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Lagrangian motion in stochastic flows
Leonid I. Piterbarg
Department of Mathematics, University of Southern California
1
Outline
• Coalescence in 1D Kraichnan
·x= u(t, x)
u(t, x) = Gaussian white noise
• Clustering and disordering in 1D and 2D Kramers (inertial particles)
·r= v·v= −v/τ + f(t, r)
f(t, r) = random force
2
Passive scalar in 1D Kraichnan (LP and V.V.Piterbarg, 2000)
∂c
∂t+ u
(
t,x
ε
)
∂c
∂x= 0, c0(x) = x 〈u(s, y)u(s + t, y + x)〉 = B(x)δ(t)
ε → 0
−10 −5 0 5 10−10
−8
−6
−4
−2
0
2
4
6
8
10
x
co(x)
c(t,x)
−4 −3 −2 −1 0 1 2 3 40
0.2
0.4
0.6
0.8
1
Tracer Correlation Function
x
• Fokker-Planck: 〈c(t, x)〉 = x
• Statistics of jumps: zn = intervals between jumps, hn = magnitudes of jumps
〈zn〉 = 〈hn〉 =√
πDt, D = B(0)
zn and hn are statistically equivalent
pz(x) =x
2Dtexp
− x2
4Dt
zn are not independent
3
Coalescence in 1D Kraichnan
• Definition
Two paths coalesce if X(s, x) = X(s, y) implies X(t, x) = X(t, y) for all t > s
• Proposition
For a smooth spatial covariance B(x), Xε converges to the Brownian coalescing flow as
ε → 0, where·
Xε= u(
t, 1
εXε
)
0 0.05 0.1 0.15 0.2 0.250
2
4
6
8
10
12
t
x
0 2 4 6 8 10−40
−20
0
20
40
60
80
100
120
140
t
X
• Coalescence in a non-smooth flow (Harris, 1984)
1D Kraichnan flow is coalescing if B(0) − B(x) ∼= xα, α < 2
• Aggregating (Deutsch,1985)
Any 1D Kraichnan flow is aggregating, i.e. |X(t, x) − X(t, y)| → 0 in probability ast → ∞ (Lyapunov exponent is negative)
4
Stochastic Flow in 1D (Kramers model, 1940)
• Equations for inertial particle position and velocity
·r= v·v= −v/τ + f(t, r)
f(t, r) is a homogeneous random force
τ = Lagrangian correlation time
• Lyapunov exponent equation
λ = limT→∞
1
T
∫ T
0σ(t)dt,
σ=separation in v/separation in r = particle velocity gradient
·σ= −σ/τ − σ2 + ξ(t)
ξ(t) = fr(t, 0)
• Terminology
λ > 0 = Disordering
λ < 0 = Clustering
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Gaussian white noise forcing
Lyapunov exponent (LP,2007, Mehlig and Wilkinson, 2003)
λ = −1 + St2/3M ′(St−4/3)/M(St−4/3)
whereM(z) =
√
Ai(z)2 + Bi(z)2,
Ai, Bi Airy functions, St = Stokes number
Realization of σ and dependence of LE on St
0 50 100 150 200 250 300 350 400−2000
−1000
0
1000
2000
3000
4000
5000
t0.1 1 10
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
ε
LE
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Telegraph noise (Falkovich, Musaccio, LP, Vucelja, 2007)
• Non-explosive case, St < 1/4
Stationary pdf of σ is given by
p(σ) = C(w − σ)m−1(σ − w)m−1
(w + σ)m+1(σ + w)m+1, σ ∈ (w, w)
and zero otherwise, where
w =(√
1 + 4St − 1)
/2, w =(√
1 − 4St − 1)
/2, m = Ku/(4w+2), m = Ku/(4w+2)
−6 −4 −2 0 2 4 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1pdf, large noise, h=1, ε =2
x−0.3 −0.2 −0.1 0 0.10
1
2
3
4
5
6h=2h=4h=8
• Lyapunov exponent
λ = w+(w − w)mF1(m + 1, m + 1, m + 1, m + m + 1, (w − w)/(w + w + 1), (w − w)/(2w + 1))
(m + m)F1(m, m + 1, m + 1, m + m, (w − w)/(w + w + 1), (w − w)/(2w + 1))< 0
7
• Explosive case, St ≥ 1/4
Stationary pdf is expressible in explicit form and satisfies
p(σ) ∼ σ−2, |σ| → ∞
• Realization of σ and pdf’s
-3
-2
-1
0
1
2
3
0 10 20 30 40 50
θ(t)
t/τ
10-4
10-3
10-2
10-1
1
101
102
-4 -3 -2 -1 0 1 2 3 4
s P(
σ)
σ/s
h=0.1h=1
h=10
• Limit St → 1/4 + 0
Core of pdf = O(1)
Tails of pdf = o(St − 1/4)
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• Dependence of LE on Ku and St
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.1 1 10 100
LE
h
ε=2ε=3ε=4
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.1 1 10 100
λ/s
ε
h=0.1h=1
h=10
-0.5
-0.25
0
10.50
• Phase transition in lg Ku and lg St
CLUSTERING
DISORDERING
UNACCESSIBLE
lg(Ku)
lg(St)
−2 −1.5 −1 −0.5 0 0.5 1 1.5−2
−1.5
−1
−0.5
0
0.5
DISODERING
CLUSTERING
UNACCESSIBLE
lg(Ku)
lg(St)
−2 −1.5 −1 −0.5 0 0.5 1 1.5−2
−1.5
−1
−0.5
0
0.5
DISODERING
CLUSTERING
UNACCESSIBLE
lg(Ku)
lg(St)
−2 −1.5 −1 −0.5 0 0.5 1 1.5−2
−1.5
−1
−0.5
0
0.5
Inertia controls transition from
clustering regime to chaotic regime
always leads to disordering
Increasing amplitude of forcing
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2D, Back to Gaussian noise (LP,2002)
• LE equation
dz = −(z + z2)dt + dw, λ = Re limT→∞
1
T
∫ T
0z(t)dt
wherew =
√2St(w1 + iΓw2), 1/3 ≤ Γ ≤ 3
x
y
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
−8 −6 −4 −2 0 2 4 6 8−15
−10
−5
0
5
x
y
PotentialSolenoidal
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
St
Γ=3 (Monte Carlo)
Γ=∞ (Asymptotic)
• LE asymptotic (Γ → ∞)
λ = −1 + St2/3Ai′(St−4/3)/Ai(St−4/3) > 0
10
Dispersion in 2D Kramers(LP,2005, 2008)
• Absolute (left) and relative (right) dispersion for different Hurst exponents α
10−1
100
101
102
10−1
100
101
102
103
104
105
time
t
10−1
100
101
102
10−2
100
102
104
106
time
t3
• Smooth space covariance (α = 2)
Three regimes in relative dispersion:
Ballistic: t2 or t3 depending on initial conditions
Exponential: exp (Λt) , Λτ(Λτ + 1)(Λτ + 2) = St
Inertial: t
12
Dispersion in 2D Kramers, cont’d
• Kramers with linear drift
·r= Gr + v·v= −v/τ+
·w (t, r)
G is constant matrix
• Existence of inertial regime for absolute dispersion
Inertial regime exists if and only if:
1. Drift is solenoidal and elliptic
2. Eigenvalues of G + 1/τ have positive real parts
3. τ <(
(g12 + g21)2/4 − g11g22
)−1/2
• Dispersion anisotropy
Absolute dispersion ellipse is the same as that of the drift.
The shape of relative dispersion ellipse depends on both, drift and normal covarianceparameters in fluctuations
• Diffusivity in a gyre
D =8σ2
uτ
1 + 4Ω2τ 2
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Concluding remarks
• 1D Kraichnan with infinitely small correlation radius seems to remain the onlyexactly solvable coalescence model (LP and VP, 2000)
• Coalescence is comprehensively studied for non-smooth multi dimensional Kraich-
nan (Gawedzki and Vergassola 2000, Le Jan and Raimond, 2005, Gabrielli andCecconi, 2008,...)
• No coalescence is proven in Kramers (inertial particles)
• Clustering, ordering, dispersion and much more are comprehensively studied for
inertial particles (Bec, Elperin, Falkovich, Horvai, LP, Wilkinson,..., 2001-2009 )
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