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

Particle transport in fluids Brownian motion observed under the microscope dispersion of pollutants in the atmosphere cosmic ray propagation through the interstellar medium We consider passive tracer particles only —- the Lagrangian viewpoint provides an alternative view of the flow structure

Single-particle turbulent diffusion mean squared displacement: �| ∆ � ∆ � X ( t ) = � X ( t ) − � X ( t ) | 2 � , X (0) Taylor’s formula (1920) 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 statistically 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 Electrically conducting fluid in a 3D periodic domain: ∂� 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, ∆ t -correlated 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) : uniformly distributed over the domain

Previous work: the 2D case 0 ∼ Rm − 1 ( Rm = UL/η ) Turbulent transport ( η T ) suppressed when B 0 > B ∗ (Vainshtein & Rosner 1991, Cattaneo & Vainshtein 1991, Cattaneo 1994, Gruzinov & Diamond 1994, Kondi´ c, Hughes & Tobias 2016) Q : Does suppression of turbulent diffusion occur in 3D ?

Eulerian fields hydrodynamic ν = η ∼ 10 − 3 � u Re = Rm ∼ 10 3 256 × 256 × 256 B 0 = 1 � u � b

Ratios of Eulerian r.m.s. velocities dynamo 0.1 1.15 1.15 0.2 0.3 1.0 1.1 1.1 u rms /w rms v rms /w rms 5.0 hydro 1.05 1.05 1 1 0.95 0.95 0.9 0.9 0 1 2 3 0 1 2 3 A A A = forcing amplitude u rms ≈ v rms ≈ 1 w rms w rms

Particle trajectories B 0 = 1 hydrodynamic transport becomes anisotropic when B 0 � = 0

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 B 0 = 1 . 0 , U rms = 0 . 29 B 0 = 0 . 2 , U rms = 1 . 42 D x /D z = 0 . 24 D x /D z = 0 . 95 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 =0.2 , L z =1 , nx=256 , ny=256 , nz=256 15 10 10 5 5 z 0 z −5 0 −10 −5 −15 −15 −10 −10 −5 15 0 0 10 2 5 15 4 5 10 10 6 5 0 8 0 15 10 −5 −5 20 −10 y y x x

Particle trajectories B 0 = 0 . 2 , U rms = 0 . 25 B 0 = 1 . 0 , U rms = 1 . 39 D x /D z = 0 . 34 D x /D z = 0 . 34 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 =1 , L z =1 , nx=256 , ny=256 , nz=256 25 10 20 5 15 0 10 z z −5 5 −10 0 −15 −5 −20 −5 0 0 5 5 10 5 5 10 10 0 0 −5 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 10 2 2 10 10 τ x B 0 τ x 1 10 1 10 dynamo 0.1 0.2 0 10 1.0 5.0 hydro 0 10 −1 0 0 10 10 10 u r m s u r m s /B 0

Velocity decorrelation time 3 10 −2 2 −2 2 10 10 τ x B 0 τ x 1 10 1 10 dynamo −1 0.1 0.2 0 10 1.0 −1 5.0 hydro 0 10 −1 0 0 10 10 10 u r m s u r m s /B 0 u rms /B 0 > 1 : τ x ∼ ( u rms ) − 1 [ τ x B 0 ∼ ( u rms /B 0 ) − 1 ] u rms /B 0 < 1 : τ x ∼ B 0 ( u rms ) − 2 [ τ x B 0 ∼ ( u rms /B 0 ) − 2 ] ongoing work: ensemble averaging to get better statistics

A tentative physical picture . . . wave induces memory into the system wave time scale: τ A ∼ B − 1 0 background turbulence removes memory turbulent decorrelation time: τ u ∼ ( U rms ) − 1 a competition between τ A and τ u anisotropic diffusion: B 0 /U rms � 1 τ A � τ u

A tentative physical picture . . . Iroshnikov–Kraichnan picture of weak MHD turbulence V A ∼ B 0 Alfven wave speed τ A ∼ ℓ wave packet interaction time V A ∼ u 2 ∆ u distortion each interaction τ A ℓ √ u ∼ N ∆ u distortion after N interactions τ cas ∼ Nτ A ∼ ℓB 0 ⇒ cascade time u 2

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