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

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

Kolmogorov energy cascade local energy flux Energy 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 the Kolmogorov spectrum!

MHD turbulence ? No exact Kolmogorov relation. Phenomenology: is conserved, and cascades toward small scales. Energy No, since dimensional arguments do not work! ? Is energy transfer time Non-dimensional parameter can 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, ... 1960’s]

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

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

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

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

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

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

Strong Magnetic Filed, Numerics Contradictions with Goldreich-Sridhar model B-parallel scaling B-perp scaling Scaling of field-parallel and field-perpendicular structure functions for different large-scale magnetic fields. [Müller, Biskamp, Grappin PRE, 67, 066302, 2003] Weak field, B→0: Goldreich-Sridhar (Kolmogorov) scaling 2 Strong field, B>>ρV : Iroshnikov-Kraichnan scaling

New Model for MHD Turbulence Analytic Introduction [S.B., ApJ, 626, L37,2005] Depletion of nonlinear interaction: 1 2 Nonlinear interaction is depleted Interaction time is increased For perturbation cannot propagate along the B-line faster than V , therefore, correlation length along the line is A This balances terms and in the MHD equations, as in the Goldreich-Sridhar picture, however, the geometric meaning is different. 1 2

New Model for MHD Turbulence Analytic Introduction [S.B., ApJ, 626, L37,2005] Nonlinear interaction is depleted Interaction time is increased Constant energy flux, Goldreich-Sridhar scaling corresponds to α=0: Explains numerically observed scalings for strong B-field ! “Iroshnikov-Kraichnan” scaling is reproduced for α=1: [Maron & Goldreich, ApJ 554, 1175, 2001] [Müller, Biskamp, Grappin PRE, 67, 066302, 2003]

New Model for MHD Turbulence Geometric Meaning Goldreich-Sridhar 1995 “eddy”: line displacement: S.B. (2005) “eddy”: line displacement: As the scale decreases, λ→0, turns into filament turns into current sheet agrees with numerics!

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

Summary and Discussions 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 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

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 -3/2 ┴ ┴