Phase resolving models based on either the mild-slope equation or

Boussinesq-type equations are better suited for problems involving the

reflection/diffraction of waves such as in coastal entrances and harbors. The

mild-slope and Boussinesq equations are vertically integrated equations for wave

propagation in the two-dimensional horizontal plane with different assumptions

made for the variation of fluid motion over the water depth. The mild-slope

equation derivation assumes a hyperbolic cosine variation of the velocity poten-

tial over depth, consistent with linear monochromatic waves in water of arbitrary

depth, while the Boussinesq equation derivation assumes a quadratic profile,

valid for shallow-water waves with wavelengths much longer than the water

depth.

This report describes BOUSS-2D, a comprehensive numerical model based

on a time-domain solution of Boussinesq-type equations. The classical form of

the Boussinesq equations for wave propagation over water of variable depth was

derived by Peregrine (1967). The equations were restricted to relatively shallow-

water depths, i.e., the water depth, *h, *had to be less than one-fifth of the wave-

length, *L*, in order to keep errors in the phase velocity to less than 5 percent.

Nwogu (1993) extended the range of applicability of Boussinesq-type equations

to deeper water by recasting the equations in terms of the velocity at an arbitrary

distance *z*α from the still-water level, instead of the depth-averaged velocity. The

elevation of the velocity variable *z*α becomes a free parameter, which is chosen to

optimize the linear dispersion characteristics of the equations. The optimized

form of the equations has errors of less than 2 percent in the phase velocity from

shallow-water depths up to the deepwater limit (*h/L *= 0.5).

Despite the improvement in the frequency dispersion characteristics,

Nwogu's (1993) equations are based on the assumption that the wave heights

were much smaller than the water depth. This limits the ability of the equations

to describe highly nonlinear waves in shallow water and led Wei et al. (1995) to

derive a fully nonlinear form of the equations. The fully nonlinear equations are

particularly useful for simulating highly asymmetric waves in shallow water,

wave-induced currents, wave setup close to the shoreline, and wave-current

interaction.

As ocean waves approach the shoreline, they steepen and ultimately break.

The turbulence and currents generated by breaking waves are important driving

mechanisms for the transport of sediments and pollutants. Nwogu (1996)

extended the fully nonlinear form of the Boussinesq equations to the surf zone,

by coupling the mass and momentum equations with a one-equation model for

the temporal and spatial evolution of the turbulent kinetic energy produced by

wave breaking. The equations have also been modified to include the effects of

bottom friction and flow through porous structures. The modified equations can

simulate most of the hydrodynamic phenomena of interest in coastal regions and

harbor basins including:

2

Chapter 1 Introduction

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