameters as follows. In the case of super-Alfvenic turbulence (that is, when the Alfvenic Mach num-ber, MA = Uturb/vA [Uturb is the turbulent velocity and vAis the local Alfven velocity] is larger than one), it coincideswith the standard turbulent mixing coefficient,

ηRD ∼ LturbUturb, (1)

where Lturb and Uturb are the length and the velocity of theturbulence at the injection scale, respectively. On the otherhand, in the regime of sub-Alfvenic turbulence (MA < 1),this value is reduced by a factor proportional to the thirdpower of MA (Lazarian 2006, 2011):

ηRD ∼ LturbUturbM3A. (2)

Therefore, according to the RD theory, the efficiency of

MNRAS 000, 1–22 (2020)

Diffusion of large-scale magnetic fields by reconnection in MHD turbulence 3

the magnetic flux transport strongly depends on the localturbulence regime. In the context of star formation, theturbulence parameters in scales below sub-parsec (downto dozens of A.U.) can be difficult to infer with precision.Nonetheless, we reinforce that RD process is always presentat some level during all the star formation process. Itis a natural consequence of the ubiquitous presence ofturbulence in astrophysical environments (see for exam-ple Krumholz & McKee 2005; Ballesteros-Paredes et al.2007; Hennebelle & Chabrier 2011; Padoan & Nordlund2011; Federrath & Klessen 2012, 2013; Hull, et al.2017). In previous work (Santos-Lima, et al.2010; Santos-Lima, de Gouveia Dal Pino & Lazarian2012, 2013; Leao, et al. 2013;Gonzalez-Casanova, Lazarian & Santos-Lima 2016), weinvestigated numerically the removal of magnetic flux fromcollapsing turbulent molecular clouds and protostellar disks,considering an “ideal” MHD approach (i.e., the microscopicmagnetic dissipation term was not considered explicitlyin the induction equation, although an effective value isalways present due to the numerical discretization of theequations). We found that the magnetic flux removal byRD is efficient in these systems, and helps the gravitationalcollapse of the structures (see also Myers, et al. 2013).However, these works focused mostly on the super andtrans-Alfvenic regimes of the turbulence where the recon-nection diffusion coefficient is controlled by Eq. (1). Theaim of this work is to test quantitatively the prediction ofEq. (2) by using three-dimensional (3D) MHD simulations.It is also the first attempt to generate simulations ofstationary weak MHD turbulence (the cenario invoked bythe RD theory) in the presence of finite compressibility,which is more realistic for astrophysical environments.

This work in organized as follows. In §2 we present themain predictions of the RD theory. The numerical methodsand setup for the numerical simulations of this study aredescribed in §3. The results are presented and discussed in§4 and §5, respectively. Our major findings are finally sum-marized in §6.

2 BASICS OF RECONNECTION DIFFUSION THEORY

To understand the process of reconnection diffusion wepresent here some basic facts of MHD turbulence theory.For simplicity, we consider only the case of incompressibleMHD turbulence.

Traditionally, Alfvenic turbulence is described in termsof non-linearly interacting wave packets in Fourier space(Iroshnikov 1963; Kraichnan 1965). The non-linear cascad-ing rate is given in this case by

τ−1casc ≈

(δuℓ/ℓ⊥)2

ωA, (3)

where the angular frequency, ω = VA/ℓ‖, and ℓ‖ is the par-allel wavelength. If the injection velocity UL is less than theAlfven velocity, the cascade is evolving decreasing only per-pendicular wavelengths ℓ⊥. This is the regime of weak turbu-lence with the parallel scale being all the time equal to theinjection scale and δuℓ ∼ ℓ

1/2⊥ (Lazarian & Vishniac 1999

[LV99 hereafter]; Galtier, et al. 2000). However the weak

cascade changes its nature at a small scale. Below we ex-plain the nature of this change.

The theory of strong MHD turbulence was formu-lated by Goldreich & Sridhar (1995, henceforth GS95) fortransAlfvenic turbulence, i.e. UL = VA. For this turbulenceGS95 formulated the condition of the critical balance that re-lates the scale of parallel and perpendicular motions, namely,δuℓ/ℓ⊥ ≈ VA/ℓ‖. This condition means that the parallelscale should change with the decrease of the perpendicularscale.

The transfer to the strong MHD turbulence regime canhappen if the turbulence is injected at UL < VA and is weakat its origin. The key to that is the increase of the strengthof non-linear interactions with the decrease of perpendicularscale. The transfer to the strong turbulence regime changesthe nature of the turbulent motions. In fact, due to fast tur-bulent reconnection predicted in LV99 the turbulent motionsbecome similar to the hydrodynamic eddies that mix mag-netic field perpendicular to the direction of magnetic field.For this eddies it is natural to assume that δuℓ/ℓ ∼ VA/ℓ‖,i.e., that the rate of turbulent mixing motions is equal to therate of perturbation propagating along the magnetic field.The difference between the original critical balance in GS95and the above relation between the perpendicular scale ℓand the parallel scale ℓ‖ is that both scales are measuredin terms of the magnetic field of the eddies, i.e. local mag-netic field, compared to the wavenumbers k‖ and k⊥ thatare given in the global mean magnetic field reference sys-tem. The concept of local eddies mixing up magnetic fieldis an essential component for the understanding of the RDprocess. Note that numerical research does indicate that thecritical balance relations are valid only in the local system ofreference (Cho & Vishniac 2000; Maron & Goldreich 2001;Cho, Lazarian & Vishniac 2002).

The generalization of GS95 theory for subAlfvenic tur-bulence provided in LV99 as well as the analogy between theeddy turbulence in MHD and ordinary hydrodynamic turbu-lence helps to understand the nature of the RD. The predic-tions for RD when turbulence is in the sub-Alfvenic regime(Lazarian 2005, 2011) can be recovered from the statis-tical calculations presented in Eyink, Lazarian & Vishniac(2011), and are summarized bellow.

The magnetic field is assumed to be diffused by tur-bulence at a rate similar to the diffusion of trace particlesin the direction perpendicular to the (locally) uniform mag-netic field. A simple statistical analysis can show that theperpendicular diffusivity of trace particles due to the turbu-lent velocity field δu is given by:

D⊥ =

∫ ∞

−∞

dt〈δu′⊥(0) · δu

′⊥(t)〉, (4)

where δu′(t) is the turbulent velocity at the position of theparticle at time t, and the angle brackets, 〈·〉, denote an en-semble average over all the particles, which are assumed tobe distributed randomly in space. Considering turbulencecomposed by Alfven waves with random phases and turbu-lence with a single scale, ℓ, perpendicular to the field lines(and corresponding parallel scale, ℓ‖). The expression insidethe last integral is

〈δu′ℓ,⊥(0) ·δu

′ℓ,⊥(t)〉 ∼ δu2

ℓℜ{

exp(

iωA,ℓ‖t− |t|/τdec)}

, (5)

where δuℓ is the perpendicular turbulent velocity at scale ℓ,

MNRAS 000, 1–22 (2020)


ωA,ℓ‖ ∼ VA/ℓ‖ is the local Alfven frequency, and τdec is thetime-scale for decorrelation of the velocity eddies (the cor-relation is simply assumed to decay exponentially in time).This leads to

D⊥(ℓ) ∼ δu2ℓ

τdec(ωA,ℓ‖τdec)

2 + 1. (6)

We expect that motions at the largest scales producethe dominant diffusivity. We will assume isotropy at the in-jection scale, that is, ℓ‖ ∼ ℓ.

When the time-scale for the microscopic diffusivity islarger than the dynamical time-scales of the system (which isthe case for most astrophysical environments), the decorrela-tion time for the velocity eddies at the injection scale shouldbe of the order of the energy cascade time for these eddies.In the regime of weak turbulence (i.e., when the energy cas-cading time is much longer than the linear wave time τw ∼ω−1A,ℓ‖

), we have τcasc ∼ (ℓ/δuℓ)M−1A (Lazarian & Vishniac

1999; Galtier, et al. 2000). Substituting this value in Eq. (6),we obtain the RD prediction for the magnetic diffusivity ηRD

(Lazarian 2005):

ηRD ∼ D⊥ ∼ ℓδuℓ min(

1,M3A

)

. (7)

In situations where the energy cascading time at theinjection scale is larger than the molecular or numerical vis-cous time (implying a low effective Reynold’s number), thedecorrelation time of the velocity, τdec, can be more closelyrelated to the dissipation time, τdiss ∼ ℓ2/ν, where ν is themolecular viscosity. The magnetic field diffusion driven byturbulence in this case will depend on the microscopic dif-fusion and, if τdec ∼ τdiss, then the diffusivity becomes de-pendent on the molecular viscosity, ν.

In our discussion we considered only one component ofMHD turbulence, namely, Alfven modes and disregarded theslow and fast modes (see Cho & Lazarian 2003). This is dueto the fact that Alfven modes are the most important inmixing the medium.

3 NUMERICAL MODELS

In order to test the dependency of the magnetic diffusion co-efficient, ηRD, with the Alfvenic Mach number, MA, of theturbulence in the sub-Alfvenic regime, we employed three-dimensional MHD numerical simulations of forced turbu-lence in a Cartesian domain, in the presence of an externaluniform magnetic field of intensity B0 in the x-direction. Weused the Pencil Code

1 for numerically solving the set ofcompressible, isothermal, MHD equations:

D ln ρ

Dt= −∇ · u, (8)

Du

Dt= −c2s∇ ln ρ+

1

ρJ× (B0 +B) + ν3∇

6u+ f , (9)

∂A

∂t= u× (B0 +B) + η3∇

6A, (10)

1 http://pencil-code.googlecode.com/

where D/Dt = ∂/∂t + u · ∇ is the lagrangian derivative,A is the magnetic potential vector, B = ∇×A is the mag-netic field generated by the internal currents, J = ∇×B/µ0

is the current density, µ0 is the magnetic permeability, ν3and η3 are, respectively, the coefficients of hyper-viscosityand magnetic hyper-diffusivity, cs is the isothermal soundspeed, u is the velocity, ρ is the density, and f representsthe force responsible for the turbulence injection. We usethe hyper-viscosity and magnetic hyper-diffusivity schemeswith the aim of obtaining a turbulent spectra with an ex-tension as large as possible for each considered resolution(Borue & Orszag 1995; Haugen & Brandenburg 2004). Thevalue of the coefficients were minimized such that numericalstability is guaranteed.

We have also performed simulations without thehyper-viscosity and the magnetic hyper-diffusivity.In these cases, we employed the usual viscosityand Ohmic resistivity, namely, the terms ν3∇

6u

in eq. 9 and η3∇6A in eq. 10 are replaced by

η1{

∇2u+ (1/3)∇∇ · u+ 2S · ∇ ln ρ

}

and η1∇2A, re-

spectively, where ν1 is the constant viscosity, η1 is themagnetic diffusivity, and S is the rate of strain tensor givenby Sij = {(1/2)(∂ui/∂xj + ∂uj/∂xi)− (1/3)δij∇ · u}.These models are explicitly mentioned in the text wheneverthey appear.

The turbulence is constantly forced by the increment ofthe velocity field with a spectrum of Fourier modes. Thesemodes are purely solenoidal, and the phases are randomlychanged at every iteration during the numerical integration.The turbulence is, therefore, nearly statistically homoge-neous, non-helical, and delta correlated in time. In § 3.1we provide more details about the spectrum of the excitedvelocity modes.

The value of the diffusion coefficient, ηRD, is extractedfrom the simulations through the Test-Field method. Itemploys a set of passive test magnetic fields in order tocalculate unambiguously the coefficients of the turbulentmean-fields (including the diffusion tensor; see the AppendixA and more details in Brandenburg & Subramanian 2005;Schrinner, et al. 2007; Brandenburg, et al. 2010).

3.1 Setup and parameters

The Reconnection Diffusion theory (§ 2) is formulated in theincompressible limit, assuming that when sub-Alfvenic tur-bulence is forced isotropically in the presence of an uniformmagnetic field (at least locally), it will develop a cascade inthe regime predicted by the weak turbulence theory. There-fore, we restricted our simulations setup and parameters tothe subsonic regime (yet using finite sound speed cs in thesimulations) and favoring conditions under which weak tur-bulence could develop.

Table 1 lists the parameters employed in the simula-tions presented in the next Section. For each set of simula-tions (identified by its name in the first column of Table 1),we spanned a range of values of MA by changing the mag-netic field strength B0 and keeping the rms value of theturbulence velocity, vrms, constant, near to a fixed referencevalue, v0. We do not control directly vrms in our simulations,but instead, the amplitude of the forcing.

The turbulent diffusivity is controlled by the largestscale motions of the system (the motions at the injection

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Table 1. Runs parameters

runs set L‖xL⊥ res. MSa MA

b vrms/v0 forcing k‖L/2π k⊥L/2πc [t0, t1]d ν3, η3e

16L-Ms0.32-A 16Lx1L 2048, 1282 0.32 0.8, 0.57, 1.10, 1.18, A [0, 4] [3, 4] [3, 8] 1.8× 10−9

0.4, 0.2 1.18, 1.220.1 1.33

16L-Ms0.08 16Lx1L 2048, 1282 0.08 0.8, 0.4, 1.09, 1.17, A [0, 4] [3, 4] [3, 8], [4, 9] 1.8× 10−9

0.28, 0.2 1.11, 1.10 [7, 12], [7, 12]0.1 1.11 [9, 14]

16L-Ms0.02-A 16Lx1L 2048, 1282 0.02 0.8, 0.4, 1.08, 1.17, A [0, 4] [3, 4] [3, 8], [4, 9] 1.8× 10−9

0.28, 0.2 1.27, 1.04 [4, 9], [13, 18]

0.1 1.04 [15, 20]

8L-Ms0.02-A 8Lx1L 1024, 1282 0.02 0.8, 0.4, 1.09, 1.17, A [0, 4] [3, 4] [3, 8], [4, 9] 1.8× 10−9

0.2, 0.1 0.99, 1.01 [13, 18], [15, 20]

4L-Ms0.02-A 4Lx1L 512, 1282 0.02 0.8, 0.4, 1.10, 1.19, A [0, 4] [3, 4] [3, 8], [4, 9] 1.8× 10−9

0.2, 0.1 0.94, 1.01 [13, 18], [15, 20]

1L-Ms0.02-A 1Lx1L 128, 1282 0.02 0.8, 0.4, 1.22, 1.19, A [0, 4] [3, 4] [3, 8], [4, 9] 1.8× 10−9

0.2, 0.1 1.04, 1.11 [7, 12], [7, 12]

8L-Ms0.02-Ab 8Lx1L 1024, 1282 0.02 0.8, 0.4, 1.07, 1.10, Ab − [4, 5] [3, 8], [4, 9] 1.8× 10−9

0.2, 0.1 1.09, 0.93 [10, 15], [15, 20]

8L-Ms0.02-I 8Lx1L 1024, 1282 0.02 0.8, 0.4, 1.07, 1.11, I − [3, 4] [3, 8], [4, 9] 1.8× 10−9

0.2, 0.1 1.19, 1.19 [4, 9], [4, 9]

16L-Ms0.02-low-A 16Lx1L 1024, 642 0.02 0.8, 0.4, 1.02, 1.12, A [0, 4] [3, 4] [11, 15] 6.2× 10−8

0.2, 0.14, 1.31, 1.430.1, 0.05 1.44, 1.400.025 1.32

16L-Ms0.02-low-A-diff2 16Lx1L 1024, 642 0.02 0.8, 0.4, 1.18, 1.28, A [0, 4] [3, 4] [4, 8] (1.2× 10−2)f

0.2, 0.1, 1.37, 1.43

16Lx2L-Ms0.02-low-A 16Lx2L 1024, 1282 0.02 0.2 1.30 A [0, 4] [3, 4] [11, 15] 6.2× 10−8

16L-Ms0.02-hi-A 16Lx1L 2048, 2562 0.02 0.2 0.90 A [0, 4] [3, 4] [3, 5]g (5.5× 10−11)h

a MS ≡ v0/cs is the approximate sonic Mach number of the simulations.b MA ≡ v0/vA,0 is the approximate Alfvenic Mach number of the simulations.c For the models with forcings ‘Ab’ and ‘I’, this column shows the absolute values of the vector k of the forced modes.d [t0, t1] is the time interval used for the averages in time, in units of ℓ⊥/v0.e ν3, η3 are the hyper-viscosity and hyper-resistivity in units of ℓ5⊥v0.f No hyper-viscosity or hyper-resistivity were used in these runs. This column shows the values of ν1 and η1, the viscosity and magnetic diffusivity,respectively, in units of ℓ⊥v0. See Section 3 for more details.

g This simulation used as initial condition a previously evolved simulation with resolution 2048, 1282.h Due to the anisotropic resolution in this simulation, these values of ν3 and η3 refer to the terms in the hyper-viscosity and hyper-resistivity containingthe derivatives in the directions perpendicular to the mean magnetic field. The values of ν3 and η3 used in the terms containing the derivatives in theparallel direction are 1.8× 10−9.

scale), where the universal laws of the inertial range arenot formally valid, and could be affected by the details ofthe forcing mechanism. At the same time, the forcing canalso determine the MHD turbulence regime. Because thiswork focuses on the subsonic turbulence, we employed forc-ing schemes purely solenoidal, and with approximately thesame coherence length in the directions parallel and perpen-dicular to the mean magnetic field. The forcing was chosendelta correlated in time for two reasons: first, if the cascad-ing time follows the weak turbulence theory, the correlationof the waves at the injection scale should persist for a timesimilar to the cascading time, even if the forcing is generat-ing low energy random waves continuously. Second, forcingwith time correlation proportional to M−1

A (e.g., Alexakis2011) must induce Alfven waves with approximately thesame decorrelation time, which could determine trivially thediffusion coefficient.

We employed three forcing schemes, which differ in thediscrete spectrum of the velocity modes excited. In one of

them, all the modes inside a spherical shell in the k-spaceare excited with the same amplitude. This k-isotropic dis-tribution of amplitudes is the most usual choice in numeri-cal simulations of forced turbulence. The models using thisscheme are identified by ‘I’ in the column “forcing” of Ta-ble 1. Alternatively, in order to constrain the parallel andperpendicular injection scales to well defined values, ℓ‖ andℓ⊥, respectively, and at the same time avoid the forcing ofpurely 2D modes (i.e., with k‖ = 0) as well as waves of largewavelength in the direction parallel to the imposed uniformmagnetic field, we also used an k-anisotropic scheme (iden-tified by ‘A’ in Table 1). It forces all the waves with thecomponents k⊥ and k‖ inside a cylindrical shell in the k-space. The amplitude of the spectrum is modulated by afactor ∝ k2

‖. The third forcing scheme we used excites all themodes inside a spherical shell in the k-space, but modulatesthe amplitudes by a factor ∝ sin(2θ), where cos θ = k‖/k.This last scheme is identified by ‘Ab’ in Table 1. The ‘A’ and‘Ab’ schemes favor distributions with reduced amplitude for

MNRAS 000, 1–22 (2020)


the wavevectors corresponding to wavelengths far from thefixed injection scale. The forcing scheme ‘I’, on the otherside, generates waves elongated in both parallel and per-pendicular directions. We emphasize that, statistically, allthe three schemes force velocity fluctuations in an (nearly)isotropic form in the physical space.

The injection scales are indicated in the columnsk‖L/2π and k⊥L/2π, where L is the shortest side of thedomain, perpendicular to the mean uniform magnetic field,in Table 1 (except for the runs set 16Lx2L-Ms0.02-low-A,where this side has length 2L). These values were chosenin order to maximize the turbulence inertial range. Note,however, that although the separation of scales between thelargest turbulence eddies and the mode employed in the testfields (k⊥,tfL/2π = kz,tfL/2π = 1; see Appendix A) is stilllimited.

Unlike real extended astrophysical environments, thefiniteness of the computational box introduces effects onthe wave turbulence due to insufficient density of large-scalemodes represented in the discrete Fourier space. A theoreti-cal constraint on the validity of the classical weak turbulenceregime in a finite box domain, which was derived both un-der the Reduced MHD approximation and the assumptionof k⊥ ≫ k‖ (Nazarenko 2007), is approximately given by√

ℓ‖L‖

≪δuℓ

B0

ℓ‖ℓ⊥

≪ 1, (11)

where δuℓ is the velocity amplitude corresponding to themode with scales ℓ‖ and ℓ⊥. Therefore, to fullfill this con-dition at the injection scale where ℓ‖ ≈ ℓ⊥, it is needed to

ensure√

ℓ/L‖ ≪ MA ≪ 1.Another effect due to the finite box size is the 2D “en-

slaving” of the 3D MHD turbulence if

ℓ⊥L⊥

√

ℓ‖L‖

≫δuℓ

B0

ℓ‖ℓ⊥

, (12)

(Nazarenko 2007), which at the injection scales becomes(ℓ/L⊥)

√

ℓ/L‖ ≪ MA. It should be noted that if the left-most inequality in Eq. (11) is satisfied, then the inequalityin Eq. (12) is automatically false (ℓ/L⊥ < 1).

To evaluate the finite size effects of the computationaldomain on the RD coefficient we ran simulations with fourdifferent domain sizes in the direction parallel to the meanmagnetic field (x-direction): L‖ = 16L, 8L, 4L, and 1L, keep-ing fixed the domain size perpendicular L⊥ = 1L. To verifya possible influence of the perpendicular domain size on theRD coefficients, we ran a comparative model with L⊥ = 2L,keeping L‖ = 16L (runs set 16Lx2L-Ms0.02-low-A).

In the results presented in the next Section the analy-sis of the simulations is performed after the turbulence hasreached the statistically stationary state. The analyses av-erage quantities between times t0 and t1 shown in Table 1.Four complete snapshots with equal time separation are ex-tracted from the simulations during this time interval (theonly exception is the model 16Lx1L-Ms0.02-hi-A, which hasonly two complete snapshots).

Figure 1 compares the distribution of the velocity mod-ulus on the central xy-plane for three selected runs fromTable 1 at time t1 (see Table 1). The three models differonly by the forcing scheme: A (top), Ab (middle), and I(bottom). All the runs in Figure 1 have MA = 0.4. Fig-

ure 2 depicts the same quantity for simulations that haveMA = 0.2. Observe that the differences between the turbu-lent structures generated by the different forcings becomemore pronounced for smaller values of MA (Figure 2). ForMA = 0.8 (not shown here for compactness), the velocitydistribution resulting from the three forcing schemes is in-distinguishable.

Below, we define the 2D power spectrum of the turbu-lence E(k‖, k⊥) in terms of the velocity u and magnetic fieldB as

E(k‖, k⊥) =∑

k′

(

1

2ρ0u

∗k′ · uk′ +

1

8πB

∗k′ ·Bk′

)

, (13)

where uk = Fk {u} and Bk = Fk {B} are the k = (k‖+k⊥)components of the discrete Fourier transform of u and B,respectively, k‖ = (k ·B0)B0/B

20 , ρ0 is the mean density,

and k‖,⊥ = |k‖,⊥|. The superscript ∗ means the complexconjugate, and the sum extends for all the discrete modes,k′, with components in the interval k‖ 6 |k′

‖| < (k‖ + 1),and k⊥ 6 |k′

⊥| < (k⊥ + 1). The 1D power spectrum E(k⊥)is defined by

E(k⊥) =

k‖,max

∑

k‖=0

E(k‖, k⊥), (14)

such that

Eturb =1

2ρ0〈u

2〉+1

8π〈B2〉 =

k⊥,max

∑

k⊥=1

E(k⊥), (15)

is the total turbulent energy in the system, and the brackets,〈·〉, represent average in space.

The transfer spectrum is obtained from the followingprocedure: we multiply u

∗k to the Fourier transform of the

momentum equation (assuming the incompressible limit byignoring the density variations), and we add the Fouriertransform of the induction equation multiplied by B

∗k/(4π).

We denote by Tk the time derivative of (ρ0u∗k · uk/2 +B

∗k ·

Bk/8π), when we neglect the forcing and dissipation terms:

Tk = ℜ

{

ρ0u∗k · Fk

[

(u · ∇)u−1

4π(∇×B)×B

]}

−1

4πℜ{B∗

k · Fk [∇× (u×B)]} , (16)

with ℜ denoting the real part of.The perpendicular transfer spectrum T (k⊥) is then de-

fined by

T (k⊥) =∑

k′

Tk′ , (17)

with the sum extending over all the modes k′, with the

perpendicular components, k′⊥, in the interval 0 6 |k′

⊥| <(k⊥ + 1) (see, for example, Alexakis, et al. 2007; Alexakis2011) . The turbulence energy transfer is given by the max-imum value of T (k⊥),

Tturb = max {T (k⊥)} . (18)

As we are going to see in Section 4.2, Figure 11 showsthe normalized 2D power spectrum distribution E(k‖, k⊥)for the same models presented in Figures 1 and 2, in theleft and right column, respectively. By observing the distri-bution of modes with the highest values of energy (around

MNRAS 000, 1–22 (2020)


8L-Ms0.02-A, MA = 0.4

0.0

0.5

1.0

1.5

2.0

2.5

|v|/v0

8L-Ms0.02-Ab, MA = 0.4

0.0

0.5

1.0

1.5

2.0

2.5

|v|/v0

8L-Ms0.02-I, MA = 0.4

0.0

0.5

1.0

1.5

2.0

2.5

|v|/v0

Figure 1. Central slice (xy-plane) showing the velocity modulus distribution at the final time of the simulations. Models with identicalparameters but different forcing distributions in the k-space are compared. From top to bottom: A-forcing, Ab-forcing, and I-forcing. Allsimulations have the same Alfvenic Mach number MA ≡ v0/vA,0 = 0.4 and sonic Mach number MS = 0.02. See Table 1 for the completedescription of the simulations parameters.

8L-Ms0.02-A, MA = 0.2

0.0

0.5

1.0

1.5

2.0

2.5

|v|/v0

8L-Ms0.02-Ab, MA = 0.2

0.0

0.5

1.0

1.5

2.0

2.5

|v|/v0

8L-Ms0.02-I, MA = 0.2

0.0

0.5

1.0

1.5

2.0

2.5

|v|/v0

Figure 2. Same as Figure 1, but for simulations with the Alfvenic Mach number MA ≡ v0/vA,0 = 0.2.

MNRAS 000, 1–22 (2020)


0 < k‖L/2π, k⊥L/2π < 4), we can qualitatively assess thedifferences between the three different forcing schemes.

Observe that in the above definitions of energy andtransfer spectrum, we are neglecting any density fluctua-tions. We choose to do so, in order to simplify the analysisand because we expect the Alfven modes to dominate theturbulence spectrum for the subsonic simulations presentedin this work.

4 RESULTS

4.1 Compressibility and domain size effects

Left panels of Figures 3 and 4 show the dependence of ηtf ,computed with the test-field method, with the MA resultingin the simulations (see for example Schrinner, et al. 2005;Brandenburg, Radler & Schrinner 2008). Each point in thecurves corresponds to one model in Table 1 with anisotropicforcing (A), and points with the same shape and color cor-respond to models with the same sonic Mach number inFigure 3, and same box size in Figure 4. The values of thediffusivity are normalized by the estimate of the hydrody-namical turbulent diffusivity ηhyd = (1/3)ℓvrms, where ℓ isthe injection scale of the simulation (corresponding to theminimum wavenumber k⊥L/2π indicated in Table 1). TheAlfvenic Mach number measured from the simulations is de-fined by, MA = vrms/〈vA〉, where 〈vA〉 = 〈B/ (4πρ)1/2〉 isthe Alfven velocity averaged in the domain. The quantitiesvrms, 〈vA〉, and ηtf represent time averaged values of therespective space averaged quantities.

The left panel of Figure 3 compares models with thesame domain size 16Lx1L, but three different values of sonicMach number, MS = v0/cs (where cs is the isothermalsound speed of the simulation). For MA larger than a cer-tain value M∗

A (M∗A ≈ 0.5, 0.4, and 0.2 for MS = 0.32, 0.08,

and 0.02, respectively), we observe the approximate relationηtf/ηhyd ∼ M3

A. For MA < M∗A, this power-law dependence

with MA seems to change asymptotically to ηtf/ηhyd ∼ M2A,

with the constant of proportionality increasing with MS . Forthe models with the lowest Mach number (MS = 0.02) thevalue of M∗

A is closer to the theoretical lower limit for va-lidity of the classical weak turbulence (Equation 11). Thislimit is indicated by the vertical solid line in Figure 3.

The turbulent diffusion is naturally dominated by themotions of the injection scales, and the decorrelation timeof the velocity fields at these scales can be expected to bedirectly related to the energy transfer time shown in theright panels of Figures 3 and 4. If the break of the de-pendence ηtf/ηhyd ∼ M3

A around some M∗A is caused by a

change in the regime of the turbulence cascade (see Equa-tion 6), the behavior of the energy transfer time τener (i.e.,the time-scale for the energy in the injection scale to cascadeto smaller scales) could also occur around the same valuesof M∗

A. We estimated τener dividing the total turbulence en-ergy Eturb = Ev +Eb, where Ev and Eb are the kinetic andmagnetic energies (in the incompressible limit), by the max-imum value of the energy transfer spectrum, Tturb (which isapproximately the energy transfer rate at the injection scaleor the turbulence injection power; see below the descriptionof the energy transfer spectrum, T (k⊥), in the right columnsof Figures 5 and 10). Right panel of Figure 3 shows the de-

pendence of τener (normalized by the estimate of the non-linear turbulence time τnl = ℓ/vrms at the injection scale),as a function of MA, for the same models shown in the leftpanel. In fact, τener/τnl follows approximately a well definedpower law in MA for MA > M∗

A, and becomes flatter forsmaller values of MA. The resulting power law does not co-incide with that in the inertial range predicted for the weakturbulence regime, τener/τnl ∼ M−1

A . Instead, we obtain adependency ∼ M−2

A .

Models with the same sonic Mach number, MS = 0.02,but different domain sizes are compared in the left panel ofFigure 4. The shorter the parallel extension L‖ of the do-main, the larger the departure from the relation ηtf/ηhyd ∼M3

A for MA smaller than some M∗A, which increases with

the decrease of the domain size. The theoretical limits forthe validity of the classical weak turbulence indicated by thevertical lines in Figure 3 are also shown in Figure 4. Eachcolor corresponds to a different domain size (following thesame color scheme for each set of simulations). We also ob-serve in the right panel of Figure 4 that for simulations withL‖ < 16L, the curve for τener/τnl deviates from a powerlawfor increasing values of M∗

A.

In the left panels of Figure 5 we present the 1D powerspectrum of the total energy, E(k⊥), for simulations withdifferent MA. Inside these same panels we also show the ra-tio between the velocity and the magnetic field power spec-tra, Eu(k⊥)/EB(k⊥). On the right column we present theenergy transfer spectrum, T (k⊥), for the same models, nor-malized by Eturb/τnl, which is the expected value of theenergy transfer spectrum in the case of strong turbulencecascade. The presence of a plato in the energy transfer spec-trum is indicative of an inertial range, where the energy fluxbetween scales is constant. The flatness of T (k⊥) just afterthe injection scale k⊥L/2π ≈ 3-4 is important to guaranteethat with the current resolution, the energy transfer timefrom this scale (and therefore the velocity correlation time)is shorter than the dissipation time. If this were not the case,the measured ηtf could be dominated by the numerical effec-tive viscosity, as the velocity decorrelation time would be ofthe order of the dissipation time (see Equation 6). The toppanels show simulations with the highest MS = 0.32, anddomain size 16Lx1L. The middle panels show models withthe smallest MS = 0.02. In both sets, the power spectrumof the model with MA closer to unity shows a poor inertialrange (4 . k⊥L/2π . 10). For this wavenumber interval,the normalized transfer spectrum decays roughly ten per-cent for the model with MS = 0.02 (middle panel). As MA

decreases, a gradual steepening in the spectrum is accom-panied by the decrease on the energy transfer values at theinjection scale, which is consistent with the increase of theenergy transfer time. The power law index seems to becomesteeper than −2 (the value predicted for the weak turbulenceregime) for the models with MS = 0.02 (middle panels). Thetransfer spectra for the simulations with MS = 0.02 (middlepanel) do not reveal a clear inertial range for the modelswith MA = 0.2 and MA = 0.1. Nonetheless, the microscopic(or numerical) dissipation does not seem to be predominantat the perpendicular wavenumbers just above the injectionwavenumber k⊥L/2π ≈ 4. We see that the transfer spectrahave not been reduced significantly until the wavenumbersabove k⊥L/2π ≈ 7. The impact of the dissipation level closeto the injection scale will be analyzed in Section 4.1 through

MNRAS 000, 1–22 (2020)

Diffusion of large-scale magnetic fields by reconnection in MHD turbulence 9ηtf/(

1 3ℓv

rm

s

)

vrms/〈vA〉

16L-Ms0.02-A

16L-Ms0.08-A

16L-Ms0.32-A10−3

10−2

10−1

100

10−1 100

M2A

M3A

τ ener/(ℓ/vrm

s)

vrms/〈vA〉

100

101

10−1 100

M−1A

M−2A M−3

A

Figure 3. Magnetic diffusion coefficients ηtf measured by the test-field method (left) and the energy transfer time τener ≡ Eturb/Tturb

(right), as a function of the Alfvenic Mach number MA = vrms/〈vA〉. Simulations with the same A-forcing, the domain sizes, butdifferent sonic Mach number MS = vrms/cs are compared. Each point corresponds to one run in Table 1. For the parameters used inthese simulations the vertical solid line indicates the lower limit of MA given by Eq. 11 at the injection scale ℓ.

ηtf/(

1 3ℓv

rm

s

)

vrms/〈vA〉

16L-Ms0.02-A

8L-Ms0.02-A

4L-Ms0.02-A

1L-Ms0.02-A10−3

10−2

10−1

100

10−1 100

M2A

M3A

τ ener/(ℓ/vrm

s)

vrms/〈vA〉

100

101

10−1 100

M−1A

M−2A M−3

A


(right), as a function of the Alfvenic Mach number MA = vrms/〈vA〉. Simulations with the same A-forcing, the same sonic Mach numberMS = 0.02, but different domain sizes are compared. Each point corresponds to one run in Table 1. For each domain size (correspondingto different color curves), a vertical solid line indicates the lower limit of MA given by Eq. 11 at the injection scale ℓ.

the comparison of models with different perpedicular resolu-tions. The bottom panels show models with MS = 0.02 anda domain size 1Lx1L, for which the validity of the classi-cal weak turbulence has the theoretical lowest limit aroundMA = 0.5. Below this value, the power spectrum does showthe steepening seen in the model with extended domain(middle panel) with the decrease of MA, and the transferspectrum does not reduce substantially, which is consistentwith a non increasing energy transfer time, confirming theprevious analysis for these models (see the right panel ofFigure 4).

The ratio between the kinetic and the magnetic powerspectra shown in the left panels of Figure 5 is approxi-

mately constant and close to unity, except for the largestwavenumbers, inside the dissipation range, where numericaleffects dominate. When the energy spectrum is dominatedby Alfven modes, we should expect equipartition betweenthe magnetic and kinetic energy spectra inside the inertialrange, following the equality between these two amplitudesin each individual Alfven mode. We observe this equiparti-tion even for our models with larger compressibility (models16L-Ms0.32-A; see top left panel in Figure 5).

Figure 6 compares the 2D energy spectrum for simu-lations from the same sets shown in Figure 5. On the leftcolumn the simulations have MA = 0.4, and on the rightcolumn MA = 0.2. Top and middle panels correspond to

MNRAS 000, 1–22 (2020)

10 Santos-Lima, Guerrero, de Gouveia Dal Pino & LazarianE(k

⊥)/E

turb,E

u(k

⊥)/E

B(k

⊥)

k⊥L/2π

16L-Ms0.32-A

MA = 0.8

MA = 0.4

MA = 0.2

MA = 0.110−4

10−3

10−2

10−1

100

100 101

k−5/3⊥

k−2⊥

T(k

⊥)/

[Eturb/(ℓ/vrm

s)]

k⊥L/2π

16L-Ms0.32-A

0.0

0.2

0.4

0.6

0.8

1.0

100 101

E(k

⊥)/E

turb,E

u(k

⊥)/E

B(k

⊥)

k⊥L/2π

16L-Ms0.02-A

MA = 0.8

MA = 0.4

MA = 0.2

MA = 0.110−4

10−3

10−2

10−1

100

100 101

k−5/3⊥

k−2⊥

T(k

⊥)/

[Eturb/(ℓ/vrm

s)]

k⊥L/2π

16L-Ms0.02-A

0.0

0.2

0.4

0.6

0.8

1.0

100 101

E(k

⊥)/E

turb,E

u(k

⊥)/E

B(k

⊥)

k⊥L/2π

1L-Ms0.02-A

MA = 0.8

MA = 0.4

MA = 0.2

MA = 0.110−4

10−3

10−2

10−1

100

100 101

k−5/3⊥

k−2⊥

T(k

⊥)/

[Eturb/(ℓ/vrm

s)]

k⊥L/2π

1L-Ms0.02-A

0.0

0.2

0.4

0.6

0.8

1.0

100 101

Figure 5. The energy spectrum E(k⊥) (left column) and the energy transfer spectrum T (k⊥) (right panel). The ratio between thevelocity and the magnetic power spectra is also shown in the energy spectrum plots. In each panel, models from Table 1 with differentAlfvenic Mach numbers MA are represented by curves with different colors. Top: models with sonic Mach number MS = 0.32 anddomain size 16Lx1L; middle: models with MS = 0.02 and domain size 16Lx1L; bottom: models with MS = 0.02 and domain size 1Lx1L.

The power laws ∝ k−5/3⊥ and ∝ k−2 are also depicted for comparison in the left panels. The gray area covers the wavenumbers range

4 < k⊥L/2π < 10 for which the transfer spectrum is approximately constant and close to its maximum value.

MNRAS 000, 1–22 (2020)


simulations with fixed domain size, L‖ = 16L, but dif-ferent sonic Mach numbers, MS = 0.32 (top panels), andMS = 0.02 (middle panels). The less compressible simula-tions (MS = 0.02) reveal two features which are not visiblein the most compressible simulations (MS = 0.32): a sup-pression in the energy cascade in the parallel direction (ver-tical axis and steepening of the energy spectrum in the per-pendicular direction (horizontal axis) when MA decreases.Both features are expected to emerge in the weak turbulenceregime in the limit of incompressible MHD (Afven waves tur-bulence). The bottom panels of Figure 6 correspond to sim-ulations with the same compressibility as the middle panels(MS = 0.02), but with a shorter domain size in the paralleldirection L‖ = 1L. The two features described above for thesimulations with larger domain size are clearly weaker in thesimulations with shorter domain, which are out of the limitgiven by Equation 11 (at least near the injection scale).

We note in Figure 6 that the energy distributions inthe parallel direction (vertical axis) present some peaks or“steps”. This effect becomes stronger in the right panels, forwhich the mean magnetic field is stronger (smaller MA).These peaks appear around the parallel wavenumbers whichare harmonics of the wavenumber where the forcing ampli-tude is maximum, k‖L/2π = 4 (see Table 1). We attributethis feature to the peaked distribution of the forced modeson the parallel wavenumbers for the A-forcing scheme. Atthe same time, the wave-turbulence character becomes morepronounced when the mean magnetic field is stronger. Non-linear coupling between triads of Alfven waves naturally gen-erates the higher order harmonics. The presence of thesesame harmonics in the energy spectrum was also pointedout in Ghosh et al. (2009), for the turbulence produced bya spectrum of Alfven waves containing only one discreteparallel frequency (monochromatic) combined with quasi-2D MHD modes (nearly zero Alfven frequency) which hasno broad enough spectrum around k‖ = 0. In Figure 11(discussed in the next subsection), it is shown that thesesteps are much less pronounced in the simulations with theAb-forcing scheme, which has a broader and smoother dis-tribution of amplitudes in the parallel wavenumbers. In thiscase, the non-linear coupling of waves involving a broaderspectrum is enough to “fill” the gaps around the peaks seenbefore. Therefore, while the presence of the steps in the par-allel spectrum of the simulations with the A-forcing makesthe wave character of the turbulence visible, the resultingmagnetic field diffusion and energy transfer time are notdifferent from the models employing the Ab-forcing scheme,where the parallel spectrum is smooth.

Figure 7 shows, for the same set of simulations presentedin the Figures 3 and 4, the dependence of the ratio betweenthe amplitude of the 2D component of the solenoidal veloc-ity field, 〈v2D,sol〉 and the total rms velocity vrms, with MA.We calculate v2D,sol from the Fourier components of the ve-locity field, vk, by removing the vector components parallelto both k and B0 (that is, the vector components which areeither potential or parallel to the mean magnetic field), andfinally keeping only the modes for which k‖ = 0). The 2D ve-locity components are not excited by the A-forcing scheme,but they naturally develop in the system from the wave in-teractions. In the weak turbulence theory, the 2D modesare required as one component in the three-wave resonantinteractions (although they can exist only in finite sized do-

mains). As these velocities do not bend the magnetic field,they can easily mix the field lines in the perpendicular direc-tion, similar to hydrodynamical motions (η2D ∼ ℓv2D), dom-inating the diffusion rate. Similar to Figure 3, the left panelof Figure 7 compares simulations with the same domain size16Lx1L, but different compressibility (MS). Analogous tothe behavior of ηtf for MA > M∗

A, the values of 〈v2D,sol〉/vrms

converge to an approximate power law in MA, and deviatefrom this trend for MA < M∗

A (M∗A ≈ 0.5, 0.4, and 0.2 for

MS = 0.32, 0.08, and 0.02, respectively; we see that thesevalues of M∗

A correspond to the value of MA above whichthe energy transfer time follows the dependence τ ∝ M−2

A ,as seen in the right panel of Figure 3). Nonetheless, thisanalysis should be taken with caution because of the shortrange of values of the ratio 〈v2D,sol〉/vrms and the statisti-cal uncertainties generated by the fluctuations in the curves.Following the trend of ηtf , 〈v2D,sol〉/vrms also increases withMS. The right panel of Figure 7 compares simulations withthe same MS = 0.02, but different domain sizes. In this caseall the sets of simulations show a similar qualitative behav-ior, i.e., 〈v2D,sol〉/vrms increases with MA. The magnitude ofeach curve is inversely proportional to the square root of L‖.In summary, the magnitude of 〈v2D,sol〉 follows qualitativelythat of ηtf , but the dependence of the diffusivity with MA

is stronger for ηtf . There is no direct evidence that the mixcaused by the 2D motions dominates the turbulent diffusiv-ity, even for MA below M∗

A, for the simulations presented inFigures 3 and 4.

In order to quantify the amount of turbulent energy incompressible modes for the simulations, for each wave vec-tor in the Fourier space we performed the projection of theMHD variables onto the magnetosonic slow and fast eigen-vectors. Figure 8 compares, for the same set of models pre-sented in Figure 7, the ratio between the turbulent energy ineach magnetosonic mode (slow and fast) and the total tur-bulent energy, as a function of MA. The left panel comparesmodels with different MS and fixed domain size 16Lx1L.For the most incompressible run set (MS = 0.02), the rel-ative energy in the slow modes is nearly constant with thedifferent values of MA, keeping close to 0.5, and increasesslightly with the decrease of MA. The relative energy in thefast modes is about two orders of magnitude below that inthe slow modes, and decreases slowly with the decrease ofMA. Only for the simulation with the smallest value of MA

there is a sudden increase in the relative energy of the fastmodes. The orther sets of simulations, with larger values ofMS, show a trend for the decrease in the relative energy ofthe slow modes with the decrease of MA, at the same timethat the relative energy of the fast modes increases. For themost compressible simulation (MS = 0.32), for the smallestvalues of MA the energy in the fast modes overpass the en-ergy of the slow modes. The right panel of Figure 8 comparesmodels with fixed MS = 0.02 but different domain sizes. Allsets of simulations behave similarly to the model with thelargest domain size shown in the left panel.

4.2 Forcing effects

All the results described so far were derived from simulationsusing the turbulence forcing anisotropically distributed inthe Fourier space (A-forcing). In order to test the sensitiv-ity of these results to the forcing scheme, we have repeated

MNRAS 000, 1–22 (2020)

12 Santos-Lima, Guerrero, de Gouveia Dal Pino & Lazariank‖L/2π

k⊥L/2π

16L-Ms0.32-A, MA = 0.4

0

4

8

12

16

20

24

4 8 12 16 20 24

10−8

10−6

10−4

10−2

E(k

‖,k

⊥)/E

turb

k‖L/2π

k⊥L/2π

16L-Ms0.32-A, MA = 0.2

0

4

8

12

16

20

24

4 8 12 16 20 24

10−8

10−6

10−4

10−2

E(k

‖,k

⊥)/E

turb

k‖L/2π

k⊥L/2π

16L-Ms0.02-A, MA = 0.4

0

4

8

12

16

20

24

4 8 12 16 20 24

10−8

10−6

10−4

10−2

E(k

‖,k

⊥)/E

turb

k‖L/2π

k⊥L/2π

16L-Ms0.02-A, MA = 0.2

0

4

8

12

16

20

24

4 8 12 16 20 24

10−8

10−6

10−4

10−2

E(k

‖,k

⊥)/E

turb

k‖L/2π

k⊥L/2π

1L-Ms0.02-A, MA = 0.4

0

4

8

12

16

20

24

4 8 12 16 20 24

10−8

10−6

10−4

10−2

E(k

‖,k

⊥)/E

turb

k‖L/2π

k⊥L/2π

1L-Ms0.02-A, MA = 0.2

0

4

8

12

16

20

24

4 8 12 16 20 24

10−8

10−6

10−4

10−2

E(k

‖,k

⊥)/E

turb

Figure 6. The 2D energy spectrum E(k‖, k⊥) distribution in the (k‖, k⊥) plane. Each column corresponds to a different nominal AlfvenicMach number MA,0 ≡ v0/vA,0. Left column: MA,0 = 0.4. Right column: MA,0 = 0.2. Top row: domain size L‖ = 16L and sonic Machnumber MS = 0.32. Middle row: domain size L‖ = 16L and sonic Mach number MS = 0.02. Bottom row: domain size L‖ = 1L and sonicMach number MS = 0.02. See Table 1 for the complete description of the simulations parameters.

MNRAS 000, 1–22 (2020)

Diffusion of large-scale magnetic fields by reconnection in MHD turbulence 13〈v

2D,sol〉/vrm

s

vrms/〈vA〉

16L-Ms0.02-A

16L-Ms0.08-A

16L-Ms0.32-A10−2

10−1

10−1 100

〈v2D,sol〉/vrm

s

vrms/〈vA〉

16L-Ms0.02-A

8L-Ms0.02-A

4L-Ms0.02-A

1L-Ms0.02-A

10−2

10−1

100

10−1 100

Figure 7. Ratio between the rms value of the 2D component of the solenoidal velocity 〈v2D,sol〉 and the total rms velocity vrms, as afunction of the Alfvenic Mach number MA = vrms/〈vA〉. Left: simulations with different sonic Mach numbers MS = v0/cs are compared.Right: simulations with the same MS = 0.02, but different domain sizes are compared. Each point in the curves corresponds to a modelin Table 1. For each domain size (corresponding to different color curves on the right panel), a vertical solid line indicates the lower limitof MA given by Eq. 11 at the injection scale ℓ.

(Eslow/E

turb),

(Efast/E

turb)

vrms/〈vA〉

16L-Ms0.02-A

16L-Ms0.08-A

16L-Ms0.32-A

10−4

10−3

10−2

10−1

100

10−1 100

(Eslow/E

turb),

(Efast/E

turb)

vrms/〈vA〉

16L-Ms0.02-A

8L-Ms0.02-A

4L-Ms0.02-A

1L-Ms0.02-A

10−4

10−3

10−2

10−1

100

10−1 100

Figure 8. Ratio between the energy in each magnetosonic component of the turbulence and the total turbulent energy Eturb as a functionof the Alfvenic Mach number MA = vrms/〈vA〉. The continuous lines are for the energy in the slow modes Eslow, and the dashed lines arefor the energy in the fast modes Efast. Left: simulations with different sonic Mach numbers MS = v0/cs are compared. Right: simulationswith the same MS = 0.02, but different domain sizes are compared. Each point corresponds to a model in Table 1. For each domain size(corresponding to different color curves on the right panel), a vertical solid line indicates the lower limit of MA given by Eq. 11 at theinjection scale ℓ.

the simulations with MS = 0.02 and domain size 8Lx1L us-ing two alternative forcings, one isotropically distributed in-side a spherical shell in the k-space (I-forcing), and anotherwhere the amplitude of the modes inside a spherical shellare concentrated around k‖/k⊥ = 1 using the modulationfactor ∝ (k‖k⊥/k

2) (Ab-forcing). This last forcing schemecan be thought as intermediate between the extreme A- andI-forcing cases (such as described in Section 3.1).

The left panel of Figure 9 shows that for simulationswith the I-forcing, ηtf is essentially independent of MA for

MA < M∗A = 0.2. Above M∗

A, this dependence follows apower law, ηtf/ηhyd ∼ M−1

A , which is much weaker than theobserved for the A-models, where the power low dependenceis ∼ M−3

A . The Ab-models behave similar to the A-models,with scaling of ηtf/ηhyd between ∼ M−3

A and ∼ M−2A . In

the right panel of Figure 9 we see that the energy cascadingtimes, τener, around the injection scale, ℓ, for the Ab-modelsare almost identical to those of A-models, at least for MA >M∗

A. In contrast, the increase of τener with the decrease of

MNRAS 000, 1–22 (2020)

14 Santos-Lima, Guerrero, de Gouveia Dal Pino & Lazarianηtf/(

1 3ℓv

rm

s

)

vrms/〈vA〉

8L-Ms0.02-A

8L-Ms0.02-Ab

8L-Ms0.02-I10−3

10−2

10−1

100

10−1 100

M2A

M3A

τ ener/(ℓ/vrm

s)

vrms/〈vA〉

100

101

10−1 100

M−1A

M−2A M−3

A


(right), as a function of the Alfvenic Mach number MA = vrms/〈vA〉. Simulations with different turbulence forcing schemes are compared.All the runs have the same sonic Mach number MS = 0.02 and domain size 8Lx1L. Each point corresponds to a model in Table 1. Forthe parameters used in these simulations the vertical solid line indicates the lower limit of MA given by Eq. 11 at the injection scale ℓ.

MA for the I-models is much slower than that for the A-models.

The turbulence energy spectrum E(k⊥) for the simula-tions with I- and Ab-forcing are shown in the left column ofFigure 10. Their respective energy transfer spectrum, T (k⊥),are shown in the right column. We do not notice the steepen-ing of the energy spectrum with the decrease of MA for theI-models as seen in the Ab-models and A-models (Figure 5).Considering the analysis of both, the energy cascading time(right panel of Figure 9) and the energy transfer spectrum(right panel of Figure 10), there is a difference between theI-models and the k-anisotropic forcing models A and Ab.Notice that all the forcing schemes produce similar energyspectrum for MA = 0.8, therefore, the difference is mani-fested only in the presence of strong magnetic fields.

The 2D energy spectrum, E(k‖, k⊥) for simulations withthe different forcing schemes are compared in Figure 11. Thetop, middle and bottom panels correspond to the I, Ab, andA forcings, respectively. Simulations in the left column haveMA = 0.4, and in the right column MA = 0.2. We note thatsubstantial differences between models with different forcingbecome evident for smaller values of MA. The 2D energyspectrum for values of MA closer to unity are almost indis-tinguishable (not shown). The k-anisotropic models (middleand bottom panels) show steepening in the energy distribu-tion in the perpendicular direction (horizontal axis), whilethis effect is not observed in the simulations with I forcing(top panels). In all the cases we observe a sharp steepeningof the spectrum in the parallel direction (vertical axis) forMA = 0.2.

With exception of the I-cases, the dependence of thediffusivity ηtf with MA for all the simulations with differentforcings seems to be closely related to the behavior of τenerwith MA. Indeed, the left panel of Figure 12 shows thatthe values of 〈v2D,sol〉/vrms for the I-models are much largercompared to the other forcing schemes. There is no surprisein this fact, as the 2D velocity modes are only forced in the

I-models. These high 2D velocities can explain the diffusivityobserved in the I-models, as ηtf/ηhyd ∼ 〈v2D,sol/vrms〉.

Finally, the right panel of Figure 12 compares the rel-ative turbulent energies in the magnetosonic slow and fastmodes for the sets of simulations with the several forcings.No appreciable difference is observed between them.

4.3 Resolution effects and convergence

Figure 13 compares the model 16L-Ms0.02-A having res-olution 2048x1282 with the models 16L-Ms0.02-low-A and16L-Ms0.02-hi-A, with identical parameters but resolutions1024x642 and 2048x2562 , respectively. In order to make thehigher resolution run feasible with the computational powerwe have available, the models 16L-Ms0.02-hi-A have twicethe resolution only in the perpendicular direction to the uni-form magnetic field. In the regime of weak turbulence, weexpect (based on the theory) the non-linear energy transfer(turbulence cascade) to be more important in this direction,at least close to the injection scale.

Moreover, in Figure 13, we compare the low resolutionmodels 16L-Ms-0.02-low-A with other two models, 16Lx2L-Ms-0.02-low-A and 16L-Ms0.02-low-A-diff2. The first ofthese models has double domain size in the direction perpen-dicular to the mean magnetic field L⊥, that is, it has the ra-tio L⊥/ℓ⊥ increased by a factor of two compared to the mod-els 16L-Ms0.02-low-A. The other models, 16-Ms-0.02-low-A-diff2, have the hyper-viscosity and hyper-diffusivity re-placed by the standard viscosity and resistivity, respectively(see the description of these microscopic diffusive terms inSec. 3).

The minimum MA below which the 3D MHD turbu-lence becomes “enslaved” to the 2D modes is theoreticallypredicted to depend on the ratio L⊥/ℓ⊥ (Nazarenko 2007;see Eq. 12). This minimum MA is indeed below the value de-limiting the validity of the weak turbulence theory, which wehave marked in our plots. Therefore, based on theory only,

MNRAS 000, 1–22 (2020)

Diffusion of large-scale magnetic fields by reconnection in MHD turbulence 15E(k

⊥)/E

turb,E

u(k

⊥)/E

B(k

⊥)

k⊥L/2π

8L-Ms0.02-I

MA = 0.8

MA = 0.4

MA = 0.2

MA = 0.110−4

10−3

10−2

10−1

100

100 101

k−5/3⊥

k−2⊥

T(k

⊥)/

[Eturb/(ℓ/vrm

s)]

k⊥L/2π

8L-Ms0.02-I

0.0

0.2

0.4

0.6

0.8

1.0

100 101

E(k

⊥)/E

turb,E

u(k

⊥)/E

B(k

⊥)

k⊥L/2π

8L-Ms0.02-Ab

MA = 0.8

MA = 0.4

MA = 0.2

MA = 0.110−4

10−3

10−2

10−1

100

100 101

k−5/3⊥

k−2⊥

T(k

⊥)/

[Eturb/(ℓ/vrm

s)]

k⊥L/2π

8L-Ms0.02-Ab

0.0

0.2

0.4

0.6

0.8

1.0

100 101

Figure 10. The energy spectrum E(k⊥) (left column) and the energy transfer spectrum T (k⊥) (right panel) for models with differentturbulence forcing schemes. The ratio between the velocity and the magnetic field power spectra for each model is shown inside theenergy spectrum plots. In each panel, models from Table 1 with different Alfvenic Mach numbers MA are represented by curves withdifferent colors. All the runs have the same sonic Mach number MS = 0.02 and domain size 8Lx1L. Top: isotropic forcing scheme I;

bottom: anisotropic forcing scheme Ab. The power laws ∝ k−5/3⊥ and ∝ k−2 are also depicted for comparison in the left panels. The

gray area covers the wavenumbers range 4 < k⊥L/2π < 10 for which the transfer spectrum is approximately constant and close to itsmaximum value.

we should not expect differences in the turbulence regime be-tween the models 16L-Ms-0.02-low-A and 16Lx2L-Ms-0.02-low-A. Nonetheless, because these theoretical expressionsgive only order of magnitude estimates, we present this con-vergence test for the perpendicular size of the domain, inorder to check whether or not the length used in all theprevious analyses influences our results.

The transport rate of large scale fields via reconnectiondiffusion (RD) is expected to be dominated by the statis-tics of the larger scale motions of the turbulence. Since theReynolds and the magnetic Reynolds numbers of the tur-bulence flow are much larger than unity, the microphysicsdescribing the diffusion mechanism (as for example ambipo-lar diffusion, ohmic resistivity, anomalous resistivity, or even“numerical diffusivity”) should not have impact on the mag-netic field diffusion coefficient (see Eq. 6). However, if the dif-

fusion mechanism can somehow change the statistics of theturbulence close to the injection scale, then it can influencethe RD process. The comparison between models 16L-Ms-0.02-low-A and 16L-Ms0.02-low-A-diff2 aims to check if thehyper-viscosity and hyper-resistivity, used in all the othermodels in this work, could have some effect on the magneticdiffusivity coefficients.

The upper-left panel Fig. 13 shows the diffusion coeffi-cients ηtf as a function ofMA (see also left panel of Figure 3).The scaling laws are similar for the different resolutions. Thechange in the domains size L⊥ also does not results in impor-tant changes in ηtf . Finally, the results of the models usingstandard viscosity and hyperdifusivity (16L-Ms0.02-low-A-diff2) show no noticieable differences.

The low resolution models 16L-Ms-0.02-low-A containruns with values of MA below the line indicating the lower

MNRAS 000, 1–22 (2020)

16 Santos-Lima, Guerrero, de Gouveia Dal Pino & Lazariank‖L/2π

k⊥L/2π

8L-Ms0.02-I, MA = 0.4

0

4

8

12

16

20

24

4 8 12 16 20 24

10−8

10−6

10−4

10−2

E(k

‖,k

⊥)/E

turb

k‖L/2π

k⊥L/2π

8L-Ms0.02-I, MA = 0.2

0

4

8

12

16

20

24

4 8 12 16 20 24

10−8

10−6

10−4

10−2

E(k

‖,k

⊥)/E

turb

k‖L/2π

k⊥L/2π

8L-Ms0.02-Ab, MA = 0.4

0

4

8

12

16

20

24

4 8 12 16 20 24

10−8

10−6

10−4

10−2

E(k

‖,k

⊥)/E

turb

k‖L/2π

k⊥L/2π

8L-Ms0.02-Ab, MA = 0.2

0

4

8

12

16

20

24

4 8 12 16 20 24

10−8

10−6

10−4

10−2

E(k

‖,k

⊥)/E

turb

k‖L/2π

k⊥L/2π

8L-Ms0.02-A, MA = 0.4

0

4

8

12

16

20

24

4 8 12 16 20 24

10−8

10−6

10−4

10−2

E(k

‖,k

⊥)/E

turb

k‖L/2π

k⊥L/2π

8L-Ms0.02-A, MA = 0.2

0

4

8

12

16

20

24

4 8 12 16 20 24

10−8

10−6

10−4

10−2

E(k

‖,k

⊥)/E

turb

Figure 11. The 2D energy spectrum E(k‖, k⊥) distribution in the (k‖, k⊥) plane for models with different turbulence forcing schemes. Allthe runs have the same sonic Mach number MS = 0.02 and domain size 8Lx1L. Each column corresponds to a different nominal AlfvenicMach number MA,0 ≡ v0/vA,0. Left column: MA,0 = 0.4. Right column: MA,0 = 0.2. From top to bottom: I-forcing, Ab-forcing, andA-forcing. See Table 1 for the complete description of the simulations parameters.

MNRAS 000, 1–22 (2020)

Diffusion of large-scale magnetic fields by reconnection in MHD turbulence 17〈v

2D,sol〉/vrm

s

vrms/〈vA〉

8L-Ms0.02-A

8L-Ms0.02-Ab

8L-Ms0.02-I

10−2

10−1

100

10−1 100

(Eslow/E

turb),

(Efast/E

turb)

vrms/〈vA〉

8L-Ms0.02-A

8L-Ms0.02-Ab

8L-Ms0.02-I

10−4

10−3

10−2

10−1

100

10−1 100

Figure 12. Ratio between the rms value of the 2D component of the solenoidal velocity 〈v2D,sol〉 and the total rms velocity vrms (left),and the ratio between the energy in each magnetosonic component of the turbulence and the total turbulent energy Eturb (right), as afunction of the Alfvenic Mach number MA = vrms/〈vA〉. In the right panel, the continuous lines are for the energy in the slow modesEslow, and the dashed lines are for the energy in the fast modes Efast. Models with different turbulence forcing schemes are comparedin each panel. All the runs have the same sonic Mach number MS = 0.02 and domain size 8Lx1L. Each point corresponds to a modelin Table 1. For the parameters used in these simulations the vertical solid line indicates the lower limit of MA given by Eq. 11 at theinjection scale ℓ.

limit of MA given by Eq. 11 at the injection scale ℓ. Thisextension shows more clearly the asymptotic change of thepower-law dependence with MA from ηtf/ηhyd ∼ M3

A toηtf/ηhyd ∼ M2

A, as seen in the models with standard res-olution of this work (see Figures 3 and 4).

The top right panel of Fig. 13 compares the normalizedenergy transfer time from the injection scale, τener, for thesesimulation sets (see also right panel of Figure 3). We observethat the curve for the set of simulations with lower resolutionis slightly steeper, with the power law index in MA between−2 and −3 (for MA larger than some M∗

A, below which thecurves become almost flat). This difference of power lawsin τener probably reflects the almost imperceptible steeperpower law in ηtf for the lower resolution model (top leftpanel). The break in the curve showing the change in thedependence of τener with MA (at M∗

A) is not evident in theruns set employing standard viscosity and resistivity.

Some small differences between the compared modelsare also visible in the lower left panel of Figure 13, whichdepicts the relation between 〈v2D,sol〉/vrms and MA (see alsoFigure 7). The higher resolution models have relatively moreenergy in the 2D modes (although the maximum differenceis still less than a factor of 2). Interestingly, when comparingthe energies in the magnetosonic modes for the two resolu-tions (bottom right panel of Figure 13), we notice that thelower resolution runs show the same increase in the energyof the fast modes seen in the medium resolution ones for thesmallest values ofMA. Nonetheless, this increase happens forhigher values of MA in the lower resolution run, contrary tothe reduction in the 2D solenoidal velocity fields.

All the comparisons in Figure 13 suggest that the re-sults for the simulations with the standard resolution (mod-els 16L-Ms0.02-A, having the same cells-size employed inall the simulations presented before in this section) may be

close to convergence with respect to the numerical resolu-tion, at least for ηtf in the interval of MA considered. Also,the results discussed in the previous sections seem not tobe sensitive to changes in the domain size in the directionperpendicular to the mean magnetic field or to the use ofthe microscopic hyper-diffusivities instead of the standarddissipative terms.

We should also mention that the microscopic resistiv-ity employed in the induction equation of the test-fields (seeAppendix A) is the standard resistivity instead of the hyper-resistivity employed in the main induction equation. Thetest-field magnetic diffusivity was kept fixed for all our sim-ulations η1 = 1.2×10−2 ℓ⊥v0 (a value relatively high in orderto keep the test-fields smooth for longer times). As discussedin the beginning of this section, we do not expect the mi-croscopic terms to influence the diffusion rate of the largescale magnetic field. Nonetheless, in order to verify if thisinconsistency between the microscopic resistive terms couldinfluence the measurement of the turbulent magnetic diffu-sivity ηtf , we repeated the run 16L-Ms0.02-low-A, MA = 0.4,using hyper-resisitivity for the test-fields. The result for ηtf(not shown here) revealed no difference.

5 DISCUSSION

The diffusion coefficient predicted by the reconnection dif-fusion (RD) theory in the sub-Alfvenic regime has been de-rived for the scenario of purely Alfvenic turbulence (i.e.,the incompressible limit) in the weak regime. In addition,the scaling laws of the inertial range are assumed to bevalid also at the injection scale, as the largest scale mo-tions are responsible for the diffusion rate. The analysis ofour simulations does not evince, entirely, the development

MNRAS 000, 1–22 (2020)

18 Santos-Lima, Guerrero, de Gouveia Dal Pino & Lazarianηtf/(

1 3ℓv

rm

s

)

vrms/〈vA〉

16L-Ms0.02-A

16L-Ms0.02-low-A

16L-Ms0.02-low-A-diff2

16Lx2L-Ms0.02-low-A

16Lx1L-Ms0.02-hi-A10−5

10−4

10−3

10−2

10−1

100

10−1 100

M2A

M3A

τ ener/(ℓ/vrm

s)

vrms/〈vA〉

100

101

10−1 100

M−1A

M−2A M−3

A

〈v2D,sol〉/vrm

s

vrms/〈vA〉

16L-Ms0.02-A

16L-Ms0.02-low-A

16L-Ms0.02-low-A-diff2

16Lx2L-Ms0.02-low-A

16Lx1L-Ms0.02-hi-A10−3

10−2

10−1

10−1 100

(Eslow/E

turb),

(Efast/E

turb)

vrms/〈vA〉

10−5

10−4

10−3

10−2

10−1

100

10−1 100

Figure 13. Comparison between models with different numerical resolution (16L-Ms0.02-A, 16L-Ms0.02-low-A, 16L-Ms0.02-hi-A). In addi-tion, models 16L-Ms0.02-low-A are compared with identical counterpart models differing only by the length of the domain perpendicularto the uniform magnetic field (16x2L-Ms0.02-low-A), or by the replacement of the hyper-viscosity and hyper-resistivity by the standardviscosity and resistivity (16L-Ms0.02-low-A-diff2). The plots show different quantities versus the Alfvenic Mach number MA = vrms/〈vA〉.Top left: magnetic diffusion coefficients ηtf measured by the test-field method. Top right: energy transfer time τener ≡ Eturb/Tturb. Bot-tom left: ratio between the rms value of the 2D component of the solenoidal velocity 〈v2D,sol〉 and the total rms velocity vrms. Bottomright: ratio between the energy in each magnetosonic component of the turbulence and the total turbulent energy Eturb. The continuouslines are for the energy in the slow modes Eslow, and the dashed lines are for the energy in the fast modes Efast. All the runs have thesame sonic Mach number MS = 0.02 and domain size 16Lx1L. Each point corresponds to a model in Table 1. For the parameters usedin these simulations the vertical solid line indicates the lower limit of MA given by Eq. 11 at the injection scale ℓ.

of weak turbulence. The resolution available does not al-low the analysis of the power law index of the turbulencespectrum, usually employed to characterize the turbulence.However, our estimate of the energy transfer rate at theinjection scales shows a dependence ∝ M−α

A converging toα = 2 − 3, which is much stronger than what is expectedby the weak turbulence theory (WTT) at the inertial range,α = 1. It should be pointed out that, to our knowledgethere are no results in the literature showing a simulationof forced MHD turbulence reproducing clearly the results ofthe WTT. However, Meyrand, Kiyani & Galtier (2014) (seealso Meyrand, et al. 2018) identified theWT regime at scaleswell below the outer scale in a simulation of decaying turbu-

lence. Employing a spectral MHD code, Perez & Boldyrev(2008) performed simulations of forced reduced MHD tur-bulence and observed a perpendicular power spectrum con-sistent with the WTT when the injection is performed ina broad range of parallel wave-numbers, although far fromthe scenario of isotropic injection. From the observationalside there is no clear evidence of the MHD WT until thedate. The only observation that could be consistent withthe WTT is the power spectrum of one of the magnetic fieldcomponents in the Jupiter’s magnetosphere (see Saur, et al.2002).

Despite of the lack of numerical simulations of forcedMHD turbulence demonstrating the development of the WT

MNRAS 000, 1–22 (2020)


for the cascade immediately from the isotropic injection scaleand below, there are several studies (for example Alexakis2011, 2012; Bigot & Galtier 2011) exploring forced turbu-lence in the sub-Alfvenic regime, with isotropic forcing (i.e.,strong turbulence is not imposed by anisotropic forcing).These studies show the effects of limited box sizes, such asthe development of a 2D dominated cascade, inverse cascade,and the influence of different types of forcing (selection ofmodes in the k-space), but none of them was able to repro-duce the WT power spectrum.

In summary, although the WTT for Alfven waves isrelatively well founded, it seems challenging to produce sim-ulations of forced MHD turbulence in this regime in order totest theories based on the WT scenario (as for example thepredictions related to turbulent reconnection, RD, and cos-mic ray diffusion). Besides, a robust observational evidenceof the MHD WT in nature is still missing.

The simulations presented in this study are not incom-pressible. However, we kept the induced turbulence subsonicand, although the dominance of the Alfven waves in the en-ergy spectrum, some level of magneto-sonic waves is seen.The compressible code employed (the Pencil Code) hasthe advantage to have implemented the test-field methodtested in a large number of studies to measure with preci-sion the turbulent diffusion coefficient of the magnetic field.Although the scenario of the RD theory in purely incom-pressible, some degree of compressibility is expected in theturbulence in astrophysical environments (as for example, instar forming regions of molecular clouds). There is still notheory of weak turbulence for a compressible MHD gas. Thecascade of fast modes is weak 2 at least in subsonic turbu-lence (see Cho & Lazarian 2002). Slow modes only cascadefast in the presence of strong Alfvenic turbulence. In thefuture, it is important to confirm our conclusions for thevalidity of the RD diffusivity prediction in the asymptoticincompressible limit using for example a spectral MHD code.

In view of all the points raised above, we cannot discardthat the turbulence regime developed in our simulations isintermediary between the asymptotic weak and strong lim-its. This issue can only be investigated by further analy-sis of simulations with improved resolution. At present, oursimulations do not have a clearly identifiable inertial range,although our analysis of the energy transfer spectrum indi-cates that the numerical dissipation at the injection scales(where the magnetic diffusivity is produced) operates at arate lower than the cascade/decorrelation rate.

It is therefore surprising that, despite the turbulencestatistics of our simulations cannot be directly related tothe scenario of the RD theory, we still obtain results closeto the predictions (at least when we consider the setups moresuitable for the development of the WT).

It is also clear that compressibility modifies the depen-dence of the diffusion with MA, making the diffusion ratecloser to the dependence expected when turbulence is in the

2 Weak here means that the interactions decrease with the am-plitude of the waves, i.e. ∼ vℓ/

√

c2s + V 2A, where cs is the sound

velocity. The corresponding spectrum corresponds to ∼ k−3/2.The simulations in Kowal & Lazarian (2010) show that the powerspectrum of fast modes can be shock-like, i.e. ∼ k−2. The latterresult still requires further studies with other codes.

strong regime (i.e., when the critical balance ℓ‖/va ∼ ℓ/δuℓ

is satisfied), and we have τcasc ∼ (ℓ/δuℓ) in eq. (6). Inthis situation the RD prediction would modify eq. (7) toηRD ∼ D⊥ ∼ ℓδuℓ min

(

1,M2A

)

. However, because the am-plitude of the 2D modes of the velocity field increases withthe sonic Mach number, it was not possible to disentangletheir effects. It remains to explore the turbulent diffusivityin the presence of turbulence more strongly compressible,which is more realistic, e.g., for the interstellar medium. Itcould be performed by implementing the test-field methodin a compressible, shock-capturing MHD code.

Two different forcing schemes for the anisotropic am-plitude distribution produced rather similar results. There-fore, the results are apparently not very sensitive to the de-tails of the “k-anisotropic” forcing, provided that the ampli-tude of the forced modes decreases to zero when the parallelwavenumber goes to zero. One way to strengthen this con-clusion in the future would be to repeat this study using adifferent scheme for driving the turbulence. One particularlyinteresting scheme consists in inducing random distributionsof finite eddies directly in real space. This method was intro-duced in Kowal, et al. (2012), where it was shown to producethe same turbulent reconnection rates as numerical simula-tions with forcing controled by their Fourier components (asis done in the present work).

Our results show that the power in the two-dimensionalmodes, which can eventually dominate the turbulence cas-cade or dominate the diffusion of the magnetic field, areonly controlled when using anisotropic (in k-space) forc-ing schemes with domains sufficiently elongated in the di-rection parallel to the mean magnetic field. These resultsare in agreement with the theoretical limits presented inNazarenko (2007).

A systematic study of the turbulent diffusivity for sub-Alfvenic turbulence with a mean magnetic field has beencarried out in Karak, et al. (2014, K14). This work alsoemployed the Pencil Code with the test-field method tomeasure the diffusion coefficients. One of the setups stud-ied in K14 uses non-helical random forcing, exciting modesisotropically in the k-space. With a square domain and val-ues extremely low of MA, the turbulent regime in their sim-ulations could, at least theoretically, be in the discrete WTregime and enslaved to the 2D cascade (Nazarenko 2007; seeEqs. 11-12). The dependence of the diffusion coefficient withMA observed in K14 (described as a quenching) is close toM−1

A for 0.05 . MA . 1, i.e., similar to that we observedin our simulations with isotropic forcing. For the smallestvalues of MA, the dependence with of ηturb with MA dis-appears, which is expected when the diffusion is dominatedby the 2D velocity modes. We can interpret the results ofthese authors as effects of the simulation setup, specially thedomain size.

In the context of numerical simulations of turbulent re-connection, the increase of the reconnection rate could alsoarise as an artifact of the limited domain size. According tothe discussion in the previous paragraphs, in order to avoidthis effect, the domain size in the direction parallel to thelarge-scale magnetic field has to be large enough in thesenumerical studies.

The present study also calls for a re-evaluation of thetransport coefficients in the large scale dynamo context (forexample for helical turbulence), taking into account all role

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of compressibility and domain size of the simulation. It isnot clear if the same effects also affect the local simulationsaimed to extract the mean field coefficients in the convectivelayer of the Sun or in accretion disks.

The present study supports the predictions of the RDtheory at least in the incompressible limit, but it also pointsto an increase of the diffusivity rate due to the compress-ibility of the turbulence, alleviating the strong suppressioncaused by the mean magnetic field. The RD mechanismhas been proposed (Lazarian 2005) and tested successfully(at least qualitatively) to solve problems related to starformation: the magnetic flux problem (Santos-Lima, et al.2010; Leao, et al. 2013) and the magnetic braking catastro-phe (Santos-Lima, de Gouveia Dal Pino & Lazarian 2012,2013; Gonzalez-Casanova, Lazarian & Santos-Lima 2016).Lazarian, Esquivel & Crutcher (2012) conclude from obser-vations of molecular cloud cores that the mass-to-flux ra-tio of super-critical cores, compared to their envelopes,are more consistent with the transport of magnetic fluxvia RD than ambipolar diffusion (AD). More recent workson simulations of star formation processes, encompassingscales from molecular cloud clumps down to protostellardisks through the use of adaptive mesh, try to disentan-gle the role of other mechanisms that could solve theseproblems as well, like the misalignment of the angular mo-mentum of the protostellar disk and the mean magneticfield (Joos, Hennebelle, & Ciardi 2012; Joos et al. 2013), theinterchange instability, and the ambipolar diffusion (see,for example, Hennebelle & Inutsuka 2019 and referencestherein). These studies in general do not quantify the turbu-lent magnetic flux transport (due to the inherent difficult toperform this measurement), and it is far from clear if theirresolution is able to represent the RD in these global simu-lations. For example, Lam, et al. (2019) concluded that onlya combination of turbulence and AD (each one dominatingin different phases and/or regions) can allow the formationof rotationally supported protostellar disks which persist forsufficiently long times. We remark, however, that the com-bination “turbulent ambipolar diffusion” is an inconsistentconcept: when there is turbulence, this means that the ADis subdominant; if AD is very strong, there is no turbu-lence (see a more complete discussion in the recent reviewby Lazarian, et al. 2020). The setting and interpretation ofsuch global simulations depend on a full understanding ofthe RD mechanism driven by turbulence and the numericalsimulation setup effects that can interfere in the RD dif-fusivity. The present study provides a contribution in thisdirection. Further study of the RD in the presence of gravityare required as the properties of turbulence can be modifiedby gravity. In fact, Santos-Lima, et al. (2010) found some ev-idences that the transport of magnetic flux via RD increaseswith the intensity of the gravitational potential.

6 SUMMARY AND CONCLUSIONS

In this work we tested numerically the dependence of themagnetic diffusion coefficient provided by reconnectioninduced by turbulence (Reconnection Diffusion, RD)with the Alfvenic Mach number of the turbulence. Inall our 3D MHD simulations we imposed an initiallyuniform magnetic field and focused on the sub-Alfvenic

regime. The turbulence is forced isotropically in the space.Although we envision applications of the results, e.g.,to studies of the role of the RD during star-formation,inside molecular clouds where turbulence is expectedto be trans-sonic or supersonic (Santos-Lima, et al.2010; Santos-Lima, de Gouveia Dal Pino & Lazarian2012, 2013; Leao, et al. 2013;Gonzalez-Casanova, Lazarian & Santos-Lima 2016), inthis study we have focused only in the subsonic case. Themotivation is the direct comparison with the current RDtheory, built on the scaling laws provided by the Alfvenicturbulence in the weak regime (for sub-Alfvenic isotropicinjection). We employed the Pencil Code with the Test-Field method to extract the average diffusion coefficientfrom the simulations.

The RD theory assumes that the inertial range scalelaws of the weak turbulence theory can be extended tothe injection scales, leading to a diffusion coefficient η pro-portional to the hydrodynamical value multiplied by thethird power of MA, when MA < 1. We found no clear evi-dence of the development of the weak turbulence regime inour numerical simulations. In particular, the cascading time∝ M−1

A from the weak turbulence theory at the injectionscale is not observed in any of our model sets. Due to limitedresolution and the fast increase of the cascading time withthe increase of the magnetic field intensity, our simulationsdo not show appreciable inertial range to allow a robust de-termination of the power law index of the power spectrum.Nonetheless, the diffusion coefficients we obtain seem to beconsistent with the RD prediction η ∝ M3

A when the domainsize parallel to the uniform magnetic field is large enough toavoid the finite box size effects (see Nazarenko 2007 in theframework of reduced MHD) and the sonic Mach numbersmall enough (MS . 0.02). For smaller boxes and biggervalues of MS , we observed a dependence of η more consis-tent with M2

A, which could be the expected dependency inthe strong cascading regime.

In the future, we will investigate both the incompress-ible limit to confirm the validity of the RD theory in a largerinterval of MA, and also the diffusivity provided by super-sonic turbulence which is more realistic for star-forming en-vironments. At the same time, more numerical investigationis necessary for the weak turbulence regime, as it is not yetclearly reproducible in direct simulations of forced turbu-lence.

Due to the omnipresence of MHD turbulence in astro-physics, the proper understanding of the turbulent diffusiv-ity is of fundamental importance not only in the context ofstar-formation, but it has also important consequences forthe more general reconnection problem, for the large scaleturbulent dynamo operating in all scales (stars, accretiondisks, galaxies; see for example Xu & Lazarian 2020 for arecent study on the nonlinear turbulent dynamo in a grav-itationally collapsing system which accounts for the RD ef-fects), and for the propagation and acceleration of cosmicrays in astrophysical media.

ACKNOWLEDGEMENTS

The authors thank A. Brandenburg, J. Cho, G. Eyink,B. Raphaldini, M. Rheinhardt, M. V. del Valle, and S.

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Xu for very useful discussions and comments. The authorsalso thank the anonymous referee who helped to improvethis work with her/his comments. RSL acknowledges par-tial support from a grant of the Brazilian Agency FAPESP(2013/15115-8), EMGDP from FAPESP (2013/10559-5) andCNPq (306598/2009-4) grants. AL acknowledges the sup-port by NASA TCAN 144AAG1967 grant and NSF AST1816234 award. The Flatiron Institute is supported by theSimons Foundation. The numerical simulations presentedhere were performed in the cluster of the Group of Plasmasand High-Energy Astrophysics (GAPAE), acquired withsupport from FAPESP (grant 2013/10559-5). This work alsomade use of the computing facilities of the Laboratory of As-troinformatics (IAG/USP, NAT/Unicsul), whose purchasewas made possible also by FAPESP (grant 2009/54006-4).

DATA AVAILABILITY

The data underlying this article will be shared on reasonablerequest to the corresponding author.

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APPENDIX A: THE TEST-FIELD METHOD

The test-field method, developed initially for spherical geo-dynamo simulations (Schrinner, et al. 2005), allows to com-pute unambiguously the contribution of the small scale onthe large scale dynamics. In mean-field analysis this con-tribution is described by the electromotive force, 〈E〉 =

〈u′ × B′〉, where the primes denote small-scale fields. Con-

sider that u′ is the turbulent part of a velocity field, u, in

a given simulation, and also a set of test-fields, BT (thesefields are independent between themselves and of the mag-netic field of the simulation B), such that the evolution ofB

′ may be computed in a set of partial differential equa-tions which depends only on u and B

T . Thus, it is possibleto compute the electromotive force due to the velocity fieldas 〈ET 〉 = 〈u′×B

T ′〉. For mean magnetic field varying slowlyin space and time, and systems where averages in the x andy directions are meaningful, it is possible to write the elec-tromotive force as

〈Ei〉 = αij〈Bj〉 − ηij〈Jj〉, (A1)

with, i, j = 1, 2. Thus, with the use of 4 test fields it ispossible to obtain the 4+4 components of α and η (seeBrandenburg, Radler & Schrinner 2008).

For the simulations presented in this work, without ro-tation and with non-helical unstratified turbulence, the in-ductive terms in the electromotive force, αij , must be zeroon average. Therefore, only 2 test fields would be sufficientto determine the turbulent diffusivity. Nevertheless, we use4 test fields to verify the existence of other turbulent effects.These are given by

B1c

= B0(cos kz, 0, 0), B2c

= B0(0, cos kz, 0), (A2)

B1s

= B0(sin kz, 0, 0), B2s

= B0(0, sin kz, 0),

where k = kz,tf = 2π/L⊥. Thus, the α-effect is given bythe diagonal components of αij , αxx = α11 and αyy = α22,while the turbulent diffusion is given by ηt = 1

2(η11 + η22).

In addition, the turbulent driven advection of the magneticfield, often called turbulent pumping, can be measured byγ = 1

2(α21−α12). As expected, all the coefficients but ηt are

consistent with zero.

This paper has been typeset from a TEX/LATEX file prepared bythe author.

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