B.9 Wind Stress
Wind stress is applied as a surface shear stress in the depth-integrated momentum equations. The wind stress vector is defined as:
\[\begin{equation} \boldsymbol{\tau}_w = \rho_a \, C_D \, |\mathbf{U}_{10}| \, \mathbf{U}_{10} \tag{B.107} \end{equation}\]
Where:
- \(\boldsymbol{\tau}_w\) is wind stress (N/m2)
- \(\rho_a\) is air density (kg/m3)
- \(C_D\) is the bulk momentum transfer coefficient (-)
- \(\mathbf{U}_{10}\) is the wind velocity vector at 10 m elevation (m/s)
In the depth integrated shallow water momentum equations, wind stress contributes as:
\[\begin{equation} \frac{\tau_{wx}}{\rho_w h}, \quad \frac{\tau_{wy}}{\rho_w h} \tag{B.108} \end{equation}\]
Where:
- \(\rho_w\) is water density (kg/m3)
- \(h\) is water depth (m)
The selected wind stress model defines the calculation of the bulk momentum transfer coefficient \(C_D\). Three models are available.
B.9.1 Wu
The Wu model defines a wind-speed-dependent bulk transfer coefficient using a piecewise linear formulation.
\[\begin{equation} C_D = \begin{cases} C_a & W_{10} < W_a \\ C_a + \dfrac{C_b - C_a}{W_b - W_a}(W_{10} - W_a) & W_a \le W_{10} < W_b \\ C_b & W_{10} \ge W_b \end{cases} \tag{B.109} \end{equation}\]
Where:
- \(U_{10}\) is the mean wind speed at 10m above the water surface (m/s)
- \(W_a\), \(W_b\) are wind speed thresholds (m/s)
- \(C_a\), \(C_b\) are bulk momentum transfer coefficients (-)
This formulation produces increasing drag with wind speed and is applied uniformly across the model domain. The Wu formulation is recommended for general coastal, estuarine and riverine modelling applications.
B.9.2 Constant
The Constant model applies a spatially and temporally constant bulk momentum transfer coefficient.
\[\begin{equation} C_D = C_{DN} \tag{B.110} \end{equation}\]
Where:
- \(C_{DN}\) is the specified constant bulk momentum transfer coefficient (-)
Wind stress is therefore directly proportional to wind speed magnitude. The constant implementation may be used for simplified studies, calibration comparison with legacy configurations, or where a fixed transfer coefficient is required.
B.9.3 Kondo
This model implements Kondo (1975), of the general form
\[\begin{equation} C_D = \alpha C_{D,Kondo} \tag{B.111} \end{equation}\]
Where:
- \(\alpha\) is the scale factor
- \(C_{D,Kondo}\) is the empirical transfer coefficient:
\[\begin{equation} C_{D,Kondo} = \begin{cases} \left[1.080\left({\max{\left(0.3,U_{10}\right)}}^{-0.150}\right)\right] \times 10^{-3} & 0.0 \text{ ms}^{-1} \le U_{10} < 2.2 \text{ ms}^{-1} \\[10pt] \left[0.771+0.0858U_{10}\right] \times 10^{-3} & 2.2 \text{ ms}^{-1} \le U_{10} < 5.0 \text{ ms}^{-1} \\[10pt] \left[0.867+0.0667U_{10}\right] \times 10^{-3} & 5.0 \text{ ms}^{-1} \le U_{10} < 8.0 \text{ ms}^{-1} \\[10pt] \left[1.200+0.0250U_{10}\right] \times 10^{-3} & 8.0 \text{ ms}^{-1} \le U_{10} < 25.0 \text{ ms}^{-1} \\[10pt] \left[0.000+0.0730U_{10}\right] \times 10^{-3} & 25.0 \text{ ms}^{-1} \le U_{10} \end{cases} \tag{B.111} \end{equation}\]
where:
- \(U_{10}\) is the wind speed at 10m above the water surface (m/s)
If atmospheric stability is activated, this valuie of \(C_{D,Kondo}\) is further modified by an atmospheric stability parameter, \(S_W\) (Equation (B.92)) following Kondo (1975).
\[\begin{equation} C_{D,Kondo} = S_W \times C_{D,Kondo} \tag{B.112} \end{equation}\]