What happens when the Kolmogorov-Zakharov spectrum is nonlocal?

What happens when the Kolmogorov-Zakharovspectrum is nonlocal?

A solvable model from the kinetics of cluster-clusteraggregation

Colm Connaughton

Mathematics Institute and Centre for Complexity Science,University of Warwick, UK

Collaborators: R. Ball (Warwick), R. Rajesh (Chennai), T. Stein (Reading), O.Zaboronski (Warwick).

Turbulence d’ondes - Wave turbulenceEcole de Physique des Houches

29 March 2012

http://www.slideshare.net/connaughtonc Nonlocal cascades

The 3-wave kinetic equation in frequency space

The frequency-space kinetic equation can be written:∂Nω

∂t= S1[Nω] + S2[Nω] + S3[Nω] + J δ(ω − 1).

Forward-transfer (S1[Nω]) and backscatter (S2[Nω] and S3[Nω]).

The forward term, S1[Nω], is:

S1[Nω] =

∫∫L(ω1, ω2) Nω1 Nω2 δ(ω − ω1 − ω2) dω1dω2

−∫∫

L(ω2, ω) Nω2 Nω δ(ω1 − ω2 − ω) dω1dω2

−∫∫

L(ω, ω1) Nω Nω1 δ(ω2 − ω − ω1) dω1dω2,

S2[Nω] and S3[Nω] obtained by permutation.http://www.slideshare.net/connaughtonc Nonlocal cascades

The Kolmogorov-Zakharov spectrum and locality

circles: µ = ν = ζ2 . squares: µ = 0, ν = ζ.

Model interaction:

L(ω1, ω2) =12

(ωµ1 ων2+ων1ω

µ2 ).

K-Z spectrum (ζ = µ+ ν):

Nω = C√

J ω−ζ+3

2

Conditions for locality:

|ν − µ| < 3xKZ > xT .

C =√

2 J

(dIdx

∣∣∣∣x= ζ+3

2

)−1/2

where

I(x) =12

∫ 1

0L(y ,1− y) (y(1− y))−x (1− yx − (1− y)x )

(1− y2x−ζ−2 − (1− y)2x−ζ−2) dy .


Cluster-cluster aggregation: 3-wave turbulencewithout backscatter

Smoluchowski equation :

∂Nm(t)∂t

=

∫ ∞0

dm1dm2K (m1,m2)Nm1Nm2δ(m −m1 −m2)

− 2∫ ∞

0dm1dm2K (m,m1)NmNm1δ(m2 −m −m1)

+ J δ(m −m0)

Describes the mean-field kinetics of aggregating particles:frequency, ω -> cluster mass, m.Frequency spectrum, Nω(t) -> cluster size distribution,Nm(t).Wave interaction L(ω1, ω2) -> collision rate K (m1,m2).


The KZ spectrum for the Smoluchowski equation

circles: µ = a, ν = −a.

Model interaction:

L(ω1, ω2) =12

(ωµ1 ων2+ων1ω

µ2 ).

K-Z spectrum:Nω = C

√J ω−

µ+ν+32

Conditions for locality:

|ν − µ| < 1

C =

√J [1− (ν − µ)2] cos[π(ν − µ)/2]

4π

But what happens in the nonlocal regime?


A regularised Smoluchowski equation

Let us now consider an explicit regularisation of the problemwhich forces all integrals to be finite:

∂tNm =12

∫ m

1dm1K (m1,m −m1)Nm1Nm−m1

− Nm

∫ M−m

1dm1K (m,m1)Nm1 + J δ(m − 1)

− DM [Nm(t)]

where

DM [Nm(t)] = Nm

∫ M

M−mdm1K (m,m1)Nm1 (1)

Regularisation corresponds to removing masses having m > Mand enforcing a minimum mass which we can take to be 1.Acts like dissipation.


Nonlocal kinetic equation (Horvai et al [2007])

Smoluchowski equation can be written (ignore source and sinkterms for now):

∂Nm

∂t=

∫ m2

1[F (m −m1,m1)− F (m,m1)] d m1

−∫ M

m2

F (m,m1) dm1

whereF (m1,m2) = K (m1,m2) N(m1, t) N(m2, t)

Taylor expand the first term to first order in m1:

[F (m −m1,m1)− F (m,m1)] = −m1N(m1, t)∂

∂m[K (m,m1) N(m, t)]

+O(m21).

We obtain an (almost) PDE.http://www.slideshare.net/connaughtonc Nonlocal cascades

Nonlocal kinetic equation (Horvai et al [2007])

∂Nm

∂t= −Dµ+1

∂

∂m[mνN(m, t)]− Dν N(m, t)

where

Dµ+1 =∫ m

21 mµ+1

1 N(m1, t)dm1 →∫ M

1mµ+1

1 N(m1, t)dm1 = Mµ+1

Dν =∫ M

m mν1N(m1, t)dm1 →

∫ M

1mν

1N(m1, t)dm1 = Mν

Solving for stationary state:

Nm = C exp[β

γm−γ

]m−ν

where γ = ν − µ− 1, C is an arbitrary constant and

β =Mν

Mµ+1.


Asymptotic solution of the nonlocal kinetic equation

β can be determined self-consistently since the momentsMµ+1 and Mν become Γ-functions.Resulting consistency condition gives:

β ∼ γ log M for M � 1.

Constant, C, is fixed by requiring global mass balance:

J =

∫ M

1dm m DM [Nm] ,

giving

C =

√J log Mγ

Mfor M � 1

Various assumptions are now verifiable a-posteriori.


Asymptotic solution of the nonlocal kinetic equation

Nonlocal stationary state is not the KZ spectrum

Stationary state (theory vs numerics).

Stationary state has theasymptotic form for M � 1:

Nm =

√2Jγ log M

MMm−γ

m−ν .

Stretched exponential for smallm, power law for large m.Stationary particle density:

N =

√J(

M −MM−γ)

M√γ log M

∼

√J

γ log Mas M →∞.


Dynamical approach to the nonlocal stationary state

Total density, N(t), vs timefor ν = 3

2 , µ = − 32 .

ν = 3/4, µ = −3/4, M = 104.

Numerics indicate that dynamicsare non-trivial when the cascadeis nonlocal (|ν − µ| > 1).Observe collective oscillations ofthe total density of clusters forlarge times.Are these oscillations slowlydecaying transients or is thestationary state unstable?Starting near the stationary statesuggests an instability.


Linear instability of the stationary state

We developed an iterativeprocedure which allowed us tocalculate the stationaryspectrum, N∗m, exactly.Using this we did a(semi-analytic) linear stabilityanalysis of the exactstationary state.Concluded that the nonlocalstationary state is linearlyunstable.Movie of density contrast,Nm(t)/N∗m.


Instability has a nontrivial dependence on parameters

Instability growth rate vs M

Instability growth rate vs ν.

Linear stability analysis of thestationary state for |ν − µ| > 1reveals presence of a Hopfbifurcation as M is increased.Contrary to intuition, dependence ofthe growth rate on the exponent ν isnon-monotonic.Oscillatory behaviour seemingly dueto an attracting limit cycle embeddedin this very high-dimensionaldynamical system.


Conclusions and open questions

Summary:The Smoluchowski equation can be thought of as a modelof 3-wave turbulence without backscatter. KZ spectrum isnonlocal when |ν − µ| > 1.The spectrum of nonlocal cascades can be calculated andhas a novel functional form.The nonlocal stationary state vanishes as the dissipationscale grows.The nonlocal stationary state can become unstable so thelong-time behaviour of the cascade dynamics is oscillatory.

Questions:Do any physical interacting particle systems really behavelike this?How much of these phenomena survive when thebackscatter terms are turned back on?


What happens when the Kolmogorov-Zakharov spectrum is nonlocal?

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