TR-08-13:Chapter2: Difference between revisions

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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 wave-action balance equation as (Mase 2001)
Taking into account the effect of an ambient horizontal current or wave behavior, CMS-Wave is based on the wave-action balance equation as (Mase 2001)
\begin{align} \tag{1}
{{Equation|
<math>\begin{align}
  \frac{\partial \left( {{C}_{x}}N \right)}{\partial x}+\frac{\partial \left( {{C}_{y}}N \right)}{\partial y}+\frac{\partial \left( {{C}_{\theta }}N \right)}{\partial \theta }=\frac{\kappa }{2\sigma }\left[ {{\left( C{{C}_{g}}\,{{\cos }^{2}}\theta {{N}_{y}} \right)}_{y}}-\frac{C{{C}_{g}}}{2}\,\,{{\cos }^{2}}\theta {{N}_{yy}} \right]-{{\varepsilon }_{b}}N-S
  \frac{\partial \left( {{C}_{x}}N \right)}{\partial x}+\frac{\partial \left( {{C}_{y}}N \right)}{\partial y}+\frac{\partial \left( {{C}_{\theta }}N \right)}{\partial \theta }=\frac{\kappa }{2\sigma }\left[ {{\left( C{{C}_{g}}\,{{\cos }^{2}}\theta {{N}_{y}} \right)}_{y}}-\frac{C{{C}_{g}}}{2}\,\,{{\cos }^{2}}\theta {{N}_{yy}} \right]-{{\varepsilon }_{b}}N-S
\end{align}
\end{align}
</math>|1}}


where
where
\begin{align} \tag{2}
{{Equation|
  N=\frac{(\sigma,\theta)}{\sigma}
<math>\begin{align}
  N=\frac{E(\sigma,\theta)}{\sigma}
\end{align}
\end{align}
</math>|2}}


is the wave-action density to be solved and is a function of frequency and direction . ''E''(,) 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 ''C''<sub>''g''</sub> are wave celerity and group velocity, respectively; ''x'' and ''y'' are the horizontal  coordinates; ''C''<sub>''x''</sub>, ''C''<sub>''y''</sub>, and ''C''<sub></sub> are the characteristic velocity with respect to ''x'', ''y'', and, respectively; ''N''<sub>''y''</sub> and ''N''<sub>''yy''</sub> denote the first and second derivatives of ''N'' with respect to ''y'', respectively; is an empirical parameter representing the intensity of diffraction effect; <sub>''b''</sub> is the parameterization of wave breaking energy dissipation; ''S'' denotes additional source ''S''<sub>''in''</sub> and sink ''S''<sub>''ds''</sub> (e.g., wind forcing, bottom friction loss, etc.) and nonlinear wave-wave interaction term.
is the wave-action density to be solved and is a function of frequency <math>\sigma</math> and direction <math>\theta</math>. E(<math>\sigma,\theta</math>) 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 ''C''<sub>''g''</sub> are wave celerity and group velocity, respectively; ''x'' and ''y'' are the horizontal  coordinates; ''C''<sub>''x''</sub>, ''C''<sub>''y''</sub>, and ''C''<sub><math>\theta</math></sub> are the characteristic velocity with respect to ''x'', ''y'', and, <math>\theta</math> respectively; ''N''<sub>''y''</sub> and ''N''<sub>''yy''</sub> denote the first and second derivatives of ''N'' with respect to ''y'', respectively; <math>\kappa</math> is an empirical parameter representing the intensity of diffraction effect; <math>\varepsilon</math><sub>''b''</sub> is the parameterization of wave breaking energy dissipation; ''S'' denotes additional source ''S''<sub>''in''</sub> and sink ''S''<sub>''ds''</sub> (e.g., wind forcing, bottom friction loss, etc.) and nonlinear wave-wave interaction term.


===Wave diffraction===
===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 values ( 0) needs to be calibrated and optimized for structures. The model omits the diffraction effect for = 0 and calculates diffraction for > 0. Large &nbsp;(> 15) should be avoided as it can cause artificial wave energy losses (Mase 2001). In practice, values of between 0 (no diffraction) and 4 (strong diffraction) have been determined in comparison to measurements. A default value of &nbsp;= 2.5 was used by Mase et al. (2001, 2005a, 2005b) to simulate wave diffraction for both narrow and wide gaps between breakwaters. In CMS-Wave, the default value of 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, &nbsp;= 4 (maximum diffraction allowed in the model) is recommended. For a relatively wider gap, with an opening greater than one wavelength, &nbsp;=&nbsp;3 is recommended. The exact value of in an application is dependent on the structure geometry and adjacent bathymetry, and should to be verified with measurements.
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 <math>\kappa</math> (values <math>\ge</math> 0) needs to be calibrated and optimized for structures. The model omits the diffraction effect for <math>\kappa</math> = 0 and calculates diffraction for <math>\kappa</math> > 0. Large <math>\kappa</math> (values > 15) should be avoided as it can cause artificial wave energy losses (Mase 2001). In practice, values of <math>\kappa</math> between 0 (no diffraction) and 4 (strong diffraction) have been determined in comparison to measurements. A default value of <math>\kappa</math> = 2.5 was used by Mase et al. (2001, 2005a, 2005b) to simulate wave diffraction for both narrow and wide gaps between breakwaters. In CMS-Wave, the default value of <math>\kappa</math> 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, <math>\kappa</math> = 4 (maximum diffraction allowed in the model) is recommended. For a relatively wider gap, with an opening greater than one wavelength, <math>\kappa</math> = 3 is recommended. The exact value of <math>\kappa</math> in an application is dependent on the structure geometry and adjacent bathymetry, and should to be verified with measurements.


===Wave-current interaction===
===Wave-current interaction===
The characteristic velocties ''C''<sub>''x''</sub>,'' C''<sub>''y''</sub>, and'' C''<sub></sub> in Equation 1 can be expressed as:
The characteristic velocties ''C''<sub>''x''</sub>,'' C''<sub>''y''</sub>, and'' C''<sub><math>\theta</math></sub> in Equation 1 can be expressed as:


\begin{align} \tag{3}
{{Equation|<math>\begin{align}
  {{C}_{x}}={{C}_{g}}\cos \theta +U
  {{C}_{x}}={{C}_{g}}\cos \theta +U
\end{align}
\end{align}
</math>|3}}


\begin{align} \tag{4}
{{Equation|<math>\begin{align}
  {{C}_{y}}={{C}_{g}}\sin \theta +V
  {{C}_{y}}={{C}_{g}}\sin \theta +V
\end{align}
\end{align}
</math>|4}}


\begin{align} \tag{5}
{{Equation|<math>\begin{align}
   & {{C}_{\theta }}=\frac{\sigma }{\sinh 2kh}\left( \sin \theta \frac{\partial h}{\partial x}-\cos \theta \frac{\partial h}{\partial y} \right) \\
   & {{C}_{\theta }}=\frac{\sigma }{\sinh 2kh}\left( \sin \theta \frac{\partial h}{\partial x}-\cos \theta \frac{\partial h}{\partial y} \right) \\
   & \begin{matrix}
   & \begin{matrix}
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\end{matrix} \\
\end{matrix} \\
\end{align}
\end{align}
</math>|5}}


where ''U'' and ''V'' are the depth-averaged horizontal current velocity components along the ''x'' and ''y'' axes, ''k'' is the wave number, and <font size = "4"><math>h</math></font> is the water depth. The dispersion relationships between the relative angular frequency , the absolute angular frequency , the wave number vector <math>\vec{k}</math>, and the current velocity vector <math>\left| {\vec{U}} \right|=\sqrt{{{U}^{2}}+{{V}^{2}}}</math>
where ''U'' and ''V'' are the depth-averaged horizontal current velocity components along the ''x'' and ''y'' axes, ''k'' is the wave number, and <font size = "4"><math>h</math></font> is the water depth. The dispersion relationships between the relative angular frequency <math>\sigma</math>, the absolute angular frequency <math>\omega</math>, the wave number vector <math>\vec{k}</math>, and the current velocity vector <math>\left| {\vec{U}} \right|=\sqrt{{{U}^{2}}+{{V}^{2}}}</math>
are (Jonsson 1990)
are (Jonsson 1990)


\begin{align} \tag{6}
{{Equation|<math>\begin{align}
  \sigma=\omega=\vec{k}\cdot\vec{U}
  \sigma=\omega=\vec{k}\cdot\vec{U}
\end{align}
\end{align}
</math>|6}}


and
and


\begin{align} \tag{7}
{{Equation|<math>\begin{align}
  {{\sigma}^{2}}=gk \tanh(kh)
  {{\sigma}^{2}}=gk \tanh(kh)
\end{align}
\end{align}
</math>|7}}


where
where
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<math>\vec{k}\cdot \overrightarrow{U}</math>
<math>\vec{k}\cdot \overrightarrow{U}</math>


is the Doppler-shifting term, and'' g'' is the acceleration due to gravity. The main difference between the wave transformation models with and without ambient currents lies in the solution of the intrinsic frequency. In treatment of the dispersion relation with the Doppler shift, there is no solution corresponding to wave blocking, if intrinsic group velocity ''C''<sub>''g''</sub> is weaker than an opposing current (Smith et al. 1998; Larson and Kraus 2002):
is the Doppler-shifting term, and ''g'' is the acceleration due to gravity. The main difference between the wave transformation models with and without ambient currents lies in the solution of the intrinsic frequency. In treatment of the dispersion relation with the Doppler shift, there is no solution corresponding to wave blocking, if intrinsic group velocity ''C''<sub>''g''</sub> is weaker than an opposing current (Smith et al. 1998; Larson and Kraus 2002):


\begin{align} \tag{8}
{{Equation|<math>\begin{align}
  {{C}_{g}}=\frac{d\sigma }{dk}<\overrightarrow{U}\cdot \vec{k}/k
  {{C}_{g}}=\frac{d\sigma }{dk}<\overrightarrow{U}\cdot \vec{k}/k
\end{align}
\end{align}
 
</math>|8}}


Under the wave blocking condition, waves cannot propagate into a strong opposing current. The wave energy is most likely to dissipate through breaking with a small portion of energy either reflected or transformed to lower frequency components in the wave blocking condition. In CMS-Wave, the wave action corresponding to the wave blocking is set to zero for the corresponding frequency and direction bin.
Under the wave blocking condition, waves cannot propagate into a strong opposing current. The wave energy is most likely to dissipate through breaking with a small portion of energy either reflected or transformed to lower frequency components in the wave blocking condition. In CMS-Wave, the wave action corresponding to the wave blocking is set to zero for the corresponding frequency and direction bin.
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The wave energy reflected at a beach (or upon the surface of a structure) is calculated under assumptions that the incident and reflected wave angles, relative to the shore-normal direction, are equal in magnitude and that the reflected energy is a given fraction of the incident wave energy. The reflected wave action ''N''<sub>''r''</sub> is assumed to be linearly proportional to the incident wave action ''N''<sub>''i ''</sub>:
The wave energy reflected at a beach (or upon the surface of a structure) is calculated under assumptions that the incident and reflected wave angles, relative to the shore-normal direction, are equal in magnitude and that the reflected energy is a given fraction of the incident wave energy. The reflected wave action ''N''<sub>''r''</sub> is assumed to be linearly proportional to the incident wave action ''N''<sub>''i ''</sub>:


\begin{align} \tag{9}
{{Equation|
<math>
\begin{align}
  {{N}_{r}}=K_{r}^{2}{{N}_{i}}
  {{N}_{r}}=K_{r}^{2}{{N}_{i}}
\end{align}
\end{align}
</math>|9}}


where ''K''<sub>''r''</sub> is a reflection coefficient (0 for no reflection and 1 for full reflection) defined as the ratio of reflected to incident wave height (Dean and Dalrymple 1984).
where ''K''<sub>''r''</sub> is a reflection coefficient (0 for no reflection and 1 for full reflection) defined as the ratio of reflected to incident wave height (Dean and Dalrymple 1984).
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The simulation of depth-limited wave breaking is essential in nearshore wave models. A simple wave breaking criterion that is commonly used as a first approximation in shallow water, especially in the surf zone, is a linear function of the ratio of wave height to depth. For random waves, the criterion is (Smith et al. 1999)
The simulation of depth-limited wave breaking is essential in nearshore wave models. A simple wave breaking criterion that is commonly used as a first approximation in shallow water, especially in the surf zone, is a linear function of the ratio of wave height to depth. For random waves, the criterion is (Smith et al. 1999)


\begin{align} \tag{10}
{{Equation|
<math>
\begin{align}
  \frac{{{H}_{b}}}{h}\le 0.64
  \frac{{{H}_{b}}}{h}\le 0.64
\end{align}
\end{align}
</math>|10}}


where ''H''<sub>''b''</sub> denotes the significant breaking wave height. A more comprehensive criterion is based on the limiting steepness by Miche (1951) for random waves as
where ''H''<sub>''b''</sub> denotes the significant breaking wave height. A more comprehensive criterion is based on the limiting steepness by Miche (1951) for random waves as


\begin{align} \tag{11}
{{Equation|
<math>
\begin{align}
  {{H}_{b}}\le \frac{0.64}{{{k}_{p}}}\text{ }\tanh ({{k}_{p}}h)
  {{H}_{b}}\le \frac{0.64}{{{k}_{p}}}\text{ }\tanh ({{k}_{p}}h)
\end{align}
\end{align}
</math>|11}}


where ''k''<sub>''p''</sub> is the wave number corresponding to the spectral peak. In the shallow water condition (''k''<sub>''p''</sub>''h'' small), Equation 11 reduces asymptotically to Equation 10. Iwagaki et al. (1980) verified that Miche's breaker criterion could replicate laboratory measurements over a sloping beach with a current present, provided that the wavelength was calculated with the current included in the dispersion equation.
where ''k''<sub>''p''</sub> is the wave number corresponding to the spectral peak. In the shallow water condition (''k''<sub>''p''</sub>''h'' small), Equation 11 reduces asymptotically to Equation 10. Iwagaki et al. (1980) verified that Miche's breaker criterion could replicate laboratory measurements over a sloping beach with a current present, provided that the wavelength was calculated with the current included in the dispersion equation.
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====Extended Goda formula====
====Extended Goda formula====
Goda (1970) developed a breaker criterion, based on laboratory data, taking into account effects of the bottom slope and wave steepness in deep water. This criterion is used widely in Japan. Goda's formula was modified later by Sakai et al. (1989) to include the action of an opposing current in which a coefficient accounted for the combined effects of current, depth, bottom slope angle , and deepwater wavelength ''L''<sub>o</sub>
Goda (1970) developed a breaker criterion, based on laboratory data, taking into account effects of the bottom slope and wave steepness in deep water. This criterion is used widely in Japan. Goda's formula was modified later by Sakai et al. (1989) to include the action of an opposing current in which a coefficient accounted for the combined effects of current, depth, bottom slope angle <math>\beta</math>, and deepwater wavelength ''L''<sub>o</sub>


\begin{equation} \tag{12}
 
{{Equation|1=<math>
{{H}_{b}}=\left\{ \begin{align}
{{H}_{b}}=\left\{ \begin{align}
   & A\cdot {{L}_{0}}\left\{ 1-\exp \left[ -1.5\frac{\pi h}{{{L}_{0}}}\left( 1+15{{\tan }^{4/3}}\beta  \right) \right] \right\}c\left( {{\varepsilon }_{d}} \right) \\  
   & A\cdot {{L}_{0}}\left\{ 1-\exp \left[ -1.5\frac{\pi h}{{{L}_{0}}}\left( 1+15{{\tan }^{4/3}}\beta  \right) \right] \right\}c\left( {{\varepsilon }_{d}} \right) \\
  & A\cdot {{L}_{0}}\left\{ 1-\exp \left[ -1.5\frac{\pi h}{{{L}_{0}}} \right] \right\} \\  
  & A\cdot {{L}_{0}}\left\{ 1-\exp \left[ -1.5\frac{\pi h}{{{L}_{0}}} \right] \right\} \\
\end{align} \right.\begin{matrix}
\end{align} \right.\begin{matrix}
   {} & \begin{align}
   {} & \begin{align}
   & \tan \beta \ge 0 \\  
   & \tan \beta \ge 0 \\
  &  \\  
  &  \\
  & \tan \beta <0 \\  
  & \tan \beta < 0 \\
\end{align}  \\
\end{align}  \\
\end{matrix}
\end{matrix}
\end{equation}
</math>|2=12}}
 


where ''A&nbsp;''=&nbsp;0.17 is a proportional constant, and
where ''A&nbsp;''=&nbsp;0.17 is a proportional constant, and


{{Equation|1=<math>c\left( {{\varepsilon }_{d}} \right)=\left\{ \begin{align} & 0.506 \\  & 1.13-260{{\varepsilon }_{d}} \\  & 1.0 \\ \end{align} \right.\begin{matrix} {} & \begin{align} & {{\varepsilon }_{d}}\ge 0.0024 \\ & 0.0024>{{\varepsilon }_{d}}\ge 0.0005 \\  & {{\varepsilon }_{d}}<0.0005 \\ \end{align}  \\ \end{matrix}</math>|2=13}}
{{Equation|1=<math>
c\left( {{\varepsilon }_{d}} \right)=\left\{ \begin{align}
  & 0.506 \\  
  & 1.13-260{{\varepsilon }_{d}} \\  
  & 1.0 \\  
\end{align} \right.\begin{matrix}
  {} & \begin{align}
  & {{\varepsilon }_{d}}\ge 0.0024 \\  
& 0.0024>{{\varepsilon }_{d}}\ge 0.0005 \\  
  & {{\varepsilon }_{d}}<0.0005 \\  
\end{align}  \\
\end{matrix}</math>|2=13}}


with
with


{{Equation|1=<math>{{\varepsilon }_{d}}=\frac{|\vec{U}|{{L}_{0}}}{{{g}^{2}}{{T}_{p}}^{3}}({{\tan }^{1/4}}\beta )</math>|2=14}}
{{Equation|<math>
{{\varepsilon }_{d}}=\frac{|\vec{U}|{{L}_{0}}}{{{g}^{2}}{{T}_{p}}^{3}}({{\tan }^{1/4}}\beta )
</math>|14}}
 


in which ''L''<sub>o</sub><math>=</math>''g''<math>T_{p}^{2}\text{ /2 }\!\!\pi\!\!\text{ }</math>, and ''T''<sub>''p''</sub> is the spectral peak period. The change in breaker height with respect to the cell length ''dx'' is defined as
in which ''L''<sub>o</sub><math>=</math>''g''<math>T_{p}^{2}\text{ /2 }\!\!\pi\!\!\text{ }</math>, and ''T''<sub>''p''</sub> is the spectral peak period. The change in breaker height with respect to the cell length ''dx'' is defined as
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{{Equation|1=<math>\frac{{{H}_{b}}}{{{L}_{b}}}=a\cdot \tanh \left( \frac{\gamma }{0.88}\frac{2\pi h}{{{L}_{b}}} \right)</math>|2=19}}
{{Equation|1=<math>\frac{{{H}_{b}}}{{{L}_{b}}}=a\cdot \tanh \left( \frac{\gamma }{0.88}\frac{2\pi h}{{{L}_{b}}} \right)</math>|2=19}}


where ''L''<sub>''b''</sub> is the wavelength at the breaking location including the current, &nbsp;is an adjustable coefficient varying with the beach slope, and ''a'' = 0.14. This formula reduces to a steepness limit in deep water and to a depth limit in shallow water, thus incorporating both wave breaking limits in a simple form. The coefficient has been treated as a constant value of 0.8 in application for random waves (Battjes and Janssen 1978). Based on field and laboratory data, Ostendorf and Madsen (1979) suggested that
where ''L''<sub>''b''</sub> is the wavelength at the breaking location including the current, <math>\gamma</math> is an adjustable coefficient varying with the beach slope, and ''a'' = 0.14. This formula reduces to a steepness limit in deep water and to a depth limit in shallow water, thus incorporating both wave breaking limits in a simple form. The coefficient <math>\gamma</math> has been treated as a constant value of 0.8 in application for random waves (Battjes and Janssen 1978). Based on field and laboratory data, Ostendorf and Madsen (1979) suggested that


{{Equation|1=<math>\gamma =\left\{ \begin{align}
{{Equation|1=<math>\gamma =\left\{ \begin{align}
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\end{matrix}</math>|2=21}}
\end{matrix}</math>|2=21}}


Equations 19 to 21 are incorporated in Equations 16 to 18 to calculate the spectral energy dissipation rate <sub>''b''</sub>.
Equations 19 to 21 are incorporated in Equations 16 to 18 to calculate the spectral energy dissipation rate <math>\varepsilon</math><sub>''b''</sub>.


====Battjes and Janssen formula====
====Battjes and Janssen formula====
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<math>\overline{D}</math> in a bore of the same height as a depth-induced breaking wave as
<math>\overline{D}</math> in a bore of the same height as a depth-induced breaking wave as


\begin{equation} \label
{{Equation|<math>
  \overline{D}=\frac{\alpha \rho g}{4}{{Q}_{b}}\overline{f}H_{b}^{2}
  \overline{D}=\frac{\alpha \rho g}{4}{{Q}_{b}}\overline{f}H_{b}^{2}
\end{equation}
</math>|22}}


where <math>\alpha</math> is an empirical coefficient of order one, <math>\rho</math> is the sea water density,
where <math>\alpha</math> is an empirical coefficient of order one, <math>\rho</math> is the sea water density,
<math>\overline{f}</math> is the spectral mean frequency, and ''Q''<sub>''b''</sub> is the probability that at a given location the wave is breaking. By assuming the wave height has a Rayleigh distribution, the probability of wave breaking can be determined from the following expression
<math>\overline{f}</math> is the spectral mean frequency, and ''Q''<sub>''b''</sub> is the probability that at a given location the wave is breaking. By assuming the wave height has a Rayleigh distribution, the probability of wave breaking can be determined from the following expression


{{Equation|1=<math>\frac{1-{{Q}_{b}}}{\ln {{Q}_{b}}}=-{{\left( \frac{{{H}_{\text{rms}}}}{{{H}_{b}}} \right)}^{2}}</math>|2=23}}
{{Equation|<math>
\frac{1-{{Q}_{b}}}{\ln {{Q}_{b}}}=-{{\left( \frac{{{H}_{\text{rms}}}}{{{H}_{b}}} \right)}^{2}}
</math>|23}}


where <math>{{H}_{\text{rms}}}={{H}_{1/3}}/\sqrt{2}</math> is the root-mean-square wave height. Battjes and Janssen (1978) calculated the maximum possible height from Equation&nbsp;19 using a constant breaker value <math>\gamma</math> = 0.8. Booij et al. (1999) and Chen et al. (2005) investigated Battjes and Janssen formula and obtained a better wave breaking estimate with <math>\gamma</math> = 0.73. In CMS-Wave, Equation 23 is adapted to parameterize the wave breaking energy dissipation by applying the Battjes and Janssen formula with <math>\gamma</math>&nbsp;=&nbsp;0.73. The calculation of the wave breaking dissipation rate is from:
where <math>{{H}_{\text{rms}}}={{H}_{1/3}}/\sqrt{2}</math> is the root-mean-square wave height. Battjes and Janssen (1978) calculated the maximum possible height from Equation&nbsp;19 using a constant breaker value <math>\gamma</math> = 0.8. Booij et al. (1999) and Chen et al. (2005) investigated Battjes and Janssen formula and obtained a better wave breaking estimate with <math>\gamma</math> = 0.73. In CMS-Wave, Equation 23 is adapted to parameterize the wave breaking energy dissipation by applying the Battjes and Janssen formula with <math>\gamma</math>&nbsp;=&nbsp;0.73. The calculation of the wave breaking dissipation rate is from:


\begin{equation} \label{eq:24}
{{Equation|<math>
  \varepsilon_b=\frac{\overline{D}}{\left( \rho g{H}^2_{\text{rms}}/8 \right)2\pi \overline{f}}  
  \varepsilon_b=\frac{\overline{D}}{\left( \rho g{H}^2_{\text{rms}}/8 \right)2\pi \overline{f}}  
\end{equation}
</math>|24}}


====Chawla and Kirby formula====
====Chawla and Kirby formula====
Line 185: Line 221:
<math>\overline{D}</math> was defined as
<math>\overline{D}</math> was defined as


{{Equation|1=<math>\overline{D}=\frac{3b\text{ }\!\!\rho\!\!\text{ }}{32\sqrt{\text{ }\!\!\pi\!\!\text{ }}}\sqrt{\frac{{{(g\bar{k})}^{3}}}{\tanh \bar{k}h}}{{\left( \frac{{\bar{k}}}{\text{ }\!\!\gamma\!\!\text{ }\tanh \bar{k}h} \right)}^{2}}H_{\text{rms}}^{5}\left\{ 1-{{\left[ 1+{{\left( \frac{\bar{k}\text{ }{{H}_{\text{rms}}}}{\text{ }\!\!\gamma\!\!\text{ }\tanh \bar{k}h} \right)}^{2}} \right]}^{-\frac{5}{2}}} \right\}</math>|2=25}}
{{Equation|<math>
\overline{D}=\frac{3b\text{ }\!\!\rho\!\!\text{ }}{32\sqrt{\text{ }\!\!\pi\!\!\text{ }}}\sqrt{\frac{{{(g\bar{k})}^{3}}}{\tanh \bar{k}h}}{{\left( \frac{{\bar{k}}}{\text{ }\!\!\gamma\!\!\text{ }\tanh \bar{k}h} \right)}^{2}}H_{\text{rms}}^{5}\left\{ 1-{{\left[ 1+{{\left( \frac{\bar{k}\text{ }{{H}_{\text{rms}}}}{\text{ }\!\!\gamma\!\!\text{ }\tanh \bar{k}h} \right)}^{2}} \right]}^{-\frac{5}{2}}} \right\}
</math>|25}}


where <math>\overline{k}</math> is the wave number corresponding to the spectral mean frequency <math>\overline{f}</math>, and scaling parameters ''b'' and are equal to 0.4 and 0.6, respectively. The rate of wave breaking energy dissipation is calculated by Equation 24.
where <math>\overline{k}</math> is the wave number corresponding to the spectral mean frequency <math>\overline{f}</math>, and scaling parameters ''b'' and <math>\gamma</math> are equal to 0.4 and 0.6, respectively. The rate of wave breaking energy dissipation is calculated by Equation 24.


===Wind forcing and whitecapping dissipation===
===Wind forcing and whitecapping dissipation===
The evolution of waves in the large-scale, open coast is more affected by wind-ocean-wave interactions than on the nearshore wave-current-bottom processes. The result is a nonlinear wave field that is balanced between wind forcing, whitecapping, and wave growth. The surface wind can feed energy into the existing waves and can also generate new waves. On the other hand, the energy can dissipate through whitecapping from turbulence-wave interactions and air-wave-water interactions. In CMS-Wave, these wind forcing and whitecapping processes are modeled as separate sink and source terms (Lin and Lin 2004a and b).
The evolution of waves in the large-scale, open coast is more affected by wind-ocean-wave interactions than on the nearshore wave-current-bottom processes. The result is a nonlinear wave field that is balanced between wind forcing, whitecapping, and wave growth. The surface wind can feed energy into the existing waves and can also generate new waves. On the other hand, the energy can dissipate through whitecapping from turbulence-wave interactions and air-wave-water interactions. In CMS-Wave, these wind forcing and whitecapping processes are modeled as separate sink and source terms (Lin and Lin 2004a and b).
====Wind input function====
====Wind input function====
The wind-input source ''S''<sub>''in''</sub> is formulated as functions of the ratio of wave celerity ''C ''to wind speed ''W'', the ratio of wave group velocity to wind speed, the difference of wind speed and wave celerity, and the difference between wind direction <sub>wind</sub> and wave direction (Lin and Lin 2006b):
The wind-input source ''S''<sub>''in''</sub> is formulated as functions of the ratio of wave celerity ''C ''to wind speed ''W'', the ratio of wave group velocity to wind speed, the difference of wind speed and wave celerity, and the difference between wind direction <math>\theta</math><sub>wind</sub> and wave direction <math>\theta</math> (Lin and Lin 2006b):


{{Equation|1=<math>{{S}_{in}}=\frac{{{a}_{1}}\sigma }{g}{{F}_{1}}(\overrightarrow{W}-{{\vec{C}}_{g}}){{F}_{2}}(\frac{{{C}_{g}}}{W})E_{\text{PM}}^{*}(\sigma )\Phi (\text{ }\!\!\theta\!\!\text{ })+\frac{{{a}_{2}}{{\sigma }^{2}}}{g}{{F}_{1}}(\overrightarrow{W}-{{\vec{C}}_{g}})\text{ }{{F}_{2}}(\frac{{{C}_{g}}}{W}){{F}_{3}}(\frac{{{C}_{g}}}{W})N</math>|2=26}}
{{Equation|<math>
{{S}_{in}}=\frac{{{a}_{1}}\sigma }{g}{{F}_{1}}(\overrightarrow{W}-{{\vec{C}}_{g}}){{F}_{2}}(\frac{{{C}_{g}}}{W})E_{\text{PM}}^{*}(\sigma )\Phi (\text{ }\!\!\theta\!\!\text{ })+\frac{{{a}_{2}}{{\sigma }^{2}}}{g}{{F}_{1}}(\overrightarrow{W}-{{\vec{C}}_{g}})\text{ }{{F}_{2}}(\frac{{{C}_{g}}}{W}){{F}_{3}}(\frac{{{C}_{g}}}{W})N
</math>|26}}


where
where


{{Equation|1=[[File:TR-08-13_Eq27.png]]|2=27}}
{{Equation|<math>
{{F}_{1}}(\vec{W}-{{\vec{C}}_{g}})=\left\{ \begin{array}{*{35}{l}}
  W\cos ({{\theta }_{wind}}-\theta )-{{C}_{g}}, & \text{if}\quad  & {{C}_{g}}<W  \\
  0, & \text{if}\quad  & {{C}_{g}}\ge W  \\
\end{array} \right.
</math>|27}}


{{Equation|1=[[File:TR-08-13_Eq28.png]]|2=28}}
{{Equation|<math>
{{F}_{1}}(\vec{W}-{{{\vec{C}}}_{g}})=\left\{ \begin{array}{*{35}{l}}
  {{\left( \frac{{{C}_{g}}}{W} \right)}^{1.15}}, & \text{if}\quad  & {{C}_{g}}<W  \\
  0, & \text{if}\quad  & {{C}_{g}}\ge W  \\
\end{array} \right.
</math>|2=28}}


{{Equation|1=[[File:TR-08-13_Eq29.png]]|2=29}}
{{Equation|<math>
{{F}_{3}}(\frac{{{C}_{g}}}{W})=\left\{ \begin{array}{*{35}{l}}
  {{\log }_{10}}\left[ {{\left( \frac{{{C}_{g}}}{W} \right)}^{-1}} \right], & \text{if}\quad  & {{C}_{g}}<W  \\
  0, & \text{if}\quad  & {{C}_{g}}\ge W  \\
\end{array} \right.
</math>|2=29}}


and
and


{{Equation|1=<math>E_{\text{PM}}^{*}(\sigma )=\frac{{{g}^{2}}}{{{\sigma }^{5}}}\exp (-0.74\frac{\sigma _{0}^{4}}{{{\sigma }^{4}}})</math>|2=30}}
{{Equation|<math>
E_{\text{PM}}^{*}(\sigma )=\frac{{{g}^{2}}}{{{\sigma }^{5}}}\exp (-0.74\frac{\sigma _{0}^{4}}{{{\sigma }^{4}}})
</math>|2=30}}


<math>E_{\text{PM}}^{*}(\sigma )</math> is the functional form of the Pierson-Moskowitz (PM) spectrum, <sub>o&nbsp;</sub>=&nbsp;''g''/''W'' is the Phillips constant, and
<math>E_{\text{PM}}^{*}(\sigma )</math> is the functional form of the Pierson-Moskowitz (PM) spectrum, <math>\sigma</math><sub>o&nbsp;</sub>=&nbsp;''g''/''W'' is the Phillips constant, and


{{Equation|1=<math>\Phi (\text{ }\!\!\theta\!\!\text{ })=\frac{8}{3\text{ }\!\!\pi\!\!\text{ }}{{\cos }^{4}}\left( \text{ }\!\!\theta\!\!\text{  - }{{\text{ }\!\!\theta\!\!\text{ }}_{\text{wind}}} \right)
{{Equation|<math>
\text{ ,for   } \text{ }\!\!\theta\!\!\text{  - }{{\text{ }\!\!\theta\!\!\text{ }}_{\text{wind}}}
\phi (\theta )=\frac{8}{3\pi }{{\cos }^{4}}(\theta -{{\theta }_{wind}}),\quad \text{for}\quad \left| \theta -\theta wind \right|\quad \le \frac{\pi }{2}
\le \frac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}</math> |2=31}}
</math>|2=31}}




is a normalized directional spreading. The function ''F''<sub>1</sub> presents the wind stress effect, ''F''<sub>2</sub> designates Phillips' mechanisms (Phillips 1957) and ''F''<sub>3</sub> accounts for the wave age effect. For swell or long waves, the wave group velocity ''C''<sub>''g''</sub> is generally large and ''F''<sub>3</sub> < 1. If'' C''<sub>''g''</sub> ''W'', then ''F''<sub>3</sub>&nbsp;=&nbsp;0. For short waves, the phase velocity is generally small and ''F''<sub>3</sub> > 1.
is a normalized directional spreading. The function ''F''<sub>1</sub> presents the wind stress effect, ''F''<sub>2</sub> designates Phillips' mechanisms (Phillips 1957) and ''F''<sub>3</sub> accounts for the wave age effect. For swell or long waves, the wave group velocity ''C''<sub>''g''</sub> is generally large and ''F''<sub>3</sub> < 1. If'' C''<sub>''g''</sub> '' <math>\ge</math> W'', then ''F''<sub>3</sub>&nbsp;=&nbsp;0. For short waves, the phase velocity is generally small and ''F''<sub>3</sub> > 1.


====Whitecapping dissipation function====
====Whitecapping dissipation function====
The wave energy dissipation (sink) ''S''<sub>''ds''</sub> (Lin and Lin 2006b) for whitecapping including current and turbulent viscous effect is
The wave energy dissipation (sink) ''S''<sub>''ds''</sub> (Lin and Lin 2006b) for whitecapping including current and turbulent viscous effect is


{{Equation|1=<math>{{S}_{ds}}=-{{c}_{ds}}{{({{a}_{e}}k)}^{1.5}}\frac{{{\sigma }^{2}}}{g}{{C}_{g}}(\sigma ,\text{ }\!\!\theta\!\!\text{ })\text{ }{{F}_{4}}(\overrightarrow{W},\overrightarrow{U},{{\vec{C}}_{g}})\text{ }{{F}_{5}}(kh)\text{ }N</math>|2=32}}
{{Equation|<math>
{{S}_{ds}}=-{{c}_{ds}}{{({{a}_{e}}k)}^{1.5}}\frac{{{\sigma }^{2}}}{g}{{C}_{g}}(\sigma ,\theta ){{F}_{4}}(\vec{W},\vec{U},{{\vec{C}}_{g}}){{F}_{5}}(kh)N
</math>|2=32}}


with
with


{{Equation|1=<math>{{F}_{4}}(\overrightarrow{W},\text{ }\overrightarrow{U},{{\vec{C}}_{g}})=\frac{v+W}{|\overrightarrow{W}+\overrightarrow{U}+{{{\vec{C}}}_{g}}|}</math>|2=33}}
{{Equation|<math>
 
{{F}_{4}}(\vec{W},\text{ }\vec{U},{{\vec{C}}_{g}})=\frac{v+W}{|\vec{W}+\vec{U}+{{{\vec{C}}}_{g}}|}
</math>|2=33}}


and
and


{{Equation|1=<math>{{F}_{5}}(kh)=\frac{1}{\tanh kh}</math>|2=34}}
{{Equation|<math>
{{F}_{5}}(kh)=\frac{1}{\tanh kh}
</math>|2=34}}


where ''c''<sub>''ds''</sub> is a proportionality coefficient, and is for the turbulent viscous dissipation. The wave amplitude <math>{{a}_{e}}=\sqrt{E(\sigma ,\text{ }\!\!\theta\!\!\text{ })d\sigma d\text{ }\!\!\theta\!\!\text{ }}</math> is calculated at each grid cell. To avoid numerical instability and considering the physical constraint of energy loss for the dissipation, the function ''F''<sub>4</sub> is set to 1 if the computed value is greater than 1.
where ''c''<sub>''ds''</sub> is a proportionality coefficient, and <math>{v}</math> is for the turbulent viscous dissipation. The wave amplitude <math>{{a}_{e}}=\sqrt{E(\sigma ,\text{ }\!\!\theta\!\!\text{ })d\sigma d\text{ }\!\!\theta\!\!\text{ }}</math> is calculated at each grid cell. To avoid numerical instability and considering the physical constraint of energy loss for the dissipation, the function ''F''<sub>4</sub> is set to 1 if the computed value is greater than 1.


===Wave generation with arbitrary wind direction===
===Wave generation with arbitrary wind direction===
Line 238: Line 300:
The bottom friction loss (sink) ''S''<sub>''ds''</sub> is calculated by a drag law model (Collins 1972)
The bottom friction loss (sink) ''S''<sub>''ds''</sub> is calculated by a drag law model (Collins 1972)


{{Equation|1=<math>{{S}_{ds}}=-{{c}_{f}}\frac{{{\text{ }\!\!\sigma\!\!\text{ }}^{2}}}{g}\frac{\langle {{u}_{b}}\rangle }{{{\sinh }^{2}}kh}N</math>|2=35}}
{{Equation|<math>
{{S}_{ds}}=-{{c}_{f}}\frac{{{\text{ }\!\!\sigma\!\!\text{ }}^{2}}}{g}\frac{\langle {{u}_{b}}\rangle }{{{\sinh }^{2}}kh}N
</math>|2=35}}


with
with


{{Equation|1=<math>\langle {{u}_{b}}\rangle \,=\frac{1}{2}\sqrt{\frac{g}{h}{{E}_{total}}}</math>|2=36}}
{{Equation|<math>
\langle {{u}_{b}}\rangle \,=\frac{1}{2}\sqrt{\frac{g}{h}{{E}_{total}}}
</math>|2=36}}


where ''u''<sub>''b''</sub>presents the ensemble mean of horizontal wave orbital velocity at the sea bed, ''E''<sub>''total''</sub> is the total energy density at a grid cell, and ''c''<sub>''f ''</sub>is the Darcy-Weisbach type friction coefficient. The relationship between ''c''<sub>''f''</sub> and the Darcy-Weisbach friction factor ''f''<sub>''DW''</sub> is ''c''<sub>''f''</sub> ='' f''<sub>''DW''</sub>/8.
where <math>\langle{u}</math><sub>''b''</sub><math>\rangle</math> presents the ensemble mean of horizontal wave orbital velocity at the sea bed, ''E''<sub>''total''</sub> is the total energy density at a grid cell, and ''c''<sub>''f ''</sub>is the Darcy-Weisbach type friction coefficient. The relationship between ''c''<sub>''f''</sub> and the Darcy-Weisbach friction factor ''f''<sub>''DW''</sub> is ''c''<sub>''f''</sub> ='' f''<sub>''DW''</sub>/8.
Typical values of ''c''<sub>''f''</sub> for sandy bottoms range from 0.004 to 0.007 based on the JONSWAP experiment and North Sea measurements (Hasselmann et&nbsp;al. 1973; Bouws and Komen 1983). Values of ''c''<sub>''f''</sub> applied for coral reefs range from 0.05 to 0.40 (Hardy 1993; Hearn 1999; Lowe et al. 2005). Application of this model capability to a specific site requires validation to field data.
Typical values of ''c''<sub>''f''</sub> for sandy bottoms range from 0.004 to 0.007 based on the JONSWAP experiment and North Sea measurements (Hasselmann et&nbsp;al. 1973; Bouws and Komen 1983). Values of ''c''<sub>''f''</sub> applied for coral reefs range from 0.05 to 0.40 (Hardy 1993; Hearn 1999; Lowe et al. 2005). Application of this model capability to a specific site requires validation to field data.
If the Manning friction coefficient ''n'' is used instead of the Darcy-Weisbach type coefficient, the relationship between the two drag coefficients is
If the Manning friction coefficient ''n'' is used instead of the Darcy-Weisbach type coefficient, the relationship between the two drag coefficients is


{{Equation|1=<math>{{c}_{f}}=\frac{g{{n}^{2}}}{{{h}^{1/3}}}</math>|2=37}}
{{Equation|<math>
{{c}_{f}}=\frac{g{{n}^{2}}}{{{h}^{1/3}}}
</math>|2=37}}


Estimates of Manning coefficient ''n'' are available in most fluid mechanics reference books (e.g., 0.01 to 0.05 for smooth to rocky/weedy channels).
Estimates of Manning coefficient ''n'' are available in most fluid mechanics reference books (e.g., 0.01 to 0.05 for smooth to rocky/weedy channels).
Line 273: Line 341:
{{Equation|1=<math>R\text{2}={{\text{2}}_{{}}}{{\eta }_{\max }}</math>|2=43}}
{{Equation|1=<math>R\text{2}={{\text{2}}_{{}}}{{\eta }_{\max }}</math>|2=43}}


In CMS-Wave, ''R2'' is calculated at the land-water interface and averaged with the local depth to determine if the water can flood the proceeding dry cell. If the wave runup level is higher than the adjacent land cell elevation, CMS-Wave can flood the dry cells and simulate wave overtopping and overwash at them. The feature is useful in coupling CMS-Wave to CMS-Flow (Buttolph et al. 2006) for calculating beach erosion or breaching. Calculated quantities of ''S''<sub>''xx''</sub>/<sub>''x''</sub>, ''S''<sub>''xy''</sub>/''x'', ''S''<sub>''xy''</sub>/''y'', and ''S''<sub>''yy''</sub>/''y'' are saved as input to CMS-Flow. CMS-Wave reports the calculated fields of wave setup and maximum water level defined as
In CMS-Wave, ''R2'' is calculated at the land-water interface and averaged with the local depth to determine if the water can flood the proceeding dry cell. If the wave runup level is higher than the adjacent land cell elevation, CMS-Wave can flood the dry cells and simulate wave overtopping and overwash at them. The feature is useful in coupling CMS-Wave to CMS-Flow (Buttolph et al. 2006) for calculating beach erosion or breaching. Calculated quantities of <math>\partial</math>''S''<sub>''xx''</sub>/<math>\partial</math>''x'', <math>\partial</math>''S''<sub>''xy''</sub>/<math>\partial</math>''x'', <math>\partial</math>''S''<sub>''xy''</sub>/<math>\partial</math>''y'', and <math>\partial</math>''S''<sub>''yy''</sub>/<math>\partial</math>''y'' are saved as input to CMS-Flow. CMS-Wave reports the calculated fields of wave setup and maximum water level defined as


{|border="0" cellspacing="2" width="100%" style="font-size:120%"
{|border="0" cellspacing="2" width="100%" style="font-size:120%"
|&nbsp;
|&nbsp;
|Maximum water level = Max (R2, + H1/3/2)
|Maximum water level = Max (R2, <math>\eta</math> + H1/3/2)
|align = "right"|(44)
|align = "right"|(44)
|}
|}
Line 284: Line 352:
CMS-Wave applies a simple analytical formula to compute the wave transmission coefficient ''K''<sub>''t</sub> ''of a rigidly moored rectangular breakwater of width ''B''<sub>''c''</sub> and draft ''D''<sub>''c''</sub> (Macagno 1953)
CMS-Wave applies a simple analytical formula to compute the wave transmission coefficient ''K''<sub>''t</sub> ''of a rigidly moored rectangular breakwater of width ''B''<sub>''c''</sub> and draft ''D''<sub>''c''</sub> (Macagno 1953)


{{Equation|1=<math>{{K}_{t}}={{\left[ 1+{{\left( \frac{k{{B}_{c}}\sinh \frac{kh}{2\text{ }\!\!\pi\!\!\text{ }}}{2\cosh k(h-{{D}_{c}})} \right)}^{2}} \right]}^{-\frac{1}{2}}}</math>|2=45}}
{{Equation|<math>
{{K}_{t}}={{\left[ 1+{{\left( \frac{k{{B}_{c}}\sinh \frac{kh}{2\pi}}{2\cosh k(h-{{D}_{c}})} \right)}^{2}} \right]}^{-\frac{1}{2}}}
</math>|2=45}}


Wave transmission over a structure or breakwater is caused mainly by the fall of the overtopping water mass. Therefore, the ratio of the structure crest elevation to the incident wave height is the prime parameter governing the wave transmission. CMS calculates the rate of overtopping of a vertical breakwater based on the simple expression (Goda 1985) as
Wave transmission over a structure or breakwater is caused mainly by the fall of the overtopping water mass. Therefore, the ratio of the structure crest elevation to the incident wave height is the prime parameter governing the wave transmission. CMS calculates the rate of overtopping of a vertical breakwater based on the simple expression (Goda 1985) as


{{Equation|1=<math>{{K}_{t}}={{0.3}_{{}}}(1.5-\frac{{{h}_{c}}}{{{H}_{i}}}),\text{  for  }0\le \frac{{{h}_{c}}}{{{H}_{i}}}\le 1.25</math>|2=46}}
{{Equation|<math>
{{K}_{t}}={{0.3}_{{}}}(1.5-\frac{{{h}_{c}}}{{{H}_{i}}}),\text{  for  }0\le \frac{{{h}_{c}}}{{{H}_{i}}}\le 1.25
</math>|2=46}}


where ''h''<sub>''c''</sub> is the crest elevation of the breakwater above the still-water level, and ''H''<sub>''i''</sub> is the incident wave height. Equation 46 is modified for a composite breakwater, protected by a mound of armor units at its front, as
where ''h''<sub>''c''</sub> is the crest elevation of the breakwater above the still-water level, and ''H''<sub>''i''</sub> is the incident wave height. Equation 46 is modified for a composite breakwater, protected by a mound of armor units at its front, as


{{Equation|1=<math>{{K}_{t}}={{0.3}_{{}}}(1.1-\frac{{{h}_{c}}}{{{H}_{i}}}),\text{    for  }0\le \frac{{{h}_{c}}}{{{H}_{i}}}\le 0.75</math>|2=47}}
{{Equation|<math>
{{K}_{t}}={{0.3}_{{}}}(1.1-\frac{{{h}_{c}}}{{{H}_{i}}}),\text{    for  }0\le \frac{{{h}_{c}}}{{{H}_{i}}}\le 0.75
</math>|2=47}}


For rubble-mound breakwaters, the calculation of wave transmission is more complicated because the overtopping rate also depends on the specific design of the breakwater (e. g., toe apron protection, front slope, armor unit shape and size, thickness of armor layers). In practice, Equation 47 still can be applied using a finer spatial resolution with the proper bathymetry and adequate bottom friction coefficients to represent the breakwater.
For rubble-mound breakwaters, the calculation of wave transmission is more complicated because the overtopping rate also depends on the specific design of the breakwater (e. g., toe apron protection, front slope, armor unit shape and size, thickness of armor layers). In practice, Equation 47 still can be applied using a finer spatial resolution with the proper bathymetry and adequate bottom friction coefficients to represent the breakwater.

Latest revision as of 16:25, 14 March 2017

Model Description

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 wave-action balance equation as (Mase 2001)

 

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial \left( {{C}_{x}}N \right)}{\partial x}+\frac{\partial \left( {{C}_{y}}N \right)}{\partial y}+\frac{\partial \left( {{C}_{\theta }}N \right)}{\partial \theta }=\frac{\kappa }{2\sigma }\left[ {{\left( C{{C}_{g}}\,{{\cos }^{2}}\theta {{N}_{y}} \right)}_{y}}-\frac{C{{C}_{g}}}{2}\,\,{{\cos }^{2}}\theta {{N}_{yy}} \right]-{{\varepsilon }_{b}}N-S \end{align} }

(1)

where

 

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} N=\frac{E(\sigma,\theta)}{\sigma} \end{align} }

(2)

is the wave-action density to be solved and is a function of frequency Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sigma} and direction Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta} . E(Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sigma,\theta} ) 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 CFailed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta} are the characteristic velocity with respect to x, y, and, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta} respectively; Ny and Nyy denote the first and second derivatives of N with respect to y, respectively; Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa} is an empirical parameter representing the intensity of diffraction effect; Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varepsilon} b 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa} (values Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \ge} 0) needs to be calibrated and optimized for structures. The model omits the diffraction effect for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa} = 0 and calculates diffraction for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa} > 0. Large Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa} (values > 15) should be avoided as it can cause artificial wave energy losses (Mase 2001). In practice, values of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa} between 0 (no diffraction) and 4 (strong diffraction) have been determined in comparison to measurements. A default value of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa} = 2.5 was used by Mase et al. (2001, 2005a, 2005b) to simulate wave diffraction for both narrow and wide gaps between breakwaters. In CMS-Wave, the default value of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa} 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, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa} = 4 (maximum diffraction allowed in the model) is recommended. For a relatively wider gap, with an opening greater than one wavelength, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa} = 3 is recommended. The exact value of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa} in an application is dependent on the structure geometry and adjacent bathymetry, and should to be verified with measurements.

Wave-current interaction

The characteristic velocties Cx, Cy, and CFailed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta} in Equation 1 can be expressed as:

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {{C}_{x}}={{C}_{g}}\cos \theta +U \end{align} } (3)
  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {{C}_{y}}={{C}_{g}}\sin \theta +V \end{align} } (4)
  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & {{C}_{\theta }}=\frac{\sigma }{\sinh 2kh}\left( \sin \theta \frac{\partial h}{\partial x}-\cos \theta \frac{\partial h}{\partial y} \right) \\ & \begin{matrix} {} & +\cos \theta \,\sin \theta \frac{\partial U}{\partial x}-{{\cos }^{2}}\theta \frac{\partial U}{\partial y}+{{\sin }^{2}}\theta \frac{\partial V}{\partial x}-\sin \theta \,\cos \theta \frac{\partial V}{\partial y} & {} \\ \end{matrix} \\ \end{align} } (5)

where U and V are the depth-averaged horizontal current velocity components along the x and y axes, k is the wave number, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h} is the water depth. The dispersion relationships between the relative angular frequency Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sigma} , the absolute angular frequency Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega} , the wave number vector Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \vec{k}} , and the current velocity vector Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \left| {\vec{U}} \right|=\sqrt{{{U}^{2}}+{{V}^{2}}}} are (Jonsson 1990)

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sigma=\omega=\vec{k}\cdot\vec{U} \end{align} } (6)

and

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {{\sigma}^{2}}=gk \tanh(kh) \end{align} } (7)

where

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \vec{k}\cdot \overrightarrow{U}}

is the Doppler-shifting term, and g is the acceleration due to gravity. The main difference between the wave transformation models with and without ambient currents lies in the solution of the intrinsic frequency. In treatment of the dispersion relation with the Doppler shift, there is no solution corresponding to wave blocking, if intrinsic group velocity Cg is weaker than an opposing current (Smith et al. 1998; Larson and Kraus 2002):

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {{C}_{g}}=\frac{d\sigma }{dk}<\overrightarrow{U}\cdot \vec{k}/k \end{align} } (8)

Under the wave blocking condition, waves cannot propagate into a strong opposing current. The wave energy is most likely to dissipate through breaking with a small portion of energy either reflected or transformed to lower frequency components in the wave blocking condition. In CMS-Wave, the wave action corresponding to the wave blocking is set to zero for the corresponding frequency and direction bin.

Wave reflection

The wave energy reflected at a beach (or upon the surface of a structure) is calculated under assumptions that the incident and reflected wave angles, relative to the shore-normal direction, are equal in magnitude and that the reflected energy is a given fraction of the incident wave energy. The reflected wave action Nr is assumed to be linearly proportional to the incident wave action Ni :

 

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {{N}_{r}}=K_{r}^{2}{{N}_{i}} \end{align} }

(9)

where Kr is a reflection coefficient (0 for no reflection and 1 for full reflection) defined as the ratio of reflected to incident wave height (Dean and Dalrymple 1984).

CMS-Wave calculates the wave energy reflection toward the shore, e.g., reflection from a sidewall or jetty, within the wave transformation routine. It can also calculate reflection toward the sea boundary, e.g., wave reflection off the beach or detached breakwater, with a backward marching calculation routine (Mase et al. 2005a). Users should be aware that although the computer execution time calculating forward reflection is relatively small, the time is almost double for the backward reflection routine.

Wave breaking formulas

The simulation of depth-limited wave breaking is essential in nearshore wave models. A simple wave breaking criterion that is commonly used as a first approximation in shallow water, especially in the surf zone, is a linear function of the ratio of wave height to depth. For random waves, the criterion is (Smith et al. 1999)

 

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{{{H}_{b}}}{h}\le 0.64 \end{align} }

(10)

where Hb denotes the significant breaking wave height. A more comprehensive criterion is based on the limiting steepness by Miche (1951) for random waves as

 

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {{H}_{b}}\le \frac{0.64}{{{k}_{p}}}\text{ }\tanh ({{k}_{p}}h) \end{align} }

(11)

where kp is the wave number corresponding to the spectral peak. In the shallow water condition (kph small), Equation 11 reduces asymptotically to Equation 10. Iwagaki et al. (1980) verified that Miche's breaker criterion could replicate laboratory measurements over a sloping beach with a current present, provided that the wavelength was calculated with the current included in the dispersion equation.

In CMS-Wave, the depth-limited spectral energy dissipation can be selected from four different formulas: (a) Extended Goda formulation (Sakai et al. 1989), (b) Extended Miche (Battjes 1972; Mase et al. 2005b), (c) Battjes and Janssen (1978), and (d) Chawla and Kirby (2002). These formulas, considered more accurate for wave breaking on a current, can be divided into two generic categories (Zheng et al. 2008). The first class of formulations attempts to simulate the energy dissipation due to wave breaking by truncating the tail of the Rayleigh distribution of wave height on the basis of some breaker criterion. The Extended Goda and Extended Miche formulas belong to this class. The second category of wave breaking formulas uses a bore model analogy (Battjes and Janssen 1978) to estimate the total energy dissipation. The Battjes and Janssen formula and Chawla and Kirby formula are in this class. The spectral energy dissipation is calculated based on one of these four wave breaking formulas, and the computed wave height is limited by both Equations 10 and 11.

Extended Goda formula

Goda (1970) developed a breaker criterion, based on laboratory data, taking into account effects of the bottom slope and wave steepness in deep water. This criterion is used widely in Japan. Goda's formula was modified later by Sakai et al. (1989) to include the action of an opposing current in which a coefficient accounted for the combined effects of current, depth, bottom slope angle Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta} , and deepwater wavelength Lo


  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{H}_{b}}=\left\{ \begin{align} & A\cdot {{L}_{0}}\left\{ 1-\exp \left[ -1.5\frac{\pi h}{{{L}_{0}}}\left( 1+15{{\tan }^{4/3}}\beta \right) \right] \right\}c\left( {{\varepsilon }_{d}} \right) \\ & A\cdot {{L}_{0}}\left\{ 1-\exp \left[ -1.5\frac{\pi h}{{{L}_{0}}} \right] \right\} \\ \end{align} \right.\begin{matrix} {} & \begin{align} & \tan \beta \ge 0 \\ & \\ & \tan \beta < 0 \\ \end{align} \\ \end{matrix} } (12)


where A = 0.17 is a proportional constant, and

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c\left( {{\varepsilon }_{d}} \right)=\left\{ \begin{align} & 0.506 \\ & 1.13-260{{\varepsilon }_{d}} \\ & 1.0 \\ \end{align} \right.\begin{matrix} {} & \begin{align} & {{\varepsilon }_{d}}\ge 0.0024 \\ & 0.0024>{{\varepsilon }_{d}}\ge 0.0005 \\ & {{\varepsilon }_{d}}<0.0005 \\ \end{align} \\ \end{matrix}} (13)

with

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\varepsilon }_{d}}=\frac{|\vec{U}|{{L}_{0}}}{{{g}^{2}}{{T}_{p}}^{3}}({{\tan }^{1/4}}\beta ) } (14)


in which LoFailed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =} gFailed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_{p}^{2}\text{ /2 }\!\!\pi\!\!\text{ }} , and Tp is the spectral peak period. The change in breaker height with respect to the cell length dx is defined as

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d{{H}_{b}}}{dx}=\left\{ \begin{align} & -\frac{3A\pi }{2}\tan \beta \left( 1+15{{\tan }^{4/3}}\beta \right)c\left( {{\varepsilon }_{d}} \right)/\exp \left[ \frac{3\pi h}{2{{L}_{0}}}\left( 1+15{{\tan }^{4/3}}\beta \right) \right] \\ & 0 \\ \end{align} \right.\begin{matrix} , & \begin{align} & \tan \beta \ge 0 \\ & \\ & \tan \beta <0 \\ \end{align} \\ \end{matrix}} (15)

In CMS-Wave, the breaking heights at the seaward and landward sides, denoted as Hbi and Hbo, respectively, of a grid cell are given by

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{H}_{bi}}={{H}_{b}}-\frac{1}{2}d{{H}_{b}}} (16)
  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{H}_{bo}}={{H}_{b}}+\frac{1}{2}d{{H}_{b}}} (17)

and the rate of wave breaking energy is (Mase et al. 2005b)

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\varepsilon }_{b}}=\left\{ 1-\frac{1-\left[ 1+\frac{\pi }{4}{{\left( 1.6\frac{{{H}_{bo}}}{{{H}_{1/3}}} \right)}^{2}} \right]\exp \left[ -\frac{\pi }{4}{{\left( 1.6\frac{{{H}_{bo}}}{{{H}_{1/3}}} \right)}^{2}} \right]}{1-\left[ 1+\frac{\pi }{4}{{\left( 1.6\frac{{{H}_{bi}}}{{{H}_{1/3}}} \right)}^{2}} \right]\exp \left[ -\frac{\pi }{4}{{\left( 1.6\frac{{{H}_{bi}}}{{{H}_{1/3}}} \right)}^{2}} \right]} \right\}\times \frac{C}{dx}} (18)

where H1/3 is the significant wave height defined as the average of the highest one-third waves in a wave spectrum.

Extended Miche formula

Battjes (1972) extended Miche's criterion, Equation 11, to water of variable depth as

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{{{H}_{b}}}{{{L}_{b}}}=a\cdot \tanh \left( \frac{\gamma }{0.88}\frac{2\pi h}{{{L}_{b}}} \right)} (19)

where Lb is the wavelength at the breaking location including the current, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \gamma} is an adjustable coefficient varying with the beach slope, and a = 0.14. This formula reduces to a steepness limit in deep water and to a depth limit in shallow water, thus incorporating both wave breaking limits in a simple form. The coefficient Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \gamma} has been treated as a constant value of 0.8 in application for random waves (Battjes and Janssen 1978). Based on field and laboratory data, Ostendorf and Madsen (1979) suggested that

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \gamma =\left\{ \begin{align} & 0.8+5\tan \beta \\ & 1.3 \\ \end{align} \right.\begin{matrix} {} & \begin{align} & \tan \beta <0.1 \\ & \tan \beta \ge 0.1 \\ \end{align} \\ \end{matrix}} (20)

which is applied in CMS-Wave. The change of breaker height with respect to the cell length dx can be obtained as follows (Mase et al. 2005b):

  (21)

Equations 19 to 21 are incorporated in Equations 16 to 18 to calculate the spectral energy dissipation rate b.

Battjes and Janssen formula

Battjes and Janssen (1978) developed a formula for predicting the mean energy dissipation in a bore of the same height as a depth-induced breaking wave as

  (22)

where is an empirical coefficient of order one, is the sea water density, is the spectral mean frequency, and Qb is the probability that at a given location the wave is breaking. By assuming the wave height has a Rayleigh distribution, the probability of wave breaking can be determined from the following expression

  (23)

where is the root-mean-square wave height. Battjes and Janssen (1978) calculated the maximum possible height from Equation 19 using a constant breaker value = 0.8. Booij et al. (1999) and Chen et al. (2005) investigated Battjes and Janssen formula and obtained a better wave breaking estimate with = 0.73. In CMS-Wave, Equation 23 is adapted to parameterize the wave breaking energy dissipation by applying the Battjes and Janssen formula with  = 0.73. The calculation of the wave breaking dissipation rate is from:

  (24)

Chawla and Kirby formula

Chawla and Kirby (2002) proposed an alternative expression for the bulk dissipation in random waves assuming the probability of wave breaking is dependent on the wave slope and a bore type of dissipation. Their modified bore dissipation formula worked well for wave breaking under a strong opposing current. The rate of energy dissipation was defined as

  (25)

where is the wave number corresponding to the spectral mean frequency , and scaling parameters b and are equal to 0.4 and 0.6, respectively. The rate of wave breaking energy dissipation is calculated by Equation 24.

Wind forcing and whitecapping dissipation

The evolution of waves in the large-scale, open coast is more affected by wind-ocean-wave interactions than on the nearshore wave-current-bottom processes. The result is a nonlinear wave field that is balanced between wind forcing, whitecapping, and wave growth. The surface wind can feed energy into the existing waves and can also generate new waves. On the other hand, the energy can dissipate through whitecapping from turbulence-wave interactions and air-wave-water interactions. In CMS-Wave, these wind forcing and whitecapping processes are modeled as separate sink and source terms (Lin and Lin 2004a and b).

Wind input function

The wind-input source Sin is formulated as functions of the ratio of wave celerity C to wind speed W, the ratio of wave group velocity to wind speed, the difference of wind speed and wave celerity, and the difference between wind direction wind and wave direction Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta} (Lin and Lin 2006b):

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{S}_{in}}=\frac{{{a}_{1}}\sigma }{g}{{F}_{1}}(\overrightarrow{W}-{{\vec{C}}_{g}}){{F}_{2}}(\frac{{{C}_{g}}}{W})E_{\text{PM}}^{*}(\sigma )\Phi (\text{ }\!\!\theta\!\!\text{ })+\frac{{{a}_{2}}{{\sigma }^{2}}}{g}{{F}_{1}}(\overrightarrow{W}-{{\vec{C}}_{g}})\text{ }{{F}_{2}}(\frac{{{C}_{g}}}{W}){{F}_{3}}(\frac{{{C}_{g}}}{W})N } (26)

where

  (27)
  (28)
  (29)

and

  (30)

is the functional form of the Pierson-Moskowitz (PM) spectrum, o = g/W is the Phillips constant, and

  (31)


is a normalized directional spreading. The function F1 presents the wind stress effect, F2 designates Phillips' mechanisms (Phillips 1957) and F3 accounts for the wave age effect. For swell or long waves, the wave group velocity Cg is generally large and F3 < 1. If Cg W, then F3 = 0. For short waves, the phase velocity is generally small and F3 > 1.

Whitecapping dissipation function

The wave energy dissipation (sink) Sds (Lin and Lin 2006b) for whitecapping including current and turbulent viscous effect is

  (32)

with

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{F}_{4}}(\vec{W},\text{ }\vec{U},{{\vec{C}}_{g}})=\frac{v+W}{|\vec{W}+\vec{U}+{{{\vec{C}}}_{g}}|} } (33)

and

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{F}_{5}}(kh)=\frac{1}{\tanh kh} } (34)

where cds is a proportionality coefficient, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {v}} is for the turbulent viscous dissipation. The wave amplitude Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{a}_{e}}=\sqrt{E(\sigma ,\text{ }\!\!\theta\!\!\text{ })d\sigma d\text{ }\!\!\theta\!\!\text{ }}} is calculated at each grid cell. To avoid numerical instability and considering the physical constraint of energy loss for the dissipation, the function F4 is set to 1 if the computed value is greater than 1.

Wave generation with arbitrary wind direction

In the case of wind forcing only, with zero wave energy input at the sea boundary, CMS-Wave can assimilate the full-plane wave generation. The model will execute an internal grid rotation, based on the given wind direction, to calculate the wave field and map the result back to the original grid. This feature is convenient for the local wave generation by wind in a lake, bay, or estuary, neglecting swell from the ocean.

Bottom friction loss

The bottom friction loss (sink) Sds is calculated by a drag law model (Collins 1972)

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{S}_{ds}}=-{{c}_{f}}\frac{{{\text{ }\!\!\sigma\!\!\text{ }}^{2}}}{g}\frac{\langle {{u}_{b}}\rangle }{{{\sinh }^{2}}kh}N } (35)

with

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle {{u}_{b}}\rangle \,=\frac{1}{2}\sqrt{\frac{g}{h}{{E}_{total}}} } (36)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle{u}} bFailed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rangle} presents the ensemble mean of horizontal wave orbital velocity at the sea bed, Etotal is the total energy density at a grid cell, and cf is the Darcy-Weisbach type friction coefficient. The relationship between cf and the Darcy-Weisbach friction factor fDW is cf = fDW/8. Typical values of cf for sandy bottoms range from 0.004 to 0.007 based on the JONSWAP experiment and North Sea measurements (Hasselmann et al. 1973; Bouws and Komen 1983). Values of cf applied for coral reefs range from 0.05 to 0.40 (Hardy 1993; Hearn 1999; Lowe et al. 2005). Application of this model capability to a specific site requires validation to field data. If the Manning friction coefficient n is used instead of the Darcy-Weisbach type coefficient, the relationship between the two drag coefficients is

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{c}_{f}}=\frac{g{{n}^{2}}}{{{h}^{1/3}}} } (37)

Estimates of Manning coefficient n are available in most fluid mechanics reference books (e.g., 0.01 to 0.05 for smooth to rocky/weedy channels).

Wave runup

Wave runup is the maximum shoreward wave swash on the beach face for engineering structures such as jetties and breakwaters by wave breaking at the shore. Wave runup is significant for beach erosion as well as wave overtopping of seawalls and jetties. The total wave runup consists of two components: (a) rise of the mean water level by wave breaking at the shore, known as the wave setup, and (b) swash of incident waves. In CMS-Wave, the wave setup is computed based on the horizontal momentum equations, neglecting current, surface wind drag and bottom stresses

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \eta }{\partial x}=-\frac{1}{\rho gh}\left( \frac{\partial {{S}_{xx}}}{\partial x}+\frac{\partial {{S}_{xy}}}{\partial y} \right)} (38)
  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \eta }{\partial y}=-\frac{1}{\rho gh}\left( \frac{\partial {{S}_{xy}}}{\partial x}+\frac{\partial {{S}_{yy}}}{\partial y} \right)} (39)

where  is the water density and Sxx, Sxy, and Syy are radiation components from the excess momentum flux caused by waves. By using the linear wave theory (Dean and Dalrymple 1984), Sxx, Sxy, and Syy can be expressed as

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{S}_{xx}}=E(\sigma ,\text{ }\!\!\theta\!\!\text{ )}\ \int{\left[ {{n}_{k}}({{\cos }^{2}}\text{ }\!\!\theta\!\!\text{ }+1)-\frac{1}{2} \right]}\ d\text{ }\!\!\theta\!\!\text{ }} (40)


  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{S}_{xx}}=E(\sigma ,\text{ }\!\!\theta\!\!\text{ })\text{ }{{\int{\left[ {{n}_{k}}({{\sin }^{2}}\text{ }\!\!\theta\!\!\text{ }+1)-\frac{1}{2} \right]}}_{{}}}d\text{ }\!\!\theta\!\!\text{ }} (41)
  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{S}_{yy}}=\frac{E}{2}{{n}_{k}}\sin 2\text{ }\!\!\theta\!\!\text{ }} (42)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{n}_{k}}=\frac{1}{2}+\frac{kh}{\sinh kh}} . Equations 38 and 39 also calculate the water level depression from the still-water level resulting from waves known as wave setdown outside the breaker zone. Because CMS-Wave is a half-plane model, Equation 38 controls mainly wave setup and setdown calculations, whereas Equation 39 acts predominantly to smooth the water level alongshore. The swash oscillation of incident natural waves on the beach face is a random process. The most landward swash excursion corresponds to the maximum wave runup. In the engineering application, a 2% exceedance of all vertical levels, denoted as R2, from the swash is usually estimated for the wave runup (Komar 1998). This quantity is approximately equal to the local wave setup on the beach or at structures such as seawalls and jetties, or the total wave runup is estimated as

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R\text{2}={{\text{2}}_{{}}}{{\eta }_{\max }}} (43)

In CMS-Wave, R2 is calculated at the land-water interface and averaged with the local depth to determine if the water can flood the proceeding dry cell. If the wave runup level is higher than the adjacent land cell elevation, CMS-Wave can flood the dry cells and simulate wave overtopping and overwash at them. The feature is useful in coupling CMS-Wave to CMS-Flow (Buttolph et al. 2006) for calculating beach erosion or breaching. Calculated quantities of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial} Sxx/Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial} x, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial} Sxy/Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial} x, Sxy/y, and Syy/y are saved as input to CMS-Flow. CMS-Wave reports the calculated fields of wave setup and maximum water level defined as

  Maximum water level = Max (R2, + H1/3/2) (44)

Wave transmission and overtopping at structures

CMS-Wave applies a simple analytical formula to compute the wave transmission coefficient Kt of a rigidly moored rectangular breakwater of width Bc and draft Dc (Macagno 1953)

  (45)

Wave transmission over a structure or breakwater is caused mainly by the fall of the overtopping water mass. Therefore, the ratio of the structure crest elevation to the incident wave height is the prime parameter governing the wave transmission. CMS calculates the rate of overtopping of a vertical breakwater based on the simple expression (Goda 1985) as

  (46)

where hc is the crest elevation of the breakwater above the still-water level, and Hi is the incident wave height. Equation 46 is modified for a composite breakwater, protected by a mound of armor units at its front, as

  (47)

For rubble-mound breakwaters, the calculation of wave transmission is more complicated because the overtopping rate also depends on the specific design of the breakwater (e. g., toe apron protection, front slope, armor unit shape and size, thickness of armor layers). In practice, Equation 47 still can be applied using a finer spatial resolution with the proper bathymetry and adequate bottom friction coefficients to represent the breakwater.

Grid nesting

Grid nesting is applied by saving wave spectra at selected locations from a coarse grid (parent grid) and inputting them along the offshore boundary of the smaller fine grid (child grid). For simple and quick applications, a single-location spectrum saved from the parent grid can be used as the wave forcing for the entire sea boundary of the child grid. If multi-location spectra were saved from the parent grid, they are then interpolated as well as extrapolated for more realistic wave forcing along the sea boundary of the child grid. Multiple grid nesting (e.g., several co-existing child grids and grandchild grids) is supported by CMS-Wave. The parent and child grids can have different orientations, but need to reside in the same horizontal coordinate system. Because CMS-Wave is a half-plane model, the difference between grid orientations between parent and child grids should be small (no greater than 45 deg) for passing sufficient wave energy from the parent to child grids.

Variable-rectangular-cell grid

CMS-Wave can run on a grid with variable rectangular cells. This feature is suited to large-domain applications in which wider spacing cells can be specified in the offshore, where wave property variation is small and away from the area of interest, to save computational time. A limit on the shore-normal to shore-parallel spacing ratio in a cell is not required as long as the calculated shoreward waves are found to be numerically stable.

Non-linear wave-wave interaction

Non-linear wave-wave interactions are a conserved energy transfer from higher to lower frequencies. They can produce transverse waves and energy diffusion in the frequency and direction domains. The effect is more pronounced in shallower water. Directional spreading of the wave spectrum tends to increase as the wavelength decreases. The exact computation of the nonlinear energy transfer involves six-dimensional integrations. This is computationally too taxing to be used in practical engineering nearshore wave transformation models. Mase et al. (2005a) have shown that calculated wave fields differ with and without nonlinear energy transfer. Jenkins and Phillips (2001) proposed a simple formula as an approximation of the nonlinear wave-wave interaction. Testing of this formula in CMS-Wave is underway.

Fast-mode calculation

CMS-Wave can run in a fast mode for simple and quick applications. The fast mode calculates the half-plane spectral transformation on either five directional bins (each 30-deg angle for a broad-band input spectrum) or seven directional bins (each 5-deg angle for a narrow-band input spectrum or 25-deg angle with wind input) to minimize simulation time. It runs at least five times faster than the normal mode, which operates with 35 directional bins. The fast-mode option is suited for a long or time-pressing simulation if users are seeking preliminary solutions. The wave direction estimated in the fast mode is expected to be less accurate than the standard mode because the directional calculation is based on fewer bins.

Table of Contents Chapter 3 - CMS-Wave Interface
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