∂V V ∂V tan φ
∂V
1
+
+
+
U + f U =
U
∂t R cosφ ∂λ R ∂φ R
(3)
τ sφ
1 ∂ ps
+ g (ξ  αη ) +
 τ *V

R ∂φ ρ0
ρ0 H
Equation 1 is the primitive continuity equation, and Equations 2 and 3 are the
λ (degrees longitude) and φ (degrees latitude) direction primitive momentum
equations in nonconservative form. The variables are defined as: ξ = free
surface elevation relative to the geoid; U, V = depthaveraged horizontal veloci
ties; H = ξ + h = total water column; h = bathymetric depth relative to the geoid;
ƒ = 2Ω sin φ = Coriolis parameter; Ω = angular speed of the Earth; ps =
atmospheric pressure at the free surface; g = acceleration due to gravity; η =
reference density of water; τsλ, τsφ = applied freesurface stress; τ* = Cf (U2 +
V2)1/2/H, and Cf = bottom friction coefficient. For this application, a hybrid form
of the standard quadratic parameterization for bottom stress is used that provides
a friction factor that increases as the depth decreases in shallow water, similar to
tide potential is given by Reid (1990).
Prior to the application of the numerical discretization, Equations 13 are
extensively rearranged for reasons of convenience and improved numerical
properties. First in order to facilitate a Finite Element solution, these equations
are mapped from spherical form into a rectilinear coordinate system using a Carte
Parallelogrammatique projection. Furthermore the equations are cast into the
Generalized Wave Continuity Equation (GWCE) form instead of their primitive
form. The GWCE is derived by substituting the rearranged, spatially differenti
ated primitive conservative momentum equations into the timedifferentiated
primitive continuity equation. Then the primitive continuity equation multiplied
by the GWCE weighting parameter, τ0, is added and the advective terms are
transformed to nonconservative form. It is noted that it is important to formulate
the advective terms in the GWCE in nonconservative form to obtain a consistent
solution with good local mass conservation properties (Kolar et al. 1994a). The
GWCE weighting parameter, τ0, is a purely numerical constant that sets the
balance between the wave equation and primitive continuity equations. An
appropriate choice of the weighting parameter τ0 is essential for the GWCE to
perform well. A large value of τ0 leads to artificial spurious modes associated
with a folded dispersion relationship, a small value of τ0 leads to poor localized
mass conservation properties. When τ0 is properly chosen, the solution exhibits a
solution free of spurious numerical oscillations while maintaining minimal local
and global mass balance errors. Our past experience indicates the optimal value
of τ0 is two to 10 times that of τ* (Kolar et al. 1994a). However, since τ* varies
linearly with the flow speed and friction factor, Cf, and varies inversely with the
total depth, H, it can vary dramatically throughout a domain. It is therefore
difficult to select a single GWCE weighting parameter value for a domain that
has large regions of both deep and shallow water. To address this problem
ADCIRC2DDI has been implemented to permit a spatially varying τ0. The
change in ADCIRC to a nodally varying weighting parameter τ0 allows for
5
Chapter 2 Governing Equations and 2D Modeling