First- and second-generation model formulations in SWAN

The source term $S_{\rm tot}$ for the first- and second-generation formulation (relaxation model) of SWAN is (Holthuijsen and de Boer, 1988):

  $$S_{\rm tot} = \left\{ \begin{array}{lll} S_{\rm in} = A + B E ... ...box{and } \vert\theta - \theta_w\vert > \frac{\pi}{2} \\ \end{array} \right.$$ (2.158)

where $S_{\rm in}$ and $S_{\rm ds,w}$ represent input by wind and decay for over-developed sea states, respectively, $A$ and $BE$ are a linear and exponential growth term, respectively, $E$ is the spectral density, $E_{\rm lim}$ is the saturated spectrum, $\tau$ is a time scale and $\theta$ and $\theta_w$ are the discrete spectral wave direction and the wind direction, respectively. The expressions for the source terms $S_{\rm in}$ and $S_{\rm ds,w}$ have been modified for shallow water applications (N. Booij and L.H. Holthuijsen, personal communication, 1996) and are given below.


The distinction between first- and second-generation is only in the formulation of the saturated spectrum $E_{\rm lim}$ as outlined below.


Linear and exponential growth


The linear growth term $A$ is given by an expression due to Cavaleri and Malanotte-Rizolli (1981) as adapted by Holthuijsen and de Boer (1988) and Holthuijsen et al. (1996):

  $$A = \left\{ \begin{array}{ll} \frac{\beta_1}{2\pi} \frac{\pi}{... ...,d} \\ \\ 0 \, , & \sigma < 0.7 \sigma_{\rm PM,d} \\ \end{array} \right.$$ (2.159)

where $\beta_1$ is a coefficient that has been tuned to be $\beta_1 = 188$, $C_{\rm drag}$ is a drag coefficient equal to $C_{\rm drag} = 0.0012$ and $\sigma_{\rm PM,d}$ is the fully developed peak frequency including the effect of shallow water and is estimated from the depth dependent relation of the Shore Protection Manual (1973):

  $$\sigma_{\rm PM,d} = \frac{\sigma_{\rm PM}}{\tanh(0.833 {\tilde d}^{0.375})}$$ (2.160)

with the dimensionless depth

  $${\tilde d} = \frac{gd}{U^2_{10}}$$ (2.161)

The Pierson-Moskowitz (1964) frequency is

  $$\sigma_{\rm PM} = \frac{0.13g}{U_{10}} 2\pi$$ (2.162)

The exponential growth term $BE$ is due to Snyder et al. (1981) rescaled in terms of $U _{10}$ as adapted by Holthuijsen and de Boer (1988) and Holthuijsen et al. (1996):

  $$B = \max [0, \beta_2 \frac{5}{2\pi} \frac{\rho_a}{\rho_w} (\frac{U_{10}}{\sigma/k}\cos(\theta-\theta_w) -\beta_3)]\sigma$$ (2.163)

in which the coefficients $\beta_2$ and $\beta_3$ have been tuned to be $\beta_2 = 0.59$ and $\beta_3 = 0.12$.


Decay


If the spectral densities are larger than the wind-dependent saturation spectrum $E_{\rm lim}$, (e.g., when the wind decreases), energy is dissipated with a relaxation model:

  $$S_{\rm ds,w} (\sigma,\theta)= \frac{E_{\rm lim}(\sigma,\theta) - E(\sigma,\theta)}{\tau(\sigma)}$$ (2.164)

where $\tau(\sigma)$ is a time scale given by

  $$\tau(\sigma) = \beta_4 \left(\frac{2\pi}{\sigma}\right)^2 \frac{g}{U_{10}\cos(\theta - \theta_w)}$$ (2.165)

in which the coefficient $\beta_4$ has been tuned to be $\beta_4 = 250$.


Saturated spectrum


The saturated spectrum has been formulated in term of wave number with a $\cos^2-$directional distribution centred at the local wind direction $\theta_w$. It is essentially an adapted Pierson-Moskowitz (1964) spectrum:

  $$S_{\rm tot} = \left\{ \begin{array}{ll} \frac{\alpha k^{-3}}{2... ...{for } \vert\theta - \theta_w\vert \geq \frac{\pi}{2} \\ \end{array} \right.$$ (2.166)

For the first-generation formulation, the scale factor $\alpha$ is a constant and equals

  $$\alpha = 0.0081$$ (2.167)

For the second-generation formulation, the scale factor $\alpha$ depends on the total dimensionless wave energy ${\tilde E}_{\rm tot,sea}$ of the wind sea part of the spectrum and the dimensionless depth ${\tilde d}$:

  $$\alpha = \max [ (0.0081 + (0.013 - 0.0081)e^{-{\tilde d}}), 0.0023 {\tilde E}^{-0.223}_{\rm tot,sea} ]$$ (2.168)

where the total dimensionless wind sea wave energy ${\tilde E}_{\rm tot,sea}$ is given by

  $${\tilde E}_{\rm tot,sea} = \frac{g^2 E_{\rm tot,sea}}{U^4_{10}}$$ (2.169)

with

  $$E_{\rm tot,sea} = \int_{\theta_w - \pi/2}^{\theta_w + \pi/2} \int_{0.7 \sigma_{\rm PM,d}}^{\infty} E(\sigma,\theta) d\sigma d\theta$$ (2.170)

The maximum value of $\alpha$ is taken to be 0.155. This dependency of $\alpha$ on the local dimensionless energy of the wind sea permits an overshoot in the wave spectrum under wave generation conditions. For deep water $\alpha = 0.0081$ as proposed by Pierson and Moskowitz (1964).