Particle diffusion in magnetohydrodynamic turbulence Yue-Kin Tsang - PowerPoint PPT Presentation

Particle diffusion in magnetohydrodynamic turbulence Yue-Kin Tsang Centre for Astrophysical and Geophysical Fluid Dynamics Mathematics, University of Exeter Joanne Mason

Single-particle diffusion transport properties in fusion experiments astrophysical pheonomena: cosmic ray propagation thermal conductivity in galaxy-cluster plasma mean scalar φ evolution: � � φ ( � x, t ) � = d � α � φ 0 ( � α ) � P ( � x, t | � α )

Diffusive turbulent transport 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 Lagrangian velocity correlation: C L ( τ ) = � � V ( τ ) · � V (0) � diffusion coefficient: � ∞ d τ � � V ( τ ) · � D = V (0) � 0

MHD turbulence The governing equations: ∂� u u = −∇ p + ( ∇ × � B ) × � u + � B + ν ∇ 2 � ∂t + ( � u · ∇ ) � f ∂ � 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 ) = � V ( X ( t ) , 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

Previous work: the 2D case 2. field-perpendicular transport is not diffusive 3. the system has long-term memory: slow decay of C L ( τ ) “ . . . it is unlikely that in three dimensions the turbulent diffusivity becomes suppressed . . . in three dimensions, motions that interchange field lines can bring together oppositely directed field lines without bending them. ”

The hydrodynamic case, � B = 0 system is homogeneous and isotropic

The field-guided case, � B = B 0 ˆ z anisotropic: elongation in the along-field direction

Particle tracking

The hydrodynamic case, � B = 0 ν =5.00e−03 , η =5.00e−03 , B0 z =0 , L z =5 , nx=128 , ny=128 , nz=256 20 50 10 x(t) − x 0 0 −10 40 −20 −30 400 450 500 550 30 time 10 20 5 z y(t) − y 0 0 10 −5 −10 −15 0 400 450 500 550 time 30 −10 20 z(t) − z 0 10 −20 0 −10 0 −10 10 10 0 −20 400 450 500 550 −10 time y x

The field-guided case, � B = B 0 ˆ z ν =5.00e−03 , η =5.00e−03 , B0 z =5 , L z =5 , nx=128 , ny=128 , nz=256 20 50 10 x(t) − x 0 0 −10 40 −20 −30 200 250 300 30 time 10 20 5 z y(t) − y 0 0 10 −5 −10 −15 0 200 250 300 time 30 −10 20 z(t) − z 0 10 −20 0 −10 0 −10 10 10 0 −20 200 250 300 −10 time y x transport suppressed in the field-perpendicular direction!

Scaling of mean-squared displacement hydrodynamic field-guided 250 250 <( ∆ x ) 2 > 200 200 <( ∆ y ) 2 > <( ∆ z ) 2 )> 150 150 100 100 50 50 0 0 0 100 0 100 50 150 50 150 t 2 2 10 10 t 1 1 10 10 0 0 10 10 t 2 t 2 -1 -1 10 10 -2 -2 10 10 -1 0 1 2 -1 0 1 2 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

Lagrangian velocity correlation function C L ( τ ) = � � V ( τ ) · � V (0) � hydrodynamic field-guided 0.5 0.4 0.4 C L,u C L,v 0.3 0.3 C L,w 0.2 0.2 0.1 0 0.1 -0.1 0 -0.2 0 5 10 15 20 25 30 0 5 10 15 20 25 30 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 suppression of turbulent transport in the field-perpendicular direction transport shows diffusive scaling at large time Is the mechanism of transport suppression the same or different in 2D and 3D?