Presentation on theme: "Spectrum of MHD turbulence"— Presentation transcript:
1Spectrum of MHD turbulence Stanislav BoldyrevUniversity of Chicago(June 20, 2005)Ref: astro-ph/ ; ApJ 626, L37, 2005
2Introduction: Kolmogorov turbulence Random flow of incompressible fluidReynolds number:vLRe=Lv/η>>1η-viscosityIf there is no intermittency, then:andKolmogorov spectrum [Kolmogorov 1941]
3Kolmogorov energy cascade local energy fluxEnergy of an eddy of size is ;it is transferred to a smaller-size eddy during time:- “eddy turn-over” time.The energy flux,, is constant for theKolmogorov spectrum!
4MHD turbulence ? No exact Kolmogorov relation. Phenomenology: is conserved, and cascadestoward small scales.EnergyNo, since dimensionalarguments do not work!?Is energy transfer timeNon-dimensional parametercan enter the answer.Need to investigate interaction of “eddies” in detail!This is also the main problem in the theory of weak (wave) turbulence.(waves is plasmas, water, solid states, liquid helium, etc…)[Kadomtsev, Zakharov, ’s]
5Iroshnikov-Kraichnan spectrum wzwzAfter interaction, shape of each packet changes, but energy does not.
6Iroshnikov-Kraichnan spectrum during one collision:number of collisions required todeform packet considerably:λλConstant energy flux:[Iroshnikov (1963); Kraichnan (1965)]
7Goldreich-Sridhar theory Anisotropy of “eddies”BλLλL>>Shear Alfvén wavesdominate the cascade:┴BCritical Balance[Goldreich & Sridhar (1995)]
8Spectrum of MHD Turbulence in Numerics [Müller & Biskamp, PRL 84 (2000) 475]
9Goldreich-Sridhar Spectrum in Numerics Cho & Vishniac, ApJ, 539, 273, 2000Cho, Lazarian & Vishniac, ApJ, 564, 291, 2002
10Strong Magnetic Filed, Numerics Contradictions with Goldreich-Sridhar modelIroshnikov-Kraichnanscaling[Maron & Goldreich, ApJ 554, 1175, 2001]
11Strong Magnetic Filed, Numerics Contradictions with Goldreich-Sridhar modelB-parallel scalingB-perp scalingScaling of field-parallel andfield-perpendicular structurefunctions for differentlarge-scale magnetic fields.[Müller, Biskamp, GrappinPRE, 67, , 2003]Weak field, B→0:Goldreich-Sridhar (Kolmogorov)scaling2Strong field, B>>ρV :Iroshnikov-Kraichnan scaling
12New Model for MHD Turbulence Analytic Introduction[S.B., ApJ, 626, L37,2005]Depletion of nonlinear interaction:12Nonlinear interactionis depletedInteraction timeis increasedFor perturbation cannot propagate along the B-line fasterthan V , therefore, correlation length along the line isAThis balances terms and in the MHD equations, as in theGoldreich-Sridhar picture, however, the geometric meaning is different.12
13New Model for MHD Turbulence Analytic Introduction[S.B., ApJ, 626, L37,2005]Nonlinear interactionis depletedInteraction timeis increasedConstant energy flux,Goldreich-Sridhar scaling corresponds toα=0:Explainsnumericallyobserved scalingsfor strong B-field !“Iroshnikov-Kraichnan” scaling is reproduced forα=1:[Maron & Goldreich, ApJ 554, 1175, 2001][Müller, Biskamp, Grappin PRE, 67, , 2003]
14New Model for MHD Turbulence Geometric MeaningGoldreich-Sridhar 1995 “eddy”:line displacement:S.B. (2005) “eddy”:line displacement:As the scale decreases,λ→0,turns into filamentturns into current sheetagrees with numerics!
15New Model for MHD Turbulence Depletion of nonlinearityS.B. (2005) “eddy”:line displacement:In our “eddy”, w and z are aligned within small angle One can check that:θλθIn our theory, this angle is:Remarkably, we reproduced thereduction factor in the original formula:The theory is self-consistent.
16Summary and Discussions 1. Weak large-scale field:dissipative structures: filamentsenergy spectrum: E(K)~K┴-5/3[Goldreich & Sridhar’ 95]2. Strong large-scale field:dissipative structures: current sheetsenergy spectrum: E(K)~K┴-3/2scale-dependentdynamic alignment3. The spectrum of MHD turbulence may be non-universal.Alternatively, it may always be E~K , but in case 1, resolutionof numerical simulations is not large enough to observe it.┴-3/2
17Conclusions Theory is proposed that explains contradiction between Goldreich-Sridhar theory and numerical findings.In contrast with GS theory, we predict that turbulent eddiesare three-dimensionally anisotropic, and that dissipativestructures are current sheets.For strong large-scale magnetic field, the energy spectrumis E~K It is quite possible that spectrum is always E~K ,but for weak large-scale field, the resolution of numericalsimulations is not large enough to observe it.-3/2-3/2┴┴