Instabilities are common phenomena frequently observed in nature, sometimes leading to unexpected catastrophes and disasters in seemingly normal conditions. One prominent form of instability in a distributed system is its response to a harmonic modulation. Such instability has special names in various branches of physics and is generally known as modulation instability (MI). The MI leads to a growth-decay cycle of unstable waves and is therefore related to Fermi-Pasta-Ulam (FPU) recurrence since breather solutions of the nonlinear Schrodinger equation (NLSE) are known to accurately describe growth and decay of modulationally unstable waves in conservative systems. Here, we report theoretical, numerical and experimental evidence of the effect of dissipation on FPU cycles in a super wave tank, namely their shift in a determined order. In showing that ideal NLSE breather solutions can describe such dissipative nonlinear dynamics, our results may impact the interpretation of a wide range of new physics scenarios.

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Modulation Instability and Phase-Shifted Fermi-Pasta-Ulam Recurrence.

http://personalpages.to.infn.it/~onorato/publications_files/2012_Onorato_etal_PLA.pdf

Approximate rogue wave solutions of the forced and damped nonlinear Schrödinger equation for water waves

Being considered as a prototype for description of oceanic rogue waves, the Peregrine breather solution of the nonlinear Schrodinger equation has been recently observed and intensely investigated experimentally in particular within the context of water waves. Here, we report the experimental results showing the evolution of the Peregrine solution in the presence of wind forcing in the direction of wave propagation. The results show the persistence of the breather evolution dynamics even in the presence of strong wind and chaotic wave field generated by it. Furthermore, we have shown that characteristic spectrum of the Peregrine breather persists even at the highest values of the generated wind velocities thus making it a viable characteristic for prediction of rogue waves.

/pdf/experiments-on-wind-perturbed-rogue-wave-hydrodynamics-using-27zy3k8pce.pdf

Experiments on wind-perturbed rogue wave hydrodynamics using the Peregrine breather model

Nonlinear fast growth of water waves under wind forcing

Interaction with an opposing current amplifies wave modulation and accelerates nonlinear wave focusing in regular wavepackets. This results in large-amplitude waves, usually known as rogue waves, even if the wave conditions are less prone to extremes. Laboratory experiments in three independent facilities are presented here to assess the role of opposing currents in changing the statistical properties of unidirectional and directional mechanically generated random wavefields. The results demonstrate in a consistent and robust manner that opposing currents induce a sharp and rapid transition from weakly to strongly non-Gaussian properties. This is associated with a substantial increase in the probability of occurrence of rogue waves for unidirectional and directional sea states, for which the occurrence of extreme and rogue waves is normally the least expected.

/pdf/rogue-waves-in-opposing-currents-an-experimental-study-on-1v8twf6p4n.pdf

Rogue waves in opposing currents: an experimental study on deterministic and stochastic wave trains

The modulational instability of gravity wave trains on the surface of water acted upon by wind and under influence of viscosity is considered. The wind regime is that of validity of Miles' theory and the viscosity is small. By using a perturbed nonlinear Schrodinger equation describing the evolution of a narrow-banded wavepacket under the action of wind and dissipation, the modulational instability of the wave group is shown to depend on both the frequency (or wavenumber) of the carrier wave and the strength of the friction velocity (or the wind speed). For fixed values of the water-surface roughness, the marginal curves separating stable states from unstable states are given. It is found in the low-frequency regime that stronger wind velocities are needed to sustain the modulational instability than for high-frequency water waves. In other words, the critical frequency decreases as the carrier wave age increases. Furthermore, it is shown for a given carrier frequency that a larger friction velocity is needed to sustain modulational instability when the roughness length is increased.

/pdf/the-modulational-instability-in-deep-water-under-the-action-3qo1y1gpfg.pdf

The modulational instability in deep water under the action of wind and dissipation

A nonlinear Schr\"odinger equation for the envelope of two dimensional surface water waves on finite depth with non zero constant vorticity is derived, and the influence of this constant vorticity on the well known stability properties of weakly nonlinear wave packets is studied. It is demonstrated that vorticity modifies significantly the modulational instability properties of weakly nonlinear plane waves, namely the growth rate and bandwidth. At third order we have shown the importance of the coupling between the mean flow induced by the modulation and the vorticity. Furthermore, it is shown that these plane wave solutions may be linearly stable to modulational instability for an opposite shear current independently of the dimensionless parameter kh, where k and h are the carrier wavenumber and depth respectively.

Roland Thomas

Papers

The modulational instability in deep water under the action of wind and dissipation

A nonlinear Schr\"odinger equation for water waves on finite depth with constant vorticity