Particle diffusion in magnetohydrodynamic turbulence: effects of a - PowerPoint PPT Presentation

Particle diffusion in magnetohydrodynamic turbulence: effects of a guiding magnetic field 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

MHD turbulence Motion of a electrically conducting fluid: ∂� u u = − 1 ∇ p + ( ∇ × � B ) × � u + � B + ν ∇ 2 � ∂t + ( � u · ∇ ) � f ρ 0

MHD turbulence 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

MHD turbulence 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 : random forcing at the largest scales

MHD turbulence 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 : random forcing at the largest scales Evolution of passive tracer particles: d � X ( t ) u ( � X ( t ) , t ) = � = � V ( t ) d t � X (0) = � α Field-guided MHD turbulence: � z + � B ( � x, t ) = B 0 ˆ b ( � x, t )

Previous work: the 2D case 1. transport suppressed in direction ⊥ to B 0 ˆ y when B 0 > B ∗ 0

Previous work: the 2D case 2. as Re m = UL/η increases, the critical B ∗ 0 decreases 3. the system has long-term memory: slow decay of C L ( τ ) Whether such suppression of turbulent diffusion occurs in 3D is not clear.

The hydrodynamic case, � B = 0

The field-guided case, � B = B 0 ˆ z

Particle tracking

The hydrodynamic case, � B = 0 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 = B 0 ˆ z 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

Summary study single-particle diffusion in 3D MHD turbulence strong field-guided case versus the hydrodynamics case transport shows diffusive scaling at large time suppression of turbulent diffusion transport in the field-perpendicular direction Check Re m dependence? What is the suppression mechanism in 3D? 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 30 x(t) − x 0 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 z(t) − z 0 0 −10 0 0 −5 −20 0 20 15 10 −20 10 5 10 5 10 −40 20 0 0 −5 200 300 400 500 −10 −40 30 time 200 300 400 500 −20 y time x y x