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With increasing subsea oil and gas activities, the safety is challenged by accidental gas release. This can be caused by leakage from gas transport pipelines or blowouts from oil and gas wells. Risk assessment of such events is associated to the correct prediction of gas flux and gas distribution through the ocean surface and the resulting surface flows. A quantitative multiphase CFD model can satisfy such needs. Bubbles can be tracked by discrete phase model (DPM), using a parcel-based Lagrangian approach. Capturing the free surface formed by surfacing bubble plumes can be handled by a volume of fluid (VOF) model. This constitutes an Eulerian-Lagrangian model framework combining the DPM and VOF models. The model is presented and validated by experiments of a gas release in 7 m deep test basin. Results from modelling and experiments are consistent.

Discrete dynamical systems constitute an elegant branch of nonlinear science, where ingenious techniques provide penetrating insight for vibrations and wave motion on lattices. In terms of applications, such systems can model oscillators with hard quartic nonlinearities and switching of optical pulses on discrete arrays. A two-component Hirota system is investigated as an extension of the widely studied Ablowitz-Ladik equation by including discrete third order dispersion. Breathers (periodic pulsating modes) are derived analytically, and are used to establish conservation laws. Rogue waves (unexpectedly large displacements from equilibrium configurations) exhibit unusual features in undergoing oscillations above and below the mean level, and may even reverse polarity. Coupling produces new regimes of modulation instabilities for discrete evolution equations. The robustness of these novel rogue waves, in terms of sensitivity to initial conditions, is elucidated by numerical simulations. Self-phase modulations and cross-phase modulations of the same or opposite signs will generate nonlinear corrections of the frequency, due to the intensity of the wave train itself and the one in the accompanying waveguide respectively. Such effects have a crucial influence on the evolution of discrete and continuous multi-component dynamical systems.

Modulation instability, breather formation, and the Fermi–Pasta–Ulam–Tsingou recurrence (FPUT) phenomena are studied in this article. Physically, such nonlinear systems arise when the medium is slightly anisotropic, e.g., optical fibers with weak birefringence where the slowly varying pulse envelopes are governed by these coherently coupled Schrödinger equations. The Darboux transformation is used to calculate a class of breathers where the carrier envelope depends on the transverse coordinate of the Schrödinger equations. A “cascading mechanism” is utilized to elucidate the initial stages of FPUT. More precisely, higher order nonlinear terms that are exponentially small initially can grow rapidly. A breather is formed when the linear mode and higher order ones attain roughly the same magnitude. The conditions for generating various breathers and connections with modulation instability are elucidated. The growth phase then subsides and the cycle is repeated, leading to FPUT. Unequal initial conditions for the two waveguides produce symmetry breaking, with “eye-shaped” breathers in one waveguide and “four-petal” modes in the other. An analytical formula for the time or distance of breather formation for a two-waveguide system is proposed, based on the disturbance amplitude and instability growth rate. Excellent agreement with numerical simulations is achieved. Furthermore, the roles of modulation instability for FPUT are elucidated with illustrative case studies. In particular, depending on whether the second harmonic falls within the unstable band, FPUT patterns with one single or two distinct wavelength(s) are observed. For applications to temporal optical waveguides, the present formulation can predict the distance along a weakly birefringent fiber needed to observe FPUT.