Energy balance for undular bores

2012

J Nonlinear Sci

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Depth-integrated long-wave models, such as the shallow-water and Boussinesq equations, are standard fare in the study of small amplitude surface waves in shallow water. While the shallow-water theory features conservation of mass, momentum and energy for smooth solutions, mechanical balance equations are not widely used in Boussinesq scaling, and it appears that the expressions for many of these quantities are not known. This work presents a systematic derivation of mass, momentum and energy densities and fluxes associated with a general family of Boussinesq systems. The derivation is based on a reconstruction of the velocity field and the pressure in the fluid column below the free surface, and the derivation of differential balance equations which are of the same asymptotic validity as the evolution equations. It is shown that all these mechanical quantities can be expressed in terms of the principal dependent variables of the Boussinesq system: the surface excursion η and the horizontal velocity w at a given level in the fluid.

Section: Higher-order Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mechanical Balance Laws for Boussinesq Models of Surface Water Waves

Ali

2012

J Nonlinear Sci

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“…This approach is made possible by a recent advance in the interpretation of Boussinesq type model equations in the context of approximate mass, momentum and energy conservation (Ali and Kalisch, 2012). This theory has proved successful in the study of the energy budget in an undular bore (Ali and Kalisch, 2010), and is now brought to bear on the problem of wave shoaling in the context of the KdV equation. The KdV equation is given in dimensional variables by…”

Section: Introductionmentioning

confidence: 99%

On the Shoaling of Solitary Waves in the KDV Equation

Khorsand

Int. Conf. Coastal. Eng.

2014

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The waveheight change in surface waves with a sufficiently slow variation in depth is examined. Using a new formulation of the energy flux associated to waves modeled by the Korteweg-de Vries equation, a system of three coupled equations is derived for the determination of the local wave properties as waves propagate over gently changing depth. The system of equations is solved numerically, and the resulting shoaling curves are compared to previous results on long wave shoaling.

“…Indeed, there are situations where the internal dynamics of the flow are an important factor. An example of such an application is the study of the energy balance in undular bores [2]. The energy flux q E was also used in a decisive way in a study of nonlinear shoaling in [13].…”

Section: Discussionmentioning

confidence: 99%

Approximate conservation laws in the KdV equation

Israwi

2019

Physics Letters A

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The Korteweg-de Vries equation is known to yield a valid description of surface waves for waves of small amplitude and large wavelength. The equation features a number of conserved integrals, but there is no consensus among scientists as to the physical meaning of these integrals. In particular, it is not clear whether these integrals are related to the conservation of momentum or energy, and some researchers have questioned the conservation of energy in the dynamics governed by the equation. In this note it is shown that while exact energy conservation may not hold, if momentum and energy densities and fluxes are defined in an appropriate way, then solutions of the Korteweg-de Vries equation give rise to approximate differential balance laws for momentum and energy. * s