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 turbulence
Ecole 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 in
the physical sciences.
Most are dispersive.
All become nonlinear at
ﬁnite amplitude.
In most situations, waves
are excited and damped
by “external" processes.

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


What is wave turbulence?
Working deﬁnition
Wave turbulence is the non-equilibrium statistical dynamics of
ensembles of interacting dispersive waves.

non-equilibrium : forcing and dissipation are key so
equipartition of energy has limited relevance.
statistical : many degrees of freedom are active.
interacting : nonlinearity cannot be neglected.
dispersive : non-dispersive waves are much tougher
theoretically.
Applications
Interfacial 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):
α
∂2h ∂2
= −c 2 − 2 h.
∂t 2 ∂x

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 simple
harmonic motion,

d 2 hk
= −c 2 k 2α hk ≡ −ωk hk .
2
dt 2
with frequency ωk = c k α . At the linear level, different types of
waves differ only in their dispersion relation.

The simple harmonic motion equation,

d 2 hk 2
= −ωk hk ,
dt 2
can be represented as a pair of ﬁrst order equations for
(xk , yk ) = (hk , ∂hk ):
∂t

dxk
= yk
dt
dyk 2
= −ωk xk .
dt
These are equivalent to a single ﬁrst order equation for the
complex coordinate ak = yk + i ωk xk :

dak
= i ωk ak .
dt



The wave equation in complex coordinates,

dak
= i ωk ak ,
dt
is probably the simplest example of a Hamiltonian system:

∂ak δH2 ∗
=i ∗ H2 = dk ωk ak ak , (1)
∂t δak

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 of
ak 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 = Vk1 k2 k3 (ak1 ak2 ak3 +ak1 ak2 ak3 )δ(k1 −k2 −k3 )dk1 dk2 dk3 .

Gives equations of motion:
∂ak
= iωk ak + Vkk2 k3 ak2 ak3 δ(k − k2 − k3 )dk2 dk3
∂t
∗
+ 2 Vk2 kk3 ak2 ak3 δ(k2 − k − k3 )dk2 dk3

RHS couples all modes together in groups of 3 (triads).
Projection of RHS onto a single triad, k3 = k1 + k2 , which is
resonant (ωk3 = ωk1 + ωk2 ) gives ODEs:
dB1 ∗ dB2 ∗ dB3
= B2 B3 = B1 B3 = −B1 B2 .
dt dt dt
(see talks by Bustamante and Kartashova).

Models of nonlinear wave interactions
∗
H2 = dk ωk ak ak

∗ ∗ ∗
H3 = Vk1 k2 k3 (ak1 ak2 ak3 + ak1 ak2 ak3 )δ(k1 − k2 − k3 )dk1 dk2 dk3 .

What should we take for the nonlinear interaction coefﬁcient,
Vk1 k2 k3 and the linear frequency ωk ?
Many interesting cases possess scale invariance :
ωhk = hα ωk Vhk1 hk2 hk3 = hβ Vk1 k2 k3 .
Detailed formulae for Vk1 k2 k3 can be quite complex.
3 9
Example: capillary waves: d = 2, α = 2 and β = 4 .
Model systems (cf MMT) are often studied, especially
numerically. For example:
ωk = c k α (2)
β
Vk1 k2 k3 = λ (k1 k2 k3 ) 3 (3)


Conservation laws in turbulence
In turbulent systems, quantities which are conserved by the
nonlinear terms in the equations of motion are very important.
Navier-Stokes turbulence
∂v
+ (v · )v = − p + ν∆v + f
∂t
·v=0
1
Nonlinear terms conserve kinetic energy, 2 v2 . Energy is
added by forcing and removed by the viscous terms.

Wave turbulence
∂ak δH
= i ∗ + fk + D [ak ]
∂t δak

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


Kolmogorov phenomology of Navier-Stokes turbulence

LU
Reynolds number R = ν .
1
Energy, 2 v2 , injected at large
scales and dissipated at small
scales.
Conservative energy transfer
in between due to interaction
between scales.
Concept of inertial range and
energy cascade.
K41 : As R → ∞, small scale statistics depend only on the
scale, k , and the energy ﬂux, J. Dimensionally :
2 5
E(k ) = c J 3 k − 3 Kolmogorov spectrum

Similar phenomenology for wave turbulence.

What can we learn about WT by dimensional analysis?
For the purposes of power counting:
2
H = c k α ak dk + λ k β ak δ(k) (dk)3
3

2
nk δ(k) = ak

Dimensions:

[c] = T −1 K −d
1 1 d
[ak ] = H 2 T 2 K − 2
1 1 d
[λ] = H 2 T 2 K − 2
[nk ] = HT

One more dimensional parameter, the ﬂux:

[J] = HT −1 K d .

Flux is "H per unit volume per unit time".

What can we learn about WT by dimensional analysis?
We have enough dimensional parameters to get any spectum:
nk = c w J x λy k −z
Matching powers of H, T and K gives:
2(β + d − z)
w =
2α − β
2β − 2α + d − z
x = −
2α − β
2α + d − z
y = − .
2α − 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 ) = ak1 ak2
∗
C3 (k1 , k2 , k3 ) = ak1 ak2 ak3

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


The wave spectrum, nk
The second order correlation function is of particular interest:
∗
ak1 ak2 = a(x1 ) a∗ (x2 ) e−i k1 ·x1 +i k2 ·x2 dx1 d x2

= f (x1 − x2 ) e−i (k1 −k2 )·x1 e−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 ) ∗
= 2 Vk1 k2 k3 Im ak1 ak2 ak3 δ(k1 − k2 − k3 )dk2 dk3
∂t
∗
− 2 Vk2 k3 k1 Im ak2 ak3 ak1 δ(k2 − k3 − k1 )dk2 dk3

∗
− 2 Vk3 k1 k2 Im ak3 ak1 ak2 δ(k3 − k1 − k2 )dk2 dk3


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

H3 /H2 = 1

provides an expansion parameter.
Strategy (cf lecture by Newell):
1 Write moment hierarchy in terms of cumulants and solve
perturbatively in .
2 Higher order cumulants decay as t → ∞ (asymptotic
closure) but resonant triads lead to secular terms:

2 (2) (2)
n(k, t) = n(0) (k, t)+ n(1) (k, t)+ t nsec (k, t) + nnon−sec (k, t) +. . .

3 Allow slow variation of lower order terms: n(0) (k, t, 2 t).

4 Choose dependence on 2t to cancel secular terms.

The wave kinetic equation for 3-wave systems

At leading order, this procedure tells us that n(0) (k, t), satisﬁes
the following equation:
The 3-wave kinetic equation:

2
∂t nk = (Rkk1 k2 − Rk1 k2 k − Rk2 kk1 )dk1 dk2
2
Rkk1 k2 = Vkk1 k2 (nk1 nk2 − nk nk2 − nk nk1 )δ(k − k1 − k2 )
δ(ωk − ωk1 − ωk2 )

Closed kinetic equation.
Interactions are restricted to resonant triads.
Quadratic energy, H2 = ωk nk dk is conserved (to leading
order).


The wave kinetic equation in frequency space
For isotropic systems, it is helpful to write the KE in frequency
space using the change of variables
1 d−α
k = ωα k d−1 dk = ω α dω.
We deﬁne the angle averaged frequency spectrum, Nω , by
Ωd d−α
nk dk = nk k d−1 dΩd dk = nω ω α dω = Nω dω.
α

Translation between nk and Nω

Nω ∼ w −z ⇐⇒ nk ∼ k −αz+α−d .

Nω satisﬁes the kinetic equation

∂t Nω = (Rωω1 ω2 − Rω1 ω2 ω − Rω2 ωω1 )dω1 dω2
Rωω1 ω2 = Lω1 ω2 (Nω1 Nω2 − Nω Nω2 − Nω Nω1 ) δ(ω − ω1 − ω2 )

The wave kinetic equation in frequency space
Details are hidden in the (very messy) kernel L(ω1 , ω2 ). Scaling:

2β − α
L1 (ω1 , ω2 ) ∼ ω ζ , ζ=
α
The frequency-space kinetic equation can be written:

∂Nω
= S1 [Nω ] + S2 [Nω ] + S3 [Nω ].
∂t
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 the
kinetics 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:
ωω1 ω 2
(ω1 , ω2 ) → ( , )
ω2 ω2
ω 2 ωω2
(ω1 , ω2 ) → ( , ).
ω1 ω1
∞
0 = C2 dω1 dω2 L(ω1 , ω2 ) (ωω1 ω2 )−x ω 2−ζ−2x δ(ω − ω1 − ω2 )
0
2x−ζ−2 2x−ζ−2
(ω x − ω1 − ω2 ) ω 2x−ζ−2 − ω1
x x
− ω2

ζ+3
Stationary solutions: x = 1 (thermodynamic) and x = 2
(constant ﬂux).
ζ+3
(Translation: Nω ∼ ω − 2 gives nk ∼ k −β−d .
Nω ∼ ω −ζ gives nk ∼ k −2β+2α−d .)

Calculation of the Kolmogorov constant

Amplitude C can be
calculated exactly.
Product kernel: ζ
L(ω1 , ω2 ) = (ω1 ω2 ) 2 .

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


Locality of the KZ spectrum
ζ+3
The integral I(x) can vanish or diverge when x = 2 and
the KZ spectrum then runs into trouble.
For a kernel having asymptotics:
ν µ
L(ω1 , ω2 ) ∼ ω1 ω2 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 the
spectrum in the inertial range continues to depend on Ω in
this limit. Hence the term “non-local”.
Not much known about nonlocal cascades (see
Connaughton talk).

Breakdown of the KZ spectrum and the generalised
Phillips (critical balance) spectrum
Weak turbulence requires the linear timescale, τL , associated
with 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:
−1 1
τNL ∼ Nω ∂Nω .
∂t
If Nω ∼ ω −x , the ratio τL /τNL is:
τL
∼ ω ζ−x .
τNL
If x = ζ this ratio is uniform in ω (Phillips spectrum).
If x = ζ+3 this ratio becomes
2
τL ζ−3
∼ω 2 (6)
τNL

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

The concept of capacity

√ ζ+3
Nω = C J ω− 2 .
Total quadratic energy contained in the spectrum:
√ Ω ζ+1
E =C J dω ω − 2 .
1

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 the
dissipation 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. There
are 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 of
resonances.


Summary (before we get on to the interesting stuff)

This lecture has introduced most of the basic concepts of
wave turbulence theory in the context of a single cascade
of energy in a 3-wave system.
Many properties are in common with hydrodynamic
turbulence but some important differences.
For weakly nonlinear waves, the statistical dynamics has a
natural closure which leads to the wave kinetic equation for
the wave spectrum.
For isotropic systems it is much neater to study the kinetic
equation in frequency space.
Under certain conditions it is possible to ﬁnd the stationary
Kolmogorov-Zakharov solution of the kinetic equation
exactly.
Concepts of locality, capacity and breakdown were
introduced.

Introduction to (weak) wave turbulence

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