input wave conditions must be known, and proceeds onshore until dry land is
encountered (taking into account wave setup). A major difference from
NMLong is the additional iteration procedures that are required to solve the
governing equations. In comparison to NMLong, NMLong-CW requires the
following iterative steps:
a. Iteration between the wave transformation and longshore current
calculations. The waves are computed first, implying that the total
longshore current is not known (which could be a current resulting
from waves, wind, and an external current). Thus, after the
longshore current has been calculated by means of the longshore
momentum equation, the wave calculations have to be redone,
followed by a new current calculation. This iteration between the
waves and current continues until convergence is achieved at
predefined level of accuracy.
b. Iteration to solve the dispersion relation with a current present
Equation 6 is solved via a Newton-Raphson technique, as opposed to
in NMLong where the dispersion relation (without a current) is
solved explicitly employing a Pad approximation.
. Iteration to solve Snell's law
. In solving Snell's law for refraction
(Equation 11), proceeding from one grid point to next, both the wave
angle and wavelength are unknown at the new grid point. These
quantities are coupled through Equation 6 (in NMLong the dispersion
relation can be solved independently of Snell's law, and vice versa).
Overall, these iterative requirements might make NMLong-CW considerably
more time demanding to run than NMLong, depending on the computer
In calculating for random waves, Monte-Carlo simulation is employed by
simulating a large number of individual waves belonging to a certain
probability density function (pdf), typically taken to be a Rayleigh
distribution in deep water. Computations are performed with the governing
equations for each individual wave, and the statistical wave properties are
derived from the series of waves obtained at respective cross-shore locations.
The number of waves selected should be large enough to yield statistically
stable values on the mean wave properties when averaging for all the waves.
The advantage of a Monte-Carlo simulation technique is that no inference for
the shape of the pdf in the nearshore is necessary; the shape is obtained in the
simulations. The disadvantage of the method is that possible wave-wave
interaction and associated energy transfer are neglected. For random waves,
the wave forcing terms (radiation stresses and roller momentum fluxes) are
determined as averages for the selected number of waves in the Monte-Carlo
simulation before they are used in the momentum equations.
The wave energy balance equation employed to calculate the roller
properties across-shore (Equation 32) is discretized according to,
2PD,i∆x + mR,i+1 cos αi +1Cr2,i+1 - gβ D mR,i+1∆x
cos αiCr2,i + gβ D∆x
i = an index to denote the grid point number
∆x = grid cell length
Chapter 3 Wave Model