Frontiers in Nonlinear Waves University of Arizona March 26, 2010

1. Frontiers in Nonlinear WavesUniversity of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

2. The modulational instability Discovered by several people, in different scientific disciplines, in different countries, using different methods: Lighthill (1965), Whitham (1965, 1967), Zakharov (1967, 1968), Ostrovsky (1967), Benjamin & Feir (1967), Benney & Newell (1967),… See Zakharov & Ostrovsky (2008) for a historical review of this remarkable period.

3. The modulational instability A central concept in these discoveries: Nonlinear Schrödinger equation For gravity-driven water waves: surfaceslow modulation fast oscillations elevation

4. Modulational instability • Dispersive medium: waves at different frequencies travel at different speeds • In a dispersive medium without dissipation, a uniform train of plane waves of finite amplitude is likely to be unstable

5. Modulational instability • Dispersive medium: waves at different frequencies travel at different speeds • In a dispersive medium without dissipation, a uniform train of plane waves of finite amplitude is likely to be unstable • Maximum growth rate of perturbation:

6. Experimental evidence of modulational instability in deep water - Benjamin (1967) near the wavemaker 60 m downstream frequency = 0.85 Hz, wavelength = 2.2 m, water depth = 7.6 m

7. Experimental evidence of modulational instability in an optical fiber Hasegawa & Kodama “Solitons in optical communications” (1995)

8. Experimental evidence of apparently stable wave patterns in deep water -(www.math.psu.edu/dmh/FRG) 3 Hz wave 4 Hz wave 17.3 cm wavelength 9.8 cm

9. More experimental results(www.math.psu.edu/dmh/FRG) 3 Hz wave 2 Hz wave (old water) (new water)

10. Q: Where did the modulational instability go?

11. Q: Where did the modulational instability go? • The modulational (or Benjamin-Feir) instability is valid for waves on deep water without dissipation

12. Q: Where did the modulational instability go? • The modulational (or Benjamin-Feir) instability is valid for waves on deep water without dissipation • But any amount of dissipation stabilizes the instability (Segur et al., 2005)

13. Q: Where did the modulational instability go? • This dichotomy exists both for (1-D) plane waves and for 2-D wave patterns of (nearly) permanent form. The logic is nearly identical. (Carter, Henderson, Segur, JFM, to appear) • Controversial

14. Q: How can small dissipation shut down the instability? Usual (linear) instability: Ordinarily, the (non-dissipative) growth rate must exceed the dissipation rate in order to see an instability. So very small dissipation does not stop an instability.

15. Q: How can small dissipation shut down the instability? Set

16. Q: How can small dissipation shut down the instability? Set Recall maximum growth rate:

17. Experimental verification of theory 1-D tank at Penn State

18. Experimental wave records x1 x8

19. Amplitudes of seeded sidebands(damping factored out of data) (with overall decay factored out) ___ damped NLS theory - - - Benjamin-Feir growth rate    experimental data

20. Q: What about a higher order NLS model (like Dysthe) ? __, damped NLS ----, NLS - - -, Dysthe   , experimental data

21. Numerical simulations of full water wave equations, plus damping Wu, Liu & Yue, J Fluid Mech, 556, 2006

22. Inferred validation Dias, Dyachenko & Zakharov (2008) derived the dissipative NLS equation from the equations of water waves See also earlier work by Miles (1967) Both papers provide analytic formulae for 

23. How to measure  Integral quantities of interest: , ,

24. Dissipationin wavetankmeasuredafter waiting a timeinterval 15 min. 45 min. 60 min. 80 min. 120 min. 1 day 2 days 6 days

25. Open questions What is the correct boundary condition at the water’s free surface? Do we need a damping rate that varies over days? If so, why?

26. Thank you for your attention