Effects of a guided-field on particle diffusion in - PowerPoint PPT Presentation

Effects of a guided-field on particle diffusion in magnetohydrodynamic turbulence Yue-Kin Tsang Centre for Astrophysical and Geophysical Fluid Dynamics Mathematics, University of Exeter Joanne Mason

Particle transport in fluids Brownian motion observed under the microscope dispersion of pollutants in the atmosphere cosmic ray propagation through the interstellar medium tracing particle trajectories gives alternative view of the structure of the fluid flow — the Lagrangian viewpoint

Single-particle turbulent diffusion mean squared displacement: �| ∆ � ∆ � X ( t ) = � X ( t ) − � X ( t ) | 2 � , X (0) Taylor’s formula (1921) for large t : � t X ( t ) = � � d τ � X (0) + V ( τ ) 0 � ∞ �| ∆ � d τ � � V ( τ ) · � X ( t ) | 2 ] � = 2 t V (0) � = 2 tD 0 assume system is homogeneous and stationary and the integral exists Lagrangian velocity correlation: C L ( τ ) = � � V ( τ ) · � V (0) � diffusion coefficient: � ∞ d τ � � V ( τ ) · � D = V (0) � 0

Field-guided MHD turbulence + tracers Motion of a electrically conducting fluid: ∂� u u = − 1 ∇ p + ( ∇ × � B ) × � u + � B + ν ∇ 2 � ∂t + ( � u · ∇ ) � f ρ 0 ∂ � B B ) + η ∇ 2 � u × � ∂t = ∇ × ( � B u = ∇ · � ∇ · � B = 0 � f : isotropic random forcing at the largest scales Field-guided MHD turbulence: � z + � B ( � x, t ) = B 0 ˆ b ( � x, t ) Evolution of passive tracer particles: d � X ( t ) = � u ( � V ( t ) = � X ( t ) , t ) d t � X (0) = � α

Typical velocity and magnetic fields ( ν = η ∼ 10 − 3 ) hydrodynamic case B 0 = 1

The hydrodynamic case ν =1.25e−03 , η =1.25e−03 , B0 z =0 , L z =1 , nx=256 , ny=256 , nz=256 40 20 x(t) − x 0 0 30 −20 −40 20 200 300 400 500 time 30 10 20 z 10 y(t) − y 0 0 0 −10 −10 −20 −30 200 300 400 500 −20 time 40 20 −20 z(t) − z 0 −10 0 0 20 10 −20 10 20 0 −10 −40 30 200 300 400 500 −20 time x y

The field-guided case ( B 0 = 1) ν =5.00e−03 , η =5.00e−03 , B0 z =1 , L z =1 , nx=128 , ny=128 , nz=128 15 10 5 x(t) − x 0 30 0 −5 −10 20 −15 200 300 400 500 time 10 15 10 0 z y(t) − y 0 5 0 −10 −5 −10 200 300 400 500 time −20 40 20 −30 z(t) − z 0 0 −5 −20 0 15 10 5 5 10 −40 0 −5 200 300 400 500 time y x transport suppressed in the field-perpendicular direction!

Scaling of mean-squared displacement field-guided hydrodynamic 200 200 D x =0.24 D x =0.04 D y =0.25 D y =0.04 150 150 D z =0.25 D z =0.26 100 100 <( ∆ x ) 2 > <( ∆ y ) 2 > 50 50 <( ∆ z ) 2 )> 0 0 0 100 200 300 400 0 100 200 300 400 t t 2 2 10 10 0 0 10 10 -2 -2 t 2 10 10 t 2 -4 -4 10 10 -2 -1 0 1 2 -2 -1 0 1 2 10 10 10 10 10 10 10 10 10 10 elapsed time, t elapsed time, t ballistic limit: ∼ t 2 at small time diffusive scaling: ∼ t at large time, � (∆ x ) 2 � ∼ 2 D x t , etc

Lagrangian velocity correlation function C L ( τ ) = � � V ( τ ) · � V (0) � hydrodynamic field-guided 0.20 0.20 C L,u 0.15 0.15 C L,v C L,w 0.10 0.10 0.05 0.05 0.00 0.00 -0.05 -0.05 0 10 20 30 40 50 0 10 20 30 40 50 τ τ hydrodynamic: ∼ exp( − τ ) , short correlation time field-guided: oscillatory, long correlation time how things depend on the guided-field strength B 0 ?

Diffusivity at different (weak) B 0 � U rms 1.2 1.2 1.2 hydro 0.1 1 1 1 0.2 0.3 0.8 0.8 0.8 D y D x D z 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 U rms U rms U rms 1 1 0.8 0.8 D x /D z D y /D z 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 U rms U rms diffusion is reduced by B 0 , including the z -direction anisotropic suppression: D x , D y � D z strong U rms ( � B 0 ) reduces the anisotropy in D ’s

Diffusivity at different B 0 hydro 1.2 1.2 1.2 0.1 0.2 1 1 1 0.3 1.0 0.8 0.8 0.8 D y D x D z 5.0 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 U rms U rms U rms 1 1 0.8 0.8 D x /D z D y /D z 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 U rms U rms At strong guided-field strength, B 0 � U rms D x , D y are strong suppressed, anomalous behavior of D z D x /D z , D y /D z ≪ 1 for the values of U rms studied

Anisotropic turbulent diffusion 1 1 0.8 0.8 D x /D z D y /D z 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0.7 0.8 0.9 1 0.7 0.8 0.9 1 b rms /U rms b rms /U rms 1 1 0.8 0.8 D x /D z D y /D z 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 log( B 0 z /U rms ) log( B 0 z /U rms )

Particle trajectories amp=0.1 , ν =1.25e−03 , η =1.25e−03 , B0 z =0.2 , L z =1 , nx=256 , ny=256 , nz=256 amp=3 , ν =1.25e−03 , η =1.25e−03 , B0 z =0.2 , L z =1 , nx=256 , ny=256 , nz=256 25 15 20 10 15 B 0 = 0 . 2 , U rms = 0 . 25 B 0 = 0 . 2 , U rms = 1 . 42 5 10 z z 0 D x /D z = 0 . 34 D x /D z = 0 . 95 5 −5 −10 0 −15 −5 −15 −10 −5 0 0 5 15 5 10 5 10 5 10 0 0 15 −5 −5 20 −10 y y x x amp=0.1 , ν =1.25e−03 , η =1.25e−03 , B0 z =1 , L z =1 , nx=256 , ny=256 , nz=256 amp=3 , ν =1.25e−03 , η =1.25e−03 , B0 z =1 , L z =1 , nx=256 , ny=256 , nz=256 10 10 5 0 5 B 0 = 1 . 0 , U rms = 0 . 29 B 0 = 1 . 0 , U rms = 1 . 39 z −5 z 0 D x /D z = 0 . 24 D x /D z = 0 . 34 −10 −5 −15 −20 −10 15 −5 0 10 2 0 4 5 5 6 10 0 8 5 10 10 −5 0 15 −5 y y x x

Lagrangian velocity correlation B 0 z =0.2 , U rms =0.25 B 0 z =0.2 , U rms =1.42 0.025 0.8 C L,u 0.7 C L,v 0.02 0.6 C L,w 0.015 0.5 0.4 0.01 0.3 0.005 0.2 0.1 0 0 −0.005 −0.1 0 20 40 60 80 100 0 5 10 15 20 B 0 z =1.0 , U rms =0.29 B 0 z =1.0 , U rms =1.39 0.04 1.2 1 0.03 0.8 0.02 0.6 0.01 0.4 0 0.2 −0.01 0 −0.02 −0.2 0 20 40 60 80 100 0 5 10 15 20 elapsed time elapsed time

Velocity decorrelation time 3 3 3 10 10 10 −1 −1 −1 τ x /B 0 τ y /B 0 τ z /B 0 2 2 2 10 10 10 dynamo 0.1 0.2 0.3 1 1 1 −2 −2 −2 10 10 10 1.0 5.0 hydro −1 0 −1 0 −1 0 10 10 10 10 10 10 U r m s U r m s U r m s 3 3 3 10 10 10 −1 −1 −1 τ x /B 0 τ y /B 0 τ z /B 0 2 2 2 10 10 10 dynamo 0.1 0.2 0.3 1 1 1 10 −2 10 −2 10 −2 1.0 5.0 hydro −1 0 −1 0 −1 0 10 10 10 10 10 10 u r m s v r m s w r m s

A physical picture . . . wave induces memory into the system wave frequency: τ − 1 A ∼ B 0 background turbulence removes memory decorrelation time: τ u a competition between τ A and τ u anisotropic diffusion: b rms /U rms ≈ 1 τ A ≪ τ u E u ( k ) ≈ E b ( k ) quantitative theory in development

Summary study single-particle diffusion in 3D MHD turbulence transport mostly shows diffusive scaling at large time anisotropic suppression of turbulent diffusion by a guided-field ( D x , D y � D z ) competition between waves and background turbulence ν =5.00e−03 , η =5.00e−03 , B0 z =1 , L z =1 , nx=128 , ny=128 , nz=128 ν =1.25e−03 , η =1.25e−03 , B0 z =0 , L z =1 , nx=256 , ny=256 , nz=256 15 40 10 5 20 x(t) − x 0 30 0 x(t) − x 0 0 −5 30 −10 20 −20 −15 200 300 400 500 −40 20 time 200 300 400 500 time 10 15 30 10 10 20 0 z z y(t) − y 0 5 10 y(t) − y 0 0 0 0 −10 −10 −5 −10 −20 −10 200 300 400 500 −30 time 200 300 400 500 −20 time −20 40 40 20 −30 20 z(t) − z 0 −20 0 z(t) − z 0 −10 0 0 −5 −20 0 20 15 10 −20 10 5 10 5 10 0 −40 20 0 −5 200 300 400 500 −10 −40 30 time 200 300 400 500 −20 y time x x y