CMS-Wave:Governing Equations: Difference between revisions
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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) | 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} | |||
<math> \frac{\partial (c_x N) }{\partial x} | |||
+ \frac{\partial (c_y N) }{\partial y} | + \frac{\partial (c_y N) }{\partial y} | ||
+ \frac{\partial (c_{\theta} N) }{\partial \theta} | + \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] | = \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 </math> | - \epsilon_b N - S </math> | ||
\end{equation} | |||
where <math> N = E(\sigma,\theta)/\sigma </math> 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. | where <math> N = E(\sigma,\theta)/\sigma </math> 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. | ||
Revision as of 14:48, 14 September 2011
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}
![{\displaystyle {\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}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a2a04067ced0f441ca6865c1eb0fd6ec4f2f2d04)
\end{equation}
where 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 |
|---|---|
| Wave frequency | |
| Wave action | |
| Spectral wave density | |
| Wave celerity | |
| Wave group velocity |
