TR-08-13:Chapter2

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)

${\begin{aligned}{\frac {\partial \left({{C}_{x}}N\right)}{\partial x}}+{\frac {\partial \left({{C}_{y}}N\right)}{\partial y}}+{\frac {\partial \left({{C}_{\theta }}N\right)}{\partial \theta }}={\frac {\kappa }{2\sigma }}\left[{{\left(C{{C}_{g}}\,{{\cos }^{2}}\theta {{N}_{y}}\right)}_{y}}-{\frac {C{{C}_{g}}}{2}}\,\,{{\cos }^{2}}\theta {{N}_{yy}}\right]-{{\varepsilon }_{b}}N-S\end{aligned}}$

(1)

where

${\begin{aligned}N={\frac {E(\sigma ,\theta )}{\sigma }}\end{aligned}}$

(2)

is the wave-action density to be solved and is a function of frequency $\sigma$ and direction $\theta$. E($\sigma ,\theta$) 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 C_g are wave celerity and group velocity, respectively; x and y are the horizontal coordinates; C_x, C_y, and C_$\theta$ are the characteristic velocity with respect to x, y, and, $\theta$ respectively; N_y and N_yy denote the first and second derivatives of N with respect to y, respectively; $\kappa$ is an empirical parameter representing the intensity of diffraction effect; $\varepsilon$_b is the parameterization of wave breaking energy dissipation; S denotes additional source S_in and sink S_ds (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 $\kappa$ (values $\geq$ 0) needs to be calibrated and optimized for structures. The model omits the diffraction effect for $\kappa$ = 0 and calculates diffraction for $\kappa$ > 0. Large $\kappa$ (values > 15) should be avoided as it can cause artificial wave energy losses (Mase 2001). In practice, values of $\kappa$ between 0 (no diffraction) and 4 (strong diffraction) have been determined in comparison to measurements. A default value of $\kappa$ = 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 $\kappa$ 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, $\kappa$ = 4 (maximum diffraction allowed in the model) is recommended. For a relatively wider gap, with an opening greater than one wavelength, $\kappa$ = 3 is recommended. The exact value of $\kappa$ 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 C_x, C_y, and C_$\theta$ in Equation 1 can be expressed as:

${\begin{aligned}{{C}_{x}}={{C}_{g}}\cos \theta +U\end{aligned}}$

(3)

${\begin{aligned}{{C}_{y}}={{C}_{g}}\sin \theta +V\end{aligned}}$

(4)

${\begin{aligned}&{{C}_{\theta }}={\frac {\sigma }{\sinh 2kh}}\left(\sin \theta {\frac {\partial h}{\partial x}}-\cos \theta {\frac {\partial h}{\partial y}}\right)\\&{\begin{matrix}{}&+\cos \theta \,\sin \theta {\frac {\partial U}{\partial x}}-{{\cos }^{2}}\theta {\frac {\partial U}{\partial y}}+{{\sin }^{2}}\theta {\frac {\partial V}{\partial x}}-\sin \theta \,\cos \theta {\frac {\partial V}{\partial y}}&{}\\\end{matrix}}\\\end{aligned}}$

(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 $h$ is the water depth. The dispersion relationships between the relative angular frequency $\sigma$, the absolute angular frequency $\omega$, the wave number vector ${\vec {k}}$, and the current velocity vector $\left|{\vec {U}}\right|={\sqrt {{{U}^{2}}+{{V}^{2}}}}$ are (Jonsson 1990)

${\begin{aligned}\sigma =\omega ={\vec {k}}\cdot {\vec {U}}\end{aligned}}$

(6)

and

${\begin{aligned}{{\sigma }^{2}}=gk\tanh(kh)\end{aligned}}$

(7)

where

${\vec {k}}\cdot {\overrightarrow {U}}$

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 C_g is weaker than an opposing current (Smith et al. 1998; Larson and Kraus 2002):

${\begin{aligned}{{C}_{g}}={\frac {d\sigma }{dk}}<{\overrightarrow {U}}\cdot {\vec {k}}/k\end{aligned}}$

(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 N_r is assumed to be linearly proportional to the incident wave action N_i :

${\begin{aligned}{{N}_{r}}=K_{r}^{2}{{N}_{i}}\end{aligned}}$

(9)

where K_r 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)

${\begin{aligned}{\frac {{H}_{b}}{h}}\leq 0.64\end{aligned}}$

(10)

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

${\begin{aligned}{{H}_{b}}\leq {\frac {0.64}{{k}_{p}}}{\text{ }}\tanh({{k}_{p}}h)\end{aligned}}$

(11)

where k_p is the wave number corresponding to the spectral peak. In the shallow water condition (k_ph 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 $\beta$, and deepwater wavelength L_o

${{H}_{b}}=\left\{{\begin{aligned}&A\cdot {{L}_{0}}\left\{1-\exp \left[-1.5{\frac {\pi h}{{L}_{0}}}\left(1+15{{\tan }^{4/3}}\beta \right)\right]\right\}c\left({{\varepsilon }_{d}}\right)\\&A\cdot {{L}_{0}}\left\{1-\exp \left[-1.5{\frac {\pi h}{{L}_{0}}}\right]\right\}\\\end{aligned}}\right.{\begin{matrix}{}&{\begin{aligned}&\tan \beta \geq 0\\&\\&\tan \beta <0\\\end{aligned}}\\\end{matrix}}$

(12)

where A = 0.17 is a proportional constant, and

$c\left({{\varepsilon }_{d}}\right)=\left\{{\begin{aligned}&0.506\\&1.13-260{{\varepsilon }_{d}}\\&1.0\\\end{aligned}}\right.{\begin{matrix}{}&{\begin{aligned}&{{\varepsilon }_{d}}\geq 0.0024\\&0.0024>{{\varepsilon }_{d}}\geq 0.0005\\&{{\varepsilon }_{d}}<0.0005\\\end{aligned}}\\\end{matrix}}$

(13)

with

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

(14)

in which L_o$=$g$T_{p}^{2}{\text{ /2 }}\!\!\pi \!\!{\text{ }}$, and T_p is the spectral peak period. The change in breaker height with respect to the cell length dx is defined as

${\frac {d{{H}_{b}}}{dx}}=\left\{{\begin{aligned}&-{\frac {3A\pi }{2}}\tan \beta \left(1+15{{\tan }^{4/3}}\beta \right)c\left({{\varepsilon }_{d}}\right)/\exp \left[{\frac {3\pi h}{2{{L}_{0}}}}\left(1+15{{\tan }^{4/3}}\beta \right)\right]\\&0\\\end{aligned}}\right.{\begin{matrix},&{\begin{aligned}&\tan \beta \geq 0\\&\\&\tan \beta <0\\\end{aligned}}\\\end{matrix}}$

(15)

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

${{H}_{bi}}={{H}_{b}}-{\frac {1}{2}}d{{H}_{b}}$

(16)

${{H}_{bo}}={{H}_{b}}+{\frac {1}{2}}d{{H}_{b}}$

(17)

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

${{\varepsilon }_{b}}=\left\{1-{\frac {1-\left[1+{\frac {\pi }{4}}{{\left(1.6{\frac {{H}_{bo}}{{H}_{1/3}}}\right)}^{2}}\right]\exp \left[-{\frac {\pi }{4}}{{\left(1.6{\frac {{H}_{bo}}{{H}_{1/3}}}\right)}^{2}}\right]}{1-\left[1+{\frac {\pi }{4}}{{\left(1.6{\frac {{H}_{bi}}{{H}_{1/3}}}\right)}^{2}}\right]\exp \left[-{\frac {\pi }{4}}{{\left(1.6{\frac {{H}_{bi}}{{H}_{1/3}}}\right)}^{2}}\right]}}\right\}\times {\frac {C}{dx}}$

(18)

where H_1/3 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

${\frac {{H}_{b}}{{L}_{b}}}=a\cdot \tanh \left({\frac {\gamma }{0.88}}{\frac {2\pi h}{{L}_{b}}}\right)$

(19)

where L_b is the wavelength at the breaking location including the current, $\gamma$ 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 $\gamma$ 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

$\gamma =\left\{{\begin{aligned}&0.8+5\tan \beta \\&1.3\\\end{aligned}}\right.{\begin{matrix}{}&{\begin{aligned}&\tan \beta <0.1\\&\tan \beta \geq 0.1\\\end{aligned}}\\\end{matrix}}$

(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):

${\frac {d{{H}_{b}}}{dx}}=\left\{{\begin{aligned}&-a\cdot \tan \beta {\frac {\gamma }{0.88}}2\pi {{\cosh }^{-2}}\left({\frac {\gamma }{0.88}}{\frac {2\pi h}{{L}_{b}}}\right)\\&0\\\end{aligned}}\right.{\begin{matrix}{}&{\begin{aligned}&\tan \beta \geq 0\\&\tan \beta <0\\\end{aligned}}\\\end{matrix}}$

(21)

Equations 19 to 21 are incorporated in Equations 16 to 18 to calculate the spectral energy dissipation rate $\varepsilon$_b.

Battjes and Janssen formula

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

${\overline {D}}={\frac {\alpha \rho g}{4}}{{Q}_{b}}{\overline {f}}H_{b}^{2}$

(22)

where $\alpha$ is an empirical coefficient of order one, $\rho$ is the sea water density, ${\overline {f}}$ is the spectral mean frequency, and Q_b 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

${\frac {1-{{Q}_{b}}}{\ln {{Q}_{b}}}}=-{{\left({\frac {{H}_{\text{rms}}}{{H}_{b}}}\right)}^{2}}$

(23)

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

$\varepsilon _{b}={\frac {\overline {D}}{\left(\rho g{H}_{\text{rms}}^{2}/8\right)2\pi {\overline {f}}}}$

(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 ${\overline {D}}$ was defined as

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

(25)

where ${\overline {k}}$ is the wave number corresponding to the spectral mean frequency ${\overline {f}}$, and scaling parameters b and $\gamma$ 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 S_in 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 $\theta$_wind and wave direction $\theta$ (Lin and Lin 2006b):

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

(26)

where

${{F}_{1}}({\vec {W}}-{{\vec {C}}_{g}})=\left\{{\begin{array}{*{35}{l}}W\cos({{\theta }_{wind}}-\theta )-{{C}_{g}},&{\text{if}}\quad &{{C}_{g}}<W\\0,&{\text{if}}\quad &{{C}_{g}}\geq W\\\end{array}}\right.$

(27)

${{F}_{1}}({\vec {W}}-{{\vec {C}}_{g}})=\left\{{\begin{array}{*{35}{l}}{{\left({\frac {{C}_{g}}{W}}\right)}^{1.15}},&{\text{if}}\quad &{{C}_{g}}<W\\0,&{\text{if}}\quad &{{C}_{g}}\geq W\\\end{array}}\right.$

(28)

${{F}_{3}}({\frac {{C}_{g}}{W}})=\left\{{\begin{array}{*{35}{l}}{{\log }_{10}}\left[{{\left({\frac {{C}_{g}}{W}}\right)}^{-1}}\right],&{\text{if}}\quad &{{C}_{g}}<W\\0,&{\text{if}}\quad &{{C}_{g}}\geq W\\\end{array}}\right.$

(29)

and

$E_{\text{PM}}^{*}(\sigma )={\frac {{g}^{2}}{{\sigma }^{5}}}\exp(-0.74{\frac {\sigma _{0}^{4}}{{\sigma }^{4}}})$

(30)

$E_{\text{PM}}^{*}(\sigma )$ is the functional form of the Pierson-Moskowitz (PM) spectrum, $\sigma$_o = g/W is the Phillips constant, and

$\phi (\theta )={\frac {8}{3\pi }}{{\cos }^{4}}(\theta -{{\theta }_{wind}}),\quad {\text{for}}\quad \left|\theta -\theta wind\right|\quad \leq {\frac {\pi }{2}}$

(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 C_g is generally large and F₃ < 1. If C_g $\geq$ W, then F₃ = 0. For short waves, the phase velocity is generally small and F₃ > 1.

Whitecapping dissipation function

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

${{S}_{ds}}=-{{c}_{ds}}{{({{a}_{e}}k)}^{1.5}}{\frac {{\sigma }^{2}}{g}}{{C}_{g}}(\sigma ,\theta ){{F}_{4}}({\vec {W}},{\vec {U}},{{\vec {C}}_{g}}){{F}_{5}}(kh)N$

(32)

with

${{F}_{4}}({\vec {W}},{\text{ }}{\vec {U}},{{\vec {C}}_{g}})={\frac {v+W}{|{\vec {W}}+{\vec {U}}+{{\vec {C}}_{g}}|}}$

(33)

and

${{F}_{5}}(kh)={\frac {1}{\tanh kh}}$

(34)

where c_ds is a proportionality coefficient, and ${v}$ is for the turbulent viscous dissipation. The wave amplitude ${{a}_{e}}={\sqrt {E(\sigma ,{\text{ }}\!\!\theta \!\!{\text{ }})d\sigma d{\text{ }}\!\!\theta \!\!{\text{ }}}}$ 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) S_ds is calculated by a drag law model (Collins 1972)

${{S}_{ds}}=-{{c}_{f}}{\frac {{{\text{ }}\!\!\sigma \!\!{\text{ }}}^{2}}{g}}{\frac {\langle {{u}_{b}}\rangle }{{{\sinh }^{2}}kh}}N$

(35)

with

$\langle {{u}_{b}}\rangle \,={\frac {1}{2}}{\sqrt {{\frac {g}{h}}{{E}_{total}}}}$

(36)

where $\langle {u}$_b$\rangle$ presents the ensemble mean of horizontal wave orbital velocity at the sea bed, E_total is the total energy density at a grid cell, and c_f is the Darcy-Weisbach type friction coefficient. The relationship between c_f and the Darcy-Weisbach friction factor f_DW is c_f = f_DW/8. Typical values of c_f 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 c_f 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

${{c}_{f}}={\frac {g{{n}^{2}}}{{h}^{1/3}}}$

(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

${\frac {\partial \eta }{\partial x}}=-{\frac {1}{\rho gh}}\left({\frac {\partial {{S}_{xx}}}{\partial x}}+{\frac {\partial {{S}_{xy}}}{\partial y}}\right)$

(38)

${\frac {\partial \eta }{\partial y}}=-{\frac {1}{\rho gh}}\left({\frac {\partial {{S}_{xy}}}{\partial x}}+{\frac {\partial {{S}_{yy}}}{\partial y}}\right)$

(39)

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

${{S}_{xx}}=E(\sigma ,{\text{ }}\!\!\theta \!\!{\text{ )}}\ \int {\left[{{n}_{k}}({{\cos }^{2}}{\text{ }}\!\!\theta \!\!{\text{ }}+1)-{\frac {1}{2}}\right]}\ d{\text{ }}\!\!\theta \!\!{\text{ }}$

(40)

${{S}_{xx}}=E(\sigma ,{\text{ }}\!\!\theta \!\!{\text{ }}){\text{ }}{{\int {\left[{{n}_{k}}({{\sin }^{2}}{\text{ }}\!\!\theta \!\!{\text{ }}+1)-{\frac {1}{2}}\right]}}_{}}d{\text{ }}\!\!\theta \!\!{\text{ }}$

(41)

${{S}_{yy}}={\frac {E}{2}}{{n}_{k}}\sin 2{\text{ }}\!\!\theta \!\!{\text{ }}$

(42)

where ${{n}_{k}}={\frac {1}{2}}+{\frac {kh}{\sinh kh}}$. 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

$R{\text{2}}={{\text{2}}_{}}{{\eta }_{\max }}$

(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 $\partial$S_xx/$\partial$x, $\partial$S_xy/$\partial$x, $\partial$S_xy/$\partial$y, and $\partial$S_yy/$\partial$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, $\eta$ + H1/3/2)

(44)

Wave transmission and overtopping at structures

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

${{K}_{t}}={{\left[1+{{\left({\frac {k{{B}_{c}}\sinh {\frac {kh}{2\pi }}}{2\cosh k(h-{{D}_{c}})}}\right)}^{2}}\right]}^{-{\frac {1}{2}}}}$

(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

${{K}_{t}}={{0.3}_{}}(1.5-{\frac {{h}_{c}}{{H}_{i}}}),{\text{ for }}0\leq {\frac {{h}_{c}}{{H}_{i}}}\leq 1.25$

(46)

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

${{K}_{t}}={{0.3}_{}}(1.1-{\frac {{h}_{c}}{{H}_{i}}}),{\text{ for }}0\leq {\frac {{h}_{c}}{{H}_{i}}}\leq 0.75$

(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.

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Table of Contents

Chapter 3 - CMS-Wave Interface