CMS-Wave:Governing Equations: Difference between revisions
(Created page with '<big> == 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-a…') |
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== Wave-action balance equation with diffraction == | == 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) | 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) | ||
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 | - \epsilon_b N - S /math | ||
where | 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. | ||
{| border= | {| border=1 | ||
! Symbol !! Description | ! Symbol !! Description | ||
|- | |- | ||
| | | math \sigma /math || Wave frequency | ||
|- | |- | ||
| | | math N /math || Wave action | ||
|- | |- | ||
| | | math E /math || Spectral wave density | ||
|- | |- | ||
| | | math c /math || Wave celerity | ||
|- | |- | ||
| | | math c_g /math || Wave group velocity | ||
|} | |} | ||
---- | ---- | ||
/big | |||
[[CMS#Documentation_Portal | Documentation Portal]] | [[CMS#Documentation_Portal | Documentation Portal]] | ||
Revision as of 14:23, 19 May 2010
big
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)
math \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 /math
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.
| Symbol | Description |
|---|---|
| math \sigma /math | Wave frequency |
| math N /math | Wave action |
| math E /math | Spectral wave density |
| math c /math | Wave celerity |
| math c_g /math | Wave group velocity |
/big Documentation Portal