[PPT] - Effects of a guided-field on particle diffusion in PowerPoint Presentation

SLIDE 1

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 Joanne Mason

SLIDE 2

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

SLIDE 3

Single-particle turbulent diffusion

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

assume system is homogeneous and stationary and the integral exists

Lagrangian velocity correlation: CL(τ) = V (τ) · V (0) diffusion coefficient: D = ∞ dτ V (τ) · V (0)

SLIDE 4

Field-guided MHD turbulence + tracers

Motion of a electrically conducting fluid: ∂ u ∂t + ( u · ∇) u = − 1 ρ0 ∇p + (∇ × B) × B + ν∇2 u + f ∂ B ∂t = ∇ × ( u × B) + η∇2 B ∇ · u = ∇ · B = 0

f : isotropic random forcing at the largest scales

Field-guided MHD turbulence:

B(

x, t) = B0ˆ z + b( x, t) Evolution of passive tracer particles: d X(t) dt = V (t) = u( X(t), t)

X(0) =

α

SLIDE 5

Typical velocity and magnetic fields

hydrodynamic case (ν = η ∼ 10−3) B0 = 1

SLIDE 6

The hydrodynamic case

−20 −10 10 20 30 −20 −10 10 20 −20 −10 10 20 30 y ν=1.25e−03 , η=1.25e−03 , B0z=0 , Lz=1 , nx=256 , ny=256 , nz=256 x z 200 300 400 500 −40 −20 20 40 time x(t) − x0 200 300 400 500 −30 −20 −10 10 20 30 time y(t) − y0 200 300 400 500 −40 −20 20 40 time z(t) − z0

SLIDE 7

The field-guided case (B0 = 1)

−5 5 10 −5 5 10 15 −30 −20 −10 10 20 30 y ν=5.00e−03 , η=5.00e−03 , B0z=1 , Lz=1 , nx=128 , ny=128 , nz=128 x z 200 300 400 500 −15 −10 −5 5 10 15 time x(t) − x0 200 300 400 500 −10 −5 5 10 15 time y(t) − y0 200 300 400 500 −40 −20 20 40 time z(t) − z0

transport suppressed in the field-perpendicular direction!

SLIDE 8

Scaling of mean-squared displacement

100 200 300 400 50 100 150 200

<(∆x)2> <(∆y)2> <(∆z)2)>

hydrodynamic 100 200 300 400 50 100 150 200 field-guided 10

2

10

1

10 10

1

10

2

elapsed time, t 10

4

10

2

10 10

2

10

2

10

1

10 10

1

10

2

elapsed time, t 10

4

10

2

10 10

2

t2 t2 t t

Dx=0.24 Dy=0.25 Dz=0.25 Dx=0.04 Dy=0.04 Dz=0.26

ballistic limit: ∼ t2 at small time diffusive scaling: ∼ t at large time, (∆x)2 ∼ 2Dxt , etc

SLIDE 9

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

10 20 30 40 50

τ

0.05

0.00 0.05 0.10 0.15 0.20

CL,u CL,v CL,w hydrodynamic

10 20 30 40 50

τ

0.05

0.00 0.05 0.10 0.15 0.20

field-guided

hydrodynamic: ∼ exp(−τ), short correlation time field-guided: oscillatory, long correlation time how things depend on the guided-field strength B0?

SLIDE 10

Diffusivity at different (weak) B0 Urms

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 1.2

Urms Dx

hydro 0.1 0.2 0.3 0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 1.2

Urms Dy

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 1.2

Urms Dz

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1

Urms Dx/Dz

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1

Urms Dy/Dz

diffusion is reduced by B0, including the z-direction anisotropic suppression: Dx, Dy Dz strong Urms( B0) reduces the anisotropy in D’s

SLIDE 11

Diffusivity at different B0

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 1.2

Urms Dx

hydro 0.1 0.2 0.3 1.0 5.0 0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 1.2

Urms Dy

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 1.2

Urms Dz

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1

Urms Dx/Dz

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1

Urms Dy/Dz

At strong guided-field strength, B0 Urms Dx, Dy are strong suppressed, anomalous behavior of Dz Dx/Dz , Dy/Dz ≪ 1 for the values of Urms studied

SLIDE 12

Anisotropic turbulent diffusion

0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1

brms /Urms Dx/Dz

0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1

brms /Urms Dy/Dz

−1.5 −1 −0.5 0.5 1 0.2 0.4 0.6 0.8 1

log(B 0z /Urms) Dx/Dz

−1.5 −1 −0.5 0.5 1 0.2 0.4 0.6 0.8 1

log(B 0z /Urms) Dy/Dz

SLIDE 13

Particle trajectories

5 10 −5 5 −5 5 10 15 20 25 y amp=0.1 , ν=1.25e−03 , η=1.25e−03 , B0z=0.2 , Lz=1 , nx=256 , ny=256 , nz=256 x z −15 −10 −5 5 10 15 20 −10 −5 5 10 15 −15 −10 −5 5 10 15 y amp=3 , ν=1.25e−03 , η=1.25e−03 , B0z=0.2 , Lz=1 , nx=256 , ny=256 , nz=256 x z 2 4 6 8 10 −5 5 10 15 −10 −5 5 10 y amp=0.1 , ν=1.25e−03 , η=1.25e−03 , B0z=1 , Lz=1 , nx=256 , ny=256 , nz=256 x z −5 5 10 15 −5 5 10 −20 −15 −10 −5 5 10 y amp=3 , ν=1.25e−03 , η=1.25e−03 , B0z=1 , Lz=1 , nx=256 , ny=256 , nz=256 x z

B0 = 0.2, Urms = 0.25 Dx/Dz = 0.34 B0 = 0.2, Urms = 1.42 Dx/Dz = 0.95 B0 = 1.0, Urms = 0.29 Dx/Dz = 0.24 B0 = 1.0, Urms = 1.39 Dx/Dz = 0.34

SLIDE 14

Lagrangian velocity correlation

20 40 60 80 100 −0.005 0.005 0.01 0.015 0.02 0.025

B0z =0.2 , Urms =0.25 CL,u CL,v CL,w

5 10 15 20 −0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

B0z =0.2 , Urms =1.42

20 40 60 80 100 −0.02 −0.01 0.01 0.02 0.03 0.04

B0z =1.0 , Urms =0.29 elapsed time

5 10 15 20 −0.2 0.2 0.4 0.6 0.8 1 1.2

B0z =1.0 , Urms =1.39 elapsed time

SLIDE 15

Velocity decorrelation time

10

−1

10 10

1

10

2

10

3

U r m s τx/B0

−1 −2

dynamo 0.1 0.2 0.3 1.0 5.0 hydro

10

−1

10 10

1

10

2

10

3

U r m s τy/B0

−1 −2 10

−1

10 10

1

10

2

10

3

U r m s τz/B0

−1 −2 10

−1

10 10

1

10

2

10

3

u r m s τx/B0

−1 −2

dynamo 0.1 0.2 0.3 1.0 5.0 hydro

10

−1

10 10

1

10

2

10

3

v r m s τy/B0

−1 −2 10

−1

10 10

1

10

2

10

3

wr m s τz/B0

−1 −2

SLIDE 16

A physical picture . . .

wave induces memory into the system wave frequency: τ −1

A ∼ B0

background turbulence removes memory decorrelation time: τu a competition between τA and τu anisotropic diffusion:

brms/Urms ≈ 1 τA ≪ τu Eu(k) ≈ Eb(k)

quantitative theory in development

SLIDE 17

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 (Dx , Dy Dz) competition between waves and background turbulence

−20 −10 10 20 30 −20 −10 10 20 −20 −10 10 20 30 y ν=1.25e−03 , η=1.25e−03 , B0z=0 , Lz=1 , nx=256 , ny=256 , nz=256 x z 200 300 400 500 −40 −20 20 40 time x(t) − x0 200 300 400 500 −30 −20 −10 10 20 30 time y(t) − y0 200 300 400 500 −40 −20 20 40 time z(t) − z0 −5 5 10 −5 5 10 15 −30 −20 −10 10 20 30 y ν=5.00e−03 , η=5.00e−03 , B0z=1 , Lz=1 , nx=128 , ny=128 , nz=128 x z 200 300 400 500 −15 −10 −5 5 10 15 time x(t) − x0 200 300 400 500 −10 −5 5 10 15 time y(t) − y0 200 300 400 500 −40 −20 20 40 time z(t) − z0