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
Top Related