Angular momentum transport in convectively unstable shear flows

arXiv:1003.0900v2 [astro-ph.SR] 6 Jul 2011 Draft version November 17, 2021 Preprint typeset using LATEX style emulateapj v. 6/22/04 ANGULAR MOMENTUM TRANSPORT IN CONVECTIVELY UNSTABLE SHEAR FLOWS Petri J. K¨apyl¨a1,2, Axel Brandenburg2,3, Maarit J. Korpi1, Jan E. Snellman1, and Ramesh Narayan4 Draft version November 17, 2021 ABSTRACT Angular momentum transport owing to hydrodynamic turbulent convection is studied using local three di- mensional numerical simulations employing the shearing box approximation. We determine the turbulent vis- cosity from non-rotating runs over a range of values of the shear parameter and use a simple analytical model in order to extract the non-diffusive contribution (Λ-effect) to the stress in runs where rotation is included. Our results suggest that the turbulent viscosity is of the order of the mixing length estimate and weakly affected by rotation. The Λ-effect is non-zero and a factor of 2–4 smaller than the turbulent viscosity in the slow rotation regime. We demonstrate that for Keplerian shear, the angular momentum transport can change sign and be outward when the rotation period is greater than the turnover time, i.e. when the Coriolis number is below unity. This result seems to be relatively independent of the value of the Rayleigh number. Subject headings: accretion, accretion disks – convection – stars: rotation – Sun: rotation – turbulence 1. INTRODUCTION Turbulence due to the convective instability is thought to account for much of the angular momentum transport in the outer layers of the Sun and other stars with convection zones (e.g. R¨udiger 1989; R¨udiger & Hollerbach 2004). In the pres- ence of turbulence the fluid mixes efficiently and diffusion processes occur much faster than in its absence. This effect is usually parameterized by a turbulent viscosity νt that is much larger than the molecular viscosity ν. Often the value of νt is estimated using simple mixing length arguments with νt = urmsl/3, where urms is the rms velocity of the turbulence and l = αMLTH where αMLT is a parameter of the order unity and H is the vertical pressure scale height. Numerical results from simpler fully periodic isotropically forced systems sug- gest that the mixing length estimate gives the correct order of magnitude of turbulent viscosity (e.g. Yousef et al. 2003; K¨apyl¨a et al. 2009a; Snellman et al. 2009). However, it is important to compute νt from convection simulations in order to see whether the results of the simpler systems carry over to convection. Furthermore, it is of interest to study whether the small-scale turbulent transport can be understood in the light of simple analytical closure models that can be used in subgrid-scale modeling. Measuring νt and its relation to aver- aged quantities, such as correlations of turbulent velocities, is one of the main purposes of our study. In addition to enhanced viscosity, turbulence can also lead to non-diffusive transport. The α-effect (e.g. Krause & R¨adler 1980), responsible for the generation of large-scale magnetic fields by helical turbulence, is one of the most well-known non-diffusive effects of turbulence. An analogous effect ex- ists in the hydrodynamical regime and is known as the Λ- effect (Krause & R¨udiger 1974). The Λ-effect is proportional to the local angular velocity and occurs if the turbulence is anisotropic in the plane perpendicular to the rotation vector 1 Department of Physics, Division of Geophysics and Astronomy, FI- 00014 University of Helsinki, Finland 2 NORDITA, AlbaNova University Center, Roslagstullsbacken 23, SE- 10691 Stockholm, Sweden 3 Department of Astronomy, Stockholm University, SE-10691 Stockholm, Sweden 4 Harvard-Smithsonian Center for Astrophysics 60 Garden Street, MS-51 Cambridge, MA 02138, USA Revision: 1.143 (November 17, 2021) (R¨udiger 1989). The existence of the Λ-effect has been estab- lished numerically from convection simulations (e.g. Pulkki- nen et al. 1993; Chan 2001; K¨apyl¨a et al. 2004; R¨udiger et al. 2005) and simpler homogeneous systems (K¨apyl¨a & Bran- denburg 2008). If, however, both shear and rotation are present it is difficult to disentangle the diffusive and non-diffusive contributions. This is particularly important in the case of accretion disks where the sign of the stress determines whether angular mo- mentum is transported inward or outward. Convection is com- monly not considered as a viable angular momentum transport mechanism in accretion disks since several studies have indi- cated that the transport owing to convection occurs inward (e.g. Cabot & Pollack 1992; Ryu & Goodman 1992; Stone & Balbus 1996; Cabot 1996; R¨udiger et al. 2002). Furthermore, in an influential paper, Stone & Balbus (1996, hereafter SB96) presented numerical simulations of hydrodynamic convection where the transport was indeed found to be small and directed inward on average. This result was used to provide additional evidence for the importance of the magneto-rotational insta- bility (Balbus & Hawl

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