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 φ0( α) P( x, t| α)

Diffusive turbulent transport

mean squared displacement: |∆ X(t)|2 , ∆ X(t) = X(t) − X(0) Taylor’s formula (1921) for large t:

• X(t) =

X(0) + t dτ V (τ) |∆ X(t)|2] = 2 t ∞ dτ V (τ) · V (0) = 2tD Lagrangian velocity correlation: CL(τ) = V (τ) · V (0) diffusion coefficient: D = ∞ dτ V (τ) · V (0)

MHD turbulence

The governing equations: ∂ u ∂t + ( u · ∇) u = −∇p + (∇ × B) × B + ν∇2 u + f ∂ B ∂t = ∇ × ( u × B) + η∇2 B ∇ · u = ∇ · B = 0

• f : random forcing at the largest scales

Evolution of passive tracer particles: d X(t) dt = V (X(t), t)

• X(0) =

α Field-guided MHD turbulence:

• B(

x, t) = B0ˆ z + b( x, t)

Previous work: the 2D case

• 1. transport suppressed in direction ⊥ to B0ˆ

y

Previous work: the 2D case

• 2. field-perpendicular transport is not diffusive
• 3. the system has long-term memory: slow decay of CL(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 = B0ˆ z

anisotropic: elongation in the along-field direction

Particle tracking

The hydrodynamic case, B = 0

−20 −10 10 −10 10 −10 10 20 30 40 50 y ν=5.00e−03 , η=5.00e−03 , B0z=0 , Lz=5 , nx=128 , ny=128 , nz=256 x z 400 450 500 550 −30 −20 −10 10 20 time x(t) − x0 400 450 500 550 −15 −10 −5 5 10 time y(t) − y0 400 450 500 550 −20 −10 10 20 30 time z(t) − z0

The field-guided case, B = B0ˆ z

−20 −10 10 −10 10 −10 10 20 30 40 50 y ν=5.00e−03 , η=5.00e−03 , B0z=5 , Lz=5 , nx=128 , ny=128 , nz=256 x z 200 250 300 −30 −20 −10 10 20 time x(t) − x0 200 250 300 −15 −10 −5 5 10 time y(t) − y0 200 250 300 −20 −10 10 20 30 time z(t) − z0

transport suppressed in the field-perpendicular direction!

Scaling of mean-squared displacement

50 100 150 50 100 150 200 250

<(∆x)2> <(∆y)2> <(∆z)2)>

hydrodynamic 50 100 150 50 100 150 200 250 field-guided 10

• 1

10 10

1

10

2

elapsed time, ∆t 10

• 2

10

• 1

10 10

1

10

2

10

• 1

10 10

1

10

2

elapsed time, ∆t 10

• 2

10

• 1

10 10

1

10

2

t2 t2 t t

ballistic limit: ∼ t2 at small time diffusive scaling: ∼ t at large time

Lagrangian velocity correlation function CL(τ) = V (τ) · V (0)

5 10 15 20 25 30

τ

0.1 0.2 0.3 0.4

CL,u CL,v CL,w hydrodynamic

5 10 15 20 25 30

τ

• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5

field-guided

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?