1 (V + w + 2wV sin α) +
2
2
1/ 2
Z= 2
(39)
2 (V + w2 - 2wV sin α)1/ 2
2
w=
um
(40)
π
To be consistent with previous assumptions, if a cross-shore current is
calculated from:
Cgr
1
S xy =
ρgH 2
sin 2α
(41)
Cr
16
Numerical Solution
The numerical solution of Equation 34 follows Kraus and Larson (1991).
A staggered grid is employed where most wave-related quantities are
evaluated at the boundaries of the calculation cells, and the longshore current
is evaluated in the middle of cells. A tridiagonal system of equations is
obtained that is efficiently solved through a double-sweep algorithm, which is
also highly stable with little numerical dispersion. The boundary conditions
are accommodated in the same way as for NMLong, with the exception that
the external current is included in the solution. Also, as discussed in
Chapter 3, iterations are performed between the wave and current
calculations to represent the wave-current interaction.
Modification of Roller Model
It was observed during implementation of the roller model that
application of Equation 32 directly for monochromatic waves could cause
some unphysical behavior. Just after the wave started breaking, the roller
would grow too quickly, inducing a gradient in the momentum fluxes (cross-
shore and alongshore) that could overpower the radiation stress gradients.
Thus, the gradient in the roller momentum fluxes would not simply balance
the gradient in the radiation stresses to yield the desired shoreward translation
in the total forcing. Rather, because the gradient arising from the roller was
larger, a longshore current would be generated that was going opposite to the
longshore component of the waves. For random waves where the radiation
stresses are ensemble-averages over many waves, the growth of the roller
will be more gradual, and this problem does not occur.
To remedy this situation, an algorithm was implemented that limits the
growth of the roller so that the gradient in the roller momentum flux does not
exceed the gradient in the radiation stress (with consideration of the signs).
This algorithm is only activated during the phase where the roller is growing;
after the roller reaches maximum size, the gradient in the roller momentum
flux will change sign, and this term will be the main driving force for the
longshore current and mean water level. If the gradient in the roller
momentum flux exceeds the radiation stress gradient during the roller growth
phase at any given location and time-step, the roller size is determined from
the condition that the two gradients are equal, instead of from Equation 32.
43
Chapter 5 Longshore Current Model
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