Asymptotic models for the generation of internal waves by a moving ship, and the dead-water phenomenon

This paper deals with the dead-water phenomenon, which occurs when a ship sails in a stratified fluid, and experiences an important drag due to waves below the surface. More generally, we study the generation of internal waves by a disturbance moving at constant speed on top of two layers of fluids of different densities. Starting from the full Euler equations, we present several nonlinear asymptotic models, in the long wave regime. These models are rigorously justified by consistency or convergence results. A careful theoretical and numerical analysis is then provided, in order to predict the behavior of the flow and in which situations the dead-water effect appears.Comment: To appear in Nonlinearit

Generation of upstream advancing solitons by moving disturbances

This study investigates the recently identified phenomenon whereby a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, periodically, a succession of solitary waves, advancing upstream of the disturbance in procession, while a train of weakly nonlinear and weakly dispersive waves develops downstream of a region of depressed water surface trailing just behind the disturbance. This phenomenon was numerically discovered by Wu & Wu (1982) based on the generalized Boussinesq model for describing two-dimensional long waves generated by moving surface pressure or topography. In a joint theoretical and experimental study, Lee (1985) found a broad agreement between the experiment and two theoretical models, the generalized Boussinesq and the forced Korteweg de Vries (fKdV) equations, both containing forcing functions. The fKdV model is applied in the present study to explore the basic mechanism underlying the phenomenon. To facilitate the analysis of the stability of solutions of the initial-boundary-value problem of the fKdV equation, a family of forced steady solitary waves is found. Any such solution, if once established, will remain permanent in form in accordance with the uniqueness theorem shown here. One of the simplest of the stationary solutions, which is a one-parameter family and can be scaled into a universal similarity form, is chosen for stability calculations. As a test of the computer code, the initially established stationary solution is found to be numerically permanent in form with fractional uncertainties of less than 2% after the wave has traversed, under forcing, the distance of 600 water depths. The other numerical results show that when the wave is initially so disturbed as to have to rise from the rest state, which is taken as the initial value, the same phenomenon of the generation of upstream-advancing solitons is found to appear, with a definite time period of generation. The result for this similarity family shows that the period of generation, T[sub]S, and the scaled amplitude [alpha] of the solitons so generated are related by the formula T[sub]S = const [alpha]^-3/2. This relation is further found to be in good agreement with the first-principle prediction derived here based on mass, momentum and energy considerations of the fKdV equation

1980 summer study program in geophysical fluid dynamics : coherent features in geophysical flows

Four principal lecturers shored the task of presenting the subject "Coherent Features in Geophysical Flows" to the participants of the twenty-second geophysical fluid dynamics summer program. Glenn Flierl introduced the topic and the Kortweg-de Vries equation via a model of finite amplitude motions on the beta plane. He extended the analysis to more complex flows in the ocean and the atmosphere and in the process treated motions of very large amplitude. Larry Redekopp's three lectures summarized an extensive body of the mathematical literature on coherent features. Andrew Ingersoll focussed on the many fascinating features in Jupiter's atmosphere. Joseph Keller supplemented an interesting summary of laboratory observations with suggestive models for treating the flows.Office of Naval Research under Contract N00014-79-C-067

Free surface flows over submerged obstructions

Steady and unsteady two-dimensional free surface flows subjected to one or multiple disturbances are considered. Flow configurations involving either a single fluid or two layers of fluid of different but constant densities, are examined. Both the effects of gravity and surface tension are included. Fully nonlinear boundary integral equation techniques based on Cauchy’s integral formula are used to derive integro-differential equations to model the problem. The integro-differential equations are discretised and solved iteratively using Newton’s method. Both forced solitary waves and critical flow solutions, where the flow upstream is subcritical and downstream is supercritical, are obtained. The behaviour of the forced wave is determined by the Froude and Bond numbers and the orientation of the disturbance. When a second disturbance is placed upstream in the pure gravity critical case, trapped waves have been found between the disturbances. However, when surface tension is included, trapped waves are shown only to exist for very small values of the Bond number. Instead, it is shown that the disturbance must be placed downstream in the gravity-capillary case to see trapped waves. The stability of these critical hydraulic fall solutions is examined. It is shown that the hydraulic fall is stable, but the trapped wave solutions are only stable in the pure gravity case. Critical, flexural-gravity flows, where a thin sheet of ice rests on top of the fluid are then considered. The flows in the flexural-gravity and gravity-capillary cases are shown to be similar. These similarities are investigated, and the physical significance of both configurations, examined. When two fluids are considered, the situation is more complex. The rigid lid approximation is assumed, and four types of critical flow are investigated. Trapped wave solutions are found to exist in some cases, depending on the Froude number in the lower layer

Mathematical Theory of Water Waves

Water waves, that is waves on the surface of a fluid (or the interface between different fluids) are omnipresent phenomena. However, as Feynman wrote in his lecture, water waves that are easily seen by everyone, and which are usually used as an example of waves in elementary courses, are the worst possible example; they have all the complications that waves can have. These complications make mathematical investigations particularly challenging and the physics particularly rich. Indeed, expertise gained in modelling, mathematical analysis and numerical simulation of water waves can be expected to lead to progress in issues of high societal impact (renewable energies in marine environments, vorticity generation and wave breaking, macro-vortices and coastal erosion, ocean shipping and near-shore navigation, tsunamis and hurricane-generated waves, floating airports, ice-sea interactions, ferrofluids in high-technology applications, ...). The workshop was mostly devoted to rigorous mathematical theory for the exact hydrodynamic equations; numerical simulations, modelling and experimental issues were included insofar as they had an evident synergy effect

On the Korteweg-de Vries approximation for uneven bottoms

In this paper we focus on the water waves problem for uneven bottoms on a two-dimensionnal domain. Starting from the symmetric Boussinesq systems derived in [Chazel, Influence of topography on long water waves, 2007], we recover the uncoupled Korteweg-de Vries (KdV) approximation justified by Schneider and Wayne for flat bottoms, and by Iguchi in the context of bottoms tending to zero at infinity at a substantial rate. The goal of this paper is to investigate the validity of this approximation for more general bathymetries. We exhibit two kinds of topography for which this approximation diverges from the Boussinesq solutions. A topographically modified KdV approximation is then proposed to deal with such bathymetries. Finally, all the models involved are numerically computed and compared

Quasi-Two-Dimensional Dynamics of Plasmas and Fluids

In the lowest order of approximation quasi-twa-dimensional dynamics of planetary atmospheres and of plasmas in a magnetic field can be described by a common convective vortex equation, the Charney and Hasegawa-Mirna (CHM) equation. In contrast to the two-dimensional Navier-Stokes equation, the CHM equation admits "shielded vortex solutions" in a homogeneous limit and linear waves ("Rossby waves" in the planetary atmosphere and "drift waves" in plasmas) in the presence of inhomogeneity. Because of these properties, the nonlinear dynamics described by the CHM equation provide rich solutions which involve turbulent, coherent and wave behaviors. Bringing in non ideal effects such as resistivity makes the plasma equation significantly different from the atmospheric equation with such new effects as instability of the drift wave driven by the resistivity and density gradient. The model equation deviates from the CHM equation and becomes coupled with Maxwell equations. This article reviews the linear and nonlinear dynamics of the quasi-two-dimensional aspect of plasmas and planetary atmosphere starting from the introduction of the ideal model equation (CHM equation) and extending into the most recent progress in plasma turbulence.U. S. Department of Energy DE-FG05-80ET-53088Ministry of Education, Science and Culture of JapanFusion Research Cente

2009 program of studies : nonlinear waves

The fiftieth year of the program was dedicated to Nonlinear Waves, a topic with many applications in geophysical fluid dynamics. The principal lectures were given jointly by Roger Grimshaw and Harvey Segur and between them they covered material drawn from fundamental theory, fluid experiments, asymptotics, and reaching all the way to detailed applications. These lectures set the scene for the rest of the summer, with subsequent daily lectures by staff and visitors on a wide range of topics in GFD. It was a challenge for the fellows and lecturers to provide a consistent set of lecture notes for such a wide-ranging lecture course, but not least due to the valiant efforts of Pascale Garaud, who coordinated the write-up and proof-read all the notes, we are very pleased with the final outcome contained in these pages. This year’s group of eleven international GFD fellows was as diverse as one could get in terms of gender, origin, and race, but all were unified in their desire to apply their fundamental knowledge of fluid dynamics to challenging problems in the real world. Their projects covered a huge range of physical topics and at the end of the summer each student presented his or her work in a one-hour lecture. As always, these projects are the heart of the research and education aspects of our summer study.Funding was provided by the National Science Foundation through Grant No. OCE-0824636 and the Office of Naval Research under Contract No. N00014-09-10844

On the critical free-surface flow over localised topography

Flow over bottom topography at critical Froude number is examined with a focus on steady, forced solitary wave solutions with algebraic decay in the far-field, and their stability. Using the forced Korteweg-de Vries (fKdV) equation the weakly-nonlinear steady solution space is examined in detail for the particular case of a Gaussian dip using a combination of asymptotic analysis and numerical computations. Non-uniqueness is established and a seemingly infinite set of steady solutions is uncovered. Non-uniqueness is also demonstrated for the fully nonlinear problem via boundary-integral calculations. It is shown analytically that critical flow solutions have algebraic decay in the far-field both for the fKdV equation and for the fully nonlinear problem and, moreover, that the leading-order form of the decay is the same in both cases. The linear stability of the steady fKdV solutions is examined via eigenvalue computations and by a numerical study of the initial value fKdV problem. It is shown that there exists a linearly stable steady solution in which the deflection from the otherwise uniform surface level is everywhere negative

Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws

We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.Comment: Revision from v2; 57 pages, 19 figure