Energy spectrum of isotropic magnetohydrodynamic turbulence in the - PowerPoint PPT Presentation

01234 19/Jul/2006, Warwick Turbulence Symposium 56789 Energy spectrum of isotropic magnetohydrodynamic turbulence in the Lagrangian renormalized approximation Kyo Yoshida (Univ. Tsukuba) In collaboration with: Toshihico Arimitsu (Univ. Tsukuba) START: ⊲

01234 0 Abstract 56789 Quantitative estimates of the inertial-subrange statistics of MHD turbulence are given by using the Lagrangian renormalized approximation (LRA). The estimate of energy spectrum is verified by DNS of forced MHD turbulence. Outline of the talk 1 Introduction (Statistical theory of turbulence) 2 Lagrangian renormalized approximation (LRA) 3 LRA of MHD turbulence 4 Verification by DNS ⊲ ⊳ 2

01234 56789 1 Introduction (Statistical theory of turbulence) ⊲ ⊳ 3

01234 1.1 Governing equations of turbulence 56789 Navier-Stokes equations ( in real space ) ∂ u −∇ p + ν ∇ 2 u + f , ∂t + ( u · ∇ ) u = ∇ · u = 0 u ( x , t ) :velocity field, p ( x , t ) : pressure field, ν : viscosity, f ( x , t ) : force field. Navier-Stokes equations ( in wavevector space ) � ∂ � � u i d p d q δ ( k − p − q ) M iab k u a p u b q + f i ∂t + νk 2 k = k = − i = δ ij − k i k j M iab k a P ib k + k b P ia P ab � � , k 2 . k k k 2 Symbolically, � ∂ � ∂t + νL u = Muu + f ⊲ ⊳ 4

01234 1.2 Turbulence as a dynamical System 56789 Characteristics of turbulence as a dynamical system • Large number of degrees of freedom • Nonlinear ( modes are strongly interacting ) • Non-equilibrium ( forced and dissipative ) Statistical mechanics of thermal equilibrium states can not be applied to turbulence. • The law of equipartition do not hold. • Probability distribution of physical variables strongly deviates from Gaussian (Gibbs distribution). ⊲ ⊳ 5

01234 1.3 Statistical Theory of Turbulence 56789 cf. (for thermal equilibrium states) Statistical mechanics Thermodynamics Macroscopic variables are related to The macroscopic state is completely microscopic characteristics characterized by the free energy, (Hamiltonian). F ( T, V, N ) . F ( T, V, N ) = − kT log Z ( T, V, N ) Statistical theory of turbulence ? What are the set of variables that characterize the statistical state of How to relate statistical variables to turbulence? Navier-Stokes equations? • ǫ ? (Kolmogorov Theory ?) • Lagrangian Closures? • Fluctuation of ǫ ? (Multifractal models?) ⊲ ⊳ 6

01234 56789 2 Lagrangian renormalized approximation (LRA) ⊲ ⊳ 7

01234 2.1 Closure problem 56789 Symbolically, du dt = λMuu + νu λ := 1 is introduced for convenience. d dt � u � = λM � uu � + ν � u � , d dt � uu � = λM � uuu � + ν � uu � , · · · Equations for statistical quantities do not close. M � uuu � should be expressed in terms of known quantity. ⊲ ⊳ 8

01234 2.2 Solvable cases 56789 • Weak turbulence (Wave turbulence) du � d ˜ u � dt = λ ˜ u ( t ) := e − iLt u ( t ) dt = λMuu + iLu, M ˜ u ˜ u, ˜ The linear term iLu is dominant and the primitive λ -expansion may be justified in estimating λM � uuu � . • Randomly advected passive scalar (or vector) model du dt = λMvu + νu. ( v : advecting velocity field with given statistics) When the correlation time scale τ v of v tends to 0 , the leading order of the primitive λ -expansion of λM � vuu � becomes exact. (One can also obtain closed equations for higher moments.) ⊲ ⊳ 9

01234 2.3 Closure for Navier-Stokes turbulence 56789 Various closures are proposed for NS turbulence, but their mathematical foundations are not well established. • Quasi normal approximation λM � uuu � = λ 2 F [ Q ( t, t )] Q ( t, s ) := � u ( t ) u ( s ) � correlation function. – Inappropriate since the closed equation derives negative energy spectrum. • Direct interaction approximation (DIA) (Kraichnan, JFM 5 497(1959)) λM � uuu � = λ 2 F [ Q ( t, s ) , G ( t, s )] G ( t, s ) response function. – Derives an incorrect energy spectrum E ( k ) ∼ k − 3 / 2 . This is due to the inclusion of the sweeping effect of large eddies. ⊲ ⊳ 10

01234 2.4 Lagrangian closures 56789 • Abridged Lagrangian history direct interaction approximation (ALHDIA) (Kraichnan, Phys. Fluids 8 575 (1965)) • Lagrangian renormalized approximation (LRA) (Kaneda, JFM 107 131 (1981)) Key ideas of LRA 1. Lagrangian representatives Q L and G L . M � vvv � = F [ Q L , G L ] . • Representatives are different between ALHDIA and LRA. 2. Mapping by the use of Lagrangian position function ψ . 3. Renormalized expansion. ⊲ ⊳ 11

01234 2.5 Generalized velocity 56789 Generalized Velocity u ( x , s | t ) : velocity at time t of a fluid particle which passes x at u ( x , s | t ) u ( x , s | s ) time s . s : labeling time t : measuring time Lagrangian Position function ψ ( y , t ; x , s ) = δ (3) ( y − z ( x , s | t )) z ( x , s | t ) : position at time t of a fluid particle which passes x at time s . � d 3 y u ( y , t ) ψ ( y , t ; x , s ) u ( x , s | t ) = D ⊲ ⊳ 12

01234 2.6 Two-time two-point correlations 56789 Representative Q (or Q L ) ALHDIA � u ( x , t | t ) u ( y , s | s ) � (DIA) (labeling time) t ( t | s ) ( t | t ) � u ( x , t | t ) u ( y , t | s ) � (ALHDIA) DIA �P u ( x , s | t ) u ( y , s | s ) � (LRA) s ( s | s ) ( s | t ) P u : solenoidal component of u . LRA Similarly for G (or G L ). s t (measuring time) ⊲ ⊳ 13

01234 2.7 Derivation of LRA 56789 (i) Primitive λ -expansion λM � uuu � = λ 2 F (2) [ Q (0) , G (0) ] + λ 3 F (3) [ Q (0) , G (0) ] + O ( λ 4 ) , ∂ ∂tQ L ( x, t ; y, s ) = λ 2 I (2) [ Q (0) , G (0) ] + λ 3 I (3) [ Q (0) , G (0) ] + O ( λ 4 ) , ∂ ∂tG L ( x, t ; y, s ) = λ 2 J (2) [ Q (0) , G (0) ] + λ 3 J (3) [ Q (0) , G (0) ] + O ( λ 4 ) , (ii) Inverse expansion Q (0) = Q L + λ K (1) [ Q L , G L ] + O ( λ 2 ) , G (0) = G L + λ L (1) [ Q L , G L ] + O ( λ 2 ) (iii) Substitute (ii) into (i) (Renormalized expansion). λM � uuu � = λ 2 F (2) [ Q L , G L ] + O ( λ 3 ) , ∂ ∂tQ L ( x, t ; y, s ) = λ 2 I (2) [ Q L , G L ] + O ( λ 3 ) , ∂ ∂tG L ( x, t ; y, s ) = λ 2 J (2) [ Q L , G L ] + O ( λ 3 ) , (iv) Truncate r.h.s.’s at the leading orders. (One may expect that λM � uuu � depends on representatives gently when representatives are appropriately chosen.) ⊲ ⊳ 14

01234 2.8 Consequences of LRA (1) 56789 3D turbulence • Kolmogorov energy spectrum E ( k ) = K o ǫ 2 / 3 k − 5 / 3 , C K ≃ 1 . 72 . (Kaneda, Phys. Fluids 29 701 (1986)) 2D turbulence • Enstrophy cascade range  C K η 2 / 3 k − 3 [ln( k/k 1 )] − 1 / 3 , C K ≃ 1 . 81  E ( k ) = , C L k − 3 ( C L is not a universal constant)  depending on the large-scale flow condition. • Inverse energy cascade range E ( k ) = C E ǫ 2 / 3 k − 5 / 3 , C E ≃ 7 . 41 . (Kaneda, PF 30 2672 (1987), Kaneda and Ishihara, PF 13 1431 (2001)) ⊲ ⊳ 15

01234 56789 Tsuji (2002) ⊲ ⊳ 16

01234 2.9 Consequences of LRA (2) 56789 LRA is also applied to • Spectrum of passive scalar field advected by turbulence (3D / 2D) (Kaneda (1986), Kaneda (1987), Gotoh, J. Phys. Soc. Jpn. 58 , 2365 (1989)) . • Anisotropic modification of the velocity correlation spectrum due to homogeneous mean flow (Yoshida et al. , Phys. Fluids, 15 , 2385 (2003)) . Merits of LRA • Fluctuation-dissipation relation Q ∝ G holds formally. • The equations are simpler than ALHDIA. ⊲ ⊳ 17

01234 56789 3 LRA for MHD ⊲ ⊳ 18

01234 3.1 Magnetohydrodynamics (MHD) 56789 • Interaction between a conducting fluid and a magnetic field. • Geodynamo theory, solar phenomena, nuclear reactor, ... Equations of incompressible MHD ∂ t u i + u j ∂ j u i = B j ∂ j B i − ∂ i P + ν u ∂ j ∂ j u i , ∂ i u i = 0 , ∂ t B i + u j ∂ j B i = B j ∂ j u i + ν B ∂ j ∂ j B i , ∂ i B i = 0 , u ( x , t ) : velocity field B ( x , t ) : magnetic field ν u : kinematic viscosity ν B : magnetic diffusivity ⊲ ⊳ 19

Energy Spectrum: k − 3 / 2 or k − 5 / 3 or else? 01234 3.2 56789 • Iroshnikov(1964) and Kraichnan(1965) derived IK spectrum 1 1 0 k − 3 E u ( k ) = E B ( k ) = Aǫ 2 B 2 , 2 � 1 3 �| B | 2 � ǫ : total-energy dissipation rate, B 0 = based on a phenomenology which includes the effect of the Alfv´ en wave. • Other phenomenologies (local anisotropy), including weak turbulence picture. (Goldreich and Sridhar (1994–1997), Galtier et al. (2000), etc.) • Some results from direct numerical simulations (DNS) are in support of Kolmogorov-like k − 5 / 3 -scaling. (Biskamp and M¨ uller (2000), M¨ uller and Grappin (2005)) ⊲ ⊳ 20

01234 3.3 Closure analysis for MHD turbulence 56789 • Eddy-damped quasi-normal Markovian (EDQNM) approximation – Eddy-damping rate is so chosen to be consistent with the IK spectrum. – Incapable of quantitative estimate of nondimensional constant A . � – Analysis of turbulence with magnetic helicity V d xB · A or cross � V d xu · B . helicity (Pouquet et al. (1976), Grappin et al. (1982,1983)) • LRA – A preliminary analysis suggests that LRA derives IK spectrum. (Kaneda and Gotoh (1987)) – Present study ∗ Quantitative analysis including the estimate of A . ∗ Verification of the estimate by DNS. ⊲ ⊳ 21