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tokens: the density and dynamic range of the frequency sampling may be computationally attainable via offloading this part of the workload to multicore architectures within the MHD code in the future, even though the attainable speedup may in fact be limited by restrictions in the memory bandwidth. Even without these technological advancements, we have demonstrated that the spectral TF method of Brandenburg et al. ( 2008 ) and its time-dependent counterpart introduced by Hubbard Brandenburg ( 2009 ) are highly useful tools and indeed produce interesting results for advanced applications, such as our investigation of realistic high-Mach-number turbulence driven by SNe in the multiphase ISM, where mean-field effects occur naturally as an outcome of the combined action of gravity and the Coriolis force. Notwithstanding the challenges of understanding the quenching in the strong-field limit (Gressel, Bendre Elstner 2013a ) and covering the parameter space required by genuinely global models (Gressel et al. 2013b ), combining these approaches (as pioneered by Rheinhardt Brandenburg 2012 ) will allow to derive the most comprehensive mean-field closures achievable. APPENDIX A: BENCHMARK USING HELICAL FORCING IN THE KINEMATIC LIMIT We verify our implementation of the spectral TF method for the simple test case of helically forced (mildly compressible) hydrodynamic turbulence, that is, in the strict kinematic limit of the mean-field dynamo, where the back reaction via the Lorentz force is ignored. To this end, we perform a hydrodynamic DNS in a cubic domain, solving along with the TF induction equation ( 11 ), and using as the equation of state. We closely follow Brandenburg ( 2001 ) and adopt a helical forcing function , with the normalization with respect to the numerical time-step, δ t , and f 0 ≲ 1 a dimensionless forcing amplitude. The forcing obtains its helical character via with an isotropically sampled arbitrary unit vector required for constructing a vector . Note that , and – being an eigenfunction of the curl operator – we have , such that the helicity of the forcing, , is positive definite. The wave vector is chosen from a list of 66 (158) pre-computed discrete wave vectors obeying the periodicity of the box, and that have , with k f = 3 ( k f = 5), and Δ k = 1/4. Note that both and the random phase φ( t ) are updated after each time-step, δ t , of the DNS. A1 Non-locality in space In Fig. A1 , we show the spectral dependence of and , adopting units of and . Both curves are very well approximated by a Lorentzian with a characteristic length scale . This is in contrast to the results of Brandenburg et al. ( 2008 ), who find k c = k f for , and for , respectively, for a very similar test case with k f = 5 and Rm = 10. While the qualitative trend is the same, the difference is much less pronounced in our case. Figure A1. Spectral amplitudes for the α effect (top panel) and turbulent diffusion, η T (lower panel) for helically forced isothermal