# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

2010 , Volume 13 , Issue 4

Special Issue on Oceanography and Mathematics

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2010, 13(4): i-ii doi: 10.3934/dcdsb.2010.13.4i +[Abstract](285) +[PDF](30.2KB)
Abstract:
The motivation for this special edition arose during the conference "Oceanography and Mathematics" organized at the Ecole Normale Supérieure of Paris, in January 2009. The goal of this conference was to bring together mathematicians and physicists working on research fields related to oceanography, and to offer the opportunity for cooperations between researchers of different cultures.
If one thing was certainly felt by all the participants to this conference, it is that the range of open problems that stem from the study of oceanographic phenomena is huge. The difficulties that remain to be understood offer very challenging problems in physical modelling, numerical simulation, and analysis of partial differential equations.

2010, 13(4): 729-738 doi: 10.3934/dcdsb.2010.13.729 +[Abstract](432) +[PDF](604.2KB)
Abstract:
In this paper, we investigate the mechanisms which control the generation of wave-induced mean current vorticity in the surf zone. From the vertically-integrated and time-averaged momentum equations given recently by Smith [21], we obtain a vorticity forcing term related to differential broken-wave energy dissipation. Then, we derive a new equation for the mean current vorticity, from the nonlinear shallow water shock-wave theory. Both approaches are consistent, under the shallow water assumption, but the later gives explicitly the generation term of vorticity, without any ad-hoc parametrization of the broken-wave energy dissipation.
2010, 13(4): 739-758 doi: 10.3934/dcdsb.2010.13.739 +[Abstract](424) +[PDF](294.7KB)
Abstract:
The numerical resolution of the multi-layer shallow water system encounters two additional difficulties with respect to the one-layer system. The first is that the system involves nonconservative terms, and the second is that it is not always hyperbolic. A splitting scheme has been proposed by Bouchut and Morales, that enables to ensure a discrete entropy inequality and the well-balanced property, without any theoretical difficulty related to the loss of hyperbolicity. However, this scheme has been shown to often give wrong solutions. We introduce here a variant of the splitting scheme, that has the overall property of being conservative in the total momentum. It is based on a source-centered hydrostatic scheme for the one-layer shallow water system, a variant of the hydrostatic scheme. The final method enables to treat an arbitrary number $m$ of layers, with arbitrary densities $\rho_1$,...,$\rho_m$, and arbitrary topography. It has no restriction concerning complex eigenvalues, it is well-balanced and it is able to treat vacuum, it satisfies a semi-discrete entropy inequality. The scheme is fast to execute, as is the one-layer hydrostatic method.
2010, 13(4): 759-772 doi: 10.3934/dcdsb.2010.13.759 +[Abstract](399) +[PDF](2328.2KB)
Abstract:
In this paper, shape optimization tools are applied to coastal engineering. A barred beach protection study is described, using a new type of water wave attenuator device, namely geotextile tube, to reduce the mobilization of sediments. A global optimization algorithm, able to pursue beyond local minima, is used to search for the optimum properties of the geotextile tube.The optimal configuration is then post-validated for generated currents.
2010, 13(4): 773-782 doi: 10.3934/dcdsb.2010.13.773 +[Abstract](317) +[PDF](1302.1KB)
Abstract:
The influence of wind on extreme wave events in shallow water is investigated numerically. A series of numerical simulations using a pressure distribution over the steep crests given by the modified Jeffreys' sheltering theory shows that wind blowing over a strongly modulated wave group due to the dispersive focusing of a chirped long wave packet increases the time duration and maximal amplitude of the extreme wave event. These results are coherent with those obtained within the framework of deep water. However, steep wave events are less unstable to wind perturbation in shallow water than in deep water.
2010, 13(4): 783-798 doi: 10.3934/dcdsb.2010.13.783 +[Abstract](410) +[PDF](236.0KB)
Abstract:
We study the low Mach number limit for the compressible Navier-Stokes system supplemented with Navier's boundary condition on an unbounded domain with compact boundary. Our main result asserts that the velocities converge pointwise to a solenoidal vector field - a weak solution of the incompressible Navier-Stokes system - while the fluid density becomes constant. The proof is based on a variant of local energy decay property for the underlying acoustic equation established by Kato.
2010, 13(4): 799-818 doi: 10.3934/dcdsb.2010.13.799 +[Abstract](459) +[PDF](619.1KB)
Abstract:
The classical dam break problem has become the de factostandard in validating the nonlinear shallow water equations solvers. Moreover, the NSWE are widely used for flooding simulations. While applied mathematics community is essentially focused on developing new numerical schemes, we tried to examine the validity of the mathematical model under consideration. The main purpose of this study is to check the pertinence of the NSWE for flooding processes. From the mathematical point of view, the answer is not obvious since all derivation procedures assumes the total water depth positivity. We performed a comparison between the two-fluid Navier-Stokes simulations and the NSWE solved analytically and numerically. Several conclusions are drawn out and perspectives for future research are outlined.
2010, 13(4): 819-840 doi: 10.3934/dcdsb.2010.13.819 +[Abstract](475) +[PDF](4033.2KB)
Abstract:
This review paper is devoted to a presentation of recent progress in wave turbulence. I first present the context and state of the art of this field of research both experimentally and theoretically. I then focus on the case of wave turbulence on the surface of a fluid, and I discuss the main results obtained by our group: caracterization of the gravity and capillary wave turbulence regimes, the first observation of intermittency in wave turbulence, the occurrence of strong fluctuations of injected power in the fluid, the observation of a pure capillary wave turbulence in low gravity environment and the observation of magnetic wave turbulence on the surface of a ferrofluid. Finally, open questions in wave turbulence are discussed.
2010, 13(4): 841-857 doi: 10.3934/dcdsb.2010.13.841 +[Abstract](294) +[PDF](399.9KB)
Abstract:
This works aims at studying the impact of the cosine terms of the Coriolis force, that are usually neglected in geophysical fluid dynamics, leading to the so-called traditional approximation. Mathematical well-posedness arguments for simplified models, as well as numerical simulations, are presented in order to suggest the use of these terms in large scale ocean modelling.
2010, 13(4): 859-872 doi: 10.3934/dcdsb.2010.13.859 +[Abstract](299) +[PDF](205.1KB)
Abstract:
This text represents the content of a talk given by the second author at ENS Paris on January 28, 2009 at the conference "Mathematics and Oceanography". We are grateful to David Gérard-Varet, David Lannes and Laure Saint-Raymond for the kind invitation to present our recent work on the water waves. The proofs of the results announced here will appear in [18].

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