GSE. In our investigations, there was a strong emphasis on addressing the GSE without significantly compromising
the existing computational speed of the model. The concern is that if computational requirements became too
lengthy, this would limit the degree of investigation and model testing that could be applied in other areas of the
study, such as wind field development. Thus, the accuracy requirements of any given scheme were carefully
weighed against the computational penalty.
Booij and Holthuijsen (1987) recognized the need for a diffusion operator to counteract the limitations of a discrete
frequency and direction spectrum, and introduced a diffusion correction into the basic equations defining wave
propagation. This approach did have the requirement of defining an appropriate "wave age", or time lapsed since
generation of the waves, for the wave fields under consideration, and can have the undesirable effect of potentially
influencing the model time step, depending on the model resolution. Tolman (2001) explored other approaches to
correcting for GSE, including the use of both a simple box averaging scheme and an approach in which divergence
was added to the advection velocities.
Both of these previous studies made the point of starting with a highly accurate propagation scheme to which the
diffusion operator was then applied that "smears" this highly accurate solution in space to address GSE issues. This
has the important effect of giving a controllable diffusion. Given our concerns with model computational speed, we
sought an approach that improved the numerics of WAVAD beyond first order but did not come the computational
demands of a high order scheme. That is, a certain degree of numerical diffusion was acceptable in the propagation
scheme given that a diffusion operator would subsequently be applied to address GSE.
Some of the numerical schemes investigated in our study included:
First order downwind interpolation.
Modified second order Lax-Wendroff.
Third order QUICKEST/ULTIMATE.
The ULTIMATE scheme, which has been employed by Tolman (2001) in WAVEWATCH III, was the most
accurate of the schemes considered and was used as the starting point in the study and as a base case for the
comparisons. This scheme, although efficient, placed significant computational demands that other alternatives
were explored.
The Garden Sprinkler Effect was addressed using Tolman's averaging scheme due to its efficiency. In this
approach, the energy at each grid point was estimated by averaging the wave energy at the four corners of a box of
dimensions α∆Cg∆t and βCg∆θ∆t surrounding the grid point, with the corner values estimated by means of bi-linear
interpolation. Cg is defined as the wave group speed; ∆θ, the directional resolution and ∆t, the time step. Two
tunable parameters, α and β, were ultimately set to 2.0 and 4.0, respectively.
One of the primary numerical tests was based on the work of Booij and Holthuijsen (1987) and subsequent
investigations of Tolman (2001). A wave model grid of dimensions of 100 degrees by 100 degrees (11,100 x 11,100
km) was established with a 1-degree resolution. An initial maximum wave height of 7.0 m was applied in the model
in the lower left region of the grid (Point 12,12). The waves had a mean direction of 30 with cos2 directional
distribution and a peak period of 10 seconds. The wave height distribution was Gaussian in space with an overall
radius of 200 km. A time step of 1 hour was employed.
Figure 6 provides a summary of results for a few selected cases. Figure 6a shows a solution for the test case after a
7 day time period if a one-degree directional resolution is utilized in the wave model. Figure 6b illustrates the
fragmentation of the GSE when a reduced directional resolution, 15 degrees, is utilized.
The results of a simulation using the Lax-Wendroff scheme in conjunction with the Tolman smoothing approach is
shown in Figure 6c. Some of the GSE is still discernable but the solution is much improved as compared to the
solution without smoothing.
It was observed, based on these and other numerical experiments, that the GSE could be reduced to some degree by
means of a higher order numerical propagation scheme in conjunction with the Tolman smoothing. In terms of
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