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

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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

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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.

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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]

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

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

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

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

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

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

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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

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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

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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]

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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!

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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

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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

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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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