6.—Smooth Fronted Waves in the Shallow Water Approximation

1975 ◽  
Vol 73 ◽  
pp. 107-116 ◽  
Author(s):  
Alan Jeffrey
Keyword(s):  
Shallow Water ◽  
Mathematical Problem ◽  
Water Velocity ◽  
Horizontal Component ◽  
Second Derivative ◽  
Surface Slope ◽  
Shallow Water Theory ◽  
Discontinuity Line ◽  
Shallow Water Approximation

SynopsisThis paper examines the mathematical problem of the propagation of a smooth fronted wavein the context of shallow water theory. Here, a smooth fronted wave will be taken to be one in which the surface slope is continuous across a line in the free-surface, while the second derivative of the surface slope is discontinuous across that same line. This discontinuity line in the surface then plays the role of the wavefront. After establishing that such wavefronts propagate along the characteristics, and deriving the appropriate transport equations, the explicit form is found for the acceleration with respect to distance of the horizontal component of the water velocity of the surface immediately behind the wavefront as a function of position and seabed profile when the wave propagates into still water. The result is then used to prove that in this approximation such a wave cannever break immediately behind the wavefront before the shore line is reached.

On the excitation of edge waves on beaches

1977 ◽  
Vol 79 (2) ◽  
pp. 273-287 ◽  
Author(s):  
A. A. Minzoni ◽  
G. B. Whitham
Keyword(s):  
Shallow Water ◽  
Incident Wave ◽  
Water Wave ◽  
Wave Train ◽  
Wave Theory ◽  
Edge Waves ◽  
Shallow Water Theory ◽  
Shallow Water Approximation ◽  
Complete Discussion ◽  
Standing Edge Waves

The excitation of standing edge waves of frequency ½ω by a normally incident wave train of frequency ω has been discussed previously (Guza & Davis 1974; Guza & Inman 1975; Guza & Bowen 1976) on the basis of shallow-water theory. Here the problem is formulated in the full water-wave theory without making the shallow-water approximation and solved for beach angles β = π/2N, where N is an integer. The work confirms the shallow-water results in the limit N [Gt ] 1, shows the effect of larger beach angles and allows a more complete discussion of some aspects of the problem.

Evolution and decay of a rotating flow over random topography

2009 ◽  
Vol 642 ◽  
pp. 159-180 ◽  
Author(s):  
L. ZAVALA SANSÓN ◽  
A. GONZÁLEZ-VILLANUEVA ◽  
L. M. FLORES
Keyword(s):  
Shallow Water ◽  
Two Dimensional ◽  
Rotating System ◽  
Rotation Periods ◽  
Horizontal Length ◽  
Shallow Water Approximation ◽  
Vorticity Production ◽  
The Right ◽  
Theoretical Considerations

The evolution and decay of a homogeneous flow over random topography in a rotating system is studied by means of numerical simulations and theoretical considerations. The analysis is based on a quasi-two-dimensional shallow-water approximation, in which the horizontal divergence is explicitly different from zero, and topographic variations are not restricted to be much smaller than the mean depth, as in quasi-geostrophic dynamics. The results are examined by comparing the evolution of a turbulent flow over different random bottom topographies characterized by a specific horizontal scale, or equivalently, a given mean slope. As in two-dimensional turbulence, the energy of the flow is transferred towards larger scales of motion; after some rotation periods, however, the process is halted as the flow pattern becomes aligned along the topographic contours with shallow water to the right. The quasi-steady state reached by the flow is characterized by a nearly linear relationship between potential vorticity and transport function in most parts of the domain, which is justified in terms of minimum-enstrophy arguments. It is found that global energy decays faster for topographies with shorter horizontal length scales due to more effective viscous dissipation. In addition, some comparisons between simulations based on the shallow-water and quasi-geostrophic formulations are carried out. The role of solid boundaries is also examined: it is shown that vorticity production at no-slip walls contributes for a slight disorganization of the flow.

scholarly journals Shallow water waves in a rotating rectangular basin

2000 ◽  
Vol 24 (10) ◽  
pp. 649-661 ◽  
Author(s):  
Mohamed Atef Helal
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Equations Of Motion ◽  
System Of Nonlinear Equations ◽  
System Of Equations ◽  
Order Of Approximation ◽  
Shallow Water Waves ◽  
Nonlinear System Of Equations ◽  
Rectangular Basin ◽  
Shallow Water Approximation

This paper is mainly concerned with the motion of an incompressible fluid in a slowly rotating rectangular basin. The equations of motion of such a problem with its boundary conditions are reduced to a system of nonlinear equations, which is to be solved by applying the shallow water approximation theory. Each unknown of the problem is expanded asymptotically in terms of the small parameterϵwhich generally depends on some intrinsic quantities of the problem of study. For each order of approximation, the nonlinear system of equations is presented successively. It is worthy to note that such a study has useful applications in the oceanography.

scholarly journals A Shallow-Water Theory for the Sun's Active Longitudes

2005 ◽  
Vol 635 (2) ◽  
pp. L193-L196 ◽  
Author(s):  
Mausumi Dikpati ◽  
Peter A. Gilman
Keyword(s):  
Shallow Water ◽  
Shallow Water Theory

scholarly journals A shallow water analogue of asymmetric core-collapse, and neutron star kick/spin

2011 ◽  
Vol 7 (S279) ◽  
pp. 134-137
Author(s):  
Thierry Foglizzo ◽  
Frédéric Masset ◽  
Jérôme Guilet ◽  
Gilles Durand
Keyword(s):  
Neutron Star ◽  
Shallow Water ◽  
Acoustic Waves ◽  
Large Scale ◽  
Massive Stars ◽  
Core Collapse ◽  
Hydraulic Jumps ◽  
Core Collapse Supernova ◽  
Accretion Shock

AbstractMassive stars end their life with the gravitational collapse of their core and the formation of a neutron star. Their explosion as a supernova depends on the revival of a spherical accretion shock, located in the inner 200km and stalled during a few hundred milliseconds. Numerical simulations suggest that the large scale asymmetry of the neutrino-driven explosion is induced by a hydrodynamical instability named SASI. Its non radial character is able to influence the kick and the spin of the resulting neutron star. The SWASI experiment is a simple shallow water analog of SASI, where the role of acoustic waves and shocks is played by surface waves and hydraulic jumps. Distances in the experiment are scaled down by a factor one million, and time is slower by a factor one hundred. This experiment is designed to illustrate the asymmetric nature of core-collapse supernova.

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