the Generalized WaveContinuity Equation (GWCE). Fundamental components
of the GWCE are the depthintegrated continuity and NavierStokes equations
for conservation of mass and momentum. The assumption of incompressibility
and the Boussinesq and hydrostatic pressure approximations were applied. The
primitive, nonconservative form of the governing equations, given in spherical
coordinates, as applied in the model are (Flather 1988; Kolar et al. 1993)
∂UD ∂ [UV cos (φ)]
∂ζ
1
+
+
=0
(2)
∂t R cos (φ) ∂ϕ
∂φ
∂U 1 ∂U tan (φ)
∂U
1
+
+ V

U+ f V
U
∂t R cos (φ) ∂ϕ R ∂φ R
(3)
τSϕ
∂ PS
1
+ g (ζ  ξ ) +
=
 τ*U
R cos (φ) ∂ϕ ρ0
ρ0 D
∂V 1 ∂V tan (φ)
∂V
1
+
+ V

U+ f U
U
∂t R cos (φ) ∂ϕ R ∂φ R
(4)
τ
1 ∂ PS
+ g (ζ  ξ ) + Sφ  τ*V
=
R ∂ϕ ρ0
ρ0 D
where t is time, ϕ is degrees longitude (east of Greenwich is taken positive), φ is
degrees latitude (north of the equator is taken positive), ζ is the freesurface
elevation relative to the geoid, U is the depthaveraged velocity component
parallel to the eastwest axis, V is the depthaveraged velocity component
parallel to the northsouth axis, R is the radius of the earth, D = ζ + h is the total
watercolumn depth, h is the ambient depth relative to the geoid, f = 2Ωcos(φ) is
the Coriolis parameter, Ω is the angular speed of the earth's rotation, Ps is the
atmospheric pressure at the free surface, g is the acceleration due to gravity, ξ is
the effective Newtonian equilibrium tide potential, ρ0 is the reference density of
water, τsϕ and τsφ are the applied freesurface stresses, and τ* is the bottom stress
given by Cf (U2 + V2 )1/2 /D where Cf is the bottomfriction coefficient.
The timedifferentiated form of the conservation of mass equation is
combined with a spacedifferentiated form of the conservation of momentum
equation to develop the GWCE (Westerink et al. 1992) given by
13
Chapter 3
Modeling Approach