Hydrodynamic Model Description

Governing Equations and 2-D Modeling

Model Input Parameters

TR-02-24
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∂t R cosφ ∂λ R ∂φ  R

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 = depth-averaged 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 =

Newtonian equilibrium tide potential; α = effective Earth elasticity factor; ρ0 =

reference density of water; τsλ, τsφ = applied free-surface 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

a Manning relationship. A practical expression for the Newtonian equilibrium

tide potential is given by Reid (1990).

Prior to the application of the numerical discretization, Equations 1-3 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 time-differentiated

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

ADCIRC-2DDI has been implemented to permit a spatially varying τ0. The

change in ADCIRC to a nodally varying weighting parameter τ0 allows for

Chapter 2 Governing Equations and 2-D Modeling

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