CMS-Flow:Wave Eqs

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)

        ${\frac {\partial (c_{x}N)}{\partial x}}+{\frac {\partial (c_{y}N)}{\partial y}}+{\frac {\partial (c_{\theta }N)}{\partial \theta }}={\frac {\kappa }{2\sigma }}{\biggl [}(cc_{g}\cos ^{2}\theta N_{y})_{y}-{\frac {cc_{g}}{2}}\cos ^{2}\theta N_{yy}{\biggr ]}-\epsilon _{b}N-S$

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.

Wave diffraction

The first term on the right side of Equation 1 is the wave diffraction term formulated from a parabolic approximation wave theory (Mase 2001). In applications, the diffraction intensity parameter κ values (≥ 0) needs to be calibrated and optimized for structures. The model omits the diffraction effect for κ = 0 and calculates diffraction for κ > 0. Large κ (> 15) should be avoided as it can cause artificial wave energy losses (Mase 2001). In practice, values of κ between 0 (no diffraction) and 4 (strong diffraction) have been determined in comparison to measurements. A default value of κ = 2.5 was used by Mase et al. (2001, 2005a, 2005b) to simulate wave diffraction for both narrow and wide gaps between breakwaters. In CMSWave, the default value of κ assigned by SMS is 4, corresponding to strong diffraction. For wave diffraction at a semi-infinite long breakwater or at a narrow gap, with the opening equal or less than one wavelength, κ = 4 (maximum diffraction allowed in the model) is recommended. For a relatively wider gap, with an opening greater than on wavelength, κ = 3 is recommended. The exact value of κ in an application is dependent on the structure geometry and adjacent bathymetry, and should to be verified with measurements.

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

---

Documentation Portal