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.helicity | fluid topology | vortex reconnections | superfluid vortices | topological fields

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/e 8 , where e 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 (flipover) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.T he Fermi-Pasta-Ulam (FPU) chains is a simple mathematical model introduced in the 1950s to study the thermal equipartition in crystals (1). The model consists of N identical masses, each one connected by a nonlinear spring; the elastic force can be expressed as a power series in the spring deformation Δx:where γ; α, and β are elastic, spring-dependent, constants. The α-FPU chain, the system studied herein, corresponds to the case of α ≠ 0 and β = 0. Fermi, Pasta, and Ulam integrated numerically the equation of motion and conjectured that, after many iterations, the system would exhibit a thermalization, i.e., a state in which the influence of the initial modes disappears and the system becomes random, with all modes excited equally (equipartition of energy) on average. Contrary to their expectations, the system exhibited a very complicated quasiperiodic behavior. This phenomenon has been named "FPU recurrence," and this finding has spurred many great mathematical and physical discoveries such as integrability (2) and soliton physics (3). More recently, very long numerical simulations have shown clear evidence of the phenomenon of equipartition (see, for instance, ref. 4 and references therein). However, despite substantial progress on the subject (5-10), to our knowledge no complete understanding of the original problem has been achieved so far, and the numerical results of the original α-FPU system remain largely unexplained from a theoretical point of view. More precisely, the physical mechanism responsible for a first "metastable state" (4) and the observation of equipartition for very large times have not been understood.In this manuscript, we study the FPU problem using an approach based on the nonlinear interaction of weakly nonlinear dispersive waves. Our main assumption is that the irreversible transfer of energy in the spectrum in a weakly nonlinear system is achieved by exact resonant wave-w...

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, U0, and the wave group velocity, cg. 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 Schrodinger equation with non constant coefficient is applicable. In particular, nonlinear optics laboratory experiments are natural candidates for verifying experimentally our results. PACS numbers:In the ocean, rogue waves are often observed in regions characterized by strong currents like the Gulf Stream, Agulhas Current and the Kuroshio Current [1, 2]. Several ship accidents have been reported in these regions as being due to the impact with very large waves. One of these occurred in February 1986 to the SS Spray, which was travelling along the East coast of the USA. The ship was hit by a wave with a height of approximately 17 m (estimated by eyes from the deck of the ship), which was the second of a system of three consecutive large waves, commonly known as the three sisters. This particular wave system is usually observed in the nonlinear stages of the modulational instability process. Such instability was discovered in the late sixties independently by Zakharov [3] and Benjamin and Feir [4] (an interesting and stimulating review on the subject can be found in [5]). The theory is based on the linear stability analysis of a plane wave and predicts that a small perturbation may grow exponentially when εN > 1/ √ 2, where ε = k 0 A 0 is the steepness of the plane wave, with k 0 its wave number and A 0 its amplitude; N = ω 0 /∆Ω is the number of waves under the modulation, with ω 0 the angular frequency corresponding to the wave number k 0 and ∆Ω the angular frequency of the modulation.

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 stability 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 unstable vortex knots break up into vortex rings.

Insight into vortex reconnections in superfluids is presented, making use of analytical results and numerical simulations of the Gross-Pitaevskii model. Universal aspects of the reconnection process are investigated by considering different initial vortex configurations and making use of a recently developed tracking algorithm to reconstruct the vortex filaments. We show that during a reconnection event the vortex lines approach and separate always according to the time scaling $ \delta \sim t^{1/2} $ with prefactors that depend on the vortex configuration. We also investigate the behavior of curvature and torsion close to the reconnection point, demonstrating analytically that the curvature can exhibit a self-similar behavior that might be broken by the development of shocklike structures in the torsion

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.

We present the first ever observation of dark solitons on the surface of water. It takes the form of an amplitude drop of the carrier wave which does not change shape in propagation. The shape and width of the soliton depend on the water depth, carrier frequency, and the amplitude of the background wave. The experimental data taken in a water tank show an excellent agreement with the theory. These results may improve our understanding of the nonlinear dynamics of water waves at finite depths.

After discussing the possible mechanisms of formation of extreme waves on the ocean surface, we consider the modulational instability in crossing seas as a potential mechanism of formation of freak waves. The problem is discussed in terms of a system of two coupled Nonlinear Schroedinger equations. The asymptotic validity of such system is discussed. We show that for some specific angles between the two wave trains, the equations reduce to an integrable system. Besides studying the standard stability analysis, we also consider analytically the maximum amplification factor for an unstable plane wave solution. We find that angles between 20 0 -30 0 are potentially the most probable for establishing a freak wave sea. We show that theoretical expectations are consistent with numerical simulations of the Euler equations. a