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Turbulent Magnetic Prandtl Number and Magnetic Diffusivity Quenching from Simulations

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Turbulent Magnetic Prandtl Number and Magnetic Diffusivity Quenching from Simulations A&A 411, 321–327 (2003) Astronomy DOI: 10.1051/0004-6361:20031371 & c ESO 2003 Astrophysics Turbulent magnetic Prandtl number and magnetic diffusivity quenching from simulations T. A. Yousef1, A. Brandenburg2,andG.R¨udiger3 1 Department of Energy and Process Engineering, Norwegian University of Science and Technology, Kolbjørn Hejes vei 2B, 7491 Trondheim, Norway 2 NORDITA, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark 3 Astrophysical Institute Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany Received 21 February 2003 / Accepted 25 August 2003 Abstract. Forced turbulence simulations are used to determine the turbulent kinematic viscosity, νt, from the decay rate of a large scale velocity field. Likewise, the turbulent magnetic diffusivity, ηt, is determined from the decay of a large scale magnetic field. In the kinematic regime, when the field is weak, the turbulent magnetic Prandtl number, νt/ηt, is about unity. When the field is nonhelical, ηt is quenched when magnetic and kinetic energies become comparable. For helical fields the quenching is stronger and can be described by a dynamical quenching formula. Key words. magnetohydrodynamics (MHD) – turbulence 1. Introduction however, this instability is suppressed (R¨udiger & Shalybkov 2002). On the other hand, the Reynolds number of the flow is The concept of turbulent diffusion is often invoked when mod- quite large (105 :::106) and the flow therefore most certainly eling large scale flows and magnetic fields in a turbulent turbulent. This led Noguchi et al. (2002) to invoke a turbulent medium. Turbulent magnetic diffusion is similar to turbulent kinematic viscosity, νt, but to retain the microscopic value of η. thermal diffusion which characterizes the turbulent exchange The resulting effective magnetic Prandtl number they used was 2 of patches of warm and cold gas. This concept is also applied to 10− – big enough for the magnetorotational instability to de- turbulent magnetic diffusion which describes the turbulent ex- velop. On may wonder, of course, why one should not instead change of patches of magnetic field with different strengths and use turbulent values for both coefficients, i.e. νt/ηt 1. This direction. Reconnection of magnetic field lines is not explicitly would lead to even more favorable conditions for the≈ magne- required, but in the long run unavoidable if the magnetic power torotational instability (R¨udiger et al. 2002). spectrum is to decrease toward small scales. The idea of Prandtl Similar constraints have also been reported for the is that only the energy carrying eddies contribute to the mixing convection-driven geodynamo: Christensen et al. (1999) found of large scale distributions of velocity and magnetic field struc- that there is a minimum value of Pm of about 0.25 below which tures. This leads to a turbulent magnetic diffusion coefficient dynamo action does not occur at all. Similar results have also η 1 U`,whereU is the typical velocity and ` the scale of the t ≈ 3 been reported by Cattaneo (2003). These results are disturb- energy carrying eddies. For the kinematic turbulent viscosity ing, because both for the sun and for the earth, P 1. m one expects similar values. Analytic theory based on the quasi- For Pm of order unity, on the other hand, earth-like magnetic linear approximation also produces similar (but not identical) configurations can more easily be reproduced (see Kutzner & values of ηt and νt (e.g. Kitchatinov et al. 1994). Christensen 2002). It is usually assumed that the values of ηt and νt are Because of these restrictions, one wonders whether the ef- independent of the molecular (microscopic) viscosity and mag- fective magnetic Prandtl number to be used is not Pm,butrather netic diffusivity, ν and η. However, in the context of the geody- the turbulent value, Pm;t = νt/ηt. This raises the important ques- namo or in laboratory liquid metals the microscopic magnetic tions whether Pm;t is actually of order unity and whether it is 5 Prandtl number, P = ν/η is very small ( 10 ). This has independent of the microscopic value, Pm. The aim of this pa- m ≈ − dramatic consequences for the magnetorotational instability per is to estimate the value of Pm;t using turbulence simulations. (Balbus & Hawley 1991). This instability is generally accepted The knowledge of the value of Pm;t is also important for as the main mechanism producing turbulence in accretion discs the solar dynamo. The qualitative properties of the dynamo de- (Balbus & Hawley 1998). For sufficiently small values of Pm, pend on the relative importance of the large scale flows and Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20031371 322 T. A. Yousef et al.: Turbulent magnetic Prandtl number hence on the magnitude of ηt.Ifηt is too large, the influence of and a meridional flow of say 10 m/s is small so that only little mod- ification can be expected for the basic αΩ-dynamo (Roberts & 1 ∞ η = u0(x; t) u0(x; t + τ) dτ, (6) Stix 1972). In this case, however, we know that conventional t 6 Z h · i dynamo models of the solar activity cycle have difficulty to ex- −∞ plain Sp¨orer’s law of equatorward sunspot migration. The al- so that for the turbulent magnetic Prandtl number is ternative that the resulting poleward migration can be overcom- pensated by an internal equatorward flow requires a sufficiently νt 2 Pm;t = = = 0:4: (7) small value of ηt, which implies that Pm;t > 1 (Choudhuri et al. ηt 5 1995; Dikpati & Charbonneau 1999; Bonanno et al. 2002). This results is similar to that of Nakano et al. (1979) for the Given the importance of the value of the turbulent magnetic thermal Prandtl number. Prandtl number it is useful to assess the problem using three- R¨udiger (1989) lists a number of other approaches for cal- dimensional simulations of turbulent flows. We determine ν t culating turbulent transport coefficients, which all yield Prandtl and η by measuring the decay rate of a large scale (mean) ve- t numbers around or below unity. One particular approach is the locity and magnetic field, u and B, respectively. We emphasize renormalization group analysis which was applied to turbu- that we are not addressing the question whether ν and η can re- t t lence by Forster et al. (1977) for the case of a passive scalar, ally be used in studies of the dynamo or the magnetorotational and later by Fournier et al. (1982) to the case with magnetic instability, for example. fields. These results are valid in the long-time large-scale limit, We consider weakly compressible nonhelically forced tur- and the value of P turned out to be close to 0.7; see Eq. (23) bulence and use a model similar to that of Brandenburg (2001), m;t of Fournier et al. (1982). but with kinetic helicity fluctuating about zero. Dynamo action Kitchatinov et al. (1994) use a mixing length approxima- for such a model has recently been considered by Haugen et al. tion where terms of the form d=dt ν 2 are replaced by τ 1 , (2003), but it sets in only at magnetic Reynolds numbers above corr− where τ is the correlation time of− the∇ turbulence. They find 30, which is not the case in the present simulations. We begin corr ν = (4=15)τ u2 and η = (1=3)τ u2 ,soP = 4=5 = however∼ by first reviewing the basic results for the values of t corr rms t corr rms m;t 0:8. Yet another approach is the τ-approximation where triple ν and η within the framework of the quasilinear (Roberts & t t correlations are replaced by a damping term that is propor- Soward 1975; R¨udiger 1989) and other approximations. tional to the quadratic moments (e.g. Kleeorin et al. 1996; Blackman & Field 2002). Here no Fourier transformation in 2 2. Results from quasilinear approximation time is used. This gives, as before, ηt = (1=3)τurms (where τ 2 is now interpreted as a relaxation time), but νt = (2=15)τurms, For steady homogeneous isotropic turbulence the correlation so Pm;t = 2=5 = 0:4. This is half the value obtained from the tensor is independent of x and t,i.e. mixing length approximation, but the same as in Eq. (7). The fact that in all these cases Pm;t is less than unity can be u0(x; t)u0 (x + ξ; t + τ) = Q (ξ,τ); (1) h i j i ij traced back to the presence of the pressure term in the momen- where angular brackets denote an ensemble average and primes tum equation. If this term is ignored (as in pressureless Burgers fluctuations about the average. In the quasilinear approxima- turbulence or “burgulence”) one always gets Pm;t = 1. tion the transport coefficients are conveniently expressed in It is tempting to speculate that the discrepancy between the terms of the Fourier transformed correlation tensor, Qˆij(k;!), different analytic approaches is related to the validity of some which is normalized such that idealizing assumptions made in order to apply the quasilinear and other approximations. Clearly, additional approaches are Q (ξ,τ) = Qˆ (k;!)ei(k ξ ωτ)dk d!: (2) needed to get a more complete picture regarding the correct ij " ij · − value of Pm;t. It is nevertheless encouraging that Pm;t does not For the turbulent viscosity and the turbulent magnetic diffusiv- strongly deviate from unity. ity one finds respectively (R¨udiger 1989) In the remainder of this paper we estimate νt and ηt numer- ically by considering the decay of an initial large scale velocity 4 ν3k6Qˆ (k;!) or magnetic field, respectively, in the presence of small scale ν = ll dk d!; (3) t 15 " (!2 + ν2k4)2 turbulence.
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