50
3 Numerical Implementation of the Wave Energy Balance Equation
Among the different numerical methods used in wind wave models, the
numerical implementation of the WAM model of the wave energy balance
equation should be pointed out (Komen et. al., 1994). An attempt was made
to solve this equation for a spherical surface, taking into account wave refraction in shallow water and currents. The WAM model is probably the
first model to implement the equation most completely. A scheme of directed
first- and second-order differences is used in the WAM model to approximate
the advective terms in (2.1). The central-difference second-order scheme is
applied for approximating the term characterizing the spectral density variation as the direction function (3.
However, the numerical scheme chosen in the WAM model for solving
(2.1) does not seem to be sufficiently optimal. The problem is that due to
large numerical diffusion the error of estimating the term as 1 8f3 . df3 1 dt becomes significant when calculating the wave propagation of a narrow-directed
spectrum. Thus, an additional source of errors appears as compared with the
balance equation for the plane. In order to minimize them it is necessary to
increase considerably the number of calculations.
It is interesting to mention the numerical implementation of the wave energy balance equation for a spherical surface proposed by V. Ryvkin (1990).
He suggested that it is possible to choose specially discrete directions at each
latitude to achieve the maximum approximation in order to get a precise solution. This can be done if a special set of characteristics for the entire grid area
is prescribed, so that these characteristics pass through the latitude at different grid points. The sets of directions at different latitudes are not the same.
The solutions are determined in terms of the characteristics. That is why
there is no need for interpolation in order to calculate the spectral density S
depending on the angle (3. It should be pointed out that in spite of some advantages of this method, it is rather difficult to generalize the proposed choice
of the special direction grid for non-uniform current and uneven bottom.
Another most typical cause of errors in wave energy propagation should
be noted. In numerical calculations, the continuous frequency-angular wave
spectrum is prescribed as a specific number of spectral components. The finite
width of the spectral frequency and angular bands introduces some complications for the numerical simulation of wave propagation. Ideally, the initial
wave energy contained originally within some area should propagate quite
smoothly over the ocean surface in time. However, in most wind wave models the spectral resolution is so rough that it induces the so-called "gardensprinkler effect" (Booij & Holtuijsen, 1987). This results in the energy spreading from the source along the directions prescribed in advance by the discrete
spectrum representation in the initial area. At some distances from it, an
anomalous increased concentration of wave energy is manifested in these directions, but is explicitly insufficient in the other directions. Thus, a limited
angular resolution of the model introduces artificial anisotropy in the spatial
wave energy distribution. As a consequence of this phenomenon, the model
prediction of swell propagation from a distant storm is unsatisfactory.
3 Numerical Implementation of the Wave Energy Balance Equation
Among the different numerical methods used in wind wave models, the
numerical implementation of the WAM model of the wave energy balance
equation should be pointed out (Komen et. al., 1994). An attempt was made
to solve this equation for a spherical surface, taking into account wave refraction in shallow water and currents. The WAM model is probably the
first model to implement the equation most completely. A scheme of directed
first- and second-order differences is used in the WAM model to approximate
the advective terms in (2.1). The central-difference second-order scheme is
applied for approximating the term characterizing the spectral density variation as the direction function (3.
However, the numerical scheme chosen in the WAM model for solving
(2.1) does not seem to be sufficiently optimal. The problem is that due to
large numerical diffusion the error of estimating the term as 1 8f3 . df3 1 dt becomes significant when calculating the wave propagation of a narrow-directed
spectrum. Thus, an additional source of errors appears as compared with the
balance equation for the plane. In order to minimize them it is necessary to
increase considerably the number of calculations.
It is interesting to mention the numerical implementation of the wave energy balance equation for a spherical surface proposed by V. Ryvkin (1990).
He suggested that it is possible to choose specially discrete directions at each
latitude to achieve the maximum approximation in order to get a precise solution. This can be done if a special set of characteristics for the entire grid area
is prescribed, so that these characteristics pass through the latitude at different grid points. The sets of directions at different latitudes are not the same.
The solutions are determined in terms of the characteristics. That is why
there is no need for interpolation in order to calculate the spectral density S
depending on the angle (3. It should be pointed out that in spite of some advantages of this method, it is rather difficult to generalize the proposed choice
of the special direction grid for non-uniform current and uneven bottom.
Another most typical cause of errors in wave energy propagation should
be noted. In numerical calculations, the continuous frequency-angular wave
spectrum is prescribed as a specific number of spectral components. The finite
width of the spectral frequency and angular bands introduces some complications for the numerical simulation of wave propagation. Ideally, the initial
wave energy contained originally within some area should propagate quite
smoothly over the ocean surface in time. However, in most wind wave models the spectral resolution is so rough that it induces the so-called "gardensprinkler effect" (Booij & Holtuijsen, 1987). This results in the energy spreading from the source along the directions prescribed in advance by the discrete
spectrum representation in the initial area. At some distances from it, an
anomalous increased concentration of wave energy is manifested in these directions, but is explicitly insufficient in the other directions. Thus, a limited
angular resolution of the model introduces artificial anisotropy in the spatial
wave energy distribution. As a consequence of this phenomenon, the model
prediction of swell propagation from a distant storm is unsatisfactory.