Issue 
A&A
Volume 636, April 2020



Article Number  A93  
Number of page(s)  12  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201935012  
Published online  24 April 2020 
Turbulent viscosity and magnetic Prandtl number from simulations of isotropically forced turbulence
^{1}
GeorgAugustUniversität Göttingen, Institut für Astrophysik, FriedrichHundPlatz 1, 37077 Göttingen, Germany
email: pkaepyl@unigoettingen.de
^{2}
ReSoLVE Centre of Excellence, Department of Computer Science, Aalto University, PO Box 15400, 00076 Aalto, Finland
^{3}
Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden
^{4}
Department of Astronomy, Stockholm University, 10691 Stockholm, Sweden
^{5}
JILA and Department of Astrophysical and Planetary Sciences, Box 440, University of Colorado, Boulder, CO 80303, USA
^{6}
Laboratory for Atmospheric and Space Physics, 3665 Discovery Drive, Boulder, CO 80303, USA
^{7}
MaxPlanckInstitut für Sonnensystemforschung, JustusvonLiebigWeg 3, 37077 Göttingen, Germany
Received:
3
January
2019
Accepted:
25
January
2020
Context. Turbulent diffusion of largescale flows and magnetic fields plays a major role in many astrophysical systems, such as stellar convection zones and accretion discs.
Aims. Our goal is to compute turbulent viscosity and magnetic diffusivity which are relevant for diffusing largescale flows and magnetic fields, respectively. We also aim to compute their ratio, which is the turbulent magnetic Prandtl number, Pm_{t}, for isotropically forced homogeneous turbulence.
Methods. We used simulations of forced turbulence in fully periodic cubes composed of isothermal gas with an imposed largescale sinusoidal shear flow. Turbulent viscosity was computed either from the resulting Reynolds stress or from the decay rate of the largescale flow. Turbulent magnetic diffusivity was computed using the testfield method for a microphysical magnetic Prandtl number of unity. The scale dependence of the coefficients was studied by varying the wavenumber of the imposed sinusoidal shear and test fields.
Results. We find that turbulent viscosity and magnetic diffusivity are in general of the same order of magnitude. Furthermore, the turbulent viscosity depends on the fluid Reynolds number (Re) and scale separation ratio of turbulence. The scale dependence of the turbulent viscosity is found to be well approximated by a Lorentzian. These results are similar to those obtained earlier for the turbulent magnetic diffusivity. The results for the turbulent transport coefficients appear to converge at sufficiently high values of Re and the scale separation ratio. However, a weak trend is found even at the largest values of Re, suggesting that the turbulence is not in the fully developed regime. The turbulent magnetic Prandtl number converges to a value that is slightly below unity for large Re. For small Re we find values between 0.5 and 0.6 but the data are insufficient to draw conclusions regarding asymptotics. We demonstrate that our results are independent of the correlation time of the forcing function.
Conclusions. The turbulent magnetic diffusivity is, in general, consistently higher than the turbulent viscosity, which is in qualitative agreement with analytic theories. However, the actual value of Pm_{t} found from the simulations (≈0.9−0.95) at large Re and large scale separation ratio is higher than any of the analytic predictions (0.4−0.8).
Key words: turbulence / Sun: rotation / stars: rotation
© ESO 2020
1. Introduction
Turbulent transport is often invoked to explain phenomena in astrophysical systems such as accretion (e.g. Shakura & Sunyaev 1973; Frank et al. 2002), maintenance of stellar differential rotation (Rüdiger 1980, 1989; Rüdiger et al. 2013), and largescale magnetic field generation (Moffatt 1978; Krause & Rädler 1980). Turbulence is typically thought to diffuse largescale structures analogously to molecular diffusion but at a rate that is several orders of magnitude higher (e.g. Väisälä et al. 2014).
Turbulent diffusion coefficients, such as turbulent viscosity (ν_{t}) and magnetic diffusivity (η_{t}), are often estimated using arguments from the mixing length theory (MLT) according to which ν_{t} ≈ η_{t} ≈ ul/3, where u and l are the characteristic velocity and length scale of the turbulence. Such estimates yield values of the order of 10^{8} − 10^{9} m^{2} s^{−1} for the solar convection zone, which coincide with values estimated for the turbulent magnetic diffusivity η_{t} from sunspot decay in the quenched case (Krause & Rüdiger 1975; Petrovay & van DrielGesztelyi 1997; Rüdiger & Kitchatinov 2000) and from cross helicity measurements in the unquenched (quiet Sun) case (Rüdiger et al. 2011). With the advent of the testfield method (Schrinner et al. 2005, 2007), it has become possible to measure turbulent transport coefficients that are relevant for the electromotive force (e.g., the turbulent magnetic diffusivity) from simulations. Detailed studies using this method indicate that the MLT estimate yields the correct order of magnitude in the kinematic regime (e.g. Sur et al. 2008; Käpylä et al. 2009a), provided that l is identified with the inverse of the wavenumber k_{f} of the energycarrying eddies. This result can further be affected by other physical properties, such as the presence of kinetic helicity in the flow, which can reduce the value of η_{t} (Brandenburg et al. 2017). The testfield method also revealed an approximately Lorentzian dependence on the wavenumber of the mean field (Brandenburg et al. 2008a).
In the absence of a corresponding testfield method for hydrodynamics, the estimates of ν_{t} are typically much less accurate than those obtained for η_{t} from such methods. Estimates of turbulent viscosity from shearing box simulations, however, also indicate a value of the order of the MLT estimate (e.g. Snellman et al. 2009; Käpylä et al. 2010). Computing ν_{t} from simulations with imposed linear shear flows is problematic due to hydrodynamical instabilities that can be excited (e.g. Elperin et al. 2003; Käpylä et al. 2009b). Furthermore, also nondiffusive contributions to the turbulent stress exist. First, the anisotropic kinetic alpha (AKA) effect can occur in the presence of Galilean noninvariant flows^{1} and can give rise to hydrodynamic instabilities analogous to the electromagnetic dynamo (e.g. Frisch et al. 1987; Brandenburg & von Rekowski 2001; Käpylä et al. 2018)^{2}. Second, anisotropic turbulence with global rotation leads to a Λ effect, which is relevant for causing differential rotation (e.g. Rüdiger 1989; Kitchatinov & Rüdiger 2005; Käpylä & Brandenburg 2008; Käpylä 2019a). Typically, these effects cannot easily be disentangled from the contribution of turbulent viscosity. Additionally, a spatially nonuniform kinetic helicity Yokoi & Brandenburg (2016) in rotating nonmirror symmetric flows leads to the generation of largescale flows.
Contrary to the microphysical magnetic Prandtl number, which can vary over tens of orders of magnitude in the astrophysical context, depending on the physical characteristics of the system under study (e.g. Brandenburg & Subramanian 2005), the ratio of ν_{t} to η_{t}, that is the turbulent magnetic Prandtl number Pm_{t}, is thought to be of the order of unity in the astrophysically relevant regime of high Reynolds numbers. Nevertheless, astrophysical applications of the possibility of Pm_{t} being different from unity have been discussed. These include both accretion disc turbulence and solar convection. In the context of accretion onto a magnetised star, one often assumes that the field lines of the star’s magnetic field are being dragged with the flow towards the star, so as to achieve a pitch angle suitable for jet launching (Blandford & Payne 1982). This requires the turbulent magnetic diffusivity to be small (Elstner & Rüdiger 2000), while subsequent work has shown that Pm_{t} has only a weak influence on the pitch angle (Rüdiger & Shalybkov 2002).
Another application has been suggested in the context of the solar convection zone. For flux transport dynamos to explain the equatorward migration of the sunspot belts, one must assume the turbulent magnetic diffusivity to be of the order of 10^{7} m^{2} s^{−1} (Chatterjee et al. 2004). On the other hand, to prevent the contours of constant angular velocity from being constant on cylinders, the turbulent viscosity must be around 10^{9} m^{2} s^{−1}, or even larger (Brandenburg et al. 1990). Thus, again, a turbulent magnetic Prandtl number in excess of unity is required for this model to be successful. A large turbulent viscosity is sometimes argued to be a consequence of the magnetic stress from smallscale dynamo action (Karak et al. 2018). Whether this idea has a solid foundation remains open, however.
The analytic estimates of the turbulent magnetic Prandtl number range between 0.4 under the firstorder smoothing approximation (FOSA) to 0.8 under various versions of the τ approximation (Yousef et al. 2003; Kitchatinov et al. 1994), of which the spectral minimal τ approximation (MTA) applied to fully developed turbulent convection yields values in the range 0.23−0.46 (Rogachevskii & Kleeorin 2006). Different renormalisation group analyses yield Pm_{t} ≈ 0.42−0.79 (e.g. Fournier et al. 1982; Kleeorin & Rogachevskii 1994; Verma 2001; Jurčišinová et al. 2011). Furthermore, the turbulent magnetic Prandtl number has been studied from simulations of forced turbulence with a decaying largescale field component by Yousef et al. (2003) who found that Pm_{t} is approximately unity irrespective of the microphysical magnetic Prandtl and Reynolds numbers. However, their dataset is limited to a few representative cases that do not probe the Reynolds number or scale dependences systematically.
Our aim is to compute the turbulent viscosity and turbulent magnetic Prandtl number from direct simulations of homogeneous isotropically forced turbulence where we systematically vary the Reynolds number and scale separation ratio and compare the obtained results with analytic ones. To achieve this, we impose a largescale shear flow with a harmonic profile on the (nonrotating) flow and determine the turbulent viscosity either from the generated Reynolds stresses or from the decay rate of the largescale flow. For obtaining the turbulent magnetic diffusivity we employ the testfield method.
2. Model
2.1. Basic equations
We model a compressible gas in a triply periodic cube with edge length L. It obeys an isothermal equation of state defined by , with pressure p, density ρ and constant speed of sound c_{s}. Hence, we solve the continuity and Navier–Stokes equations with both an imposed random and largescale shear forcing
where D/Dt = ∂/∂t + U ⋅ ∇ is the advective time derivative, U is the velocity, ν is the constant kinematic viscosity, is the traceless rate of strain tensor, and the commas denote spatial derivatives. The forcing function f is given by
where k(t) is a random wavevector and
is used to produce a nonhelical transversal sinusoidal f, where e(t) is an arbitrary random unit vector, not aligned with k, and ϕ(t) is a random phase. is a normalisation factor, k = k, δt is the length of the integration time step and f_{0} is a constant dimensionless scaling factor. The quantities k, e, and ϕ change at every time step, so that the external force is deltacorrelated (white) in time. Numerically, we integrate the forcing term by using the Euler–Maruyama scheme (Higham 2001). We consider models where k is within a narrow shell of wavevectors with k close to a chosen k_{f}, and determined such that the forcing always obeys the periodic boundary conditions.
The last term in Eq. (2) maintains a largescale shear flow on top of the forced background turbulence via relaxing the horizontally (xy) averaged part of the y velocity, indicated by the overbar, towards the temporally constant profile ; is the unit vector in the ydirection. The relaxation time scale τ is chosen to match the turnover time (u_{rms}k_{f})^{−1} of the turbulence, where u_{rms} is the rms value of the fluctuating velocity, , with the average taken over the full volume as indicated by the angle brackets, and over the statistically steady part of the simulations, indicated by the subscript t. Our results are not sensitive to the relaxation time τ in the range 0.1 < τu_{rms}k_{f} < 10 so the (arbitrary) choice τu_{rms}k_{f} = 1 is justified. We choose a simple harmonic form for the shear flow according to
where U_{0} is the flow amplitude, and , k_{1} = 2π/L.
2.2. Input and output quantities
We measure density in terms of its initially uniform value ρ_{0}, velocity in units of the sound speed c_{s}, and length in terms of . Furthermore, in the cases with the testfield method employed, we choose a system of electromagnetic units in which μ_{0} = 1, where μ_{0} is the permeability of vacuum. The simulations are fully defined by choosing the forcing amplitude f_{0} and scale k_{f}/k_{1}, kinematic viscosity ν, microscopic magnetic Prandtl number
where η is the microscopic magnetic diffusivity in the testfield method, and the shear parameter
We further assume that the scale of the test fields always equals that of the imposed largescale flow, that is k_{B} = k_{U}, and that the value of Pm for the testfield simulations equals unity. For the scale separation ratio 𝒮 we employ the definition
The following quantities are used as diagnostics of our models. We quantify the level of turbulence in the simulations by the fluid and magnetic Reynolds numbers
The strength of the imposed shear is measured by the dynamic shear number
Guided by MLT and FOSA, we normalise both the turbulent viscosity and magnetic diffusivity by
while the turbulent magnetic Prandtl number is given by
3. Computation of ν_{t} and η_{t}
3.1. Meanfield analysis
In what follows, we rely upon Reynolds averaging, specifically defining the mean quantities as averages over x and y. Hence, they can only depend on z and time. Averages are indicated by overbars and fluctuations by lowercase or primed quantities, thus , etc.
3.1.1. Hydrodynamics
In the incompressible case all turbulent effects can be subsumed in the Reynolds stress tensor whose divergence appears in the evolution equation of the mean flow. Including compressibility and starting from
where the dots stand for viscous and external forces, one obtains after averaging
The contributions proportional to the Reynolds stresses, , no longer cover all turbulent effects originating from the inertial terms. However, in our weakly compressible setups with Ma ≈ 0.1 the difference between, for example and is 𝒪(10^{−2})^{3}. Thus, we will consider, as in the incompressible case, only the Reynolds stresses^{4}. When restricting to first order in the mean quantities, they can be decomposed into three contributions,
where is already present in the absence of both a mean flow and a gradient of , is occurring due to the presence of , and occurs due to the presence of (for a justification see Appendix A). As in our simulations no significant occurs, we disregard . Further, as the fluctuations are isotropically forced, the only nonzero components of are . Apart from small fluctuations, they do not depend on z and thus do not act onto the mean flow. Note that due to the absence of a global rotation there is also no contribution of the Λ effect in Q_{ij}. In what follows we drop the superscript for brevity.
For sufficiently slowly varying mean flows and sufficient scale separation, Q_{ij} can be approximately represented by the truncated Taylor expansion
with the symmetry requirements
Here, A_{ijk} describes the AKA effect, while N_{ijkl} comprises turbulent viscosity (amongst other effects)^{5}. For isotropic (and hence homogeneous) fluctuations, that is in the kinematic limit , A_{ijk} = 0, and N_{ijkl} must have the form
where the constants ν_{t} and ζ_{t} are the turbulent shear and bulk viscosities, respectively. The Reynolds stresses appear then correspondingly as
with the first term reproducing the Boussinesq ansatz. Although our turbulence is isotropically forced, the presence of finite shear causes it to be anisotropic with preferred directions given by the direction of the mean flow and, say, its curl, . Given that it is the divergence of Q_{ij} which enters the mean momentum equation and mean quantities depend only on z, merely the components A_{i3k} and N_{i3k3} matter in (16). As needs not to be solenoidal, might in general depend on z and the turbulent bulk viscosity is then of interest.
Further simplification is obtained when assuming that the mean velocity has only one component. In our setup, the mean flow is always very close to the maintained one, that is, . Then we have
where we have introduced new coefficients a_{i} = A_{i32} and n_{i} = N_{i323}. Comparison with (19) reveals that for with isotropic forcing n_{2} → −ν_{t} while a_{i} and n_{1, 3} should approach zero.
We note that the AKAeffect can only be expected to appear in Galilean noninvariant flows (Frisch et al. 1987). This is not the case for the flows considered here, because the forcing is δ correlated in time, so there is no difference between a forcing defined in an arbitrary inertial frame and a forcing defined in the (resting) lab frame.
3.1.2. Magnetohydrodynamics
We consider only z dependent mean fields in which case the mean electromotive force , when truncated in analogy to (16), can be represented by two rank–2 tensors
where J = ∇ × B is the current density. Given that all quantities depend only on z, we have and because const. by virtue of , has no effect on the evolution of . Hence we set and restrict our interest to the components α_{ij} and η_{ij} with i, j = 1, 2. As the pseudotensor α_{ij} can for nonhelical forcing merely be constructed from the building blocks and by the products and , within its restricted part, only the components α_{12} and α_{21} can be nonzero for our setup. Building blocks for the anisotropic part of the restricted η_{ij} are here
hence the offdiagonal components η_{12, 21} need to vanish. So all the relevant components, except an isotropic contribution to η, have leading order in U_{0} of at least 2. In the limit U_{0} → 0 we have α_{ij} → 0 while η_{11, 22} → η_{t}.
3.2. Imposed shear method
We apply three methods to extract the meanfield coefficients from the simulation data:

M1:
The mean flow depends on z, and as it is approximately harmonic, its zeros do not coincide with those of its derivative . Hence the coefficients a_{i} and n_{i} can be isolated by
where and are the zeros of and , respectively. a_{i} and n_{i} are then further subjected to temporal averaging. In general, their values at the different zeros will only coincide in the limit U_{0} → 0, but in our case the differences turned out to be smaller than the error bars.

M2a:
We use constant fit coefficients a_{i} and n_{i} in the time averaged simulation data of Q_{iz}, , and :

M2b:
Alternatively, we drop the nondiffusive contribution and use only a single coefficient n_{i} as a fit parameter:
For method M1 we divide the time series of a_{i} and n_{i} into three parts and define the largest deviation from the average, taken over the whole time series, as the error. For M2a,b we similarly perform the fit for data averaged over three equally long parts of the time series and take the error to be the largest deviation from the fitted values obtained from a time average over the full time series. Our results indicate that only the Reynolds stress component Q_{yz} shows a significant signal that can be related to the meanfield effects discussed above.
Figure 1 shows the horizontally averaged mean flow , Reynolds stress component Q_{yz}(z, t), and the z profiles of its temporal average along with from method M2b. The imposed velocity profile induces a largescale pattern in the Reynolds stress with the same vertical wavenumber, but with a vertical shift of π/2.
Fig. 1. Horizontally averaged velocity (top), (middle), and its temporal average in comparison with (bottom) from Run E9 (see Table 2) with , , Sh ≈ 0.04, and Re ≈ 497. ν_{t} from method M2b. 
3.3. Decay experiments
Apart from measuring the response of the system to imposed shear, it is possible to measure the turbulent viscosity independently from the decay of largescale flows. We refer to this procedure as M3. We employ this method to check the consistency of methods M1 and M2 in a few cases.
The dispersion relation for the largescale flow is given by
where ν_{T} = ν + ν_{t} and k_{z} is the wavenumber of the flow. Equation (27) is valid if largescale velocities other than , the pressure gradient, and the effects of compressibility are negligible. We measure the decay rate of the k_{z} = k_{U} constituent of the flow by extracting its amplitude using Fourier transform and fitting an exponential function to the data. The clear exponential decay is drowned out by the random signal from the turbulence after a time that depends on the amplitude of the initial largescale flow and other characteristics of the simulations. Thus we limit the fitting to the clearly decaying part of the time series which typically covers roughly 300 turnover times.
To reduce the effect of the stochastic fluctuations of the turbulence, we perform N independent realisations of the decay and measure ν_{t} from the decay rate in each case. This is achieved by using N uncorrelated snapshots from the fiducial run with imposed shear flow as initial conditions for decay experiments, see Fig. 6 for representative results where N = 10. Such snapshots are separated by at least 80 turbulent eddy turnover times. An error estimate is obtained by dividing the obtained values of ν_{t} into two groups and considering the largest deviation of averages over these from the average over the full set.
3.4. Testfield method
We use the testfield method, originally described in Schrinner et al. (2005, 2007), to determine the turbulent transport coefficients α_{ij} and η_{ij}. Our formulation is essentially the same as in Brandenburg et al. (2008b). The fluctuating magnetic fields are evolved with the flow taken from the simulation by
where b^{T} = ∇ × a^{T}, η is the magnetic diffusivity, and is one out of a set of largescale test fields. Neither the fluctuating fields a^{T} nor the test fields B^{T} act back on the flow. Each of the test fields yields an electromotive force (EMF)
Assuming that the mean field varies slowly in space and time, the electromotive force can be written as
where α_{ij} and β_{ijk} represent the α effect and turbulent diffusion, respectively. These coefficients can be unambiguously inverted from Eq. (30) by choosing an appropriate number of independent test fields.
We use four stationary z dependent test fields
where k_{B} is a wavenumber. As explained in Sect. 3.1.2, Eq. (30) simplifies here to Eq. (21) with η_{i1} = β_{i23} and η_{i2} = −β_{i13}. Because of α_{11, 22} = 0, α_{12, 21} → 0 for U_{0} → 0 we neglect the latter for weak imposed shear flows and simplify Eq. (21) further to
We are interested in the diagonal components of η_{ij} which we represent in terms of the turbulent diffusivity by
In the case of homogeneous isotropic turbulence, the turbulent transport coefficients are uniform across the system and volume averages are appropriate. In the present case, however, the turbulence can neither be considered fully isotropic nor homogeneous due to the imposed z dependent shear flow, which makes the coefficients also anisotropic and z dependent. Both effects are weak though in the computed η_{t}; see Sect. 4.2 for the effect of anisotropy.
Exponential growth of the test solutions b^{T} at high Rm is a known issue in the testfield method (Sur et al. 2008). To circumvent it, we reset the b^{T} periodically to zero with a resetting time that is roughly inversely proportional to the magnetic Reynolds number.
The error of the turbulent magnetic Prandtl number is computed from
where δν_{t} and δη_{t} are the errors of turbulent viscosity and diffusivity, respectively.
4. Results
We perform several sets of simulations where we vary the forcing wavenumber k_{f}, determining the scale separation ratio, fluid and magnetic Reynolds numbers Re and Rm, respectively, and the wavenumber of the largescale flow k_{U}. Representative examples of the flow patterns realised in runs with small, medium, and high Reynolds numbers (from left to right) and forcing wavenumbers (from top to bottom, Sets D−G) are shown in Fig. 2. We also typically evolve the testfield equations in our runs so the results pertaining to ν_{t} and η_{t} are always obtained from the same simulation. All of our runs are listed in Tables 1–3.
Fig. 2. Normalised streamwise velocity component at the periphery of the computational domain for increasing scale separation ratio (from top to bottom), and increasing Reynolds number (left to right) for selected runs from D1 to G6. 
Summary of the runs with varying shear.
Summary of the runs with varying Reynolds numbers.
Summary of the runs with varying scale separation ratio 𝒮 = k_{f}/k_{U}.
In Fig. 3 we show representative results for a_{2} and ν_{t} obtained with the methods M1 and M2a from Sets A–C (see Table 1) with forcing wavenumbers 3, 5, and 10. The coefficient a_{2}, corresponding to the AKAeffect, is consistent with zero for all values of shear and with both methods that can detect it. This conclusion applies to all of our models and is consistent with the Galilean invariance of our forcing.
Fig. 3. AKA–effect coefficient a_{2} (top) and turbulent viscosity ν_{t} (bottom) as functions of Sh for three scale separation ratios, 𝒮 = 3 (Set A, left), 5 (Set B, middle), and 10 (Set C, right). The colours refer to methods M1 (blue and red), and M2a (black). M1_{1} and M1_{2} refer to the two zeros employed in Eqs. (23) and (24). 
We note that, as a_{2} from method M2a is always very small, it has a negligible effect on the quality of the fit and the value of ν_{t} in comparison to method M2b. For simplicity, we present results obtained using M2b in what follows. Overall, no statistically significant values were obtained for the coefficients a_{1}, a_{3}, n_{1}, and n_{3}.
4.1. Turbulent viscosity
4.1.1. Dependence on position and sensitivity to shear
The turbulent viscosities obtained for the two zeros employed in M1 (M1_{1} and M1_{2} in Fig. 3) agree within error estimates and agree also with those obtained from M2a and M2b. This suggests that ν_{t} has only a weak dependence on z or that its spatial profile is such that it is not captured by this method.
When the amplitude of the shear flow is varied, the values of ν_{t} start to increase rapidly at the largest values of Sh; see Fig. 3. This is because the Navier–Stokes equations are inherently nonlinear. Therefore, imposing a largescale flow has an impact on the turbulence. However, if the shear is sufficiently weak, such feedback is small and reliable results for ν_{t} can be obtained. To assess this question, we perform simulations at fixed kinematic viscosity and given forcing wavenumber k_{f} while varying the shear systematically. With the other quantities held unchanged, the fluid Reynolds number is a measure of the rms–velocity of the turbulence. In Fig. 4, we show the Reynolds numbers realised in the same sets as in Fig. 3. We find that Re increases mildly as a function of Sh for weak shear (Sh ≲ 0.1) and starts to increase sharply at higher values while the location of the transition depends weakly on the forcing wavenumber such that the larger the k_{f}, the smaller Sh is needed for the increase to occur.
Fig. 4. Reynolds number as a function of Sh for three scale separation ratios, 𝒮 = 3 (blue), 5 (red), and 10 (black), or Sets A, B, and C, respectively. 
The increase of Re is due to the fact that the turbulence becomes increasingly affected by the imposed shear and attains significant anisotropy. In some cases with the highest values of Sh, we also see largescale vorticity generation, which is likely related to what is known as vorticity dynamo (e.g. Käpylä et al. 2009b). Such hydrodynamic instability can be excited by the offdiagonal components of the turbulent viscosity tensor in anisotropic turbulence in the presence of shear (Elperin et al. 2003, 2007).
These tests suggest that values of Sh below 0.1 are needed for the influence of the shear on the turbulence to remain weak. However, the excitation condition of the vorticity dynamo manifestly depends on the scale separation ratio and likely also on the Reynolds number. In our runs, we choose a constant value of Sh_{c} for which Sh remains clearly below the excitation threshold (note that Sh_{c} = Sh Ma). Another factor supposedly contributing at large Reynolds numbers is shearproduced turbulence – possibly through some sort of finite amplitude instability. Given that the shear strengths (in terms of Sh) considered here are relatively small, this effect is likely to be weak in comparison to the turbulence production due to the applied forcing.
4.1.2. Dependence on Re
Results for the turbulent viscosity as a function of the fluid Reynolds number are shown in Fig. 5 for Sets D−G (see Table 2). Here the value of the shear parameter Sh_{c} is constant in each set. Additionally, the normalized relaxation time τu_{rms}k_{f} = 1 is kept fixed by adjusting τ, and is varied between 3 (Set D) and 30 (Set G). Furthermore, these runs use .
Fig. 5. Turbulent viscosity ν_{t}, normalised by ν_{t0}, as a function of the Reynolds number Re for sets of runs with different scale separation ratio 𝒮, but Sh_{c} = const. within each of the Sets D (blue), E (red), F (black), and G (purple). The dotted black line is proportional to Re. 
We find that for low Re and poor scale separation the signal is noisy and produces large errors in ν_{t} unless very long time series are produced. The runs with k_{f} ≈ 3 and Re ≈ 1 were in all sets typically run for several thousand turnover times whereas for larger Reynolds numbers and scale separations the integration times can be an order of magnitude shorter. The results in the low Reynolds number regime are in agreement with ν_{t} ∝ Re as expected from analytic studies using FOSA (Krause & Rüdiger 1974). The value of ν_{t} increases until Re ≈ 10 after which it saturates roughly to a constant between one and two times ν_{t0} depending on the scale separation ratio 𝒮. However, we still see a slow decrease for the highest values of Re which likely indicates that even the highest resolution simulations are not in the regime of fully developed turbulence. We note that the Mach number changes by a factor between roughly two (Set D) to four (Set F) between the extreme runs in each set. However, Ma saturates in the highRe runs so compressibility effects are unlikely to explain the slow declining trend of ν_{t}.
There is also a dependence on the scale separation ratio such that higher values of 𝒮 result in larger values of . In theory ν_{t} should converge towards the value at infinite scale separation. This is confirmed by Sets F and G where and 30, respectively.
4.1.3. Results from M3
As a representative example, Fig. 6 shows the decay of the k = k_{1} constituent of for ten independent initial conditions derived from Run F8. We compare the results for ν_{t} from methods M2b and M3 in Fig. 7 for Sets F and Fd. The runs of the latter were set up such that N = 10 snapshots from each of the runs in Set F with imposed shear were used as initial conditions. Thus each run in Set F works as a progenitor to ten decay experiments with the same system parameters in Set Fd. We find that the results from methods M2b and M3 coincide within the error estimates for low and intermediate Reynolds numbers (Re ≲ 20). However, there is a systematic tendency for the ν_{t} from the decay experiments to exceed the value from the Reynolds stress method for Re ≳ 30 by 10–20%.
Fig. 6. Amplitude of the k = k_{1} constituent of normalised by u_{rms} in Run F8d as a function of time from ten independent realisations of the decay. The solid red lines show exponential fits to the data. 
Fig. 7. Turbulent viscosity as a function of Re from method M2b, Set F (black) and from corresponding decay experiments (method M3), Set Fd (red). 
4.1.4. Dependence on scale separation ratio
The dependence of ν_{t} on the scale separation ratio 𝒮 for four different forcing scales is given in Table 3 (Sets H–K). We fit the data to a Lorentzian as a function of the inverse scale separation ratio 𝒮^{−1}
where ν_{t}(0) and σ are fit parameters^{6}. We also fit the data to a more general function where the exponent c of k_{U}/k_{f} is another fit parameter:
The inverse relative errors ν_{t}/δν_{t} are used as weights in the fit. The data and the fits are shown in Fig. 8. While the data is in reasonable agreement with a Lorentzian with ν_{t}(0) = 1.90 and σ = 1.80, the more general function with ν_{t}(0) = 2.00, σ = 1.61, and c = 1.44 yields a somewhat better fit. Data consistent with Lorentzian behaviour has been found earlier for η_{t} in low Reynolds number turbulence (Brandenburg et al. 2008a); see Table for an overview of the σ values found previously in various cases ranging from magnetic diffusion in isotropic turbulence to passive scalar diffusion in shear flows, in which σ was typically below unity. However, a value of c close to 4/3, as found here, is indeed expected for fully developed Kolmogorov turbulence; see the discussion in the conclusions of Madarassy & Brandenburg (2010).
Fig. 8. Turbulent viscosity as a function of the inverse scale separation ratio k_{U}/k_{f} for the four normalised forcing wavenumbers (blue), 5 (red), 10 (black), and 30 (purple), corresponding to Sets H–K, respectively. Fits according to Eqs. (35) and (36) are shown by dotted and dashed lines, respectively. 
Note that σ was found to be larger under the secondorder correlation approximation (SOCA), but the reason for this departure is unclear. Also, looking at Table 4, there is no obvious connection between the values of σ in different physical circumstances. More examples are needed to assess the robustness of the results obtained so far.
Examples of σ values found previously in other cases.
Knowing the value of σ is important for more accurate meanfield modelling. In physical space, a prescription like Eq. (35) corresponds to a convolution, which makes the Reynolds stress at a given position dependent on the mean velocity within a certain neighbourhood. In that way, nonlocality is modelled. This is generally ignored in the common use of turbulent viscosity, although some attempts have been made to include such affects to leading order (Rüdiger 1982). Ignoring nonlocality corresponds to the limit σ → 0 or k_{U}/k_{f} → 0, which is often a questionable assumption; see Brandenburg & Chatterjee (2018) for a discussion in the context of spherical meanfield dynamos.
4.2. Turbulent diffusivity η_{t}
The turbulent magnetic diffusivity η_{t} from Sets D–G is shown in Fig. 9. We find a similar qualitative behaviour as for ν_{t} so that for small magnetic Reynolds numbers the value of η_{t} is proportional to Rm and the results converge when the scale separation ratio is increased. As in the case of the turbulent viscosity, we find a weak declining trend as a function of Rm at its highest values which was neither observed by Sur et al. (2008) in similar simulations without shear nor by Brandenburg et al. (2008b) and Mitra et al. (2009) in runs where the largescale flow was imposed via the shearingbox approximation. However, the error estimates in the aforementioned studies are clearly greater than in the present one and thus a weak decreasing trend as a function of Rm cannot be ruled out. Furthermore, the shear flows in the present simulations are significantly weaker than in the cases of Brandenburg et al. (2008b) and Mitra et al. (2009), such that their influence on the turbulent transport coefficients is also weaker.
Fig. 9. Turbulent diffusivity η_{t}, normalised by η_{t0}, as a function of the magnetic Reynolds number Rm for runs with Sh_{c} = const. within each of the Sets D (blue), E (red), F (black), and G (purple). The dotted black line is proportional to Rm and 𝒮_{B} = k_{f}/k_{B}. 
We assess the effect of the shear flow on the results by performing an additional set of simulations in which it is omitted, but otherwise the same parameters as in Set F are employed. We show the results for the difference of η_{t} in these sets in Fig. 10. The difference is typically of the order of a few per cent such that in most cases the value from the case with shear is greater. This is of the same order of magnitude as the error estimates for η_{t}. Thus we conclude that the systematic error due to the largescale anisotropy induced by the shear flow is insignificant in the determination of the turbulent diffusivity.
Fig. 10. Relative difference of turbulent magnetic diffusivity from Set F and a corresponding set without shear, here denoted by superscript zero. The vertical bars indicate the error estimates of from Set F. 
4.3. Turbulent magnetic Prandtl number
Our results for Pm_{t} as a function of Reynolds number and scale separation ratio 𝒮 are shown in Fig. 11. We find that Pm_{t} for Re ≳ 20 is roughly a constant for each value of while increasing from roughly 0.8 for to 0.95 for . Especially at low Re, the convergence with respect to the scale separation is not as clear as for ν_{t} and η_{t} individually. With respect to low Reynolds numbers, we see an increasing trend starting from values between 0.55 and 0.65 at Re ≈ 5 until Re ≈ 20. At even lower Re the uncertainty in the determination of ν_{t} becomes larger and the values of Pm_{t} have substantial error margins.
Fig. 11. Turbulent magnetic Prandtl number Pm_{t} as a function of the Reynolds number Re for the same sets of runs as in Fig. 6. Pm = 1 is used in all runs. The dotted horizontal lines indicate the extrema of analytical results from different methods; see Table 5. 
The turbulent magnetic Prandtl number has been computed with various analytical techniques, see Table 5. Considering the limit of ν → 0 or Re → ∞, different flavours of FOSA yield either Pm_{t} = 0.8 (Kitchatinov et al. 1994) or 0.4 (Yousef et al. 2003), whereas MTA results for fully developed turbulent convection favours lower values (Rogachevskii & Kleeorin 2006). A similar spread of values from Pm_{t} ≈ 0.42 (Verma 2001) to ≈0.7−0.8 (Fournier et al. 1982; Kleeorin & Rogachevskii 1994; Jurčišinová et al. 2011) has been reported using renormalisation group methods for the case of three spatial dimensions and weak magnetic fields.
Comparison of values of Pm_{t} from analytic and numerical studies.
Particularly at high scale separation, our results are not compatible with any of the analytic results but indicate a higher value than all of the theories. This can be due to the fact that the turbulence in the simulations is not in the fully developed regime and because the scale separation achieved is still insufficient. Furthermore, analytic theories must resort to approximations that cannot be justified in highReynolds number turbulence.
4.4. Dependence on forcing time scale
The bulk of the simulations considered here use a δ correlated random (white) forcing, see Eq. (3), such that a new wavevector is chosen at every time step. It cannot, however, be ruled out a priori that the results depend on the correlation time of the forcing. Here we test the sensitivity of the results with respect to this correlation time by comparing the default case of white–in–time forcing with cases where the forcing wavevector is held constant for a time δt_{f}. We take Run C4 as our fiducial model where δt_{f}k_{f}u_{rms} ≈ 0.02 and δt_{f} is equal to the time step of the simulation. We increased δt_{f}k_{f}u_{rms} in steps by a factor of 50 altogether and computed turbulent viscosity, magnetic diffusivity, and magnetic Prandtl number; see Fig. 12. We find that ν_{t}, η_{t}, and Pm_{t} are essentially constant in this range of parameters. Our method of switching the forcing with a period δt_{f} is crude because it induces discontinuities, but the approximate constancy of the coefficients suggests that robust results for the turbulent transport coefficients are obtained nevertheless.
Fig. 12. Coefficients ν_{t} (black) and η_{t} (red), normalised by ν_{t0} and η_{t0}, respectively; and Pm_{t} (blue) as functions of δt_{f}k_{f}u_{rms} using Run C4 as the fiducial model. 
5. Conclusions
We have computed the turbulent viscosity (ν_{t}) and magnetic diffusivity (η_{t}) from simulations of forced turbulence using imposed shear flows and the testfield method, respectively. As expected, ν_{t} and η_{t} are found to be proportional to the respective Reynolds number at low Re and Rm. With increasing values of Re and Rm, the turbulent transport coefficients saturate at around Re ≈ Rm ≈ 10, but show a weakly decreasing trend beyond. The value of the turbulent viscosity estimated from the Reynolds stress, which is interpreted to reflect the response of the system to a largescale flow, and from the decay of a mean flow in the presence of turbulence are in fair agreement. However, the latter yields systematically slightly higher values for high Reynolds numbers by less than 10%.
The turbulent magnetic Prandtl number Pm_{t} saturates between 0.8 and 0.95 for Re ≳ 10 depending on the scale separation ratio. We note that these values are somewhat higher than those from the renormalisation group approach and, especially, the firstorder smoothing approach. The value of Pm_{t} computed here corresponds to the kinematic case where the magnetic field is weak, which is often not the case in astrophysical systems. Analytic studies predict quenching of turbulent viscosity and magnetic diffusivity when the magnetic fields are dynamically significant (e.g. Kitchatinov et al. 1994). The quenching of η_{t} has also been computed from numerical simulations (e.g. Käpylä & Brandenburg 2009; Brandenburg et al. 2008c; Karak et al. 2014). Similar studies for turbulent viscosity are so far lacking. Such results will be reported elsewhere.
One of the other remaining issues to be addressed in the future is the role of compressibility effects, in particular that of fluctuations of ρ. In addition to making analytic progress by identifying potentially new effects owing to their presence, it would be useful to extend our simulations to the regime of larger Mach numbers. Another possible extension of our work is to study the potential of a smallscale dynamo to give rise of magnetic stresses that could enhance the turbulent viscosity, as has been suggested in the solar context to alleviate the discrepancies between observations and simulations of differential rotation and convective velocities (Karak et al. 2018).
Neglecting density fluctuations may not be rigorously justified, given that the variety of potentially new effects owing to compressibility has not yet been fully explored, but see the recent studies by Rogachevskii et al. (2018) and Yokoi (2018) for the electromotive force. However, recent hydrodynamic results for the Λ effect suggest that the effect of compressibility is weak up to Ma ≈ 0.8 (Käpylä 2019b).
Further turbulence effects result from the term , but are not considered here either because of our assumption of weak compressibility. We recall that S is the traceless rate of strain tensor used in Eq. (2).
Note that relation (16) is yielding the stresses without truncation when interpreted to be a representation of the Fouriertransformed kernel of a general convolutionlike relationship between Q_{ij} and (cf. Brandenburg et al. 2008a).
Acknowledgments
We thank the anonymous referee for critical and constructive comments on the manuscript. The simulations were performed using the supercomputers hosted by CSC – IT Center for Science Ltd. in Espoo, Finland, who are administered by the Finnish Ministry of Education. This work was supported by the Deutsche Forschungsgemeinschaft Heisenberg programme (grant No. KA 4825/21; PJK), the Academy of Finland ReSoLVE Centre of Excellence (Grant No. 307411; PJK, MR, MJK), the NSF Astronomy and Astrophysics Grants Program (Grant 1615100), and the University of Colorado through its support of the George Ellery Hale visiting faculty appointment. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (project “UniSDyn”, Grant agreement No. 818665).
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Appendix A: Justification for Eq. (15)
In the isothermal case, it is convenient to consider the quantity H = ln ρ instead of the density itself resulting in equations linear in and free of triple correlations of fluctuating quantities. We denote the turbulence present under the condition by u^{(0)}. If u^{(0)} describes isotropic turbulence, only the diagonal components Q_{ii} exist. Proceeding to the situation , indicated by the superscript the fluctuations u, h can be written as , . Correspondingly, the mean force is
with , and
Restricting to first order in the mean quantities, we get
Analogously, for , we have
with
in first order. If now both and do not vanish, one can see from the equations for the fluctuating quantities, again restricted to first order in the mean quantities,
that due to their linearity in , , and the additivity of their inhomogeneities depending on and , respectively,
holds and hence
to first order in and . Given that the major part of ℱ is ∇ ⋅ Q, an equivalent relationship can be assumed for Q.
All Tables
All Figures
Fig. 1. Horizontally averaged velocity (top), (middle), and its temporal average in comparison with (bottom) from Run E9 (see Table 2) with , , Sh ≈ 0.04, and Re ≈ 497. ν_{t} from method M2b. 

In the text 
Fig. 2. Normalised streamwise velocity component at the periphery of the computational domain for increasing scale separation ratio (from top to bottom), and increasing Reynolds number (left to right) for selected runs from D1 to G6. 

In the text 
Fig. 3. AKA–effect coefficient a_{2} (top) and turbulent viscosity ν_{t} (bottom) as functions of Sh for three scale separation ratios, 𝒮 = 3 (Set A, left), 5 (Set B, middle), and 10 (Set C, right). The colours refer to methods M1 (blue and red), and M2a (black). M1_{1} and M1_{2} refer to the two zeros employed in Eqs. (23) and (24). 

In the text 
Fig. 4. Reynolds number as a function of Sh for three scale separation ratios, 𝒮 = 3 (blue), 5 (red), and 10 (black), or Sets A, B, and C, respectively. 

In the text 
Fig. 5. Turbulent viscosity ν_{t}, normalised by ν_{t0}, as a function of the Reynolds number Re for sets of runs with different scale separation ratio 𝒮, but Sh_{c} = const. within each of the Sets D (blue), E (red), F (black), and G (purple). The dotted black line is proportional to Re. 

In the text 
Fig. 6. Amplitude of the k = k_{1} constituent of normalised by u_{rms} in Run F8d as a function of time from ten independent realisations of the decay. The solid red lines show exponential fits to the data. 

In the text 
Fig. 7. Turbulent viscosity as a function of Re from method M2b, Set F (black) and from corresponding decay experiments (method M3), Set Fd (red). 

In the text 
Fig. 8. Turbulent viscosity as a function of the inverse scale separation ratio k_{U}/k_{f} for the four normalised forcing wavenumbers (blue), 5 (red), 10 (black), and 30 (purple), corresponding to Sets H–K, respectively. Fits according to Eqs. (35) and (36) are shown by dotted and dashed lines, respectively. 

In the text 
Fig. 9. Turbulent diffusivity η_{t}, normalised by η_{t0}, as a function of the magnetic Reynolds number Rm for runs with Sh_{c} = const. within each of the Sets D (blue), E (red), F (black), and G (purple). The dotted black line is proportional to Rm and 𝒮_{B} = k_{f}/k_{B}. 

In the text 
Fig. 10. Relative difference of turbulent magnetic diffusivity from Set F and a corresponding set without shear, here denoted by superscript zero. The vertical bars indicate the error estimates of from Set F. 

In the text 
Fig. 11. Turbulent magnetic Prandtl number Pm_{t} as a function of the Reynolds number Re for the same sets of runs as in Fig. 6. Pm = 1 is used in all runs. The dotted horizontal lines indicate the extrema of analytical results from different methods; see Table 5. 

In the text 
Fig. 12. Coefficients ν_{t} (black) and η_{t} (red), normalised by ν_{t0} and η_{t0}, respectively; and Pm_{t} (blue) as functions of δt_{f}k_{f}u_{rms} using Run C4 as the fiducial model. 

In the text 
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