Large- and small-scale interactions and quenching in an  2 -dynamo Peter Frick and Rodion Stepanov Institute of Continuous Media Mechanics, 1, Korolev, Perm, 614013, Russia Dmitry Sokoloff Department of Physics, Moscow State University, 119899, Moscow, Russia Received 26 February 2006; revised manuscript received 5 October 2006; published 22 December 2006 The evolution of the large-scale magnetic field in a turbulent flow of conducting fluid is considered in the framework of a multiscale  2 -dynamo model, which includes the poloidal and the toroidal components for the large-scale magnetic field and a shell model for the small-scale magnetohydrodynamical turbulence. The conjugation of the mean-field description for the large-scale field and the shell formalism for the small-scale turbulence is based on strict conformity to the conservation laws. The model displays a substantial magnetic contribution to the  effect. It was shown that a large-scale magnetic field can be generated by current helicity even solely. The  quenching and the role of the magnetic Prandtl number  P m  are studied. We have deter- mined the dynamic nature of the saturation mechanism of dynamo action. Any simultaneous cross correlation of  and large-scale magnetic field energy E B is negligible, whereas coupling between  and E B becomes substantial for moderate time lags. An unexpected result is the behavior of the large-scale magnetic energy with variation of the magnetic Prandtl number. Diminishing of P m does not have an inevitable ill effect on the magnetic field generation. The most efficient large-scale dynamo operates under relatively low Prandtl numbers—then the small-scale dynamo is suppressed and the decrease of P m can lead even to superequipar- tition of the large-scale magnetic field i.e., E B  E u . In contrast, the growth of P m does not promote the large-scale magnetic field generation. A growing counteraction of the magnetic  effect reduces the level of mean large-scale magnetic energy at the saturated state. DOI: 10.1103/PhysRevE.74.066310 PACS numbers: 47.27.E-, 52.30.Cv, 95.30.Qd, 91.25.Cw I. INTRODUCTION Magnetic fields of celestial bodies are supposed to be gen- erated by a hydromagnetic dynamo operating in a moving electrically conductive medium see, e.g. 1. The corre- sponding flows are characterized by huge values of govern- ing parameters, so a direct numerical simulation DNS of the corresponding magnetohydrodynamical MHD problem with realistic governing parameters becomes impossible even with modern numerical facilities. Due to rapid progress of numerical facilities the parameter range accessible for DNS is becoming increasingly wide. Some high-resolution simulations of MHD turbulence up to 1024 3 grid points have been performed 2,3, which allows one to study the small-scale dynamo processes in detail but does not elimi- nate the difference of the accessible range from that of the cosmic media e.g., the magnetic Reynolds number for the interstellar medium is R m =10 6 even if the estimation of mag- netic field losses is based on ambipolar diffusion. Moreover, even if a direct numerical simulation is performed, identifi- cation of the large-scale features comparable with observa- tional data is a nontrivial task 4. The theory of astrophysical dynamos was mainly devel- oped in the framework of the mean-field approach 5. The mean-field theory is based on the so-called two-scale ap- proximation, which suggests that the magnetic field consists of the large-scale field B and the small-scale fluctuations b, and that the velocity field is represented as the sum of the mean-field motion V and velocity fluctuations u. Bearing this in mind some governing equations for large-scale com- ponents are evaluated. The transport coefficients in the equa- tions contain some averaged information concerning the small-scale components parametrized in terms of B and V. This approach yields various models for large-scale magnetic fields of particular celestial bodies, such as galaxies, stars, and planets, which reproduce to some extent the available phenomenology 6. Various approaches to proper parametri- zation were suggested see for review 7,8. However, the mean-field approach seems to be insufficient because it does not provide an in-depth insight into the evolution of the small-scale magnetic and velocity fields, which are replaced by turbulence parametrization. It is desirable that the simplicity of the mean-field ap- proach be combined with a relatively simple model of small- scale MHD turbulence, which nevertheless would provide a proper description of interactions of the large-scale magnetic field with the MHD turbulence. This would allow us to avoid the time and resource consuming calculations involved in DNS as well as the insecure parametrizations involved in the mean-field theories. The shell models were suggested to describe the spectral energy transfer 9,10. After numerous refinements they be- came an effective tool for description of the spectral proper- ties of the small-scale turbulence see for review 11. These models do not describe the dynamics of turbulent fluctua- tions b and u in all details, but replace the full system of governing partial differential equations by some simple ordi- nary differential equations describing the spectral transfer of energy and other relevant quantities. The shell models for MHD turbulence were introduced in 12–14. This approach reveals many intrinsic features of the small-scale dynamo action in the fully developed turbulence of conducting fluids 15. It also allows study of a specific case of a rotating PHYSICAL REVIEW E 74, 066310 2006 1539-3755/2006/746/06631012 ©2006 The American Physical Society 066310-1