arXiv:1005.5700v1 [astro-ph.SR] 31 May 2010 Draft version June 1, 2010 Preprint typeset using L A T E X style emulateapj v. 6/22/04 EFFECT OF STRATIFIED TURBULENCE ON MAGNETIC FLUX CONCENTRATIONS Axel Brandenburg,Koen Kemel NORDITA, AlbaNova University Center, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden; Department of Astronomy, AlbaNova University Center, Stockholm University, SE-10691 Stockholm, Sweden and Nathan Kleeorin,Igor Rogachevskii Department of Mechanical Engineering, Ben-Gurion University of the Negev, POB 653, Beer-Sheva 84105, Israel (Dated: Revision: 1.72 ) Draft version June 1, 2010 ABSTRACT According to conventional view, sunspots form when coherent magnetic flux tubes intersect the surface. Such tubes would rise from the bottom of the convection zone as a result of an instability. In this study an alternative view is advanced where the field is produced throughout the entire convection zone and sunspots may form from local flux concentrations near the surface. In order to understand the basic mechanism of the formation of magnetic flux concentrations, we determine by direct numerical simulations the turbulence contributions to the mean magnetic pressure in a strongly stratified isothermal layer, where a weak uniform horizontal mean magnetic field is applied. In a first setup, the turbulent intensity is nearly constant in height, so the kinetic energy density decreases with height due to the decrease in density, while in a second series of nu- merical experiments, the turbulent intensity increases with height such that the kinetic energy density is nearly independent of height. Turbulent magnetic diffusivity and turbulent pumping velocity are determined with the test-field method for both cases. Corresponding mean-field numerical models are used to assess whether or not a large-scale instability is to be expected. A negative turbulence contribution to the effective mean magnetic pressure is confirmed and found to be in agreement with results of earlier work. The vertical profile of the turbulent magnetic diffusivity is found to agree with what is expected based on simple mixing length expres- sions, but the turbulent pumping velocity is found to be equal to the negative gradient of turbulent magnetic diffusivity without the 1/2 factor expected from the kinematic mean-field theory. Mean-field numerical mod- elling confirms the excitation of the instability for both setups, although no large-scale instability is found in the direct numerical simulations. Subject headings: MHD – Sun: magnetic fields – sunspots – Turbulence 1. INTRODUCTION In a stratified layer, magnetic fields do not normally stay in equilibrium but tend to become buoyantly unstable (e.g., Parker 1966, 1979a; Gilman 1970a, 1970b; Hughes & Proc- tor 1988). This mechanism is generally invoked in order to understand magnetic flux emergence at the solar surface (e.g. Hood et al. 2009). The mechanism does not explicitly rely upon the existence of turbulence, except that the origin of the Sun’s magnetic field is generally believed to be related to a turbulent dynamo operating in the convection zone, or possi- bly beneath it; see Solanki et al. (2006) for a recent review. Turbulent dynamos work in a variety of circumstances and are able to produce large-scale magnetic fields (see Bran- denburg & Subramanian 2005 for a review). At first glance this generation process is counter-intuitive, because it works against the well-known concept of turbulent mixing (Taylor 1921; Prandtl 1925). However, it is now well established that turbulence can also have non-diffusive effects. In addition to the well-known α effect that is generally believed to be re- sponsible for the Sun’s large-scale field (Moffatt 1978; Parker 1979b; Krause & R¨ adler 1980), there is also the Λ effect that is responsible for driving the differential rotation of the Sun (R¨ udiger 1980, 1989; R¨ udiger & Hollerbach 2004). When invoking the concept of magnetic buoyancy, one must ask what the effect of turbulence is in this context. The turbulent pressure associated with the convective fluid motions is certainly not negligible and reacts sensitively to changes in the background magnetic field. The main reason for this is that the kinetic energy density in isotropic turbu- lence contributes to the total turbulent dynamic pressure twice as much as turbulent magnetic energy density, i.e. P turb = 1 3 ρu 2 + 1 6 b 2 /μ 0 . (1) Here, P turb is the total turbulent dynamic pressure caused by velocity and magnetic fluctuations, u and b, respectively, μ 0 is the vacuum permeability, ρ is the fluid density, and the overbar indicates ensemble averaging. On the other hand, any rise in local turbulent magnetic energy density must be accompanied by an equal and opposite change of turbulent kinetic energy density in order to obey approximate energy conservation, i.e. 1 2 ρu 2 + 1 2 b 2 /μ 0 ≡ E tot ≈ const. (2) This relation is known to hold quite well even in open sys- tems with boundaries, as was demonstrated by direct numeri- cal simulations (Brandenburg et al. 2010, hereafter referred to as BKR). This clearly implies that, upon generation of mag- netic fluctuations, the total turbulent dynamic pressure shows a reversed feedback, i.e. P turb = − 1 6 b 2 /μ 0 + const, (3) where the constant is 2E tot /3 (Kleeorin et al. 1990). For a strongly anisotropic turbulence Eq. (3) is also valid except for the change of the 1/6 factor by 1/2 (Rogachevskii & Klee- orin 2007). This phenomenology was confirmed by analyti- cal studies using the spectral τ approximation and the renor- malization approach and led Kleeorin & Rogachevskii (1994)