Particle diffusion in magnetohydrodynamic turbulence

Yue-Kin Tsang

Centre for Astrophysical and Geophysical Fluid DynamicsMathematics, 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

0

dτ ~V (τ)

〈|∆ ~X(t)|2]〉 = 2 t

∫∞

0

dτ 〈~V (τ) · ~V (0)〉 = 2tD

Lagrangian velocity correlation:

CL(τ) = 〈~V (τ) · ~V (0)〉

diffusion coefficient:

D =

∫∞

0

dτ 〈~V (τ) · ~V (0)〉

MHD turbulenceThe 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) = B0ẑ +~b(~x, t)

Previous work: the 2D case

1. transport suppressed in direction ⊥ to B0ŷ

Previous work: the 2D case2. 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 = B0ẑ

anisotropic: elongation in the along-field direction

Particle tracking

The hydrodynamic case, ~B = 0

−20

−10

0

10

−100

10

−10

0

10

20

30

40

50

y

ν=5.00e−03 , η=5.00e−03 , B0z=0 , L

z=5 , nx=128 , ny=128 , nz=256

x

z

400 450 500 550−30

−20

−10

0

10

20

time

x(t)

− x

0

400 450 500 550−15

−10

−5

0

5

10

time

y(t)

− y

0

400 450 500 550−20

−10

0

10

20

30

time

z(t)

− z

0

The field-guided case, ~B = B0ẑ

−20

−10

0

10

−100

10

−10

0

10

20

30

40

50

y

ν=5.00e−03 , η=5.00e−03 , B0z=5 , L

z=5 , nx=128 , ny=128 , nz=256

x

z

200 250 300−30

−20

−10

0

10

20

time

x(t)

− x

0

200 250 300−15

−10

−5

0

5

10

time

y(t)

− y

0

200 250 300−20

−10

0

10

20

30

time

z(t)

− z

0

transport suppressed in the field-perpendicular direction!

Scaling of mean-squared displacement

0 50 100 1500

50

100

150

200

250

hydrodynamic

0 50 100 1500

50

100

150

200

250field-guided

10-1

100

101

102

elapsed time, ∆t

10-2

10-1

100

101

102

10-1

100

101

102

elapsed time, ∆t

10-2

10-1

100

101

102

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)〉

0 5 10 15 20 25 30τ0

0.1

0.2

0.3

0.4

CL,uCL,vCL,w

hydrodynamic

0 5 10 15 20 25 30τ-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5field-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?

Single-particle diffusionDiffusive turbulent transportMHD turbulencePrevious work: the 2D casePrevious work: the 2D caseThe hydrodynamic case, $vec B=0$The field-guided case, $vec B=B_0hat z$Particle trackingThe hydrodynamic case, $vec B=0$The field-guided case, $vec B=B_0hat z$Scaling of mean-squared displacementLagrangian velocity correlation functionSummary