Ann. Henri Poincaré 4, Suppl. 1 (2003) S395 – S400
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/01S395-6
DOI 10.1007/s00023-003-0930-1
Annales Henri Poincaré
Passive Sliders on Growing Surface
and (Anti-)Advection in Burger’s Flows
Barbara Drossel and Mehran Kardar
1 Introduction
Passive scalar advection: Dust particles or dye or a hot fluid element entrained by
a flow, but having no effect on the flow.
Famous problem in turbulence theory. The advected scalar shows a complex
scaling behavior that stands in no simple relation to the scaling behavior of the
flow.
Passive sliders: Particles sliding on stochastically growing surface, e.g., domain walls, particles driven by gravity, . . .
Noisy Burger’s Equation
∂v
+ λ(v · ∇)v = ν∇2v + ∇ζh ;
∂t
v = −∇h
Compressible; vortex-free.
Advection of particles of density ρ:
∂ρ
= κ∇2 ρ − a∇(ρv ) + ζρ
∂t
ζh and ζρ : white noise, delta-correlated
Equivalent: KPZ equation for interface growth
∂h
λ
= ν∇2 h + (∇h)2 + ζh
∂t
2
Particles riding on the surface
ẋ = −a∇h + ζx
Scaling behavior
2
2chi
[h(x, t) − h(x , t )] ∼ |x − x |
g
t − t
z
|x − x |
z+χ=2
in d = 1: χ = 1/2 (roughness exponent), z = 3/2 (dynamical critical exponent).
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B. Drossel and M. Kardar
2χρ
ρ(x, t)ρ(x , t ) − ρ2 ∼ |x − x |
− 2g
Ann. Henri Poincaré
t − t
z
|x − x | ρ
Our goal: find χρ and zρ .
1. Renormalization group
2. Computer simulations in 1d
Renormalization group in 1-loop
Flow equations (with α = a/λ and k = κ/ν):
dα
dl
dk
dl
dDρ
dl
=
=
=
2(α − α2 )
α zρ − 3/2 +
(1 + k)2
2
α (3 − k)
k zρ − 2 +
k(1 + k)2
α2
Dρ zρ − 2χρ − 1 + 2
k(1 + k)
Fixed points:
• a/λ > 0: attractive fixed point α = k = 1: Here the equations are Galilei
invariant (set x = x − λ
t, t = t, h = h + x). χρ = 3/4.
• a/λ < 0: flow goes to α = k = 0, with α2 /k = 1/6, χρ = 2/3 for zρ = z = 3/2.
For zρ ∈ ]3/2, 2], nonzero fixed points exist. Particles cannot follow surface
fluctuations.
Computer simulations
• Choose a random surface site and add a particle if no overhangs are generated. → KPZ growth.
• Choose the color of the new particle with probability (1+K)/2 in accordance
with the majority of the neighbors → moving domain wall.
Vol. 4, 2003
Passive Sliders on Growing Surface and (Anti-)Advection
S397
Particle diffusion
6
mean square distance travelled
10
K=1, z=1.74
K=0.25, z=1.83
K=0.125, z=1.91
K=0.025, z=1.98
a < 0, z=1.5
4
10
2
10
0
10
10
100
1000
time
10000
100000
Nonuniversal exponent for a/λ < 0.
Particle positions
• a/λ > 0
Mean number of particles that are in the same bin as a given particle
divided by L
as measured
4
4
10
3
L=4096
L=2048
L=1024
L=512
L=256
L=128
2
3
10
1
2
10
0
10
1
10
2
10
binsize
3
10
0
10
10
1
10
binsize
2
3
10
Almost all particles at the same site. ρ is not a smoothly varying quantity.
χρ = 1/2.
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B. Drossel and M. Kardar
Ann. Henri Poincaré
• a/λ < 0
Mean number of particles that are in the same bin as a given particle
as measured
divided by L^0.3
L=4096
L=2048
L=1024
L=512
L=256
3
10
slope 0.7
2
10
0
10
1
10
10
binsize
2
34
0
10 10
1
10
2
10
binsize
10
3
10
4
Not all particles can follow the height fluctuations. χρ = 0.85. (For K < 1
different exponents → no universality.)
Particle velocities
Fix the height difference between the two ends of the system and measure the
mean velocity
• a/λ > 0, fixed L
Vol. 4, 2003
Passive Sliders on Growing Surface and (Anti-)Advection
S399
• a/λ < 0, fixed L
velocity satisfies scaling law
√
v = L−y ∇h C(∇h L)
with y 0.14
0.3
C(x)
L=128
L=256
L=512
L=1024
0.2
0.25
0.5
1
2
x
asymptotic behavior:
C(x) ∼ x2y for large x.
√
For sufficiently small ∇h (< 1/2 L):
v = a(L)∇(h)
with
a(L) ∼ L−y
and y 0.14. (RG: y = 0).
Relation to particle diffusion exponent zρ :
After time t: slope on length l ∼ t2/3 has not yet changed. Particle has moved
a distance
x ∼ a(l)l−1/2 t ∼ t2(1−y)/3
→ zρ = 3/2(1 − y)
For a/λ > 0: y = 0 and zρ = 3/2.
For a/λ < 0: y 0.14 and zρ 1.74.
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B. Drossel and M. Kardar
Ann. Henri Poincaré
Conclusions
• Even in one dimension nontrivial scaling behavior
• Fundamental difference between a/λ > 0 (Galilei-invariant case) and
a/λ < 0.
• There seems to be nonuniversal scaling for the case a/λ < 0.
Literature: B. Drossel and M. Kardar, Phys. Rev. B. 66, 195914 (2002).
Barbara Drossel
TU Darmstadt
and
Mehran Kardar
MIT