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function of the total depth and the depth averaged mean currents, and then computing the
radiation stresses from the complex amplitudes. The model starts with initial conditions
of zero mean currents and zero mean water surface elevation. The offshore incident wave
height is initially zero and then is smoothly increased up to its given value using a hyperbolic
tangent curve. This is done so as to minimize any transients due to a shock start-up.
The model iterates between the wave model and the circulation model until steady state
convergence is obtained. Each of the two parts of the model will be described separately.
The circulation model uses an alternating direction implicit (ADI) method of solution.
Three equations are solved for three unknowns. The unknowns are fj the mean water surface
level, and U and V, the two components of the depth averaged mean current. The equations
are the continuity equation and the x- and y-direction momentum equations. Each iteration
of the circulation model solves for the three unknowns on the next time level. In other words
each iteration advances the values of fj, U and V by a time increment, At. For the purposes
of this investigation the model solves for steady state conditions using the time marching
character of the circulation model. The model can be used to solve for time dependent
wave fields, but this would require the substitution of a time dependent wave equation in
the wave model. The ADI method first solves, grid row by grid row, for U at the next time
level and an intermediate value of fj, by solving the x direction equation of motion and the
continuity equation using an implicit numerical scheme. This solution is in terms of the
known values of 17,V,and fj at the present time level. Once all the grid rows are solved in
the x direction, the y equation of motion and the continuity equation are solved, again by
an implicit scheme, grid column by grid column, for V and fj at the next time level in terms
of U at the new time level, fj at the intermediate level and V at the present level. For each
of the implicit schemes in the two directions, the governing equations yield a tridiagonal
matrix. The tridiagonal matrix and the method of solution will be discussed later.
The wave model also yields a tridiagonal matrix equation. The governing equation is
a parabolic equation for the complex amplitude A. The complex amplitude contains phase