CMS-Wave:Governing Equations

Wave-action balance equation with diffraction

Taking into account the effect of an ambient horizontal current or wave behavior, CMS-Wave is based on the steady wave-action balance equation (Mase 2001)

\begin{equation}

\frac{\partial (c_x N)  }{\partial x}

+ \frac{\partial (c_y N) }{\partial y} + \frac{\partial (c_{\theta} N) }{\partial \theta} = \frac{\kappa}{2 \sigma} \biggl[ (c c_g \cos ^2 \theta N_y)_y - \frac{c c_g}{2} \cos ^2 \theta N_{yy} \biggr] - \epsilon_b N - S \end{equation}

where $N=E(\sigma ,\theta )/\sigma$ is the wave-action density to be solved and is a function of frequency σ and direction θ. E(σ,θ) is spectral wave density representing the wave energy per unit water-surface area per frequency interval. In the presence of an ambient current, the wave-action density is conserved, whereas the spectral wave density is not (Bretherton and Garrett 1968; Whitham 1974). Both wave diffraction and energy dissipation are included in the governing equation. Implementation of the numerical scheme is described elsewhere in the literature (Mase 2001; Mase et al. 2005a). C and Cg are wave celerity and group velocity, respectively; x and y are the horizontal coordinates; Cx, Cy, and Cθ are the characteristic velocity with respect to x, y, and, θ respectively; Ny and Nyy denote the first and second derivatives of N with respect to y, respectively; κ is an empirical parameter representing the intensity of diffraction effect; εb is the parameterization of wave breaking energy dissipation; S denotes additional source Sin and sink Sds (e.g., wind forcing, bottom friction loss, etc.) and nonlinear wave-wave interaction term.

Symbol	Description
$\sigma$	Wave frequency
$N$	Wave action
$E$	Spectral wave density
$c$	Wave celerity
$c_{g}$	Wave group velocity

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