Discrete effects in Wave Turbulence

S. Nazarenko @ Warwick-Hull WTS workshop, 20/9/7

Discrete effects in Wave Turbulence

Sergey Nazarenko (Warwick)

Collaborators: Yeontaek Choi, Colm Connaughton, Petr Denissenko, Uriel Frisch, Sergei Lukaschuk, Yuri and Victor Lvov, Elena Kartashova, Dhuba Mitra, Boris Pokorni, Andrei Pushkarev, Vladimir Zakharov.


What is Wave Turbulence?WT describes a stochastic field of weakly

interacting dispersive waves.


Other Examples of Wave Turbulence:

Sound waves, Plasma waves, Waves in Bose-Einstein condensates, Kelvin waves on quantised vortex

filaments, Interstellar turbulence & solar wind, Waves in Semi-conductor Lasers, Spin waves….


How can we describe WT?

Deterministic equations for the wave field.

Weak nonlinearity expansion. Statistical averaging. Large-box limit. Long-time limit.


The order of the limits is essential

Nonlinear resonance broadening must be much wider than the spacing of the discrete (because of the finite box) Fourier modes.


Drift waves in plasma and Rossby waves in GFD: Charney-Hasegawa-Mima equation

• Ψ -- electrostatic potential (stream-function)• ρ -- ion Larmor radius (by Te) (Rossby radius)• β -- drift velocity (Rossby velocity)• x -- poloidal arc-length (east-west)• y -- radial length (south-north)


Weakly nonlinear drift waves with random phases→ wave kinetic equation (Longuet-Higgens &Gill, 1967)

Resonant three-wave interactions


Characteristic evolution times Deterministic: T~1/Ψ, Stochastic: T~1/Ψ2 – due to

cancellations of contributions of random-phased waves.

What happens if the box is finite and the resonance broadening is of the order or smaller than the spacing of discrete k-modes?


3 possibilities: Exact frequency resonances are absent in

for discrete k on the grid e.g. capillary waves (Kartashova 91).

Some (usually small) number of resonances survives e.g. deep water surface waves (Kartashova 94,07, Lvov et al 05).

All resonances survive e.g. Alfven waves (Nazarenko’ 07)


Capillary waves (Connaughton et al 2001)

Quasi-resonant interactions. δ depends on the wave intensities. Exists δcrit for solutions to appear.


Turbulent cascades

Start with a set of modes at small k’s.

Find quasi-resonances and add the new modes to the original set.

Continue like this to build further cascade steps.


Cascade stages


Critical intensity for initiating the cascade to infinite k’s.

Cascade dies out in a finite number of steps if δ< 2nd crit value, and it continues to infinite k otherwise.


Deep water gravity waves (Choi et al 2004, Korotkevich et al 2005)

Surviving resonances cause k-space intemittency Low k modes are more intermittent – discrete effects. Theory in Choi et al predicts power-law PDF tails.


Only one critical δ: once started the cascade proceeds to infinity


Evolving turbulence – VERY weak field (Rossby- Kartashova et al 89, water - Craig 90’s, Alfven – Nazarenko’06)

Only modes which are in exact resonance are active Sometimes sets of resonant modes are finite – no cascade

to high k, deterministic recursive (periodic or chaotic) dynamics – e.g. Rossby (Kartashova & Lvov’ 07).

Sometimes resonant triads (or quartets) form chains leading to infinite k’s – e.g. Alfven or NLS – cascades possible – poorly studied.

But… Most interesting question is how discrete/deterministic

evolution at low k gets transformed into continuous/random process at high k. Mesoscopic turbulence. Some ideas suggested but a lot of work remains to be done.


Mesoscopic turbulence – sandpile model (Nazarenko 05)

Put weak forcing at low k’s. Originally, the wave intensity is weak -> no quasi-resonances -> accumulation of wave energy at low k

until δ reaches the critical value -> initiation of cascade as an

“avalanche” spill toward larger k. -> value of δ drops to its critical value -> repeat the process


Weak forcing -> critical spectrum

δ ~ δcrit ->”sandpile” - E(ω) ~ ω-6. For reference: Phillips spectrum E(ω) ~ ω-5 Zakharov-Filonenko - E(ω) ~ ω-4.Kuznetsov (modified Phillips) - E(ω) ~

ω-4.


Wave-tank experiments(Denissenko et al 06, Falcon et al 06).

Paris setup -> 4cm to 1cm gravity range + capillary range

Hull setup -> 1m to 1cm of gravity-wave range.

S. Nazarenko @ Warwick-Hull WTS workshop, 20/9/76 metres

12 m

etre

s

90 c

m8 Panel Wave Generator

Laser

Capacity Probes

Rain Generator


Spectra

Non-universality: Steeper spectra for low intensities


Exponents

Agreement with the critical spectrum at low intensity. Phillips and Kuznetsov spectra at higher intensities (but not

ZF) Forcing-independent intensity – avalanches?


Cascade “sandpiles” in numerics (Choi et al’ 05)

Flux(t) at two different k’s.


Alfven wave turbulence Weak weak – dynamics of only

modes who are in exact resonances -> enslaving to the 2D component (Nazarenko 06).

Strong weak – classical WT (Galtier et al 2000).

Intermediate weak – two component system? Avalanches?


Summary

Discreteness causes selective dynamics of k-modes

-> intermittency -> anisotropic avalanches -> need better theory, numerics

and experiment for mesoscopic wave systems

Discrete effects in Wave Turbulence

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