MHD Turbulence: Nonlocal, Anisotropic, Nonuniversal?

Transcription

1 MHD Turbulence: Nonlocal, Anisotropic, Nonuniversal? Alexander A Schekochihin 1,2,3, Steven C Cowley 1,4, and Tarek A Yousef 3 1 Blackett Laboratory, Imperial College, London SW7 2BW, UK 2 King s College, Cambridge CB2 1ST, UK 3 DAMTP, University of Cambridge, Cambridge CB3 0WA, UK 4 Department of Physics, UCLA, Los Angeles, California , USA Summary. Kolmogorov s theory and philosophy of turbulence are based on a number of assumptions that have become standard notions with which one approaches turbulence in many, including non-hydrodynamic, systems. However, it turns out that in MHD turbulence, locality of interactions in scale space, isotropy of small scales or even universality cannot be taken for granted and, in fact, can be shown to fail. This note focuses on these unconventional aspects of MHD turbulence and on the related phenomenon of small-scale dynamo using a combination of simple physical reasoning and numerical evidence. MHD turbulence is the turbulence of a highly conducting fluid in which chaotic velocity and magnetic-field fluctuations exist and interact in a broad range of spatial and temporal scales. It is the prevailing form of turbulence that is presumed to occur and in many cases is, indeed, observed in astrophysical plasmas: examples are solar wind, interstellar medium, accretion discs, galaxy clusters etc. (see, e.g., a recent review [22] and references therein). It is important to remember that the MHD approximation is strictly only appropriate for low-frequency fluctuations at collisional scales, although for strongly magnetised plasmas, it can under certain assumptions be extended to collisionless scales as well [27]. In this short note, we shall not venture beyond the incompressible MHD equations, which read t u + u u = p + B B + ν 2 u + f, (1) t B + u B = B u + η 2 B, (2) where u is velocity, B magnetic field, p the total pressure determined by the incompressibility condition u = 0, ν viscosity, η magnetic diffusivity, and f a body force (we use units in which p is scaled by ρ and B by 4πρ, where ρ = const is the density of the medium). The body force models some system-specific energy input at the outer scale l 0. As usual, we assume that the properties of turbulence at scales much smaller than l 0 are not sensitive

2 2 Alexander A Schekochihin, Steven C Cowley, and Tarek A Yousef to the particular choice of f and only depend on the average injected power per unit volume ɛ = u f, where... means volume averaging. The first theories of MHD turbulence date back to mid-20 century [2, 29, 8, 10] and a vast amount of literature has since been generated. However, our current understanding of this type of turbulence has yet to reach the level comparable to that which exists in the area of neutral-fluid turbulence. Not surprisingly, most theories of MHD turbulence have, in one form or another, attempted to generalise the spirit and the method of Kolmogorov s 1941 dimensional theory (K41) to the MHD case. This course of action involves making assumptions that, while certainly not automatically valid, might appear natural. Roughly speaking, here are the notions that have to various degrees dominated the study of MHD turbulence: there is only one universal kind of MHD turbulence; velocity and magnetic fluctuations are in scale-by-scale equipartition; interactions are local in scale space; there is isotropy at sufficiently small scales; numerical simulations confirm all of the above, while astronomical observational evidence is consistent with it. In this short note, we use simple physical reasoning and numerical evidence to argue that all of these statements are in general untrue. The non-universality of MHD turbulence has to do with the presence or absence of a strong mean (large-scale) field B 0. When B 0 is zero or dynamically insignificant, the magnetic field mostly consists of randomly tangled fluctuations, δb/b 0 1. Galaxy-cluster plasmas [21] and some parts of the interstellar medium [3, 7, 27] appear to be in this regime. In the opposite asymptotic situation when B 0 is imposed externally and is such that its energy is larger than the energy of the turbulence motions, B 0 u rms, the physics is clearly different: only small perturbations of the mean field are possible, so δb/b 0 u rms /B 0 1. These perturbations take the form of Alfvén and slow MHD waves, with the dispersion relation ω = ±k v A, where v A = B 0 is the Alfvén speed. The MHD turbulence in this regime, which is realised most clearly in the solar wind [18, 27], is a turbulence of these waves, or, more precisely, wave packets that interact nonlinearly and give rise to an energy cascade. The assumption of locality of interactions is probably acceptable here and the velocity and magnetic field should be at least approximately in a scale-by-scale equipartition because δb δu in an Alfvén wave. However, the isotropy assumption, originally considered plausible [8, 10] on the same philosophical grounds as in K41, turns out to be incorrect, as evidenced both by numerical simulations [11, 6] and solar-wind measurements [18]. Instead, the Alfvénic turbulence seems to be correctly described if one assumes that the Alfvén-wave-propagation time and the nonlinear interaction time are balanced at each scale, k v A k u, i.e., interactions are strong an assumption due to Goldreich & Sridhar [16]. This leads to a Kolmogorov spectrum in field-perpendicular wave numbers, k 5/3, but also to a nontrivial

3 MHD Turbulence: Nonlocal, Anisotropic, Nonuniversal? 3 relation between the parallel and perpendicular scales: k k 2/3 (see [22] for a short review). Thus, the turbulent cascade has an intrinsic anisotropy that increases at small scales a unique situation in turbulence theory. It is often thought that this is, in fact, the only type of MHD turbulence and that in systems where no mean field is present, large-scale magnetic fluctuations will dominate energetically and play the role of the mean field to the small-scale Alfvénic fluctuations an assumption originally proposed by Kraichnan [10]. This may indeed be what happens in decaying MHDturbulence simulations, which seem to be dominated by large-scale force-free magnetic structures [4, 13]. The situation that emerges in the numerical simulations of the forced case [12, 26, 17] is very different: while the velocity spectra suggest a cascade of a usual kind, the magnetic field is dominated by long folded flux sheets (or ribbons) with direction reversals at the diffusive scale (see Fig. 2b) spectrally, this folded structure manifests itself as a large energy excess at small scales (see Fig. 3a, inset). The key to making sense of these results is to realise that in MHD turbulence without a mean field, magnetic field has to be generated self-consistently by the action of the velocity field [the induction equation (2) is not forced a modeling choice mandated by the physical constraint of magnetic flux conservation]. As long as the (random) forcing in (1) is spatially homogeneous and nonhelical, no mean (large-scale) field is produced by the turbulent velocity and the only generation process is the small-scale dynamo stretching of the magnetic field by the random velocity gradients [2, 9]. This is a fundamentally nonlocal type of interaction (cf. [14]): indeed, even a linear-in-space (i.e., infinite-scale) velocity field will, in 3D, lead to exponential growth of magnetic energy from any weak initial field and the magnetic structures created in the process are small-scale in the sense that they are characterised by direction reversals at the diffusive scale [15, 31, 19, 22] (see inset in Fig. 1a). In what follows, it will be convenient to concentrate on the asymptotic limit of large magnetic Prandtl number, Pm = ν/η 1, i.e., Rm Re 1, where Re = u rms l 0 /ν and Rm = u rms l 0 /η are the Reynolds and magnetic Reynolds numbers, respectively. The favourite numerical case of Pm = 1, while nonasymptotic and, therefore, obscure in many ways, belongs to the same universality class (the limit Pm 1, which occurs in liquid-metal applications and in the stellar convective zones, is fundamentally different [28] and will not be considered here). When Pm 1, the random but spatially smooth motions at the viscous scale l ν Re 3/4 l 0 are the ones that contribute dominantly to the random stretching of an initially weak magnetic field because they have the largest turnover rate, γ (ɛ/ν) 1/2 (u rms /l 0 )Re 1/2, in Kolmogorov turbulence. As a result, the magnetic field grows exponentially at the rate γ and is organised into folded structures with direction reversals at the diffusive scale l η (η/γ) 1/2 Pm 1/2 l ν l ν. While the reversal scale is small, the field remains straight at scales below the scale of the velocity field, so l l ν l l η a case of local anisotropy (the orientation of the folds is random, so the turbulence is isotropic globally, in a strict statistical sense).

4 4 Alexander A Schekochihin, Steven C Cowley, and Tarek A Yousef (a) (b) (c) Fig. 1. Schematic illustration of the three stages of the evolution of MHD turbulence with B 0 = 0 and Rm Re 1: (a) kinematic (weak-field) stage: small-scale dynamo, exponential growth, magnetic field is stretched by the viscous-scale motions (the amplification mechanism and the emerging folded structure with direction reversals are illustrated by the inset); (b) intermediate stage: secular growth, field is stretched by the inertial-range motions at scale l s(t); (c) saturated state: balance between stretching by the outer-scale motions and back reaction from folds, possibly a cascade of Alfvénic perturbations of the folded structure (illustrated by the inset).

5 MHD Turbulence: Nonlocal, Anisotropic, Nonuniversal? 5 (a) (b) Fig. 2. Example of saturated state of MHD turbulence (B 0 = 0) dominated by nonlocal interactions. These are cross sections of (a) u and (b) B for a simulation with Pm = 1250, Re 1 (run S5 of [26], ). Velocity is random in time due to random forcing, but spatially smooth (single-scale), while magnetic field has folded structure. Arrows indicate the in-plane direction of the fields. This geometric property of the field, which can be analytically and numerically demonstrated by considering the statistics of the field-line curvature [23, 26], has a direct implication for the physics of nonlinear back reaction that the field exerts on the fluid: since the Lorentz tension force in (1) is quadratic in B and depends only on its variation along itself, it is oblivious to the direction reversals and can act on the scale of the velocity field another nonlocal interaction process, which ends up saturating the smallscale dynamo. In order for saturation to be achieved, the stretching action of the turbulent motions should be suppressed in some way that is not quantitatively fully understood yet but probably amounts to some form of partial statistical anisotropisation of the velocity gradients, making them locally more two-dimensional and unable to act as small-scale dynamo [25, 26] the precise diagnosing of this effect is a subject of ongoing investigations. Figure 2 presents such a saturated state in a numerical simulation with Rm Re 1 (the velocity field is random in time due to random forcing but is effectively single-scale): the folded structure and a large-scale spatially smooth velocity field coexist, while a balance of the two nonlocal interaction processes small-scale dynamo and the field s back reaction maintains a statistically stationary field level B 2 u 2 rms. In the more realistic case of Re 1, the saturation is preceded by an intermediate nonlinear stage [24, 26, 22] (see Fig. 1b). The back reaction from the field on the viscous-scale motions should become important when the energy of the field approaches the energy of these motions: B 2 δu 2 l ν u 2 rmsre 1/2 (then B B B 2 /l δu 2 l ν /l ν u u). Once this happens, the ability

6 6 Alexander A Schekochihin, Steven C Cowley, and Tarek A Yousef of the viscous-scale motions to amplify the field must be suppressed. During the period the follows, the back reaction gradually suppresses also the inertialrange motions these have slower turnover rates than the viscous-scale ones, so amplify the field less quickly, but are more energetic and, therefore, require larger magnetic energy to be suppressed. If at any given time t during this period, l s (t) (l ν, l 0 ) is the scale in the inertial range such that the motions associated with it have energy comparable to the energy of the magnetic fields, δu 2 l s(t) B2 (t), we may argue that the magnetic field is amplified exponentially at the instantaneous rate γ(t) δu ls(t)/l s (t): d dt B2 γ(t) B 2 (t) δu3 l s l s ɛ = const. (3) Thus, the magnetic energy grows secularly: B 2 ɛt (cf. [29]). Assuming that the turbulence is still Kolmogorov at scales above l s, we have δu ls (ɛl s ) 1/3, whence the scale l s (t) ɛ 1/2 t 3/2 grows until it is comparable to the outer scale l 0 after time t l 2/3 0 ɛ 1/3 l 0 /u rms (the turnover time at the outer scale). At this point, B 2 u 2 rms and the saturated state is reached. By the logic of the above, at any given time t, the field amplification process is a small-scale dynamo done by the motions of scale l s (t), so the length of the folds increases during the intermediate stage, l (t) l s (t), and is comparable to the outer scale l 0 in saturation. The diffusive (field-reversal) scale also increases, but slower: l η (t) (η/γ(t)) 1/2 (ηt) 1/2 until it is l η Rm 1/2 l 0 in saturation. Figure 3 shows quantitatively how the evolution described above appears in a (nonasymptotic) numerical simulation with Pm = 1 (see [26, 30] for further numerical tests and discussion). Thus, the fully developed state of forced MHD turbulence in the absence of a strong externally imposed mean field is the saturated state of the small-scale dynamo, in which statistically stationary level of magnetic energy B 2 u 2 rms is achieved via a highly nonlocal interaction (stretching balanced by back reaction) between the outer-scale velocity field and the small-scale directionreversing folded magnetic fields (see Fig. 1c). A detailed numerical investigation of energy transfer in the wave-number space [1, 14] lends further support to this view (see also A Pouquet s contribution in these Proceedings). Another way to corroborate it is to examine the scale-by-scale budgets of MHD turbulence in terms of the exact laws analogous to the 4/5 and 4/3 laws of turbulence [5, 20] this is done in [30], where it becomes evident that the energy cascade is short-circuited at the outer scale with most or, at least, a large fraction of the kinetic energy injected by the forcing diverted directly into maintaining the small-scale folded structure against magnetic diffusion. Whether there is also a significant amount of local energy transfer is still unclear. It is possible that a local cascade does exist and consists of Alfvénic perturbations of the folded structure (see inset in Fig. 1c) these are mathematically possible [24] and physically allowed because they do not stretch the field, but remain a conjecture as it is unclear whether they can support a

7 MHD Turbulence: Nonlocal, Anisotropic, Nonuniversal? 7 (a) (b) Fig. 3. Evolution of MHD turbulence (B 0 = 0) in a simulation with Pm = 1, Re 400 (Re λ 155 during the weak-field stage, Re λ 116 in the saturated state; this is run A of [26], ): (a) kinetic and magnetic energies vs. time (inset shows kinetic- and magnetic-energy spectra in the saturated state red and blue lines, respectively); (b) characteristic wave numbers k rms = ( B 2 / B 2 ) 1/2 (measure of inverse reversal scale, blue line), k λ = ( u 2 / u 2 ) 1/2 (Taylormicroscale wave number, red line), k = ( B B 2 / B 4 ) 1/2 (measure of inverse fold length, black line). Note the manifest presence of the intermediate nonlinear stage characterised by secular field growth and elongation of folds. viable turbulent cascade and also whether there is a practical way of detecting them at currently accessible numerical resolutions. Another unsolved problem is the precise shape of the magnetic-energy spectrum in the saturated state (see Fig. 3a, inset). A simplistic model of the folds as uniform bundles of stripy, direction-reversing straight fields gives a k 1 spectrum: this follows from the fact that for such a field an average field increment across some distance l (l η l l 0 ) is constant: δb l B rms const [30]. While it seems clear that the spectrum cannot be steeper than k 1 if it is energetically dominated by folds, it may be shallower and possibly have a small positive exponent [25, 26] similar to its form in the small-scale-dynamo (weak-field) growth stage when the scaling is k +3/2 [9]. As genuinely high-resolution numerical simulations are now becoming possible, a full quantitative picture of MHD turbulence is starting to emerge. Much, however, remains to be understood indeed, a true quantification and verification of the ideas proposed above will require much more than the runs quoted here [26]; even the record achieved in [17] appears to be insufficient. However, progress is inevitable in the near future. Our motivation in writing this short note has been the view that this progress will be greatly accelerated if numerical and analytical studies are done in full under-

8 8 Alexander A Schekochihin, Steven C Cowley, and Tarek A Yousef standing that none of the standard received notions that one brings into the field from the conventional turbulence theory such as the expectation of locality, isotropy etc. is automatically guaranteed to be justified and that many of them can, in fact, be shown to fail by very simple physical arguments. This paper has benefited from discussions with P Mininni, A Tsinober, F Rincon and K Yoshida. AAS was supported by a PPARC Advanced Fellowship. He also thanks the Nagoya conference organisers for travel support. TAY was supported by a UKAFF Fellowship and the Newton Trust. References 1. Alexakis A, Mininni PD, Pouquet A (2005) Phys Rev E 72: Batchelor GK (1950) Proc Roy Soc London A 201: Beck R (2006) in: Boulanger F, Miville-Deschenes MA (eds) Polarisation EAS Pub Ser, in press (e-print astro-ph/ ) 4. Biskamp D, Müller W-C (2000) Phys Plasmas 7: Chandrasekhar S (1951) Proc Roy Soc London Ser A 204: Cho J, Lazarian A, Vishniac ET (2002) Astrophys J 564: Haverkorn M, Gaensler BM et al (2005) Astrophys J 637:L33 L35 8. Iroshnikov RS (1963) Astron Zh 40: Kazantsev AP (1967) Zh Eksp Teor Fiz 53: Kraichnan RH (1965) Phys Fluids 8: Maron J, Goldreich P (2001) Astrophys J 554: Meneguzzi M, Frisch U, Pouquet A (1981) Phys Rev Lett 47: Mininni PD (2006) private communication 14. Mininni P, Alexakis A, Pouquet A (2005) Phys Rev E 72: Moffatt HK, Saffman PG (1964) Phys Fluids 7: Goldreich P, Sridhar S (1995) Astrophys J 438: Haugen NEL, Brandenburg A, Dobler W (2004) Phys Rev E 70: Horbury TS, Forman MA, Oughton S (2005) Plasma Phys Control Fusion 47:B703 B Ott E (1998) Phys Plasmas 5: Politano H, Pouquet A (1998) Geophys Res Lett 25: Schekochihin AA, Cowley SC (2006) Phys Plasmas 13: Schekochihin AA, Cowley SC (2007) in: Molokov S, Moreau R, Moffatt HK (eds) Magnetohydrodynamics: Historical Evolution and Trends. Springer, Berlin (e-print astro-ph/ ) 23. Schekochihin A, Cowley S et al (2002) Phys Rev E 65: Schekochihin AA, Cowley CS et al (2002) New J Phys 4: Schekochihin AA, Cowley CS et al (2004) Phys Rev Lett 92: Schekochihin AA, Cowley CS et al (2004) Astrophys J 612: Schekochihin AA, Cowley CS, Dorland W (2007) Plasma Phys Control Fusion, in press (e-print astro-ph/ ) 28. Schekochihin AA, Haugen NEL et al (2005) Astrophys J 625:L115 L Schlüter A, Biermann L (1950) Z Naturforsch 5a: Yousef TA, Rincon F, Schekochihin AA (2006) J Fluid Mech, in press (e-print astro-ph/ ) 31. Zeldovich YaB, Ruzmaikin AA et al (1984) J Fluid Mech 144:1 11