The BenjaminFeir instabilitya popular and debated issue Dysthe
The Benjamin-Feir instability-a popular and debated issue. ( Dysthe and Trulsen) • • The beginning, selfaction effects in various waves. Initial growth of side bands. Recurrence, and the NLS. Frequency downshift. Longtime evolution. How to make exstreme waves. BF instability of stokastic waves. 3 D development
Self-action of waves The LASER-induced revolution in optics led to experimental demonstration of self-action phenomena: self-focusing Lallemand & Bloembergen (1965) and self-trapping Garmire et. al. (1966). Self-modulation of gravity waves demonstrated at roughly the same time (Benjamin & Feir 1967), was not induced by advances in technology. One wanders why it was not realized before? There were, however, corroborating theoretical results obtained at the same time : Lighthill (1965), Whitham (1967), Ostrovsky (1963, 1967), Zakharov (1967) and Benney & Newell (1967).
Early history of self-modulation • It is a remakable fact that the opening of this field of reseach happened at the same time in US, England USSR. • The observations were in a number of physical situations: water waves, plasma waves, laser beams and electromagnetic transmission lines. • For an excellent overview of the early development, see: ”Modulation instability: The beginning” Zakharov & Ostrovsky: Physica D 238, 2009
Benjamin &Feir (1967) • A short glimpse of their experiment. • There must have been previous experimenters seeing the same, when trying to produce regular Stokes waves. • They probably conceived it as a nuisance (possibly caused by a poor wavemaker? ? ) rather than a scientific challenge.
Benjamin and Feir saw the selfmodulation as a nonlinear interaction between 4 waves, mediated by the second harmonic ”virtual wave”.
First major experimental verification of the BF instability (10 years later). Lake, Yuen, Rungaldier and Ferguson: J. Fluid Mech. 1977
Growth of upper (open symbols) and lower sidebands (solid symbols) • Growth of the most unstable sidebands, Lake et. al. 1977 • The good agrement with B&F’s theoretical growth rate is due to the fact that they used a nominal steepness, only 80% of the measured one. • If instead the measured steepness is used, their growth rate agree with later measurement (e. g. Waseda & Tulin 1999)
Waseda & Tulin (1999)
New opening for theory of dispersive waves at the end of the 1960 th. • The development of the NLS equation has proved to be of great value, because: • It predicts qualitatively correct many of the new and surprising phenomena observed for nonlinear dispersive waves, such as the BF instability. • Its 2 D-version can be solved by the inverse scattering method (Zakharov & Shabat 1971) like the Kd. V equation, and many explicite solutions are known.
Long-time evolution of the BF instability according to the NLS equation (Yuen & Lake 1980)
The ”breather” family of solutions of the NLS equation
Asymptotic state of the BF instability according to the NLS equation. • Ma (1979) used the inverse scattering method on the NLS equation. He found that : • the asymptotic state of a Stokes wave that is perturbed over a finite domain, is a series of Kuznetsov breathers and dispersive waves.
Simulation of a periodically perturbed wavetrain with the NLS equation.
Simulation of a periodically perturbed wavetrain with higher order MNLS equations.
First glimpse of a ”downshift” A closer look at the B-F evolution from start to a ”quasi- recurrence ”. (Lake et. al. 1977) • Counting the number of waves at 5 ft (13), and 30 ft (10), a frequency downshift is seen to occur. Not mentioned by the authors. • However, in a later paper by Yuen and Lake: (Ann. Rev. of Fluid Mech. 1980), it was pointed out. • Melville (1982) reported that this phenomenon is happening when the BF modulation is deep enough to produce breaking.
Frequency ”separation”. Experiment at Marintek, Trulsen and Stansberg (2001). • Bichromatic waves evolve • 10 m from wavemaker frequency: • 80 m from wavemaker, ”separation” have separated
before breaking after breaking. Hwung et. al. : ”Observation of the evolution of wave modulation” Proc. R. Soc 2007
Why dissipation stabilize the BF in the strict mathematical sence (Segur et. al. 2005) (A pedestrian approach ) • As the amplitude decreases due to dissipation, the domain of instability for a sideband perturbation shrinks. • Consequently the growth of that sideband is limited in time. • Therefore its amplitude can in principle stay arbitrarily small, provided that its initial value is sufficiently small. • Thus the damped Stokes wave is modulationally stabel (in the strict mathematical sence). SCALED GROWTH-RATE
The pursuite of extreme waves • Starting et the end of the 19 -hundreds. Extreme waves were much in the news and scientists soon got very interrested. • The main question: is there a new and overlooked physical mechanism responsible for the ”freak-”or ”rogue” waves?
Henderson, Peregrine and Dold (1999). Long time history of the BF instability and Steep Wave Events (SWE) • Development of BF instability. Full nonlinear equations. • The maximum surface elevation of the wavetrain as a function of time. • The SWE’s were found to have a conspicuous likeness to the breather solution of the NLS. Particularly to the Kuznetsov breather. ~3 a 4 a~
”Explosion” of a group. 2 D simulation with fully nonlinear equations. • Initial condition in the shape of a NLS-soliton with maximum steepnes ka=0. 14 • After a couple of dozen periods this happens! Zakharov, Dyachenko and Prokofiev (2005) • Here a little less spectacular. Max initial steepness ka=0. 09 Clamond et. al. (2006)
MNLS animation 1
MNLS animation 3
MNLS animation 2
Alber (1978) found that even an irregular (stokastic) wave-train may be unstable if the spectrum is sufficiently narrow. • The condition for instability is that the relative half-width Δω/ωp of the frequency spectrum at the spectral peak ωp , is less than the wave steepness s i. e. when : s > Δω/ωp
Development of narrow wave spectra due to the Benjamin-Feir instability (simulations, Dysthe et. al. 2003)
Development of the angular averaged spectrum.
Experimental development of an irregular wavetrain with BF-index ~1, Onorato et. al. (2005) • The wavemaker produces 32 min. irregular wavetrains having a JONSWAP spectrum with random phases. • The BF index is 0. 9 with BFI= s/d where d is the spectral half width at half the peak value and s the steepness.
Wave statistics by large scale simulation by the MNLS equations. Socquet-Juglard et. al. (2005) • Initial conditions: JONSWAP spectrum with BFI >1. Random phases. Three different angular distributions. Average steepness s=0. 1 • The computational domain contains appr. 10. 000 typical waves at any time.
Steepness • Scatter-diagram of peak period Tp and significant wave height Hs. • Data from the northern North Sea collected from 5 platforms, 1973 -2001. (Haver, Eik) • Each point extracted from 20 min. timeseries.
Angular distribution
The probability of a crest height exceeding x standard deviations
Does the B-F instability increase the population of rogue waves? ? • 2 D: Yes, when Δω/ωp<s. This has been verified in a wave flume (Marintek, Trondheim) Onorato et. al. (2004, 2006). • 3 D: Simulations indicate that it does not work when the average crest length is less than appr. 10 wave lengths. Socquet-Juglard et. al. (2005), Gramstad & Trulsen (2007). • Independent experiments in two wave basins (at Marintek, Trondheim and University of Tokyo) have shown these conclutions to be at least qualitatively correct. • Onorato et. al. (2009)
Development of Zig-Zak patterns. 3 D BF effect). Ruban (PRL 2007) • Ruban used a fully nonlinear and ”weakly 3 D” numerical code to analyse the late stages of a plane Stokes wave. The phase, on entering the computational area has been randomly perturbed.
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