Boundary Conditions

ADCIRC implements various boundary conditions to represent physical constraints and forcing at domain boundaries. These boundary conditions are essential for accurately simulating coastal and ocean processes.

External Boundary Conditions

External boundaries are the outer boundaries of the computational domain. ADCIRC supports these types of external boundary conditions:

Elevation Boundary Condition

At an elevation-specified boundary, the water surface elevation is prescribed as a function of time:

\[\zeta(t) = \sum_{k=1}^{N_{tides}} f_k \alpha_k \cos(\omega_k t - \phi_k)\]

where:

\(N_{tides}\) = number of tidal constituents
\(f_k\) = nodal factor
\(\alpha_k\) = tidal amplitude
\(\omega_k\) = tidal frequency
\(\phi_k\) = tidal phase

Flow Boundary Condition

At flow-specified boundaries, the normal component of flow is prescribed:

\[U_n = Q_n(t)\]

where \(U_n\) is the normal component of velocity and \(Q_n(t)\) is the specified discharge per unit width.

Radiation Boundary Condition

Radiation boundaries allow waves to exit the domain with minimal reflection. ADCIRC implements the Sommerfeld radiation condition:

\[\frac{\partial \zeta}{\partial t} + c \frac{\partial \zeta}{\partial n} = 0\]

where \(c = \sqrt{gH}\) is the shallow water wave speed and \(n\) is the outward normal direction.

Combined Radiation-Flow Boundary Condition

For cases where both wave radiation and external flow need to be accommodated:

\[\frac{\partial \zeta}{\partial t} + c \frac{\partial \zeta}{\partial n} = \frac{Q_n(t)}{H}\]

Internal Boundary Conditions

Internal boundaries represent features within the computational domain, such as islands, barriers, and hydraulic structures.

No-Flow Boundary Condition

Applied at land boundaries where fluid cannot penetrate:

\[U_n = 0\]

Cross-Barrier Boundary Condition

For hydraulic structures like weirs and barriers, the flow across the structure is modeled as:

\[Q = C_d b H_u^{3/2} \sqrt{2g(H_u - H_d)}\]

where:

\(Q\) = flow rate across the structure
\(C_d\) = discharge coefficient
\(b\) = structure width
\(H_u\) = upstream water depth
\(H_d\) = downstream water depth

When the structure is submerged, the equation is modified to account for flow over the top of the structure.

Surface Boundary Conditions

Wind Stress

The wind stress at the water surface is parametrized as:

\[\frac{\tau_{s\lambda}}{\rho_0} = C_D \frac{\rho_a}{\rho_0} |W| W_\lambda\]

\[\frac{\tau_{s\phi}}{\rho_0} = C_D \frac{\rho_a}{\rho_0} |W| W_\phi\]

where:

\(C_D\) = wind drag coefficient
\(\rho_a\) = air density
\(|W|\) = wind speed magnitude at 10m height
\(W_\lambda, W_\phi\) = components of the wind velocity

The wind drag coefficient is typically modeled as a function of wind speed:

\[\begin{split}C_D = \begin{cases} C_{D1} & \text{if } |W| \leq W_1 \\ C_{D1} + (C_{D2} - C_{D1}) \frac{|W| - W_1}{W_2 - W_1} & \text{if } W_1 < |W| < W_2 \\ C_{D2} & \text{if } |W| \geq W_2 \end{cases}\end{split}\]

Atmospheric Pressure

The effect of atmospheric pressure is included in the momentum equations as the inverse barometer effect:

\[\frac{1}{\rho_0} \frac{\partial p_s}{\partial \lambda}, \frac{1}{\rho_0} \frac{\partial p_s}{\partial \phi}\]

where \(p_s\) is the atmospheric pressure at the sea surface.