1 (*V *+ *w *+ 2*wV *sin α) +

2

2

1/ 2

(39)

2 (*V *+ *w*2 - 2*wV *sin α)1/ 2

2

(40)

π

To be consistent with previous assumptions, if a cross-shore current is

calculated from:

1

ρ*gH * 2

sin 2α

(41)

16

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.

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