Paul A. Milewski

Dynamics of steep two-dimensional gravity-capillary solitary waves

In this paper, the unsteady evolution of two-dimensional fully nonlinear free-surface gravity–capillary solitary waves is computed numerically in infinite depth. Gravity–capillary wavepacket-type solitary waves were found previously for the full Euler equations, bifurcating from the minimum of the linear dispersion relation. Small and moderate amplitude elevation solitary waves, which were known to be linearly unstable, are shown to evolve into stable depression solitary waves, together with a radiated wave field. Depression waves and certain large amplitude elevation waves were found to be robust to numerical perturbations. Two kinds of collisions are computed: head-on collisions whereby the waves are almost unchanged, and overtaking collisions which are either almost elastic if the wave amplitudes are both large or destroy the smaller wave in the case of a small amplitude wave overtaking a large one.

Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations

After we derive the Serre system of equations of water wave theory from a generalized variational principle, we present some of its structural properties. We also propose a robust and accurate finite volume scheme to solve these equations in one horizontal dimension. The numerical discretization is validated by comparisons with analytical, experimental data or other numerical solutions obtained by a highly accurate pseudo-spectral method.

A PseudoSpectral Procedure for the Solution of Nonlinear Wave Equations with Examples from Free-Surface Flows

An algorithm for the solution of general isotropic nonlinear wave equations is presented. The algorithm is based on a symmetric factorization of the linear part of the wave operator, followed by its exact integration through an integrating factor in spectral space. The remaining nonlinear and forcing terms can be handled with any standard pseudospectral procedure. Solving the linear part of the wave operator exactly effectively eliminates the stiffness of the original problem, characterized by a wide range of temporal scales. The algorithm is tested and applied to several problems of three-dimensional long surface waves: solitary wave propagation, interaction, diffraction, and the generation of waves by flow over slowly varying bottom topography. Other potential applications include waves in rotating and stratified flows and wave interaction with more pronounced topographic features.

The Generation and Evolution of Lump Solitary Waves in Surface-Tension-Dominated Flows

Three-dimensional solitary waves or lump solitons are known to be solutions to the Kadomtsev-Petviashvili I equation, which models small-amplitude shallow-water waves when the Bond number is greater than 1 . Recently, Pego and Quintero presented a proof of the existence of such waves for the Benney-Luke equation with surface tension. Here we establish an explicit connection between the lump solitons of these two equations and numerically compute the Benney- Luke lump solitons and their speed-amplitude relation. Furthermore, we numerically collide two Benney-Luke lump solitons to illustrate their soliton wave character. Finally, we study the flow over an obstacle near the linear shallow-water speed and show that three-dimensional lump solitons are periodically generated.

Dynamics of gravity-capillary solitary waves in deep water

The dynamics of solitary gravity–capillary water waves propagating on the surface of a three-dimensional fluid domain is studied numerically. In order to accurately compute complex time-dependent solutions, we simplify the full potential flow problem by using surface variables and taking a particular cubic truncation possessing a Hamiltonian with desirable properties. This approximation agrees remarkably well with the full equations for the bifurcation curves, wave profiles and the dynamics of solitary waves for a two-dimensional fluid domain, and with higher-order truncations in three dimensions. Fully localized solitary waves are then computed in the three-dimensional problem and the stability and interaction of both line and localized solitary waves are investigated via numerical time integration of the equations. There are many solitary wave branches, indexed by their finite energy as their amplitude tends to zero. The dynamics of the solitary waves is complex, involving nonlinear focusing of wavepackets, quasi-elastic collisions, and the generation of propagating, spatially localized, time-periodic structures akin to breathers.

Merging and wetting driven by surface tension

The surface tension driven merging of two wedge-shaped regions of fluid, and the wetting of a wedge shaped solid, are analyzed. Following the work of Keller and Miksis in 1983, initial conditions are chosen so that the flows and their free surfaces are self-similar at all times after the init ial contact. Then the configuration magnifies by the factor t 2=3 and the fluid velocity at the point x=t 2=3 decays like t 1=3 , where the origin of x andt are the point and time of contact. In the merging problem the vertices of the two wedges of fluid are initially in contact. In the wetting problem, the vertex of a wedge of fluid is initially at the corner of the solid. The motions and free surfaces are found numerically. These results complement those of Keller and Miksis for the wetting of a single flat surface and for the rebound of a wedge of fluid after it pinches off from another body of fluid.

Resonant Wave Interactions in the Equatorial Waveguide

Weakly nonlinear interactions among equatorial waves have been explored in this paper using the adiabatic version of the equatorial � -plane primitive equations in isobaric coordinates. Assuming rigid lid vertical boundary conditions, the conditions imposed at the surface and at the top of the troposphere were expanded in a Taylor series around two isobaric surfaces in an approach similar to that used in the theory of surface–gravity waves in deep water and capillary–gravity waves. By adopting the asymptotic method of multiple time scales, the equatorial Rossby, mixed Rossby–gravity, inertio-gravity, and Kelvin waves, as well as their vertical structures, were obtained as leading-order solutions. These waves were shown to interact resonantly in a triad configuration at the O(�) approximation. The resonant triads whose wave components satisfy a resonance condition for their vertical structures were found to have the most significant interactions, although this condition is not excluding, unlike the resonant conditions for the zonal wavenumbers and meridional modes. Thus, the analysis has focused on such resonant triads. In general, it was found that for these resonant triads satisfying the resonance condition in the vertical direction, the wave with the highest absolute frequency always acts as an energy source (or sink) for the remaining triad components, as usually occurs in several other physical problems in fluid dynamics. In addition, the zonally symmetric geostrophic modes act as catalyst modes for the energy exchanges between two dispersive waves in a resonant triad. The integration of the reduced asymptotic equations for a single resonant triad shows that, for the initial mode amplitudes characterizing realistic magnitudes of atmospheric flow perturbations, the modes in general exchange energy on low-frequency (intraseasonal and/or even longer) time scales, with the interaction period being dependent upon the initial mode amplitudes. Potential future applications of the present theory to the real atmosphere with the inclusion of diabatic forcing, dissipation, and a more realistic background state are also discussed.

EVOLUTION OF PERIODICITY IN PERIODICAL CICADAS

Periodical cicadas present numerous puzzles for biologists. First, their period is fixed, with individuals emerging as adults precisely after either 13 or 17 years (depending on species). Second, even when there are multiple species of either 13- or 17-year cicadas at the same location, only one or rarely two broods (cohorts) co-occur, so that periodical cicada adults appear episodically. Third, the 13- or 17-year periods of cicadas suggest there is something important about prime numbers. Finally, single broods can dominate large areas, with geographical boundaries of broods remaining generally stable through time. While previous mathematical models have been used to investigate some of these puzzles individually, here we investigate them all simultaneously. Unlike previous models, we take an explicitly evolutionary approach. Although not enough information is known about periodical cicadas to draw firm conclusions, the theoretical arguments favor a combination of predator satiation and nymph competition as being key to the evolution of strictly fixed periods and occurrence of only one brood at most geographical locations. Despite ecological mechanisms that can select for strictly fixed periods, there seem to be no plausible ecological mechanisms that select for periods being prime numbers. This suggests that the explanation for prime-numbered periods, rather than just fixed periods, may reside in physiological or genetic mechanisms or constraints.

SIMULATION OF WAVE INTERACTIONS AND TURBULENCE IN ONE-DIMENSIONAL WATER WAVES ∗

The weak- or wave-turbulence problemconsists of finding statistical states of a large number of interacting waves. These states are obtained by forcing and dissipating a conservative dispersive wave equation at disparate scales to model physical forcing and dissipation, and by pre- dicting the spectrum, often as a Kolmogorov-like power law, at intermediate scales. The mechanism for energy transfer in such systems is usually triads or quartets of waves. Here, we first derive a small-amplitude nonlinear dispersive equation (a finite-depth Benney-Luke-type equation), which we validate, analytically and numerically, by showing that it correctly captures the main determinis- tic aspects of gravity wave interactions: resonant quartets, Benjamin-Feir-type wave-packet stability, and wave-mean flow interactions. Numerically, this equation is easier to integrate than either the full problemor the Zakharov integral equation. Som e additional features of wave interaction are discussed such as harmonic generation in shallow water. We then perform long time computations on the forced-dissipated model equation and compute statistical quantities of interest, which we compare to existing predictions. The forward cascade yields a spectrum close to the prediction of Zakharov, and the inverse cascade does not.

Numerical study of interfacial solitary waves propagating under an elastic sheet.

Steady solitary and generalized solitary waves of a two-fluid problem where the upper layer is under a flexible elastic sheet are considered as a model for internal waves under an ice-covered ocean. The fluid consists of two layers of constant densities, separated by an interface. The elastic sheet resists bending forces and is mathematically described by a fully nonlinear thin shell model. Fully localized solitary waves are computed via a boundary integral method. Progression along the various branches of solutions shows that barotropic (i.e. surface modes) wave-packet solitary wave branches end with the free surface approaching the interface. On the other hand, the limiting configurations of long baroclinic (i.e. internal) solitary waves are characterized by an infinite broadening in the horizontal direction. Baroclinic wave-packet modes also exist for a large range of amplitudes and generalized solitary waves are computed in a case of a long internal mode in resonance with surface modes. In contrast to the pure gravity case (i.e without an elastic cover), these generalized solitary waves exhibit new Wilton-ripple-like periodic trains in the far field.