© ESO, 2012

1. Introduction

Stellar mixing length theory is a rudimentary description of turbulent convective energy transport. The mixing length theory of turbulent transport goes back to Prandtl (1925) and, in the stellar context, to Vitense (1953). The simplest form of turbulent transport is turbulent diffusion, which quantifies the mean flux of a given quantity, e.g., momentum, concentration of chemicals, specific entropy or magnetic fields, down the gradient of its mean value. In all these cases essentially a Fickian diffusion law is established, where the turbulent diffusion coefficient is proportional to the rms velocity of the turbulent eddies and the effective mean free path of the eddies or their correlation length.

Mean-field theories, which have been elaborated, e.g., for the behavior of magnetic fields or of passive scalars in turbulent media, go beyond this concept. In the case of magnetic fields, the effects of turbulence occur in a mean electromotive force, which is related to the mean magnetic field and its derivatives in a tensorial fashion. Examples for effects described by the mean magnetic field alone, without spatial derivatives, are the α-effect (Steenbeck et al. 1966) and the pumping of mean magnetic flux (Rädler 1966, 1968; Roberts & Soward 1975); for more information on these topics see, e.g., Krause & Rädler (1980) or Brandenburg & Subramanian (2005). Likewise the mean passive scalar flux contains a pumping effect (Elperin et al. 1996). In both the magnetic and the passive scalar cases turbulent diffusion occurs, which is in general anisotropic. The coupling between the mean electromotive force and the magnetic field and its derivatives, or mean passive scalar flux and the mean scalar and its derivatives, is given by turbulent transport coefficients.

On the analytic level of the theory the determination of these transport coefficients is only possible with some approximations. The most often used one is the second-order correlation approximation (SOCA), which has delivered so far many important results. Its applicability is however restricted to certain ranges of parameters like the magnetic Reynolds number or the Péclet number. In spite of this restriction, SOCA is an invaluable tool, because it allows a rigorous treatment within the limits of its applicability. It is in particular important for testing numerical methods that apply in a wider range.

In recent years it has become possible to compute the full set of turbulent transport coefficients numerically from simulations of turbulent flows. The most accurate method for that is the test-field method (Schrinner et al. 2005, 2007). In addition to the equations describing laminar and turbulent flows, one solves a set of evolution equations for the small-scale magnetic or scalar fields which result from given mean fields, the test fields. By selecting a sufficient number of independent test fields, one obtains a corresponding number of mean electromotive forces or mean scalar fluxes and can then compute in a unique way all the associated transport coefficients.

Most of the applications of the test-field method are based on spatial averages that are taken over two coordinates. In the magnetic case this approach has been applied to a range of different flows including isotropic homogeneous turbulence (Sur et al. 2008; Brandenburg et al. 2008a), homogeneous shear flow turbulence (Brandenburg et al. 2008b) without and with helicity (Mitra et al. 2009), and turbulent convection (Käpylä et al. 2009). One of the main results is that in the isotropic case, for magnetic Reynolds numbers R_m larger than unity, the turbulent diffusivity is given by $\frac{\mathrm{1}}{\mathrm{3}}\mathit{\tau }{\mathit{u}}_{\mathrm{rms}}^{\mathrm{2}}$, where the correlation time τ is, to a good approximation, given by τ = (u_rmsk_f)^-1. Here, u_rms is the rms velocity of the turbulent small-scale flow and k_f is the wavenumber of the energy-carrying eddies. For smaller R_m, the turbulent diffusivity grows linearly with R_m. Furthermore, if the turbulence is driven isotropically by polarized waves, the flow becomes helical and there is an α effect. In the kinematic regime (for weak magnetic fields), the α coefficient is proportional to $\overline{{\omega }\mathrm{·}{u}}$, where ω = ∇ × u is the vorticity of the small-scale flow, u. In the passive scalar case, test scalars are used to determine the transport coefficients. Results have been obtained for anisotropic flows in the presence of rotation or strong magnetic fields (Brandenburg et al. 2009), linear shear (Madarassy & Brandenburg 2010), and for irrotational flows (Rädler et al. 2011).

The present paper deals with the magnetic and the passive scalar case in the above sense. Its goal is to compute the transport coefficients for axisymmetric turbulence, that is, turbulence with one preferred direction, given by the presence of either rotation or density stratification or, if the relevant directions coincide, of both. (Axisymmetric turbulence can be defined by requiring that any averaged quantity depending on the turbulent velocity field is invariant under any rotation of this field about the preferred axis.) Note that a dynamo-generated magnetic field will in general violate the assumption of axisymmetric turbulence. To avoid this problem while still being able to investigate the general effects arising from only one preferred direction, we assume such fields to be weak so as not to affect the assumption of axisymmetry of the turbulence. An imposed uniform magnetic field in the preferred direction would still be allowed, but this case will not be investigated in this paper; see Brandenburg et al. (2009) for numerical investigations of passive scalar transport with a uniform field.

Except for a few comparison cases, we always consider flows in a slab between stress-free boundaries. This is the simplest example of flows that are non-vanishing on the boundary and compatible with axisymmetric turbulence. To facilitate comparison with earlier work on forced turbulence, we consider an isothermal layer even in the density-stratified case, i.e., there is no convection, and the flow is driven by a prescribed random forcing. This is similar to earlier work on forced homogeneous turbulence (Brandenburg et al. 2008a,b, 2009), but now we will be able to address questions regarding vertical pumping as well as helicity production and α effect in the presence of rotation. This setup allows us to isolate effects of density stratification from those originating from the nonuniformities of turbulence intensity and local correlation length. In addition to isothermal stratification, we assume an isothermal equation of state and thus do not consider an equation for the specific entropy. Hence, no Brunt-Väisälä oscillations can occur. This assumption would need to be relaxed for studying turbulent convection, which will be the subject of a future investigation.

2. Mean-field concept in turbulent transport

2.1. Mean electromotive force

The evolution of the magnetic field B in an electrically conducting fluid is assumed to obey the induction equation, $\frac{\mathit{\partial }{B}}{\mathit{\partial t}}\mathrm{=}\mathbf{\nabla }\mathrm{\times }\left({U}\mathrm{\times }{B}\mathrm{-}\mathit{\eta }{J}\right)\mathit{,}$(1)where U is the velocity and η the microscopic magnetic diffusivity of the fluid, and J is defined by J = ∇ × B (so that J/μ₀ with μ₀ being the magnetic permeability is the electric current density). We define mean fields as averages, assume that the averaging satisfies (exactly or approximately) the Reynolds rules, and denote averaged quantities by overbars¹. The mean magnetic field $\overline{{B}}$ is then governed by $\frac{\mathit{\partial }\overline{{B}}}{\mathit{\partial t}}\mathrm{=}\mathbf{\nabla }\mathrm{\times }\mathrm{(}\overline{{U}}\mathrm{\times }\overline{{B}}\mathrm{+}\overline{\mathbf{ℰ}}\mathrm{-}\mathit{\eta }\overline{{J}}{\mathrm{)}}^{}\mathit{,}$(2)where $\overline{\mathbf{ℰ}}\mathrm{=}\overline{{u}\mathrm{\times }{b}}$ is the mean electromotive force resulting from the correlation of velocity and magnetic field fluctuations, ${u}\mathrm{=}{U}\mathrm{-}\overline{{U}}$ and ${b}\mathrm{=}{B}\mathrm{-}\overline{{B}}$.

We focus attention on the mean electromotive force $\overline{\mathbf{ℰ}}$ in cases in which the velocity fluctuations u constitute axisymmetric turbulence, that is, turbulence with one preferred direction, which we describe by the unit vector $\hat{{e}}$. Until further notice we accept the traditional assumption according to which $\overline{\mathbf{ℰ}}$ in a given point in space and time is a linear homogeneous function of $\overline{{B}}$ and its first spatial derivatives in this point. Then, $\overline{\mathbf{ℰ}}$ can be represented in the form $\begin{array}{ccc}\overline{\mathbf{ℰ}}& \mathrm{=}& \mathrm{-}{\mathit{\alpha }}_{\mathrm{\perp }}\overline{{B}}\mathrm{-}\mathrm{(}{\mathit{\alpha }}_{\mathrm{\parallel }}\mathrm{-}{\mathit{\alpha }}_{\mathrm{\perp }}\mathrm{)}\mathrm{(}\hat{{e}}\mathrm{·}\overline{{B}}\mathrm{)}\hat{{e}}\mathrm{-}\mathit{\gamma }\hat{{e}}\mathrm{\times }\overline{{B}}\\ & & \\ & & \mathrm{-}{\mathit{\kappa }}_{\mathrm{\perp }}\overline{{K}}\mathrm{-}\mathrm{(}{\mathit{\kappa }}_{\mathrm{\parallel }}\mathrm{-}{\mathit{\kappa }}_{\mathrm{\perp }}\mathrm{)}\mathrm{(}\hat{{e}}\mathrm{·}\overline{{K}}\mathrm{)}\hat{{e}}\mathrm{-}\mathit{\mu }\hat{{e}}\mathrm{\times }\overline{{K}}\end{array}$(3)with nine coefficients α _⊥ , α _∥ , ..., μ². Like $\overline{{J}}\mathrm{=}\mathbf{\nabla }\mathrm{\times }\overline{{B}}$, also $\overline{{K}}$ is determined by the gradient tensor $\mathbf{\nabla }\overline{{B}}$. While $\overline{{J}}$ is given by its antisymmetric part, $\overline{{K}}$ is a vector defined by $\overline{{K}}\mathrm{=}\hat{{e}}\mathrm{·}\mathrm{(}\mathbf{\nabla }\overline{{B}}{\mathrm{)}}^{\mathrm{S}}$ with $\mathrm{(}\mathbf{\nabla }\overline{{B}}{\mathrm{)}}^{\mathrm{S}}$ being the symmetric part of $\mathbf{\nabla }\overline{{B}}$. A more detailed explanation of (3) is given in Appendix A. If $\hat{{e}}$ is understood as polar vector (for example $\mathbf{\nabla }\overline{\mathit{\rho }}\mathit{/}\mathrm{\left|}\mathbf{\nabla }\overline{\mathit{\rho }}\mathrm{\right|}$, where $\overline{\mathit{\rho }}$ is the mean mass density), then $\overline{{K}}$ is axial and γ, β_⊥, β_∥ and μ are true scalars, but α_⊥, α_∥, δ, κ_⊥ and κ_∥ pseudoscalars. (Scalars are invariant but pseudoscalars change sign if the turbulent velocity field is reflected at a point or at a plane containing the preferred axis.) Sometimes it is useful to interpret $\hat{{e}}$ as an axial vector (for example Ω/|Ω| with Ω being an angular velocity). Then, $\overline{{K}}$ is a polar vector, β_⊥, β_∥, δ, κ_⊥, κ_∥ and μ are true scalars but α_⊥, α_∥ and γ pseudoscalars.

We may split $\overline{\mathbf{ℰ}}$ and $\overline{{B}}$ into parts $\overline{\mathbf{ℰ}}{\mathrm{\perp }}_{}$ and $\overline{{B}}{\mathrm{\perp }}_{}$ perpendicular to $\hat{{e}}$ and parts $\overline{\mathbf{ℰ}}{\mathrm{\parallel }}_{}$ and $\overline{{B}}{\mathrm{\parallel }}_{}$ parallel to it. Then (3) can be written in the form $\begin{array}{ccc}\overline{\mathbf{ℰ}}{}_{\mathrm{\perp }}& \mathrm{=}& \mathrm{-}{\mathit{\alpha }}_{\mathrm{\perp }}\overline{{B}}{}_{\mathrm{\perp }}\mathrm{-}\mathit{\gamma }\hat{{e}}\mathrm{\times }\overline{{B}}{}_{\mathrm{\perp }}\mathrm{-}{\mathit{\beta }}_{\mathrm{\perp }}\overline{{J}}{}_{\mathrm{\perp }}\mathrm{-}\mathit{\delta }\hat{{e}}\mathrm{\times }\overline{{J}}{}_{\mathrm{\perp }}\\ & \mathrm{-}& \\ \overline{\mathbf{ℰ}}{}_{\mathrm{\parallel }}& \mathrm{=}& \mathrm{-}{\mathit{\alpha }}_{\mathrm{\parallel }}\overline{{B}}{}_{\mathrm{\parallel }}\mathrm{-}{\mathit{\beta }}_{\mathrm{\parallel }}\overline{{J}}{}_{\mathrm{\parallel }}\mathrm{-}{\mathit{\kappa }}_{\mathrm{\parallel }}\overline{{K}}{}_{\mathrm{\parallel }}\mathit{.}\end{array}$(4)Let us return to (3). In the simple case of homogeneous isotropic turbulence we have α_⊥ = α_∥ and β _⊥  = β_∥, and all remaining coefficients vanish. Then, (3) takes the form $\overline{\mathbf{ℰ}}\mathrm{=}\mathit{\alpha }\overline{{B}}\mathrm{-}{\mathit{\eta }}_{\mathrm{t}}\overline{{J}}$ with properly defined α and η_t. These two coefficients have been determined by test-field calculations (Sur et al. 2008; Brandenburg et al. 2008a).

In several previous studies of $\overline{\mathbf{ℰ}}$, more general kinds of turbulence (that is, not only axisymmetric turbulence) have been considered, but with a less general definition of mean fields, which were just horizontal averages. More precisely, Cartesian coordinates (x,y,z) were adopted and the averages were taken over all x and y so that they depend on z and t only (Brandenburg et al. 2008a,b). This definition implies remarkable simplifications. Of course, we then have ${\overline{\mathit{J}}}_{\mathit{z}}\mathrm{=}\mathrm{0}$. Further, there are no non-zero components of $\mathbf{\nabla }\overline{{B}}$ other than ${\overline{\mathit{B}}}_{\mathit{x,z}}$ and ${\overline{\mathit{B}}}_{\mathit{y,z}}$, for $\mathbf{\nabla }\mathrm{·}\overline{{B}}\mathrm{=}\mathrm{0}$ requires ${\overline{\mathit{B}}}_{\mathit{z,z}}\mathrm{=}\mathrm{0}$, and these components can be expressed as components of $\overline{{J}}$, viz. ${\overline{\mathit{B}}}_{\mathit{x,z}}\mathrm{=}{\overline{\mathit{J}}}_{\mathit{y}}$ and ${\overline{\mathit{B}}}_{\mathit{y,z}}\mathrm{=}\mathrm{-}{\overline{\mathit{J}}}_{\mathit{x}}$. (Here and in what follows, commas denote partial derivatives.) This again implies $\overline{{K}}\mathrm{=}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\hspace{0.17em}\hat{{e}}\mathrm{\times }\overline{{J}}$. As a consequence, this definition of mean fields reduces (3) to $\begin{array}{ccc}\overline{\mathbf{ℰ}}& \mathrm{=}& \mathrm{-}{\mathit{\alpha }}_{\mathrm{\perp }}\overline{{B}}\mathrm{-}\mathrm{(}{\mathit{\alpha }}_{\mathrm{\parallel }}\mathrm{-}{\mathit{\alpha }}_{\mathrm{\perp }}\mathrm{)}\mathrm{(}\hat{{e}}\mathrm{·}\overline{{B}}\mathrm{)}\hat{{e}}\mathrm{-}\mathit{\gamma }\hat{{e}}\mathrm{\times }\overline{{B}}\\ & & \mathrm{-}{\mathit{\beta }}^{†}\overline{{J}}\mathrm{-}{\mathit{\delta }}^{†}\hspace{0.17em}\hat{{e}}\mathrm{\times }\overline{{J}}\mathit{,}\end{array}$(5)where and . Of course, α _⊥ , α _∥ , γ, and are independent of x or y. Clearly, β _⊥  and μ as well as δ and κ _⊥  have no longer independent meanings. From (2) we may conclude that $\mathit{\partial }{\overline{\mathit{B}}}_{\mathit{z}}\mathit{/}\mathit{\partial t}\mathrm{=}\mathrm{0}$. If we restrict ourselves to applications in which ${\overline{\mathit{B}}}_{\mathit{z}}$ vanishes initially, it does so at all times and the term with α _∥  − α _⊥  in (5) disappears. Then, only the four coefficients α _⊥ , γ, and are of interest. They can be determined by test-field calculations using two test fields independent of x and y (Brandenburg et al. 2008a,b).

In this paper we go beyond the aforementioned assumptions in the following respects. Firstly, we relax the assumption that $\overline{\mathbf{ℰ}}$ in a given point in space is a homogeneous function of $\overline{{B}}$ and its first spatial derivatives in this point. Instead, we admit a non-local connection between $\overline{\mathbf{ℰ}}$ and $\overline{{B}}$. For simplicity, however, we further on assume that $\overline{\mathbf{ℰ}}$ at a given time depends only on $\overline{{B}}$ at the same time, that is, we remain with an instantaneous connection between $\overline{\mathbf{ℰ}}$ and $\overline{{B}}$. This approximation requires that the mean field varies slowly on a time scale much longer than the turnover time of the turbulence; see Hubbard & Brandenburg (2009) for a more general treatment of rapidly changing fields. Secondly, we consider mean fields no longer as averages over all x and y. We define $\overline{{B}}$ at a point (x,y) in a plane z = const by averaging over some surroundings of this point in this plane so that it still depends on x and y. In that sense we generalize (3) so that $\begin{array}{ccc}\overline{\mathbf{ℰ}}\mathrm{\left(}{x}\mathrm{\right)}& \mathrm{=}& \mathrm{-}\mathrm{\int }({\mathit{\alpha }}_{\mathrm{\perp }}\mathrm{\left(}{x}\mathit{,}{\xi }\mathrm{\right)}\overline{{B}}\mathrm{(}{x}\mathrm{-}{\xi }\mathrm{)}\\ & & \mathrm{+}({\mathit{\alpha }}_{\mathrm{\parallel }}\mathrm{\left(}{x}\mathit{,}{\xi }\mathrm{\right)}\mathrm{-}{\mathit{\alpha }}_{\mathrm{\perp }}\mathrm{\left(}{x}\mathit{,}{\xi }\mathrm{\right)})(\hat{{e}}\mathrm{·}\overline{{B}}\mathrm{(}{x}\mathrm{-}{\xi }\mathrm{)})\hat{{e}}\\ & & \mathrm{+}\mathit{\gamma }\mathrm{\left(}{x}\mathit{,}{\xi }\mathrm{\right)}\hspace{0.17em}\hat{{e}}\mathrm{\times }\overline{{B}}\mathrm{(}{x}\mathrm{-}{\xi }\mathrm{)}\\ & & \mathrm{+}{\mathit{\beta }}_{\mathrm{\perp }}\mathrm{\left(}{x}\mathit{,}{\xi }\mathrm{\right)}\hspace{0.17em}\overline{{J}}\mathrm{(}{x}\mathrm{-}{\xi }\mathrm{)}\\ & & \mathrm{+}({\mathit{\beta }}_{\mathrm{\parallel }}\mathrm{\left(}{x}\mathit{,}{\xi }\mathrm{\right)}\mathrm{-}{\mathit{\beta }}_{\mathrm{\perp }}\mathrm{\left(}{x}\mathit{,}{\xi }\mathrm{\right)})(\hat{{e}}\mathrm{·}\overline{{J}}\mathrm{(}{x}\mathrm{-}{\xi }\mathrm{)})\hat{{e}}\\ & & \\ & & \mathrm{+}{\mathit{\kappa }}_{\mathrm{\perp }}\mathrm{\left(}{x}\mathit{,}{\xi }\mathrm{\right)}\hspace{0.17em}\overline{{K}}\mathrm{(}{x}\mathrm{-}{\xi }\mathrm{)}\\ & & \mathrm{+}({\mathit{\kappa }}_{\mathrm{\parallel }}\mathrm{\left(}{x}\mathit{,}{\xi }\mathrm{\right)}\mathrm{-}{\mathit{\kappa }}_{\mathrm{\perp }}\mathrm{\left(}{x}\mathit{,}{\xi }\mathrm{\right)})(\hat{{e}}\mathrm{·}\overline{{K}}\mathrm{(}{x}\mathrm{-}{\xi }\mathrm{)})\hat{{e}}\\ & & \mathrm{+}\mathit{\mu }\mathrm{\left(}{x}\mathit{,}{\xi }\mathrm{\right)}\hspace{0.17em}\hat{{e}}\mathrm{\times }\overline{{K}}\mathrm{(}{x}\mathrm{-}{\xi }\mathrm{)})\hspace{0.17em}\mathrm{d}{}^{\mathrm{3}}\mathit{\xi }\mathit{.}\end{array}$(6)As a consequence of the axisymmetry of the turbulence, the coefficients α _⊥ , α _∥ , ..., μ depend only via ${\mathit{\xi }}_{\mathit{x}}^{\mathrm{2}}\mathrm{+}{\mathit{\xi }}_{\mathit{y}}^{\mathrm{2}}$ on ξ_x and ξ_y. We consider them also as symmetric in ξ_z. The integration is over all ξ space. Of course, $\overline{\mathbf{ℰ}}$, $\overline{{B}}$, $\overline{{J}}$, and $\overline{{K}}$ may depend on t. For simplicity, however, the argument t has been dropped.

Let us subject (6) to a Fourier transformation with respect to ξ. We define it by $\mathit{F}\mathrm{\left(}{\xi }\mathrm{\right)}\mathrm{=}\mathrm{\left(}\mathrm{2}\mathit{\pi }{\mathrm{)}}^{-3}\mathrm{\int }\begin{array}{}\mathrm{˜}\\ \mathit{F}\end{array}\mathrm{\right(}{k}\mathrm{)}\hspace{0.17em}\exp \mathrm{(}\mathrm{i}{k}\mathrm{·}{\xi }\mathrm{)}\hspace{0.17em}\mathrm{d}{\mathrm{3}}^{}\mathit{k}\mathit{.}$(7)Remembering the convolution theorem we obtain $\begin{array}{ccc}\overline{\mathbf{ℰ}}\mathrm{\left(}{x}\mathrm{\right)}& \mathrm{=}& \mathrm{-}\mathrm{\left(}\mathrm{2}\mathit{\pi }{\mathrm{)}}^{-3}\mathrm{\int }(\begin{array}{}\mathrm{˜}\\ {\mathit{\alpha }}_{\mathrm{\perp }}\end{array}\mathrm{\right(}{x}\mathit{,}{k}\mathrm{\left)}\begin{array}{}\mathrm{˜}\\ \overline{{B}}\end{array}\mathrm{\right(}{k}\mathrm{)}\\ & \mathrm{+}& (\begin{array}{}\mathrm{˜}\\ {\mathit{\alpha }}_{\mathrm{\parallel }}\end{array}\mathrm{\left(}{x}\mathit{,}{k}\mathrm{\right)}\mathrm{-}\begin{array}{}\mathrm{˜}\\ {\mathit{\alpha }}_{\mathrm{\perp }}\end{array}\mathrm{\left(}{x}\mathit{,}{k}\mathrm{\right)})(\hat{{e}}\mathrm{·}\begin{array}{}\mathrm{˜}\\ \overline{{B}}\end{array}\mathrm{\left(}{k}\mathrm{\right)})\hspace{0.17em}\hat{{e}}\\ & \mathrm{+}& \begin{array}{}\mathrm{˜}\\ \mathit{\gamma }\end{array}\mathrm{\left(}{x}\mathit{,}{k}\mathrm{\right)}\hspace{0.17em}\hat{{e}}\mathrm{\times }\begin{array}{}\mathrm{˜}\\ \overline{{B}}\end{array}\mathrm{\left(}{k}\mathrm{\right)}\\ & \mathrm{+}& \begin{array}{}\mathrm{˜}\\ {\mathit{\beta }}_{\mathrm{\perp }}\end{array}\mathrm{\left(}{x}\mathit{,}{k}\mathrm{\right)}\hspace{0.17em}\begin{array}{}\mathrm{˜}\\ \overline{{J}}\end{array}\mathrm{\left(}{k}\mathrm{\right)}\mathrm{+}(\begin{array}{}\mathrm{˜}\\ {\mathit{\beta }}_{\mathrm{\parallel }}\end{array}\mathrm{\left(}{x}\mathit{,}{k}\mathrm{\right)}\mathrm{-}\begin{array}{}\mathrm{˜}\\ {\mathit{\beta }}_{\mathrm{\perp }}\end{array}\mathrm{\left(}{x}\mathit{,}{k}\mathrm{\right)})(\hat{{e}}\mathrm{·}\begin{array}{}\mathrm{˜}\\ \overline{{J}}\end{array}\mathrm{\left(}{k}\mathrm{\right)})\hspace{0.17em}\hat{{e}}\\ & \mathrm{+}& \\ & \mathrm{+}& \begin{array}{}\mathrm{˜}\\ {\mathit{\kappa }}_{\mathrm{\perp }}\end{array}\mathrm{\left(}{x}\mathit{,}{k}\mathrm{\right)}\begin{array}{}\mathrm{˜}\\ \overline{{K}}\end{array}\mathrm{\left(}{k}\mathrm{\right)}\mathrm{+}(\begin{array}{}\mathrm{˜}\\ {\mathit{\kappa }}_{\mathrm{\parallel }}\end{array}\mathrm{\left(}{x}\mathit{,}{k}\mathrm{\right)}\mathrm{-}\begin{array}{}\mathrm{˜}\\ {\mathit{\kappa }}_{\mathrm{\perp }}\end{array}\mathrm{\left(}{x}\mathit{,}{k}\mathrm{\right)})(\hat{{e}}\mathrm{·}\begin{array}{}\mathrm{˜}\\ \overline{{K}}\end{array}\mathrm{\left(}{k}\mathrm{\right)})\hspace{0.17em}\hat{{e}}\\ & \mathrm{+}& \begin{array}{}\mathrm{˜}\\ \mathit{\mu }\end{array}\mathrm{\left(}{x}\mathit{,}{k}\mathrm{\right)}\hspace{0.17em}\hat{{e}}\mathrm{\times }\begin{array}{}\mathrm{˜}\\ \overline{{K}}\end{array}\mathrm{\left(}{k}\mathrm{\right)})\hspace{0.17em}\exp \mathrm{(}\mathrm{i}{k}\mathrm{·}{x}\mathrm{)}\hspace{0.17em}\mathrm{d}{}^{\mathrm{3}}{k}\hspace{0.17em}\mathrm{;}\end{array}$(8)see Chatterjee et al. (2011) for a corresponding relation in the case of horizontally averaged magnetic fields that depend only on z. Like α_⊥, α_∥, ..., μ, the $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\alpha }}_{\mathrm{\perp }}\end{array}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\alpha }}_{\mathrm{\parallel }}\end{array}$, ..., $\begin{array}{}{\mathrm{˜}}_{}\\ \mathit{\mu }\end{array}$ are real quantities. They depend only via ${\mathit{k}}_{\mathrm{\perp }}\mathrm{=}\mathrm{(}{\mathit{k}}_{\mathit{x}}^{\mathrm{2}}\mathrm{+}{\mathit{k}}_{\mathit{y}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}$ on k_x and k_y and are symmetric in k_z, i.e., depend only via k_∥ = |k_z| on k_z. Due to the reality of the α _⊥ , α_∥, ..., μ and their symmetry in ξ_x, ξ_y and ξ_z we have $\begin{array}{}\mathrm{˜}\\ {\mathit{\alpha }}_{\mathrm{\perp }}\end{array}\mathrm{\left(}{x}\mathit{,}{k}\mathrm{\right)}\mathrm{=}\mathrm{\int }{\mathit{\alpha }}_{\mathrm{\perp }}\mathrm{\left(}{x}\mathit{,}{\xi }\mathrm{\right)}\hspace{0.17em}\cos {\mathit{k}}_{\mathit{x}}{\mathit{\xi }}_{\mathit{x}}\hspace{0.17em}\cos {\mathit{k}}_{\mathit{y}}{\mathit{\xi }}_{\mathit{y}}\hspace{0.17em}\cos {\mathit{k}}_{\mathit{z}}{\mathit{\xi }}_{\mathit{z}}\hspace{0.17em}\mathrm{d}{\mathrm{3}}^{}\mathit{\xi }$(9)and analogous relations for $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\alpha }}_{\mathrm{\parallel }}\end{array}$, ..., $\begin{array}{}{\mathrm{˜}}_{}\\ \mathit{\mu }\end{array}$. We note that $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\alpha }}_{\mathrm{\perp }}\end{array}$, ..., $\begin{array}{}{\mathrm{˜}}_{}\\ \mathit{\mu }\end{array}$, taken at k = 0, agree with α_⊥, ..., μ in Eq. (3).

2.2. Mean passive scalar flux

There are interesting analogies between turbulent transport of magnetic flux and that of a passive scalar (cf. Rädler et al. 2011). Assume that the evolution of a passive scalar C, e.g., the concentration of an admixture in a fluid, is given by $\frac{\mathit{\partial C}}{\mathit{\partial t}}\mathrm{=}\mathrm{-}\mathbf{\nabla }\mathrm{·}\mathrm{(}{U}\mathit{C}\mathrm{-}\mathit{D}\mathbf{\nabla }\mathit{C}\mathrm{)}\mathit{,}$(10)where D is the microscopic (molecular) diffusivity. Then the mean scalar $\overline{\mathit{C}}$ has to satisfy $\frac{\mathit{\partial }\overline{\mathit{C}}}{\mathit{\partial t}}\mathrm{=}\mathrm{-}\mathbf{\nabla }\mathrm{·}\mathrm{(}\overline{{U}}\hspace{0.17em}\overline{\mathit{C}}\mathrm{+}\overline{\mathbf{ℱ}}\mathrm{-}\mathit{D}\mathbf{\nabla }\overline{\mathit{C}}\mathrm{)}\mathit{,}$(11)where $\overline{\mathbf{ℱ}}\mathrm{=}\overline{{u}\mathit{c}}$ is the mean passive scalar flux, u stands again for the fluctuations of the velocity and $\mathit{c}\mathrm{=}\mathit{C}\mathrm{-}\overline{\mathit{C}}$ for the fluctuations of C. Consider again axisymmetric turbulence with a preferred direction given by the unit vector $\hat{{e}}$. Assume that $\overline{\mathbf{ℱ}}$ in a given point in space and time is determined by $\overline{\mathit{C}}$ and its gradient $\overline{{G}}\mathrm{=}\mathbf{\nabla }\overline{\mathit{C}}$ in this point. Then we have $\overline{\mathbf{ℱ}}\mathrm{=}\mathrm{-}{\mathit{\gamma }}^{\mathit{C}}\overline{\mathit{C}}\hat{{e}}\mathrm{-}{\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}\overline{{G}}\mathrm{-}\mathrm{(}{\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}\mathrm{-}{\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}\mathrm{)}\mathrm{(}\hat{{e}}\mathrm{·}\overline{{G}}\mathrm{)}\hat{{e}}\mathrm{-}{\mathit{\delta }}^{\mathit{C}}\hat{{e}}\mathrm{\times }\overline{{G}}\mathit{,}$(12)with coefficients γ^C, ${\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}$, ${\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}$ and δ^C. If $\hat{{e}}$ is a polar vector, γ^C is a scalar but δ^C a pseudoscalar, and if $\hat{{e}}$ is an axial vector, γ^C is a pseudoscalar but δ^C a scalar, while ${\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}$ and ${\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}$ are always scalars. We note that $\mathbf{\nabla }\mathrm{·}\mathrm{(}{\mathit{\delta }}^{\mathit{C}}\hat{{e}}\mathrm{\times }\overline{{G}}\mathrm{)}$ is only unequal zero if δ^C is not constant but varies in the direction of $\hat{{e}}\mathrm{\times }\overline{{G}}$.

We may split $\overline{\mathbf{ℱ}}$ and $\overline{{G}}$ into parts $\overline{\mathbf{ℱ}}{\mathrm{\perp }}_{}$ and $\overline{{G}}{\mathrm{\perp }}_{}$ perpendicular to $\hat{{e}}$, and parts $\overline{\mathbf{ℱ}}{\mathrm{\parallel }}_{}$ and $\overline{{G}}{\mathrm{\parallel }}_{}$ parallel to it, and give (12) the form $\begin{array}{ccc}\overline{\mathbf{ℱ}}{}_{\mathrm{\perp }}& \mathrm{=}& \mathrm{-}{\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}\overline{{G}}{}_{\mathrm{\perp }}\mathrm{-}{\mathit{\delta }}^{\mathit{C}}\hat{{e}}\mathrm{\times }\overline{{G}}{}_{\mathrm{\perp }}\\ \overline{\mathbf{ℱ}}{}_{\mathrm{\parallel }}& \mathrm{=}& \end{array}$(13)Let us now relax the assumption that $\overline{\mathbf{ℱ}}$ in a given point in space and time is determined by $\overline{\mathit{C}}$ and $\overline{{G}}$ in this point. Analogously to the magnetic case we consider a non-local but instantaneous connection between $\overline{\mathbf{ℱ}}$ and $\overline{\mathit{C}}$. Then we have $\begin{array}{ccc}\overline{\mathbf{ℱ}}\mathrm{\left(}{x}\mathrm{\right)}& \mathrm{=}& \mathrm{-}\mathrm{\int }({\mathit{\gamma }}^{\mathit{C}}\mathrm{\left(}{x}\mathit{,}{\xi }\mathrm{\right)}\hspace{0.17em}\hat{{e}}\hspace{0.17em}\overline{\mathit{C}}\mathrm{(}{x}\mathrm{-}{\xi }\mathrm{)}\\ & & \mathrm{+}{\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}\mathrm{\left(}{x}\mathit{,}{\xi }\mathrm{\right)}\overline{{G}}\mathrm{(}{x}\mathrm{-}{\xi }\mathrm{)}\\ & & \\ & & \mathrm{+}{\mathit{\delta }}^{\mathit{C}}\mathrm{\left(}{x}\mathit{,}{\xi }\mathrm{\right)}\hspace{0.17em}\hat{{e}}\mathrm{\times }\overline{{G}}\mathrm{(}{x}\mathrm{-}{\xi }\mathrm{)})\hspace{0.17em}\mathrm{d}{}^{\mathrm{3}}\mathit{\xi }\mathit{.}\end{array}$(14)As α_⊥, α_∥, ..., μ in the magnetic case, γ^C, ${\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}$, ${\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}$ and δ^C depend only via ${\mathit{\xi }}_{\mathit{x}}^{\mathrm{2}}\mathrm{+}{\mathit{\xi }}_{\mathit{y}}^{\mathrm{2}}$ on ξ_x and ξ_y, and we consider them also as symmetric in ξ_z. The integration is again over all ξ space. Note that $\overline{\mathbf{ℱ}}$, $\overline{\mathit{C}}$, and $\overline{{G}}$ may, even if it is not explicitly indicated, depend on t. Applying the Fourier transformation defined by (7) on (14), we arrive at $\begin{array}{ccc}\overline{\mathbf{ℱ}}\mathrm{\left(}{x}\mathrm{\right)}& \mathrm{=}& \mathrm{-}\mathrm{\left(}\mathrm{2}\mathit{\pi }{\mathrm{)}}^{-3}\mathrm{\int }(\stackrel{}{{{\mathit{\gamma }}^{\mathit{C}}}_{\mathrm{˜}}}\mathrm{\right(}{x}\mathit{,}{k}\mathrm{\left)}\hspace{0.17em}\hat{{e}}\hspace{0.17em}\begin{array}{}\mathrm{˜}\\ \overline{\mathit{C}}\end{array}\mathrm{\right(}{k}\mathrm{)}\\ & & \mathrm{+}\stackrel{}{{{\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}}_{\mathrm{˜}}}\mathrm{\left(}{x}\mathit{,}{k}\mathrm{\right)}\begin{array}{}\mathrm{˜}\\ \overline{{G}}\end{array}\mathrm{\left(}{k}\mathrm{\right)}\\ & & \\ & & \mathrm{+}\stackrel{}{{{\mathit{\delta }}^{\mathit{C}}}_{\mathrm{˜}}}\mathrm{\left(}{x}\mathit{,}{k}\mathrm{\right)}\hspace{0.17em}\hat{{e}}\mathrm{\times }\begin{array}{}\mathrm{˜}\\ \overline{{G}}\end{array}\mathrm{\left(}{k}\mathrm{\right)})\hspace{0.17em}\exp \mathrm{(}\mathrm{i}{k}\mathrm{·}{x}\mathrm{)}\hspace{0.17em}\mathrm{d}{}^{\mathrm{3}}\mathit{k,}\end{array}$(15)where $\stackrel{}{{{\mathit{\gamma }}_{\mathrm{\perp }}^{\mathit{C}}}_{\mathrm{˜}}}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}\end{array}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}\end{array}$ and $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\delta }}^{\mathit{C}}\end{array}$ are real quantities. They depend only via ${\mathit{k}}_{\mathit{x}}^{\mathrm{2}}\mathrm{+}{\mathit{k}}_{\mathit{y}}^{\mathrm{2}}$ on k_x and k_y, and only via k_∥ on k_z, and they satisfy relations analogous to (9). We note that $\stackrel{}{{{\mathit{\gamma }}^{\mathit{C}}}_{\mathrm{˜}}}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}\end{array}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}\end{array}$, and $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\delta }}^{\mathit{C}}\end{array}$ at k = 0 agree with γ^C, ${\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}$, ${\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}$, and δ^C in (12).

3. Simulating the turbulence

We assume that the fluid is compressible and its flow is governed by the equations $\begin{array}{ccc}\frac{\mathrm{D}{U}}{\mathrm{D}\mathit{t}}& \mathrm{=}& {f}\mathrm{+}{g}\mathrm{-}\mathbf{\nabla }\mathit{h}\mathrm{-}\mathrm{2}\mathbf{\Omega }\mathrm{\times }{U}\mathrm{+}{\mathit{\rho }}^{-1}\mathbf{\nabla }\mathrm{·}\mathrm{\left(}\mathrm{2}\mathit{\nu \rho }\mathbf{S}\mathrm{\right)}\\ \frac{\mathrm{D}\mathit{h}}{\mathrm{D}\mathit{t}}& \mathrm{=}& \end{array}$(16)Here, f means a random force which primarily drives isotropic turbulence (e.g., Haugen et al. 2004), g the gravitational force, and h the specific enthalpy. An isothermal equation of state, $\mathit{p}\mathrm{=}\mathit{\rho }{\mathit{c}}_{\mathrm{s}}^{\mathrm{2}}$, has been adopted with a constant isothermal sound speed c_s. In general a fluid flow in a rotating system is considered, Ω is the angular velocity which defines the Coriolis force. As usual ρ means the mass density, ν the kinematic viscosity and S the trace-free rate of strain tensor, ${\mathrm{S}}_{\mathit{ij}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{(}{\mathit{U}}_{\mathit{i,j}}\mathrm{+}{\mathit{U}}_{\mathit{j,i}}\mathrm{)}\mathrm{-}\frac{\mathrm{1}}{\mathrm{3}}{\mathit{\delta }}_{\mathit{ij}}\mathbf{\nabla }\mathrm{·}{U}$. The influence of the magnetic field on the fluid motion, that is the Lorentz force, is ignored throughout the paper.

The numerical simulation is carried out in a cubic domain of size L³, so the smallest wavenumber is k₁ = 2π/L. In most of the cases a density stratification is included with g = (0,0, −g), so the density scale height is ${\mathit{H}}_{\mathit{\rho }}\mathrm{=}{\mathit{c}}_{\mathrm{s}}^{\mathrm{2}}\mathit{/}\mathit{g}$. The number of scale heights across the domain is equal to Δlnρ, where Δ denotes the difference of values at the two edges of the domain. The forcing is assumed to work with an average wavenumber k_f. The scale separation ratio is then given by k_f/k₁, for which we usually adopt the value 5. This means that we have about 5 eddies in each of the three coordinate directions.

The flow inside the considered domain depends on the boundary conditions. Unless indicated otherwise we take the top and bottom surfaces z = z₁ and z = z₂ with z₂ =  −z₁ = L/2 as stress-free and adopt periodic boundary conditions for the other surfaces.

4. Computing the transport coefficients

4.1. Test-field method

In the magnetic case the coefficients α_⊥, α_∥, ..., μ are determined by the test-field method (Schrinner et al. 2005, 2007; Brandenburg et al. 2008a). This method works with a set of test fields $\overline{{B}}$, called $\overline{{B}}{\mathrm{T}}^{}$, and the corresponding mean electromotive forces $\overline{\mathbf{ℰ}}$, called $\overline{\mathbf{ℰ}}{\mathrm{T}}^{}$. For the latter we have $\overline{\mathbf{ℰ}}{\mathrm{T}}^{}\mathrm{=}\overline{{u}\mathrm{\times }{{b}}^{\mathrm{T}}}$, where the b^T obey $\begin{array}{ccc}{{b}}^{\mathrm{T}}& \mathrm{=}& \mathbf{\nabla }\mathrm{\times }{a}{}^{\mathrm{T}}\\ \frac{\mathit{\partial }{a}{}^{\mathrm{T}}}{\mathit{\partial t}}& \mathrm{=}& \end{array}$(17)with $\overline{{U}}$ and u taken from the solutions of (16). For the boundaries z = const we choose conditions which correspond to an adjacent perfect conductor, for the x and y directions periodic boundary conditions.

We define four test fields by $\begin{array}{ccc}\overline{{B}}{}^{\mathrm{1}\mathrm{s}}& \mathrm{=}& \mathrm{\left(}{\mathit{B}}_{\mathrm{0}}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\hspace{0.17em}\mathrm{s}\mathit{z,}\mathrm{0}\mathit{,}\mathrm{0}\mathrm{\right)}\mathit{,} \overline{{B}}{}^{\mathrm{1}\mathrm{c}}\mathrm{=}\mathrm{\left(}{\mathit{B}}_{\mathrm{0}}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\hspace{0.17em}\mathrm{c}\mathit{z,}\mathrm{0}\mathit{,}\mathrm{0}\mathrm{\right)}\\ \overline{{B}}{}^{\mathrm{2}\mathrm{s}}& \mathrm{=}& \end{array}$(18)with a constant B₀. Here and in what follows we use the abbreviations $\begin{array}{ccc}\hspace{0.17em}\mathrm{s}\mathit{x}& \mathrm{=}& \sin {\mathit{k}}_{\mathit{x}}\mathit{x,} \hspace{0.17em}\mathrm{c}\mathit{x}\mathrm{=}\cos {\mathit{k}}_{\mathit{x}}\mathit{x}\\ \hspace{0.17em}\mathrm{s}\mathit{y}& \mathrm{=}& \\ \hspace{0.17em}\mathrm{s}\mathit{z}& \mathrm{=}& \sin {\mathit{k}}_{\mathit{z}}\mathit{z,} \hspace{0.17em}\mathrm{c}\mathit{z}\mathrm{=}\cos {\mathit{k}}_{\mathit{z}}\mathit{z}\mathit{.}\end{array}$(19)We recall that test-fields need not to be solenoidal (see Schrinner et al. 2005, 2007).

We denote the mean electromotive forces which correspond to the test fields (18) by $\overline{\mathbf{ℰ}}{\mathrm{1}\mathrm{s}}^{}$, $\overline{\mathbf{ℰ}}{\mathrm{1}\mathrm{c}}^{}$, $\overline{\mathbf{ℰ}}{\mathrm{2}\mathrm{s}}^{}$, and $\overline{\mathbf{ℰ}}{\mathrm{2}\mathrm{c}}^{}$. With the presentation (6) and relations like (9) we find $\begin{array}{ccc}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{1}\mathrm{s}}& \mathrm{=}& \mathrm{-}{\mathit{B}}_{\mathrm{0}}\left(\begin{array}{}\mathrm{˜}\\ {\mathit{\alpha }}_{\mathrm{\perp }}\end{array}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\hspace{0.17em}\mathrm{s}\mathit{z}\mathrm{-}\left(\begin{array}{}\mathrm{˜}\\ \mathit{\delta }\end{array}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\begin{array}{}\mathrm{˜}\\ {\mathit{\kappa }}_{\mathrm{\perp }}\end{array}\right){\mathit{k}}_{\mathit{z}}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\hspace{0.17em}\mathrm{c}\mathit{z}\right)\\ \overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{1}\mathrm{s}}& \mathrm{=}& \mathrm{-}{\mathit{B}}_{\mathrm{0}}\left(\begin{array}{}\mathrm{˜}\\ \mathit{\gamma }\end{array}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\hspace{0.17em}\mathrm{s}\mathit{z}\mathrm{+}\left(\begin{array}{}\mathrm{˜}\\ {\mathit{\beta }}_{\mathrm{\perp }}\end{array}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\begin{array}{}\mathrm{˜}\\ \mathit{\mu }\end{array}\right){\mathit{k}}_{\mathit{z}}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\hspace{0.17em}\mathrm{c}\mathit{z}\right)\\ \overline{\mathrm{ℰ}}{}_{\mathit{z}}^{\mathrm{1}\mathrm{s}}& \mathrm{=}& \\ \overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{2}\mathrm{s}}& \mathrm{=}& \mathrm{-}{\mathit{B}}_{\mathrm{0}}\left(\left(\begin{array}{}\mathrm{˜}\\ {\mathit{\beta }}_{\mathrm{\perp }}\end{array}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\begin{array}{}\mathrm{˜}\\ \mathit{\mu }\end{array}\right){\mathit{k}}_{\mathit{y}}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{c}\mathit{y}\hspace{0.17em}\mathrm{s}\mathit{z}\mathrm{+}\left(\begin{array}{}\mathrm{˜}\\ \mathit{\delta }\end{array}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\begin{array}{}\mathrm{˜}\\ {\mathit{\kappa }}_{\mathrm{\perp }}\end{array}\right){\mathit{k}}_{\mathit{x}}\hspace{0.17em}\mathrm{c}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\hspace{0.17em}\mathrm{s}\mathit{z}\right)\\ \overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{2}\mathrm{s}}& \mathrm{=}& {\mathit{B}}_{\mathrm{0}}\left(\left(\begin{array}{}\mathrm{˜}\\ {\mathit{\beta }}_{\mathrm{\perp }}\end{array}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\begin{array}{}\mathrm{˜}\\ \mathit{\mu }\end{array}\right){\mathit{k}}_{\mathit{x}}\hspace{0.17em}\mathrm{c}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\hspace{0.17em}\mathrm{s}\mathit{z}\mathrm{-}\left(\begin{array}{}\mathrm{˜}\\ \mathit{\delta }\end{array}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\begin{array}{}\mathrm{˜}\\ {\mathit{\kappa }}_{\mathrm{\perp }}\end{array}\right){\mathit{k}}_{\mathit{y}}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{c}\mathit{y}\hspace{0.17em}\mathrm{s}\mathit{z}\right)\\ \overline{\mathrm{ℰ}}{}_{\mathit{z}}^{\mathrm{2}\mathrm{s}}& \mathrm{=}& \mathrm{-}{\mathit{B}}_{\mathrm{0}}\left(\begin{array}{}\mathrm{˜}\\ {\mathit{\alpha }}_{\mathrm{\parallel }}\end{array}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\hspace{0.17em}\mathrm{s}\mathit{z}\mathrm{+}\begin{array}{}\mathrm{˜}\\ {\mathit{\kappa }}_{\mathrm{\parallel }}\end{array}{\mathit{k}}_{\mathit{z}}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\hspace{0.17em}\mathrm{c}\mathit{z}\right)\end{array}$(20)and corresponding relations for $\overline{\mathrm{ℰ}}{\mathit{x}}_{\mathrm{1}\mathrm{c}}^{}\mathit{,}\mathit{...}\mathit{,}\overline{\mathrm{ℰ}}{\mathit{z}}_{\mathrm{2}\mathrm{c}}^{}$, whose right-hand sides can be derived from those in (20) simply by replacing    sz and    cz by    cz and  −    sz, respectively.

In view of the assumed axisymmetry of the turbulence, we consider α_⊥, α_∥, ..., μ in what follows as independent of x and y but admit a dependence on z. When multiplying both sides of Eqs. (20) and of the corresponding ones for $\overline{\mathrm{ℰ}}{\mathit{x}}_{\mathrm{1}\mathrm{c}}^{}\mathit{,}\mathit{...}\mathit{,}\overline{\mathrm{ℰ}}{\mathit{z}}_{\mathrm{2}\mathrm{c}}^{}$ with    sx   sy,    sx   cy or    cy   sy and averaging over all x and y, we obtain a system of equations, which can be solved for $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\alpha }}_{\mathrm{\perp }}\end{array}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\alpha }}_{\mathrm{\parallel }}\end{array}$, ..., $\begin{array}{}{\mathrm{˜}}_{}\\ \mathit{\mu }\end{array}$. The result reads $\begin{array}{ccc}\begin{array}{}\mathrm{˜}\\ {\mathit{\alpha }}_{\mathrm{\perp }}\end{array}& \mathrm{=}& \mathrm{-}\mathrm{⟨}{\mathit{b}}^{\mathit{ss}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{1}\mathrm{s}}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{1}\mathrm{c}}\mathrm{)}\mathrm{⟩}\\ \begin{array}{}\mathrm{˜}\\ {\mathit{\alpha }}_{\mathrm{\parallel }}\end{array}& \mathrm{=}& \mathrm{-}\mathrm{⟨}{\mathit{b}}^{\mathit{ss}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{z}}^{\mathrm{2}\mathrm{s}}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{z}}^{\mathrm{2}\mathrm{c}}\mathrm{)}\mathrm{⟩}\\ \begin{array}{}\mathrm{˜}\\ \mathit{\gamma }\end{array}& \mathrm{=}& \mathrm{-}\mathrm{⟨}{\mathit{b}}^{\mathit{ss}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{1}\mathrm{s}}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{1}\mathrm{c}}\mathrm{)}\mathrm{⟩}\\ \begin{array}{}\mathrm{˜}\\ {\mathit{\beta }}_{\mathrm{\perp }}\end{array}& \mathrm{=}& \mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{⟨}{\mathit{B}}^{\mathit{ss}}\mathrm{(}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{1}\mathrm{s}}\mathrm{-}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{1}\mathrm{c}}\mathrm{)}\mathrm{+}{\mathit{B}}^{\mathit{sc}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{2}\mathrm{s}}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{2}\mathrm{c}}\mathrm{)}\mathrm{⟩}\\ & \mathrm{=}& \mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{⟨}{\mathit{B}}^{\mathit{ss}}\mathrm{(}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{1}\mathrm{s}}\mathrm{-}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{1}\mathrm{c}}\mathrm{)}\mathrm{-}{\mathit{B}}^{\mathit{cs}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{2}\mathrm{s}}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{2}\mathrm{c}}\mathrm{)}\mathrm{⟩}\\ \begin{array}{}\mathrm{˜}\\ {\mathit{\beta }}_{\mathrm{\parallel }}\end{array}& \mathrm{=}& \\ \begin{array}{}\mathrm{˜}\\ \mathit{\delta }\end{array}& \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\mathrm{⟨}{\mathit{B}}^{\mathit{ss}}\mathrm{(}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{1}\mathrm{s}}\mathrm{-}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{1}\mathrm{c}}\mathrm{)}\mathrm{-}{\mathit{B}}^{\mathit{cs}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{2}\mathrm{s}}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{2}\mathrm{c}}\mathrm{)}\mathrm{⟩}\\ & \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\mathrm{⟨}{\mathit{B}}^{\mathit{ss}}\mathrm{(}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{1}\mathrm{s}}\mathrm{-}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{1}\mathrm{c}}\mathrm{)}\mathrm{-}{\mathit{B}}^{\mathit{sc}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{2}\mathrm{s}}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{2}\mathrm{c}}\mathrm{)}\mathrm{⟩}\\ \begin{array}{}\mathrm{˜}\\ {\mathit{\kappa }}_{\mathrm{\perp }}\end{array}& \mathrm{=}& \mathrm{-}\mathrm{⟨}{\mathit{B}}^{\mathit{ss}}\mathrm{(}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{1}\mathrm{s}}\mathrm{-}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{1}\mathrm{c}}\mathrm{)}\mathrm{+}{\mathit{B}}^{\mathit{cs}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{2}\mathrm{s}}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{2}\mathrm{c}}\mathrm{)}\mathrm{⟩}\\ & \mathrm{=}& \mathrm{-}\mathrm{⟨}{\mathit{B}}^{\mathit{ss}}\mathrm{(}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{1}\mathrm{s}}\mathrm{-}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{1}\mathrm{c}}\mathrm{)}\mathrm{+}{\mathit{B}}^{\mathit{sc}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{2}\mathrm{s}}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{2}\mathrm{c}}\mathrm{)}\mathrm{⟩}\\ \begin{array}{}\mathrm{˜}\\ {\mathit{\kappa }}_{\mathrm{\parallel }}\end{array}& \mathrm{=}& \mathrm{-}\mathrm{⟨}{\mathit{B}}^{\mathit{ss}}\mathrm{(}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{z}}^{\mathrm{2}\mathrm{s}}\mathrm{-}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{z}}^{\mathrm{2}\mathrm{c}}\mathrm{)}\mathrm{⟩}\\ \begin{array}{}\mathrm{˜}\\ \mathit{\mu }\end{array}& \mathrm{=}& \mathrm{-}\mathrm{⟨}{\mathit{B}}^{\mathit{ss}}\mathrm{(}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{1}\mathrm{s}}\mathrm{-}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{1}\mathrm{c}}\mathrm{)}\mathrm{-}{\mathit{B}}^{\mathit{sc}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{2}\mathrm{s}}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{x}}^{\mathrm{2}\mathrm{c}}\mathrm{)}\mathrm{⟩}\\ & \mathrm{=}& \mathrm{-}\mathrm{⟨}{\mathit{B}}^{\mathit{ss}}\mathrm{(}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{1}\mathrm{s}}\mathrm{-}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{1}\mathrm{c}}\mathrm{)}\mathrm{+}{\mathit{B}}^{\mathit{cs}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{2}\mathrm{s}}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℰ}}{}_{\mathit{y}}^{\mathrm{2}\mathrm{c}}\mathrm{)}\mathrm{⟩}\mathit{,}\end{array}$(21)where $\begin{array}{ccc}{\mathit{b}}^{\mathit{ss}}& \mathrm{=}& \mathrm{4}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\mathit{/}{\mathit{B}}_{\mathrm{0}}\mathit{,} {\mathit{B}}^{\mathit{ss}}\mathrm{=}{\mathit{b}}^{\mathit{ss}}\mathit{/}{\mathit{k}}_{\mathit{z}}\\ {\mathit{B}}^{\mathit{cs}}& \mathrm{=}& \end{array}$(22)The angle brackets indicate averaging over x and y. Although the relations (21) and (22) contain k_x, k_y and k_z as independent variables, the $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\alpha }}_{\mathrm{\perp }}\end{array}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\alpha }}_{\mathrm{\parallel }}\end{array}$, ..., $\begin{array}{}{\mathrm{˜}}_{}\\ \mathit{\mu }\end{array}$ should vary only via ${\mathit{k}}_{\mathrm{\perp }}\mathrm{=}\mathrm{(}{\mathit{k}}_{\mathit{x}}^{\mathrm{2}}\mathrm{+}{\mathit{k}}_{\mathit{y}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}$ with k_x and k_y, and only via k _∥  with k_z.

4.2. Test-scalar method

In the passive-scalar case the coefficients γ^C, ${\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}$, ${\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}$, and δ^C are determined by the test-scalar method with test scalars ${\overline{\mathit{C}}}^{\mathrm{T}}$ and the corresponding fluxes $\overline{\mathbf{ℱ}}{\mathrm{T}}^{}$. For the latter, we have $\overline{\mathbf{ℱ}}{\mathrm{T}}^{}\mathrm{=}\overline{{u}{\mathit{c}}^{\mathrm{T}}}$, where c^T obeys $\frac{\mathit{\partial }{\mathit{c}}^{\mathrm{T}}}{\mathit{\partial t}}\mathrm{=}\mathrm{-}\mathbf{\nabla }\mathrm{·}\left(\overline{{U}}{\mathit{c}}^{\mathrm{T}}\mathrm{+}{u}{\overline{\mathit{C}}}^{\mathrm{T}}\mathrm{+}\mathrm{(}{u}{\mathit{c}}^{\mathrm{T}}{\mathrm{)}}^{\mathrm{\prime }}\mathrm{-}\mathit{D}\mathbf{\nabla }{\mathit{c}}^{\mathrm{T}}\right)\mathit{.}$(23)Again $\overline{{U}}$ and u are taken from the solutions of (16).

We define two test-scalars ${\overline{\mathit{C}}}^{\mathrm{Ts}}$ and ${\overline{\mathit{C}}}^{\mathrm{Tc}}$ by ${\overline{\mathit{C}}}^{\mathrm{s}}\mathrm{=}{\mathit{C}}_{\mathrm{0}}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\hspace{0.17em}\mathrm{s}\mathit{z,} {\overline{\mathit{C}}}^{\mathrm{c}}\mathrm{=}{\mathit{C}}_{\mathrm{0}}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\hspace{0.17em}\mathrm{c}\mathit{z,}$(24)where C₀ is a constant and the abbreviations (19) are used. From (14) we then have $\begin{array}{ccc}\overline{\mathrm{ℱ}}\begin{array}{}\mathrm{s}\\ \mathit{x}\end{array}& \mathrm{=}& \mathrm{-}{\mathit{C}}_{\mathrm{0}}\mathrm{(}\stackrel{}{{{\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}}_{\mathrm{˜}}}{\mathit{k}}_{\mathit{x}}\hspace{0.17em}\mathrm{c}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\hspace{0.17em}\mathrm{s}\mathit{z}\mathrm{-}\stackrel{}{{{\mathit{\delta }}^{\mathit{C}}}_{\mathrm{˜}}}{\mathit{k}}_{\mathit{y}}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{c}\mathit{y}\hspace{0.17em}\mathrm{s}\mathit{z}\mathrm{)}\\ \overline{\mathrm{ℱ}}\begin{array}{}\mathrm{s}\\ \mathit{y}\end{array}& \mathrm{=}& \\ \overline{\mathrm{ℱ}}\begin{array}{}\mathrm{s}\\ \mathit{z}\end{array}& \mathrm{=}& \mathrm{-}{\mathit{C}}_{\mathrm{0}}\mathrm{(}\stackrel{}{{{\mathit{\gamma }}_{\mathrm{\perp }}^{\mathit{C}}}_{\mathrm{˜}}}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\hspace{0.17em}\mathrm{s}\mathit{z}\mathrm{+}\stackrel{}{{{\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}}_{\mathrm{˜}}}{\mathit{k}}_{\mathit{z}}\hspace{0.17em}\mathrm{s}\mathit{x}\hspace{0.17em}\mathrm{s}\mathit{y}\hspace{0.17em}\mathrm{c}\mathit{z}\mathrm{)}\end{array}$(25)and analogous relations for $\overline{\mathrm{ℱ}}\begin{array}{}\mathrm{c}\\ \mathit{x}\end{array}\mathit{,}\mathit{...}\mathit{,}\overline{\mathrm{ℱ}}\begin{array}{}\mathrm{c}\\ \mathit{z}\end{array}$ with sz and cz replaced by cz and  − sz, respectively.

Analogous to the magnetic case, we assume that γ^C, ${\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}$, ${\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}$, and δ^C are independent of x and y but may depend on z. Analogous to (21) we find here $\begin{array}{ccc}\stackrel{}{{{\mathit{\gamma }}^{\mathit{C}}}_{\mathrm{˜}}}& \mathrm{=}& \mathrm{-}\mathrm{⟨}{\mathit{c}}^{\mathit{ss}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℱ}}\begin{array}{}\mathrm{s}\\ \mathit{z}\end{array}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℱ}}\begin{array}{}\mathrm{c}\\ \mathit{z}\end{array}\mathrm{)}\mathrm{⟩}\\ \stackrel{}{{{\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}}_{\mathrm{˜}}}& \mathrm{=}& \mathrm{-}\mathrm{⟨}{\mathit{C}}^{\mathit{cs}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℱ}}\begin{array}{}\mathrm{s}\\ \mathit{x}\end{array}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℱ}}\begin{array}{}\mathrm{c}\\ \mathit{x}\end{array}\mathrm{)}\mathrm{⟩}\mathrm{=}\mathrm{-}\mathrm{⟨}{\mathit{C}}^{\mathit{sc}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℱ}}\begin{array}{}\mathrm{s}\\ \mathit{y}\end{array}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℱ}}\begin{array}{}\mathrm{c}\\ \mathit{y}\end{array}\mathrm{)}\mathrm{⟩}\\ \stackrel{}{{{\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}}_{\mathrm{˜}}}& \mathrm{=}& \\ \stackrel{}{{{\mathit{\delta }}^{\mathit{C}}}_{\mathrm{˜}}}& \mathrm{=}& \mathrm{⟨}{\mathit{C}}^{\mathit{sc}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℱ}}\begin{array}{}\mathrm{s}\\ \mathit{x}\end{array}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℱ}}\begin{array}{}\mathrm{c}\\ \mathit{x}\end{array}\mathrm{)}\mathrm{⟩}\mathrm{=}\mathrm{-}\mathrm{⟨}{\mathit{C}}^{\mathit{cs}}\mathrm{(}\hspace{0.17em}\mathrm{s}\mathit{z}\overline{\mathrm{ℱ}}\begin{array}{}\mathrm{s}\\ \mathit{y}\end{array}\mathrm{+}\hspace{0.17em}\mathrm{c}\mathit{z}\overline{\mathrm{ℱ}}\begin{array}{}\mathrm{c}\\ \mathit{y}\end{array}\mathrm{)}\mathrm{⟩}\mathit{,}\end{array}$(26)where c^ss, C^ss, C^sc, and C^cs are defined like b^ss, B^ss, B^sc, and B^cs, with C₀ at the place of B₀. The angle brackets indicate again averaging over x and y. Note that $\stackrel{}{{{\mathit{\gamma }}^{\mathit{C}}}_{\mathrm{˜}}}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}\end{array}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}\end{array}$, and $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\delta }}^{\mathit{C}}\end{array}$ should depend only via ${\mathit{k}}_{\mathrm{\perp }}\mathrm{=}\mathrm{(}{\mathit{k}}_{\mathit{x}}^{\mathrm{2}}\mathrm{+}{\mathit{k}}_{\mathit{y}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}$ on k_x and k_y, and only via k_∥ on k_z.

4.3. Validation using the Roberts flow

For a validation of our test-field procedure for the determination of the coefficients occurring in (3) we rely on the Roberts flow. We define it here by $\begin{array}{ccc}{u}& \mathrm{=}& {\mathit{u}}_{\mathrm{0}}\mathrm{(}\mathrm{-}\cos {\mathit{k}}_{\mathrm{0}}\mathit{x}\hspace{0.17em}\sin {\mathit{k}}_{\mathrm{0}}\mathit{y,}\hspace{0.17em}\sin {\mathit{k}}_{\mathrm{0}}\mathit{x}\hspace{0.17em}\cos {\mathit{k}}_{\mathrm{0}}\mathit{y,}\\ & & \end{array}$(27)with some wavenumber k₀ and a factor f which characterizes the ratio of the magnitude of u_z to that of u_x and u_y. We further define mean fields as averages over x and y with an averaging scale which is much larger than the period length 2π/k₀ of the flow pattern. When calculating the mean electromotive force $\overline{\mathbf{ℰ}}$ for this flow, we assume that it is a linear homogeneous function of $\overline{{B}}$ and its first spatial derivatives and adopt the second-order correlation approximation. Although the Roberts flow is far from being axisymmetric, the result for $\overline{\mathbf{ℰ}}$ can be written in the form (3), and we have $\begin{array}{ccc}{\mathit{\alpha }}_{\mathrm{\perp }}& \mathrm{=}& \frac{{\mathit{u}}_{\mathrm{0}}^{\mathrm{2}}\mathit{f}}{\mathrm{2}\mathit{\eta }{\mathit{k}}_{\mathrm{0}}}\mathit{,} {\mathit{\alpha }}_{\mathrm{\parallel }}\mathrm{=}\mathit{\gamma }\mathrm{=}\mathrm{0}\\ {\mathit{\beta }}_{\mathrm{\perp }}& \mathrm{=}& \\ {\mathit{\kappa }}_{\mathrm{\perp }}& \mathrm{=}& {\mathit{\kappa }}_{\mathrm{\parallel }}\mathrm{=}\mathrm{0}\mathit{,} \mathit{\mu }\mathrm{=}\mathrm{-}\frac{{\mathit{u}}_{\mathrm{0}}^{\mathrm{2}}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{4}{\mathit{f}}^{\mathrm{2}}\mathrm{)}}{\mathrm{8}\mathit{\eta }{\mathit{k}}_{\mathrm{0}}^{\mathrm{2}}}\mathrm{=}\mathrm{2}\mathrm{(}{\mathit{\beta }}_{\mathrm{\perp }}\mathrm{-}{\mathit{\beta }}_{\mathrm{\parallel }}\mathrm{)}\mathit{.}\end{array}$(28)It agrees with and can be deduced from results reported in Rädler et al. (2002a,b). As for the passive scalar case, an analogous analytical calculation of the mean scalar flow $\overline{\mathrm{ℱ}}$ leads to (12) with ${\mathit{\gamma }}^{\mathit{C}}\mathrm{=}\mathrm{0}\mathit{,} {\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}\mathrm{=}\frac{{\mathit{u}}_{\mathrm{0}}^{\mathrm{2}}}{\mathrm{8}\mathit{D}{\mathit{k}}_{\mathrm{0}}^{\mathrm{2}}}\mathit{,} {\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}\mathrm{=}\frac{{\mathit{u}}_{\mathrm{0}}^{\mathrm{2}}{\mathit{f}}^{\mathrm{2}}}{\mathrm{2}\mathit{D}{\mathit{k}}_{\mathrm{0}}^{\mathrm{2}}}\mathit{,} {\mathit{\delta }}^{\mathit{C}}\mathrm{=}\mathrm{0.}$(29)We may proceed from the local connection of $\overline{\mathbf{ℰ}}$ with $\overline{{B}}$ and its derivatives considered in (3) to the non-local ones given by (6) or (8). As a consequence of the deviation of the flow from axisymmetry, we can then no longer justify that coefficients like α _⊥ (ξ) depend only via ${\mathit{\xi }}_{\mathit{x}}^{\mathrm{2}}\mathrm{+}{\mathit{\xi }}_{\mathit{y}}^{\mathrm{2}}$ on ξ_x and ξ_y, and coefficients like $\begin{array}{}{}_{\mathrm{˜}}\\ {\mathit{\alpha }}_{\mathrm{\perp }}\end{array}\mathrm{\left(}{k}\mathrm{\right)}$ only via k _⊥  on k_x and k_y. This applies analogously to the connection of $\overline{\mathrm{ℱ}}$ with $\overline{\mathit{C}}$ and its derivatives and to coefficients like β _⊥ (ξ) and $\begin{array}{}{}_{\mathrm{˜}}\\ {\mathit{\beta }}_{\mathrm{\perp }}\end{array}\mathrm{\left(}{k}\mathrm{\right)}$.

A test-field calculation of the coefficients $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\alpha }}_{\mathrm{\perp }}\end{array}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\alpha }}_{\mathrm{\parallel }}\end{array}$, ..., $\begin{array}{}{\mathrm{˜}}_{}\\ \mathit{\mu }\end{array}$, as well as $\stackrel{}{{{\mathit{\gamma }}^{\mathit{C}}}_{\mathrm{˜}}}$, ..., $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\delta }}^{\mathit{C}}\end{array}$, has been carried out under the conditions of the second-order correlation approximation with u given by (27) and f = 1/$\sqrt{\mathrm{2}}$. Figure 1 shows the results obtained for $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\alpha }}_{\mathrm{\perp }}\end{array}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\perp }}\end{array}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\parallel }}\end{array}$ and $\begin{array}{}{\mathrm{˜}}_{}\\ \mathit{\mu }\end{array}$, as well as $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}\end{array}$ and $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}\end{array}$, as functions of k _⊥ /k_f, with ${\mathit{k}}_{\mathrm{f}}\mathrm{=}\sqrt{\mathrm{2}}{\mathit{k}}_{\mathrm{0}}$, for two fixed ratios k_∥/k_⊥. In the limit k_⊥/k_f ≪ 1 these coefficients take just the values of α_⊥, β_⊥, β_∥, μ, ${\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}$ and ${\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}$ given in (28) and (29). For larger values of k_⊥/k_f, as to be expected, the $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\alpha }}_{\mathrm{\perp }}\end{array}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\perp }}\end{array}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\parallel }}\end{array}$, $\begin{array}{}{\mathrm{˜}}_{}\\ \mathit{\mu }\end{array}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}\end{array}$ and $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}\end{array}$ depend also on the ratio of k_x and k_y.

Fig. 1 The coefficients $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\alpha }}_{\mathrm{\perp }}\end{array}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\perp }}\end{array}$, $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\parallel }}\end{array}$, and $\begin{array}{}{\mathrm{˜}}_{}\\ \mathit{\mu }\end{array}$, as well as $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}\end{array}$ and $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\parallel }}^{\mathit{C}}\end{array}$ for the Roberts flow, calculated in the second-order correlation approximation, as functions of k ⊥ /kf, where ${\mathit{k}}_{\mathrm{f}}\mathrm{=}\sqrt{\mathrm{2}}{\mathit{k}}_{\mathrm{0}}$ is the effective wavenumber of the flow. Results obtained with kx = ky and ${\mathit{k}}_{\mathrm{\parallel }}\mathit{/}{\mathit{k}}_{\mathrm{\perp }}\mathrm{=}\mathrm{1}\mathit{/}\sqrt{\mathrm{2}}\mathrm{\approx }\mathrm{0.7}$ or ${\mathit{k}}_{\mathrm{\parallel }}\mathit{/}{\mathit{k}}_{\mathrm{\perp }}\mathrm{=}\mathrm{1}\mathit{/}\mathrm{16}\sqrt{\mathrm{2}}\mathrm{\approx }\mathrm{0.004}$ are represented by open squares and dotted lines or by open diamonds and dashed lines, respectively. Results with kx/ky = 0.75 [k ⊥  = (3,4,0)k1] or kx/ky = 5 [k ⊥  = (5,1,0)k1] and k∥/k ⊥  = 0.2 are indicated by open or filled circles, respectively. Orange and black symbols correspond to the first and second expressions for $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\perp }}\end{array}$ and $\begin{array}{}{\mathrm{˜}}_{}\\ \mathit{\mu }\end{array}$ in (21) or for $\begin{array}{}{\mathrm{˜}}_{}\\ {\mathit{\beta }}_{\mathrm{\perp }}^{\mathit{C}}\end{array}$ in (26).