CMS-Flow:Wave Eqs: 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)


         <math> \frac{\partial c_x N  }{\partial x}
         <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>


for <math> j=1,</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.
 
        <math> \frac{\partial ( h U_i ) }{\partial t} + \frac{\partial (h U_i U_j )}{\partial x_j}
- \epsilon_{ij3} f_c U_j h = - g h \frac{\partial \eta }{\partial x_j}
- \frac{h}{\rho_0} \frac{\partial p_a }{\partial x_j}
+ \frac{\partial }{\partial x_j} \biggl ( \nu_t  h \frac{\partial U_i }{\partial x_j} \biggr )
+ \frac{\tau_i }{\rho}
</math> 
 
for <math> i=1,2 </math> and <math> j=1,2 </math>


{| border="1"
{| border="1"

Latest revision as of 21:56, 8 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)

        

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

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