CMS-Wave:Governing Equations: Difference between revisions

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<big>
== 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)
 
{{Equation|<math>
        <math> \frac{\partial (c_x N)  }{\partial x}
\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>|1}}


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.
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----
=References=
</big>
*
Mase, H., K. Oki, T. S. Hedges, and H. J. Li. 2005. Extended energy-balance-equation wave model for multidirectional random wave transformation. Ocean Engineering 32(8–9):961–985.----
[[CMS#Documentation_Portal | Documentation Portal]]
[[CMS#Documentation_Portal | Documentation Portal]]
[[category:CMS-Wave]]

Latest revision as of 16:12, 23 January 2023

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)

  (1)

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

References

Mase, H., K. Oki, T. S. Hedges, and H. J. Li. 2005. Extended energy-balance-equation wave model for multidirectional random wave transformation. Ocean Engineering 32(8–9):961–985.---- Documentation Portal