Magnetohydrodynamic Turbulence Wolf-Christian Mller - PowerPoint PPT Presentation

Magnetohydrodynamic Turbulence Wolf-Christian Müller Max-Planck-Institut für Plasmaphysik, Garching, Germany Dieter Biskamp, Max-Planck-Institut für Plasmaphysik, Garching, Germany Roland Grappin, Observatoire de Paris-Meudon, Meudon, France James Merrifield, Sandra Chapman, University of Warwick, Warwick, United Kingdom Richard Dendy, UKAEA Culham Division, Abingdon, United Kingdom

Turbulence Turbulent flows: ensemble of random fluctuations without apparent structure/order Systems appears to be ’smooth’ (no specific feature/symmetry to cling to). Under idealized conditions (statistical stationarity/homogeneity, no boundaries, no friction) − → (generalized) scale-covariance Self-similar function f ( ℓ ) = A · ℓ γ − → f ( λ ℓ ) ∼ λ γ f ( ℓ ) Function f ( ℓ ) under magnifying glass ( ℓ → λ ℓ ) looks identical (neglecting constant factor) For simplificity: statistical isotropy, i.e. ensemble average �•� independent of direction implies stat. homogeneity (independence of position). Turbulent fields exhibit statistical (self-)similarity !

Tackling the problem Starting point for mostly phenomenological theories dealing with ◮ temporal/spectral evolution of low-order statistical moments, e.g. magnetic and kinetic energies, helicities, associated spectral fluxes ◮ spatially intermittent structure of turbulent fields New development (emerging from turbulent passive-scalar transport): ◮ Lagrangian statistics and invariants Applications: ◮ lifetime of/structure formation in interstellar molecular clouds (star-formation) ◮ transport/dispersion/acceleration of substances/particles (nuclear fusion/environmental sciences/cosmic rays) ◮ magnetic field amplification (turbulent dynamo)/formation of large-scale structures (meteorology) ◮ friction/mixing/flow control (engineering)

Ideal Invariants and Cascade directions V dV ( v 2 + b 2 ) ◮ total energy E = � no dissipation ◮ cross helicity H C = � V dV v · b frozen-in field lines ◮ magnetic helicity H M = b = ∇ × a � V dV a · b , no reconnection Ideal invariants satisfy detailed balance relations, e.g., triad interactions (quadratic nonlinearities) E k 1 + ˙ ˙ E k 2 + ˙ E k 3 = 0 , k 1 + k 2 + k 3 = 0 ⇐ = small k = ⇒ large k Inverse cascade direct cascade inverse cascade: formation of large-scale coherent structures. Detailed balance prerequisite for cascade/power-law scaling.

Kolmogorov-Richardson Picture Large Energy Small-scale 0 10 eddies (arb. units) structures Direct cascade -1 10 Inverse cascade Dissipation -2 Inertial Drive 10 range range range -2 -1 0 10 10 10 k (arb. units)

Energy Cascade Phenomenology ◮ Kolmogorov (K41) Turbulent eddies break up in successively smaller structures Time-scale: τ NL ∼ ℓ/ v ℓ , ε ∼ v 2 l / τ NL , v 2 ℓ ∼ kE k → Energy spectrum E ( k ) ∼ k − 5 / 3 ◮ Iroshnikov-Kraichnan (IK) Alfvén waves interact nonlinearly along magnetic field τ ∗ ∼ τ NL Time-scale: τ A ∼ ℓ/ B 0 , ε ∼ v 2 l / τ ∗ , τ A τ NL → Energy spectrum E ( k ) ∼ k − 3 / 2 ◮ Goldreich-Sridhar Magnetic field causes local anisotropy → Field-parallel: transfer negligible → Field-perpendicular: Kolmogorov cascade → Perpendicular energy spectrum E ( k ⊥ ) ∼ k − 5 / 3 ⊥

Doradus 30

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Probing the Solar Wind

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Experimental Observation 10 2 f -1.7 E f 10 0 f -4 10 -2 10 -4 10 -3 10 -2 10 -1 10 0 10 1 f Leamon et al. JGR ’98 Solar wind fluctuations measured by WIND probe at ≃ 1 A . U . ⇒ K41 scaling ∼ k − 5 / 3

Incompressible Magnetohydrodynamics (MHD) Simplified incompressible fluid model: ∂ t v = − ( v · ∇ ) v − ∇ p − b × ( ∇ × b )+ Re − 1 ∆ v , ∂ t b = ∇ × ( v × b )+ Rm − 1 ∆ b , ∇ · v = ∇ · b = 0 . ◮ Kinetic and magnetic Reynolds number: Re : = ℓ 0 v 0 Rm : = ℓ 0 v 0 η µ ◮ Kinematic viscosity µ , magnetic diffusivity η ◮ Turbulence, if Re,Rm ≫ 1 – Solar convection zone (Re ∼ 10 15 , Rm ∼ 10 8 ) – Black hole accretion disk (Re ∼ 10 11 , Rm ∼ 10 10 ) – Earth’s liquid core (Re ∼ 10 9 , Rm ∼ 10 2 )

Turbulent Magnetic Field (Isotropic)

Numerical Simulation (Isotropic) Pseudospectral direct numerical simulation ( 1024 3 collocation points) Three-dimensional periodic cube Initially: nonhelical isotropic random fields with amplitudes ∼ exp [ − k 2 / ( 2 k 2 0 )] , k 0 = 4

Introducing Anisotropy Switching from isotropic K41 to anisotropic Goldreich-Sridhar configuration by imposed mean magnetic field B 0 = B 0 e z ( B 0 ≃ 5 | b | rms )

Turbulent Magnetic Field (Anisotropic)

Numerical Simulation (Anisotropic) Three-dimensional forced anisotropic turbulence ( 1024 2 × 256 collocation points) displays IK-scaling ∼ k − 3 / 2

Closure Theory Regarding statistical moments of fluid equations schematically: ∂ t � u � = � uu � ∂ t � uu � = � uuu � ∂ t � uuu � = � uuuu � . . . Closure (Quasi-normal approximation): 4 th and higher order moments → Expressed via second-order moments Problem: Unphysical, negative energy spectra possible Solution: Introduction of damping term on 3rd order level (Eddy-damped-quasi-normal-Markovian (EDQNM) approximation)

Spectral EDQNM Equations Equation for energy spectrum E k : � � ( ∂ t + 2 Re − 1 k 2 ) E k = △ d p d q Θ kpq T kpq ◮ ‘ △ ’: Integration over modes with k + p + q = 0 ◮ T kpq = T kpq ( E p , E q ,... ) complicated energy transfer function ◮ Θ kpq phenomenological relaxation time of triad interactions (remains of Green’s function after Markovianization) Inertial range: Constant spectral energy flow ε towards small-scales (direct cascade) � � � d k d p d q Θ kpq T kpq ∼ Θ k k 4 E 2 ∂ t E = ε = k � − 1 ⇒ Quartic equation in E k With Θ k = τ − 1 NL + τ − 1 � A τ NL ≪ τ A ⇒ E k ∼ k − 5 / 3 � K41 Phenomenological dead-end τ A ≪ τ NL ⇒ E k ∼ k − 3 / 2 IK Matthaeus & Zhou, Phys.Fluids B, ’89

Inertial-Range Energetics EDQNM equation for residual energy spectrum, E R k = E M k − E K k : � � ( ∂ t + 2 Re − 1 k 2 ) E R △ d p d q Θ kpq R kpq k = Right-hand side complicated function with two types of contributions: ◮ Spectrally local interactions ( k ∼ p ∼ q ): – fluid scrambling on time scale τ NL ∼ ℓ ∼ ( k 3 E k ) − 1 / 2 √ (Dynamo effect) v 2 ℓ + b 2 ℓ – R Dyn ∼ Θ k k 3 E 2 k ◮ Spectrally non-local interactions (e.g. k ≪ p ∼ q ): – Alfvén-wave scattering on time scale τ A ∼ ( kB 0 ) − 1 ≃ ( k 2 E M ) − 1 / 2 (Alfvén effect) – R Alf ∼ Θ k k 2 E M E R k

Residual Energy Assuming equilibrium between — magnetic field amplification by field line streching (small-scale dynamo) — energy equipartition by Alfvén wave effect � 2 � τ A ⇒ E R E k ∼ kE 2 k ∼ τ NL k Anisotropic 1024 2 × 256 simulation, B 0 = 5 Isotropic 1024 3 simulation, B 0 = 0 K41: E k ∼ k − 5 / 3 ⇒ E R IK: E k ∼ k − 3 / 2 ⇒ E R k ∼ k − 7 / 3 k ∼ k − 2

Two-Dimensional Simulations (MHD) Left: Total energy spectrum × k 3 / 2 Right: Residual energy spectrum × k 2 2048 2 spectral MHD turbulence simulations Biskamp & Schwartz Chaos, Solitons & Fractals ’91

Energy Contours in Plane along B 0 Strong anisotropy visible. As opposed to isotropic simulation (nearly perfect circles). Cho & Vishniac ApJ, ’00

k ⊥ - k � Scaling Consequence of τ NL ∼ τ A (’critical balance’) Distortion of field line by eddy of size ℓ on time-scale τ NL triggers Alfvén wave of length λ ∼ b 0 τ A ⇒ k � ∼ k 2 / 3 Goldreich & Sridhar ApJ ’94, Galtier et al. ’05 ⊥

Spatial Structure of Dissipation (Hydrodynamics)

Measuring Structure ◮ Regard turbulent field difference over distance ℓ , δ v ℓ = [ v ( x ) − v ( x + ℓ )] · ˆ ℓ ℓ � ∼ ℓ ζ p display power-law scaling ◮ Statistical moments � δ v p ◮ Change of scaling exponents ζ p indicates deviation from self-similarity

Third-Order Structure Function 1.00 Slope 0.10 + 3 S 1.6 1.4 1.2 1.0 0.01 0.8 0.6 0.4 100 10 100 1000 L Hydrodynamics: S 3 = 4 5 ε ℓ Kolmogorov, ’41 MHD: ∑ 3 i = 1 � δ z ∓ ℓ ( δ i z ± 3 ε ± ℓ ℓ ) 2 � = − 4 Politano & Pouquet PRE & GRL ’98

Extended Self-Similarity (ESS) 1.0 Slope: 0.39 Slope: 0.72 1.0 S+ S+ 1 2 0.1 0.1 0.01 0.10 1.00 0.01 0.10 1.00 S+ S+ 3 3 Slope: 1.23 10.000 Slope: 1.42 1.00 1.000 S+ 4 0.10 S+ 5 0.100 0.010 0.01 0.001 0.01 0.10 1.00 0.01 0.10 1.00 S+ S+ 3 3 Observe extended scaling-range by plotting structure functions, S q ∼ ℓ ζ q , against reference structure function, S q 0 ∼ ℓ ζ q 0 : ⇒ S q ( S q 0 ) ∼ ℓ ζ q ζ q 0 ∼ ℓ ξ q ⇒ ζ q = ξ q / ζ q 0 Benzi et al. PRE ’93

Spatial Structure of Dissipation (MHD) Left: Dissipative current sheets in isotropic MHD turbulence Right: Same picture with strong mean magnetic field pointing upwards