ηt + ∇ ⋅ *u * f

= 0

(4)

(5)

1

+ ( zα + *h *) - *h*2 ∇ (∇ ⋅ *u*α,*t *) = 0

2

2

by:

η

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

∫

=

2

-*h*

(6)

( zα + *h *)2 h2

+ * h*

- ∇(∇ ⋅ *u*α )

2

6

The depth-integrated equations are able to describe the propagation and trans-

formation of irregular multidirectional waves over water of variable depth. The

elevation of the velocity variable *z*α is a free parameter and is chosen to minimize

the differences between the linear dispersion characteristics of the model and the

exact dispersion relation for small amplitude waves. The optimal value,

For steep near-breaking waves in shallow water, the wave height becomes of

the order of the water depth and the weakly nonlinear assumption made in deriv-

ing Equations 4 and 5 is no longer valid. Wei et al. (1995) derived a fully non-

linear form of the equations from the dynamic free surface boundary condition by

retaining all nonlinear terms, up to the order of truncation of the dispersive terms.

Nwogu (1996) derived a more compact form of the equations by expressing some

of the nonlinear terms as a function of the velocity at the free surface, *u*η, instead

of *u*α. Additional changes have also been made to the equations to allow for

weakly rotational flows in the horizontal plane and ensure that *z*α remains in the

water column for steep waves near the shoreline and during the wave runup

process. The revised form of the fully nonlinear equations can be written as:

ηt + ∇ ⋅ *u * f

= 0

(7)

1

( zα + *h *) - (*h *+ η)2 ∇ (∇ ⋅ *u*α,*t *)

2

+

2

(8)

- (uα,*t *⋅ ∇*h *) + (*h *+ η)∇ ⋅ *u*α,*t * ∇η

+ ∇ (uα,*t *⋅ ∇*h *) + (∇ ⋅ *u*α,*t *)∇*h *+ ( zα + *h *)∇ (∇ ⋅ *u*α ) *z*α,*t*

=

0

5

Chapter 2 Theoretical Background

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