Davide Proment

Triggering Rogue Waves in Opposing Currents

We show that rogue waves can be triggered naturally when a stable wave train enters a region of an opposing current flow. We demonstrate that the maximum amplitude of the rogue wave depends on the ratio between the current velocity U(0) and the wave group velocity c(g). We also reveal that an opposing current can force the development of rogue waves in random wave fields, resulting in a substantial change of the statistical properties of the surface elevation. The present results can be directly adopted in any field of physics in which the focusing nonlinear Schrödinger equation with nonconstant coefficient is applicable. In particular, nonlinear optics laboratory experiments are natural candidates for verifying experimentally our results.

Rogue Waves: From Nonlinear Schrödinger Breather Solutions to Sea-Keeping Test

Under suitable assumptions, the nonlinear dynamics of surface gravity waves can be modeled by the one-dimensional nonlinear Schrödinger equation. Besides traveling wave solutions like solitons, this model admits also breather solutions that are now considered as prototypes of rogue waves in ocean. We propose a novel technique to study the interaction between waves and ships/structures during extreme ocean conditions using such breather solutions. In particular, we discuss a state of the art sea-keeping test in a 90-meter long wave tank by creating a Peregrine breather solution hitting a scaled chemical tanker and we discuss its potential devastating effects on the ship.

Vortex knots in a Bose-Einstein condensate.

We present a method for numerically building a vortex knot state in the superfluid wave function of a Bose-Einstein condensate. We integrate in time the governing Gross-Pitaevskii equation to determine evolution and shape preservation of the two (topologically) simplest vortex knots which can be wrapped over a torus. We find that the velocity of a vortex knot depends on the ratio of poloidal and toroidal radius: for smaller ratio, the knot travels faster. Finally, we show how vortex knots break up into vortex rings.

Helicity conservation by flow across scales in reconnecting vortex links and knots

Significance Ideal fluids have a conserved quantity—helicity—which measures the degree to which a fluid flow is knotted and tangled. In real fluids (even superfluids), vortex reconnection events disentangle linked and knotted vortices, jeopardizing helicity conservation. By generating vortex trefoil knots and linked rings in water and simulated superfluids, we observe that helicity is remarkably conserved despite reconnections: vortex knots untie and links disconnect, but in the process they create helix-like coils with the same total helicity. This result establishes helicity as a fundamental building block, like energy or momentum, for understanding the behavior of complex knotted structures in physical fields, including plasmas, superfluids, and turbulent flows. The conjecture that helicity (or knottedness) is a fundamental conserved quantity has a rich history in fluid mechanics, but the nature of this conservation in the presence of dissipation has proven difficult to resolve. Making use of recent advances, we create vortex knots and links in viscous fluids and simulated superfluids and track their geometry through topology-changing reconnections. We find that the reassociation of vortex lines through a reconnection enables the transfer of helicity from links and knots to helical coils. This process is remarkably efficient, owing to the antiparallel orientation spontaneously adopted by the reconnecting vortices. Using a new method for quantifying the spatial helicity spectrum, we find that the reconnection process can be viewed as transferring helicity between scales, rather than dissipating it. We also infer the presence of geometric deformations that convert helical coils into even smaller scale twist, where it may ultimately be dissipated. Our results suggest that helicity conservation plays an important role in fluids and related fields, even in the presence of dissipation.

Excitation of rogue waves in a variable medium:An experimental study on the interaction of water waves and currents

We show experimentally that a stable wave propagating into a region characterized by an opposite current may become modulationally unstable. Experiments have been performed in two independent wave tank facilities; both of them are equipped with a wavemaker and a pump for generating a current propagating in the opposite direction with respect to the waves. The experimental results support a recent conjecture based on a current-modified nonlinear Schrödinger equation which establishes that rogue waves can be triggered by a nonhomogeneous current characterized by a negative horizontal velocity gradient.

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

Abstract We consider the effect of the wind and the dissipation on the nonlinear stages of the modulational instability. By applying a suitable transformation, we map the forced/damped nonlinear Schrodinger (NLS) equation into the standard NLS with constant coefficients. The transformation is valid as long as | Γ t | ≪ 1 , with Γ the growth/damping rate of the waves due to the wind/dissipation. Approximate rogue wave solutions of the equation are presented and discussed. The results shed some lights on the effects of wind and dissipation on the formation of rogue waves.

Quantum turbulence cascades in the Gross-Pitaevskii model

We present a numerical study of turbulence in Bose-Einstein condensates within the 3D GrossPitaevskii equation. We concentrate on the direct energy cascade in forced-dissipated systems. We show that behavior of the system is very sensitive to the properties of the model at the scales greater than the forcing scale, and we identify three different universal regimes: (1) a non-stationary regime with condensation and transition from a four-wave to a three-wave interaction process when the largest scales are not dissipated, (2) a steady weak wave turbulence regime when largest scales are dissipated with a friction-type dissipation, (3) a state with a scale-by-scale balance of the linear and the nonlinear timescales when the large-scale dissipation is a hypo-viscosity. PACS numbers: 03.75.Kk, 94.05.Lk, 05.45.-a, 94.05.Pt ∗Electronic address: davideproment@gmail.com 1 ar X iv :0 90 5. 32 63 v1 [ nl in .C D ] 2 0 M ay 2 00 9 Experimental discovery of Bose-Einstein condensates (BEC) in 1995 [1, 2], some sixty years after their theoretical prediction [3, 4], sparked a renewed interest in this subject. Besides the obvious importance of such systems for fundamental physics, BEC experiments provide an excellent opportunity to build and study new nonlinear dynamical systems fabricated with high degree of control and flexibility supplied by optical means. For the nonlinear science and applied mathematics such an opportunity is extremely valuable, because it allows to implement and test dynamical and statistical regimes previously predicted theoretically and to gain insights about new ones for which the theory is yet to be developed. This is because BEC can be described by one of the most important and universal PDE’s, the nonlinear Schrödinger equation, called in this case Gross-Pitaevskii equation (GPE) [5]: i ∂ψ ∂t +∇ψ − |ψ|ψ = F +D, (1) where ψ is the order parameter indicating the condensate wave function, F and D represent possible external forcings and dissipation mechanisms. In general, when F = 0 and D = 0, GPE conserves total energy and particles H = ∫ 1 2 |∇ψ|dx + ∫ 1 4 |ψ|dx = HLIN +HNL, (2a) N = ∫ 1 2 |ψ|dx. (2b) As GPE model describes a Bose gas at very low temperature, it has been used to study the formation of a condensate in [6, 7, 8]. Moreover GPE can be mapped, using the Madelung transformation, to the Euler equation for ideal fluid flows with the extra quantum pressure term. This is why many concepts arising from the fluid dynamics have been discussed and studied with GPE, for example vortices and their reconnection [9]. It was also suggested that this model allows statistical motions similar to classical fluid turbulence and a number of papers was devoted to finding Kolmogorov spectrum in such GPE turbulence [10, 11, 12, 13, 14]. On the other hand, GPE solutions also include dispersive waves which may be involved in nonlinear interactions, and an approach known as weak wave turbulence (WWT) can be made for GPE. Generally, WWT describes statistics on large ensembles of weakly nonlinear waves in different applications, i.e. water waves or waves in plasmas [15]. Such waves interact with each other in a resonant way, e.g. in triads ar quartets, thereby transferring energy

Route to thermalization in the α-Fermi-Pasta-Ulam system

Significance Despite the fact that more than 60 years have passed, the α-Fermi–Pasta–Ulam (FPU) system has not yet been fully understood. Their seminal work stimulated many interdisciplinary research topics in mathematics and physics like integrable systems, soliton theory, ergodic theory, and chaos. In this article, we theoretically investigate the original problem by applying the wave–wave interaction theory. By using this mathematical approach, we are able to explain why the emergence of equipartition requires very long times (inaccessible when the original numerical experiments were performed but nowadays recently observed using computer power). Our approach is general and can be used to attack other problems of weakly nonlinear dispersive waves. We study the original α-Fermi–Pasta–Ulam (FPU) system with N = 16, 32, and 64 masses connected by a nonlinear quadratic spring. Our approach is based on resonant wave–wave interaction theory; i.e., we assume that, in the weakly nonlinear regime (the one in which Fermi was originally interested), the large time dynamics is ruled by exact resonances. After a detailed analysis of the α-FPU equation of motion, we find that the first nontrivial resonances correspond to six-wave interactions. Those are precisely the interactions responsible for the thermalization of the energy in the spectrum. We predict that, for small-amplitude random waves, the timescale of such interactions is extremely large and it is of the order of 1/ϵ8, where ϵ is the small parameter in the system. The wave–wave interaction theory is not based on any threshold: Equipartition is predicted for arbitrary small nonlinearity. Our results are supported by extensive numerical simulations. A key role in our finding is played by the Umklapp (flip-over) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.

Experimental evidence of the modulation of a plane wave to oblique perturbations and generation of rogue waves in finite water depth

We present a laboratory experiment in a large directional wave basin to discuss the instability of a plane wave to oblique side band perturbations in finite water depth. Experimental observations, with the support of numerical simulations, confirm that a carrier wave becomes modulationally unstable even for relative water depths k0h < 1.36 (with k the wavenumber of the plane wave and h the water depth), when it is perturbed by appropriate oblique disturbances. Results corroborate that the underlying mechanism is still a plausible explanation for the generation of rogue waves in finite water depth.

Sustained turbulence in the three-dimensional Gross–Pitaevskii model

We study the three-dimensional forced–dissipated Gross–Pitaevskii equation. We force at relatively low wave numbers, expecting to observe a direct energy cascade and a consequent power-law spectrum of the form k−α. Our numerical results show that the exponent α strongly depends on how the inverse particle cascade is attenuated at ks lower than the forcing wave-number. If the inverse cascade is arrested by a friction at low ks, we observe an exponent which is in good agreement with the weak wave turbulence prediction k−1. For a hypo-viscosity, a k−2 spectrum is observed which we explain using a critical balance argument. In simulations without any low k dissipation, a condensate at k=0 is growing and the system goes through a strongly turbulent transition from a 4-wave to a 3-wave weak turbulence acoustic regime with evidence of k−3/2 Zakharov–Sagdeev spectrum. In this regime, we also observe a spectrum for the incompressible kinetic energy which formally resembles the Kolmogorov k−5/3, but whose correct explanation should be in terms of the Kelvin wave turbulence. The probability density functions for the velocities and the densities are also discussed.