Weak Isotropic three-wave turbulence, Fondation des Treilles, July 16 2010

Intro: 3-Wave Kinetic Equation
Part 1: Developing Wave Turbulence
Part 2: Cascades Without Backscatter
Part 3: Bottlenecks and Thermalisation

Weak isotropic three-wave turbulence

Colm Connaughton

Centre for Complexity Science and Mathematics Institute
University of Warwick.

“Wave Turbulence”, Fondation des Treilles, Jul 16 2010

Colm Connaughton 3 Wave Kinetic Equation


Outline

1 Intro: 3-Wave Kinetic Equation

2 Part 1: Developing Wave Turbulence

3 Part 2: Cascades Without Backscatter

4 Part 3: Bottlenecks and Thermalisation



The 3-wave kinetic equation
Evolution of wave spectrum, nk , given by:
∂nk1 2
= π Vk1 k2 k3 (nk2 nk3 − nk1 nk2 − nk1 nk3 )
∂t
δ(ωk1 − ωk2 − ωk3 ) δ(k1 − k2 − k3 ) dk2 dk3
2
+π Vk2 k1 k3 (nk2 nk3 + nk1 nk2 − nk1 nk3 )
2
+π Vk3 k1 k2 (nk2 nk3 − nk1 nk2 + nk1 nk3 )
Scaling parameters : Dimension, d: k ∈ Rd
(d, α, γ) Dispersion, α: ωk ∼ k α
Nonlinearity, γ: Vk1 k2 k3 ∼ k γ


Isotropic case: frequency space representation
We consider only isotropic systems: all functions of k depend
only on k = |k|.
Dispersion relation, ωk = ω(k ), can be used to pass to a
frequency-space description where the kinetic equation is:
∂Nω
= S1 [Nω ] + S2 [Nω ] + S3 [Nω ].
∂t
d−α
Nω = Ωd ω α nω is the frequency spectrum.
α
The RHS has been split into forward-transfer terms
(S1 [Nω ]) and backscatter terms (S2 [Nω ] and S3 [Nω ]).



Isotropic case: frequency space representation
The forward term, S1 [Nω ] (S2 [Nω ] and S3 [Nω ] look similar) is:

S1 [Nω1 ] = L1 (ω2 , ω3 ) Nω2 Nω3 δ(ω1 − ω2 − ω3 ) dω2 dω3

− L1 (ω3 , ω1 ) Nω3 Nω1 δ(ω2 − ω3 − ω1 ) dω2 dω3

− L1 (ω1 , ω2 ) Nω1 Nω2 δ(ω3 − ω1 − ω2 ) dω2 dω3 ,

Advantage of this notation: only 1 scaling parameter, λ:
2γ − α
L1 (ω1 , ω2 ) ∼ ω λ , λ=
α
Disadvantage: S1 [Nω ], S2 [Nω ] and S3 [Nω ] have slightly
different L(ω1 , ω2 )’s.


“Cheat-sheet”: 3-wave turbulence on one slide
Spectra:
√ λ+3
Kolmogorov–Zakharov: Nω = cKZ J ω− 2 .
Generalised Phillips (critical balance): Nω = cP ω −λ .
d
Thermodynamic: Nω ∼ ω −2+ α .
Phase transitions:
Inﬁnite capacity: λ < 1.
Finite capacity: λ > 1.
Breakdown at small scales: λ > 3.
Breakdown at large scales: λ < 3.
Locality: if L1 (ωi , ωj ) has asymptotics K1 (ωi , ωj ) ∼ ωiµ ωjν with
µ + ν = λ for ω1 ω2 , the KZ spectrum is local if
µ < ν + 3 and xKZ > xT .


Some model problems (d = α):

Sometimes cKZ can be
calculated exactly.
Product kernel: λ
L(ω1 , ω2 ) = (ω1 ω2 ) 2 .

Sum kernel:
1
L(ω1 , ω2 ) = ω λ + ω2 .
λ
2 1
−1/2
√ dI
cKZ = 2 where
dx x= λ+3
1 2
1
I(x) = L(y , 1 − y ) (y (1 − y ))−x (1 − y x − (1 − y )x )
2 0
(1 − y 2x−λ−2 − (1 − y )2x−λ−2 ) dy .


Developing wave turbulence

Developing wave turbulence refers to the evolution of the
spectrum before the onset of dissipation.
We shall focus on the unforced case.



Distinguish Finite and Infinite Capacity Cases

Stationary KZ spectrum:
√ λ+3
Nω = cKZ J ω− 2 .

Total energy contained in the spectrum:
√ Ω λ+1
E = cKZ J dω ω − 2 .
1

E diverges as Ω → ∞ if λ ≤ 1 : Infinite Capacity .
E finite as Ω → ∞ if λ > 1 : Finite Capacity .
Transition occurs at λ = 1.



Dissipative Anomaly

Finite capacity systems exhibit a dissipative anomaly as the
dissipation scale → ∞, inﬁnite capacity systems do not:

λ = 3/4 λ = 3/2



Transient solutions (Falkovich and Shafarenko, 1991)
Scaling hypothesis: there exists a typical scale, s(t), such
that Nω (t) is asymptotically of the form
ω
Nω (t) ∼ s(t)−a F (ξ) ξ= .
s(t)

Typical frequency, s(t), and the scaling function, F (ξ), satisfy:

ds
= sλ−a+2
dt
dF
−a F − ξ = S1 [F (ξ)] + S2 [F (ξ)] + S3 [F (ξ)].
dξ

Transient scaling exponent given by the small ξ divergence of
the scaling function, F (ξ) ∼ A ξ −x .



Self-similarity: forced case
Forced case: energy grows linearly:
∞ ∞
Jt = ω Nω dω = s(t)a+2 ξ F (ξ) dξ
0 0

Assuming F (ξ) ∼ A ξ −x , and bal-
ancing leading order terms in scaling
equation leads to:

λ+3
Self-similar evolution for the forced x =
case with L(ω1 , ω2 ) = 1. Inset
2
−1/2
shows scaling function F (ξ) compen- √ dI
sated by x 3/2 and predicted ampli-
A = 2 = cKZ
dx x= λ+3
2
tude.

Convergence problem at λ = 1.


Self-similarity: unforced case

Unforced case: energy conserved:
∞ ∞
1= ω Nω dω = s(t)a+2 ξ F (ξ) dξ
0 0

Assuming F (ξ) ∼ A ξ −x , and bal-
ancing leading order terms in scaling
equation leads to:

x= λ+1
λ−1
A = .
I(λ + 1)

Convergence problem at λ = 1.



Logarithmic corrections to scaling at λ = 0
For λ = 0 the transient exponent,
x = λ + 1, coincides with the
equipartition exponent x = 1 and
A diverges.
Correct leading order balance is:
F (ξ) ∼ ξ −1 ln(1/ξ) as ξ → 0,

c.f. 2D optical turbulence
(Dyachenko et al, 1992)
Scaling function F (ξ) compen- Anomalously slow development
sated by x ln(1/ξ) for the case of the turbulence. For example,
L(ω1 , ω2 ) = 1. the total wave action is:
3 (ln t)2
N(t) ∼ 2 .
π t


Finite capacity case

Previous argument fails at λ = 1 for both forced and
unforced cases (boundary of finite capacity regime).
∞
0 ξ F (ξ) dξ
diverges at 0 so that conservation of energy
does not determine dynamical exponent, a.
For finite capacity systems, s(t) → ∞ as t → t ∗ . If
F (ξ) ∼ ξ −x as ξ → 0 then scaling implies that
−x
ω
N(ω, t) ∼ s(t)a = s(t)a+x ω −x
s(t)

as t → t ∗ . Hence the transient exponent, x must be taken
equal to −a if the spectrum is to remain finite as t → t ∗ .
This is true independent of whether we force or not.



The ﬁnite capacity anomaly
Dynamical exponent, a, must be determined by self
consistently solving the scaling equation:

dF
−a F − ξ = S1 [F (ξ)] + S2 [F (ξ)] + S3 [F (ξ)].
dξ

Dimensionally, one might guess a = − λ+3 , the KZ value.
2
Overwhelming numerical evidence suggests that this is not so:
Cluster-cluster aggregation (Lee, 2000).
MHD turbulence (Galtier et al., 2000).
Non-equilibrium BEC (Lacaze et al, 2001)
Leith model (CC & Nazarenko, 2004)
Generic 3-wave turbulence (CC & Newell, 2010)



Finite capacity anomaly in 3WKE
For 3WKE with
K (ω1 , ω2 ) = (ω1 ω2 )λ/2 , the
transient spectrum is consistently
(slightly) steeper than the KZ
value.
KZ spectrum is set up from right
to left.
Finite capacity in itself is not
sufﬁcient for the anomaly (eg
Smoluchowski eqn with product
kernel).
Anisotropy is not necessary but it
might help! (c.f. Galtier et al.
2005).


Entropy production on transient spectra
Formal entropy-like quantity,

S(t) = Sω (t) dω = log(Nω (t)) dω

diverges on Nω = c ω −x .
∂S
Entropy production, ∂t , is
well-deﬁned:
∂Sω (t) 1 ∂Nω (t)
=
Entropy production as a function of x
∂t Nω (t) ∂t
for spectrum Nω = ω −x for several = c ω 2 xKZ −x−2 I(x)
different values of λ.
Transient spectrum requires
positive entropy production?


Anomalous transient scaling in NS turbulence?

The value of the dynamical
scaling exponent in NS
turbulence remains open.
Recent studies of EDQNM
closure equations give a transient
spectrum of 1.88 ± 0.04 (Bos et
al, 2010)
DNS measurements were
inconclusive.



Cascades without backscatter (cluster aggregation)

3WKE → Smoluchowski eqn:
m
∂t Nm = dm1 K (m1 , m − m1 )Nm1 Nm−m1
0
∞
− 2Nm dm1 K (m, m1 )Nm1 + J δ(m − 1)
0

Nm (t): cluster mass distribution.
K (m1 , m2 ): kernel.
K (am1 , am2 ) = aµ+ν K (m1 , m2 )
µ ν
K (m1 , m2 ) ∼ m1 m2 m1 m2
Typical size, s(t):
N(m, t) ∼ s(t)a F (m/s(t))
F (x) ∼ x −y x 1


Gelation Transition (Lushnikov [1977], Ziff [1980])
∞
M(t) = 0 m N(m, t) dm is
formally conserved. However for
µ + ν > 1:
∞
M(t) < m N(m, 0) dm t > t ∗
0

Best studied by introducing
cut-off, Mmax , and studying limit
M(t) for K (m1 , m2 ) = (m1 m2 )3/4 . Mmax → ∞. (Laurencot [2004])
“Instantaneous” Gelation
If ν > 1 then t ∗ = 0. (Van Dongen & Ernst [1987])
May be complete: M(t) = 0 for t > 0. Example :
1+ 1+
K (m1 , m2 ) = m1 + m2 for > 0.
Mathematically pathological?


Instantaneous Gelation in Regularised System
Yet there are applications with ν > 1: differential sedimentation.
With cut-off, Mmax , regularized
∗
gelation time, tMmax , is clearly
identiﬁable.
∗
tMmax decreases as Mmax
increases.
Van Dongen & Ernst recovered in
limit Mmax → ∞.
M(t) for K (m1 , m2 ) = (m1 m2 )3/4 .
∗
Decrease of tMmax as Mmax is very slow (numerics suggest
logarithmic decrease). This suggests such models are
physically reasonable.
Consistent with related results of Krapivsky and Ben-Naim
and Krapivsky [2003] on exchange-driven growth.


Nonlocal Interactions Drive Instantaneous Gelation
Instantaneous gelation is driven
by the runaway absorbtion of
small clusters by large ones.
This is clear from the analytically
tractable (but non-gelling)
marginal kernel, m1 + m2 , with
source of monomers.
R∞
Total density, 0 N(m, t) dm for M(t) ∼ t (due to source) but
K (m1 , m2 ) = m1 + m2 and source. N(t) ∼ 1/t.

If the exponent ν > 1 then big clusters are so “hungry" that they
eat all smaller clusters at a rate which diverges with the cut-off,
Mmax . (cf “addition model" (Brilliantov & Krapivsky [1992],
Laurencot [1999])).


Stationary State in the presence of a source
With cut-off, a stationary state may be reached if a source of
monomers is present (Horvai et al [2007]) even though no such
state exists in the unregularized system.
Introduce model kernel
ν ν
K (m1 , m2 ) = max(m1 , m2 ).
Stationary state expressible via a
recursion relation.
Asymptotics:

N(m) ∼ A m−ν m 1

A → 0 slowly as Mmax → ∞
Non-local stationary state (theory vs

numerics) for ν = 3/2.
Decay near monomer scale
seems exponential.



Approach to Stationary State is non-trivial

Numerics indicate that the
approach to stationary state is
non-trivial.
Collective oscillations of the total
density of clusters.
Total density vs
1+
time for
1+
Numerical measurements of the
K (m1 , m2 ) = m1 + m2 .
Q-factor of these oscillations
suggests that they are long-lived
transients. Last longer as Mmax
increases.
Heuristic explanation in terms of
“reset” mechanism.
“Q-factor" for ν = 0.2.


Spectral truncation of the 3WKE

Introduce a maximum frequency ω = Ω: modes having
ω > Ω have Nω = 0.
In sum over triads we only include ωj ≤ ωi < ωk ≤ Ω.
However we must choose what to do with triads having
ωj ≤ ωi < Ω < ωk (only relevant for S1 [Nω ]).
Include them: open truncation (dissipative)
Remove them: closed truncation (conservative)
Damp them by a factor 0 < ν < 1: partially open truncation
(dissipative)
“Boundary conditions” on the energy ﬂux are not local for
integral collision operators.



Open Truncation : Bottleneck Phenomenon

Product kernel:
L(ω1 , ω2 ) = (ω1 ω2 )λ/2 .
Open truncation can
produce a bottleneck as
the solution approaches
the dissipative cut-off
(Falkovich 1994).
Bottleneck does not
occur for all L(ω1 , ω2 ).
Compensated stationary Why?
spectra with open truncation. Energy ﬂux at Ω is 1.



Closed Truncation: Thermalisation

Product kernel:
L(ω1 , ω2 ) = (ω1 ω2 )λ/2 .
Closed truncation
produces thermalisation
near the cut-off (CC and
Nazarenko (2004),
Cichowlas et al (2005) ).
Thermalisation occurs
for all L(ω1 , ω2 ).
Quasi-stationary spectra with closed truncation. Energy ﬂux at Ω is 0.



Closed Truncation: Thermalisation

Product kernel:
L(ω1 , ω2 ) = (ω1 ω2 )λ/2 .
Closed truncation
produces thermalisation
near the cut-off (CC and
Nazarenko (2004),
Cichowlas et al (2005) ).
Thermalisation occurs
for all L(ω1 , ω2 ).
Quasi-stationary spectra with closed truncation com-
Energy ﬂux at Ω is 0.
pensated by KZ scaling.



Conclusions
Transient scaling of unforced inﬁnite capacity turbulence is
different from KZ.
Logarithmic corrections to scaling are obtained when thge
transient exponent coincides with equipartition exponent.
Dynamical scaling exponents for ﬁnite capacity turbulence
show a small but systematic dynamical scaling anomaly.
Curious effects seen for cascades without backscatter.
Choice of spectral truncation allows one to produce
bottleneck and / or thermalisation phenomena.
Thanks to: Robin Ball (Warwick), Wouter Bos (EC-Lyon)
Fabien Godeferd (EC-Lyon) Paul Krapivsky (Boston),
Sergey Nazarenko (Warwick), Alan Newell (Arizona), R.
Rajesh (Chennai), Thorwald Stein (Reading), Oleg
Zaboronski (Warwick),


References:
C. Connaughton and S. Nazarenko, Warm cascades and
anomalous scaling in a diffusion model of turbulence,
Phys. Rev. Lett. 92:044501, 2004
C. Connaughton, Numerical Solutions of the Isotropic
3-Wave Kinetic Equation, Physica D, 238(23), 2009.
C. Connaughton and A. C. Newell, Dynamical Scaling and
the Finite Capacity Anomaly in 3-Wave Turbulence, Phys.
Rev. E, 81:036303, 2010.
C. Connaughton and P.L. Krapivsky, Aggregation
fragmentation processes and decaying 3-wave turbulence,
Phys. Rev. E, 81:035303(R), 2010
W. Bos, C. Connaughton and F. Godeferd, Developing
Homogeneous Isotropic Turbulence, arXiv:1006.0798,
2010

Weak Isotropic three-wave turbulence, Fondation des Treilles, July 16 2010

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