Discrete effects in Wave Turbulence

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

2. What is Wave Turbulence? WT describes a stochastic field of weakly interacting dispersive waves. S. Nazarenko @ Warwick-Hull WTS workshop, 20/9/7

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

4. How can we describe WT? • Deterministic equations for the wave field. • Weak nonlinearity expansion. • Statistical averaging. • Large-box limit. • Long-time limit. S. Nazarenko @ Warwick-Hull WTS workshop, 20/9/7

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

6. Drift waves in plasma and Rossby waves in GFD: Charney-Hasegawa-Mimaequation • 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) S. Nazarenko @ Warwick-Hull WTS workshop, 20/9/7

7. Weakly nonlinear drift waves with random phases→ wave kinetic equation (Longuet-Higgens &Gill, 1967) Resonant three-wave interactions S. Nazarenko @ Warwick-Hull WTS workshop, 20/9/7

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

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

10. Capillary waves (Connaughton et al 2001) • Quasi-resonant interactions. • δ depends on the wave intensities. • Exists δcrit for solutions to appear. S. Nazarenko @ Warwick-Hull WTS workshop, 20/9/7

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

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

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

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

15. Only one critical δ: once started the cascade proceeds to infinity S. Nazarenko @ Warwick-Hull WTS workshop, 20/9/7

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

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

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

19. 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, 20/9/7

20. Rain Generator 8 Panel Wave Generator Capacity Probes Laser 12 metres 90 cm 6 metres S. Nazarenko @ Warwick-Hull WTS workshop, 20/9/7

21. Spectra • Non-universality: Steeper spectra for low intensities S. Nazarenko @ Warwick-Hull WTS workshop, 20/9/7

22. Exponents • Agreement with the critical spectrum at low intensity. • Phillips and Kuznetsov spectra at higher intensities (but not ZF) • Forcing-independent intensity – avalanches? S. Nazarenko @ Warwick-Hull WTS workshop, 20/9/7

23. Cascade “sandpiles” in numerics (Choi et al’ 05) • Flux(t) at two different k’s. S. Nazarenko @ Warwick-Hull WTS workshop, 20/9/7

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

25. Summary • Discreteness causes selective dynamics of k-modes • -> intermittency • -> anisotropic avalanches • -> need better theory, numerics and experiment for mesoscopic wave systems S. Nazarenko @ Warwick-Hull WTS workshop, 20/9/7