GenCade:Sand Transport Rates

As stated on the Governing Equations page, the governing equation of GenCade is

${\frac {\partial y}{\partial t}}+{\frac {1}{(D_{B}+D_{C})}}({\frac {\partial Q}{\partial x}}-q)=0$

This equation is solved with the inputs of boundary conditions and values for Q, q, DB and DC given.

The sand transport rates, Q, is taken from the 'CERC' equation for calculating longshore sediment transport. As formulated in Komar (1969):

$Q=\left(H^{2}C_{g}\right)_{b}a_{1}\sin 2a_{b}$

where H is the wave height (meters), $C_{g}$ is the group wave celerity (meters/second), $a_{b}$ is the angle of the breaking waves to the shoreline, with the subscript b indicating the wave breaker position. Lastly, $a_{1}$ is a non-dimensional parameter:

$a_{1}={\frac {K_{1}}{16\left({\frac {\rho _{s}}{\rho -1}}\right)(1-p)1.416^{5/2}}}$

where K₁ is an empirical coefficient with a nominal value of 0.77, p_s and p are the density of sand and water, respectively, and p is the porosity of sand. In the GENESIS model (Hanson 1987) used an extended version of this relation (Kraus and Harikai 1983):

$Q=\left(H^{2}C_{g}\right)_{b}\left(a_{1}\sin 2a_{b}-a_{2}\cos a_{b}{\frac {\partial H_{b}}{\partial x}}\right)$

where $a_{2}$ is a non dimensional parameter given by:

$a_{2}={\frac {K_{2}}{8\left({\frac {\rho _{s}}{\rho -1}}\right)(1-p)\tan \beta 1.416^{5/2}}}$

where $\tan \beta$ is the average bottom slope from the shoreline to the "maximum depth of longshore transport" (See Empirical Parameters). The nominal value of $K_{1}$ is 0.39 if waves are specified in terms of RMS wave heights (Komar 1976) and 0.77 when using significant wave heights. As a rule of thumb, based on modeling experience, Hanson and Kraus (1989) recommend $0.5K_{1}<K_{2}<1.5K_{1}$.

In the calibration and verification process, $K_{1}$ and $K_{2}$ values are determined by reproducing changes in shoreline position measured over a certain time interval. Adjusting the $K_{1}$ coefficient will effect the entire modeling domain while $K_{2}$ will only affect the evolution in areas influenced by wave diffraction near structures. In the calibration process, it is recommended that the $K_{1}$ value is adjusted first to get a reasonable agreement with respect to annual transport rates and overall shoreline evolution. Then, the $K_{2}$ value may be altered to improve predicted shoreline response near structures.

Using the immersed weight transport rate to update the volumetric transport rate results in:

$Q=\left(H^{2}C_{g}\right)_{b}\left(a_{1}\sin 2a_{b}+(a_{3}{\frac {\bar {v_{t}}}{u_{m}}}-a_{2}{\frac {\partial H_{b}}{\partial x}})\cos a_{b}\right)$

where $a_{3}$ is a non-dimensional parameter given by:

$a_{3}={\frac {K_{3}}{8({\frac {\rho _{s}}{\rho }}-1)(1-p)1.415^{5/2}}}$

In GenCade, the calculated breaking wave condition at each grid point is converted to a wave-driven longshore current velocity from which the associated longshore sediment transport rate is calculated. A tidal or wind driven current (or both) are read from files and linearly added to the wave generated current before calculating the transport rate. GenCade at present does not calculate the current produced from the tide or the wind. These currents should be represented by an average through the surf zone and must be obtained from an external source, such as from a model or measurements. For further guidance on modeling wind-driven surf zone currents, see Kraus and Larson (1991) and Long and Hubertz (1988). Values of the external current are stored in a file that provides tide and wind currents at each calculation cell wall. These currents then serve as input for continued transport calculations.

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