2. MHD equations and the mean-field dynamo problem

To study the MHD turbulence in an incompressible conducting fluid (plasma) we consider the following dynamical equations describing the evolution of the velocity field of the fluid flow $\boldsymbol {U}(\boldsymbol {x},t)$ and the magnetic field $\boldsymbol {B}(\boldsymbol {x}.t)$

(2.1a)\begin{gather} \frac{\partial\boldsymbol{U}}{\partial t}+ (\boldsymbol{U}\boldsymbol{\cdot} \boldsymbol{\nabla} )\boldsymbol{U}=\boldsymbol{f}-\boldsymbol{\nabla} \varPi+ \frac{1}{\mu_{0}\rho}(\boldsymbol{B}\boldsymbol{\cdot} \boldsymbol{\nabla} ) \boldsymbol{B}+\nu\nabla^{2}\boldsymbol{U}, \end{gather}

(2.1b)\begin{gather}\frac{\partial\boldsymbol{B}}{\partial t}+(\boldsymbol{U}\boldsymbol{\cdot} \boldsymbol{\nabla} ) \boldsymbol{B}=(\boldsymbol{B}\boldsymbol{\cdot} \boldsymbol{\nabla} )\boldsymbol{U}+ \eta\nabla^{2}\boldsymbol{B}, \end{gather}

(2.1c)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{U}=0 \quad \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{B}=0, \end{gather}

where $\nu$ is the fluid's viscosity, $\eta$ the magnetic diffusivity and

(2.2)\begin{equation} \varPi=\frac{p}{\rho}+\frac{B^{2}}{2\mu_{0}\rho},\end{equation}

is the total pressure, with p denoting the thermodynamic pressure, $\rho$ the density and $\mu_0$ the vacuum permeability. Without loss of generality we may assume that the forcing is solenoidal

(2.3)\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{f}=0.\end{equation}

The first (2.1a) is the well-known Navier–Stokes equation with the Lorentz force, whereas the second one is the induction equation derived from Maxwell's laws and the material Ohm's law (the latter also utilizes the assumption that the time scale associated with electromagnetic waves $L/c$ is much shorter than any time scale in the fluid flow). The solenoidal constraints for the dynamical fields (2.1c) simply express the law of mass conservation and Gauss’ law for magnetism. For the purpose of simplicity we rescale the magnetic field in the following way:

(2.4)\begin{equation} \frac{\boldsymbol{B}}{\sqrt{\mu_{0}\rho}}\rightarrow\boldsymbol{B}, \end{equation}

so that the prefactor $1/\mu _{0}\rho$ in the Lorentz-force term in the Navier–Stokes equation is lost.

Next, denoting by angular brackets the ensemble mean,

(2.5)\begin{equation} \langle \boldsymbol{\cdot} \rangle -\textrm{ensemble mean}, \end{equation}

let us assume

(2.6)\begin{gather} \langle \,\boldsymbol{f}\rangle =0, \end{gather}

(2.7a–c)\begin{gather}\boldsymbol{U}=\langle \boldsymbol{U}\rangle +\boldsymbol{u},\quad\boldsymbol{B}=\langle \boldsymbol{B}\rangle +\boldsymbol{b},\quad p=\langle p\rangle +p', \end{gather}

and write down separately the equations for the mean fields $\langle \boldsymbol {U}\rangle$ and $\langle \boldsymbol {B}\rangle$ and the turbulent fluctuations $\boldsymbol {u}$ and $\boldsymbol {b}$; this yields

(2.8a)\begin{gather} \frac{\partial\langle \boldsymbol{U}\rangle }{\partial t}+(\langle \boldsymbol{U}\rangle \boldsymbol{\cdot} \boldsymbol{\nabla} )\langle \boldsymbol{U}\rangle ={-}\boldsymbol{\nabla} \langle \varPi\rangle +(\langle \boldsymbol{B}\rangle \boldsymbol{\cdot} \boldsymbol{\nabla} )\langle \boldsymbol{B}\rangle +\nu\nabla^{2}\langle \boldsymbol{U}\rangle \nonumber\\ \hspace{6.3pc}-\boldsymbol{\nabla} \boldsymbol{\cdot} (\langle \boldsymbol{u}\boldsymbol{u}\rangle - \langle \boldsymbol{b}\boldsymbol{b}\rangle ), \end{gather}

(2.8b)\begin{gather} \frac{\partial\langle \boldsymbol{B}\rangle }{\partial t}=\boldsymbol{\nabla} \times (\langle \boldsymbol{U}\rangle \times\langle \boldsymbol{B}\rangle ) +\boldsymbol{\nabla} \times\langle \boldsymbol{u}\times\boldsymbol{b}\rangle + \eta\nabla^{2}\langle \boldsymbol{B}\rangle , \end{gather}

(2.8c)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \langle \boldsymbol{B}\rangle =0,\quad\boldsymbol{\nabla} \boldsymbol{\cdot} \langle \boldsymbol{U}\rangle =0, \end{gather}

where $\boldsymbol {\mathcal {E}}=\langle \boldsymbol {u}\times \boldsymbol {b}\rangle$ is the large-scale EMF and

(2.9a)\begin{align} &\frac{\partial\boldsymbol{u}}{\partial t}-\nu\nabla^{2}\boldsymbol{u}+(\langle \boldsymbol{U}\rangle \boldsymbol{\cdot} \boldsymbol{\nabla} )\boldsymbol{u}+(\boldsymbol{u}\boldsymbol{\cdot} \boldsymbol{\nabla} )\langle \boldsymbol{U}\rangle -(\langle \boldsymbol{B}\rangle \boldsymbol{\cdot} \boldsymbol{\nabla} )\boldsymbol{b}-(\boldsymbol{b}\boldsymbol{\cdot} \boldsymbol{\nabla} )\langle \boldsymbol{B}\rangle +\boldsymbol{\nabla} \varPi' \nonumber\\ &\quad =\boldsymbol{f}-\boldsymbol{\nabla} \boldsymbol{\cdot} (\boldsymbol{u}\boldsymbol{u}- \boldsymbol{b}\boldsymbol{b})+\boldsymbol{\nabla} \boldsymbol{\cdot} (\langle \boldsymbol{u} \boldsymbol{u}\rangle -\langle \boldsymbol{b}\boldsymbol{b}\rangle ), \end{align}

(2.9b)$$\begin{align} &\frac{\partial\boldsymbol{b}}{\partial t}-\eta\nabla^{2} \boldsymbol{b}+(\langle \boldsymbol{U}\rangle \boldsymbol{\cdot} \boldsymbol{\nabla} )\boldsymbol{b}-(\langle \boldsymbol{B}\rangle \boldsymbol{\cdot} \boldsymbol{\nabla} )\boldsymbol{u}+(\boldsymbol{u}\boldsymbol{\cdot} \boldsymbol{\nabla} )\langle \boldsymbol{B}\rangle -(\boldsymbol{b}\boldsymbol{\cdot} \boldsymbol{\nabla} )\langle \boldsymbol{U}\rangle\nonumber\\ &\quad =\boldsymbol{\nabla} \times(\boldsymbol{u}\times\boldsymbol{b}-\langle \boldsymbol{u}\times\boldsymbol{b}\rangle), \end{align}$$

(2.9c)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{b}=0,\quad\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u}=0. \end{gather}

Furthermore, we assume scale separation between the means and the fluctuations, and express the dynamical fields using a Fourier transform defined in the following way:

(2.10a)\begin{gather} u_{i}(\boldsymbol{x},t)=\int_{\varLambda_{L}}^{\varLambda_{\nu}}\, \textrm{d}^{3}k \int_{-\infty}^{\infty}\, \textrm{d} \omega\hat{u}_{i}(\boldsymbol{k},\omega)\, \mathrm{e}^{\mathrm{i}(\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{x}-\omega t)}, \end{gather}

(2.10b)\begin{gather}b_{i}(\boldsymbol{x},t),=\int_{\varLambda_{L}}^{\varLambda_{\eta}}\, \textrm{d}^{3}k \int_{-\infty}^{\infty}\, \textrm{d} \omega\hat{b}_{i}(\boldsymbol{k},\omega)\, \mathrm{e}^{\mathrm{i}(\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{x}-\omega t)}, \end{gather}

(2.10c)\begin{gather}\langle U\rangle_{i}(\boldsymbol{x},t)=\int_{0}^{\kappa_{m}}\, \textrm{d}^{3} \kappa\int_{-\infty}^{\infty}\, \textrm{d} \omega\widehat{\langle U\rangle }_{i} (\boldsymbol{\kappa},\omega)\,\mathrm{e}^{\mathrm{i}(\boldsymbol{\kappa} \boldsymbol{\cdot} \boldsymbol{x}-\omega t)}, \end{gather}

(2.10d)\begin{gather}\langle B\rangle_{i}(\boldsymbol{x},t)=\int_{0}^{\kappa_{m}}\, \textrm{d}^{3}\kappa\int_{-\infty}^{\infty}\, \textrm{d} \omega \widehat{\langle B\rangle }_{i}(\boldsymbol{\kappa},\omega)\, \mathrm{e}^{\mathrm{i}(\boldsymbol{\kappa}\boldsymbol{\cdot} \boldsymbol{x}-\omega t)}, \end{gather}

where the upper cutoff for the mean-field Fourier modes $\kappa _{m}$ and the lower cutoff for the fluctuations $\varLambda _{L}$ satisfy

(2.11)\begin{equation} \kappa_{m}\ll\varLambda_{L}.\end{equation}

In the above we have also introduced upper cutoffs for the Fourier modes in the velocity fluctuations $\varLambda _{\nu }$ and in the magnetic fluctuations $\varLambda _{\eta }$, which in natural systems appear due to viscous kinetic energy dissipation and resistive dissipation of magnetic energy; for generality the magnetic and kinetic cutoffs are assumed unequal, $\varLambda _{v}\neq \varLambda _{\eta }$. The assumption of separation of spatial scales is standard in mean-field theories, put forward in order to allow for analytic formulation of the dynamics of large-scale fields. Such a clear separation is often absent in natural systems and energy is present at all scales. Still, there are many cases when scale separation is very apparent. Hughes & Tobias (Reference Hughes and Tobias2010) have estimated that a reasonable scale separation of approximately a decade between domain size and the size of the most energetic turbulent eddy, is achieved in numerical simulations with $1000^{3}$ spectral modes, which accurately describe the dynamics at magnetic Reynolds numbers of the order $O(10^{3})$. Even more importantly, natural scale separation is introduced by rapid background rotation in convective flows, such as e.g. in the Earth's liquid core, where the large length scale is defined by $E^{-1/3}L$, with $E\ll 1$ being the Ekman number and $L$ the thickness of the convection zone. There are numerous examples of studies which utilize the multiple-scale approach for MHD turbulent convection, such as, Childress & Soward (Reference Childress and Soward1972) and Soward (Reference Soward1974), or more recently Mizerski & Tobias (Reference Mizerski and Tobias2013), Calkins et al. (Reference Calkins, Julien, Tobias and Aurnou2015) and Calkins (Reference Calkins2018). In the mean-field theory a fluctuational wave vector $k=2{\rm \pi} /\mathscr {L}$ is associated with the length scale $\mathscr {L}$, which is regarded as the typical size of the energetic eddies and $\varLambda _{L}=2{\rm \pi} /\mathscr {L}_{L}$ is defined by the size of the largest, most energetic turbulent eddies, $\mathscr {L}_{L}$.

The aim of this analysis is to study the effect of the statistically homogeneous, stationary and isotropic MHD turbulence. However, the MHD turbulence may in particular involve the very important process of the generation of large-scale magnetic fields by the complex flow (cf. Moffatt & Dormy Reference Moffatt and Dormy2019). This requires a lack of reflectional symmetry in the flow, thus, for the sake of generality, we will study the general case of helical turbulence. In other words, we consider an MHD system driven by a non-reflectionally symmetric (helical) forcing $\boldsymbol {f}$ in the Navier–Stokes equation. The forcing is assumed Gaussian with zero mean (cf. (2.6)) and is fully defined by the following correlation function:

(2.12)\begin{equation} \langle \,\hat{f}_{i}(\boldsymbol{k},\omega)\hat{f}_{j}(\boldsymbol{k}',\omega')\rangle =\left[\frac{D_{0}}{k^{\sigma_{0}}}P_{ij}(\boldsymbol{k})+ \mathrm{i}\frac{D_{1}}{k^{\sigma_{1}}}\epsilon_{ijk}k_{k}\right] \delta(\boldsymbol{k}+\boldsymbol{k}')\delta(\omega+\omega'),\end{equation}

where

(2.13)\begin{equation} P_{ij}(\boldsymbol{k})=\delta_{ij}-\frac{k_{i}k_{j}}{k^{2}},\end{equation}

is the projection operator and $D_{0}, D_{1}$ and $\sigma _{0}, \sigma _{1}$ are real constants; the correlations function satisfies $k_{j}\langle \hat {f}_{i}(\boldsymbol {k},\omega )\hat {f}_{j}(\boldsymbol {k}',\omega ')\rangle =0$ and $k_{i}\langle \hat {f}_{i}(\boldsymbol {k},\omega )\hat {f}_{j}(\boldsymbol {k}',\omega ')\rangle =0$. The term proportional to $D_{1}$ introduces lack of reflectional symmetry (for which helicity can be used as a measure), thus it is crucial for the large-scale dynamo process. The value of $\sigma _{0}=-2$ (at $D_{1}=0$) corresponds to fluid in thermal equilibrium (cf. Landau & Lifshitz Reference Landau and Lifshitz1987) whereas $\sigma _{0}=3$ was shown by Yakhot & Orszag (Reference Yakhot and Orszag1986) to correspond to the Kolmogorov-type turbulence in the absence of a magnetic field. We will, therefore, assume $\sigma _{0}>-2$ and consider non-equilibrium flows. The $\sigma _{1}$ exponent will turn out to influence the turbulent magnetic energy and helicity spectra and we will demonstrate that the value of $\sigma _{1}=5$ corresponds to $k^{-5/3}$ spectral scaling for turbulent helicity, reported for the inertial range of helical isotropic turbulence in a number of works, e.g. Brissaud et al. (Reference Brissaud, Frisch, Leorat, Lesieur and Mazure1973) and Chen et al. (Reference Chen, Chen, Eyink and Holm2003). Furthermore, we assume that the turbulence is forced only at small scales, i.e. within the wavenumber band $k>\varLambda _{L}$.

We can calculate a positive definite quantity

(2.14)\begin{equation} F^{2}(k,\omega)=\int k^{2}\,\mathrm{d}\mathring{\varOmega}_{\boldsymbol{k}}\int\,\mathrm{d}^{4}q'\langle \,\hat{f}_{i}(\boldsymbol{k},\omega)\hat{f}_{i}(\boldsymbol{k}',\omega')\rangle =\frac{8{\rm \pi} D_{0}}{k^{\sigma_{0}-2}}>0,\end{equation}

where $\mathring {\varOmega }_{\boldsymbol {k}}$ denotes a solid angle associated with the vector $\boldsymbol {k}$ and to simplify notation we have introduced a four-component vector notation

(2.15a,b)\begin{equation} \boldsymbol{q}'=(\boldsymbol{k}',\omega'),\quad \int\,\mathrm{d}^{3}k'\int_{-\infty}^{\infty}\,\mathrm{d}\omega'(\boldsymbol{\cdot} )=\int \, \textrm{d}^{4}q'(\boldsymbol{\cdot} ).\end{equation}

This implies

(2.16)\begin{equation} D_{0}>0.\end{equation}

The renormalization group technique is an iterative procedure based on successive elimination of thin wavenumber bands from the Fourier spectrum of the fluctuating fields. In this way, the effect of thin bands of modes with the shortest wavelengths on the remaining modes is calculated at each step of the procedure. The final aim of this approach is to obtain recursion equations for coefficients describing the effective mean Reynolds stress, the mean EMF and the mean Lorentz force (see (4.30a,b)) as functions of the wavenumber at each step of the procedure. The Reynolds stresses are responsible for the creation of the turbulent viscosity, and the mean EMF for the creation of the turbulent magnetic diffusivity and what is traditionally called the $\alpha$-effect, which involves the part of the EMF that is linear in the mean magnetic field. The recursion equations (provided in (A91a–e)) are then solved for $k\rightarrow \varLambda _{L}$ in order to obtain the final forms of the large-scale viscosity, EMF and the Lorentz force which appear in the renormalized mean-field equations and include the effect of nonlinear evolution of turbulent fluctuations. However, for the sake of clarity, we start by considering the simplified limit of weak turbulence, with linearized evolution equations for fluctuations, rather common in the literature on mean-field dynamo theory. This does not involve renormalization, however, it will set the grounds for § 4, where the renormalization method is applied. Moreover, the linear regime, considered in the next section, demonstrates at a simple level the basic ideas of the mathematical approach undertaken to provide a comprehensive picture of the isotropic, homogeneous and stationary MHD turbulence, which may amplify mean magnetic fields.

3. Introductory problem of forced system with very weak turbulent fluctuations – linearization

Utilizing the scale separation assumption and introducing

(3.1a,b)\begin{equation} \varepsilon=\frac{\kappa_{m}}{\varLambda_{L}},\quad \boldsymbol{X}=\varepsilon\boldsymbol{x},\end{equation}

one can write

(3.2a,b)\begin{equation} \langle U\rangle_{i}\approx\langle U\rangle_{0i}+\varepsilon x_{j}G_{ij},\quad \langle B\rangle_{i}\approx\langle B\rangle_{0i}+\varepsilon x_{j} \varGamma_{ij},\end{equation}

where

(3.3a,b)\begin{gather} \langle \boldsymbol{U}\rangle_{0}=\langle \boldsymbol{U}\rangle (\varepsilon=0)=O (1),\quad G_{ij}= \frac{\partial\langle U\rangle_{i}}{\partial X_{j}}=O (1), \end{gather}

(3.4a,b)\begin{gather}\langle \boldsymbol{B}\rangle_{0}=\langle \boldsymbol{B}\rangle (\varepsilon=0)=O (1),\quad \varGamma_{ij}= \frac{\partial\langle B\rangle_{i}}{\partial X_{j}}=O (1). \end{gather}

Of course, formally, the gradient matrices are defined at $\varepsilon =0$ in the expansions, however, we neglect terms of the order $O(\varepsilon ^{2})$, which allows us to substitute for $G_{ij}(\varepsilon =0)$ and $\varGamma _{ij}(\varepsilon =0)$ with $G_{ij}$ and $\varGamma _{ij}$; note that, $\langle \boldsymbol {U}\rangle _{0}$ and $\langle \boldsymbol {B}\rangle _{0}$ are still allowed to vary on length scales significantly larger than $\varepsilon ^{-1}$. The latter expansion allows us to express the terms $(\langle \boldsymbol {U}\rangle \boldsymbol {\cdot } \boldsymbol {\nabla } )\boldsymbol {u}$ and $(\langle \boldsymbol {U}\rangle \boldsymbol {\cdot } \boldsymbol {\nabla } )\boldsymbol {b}$ in the following way:

(3.5)\begin{align} (\langle \boldsymbol{U}\rangle \boldsymbol{\cdot} \boldsymbol{\nabla} )\boldsymbol{u}&= \mathrm{i}\int^{\varLambda_{\nu}}\, \textrm{d}^{4}q(\langle U\rangle_{0m}+ \varepsilon G_{mj}x_{j})k_{m}\hat{u}_{i}(\boldsymbol{q})\, \mathrm{e}^{\mathrm{i}(\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{x}-\omega t)} \nonumber\\ &= \mathrm{i}\int^{\varLambda_{\nu}}\, \textrm{d}^{4}q \left[\langle U\rangle_{0m}+\mathcal{U}\sin\left(\varepsilon \frac{G_{mj}}{\mathcal{U}}x_{j}\right)\right]k_{m}\hat{u}_{i}(\boldsymbol{q}) \,\mathrm{e}^{\mathrm{i}(\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{x}-\omega t)}+ O \left(\varepsilon^{3}\frac{\mathcal{U}^{2}}{\mathscr{L}}\right) \nonumber\\ &= \mathrm{i}\int^{\varLambda_{\nu}}\, \textrm{d}^{4}q\langle U\rangle_{0m}k_{m} \hat{u}_{i}(\boldsymbol{q})\,\mathrm{e}^{\mathrm{i}(\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{x}-\omega t)}\nonumber\\ &\quad +\frac{\mathcal{U}}{2}\int^{\varLambda_{\nu}}\, \textrm{d}^{4}qk_{m}\hat{u}_{i} (\boldsymbol{q})\,\mathrm{e}^{-\mathrm{i}\omega t} [\mathrm{e}^{\mathrm{i}(k_{j}+\varepsilon({G_{mj}}/{\mathcal{U}}))x_{j}}- \mathrm{e}^{\mathrm{i}(k_{j}-\varepsilon({G_{mj}}/{\mathcal{U}}))x_{j}}] +O \left(\varepsilon^{3}\frac{\mathcal{U}^{2}}{\mathscr{L}} \right) \nonumber\\ &= \int^{\varLambda_{\nu}}\, \textrm{d}^{4}q\left\{ \vphantom{\frac{\mathcal{U}}{2}}(\mathrm{i}\langle U \rangle_{0m}k_{m}-\varepsilon G_{mm})\hat{u}_{i}(\boldsymbol{q})\right.\nonumber\\ & \quad \left.+\frac{\mathcal{U}}{2}k_{m} \left[\hat{u}_{i}\left(\boldsymbol{k}-\varepsilon\frac{\boldsymbol{G}_{m}}{\mathcal{U}}, \omega\right)-\hat{u}_{i}\left(\boldsymbol{k}+\varepsilon \frac{\boldsymbol{G}_{m}}{\mathcal{U}},\omega\right)\right]\right\} \,\mathrm{e}^{\mathrm{i}(\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{x}-\omega t)} +O \left(\varepsilon^{2}\frac{\mathcal{U}^{2}}{\mathscr{L}}\right), \end{align}

where $\mathcal {U}$ is the velocity scale and $\mathscr {L}$ is the length scale of variation of turbulent fluctuations and we have defined the vector $(\boldsymbol {G}_{m})_{j}=G_{mj}$ for $j=1,2,3$, composed of rows of the matrix $G_{mj}$; of course, $G_{mm}=\bar {\boldsymbol {\nabla }}\boldsymbol {\cdot } \langle \boldsymbol {U}\rangle =0$. For the purpose of this section, to simplify the notation we will put $\mathcal {U}=1$.

As explained above, for the sake of clarity, we start by considering the simple problem of weak turbulent fluctuations and the dynamical equations linearized about the means

(3.6a)\begin{gather} (-\mathrm{i}\omega+\nu k^{2})\hat{u}_{i}(\boldsymbol{q})- \mathrm{i}\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}\hat{b}_{i} (\boldsymbol{q})+\mathrm{i}k_{i}\hat{\varPi}+\mathcal{R}_{i}^{(u)}= \hat{f}_{i}(\boldsymbol{q}), \end{gather}

(3.6b)\begin{gather}(-\mathrm{i}\omega+\eta k^{2})\hat{b}_{i}(\boldsymbol{q})-\mathrm{i} \boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}\hat{u}_{i} (\boldsymbol{q})+\mathcal{R}_{i}^{(b)}=0, \end{gather}

where

(3.7a)\begin{align} \mathcal{R}_{i}^{(u)}&= \varepsilon G_{ij}\hat{u}_{j}(\boldsymbol{q})- \varepsilon\varGamma_{ij}\hat{b}_{j}(\boldsymbol{q})+\tfrac{1}{2}k_{m} [\hat{u}_{i}(\boldsymbol{k}-\varepsilon\boldsymbol{G}_{m},\omega)- \hat{u}_{i}(\boldsymbol{k}+\varepsilon\boldsymbol{G}_{m},\omega)]\nonumber\\ &\quad -\tfrac{1}{2}k_{m}[\hat{b}_{i}(\boldsymbol{k}-\varepsilon \boldsymbol{\varGamma}_{m},\omega)-\hat{b}_{i}(\boldsymbol{k}+\varepsilon \boldsymbol{\varGamma}_{m},\omega)]=O (\varepsilon), \end{align}

(3.7b)\begin{align} \mathcal{R}_{i}^{(b)}&= \varepsilon\varGamma_{ij}\hat{u}_{j}(\boldsymbol{q}) -\varepsilon G_{ij}\hat{b}_{j}(\boldsymbol{q})+\tfrac{1}{2}k_{m} [\hat{b}_{i}(\boldsymbol{k}-\varepsilon\boldsymbol{G}_{m},\omega)- \hat{b}_{i}(\boldsymbol{k}+\varepsilon\boldsymbol{G}_{m},\omega)]\nonumber\\ &\quad -\tfrac{1}{2}k_{m}[\hat{u}_{i}(\boldsymbol{k}-\varepsilon \boldsymbol{\varGamma}_{m},\omega)-\hat{u}_{i}(\boldsymbol{k}+\varepsilon \boldsymbol{\varGamma}_{m},\omega)]=O (\varepsilon). \end{align}

Since the mean velocity $\langle \boldsymbol {U}\rangle _{0}$ only creates a shift of the frequency $\omega \rightarrow \omega -\boldsymbol {k}\boldsymbol {\cdot } \langle \boldsymbol {U}\rangle _{0}$, we simply absorb it into the frequency. This is allowed here, since we do not introduce into the problem the time-scale separation and the Fourier frequencies take all real values from $-\infty$ to $\infty$, whereas the norm of the wave vectors is bounded from above. For the velocity and magnetic fields, likewise the forcings are solenoidal

(3.8a–c)\begin{equation} \boldsymbol{k}\boldsymbol{\cdot} \hat{\boldsymbol{u}}=0,\quad\boldsymbol{k}\boldsymbol{\cdot} \hat{\boldsymbol{b}}=0,\quad\boldsymbol{k}\boldsymbol{\cdot} \hat{\boldsymbol{f}}=0.\end{equation}

Thus, by applying the projection operator to both sides of the equation (2.13) and after some simple algebra we get

(3.9a)\begin{equation} \hat{u}_{i}(\boldsymbol{q})=\frac{\mathrm{i}\hat{f}_{i} (\boldsymbol{q})}{\sigma(\boldsymbol{q})}-\frac{\mathrm{i}}{\sigma (\boldsymbol{q})}P_{ij}\mathcal{R}_{j}^{(u)}+\frac{\mathrm{i}\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle }{(\omega+\mathrm{i}\eta k^{2}) \sigma(\boldsymbol{q})}\mathcal{R}_{i}^{(b)},\\ \end{equation}

(3.9b)\begin{align} \hat{b}_{i}(\boldsymbol{q})&={-}\frac{\mathrm{i}\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}}{(\omega+\mathrm{i}\eta k^{2}) \sigma(\boldsymbol{q})}\hat{f}_{i}(\boldsymbol{q})- \frac{\mathrm{i}}{\omega+\mathrm{i}\eta k^{2}} \left[1+\frac{(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2}}{(\omega +\mathrm{i}\eta k^{2})\sigma(\boldsymbol{q})}\right]\mathcal{R}_{i}^{(b)}\nonumber\\ &\quad +\frac{\mathrm{i}\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle }{(\omega+ \mathrm{i}\eta k^{2})\sigma(\boldsymbol{q})}P_{ij}\mathcal{R}_{j}^{(u)}, \end{align}

with

(3.10)\begin{equation} \sigma(\boldsymbol{q})=\omega+\mathrm{i}\nu k^{2}-\frac{(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2}}{\omega+\mathrm{i}\eta k^{2}}.\end{equation}

Of course the problem, although linearized in fluctuations, remains nonlinear because of the presence of the forcing $\boldsymbol {f}$, which allows us to calculate the turbulent transport coefficients and the $\alpha$-effect.

We consider the simplest case of a statistically stationary, helical forcing defined by (2.12). The small corrections from the gradients of means $\boldsymbol {\mathcal {R}}^{(u)}$ and $\boldsymbol {\mathcal {R}}^{(b)}$ will be treated in a perturbational manner within the asymptotic limit $\varepsilon \ll 1$. A similar linearized problem that can be easily compared with the one studied here, has been solved in chapter 12.4.1 of Moffatt & Dormy (Reference Moffatt and Dormy2019), although the authors’ interests lie predominantly in the $\alpha$-effect (see also the rest of chapter 12.4 in that book, for other interesting aspects of the linearized regime in stationary turbulence).

In addition, let us also assume that the mean magnetic field is weak enough, so that terms of the order $(\boldsymbol {k}\boldsymbol {\cdot } \langle \boldsymbol {B}\rangle )^{2}$ can be neglected. By the use of (3.9a,b) and (3.10) and neglecting terms of the order $O(\varepsilon ^{2}, (\boldsymbol {k}\boldsymbol {\cdot } \langle \boldsymbol {B}\rangle )^{2}, \varepsilon \boldsymbol {k}\boldsymbol {\cdot } \langle \boldsymbol {B}\rangle )$ we can calculate the mean EMF and the mean Reynolds and Maxwell stresses in the following way:

(3.11)\begin{align} \mathcal{E}_{i} &= \epsilon_{ijk}\langle u_{j}b_{k}\rangle \approx\epsilon_{ijk}\int\,\mathrm{d}^{4}q\int\,\mathrm{d}^{4}q' \frac{\boldsymbol{k}'\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle \langle \,\hat{f}_{j}(\boldsymbol{q})\hat{f}_{k}(\boldsymbol{q}') \rangle}{(\omega'+\mathrm{i}\eta k^{\prime2}) \sigma(\boldsymbol{q}')\sigma(\boldsymbol{q})}\, \mathrm{e}^{\mathrm{i}[(\boldsymbol{k}+\boldsymbol{k}') \boldsymbol{\cdot} \boldsymbol{x}-(\omega+\omega')t]} \nonumber\\ & \quad +\epsilon_{ijk}\int\,\mathrm{d}^{4}q\int\,\mathrm{d}^{4}q' \frac{\langle \,\hat{f}_{j}(\boldsymbol{q})\mathcal{R}_{k}^{(b)} (\boldsymbol{q}')\rangle }{(\omega'+\mathrm{i}\eta k^{\prime2}) (\omega+\mathrm{i}\nu k^{2})}\,\mathrm{e}^{\mathrm{i} [(\boldsymbol{k}+\boldsymbol{k}')\boldsymbol{\cdot} \boldsymbol{x}-(\omega+\omega')t]} \nonumber\\ & \approx \eta D_{1}\langle B\rangle_{p}\epsilon_{ijk}\epsilon_{jkm}\int \frac{\mathrm{d}^{4}q}{k^{\sigma_{1}-2}}\frac{k_{p}k_{m}}{(\omega^{2}+\eta^{2}k^{4}) |\sigma(\boldsymbol{q})|^{2}} \nonumber\\ & \quad +\epsilon_{ijk}\int\,\mathrm{d}^{4}q\int\,\mathrm{d}^{4}q' \frac{\mathrm{e}^{\mathrm{i}[(\boldsymbol{k}+\boldsymbol{k}')\boldsymbol{\cdot} \boldsymbol{x}-(\omega+\omega')t]}}{(\omega'+\mathrm{i}\eta k^{\prime2}) (\omega+\mathrm{i}\nu k^{2})}\left\{ \varepsilon\varGamma_{kn}\langle \,\hat{f}_{j}(\boldsymbol{q})\hat{u}_{n}(\boldsymbol{q}')\rangle \vphantom{\tfrac{1}{2}}\right.\nonumber\\ &\quad -\varepsilon G_{kn}\langle \,\hat{f}_{j}(\boldsymbol{q}) \hat{b}_{n}(\boldsymbol{q}')\rangle \nonumber\\ & \quad +\tfrac{1}{2}k_{m}^{\prime}[\langle \,\hat{f}_{j}(\boldsymbol{q}) \hat{b}_{k}(\boldsymbol{k}'-\varepsilon\boldsymbol{G}_{m},\omega')\rangle - \langle \,\hat{f}_{j}(\boldsymbol{q})\hat{b}_{k}(\boldsymbol{k}'+ \varepsilon\boldsymbol{G}_{m},\omega')\rangle ] \nonumber\\ & \quad \left.-\tfrac{1}{2}k_{m}^{\prime}[\langle \,\hat{f}_{j}(\boldsymbol{q}) \hat{u}_{k}(\boldsymbol{k}'-\varepsilon\boldsymbol{\varGamma}_{m},\omega') \rangle -\langle \,\hat{f}_{j}(\boldsymbol{q})\hat{u}_{k}(\boldsymbol{k}' +\varepsilon\boldsymbol{\varGamma}_{m},\omega')\rangle ]\right\}, \end{align}

(3.12)\begin{align} -\langle u_{i}u_{j}\rangle &\approx \int\,\mathrm{d}^{4}q\int\,\mathrm{d}^{4}q' \frac{\mathrm{e}^{\mathrm{i}[(\boldsymbol{k}+\boldsymbol{k}')\boldsymbol{\cdot} \boldsymbol{x}-(\omega+\omega')t]}}{\sigma(\boldsymbol{q}') \sigma(\boldsymbol{q})}\left\{\vphantom{\tfrac{1}{2}}\langle \,\hat{f}_{i}(\boldsymbol{q})\hat{f}_{j}(\boldsymbol{q}') \rangle \right. \nonumber\\ & \quad \left.-\frac{P_{jk}(\boldsymbol{k}') \langle \,\hat{f}_{i}(\boldsymbol{q})\mathcal{R}_{k}^{(u)}(\boldsymbol{q}')\rangle +P_{ik}(\boldsymbol{k})\langle \,\hat{f}_{j}(\boldsymbol{q}')\mathcal{R}_{k}^{(u)} (\boldsymbol{q})\rangle}{\sigma(\boldsymbol{q}')\sigma(\boldsymbol{q})}\right\} \nonumber\\ &\approx{-}\tfrac{1}{2}D_{0}\int\frac{\mathrm{d}^{4}q}{k^{\sigma_{0}}} \frac{P_{ij}(\boldsymbol{k})}{\omega^{2}+\nu^{2}k^{4}} \nonumber\\ & \quad -\varepsilon G_{kn}\int\,\mathrm{d}^{4}q\int\,\mathrm{d}^{4}q' \frac{P_{jk}(\boldsymbol{k}')\langle \,\hat{f}_{i}(\boldsymbol{q})\hat{u}_{n} (\boldsymbol{q}')\rangle}{(\omega'+\mathrm{i}\nu k^{\prime2}) (\omega+\mathrm{i}\nu k^{2})}\,\mathrm{e}^{\mathrm{i} [(\boldsymbol{k}+\boldsymbol{k}')\boldsymbol{\cdot} \boldsymbol{x}-(\omega+\omega')t]} \nonumber\\ & \quad -\int\,\mathrm{d}^{4}q\int\,\mathrm{d}^{4}q'\frac{\mathrm{e}^{\mathrm{i} [(\boldsymbol{k}+\boldsymbol{k}')\boldsymbol{\cdot} \boldsymbol{x}-(\omega+\omega')t]} P_{jk}(\boldsymbol{k}')k_{m}^{\prime}}{2(\omega'+\mathrm{i}\nu k^{\prime2}) (\omega+\mathrm{i}\nu k^{2})} [\langle \,\hat{f}_{i}(\boldsymbol{q})\hat{u}_{k} (\boldsymbol{k}'-\varepsilon\boldsymbol{G}_{m},\omega')\rangle \nonumber\\ &\quad -\langle \,\hat{f}_{i}(\boldsymbol{q})\hat{u}_{k}(\boldsymbol{k}' +\varepsilon\boldsymbol{G}_{m},\omega')\rangle ] +(i\leftrightarrow j), \end{align}

(3.13)\begin{equation} \langle b_{i}b_{j}\rangle \approx0,\end{equation}

where $(i\leftrightarrow j)$ in (3.12) denotes terms of the same structure as all the three previous ones but with exchanged indices $i$ and $j$. Now we substitute for $\hat {\boldsymbol {u}}$ and $\hat {\boldsymbol {b}}$ the leading-order terms from (3.9a,b) and make use of (2.12) and $\epsilon _{ijk}P_{jk}=0$ to obtain

(3.14)\begin{align} \mathcal{E}_{i} &= 2\eta D_{1}\langle B\rangle_{p}\int \frac{\mathrm{d}^{4}q}{k^{\sigma_{1}-2}}\frac{k_{p}k_{i}}{(\omega^{2}+ \eta^{2}k^{4})(\omega^{2}+\nu^{2}k^{4})} \nonumber\\ &\quad -\eta D_{0}\epsilon_{ijk}\frac{\partial\langle B\rangle_{k}}{\partial x_{n}} \int\frac{\mathrm{d}^{4}q}{k^{\sigma_{0}-2}}\frac{P_{jn}(\boldsymbol{k})}{(\omega^{2} +\eta^{2}k^{4})(\omega^{2}+\nu^{2}k^{4})} \nonumber\\ &\quad -D_{1}\int\,\mathrm{d}^{4}q\frac{k_{i}k_{m}}{k^{\sigma_{1}}} \frac{1}{(\omega^{2}+\nu^{2}k^{4})}\nonumber\\ &\quad\times \left[\frac{1+\mathrm{i} \varepsilon\boldsymbol{\varGamma}_{m}\boldsymbol{\cdot} \boldsymbol{x}}{\omega- \mathrm{i}\eta(k^{2}-2\varepsilon\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{\varGamma}_{m})}- \frac{1-\mathrm{i}\varepsilon\boldsymbol{\varGamma}_{m}\boldsymbol{\cdot} \boldsymbol{x}}{\omega-\mathrm{i}\eta(k^{2}+2\varepsilon\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{\varGamma}_{m})}\right], \end{align}

(3.15)\begin{align} -\langle u_{i}u_{j}\rangle &\approx{-}\tfrac{1}{2}D_{0}\int\frac{\mathrm{d}^{4}q}{k^{\sigma_{0}}} \frac{P_{ij}(\boldsymbol{k})}{\omega^{2}+\nu^{2}k^{4}}\nonumber\\ &\quad +\nu D_{0}\frac{\partial\langle U\rangle_{k}}{\partial x_{n}}\int \frac{\mathrm{d}^{4}q}{k^{\sigma_{0}-2}}\frac{P_{jk}(\boldsymbol{k})P_{in} (\boldsymbol{k})}{(\omega^{2}+\nu^{2}k^{4})^{2}} \nonumber\\ &\quad +\mathrm{i}\tfrac{1}{2}\int\,\mathrm{d}^{4}q\frac{k_{m}}{\omega^{2}+ \nu^{2}k^{4}}\left[\frac{D_{0}}{k^{\sigma_{0}}}P_{ik}(\boldsymbol{k})+\mathrm{i} \frac{D_{1}}{k^{\sigma_{1}}}\epsilon_{ikl}k_{l}\right] \nonumber\\ & \quad \times \left\{ \frac{P_{jk}(\boldsymbol{k}- \varepsilon\boldsymbol{G}_{m})(1+\mathrm{i}\varepsilon \boldsymbol{G}_{m}\boldsymbol{\cdot} \boldsymbol{x})}{\omega-\mathrm{i}\nu (k^{2}-2\varepsilon\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{G}_{m})}- \frac{P_{jk}(\boldsymbol{k}+\varepsilon\boldsymbol{G}_{m}) (1-\mathrm{i}\varepsilon\boldsymbol{G}_{m}\boldsymbol{\cdot} \boldsymbol{x})}{\omega-\mathrm{i}\nu(k^{2}+2\varepsilon \boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{G}_{m})}\right\}\nonumber\\ & \quad+ (i\leftrightarrow j), \end{align}

where we have also made use of the fact that $\omega ^{2}+\eta ^{2}k^{4}$ and $\omega ^{2}+\nu ^{2}k^{4}$ are both even functions of $\omega$, which implies

(3.16a,b)\begin{equation} \int_{-\infty}^{\infty}\,\mathrm{d}\omega\frac{\omega}{(\omega^{2} +\nu^{2}k^{4})(\omega^{2}+\eta^{2}k^{4})}=0,\quad \int_{-\infty}^{\infty}\,\mathrm{d}\omega \frac{\omega}{(\omega^{2}+\nu^{2}k^{4})^{2}}=0.\end{equation}

The last line of (3.14) and the last two lines of (3.15), obtained by integration of the Dirac functions of the type $\delta (\boldsymbol {k}+\boldsymbol {k}'-\varepsilon \boldsymbol {G}_{m})$ over the $\boldsymbol {k}'$ domain have a non-symmetric integration domain for the wave vector $\boldsymbol {k}$. Therefore, to simplify the calculation we make a substitution $\boldsymbol {k}\mapsto \boldsymbol {k}+\frac {1}{2}\varepsilon \boldsymbol {G}_{m}$. Hence, making use of

(3.17a)\begin{gather} \int\mathrm{d}\mathring{\varOmega}\underset{N}{\underbrace{k_{m}\ldots k_{n}k_{k}}}=0,\quad\textrm{for any odd} {N} \textrm{and all }m,\ldots,n, k, \end{gather}

(3.17b)\begin{gather}\int k^{2}\,\mathrm{d}\mathring{\varOmega}\frac{k_{m}k_{i}}{k^{4}}= \frac{4{\rm \pi}}{3}\delta_{mi}, \end{gather}

(3.17c)\begin{gather}\int k^{2}\,\mathrm{d}\mathring{\varOmega}\frac{k_{m}k_{n}k_{p}k_{q}}{k^{6}}= \frac{4{\rm \pi}}{15}(\delta_{mn}\delta_{pq}+\delta_{mp}\delta_{nq}+\delta_{mq} \delta_{np}), \end{gather}

(3.17d)\begin{gather}\int_{-\infty}^{+\infty}\,\mathrm{d}\omega\frac{1}{(\omega^{\prime2}+ \nu^{2}k^{\prime4})(\omega^{\prime2}+\eta^{2}k^{\prime4})}= \frac{\rm \pi}{\nu\eta(\nu+\eta)k^{\prime6}}, \end{gather}

(3.17e)\begin{gather}\int_{-\infty}^{\infty}\frac{\mathrm{d}\omega'}{(\omega^{\prime2}+ \nu^{2}k^{\prime4})^{2}}=\frac{\rm \pi}{2\nu^{3}k^{\prime6}}, \end{gather}

(3.17f)\begin{gather}\frac{1}{(k^{\prime2}+\varepsilon\boldsymbol{k'\boldsymbol{\cdot} } \boldsymbol{G}_{m})^{\sigma_{0}/2}}=\frac{1}{k^{\prime\sigma_{0}}}- \frac{\sigma_{0}\varepsilon\boldsymbol{k'\boldsymbol{\cdot} } \boldsymbol{G}_{m}}{2k^{\prime\sigma_{0}+2}}, \end{gather}

(3.17g)\begin{gather}\frac{1}{\omega^{2}+\eta^{2}k^{4}-2\varepsilon\eta^{2}k^{2} \boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{\varGamma}_{m}}\approx\frac{1}{\omega^{2}+ \eta^{2}k^{4}}+\frac{2\varepsilon\eta^{2}k^{2}\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{\varGamma}_{m}}{(\omega^{2}+\eta^{2}k^{4})^{2}}, \end{gather}

where $\mathring {\varOmega }$ denotes a solid angle and neglecting terms of order $O(\varepsilon ^{2})$ we get

(3.18) \begin{align} \boldsymbol{\mathcal{E}} &= \frac{8{\rm \pi}^{2}D_{1}}{3\nu(\nu+\eta)} [\langle \boldsymbol{B}\rangle_{0}+(\boldsymbol{x}\boldsymbol{\cdot} \boldsymbol{\nabla} )\langle \boldsymbol{B}\rangle ]\int_{\varLambda_{L}}^{\varLambda} \frac{\mathrm{d}k}{k^{\sigma_{1}}}-\frac{8{\rm \pi}^{2}D_{0}}{3\nu(\nu+\eta)}\boldsymbol{\nabla} \times\langle \boldsymbol{B}\rangle \int_{\varLambda_{L}}^{\varLambda} \frac{\mathrm{d}k}{k^{\sigma_{0}+2}} \nonumber\\ &= \frac{8{\rm \pi}^{2}D_{1}}{3\nu(\nu+\eta)}\langle \boldsymbol{B}\rangle \int_{\varLambda_{L}}^{\varLambda}\frac{\mathrm{d}k}{k^{\sigma_{1}}}- \frac{8{\rm \pi}^{2}D_{0}}{3\nu(\nu+\eta)}\boldsymbol{\nabla} \times\langle \boldsymbol{B}\rangle \int_{\varLambda_{L}}^{\varLambda} \frac{\mathrm{d}k}{k^{\sigma_{0}+2}} \nonumber\\ &= \frac{8{\rm \pi}^{2}D_{1}}{3(\sigma_{1}-1)\nu(\nu+\eta)} \left(\frac{1}{\varLambda_{L}^{\sigma_{1}-1}}-\frac{1}{\varLambda^{\sigma_{1} -1}}\right)\langle \boldsymbol{B}\rangle \nonumber\\ & \quad -\frac{8{\rm \pi}^{2}D_{0}}{3(\sigma_{0}+1)\nu(\nu+\eta)} \left(\frac{1}{\varLambda_{L}^{\sigma_{0}+1}}-\frac{1}{\varLambda^{\sigma_{0} +1}}\right)\boldsymbol{\nabla} \times\langle \boldsymbol{B}\rangle \nonumber\\ & \overset{\varLambda_{L}\ll\varLambda}{\approx} \frac{8{\rm \pi}^{2}D_{1}}{3(\sigma_{1}-1)\nu(\nu+\eta)\varLambda_{L}^{\sigma_{1}-1}}\langle \boldsymbol{B}\rangle -\frac{8{\rm \pi}^{2}D_{0}}{3(\sigma_{0}+1)\nu(\nu+\eta) \varLambda_{L}^{\sigma_{0}+1}}\boldsymbol{\nabla} \times\langle \boldsymbol{B}\rangle, \end{align}

(3.19) \begin{align} -\langle u_{i}u_{j}\rangle & \approx \frac{14{\rm \pi}^{2}D_{0}}{15\nu^{2}} \left(\frac{\partial\langle U\rangle_{j}}{\partial x_{i}}+ \frac{\partial\langle U\rangle_{i}}{\partial x_{j}}\right) \int_{\varLambda_{L}}^{\varLambda_{\nu}}\frac{\mathrm{d}k}{k^{\sigma_{0}+2}} \nonumber\\ & \quad -\frac{2{\rm \pi}^{2}D_{0}}{15\nu^{2}}(2+\sigma_{0}) \left(\frac{\partial\langle U\rangle_{i}}{\partial x_{j}}+ \frac{\partial\langle U\rangle_{j}}{\partial x_{i}}\right) \int_{\varLambda_{L}}^{\varLambda_{\nu}}\frac{\mathrm{d}k}{k^{\sigma_{0}+2}} +\mathrm{const.} \nonumber\\ & \quad +\frac{2{\rm \pi}^{2}D_{1}}{3\nu^{2}}(\epsilon_{ijm}+\epsilon_{jim}) x_{j}\frac{\partial\langle U\rangle_{m}}{\partial x_{j}}\int \frac{\mathrm{d}k}{k^{\sigma_{1}}} \nonumber\\ & \approx \frac{2{\rm \pi}^{2}(5-\sigma_{0})}{15\nu^{2}(\sigma_{0}+1)}D_{0} \left(\frac{1}{\varLambda_{L}^{\sigma_{0}+1}}- \frac{1}{\varLambda_{\nu}^{\sigma_{0}+1}}\right) \left(\frac{\partial\langle U\rangle_{i}}{\partial x_{j}}+ \frac{\partial\langle U\rangle_{j}}{\partial x_{i}}\right)+\mathrm{const.} \nonumber\\ & \overset{\varLambda_{L}\ll\varLambda}{\approx} \frac{2{\rm \pi}^{2}(5-\sigma_{0})}{15\nu^{2}(\sigma_{0}+1)} \frac{D_{0}}{\varLambda_{L}^{\sigma_{0}+1}}\left(\frac{\partial\langle U\rangle_{i}}{\partial x_{j}}+\frac{\partial\langle U\rangle_{j}}{\partial x_{i}} \right)+\mathrm{const.} \end{align}

In the above we have also used

(3.20)\begin{equation} P_{jk}\left(\boldsymbol{k}-\tfrac{1}{2}\varepsilon\boldsymbol{G}_{m}\right) =P_{jk}(\boldsymbol{k})-\varepsilon\left(\frac{G_{mp}k_{p}k_{j}k_{k}}{k^{4}} -\frac{k_{j}G_{mk}+k_{k}G_{mj}}{2k^{2}}\right),\end{equation}

and $\varLambda =\mathrm {min}(\varLambda _{\nu }, \varLambda _{\eta })$; in principle, both cutoffs can be infinite. Therefore, in a weak turbulence with a weak mean magnetic field and in the limit $\varLambda _{L}\ll \varLambda$ the turbulent magnetic diffusivity and viscosity and the so-called $\bar {\alpha }$-coefficient defined as

(3.21)\begin{equation} \boldsymbol{\mathcal{E}}=\bar{\alpha}\langle \boldsymbol{B}\rangle - (\bar{\eta}-\eta)\boldsymbol{\nabla} \times\langle \boldsymbol{B}\rangle ,\end{equation}

take the form

(3.22)\begin{gather} \bar{\eta}=\eta\left[1+\frac{8{\rm \pi}^{2}D_{0}}{3(\sigma_{0}+1) \nu\eta(\nu+\eta)\varLambda_{L}^{\sigma_{0}+1}}\right]\approx \frac{8{\rm \pi}^{2}D_{0}}{3(\sigma_{0}+1)\nu(\nu+\eta)\varLambda_{L}^{\sigma_{0}+1}}, \end{gather}

(3.23)\begin{gather}\bar{\nu}=\nu\left[1+\frac{2{\rm \pi}^{2}(5-\sigma_{0})D_{0}}{15\nu^{3}(\sigma_{0}+1) \varLambda_{L}^{\sigma_{0}+1}}\right]\approx\frac{2{\rm \pi}^{2}(5-\sigma_{0}) D_{0}}{15\nu^{2}(\sigma_{0}+1)\varLambda_{L}^{\sigma_{0}+1}}, \end{gather}

(3.24)\begin{gather}\bar{\alpha}=\frac{8{\rm \pi}^{2}D_{1}}{3(\sigma_{1}-1)\nu(\nu+\eta) \varLambda_{L}^{\sigma_{1}-1}}. \end{gather}

It follows that models of weak turbulence with a forcing for which the correlations are defined with $\sigma _{0}>5$ are unphysical, since they lead to negative turbulent diffusion.

4. Renormalization procedure of the MHD equations

Let us introduce the non-dimensional variables

(4.1a)\begin{gather} \boldsymbol{f}\mapsto\mathcal{F}\boldsymbol{f},\quad \boldsymbol{U}\mapsto\mathcal{U}\boldsymbol{U},\quad \boldsymbol{b}\mapsto\mathcal{B}\boldsymbol{b}, \quad \langle \boldsymbol{B}\rangle \mapsto B_{0}\langle \boldsymbol{B}\rangle ,\quad \varPi\mapsto\mathscr{L}\mathcal{F}\varPi, \end{gather}

(4.1b)\begin{gather}\boldsymbol{x}\mapsto\mathscr{L}\boldsymbol{x},\quad t\mapsto\frac{\mathcal{U}}{\mathcal{F}}t, \end{gather}

(4.1c)\begin{gather}D_{0}\mapsto\frac{\mathcal{F}\mathcal{U}}{\mathscr{L}^{\sigma_{0}-3}}D_{0},\quad D_{1}\mapsto\frac{\mathcal{F}\mathcal{U}}{\mathscr{L}^{\sigma_{1} -4}}D_{1}, \end{gather}

where $\mathscr {L}$ is some length scale of variation of the fluctuations,

(4.2)\begin{equation} \frac{2{\rm \pi}}{\varLambda_{L}}\leq\mathscr{L}\leq\frac{2{\rm \pi}}{\varLambda},\quad \varLambda=min(\varLambda_{\nu,}\varLambda_{\eta}),\end{equation}

and the scales $\mathcal {U}$ and $\mathcal {B}$ are defined by the norm of the initial velocity fluctuation, i.e. $\mathcal {B}\sim \mathcal {U}=\Vert \boldsymbol {u}(t=0)\Vert _{L^{2}}$; in an analogous way we define $B_{0}=\Vert \langle \boldsymbol {B}\rangle (t=0)\Vert _{L^{2}}$ and the characteristic length scale of variation of the mean fields will be denoted by

(4.3)\begin{equation} L\lesssim\frac{2{\rm \pi}}{\kappa_{m}}.\end{equation}

Physically, we can associate the scale of the stirring force $\mathcal {F}=\sqrt {\omega _{s}D_{0}\mathscr {L}^{\sigma _{0}-3}}$, where $\omega _{s}$ is the scale of the fluctuational frequencies, with the magnitude of the driving force, thus, for example, the buoyancy force. On the other hand, the helical part proportional to $D_{1}$ can be associated with the Coriolis force, which in natural systems is typically responsible for introducing a lack of reflectional symmetry, thus $2\varOmega \mathcal {U}=\sqrt {\omega _{s}D_{1}\mathscr {L}^{\sigma _{1}-4}}$, where $\varOmega$ denotes the magnitude of the background rotation. The aim is to try to mimic in the simplest way the turbulence in astrophysical systems. Of course, this is a great simplification, because the Coriolis force introduces anisotropy (one distinguished axis of rotation), which is neglected here in order to obtain analytic results for strong isotropic turbulence. A study of the $\alpha$-effect in non-isotropic rapidly rotating turbulence via the renormalization theory has been done in Mizerski (Reference Mizerski2021), in an effectively weakly nonlinear limit.

We introduce the following non-dimensional parameters:

(4.4a–d)\begin{equation} Ro=\frac{\mathcal{U}^{2}}{\mathscr{L}\mathcal{F}},\quad E_{v}=\frac{\nu\mathcal{U}}{\mathscr{L}^{2}\mathcal{F}},\quad E_{\eta}=\frac{\eta\mathcal{U}}{\mathscr{L}^{2}\mathcal{F}},\quad \beta=\frac{B_{0}}{\mathcal{B}},\end{equation}

and

(4.5a,b)\begin{equation} H=\frac{\mathcal{B}^{2}}{\mathcal{U}^{2}},\quad \varepsilon=\frac{\mathscr{L}}{L}.\end{equation}

Defining

(4.6a,b)\begin{equation} \boldsymbol{X}=\varepsilon\boldsymbol{x},\quad \bar{\boldsymbol{\nabla}}=\boldsymbol{\nabla}_{\boldsymbol{X}},\end{equation}

and assuming

(4.7a,b)\begin{equation} \bar{\boldsymbol{\nabla}}\langle \boldsymbol{U}\rangle \sim\varepsilon \frac{\mathcal{U}}{\mathscr{L}},\quad \bar{\boldsymbol{\nabla}}\langle \boldsymbol{B}\rangle \sim\varepsilon\frac{\mathcal{B}}{\mathscr{L}}= \frac{\varepsilon}{\beta}\frac{B_{0}}{\mathscr{L}},\end{equation}

we can rewrite the dynamical equations in non-dimensional form

(4.8a)\begin{align} \frac{\partial\langle \boldsymbol{U}\rangle }{\partial t}+\varepsilon Ro (\langle \boldsymbol{U}\rangle \boldsymbol{\cdot} \bar{\boldsymbol{\nabla}})\langle \boldsymbol{U}\rangle &={-}\varepsilon\bar{\boldsymbol{\nabla}}\langle \varPi\rangle +\varepsilon Ro\beta H(\langle \boldsymbol{B}\rangle \boldsymbol{\cdot} \bar{\boldsymbol{\nabla}}) \langle \boldsymbol{B}\rangle +\varepsilon^{2}E_{\nu}\bar{\boldsymbol{\nabla}}^{2}\langle \boldsymbol{U}\rangle \nonumber\\ &\quad -\varepsilon Ro\bar{\boldsymbol{\nabla}}\boldsymbol{\cdot} (\langle \boldsymbol{u}\boldsymbol{u} \rangle -H\langle \boldsymbol{b}\boldsymbol{b}\rangle ), \end{align}

(4.8b)\begin{equation} \frac{\partial\langle \boldsymbol{B}\rangle }{\partial t}=\varepsilon Ro (\langle \boldsymbol{B}\rangle \boldsymbol{\cdot} \bar{\boldsymbol{\nabla}})\langle \boldsymbol{U}\rangle -\varepsilon\frac{Ro}{\beta}(\langle \boldsymbol{U}\rangle \boldsymbol{\cdot} \bar{\boldsymbol{\nabla}})\langle \boldsymbol{B}\rangle +\varepsilon Ro\beta^{{-}1}\bar{\boldsymbol{\nabla}}\times\langle \boldsymbol{u}\times\boldsymbol{b}\rangle +\frac{\varepsilon^{2}}{\beta}E_{\eta}\bar{\boldsymbol{\nabla}}^{2}\langle \boldsymbol{B}\rangle ,\end{equation}

(4.8c)\begin{align} &\frac{\partial\boldsymbol{u}}{\partial t}-E_{\nu}\nabla^{2} \boldsymbol{u}+Ro(\langle \boldsymbol{U}\rangle \boldsymbol{\cdot} \boldsymbol{\nabla} ) \boldsymbol{u}-Ro\beta H(\langle \boldsymbol{B}\rangle \boldsymbol{\cdot} \boldsymbol{\nabla} ) \boldsymbol{b}+\boldsymbol{\nabla} \varPi'= \boldsymbol{f}-Ro\boldsymbol{\nabla} \boldsymbol{\cdot} (\boldsymbol{u}\boldsymbol{u}-H\boldsymbol{b}\boldsymbol{b}) \nonumber\\ & \quad +Ro\varepsilon\bar{\boldsymbol{\nabla}}\boldsymbol{\cdot} (\langle \boldsymbol{u}\boldsymbol{u}\rangle -H\langle \boldsymbol{b}\boldsymbol{b}\rangle )- Ro\varepsilon(\boldsymbol{u}\boldsymbol{\cdot} \bar{\boldsymbol{\nabla}})\langle \boldsymbol{U}\rangle +RoH\varepsilon(\boldsymbol{b}\boldsymbol{\cdot} \bar{\boldsymbol{\nabla}}) \langle \boldsymbol{B}\rangle, \end{align}

(4.8d)\begin{align} \frac{\partial\boldsymbol{b}}{\partial t}-E_{\eta}\nabla^{2}\boldsymbol{b} +Ro(\langle \boldsymbol{U}\rangle \boldsymbol{\cdot} \boldsymbol{\nabla} )\boldsymbol{b}&= Ro\beta(\langle \boldsymbol{B}\rangle \boldsymbol{\cdot} \boldsymbol{\nabla} ) \boldsymbol{u}+Ro\boldsymbol{\nabla} \times(\boldsymbol{u}\times\boldsymbol{b}) -Ro\varepsilon\bar{\boldsymbol{\nabla}}\times\langle \boldsymbol{u}\times\boldsymbol{b}\rangle \nonumber\\ &\quad +Ro\varepsilon(\boldsymbol{b}\boldsymbol{\cdot} \bar{\boldsymbol{\nabla}})\langle \boldsymbol{U}\rangle -Ro\varepsilon(\boldsymbol{u}\boldsymbol{\cdot} \bar{\boldsymbol{\nabla}}) \langle \boldsymbol{B}\rangle , \end{align}

(4.8e)\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} \langle \boldsymbol{B}\rangle =0,\quad\boldsymbol{\nabla} \boldsymbol{\cdot} \langle \boldsymbol{U}\rangle =0,\quad\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{b}=0,\quad\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u}=0.\end{equation}

Note that $H\neq 0$ is associated with the presence of the Lorentz force, i.e. when $H=0$ the dynamo problem becomes kinematic. We stress here that the turbulent fluctuations are defined in such a way that they do not depend on the slow variable $\boldsymbol {X}$ since spatial scale separation has been assumed. Of course, the Reynolds and Maxwell stresses as nonlinear quantities do depend on the slow variable. For the formal two-scale direct-interaction approach see Yoshizawa (Reference Yoshizawa1998) and Yokoi (Reference Yokoi2020), where the dependencies on slow and fast variables are formally treated with the multiple-scale asymptotic method. We now introduce the following assumptions:

(4.9a)\begin{gather} Ro\ll1, \quad 1\ll\beta\ll Ro^{{-}1}, \end{gather}

(4.9b)\begin{gather}\frac{\kappa_{m}}{\varLambda_{L}}=O (Ro^{2}), \Rightarrow \varepsilon=O (Ro^{2}), \end{gather}

and

(4.10)\begin{equation} H=O (1),\end{equation}

where the assumption (4.9b) allows us to retain the effect of weak gradients of means $\boldsymbol {\nabla } \langle \boldsymbol {U}\rangle$ and $\boldsymbol {\nabla } \langle \boldsymbol {B}\rangle$ on the fluctuations at the highest order. Introducing a new shorter notation

(4.11a,b)\begin{equation} \boldsymbol{q}=(\boldsymbol{k},\omega),\quad \int_{\varLambda_{L}}^{\varLambda_{i}}\, \textrm{d}^{3}k\int_{-\infty}^{\infty}\, \textrm{d} \omega(\boldsymbol{\cdot} )=\int^{\varLambda_{i}}\, \textrm{d}^{4}q(\boldsymbol{\cdot}),\end{equation}

so that e.g.

(4.12)\begin{equation} u_{i}(\boldsymbol{x},t)=\int^{\varLambda_{\nu}}\, \textrm{d}^{4}q\hat{u}_{i}(\boldsymbol{q})\,\mathrm{e}^{\mathrm{i}(\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{x}-\omega t)},\end{equation}

and utilizing ((3.2a,b)–(3.5)) the equations for the fluctuations take the form

(4.13a)\begin{align} &(-\mathrm{i}\omega+E_{\nu}k^{2})\hat{u}_{i}(\boldsymbol{q})- \mathrm{i}Ro\beta H(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}) \hat{b}_{i}(\boldsymbol{q})+\mathrm{i}k_{i}\hat{\varPi}= \hat{f}_{i}(\boldsymbol{q})-\mathrm{i}Rok_{j} [\mathbb{I}_{ij}^{(u)}-H\mathbb{I}_{ij}^{(b)}] \nonumber\\ & \quad +\mathrm{i}Rok_{j}[\langle \mathbb{I}_{ij}^{(u)}\rangle -H \langle \mathbb{I}_{ij}^{(b)}\rangle ]-Ro\varepsilon G_{ij}\hat{u}_{j}(\boldsymbol{q}) +RoH\varepsilon\varGamma_{ij}\hat{b}_{j}(\boldsymbol{q}) \nonumber\\ & \quad -\frac{Ro}{2}k_{m}[\hat{u}_{i}(\boldsymbol{k}-\varepsilon \boldsymbol{G}_{m},\omega)-\hat{u}_{i}(\boldsymbol{k}+\varepsilon\boldsymbol{G}_{m}, \omega)] \nonumber\\ & \quad +\frac{Ro}{2}k_{m}[\hat{b}_{i}(\boldsymbol{k}- \varepsilon\boldsymbol{\varGamma}_{m},\omega)-\hat{b}_{i}(\boldsymbol{k}+ \varepsilon\boldsymbol{\varGamma}_{m},\omega)], \end{align}

(4.13b)\begin{align} (-\mathrm{i}\omega+E_{\eta}k^{2})\hat{b}_{i}(\boldsymbol{q})&= \mathrm{i}Ro\beta(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}) \hat{u}_{i}(\boldsymbol{q})+\mathrm{i}Rok_{j}\mathbb{I}_{ij}^{(ub)}- \mathrm{i}Rok_{j}\langle \mathbb{I}_{ij}^{(ub)}\rangle \nonumber\\ &\quad +Ro\varepsilon G_{ij}\hat{b}_{j}(\boldsymbol{q})-Ro\varepsilon \varGamma_{ij}\hat{u}_{j}(\boldsymbol{q}) \nonumber\\ &\quad -\frac{Ro}{2}k_{m}[\hat{b}_{i}(\boldsymbol{k}-\varepsilon\boldsymbol{G}_{m}, \omega)-\hat{b}_{i}(\boldsymbol{k}+\varepsilon\boldsymbol{G}_{m},\omega)] \nonumber\\ &\quad +\frac{Ro}{2}k_{m}[\hat{u}_{i}(\boldsymbol{k}-\varepsilon\boldsymbol{\varGamma}_{m}, \omega)-\hat{u}_{i}(\boldsymbol{k}+\varepsilon\boldsymbol{\varGamma}_{m},\omega)], \end{align}

(4.13c)\begin{equation} \boldsymbol{k}\boldsymbol{\cdot} \hat{\boldsymbol{u}}(\boldsymbol{q})=0, \quad\boldsymbol{k}\boldsymbol{\cdot} \hat{\boldsymbol{b}}(\boldsymbol{q})=0, \quad\boldsymbol{k}\boldsymbol{\cdot} \hat{\boldsymbol{f}}(\boldsymbol{q})=0,\end{equation}

where the constant term $-Ro\boldsymbol {k}\boldsymbol {\cdot } \langle \boldsymbol {U}\rangle _{0}$ has been absorbed into the frequency $\omega$ (as in the previous section) and

(4.14a,b)\begin{gather} \mathbb{I}_{ij}^{(u)}=\theta_{\varLambda_{\nu}}\int^{\varLambda_{\nu}}\, \textrm{d}^{4}q'\hat{u}_{i}(\boldsymbol{q}-\boldsymbol{q}')\hat{u}_{j}(\boldsymbol{q}'),\quad \mathbb{I}_{ij}^{(b)}=\theta_{\varLambda_{\eta}}\int^{\varLambda_{\eta}}\, \textrm{d}^{4}q'\hat{b}_{i}(\boldsymbol{q}-\boldsymbol{q}')\hat{b}_{j} (\boldsymbol{q}'), \end{gather}

(4.15)\begin{gather}\mathbb{I}_{ij}^{(ub)}=\theta_{\varLambda}\epsilon_{ijk}\epsilon_{kmn} \int^{\varLambda}\, \textrm{d}^{4}q'\hat{u}_{m}(\boldsymbol{q}-\boldsymbol{q}') \hat{b}_{n}(\boldsymbol{q}'), \end{gather}

with

(4.16a,b)\begin{equation} \varLambda=min(\varLambda_{\nu,}\varLambda_{\eta}),\quad \theta_{\varLambda_{i}}=\theta(\varLambda_{i}-k).\end{equation}

The convolution integrals possess the following symmetry properties:

(4.17a–c)\begin{equation} \mathbb{I}_{ij}^{(u)}=\mathbb{I}_{ji}^{(u)},\quad \mathbb{I}_{ij}^{(b)}=\mathbb{I}_{ji}^{(b)},\quad \mathbb{I}_{ij}^{(ub)}={-}\mathbb{I}_{ji}^{(ub)}.\end{equation}

Factors $\theta$ were added for clarity in the following calculations. It should be noted at this stage that the convolution integrals, which represent the nonlinear interactions between fluctuating turbulent fields, are not neglected in the evolution equations for the fluctuations (4.13a,b), and their role is crucial. In fact, it is the aim of the entire renormalization procedure to go beyond the weak turbulence regime and quantitatively express the effect of those nonlinearities on the dynamics of the mean fields. Therefore, contrary to the findings of the previous section, where the MHD turbulence was studied in the weak regime (i.e. under neglection of $\mathbb {I}_{ij}^{(u)}, \mathbb {I}_{ij}^{(b)}$ and $\mathbb {I}_{ij}^{(ub)}$ in (4.13a,b)), we will now investigate the nonlinear evolution of the turbulent fluctuations and its effect on the dynamics of mean fields. This corresponds to a regime when the turbulence is no longer weak and, although we have assumed $Ro\ll 1$, we will refer to the studied regime as the ‘strong-turbulence’ regime, to clearly distinguish from the simplest case of a weak, linear turbulence.

In order to eliminate pressure we apply the projection operator (2.13) to both sides of the Navier–Stokes equation (4.13a) and, after some simple algebra, we get more explicit expressions for $\hat {u}_{i}(\boldsymbol {q})$ and $\hat {b}_{i}(\boldsymbol {q})$

(4.18a)\begin{align} \hat{u}_{i}(\boldsymbol{q})&= \frac{1}{\gamma_{u}}\hat{f}_{i}(\boldsymbol{q})- \tfrac{1}{2}\mathrm{i}Ro\frac{P_{imn}(\boldsymbol{k})}{\gamma_{u}} [\mathbb{I}_{mn}^{(u)}-H\mathbb{I}_{mn}^{(b)}-\langle \mathbb{I}_{mn}^{(u)}\rangle +H\langle \mathbb{I}_{mn}^{(b)}\rangle ] \nonumber\\ & \quad -Ro(Ro\beta)H\frac{\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}}{\gamma_{u}\gamma_{\eta}}k_{j} [\mathbb{I}_{ij}^{(ub)}-\langle \mathbb{I}_{ij}^{(ub)}\rangle ] \nonumber\\ & \quad -Ro\varepsilon\frac{P_{ij}(\boldsymbol{k})}{\gamma_{u}}G_{jk}\hat{u}_{k} (\boldsymbol{q})+RoH\varepsilon\frac{P_{ij}(\boldsymbol{k})}{\gamma_{u}} \varGamma_{jk}\hat{b}_{k}(\boldsymbol{q}) \nonumber\\ &\quad +\mathrm{i}Ro(Ro\beta)H\varepsilon\frac{\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}}{\gamma_{u}\gamma_{\eta}}G_{ij}\hat{b}_{j} (\boldsymbol{q})-\mathrm{i}Ro(Ro\beta)H\varepsilon\frac{\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}}{\gamma_{u}\gamma_{\eta}} \varGamma_{ij}\hat{u}_{j}(\boldsymbol{q}) \nonumber\\ &\quad -\frac{Ro}{2\gamma_{u}}k_{m}P_{ij}(\boldsymbol{k}) [\hat{u}_{j}(\boldsymbol{k}-\varepsilon\boldsymbol{G}_{m},\omega)- \hat{u}_{j}(\boldsymbol{k}+\varepsilon\boldsymbol{G}_{m},\omega)] \nonumber\\ &\quad +\frac{Ro}{2\gamma_{u}}k_{m}P_{ij}(\boldsymbol{k}) [\hat{b}_{j}(\boldsymbol{k}-\varepsilon\boldsymbol{\varGamma}_{m},\omega)- \hat{b}_{j}(\boldsymbol{k}+\varepsilon\boldsymbol{\varGamma}_{m},\omega)] \nonumber\\ &\quad -\mathrm{i}Ro(Ro\beta)H\frac{\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}}{2\gamma_{u}\gamma_{\eta}}k_{m} [\hat{b}_{i}(\boldsymbol{k}-\varepsilon\boldsymbol{G}_{m},\omega)-\hat{b}_{i} (\boldsymbol{k}+\varepsilon\boldsymbol{G}_{m},\omega)] \nonumber\\ &\quad +\mathrm{i}Ro(Ro\beta)H\frac{\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}}{2\gamma_{u}\gamma_{\eta}}k_{m} [\hat{u}_{i}(\boldsymbol{k}-\varepsilon\boldsymbol{\varGamma}_{m},\omega)- \hat{u}_{i}(\boldsymbol{k}+\varepsilon\boldsymbol{\varGamma}_{m},\omega)], \end{align}

(4.18b)\begin{align} \hat{b}_{i}(\boldsymbol{q})&= \mathrm{i}Ro\beta\frac{\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}}{\gamma_{\eta}}\hat{u}_{i}(\boldsymbol{q})+ \mathrm{i}\frac{Ro}{\gamma_{\eta}}k_{j}[\mathbb{I}_{ij}^{(ub)}- \langle \mathbb{I}_{ij}^{(ub)}\rangle ] \nonumber\\ &\quad +\frac{Ro}{\gamma_{\eta}}\varepsilon G_{ij}\hat{b}_{j}(\boldsymbol{q})- \frac{Ro}{\gamma_{\eta}}\varepsilon\varGamma_{ij}\hat{u}_{j}(\boldsymbol{q}) \nonumber\\ &\quad -\frac{Ro}{2\gamma_{\eta}}k_{m}[\hat{b}_{i}(\boldsymbol{k}-\varepsilon \boldsymbol{G}_{m},\omega)-\hat{b}_{i}(\boldsymbol{k}+\varepsilon\boldsymbol{G}_{m}, \omega)] \nonumber\\ &\quad +\frac{Ro}{2\gamma_{\eta}}k_{m}[\hat{u}_{i}(\boldsymbol{k}- \varepsilon\boldsymbol{\varGamma}_{m},\omega)-\hat{u}_{i}(\boldsymbol{k}+ \varepsilon\boldsymbol{\varGamma}_{m},\omega)], \end{align}

where

(4.19)\begin{gather} P_{imn}(\boldsymbol{k})=k_{m}P_{in}(\boldsymbol{k})+k_{n}P_{im}(\boldsymbol{k}), \end{gather}

(4.20)\begin{gather}\gamma_{u}=\gamma_{\nu}+H(Ro\beta)^{2}\frac{(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2}}{\gamma_{\eta}},\quad\gamma_{\nu}={-}\mathrm{i}\omega+ E_{\nu}k^{2},\quad\gamma_{\eta}={-}\mathrm{i}\omega+E_{\eta}k^{2}. \end{gather}

The following simple transformation allows us to return to original dimensional variables

(4.21a–e)\begin{equation} E_{\nu}\rightarrow\nu, \quad E_{\eta}\rightarrow\eta, \quad H\rightarrow1, \quad Ro\rightarrow1, \quad \beta\rightarrow1.\end{equation}

In the non-dimensional variables the isotropic, homogeneous and stationary forcing is still defined by the same formula

(4.22)\begin{equation} \langle\hat{f}_{i}(\boldsymbol{k},\omega)\hat{f}_{j}(\boldsymbol{k}',\omega')\rangle =\left[\frac{D_{0}}{k^{\sigma_{0}}}P_{ij}(\boldsymbol{k})+\mathrm{i} \frac{D_{1}}{k^{\sigma_{1}}}\epsilon_{ijk}k_{k}\right]\delta(\boldsymbol{k}+ \boldsymbol{k}')\delta(\omega+\omega').\end{equation}

(Of course when we return to dimensional variables $D_{0}\rightarrow D_{0}, D_{1}\rightarrow D_{1}$.)

We can now comment on the second assumption in (4.9a), which states that the mean field is much stronger than the fluctuating one, but $Ro\beta \ll 1$. The former allows for proper formulation of the problem, since, for a strong mean field, the fluctuating magnetic field (4.18b) is expressed at leading order by the stirring force and the nonlinearities, which are of the order $O(Ro)$, and can be treated in a perturbational sense. The iterational procedure of renormalization is then applicable. On the other hand, the assumption $Ro\beta \ll 1$ allows for expansion of factors such as $\gamma _{u}^{-1}$ and therefore explicit calculation of Fourier integrals in the nonlinear $\mathbb {I}$-terms. In turn, the final recursion differential relations for the coefficients describing the renormalized EMF, the Reynolds stresses and the Lorentz force can be solved analytically; thus, in particular, the full leading-order form of the turbulent diffusivities and the $\alpha$-coefficient can be determined.

We now start the renormalization procedure of taking successive little bites off the Fourier spectrum from the short-wavelength side in order to obtain the final nonlinear effect of the fluctuations on the means. At the first step of the procedure we introduce a parameter $\lambda _{1}$, which satisfies

(4.23)\begin{equation} \delta\lambda=\varLambda_{max}-\lambda_{1}\ll1,\quad\text{where } \varLambda_{max}=\mathrm{max}(\varLambda_{\nu},\varLambda_{\eta}),\end{equation}

and divide the Fourier spectrum into two parts

(4.24)\begin{gather} \hat{u}_{i}^{>}(\boldsymbol{k},\omega)=\theta(k-\lambda_{1}) \hat{u}_{i}(\boldsymbol{k},\omega),\textrm{ or } \hat{u}_{i}^{>}(\boldsymbol{k},\omega)=\hat{u}_{i}(\boldsymbol{k}^{>},\omega), \quad\lambda_{1}<|\boldsymbol{k}^{>}|<\varLambda_{max}, \end{gather}

(4.25)\begin{gather}\hat{u}_{i}^{<}(\boldsymbol{k},\omega)=\theta(\lambda_{1}-k) \hat{u}_{i}(\boldsymbol{k},\omega),\textrm{ or }\hat{u}_{i}^{<} (\boldsymbol{k},\omega)=\hat{u}_{i}(\boldsymbol{k}^{<},\omega), \quad|\boldsymbol{k}^{<}|<\lambda_{1}, \end{gather}

and the same for $\hat {\boldsymbol {b}}$ and $\hat {\boldsymbol {f}}$. The equations for the fields $\hat {u}_{i}^{<}(\boldsymbol {k},\omega )$ and $\hat {b}_{i}^{<}(\boldsymbol {k},\omega )$ are obtained by projecting (4.13a) onto the direction perpendicular to $\boldsymbol {k}$ (with the use of (2.13)), and averaging both $\boldsymbol {P}(\boldsymbol {k})\cdot$ (4.13a) and (4.13b) over the first shell ($\lambda _{1}< k<\varLambda _{max}$)

(4.26a)\begin{align} (-\mathrm{i}\omega+E_{\nu}k^{2})\hat{u}_{i}^{<}(\boldsymbol{q}) &= \hat{f}_{i}^{<}(\boldsymbol{q})+\mathrm{i}Ro\beta H (\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0})\hat{b}_{i}^{<} (\boldsymbol{q})\nonumber\\ &\quad -\tfrac{1}{2}\mathrm{i}RoP_{imn}(\boldsymbol{k})[\mathbb{I}_{mn}^{(u^{<})}- H\mathbb{I}_{mn}^{(b^{<})}-\langle \mathbb{I}_{mn}^{(u^{<})}\rangle +H \langle \mathbb{I}_{mn}^{(b^{<})}\rangle ] \nonumber\\ &\quad -\tfrac{1}{2}\mathrm{i}RoP_{imn}(\boldsymbol{k}) \left\{ \theta_{\varLambda_{\nu}}\int^{\varLambda_{\nu}}\, \textrm{d}^{4}q' [\langle \hat{u}_{m}^{>}(\boldsymbol{q}')\hat{u}_{n}^{>}(\boldsymbol{q}- \boldsymbol{q}')\rangle_{c}\right.\nonumber\\ &\left.\quad-\langle \hat{u}_{m}^{>}(\boldsymbol{q}')\hat{u}_{n}^{>} (\boldsymbol{q}-\boldsymbol{q}')\rangle ]\right. \nonumber\\ &\quad -\left.H\theta_{\varLambda_{\eta}}\int^{\varLambda_{\eta}}\, \textrm{d}^{4}q' [\langle \hat{b}_{m}^{>}(\boldsymbol{q}')\hat{b}_{n}^{>}(\boldsymbol{q}- \boldsymbol{q}')\rangle_{c}-\langle \hat{b}_{m}^{>}(\boldsymbol{q}') \hat{b}_{n}^{>}(\boldsymbol{q}-\boldsymbol{q}')\rangle ]\right\} \nonumber\\ &\quad -Ro\varepsilon P_{ij}(\boldsymbol{k})G_{jk}\hat{u}_{k}^{<} (\boldsymbol{q})+RoH\varepsilon P_{ij}(\boldsymbol{k}) \varGamma_{jk}\hat{b}_{k}^{<}(\boldsymbol{q}) \nonumber\\ &\quad -\frac{Ro}{2}P_{ij}(\boldsymbol{k})k_{m} [\hat{u}_{j}(\boldsymbol{k}^{<}-\varepsilon\boldsymbol{G}_{m},\omega)- \hat{u}_{j}(\boldsymbol{k}^{<}+\varepsilon\boldsymbol{G}_{m},\omega)] \nonumber\\ &\quad +\frac{Ro}{2}P_{ij}(\boldsymbol{k})k_{m}[\hat{b}_{j}(\boldsymbol{k}^{<}- \varepsilon\boldsymbol{\varGamma}_{m},\omega)-\hat{b}_{j}(\boldsymbol{k}^{<}+ \varepsilon\boldsymbol{\varGamma}_{m},\omega)] \end{align}

(4.26b) \begin{align} (-\mathrm{i}\omega+E_{\eta}k^{2})\hat{b}_{i}^{<}(\boldsymbol{q})&= \mathrm{i}Ro\beta(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}) \hat{u}_{i}^{<}(\boldsymbol{q})+\mathrm{i}Rok_{j}[\mathbb{I}_{ij}^{(u^{<}b^{<})}- \langle \mathbb{I}_{ij}^{(u^{<}b^{<})}\rangle ] \nonumber\\ & \quad +\mathrm{i}Rok_{j}\epsilon_{ijk}\epsilon_{kmn}\int^{\varLambda}\, \textrm{d}^{4}q'[\langle \hat{u}_{m}^{>}(\boldsymbol{q}')\hat{b}_{n}^{>}(\boldsymbol{q}- \boldsymbol{q}')\rangle_{c}\nonumber\\ & \quad-\langle \hat{u}_{m}^{>}(\boldsymbol{q}')\hat{b}_{n}^{>} (\boldsymbol{q}-\boldsymbol{q}')\rangle ] +Ro\varepsilon G_{ij}\hat{b}_{j}^{<}(\boldsymbol{q})-Ro\varepsilon \varGamma_{ij}\hat{u}_{j}^{<}(\boldsymbol{q}) \nonumber\\ & \quad -\frac{Ro}{2}k_{m}[\hat{b}_{i}(\boldsymbol{k}^{<}- \varepsilon\boldsymbol{G}_{m},\omega)-\hat{b}_{i}(\boldsymbol{k}^{<}+ \varepsilon\boldsymbol{G}_{m},\omega)] \nonumber\\ & \quad +\frac{Ro}{2}k_{m}[\hat{u}_{i}(\boldsymbol{k}^{<}-\varepsilon \boldsymbol{\varGamma}_{m},\omega)-\hat{u}_{i}(\boldsymbol{k}^{<}+ \varepsilon\boldsymbol{\varGamma}_{m},\omega)]. \end{align}

On the other hand, to get equations for $\hat {u}_{i}^{>}(\boldsymbol {k},\omega )$ and $\hat {b}_{i}^{>}(\boldsymbol {k},\omega )$ we utilize (4.18a,b), however, we substitute explicitly for $\hat {u}_{i}(\boldsymbol {q})$ from (4.18a) into the equation for the fluctuating magnetic field (4.18b) but neglect terms of the order $o(Ro^{3})$ and $O(Ro^{2}(Ro\beta )^{2})$

(4.27a)\begin{align} \hat{u}_{i}^{>}(\boldsymbol{q})&= \frac{1}{\gamma_{u}}\hat{f}_{i}^{>} (\boldsymbol{q})-\tfrac{1}{2}\mathrm{i}Ro\frac{P_{imn}(\boldsymbol{k})}{\gamma_{u}} [\mathbb{I}_{mn}^{(u^{<})}-H\mathbb{I}_{mn}^{(b^{<})}-\langle \mathbb{I}_{mn}^{(u^{<})} \rangle +H\langle \mathbb{I}_{mn}^{(b^{<})}\rangle ] \nonumber\\ & \quad -HRo(Ro\beta)\frac{\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B} \rangle_{0}}{\gamma_{u}\gamma_{\eta}}k_{j}[\mathbb{I}_{ij}^{(u^{<}b^{<})}- \langle \mathbb{I}_{ij}^{(u^{<}b^{<})}\rangle ] \nonumber\\ & \quad -Ro\varepsilon\frac{P_{ij}(\boldsymbol{k})}{\gamma_{u}^{2}}G_{jk}\hat{f}_{k}^{>} (\boldsymbol{q})\nonumber\\ &\quad -\frac{Ro}{2\gamma_{u}}k_{m}P_{ij}(\boldsymbol{k})\left[\frac{\hat{f}_{j} (\boldsymbol{k}^{>}-\varepsilon\boldsymbol{G}_{m},\omega)}{\gamma_{u} (\boldsymbol{k}^{>}-\varepsilon\boldsymbol{G}_{m},\omega)}- \frac{\hat{f}_{j}(\boldsymbol{k}^{>}+\varepsilon\boldsymbol{G}_{m}, \omega)}{\gamma_{u}(\boldsymbol{k}^{>}+\varepsilon\boldsymbol{G}_{m},\omega)}\right] \nonumber\\ &\quad -\mathrm{i}\frac{Ro}{\gamma_{u}}P_{imn}[\mathbb{J}_{mn}^{(u)}-H \mathbb{J}_{mn}^{(b)}]-HRo(Ro\beta)\frac{\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}}{\gamma_{u}\gamma_{\eta}}k_{j}\mathbb{J}_{ij}^{(ub)} +R_{i}^{(u)}, \end{align}

(4.27b)\begin{align} \hat{b}_{i}^{>}(\boldsymbol{q})&= \mathrm{i}Ro\beta\frac{\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}}{\gamma_{u}\gamma_{\eta}}\hat{f}_{i}^{>} (\boldsymbol{q})+\mathrm{i}\frac{Ro}{\gamma_{\eta}}k_{j} [\mathbb{I}_{ij}^{(u^{<}b^{<})}-\langle \mathbb{I}_{ij}^{(u^{<}b^{<})}\rangle ] \nonumber\\ &\quad +\tfrac{1}{2}Ro(Ro\beta)\frac{\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}}{\gamma_{u}\gamma_{\eta}}P_{imn}(\boldsymbol{k}) [\mathbb{I}_{mn}^{(u^{<})}-H\mathbb{I}_{mn}^{(b^{<})}-\langle \mathbb{I}_{mn}^{(u^{<})}\rangle +H\langle \mathbb{I}_{mn}^{(b^{<})}\rangle ] \nonumber\\ &\quad -\frac{Ro}{\gamma_{u}\gamma_{\eta}}\varepsilon\varGamma_{ij} \hat{f}_{j}^{>}(\boldsymbol{q})+\frac{Ro}{2\gamma_{\eta}}k_{m} \left[\frac{\hat{f}_{i}(\boldsymbol{k}^{>}-\varepsilon\boldsymbol{\varGamma}_{m}, \omega)}{\gamma_{u}(\boldsymbol{k}^{>}-\varepsilon\boldsymbol{\varGamma}_{m}, \omega)}-\frac{\hat{f}_{i}(\boldsymbol{k}^{>}+\varepsilon \boldsymbol{\varGamma}_{m},\omega)}{\gamma_{u}(\boldsymbol{k}^{>}+ \varepsilon\boldsymbol{\varGamma}_{m},\omega)}\right] \nonumber\\ &\quad +\mathrm{i}\frac{Ro}{\gamma_{\eta}}k_{j}\mathbb{J}_{ij}^{(ub)} +Ro(Ro\beta)\frac{\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B} \rangle_{0}}{\gamma_{u}\gamma_{\eta}}P_{imn}[\mathbb{J}_{mn}^{(u)} -H\mathbb{J}_{mn}^{(b)}]+R_{i}^{(b)}, \end{align}

where $\langle \boldsymbol {\cdot } \rangle _{c}$ denotes the conditional average over the first shell ($\lambda _{1}< k<\varLambda _{max}$) statistical subensemble, described at the beginning of Appendix A (cf. McComb et al. Reference McComb, Roberts and Watt1992 and McComb & Watt Reference McComb and Watt1990, Reference McComb and Watt1992); furthermore, we have defined

(4.28a)\begin{gather} \mathbb{J}_{mn}^{(u)}(\boldsymbol{q})=\theta_{\varLambda_{\nu}} \int^{\varLambda_{\nu}}\, \textrm{d}^{4}q'\hat{u}_{m}^{<}(\boldsymbol{q}') \hat{u}_{n}^{>}(\boldsymbol{q}-\boldsymbol{q}'), \end{gather}

(4.28b)\begin{gather}\mathbb{J}_{mn}^{(b)}(\boldsymbol{q})=\theta_{\varLambda_{\eta}}\int^{\varLambda_{\eta}}\, \textrm{d}^{4}q'\hat{b}_{m}^{<}(\boldsymbol{q}')\hat{b}_{n}^{>}(\boldsymbol{q}-\boldsymbol{q}'), \end{gather}

(4.28c)\begin{gather}\mathbb{J}_{ij}^{(ub)}(\boldsymbol{q})=\epsilon_{ijk}\epsilon_{kmn}\int^{\varLambda}\, \textrm{d}^{4}q'[\hat{u}_{m}^{<}(\boldsymbol{q}')\hat{b}_{n}^{>}(\boldsymbol{q}- \boldsymbol{q}')+\hat{u}_{m}^{>}(\boldsymbol{q}')\hat{b}_{n}^{<}(\boldsymbol{q}- \boldsymbol{q}')], \end{gather}

and the rests in (4.27a,b) are given by

(4.29a)\begin{align} R_{i}^{(u)}&={-}\frac{Ro}{\gamma_{u}}\left\{ \tfrac{1}{2}\mathrm{i}P_{imn} [\mathbb{I}_{mn}^{(u^{>})}-H\mathbb{I}_{mn}^{(b^{>})}]+HRo\beta \frac{\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle }{\gamma_{\eta}} k_{j}\mathbb{I}_{ij}^{(u^{>}b^{>})}\right\} \nonumber\\ & \quad +\frac{Ro}{\gamma_{u}}\left\{ \tfrac{1}{2}\mathrm{i}P_{imn} [\langle \mathbb{I}_{mn}^{(u^{>})}\rangle -H\langle \mathbb{I}_{mn}^{(b^{>})} \rangle ]+HRo\beta\frac{\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B} \rangle }{\gamma_{\eta}}k_{j}\langle \mathbb{I}_{ij}^{(u^{>}b^{>})}\rangle \right\} , \end{align}

(4.29b)\begin{align} R_{i}^{(b)} &= \mathrm{i}\frac{Ro}{\gamma_{\eta}}k_{j} [\mathbb{I}_{ij}^{(u^{>}b^{>})}-\langle \mathbb{I}_{ij}^{(u^{>}b^{>})}\rangle ] \nonumber\\ &\quad +\tfrac{1}{2}Ro(Ro\beta)\frac{\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B} \rangle_{0}}{\gamma_{u}\gamma_{\eta}}P_{imn}[\mathbb{I}_{mn}^{(u^{>})} -H\mathbb{I}_{mn}^{(b^{>})}-\langle \mathbb{I}_{mn}^{(u^{>})}\rangle +H \langle \mathbb{I}_{mn}^{(b^{>})}\rangle ] \nonumber\\ &\quad -\mathrm{i}HRo(Ro\beta)^{2}\frac{(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2}}{\gamma_{u}\gamma_{\eta}^{2}}k_{j} [\mathbb{I}_{ij}^{(u^{<}b^{<})}+\mathbb{J}_{ij}^{(ub)}+\mathbb{I}_{ij}^{(u^{>}b^{>})}- \langle \mathbb{I}_{ij}^{(u^{<}b^{<})}\rangle -\langle \mathbb{I}_{ij}^{(u^{>}b^{>})} \rangle ]. \end{align}

The rests will be neglected either on the basis of generating only third-order statistical correlations, as in the case of all the terms of second order in $\boldsymbol {u}^{>}$ or $\boldsymbol {b}^{>}$, or because of the kept order of accuracy in the asymptotic limit $Ro\ll Ro\beta \ll 1$, which will allow us to neglect terms of order $O(Ro(Ro\beta )^{2})$. For details the reader is referred to Appendix A.

In what follows we provide a short description of the asymptotic renormalization procedure, described in detail in Appendix A. First, we introduce (4.27a,b) into (4.26a,b) and calculate the dynamical effect of short-wavelength components $\hat {u}_{i}^{>}(\boldsymbol {k},\omega )$ and $\hat {b}_{i}^{>}(\boldsymbol {k},\omega )$ on the evolution of $\hat {u}_{i}^{<}(\boldsymbol {k},\omega )$ and $\hat {b}_{i}^{<}(\boldsymbol {k},\omega )$ (long-wavelength modes). This results in corrections to some of the terms in (4.26a,b), but also generates terms with a new structure. Therefore, a next step is necessary, involving calculation of the effect of the next shell $\lambda _{2}=\lambda _{1}-\delta \lambda < k<\lambda _{1}$ (new short-wavelength modes) on the modes with $k<\lambda _{1}-\delta \lambda$ (new long-wavelength modes). We can then take the limit of infinitesimally narrow wavenumber bands $\delta \lambda \rightarrow 0$, which leads to differential recursion relations for all the coupling constants introduced into the equations for long-wavelength modes by couplings of the short-wavelength ones. Moffatt (Reference Moffatt1983) obtained such equations but in the kinematic case with the turbulence flow given beforehand, unaffected by the Lorentz force. In the non-magnetic case, Yakhot & Orszag (Reference Yakhot and Orszag1986) calculated the leading-order correction from short-wavelength modes in the Navier–Stokes equation which was proportional to $k^{2}\hat {u}_{i}^{<}\delta \lambda$, thus creating a viscosity correction; the turbulent viscosity was then obtained from an equation of the form $\mathrm {d}\nu _{turb}/\mathrm {d}\lambda =f(\lambda )$ with an ‘ initial’ condition $\nu _{eff}(\lambda =\varLambda )=\nu$. In the case at hand under the assumptions $Ro\ll Ro\beta \ll 1$, the explicit calculation of two initial steps of the renormalization procedure is enough to derive the final differential recursion relations with satisfactory accuracy. The details of the procedure are provided in Appendix A.

4.1. Dynamics of mean fields

The renormalized mean-field equations take the following form:

(4.30a)\begin{gather} \frac{\partial\langle \boldsymbol{U}\rangle }{\partial t}+ (\langle \boldsymbol{U}\rangle \boldsymbol{\cdot} \boldsymbol{\nabla} )\langle \boldsymbol{U}\rangle ={-}\boldsymbol{\nabla} \left(\frac{\langle p\rangle }{\rho}+\bar{Q}_{p} \frac{\langle \boldsymbol{B}\rangle ^{2}}{2}\right)+\bar{Q} (\langle \boldsymbol{B}\rangle \boldsymbol{\cdot} \boldsymbol{\nabla} )\langle \boldsymbol{B}\rangle +\bar{\nu}\nabla^{2}\langle \boldsymbol{U}\rangle, \end{gather}

(4.30b)\begin{gather}\frac{\partial\langle \boldsymbol{B}\rangle }{\partial t}=\boldsymbol{\nabla} \times (\bar{\alpha}\langle \boldsymbol{B}\rangle )+ \boldsymbol{\nabla} \times(\langle \boldsymbol{U}\rangle \times\langle \boldsymbol{B}\rangle )+\bar{\eta}\nabla^{2}\langle \boldsymbol{B}\rangle , \end{gather}

where the coefficients $\bar {\nu }, \bar {\eta }, \bar {\alpha }, \bar {Q}$ and $\bar {Q}_{p}$ include the effect of the turbulent fluctuations on the means. We note that, in the absence of chirality (in non-helical turbulence), $D_{1}=0$, the $\alpha$-effect vanishes, i.e. $\bar {\alpha }=0$. In such a case, unless the mean flow is capable of generating the mean magnetic field through advection and stretching, that is via the term $\boldsymbol {\nabla } \times (\langle \boldsymbol {U}\rangle \times \langle \boldsymbol {B}\rangle )$, the mean magnetic field can only decay from some initial state through the effective diffusivity $\bar {\eta }$ and the transfer of the magnetic energy to the kinetic one via the Lorentz force. In other words, unless the energy transfer through $\boldsymbol {\nabla } \times (\langle \boldsymbol {U}\rangle \times \langle \boldsymbol {B}\rangle )$ and the Lorentz force can account for magnetic energy gain, in the reflectionally symmetric case we are dealing with the problem of magnetic energy relaxation. On the other hand, in chiral turbulence, the $\alpha$-effect can account for rapid amplification of the energy of the mean magnetic field. The saturation of the energy occurs via the effect of the turbulent magnetic diffusivity and the effect of the Lorentz force present in the $\bar {\alpha }$ coefficient. The two coefficients $\bar {Q}$ and $\bar {Q}_{P}$ in the mean-field Navier–Stokes equation describe the renormalized Lorentz force. The general differential recursion relations for all the renormalized coefficients $\bar {\nu }, \bar {\eta }, \bar {\alpha }, \bar {Q}$ and $\bar {Q}_{P}$ are solved in Appendix A, see (A91a–d) and below.

4.2. Dynamics of turbulent fluctuations in the limit $k\rightarrow \varLambda _{L}$

The evolution of the fluctuating fields in the limit $k\rightarrow \varLambda _{L}$ is governed by the following set of equations in the Fourier space:

(4.31a)\begin{align} [-\mathrm{i}\omega+\breve{\nu}(k)k^{2}]\hat{u}_{i}(\boldsymbol{q})&= \hat{f}_{i}(\boldsymbol{q})+\mathrm{i}(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0})\hat{b}_{i}(\boldsymbol{q})\nonumber\\ &\quad -\tfrac{1}{2}\mathrm{i}P_{imn}(\boldsymbol{k}) [\mathbb{I}_{mn}^{(u)}-\mathbb{I}_{mn}^{(b)}-\langle \mathbb{I}_{mn}^{(u)} \rangle +\langle \mathbb{I}_{mn}^{(b)}\rangle ] \nonumber\\ &\quad -P_{ij}(\boldsymbol{k})\frac{\partial\langle U\rangle_{j}}{\partial x_{k}}\hat{u}_{k}(\boldsymbol{q})+P_{ij}(\boldsymbol{k}) \frac{\partial\langle B\rangle_{j}}{\partial x_{k}}\hat{b}_{k}(\boldsymbol{q}) \nonumber\\ &\quad -\tfrac{1}{2}P_{ij}(\boldsymbol{k})k_{m}[\hat{u}_{j}(\boldsymbol{k}- \boldsymbol{\nabla} \langle U\rangle_{m},\omega)-\hat{u}_{j}(\boldsymbol{k}+\boldsymbol{\nabla} \langle U\rangle_{m},\omega)] \nonumber\\ &\quad +\tfrac{1}{2}P_{ij}(\boldsymbol{k})k_{m}[\hat{b}_{j}(\boldsymbol{k}-\boldsymbol{\nabla} \langle B\rangle_{m},\omega)-\hat{b}_{j}(\boldsymbol{k}+\boldsymbol{\nabla} \langle B\rangle_{m},\omega)], \end{align}

(4.31b)\begin{align} &[-\mathrm{i}\omega+\breve{\eta}(k)k^{2}]\hat{b}_{i}(\boldsymbol{q})- \mathrm{i}\breve{\alpha}(k)\epsilon_{ijk}k_{j}\hat{b}_{k}(\boldsymbol{q})\nonumber\\&\quad= \mathrm{i}(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle_{0}) \hat{u}_{i}(\boldsymbol{q})+\mathrm{i}Rok_{j} [\mathbb{I}_{ij}^{(ub)}-\langle \mathbb{I}_{ij}^{(ub)}\rangle ] \nonumber\\ &\qquad +\frac{\partial\langle U\rangle_{i}}{\partial x_{j}} \hat{b}_{j}(\boldsymbol{q})-\frac{\partial\langle B\rangle_{i}}{\partial x_{j}}\hat{u}_{j}(\boldsymbol{q}) \nonumber\\ &\qquad -\tfrac{1}{2}k_{m}[\hat{b}_{i}(\boldsymbol{k}-\boldsymbol{\nabla} \langle U\rangle_{m},\omega)-\hat{b}_{i}(\boldsymbol{k}+\boldsymbol{\nabla} \langle U\rangle_{m},\omega)] \nonumber\\ &\qquad +\tfrac{1}{2}k_{m}[\hat{u}_{i}(\boldsymbol{k}-\boldsymbol{\nabla} \langle B\rangle_{m},\omega) -\hat{u}_{i}(\boldsymbol{k}+\boldsymbol{\nabla} \langle B\rangle_{m},\omega)]. \end{align}

The general formulae for all the renormalized coefficients $\breve {\nu }(k), \breve {\eta }(k)$ and $\breve {\alpha }(k)$ are provided in Appendix A, see (A54a,b) and below. We now turn to the problem of turbulent energy and helicity spectra and their scaling laws with respect to the wavenumber.

4.2.1. Energy and helicity spectra in the limit $k\rightarrow \varLambda _{L}$

It is of interest to determine the scaling exponents of the energy and helicity spectra and show how the scaling laws change when the turbulence shifts from the weak regime to the strong one; the former is characterized by weak fluctuations and thus the dynamical equations for fluctuations are linearized about the means as in (3.6a,b), whereas, in the strong regime, defined by (4.8c,d), the dynamics is predominantly determined by the nonlinearities $\boldsymbol {\nabla } \boldsymbol {\cdot } (\boldsymbol {u}\boldsymbol {u}-\boldsymbol {b}\boldsymbol {b})$ and $\boldsymbol {\nabla } \times (\boldsymbol {u}\times \boldsymbol {b})$ in the fluctuational equations. Naturally, the spectral scaling laws for energies and helicity will be determined by the exponents $\sigma _{0}$ and $\sigma _{1}$, which define the correlation function of the stirring force, cf. (2.12). Unfortunately, those are not well-established quantities. The only clue is provided by comparison with the Kolmogorov-like spectral scaling laws in the non-magnetic case. Such a comparison allows us to determine the spectral structure of the forcing correlations (i.e. the values of the exponents $\sigma _{0}$ and $\sigma _{1}$) which lead to the expected kinetic energy and helicity spectra in the absence of a magnetic field. Yakhot & Orszag (Reference Yakhot and Orszag1986) have demonstrated that the purely hydrodynamic (non-MHD), viscously controlled renormalized scaling of the kinetic energy spectrum agrees with the Kolmogorov $k^{-5/3}$ law for $\sigma _{0}=3$. Below, we will demonstrate that the value of $\sigma _{1}=5$ corresponds to $k^{-5/3}$ spectral scaling for turbulent helicity (expected in helical isotropic turbulence, cf. Brissaud et al. Reference Brissaud, Frisch, Leorat, Lesieur and Mazure1973 or Chen et al. Reference Chen, Chen, Eyink and Holm2003).

Within the considered asymptotic limit $Ro\ll Ro\beta \ll 1$ the fluctuating velocity and magnetic fields at each step of the renormalization procedure at the leading order are given by

(4.32a,b)\begin{gather} \hat{u}_{i}(\boldsymbol{q})\approx\frac{1}{\gamma_{\nu} (\boldsymbol{q})}\hat{f}_{i}(\boldsymbol{q}),\quad \hat{b}_{i}(\boldsymbol{q})\approx\mathrm{i}\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle \frac{K_{ij}}{\gamma_{\nu}}\hat{f}_{j}(\boldsymbol{q}), \end{gather}

(4.33)\begin{gather} K_{ik}=\frac{1}{\gamma_{\alpha}^{2}}(\gamma_{\eta}\delta_{ik}+ \mathrm{i}\breve{\alpha}\epsilon_{ijk}k_{j}), \quad \gamma_{\alpha}^{2}=\gamma_{\eta}^{2}-k^{2}\breve{\alpha}^{2}, \end{gather}

where the $\gamma$-factors are now defined using the renormalized fluctuational diffusivities for small wavenumbers

(4.34a,b)\begin{equation} \gamma_{\nu}={-}\mathrm{i}\omega+\breve{\nu}(k)k^{2}, \quad\gamma_{\eta}={-}\mathrm{i}\omega+\breve{\eta}(k)k^{2}.\end{equation}

It follows that the fluctuating vorticity at the leading order takes the simple form

(4.35)\begin{equation} \hat{w}_{i}(\boldsymbol{q})=\mathrm{i}\epsilon_{ijk}k_{j}\hat{u}_{k} (\boldsymbol{q})=\frac{\mathrm{i}}{\gamma_{\nu}(\boldsymbol{q})} \epsilon_{ijk}k_{j}\hat{f}_{k}(\boldsymbol{q}).\end{equation}

With the aid of (2.12) and (3.17a,b), it is now possible to write down explicit general formulae for the turbulent spectra of the fluctuating kinetic and magnetic energies and helicity in the following way:

(4.36a)\begin{align} \langle e_{k}\rangle &= \tfrac{1}{2}\langle u_{i}(\boldsymbol{x},t)u_{i} (\boldsymbol{x},t)\rangle =\tfrac{1}{2}\int\,\mathrm{d}^{4}q\int\,\mathrm{d}^{4}q' \frac{\mathrm{e}^{\mathrm{i}[(\boldsymbol{k}+\boldsymbol{k}')\boldsymbol{\cdot} \boldsymbol{x}-(\omega+\omega')t]}}{\gamma_{\nu}(\boldsymbol{q}') \gamma_{\nu}(\boldsymbol{q})}\langle \,\hat{f}_{i}(\boldsymbol{q}) \hat{f}_{i}(\boldsymbol{q}')\rangle \nonumber\\ &= 4{\rm \pi} D_{0}\int_{\varLambda_{L}}^{\varLambda} \frac{\mathrm{d}k}{k^{\sigma_{0}-2}}\int_{-\infty}^{\infty} \frac{\mathrm{d}\omega}{\omega^{2}+\breve{\nu}(k)^{2}k^{4}} \nonumber\\ &= 4{\rm \pi}^{2}D_{0}\int_{\varLambda_{L}}^{\varLambda} \frac{\mathrm{d}k}{\breve{\nu}(k)k^{\sigma_{0}}}, \end{align}

(4.36b)\begin{align} \langle e_{m}\rangle &= \tfrac{1}{2}\langle b_{i}(\boldsymbol{x},t)b_{i} (\boldsymbol{x},t)\rangle\nonumber\\ &={-}\tfrac{1}{2}\langle B\rangle_{m}\langle B\rangle_{n}\int\,\mathrm{d}^{4}q \int\,\mathrm{d}^{4}q'\frac{k_{m}k_{n}^{\prime}K_{ij}(\boldsymbol{q})K_{ik} (\boldsymbol{q}')}{\gamma_{\nu}(\boldsymbol{q})\gamma_{\nu}(\boldsymbol{q}')} \langle \,\hat{f}_{j}(\boldsymbol{q})\hat{f}_{k}(\boldsymbol{q}')\rangle \, \mathrm{e}^{\mathrm{i}[(\boldsymbol{k}+\boldsymbol{k}')\boldsymbol{\cdot} \boldsymbol{x}- (\omega+\omega')t]} \nonumber\\ & = \tfrac{1}{2}\langle B\rangle_{m}\langle B\rangle_{n}\int\, \mathrm{d}^{4}q\frac{k_{m}k_{n}}{(\omega^{2}+\breve{\nu}^{2}k^{4}) \gamma_{\alpha}^{2}(\boldsymbol{q})\gamma_{\alpha}^{2}(-\boldsymbol{q})} \left[|\gamma_{\eta}(\boldsymbol{q})|^{2}\frac{D_{0}}{k^{\sigma_{0}}}P_{jk}(\boldsymbol{k}) \delta_{jk}\right.\nonumber\\ &\quad +\breve{\alpha}^{2}k^{2}\frac{D_{0}}{k^{\sigma_{0}}}P_{jk}(\boldsymbol{k})P_{jk}(\boldsymbol{k}) +\left.\breve{\alpha}\frac{D_{1}}{k^{\sigma_{1}}}\epsilon_{jkl}\epsilon_{jtk}k_{l}k_{t} (\gamma_{\eta}(\boldsymbol{q})+\gamma_{\eta}(-\boldsymbol{q}))\right] \nonumber\\ &= \frac{4{\rm \pi}}{3}\langle B\rangle ^{2}\int_{\varLambda_{L}}^{\varLambda}\, \mathrm{d}k\left[\mathcal{I}_{e1}(k)\frac{D_{0}}{k^{\sigma_{0}-4}}- 2\breve{\alpha}\breve{\eta}\mathcal{I}_{e2}(k)\frac{D_{1}}{k^{\sigma_{1}-8}}\right], \end{align}

(4.36c)\begin{align} \langle h_{k}\rangle &= \langle u_{i}(\boldsymbol{x},t)w_{i}(\boldsymbol{x},t)\rangle =\mathrm{i}\epsilon_{ijk}\int\,\mathrm{d}^{4}q\int\,\mathrm{d}^{4}q' \frac{k_{j}^{\prime}\,\mathrm{e}^{\mathrm{i} [(\boldsymbol{k}+\boldsymbol{k}')\boldsymbol{\cdot} \boldsymbol{x}-(\omega+\omega')t]}}{\gamma_{\nu} (\boldsymbol{q})\gamma_{\nu}(\boldsymbol{q}')} \langle \,\hat{f}_{i}(\boldsymbol{q})\hat{f}_{k}(\boldsymbol{q}')\rangle \nonumber\\ &={-}8{\rm \pi} D_{1}\int_{\varLambda_{L}}^{\varLambda} \frac{\mathrm{d}k}{k^{\sigma_{1}-4}}\int_{-\infty}^{\infty} \frac{\mathrm{d}\omega}{\omega^{2}+\breve{\nu}(k)^{2}k^{4}} \nonumber\\ &={-}8{\rm \pi}^{2}D_{1}\int_{\varLambda_{L}}^{\varLambda}\,\mathrm{d}k \frac{1}{\breve{\nu}(k)k^{\sigma_{1}-2}}, \end{align}

where

(4.37a)\begin{gather} \mathcal{I}_{e1}(k)=\int_{-\infty}^{\infty}\,\mathrm{d}\omega \frac{\breve{\eta}^{2}k^{4}+\breve{\alpha}^{2}k^{2}+\omega^{2}}{(\omega^{2} +\breve{\nu}^{2}k^{4})[(\breve{\eta}^{2}k^{4}-\breve{\alpha}^{2}k^{2}- \omega^{2})^{2}+4\breve{\eta}^{2}k^{4}\omega^{2}]}, \end{gather}

(4.37b)\begin{gather}\mathcal{I}_{e2}(k)=\int_{-\infty}^{\infty}\,\mathrm{d}\omega\frac{1}{(\omega^{2} +\breve{\nu}^{2}k^{4})[(\breve{\eta}^{2}k^{4}-\breve{\alpha}^{2}k^{2}- \omega^{2})^{2}+4\breve{\eta}^{2}k^{4}\omega^{2}]}. \end{gather}

Note that $h_{k}$ denotes the kinetic helicity (in contrast to the magnetic one). Thus the spectral densities are given by the integrands in the final expressions for each quantity

(4.38a)\begin{gather} \hat{e}_{k}(k)=\frac{4{\rm \pi}^{2}D_{0}}{\breve{\nu}(k)k^{\sigma_{0}}},\quad \hat{e}_{m}(k)=\frac{4{\rm \pi}}{3}\langle B\rangle ^{2}\left[\mathcal{I}_{e1}(k) \frac{D_{0}}{k^{\sigma_{0}-4}}-2\breve{\alpha}\breve{\eta}\mathcal{I}_{e2}(k) \frac{D_{1}}{k^{\sigma_{1}-8}}\right], \end{gather}

(4.38b)\begin{gather}\hat{h}_{k}(k)={-}\frac{8{\rm \pi}^{2}D_{1}}{\breve{\nu}(k)k^{\sigma_{1}-2}}. \end{gather}

Recall that, in non-dimensional units, the total magnetic energy is $\langle B\rangle ^{2}+\beta ^{-2}\langle b^{2}\rangle$ where by assumption $\beta \gg 1$. Furthermore, since by the use of (4.18b) the fluctuating magnetic field satisfies $\boldsymbol {b}=O(Ro\beta )$, the fluctuating magnetic energy constitutes a $O(Ro^{2})$ correction to the total magnetic energy. Note also that the fluctuating magnetic energy spectrum is in general strongly influenced by the helical component of the forcing.

In the following sections the results for all the renormalized coefficients $\bar {\nu }, \bar {\eta }, \bar {\alpha }, \bar {Q}$ and $\bar {Q}_{p}, \breve {\nu }(k), \breve {\eta }(k), \breve {\alpha }(k)$ and the spectral densities $\hat {e}_{k}(k), \hat {e}_{m}(k), \hat {h}_{k}(k)$ will be provided. We start with the simplest case of reflectionally symmetric turbulence, $D_{1}=0$, when the $\alpha$-effect is eliminated. This will allow for a smooth introduction to the more complicated problem of helical turbulence considered in § 6. Moreover, in the next section we will explicitly consider two distinguished regimes of the MHD turbulence – the strong and weak regimes.

5. Non-helical MHD turbulence

The MHD turbulence in the absence of the flow chirality has been studied via the renormalization group method by Kleeorin & Rogachevskii (Reference Kleeorin and Rogachevskii1994). They have concluded that the MHD turbulence creates a strong negative contribution to the effective mean Lorentz force so that, in particular, the effective magnetic pressure can in fact become negative. Here, we will recall some of the results of Kleeorin & Rogachevskii (Reference Kleeorin and Rogachevskii1994), at least in a qualitative way, and study the reflectionally symmetric MHD turbulence in two distinguished regimes: weak and strong. Moreover, by inclusion of the gradients of the means in the evolution equations for the fluctuations, the correspondence between the renormalized fluctuation diffusivities $\breve {\nu }(\varLambda _{L}), \breve {\eta }(\varLambda _{L})$ and the mean transport coefficients $\bar {\nu }$ and $\bar {\eta }$ will be clearly demonstrated.

5.1. Strong turbulence

In strong turbulence the evolution of pulsations is nonlinear and to take proper account of the nonlinearities $\boldsymbol {\nabla } \boldsymbol {\cdot } (\boldsymbol {u}\boldsymbol {u}-\boldsymbol {b}\boldsymbol {b})$ and $\boldsymbol {\nabla } \times (\boldsymbol {u}\times \boldsymbol {b})$ we apply the renormalization group technique. The details of the procedure are postponed until Appendix A and, here, we only provide the final results. First, the recursions for the fluctuational diffusivities (A57) and (A61) have to be resolved; the general solution of the latter is obtained via the method of characteristics in (A63a,b) and also (A67), (A68). Including the dominant terms in the limit $\varLambda _{L}\leq k\ll \varLambda$ yields

(5.1)\begin{align} \breve{\nu}(k)=\frac{1}{a}\breve{\eta}(k)= \left[\frac{2{\rm \pi}^{2}D_{0}(5-\sigma_{0})}{5(\sigma_{0}+1)}\right]^{1/3} \frac{1}{k^{(\sigma_{0}+1)/3}}, \quad \textrm{where } a=\tfrac{1}{2}\left(\sqrt{1+\frac{240}{3(5-\sigma_{0})}}-1\right).\end{align}

Let us first examine the Fourier spectra of turbulent kinetic energy and helicity and the magnetic energy which are obtained from (4.38a,b) by substitution for $\breve {\nu }(k)$ and $\breve {\eta }(k)$ from (5.1) and $D_{1}=0 \Rightarrow \breve {\alpha }=0,$

(5.2a)\begin{gather} \hat{e}_{k}(k)=4\left[\frac{5(\sigma_{0}+1)}{2(5-\sigma_{0})}\right]^{1/3} \frac{({\rm \pi}^{2}D_{0})^{2/3}}{k^{2\sigma_{0}/3-1/3}}, \end{gather}

(5.2b)\begin{gather}\hat{e}_{m}(k)=\frac{4{\rm \pi}}{3}\langle B\rangle^{2}\frac{D_{0}}{k^{\sigma_{0}-4}} \frac{\rm \pi}{\breve{\nu}\breve{\eta}(\breve{\nu}+\breve{\eta})k^{6}}= \frac{10(\sigma_{0}+1)}{3a(1+a)(5-\sigma_{0})} \frac{\langle B\rangle ^{2}}{k}, \end{gather}

where we have used

(5.3)\begin{equation} \mathcal{I}_{e1}(k)=\int_{-\infty}^{\infty}\,\mathrm{d}\omega \frac{1}{(\omega^{2}+\breve{\nu}^{2}k^{4})(\omega^{2}+\breve{\eta}^{2}k^{4})}= \frac{\rm \pi}{\breve{\nu}\breve{\eta}(\breve{\nu}+\breve{\eta})k^{6}}.\end{equation}

Observe that we have reproduced the result of Yakhot & Orszag (Reference Yakhot and Orszag1986) for the kinetic energy spectral scaling law, namely, that it corresponds to the Kolmogorov spectral law $k^{-5/3}$ for $\sigma _{0}=3$. The $k^{-1}$ spectrum for the magnetic energy, independent of the value of $\sigma _{0}$, is in agreement with the findings of Kleeorin & Rogachevskii (Reference Kleeorin and Rogachevskii1994) and an earlier dimensional analysis of Ruzmaikin & Shukurov (Reference Ruzmaikin and Shukurov1982). Since the parameter $\sigma _{0}$ describes the forcing which generates the turbulence and we have determined that, in the non-helical case, $\sigma _{0}=3$ is consistent with the expected turbulent energy spectrum, in the following we will only consider the case with $\sigma _{0}=3$. The strong-turbulence energy spectra in the non-helical case are therefore given by

(5.4a,b)\begin{equation} \hat{e}_{k}(k)=\frac{4(\sqrt{5}{\rm \pi}^{2}D_{0})^{2/3}}{k^{5/3}},\quad \hat{e}_{m}(k)=\frac{2}{3}\frac{\langle B\rangle^{2}}{k}.\end{equation}

and the fluctuational diffusivities take the form

(5.5a,b)\begin{equation} \breve{\nu}(k)=\left(\frac{{\rm \pi}^{2}D_{0}}{5}\right)^{1/3}\frac{1}{k^{4/3}}, \quad \breve{\eta}(k)=a\left(\frac{{\rm \pi}^{2}D_{0}}{5}\right)^{1/3}\frac{1}{k^{4/3}},\quad \varLambda_{L}\leq k\ll\varLambda,\end{equation}

where

(5.6)\begin{equation} a=\frac{\sqrt{41}-1}{2}\approx2.7.\end{equation}

Having found $\breve {\nu }(k)$ and $\breve {\eta }(k)$ we can introduce them into the recursion relations (A81) and (A91a–c) for the mean-field coefficients, which can be solved to obtain in the limit $k\rightarrow \varLambda _{L}\ll \varLambda$

(5.7a)\begin{gather} \bar{Q}_{p}\approx{-}\frac{4(4a-1)}{15}\ln\frac{\varLambda}{\varLambda_{L}} \approx{-}2.6\ln\frac{\varLambda}{\varLambda_{L}} \end{gather}

(5.7b)\begin{gather}\bar{Q}\approx{-}\frac{8}{3a}\ln\frac{\varLambda}{\varLambda_{L}} \approx{-}\ln\frac{\varLambda}{\varLambda_{L}} \end{gather}

(5.7c)\begin{gather}\bar{\nu}=\frac{1}{a}\bar{\eta}=\left(\frac{{\rm \pi}^{2}D_{0}}{5}\right)^{1/3} \frac{1}{\varLambda_{L}^{4/3}}=\left(\frac{D_{0}}{80{\rm \pi}^{2}}\right)^{1/3} \mathscr{L}_{L}^{4/3}, \end{gather}

where $\mathscr {L}_{L}$ is the largest fluctuational length scale and thus could be interpreted as the size of the largest eddies in the system. It follows that

(5.8a,b)\begin{equation} \bar{\nu}=\breve{\nu}(\varLambda_{L}),\quad \bar{\eta}=\breve{\eta}(\varLambda_{L}),\end{equation}

i.e. the fluctuational diffusivities at the largest turbulent scales and the mean diffusivities are equal. Note that, in agreement with the results of Kleeorin & Rogachevskii (Reference Kleeorin and Rogachevskii1994) in the strong-turbulence limit, the contribution to the Lorentz force from interactions of turbulent pulsations, which becomes dominant when $\varLambda _{L}\ll \varLambda$, is negative, cf. (5.7a,b).

5.2. Weak turbulence

The weak turbulence corresponds to the simplest case when the amplitude of the turbulent pulsations is small enough so that their evolution can be considered linear, as in (3.6a,b); this means that the terms $\boldsymbol {\nabla } \boldsymbol {\cdot } (\boldsymbol {u}\boldsymbol {u}-\boldsymbol {b}\boldsymbol {b})$ and $\boldsymbol {\nabla } \times (\boldsymbol {u}\times \boldsymbol {b})$ are neglected in the evolution equations for the fluctuations. Tobias et al. (Reference Tobias, Cattaneo and Boldyrev2013) argued that, under certain conditions, the state of weak turbulence can survive for long periods of time without being destroyed by nonlinear interactions of wave packets, which nevertheless eventually still lead to transition into the strong-turbulence regime.

The case at hand in fact does not require renormalization, since the nonlinearities are neglected. The turbulent diffusivities can be taken from the introductory § 3 with $D_{1}=0$ and the coefficients $\bar {Q}$ and $\bar {Q}_{p}$ can be calculated in a similar way. However, we can alternatively utilize the recursion differential equations for the turbulent diffusivities and the Lorentz force coefficients obtained via the renormalization, which can be found in (A81) and (A81a–c) and, for the current case of weak turbulence, replace the fluctuational diffusivities $\breve {\nu }(k)$ and $\breve {\eta }(k)$ by the molecular ones, $\nu$ and $\eta$. The solutions in the limit $k\rightarrow \varLambda _{L}\ll \varLambda$ are provided only for the case $\sigma _{0}=3$

(5.9a)\begin{gather} \bar{Q}_{p}=1-\frac{2{\rm \pi}^{2}(4\eta-\nu)}{15\nu^{2}\eta(\nu+\eta)} \frac{D_{0}}{\varLambda_{L}^{4}}, \end{gather}

(5.9b)\begin{gather}\bar{Q}=1-\frac{2{\rm \pi}^{2}}{15\nu^{2}\eta}\frac{D_{0}}{\varLambda_{L}^{4}}, \end{gather}

(5.9c)\begin{gather}\bar{\nu} = \nu+\frac{{\rm \pi}^{2}D_{0}}{15\nu^{2}\varLambda_{L}^{4}} \approx\frac{{\rm \pi}^{2}D_{0}}{15\nu^{2}\varLambda_{L}^{4}}, \end{gather}

(5.9d)\begin{gather}\bar{\eta}=\eta+\frac{2{\rm \pi}^{2}D_{0}}{3\nu(\nu+\eta)\varLambda_{L}^{4}} \approx\frac{2{\rm \pi}^{2}D_{0}}{3\nu(\nu+\eta)\varLambda_{L}^{4}}. \end{gather}

The Fourier spectra of turbulent kinetic energy and helicity and the magnetic energy are obtained from (4.38a,b) by substitution $\breve {\nu }=\nu$ and $\breve {\eta }=\eta$ and $D_{1}=0$, which yields

(5.10a,b)\begin{equation} \hat{e}_{k}(k)=\frac{4{\rm \pi}^{2}D_{0}}{\nu k^{3}},\quad \hat{e}_{m}(k)=\frac{4{\rm \pi}^{2}D_{0}}{3\nu\eta(\nu+\eta)}\langle B\rangle ^{2}\frac{1}{k^{5}}.\end{equation}

6. Helical MHD turbulence

We now turn to the more complicated problem of MHD turbulence with chirality. In natural systems such as e.g. stellar and planetary interiors, chirality is typically introduced into the flow by rotation of the system; the latter creates the Coriolis force in the rotating frame which breaks the reflectional symmetry. However, since it also introduces significant complexity through anisotropy, we get around this difficulty by considering a non-reflectionally symmetric but statistically isotropic forcing; this allows us to perform the renormalization of the MHD equations in the strong-turbulence regime.

6.1. Strong turbulence

The case of strong turbulence is nonlinear and thus we apply the renormalization technique. As remarked below (A45d), at the leading order, the turbulent fluctuational viscosity remains unchanged with respect to the non-helical case (cf. (5.5a,b)), and thus takes the form

(6.1)\begin{equation} \breve{\nu}(k)=\left(\frac{{\rm \pi}^{2}D_{0}}{5}\right)^{1/3} \frac{1}{k^{4/3}},\quad \varLambda_{L}\leq k\ll\varLambda,\end{equation}

and we recall that this result is obtained for $\sigma _{0}=3$.

Let us also recall that, in helical, isotropic, homogeneous and stationary turbulence, the magnitude of the helicity spectrum (4.38b) is bounded by the energy spectrum (cf. e.g. Moffatt & Dormy Reference Moffatt and Dormy2019, p. 198, (7.54))

(6.2)\begin{equation} |\hat{h}_{k}(k)|\le 2k\hat{e}_{k}(k),\quad\textrm{for all }k. \end{equation}

This implies $D_1\le k^{\sigma _1-\sigma _0-1}D_0$. The recursion differential equations for the turbulent fluctuating magnetic diffusivity (A54a) and the turbulent fluctuating EMF ($\breve {\alpha }$ coefficient) (A54b) can be solved explicitly in the limit

(6.3)\begin{equation} \frac{D_{0}\varLambda_{L}^{\sigma_{1}-1-\sigma_{0}}}{D_{1}}\gg1,\end{equation}

which, for $\sigma _{1}-\sigma _{0}-1>0$, determines that the helical component of the driving force is weaker than the non-helical one at all fluctuating wavelengths $\varLambda _{L}\leq k\leq \varLambda$ (cf. (2.12)), consistently with (6.2). In other words, in such a limit the non-helical driving dominates the helical one and, roughly speaking, this could correspond to a situation expected in many natural systems where thermal/compositional buoyancy is stronger than the Coriolis force. Of course, the total effect of the helical driving remains non-negligible and significantly affects the dynamics of the mean fields through creation of the $\alpha$-effect. The solutions for $\breve {\eta }$ and $\breve {\alpha }$ in this limit take the form

(6.4a)\begin{gather} \breve{\eta}(k)=a\breve{\nu}(k)=a\left(\frac{{\rm \pi}^{2}D_{0}}{5}\right)^{1/3} \frac{1}{k^{4/3}},\quad \varLambda_{L}\leq k\ll\varLambda, \end{gather}

(6.4b)\begin{gather}\breve{\alpha}(k)=\frac{10D_{1}}{(2a+1)D_{0}}\breve{\nu}(k)=\frac{10D_{1}}{(2a+1) D_{0}}\left(\frac{{\rm \pi}^{2}D_{0}}{5}\right)^{1/3}\frac{1}{k^{4/3}},\quad \varLambda_{L}\leq k\ll\varLambda, \end{gather}

with the constant $a\approx 2.7$ given in (5.6).

Substitution of $\breve {\nu }(k), \breve {\eta }(k)$ and $\breve {\alpha }(k)$ from (6.1) and (6.4a,b) into the formulae (4.38a,b) leads to the following form of the energy spectra in helical MHD turbulence, in the limit (6.3):

(6.5a)\begin{align} \hat{e}_{k}(k)&\approx\frac{4{\rm \pi}^{2}D_{0}}{\breve{\nu}(k)k^{3}}\approx \frac{4(\sqrt{5}{\rm \pi}^{2}D_{0})^{2/3}}{k^{5/3}},\end{align}

(6.5b)\begin{align} \hat{e}_{m}(k)&\approx \frac{4{\rm \pi}^{2}}{3}\langle B\rangle ^{2} \left[\frac{D_{0}}{\breve{\nu}\breve{\eta}(\breve{\nu}+\breve{\eta})k^{5}}- \frac{D_{1}(\breve{\nu}+2\breve{\eta})\breve{\alpha}}{\breve{\nu}\breve{\eta}^{2} (\breve{\nu}+\breve{\eta})^{2}k^{\sigma_{1}+2}}\right] \nonumber\\ &\approx \frac{2}{3}\langle B\rangle ^{2}\left(\frac{1}{k}- \frac{D_{1}^{2}}{D_{0}^{2}k^{\sigma_{1}-2}}\right), \end{align}

(6.5c)\begin{align} \hat{h}_{k}(k)&\approx{-}\left(\frac{5{\rm \pi}}{D_{0}}\right)^{1/3} \frac{8D_{1}}{k^{\sigma_{1}-10/3}},\end{align}

where we have used

(6.6a,b)\begin{equation} \mathcal{I}_{e1}(k)\approx\frac{\rm \pi}{\breve{\nu}\breve{\eta} (\breve{\nu}+\breve{\eta})k^{6}},\quad \mathcal{I}_{e2}(k)\approx\frac{{\rm \pi}(\breve{\nu}+2\breve{\eta})}{2\breve{\nu} \breve{\eta}^{3}(\breve{\nu}+\breve{\eta})^{2}k^{10}}.\end{equation}

Similarly to what we have done for the exponent $\sigma _{0}$, we can now compare the obtained helicity spectrum (6.5c) with the Kolmogorov-type scaling for an isotropic, homogeneous and stationary turbulence $\hat {h}_{k}(k)\sim k^{-5/3}$ (cf. Brissaud et al. Reference Brissaud, Frisch, Leorat, Lesieur and Mazure1973; Chen et al. Reference Chen, Chen, Eyink and Holm2003) which leads to

(6.7)\begin{equation} \sigma_{1}-\tfrac{10}{3}=\tfrac{5}{3} \Rightarrow \sigma_{1}=5.\end{equation}

The parameters $\sigma _{0}$ and $\sigma _{1}$ describe the statistical properties of the forcing which generates the turbulence and since the values of $\sigma _{0}=3$ and $\sigma _{1}=5$ correspond to the expected turbulent energy and helicity spectra, in the following we will only consider the case with $\sigma _{0}=3$ and $\sigma _{1}=5$. Finally, the turbulent energy and helicity spectra are given by

(6.8a–c)\begin{equation} \hat{e}_{k}(k)\approx\frac{4(\sqrt{5}{\rm \pi}^{2}D_{0})^{2/3}}{k^{5/3}}, \quad\hat{e}_{m}(k)\approx\frac{2}{3}\langle B\rangle ^{2}\frac{1}{k}, \quad\hat{h}_{k}(k)\approx{-}\left(\frac{5{\rm \pi}}{D_{0}}\right)^{1/3} \frac{8D_{1}}{k^{5/3}}.\end{equation}

Furthermore, in the currently considered limit

(6.9)\begin{equation} D_{0}\varLambda_{L}\gg D_{1},\end{equation}

(cf. (6.3) with $\sigma _{0}=3$ and $\sigma _{1}=5$) the mean renormalized coefficients $\bar {\nu }, \bar {\eta }, \bar {Q}, \bar {Q}_{p}$ at leading order remain unaltered with respect to the non-helical case (cf. equations (A91a–c) and the discussion below)

(6.10a,b)\begin{gather} \bar{\nu}=\breve{\nu}(\varLambda_{L})=\left(\frac{{\rm \pi}^{2}D_{0}}{5}\right)^{1/3} \frac{1}{\varLambda_{L}^{4/3}},\quad \bar{\eta}=\breve{\eta} (\varLambda_{L})=a\left(\frac{{\rm \pi}^{2}D_{0}}{5}\right)^{1/3} \frac{1}{\varLambda_{L}^{4/3}}, \end{gather}

(6.11)\begin{gather} \bar{Q}(\varLambda_{L})\approx{-}\frac{8{\rm \pi}^{2}D_{0}}{15a} \int_{\varLambda_{L}}^{\varLambda}\frac{\mathrm{d}\lambda}{\lambda^{5}} \frac{1}{\breve{\nu}^{3}}\approx{-}\frac{8}{3a}\ln \frac{\varLambda}{\varLambda_{L}}, \end{gather}

(6.12)\begin{gather}\bar{Q}_{p}(\varLambda_{L})={-}\frac{8(4a-1){\rm \pi}^{2}D_{0}}{150} \int_{\varLambda_{L}}^{\varLambda}\frac{\mathrm{d}\lambda}{\lambda^{5}} \frac{1}{\breve{\nu}^{3}}\approx{-}\frac{4(4a-1)}{15}\ln \frac{\varLambda}{\varLambda_{L}}, \end{gather}

and the effect of chirality is expressed by the $\alpha$-effect

(6.13)\begin{equation} \bar{\alpha}=\breve{\alpha}(\varLambda_{L})= \frac{10D_{1}}{(2a+1)D_{0}}\left(\frac{{\rm \pi}^{2}D_{0}}{5}\right)^{1/3} \frac{1}{\varLambda_{L}^{4/3}}+O (Ro^{2}(Ro\beta)^{4}\mathcal{U}) +\mathrm{o}(Ro^{4}\mathcal{U}),\end{equation}

Note that, in the latter expression, the quenching effect of the Lorentz force vanishes at the order $Ro^{2}(Ro\beta )^{2}\langle B\rangle ^{2}$ (cf. the expansion of the $\gamma ^{-1}$ factors (A8)) and thus saturation effects for the mean magnetic field evolution can only be present at the unexplored order $Ro^{2}(Ro\beta )^{4}\langle B\rangle ^{4}$.

It is also of interest to point out that, in strongly helical strong turbulence, when the helical component of the driving force (2.12) is of comparable magnitude to the non-helical one at all wavelengths (and (6.3) is not satisfied), the $k$-dependencies of the fluctuation coefficients $\breve {\eta }(k)$ and $\breve {\alpha }(k)$ and hence also of the magnetic energy and helicity spectra are likely not to take the form of simple scaling laws. The general system of equations for the turbulent renormalized fluctuational coefficients $\breve {\eta }(k)$ and $\breve {\alpha }(k)$ obtained in the asymptotic limit $Ro\ll Ro\beta \ll 1$ is provided in (A54a,b) with the integrals $\mathcal {I}_{1}(k)$ and $\mathcal {I}_{2}(k)$ defined in (A50a,b); at the leading order of the limit $Ro\ll Ro\beta \ll 1$, the turbulent fluctuational viscosity $\breve {\nu }(k)$ is still given by (A58) in the general strongly helical case, which reduces to (6.1) for $\varLambda _{L}\leq k\ll \varLambda$. Once $\breve {\nu }(k)$ , $\breve {\eta }(k)$ and $\breve {\alpha }(k)$ are known, the mean coefficients $\bar {\eta }, \bar {\alpha }, \bar {Q}$ and $\bar {Q}_{p}$ can be computed from (A91a–d), with $\bar {\nu }$ unaltered with respect to (6.10a,b). In principle, this general case which consists of (A54a,b) and (A91a–d) with the aid of (A58), (A50a,b) and (A85a–d) and the ‘boundary’ conditions

(6.14a–f)\begin{equation} \breve{\eta}(\varLambda)=\eta,\quad\breve{\alpha}(\varLambda)=0, \quad\bar{\eta}(\varLambda)=\eta,\quad\bar{\alpha}(\varLambda)=0, \quad\bar{Q}(\lambda)=1,\quad\bar{Q}(\varLambda)=1,\end{equation}

can be solved numerically.

6.2. Weak turbulence

The case of weak turbulence is fairly simple and, as already remarked, does not require renormalization, because the nonlinearities $\boldsymbol {\nabla } \boldsymbol {\cdot } (\boldsymbol {u}\boldsymbol {u}-\boldsymbol {b}\boldsymbol {b})$ and $\boldsymbol {\nabla } \times (\boldsymbol {u}\times \boldsymbol {b})$ in the equations for the fluctuations are neglected. Therefore, the evolution of the fluctuations is not influenced by the effects of nonlinearities and thus there is no effect of turbulence in the fluctuational turbulent coefficients, i.e. $\breve {\nu }=\nu , \breve {\eta }=\eta$ and $\breve {\alpha }=0$. It follows that the mean turbulent diffusivities $\bar {\nu }$ and $\bar {\eta }$ can be taken from the introductory § 3. The Lorentz-force coefficients $\bar {Q}_{p}, \bar {Q}$ and $\bar {\alpha }$ can be calculated with the use of the recursion differential equations (A91a,b,d) obtained via the renormalization, where for the current case of weak turbulence we replace the fluctuational diffusivities $\breve {\nu }(k)$ and $\breve {\eta }(k)$ by the molecular ones $\nu$ and $\eta$ and substitute $\breve {\alpha }(k)=0$. The solutions in the limit $k\rightarrow \varLambda _{L}\ll \varLambda$ are provided for $\sigma _{0}=3$ and $\sigma _{1}=5$

(6.15a)\begin{gather} \bar{Q}_{p}=1-\frac{2{\rm \pi}^{2}(4\eta-\nu)}{15\nu^{2}\eta(\nu+\eta)} \frac{D_{0}}{\varLambda_{L}^{4}},\quad \bar{Q}=1- \frac{2{\rm \pi}^{2}}{15\nu^{2}\eta}\frac{D_{0}}{\varLambda_{L}^{4}}, \end{gather}

(6.15b)\begin{gather}\bar{\nu}=\frac{{\rm \pi}^{2}D_{0}}{15\nu^{2}\varLambda_{L}^{4}},\quad \bar{\eta}=\frac{2{\rm \pi}^{2}D_{0}}{3\nu(\nu+\eta)\varLambda_{L}^{4}},\quad \bar{\alpha}=\frac{2{\rm \pi}^{2}D_{1}}{3\nu(\nu+\eta)\varLambda_{L}^{4}}- \frac{4{\rm \pi}^{2}D_{1}\langle B\rangle ^{2}}{15\nu^{2}\eta(\nu+\eta) \varLambda_{L}^{6}}. \end{gather}

It is also notable that the effect of the Lorentz force is now present in the EMF at the order $\langle B\rangle ^{2}$, contrary to (6.13); hence, strong turbulence tends to suppress the saturation effects for the mean magnetic field, and the magnetic energy $\langle B\rangle ^{2}$ can saturate only above a certain threshold, when the saturation effects can enter the dynamics. We also emphasize that, in the current case of weak turbulence, there is no need to invoke the asymptotic limit (6.9) and the results are valid for $D_{0}\varLambda _L/D_{1}\le 1$. Furthermore, although it may seem obvious it is perhaps worth mentioning that the role of turbulent magnetic diffusivity in the large-scale dynamo process is non-trivial and naive substitution for $\nu$ and $\eta$ of their turbulent counterparts in the expression for $\bar {\alpha }$ in (6.15b) does not lead to anything similar in form to the strong-turbulence value (6.13).

The Fourier spectra of the turbulent kinetic and magnetic energies and the helicity can be obtained from (4.38a,b) again, by substitution $\breve {\nu }=\nu$ and $\breve {\eta }=\eta$ and $\breve {\alpha }=0$, which yields

(6.16a–c)\begin{equation} \hat{e}_{k}(k)=\frac{4{\rm \pi}^{2}D_{0}}{\nu k^{3}},\quad \hat{e}_{m}(k)=\frac{4{\rm \pi}^{2}D_{0}}{3\nu\eta(\nu+\eta)}\langle B \rangle^{2}\frac{1}{k^{5}},\quad \hat{h}_{k}(k)={-}\frac{8{\rm \pi}^{2}D_{1}}{\nu k^{3}}.\end{equation}

Note that the energy spectra are the same as in the case of non-helical weak turbulence (5.10a,b).

Summarizing, the physical pictures of the helical weak and strong turbulence considered here are rather simple, i.e. the pictures differ from their relevant non-helical cases only by the presence of helicity and non-zero EMF: $\bar {\alpha }$ in the case of weak turbulence and both $\breve {\alpha }$ and $\bar {\alpha }$ in the case of weakly helical strong turbulence. Note, however, that, as remarked at the end of the previous subsection, the helical turbulence outside the asymptotic limit $D_0\varLambda _L\gg D_1$ is much more complex than the non-helical case even in the isotropic case.

7. Role of diffusivities

It is very important to understand the role of viscosity and magnetic diffusivity in the presented approach. First of all we recall that the assumption

(7.1)\begin{equation} Ro\beta\ll1,\end{equation}

is utilized to facilitate the Fourier integrals by expanding the integrands in Taylor series in $Ro\beta$, such as e.g. in (A8). One must realize that this implies we assume (cf. (4.20))

(7.2)\begin{equation} |-\mathrm{i}\omega+E_{\nu}k^{2}|\gg\frac{H(Ro\beta)^{2} (\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2}}{|- \mathrm{i}\omega+E_{\eta}k^{2}|},\end{equation}

for all admissible values of $\omega$, thus, in particular, for $\omega$ close (and equal to) zero. Consequently, we require

(7.3)\begin{equation} E_{\nu}E_{\eta}\varLambda_{L}^{2}\gg H(Ro\beta)^{2}\left( \frac{\boldsymbol{\varLambda}_{L}}{\varLambda_{L}}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle \right)^{2}\sim(Ro\beta)^{2},\end{equation}

hence, the approach is valid only when both the diffusivities are non-zero, thus for example the limit $E_{\nu }\ll 1$ and/or $E_{\eta }\ll 1$ must be taken with care, incorporating (7.3). In other words, setting $\nu =0$ makes the problem ill posed, because the $\omega$-integrals cease to converge. The case of $\eta =0$ is similar and at least some of the $\omega$-integrals diverge in this limit. Hence, we must require

(7.4a,b)\begin{equation} \nu\neq0,\quad\textrm{and}\quad\eta\neq0,\end{equation}

for soundness and consistency of the mathematical approach. However, we can easily see from the analysis in § 3 that, for weak turbulence, setting $\eta =0$ leads to a vanishing $\alpha$-effect (vanishing of the part of the EMF which is not proportional to $\boldsymbol {\nabla } \langle \boldsymbol {B}\rangle$) for all values of $Ro\beta$, despite irregularity of the $\omega$-integral in the perturbation expansion for $Ro\beta \ll 1$. It is clear from the first line in (3.11) and $\langle \hat {f}_{j}(\boldsymbol {q})\hat {f}_{k}(\boldsymbol {q}')\rangle \sim \delta (\omega +\omega ')$ that, since the integrand is an odd function of $\omega$, the entire integral over $-\infty <\omega <\infty$ must vanish, even without expanding in Taylor series in $Ro\beta$. Consequently, we arrive at the well-known result that, in a weak turbulence, the mean-field dynamo does not operate when the magnetic diffusivity is too weak and negligible (cf. Moffatt & Dormy Reference Moffatt and Dormy2019).

Equation (7.3) involves non-dimensional $\varLambda _{L}$ and $\langle \boldsymbol {B}\rangle$, which by definition are order-unity quantities. It follows that the validity of the current approach can be expressed in a more straightforward manner

(7.5)\begin{equation} \frac{\nu\eta}{\mathcal{U}^{2}\mathscr{L}^{2}}\gg\beta^{2},\quad \textrm{or equivalently }\frac{Pm}{Rm_{L}^{2}\varepsilon^{2}}\gg\beta^{2},\end{equation}

where $Rm_{L}=\mathcal {U}L/\eta$ is the magnetic Reynolds number based on the global length scale $L$, a useful parameter in the description of the MHD turbulence in natural systems, in particular the theory of natural dynamos. From the point of view of applications to real astrophysical systems, the condition (7.5) is quite restrictive, since the astrophysical length scales are very large and the condition (7.5) demands that the magnetic Reynolds number $Rm_{L}$, which is proportional to $L$, is ‘not too large.’ Nevertheless, the upper bound for $Rm_{L}$ can be very large in particular for high-$Pm$ systems, and then the magnetic Reynolds number is allowed to be very high.

8. Estimates of the turbulent coefficients in natural systems

For the sake of providing sensible estimates of the turbulent coefficients $\breve {\nu }, \breve {\eta }, \breve {\alpha }$ $\bar {\nu }, \bar {\eta }$ and $\bar {\alpha }$ we associate the non-helical part of the forcing $\sim D_{0}$ with the turbulence driving which, e.g. in natural systems, could be realized by thermal/compositional buoyancy and the helical part $\sim D_{1}$ with the Coriolis force, which in natural systems is typically responsible for lack of reflexional symmetry in turbulence (cf. discussion below (4.3)). We recall here the expression for force correlations

(8.1)\begin{equation} \langle \,\hat{f}_{i}(\boldsymbol{k},\omega)\hat{f}_{j}(\boldsymbol{k}',\omega')\rangle =\left[\frac{D_{0}}{k^{3}}P_{ij}(\boldsymbol{k})+\mathrm{i} \frac{D_{1}}{k^{4}}\epsilon_{ijk}\frac{k_{k}}{k}\right] \delta(\boldsymbol{k}+\boldsymbol{k}')\delta(\omega+\omega').\end{equation}

Since

(8.2)\begin{equation} \langle \,\hat{f}_{i}(\boldsymbol{k},\omega)\hat{f}_{j}(\boldsymbol{k}',\omega')\rangle \sim\frac{\mathcal{F}^{2}\mathscr{L}^{6}}{\omega_{s}^{2}},\end{equation}

in accordance with the above statement we assume the following for the non-helical driving and the helical part of the forcing:

(8.3a,b)\begin{equation} \mathcal{F}^{2}=D_{0}\omega_{s},\quad 4\varOmega^{2}\mathcal{U}^{2}=\mathcal{F}_{helical}^{2}=D_{1} \omega_{s}\mathscr{L},\end{equation}

where the latter has been estimated with the magnitude of the Coriolis force $\mathcal {F}=2\varOmega \mathcal {U}$. Taking $\omega _{s}=\varOmega$ and fixing the length scale at the maximal fluctuational length scale $\mathscr {L}_{L}$ one obtains

(8.4a,b)\begin{equation} D_{1}=\frac{4\varOmega\mathcal{U}^{2}}{\mathscr{L}_{L}},\quad D_{0}=\frac{\mathcal{F}^{2}}{\varOmega}=16\varOmega^{3}\mathscr{L}_{L}^{2} \widetilde{Ra}^{2}E^{4},\end{equation}

where

(8.5a,b)\begin{equation} \widetilde{Ra}=\frac{\mathcal{F}\mathscr{L}_{L}^{3}}{\nu^{2}}, \quad E=\frac{\nu}{2\varOmega\mathscr{L}_{L}^{2}}.\end{equation}

Hence, for example for thermal buoyancy driving $\mathcal {F}=g\beta \varDelta T$, where $\beta$ denotes the thermal expansion coefficient and $\varDelta T$ the temperature jump across the fluid layer, the parameter $\widetilde {Ra}=g\beta \varDelta T\mathscr {L}_{L}^{3}/\nu ^{2}$ is proportional to the standard Rayleigh number; $E$ is the Ekman number based on the maximal fluctuational length scale and molecular viscosity. Let us also introduce the new Rossby number

(8.6)\begin{equation} \widetilde{Ro}=\frac{\mathcal{U}}{2\varOmega\mathscr{L}_{L}}.\end{equation}

Note that, if we assume the magnitudes of the driving $\mathcal {F}$ and the Coriolis force $2\varOmega \mathcal {U}$ to be comparable, then $\widetilde {Ra}E^{2}=O(Ro)$ and $\widetilde {Ro}=O(Ro)$.

8.1. Estimates for strong turbulence in the limit $\mathcal {F}\gg 2\varOmega \mathcal {U}$

When driving (say thermal) is strong and its magnitude significantly exceeds the magnitude of the Coriolis force the situation corresponds to that defined by (6.9); the latter is equivalent to $\mathcal {F}\gg 2\varOmega \mathcal {U}$ by (8.4a,b). Thus, in the current case, the turbulent coefficients can be estimated using (6.10a,b) and (6.13), which yields

(8.7a)\begin{align} \bar{\nu}=\breve{\nu}(\varLambda_{L})&=\left(\frac{{\rm \pi}^{2}D_{0}}{5}\right)^{1/3} \frac{1}{\varLambda_{L}^{4/3}}\approx\left(\frac{\mathcal{F}^{2} \mathscr{L}_{L}^{4}}{80{\rm \pi}^{2}\varOmega}\right)^{1/3} \nonumber\\ &\approx 0.11\varOmega\mathscr{L}_{L}^{2}\left(\frac{\mathcal{F}}{\varOmega^{2} \mathscr{L}_{L}}\right)^{2/3}\approx0.27\varOmega \mathscr{L}_{L}^{2}(\widetilde{Ra}E^{2})^{2/3}, \end{align}

(8.7b)\begin{equation} \bar{\eta}=\breve{\eta}(\varLambda_{L})=a\bar{\nu}\approx0.74 \varOmega\mathscr{L}_{L}^{2}(\widetilde{Ra}E^{2})^{2/3},\end{equation}

(8.7c)\begin{align} \bar{\alpha}=\breve{\alpha}(\varLambda_{L})&= \frac{10D_{1}}{(2a+1)D_{0}}\bar{\nu}\approx1.56\frac{4\varOmega^{2} \mathcal{U}^{2}}{\mathscr{L}_{L}}\left(\frac{\mathscr{L}_{L}^{4}}{80{\rm \pi}^{2} \varOmega\mathcal{F}^{4}}\right)^{1/3} \nonumber\\ &\approx \varOmega\mathscr{L}_{L}\frac{5.8}{{\rm \pi}^{2/3}} \widetilde{Ro}^{2}\left(\frac{\varOmega^{2}\mathscr{L}_{L}}{\mathcal{F}}\right)^{4/3} =0.43\varOmega\mathscr{L}_{L}\widetilde{Ro}^{2} (\widetilde{Ra}E^{2})^{{-}4/3}. \end{align}

Such expressions may be useful in particular applications of the theory to the MHD turbulence. For example, in natural systems such as stellar and planetary interiors or accretion disks, the observational data allow us to estimate the parameters $\varOmega , \mathscr {L}_{L}, \widetilde {Ro}, \widetilde {Ra}$ and $E$ for a particular system and in turn obtain estimates of the turbulent diffusivities and the magnitude of the $\alpha$-effect with the use of (8.7a–c).

It is also of interest to point out that similar estimates can be made for the case of weak turbulence under the assumptions $Ro\ll Ro\beta \ll 1$, in which case $\bar {\nu }\approx 1.7\times 10^{-3}\varOmega \mathscr {L}_{L}^{2}\widetilde {Ra}E^{2}, \bar {\eta }=\bar {\nu }10Pm/(1+Pm)$ and $\bar {\alpha }=-(\bar {\eta }/\mathscr {L}_{L})(\widetilde {Ro}^{2}/E^{4}\widetilde {Ra}^{2}).$ However, the case of strong MHD turbulence seems to be much more common in natural systems.

8.2. The $\alpha$-effect in weak turbulence with very weak molecular diffusivities $\nu$ and $\eta$

It is evident from § 7 that the presented theory is valid only for non-zero molecular viscosity and molecular magnetic resistivity and, moreover, there exists a lower bound for their product (7.3). However, there are known and important cases of vigorous plasma flow when the molecular diffusivities of the plasma are extremely small, such as for example in the interstellar galactic medium, intracluster medium and some accretion disks. A consistent description of the MHD turbulence in such objects requires estimates of the magnitude of the $\alpha$-effect when both $E_{\nu }$ and $E_{\eta }$ are very small and (7.3) is not satisfied in order to properly incorporate the dynamo effect. Therefore, it seems useful to provide such an estimate at least for the simplest case of weak turbulence, when we can neglect the effect of turbulence on the diffusivities, to clearly demonstrate the differences between such a case and the case of finite diffusivities characterized by (7.3). Neglecting of the mean-field gradients, the mean EMF can be calculated as follows (cf. § 3 or Moffatt & Dormy Reference Moffatt and Dormy2019; Mizerski Reference Mizerski2020):

(8.8)\begin{align} \mathcal{E}_{i} &= \epsilon_{ijk}\langle u_{j}b_{k}\rangle \nonumber\\ &=\epsilon_{ijk}\int\,\mathrm{d}^{4}q\int\,\mathrm{d}^{4}q'\frac{\boldsymbol{k}'\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle }{(\omega'+\mathrm{i}\eta k^{\prime2})\sigma(\boldsymbol{q}')\sigma(\boldsymbol{q})}\langle \,\hat{f}_{j}(\boldsymbol{q})\hat{f}_{k}(\boldsymbol{q}')\rangle \, \mathrm{e}^{\mathrm{i}[(\boldsymbol{k}+\boldsymbol{k}')\boldsymbol{\cdot} \boldsymbol{x}-(\omega+\omega')t]} \nonumber\\ &={-}\mathrm{i}\epsilon_{ijk}\epsilon_{jkl}D_{1}\int\,\mathrm{d}^{4}q\frac{\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle }{k^{5}|\sigma(\boldsymbol{q})|^{2}(\omega-\mathrm{i}\eta k^{2})}k_{l} \nonumber\\ &= 2\eta D_{1}\langle B\rangle_{m}\int\,\mathrm{d}^{4}q \frac{k_{m}k_{i}}{k^{3}|\sigma(\boldsymbol{q})|^{2}(\omega^{2} +\eta^{2}k^{4})}, \end{align}

where we recall

(8.9)\begin{equation} \sigma(\boldsymbol{q})=\omega+\mathrm{i}\nu k^{2}-\frac{(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B} \rangle)^{2}}{\omega+\mathrm{i}\eta k^{2}},\end{equation}

and since $|\sigma (\boldsymbol {q})|^{2}(\omega ^{2}+\eta ^{2}k^{4})$ is an even function of $\omega$ we have used

(8.10)\begin{equation} \int_{-\infty}^{\infty}\,\mathrm{d}\omega\omega/|\sigma(\boldsymbol{q})|^{2}(\omega^{2} +\eta^{2}k^{4})=0.\end{equation}

Next, we express $|\sigma (\boldsymbol {q})|^{2}$ in the following form:

(8.11)\begin{align} |\sigma(\boldsymbol{q})|^{2} & =\left|\omega\left(1-\frac{(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2}}{\omega^{2}+ \eta^{2}k^{4}}\right)+\mathrm{i}k^{2}\left[\nu+\eta\frac{(\boldsymbol{k} \boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2}}{\omega^{2}+\eta^{2}k^{4}} \right]\right|^{2} \nonumber\\ & =\frac{\omega^{2}(\omega^{2}+\eta^{2}k^{4}-(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2})^{2}+k^{4}[\nu(\omega^{2}+ \eta^{2}k^{4})+\eta(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2}]^{2}}{(\omega^{2}+\eta^{2}k^{4})^{2}} \nonumber\\ &=\frac{\omega^{4}+\omega^{2}[k^{4}(\nu^{2}+\eta^{2})-2(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2}]+(\nu\eta k^{4}+(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2})^{2}}{\omega^{2}+\eta^{2}k^{4}}. \end{align}

Using

(8.12)\begin{align} &\int_{-\infty}^{\infty}\,\mathrm{d}\omega\frac{1}{\omega^{4}+\omega^{2} [k^{4}(\nu^{2}+\eta^{2})-2(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2} ]+(\nu\eta k^{4}+(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2})^{2}}\nonumber\\ &\quad =\frac{\rm \pi}{k^{2}(\nu+\eta)[\nu\eta k^{4}+(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2}]}, \end{align}

and introducing spherical variables $(k, \theta , \varphi )$ with $\theta =0$ on the axis of the mean magnetic field, one obtains

(8.13)\begin{align} \mathcal{E}_{i} &= 2\eta D_{1}\langle B\rangle_{m}\int\frac{\mathrm{d}^{4}q}{k^{3}}\frac{k_{m}k_{i}}{\omega^{4}+\omega^{2} [k^{4}(\nu^{2}+\eta^{2})-2(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2}]+(\nu\eta k^{4}+(\boldsymbol{k}\boldsymbol{\cdot} \langle \boldsymbol{B}\rangle )^{2})^{2}} \nonumber\\ &= \frac{2{\rm \pi}\eta D_{1}}{(\nu+\eta)}\langle B\rangle_{m}\int_{\varLambda_{L}}^{\varLambda}\frac{\mathrm{d}k}{k}\int_{{-}1}^{1}\,\mathrm{d}(\cos\theta)\int_{0}^{2{\rm \pi}}\,\mathrm{d}\varphi\frac{k_{m}k_{i}/k^{2}}{\nu\eta k^{4}+k^{2}\langle B\rangle ^{2}\cos^{2}\theta}, \nonumber\\ &= \frac{8{\rm \pi}^{2}\eta D_{1}}{(\nu+\eta)}\frac{\langle B\rangle_{i}}{\langle B\rangle ^{2}}\int\frac{\mathrm{d}k}{k^{3}}\left[1-\sqrt{\frac{\nu\eta k^{2}}{\langle B\rangle ^{2}}}\arctan\sqrt{\frac{\langle B\rangle ^{2}}{\nu\eta k^{2}}}\right]. \end{align}

So far, we have simply calculated the EMF for the weak-turbulence case in a straightforward manner, without making any simplifying assumptions about the magnitudes of the diffusivities or the magnitude of the mean magnetic field. In particular, we have avoided the expansion in small $Ro\beta$ such as in (A8) which would require (7.3). We now make a somewhat opposite assumption to $Ro\beta \ll 1$ that the field is strong enough and/or the diffusivities are weak enough so that

(8.14)\begin{equation} \varLambda_{L}\ll\frac{\langle B\rangle }{\sqrt{\nu\eta}}.\end{equation}

This allows us to express the EMF in the following way:

(8.15) \begin{align} \mathcal{E}_{i} & \approx \frac{8{\rm \pi}^{2}\eta D_{1}}{(\nu+\eta)}\frac{\langle B\rangle_{i}}{\langle B\rangle ^{2}}\left\{ \frac{1}{2\varLambda_{L}^{2}}-\frac{{\rm \pi}\sqrt{\nu\eta}}{2\langle B\rangle \varLambda_{L}}+\frac{\nu\eta}{\langle B\rangle ^{2}}\ln\frac{\varLambda}{\varLambda_{L}}\right\} \nonumber\\ & \approx\frac{4{\rm \pi}^{2}\eta D_{1}}{(\nu+\eta)\varLambda_{L}^{2}}\frac{\langle B\rangle_{i}}{\langle B\rangle ^{2}}\left\{ 1-\frac{{\rm \pi}\sqrt{\nu\eta}\varLambda_{L}}{\langle B\rangle }\right\} \nonumber\\ & \approx \frac{4{\rm \pi}^{2}\eta D_{1}}{(\nu+\eta)\varLambda_{L}^{2}}\frac{\langle B\rangle_{i}}{\langle B\rangle ^{2}}. \end{align}

Utilizing the same approach as in (8.4a,b) we get the following estimate for the mean $\bar {\alpha }$ coefficient

(8.16)\begin{equation} \bar{\alpha}\approx\frac{16\widetilde{Ro}^{2}}{(1+Pm)} \frac{\varOmega^{3}\mathscr{L}_{L}^{3}}{\langle B\rangle ^{2}}.\end{equation}

Such an estimate is valid in the limit of extremely weak diffusivities which enter the expression only through the magnetic Prandtl number based on molecular diffusivities $Pm=\nu /\eta$. It is noteworthy that the EMF is inversely proportional to $\langle B\rangle ^{2}$ at leading order. Hence, the coefficient $\bar {\alpha }$ is singular at $\langle \boldsymbol {B}\rangle =0$ but, according to our assumption (8.14), $\langle \boldsymbol {B}\rangle =0$ is excluded in the considered regime. Still, it is of some interest to observe that the relation (8.14) could be satisfied by a relatively weak mean magnetic field and extremely weak diffusivities, which would make the magnitude of the EMF rather large and thus vivid amplification of the magnetic energy $\langle B\rangle ^{2}$ would be expected.

9. Limitations of the current theory and relations to some previous nonlinear approaches

The main limitations of the current theory result from the assumptions of isotropy, homogeneity and stationarity of the forcing (cf. (2.12)). The forcing has been assumed to act only at small scales and its non-helical and helical components to be related to buoyancy driving and the Coriolis force, respectively. In other words, the latter effects can only be considered at small scales, which seems right for the buoyancy force, but, as shown in Yokoi & Yoshizawa (Reference Yokoi and Yoshizawa1993), the large-scale Coriolis force in inhomogeneous turbulence can contribute to the so-called vortex-motive force and the creation of complex large-scale flows; such flows can then significantly alter the large-scale hydromagnetic dynamo process.

Furthermore, the gradients of the mean fields have been assumed weak in the presented theory, according to (4.6a,b) and (4.9b). This is just enough to calculate the leading-order form of the turbulent diffusion, but, in conjunction with the assumed statistical isotropy of the driving force, eliminates some effects which are based on the inhomogeneity of turbulence, in particular the effect of strong gradients of means on the small-scale turbulence, which can contribute to mean-field dynamo. For example, the effect of magnetic pumping or the shear-current effect (Krause & Rädler Reference Krause and Rädler1980; Rogachevskii & Kleeorin Reference Rogachevskii and Kleeorin2004), the so-called cross-helicity dynamo (Yokoi Reference Yokoi2013) and the cross-helicity effect coupled with the mean magnetic strain (Yoshizawa Reference Yoshizawa1990; Yokoi Reference Yokoi2013) have been excluded in the presented theory. In addition, stationarity of the forcing has recently been shown by Mizerski (Reference Mizerski2018a,Reference Mizerskib, Reference Mizerski2020) to be significantly limiting, as interactions between waves with distinct frequencies provide a powerful mechanism of mean-field dynamo generation, operating even in the absence of magnetic diffusion.

However, the advantage of the presented theory lies in the clarity of the considered model, with a given driving force that conceptually can be attributed to common physical effects. Such an approach allows for clear estimates of transport coefficients in real systems, in contrast to approaches which only relate statistical properties of the turbulent velocity and magnetic fields. The estimates provided here include the nonlinear dynamics of the MHD turbulence and in this sense are more accurate than commonly used estimates of the EMF based on weak (linear) turbulence.