Wave Continuity Formulation

The Generalized Wave Continuity Equation (GWCE) is a key feature of the ADCIRC model. This formulation provides enhanced numerical stability compared to primitive continuity-momentum formulations.

Derivation of the GWCE

The GWCE is derived by combining the primitive continuity equation with the time derivative of the primitive continuity equation. This process involves:

  1. Taking the time derivative of the continuity equation

  2. Adding it to the spatial derivative of the momentum equations, multiplied by a weighting parameter tau_0

The resulting equation is:

\[\begin{split}\frac{\partial^2 \zeta}{\partial t^2} + \tau_0 \frac{\partial \zeta}{\partial t} + \frac{1}{R \cos \phi} \frac{\partial}{\partial \lambda} \left( \frac{\partial U}{\partial t} + \tau_0 U + \frac{1}{R \cos \phi} \frac{\partial}{\partial \lambda} \left( \frac{U^2}{H} \right) + \frac{1}{R} \frac{\partial}{\partial \phi} \left( \frac{UV}{H} \right) - \frac{UV \tan \phi}{RH} - fV + \ldots \right) \\ + \frac{1}{R \cos \phi} \frac{\partial}{\partial \phi} \left( \frac{\partial V}{\partial t} + \tau_0 V + \frac{1}{R \cos \phi} \frac{\partial}{\partial \lambda} \left( \frac{UV}{H} \right) + \frac{1}{R} \frac{\partial}{\partial \phi} \left( \frac{V^2}{H} \right) + \frac{U^2 \tan \phi}{RH} + fU + \ldots \right) \cos \phi = 0\end{split}\]

In this equation, all terms from the momentum equations (except for the time derivative term) are included in the spatial derivatives, represented by ellipses above.

Advantages of the GWCE

The GWCE formulation offers several advantages:

  1. Enhanced Stability: The GWCE is less susceptible to spurious oscillations (known as “2Δx noise”) that commonly affect primitive equation models.

  2. Better Mass Conservation: The formulation better preserves mass conservation properties, especially important in coastal regions with complex bathymetry.

  3. Improved Numerical Performance: The GWCE allows for larger timesteps and provides more accurate solutions, particularly for propagating waves.

  4. Prevention of Gravitational Oscillations: The formulation damps out short-wavelength numerical noise without significantly affecting the physical solution.

Role of the Tau Parameter

The weighting parameter \(\tau_0\) in the GWCE is a critical component that controls the numerical properties of the model:

  • When \(\tau_0 = 0\), the GWCE reduces to a second-order wave equation

  • As \(\tau_0 \rightarrow \infty\), the GWCE approaches the primitive continuity equation

  • For optimal performance, ADCIRC typically uses a spatially varying \(\tau_0\) that is: - Larger in deep water regions (approaching the primitive form) - Smaller in shallow regions (approaching the wave equation form)

The spatially varying \(\tau_0\) is commonly defined as:

\[\begin{split}\tau_0 = \begin{cases} \tau_{min} & \text{if } H \leq H_{crit} \\ \tau_{min} + (\tau_{max} - \tau_{min}) \frac{H - H_{crit}}{H_{deep} - H_{crit}} & \text{if } H_{crit} < H < H_{deep} \\ \tau_{max} & \text{if } H \geq H_{deep} \end{cases}\end{split}\]

where \(H_{crit}\) and \(H_{deep}\) are user-defined depth thresholds, and \(\tau_{min}\) and \(\tau_{max}\) are minimum and maximum values for \(\tau_0\).