Introduction to (weak) wave turbulence

An introduction to (weak) wave turbulence

Colm Connaughton

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

Turbulence d’ondes - Wave turbulenceEcole de Physique des Houches

26 March 2012

http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence

Motivation from natural sciences

Waves are ubiqitous inthe physical sciences.Most are dispersive.All become nonlinear atfinite amplitude.In most situations, wavesare excited and dampedby “external" processes.

This lecture:An introduction to wave turbulence. Mostly focusing onconcepts and avoiding detailed calculations.


What is wave turbulence?

Working definition

Wave turbulence is the non-equilibrium statistical dynamics ofensembles of interacting dispersive waves.

non-equilibrium : forcing and dissipation are key soequipartition of energy has limited relevance.statistical : many degrees of freedom are active.interacting : nonlinearity cannot be neglected.dispersive : non-dispersive waves are much toughertheoretically.

ApplicationsInterfacial waves in fluids (gravity, capillary).Waves in plasmas (Alfvén, drift, sound).Nonlinear optics and BECs (NLS).Geophysical waves (Rossby, inertial).


Linear waves and complex coordinates

A linear wave equation for h(x , t):

∂2h∂t2 = −c2

(− ∂2

∂x2

)αh.

Such an equation is more natural in Fourier space:

h(x , t) =

∫hk (t)ei k x dk .

Each (complex-valued) Fourier mode, hk (t), executes simpleharmonic motion,

d2hk

dt2 = −c2k2αhk ≡ −ω2k hk .

with frequency ωk = c kα. At the linear level, different types ofwaves differ only in their dispersion relation.



The simple harmonic motion equation,

d2hk

dt2 = −ω2k hk ,

can be represented as a pair of first order equations for(xk , yk ) = (hk ,

∂hk∂t ):

dxk

dt= yk

dyk

dt= −ω2

k xk .

These are equivalent to a single first order equation for thecomplex coordinate ak = yk + i ωk xk :

dak

dt= i ωk ak .



The wave equation in complex coordinates,

dak

dt= i ωk ak ,

is probably the simplest example of a Hamiltonian system:

∂ak

∂t= i

δH2

δa∗kH2 =

∫dk ωk aka∗k , (1)

where we have reverted to treating k as a continuous variable.Hamiltonian structure is not necessary but it is convenient.Nonlinearities manifest themselves as higher order powers ofak in Eqs. (1):

H = H2 +H3 +H4 + . . . .

In this lecture we will stop at H3. (H4 in Nazarenko lecture).


Mode coupling in nonlinear wave equations

A simple model of H3 is

H3 =

∫Vk1k2k3(ak1a∗k2

a∗k3+a∗k1

ak2ak3)δ(k1−k2−k3)dk1dk2dk3.

Gives equations of motion:

∂ak

∂t= iωkak +

∫Vkk2k3ak2ak3δ(k− k2 − k3)dk2dk3

+ 2∫

Vk2kk3ak2a∗k3δ(k2 − k− k3)dk2dk3

RHS couples all modes together in groups of 3 (triads).Projection of RHS onto a single triad, k3 = k1 + k2, which isresonant (ωk3 = ωk1 + ωk2) gives ODEs:

dB1

dt= B∗2B3

dB2

dt= B∗1B3

dB3

dt= −B1B2.

(see talks by Bustamante and Kartashova).http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence

Models of nonlinear wave interactions

H2 =

∫dk ωk aka∗k

H3 =

∫Vk1k2k3(ak1a∗k2

a∗k3+ a∗k1

ak2ak3)δ(k1 − k2 − k3)dk1dk2dk3.

What should we take for the nonlinear interaction coefficient,Vk1k2k3 and the linear frequency ωk?

Many interesting cases possess scale invariance :

ωhk = hαωk Vhk1hk2hk3 = hβVk1k2k3 .

Detailed formulae for Vk1k2k3 can be quite complex.Example: capillary waves: d = 2, α = 3

2 and β = 94 .

Model systems (cf MMT) are often studied, especiallynumerically. For example:

ωk = c kα (2)

Vk1k2k3 = λ (k1k2k3)β3 (3)


Conservation laws in turbulence

In turbulent systems, quantities which are conserved by thenonlinear terms in the equations of motion are very important.

Navier-Stokes turbulence

∂v∂t

+ (v · ∇)v = −∇p + ν∆v + f

∇ · v = 0Nonlinear terms conserve kinetic energy, 1

2v2. Energy isadded by forcing and removed by the viscous terms.

Wave turbulence∂ak

∂t= i

δHδa∗k

+ fk + D [ak ]

In WT, H, is conserved by nonlinearity - not quadratic.Possibly other conserved quantities (cf Nazarenko lecture).


Kolmogorov phenomology of Navier-Stokes turbulence

Reynolds number R = LUν .

Energy, 12v2, injected at large

scales and dissipated at smallscales.Conservative energy transferin between due to interactionbetween scales.Concept of inertial range andenergy cascade.

K41 : As R →∞, small scale statistics depend only on thescale, k , and the energy flux, J. Dimensionally :

E(k) = c J23 k−

53 Kolmogorov spectrum

Similar phenomenology for wave turbulence.


What can we learn about WT by dimensional analysis?

For the purposes of power counting:

H = c kαa2k dk + λ kβ a3

k δ(k) (dk)3

nk δ(k) = a2k

Dimensions:

[c] = T−1K−d

[ak ] = H12 T

12 K−

d2

[λ] = H12 T

12 K−

d2

[nk] = HT

One more dimensional parameter, the flux:

[J] = HT−1K d .

Flux is "H per unit volume per unit time".http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence

What can we learn about WT by dimensional analysis?

We have enough dimensional parameters to get any spectum:

nk = cwJxλyk−z

Matching powers of H, T and K gives:

w =2(β + d − z)

2α− β

x = −2β − 2α + d − z2α− β

y = −2α + d − z2α− β

.

There are special scalings:

(w = 0) : z = β + d (KZ)(x = 0) : z = 2β − 2α + d (Generalised Phillips)(y = 0) : z = 2α + d (???).


What do we want from a statistical description?

Ideally want full joint PDF, P(ak1(t1),ak2(t2), . . .)!Start with (single-time) correlation functions:

C2(k1,k2) = 〈ak1a∗k2〉

C3(k1,k2,k3) = 〈ak1ak2a∗k3〉

〈.〉 is an ensemble average with respect to the statistics ofthe forcing (forced/steady turbulence) or initial condition(unforced/decaying turbulence).We almost always assume ergodicity in practice. Forsteady turbulence, this allows ensemble averages to bereplaced with time averages.We also typically assume statistical spatial homogeneity.


The wave spectrum, nk

The second order correlation function is of particular interest:

〈ak1a∗k2〉 =

∫∫〈a(x1) a∗(x2)〉 e−i k1·x1+i k2·x2 dx1 d x2

=

∫∫f (x1 − x2) e−i (k1−k2)·x1e−i k2·(x1−x2) dx1 d x2

=

∫f (r) e−i k2·r dr

∫e−i (k1−k2)·x1 dx1 r = x1 − x2

= n(k2) δ(k1 − k2).

Equations of motion lead to closure problem:

∂n(k1)

∂t= 2

∫Vk1k2k3Im〈a∗k1

ak2ak3〉δ(k1 − k2 − k3)dk2dk3

− 2∫

Vk2k3k1Im〈a∗k2ak3ak1〉δ(k2 − k3 − k1)dk2dk3

− 2∫

Vk3k1k2Im〈a∗k3ak1ak2〉δ(k3 − k1 − k2)dk2dk3


Weakly nonlinear wave kineticsInteraction Hamiltonian is small com-pared to the linear Hamiltonian.

H3/H2 = ε 1

ε provides an expansion parameter.

Strategy (cf lecture by Newell):

1 Write moment hierarchy in terms of cumulants and solveperturbatively in ε.

2 Higher order cumulants decay as t →∞ (asymptoticclosure) but resonant triads lead to secular terms:

n(k, t) = n(0)(k, t)+εn(1)(k, t)+ε2[t n(2)

sec (k, t) + n(2)non−sec(k, t)

]+. . .

3 Allow slow variation of lower order terms: n(0)(k, t , ε2t).4 Choose dependence on ε2t to cancel secular terms.


The wave kinetic equation for 3-wave systems

At leading order, this procedure tells us that n(0)(k, t), satisfiesthe following equation:

The 3-wave kinetic equation:

∂tnk = ε2∫∫

(Rkk1k2 − Rk1k2k − Rk2kk1)dk1dk2

Rkk1k2 = V 2kk1k2

(nk1nk2 − nknk2 − nknk1)δ(k− k1 − k2)

δ(ωk − ωk1 − ωk2)

Closed kinetic equation.Interactions are restricted to resonant triads.Quadratic energy, H2 =

∫ωknkdk is conserved (to leading

order).


The wave kinetic equation in frequency space

For isotropic systems, it is helpful to write the KE in frequencyspace using the change of variables

k = ω1α kd−1 dk = ω

d−αα dω.

We define the angle averaged frequency spectrum, Nω, by∫nkdk =

∫nkkd−1dΩddk =

Ωd

α

∫nωω

d−αα dω =

∫Nω dω.

Translation between nk and Nω

Nω ∼ w−z ⇐⇒ nk ∼ k−αz+α−d .

Nω satisfies the kinetic equation

∂tNω =

∫∫(Rωω1ω2 − Rω1ω2ω − Rω2ωω1)dω1dω2

Rωω1ω2 = Lω1ω2 (Nω1Nω2 − NωNω2 − NωNω1) δ(ω − ω1 − ω2)


The wave kinetic equation in frequency space

Details are hidden in the (very messy) kernel L(ω1, ω2). Scaling:

L1(ω1, ω2) ∼ ωζ , ζ =2β − αα

The frequency-space kinetic equation can be written:

∂Nω

∂t= S1[Nω] + S2[Nω] + S3[Nω].

The RHS has been split into forward-transfer terms (S1[Nω])and backscatter terms (S2[Nω] and S3[Nω]).

Each conserves energy independently. S1[Nω] describes thekinetics of cluster aggregation.


How does the solution of the kinetic equation look?

Illustrative numerical simulations for L(ω1, ω2) = 1.

Unforced turbulence Forced turbulence


The stationary Kolmogorov–Zakharov spectrum

Look for stationary solutions of 3WKE Nω = C ω−x

Zakharov Transformations:

(ω1, ω2) → (ωω′1ω′2

,ω2

ω′2)

(ω1, ω2) → (ω2

ω′1,ωω′2ω′1

).

0 = C2∫ ∞

0dω1dω2 L(ω1, ω2) (ωω1ω2)−xω2−ζ−2x δ(ω − ω1 − ω2)

(ωx − ωx1 − ωx

2)(ω2x−ζ−2 − ω2x−ζ−2

1 − ω2x−ζ−22

)Stationary solutions: x = 1 (thermodynamic) and x = ζ+3

2(constant flux).(Translation: Nω ∼ ω−

ζ+32 gives nk ∼ k−β−d .

Nω ∼ ω−ζ gives nk ∼ k−2β+2α−d .)http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence

Calculation of the Kolmogorov constant

Amplitude C can becalculated exactly.Product kernel:

L(ω1, ω2) = (ω1 ω2)ζ2 .

Sum kernel:L(ω1, ω2) =

12

(ωζ1 + ωζ2

).

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 .


Locality of the KZ spectrum

The integral I(x) can vanish or diverge when x = ζ+32 and

the KZ spectrum then runs into trouble.For a kernel having asymptotics:

L(ω1, ω2) ∼ ων1ωµ2 for ω1 ω2,

the integral is finite and non-zero provided:

|ν − µ| < 3 (4)xKZ > xT . (5)

Such cascades are referred to as “local”.Introduce a cut-off, Ω, and study what happens as Ω→∞.In systems for which Eqs. (4) and (5) are not satisfied thespectrum in the inertial range continues to depend on Ω inthis limit. Hence the term “non-local”.Not much known about nonlocal cascades (seeConnaughton talk).


Breakdown of the KZ spectrum and the generalisedPhillips (critical balance) spectrum

Weak turbulence requires the linear timescale, τL, associatedwith the waves to be much faster than the nonlinear timescale,τNL, associated with resonant energy transfer between waves.

Linear timescale: τL ∼ ω−1. Nonlinear timescale:τ−1

NL ∼1

Nω∂Nω∂t .

If Nω ∼ ω−x , the ratio τL/τNL is:τL

τNL∼ ωζ−x .

If x = ζ this ratio is uniform in ω (Phillips spectrum).If x = ζ+3

2 this ratio becomesτL

τNL∼ ω

ζ−32 (6)

Conclusion: breakdown criterionKZ spectrum breaks down as ω →∞ if ζ > 3 (or β > 2α).


The concept of capacity

Nω = C√

J ω−ζ+3

2 .

Total quadratic energy contained in the spectrum:

E = C√

J∫ Ω

1dω ω−

ζ+12 .

E diverges as Ω→∞ if ζ ≤ 1 (β < α): Infinite Capacity .E finite as Ω→∞ if ζ > 1 (β > α): Finite Capacity .

Transition occurs at ζ = 1.


Dissipative anomaly in finite capacity systems

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

ζ = 3/4 ζ = 3/2


“Snakes in the grass”

Despite the elegance of the theory which I have described,experimental observations of clean KZ scaling, are rare. Thereare many possible complications:

Insufficient inertial range.Crossover between different scaling regimes.More than one cascade leading to mixing.KZ and equilibium scaling exponents coincide.Spectrally broad forcing can contaminate the inertial range.KZ spectrum can be nonlocal.KZ spectrum can break down at scales of interest.Presence of coherent structures with their own scaling.Finite basin can invalidate the continuum description ofresonances.


Summary (before we get on to the interesting stuff)

This lecture has introduced most of the basic concepts ofwave turbulence theory in the context of a single cascadeof energy in a 3-wave system.Many properties are in common with hydrodynamicturbulence but some important differences.For weakly nonlinear waves, the statistical dynamics has anatural closure which leads to the wave kinetic equation forthe wave spectrum.For isotropic systems it is much neater to study the kineticequation in frequency space.Under certain conditions it is possible to find the stationaryKolmogorov-Zakharov solution of the kinetic equationexactly.Concepts of locality, capacity and breakdown wereintroduced.


Introduction to (weak) wave turbulence

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