6,155 research outputs found
Asymptotic shallow water models for internal waves in a two-fluid system with a free surface
In this paper, we derive asymptotic models for the propagation of two and
three-dimensional gravity waves at the free surface and the interface between
two layers of immiscible fluids of different densities, over an uneven bottom.
We assume the thickness of the upper and lower fluids to be of comparable size,
and small compared to the characteristic wavelength of the system (shallow
water regimes). Following a method introduced by Bona, Lannes and Saut based on
the expansion of the involved Dirichlet-to-Neumann operators, we are able to
give a rigorous justification of classical models for weakly and strongly
nonlinear waves, as well as interesting new ones. In particular, we derive
linearly well-posed systems in the so called Boussinesq/Boussinesq regime.
Furthermore, we establish the consistency of the full Euler system with these
models, and deduce the convergence of the solutions.Comment: 32 pages, 4 figure
Unsteady undular bores in fully nonlinear shallow-water theory
We consider unsteady undular bores for a pair of coupled equations of
Boussinesq-type which contain the familiar fully nonlinear dissipationless
shallow-water dynamics and the leading-order fully nonlinear dispersive terms.
This system contains one horizontal space dimension and time and can be
systematically derived from the full Euler equations for irrotational flows
with a free surface using a standard long-wave asymptotic expansion.
In this context the system was first derived by Su and Gardner. It coincides
with the one-dimensional flat-bottom reduction of the Green-Naghdi system and,
additionally, has recently found a number of fluid dynamics applications other
than the present context of shallow-water gravity waves. We then use the
Whitham modulation theory for a one-phase periodic travelling wave to obtain an
asymptotic analytical description of an undular bore in the Su-Gardner system
for a full range of "depth" ratios across the bore. The positions of the
leading and trailing edges of the undular bore and the amplitude of the leading
solitary wave of the bore are found as functions of this "depth ratio". The
formation of a partial undular bore with a rapidly-varying finite-amplitude
trailing wave front is predicted for ``depth ratios'' across the bore exceeding
1.43. The analytical results from the modulation theory are shown to be in
excellent agreement with full numerical solutions for the development of an
undular bore in the Su-Gardner system.Comment: Revised version accepted for publication in Phys. Fluids, 51 pages, 9
figure
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
Decoupled and unidirectional asymptotic models for the propagation of internal waves
We study the relevance of various scalar equations, such as inviscid
Burgers', Korteweg-de Vries (KdV), extended KdV, and higher order equations (of
Camassa-Holm type), as asymptotic models for the propagation of internal waves
in a two-fluid system. These scalar evolution equations may be justified with
two approaches. The first method consists in approximating the flow with two
decoupled, counterpropagating waves, each one satisfying such an equation. One
also recovers homologous equations when focusing on a given direction of
propagation, and seeking unidirectional approximate solutions. This second
justification is more restrictive as for the admissible initial data, but
yields greater accuracy. Additionally, we present several new coupled
asymptotic models: a Green-Naghdi type model, its simplified version in the
so-called Camassa-Holm regime, and a weakly decoupled model. All of the models
are rigorously justified in the sense of consistency
Fast and slow resonant triads in the two layer rotating shallow water equations
In this paper we examine triad resonances in a rotating shallow water system
when there are two free interfaces. This allows for an examination in a
relatively simple model of the interplay between baroclinic and barotropic
dynamics in a context where there is also a geostrophic mode. In contrast to
the much-studied one-layer rotating shallow water system, we find that as well
as the usual slow geostrophic mode, there are now two fast waves, a barotropic
mode and a baroclinic mode. This feature permits triad resonances to occur
between three fast waves, with a mixture of barotropic and baroclinic modes, an
aspect which cannot occur in the one-layer system. There are now also two
branches of the slow geostrophic mode with a repeated branch of the dispersion
relation. The consequences are explored in a derivation of the full set of
triad interaction equations, using a multi-scale asymptotic expansion based on
a small amplitude parameter. The derived nonlinear interaction coefficients are
confirmed using energy and enstrophy conservation. These triad interaction
equations are explored with an emphasis on the parameter regime with small
Rossby and Froude numbers
On the rigid-lid approximation for two shallow layers of immiscible fluids with small density contrast
The rigid-lid approximation is a commonly used simplification in the study of
density-stratified fluids in oceanography. Roughly speaking, one assumes that
the displacements of the surface are negligible compared with interface
displacements. In this paper, we offer a rigorous justification of this
approximation in the case of two shallow layers of immiscible fluids with
constant and quasi-equal mass density. More precisely, we control the
difference between the solutions of the Cauchy problem predicted by the
shallow-water (Saint-Venant) system in the rigid-lid and free-surface
configuration. We show that in the limit of small density contrast, the flow
may be accurately described as the superposition of a baroclinic (or slow)
mode, which is well predicted by the rigid-lid approximation; and a barotropic
(or fast) mode, whose initial smallness persists for large time. We also
describe explicitly the first-order behavior of the deformation of the surface,
and discuss the case of non-small initial barotropic mode.Comment: Compared to version 2, typos have been corrected and additional
remarks/discussion added. To appear in Journal of Nonlinear Scienc
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