arXiv:1002.2428v1 [astro-ph.GA] 11 Feb 2010 Draft version March 12, 2019 Preprint typeset using L A T E X style emulateapj v. 11/10/09 NONLOCALITY OF BALANCED AND IMBALANCED TURBULENCE. A. Beresnyak, A. Lazarian Dept. of Astronomy, Univ. of Wisconsin, Madison, WI 53706 Draft version March 12, 2019 ABSTRACT The search of ways to generalize the theory of strong MHD turbulence for the case of non-zero cross- helicity (or energy imbalance) has attracted considerable interest recently. In Beresnyak & Lazarian (2009a, BL09a) we performed numerical simulations and showed that some of existing models which require the locality of the energy transfer, including the model in Perez & Boldyrev (2009, PB09) predict the energy cascading rates which are inconsistent with numerical simulations. In a new paper Perez & Boldyrev (2010, PB10) argue that our simulations are performed for high degree of cross- helicity for which no self-similar cascade can be established and stress that the spectral slopes of their simulations are consistent with the predictions of PB09 model. In this short paper we show that PB09 predictions are inconsistent with numerics even for low degree of cross-helicity at which PB10 claims that the numerics should be accurate. Subject headings: MHD – turbulence – ISM: kinematics and dynamics 1. INTRODUCTION MHD turbulence has attracted attention of as- tronomers since long time ago. As most astrophysical media are ionized, they are coupled to the magnetic fields (see, e.g., Biskamp 2003). A simple one-fluid description known as magnetohydrodynamics or MHD is broadly ap- plicable to most astrophysical environments on macro- scopic scales. On the other hand, turbulence has been observed in various environments and with a huge range of scales (see, e.g. Armstrong et al. 1995). As with hydrodynamics which has a “standard” phe- nomenological model of energy cascade (Kolmogorov, 1941), MHD turbulence has one too. This is the Goldreich-Sridhar model (Goldreich & Sridhar 1995, henceforth GS95) that uses a concept of critical balance, which maintains that turbulence will stay marginally strong down the cascade. The spectrum of GS95 is sup- posed to follow a −5/3 Kolmogorov scaling. However, a shallower slopes has been reported by numerics, which motivated to modify GS95 (see, e.g., Boldyrev 2005, 2006, Gogoberidze 2007). The other problem of GS95 is that it is incomplete, as it does not treat the most general imbalanced, or cross- helical case. As turbulence is a stochastic phenomenon, an average zero cross helicity does not preclude a fluc- tuations of this quantity in the turbulent volume. Also, most of astrophysical turbulence is naturally imbalanced, due to the fact that it is generated by a strong localized source of perturbations, such as the Sun in case of solar wind or central engine in case of AGN jets. Several models of imbalanced turbulence appeared re- cently (e.g., Lithwick et al. (2007), henceforth LGS07, Beresnyak & Lazarian (2008), Chandran (2008), PB09, Podesta & Bhattacharjee (2009)). The full self- consistent analytical model for strong turbulence, how- ever, does not yet exist. In this situation observations and direct numerical simulations (DNS) of MHD turbu- lence will provide necessary feedback to theorists. We concentrated on two issues, namely that a) the energy andrey, lazarian@astro.wisc.edu power-law slopes of MHD turbulence can not be mea- sured directly from available numerical simulations (this issue is also covered in BL09b), b) the ratio of energy dis- sipation rates is a very robust quantity that can be used to differentiate among many imbalanced models (this is- sue is also covered in BL09a). We present new result from numerical simulations of imbalanced turbulence that complement and reinforce earlier claims made in BL09b and BL09a. In what follows in §2 we briefly describe numerical methods, §3 we discuss dissipation rates as the most ro- bust measures in numerical turbulence, in §4 we show that it is impossible to measure of spectral slopes of turbulence directly from currently available DNS, in §5 we clear some misunderstanding and confusion regard- ing our our earlier results and claims in PB10, in §6 we present our conclusions. 2. NUMERICAL SETUP We solved incompressible MHD or Navier-Stokes equa- tions: ∂ t w ± + ˆ S(w ∓ ·∇)w ± = −ν n (−∇ 2 ) n/2 w ± + f ± , (1) where ˆ S is a solenoidal projection and w ± (Elsasser variables) are w + = v + b and w − = v − b where we use velocity v and magnetic field in velocity units b = B/(4πρ) 1/2 . Navier-Stokes equation is a special case of equations (1), where b ≡ 0 and, therefore, both equa- tions are equivalent with w + ≡ w − . The RHS of this equation includes a linear dissipation term which is called viscosity or diffusivity for n = 2 and hyper-viscosity or hyper-diffusivity for n> 2 and the driving force f ± . The special case of f + = f − is a velocity driving. We solved these equations with a pseudospectral code that was de- scribed in great detail in our earlier publications BL09a, BL09b. 3. DISSIPATION RATE Due to relative nonlocality of MHD turbulence, instead of measuring spectral slopes which are fairly unreliable,