Governing Equations

The ADCIRC model solves the shallow water equations, a form of the Navier-Stokes equations with traditional hydrostatic pressure and Boussinesq approximations. This section describes the fundamental equations that form the basis of the model.

Continuity Equation

The depth-integrated continuity equation in spherical coordinates is:

\[\frac{\partial \zeta}{\partial t} + \frac{1}{R \cos \phi} \left[ \frac{\partial}{\partial \lambda} \left( U H \right) + \frac{\partial}{\partial \phi} \left( V H \cos \phi \right) \right] = 0\]

where:

  • \(\zeta\) = free surface elevation relative to the geoid

  • \(t\) = time

  • \(R\) = radius of the Earth

  • \(\phi\) = latitude

  • \(\lambda\) = longitude

  • \(H = \zeta + h\) = total water column depth

  • \(h\) = bathymetric depth relative to the geoid

  • \(U\) = depth-integrated velocity in longitude direction

  • \(V\) = depth-integrated velocity in latitude direction

Momentum Equations

The depth-integrated momentum equations in spherical coordinates are:

\[\begin{split}\frac{\partial U}{\partial t} + \frac{1}{R \cos \phi} \frac{\partial}{\partial \lambda} \left( \frac{U^2}{H} \right) + \frac{1}{R} \frac{\partial}{\partial \phi} \left( \frac{UV}{H} \right) - \frac{UV \tan \phi}{R H} - fV \\ + \frac{g}{R \cos \phi} \frac{\partial \zeta}{\partial \lambda} + \frac{\tau_{s\lambda}}{\rho_0 H} - \frac{\tau_{b\lambda}}{\rho_0 H} - \frac{1}{\rho_0 H} \left[ \frac{\partial}{\partial \lambda} \left( p_s \right) - \frac{\partial}{\partial \lambda} \int_{-h}^{\zeta} \int_{z'}^{\zeta} \frac{\partial \rho}{\partial \lambda'} g dz dz' \right] \\ + \frac{1}{R \cos \phi} \frac{\partial}{\partial \lambda} \left[ \frac{2 N_\lambda}{H} \frac{\partial U}{\partial \lambda} \right] + \frac{1}{R} \frac{\partial}{\partial \phi} \left[ \frac{N_\phi}{H} \left( \frac{\partial U}{\partial \phi} + \frac{\partial V}{\partial \lambda} - V \tan \phi \right) \right] = 0\end{split}\]
\[\begin{split}\frac{\partial V}{\partial t} + \frac{1}{R \cos \phi} \frac{\partial}{\partial \lambda} \left( \frac{UV}{H} \right) + \frac{1}{R} \frac{\partial}{\partial \phi} \left( \frac{V^2}{H} \right) + \frac{U^2 \tan \phi}{R H} + fU \\ + \frac{g}{R} \frac{\partial \zeta}{\partial \phi} + \frac{\tau_{s\phi}}{\rho_0 H} - \frac{\tau_{b\phi}}{\rho_0 H} - \frac{1}{\rho_0 H} \left[ \frac{\partial}{\partial \phi} \left( p_s \right) - \frac{\partial}{\partial \phi} \int_{-h}^{\zeta} \int_{z'}^{\zeta} \frac{\partial \rho}{\partial \phi'} g dz dz' \right] \\ + \frac{1}{R \cos \phi} \frac{\partial}{\partial \lambda} \left[ \frac{N_\lambda}{H} \left( \frac{\partial U}{\partial \phi} + \frac{\partial V}{\partial \lambda} - V \tan \phi \right) \right] + \frac{1}{R} \frac{\partial}{\partial \phi} \left[ \frac{2 N_\phi}{H} \frac{\partial V}{\partial \phi} \right] = 0\end{split}\]

where:

  • \(f = 2 \Omega \sin \phi\) = Coriolis parameter

  • \(\Omega\) = angular speed of the Earth

  • \(g\) = gravitational acceleration

  • \(\rho_0\) = reference density of water

  • \(\rho\) = perturbation density

  • \(p_s\) = atmospheric pressure at the free surface

  • \(\tau_{s\lambda}, \tau_{s\phi}\) = surface stress components

  • \(\tau_{b\lambda}, \tau_{b\phi}\) = bottom stress components

  • \(N_\lambda, N_\phi\) = lateral eddy viscosity coefficients

Bottom Stress Formulation

Bottom stress is parametrized using a quadratic friction law:

\[\frac{\tau_{b\lambda}}{\rho_0} = \frac{C_f U \sqrt{U^2 + V^2}}{H}\]
\[\frac{\tau_{b\phi}}{\rho_0} = \frac{C_f V \sqrt{U^2 + V^2}}{H}\]

where \(C_f\) is the bottom friction coefficient, which can be specified as a constant or calculated from:

\[C_f = \frac{g n^2}{H^{1/3}}\]

for Manning’s formulation, or:

\[C_f = \max \left[ C_{fmin}, \frac{\kappa^2}{\ln^2 \left( \frac{H}{z_0} \right)} \right]\]

for the logarithmic formulation, where:

  • \(n\) = Manning’s roughness coefficient

  • \(\kappa\) = von Karman constant (≈ 0.4)

  • \(z_0\) = bottom roughness length

  • \(C_{fmin}\) = minimum bottom friction coefficient