Theory and modelling of turbulent transport in astrophysical phenomena

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1 MHD 2017 Tokyo, 29 August 2017 Theory and modelling of turbulent transport in astrophysical phenomena Nobumitsu YOKOI Institute of Industrial Science (IIS), University of Tokyo In collaboration with Akira YOSHIZAWA (IIS Emeritus) Axel BRANDENBURG (CU, NORDITA) Mark MIESCH (HAO)

2 Topics I. Turbulent transport II. Turbulence modeling based on statistical theory III. Global flow generation IV. Stellar convection zone V. Summary

3 I. Turbulent transport

4 Equation of fluctuating velocity turbulence mean velocity interaction turbulence turbulence interaction Instability or wave approach Linear in and, each (Fourier) mode evolves independently Closure approach Homogeneous turbulence, no dependence on large-scale inhomogeneity

5 Nonlinear terms appear under the divergence operator Integrated over the volume of the system Fourier representations No net contribution only transfer Nonlinear term The dynamics of k mode is governed by its interaction with all other modes

6 Integral scale E(k,t) k -5/3 Kolmogorov microscale O(l -1 ) O(η -1 ) k Required grid points Walking Cars Airplanes Re NG Re NG Earth s outer core Solar convection zone Galaxies

7 Enhancement of transport Reynolds stress νt: eddy viscosity (turbulent viscosity) (Model) (Boussinesq, 1877) enhancing transport spatial and temporal dependence Turbulent Laminar

8 Transport coefficients An example: Turbulent eddy viscosity T - Parameters - Mixing length - Turbulence energy - Propagators - Transport equations T = T0 T = u` T = u 2 = K T = K = C µ K K " K = u 02 /2

9 Suppression of transport Turbulent swirling pipe flow z U z U z θ U θ r U θ r θ Large-scale structure again! Additional symmetry breakage Experimental studies (Kitoh, 1991; Steenbergen, 1995) upstream downstream circumferential velocity U θ /U M z/d = 4.3 (Re = 50,000) U θ /U M z/d = 7.7 (Re = 50,000) U θ /U M z/d = 11.5 (Re = 50,000) U θ /U M z/d = 21.3 (Re = 50,000) U θ /U M z/d = 28.1 (Re = 50,000) U θ /U M z/d = 34.9 (Re = 50,000) r /a r /a r /a r /a r /a r /a axial velocity U z /U M z/d = 4.3 (Re = 50,000) U z /U M z/d = 7.7 (Re = 50,000) U z /U M z/d = 11.5 (Re = 50,000) U z /U M z/d = 21.3 (Re = 50,000) U z /U M z/d = 28.1 (Re = 50,000) U z /U M z/d = 34.9 (Re = 50,000) r /a r /a r /a r /a r /a r /a

10 II. Turbulence modelling based on statistical theory

11 Vortex generation Vorticity cf., Biermann battery Mean vorticity vortexmotive force cf., Mean magnetic field electromotive force Reynolds stress

12 Modelling in dynamos Mean field differential rotation, Ω effect Inhomogeneous Homogeneous U U B Turbulence Ansatz

13 B U b cross helicity C R ij = R ij non di + N

14 Theoretical formulation Yoshizawa, 1984: mirror-symmetric case Yokoi & Yoshizawa, 1993: non-mirror-symmetric case DIA Multiple-scale analysis A closure theory (propagator renormalization) for homogeneous isotropic turbulence Introduction of two scales Fourier transform of the fast variables Scale-parameter expansion Introduction of the Green s function Statistical assumptions on the basic fields Fast and slowly varying fields Calculation of the statistical quantities using the DIA

15 (i) Introduction of two scales Fast and slow variables Slow variables X and T change only when x and t change much. Velocity-fluctuation equation where

16 (ii) Fourier transform of the fast variables The fluctuation fields are homogeneous with respect to the fast variables: (iii) Scale-parameter expansion Eliminating the pressure term, we have (iv) Introduction of the Green s function

17 1st-order field Green s function

18 Formal solution in terms of

19 (v) Statistical assumptions on the basic field Basic field: homogeneous isotropic but non-mirror-symmetric (vi) Calculation of the statistical quantities by DIA where mixing length Eddy viscosity Helicity-related coefficient helicity inhomogeneity is essential

20 Eddy viscosity + Helicity model Reynolds stress R u 0 u 0 = 2 3 T = C Yokoi & Yoshizawa (1993) Phys. Fluids A5, 464 = K/, @x 2 3 ( r) H Turbulence quantities K hu02 b H hu 0! b i, H a Helicity turbulence model Velocity profiles in swirl U z θ U θ z U z r U θ r θ

21 III. Global flow generation

22 DNS set-up z y z z +z 0 +z 0 0 x ω F 0 H dh dz z 0 z 0 Set-up of the turbulence and rotation ωf (left), the schematic spatial profile of the turbulent helicity H (= uʹ ωʹ ) (center) and its derivative dh/dz (right). Rotation Inhomogeneous turbulent helicity H(z) =H 0 sin( z/z 0 ) Run k f /k 1 Re Co /( T 2 ) A B B B C C C Summary of DNS results

23 Global flow generation Axial flow component U y on the periphery of the domain Turbulent helicity uʹ ωʹ (top) and mean-flow helicity U 2ωF (bottom)

24 Reynolds stress Early stage Developed stage Reynolds stress uʹyuʹz (top), helicity-effect term ( H) z 2ωF y (middle), and their correlation (bottom). Mean axial velocity U y (top), turbulent helicity multiplied by rotation 2ωFH (middle), and their correlation (bottom).

25 Spectra Run A Run B2

26 Physical origin Reynolds stress Vortexmotive force R ij hu 0i u 0j i V M hu 0! 0 i Ω * δu = τ δu δω δh δu + = τ δu δω + δω u δω + δh + δω = δu δu = τu Ω *

27 Reynolds stress evolution Local helical forcing (Inagaki, Yokoi & Hamba, submitted to Phys. Rev. Fluids)

28 Reynolds-stress budget R xy = + N GS ' Pxy GS + GS xy + GS xy + C GS xy ' 0 Production P GS xy = 2 3 B GS B GS Press. strain GS xy =2 p 0 s 0 xy Press. diff. GS hp0 u 0 xi Coriolis C GS xy =2R GS xz F x

29 IV. Stellar convection zone

30 Angular-momentum transport in the solar convection zone Angular momentum around the rotation axis Vector flux of angular momentum Miesch (2005) Liv. Rev. Sol. Phys Helicity effect

31 Helicity effect in the stellar convection zone Duarte, et al, (2016) MNRAS 456, 1708 Schematic helicity distribution H r/r r 2 H>0

32 Helicity effect in the Reynolds stress Helicity u! Azimuthal Vorticity Reynolds stress u 0r u 0 C `2 (r 2 H) Solar parameters v 200 m s 1 = cm s 1 ` 200 Mm = cm `/v 10 6 s u! u! ( H) u 0 u 0 (provided by Mark Miesch) Magnitude same as the Reynolds stress rφ component u 0r u `2 θφ component u 0 u `2 1 r with C = O(10 3 ) 10 8

33 V. Summary

34 Summary In turbulent momentum transport in hydrodynamics Mean velocity strain (symmetric part of velocity shear) + Energy Transport enhancement (structure destruction) Mean absolute vorticity (antisymmetric part of velocity shear) + (Inhomogeneous) Helicity Transport suppression (structure formation) N. Yokoi & A. Brandenburg, Phys. Rev. E 34, (2016) N. Yokoi, Geophys. Astrophys. Fluid Dyn. 107, 114 (2013) N. Yokoi & A. Yoshizawa, Phys. Fluids A 5, 464 (1993)

35 Magnetohydrodynamic Case : Mean velocity strain Only transport enhancement or structure destruction α dynamo : Mean magnetic strain Cross-helicity dynamo α and cross-helicity dynamo Energy Cross helicity Helicity Guide field Widmer, Büchner & Yokoi Phys. Plasmas, 23, (2016)

36 Helicity Thinkshop November 2017, Tokyo, Japan Institute of Industrial Science (IIS), Univ. of Tokyo and National Astronomical Observatory of Japan (NAOJ) S.O.C. Axel Brandenburg, Manolis Georgoulis, Kirill Kuzanyan, Raffaele Marino, Alexei Pevtsov, Takashi Sakurai, Dmitry Sokoloff, Nobumitsu Yokoi (Chair), Hongqi Zhang

37 References [1] Yokoi, N. & Yoshizawa, A. Statistical analysis of the effects of helicity in in homogeneous turbulence, Phys. Fluids A5, (1993). [2] Yokoi, N. & Brandenburg, A. Large-scale flow generation by inhomogeneous helicity Phys. Rev. E 93, (2016). [3] Yokoi, N. & Yoshizawa, A. Subgrid-scale model with structural effects incorporated through the helicity, in Progress in Turbulence VII, pp (2017). [4] Inagaki, H., Yokoi, N., & Hamba, F. Mechanism of mean flow generation in rotating turbulence through inhomogeneous helicity, submitted to Phys. Rev. Fluids [5] Yokoi, N. Cross helicity and related dynamo, Geophys. Atrophys. Fluid Dyn. 107, (2013).