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We investigate a new two-dimensional compressible Navier-Stokes hydrodynamic model design to explain and study large scale ice swirls formation at the surface of the ocean. The linearized model generates a basis of Bessel solutions from where various types of spiral patterns can be generated and their evolution and stability in time analyzed. By restricting the nonlinear system of equations to its quadratic terms we obtain swirl solutions emphasizing logarithmic spiral geometry. The resulting solutions are analyzed and validated using three mathematical approaches: one predicting the formation of patterns as Townes solitary modes, another approach mapping the nonlinear system into a sine-Gordon equation, and a third approach uses a series expansion. Pure radial, azimuthal and spiral modes are obtained from the fully nonlinear equations. Combinations of multiple-spiral solutions are also obtained, matching the experimental observations. The nonlinear stability of the spiral patterns is analyzed by Arnolds convexity method, and the Hamiltonian of the solutions is plotted versus some order parameters showing the existence of geometric phase transitions.
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We discover new type of interference patterns generated in the focusing nonlinear Schrodinger equation (NLSE) with localised periodic initial conditions. At special conditions, found in the present work, these patterns exhibit novel chess-board-like spatio-temporal structures which can be observed as the outcome of collision of two breathers. The infinitely extended chess-board-like patterns correspond to the continuous spectrum bands of the NLSE theory. More complicated patterns can be observed when the initial condition contains several localised periodic swells. These patterns can be observed in a variety of physical situations ranging from optics and hydrodynamics to Bose-Einstein condensates and plasma.
The multifractal theory of turbulence is used to investigate the energy cascade in the Northwestern Atlantic ocean. The statistics of singularity exponents of velocity gradients computed from in situ measurements are used to show that the anomalous scaling of the velocity structure functions at depths between 50 ad 500 m has a linear dependence on the exponent characterizing the strongest velocity gradient, with a slope that decreases with depth. Since the distribution of exponents is asymmetric about the mode at all depths, we use an infinitely divisible asymmetric model of the energy cascade, the log-Poisson model, to derive the functional dependence of the anomalous scaling with dissipation. Using this model we can interpret the vertical change of the linear slope as a change in the energy cascade.
The simulation of large open water surface is challenging for a uniform volumetric discretization of the Navier-Stokes equation. The water splashes near moving objects, which height field methods for water waves cannot capture, necessitates high resolution simulation such as the Fluid-Implicit-Particle (FLIP) method. On the other hand, FLIP is not efficient for the long-lasting water waves that propagates to long distances, which requires sufficient depth for correct dispersion relationship. This paper presents a new method to tackle this dilemma through an efficient hybridization of volumetric and surface-based advection-projection discretizations. We design a hybrid time-stepping algorithm that combines a FLIP domain and an adaptively remeshed Boundary Element Method (BEM) domain for the incompressible Euler equations. The resulting framework captures the detailed water splashes near moving objects with FLIP, and produces convincing water waves with correct dispersion relationship at modest additional cost.
We calculate the rate of ocean waves energy dissipation due to whitecapping by numerical simulation of deterministic phase resolving model for dynamics of ocean surface. Two independent numerical experiments are performed. First, we solve the $3D$ Hamiltonian equation that includes three- and four-wave interactions. This model is valid for moderate values of surface steepness only, $mu < 0.09$. Then we solve the exact Euler equation for non-stationary potential flow of an ideal fluid with a free surface in $2D$ geometry. We use the conformal mapping of domain filled with fluid onto the lower half-plane. This model is applicable for arbitrary high levels of steepness. The results of both experiments are close. The whitecapping is the threshold process that takes place if the average steepness $mu > mu_{cr} simeq 0.055$. The rate of energy dissipation grows dramatically with increasing of steepness. Comparison of our results with dissipation functions used in the operational models of wave forecasting shows that these models overestimate the rate of wave dissipation by order of magnitude for typical values of steepness.
Soliton and breather solutions of the nonlinear Schrodinger equation (NLSE) are known to model localized structures in nonlinear dispersive media such as on the water surface. One of the conditions for an accurate propagation of such exact solutions is the proper generation of the exact initial phase-shift profile in the carrier wave, as defined by the NLSE envelope at a specific time or location. Here, we show experimentally the significance of such initial exact phase excitation during the hydrodynamic propagation of localized envelope solitons and breathers, which modulate a plane wave of constant amplitude (finite background). Using the example of stationary black solitons in intermediate water depth and pulsating Peregrine breathers in deep-water, we show how these localized envelopes disintegrate while they evolve over a long propagation distance when the initial phase shift is zero. By setting the envelope phases to zero, the dark solitons will disintegrate into two gray-type solitons and dispersive elements. In the case of the doubly-localized Peregrine breather the maximal amplification is considerably retarded; however locally, the shape of the maximal focused wave measured together with the respective signature phase-shift are almost identical to the exact analytical Peregrine characterization at its maximal compression location. The experiments, conducted in two large-scaled shallow-water as well as deep-water wave facilities, are in very good agreement with NLSE simulations for all cases.
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