Input by wind ($S_{\rm in}$)

Wave growth by wind is described by

  $$S_{\rm in} (\sigma, \theta) = A + B E(\sigma,\theta)$$ (2.33)

in which $A$ describes linear growth and $BE$ exponential growth. It should be noted that the SWAN model is driven by the wind speed at 10m elevation $U _{10}$ whereas it uses the friction velocity $U _{*}$. For the WAM Cycle 3 formulation the transformation from $U _{10}$ to $U _{*}$ is obtained with

  $$U^2_* = C_D U^2_{10}$$ (2.34)

in which $C_D$ is the drag coefficient from Wu (1982):

  $$C_D(U_{10}) = \left\{ \begin{array}{ll} 1.2875 \times 10^{-3} ... ...0}) \times 10^{-3} \, , & \mbox{for } U_{10} \geq 7.5 m/s \end{array} \right.$$ (2.35)

Recent observations indicate that this parameterization overestimate the drag coefficient at high wind speeds ($U_{10} >$ 20 m/s, say). Based on many authoritative studies it appears that the drag coefficient increases almost linearly with wind speed up to approximately 20 m/s, then levels off and decreases again at about 35 m/s to rather low values at 60 m/s wind speed. We fitted a 2nd order polynomial to the data obtained from these studies, and this fit is given by

  $$C_D(U_{10}) = (0.55 + 2.97 {\tilde U} - 1.49 {\tilde U}^2) \times 10^{-3}$$ (2.36)

where ${\tilde U} = U_{10}/U_{\rm ref}$, and the reference wind speed $U_{\rm ref}$ = 31.5 m/s is the speed at which the drag attains its maximum value in this expression. These drag values are lower than in the expression of Wu (1982) by 10% $-$ 30% for high wind speeds (15 $\leq U_{10} \leq$ 30 m/s) and over 30% for hurricane wind speeds ($U_{10} >$ 30 m/s). More details can be found in Zijlema et al. (2012). Since version 41.01, the SWAN model employs the drag formulation as given by Eq. (2.36).


For the WAM Cycle 4 formulations, the computation of $U _{*}$ is an integral part of the source term.


Linear growth by wind


For the linear growth term $A$, the expression due to Cavaleri and Malanotte-Rizzoli (1981) is used with a filter to eliminate wave growth at frequencies lower than the Pierson-Moskowitz frequency (Tolman, 1992a)^2.1:

  $$A = \frac{1.5 \times 10^{-3}}{2 \pi g^2} (U_* \max [0,\cos(\theta... ...)^{-4} \right\} }\, , \quad \sigma_{\rm PM}^{*} = \frac{0.13 g}{28 U_*} 2 \pi$$ (2.37)

in which $\theta_w$ is the wind direction, $H$ is the filter and $\sigma^{*}_{\rm PM}$ is the peak frequency of the fully developed sea state according to Pierson and Moskowitz (1964) as reformulated in terms of friction velocity.


Exponential growth by wind


Two expressions for exponential growth by wind are optionally available in the SWAN model. The first expression is due to Komen et al. (1984). Their expression is a function of $U_{*}/c_{\rm ph}$:

  $$B = \max [0, 0.25 \frac{\rho_a}{\rho_w} (28 \frac{U_*}{c_{\rm ph}}\cos(\theta-\theta_w) -1)]\sigma$$ (2.38)

in which $c _{\rm ph}$ is the phase speed and $\rho_a$ and $\rho_w$ are the density of air and water, respectively. This expression is also used in WAM Cycle 3 (the WAMDI group, 1988). The second expression is due to Janssen (1989,1991a). It is based on a quasi-linear wind-wave theory and is given by

  $$B = \beta \frac{\rho_a}{\rho_w} \left(\frac{U_*}{c_{\rm ph}} \right)^2 \max[0,\cos(\theta-\theta_w)]^2\sigma$$ (2.39)

where $\beta$ is the Miles constant. In the theory of Janssen (1991a), this constant is estimated from the non-dimensional critical height $\lambda$:

  $$\left\{ \begin{array}{ll} \beta = \frac{1.2}{\kappa^2} \lambda ... ... \, , & r = \kappa c/\vert U_* \cos(\theta-\theta_w)\vert \end{array} \right.$$ (2.40)

where $\kappa=0.41$ is the Von Karman constant and $z_e$ is the effective surface roughness. If the non-dimensional critical height $\lambda>1$, the Miles constant $\beta$ is set equal 0. Janssen (1991a) assumes that the wind profile is given by

  $$U(z) = \frac{U_*}{\kappa} \ln [ \frac{z+z_e-z_0}{z_e} ]$$ (2.41)

in which $U(z)$ is the wind speed at height $z$ (10m in the SWAN model) above the mean water level, $z_0$ is the roughness length. The effective roughness length $z_e$ depends on the roughness length $z_0$ and the sea state through the wave-induced stress $\vec{\tau_w}$ and the total surface stress $\vec{\tau} = \rho_a \vert\vec{U_*}\vert \vec{U_*}$:

  $$z_e = \frac{z_0}{\sqrt{1 - \frac{\vert\vec{\tau_w}\vert}{\vert\vec{\tau}\vert}}}\, , \quad z_0 = \hat{\alpha} \frac{U_*^2}{g}$$ (2.42)

The second of these two equations is a Charnock-like relation in which $\hat{\alpha}$ is a constant equal to 0.01. The wave stress $\vec{\tau}_w$ is given by

  $$\vec{\tau}_w = \rho_w \int_{0}^{2\pi} \int_{0}^{\infty} \sigma B E (\sigma, \theta) \frac{\vec{k}}{k} d \sigma d \theta$$ (2.43)

The value of $U_*$ can be determined for a given wind speed $U _{10}$ and a given wave spectrum $E(\sigma,\theta)$ from the above set of equations. In the SWAN model, the iterative procedure of Mastenbroek et al. (1993) is used. This set of expressions (2.39) through (2.43) is also used in WAM Cycle 4 (Komen et al., 1994).