ηt + ∇ ⋅ u f
= 0
(4)
uα,t + g∇η + (uα ⋅ ∇ )uα + zα ∇ (uα,t ⋅ ∇h ) + (∇ ⋅ uα,t )∇h
(5)
1
+ ( zα + h ) - h2 ∇ (∇ ⋅ uα,t ) = 0
2
2
by:
h
η
= (h + η) uα + h zα + [∇(uα ⋅ ∇h) + (∇ ⋅ uα )∇h]
∫
=
uf
u dz
2
-h
(6)
( zα + h )2 h2
+ h
- ∇(∇ ⋅ uα )
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 zα 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,
zα = -0.535h, is close to middepth.
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ing Equations 4 and 5 is no longer valid.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, uη, instead
of uα. Additional changes have also been made to the equations to allow for
weakly rotational flows in the horizontal plane and ensure that zα 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)
uα,t + g∇η + (uη ⋅ ∇ ) uη + wη∇wη + ( zα - η) ∇ (uα,t ⋅ ∇h ) + (∇ ⋅ uα,t )∇h
1
( zα + h ) - (h + η)2 ∇ (∇ ⋅ uα,t )
2
+
2
(8)
- (uα,t ⋅ ∇h ) + (h + η)∇ ⋅ uα,t ∇η
+ ∇ (uα,t ⋅ ∇h ) + (∇ ⋅ uα,t )∇h + ( zα + h )∇ (∇ ⋅ uα ) zα,t
=
0
5
Chapter 2 Theoretical Background
Previous Page
