BOUSS-2D is based on Boussinesq-type equations derived by Nwogu (1993,

1996). The equations are depth-integrated equations for the conservation of mass

and momentum for nonlinear waves propagating in shallow and intermediate

water depths. They can be considered to be a perturbation from the shallow-water

equations, which are often used to simulate tidal flows in coastal regions. For

short-period waves, the horizontal velocities are no longer uniform over depth

and the pressure is nonhydrostatic. The vertical profile of the flow field is

obtained by expanding the velocity potential, Φ, as a Taylor series about an

arbitrary elevation, zα, in the water column. For waves with length, *L*, much

longer than the water depth, *h*, the series is truncated at second order resulting in

a quadratic variation of the velocity potential over depth:

φ α + 2 ( *z*α - *z *) [∇ φ α ⋅ ∇ *h *]

Φ ( *x*, *z*, *t*)

=

(1)

2

( zα + *h *) - ( *z *+ *h *) 2 ∇ 2 φ α + *O *( 4 )

2

+

2

dispersion. The horizontal and vertical velocities are obtained from the velocity

potential as:

1

( zα + *h *) - ( z + *h *) ∇(∇ ⋅ *u*α )

2

2

+

(2)

2

∂Φ

= - [uα ⋅ ∇*h *+ ( *z *+ *h*)∇ ⋅ *u*α ]

(3)

∂*z*

where *u*α = ∇Φ*|*zα is the horizontal velocity at *z = z*α. Given a vertical profile for

the flow field, the continuity and Euler (momentum) equations can be integrated

over depth, reducing the three-dimensional problem to two dimensions. For

weakly nonlinear waves with height, *H*, much smaller than the water depth, *h*, the

vertically integrated equations are written in terms of the water-surface elevation

η*(***x**,t) and velocity *u*α(**x**,t) as (Nwogu 1993):

4

Chapter 2 Theoretical Background

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