Non-Gaussian Generalization of the Kazantsev–Kraichnan Model for a Turbulent Dynamo

Magnetic field generation in turbulent plasma is one of the most probable mechanisms responsible for stellar, interstellar, and intergalactic magnetism (see, e.g., Moffatt 1978; Parker 1979; Zeldovich et al. 1984a; Brandenburg & Subramanian 2005; Schober et al. 2018). Small-scale turbulent dynamo problem has been the object of interest of many researchers (see, e.g., Falkovich et al. 2001; Brandenburg et al. 2012; Alexakis & Biferale 2018), as turbulence can provide an intense increase of the magnetic field (Moffatt 1978; Kulsrud & Anderson 1992). In these problems, the characteristic scale of the magnetic field fluctuations is much smaller than the scale at which turbulence is generated; this corresponds to inertial and viscous scale ranges of turbulence.

The conception of small (seed) initial magnetic field fluctuations implies that there is an important stage of a kinematic dynamo: the magnetic field is small enough to cause no feedback on the velocity distribution, so it is passively advected by the turbulent flow. The magnetic Prandtl number, i.e., the ratio of the kinematic viscosity ν to the magnetic diffusivity ϰ, is the most important characteristic of this advection process. In this paper, we consider large Prandtl numbers:

$$\begin{eqnarray*}&&{{\rm{\Pr }}}_{m}=\nu /\varkappa \,\gg 1.\end{eqnarray*}$$

Such a situation is observed, e.g., in the interstellar medium (Brandenburg & Subramanian 2005; Rincon 2019). This means that the magnetic diffusive scale length r_d is much smaller than the Kolmogorov viscous scale r_ν . We assume the characteristic scale length l of the initial magnetic field fluctuation to lie between these two scales:

$$\begin{eqnarray*}&&{r}_{d}\ll l\ll {r}_{\nu }.\end{eqnarray*}$$

The evolution of the magnetic field is described by a stochastic partial differential equation with a random velocity field acting as multiplicative noise. The velocity statistics is assumed to be stationary and known. The problem is to find the statistics of the magnetic field, in particular, its correlations.

The Kazantsev–Kraichnan model (Kraichnan & Nagarajan 1967; Kazantsev 1968) is the simplest and most natural approximation of the velocity statistics; the velocity field is assumed to be Gaussian and δ-correlated in time. In this model, all magnetic field correlators are governed by the only two-point velocity correlator.

This model is an essential simplification. Actually, unlike the additive random processes, in stochastic equations with multiplicative noise, the cumulants of all orders give comparable contributions to any statistical moment. So, the central limit theorem "does not work" for these processes, and, to calculate even the second-order correlators of the magnetic field, one should use the large deviation principle and take all velocity correlators into account. So, the replacement of an arbitrary random velocity field by a Gaussian process can change the result crucially.

Besides, in the Gaussian approximation of a velocity field—and hence, in the Kazantsev–Kraichnan model—there is no energy cascade. The energy of the magnetic field excitation comes from the energy of the turbulent flow, which is generated at large scales; thus, the cascade may be important for the dynamo. The nonzero third-order velocity correlator is responsible for the energy cascade and time asymmetry in general (Kolmogorov 1991; Frisch 1995). Indeed, the inversion of time would result in a change of sign of all velocities, and time symmetry implies that the statistics would not change; hence, the third-order correlator is zero for time-symmetric flows. Its presence indicates time asymmetry. So, the account of non-Gaussianity is highly desirable.

There are two different theoretical approaches to investigate the magnetic field statistics. One of them is based on the Lagrangian deformations statistics (see Zel'dovich et al. 1984b; Chertkov et al. 1999; Il'yn et al. 2018); it implies a direct solution of the magnetic field evolution equation by means of the T-exponential formalism. The physical meaning of this method can be formulated in terms of independent magnetic blobs, each of them undergoing evolution in the turbulent flow (Moffatt & Saffman 1964; Kolokolov 2017; Il'yn et al. 2019, 2021). This approach allows one to calculate the magnetic field correlators of all orders and to consider inhomogeneous, in particular, localized initial magnetic field distributions. In this frame, it is possible to deal with arbitrary (not necessarily Gaussian) velocity statistics. So, this approach allows one to consider velocity statistics wider than the Kazantsev–Kraichnan model.

However, this approach is restricted to the so-called Batchelor regime (Batchelor 1959); the characteristic scale of the magnetic field must lie deep inside the viscous range of turbulence, so that the velocity field can be approximated by a linear function. This means that the solutions found by this method are definitely applicable for some finite time range $t\propto \mathrm{ln}{r}_{\nu }/l$. Later on, the characteristic scale of the magnetic field continues to increase and reaches the inertial range of turbulence. The "Lagrangian deformation" approach may fail to predict the behavior of correlators at this stage. The details of the applicability of the method to the inertial stage were considered by Il'yn et al. (2021).

The other approach is based on statistical properties of pair correlators and allows one to derive a closed differential equation for the pair correlator of the magnetic field (Kazantsev equation). It was used and developed in many papers (see, e.g., Kazantsev 1968; Kraichnan 1968; Vainshtein & Kichatinov 1986; Schekochihin et al. 2002a; Malyshkin & Boldyrev 2007; Istomin & Kiselev 2013; Seshasayanan & Alexakis 2016; Kolokolov 2017); hereafter, we will refer to it as to Kazantsev approach. The advantage of the method is its applicability to any stage of the magnetic field evolution. However, it is restricted to statistically homogenous magnetic field configurations, and it allows one to calculate the second-order two-point correlator only. There is one more vice of this approach: it requires Gaussian and δ-correlated in time velocity statistics, so it is restricted to the Kazantsev–Kraichnan model of the velocity field. Only for several models with some special additional conditions has its applicability been enlarged (Schekochihin & Kulsrud 2001; Kleeorin et al. 2002; Bhat & Subramanian 2014).

The two approaches produce concordant results wherever their domains of applicability overlap (Chertkov et al. 1999; Il'yn et al. 2021); they are also verified by numerical simulations (Schekochihin et al. 2004; Mason et al. 2011; Seta et al. 2020). However, there remains a domain where neither of them can be applied: processes with non-Gaussian and/or not δ-correlated velocity statistics cannot be analyzed at a late (inertial) stage of their evolution by either the Lagrangian deformations approach or the classical Kazantsev method. The finite correlation time was taken into account in Schekochihin & Kulsrud (2001), Kleeorin et al. (2002), Mason et al. (2011), and Bhat & Subramanian (2014) for some special types of flows. The non-Gaussian velocity statistics in combination with the inertial stage has not been considered yet.

To fill this gap, in this paper, we consider the simplest non-Gaussian generalization of the Kazantsev–Kraichnan model introduced in Il'yn et al. (2016, 2019). It implies a nonzero third-order velocity correlator and thus takes into account the time asymmetry of the flow. This model allows one to investigate the long-term evolution of statistics for advection (and, more generally, multiplicative) equations for an arbitrary velocity field with a small but nonzero third-order correlator. We generalize the Kazantsev method to apply it to this "V³ model" and find the two-point pair magnetic field correlator. We show that inside the Batchelor regime, the results obtained for the V³ model by means of the Lagrangian deformations approach and the generalized Kazantsev method coincide, which verifies the new generalized method. Now, for the later (inertial) stage, we show that the V³ model is stable relative to the limit of "zero non-Gaussianity" as it turns into the Kazantsev–Kraichnan model. We calculate the magnetic field growth increment for finite time asymmetry and evaluate the correction produced by the finite Prandtl number.

It appears that a small time asymmetry decreases the magnetic field generation independently of the direction of the energy cascade. The range of Prandtl numbers that produce effective generation is also narrower as the non-Gaussian time asymmetry increases. The V³ model is shown to be a useful and effective instrument to investigate magnetic field advection in finite Prandtl non-Gaussian flows.

The paper is organized as follows. In the next section, we briefly review the basic ideas and equations of the two approaches by the example of the Kazantsev–Kraichnan model. Then we recall the formulation and restrictions of the V³ model and the results obtained for this model by means of the Lagrangian deformations method (Section 3). In Section 4, we derive the modified Kazantsev equation in order to apply the Kazantsev approach to the V³ model. In the limit of an infinite Prandtl number, it appears to be possible to solve the equation and find the increment of the pair magnetic field correlator. The results of the numerical solution of the equation for finite Prandtl numbers, validation of the method, and check of its stability are performed in Section 5. In Section 6, we analyze the obtained results and pay special attention to the applicability of the δ-correlated in time velocity distribution and the comparison with the models with a finite correlation time.

To introduce the notations and equations needed, we start from the classical problem statement. Kinematic transport of the magnetic field B (t, r ) advected by a random statistically homogeneous and isotropic nondivergent velocity field v (t, r ), ∇ · v = 0, is described by the evolution equation

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t}+\left({\boldsymbol{v}}{\rm{\nabla }}\right){\boldsymbol{B}}-\left({\boldsymbol{B}}{\rm{\nabla }}\right){\boldsymbol{v}}=\varkappa \,{\rm{\Delta }}{\boldsymbol{B}},\end{eqnarray} \tag{ 1 }$$

where ϰ is the diffusivity. The random process v ( r , t) is assumed to be stationary and have given statistical properties. The initial conditions for the magnetic field are also stochastically isotropic and homogeneous. The aim is to find the statistical characteristics of the process B , in particular, its pair correlator.

From the statistical homogeneity and isotropy and nondivergency of B , it follows that its simultaneous pair correlator has the form

$$\begin{eqnarray}&&\begin{array}{l}\left\langle {B}_{i}({\boldsymbol{R}},t){B}_{j}({\boldsymbol{R}}+{\boldsymbol{r}},t)\right\rangle =G(r,t){\delta }_{\mathrm{ij}}\\ \qquad +\,\displaystyle \frac{1}{2}{rG}^{\prime} (r,t)({\delta }_{\mathrm{ij}}-{r}_{i}{r}_{j}{/r}^{2}).\end{array}\end{eqnarray} \tag{ 2 }$$

The average here is taken over the initial conditions B ( r , 0) and the possible realizations of the velocity field v ( r , t).

So, we are interested in the time dependence of G. If it increases exponentially,

$$\begin{eqnarray}&&G\sim {e}^{\gamma t}G(r),\end{eqnarray} \tag{ 3 }$$

where one calls the process a "turbulent dynamo," and γ is called the magnetic field increment.

The Kazantsev–Kraichnan model implies that the velocity field is Gaussian, δ-correlated in time; its statistics is completely determined by the second-order correlator

$$\begin{eqnarray}&&\left\langle {v}_{i}({\boldsymbol{R}},t){v}_{j}({\boldsymbol{R}}+{\boldsymbol{r}},t+\tau )\right\rangle ={D}_{\mathrm{ij}}({\boldsymbol{r}})\delta (\tau ).\end{eqnarray} \tag{ 4 }$$

The correlator does not depend on R and t because of homogeneity and stationarity. To make contact with the finite correlation time real flows, one can define D_ij by

$$\begin{eqnarray}&&{D}_{\mathrm{ij}}({\boldsymbol{r}})=\int \left\langle {v}_{i}({\boldsymbol{R}},t){v}_{j}({\boldsymbol{R}}+{\boldsymbol{r}},t+\tau )\right\rangle d\tau .\end{eqnarray} \tag{ 5 }$$

Just as in Equation (2), nondivergency and isotropy oblige the tensor D_ij to be determined by only one scalar function of a scalar argument. For the purposes of the next subsection, it is convenient to consider the (time-integrated) longitudinal structure function

$$\begin{eqnarray}\begin{array}{rcl}\sigma (r) & = & \displaystyle \frac{1}{4}\displaystyle \int d\tau \left\langle \left(\left({\boldsymbol{v}}({\boldsymbol{r}},\tau )-{\boldsymbol{v}}(0,\tau )\right){\boldsymbol{r}}/r\right)\right.\\ & & \left.\times \left(\left({\boldsymbol{v}}({\boldsymbol{r}},0)-{\boldsymbol{v}}(0,0)\right){\boldsymbol{r}}/r\right)\right\rangle .\end{array}\end{eqnarray} \tag{ 6 }$$

Then,

$$\begin{eqnarray*}&&\sigma (r)=\displaystyle \frac{1}{2}\left({D}_{\mathrm{ij}}(0)\displaystyle \frac{{r}_{i}{r}_{j}}{{r}^{2}}-{D}_{\mathrm{ij}}(r)\displaystyle \frac{{r}_{i}{r}_{j}}{{r}^{2}}\right).\end{eqnarray*}$$

If one presents D_ij in the form analogous to Equation (2),

$$\begin{eqnarray}&&{D}_{\mathrm{ij}}=P(r){\delta }_{\mathrm{ij}}+\displaystyle \frac{1}{2}{rP}^{\prime} (r)({\delta }_{\mathrm{ij}}-{r}_{i}{r}_{j}{/r}^{2}),\end{eqnarray} \tag{ 7 }$$

then

$$\begin{eqnarray*}&&\sigma (r)=\displaystyle \frac{1}{2}(P(0)-P(r)).\end{eqnarray*}$$

In presence of viscosity, the velocity field is smooth at the smallest scales, and

$$\begin{eqnarray}&&P(r){=}_{r\to 0}P(0)-\displaystyle \frac{2}{3}{{Dr}}^{2}+O({r}^{3}),D=-\displaystyle \frac{3}{4}P^{\prime\prime} (0),\end{eqnarray} \tag{ 8 }$$

2.1. Kazantsev Equation

The equation to describe the evolution of the second-order correlator (Equation (2)) can be found from Equation (1) by means of multiplying and subsequent averaging. The cross-correlations of the magnetic field and velocity can be split by means of the Furutsu–Novikov theorem due to Gaussianity and δ-correlation (Furutsu 1963; Novikov 1965),

$$\begin{eqnarray}&&\left\langle {v}_{i}({\boldsymbol{r}},t)g[{\boldsymbol{v}}]\right\rangle =\displaystyle \frac{1}{2}\int \,{\rm{d}}{\boldsymbol{r}}^{\prime} {D}_{\mathrm{ij}}({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )\left\langle \displaystyle \frac{\delta g[{\boldsymbol{v}}]}{\delta {v}_{j}({\boldsymbol{r}}^{\prime} ,t)}\right\rangle ,\end{eqnarray} \tag{ 9 }$$

where g[ v ] is an arbitrary analytic functional of v ( r , t), and δ/δ v is a functional derivative. Thus, for G( r , t), one gets the equation

$$\begin{eqnarray}&&\displaystyle \frac{\partial G({\boldsymbol{r}},t)}{\partial t}={L}_{\mathrm{Gauss}}G,\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray*}\begin{array}{rcl}{L}_{\mathrm{Gauss}} & = & 2\sigma (r)\displaystyle \frac{{\partial }^{2}}{\partial {r}^{2}}+2\left(\sigma ^{\prime} +\displaystyle \frac{4}{r}\sigma \right)\displaystyle \frac{\partial }{\partial r}+2\left(\sigma ^{\prime\prime} +\displaystyle \frac{4}{r}\sigma ^{\prime} \right)\\ & & +\,2\varkappa \,\left(\displaystyle \frac{{\partial }^{2}}{\partial {r}^{2}}+\displaystyle \frac{4}{r}\displaystyle \frac{\partial }{\partial r}\right).\end{array}\end{eqnarray*}$$

In a turbulent hydrodynamic flow, σ has the following asymptotics:

$$\begin{eqnarray}\sigma (r)=\left\{\begin{array}{ll}\displaystyle \frac{D}{3}{r}^{2}, & r\ll {r}_{\nu },\\ {\rm{const}}\cdot {r}^{\xi }, & {r}_{\nu }\ll r\ll L,\\ \displaystyle \frac{1}{2}P(0), & r\gg L,\end{array}\right.\end{eqnarray} \tag{ 11 }$$

where r_ν is the viscous dissipation scale, and L is the integral scale of the turbulence. The timescale

$$\begin{eqnarray*}&&{D}^{-1}=\displaystyle \frac{2{r}_{\nu }}{{v}_{\nu }}\end{eqnarray*}$$

is of the order of the eddy turnover time at the viscous scale (see Brandenburg & Subramanian 2005). The first asymptote in Equation (11) corresponds to the viscous range of scales, the second presents the inertial range, and the last is for the integral range of turbulence (see Landau & Lifshitz 1987).

The well-known result (Kleeorin et al. 2002; Schekochihin et al. 2002b) is that for large Prandtl numbers; independently of the parameter ξ (characterizing the inertial range), Equation (10) has a growing mode with the increment

$$\begin{eqnarray*}&&\gamma =\displaystyle \frac{5}{2}D-O\left({\mathrm{ln}}^{-2}\displaystyle \frac{{r}_{d}}{{r}_{\nu }}\right),\end{eqnarray*}$$

where

$$\begin{eqnarray*}&&{r}_{d}=\sqrt{\varkappa /D}\propto {r}_{\nu }/\sqrt{{{\rm{\Pr }}}_{m}}.\end{eqnarray*}$$

Thus, a turbulent dynamo exists at large Prandtl numbers, and the increment is determined by the viscous range of the turbulence.

2.2. Lagrangian Deformations Approach

This alternative way can only be applied for scales deep inside the viscous range, which corresponds to the early stage of evolution of initially small-scale (l ≪ r_ν ) magnetic fluctuation (Chertkov et al. 1999). For scales much smaller than r_ν , the velocity field is smooth. So, one chooses a comoving quasi-Lagrangian reference frame (Belinicher & L'vov 1987) associated with some fluid particle r ₀(t) and, in this frame, expands the (relative) velocity into a series up to first order:

$$\begin{eqnarray*}&&\delta {v}_{i}({\boldsymbol{r}},t)={A}_{\mathrm{ij}}(t){r}_{j}.\end{eqnarray*}$$

This is called the Batchelor approximation (Batchelor 1959), and A_ij is the velocity gradient tensor:

$$\begin{eqnarray*}&&{A}_{\mathrm{ij}}={\partial }_{j}{v}_{i}({{\boldsymbol{r}}}_{0}(t),t).\end{eqnarray*}$$

The transport Equation (1) now takes the form

$$\begin{eqnarray}&&{\partial }_{t}{B}_{i}+{A}_{{jk}}{r}_{k}{\partial }_{j}{B}_{i}-{A}_{\mathrm{ij}}{B}_{j}=\varkappa {\rm{\Delta }}{B}_{i}.\end{eqnarray} \tag{ 12 }$$

In the Kazantsev–Kraichnan model, the statistics of A is determined by its second-order correlator,

$$\begin{eqnarray*}&&\left\langle {A}_{\mathrm{ij}}(t){A}_{\mathrm{kp}}(t^{\prime} )\right\rangle ={D}_{\mathrm{ijkp}}\delta (t-t^{\prime} ),\end{eqnarray*}$$

where, in accordance with Equation (4),

$$\begin{eqnarray*}&&{D}_{\mathrm{ijkp}}=-{\partial }_{j}{\partial }_{p}{D}_{\mathrm{ik}}(0).\end{eqnarray*}$$

Equation (12) can be solved explicitly by means of the Fourier transform (Zel'dovich et al. 1984b); for brevity, here we restrict ourselves to one spatial point and, hence, a one-point correlator:

$$\begin{eqnarray}&&{B}_{m}(t)={Q}_{\mathrm{mn}}^{-1}\int {B}_{n}({\boldsymbol{p}},0){e}^{-\varkappa \,{p}_{i}{p}_{j}\int {\left({{QQ}}^{T}\right)}_{\mathrm{ij}}(t^{\prime} ){dt}^{\prime} }{\rm{d}}{\boldsymbol{p}}.\end{eqnarray} \tag{ 13 }$$

Here Q(t) is the evolution matrix defined by the equation

$$\begin{eqnarray}&&\displaystyle \frac{{dQ}}{{dt}}=-{QA},\quad Q(0)=1.\end{eqnarray} \tag{ 14 }$$

It is convenient to use the polar decomposition for the evolution matrix:

$$\begin{eqnarray*}&&Q={sdR},\qquad s,R\in {SO}(3),\qquad d={\rm{diag}}\{{e}^{-{\zeta }_{i}t}\}.\end{eqnarray*}$$

From incompressibility, it follows $\det Q=1$; hence, ζ₁ + ζ₂ + ζ₃ = 0.

It is well known (Furstenberg 1963; Letchikov 1996) that the long-time asymptotic behavior of these three components is quite different. As Q obeys Equation (14), s(t) stabilizes at some random value that depends on the realization of the process; ζ_i (t) are asymptotically stationary random processes and tend (with unitary probability) to the limits λ_i ,

$$\begin{eqnarray*}&&{\lambda }_{1}\geqslant {\lambda }_{2}\geqslant {\lambda }_{3},\end{eqnarray*}$$

where the set of λ_i is called the Lyapunov spectrum (Oseledets 1968); and R(t) rotates randomly. We note that since QQ^T = sd² s^T and ${({Q}^{-1})}^{T}{Q}^{-1}={{sd}}^{-2}{s}^{T}$, the matrix R vanishes in the expression for B²(t). This simplifies the calculation of the statistical moments. The Kazantsev–Kraichnan model provides one more significant simplification: in particular, it corresponds to time-reversible flow, which means λ₂ = 0.

Without a loss of generality, the initial conditions for B( p , 0) can be chosen in the Gaussian form, with a pair correlator

$$\begin{eqnarray}&&\langle {B}_{n}(0,{\boldsymbol{p}}){B}_{m}(0,{\boldsymbol{p}}^{\prime} )\rangle =\delta ({\boldsymbol{p}}+{\boldsymbol{p}}^{\prime} ){e}^{-{p}^{2}{l}^{2}}\left({p}^{2}{\delta }_{\mathrm{mn}}-{p}_{m}{p}_{n}\right).\end{eqnarray} \tag{ 15 }$$

Raising Equation (13) to the square and taking the average over the initial conditions, one can obtain (Zel'dovich et al. 1984b; Chertkov et al. 1999)

$$\begin{eqnarray*}{\left\langle {B}^{2}\right\rangle }_{{ic}}\sim \left\{\begin{array}{ll}{d}_{2}/{d}_{1}, & \ \mathrm{ln}{d}_{2}\gt 0,\\ {d}_{3}/{d}_{2}, & \ \mathrm{ln}{d}_{2}\lt 0.\end{array}\right.\end{eqnarray*}$$

To average this expression over all possible realizations of A, one considers the probability density of ζ:

$$\begin{eqnarray}\begin{array}{rcl}P({\boldsymbol{y}},t) & = & \langle \displaystyle \prod _{j=1..3}\delta ({\zeta }_{j}(t)-{y}_{j})\rangle ,\\ {\left\langle {B}^{2}\right\rangle }_{i.c.,{\boldsymbol{v}}} & = & \displaystyle \int P({\boldsymbol{\zeta }},t)\langle {B}^{2}{\rangle }_{i.c.}d{\zeta }_{1}...d{\zeta }_{3}.\end{array}\end{eqnarray} \tag{ 16 }$$

The incompressibility condition leaves only two independent variables (e.g., ζ₁, ζ₂). The probability density of ζ_j for any (not necessarily Gaussian) A(t) can be expressed in terms of statistics of the process A(t) (Il'yn et al. 2016, 2019).

Eventually, for the Kazantsev–Kraichnan model, one gets (Chertkov et al. 1999)

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{t\to \infty }\displaystyle \frac{1}{t}\mathrm{ln}\left\langle {B}^{2}\right\rangle =\displaystyle \frac{5}{2}D,\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{t\to \infty }\displaystyle \frac{1}{t}\mathrm{ln}\left\langle {B}^{2n}\right\rangle =\left(2n+\displaystyle \frac{{n}^{2}}{2}\right)D.\end{eqnarray} \tag{ 18 }$$

We see that Equation (17) coincides with the increment obtained in the Kazantsev approach. So, it appears that the asymptote found in the Batchelor approximation continues to be valid not only during the initial stage, $t\lt \tfrac{1}{D}\mathrm{ln}({r}_{\nu }/l)$, but also at later stages of evolution. This fact is nontrivial, since, e.g., in two-dimensional flows, the Kazantsev equation shows no growing modes, and the exponential increase of the magnetic field at the initial Batchelor stage changes to a decrease at larger times (Schekochihin et al. 2002a; Kolokolov 2017).

To generalize the Kazantsev–Kraichnan model, one has to add higher-order connected correlators; in particular, to take time-asymmetric processes into account, one has to deal with third-order correlators. The isotropy and incompressibility conditions reduce the degrees of freedom of the whole tensor 〈A_ij A_km A_nl〉 to one arbitrary multiplier F. The general expression for all of the components is given by Pumir (2017); here we restrict our consideration to the correlators of the diagonal elements A_jj (no summation), since these components are the only ones needed for the calculation of 〈B²〉 (Il'yn et al. 2016, 2019):

$$\begin{eqnarray}&&\langle {A}_{\mathrm{pp}}(t){A}_{\mathrm{qq}}(t^{\prime} ){A}_{{rr}}(t^{\prime\prime} )\rangle ={{Ff}}_{\mathrm{pqr}}\delta (t-t^{\prime} )\delta (t-t^{\prime\prime} ),\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray*}\begin{array}{rcl}{f}_{111} & = & {f}_{222}={f}_{333}={f}_{123}=-\displaystyle \frac{4}{3},\\ {f}_{112} & = & {f}_{113}={f}_{221}=...=\displaystyle \frac{2}{3}.\end{array}\end{eqnarray*}$$

We note that here f_pqr is not a tensor. The right-hand side of Equation (19) is written in the form corresponding to an effective δ-process (Il'yn et al. 2016). The validity of this approximation is verified by the possibility of reducing any finite correlation time non-Gaussian process to some δ-correlated process; see Appendix A.

In the frame of the V³ model, we set all of the higher-order connected correlators to zero. A vice of this simplification is that the probability density is negative in some range of its argument (Rytov et al. 1978; Monin & Yaglom 1987), as only the second and third connected correlators are unequal to zero. This artifact can be fixed in the case of small F by the addition of negligibly small but nonzero higher-order correlators. These higher-order corrections would not affect the magnetic field increment.

So, the time asymmetry of the velocity field in the Batchelor regime is governed by only one parameter, F. The coefficient F is the index of asymmetry of the flow. The direction of the cascade observed in real three-dimensional hydrodynamic flows corresponds to F > 0 (Il'yn & Zybin 2015). The numerical simulation (Girimaji & Pope 1990) and the experiment (Luthi et al. 2005) give an estimate of the relation between the Lyapunov exponents λ₂/λ₁ ≃ 0.14. The Lyapunov exponents are related to the parameters F and D (Balkovsky & Fouxon 1999; Il'yn et al. 2016) by λ₁/λ₂ = (2D − F)/(2F). So, this result for the Lyapunov spectrum corresponds to

$$\begin{eqnarray}&&F/D\simeq 0.13.\end{eqnarray} \tag{ 20 }$$

Returning to the calculation of 〈B²〉, one makes use of the statistics of A to calculate the probability density P( ζ , t) (Il'yn et al. 2016, 2019); the long-term asymptote of the integral in Equation (16) can be found by the saddle point method, and the result is (Il'yn et al. 2019) ⁴

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{t\to \infty }\displaystyle \frac{1}{t}\mathrm{ln}\langle {B}^{2}(t)\rangle =\displaystyle \frac{4}{9}\displaystyle \frac{{D}^{3}}{{F}^{2}}+7D-\displaystyle \frac{{\left(4{D}^{2}+27{F}^{2}\right)}^{3/2}}{18{F}^{2}}.\end{eqnarray} \tag{ 21 }$$

In the limit F ≪ 1, we arrive at

$$\begin{eqnarray}&&\gamma =\displaystyle \frac{5}{2}D\left(1-\displaystyle \frac{243}{80}{\left(\displaystyle \frac{F}{D}\right)}^{2}+o{\left(\displaystyle \frac{F}{D}\right)}^{2}\right).\end{eqnarray} \tag{ 22 }$$

We see that the average coincides with that for the Kazantsev–Kraichnan model if F = 0. Analogous calculations for the higher-order increments lead to

$$\begin{eqnarray*}&&\mathop{\mathrm{lim}}\limits_{t\to \infty }\displaystyle \frac{1}{t}\mathrm{ln}\langle {B}^{2n}(t)\rangle \simeq \left(2n+\displaystyle \frac{{n}^{2}}{2}\right)D-\displaystyle \frac{3}{32}{\left(2+n\right)}^{4}\displaystyle \frac{{F}^{2}}{D}.\end{eqnarray*}$$

The dependence γ(F) for the exact and approximated Equations (21) and (22) is presented in Figure 1. One can see that the approximation of Equation (22) works well for F/D ≲ 0.1.

Zoom In Zoom Out Reset image size

Equation (22) generalizes Equation (17) for time-asymmetric flows, but it is still derived for the linear velocity field and thus is only valid at the Batchelor stage of magnetic field evolution. To investigate the dynamo generation at longer times, one should apply the Kazantsev approach to the V³ model.

In the Kazantsev–Kraichnan model, there is only one nontrivial velocity correlator:

$$\begin{eqnarray}&&\langle {v}_{i}({\boldsymbol{r}},t){v}_{j}({\boldsymbol{r}}^{\prime} ,t^{\prime} )\rangle ={D}_{\mathrm{ij}}({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )\delta (t-t^{\prime} ).\end{eqnarray} \tag{ 23 }$$

In the frame of the V³ model ideology, we add the third-order correlator: ⁵

$$\begin{eqnarray}&&\begin{array}{l}\langle {v}_{i}({\boldsymbol{r}},t){v}_{j}({\boldsymbol{r}}^{\prime} ,t^{\prime} ){v}_{k}({\boldsymbol{r}}^{\prime\prime} ,t^{\prime\prime} )\rangle \\ \qquad =\,{F}_{\mathrm{ij}k}({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} ,{\boldsymbol{r}}-{\boldsymbol{r}}^{\prime\prime} )\delta (t-t^{\prime} )\delta (t-t^{\prime\prime} ).\end{array}\end{eqnarray} \tag{ 24 }$$

The correspondence with velocity gradients statistics (Equations (8) and (19)) requires

$$\begin{eqnarray}&&D=-\displaystyle \frac{3}{4}\displaystyle \frac{{d}^{2}}{{{dr}}^{2}}{\left.\left(\displaystyle \frac{1}{{r}^{2}}{r}_{i}{r}_{j}{D}_{\mathrm{ij}}\right)\right|}_{{\boldsymbol{r}}=0},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&F=-\displaystyle \frac{3}{4}\displaystyle \frac{{\partial }^{3}}{\partial {r}_{1}\,\partial {r}_{1}^{{\prime} }\,\partial r{{\prime} }_{1}^{{\prime} }}{F}_{111}({\bf{0}},{\bf{0}}).\end{eqnarray} \tag{ 26 }$$

Again, just as in the case of velocity gradients, the requirement of isotropy and incompressibility of the flow leaves only one free parameter for the tensor F_ijk; it is the multiplier F that plays the role.

To apply the Kazantsev method to the non-Gaussian velocity field, one has to use the non-Gaussian version of the Furutsu–Novikov relation (Klyatskin 2005, Rytov et al. 1978) to take the nonzero third-order correlator into account:

$$\begin{eqnarray}&&\begin{array}{l}\langle {v}_{i}({\boldsymbol{r}},t)g[{\boldsymbol{v}}]\rangle =\displaystyle \frac{1}{2}\displaystyle \int {dr}^{\prime} \,{D}_{\mathrm{ij}}({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )\left\langle \displaystyle \frac{\delta }{\delta {v}_{j}({\boldsymbol{r}}^{\prime} ,t)}g[{\boldsymbol{v}}]\right\rangle \\ \quad +\,\displaystyle \frac{1}{6}\displaystyle \int d{\boldsymbol{r}}^{\prime} d{\boldsymbol{r}}^{\prime\prime} \,{F}_{\mathrm{ij}k}\left({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} ,{\boldsymbol{r}}-{\boldsymbol{r}}^{\prime\prime} \right)\\ \quad \times \,\left\langle \displaystyle \frac{{\delta }^{2}}{\delta {v}_{j}({\boldsymbol{r}}^{\prime} ,t)\delta {v}_{k}({\boldsymbol{r}}^{\prime\prime} ,t)}g[{\boldsymbol{v}}]\right\rangle .\end{array}\end{eqnarray} \tag{ 27 }$$

In this equation, the time integral is already calculated; see the details in Appendix B.

Taking the average of the square of Equation (1), we then arrive at the modification of Kazantsev Equation (10):

$$\begin{eqnarray}&&\displaystyle \frac{\partial G(r,t)}{\partial t}=({L}_{\mathrm{Gauss}}+\delta L)G(r,t).\end{eqnarray} \tag{ 28 }$$

The expression for δ L is very cumbersome. Shorter expressions for important special choices of F_ijk(r) will be presented in the next subsections.

We are interested in the long-term asymptotics of the magnetic field correlator, so we seek the solution in the form:

$$\begin{eqnarray*}&&G(r,t)={e}^{\gamma t}G(r),\end{eqnarray*}$$

which transforms Equation (28) into the ordinary differential equation

$$\begin{eqnarray}&&\gamma G(r)=({L}_{\mathrm{Gauss}}+\delta L)G(r).\end{eqnarray} \tag{ 29 }$$

The important feature is that Equations (28) and (29) are third-order differential equations with a small multiplier at the elder derivative. This results in the appearance of a nonphysical solution that does not coincide with the solutions of Equation (10) as F → 0; instead, it goes to infinity. This solution must not be taken into consideration (see also more details in Appendix C). Technically, this nonphysical solution is a consequence of the truncated sequence of correlators; if one takes higher-order correlators of A into consideration, the number of solutions of the Kazantsev equation would increase in accordance with the order of the highest correlator. The nonphysical solutions would grow unrestrictedly as the magnitudes of the higher-order correlators tend to zero. These solutions have to be excluded by an accurate choice of the boundary conditions.

4.1. Batchelor Regime

In the Batchelor approximation, velocity gradients are assumed to be constant in space (although dependent on time). Accordingly, the second derivatives of D_ij and third derivatives of F_ijk are assumed to be constant over the liquid volume. Then, in accordance with Equations (7) and (8),

$$\begin{eqnarray}&&\sigma (r)=\displaystyle \frac{D}{3}{r}^{2}.\end{eqnarray} \tag{ 30 }$$

The exact expression for ${F}_{\mathrm{ij}k}({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} ,{\boldsymbol{r}}-{\boldsymbol{r}}^{\prime\prime} )$ in this case is presented in Appendix B. Substituting this in Equation (27), we get

$$\begin{eqnarray}&&{L}_{\mathrm{Gauss}}^{B}=\displaystyle \frac{2}{3}D\left({r}^{2}\displaystyle \frac{{\partial }^{2}}{\partial {r}^{2}}+6r\displaystyle \frac{\partial }{\partial r}+10\right)+2\varkappa \,\left(\displaystyle \frac{{\partial }^{2}}{\partial {r}^{2}}+\displaystyle \frac{4}{r}\displaystyle \frac{\partial }{\partial r}\right)\end{eqnarray} \tag{ 31 }$$

and the additional term in Equations (28) and (29) (see the details of the derivation in Appendix B):

$$\begin{eqnarray}&&\delta {L}^{B}=\displaystyle \frac{1}{9}F\left(2{r}^{3}\displaystyle \frac{{\partial }^{3}}{\partial {r}^{3}}+21{r}^{2}\displaystyle \frac{{\partial }^{2}}{\partial {r}^{2}}+14r\displaystyle \frac{\partial }{\partial r}-70\right).\end{eqnarray} \tag{ 32 }$$

The analytic analysis of Equations (28), (30), and (32) is performed in Appendix C. We show that the fastest-growing mode corresponds to

$$\begin{eqnarray}&&\gamma =\displaystyle \frac{5}{2}D\left(1-\displaystyle \frac{243}{80}{\left(\displaystyle \frac{F}{D}\right)}^{2}+o{\left(\displaystyle \frac{F}{D}\right)}^{2}\right).\end{eqnarray} \tag{ 33 }$$

This coincides with the exponents (Equation (22)) found in the previous section. The important consequence of this expression is that, independently of the sign of F, the resulting γ is smaller than that found in the Kazantsev–Kraichnan model. This means that the magnetic field generation is weaker in time-asymmetric flows than in the flow with Gaussian velocity gradients, independently of the direction of the energy cascade. Also, γ is the monotonic function of F²; the time asymmetry of the flow decreases the generation.

The coincidence of the results obtained by means of the Lagrangian deformation method and the modified Kazantsev approach is not trivial or evident, and we will consider it in more detail in Section 6. Anyway, it proves that both methods work well as long as the Batchelor approximation is valid. However, the Kazantsev approach allows one to investigate the later stages of magnetic field evolution, when the characteristic scales of the magnetic line lengths exceed the viscous scale, and the spatial inhomogeneity of the velocity gradients cannot be neglected.

4.2. Nonlinear Velocity Field

To take this inhomogeneity into account, one should consider scales comparable to or larger than the viscous scale; outside this scale, the correlators change essentially. In Equation (11), this is expressed by means of three different ranges. We also have to cut off the third-order correlator. Since it is known (Novikov et al. 1983; Kulsrud & Anderson 1992) that (in the case of large Pr_m ) the details of the outer ranges do not significantly affect the result, we simplify Equation (11) to

$$\begin{eqnarray}\sigma (r)=\left\{\begin{array}{ll}\displaystyle \frac{D}{3}{r}^{2}, & r\leqslant {r}_{\nu },\\ \displaystyle \frac{D}{3}{r}_{\nu }^{2}, & r\gt {r}_{\nu }.\end{array}\right.\end{eqnarray} \tag{ 34 }$$

In the V³ model, we cut F_ijk at the same boundary, r = r_ν . The first term in Equation (29) then takes the form

$$\begin{eqnarray}{\tilde{L}}_{\mathrm{Gauss}}=\left\{\begin{array}{ll}{L}_{\mathrm{Gauss}}^{B},\quad r\leqslant {r}_{\nu }, & \\ \left(\displaystyle \frac{2}{3}{{Dr}}_{\nu }^{2}+2\varkappa \right)\left(\displaystyle \frac{{\partial }^{2}}{\partial {r}^{2}}+\displaystyle \frac{4}{r}\displaystyle \frac{\partial }{\partial r}\right), & r\gt {r}_{\nu },\end{array}\right.\end{eqnarray} \tag{ 35 }$$

and the second term is

$$\begin{eqnarray}&&\tilde{\delta L}=\left\{\begin{array}{l}\delta {L}^{B},r\leqslant {r}_{\nu },\\ 0,r\gt {r}_{\nu }.\end{array}\right.\end{eqnarray} \tag{ 36 }$$

Since we consider large magnetic Prandtl numbers, the viscosity is large compared to the magnetic diffusivity, and

$$\begin{eqnarray}&&{r}_{d}^{2}=\displaystyle \frac{\varkappa }{D}\ll {r}_{\nu }^{2}.\end{eqnarray} \tag{ 37 }$$

So, in addition to the small parameter F/D in the Batchelor problem statement, here we have one more small parameter:

$$\begin{eqnarray*}&&1/\mu ={r}_{d}/{r}_{\nu }.\end{eqnarray*}$$

Equation (29) with Equations (34) and (36) can be solved in special functions. However, we make an analytic estimate for the contribution of large but finite μ to the magnetic increment:

$$\begin{eqnarray}&&{\gamma }_{\max }\gtrsim D\left(\displaystyle \frac{5}{2}-\displaystyle \frac{243{F}^{2}}{32{D}^{2}}-\displaystyle \frac{2\,{\pi }^{2}}{3{\mathrm{ln}}^{2}\mu }\left(1+O({F}^{2}/{D}^{2})\right)\right)\end{eqnarray} \tag{ 38 }$$

(see Appendix C for the derivation). This estimate shows that, in the first approximation, the non-Gaussianity and finiteness of the magnetic Prandtl number (μ < ∞) act independently.

In the next section, we will consider the numerical solutions to Equation (29) for finite μ, investigate the dependence of magnetic field generation on the parameters F and μ, and check the reliability of the model.

The generalized Kazantsev Equation (28) is a third-order differential equation with a small multiplier F/D at the elder derivative. It is not evident whether its solutions are stable and converge to the solutions of Equation (10) in the limit F/D → 0. We also test our theoretical conclusion (Equation (38)) and show that the behavior of the increment does not depend on the details of the cutoff at large r.

5.1. Technical Details

We consider Equation (29) with σ and δ L determined by Equations (34) and (36). In dimensionless notations,

$$\begin{eqnarray*}&&x=r/{r}_{d},\ {\rm{\Gamma }}=\gamma /D,\ f=F/D,\ \mu ={r}_{\nu }/{r}_{d},\end{eqnarray*}$$

we get the equation for G(x):

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Gamma }}G & = & \displaystyle \frac{2}{3}\left({x}^{2}\theta (\mu -x)+{\mu }^{2}\theta (x-\mu )+3\right)G^{\prime\prime} \\ & & +\,\displaystyle \frac{4}{3}\left(3x\theta (\mu -x)+2\displaystyle \frac{{\mu }^{2}}{x}\theta (x-\mu )+\displaystyle \frac{6}{x}\right)G^{\prime} \\ & & +\,\displaystyle \frac{20}{3}\theta (\mu -x)G\\ & & +\,\displaystyle \frac{1}{9}f\left(2{x}^{3}G\prime\prime\prime +21{x}^{2}G^{\prime\prime} +14{xG}^{\prime} -70\right)\theta (\mu -x),\end{array}\end{eqnarray} \tag{ 39 }$$

where θ(y) denotes the Heaviside function.

The solution G(x) depends on two parameters, f and μ. The asymptotes of the solutions can be found analytically.

In the limit x → 0, Equation (39) has three modes:

$$\begin{eqnarray}&&{G}_{(1)}=1-{{Yx}}^{2},\qquad Y=\displaystyle \frac{1}{3}-\displaystyle \frac{7f}{18}-\displaystyle \frac{{\rm{\Gamma }}}{20},\end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray}&&{G}_{(2)}(x)\sim {x}^{-3},\quad {G}_{(3)}(x)\sim {x}^{6}\exp \left(\displaystyle \frac{9}{2{{fx}}^{2}}\right).\end{eqnarray} \tag{ 41 }$$

The first two are close to the corresponding solutions for the Kazantsev–Kraichnan model; the third is "produced" by the third-order term. The last two modes diverge as x → 0, so they must be excluded from the physical solution.

The other asymptote is x ≫ μ. The equation is significantly simplified in this limit because σ(r) becomes a constant:

$$\begin{eqnarray}&&{p}^{2}G=G^{\prime\prime} +\displaystyle \frac{4G^{\prime} }{x},\quad {p}^{2}=\displaystyle \frac{{\rm{\Gamma }}}{2(1+\tfrac{1}{3}{\mu }^{2})}.\end{eqnarray} \tag{ 42 }$$

The exact solution to this equation is

$$\begin{eqnarray}&&G(x)={Y}_{1}\displaystyle \frac{{e}^{-{px}}}{x}\left(1+\displaystyle \frac{1}{{px}}\right)+{Y}_{2}\displaystyle \frac{{e}^{{px}}}{x}\left(1-\displaystyle \frac{1}{{px}}\right).\end{eqnarray} \tag{ 43 }$$

Again, the divergency condition G(x) → _x→∞0 requires Y₂ = 0 and leaves only one of the two modes.

To solve Equation (39) numerically, we fix some Γ and the initial point x₁ ≪ 1. The initial conditions G(x₁), $G^{\prime} ({x}_{1})$, and G''(x₁) are determined by the asymptote (Equation (40)). Then we get the numerical solution G(x) up to x = μ and match it with the asymptote (Equation (43)); the condition Y₂ = 0 singles out a discrete spectrum of possible values of Γ. We are looking for the maximal value ${\rm{\Gamma }}={{\rm{\Gamma }}}_{\max }$. We also check the stability of the solution relative to the choice of x₁.

5.2. Results

As it follows from Equation (38), the theoretical prediction for γ(f, μ) is

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Gamma }} & = & \displaystyle \frac{\gamma }{D}\simeq \left(\displaystyle \frac{5}{2}-{c}_{2}{f}^{2}-\displaystyle \frac{{c}_{3}+O({f}^{2})}{{\mathrm{ln}}^{2}\mu }\right),\\ {c}_{2} & = & \displaystyle \frac{243}{32}\simeq 7.59,{c}_{3}\lesssim \displaystyle \frac{2{\pi }^{2}}{3}\simeq 6.6.\end{array}\end{eqnarray} \tag{ 44 }$$

First, we analyze the dependence Γ(f) for some fixed μ. The results for three values of μ are presented in Figure 2. We see that, in accordance with Equation (44), Γ(f) has a parabolic shape at small f for all considered values of μ. To observe the dependence of the second-order term on μ, we fit the results of the simulation by parabolic functions in the range 0 ≤ f ≤ 0.1; at larger f, the higher-order terms in Equation (44) may come into play. The results are presented in Table 1. One can see that the absolute term is smaller than 2.5 and increases as a function of μ. The magnitude of the decl. agrees with Equation (44). The absolute value of the coefficient at f² decreases as a function of μ, approaching the theoretical prediction c₂ for μ = ∞.

Zoom In Zoom Out Reset image size

Table 1. Fit of the Numerical Data (Figure 2) for 0 < f < 0.1: Γ(f) = C₁ − C₂ f²

μ	C1	C2
30	1.884 ± 0.003	11.5 ± 0.5
100	2.159 ± 0.001	9.5 ± 0.2
300	2.277 ± 0.001	8.7 ± 0.3
∞ (theory)	2.5	7.59

Download table as:  ASCIITypeset image

Second, we calculate the function Γ(μ) at some given f. We take f = 0.13 because it corresponds to the asymmetry of a real flow (Equation (20)) observed in the numerical calculation (Girimaji & Pope 1990) and experiment (Luthi et al. 2005). We also calculate Γ(μ) for a symmetric flow, f = 0; in this case, Equation (39) becomes a second-order equation. The results are presented in Figure 3. We fit the graphs by the ansatz

$$\begin{eqnarray}&&{\rm{\Gamma }}(\mu )={C}_{1}-\displaystyle \frac{{C}_{3}}{{\mathrm{ln}}^{2}\mu }.\end{eqnarray} \tag{ 45 }$$

The correspondence is good enough; for f = 0, we get C₁ = 2.50 ± 0.02 and C₃ = 6.64 ± 0.33, which coincides with the theoretical prediction c₁ and c₃ in Equation (44).

Zoom In Zoom Out Reset image size

The choice of the σ(r) profile at r ≃ r_ν is rather conditional; the details of the velocity structure function at this range of scales are not believed to significantly affect the result. This has been checked numerically (Novikov et al. 1983) for the Kazantsev–Kraichnan model, but it also has to be proved for F ≠ 0. So, apart from Equation (34), we consider a smooth function,

$$\begin{eqnarray}&&\tilde{\sigma }(r)=\displaystyle \frac{D}{3}\times \displaystyle \frac{{\left(r/{r}_{d}\right)}^{2}}{1+{\left(r/{r}_{\nu }\right)}^{2}}.\end{eqnarray} \tag{ 46 }$$

At r → ∞ and r → 0, it has the same asymptotes as σ(r). We perform the same calculations with this function. In Figure 4, the dependence Γ(μ) for f = 0.13 is presented for both choices of σ. We see that the details of the saturation do not crucially affect the increment behavior.

Zoom In Zoom Out Reset image size

Finally, from Equation (44), it follows that γ(f, μ) becomes negative outside some region in the f, μ plane. So, the range of the parameters at which the generation of the magnetic field is possible is restricted by some μ > μ_crit(f). The numerical calculations confirm this prediction. We also find the dependence μ_crit(f); it is presented in Figure 5. One can see that the presence of a non-Gaussian term decreases the range of μ that permits the generation.

Zoom In Zoom Out Reset image size

In this paper, we consider the magnetic field generation in a turbulent hydrodynamic flow. We derive the generalization of the Kazantsev equation for the case of time-asymmetric flows with a small but finite third-order velocity correlator, which is probably the general case of real hydrodynamic turbulent flows. The nonzero third-order correlator corresponds to the time asymmetry of the flow and is responsible for the energy cascade.

We use the V³ model. A slightly non-Gaussian stochastic velocity field is replaced by an effective δ-process, and the connected correlators of velocity of order higher than 3 are assumed to be zero. The validity of this model is argued in Section 3 and Appendix A. We also use one more (unessential) simplifying assumption on the shape of the velocity structure function, which is considered piecewise.

The main results are as follows.

• 1.  
  We show that the magnetic field generation weakens in the presence of time asymmetry, independently of the direction of the energy cascade (Equation (38)). The increment of the average magnetic energy density decreases proportionally to the square of the magnitude F of the third-order correlator.
• 2.  
  The range of magnetic Prandtl numbers (presented in the considered model by the parameter μ) that allow the magnetic field generation also becomes narrower for finite F; the critical Prandtl number increases as a function of F (Figure 5).
• 3.  
  We show that, despite the presence of a small higher-order derivative, the numerical solution of the generalized Kazantsev equation converges in the limit F → 0 to the solution of the Kazantsev equation. This proves that the generalized Kazantsev equation allows one to numerically investigate the magnetic field evolution in time-asymmetric flows with intermediate magnetic Prandtl numbers.

Now we proceed to the discussion of some particularly interesting consequences.

We solve the generalized Kazantsev equation analytically in the limit of the Batchelor regime, r_ν = ∞. In this limit, the resulting increment (Equation (33)) coincides with the result that follows from the Lagrangian deformation method (Equation (22)). This coincidence not only proves the reliability of both methods, it also reflects a nontrivial physical fact: the time-averaged magnetic energy density measured at some fixed point coincides with that measured along a liquid particle trajectory. Not only do the methods of calculation differ in the two approaches, but the averages 〈B²〉 are taken over different ensembles. So, the coincidence not only verifies the results but also proves the equivalence of these two averages. This coincidence is not a consequence of ergodicity, since the trajectory of a particle and the magnetic energy are functionals of the velocity field and thus not independent.

The equivalence of the two averages was found for passive scalar (Balkovsky & Fouxon 1999) and vector (Chertkov et al. 1999) advection in the case of the Kazantsev–Kraichnan velocity field. Now we see that it also holds for vector advection in time-asymmetric velocity fields.

Another important coincidence establishes the relation between the magnetic increments calculated for the Batchelor limit and the case of a finite Prandtl number. Namely, the theoretical analysis and numerical simulation confirm that, as μ → ∞, the increment γ(μ, F) converges to the value γ(F) found for the Batchelor regime. This is also not trivial; for instance, this equality does not hold for two-dimensional flows (Schekochihin et al. 2002a; Kolokolov 2017) or higher-order correlators in three-dimensional flows (Zybin et al. 2020). The Kazantsev equation corresponds to the infinite-time limit; the convergence of the solutions in the limit Pr_m → ∞ (and its coincidence with the value obtained for the Batchelor case) means that the limits Pr_m → ∞ and t → ∞ commutate. This has also been known for Gaussian flows (Kazantsev 1968; Novikov et al. 1983; Kulsrud & Anderson 1992) and now is also checked for the time-asymmetric case.

Eventually, the deformation of the magnetic energy spectrum is also an interesting and important question (Kazantsev 1968; Kulsrud & Anderson 1992; Schekochihin et al. 2002a; Bhat & Subramanian 2014; Aiyer et al. 2017). In the frame of the V³ model, it is also possible to find the spectrum evolution; however, this problem is rather complicated and deserves a separate investigation. We point the reader to Kopyev et al.(2021).

Summarizing, we stress that the asymmetry of the hydrodynamic flow statistics is an essential feature that may affect the process of magnetic field generation. The V³ model allows one to investigate the asymmetric flows and thus opens the door for investigation of transport problems in the flows with energy cascades.

The authors are grateful to Professor A. V. Gurevich for his permanent attention to their work. A.M.K. thanks Professor Ya. N. Istomin for considerable contribution to the work in its early stage. The work of A.V.K. and A.M.K. was supported by RSF grant No. 20-12-00047.

Equation (12) is a stochastic differential equation; the random velocity gradient tensor A acts as multiplicative noise (Batchelor 1959).

One can show that for any non-Gaussian process with a finite correlation time, there exists some corresponding effective δ-process (Il'yn et al. 2016). This means that

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{T\to \infty }\displaystyle \frac{1}{T}\mathrm{log}\langle {B}^{2}(T)\rangle =\mathop{\mathrm{lim}}\limits_{T\to \infty }\displaystyle \frac{1}{T}\mathrm{log}\langle {B}^{2}(T){\rangle }_{\mathrm{eff}},\end{eqnarray} \tag{ A1 }$$

where the average on the right-hand side is calculated for velocity statistics defined by the effective δ-process. The reason to replace an arbitrary finite correlation time process with the corresponding δ-process is that in the equation with multiplicative noise, the higher-order connected correlators of the noise contribute to the long-time statistical properties of the solutions only via their integrals. This allows one to substitute singular correlation functions for the real correlators.

We demonstrate this fact with a simple example. Consider a one-dimensional stochastic equation with multiplicative noise ξ(t),

$$\begin{eqnarray}&&{\partial }_{t}x(t)=\xi (t)x(t),\ x(0)=0,\end{eqnarray} \tag{ A2 }$$

where ξ(t) is a continuous stationary random process with a finite correlation time. Let it have regular fast-decaying connected correlation functions (cumulants):

$$\begin{eqnarray*}&&\langle \xi ({t}_{1})...\xi ({t}_{n}){\rangle }_{c}={W}^{(n)}({t}_{1}-{t}_{2},...,{t}_{1}-{t}_{n}).\end{eqnarray*}$$

The cumulant generating functional is defined by

$$\begin{eqnarray}&&\langle {e}^{\int \xi (t)\eta (t){dt}}\rangle ={e}^{W\left[\eta (t)\right]},\end{eqnarray} \tag{ A3 }$$

then

$$\begin{eqnarray}&&\begin{array}{l}W[\eta (t)]=\displaystyle \sum _{n}\displaystyle \frac{1}{n!}\int {W}^{(n)}({t}_{1}-{t}_{2},...,{t}_{1}-{t}_{n})\\ \qquad \times \,\eta ({t}_{1})...\eta ({t}_{n}){{dt}}_{1}...{{dt}}_{n}.\end{array}\end{eqnarray} \tag{ A4 }$$

The solution of Equation (A2) for each continuous realization of ξ(t) can be written as

$$\begin{eqnarray}&&x(T)={e}^{\underset{0}{\overset{T}{\int }}\xi (t){dt}}.\end{eqnarray} \tag{ A5 }$$

We are interested in the statistical moments

$$\begin{eqnarray*}&&\langle {x}^{m}(T)\rangle =\left\langle {e}^{m\underset{0}{\overset{T}{\int }}\xi (t){dt}}\right\rangle .\end{eqnarray*}$$

From Equation (A3), it then follows that

$$\begin{eqnarray*}&&\langle {x}^{m}(T)\rangle ={e}^{W[m\theta (t)\theta (T-t)]},\end{eqnarray*}$$

where θ is the Heaviside steplike function. According to Equation (A4), we then have

$$\begin{eqnarray*}&&\mathop{\mathrm{lim}}\limits_{T\to \infty }\displaystyle \frac{1}{T}\mathrm{log}\langle {x}^{m}(T)\rangle =w(m),\end{eqnarray*}$$

$$\begin{eqnarray*}&&w(m)=\sum _{n}\displaystyle \frac{{m}^{n}}{n!}{w}^{(n)},\end{eqnarray*}$$

$$\begin{eqnarray*}&&{w}^{(n)}=\int {W}^{(n)}({\tau }_{2},...,{\tau }_{n})d{\tau }_{2}...d{\tau }_{n}.\end{eqnarray*}$$

We see that as T → ∞, the statistical moments of x depend only on the integrals w⁽ⁿ⁾ and not on the detailed shape of the correlators. So, for any random process ξ(t) (i.e., for any given W⁽ⁿ⁾), we can consider a series of random processes ξ (t) with

$$\begin{eqnarray*}&&{W}_{\epsilon }^{(n)}=\displaystyle \frac{1}{{\epsilon }^{n-1}}{W}^{(n)}({\tau }_{2}/\epsilon ,...,{\tau }_{n}/\epsilon ),\end{eqnarray*}$$

and all of these processes would produce the same long-term asymptotes for the moments of x(t). The effective process ξ_eff(t) is defined as the formal limit of this consequence as → 0. Its connected correlation functions are

$$\begin{eqnarray*}&&\langle {\xi }_{\mathrm{eff}}({t}_{1})...{\xi }_{\mathrm{eff}}({t}_{n}){\rangle }_{c}={w}^{(n)}\delta ({t}_{1}-{t}_{2})...\delta ({t}_{1}-{t}_{n}).\end{eqnarray*}$$

In the case of multidimensional stochastic processes and stochastic fields, the expression for the exact solution looks much more complicated than Equation (A5), since the integral transforms into a continuous matrix product (T-exponent). However, the long-time asymptotes of the solutions' moments still depend only on the integrals of the connected correlators of the noise. This is the reason and justification for substitution of the effective δ-process for any random process. This tool is convenient in turbulence and turbulent transport problems, since it allows one to get closed equations for statistical moments.

We start with Equation (1); to get the equation for the pair correlator, we multiply it by B and take the average. For brevity, we denote

$$\begin{eqnarray*}\begin{array}{rcl}{B}_{\alpha } & = & {B}_{\alpha }({\boldsymbol{r}},t),\quad {B}_{\alpha }^{{\prime} }={B}_{\alpha }({\boldsymbol{r}}^{\prime} ,t),\\ {v}_{\alpha } & = & {v}_{\alpha }({\boldsymbol{r}},t),\quad {v}_{\alpha }^{{\prime} }={v}_{\alpha }({\boldsymbol{r}}^{\prime} ,t),\\ {\partial }_{\alpha } & = & \displaystyle \frac{\partial }{\partial {r}_{\alpha }},\quad {\partial }_{\alpha }^{{\prime} }=\displaystyle \frac{\partial }{\partial {r}_{\alpha }^{{\prime} }},\quad {\partial }_{\alpha }^{\rho }=\displaystyle \frac{\partial }{\partial {\left({\boldsymbol{r}}^{\prime} -{\boldsymbol{r}}\right)}_{\alpha }}.\end{array}\end{eqnarray*}$$

We also take into account the homogeneity of the flow, which results in ${\partial }_{m}\langle ...\rangle =-{\partial }_{m}^{{\prime} }\langle ...\rangle =-{\partial }_{m}^{\rho }\langle ...\rangle$ and $\langle {v}_{p}^{{\prime} }{B}_{q}^{{\prime} }{B}_{\alpha }\rangle \,=-\langle {v}_{p}{B}_{q}{B}_{\alpha }^{{\prime} }\rangle$. Then the equation for the pair correlator takes the form

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial }{\partial t}\langle {B}_{\alpha }{B}_{\beta }^{{\prime} }\rangle & = & -{\varepsilon }_{\alpha \mathrm{mn}}{\varepsilon }_{\mathrm{npq}}{\partial }_{m}^{\rho }\langle {v}_{p}{B}_{q}{B}_{\beta }^{{\prime} }\rangle \\ & & -\,{\varepsilon }_{\beta \mathrm{mn}}{\varepsilon }_{\mathrm{npq}}{\partial }_{m}^{\rho }\langle {v}_{p}{B}_{q}{B}_{\alpha }^{{\prime} }\rangle +2\varkappa {\partial }_{m}^{\rho }{\partial }_{m}^{\rho }\langle {B}_{\alpha }{B}_{\beta }^{{\prime} }\rangle .\end{array}\end{eqnarray} \tag{ B1 }$$

Now we have to split the mixed correlations by means of the generalized Furutsu–Novikov equation (Furutsu 1963, Klyatskin 2005):

$$\begin{eqnarray}&&\begin{array}{l}\langle {v}_{p}{B}_{q}{B}_{r}^{{\prime} }\rangle =\displaystyle \int \,d{{\boldsymbol{r}}}_{1}{{dt}}_{1}\langle {v}_{p}({\boldsymbol{r}},t){v}_{{i}_{1}}({{\boldsymbol{r}}}_{1},{t}_{1})\rangle \,\\ \ \times \left\langle \displaystyle \frac{\delta \left({B}_{q}({\boldsymbol{r}},t){B}_{r}({\boldsymbol{r}}^{\prime} ,t)\right)}{\delta {v}_{{i}_{1}}({{\boldsymbol{r}}}_{1},{t}_{1})}\right\rangle \\ \ +\,\displaystyle \frac{1}{2}\displaystyle \int \,d{{\boldsymbol{r}}}_{1}{{dt}}_{1}d{{\boldsymbol{r}}}_{2}{{dt}}_{2}\langle {v}_{p}({\boldsymbol{r}},t){v}_{{i}_{1}}({{\boldsymbol{r}}}_{1},{t}_{1}){v}_{{i}_{2}}({{\boldsymbol{r}}}_{2},{t}_{2})\rangle \,\\ \ \times \left\langle \displaystyle \frac{{\delta }^{2}\left({B}_{q}({\boldsymbol{r}},t){B}_{r}({\boldsymbol{r}}^{\prime} ,t)\right)}{\delta {v}_{{i}_{1}}({{\boldsymbol{r}}}_{1},{t}_{1})\delta {v}_{{i}_{2}}({{\boldsymbol{r}}}_{2},{t}_{2})}\right\rangle .\end{array}\end{eqnarray} \tag{ B2 }$$

Since the variational derivatives contain δ-functions, to avoid the products of δ-functions with coinciding arguments, we have to deal more accurately with the time coincidence in the correlators. Namely, we have to introduce a "regularized δ-function": a bell-shaped function δ (τ) satisfying the normalization condition ∫δ (τ)d τ = 1 and with a width of the order of the correlation time. Then Equations (23) and (24) can be written more precisely:

$$\begin{eqnarray}&&\langle {v}_{i}({\boldsymbol{r}},t){v}_{j}({{\boldsymbol{r}}}_{1},{t}_{1})\rangle ={D}_{\mathrm{ij}}({\boldsymbol{r}}-{{\boldsymbol{r}}}_{1}){\delta }_{\varepsilon }(t-{t}_{1})\end{eqnarray} \tag{ B3 }$$

and

$$\begin{eqnarray}&&\begin{array}{l}\langle {v}_{i}({\boldsymbol{r}},t){v}_{j}({{\boldsymbol{r}}}_{1},{t}_{1}){v}_{k}({{\boldsymbol{r}}}_{2},{t}_{2})\rangle =\displaystyle \frac{1}{3}{F}_{\mathrm{ij}k}({\boldsymbol{r}}-{{\boldsymbol{r}}}_{1},{\boldsymbol{r}}-{{\boldsymbol{r}}}_{2})\\ \quad \times \,\left({\delta }_{\varepsilon }(t-{t}_{1}){\delta }_{\varepsilon }(t-{t}_{2})+{\delta }_{\varepsilon }(t-{t}_{1}){\delta }_{\varepsilon }({t}_{1}-{t}_{2})\right.\\ \quad \left.+\,{\delta }_{\varepsilon }(t-{t}_{2}){\delta }_{\varepsilon }({t}_{1}-{t}_{2}\right).\end{array}\end{eqnarray} \tag{ B4 }$$

The isotropy and homogeneity conditions imply that the expression for F_ijk can be written as a composition of δ symbols and the components of a, b. The nondivergency condition results in the requirement ∂F_ijk /∂a_i = ∂F_ijk /∂b_j = (∂/∂a_k +∂/∂b_k )F_ijk = 0; also, the existence of viscosity means that F_ijk is proportional to r³, i.e., a third-order polynomial. These conditions determine the tensor F_ijk to a constant multiplier:

$$\begin{eqnarray}&&\begin{array}{l}{F}_{\mathrm{ij}k}({\boldsymbol{a}},{\boldsymbol{b}})=-\displaystyle \frac{1}{18}F\left(2{a}^{2}-15{b}^{2}-2\,{\boldsymbol{a}}\cdot {\boldsymbol{b}}){a}_{j}{\delta }_{\mathrm{ik}}\right.\\ \quad +\,(2{b}^{2}-15{a}^{2}-2\,{\boldsymbol{a}}\cdot {\boldsymbol{b}}){b}_{k}{\delta }_{\mathrm{ij}}\\ \quad +\,(20{b}^{2}-5{a}^{2}-16\,{\boldsymbol{a}}\cdot {\boldsymbol{b}}){a}_{i}{\delta }_{{jk}}\\ \quad +\,(20{a}^{2}-5{b}^{2}-16\,{\boldsymbol{a}}\cdot {\boldsymbol{b}}){b}_{i}{\delta }_{{jk}}\\ \quad -\,(5{a}^{2}+{b}^{2}-26\,{\boldsymbol{a}}\cdot {\boldsymbol{b}}){a}_{k}{\delta }_{\mathrm{ij}}\\ \quad -\,(5{b}^{2}+{a}^{2}-26\,{\boldsymbol{a}}\cdot {\boldsymbol{b}}){b}_{j}{\delta }_{\mathrm{ik}}\\ \quad +\,4\,{a}_{i}\,{a}_{j}\,{a}_{k}+4\,{b}_{i}\,{b}_{j}\,{b}_{k}+12\,{a}_{i}\,{a}_{j}\,{b}_{k}\\ \quad +\,12\,{b}_{i}\,{a}_{j}\,{b}_{k}-2\,{a}_{i}\,{b}_{j}\,{a}_{k}-2\,{b}_{i}\,{b}_{j}\,{a}_{k}\\ \quad -\left.\,16\,{a}_{i}\,{b}_{j}\,{b}_{k}-16\,{b}_{i}\,{a}_{j}\,{a}_{k}\right).\end{array}\end{eqnarray} \tag{ B5 }$$

Analogous calculations for the inertial range were performed by Kopyev & Zybin (2018).

B.1. Taking the Variational Derivative

To take the variational derivative, we make a formal functional consequence from Equation (1):

$$\begin{eqnarray}&&\begin{array}{l}{B}_{q}({\boldsymbol{r}},t)={\varepsilon }_{{{qi}}_{1}{j}_{1}}{\varepsilon }_{{j}_{1}{l}_{1}{k}_{1}}\displaystyle \underset{-\infty }{\overset{t}{\displaystyle \int }}{\partial }_{{i}_{1}}({v}_{{l}_{1}}({\boldsymbol{r}},\tau )\\ \ \times \,{B}_{{k}_{1}}({\boldsymbol{r}},\tau ))d\tau +\varkappa \,{\partial }_{{i}_{1}}{\partial }_{{i}_{1}}\displaystyle \underset{-\infty }{\overset{t}{\displaystyle \int }}{B}_{q}({\boldsymbol{r}},\tau )d\tau .\end{array}\end{eqnarray} \tag{ B6 }$$

The variational derivative of this expression is

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\delta {B}_{q}({\boldsymbol{r}},t)}{\delta {v}_{j}({{\boldsymbol{r}}}_{1},{t}_{1})}={\varepsilon }_{{{qi}}_{1}{j}_{1}}{\varepsilon }_{{j}_{1}{l}_{1}{k}_{1}}\\ \ \times \,\displaystyle \underset{-\infty }{\overset{t}{\displaystyle \int }}{\partial }_{{i}_{1}}\left({\delta }_{{{jl}}_{1}}\delta ({\boldsymbol{r}}-{{\boldsymbol{r}}}_{1})\delta (\tau -{t}_{1}){B}_{{k}_{1}}({\boldsymbol{r}},\tau )\right)d\tau \\ \ +\,{\varepsilon }_{{{qi}}_{1}{j}_{1}}{\varepsilon }_{{j}_{1}{l}_{1}{k}_{1}}\displaystyle \underset{{t}_{1}}{\overset{t}{\displaystyle \int }}{\partial }_{{i}_{1}}\left({v}_{{l}_{1}}({\boldsymbol{r}},\tau )\displaystyle \frac{\delta {B}_{{k}_{1}}({\boldsymbol{r}},\tau )}{\delta {v}_{j}({{\boldsymbol{r}}}_{1},{t}_{1})}\right)d\tau \\ \ +\,\varkappa {\partial }_{{i}_{1}}{\partial }_{{i}_{1}}\displaystyle \underset{{t}_{1}}{\overset{t}{\displaystyle \int }}\displaystyle \frac{\delta {B}_{q}({\boldsymbol{r}},\tau )}{\delta {v}_{j}({{\boldsymbol{r}}}_{1},{t}_{1})}d\tau .\end{array}\end{eqnarray} \tag{ B7 }$$

The δ-functions here are "real," not regularized. The lower limits of the integrals in the last two summands are changed in accordance with the causality principle; after multiplying by δ in the velocity correlators in Equation (B2), these lower limits will make no difference. Taking the second derivative, we get

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{\delta }^{2}{B}_{q}({\boldsymbol{r}},t)}{\delta {v}_{j}({{\boldsymbol{r}}}_{1},{t}_{1})\delta {v}_{k}({{\boldsymbol{r}}}_{2},{t}_{2})}\mathop{\simeq }\limits^{{t}_{1},{t}_{2}\to t-0}{\varepsilon }_{{{qi}}_{1}{j}_{1}}{\varepsilon }_{{j}_{1}{{jk}}_{1}}\\ \quad \times \,\displaystyle \underset{-\infty }{\overset{t}{\displaystyle \int }}d{\tau }_{1}\,{\partial }_{{i}_{1}}\left(\delta ({\boldsymbol{r}}-{{\boldsymbol{r}}}_{1})\delta (\tau -{t}_{1})\displaystyle \frac{\delta {B}_{{k}_{1}}({\boldsymbol{r}},t)}{\delta {v}_{k}({{\boldsymbol{r}}}_{2},{t}_{2})}\right)\\ \ =\,{\varepsilon }_{{{qi}}_{1}{j}_{1}}{\varepsilon }_{{j}_{1}{{jk}}_{1}}{\varepsilon }_{{k}_{1}{i}_{2}{j}_{2}}{\varepsilon }_{{j}_{2}{{kk}}_{2}}\,\theta (t-{t}_{1})\theta (t-{t}_{2})\\ \quad \times \,{\partial }_{{i}_{1}}\left(\delta ({\boldsymbol{r}}-{{\boldsymbol{r}}}_{1}){\partial }_{{i}_{2}}(\delta ({\boldsymbol{r}}-{{\boldsymbol{r}}}_{2}){B}_{{k}_{2}})\right).\end{array}\end{eqnarray} \tag{ B8 }$$

Here we omit the terms that become zero being multiplied by the regularized δ-function. Now we substitute these expressions into Equation (B2). For the regularized delta functions, we have

$$\begin{eqnarray}&&\begin{array}{l}\underset{-\infty }{\overset{+\infty }{\displaystyle \int }}{{dt}}_{1}\,\underset{-\infty }{\overset{t}{\displaystyle \int }}d\tau \,{\delta }_{\varepsilon }(t-{t}_{1})\delta (\tau -{t}_{1})\\ \ =\,\underset{-\infty }{\overset{+\infty }{\displaystyle \int }}{{dt}}_{1}\,{\delta }_{\varepsilon }(t-{t}_{1})\theta (t-{t}_{1})=\underset{-\infty }{\overset{t}{\displaystyle \int }}{{dt}}_{1}\,{\delta }_{\varepsilon }(t-{t}_{1})=\displaystyle \frac{1}{2},\end{array}\end{eqnarray} \tag{ B9 }$$

$$\begin{eqnarray}&&\begin{array}{l}\underset{-\infty }{\overset{t}{\displaystyle \int }}{{dt}}_{1}\,\underset{-\infty }{\overset{t}{\displaystyle \int }}{{dt}}_{2}\,{\delta }_{\varepsilon }(t-{t}_{1})\,{\delta }_{\varepsilon }(t-{t}_{2})\\ \ =\,\displaystyle \frac{1}{4}\underset{-\infty }{\overset{t}{\displaystyle \int }}{{dt}}_{1}\,\underset{-\infty }{\overset{t}{\displaystyle \int }}{{dt}}_{2}\,{\delta }_{\varepsilon }(t-{t}_{1})\,{\delta }_{\varepsilon }({t}_{1}-{t}_{2})\\ \ =\,\underset{0}{\overset{+\infty }{\displaystyle \int }}{dx}\,\underset{{t}_{1}-t}{\overset{+\infty }{\displaystyle \int }}{dy}\,{\delta }_{\varepsilon }(x)\,{\delta }_{\varepsilon }(y)=\displaystyle \frac{3}{8}.\end{array}\end{eqnarray} \tag{ B10 }$$

These relations allow one to take the time integrals in Equation (B2). Thus, we arrive at Equation (27). Taking the space integrals, we get the closed equation for the pair correlator,

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial }{\partial t}\langle {B}_{\alpha }{B}_{\beta }^{{\prime} }\rangle =2\varkappa {\partial }_{m}^{\rho }{\partial }_{m}^{\rho }\langle {B}_{\alpha }{B}_{\beta }^{{\prime} }\rangle \\ +\,\displaystyle \frac{1}{2}\left({\varepsilon }_{\alpha \mathrm{mn}}{\varepsilon }_{\mathrm{npq}}{\delta }_{\beta r}+{\varepsilon }_{\beta \mathrm{mn}}{\varepsilon }_{\mathrm{npq}}{\delta }_{\alpha r}\right){\varepsilon }_{{j}_{1}{{jk}}_{1}}{\partial }_{m}^{\rho }{\partial }_{{i}_{1}}^{\rho }\\ \ \times \,\left({\varepsilon }_{{{qi}}_{1}{j}_{1}}{D}_{{jp}}(0)\langle {B}_{{k}_{1}}{B}_{r}^{{\prime} }\rangle -{\varepsilon }_{{{ri}}_{1}{j}_{1}}{D}_{{jp}}({\boldsymbol{\rho }})\langle {B}_{q}{B}_{{k}_{1}}^{{\prime} }\rangle \right)\\ \ +\,\displaystyle \frac{1}{6}\left({\varepsilon }_{\alpha \mathrm{mn}}{\varepsilon }_{\mathrm{npq}}{\delta }_{\beta r}+{\varepsilon }_{\beta \mathrm{mn}}{\varepsilon }_{\mathrm{npq}}{\delta }_{\alpha r}\right){\varepsilon }_{{j}_{1}{{jk}}_{1}}{\varepsilon }_{{j}_{2}{{kk}}_{2}}{\partial }_{m}^{\rho }\\ \ \times \,\left(2{\varepsilon }_{{{qi}}_{1}{j}_{1}}{\varepsilon }_{{{ri}}_{2}{j}_{2}}{\partial }_{{i}_{2}}^{\rho }\left({\partial }_{{i}_{1}}^{(2)}{F}_{\mathrm{kpj}}({\boldsymbol{\rho }},{\boldsymbol{\rho }})\langle {B}_{{k}_{1}}{B}_{{k}_{2}}^{{\prime} }\rangle \right.\right.\\ \ +\,\left.\,{F}_{\mathrm{kpj}}({\boldsymbol{\rho }},{\boldsymbol{\rho }}){\partial }_{{i}_{1}}^{\rho }\langle {B}_{{k}_{1}}{B}_{{k}_{2}}^{{\prime} }\rangle \right)+{\varepsilon }_{{{ri}}_{1}{j}_{1}}{\varepsilon }_{{k}_{1}{i}_{2}{j}_{2}}{\partial }_{{i}_{1}}^{\rho }\\ \ \left.\times \,\left({\partial }_{{i}_{2}}^{(2)}{F}_{{pjk}}({\boldsymbol{\rho }},{\boldsymbol{\rho }})\langle {B}_{q}{B}_{{k}_{2}}^{{\prime} }\rangle +{F}_{{pjk}}({\boldsymbol{\rho }},{\boldsymbol{\rho }}){\partial }_{{i}_{2}}^{\rho }\langle {B}_{q}{B}_{{k}_{2}}^{{\prime} }\rangle \right)\right),\end{array}\end{eqnarray} \tag{ B11 }$$

where ${\boldsymbol{\rho }}={\boldsymbol{r}}^{\prime} -{\boldsymbol{r}}$ and ${\partial }_{n}^{(2)}$ denotes the derivative over the second argument.

Making use of Equation (7), we express all of the coefficients in Equation (B11) by means of the function P(r) and the longitudinal velocity structure function. After cumbersome calculations performed with the aid of computer algebra, we arrive at the generalized Kazantsev equation: Equations (28), (31), and (32) for the Batchelor regime and Equations (28), (35), and (36) for the nonlinear velocity field.

C.1. Kazantsev–Kraichnan Model, Linear Velocity Field

Consider first the simplest Kazantsev equation for the Kazantsev–Kraichnan model (Equation (10)) in the Batchelor regime, i.e., with σ = (D/3)r² (which corresponds to r_ν → ∞). The substitution of the ansatz

$$\begin{eqnarray*}&&G(t,r)={e}^{\gamma t}G(r)\end{eqnarray*}$$

reduces this equation to the ordinary differential equation

$$\begin{eqnarray*}&&\gamma G=\displaystyle \frac{2}{3}D\left({r}^{2}G^{\prime\prime} +6{rG}^{\prime} +10G\right)+2\varkappa \,\left(G^{\prime\prime} +\displaystyle \frac{4}{r}G^{\prime} \right).\end{eqnarray*}$$

We proceed to the dimensionless variables x = r/r_d ; with account of $\varkappa ={{Dr}}_{d}^{2}$, we get

$$\begin{eqnarray}&&\left({x}^{2}+3\right)G^{\prime\prime} +\left(6x+\displaystyle \frac{12}{x}\right)G^{\prime} +\left(10-\displaystyle \frac{3}{2}{\rm{\Gamma }}\right)G=0,\end{eqnarray} \tag{ C1 }$$

where

$$\begin{eqnarray*}&&{\rm{\Gamma }}=\gamma /D.\end{eqnarray*}$$

The two exact solutions of this equation can be written by means of hypergeometric functions. Here we restrict our consideration to the analysis of their asymptotic behavior to get the experience necessary to generalize the solutions to the cases of nonzero F and/or finite r_ν .

One of two independent solutions diverges as x → 0, so we are interested in the other one. It satisfies the boundary condition $G^{\prime} (0)=0$.

As x → ∞, Equation (C1) is simplified to a homogenous differential equation; its characteristic equation is

$$\begin{eqnarray*}&&\alpha (\alpha -1)+6\alpha +10-\displaystyle \frac{3}{2}{\rm{\Gamma }}=0,\end{eqnarray*}$$

and the solution is

$$\begin{eqnarray*}&&{G}^{({\rm{\Gamma }})}(x){\simeq }_{x\to \infty }{A}_{+}{x}^{{\alpha }_{+}}+{A}_{-}{x}^{{\alpha }_{-}},\end{eqnarray*}$$

where

$$\begin{eqnarray*}&&{\alpha }_{\pm }=-\displaystyle \frac{5}{2}\pm \displaystyle \frac{\sqrt{3}}{2}\sqrt{2{\rm{\Gamma }}-5}.\end{eqnarray*}$$

The real and imaginary parts of α₊(Γ) are illustrated in Figure 6. One can see that for all Γ < 5/2, G^(Γ)(x) decreases as x → ∞ with the same rate and oscillates, while for Γ > 5/2, it decreases slower (and for Γ > 20/3, even grows) without oscillations.

Zoom In Zoom Out Reset image size

Now let us return to evolution Equation (10). Equation (C1) can be reduced to the Sturm–Liouville equation (by the change of variables $y={\rm{arsinh}}(x/\sqrt{3})$, $q{(y)=(3+{x}^{2})}^{1/4}{x}^{2}G(x)$). So, an arbitrary initial perturbation G₀(x) can be decomposed into a sum (integral) of the eigenfunctions G^(Γ)(x). The time dependence of each summand is exponential with its own rate, G^(Γ)(t, x) ∝ e^{γ t} , so the term with the biggest γ would survive after large times. Now, if the initial function is localized (more precisely, decreases no faster than x^−5/2 as x → ∞), then all of the terms in the decomposition must oscillate or decrease at x → ∞ faster than G₀(x). Since $\mathrm{Im}{\alpha }_{+}({\rm{\Gamma }}\geqslant 5/2)=0$ and $\mathrm{Re}{\alpha }_{+}({\rm{\Gamma }}\geqslant 5/2)\gt -5/2$, we get the upper boundary of the spectrum:

$$\begin{eqnarray*}&&{{\rm{\Gamma }}}_{\max }=\displaystyle \frac{5}{2},\quad G(x,t)\propto {e}^{{{\rm{\Gamma }}}_{\max }{Dt}}.\end{eqnarray*}$$

C.2. Nonzero Time Asymmetry

Now we account for F ≠ 0 and apply the same ideas to analyze Equations (28) and (32). The equation for the eigenvalues is

$$\begin{eqnarray}\begin{array}{rcl}\gamma G & = & \displaystyle \frac{2}{3}D\left({r}^{2}G^{\prime\prime} +6{rG}^{\prime} +10G\right)+2\varkappa \,\left(G^{\prime\prime} +\displaystyle \frac{4}{r}G^{\prime} \right)\\ & & +\,\displaystyle \frac{1}{9}F\left(2{r}^{3}G\prime\prime\prime +21{r}^{2}G^{\prime\prime} +14{rG}^{\prime} -70G\right).\end{array}\end{eqnarray} \tag{ C2 }$$

In the dimensionless variables, this is equivalent to

$$\begin{eqnarray}&&\begin{array}{l}\left({x}^{2}+3\right)G^{\prime\prime} +\left(6x+\displaystyle \frac{12}{x}\right)G^{\prime} +\left(10-\displaystyle \frac{3}{2}{\rm{\Gamma }}\right)G\\ \ +\,\displaystyle \frac{1}{6}f\left(2{x}^{3}G\prime\prime\prime +21{x}^{2}G^{\prime\prime} +14{xG}^{\prime} -70G\right)=0,\end{array}\end{eqnarray} \tag{ C3 }$$

where f = F/D is the new small parameter. Again, we consider the asymptote x → ∞ and get the homogeneous differential equation with the characteristic polynomial

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{f}{3}\beta (\beta -1)(\beta -2)+\left(1+\displaystyle \frac{7}{2}f\right)\beta (\beta -1)\\ \ +\,\left(6+\displaystyle \frac{7}{3}f\right)\beta +\left(10-\displaystyle \frac{35}{3}f-\displaystyle \frac{3}{2}{\rm{\Gamma }}\right)=0.\end{array}\end{eqnarray} \tag{ C4 }$$

This equation has three solutions; two are close to α_±, and the third is real (negative) and very big:

$$\begin{eqnarray}&&{G}^{({\rm{\Gamma }})}\simeq {B}_{+}{\xi }^{{\beta }_{+}}+{B}_{-}\,{\xi }^{{\beta }_{-}}+{B}_{f}\,{\xi }^{{\beta }_{f}},\qquad \xi \to \infty ,\end{eqnarray} \tag{ C5 }$$

$$\begin{eqnarray}&&{\beta }_{f}\propto -1/f,\end{eqnarray} \tag{ C6 }$$

$$\begin{eqnarray}&&{\beta }_{\pm }-{\alpha }_{\pm }\propto f.\end{eqnarray} \tag{ C7 }$$

The third summand is an artifact of the model; it corresponds to the nonphysical solution and must be excluded by setting B_f = 0.

The internal limit x → 0 also gives three solutions, and only one of them is convergent. We assume that there exist modes that converge at both limits x → 0 and x → ∞. Unlike the Gaussian (Kazantsev–Kraichnan) case, this is not guaranteed. This supposition needs to be proved by numerical simulations. This is done in Section 5. Also, the completeness of this set of functions is not evident; however, we suppose that the asymptotic time dependence of the arbitrary solution is exponential, as it was in the case of the second-order equation, so this arbitrary solution can be presented as a sum of eigenfunctions. We are again interested in the eigenfunction with the fastest increment.

But as soon as we suppose the existence of the spectrum, we can derive the upper boundary of the eigenvalue in the decomposition of an arbitrary initial distribution G₀(x) based on the asymptotic behavior of the eigenfunctions. The eigenfunctions presented in the decomposition can either decrease faster than G₀(x) or oscillate as x → ∞. From the characteristic polynomial (Equation (C4)), we find that the condition of oscillations $\mathrm{Im}\beta \ne 0$ holds for

$$\begin{eqnarray}&&{\rm{\Gamma }}\lt {{\rm{\Gamma }}}_{\max }=\displaystyle \frac{8+126{f}^{2}-{\left(4+27{f}^{2}\right)}^{3/2}}{18{f}^{2}}.\end{eqnarray} \tag{ C8 }$$

This coincides with Equation (21). In the limit f ≪ 1, we arrive at Equation (33).

C.3. Finite Prandtl Numbers

To evaluate the effect of the nonlinearity of the velocity field, we consider the function σ(r) truncated in accordance with Equation (34). For this model, the Kazantsev equation is stepwise; it coincides with Equation (C2) for r < r_ν , while for r > r_ν , it becomes

$$\begin{eqnarray}&&\gamma G=\displaystyle \frac{2}{3}D(G^{\prime\prime} +\displaystyle \frac{4}{r}G^{\prime} )({r}_{\nu }^{2}+2{r}_{d}^{2}),r\gt {r}_{\nu }.\end{eqnarray} \tag{ C9 }$$

So, one has to match the solution of Equation (C2) with the descending solution of this equation. The coincidence of G and $G^{\prime}$ determines the spectrum of γ. But the biggest ${\gamma }_{\max }^{(\mu )}$ is still restricted by the condition $\mathrm{Im}\beta \ne 0$.

To estimate the correction to γ produced by a large but finite r_ν , we note that the solution of Equation (C2) oscillates as a function of r, and the phase of the oscillations is determined by the imaginary part of β₊:

$$\begin{eqnarray}&&G(r)\propto {r}_{+}^{\beta }\propto {e}^{i\mathrm{Im}{\beta }_{+}\mathrm{ln}\mu }.\end{eqnarray} \tag{ C10 }$$

Matching the amplitudes of both branches of the solution at r = r_ν is done by a multiplier; to match the derivatives, one has to choose the phase. However, half a period of oscillations is enough to ensure any phase that is needed. So, the correction to ${\gamma }_{\max }$ produced by finite r_ν at any rate leaves it within the range of γ in which the phase of G(r_ν ) changes by π:

$$\begin{eqnarray}&&\mathrm{Im}{\beta }_{+}({{\rm{\Gamma }}}_{\max }^{(\mu )})\mathrm{ln}\mu =\pi .\end{eqnarray} \tag{ C11 }$$

The bigger the μ, the closer ${{\rm{\Gamma }}}_{\max }^{(\mu )}$ is to the Batchelor value ${{\rm{\Gamma }}}_{\max }$.