Magnetic field in molecular clouds

# Magnetic field generation in galactic molecular clouds

Ya. N. Istomin , A. Kiselev
P.N. Lebedev Physical Institute, Leninsky Prospect 53, Moscow 119991, Russia
E-mail: istomin@lpi.ru
E-mail: kiselevalexs@gmail.com
###### Abstract

We investigate the magnetic field which is generated by turbulent motions of a weakly ionized gas. Galactic molecular clouds give us an example of such a medium. As in the Kazantsev-Kraichnan model we assume a medium to be homogeneous and a neutral gas velocity field to be isotropic and delta-correlated in time. We take into consideration the presence of a mean magnetic field, which defines a preferred direction in space and eliminates isotropy of magnetic field correlators. Evolution equations for the anisotropic correlation function are derived. Isotropic cases with zero mean magnetic field as well as with small mean magnetic field are investigated. It is shown that stationary bounded solutions exist only in the presence of the mean magnetic field for the Kolmogorov neutral gas turbulence. The dependence of the magnetic field fluctuations amplitude on the mean field is calculated. The stationary anisotropic solution for the magnetic turbulence is also obtained for large values of the mean magnetic field.

###### keywords:
magnetic fields - MHD - turbulence - ISM: magnetic fields - ISM:clouds - methods: analytical
pagerange: Magnetic field generation in galactic molecular cloudsLABEL:lastpagepubyear: 2013

## 1 Introduction

The standard theory of cosmic rays (CR) formation suggests that primary CR consist mainly of protons and contain no antimatter. During their propagation in the Galaxy primary CR interact with protons of galactic gas, resulting in production of secondary CR, including antiprotons and positrons. The secondary particles energy spectrum, calculated in the framework of this theory, falls down with energy by a power law manner. The antiparticles to particles ratio should behave in the same way (Moscalenko & Strong 1998). However, recent antimatter observations by PAMELA satellite detected an excess of positrons with energies GeV (Adriani et al. 2009). In this range, the ratio is about and increases with energy. These observations attracted much attention. Several theoretical explanations for this effect were proposed, among them the positrons generation in pulsars and in the annihilation process of dark matter particles. Another mechanism for the positrons generation in the Galaxy is also possible, that is acceleration of charged particles in giant molecular clouds and secondary CR production there. This mechanism was discussed by Dogiel et al. (1987, 2005), long before the launch of the PAMELA satellite in 2006.

The particle acceleration in molecular clouds takes place due to turbulent motion of a partially ionized gas inside them. This motion switches on the dynamo mechanism of a magnetic field generation. Besides the magnetic field the electric field also appears, . Here is plasma velocity and is the magnetic viscosity. Moving in this stochastic electric field, protons and electrons gain energy and can be accelerated up to energies  GeV (Dogiel et al. 1987). In the presence of magnetic field one can describe motion of relativistic charged particles as diffusion in coordinate and momentum spaces. To find diffusion coefficient and, after that, maximum energy and spectrum of accelerated particles one need to know properties of the magnetic field, for example, its pair correlators (see Shalchi 2009; Dogiel et al. 1987), which will be calculated below. Dogiel et al. (2005) predicted the positron excess in GeV energy range. Appearance of appropriate observations requires a detailed investigation of particle acceleration in molecular clouds, with taking into account modern data.

Molecular clouds are clusters of molecular hydrogen with a complex inhomogeneous structure (Larson 2003). Their dimensions may reach  parsecs, masses are up to . Gas concentration in molecular clouds is about , the gas temperature is . According to observations, gas is strongly turbulent. The turbulence has a power law Kolmogorov-like spectrum. In addition, a gas is partially ionized . In such a system stochastic magnetic field arises, as will be shown below. The only way to measure directly the magnetic field strength is the Zeeman effect. Zeeman observations were carried out recently for many clouds, their results are summarized in the paper by Crutcher (2010). Typical values of the magnetic field projection on the line of sight are for the molecular clouds cores. Polarization observations (Tassis et al. 2009), carried out for several molecular clouds, showed that magnetic field directions in distant points of a cloud may be similar, Fig. 1. It appears that a mean homogeneous magnetic field exists in clouds together with a stochastic field, produced by the turbulence. In papers (Dogiel et al. 1987, 2005) a mean magnetic field was assumed to be zero. In present paper we treat the problem of magnetic field generation in weakly ionized turbulent gas with a nonzero mean magnetic field. No assumptions about the ratio of a mean to a fluctuating fields are made.

This paper consists of five parts. In Section 2 we write equations describing a gas and a magnetic field in molecular clouds. Evolution equations for magnetic field pair correlators are derived in Section 3. Their stationary solutions are obtained in Section 4. Three cases are considered in detail: a) zero mean magnetic field, b) small mean field and c) large mean field. In Section 5 we compare our results with that obtained by other authors. Summary is compiled in Section 6.

## 2 Magnetohydrodynamics equations for weakly ionized gas

For description of gas motion in molecular clouds one can use two-fluid hydrodynamic equations. We denote velocities of a neutral and an ionized gas components as and respectively. Magnetic viscosity in molecular clouds is much less than the kinematic viscosity (see Dogiel et al. 1987).

We consider gas motions on the scales corresponding to the inertial range , where is determined by the size of the system, and corresponds to the viscous scale. Typical values for molecular clouds are cm, Reynolds number is , hence for the Kolmogorov turbulence cm. In this range of scales the viscosity can be neglected. Turbulent velocity at small scales is less than the sound velocity, it reaches the value of sound velocity only at large scales. So we assume gas to be incompressible, because subsonic gas motions can be considered to be incompressible. Then equations for the ionized component motion and the magnetic field are

 ∂u∂t+(u∇)u=1ρi[−∇Pi+(∇×B)×B4π]− (1) −μin(u−v), ∂B∂t=∇×(u×B), (2) ∇u=0, ∇B=0,

where and are the pressure and the density of ionized component, is ion-neutral collision rate. In contrast to the usual magnetohydrodynamics, in this case an external force in the form of friction between the ionized and the neutral gas components is present. Indeed, estimations give

 μni=10−13s−1,μin=2⋅10−6s−1, (3)

whereas the turbulent fluctuation frequencies lie in the range

 ωmin=10−12s−1,ωmax=10−8s−1. (4)

This implies an important condition

 μni≪ω≪μin. (5)

This means that the neutral gas does not feel the presence of an ionized component. Ions motion, by contrast, is completely determined by the motion of a neutral gas. Numerical estimates of the characteristic frequencies of the problem were discussed in detail by Dogiel et al. (1987).

Thus we can treat the motion of the neutral component to be known, it coincides with the ordinary hydrodynamic turbulence. For an ionized component only two forces are essential: the force of friction on a neutral gas and the Lorentz force, caused by a magnetic field. As we will see below, the pressure can be neglected in comparison with the pressure of the magnetic field. Therefore the motion equation for an ionized component becomes

 (∇×B)×B4πρi−μin(u−v)=0. (6)

Let us denote

 a=14πρiμin. (7)

As far as we consider a gas to be incompressible, const. Expressing the velocity of an ionized component in terms of velocity of neutrals and substituting it to the induction equation (2), we obtain

 ∂B∂t=∇×(v×B)−a∇×(B×(∇×B)×B). (8)

This equation gives the dependence of the magnetic field on the velocity of a neutral gas.

## 3 Derivation of evolution equation

### 3.1 Tensor structure of correlators

To describe the turbulent motion of a neutral gas and obtain closed equations for magnetic field correlators, we use solvable model proposed by Kazantsev (1968) and Kraichnan (1968). We assume neutral gas velocity to be a Gaussian stochastic process with zero mean value, . All information about it is contained in the pair correlation function . The angle brackets here and below denote averaging over an ensemble of realizations. We assume a neutral gas to be a homogeneous isotropic medium, so the pair correlation function can depend only on . We consider the velocity field to be delta-correlated in time,

 ⟨vi(x,t)vj(x+r,t′)⟩=vij(r)τcδ(t−t′), (9)

and mirror-symmetric. It possesses no helicity, and its correlation tensor is symmetric with respect to the interchange of indices . In this case one can construct the tensor structure of the correlation function from only two second-rank tensors

 vij=2V(r)δij+S(r)(δij−rirjr2). (10)

The factor in the first term of the right hand side is written for convenience. From the incompressibility condition , i.e. , we obtain

 S=rV′, (11)

where prime denotes the derivative with respect to , . This relation leads us to the general form of the correlation tensor

 vij(r)=2V(r)δij+rV′(r)(δij−rirjr2). (12)

Thus, the neutral gas turbulence is described by one scalar function . A common method to handle this problem is to pass to the Fourier space

 v(r,t)=∫v(k,t)exp(ikr)dk. (13)

Indeed, the correlation tensor structure becomes simplier

 ⟨vi(k,t)vj(k′,t′)⟩=¯¯¯¯V(k)τcδ(t−t′)(δij−kikjk2)δ(k+k′). (14)

The factor arises from homogeneity, and the tensor structure is uniquely determined by isotropy and incompressibility condition . Functions and are related by

 ¯¯¯¯V(r)=3V(r)+rV′(r), (15)

and is a Fourier transform of ,

 ¯¯¯¯V(r)=∫¯¯¯¯V(k)exp(i(kr))dk. (16)

Now let us consider correlators of the magnetic field. We assume magnetic field to be a Gaussian stochastic process too. But it has nonzero mean value. Let us denote its mean and fluctuating components by and respectively:

 B=H+b,⟨B⟩=H. (17)

We suppose mean magnetic field to be constant in space . One can look for the evolution equation for , using the technique described below, and get . Hence the value of the mean field is an external parameter of the problem.

Strictly speaking, magnetic field , generated by the Gaussian stochastic field , is not pure Gaussian. Its properties are not completely described by the second order correlation function. But significant difference appears in higher than second order moments, and for investigation of second order moment evolution one can assume to be Gaussian, see, for example, the paper by Brandenburg & Subramanian (2000).

To describe fluctuating component of the magnetic field, we introduce its pair correlator , which is similar to the velocity correlator, but the average is taken at the same time moments. Our aim is to establish the evolution equation for this correlator. Since

 ∂∂t⟨bibj⟩=⟨∂bi∂tbj⟩+⟨bi∂bj∂t⟩, (18)

one have to calculate . We suppose that . Averaging Eq. (8), subtracting the resulting equation from Eq. (8), we get

 ∂b∂t=∇×[v×H]+∇×[v×b]−a∇×(H×[j×H]+ +H×[j×b]+b×[j×H]+b×[j×b])− −∇×⟨v×b⟩+a∇×⟨H×[j×b]+b×[j×H]⟩, (19)

where we use the notation for short.

Last three terms (last line) in Eq. (3.1) give no contribution to , since so they can be ignored. Similarly we omit terms in Eq. (3.1), which are proportional to , because their contribution to the derivative is proportional to .

### 3.2 The case with zero mean field

To begin with we consider the case when the mean field is absent, . Then all correlators are isotropic. Maxwell equation is similar to the incompressibility equation , so tensor structure of the magnetic field correlator for the non-helical case is similar to the velocity correlator (12)

 ⟨bi(x,t)bj(x+r,t)⟩=2Q(r)δij+rQ′(r)(δij−rirjr2). (20)

In the Fourier space the correlator is

 ⟨bi(k,t)bj(k′,t)⟩=¯¯¯¯Q(k)(δij−kikjk2)δ(k+k′), (21)

where the relation between and is similar to (15). Due to the isotropy . To get we apply Fourier transform to Eq. (3.1), drop terms which do not contribute to the value of , put and obtain

 ∂b(k)∂t=i∫dqk×[v(q)×b(p)]++a∫dk1dk2k×{b(k1)×([k2×b(k2)]×b(k3))}. (22)

In the first term of the right hand side , in the second term . One can see the following structure of correlator’s derivative

 ∂∂t⟨bb⟩≃⟨vb2⟩+⟨b4⟩, (23)

where only the magnetic field and the velocity are shown, and all tensor indices are dropped. We assume random processes and to be Gaussian. So, to split the correlators in the first term of Eq. (23) one should use the Furutsu-Novikov formula (see Furutsu 1963; Novikov 1965; Klyatskin 2005). It states that if some functional depends on the random process as a solution of some differential equation, one can split correlator

 ⟨vi(k,t)R[v]⟩=∫dk′dt′⟨vi(k,t)vj(k′,t′)⟩⟨δR[v]δvj(k′,t′)⟩, (24)

where is a functional derivative. To compute it one should use the equation which specifies the dependence of on . In our case it is the evolution equation (22). Since the random process is assumed to be delta-correlated in time, and at causality principle states that , we need to know only the value of

 limt′→t−0δbi(k,t)δvj(k′,t′)=ikm(δijbk−k′m−δjmbk−k′i). (25)

In the right hand side the argument is written as a superscript for clarity. In the second term of Eq. (23) one can presents forth order correlator as a product of pair correlators

 ⟨b1b2b3b4⟩=⟨b1b2⟩⟨b3b4⟩+⟨b1b3⟩⟨b2b4⟩+⟨b1b4⟩⟨b2b3⟩. (26)

Let us write out the intermediate result obtained after splitting of correlators. To do this, we use the notations

 Ωkij=δij−kikjk2, (27)

and . We obtain

 ∂¯¯¯¯Q(k)∂tΩkij=2a∫dq¯¯¯¯Q(k)¯¯¯¯Q(q)(−k2ΩqisΩksj− (28) −kmksΩqmsΩkij+kmkiΩqmsΩksj)−τckm∫dq¯¯¯¯V(q)¯¯¯¯Q(k)× ×(Ωqispn(δmsΩknj−δnsΩkmj)−(i↔m))+τckm× ×∫dq¯¯¯¯V(q)¯¯¯¯Q(p)(Ωqiskn(δjsΩpnm−δnsΩpmj)−(i↔m)).

Tensors in right hand side of last equation contain vectors and , which are absent in the left hand side. To extract the factor in the right hand side one can use isotropy of and , then average first two integrals over the sphere and the third integral over a circle, determined by conditions and . After that tensor factors are cancelled and we get scalar evolution equation

 1τc∂¯¯¯¯Q(k)∂t=−23k2¯¯¯¯Q(k)∫¯¯¯¯V(q)dq++k2∫¯¯¯¯V(q)¯¯¯¯Q(p)[1−(kp)(kq)(pq)k2p2q2]dq−−23πρiμinτck2¯¯¯¯Q(k)∫¯¯¯¯Q(q)dq, (29)

where . This equation was derived firstly by Dogiel et al. (2005). Let’s introduce the notation

 λ=13πρiμinτc∫¯¯¯¯Q(q)dq. (30)

The value of is proportional to which is the energy of the fluctuating magnetic field.

Since solving the integral equation on in the -space is rather complicated problem, let us turn back to the -space and obtain the differential equation for . To do this, we apply inverse Fourier transform to the Eq. (29). One can express the value of parameter in terms of

 λ=43aτc¯¯¯¯Q(r=0)=2aτc(2Q(r)+23rQ′(r))|r→0. (31)

Also note that

 V(0)=13∫¯¯¯¯V(q)dq. (32)

We proceed from functions , to , according to Eq. (15), use the spherical symmetry of the correlation functions and get after some algebra

 12τc∂Q(r)∂t=(V(0)−V(r)+λ)(Q′′+4Q′r)−−V′Q′−1r(4V′+rV′′)Q. (33)

One of the key points of this derivation is averaging over a sphere (or circle), when we use the isotropy of the functions and . In the anisotropic case, in the presence of the mean field, for example, this averaging fails. Thus, we conclude that in general case it is necessary to derive and solve the evolution equation in -space. It is no need to apply a Fourier transform. Tensor structure of correlators (12) in this method is more complicated, but there arise no integral equations and additional vectors (as , ) in the tensor structure.

Indeed, the equation (33) can be obtained directly in the -space, by analogy with the above derivation. We will discuss its solution in the section 4.2.

### 3.3 The general case of a non-zero mean field H≠0

In this part we consider general case when the mean magnetic field is present. The correlation function of magnetic fluctuations becomes anisotropic. We assume magnetic field to have no helicity, so its correlation tensor is symmetric with respect to the interchange of indices. We suppose that it has the form

 ⟨bi(x,t)bj(x+r,t)⟩=A(r)δij+B(r)ninj++C(r)(nihj+njhi)+D(r)hihj, (34)

where , are unit vectors in the directions of and respectively.

The most general form of the correlation tensor of the second order in the presence of a preferred direction of the mean magnetic field is given in the paper by Matthaeus & Smith (1981). However, we restrict our attention to terms, specified in Eq. (34). As we will see below this assumption is consistent, in the right hand side of evolution equations only the same tensor terms arise.

All scalar functions are no longer spherically symmetric, but they are axially symmetric with respect to the direction of . We choose this direction as -axis. Below spherical coordinates are used, where is the polar angle between and .

From the symmetry of the correlator and homogeneity of space, functions should be symmetric with respect to , , whereas function should be anti-symmetric .

Condition gives us two relations

 A′r−μrA′μ+B′r+μC′r+1−μ2rC′μ+2Br−μCr=0 1rA′μ+C′r+μD′r+1−μ2rD′μ+3Cr=0. (35)

The derivation of the evolution equations is similar to that performed in the previous section, but now we work in the -space. We use Eq. (3.1) instead of Eq. (22) and the functional derivative

 δbi(r,t)δvj(r′,t−0)=∂∂rm(δ(r−r′)(Bm(r)δij−Bi(r)δjm)), (36)

where is total magnetic field, instead of the formula (25). As before, we use the relation

 ⟨bi(r)ddrbj(r)⟩=0. (37)

which is due to the symmetry of the correlator and homogeneity of space. Finally one principally can write out the bulky system of evolution equations (for the functions ), which contains the second order derivatives in the right hand side. It must be solved taking into account relations (3.3). To solve such a system would be very difficult. To simplify this system, one can use the structure of the Eq. (8), and split the system of equations with second-order spatial derivatives into two systems with first-order derivatives. Namely, we replace one of the unknown functions by

 ~D=D+H2, (38)

introduce the notations

 X = a(D(0)+H2)=a~D(0) (39) Y(r) = −2aA(0)+τc(V(r)−V(0)+12rV′)

and get

 ∂∂t⟨bk(x)bj(x+r)⟩=eksi∂s(f1hmeijm+f2[nh]ihj+f3[nh]inj+f4nmeijm)+(j↔k,r↔(−r))=0, (40)

where is the completely anti-symmetric pseudo-tensor, and

 f1 = X(μA′r+1−μ2rA′μ−μB+Cr)+Y(−1rA′μ+Cr)− (41) −(3V′+rV′′)(C+μ~D) f2 = −X(A′r−μrA′μ−Br)+(X−Y)1rC′μ− −(X−Y+rV′)~D′r+(X−Y)μr~D′μ+V′~D f3 = (X−Y)1rB′μ−(X−Y+rV′)C′r+(X−Y)μrC′μ+ +1r(X−Y+r2V′′)C−(V′−rV′′)μ~D f4 = (rV′−Y)A′r+Y(μrA′μ+Br)−V′A− −(3V′+rV′′)(A+B+μC).

Let us note that the correlator of the form (34) with function instead of in the right hand side describes the correlator of the total magnetic field . The system of evolution equations with nonzero mean field is not homogeneous, it contains a "source" term, which is proportional to . However, the system with function is homogeneous, but the term with arises in boundary conditions.

From Eq. (40) one can obtain evolution equations:

 12∂A∂t = μ∂rf1+1−μ2r∂μf1+∂rf4+f4r (42) 12∂B∂t = μ∂rf3+1−μ2r∂tf3−2μf3r−∂rf4+μr∂μf4+f4r 12∂C∂t = 12(−∂rf1+μr∂μf1+μ∂rf2+1−μ2r∂μf2−μf2r− −∂rf3−1r∂μf4) 12∂D∂t = −(1r∂μf1+∂rf2+f2r).

We restrict ourselves to search for stationary solutions. In this case, the time derivatives are equal to zero, and the system (42) can be solved exactly. Its solution depends on one arbitrary function

 f1 = −g′r (43) f2 = 1rg′μ f3 = g′r−μrg′μ−gr f4 = μg′r+1−μ2rg′μ−μgr.

From Eq. (41) and (43), taking into account two relations (3.3), we get six first-order equations for five unknown functions . The single boundary condition is when because the pair correlation function of magnetic field fluctuations should vanish at large scales.

The system of six equations for five unknown functions seems overdetermined and has no solutions at first glance. However, if this system can be simplified considerably and allows the analytical solution, which is given in section 4.4. In this limit case the system is degenerate and is not overdetermined. This suggests that the same would takes place in the general case.

### 3.4 Small mean field H

Because of the large complexity of equations in the anisotropic case, we consider firstly a simpler problem, we assume that the correlator is isotropic, i.e. it has the form (20) even in the case . This can be done in the case of small mean field , because, as we will see below, if , the amplitude of the fluctuations is greater than .

To get isotropic equations we have to replace

 hihj⟹13δij.

Evolution equation becomes

 (44)

where as before

 λ=2aτc(2Q(0)+23rQ′(r)|r→0). (45)

When Eq. (44) turns into Eq. (33).

## 4 The solution of equations

### 4.1 Preliminaries

Let us reduce our equation to dimensionless one. Before that we denote the rms velocity of the neutral gas by , , and the rms amplitude of the magnetic field fluctuations by , . The correlation time of the velocity field assumed to be equal to . That is the eddy turnover time at scale . For length unit we take the size of the molecular cloud , for unit of time we take , and for unit of magnetic field we take the value of

 Bunit=(τc2aV(0))1/2=(π3L0v0ρiμin)1/2.

In other words, we introduce new functions and variables

 r(1)=r/L0;t(1)=t/τmax;V(1)(r)=V(r)/V(0);Q(1)(r)=Q(r)/(τc2aV(0)). (46)

Thus, we reduce Eq. (44) to the form (below we omit superscript (1))

 3∂Q(r)∂t=(V(0)−V(r)+λ+13H2)(Q′′+4Q′r)−−V′Q′−1r(4V′+rV′′)(Q+16H2). (47)

where , and . For typical parameters of molecular clouds

 Nn≃103cm−3;Ni≃10−2cm−3;L0≃3⋅1018cm;v0≃1km/s;T≃50K, (48)

the unit of magnetic field is equal to . But the cloud parameters change in a wide range, so values of for them can vary significantly.

Let us estimate the characteristic scales of our problem. The inertial range of the turbulence is (in the dimensionless variables)

 lν

where is the viscous scale. Since in the stationary case the magnetic field energy on one hand is less than the kinetic energy of a neutral gas, but on the other hand is larger than the kinetic energy of an ionized gas, then

 10−6≃ωminμin=(τcμin)−1≪λ<(τcμni)−1=ωminμni≃10.

Therefore, we can consider .

Let us examine the dimensionless equation (47). The parameter contains the value of , so this equation is nonlinear. Let us suppose that the initially magnetic field fluctuations are weak (and the parameter is small). Then the term, which is proportional to , makes the positive contribution to the value of , and lead to the initial growth of the magnetic field. Further, this growth will be stopped by nonlinear terms. So we restrict ourselves looking for stationary solutions only. Investigation of a stability of these solutions, especially for anisotropic equations (41), (42), is beyond the scope of our paper.

Since for the stationary solution the parameter does not depend on time, we will initially consider it to be a constant, which is not connected with the function . The single restriction is , because of (in dimensionless variables). Under such approach the equation (47) becomes linear.

The similar linear equation arises in the problem of magnetic dynamo in a turbulent conducting media. In this problem the parameter corresponds to the magnetic viscosity and is considered to be known. The equation, coinciding with Eq. (47), was investigated in large number of papers beginning from the paper by Kazantsev (1968), where the integral equation, similar to Eq. (29), was derived. The brief literature review and necessary references are given in the Discussion. In most of the papers the mean magnetic field was considered to be zero. In order to reveal the role of the mean magnetic field we discuss separately cases and .

To resolve our equations we need to determine the function which characterizes the motion of a neutral gas. The value of corresponds to the maximum scale of the turbulence. It means that the correlation of gas velocities vanishes at the such scale. We consider the velocity spectrum of a neutral gas to be the power law. So, the correlation function of gas velocities is

 V(r)={1−rα,r<10,r>1. (50)

Hereinafter we consider the Kolmogorov turbulence. In this case velocity fluctuation on the scale is in the inertial range, hence . On the viscous scales velocity fluctuations are . Therefore the correlation function is . Because and we consider , then in this range the correlation function is approximately constant, , and the sought correlation function of the magnetic field is, , too. The exclusion is solution which have singularity at . But we do not consider such solutions (see below). Thus, the viscous range of scales does not affect the further analysis.

Let us note that in the Kazantsev-Kraichnan model the turbulent velocity is assumed to be -correlated in time (9) with some value of the correlation time . Because characterizes the total realization of the turbulent motion, it formally can not be a function of the scale . However, many authors who apply the Kazantsev-Kraichnan model to the problem of magnetic dynamo, in order to approach the physical reality, consider to be a function of the scale. They assume to be equial to the turnover time at given scale, . In this case and the Kolmogorov turbulence corresponds to the value . This assumption is justified by the comparison of theoretical results in the model with numerical simulations of forced Navier–Stokes equation (Mason et al. 2011; Tobias,Cattaneo & Boldyrev 2013).

From the theoretical point of view, Vainshtein & Kichatinov (1986) and Boldyrev & Cattaneo (2004) state that we need to know only the integral of the velocity correlation function over time, that is, the turbulent diffusivity, which in Kolmogorov turbulence scales as .

In current paper we use the Kazantsev-Kraichnan model (9) considering .

### 4.2 The case with zero mean field

To begin with we consider the case of zero mean magnetic field. We put and in Eq. (47) and we obtain second order equation for the function

 (V(0)−V(r)+λ)(Q′′+4Q′r)−V′Q′−1r(4V′+rV′′)Q=0, (51)

where . In this expression the term is essential only for solutions which have singularity at . The boundary condition is at .

Eq. (51) has two independent solutions. At () one can find the asymptotics of the solutions, and at Eq. (51) can be solved exactly

 r→0 : Q1∼1;Q2∼r−3 (52) r>1 : Q1=1;Q2=r−3.

Now let us discuss the existence of the solution which is bounded at and is decreasing at . At the same time we can consider more general nonstationary equation, where we assume to be independent of time constant as before. Making the substitution

 Q(r,t)=Ψ(r)r2(m(r))1/2e−Et, (53)

where , we get for the function the Shrödinger equation with the variable mass

 1m(r)d2Ψdr2+(E−U(r))Ψ=0. (54)

The such substitution was firstly done by Kazantsev (1968). The reduction to the Shrödinger equation was discussed in detail in the paper by Schekochihin, Boldyrev & Kulsrud (2002). The stationary solution corresponds to , the negative values of the energy mean the exponential growth of the magnetic field. For the chosen velocity correlator (50) one can obtain the analytic expression for the potential . For (that includes the Kolmogorov turbulence) the potential is positive everywhere. Hence there is no solutions exponentially growing with time. Eq. (51) also has no finite at and vanishing at solution. Consequently any solution has the same power asymptotics (52) at and at . The solution of the stationary equation, which is finite at , falls down at not to zero, but to some positive value. We demand at . Only the solution , which has singularity at , satisfies this condition. This solution has the power law behavior even at scales , up to very small scales where the magnetic viscosity becomes important. Thus, there is no finite solutions with the zero mean magnetic field for the Kolmogorov turbulence ().

For and there exists the region where . There appears the bound states, i.e. finite solutions of the Shrödinger equation with . That means that the magnetic field will grow up to the level when .

### 4.3 Small mean field H

Now we look for stationary solutions of Eq. (47) with small but nonzero mean magnetic field. This equation is inhomogeneous one, and its partial solution is a constant, . We introduce the quantity

 ~Q=Q+16H2, (55)

denote , and obtain for the exactly the homogeneous equation (51) with the replacement . In fact, the quantity is the correlation function of the total magnetic field

 ⟨Bi(x)Bj(x+r)⟩=⟨bi(x)bj(x+r)⟩+13H2δij. (56)

The last term is written in the form , because in this subsection we assume correlators to be isotropic. Therefore, the tensor structure of the correlator coincides with (20). It is possible to derive evolution equation directly for this correlator, including the mean magnetic field, as was done by Boldyrev, Cattaneo & Rosner (2005). However, the boundary conditions for are different

 2~Q(0)+23r~Q′|r→0=λ′ (57) ~Q→16H2,r→∞, (58)

because of the fluctuations correlator tends to zero at large distances, but the correlator is independent on the distance due to its homogeneity.

As was mentioned in the previous subsection, the solution of the stationary equation (51), which is finite at , tends to a positive constant at . It is in accordance with the new boundary condition (58). Now even for there exists the unique bounded solution, we will call it as the preferential solution. The value of for them is defined by unique manner. Graphs of this solution for three values of the mean magnetic field , and are presented on the Fig. 2. These graphs and all other numerical results are obtained for the Kolmogorov turbulence, . Since

 ⟨bi(x)bi(x)⟩=b20=3λ, (59)

we get the dependence of on , which is presented on the Fig. 3. For the amplitude of the fluctuating magnetic field . Thus, for we have , i.e. the fluctuating field dominates over the mean field.

Let us note that for the amplitude of the magnetic fluctuations for the preferential solution tends to zero also. For the preferential solution turns to the zero solution of equation without the mean field. It corresponds to the fact that for there are no bounded solutions vanishing at large scales.

For small values of , , the asymptotic behavior of the preferential solution is