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Spectrum of MHD turbulence Stanislav Boldyrev University of Chicago Ref: astro-ph/0503053; ApJ 626, L37, 2005 (June 20, 2005)

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Presentation on theme: "Spectrum of MHD turbulence Stanislav Boldyrev University of Chicago Ref: astro-ph/0503053; ApJ 626, L37, 2005 (June 20, 2005)"— Presentation transcript:

1 Spectrum of MHD turbulence Stanislav Boldyrev University of Chicago Ref: astro-ph/ ; ApJ 626, L37, 2005 (June 20, 2005)

2 2 Introduction: Kolmogorov turbulence L v Re=Lv/η>>1 Reynolds number: Random flow of incompressible fluid η -viscosity If there is no intermittency, then: Kolmogorov spectrum [Kolmogorov 1941] and

3 3 Kolmogorov energy cascade Energy of an eddy of size is ; it is transferred to a smaller-size eddy during time: The energy flux,, is constant for the Kolmogorov spectrum! local energy flux - eddy turn-over time.

4 4 MHD turbulence No exact Kolmogorov relation. Phenomenology: Energy is conserved, and cascades toward small scales. Is energy transfer time ? No, since dimensional arguments do not work! Need to investigate interaction of eddies in detail! Non-dimensional parametercan enter the answer. 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]

5 5 Iroshnikov-Kraichnan spectrum After interaction, shape of each packet changes, but energy does not. w zw z

6 6 Iroshnikov-Kraichnan spectrum λ λ during one collision: number of collisions required to deform packet considerably: Constant energy flux: [Iroshnikov (1963); Kraichnan (1965)]

7 7 Goldreich-Sridhar theory Anisotropy of eddies L λ Shear Alfvén waves dominate the cascade: B B Critical Balance λ L>> [Goldreich & Sridhar (1995)]

8 8 Spectrum of MHD Turbulence in Numerics [Müller & Biskamp, PRL 84 (2000) 475]

9 9 Goldreich-Sridhar Spectrum in Numerics Cho & Vishniac, ApJ, 539, 273, 2000 Cho, Lazarian & Vishniac, ApJ, 564, 291, 2002

10 10 Strong Magnetic Filed, Numerics Contradictions with Goldreich-Sridhar model [Maron & Goldreich, ApJ 554, 1175, 2001] Iroshnikov-Kraichnan scaling

11 11 Strong Magnetic Filed, Numerics Contradictions with Goldreich-Sridhar model Scaling of field-parallel and field-perpendicular structure functions for different large-scale magnetic fields. [Müller, Biskamp, Grappin PRE, 67, , 2003] Weak field, B0: Goldreich-Sridhar (Kolmogorov) scaling B-parallel scaling B-perp scaling Strong field, B>>ρV : Iroshnikov-Kraichnan scaling 2

12 12 New Model for MHD Turbulence Depletion of nonlinear interaction: Nonlinear interaction is depleted Interaction time is increased A 12 This balances terms and in the MHD equations, as in the Goldreich-Sridhar picture, however, the geometric meaning is different. 12 [S.B., ApJ, 626, L37,2005] For perturbation cannot propagate along the B-line faster than V, therefore, correlation length along the line is Analytic Introduction

13 13 New Model for MHD Turbulence Analytic Introduction Nonlinear interaction is depleted Interaction time is increased Constant energy flux, Goldreich-Sridhar scaling corresponds toα=0: Iroshnikov-Kraichnan scaling is reproduced forα=1: Explains numerically observed scalings for strong B-field ! [Maron & Goldreich, ApJ 554, 1175, 2001] [Müller, Biskamp, Grappin PRE, 67, , 2003] [S.B., ApJ, 626, L37,2005]

14 14 New Model for MHD Turbulence Geometric Meaning S.B. (2005) eddy: line displacement: Goldreich-Sridhar 1995 eddy: line displacement: As the scale decreases, λ0, turns into filament turns into current sheet agrees with numerics!

15 15 New Model for MHD Turbulence Depletion of nonlinearity S.B. (2005) eddy: line displacement: Remarkably, we reproduced the reduction factor in the original formula: θ In our eddy, w and z are aligned within small angle. One can check that: θ In our theory, this angle is: The theory is self-consistent. λ

16 16 Summary and Discussions 3. The spectrum of MHD turbulence may be non-universal. Alternatively, it may always be E~K, but in case 1, resolution of numerical simulations is not large enough to observe it. -3/2 1. Weak large-scale field: dissipative structures: filaments energy spectrum: E(K)~K -5/3 [Goldreich & Sridhar 95] 2. Strong large-scale field: dissipative structures: current sheets energy spectrum: E(K)~K -3/2 scale-dependent dynamic alignment

17 17 Conclusions Theory is proposed that explains contradiction between Goldreich-Sridhar theory and numerical findings. In contrast with GS theory, we predict that turbulent eddies are three-dimensionally anisotropic, and that dissipative structures are current sheets. For strong large-scale magnetic field, the energy spectrum is E~K. It is quite possible that spectrum is always E~K, but for weak large-scale field, the resolution of numerical simulations is not large enough to observe it. -3/2


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