Quantifying the effect of turbulent magnetic diffusion on the growth rate of the magneto−rotational instability^{⋆}
^{1} Department of Physics, Gustaf Hällströmin katu 2a, PO Box 64, 00014 University of Helsinki, Finland
email: miikka.vaisala@helsinki.fi
^{2} Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden
^{3} Department of Astronomy, AlbaNova University Center, Stockholm University, 10691 Stockholm, Sweden
^{4} Aalto University, ReSoLVE Centre of Excellence, Department of Information and Computer Science, PO Box 15400, 00076 Aalto, Finland
Received: 11 October 2013
Accepted: 9 May 2014
Context. In astrophysics, turbulent diffusion is often used in place of microphysical diffusion to avoid resolving the small scales. However, we expect this approach to break down when time and length scales of the turbulence become comparable with other relevant time and length scales in the system. Turbulent diffusion has previously been applied to the magnetorotational instability (MRI), but no quantitative comparison of growth rates at different turbulent intensities has been performed.
Aims. We investigate to what extent turbulent diffusion can be used to model the effects of smallscale turbulence on the kinematic growth rates of the MRI, and how this depends on angular velocity and magnetic field strength.
Methods. We use direct numerical simulations in threedimensional shearing boxes with periodic boundary conditions in the spanwise direction and additional random planewave volume forcing to drive a turbulent flow at a given length scale. We estimate the turbulent diffusivity using a mixing length formula and compare with results obtained with the testfield method.
Results. It turns out that the concept of turbulent diffusion is remarkably accurate in describing the effect of turbulence on the growth rate of the MRI. No noticeable breakdown of turbulent diffusion has been found, even when time and length scales of the turbulence become comparable with those imposed by the MRI itself. On the other hand, quenching of turbulent magnetic diffusivity by the magnetic field is found to be absent.
Conclusions. Turbulence reduces the growth rate of the MRI in the same way as microphysical magnetic diffusion does.
Key words: turbulence / magnetohydrodynamics (MHD) / hydrodynamics
Appendix A is available in electronic form at http://www.aanda.org
© ESO, 2014
1. Introduction
A cornerstone in the study of astrophysical fluids is linear stability theory (Chandrasekhar 1961). An important example is the magnetorotational instability (MRI, see Balbus & Hawley 1998), which will also be the focus of the present paper. However, the subject is more general, and there are other instabilities that we mention below. When studying linear stability, one typically considers a stationary solution of the full nonlinear equations, linearizes the equations about this solution, and looks for the temporal behavior of small perturbations (wavenumber k) proportional to e^{λt}, where t is time and λ(k) is generally complex. The real part of λ is the growth rate, and λ as a function of k is the dispersion relation. Linear stability theory is useful to explain why many astrophysical flows are turbulent (e.g., accretion disks through the MRI or the stellar convection zones through the convective instability).
Linear stability theory is also generalized to study the formation of largescale instabilities in the presence of turbulent flows; e.g., studies of stability of the solar tachocline where convective turbulence is expected to be present (Arlt et al. 2007; Miesch et al. 2007). We first revisit this generalization. In the case of a turbulent flow, there is no stationary state in the usual sense; we can at best expect a statistically steady state. In such a situation, the prescription is to average over, or coarsegrain, the fundamental nonlinear equations (e.g., equations of magnetohydrodynamics) to write a set of effective equations valid for large length and timescales. Typical examples of such averaging include Reynolds averaging (Moffatt 1978; Krause & Rädler 1980), the multiscale techniques (Zheligovsky 2012), and application of the dynamical renormalization group (see, e.g., Goldenfeld 1992). The effective equations themselves depend on the averaging process, and also on the length and timescales to which they are applied. The averaging process can give rise to new terms in the effective equations and it introduces new transport coefficients that are often called turbulent transport coefficients to distinguish them from their microphysical counterparts. An example of such an effective equation is the meanfield dynamo equation which, in its simplest form, has two turbulent transport coefficients: the alpha effect, α, and turbulent magnetic diffusivity, η_{t}. Once the effective equations and the turbulent transport coefficients are known, we apply the standard machinery of linear stability theory to the effective equations to obtain the exponential growth or decay rate of largescale instabilities in or even because of the presence of turbulence.
This prescription, applied to real turbulent flows turns out to be not very straightforward because of several reasons that we list below: First, any spatial averaging procedure will retain some level of fluctuations (Hoyng 1988). This automatically limits the dynamical range over which exponential growth can be obtained. The larger the size of the turbulent eddies compared with the size of the domain, i.e., the smaller the scale separation ratio, the smaller the dynamical range. A wellknown example is the α effect in meanfield electrodynamics (Moffatt 1978; Krause & Rädler 1980), which gives rise to a linear instability of the meanfield equations. In direct numerical simulations (DNS), however, the expected exponential growth can only be seen over a limited dynamical range. A second, more recent, example is the negative effective magnetic pressure instability (NEMPI; Brandenburg et al. 2011), where the magnetic pressure develops negative contributions caused by the turbulence itself (Kleeorin et al. 1989; Kleeorin & Rogachevskii 1994; Rogachevskii & Kleeorin 2007). NEMPI could be detected in DNS only for a scale separation ratio of ten or more. Second, the averaged equation, in addition to the usual diffusive terms, can have higher order derivatives in both space and time (Rheinhardt & Brandenburg 2012). Such terms become important for a small scaleseparation ratio that generally reduces the efficiency of turbulent transport (Brandenburg et al. 2008a, 2009; Madarassy & Brandenburg 2010). So, in general, a simple prescription of replacing the microphysical value of diffusivity by its turbulent counterpart may not work. Third, there are important conceptual differences between microphysical and turbulent transport coefficients. The turbulent ones must reflect the anisotropies and inhomogeneities of real flows, and they are hence, in general, tensors of rank two or higher. Moreover, a major challenge in this formalism is the actual calculation of the turbulent transport coefficients. For turbulent flows, there is at present no known analytical technique that allows us to calculate them from first principles. A recent breakthrough is the use of the testfield method (Schrinner et al. 2005, 2007; Brandenburg et al. 2008b), which allows us to numerically calculate the turbulent transport coefficients for a large class of flows. Armed with the testfield method, we are now in a position to quantify how accurately the linear stability theory applied to the meanfield equations describes the growth of largescale instabilities in a turbulent flow. This is the principal objective of this paper.
The MRI is a relatively simple axisymmetric (twodimensional) linear instability of a rotating shear flow in the presence of an imposed magnetic field along the rotation axis. The dispersion relation for MRI is well known (Balbus & Hawley 1991, 1998). We now consider the situation in which we have a turbulent flow (which may have been generated due to MRI with microphysical parameters) in a rotating box in the presence of an axial magnetic field and largescale shear. We assume that we can use the dispersion relation for MRI and simply replace the microphysical values of magnetic diffusivity (η) and kinematic viscosity (ν) by the total (turbulent plus microphysical) values, η_{T} = η_{t} + η and ν_{T} = ν_{t} + ν, respectively. The corresponding dispersion relation has been derived by Lesur & Longaretti (2007) and Pessah & Chan (2008). In the special case where η_{T} = ν_{T}, it simplifies to (1)where V_{A}(k)k is the growth rate in the nonturbulent, ideal case. For the MRI with Keplerian shear, V_{A}(k) is given in terms of with (Balbus & Hawley 1998) (2)where , v_{A} is the Alfvén speed, k is the wavenumber, and Ω is the angular velocity. The qualitative validity of turbulent diffusion in MRI was previously demonstrated by Korpi et al. (2010), who focussed attention on the Maxwell and Reynolds stresses in the nonlinear regime, following earlier work by Workman & Armitage (2008) on the combined action of MRI in the presence of forced turbulence. The effect of forced turbulence on the MRI has been studied previously in connection with quasiperiodic oscillations driven by the interaction with rotational and epicyclic frequencies (Brandenburg 2005). We also note that in Eq. (1) we have assumed that ν_{T} = η_{T} which is essentially equivalent to assuming that the turbulent magnetic Prandtl number, ν_{t}/η_{t}, is unity because in most astrophysical flows ν ≪ ν_{t} and η ≪ η_{t}. This assumption is supported by DNS studies (Yousef et al. 2003; Fromang & Stone 2009; Guan & Gammie 2009). We note in this connection that the MRI may not work at small values of the microphysical magnetic Prandtl number, ν/η (Lesur & Longaretti 2007; Fromang et al. 2007). This, however, is not directly of concern to us, because we assume turbulence to be driven by an externally applied forcing and not a consequence of the MRI itself. We note however that the technique is agnostic about the mechanism that drives the turbulence, meaning that our conclusions would be unchanged even if turbulence was driven by (microphysical) MRI.
There is another important difference between microscopic and turbulent magnetic diffusion. For any linear instability the level of the exponentially growing perturbation depends logarithmically on the strength of the initial field. However, turbulent diffusion implies the presence of turbulence, so there is always some nonvanishing projection of the random velocity and magnetic fields, which will act as a seed such that the growth of the magnetic field is independent of the initial conditions and depends just on the value of the forcing wavenumber and the forcing amplitude. This can become particularly important in connection with the largescale dynamo instability, which is an important example of an instability that operates especially well in a turbulent system. Again, in that case, turbulence can provide a seed magnetic field to the largescale dynamo through the action of the much faster smallscale dynamo. This idea was first discussed by Beck et al. (1994) in an attempt to explain the rapid saturation of a largescale magnetic field in the galactic dynamo.
2. Model
Following earlier work of Workman & Armitage (2008) and Korpi et al. (2010), we solve the threedimensional equations of magnetohydrodynamics (MHD) in a cubic domain of size L^{3} in the presence of rotation with angular velocity Ω = (0,0,Ω), a shear flow with shear , and an imposed magnetic field B_{0} = (0,0,B_{0}). We adopt shearperiodic boundary conditions in the x direction (Wisdom & Tremaine 1988) and periodic boundary conditions in the y and z directions. We generate turbulence by adding a stochastic force with amplitude f_{0} and a wavenumber k_{f}. We have varied f_{0} to achieve different rootmeansquare (rms) velocities of the turbulence. Different values of the forcing wavenumber k_{f} will also be considered.
Definitions of essential variables.
We assume an isothermal gas with sound speed c_{s}, so the pressure is linearly related to the density ρ. The hydromagnetic equations are solved in terms of the magnetic vector
potential A, the velocity U, and the logarithmic density lnρ in the form
where is the advective derivative based on the shear flow and is the advective derivative based on the full flow field that includes both the shear flow and the deviations from it, B = ∇ × A is the magnetic field expressed in terms of the magnetic vector potential A. In our units, the vacuum permeability μ_{0} = 1. The current density is given by J = ∇ × B, η is the microphysical magnetic diffusivity, ν is the microphysical kinematic viscosity, and f is the turbulent forcing function given by (6)where k(t) is a random wavevector and (7)is used to produce nonhelical transversal waves, is an arbitrary unit vector needed to generate a nonvanishing vector that is perpendicular to k, φ(t) is a random phase, and N = f_{0}c_{s}(kc_{s}/δt)^{1/2}, k =  k , and δt is the length of the time step. The quantities k(t), , and φ(t) change at every time step, i.e., the external force is whiteintime. This oftused prescription for the external force has the advantage of not introducing any new timescales into the problem. Numerically, we integrate the whiteintime term by using the Euler–Marayuma scheme (Higham 2001). We focus on the case where  k  is from a narrow band of wavenumbers around k_{f}.
The smallest wavenumber that fits into the domain is k_{1} = 2π/L, and we shall use k_{1} as our inverse unit length. Our time unit is given by Ω^{1}. Nondimensional quantities will be expressed by a tilde. For example the nondimensional growth rate is and the nondimensional rms velocity is given by , and the nondimensional Alfvén speed is given by , where is the Alfvén speed based on the strength of the imposed magnetic field and ρ_{0} is the volume averaged density. Furthermore, the nondimensional forcing wavenumber and turbulent diffusion are given by and , respectively.
We quantify our results in terms of fluid and magnetic Reynolds numbers, as well as the Coriolis number, which are respectively defined as (8)In this paper, u_{rms} is the rms velocity before the onset of MRI, when u_{rms} has not yet started to grow exponentially. This particular value of u_{rms} is also used to estimate the turbulent magnetic diffusivity of the system in Eq. (10). We characterize our solutions by measuring u_{rms} and a similarly defined b_{rms}, which refers to the departure from the imposed field, again before the onset of MRI. We also use the quantity B_{rms} to characterize the total field, given by , which we use to define the Lundquist number, (9)At small magnetic Reynolds numbers, Re_{M} ≪ 1, we would expect (Krause & Rädler 1980), but in many of our runs we have Re_{M} ≫ 1, in which case . We note that the ratio Lu/Re_{M} is then equal to the ratio of magnetic field to the equipartition value. We also consider horizontally averaged magnetic field, , as well as its rms value, which is then still a function of time.
The DNS are performed with the Pencil Code^{1}, which uses sixthorder explicit finite differences in space and a thirdorder accurate timestepping method. We use numerical resolutions of 128^{3} and 256^{3} mesh points.
In the following, we discuss the dependence of the growth rate on the anticipated turbulent magnetic diffusivity (10)This simple formula was previously found to be a good estimate of the actual value of η_{t} (Sur et al. 2008), but this ignores complications from a weak dependence on k_{f}/k_{1} (Brandenburg et al. 2008a), as well as the mean magnetic field (Brandenburg et al. 2008c), which would result in magnetic quenching of η_{t}. To shed more light onto this uncertainty, we also make use of the quasikinematic testfield method of Schrinner et al. (2005, 2007) to calculate the actual value of η_{t} based on the measured diagonal components of the magnetic diffusion tensor η_{ij}, i.e., η_{t} ≡ (η_{11} + η_{22})/2. We note that the evolution of the horizontally averaged magnetic field is governed by just four components of η_{ij} (Brandenburg et al. 2008b) and another four components of what is called the α_{ij} tensor, whose components turn out to be zero in all cases investigated in this paper.
To clarify the definitions of the most important variables, we have also collected them in Table 1 for reference. We also list several alternative ways of estimating η_{t} that will be described later in the text. One possibility is to measure η_{t} using the testfield method, and another is to determine it from the decrease in the MRI growth rate owing to the effect of turbulence, which is referred to as ; the corresponding details will be explained in Sect. 3.3.
3. Results
3.1. Turbulence as a seed of MRI
We begin by calculating the growth rate of the largescale instability from our DNS. The DNS is started with an initial condition where the velocity is initially zero. As a result of the action of the external force, smallscale velocity grows fast and then saturates. This smallscale velocity acts as a seed field for the largescale MRI. Consequently we see a second growth phase at late times. This is due to the growth (via MRI) of largescale velocity and magnetic field, both of which show exponential growth at this phase. In Fig. 1 we show this growth for different values of the amplitude of the external force. The growth rate of the largescale instability can be calculated from the exponentially growing part of these plots.
Fig. 1 Time dependence of and for runs with (Runs G1G4 and G7). In addition, sample runs with higher resolution (256^{3}, Runs P1P4), no magnetic fields, and no forcing (Run N6) are included for comparison. 

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At late times, saturates near unity, while continues to grow; see Fig. 1. Eventually, however, our DNS become numerically unstable, which is a result of the “channel solution” which is excited when a vertical field is imposed (Hawley & Balbus 1991). This solution breaks up when secondary instabilities become effective but this occurs at a point where the Mach number is on the order of unity in the present setup. This causes shocks that our numerical model could handle if shock diffusion would be enabled. Although this is technically possible in our code (Haugen et al. 2004), we do not use this option as we are not interested in the nonlinear stage of the MRI. By increasing the resolution, we have been able to continue the saturated phase for a somewhat longer time. The results of higher resolutions runs are shown with longdashed lines in Fig. 1, where we used 256^{3} mesh points. On the other hand, higher resolution is not crucial for determining the turbulence effects on the linear growth phase of MRI, which is why in the following we only present results obtained at a resolution of 128^{3} mesh points.
Compared to the nonmagnetic case, the imposed magnetic field slightly decreases the saturation level of the forced turbulence before the visible growth of MRI. In addition, the presence of a magnetic field causes to have two plateaus: first at the very beginning and second after reaches the level of . In the linear growth stage leading to saturation, and are in antiphase and vary sinusoidally with wavenumber k = k_{1}; see Fig. 2. At early times, and are shifted in phase by about 90° and have a wavenumber of 2k_{1}.
Once there is exponential growth, the growth rates of and are, as expected, the same, but they are different for different amplitudes f_{0} of the forcing function, see also Tables A.1 and A.2. We also note that the runs with the weakest forcing have a slightly faster growth, because the resulting turbulent viscosity and diffusivity are smaller, but they also show a later onset of exponential growth. This in turn is related to a weaker residual projection onto the MRI eigenfunction, simply because the amplitude of the turbulence is lower.
Fig. 2 Time dependence of (top) and (bottom) for and (Run O7). 

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The growth rate λ thus calculated is plotted in Fig. 3 as a function of nondimensionalized v_{A}. For comparison, we have also plotted the growth rate calculated from the dispersion relation of MRI, Eq. (1), with a fixed coefficient of magnetic diffusivity , where . We chose a value for from Run O3 with , (see Table A.1). Both computed runs and the dispersion relation agree reasonably everywhere except with , where linear theory predicts no MRI. The positive growth rates in the DNS results in this regime are likely due to another instability such as the incoherent αshear dynamo (Vishniac & Brandenburg 1997; Mitra & Brandenburg 2012) and/or the turbulent shear dynamo (Yousef et al. 2008a,b; Heinemann et al. 2011). Another possibility might be a hydrodynamic shear dynamo, which has been seen previously in the absence of rotation (Käpylä et al. 2009).
Fig. 3 Dependence of on for the Set O (triangles). The solid line represents the dispersion relation of Eq. (1) with . For comparison, the ideal and nonturbulent cases are shown as dashed and dotted lines, respectively. 

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In the beginning, the components of the horizontally averaged magnetic field are still randomly fluctuating, but at later times, when nonlinear effects begin to play a role, a clear pattern with wavenumber k = k_{1} develops; see Fig. 2. This is expected in this particular run (O7) where the fastest growing mode has a wavenumber close to k_{1}. However, we see the same behavior also in other runs in Set O where the theoretically predicted k_{max} varies by more than an order of magnitude, see Table A.1. By contrast, according to linear theory, the eigenfunction always settles onto the fastest growing one, which would have a wavenumber larger than k_{1}. Again, possible reasons for this could be the aforementioned incoherent αshear dynamo or a hydrodynamic shear dynamo.
The kinetic energy spectra from the forcingdominated and from the linear growth phase of the MRI from a representative model (Run P5) are shown in Fig. 4. The power falls off from the k_{f} peak to higher wavenumbers approximately as a ∝ k^{−5/3} Kolmogorov spectrum with (11)where is the total energy dissipation and C_{K} is the Kolmogorov constant. The inset of Fig. 4 shows that C_{K} ≈ 1. In the forcingdominated regime the power falls again at lower wavenumbers whereas in the MRI dominated case the largescale k = k_{1} mode has the highest power.
Fig. 4 Kinetic energy spectra from Run P5 during both the forcingdominated plateau (red dashed line) and the linear growth phase (solid line). The inset shows the same energy spectra, but compensated with ϵ^{− 2/3}k^{5/3}. 

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3.2. Different ways of varying η_{t}
To explore the dependence of the solutions on the anticipated turbulent magnetic diffusivity, we consider three sets of runs. In two of them (Sets A and B), we vary k_{f}, and in one (Set C) we vary the value of Ω, thus changing Co which was defined in Eq. (8). Given the definition of η_{t0} in Eq. (10), we have (12) This shows that increasing either Co or k_{f} or both leads to a decrease in . We note that is the value before the onset of MRI and has been estimated by measuring the height of the plateau seen in Fig. 1. We should point out that for small values of the length of the plateau becomes rather short, which leads therefore to a significant source of error. The parameters for the three sets of runs are summarized in Table A.1.
Fig. 5 versus Coriolis number for , at a resolution of 128^{3} mesh points. The solid line is a fit of Eq. (12) into the results of Set C. 

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In Fig. 5 we plot these three sets of runs in a Co– diagram. Looking at Eq. (12), and since is fixed, it is clear that the runs of Set C all fall on a line proportional to Co^{1}. For the other two sets, varies. Small values of correspond to large values of both Co and , and vice versa, which is the reason why the other two branches for Sets A and B show an increase in for increasing values of Co. Correspondingly, decreases with increasing Co for Sets A and B, while for Set C, increases with increasing Co; see Fig. 6. We note that in Sets A and B the magnetic Reynolds number is changing by an order of magnitude which is not captured by Eq. (12).
Fig. 6 Growth rate versus Coriolis number for at a resolution of 128^{3} mesh points. 

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For Sets A and B we show the dependence of the growth rate on in Fig. 7. For both sets, increases with increasing . This increase is related to the fact for increasing values of , decreases, and thus shows a mild increase. Indeed, we should expect that varies with as (13)where in the present case the best agreement with the DNS is obtained when is chosen. This theoretically expected dependency is overplotted in Fig. 7.
Fig. 7 Growth rate versus the scale separation for , at a resolution of 128^{3} mesh points. The solid line shows a fit to the theoretical dependency given by Eq. (13). 

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Fig. 8 Growth rate versus the for Sets A–G, at a resolution of 128^{3} mesh points. For the solid line we used k/k_{1} = 1. 

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We now turn to relation (1), which predicts a parabolic decline for increasing values of (η_{t} + η)k^{2}. This relation is surprisingly well obeyed; see Fig. 8, where we plot as a function of (η_{t} + η)k^{2} for models of all three sets, together with those of Sets D–G listed in Table A.1. For the solid line we used k/k_{1} = 1.
3.3. Comparison with testfield results
Our results presented so far have demonstrated that the growth of largescale perturbations is determined by the same equations that describe the growth of MRI using values of magnetic diffusivity (and viscosity) that are not their microphysical values, but turbulent values. Hence, by turning the problem on its head, we have here a new method of calculating the turbulent magnetic diffusivity by measuring the growth rate of the largescale instability. Such a method would proceed in the following manner. First we would study the growth of the largescale instability and produce a plot similar to Fig. 1 from which we can calculate the growth rate λ. Once we know λ we can read off η_{t} by using Fig. 8. We call the turbulent diffusivity, measured in this fashion, . Alternatively, we use the testfield method to calculate the turbulent magnetic diffusivity. It then behooves us to compare these two methods, for cases where they both can be applied.
Fig. 9 Dependence of (△) and (□) as a function of Lu compared to and . 

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Fig. 10 Dependence of (□) and (△) as a function of Lu. 

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To apply the testfield method (Schrinner et al. 2005, 2007; Brandenburg et al. 2008b) to the present problem, we define averaged quantities by averaging over the horizontal xy plane and choose zdependent test fields which are sines and cosines, (14)respectively. Here, with p = 1,2 are the unit vectors in the x and y directions, respectively. For each testfield, we evolve a separate evolution equation for the resulting fluctuating magnetic field, , expressed in terms of its vector potential , (15)where is the fluctuating nonlinear term that is often ignored in analytic calculations. We then calculate the corresponding electromotive force, , and express it as (16)where . We thus obtain four vector equations, each with two relevant components for the x and y directions, so we have eight equations for the eight unknowns α_{11}, α_{12}, ..., η_{22}; see Brandenburg (2005) and Brandenburg et al. (2008b) for details. The components of α_{ij} are all zero within error bars and will not be discussed further.
In principle the turbulent magnetic diffusivity thus calculated is a secondrank tensor, η_{ij}. We plot the diagonal and offdiagonal components of this tensor in Figs. 9 and 10, respectively. The offdiagonal elements are close to zero and the diagonal elements are equal to each other and also equal to η_{t0}. In Table A.1 we list . Regarding the offdiagonal elements, if any departure from zero is significant, it would be for small values of , i.e., in the kinematic regime where the effects of magnetic quenching are weak.
3.4. Is there η_{t} quenching?
The two methods we have described and compared in the previous subsection now allow us to quantify how turbulent diffusivity is quenched in the presence of the background magnetic field. Quenching of turbulent magnetic diffusivity has been computed analytically (Kitchatinov et al. 1994) and numerically (Yousef et al. 2003; Gressel et al. 2013), and it has been used in dynamo models (Tobias 1996; Guerrero et al. 2009). Here, we address this question by considering the turbulent magnetic diffusivity and as a function of Lu, as done in Fig. 9. In none of the cases do we observe any η_{t} quenching.
For Set G we see that η_{t0} shows an increase with magnetic field strength (see Table A.1), which might suggest the possibility of “antiquenching”. However, in Set G, the value of Re_{M} is also increasing, so the increase in η_{t0} is really just a consequence of too small values of Re_{M} in the runs with weak magnetic field. This is confirmed by considering the runs in Set O, where Re_{M} is approximately constant and η_{t0} is then found to be approximately independent of the imposed field strength. It should however be pointed out that the possibility of antiquenching of turbulent magnetic diffusivity (as well as antiquenching of the α effect in dynamo theory) has been invoked in the past to explain the observed increase in the ratio of dynamo frequency to rotational frequency for more active stars (Brandenburg et al. 1998). Antiquenching of both turbulent effects was also found for flows driven by the magnetic buoyancy instability (Chatterjee et al. 2011). On the other hand, regular quenching has been found both in the absence of shear (Brandenburg et al. 2008c) as well as in the presence of shear (Käpylä & Brandenburg 2009). It should therefore be checked whether earlier findings of antiquenching may also have been affected by too small magnetic Reynolds numbers.
4. Conclusions
Our work has demonstrated several unexpected aspects of turbulent mixing on the operation of the MRI. Firstly, the effect of turbulent magnetic diffusivity seems to be in all aspects equivalent to that of microphysical magnetic diffusivity. This is true even when scale separation is poor, e.g., for k_{f}/k_{1} = 1.5 or 2.2. This is rather surprising, because in such an extreme case the memory effect was previously found to be important (Brandenburg et al. 2004), which means that higher time derivatives in the meanfield parameterization need to be included (Hubbard & Brandenburg 2009). Secondly, the simple estimate given by Eq. (10) is remarkably accurate. As a consequence, Eq. (1) provides a quantitatively useful estimate for the effects of turbulence on the growth rate of the MRI. Our simple estimates also agree with results obtained from the testfield method. In principle, there could be other nondiffusive effects resulting from the socalled Ω × J effect (Rädler 1969) or the shear–current effect (Rogachevskii & Kleeorin 2003, 2004), but our present results show that this does not seem to be the case, because the signs of η_{2,1} and agree; see Brandenburg (2005) and Gressel (2010, 2013) for earlier results in the context of MRI and Brandenburg et al. (2008b) in the context of forced turbulence. One difference is, however, that in Brandenburg (2005) the component η_{12} had the opposite sign, but this term is subdominant compared with shear and unimportant for dynamo action.
It should also be pointed out that no new terms seem to appear in the momentum equation other than the turbulent viscous force. Of course, this could change if we were to allow for extra effects such as strong density stratification, which could lead to the development of the negative effective magnetic pressure instability (see Brandenburg et al. 2011, and references therein). Furthermore, if there is crosshelicity, there can be new terms in the momentum equation that are linear in the mean magnetic field (Rheinhardt & Brandenburg 2010). Also kinetic and magnetic helicity could affect our results, although there have not yet been any indications for this from purely hydrodynamic shear flow turbulence (Madarassy & Brandenburg 2010). Neither the negative effective magnetic pressure instability nor the α effect dynamo instability are possible in the simple example studied here, because stratification is absent. However, as alluded to in the introduction, they both are examples that have contributed to the motivation of the work presented here.
Acknowledgments
The authors thank Nordita for hospitality during their visits. Financial support from Jenny and Antti Wihuri Foundation and Finnish Cultural Foundation grants (M.V.), the Academy of Finland grants No. 136189, 140970, 272786 (P.J.K.), the Academy of Finland Centre of Excellence ReSoLVE No. 272157 (M.J.M.), as well as the Swedish Research Council grants 62120115076 and 20125797, and the European Research Council under the AstroDyn Research Project 227952 are acknowledged. We acknowledge CSC – IT Center for Science Ltd., who are administered by the Finnish Ministry of Education, for the allocation of computational resources. This research has made use of NASA’s Astrophysics Data System.
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Online material
Results for all 128^{3} datasets.
Results for 256^{3} runs.
All Tables
All Figures
Fig. 1 Time dependence of and for runs with (Runs G1G4 and G7). In addition, sample runs with higher resolution (256^{3}, Runs P1P4), no magnetic fields, and no forcing (Run N6) are included for comparison. 

Open with DEXTER  
In the text 
Fig. 2 Time dependence of (top) and (bottom) for and (Run O7). 

Open with DEXTER  
In the text 
Fig. 3 Dependence of on for the Set O (triangles). The solid line represents the dispersion relation of Eq. (1) with . For comparison, the ideal and nonturbulent cases are shown as dashed and dotted lines, respectively. 

Open with DEXTER  
In the text 
Fig. 4 Kinetic energy spectra from Run P5 during both the forcingdominated plateau (red dashed line) and the linear growth phase (solid line). The inset shows the same energy spectra, but compensated with ϵ^{− 2/3}k^{5/3}. 

Open with DEXTER  
In the text 
Fig. 5 versus Coriolis number for , at a resolution of 128^{3} mesh points. The solid line is a fit of Eq. (12) into the results of Set C. 

Open with DEXTER  
In the text 
Fig. 6 Growth rate versus Coriolis number for at a resolution of 128^{3} mesh points. 

Open with DEXTER  
In the text 
Fig. 7 Growth rate versus the scale separation for , at a resolution of 128^{3} mesh points. The solid line shows a fit to the theoretical dependency given by Eq. (13). 

Open with DEXTER  
In the text 
Fig. 8 Growth rate versus the for Sets A–G, at a resolution of 128^{3} mesh points. For the solid line we used k/k_{1} = 1. 

Open with DEXTER  
In the text 
Fig. 9 Dependence of (△) and (□) as a function of Lu compared to and . 

Open with DEXTER  
In the text 
Fig. 10 Dependence of (□) and (△) as a function of Lu. 

Open with DEXTER  
In the text 