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Effects of a guided-field on particle diffusion in - - PowerPoint PPT Presentation
Effects of a guided-field on particle diffusion in - - PowerPoint PPT Presentation
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 Particle transport in fluids Brownian motion observed
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Single-particle turbulent diffusion
mean squared displacement: |∆ X(t)|2 , ∆ X(t) = X(t) − X(0) Taylor’s formula (1920) 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 statistically homogeneous and stationary and the integral exists
Lagrangian velocity correlation: CL(τ) = V (τ) · V (0) diffusion coefficient: D = ∞ dτ V (τ) · V (0)
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Field-guided MHD turbulence + tracers
Electrically conducting fluid in a 3D periodic domain: ∂ 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, ∆t-correlated
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) : uniformly distributed over the domain
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Previous work: the 2D case
Turbulent transport (ηT ) suppressed when B0 > B∗
0 ∼ Rm−1 (Rm = UL/η)
(Vainshtein & Rosner 1991, Cattaneo & Vainshtein 1991, Cattaneo 1994, Gruzinov & Diamond 1994, Kondi´ c, Hughes & Tobias 2016)
Q : Does suppression of turbulent diffusion occur in 3D ?
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Eulerian fields
hydrodynamic
- u
B0 = 1
- u
- b
ν = η ∼ 10−3 Re = Rm ∼ 103 256 × 256 × 256
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Ratios of Eulerian r.m.s. velocities
1 2 3 0.9 0.95 1 1.05 1.1 1.15
A urms/wrms
dynamo 0.1 0.2 0.3 1.0 5.0 hydro
1 2 3 0.9 0.95 1 1.05 1.1 1.15
A vrms/wrms
A = forcing amplitude urms wrms ≈ vrms wrms ≈ 1
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Particle trajectories
hydrodynamic B0 = 1 transport becomes anisotropic when B0 = 0
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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
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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?
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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
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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
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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
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Particle trajectories B0 = 0.2, Urms = 1.42 Dx/Dz = 0.95
−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
B0 = 1.0, Urms = 0.29 Dx/Dz = 0.24
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
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Particle trajectories B0 = 0.2, Urms = 0.25 Dx/Dz = 0.34
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
B0 = 1.0, Urms = 1.39 Dx/Dz = 0.34
−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
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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
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Velocity decorrelation time
10
−1
10 10 10
1
10
2
u r m s τx
dynamo 0.1 0.2 1.0 5.0 hydro
10 10 10
1
10
2
10
3
u r m s/B0 τxB0
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Velocity decorrelation time
10
−1
10 10 10
1
10
2
u r m s τx
−1 −2
dynamo 0.1 0.2 1.0 5.0 hydro
10 10 10
1
10
2
10
3
u r m s/B0 τxB0
−1 −2
urms/B0 > 1 : τx ∼ (urms)−1 [τxB0 ∼ (urms/B0)−1] urms/B0 < 1 : τx ∼ B0(urms)−2 [τxB0 ∼ (urms/B0)−2]
- ngoing work: ensemble averaging to get better statistics
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A tentative physical picture . . .
wave induces memory into the system wave time scale: τA ∼ B−1 background turbulence removes memory turbulent decorrelation time: τu ∼ (Urms)−1 a competition between τA and τu anisotropic diffusion:
B0/Urms 1 τA τu
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A tentative physical picture . . .
Iroshnikov–Kraichnan picture of weak MHD turbulence
VA ∼ B0
Alfven wave speed
τA ∼ ℓ VA
wave packet interaction time
∆u τA ∼ u2 ℓ
distortion each interaction
u ∼ √ N∆u
distortion after N interactions
⇒ τcas ∼ NτA ∼ ℓB0 u2
cascade time
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