3 Numerical Implementation
of the Wave Energy Balance Equation
3.1 Statement of the Problem
Successful solution of the problem of hindcasting and forecasting a sea wind
wave depends on the quality of the physical model, the numerical implementation of the wave energy balance equation and the accuracy of the wind
field data. It should be noted that in general most modern wind wave models are not characterized by their high-level numerical implementation. The
numerical imeplementation of the wave energy balance equation is principally important for accuracy and model efficiency (Lavrenov & Ryvkin, 1989;
Ryvkin, 1990; Ryabinin, 1991a,b; Tolman, 1992). Numerical implementation
errors are comparable with those obtained due to inaccuracy of wind data
and imperfect parameterization of the mechanisms of the physical model.
The method of characteristics used for solving (2.1) allows one to obtain
a solution accurately reproducing wave propagation. In order to define the
frequency-angular spectrum at one point, all spectral components arriving
from the entire water basin should be collected.
However, it is unreasonable to use the method of characteristics in the
previous form to estimate wind waves over vast oceanic areas. Firstly, it does
not allow the calculation of the non-linear wave interaction, since it is necessary to obtain information about all spectral harmonics at every grid point,
whereas they are gathered only at one point. Secondly, it is important to have
complete information of the integral wave parameters and the spectrum at
all grid points for the prediction of ocean operational conditions. The application of the method of characteristics for large areas is not justified, since all
rays must be collected from the entire area at each calculating point at every
time step (there are about 400 points in the North Atlantic for a numerical
grid with 2.5° X 2.5°).
For numerically solving (2.1), the widely-used finite-difference method
can be employed. A review of these methods for the plane case is given by
Rogers et al. (1999). However, numerical implementation of (2.1) on a sphere
is rather difficult. Unlike the solution of the similar problem in a plane, it
is necessary to approximate the additional term as 1 8f3 . df3 1 dt, increasing
the dimension of the equation. The problem becomes more complicated when
it is implemented in full form, taking into account a spatially non-uniform
current and an uneven bottom.
of the Wave Energy Balance Equation
3.1 Statement of the Problem
Successful solution of the problem of hindcasting and forecasting a sea wind
wave depends on the quality of the physical model, the numerical implementation of the wave energy balance equation and the accuracy of the wind
field data. It should be noted that in general most modern wind wave models are not characterized by their high-level numerical implementation. The
numerical imeplementation of the wave energy balance equation is principally important for accuracy and model efficiency (Lavrenov & Ryvkin, 1989;
Ryvkin, 1990; Ryabinin, 1991a,b; Tolman, 1992). Numerical implementation
errors are comparable with those obtained due to inaccuracy of wind data
and imperfect parameterization of the mechanisms of the physical model.
The method of characteristics used for solving (2.1) allows one to obtain
a solution accurately reproducing wave propagation. In order to define the
frequency-angular spectrum at one point, all spectral components arriving
from the entire water basin should be collected.
However, it is unreasonable to use the method of characteristics in the
previous form to estimate wind waves over vast oceanic areas. Firstly, it does
not allow the calculation of the non-linear wave interaction, since it is necessary to obtain information about all spectral harmonics at every grid point,
whereas they are gathered only at one point. Secondly, it is important to have
complete information of the integral wave parameters and the spectrum at
all grid points for the prediction of ocean operational conditions. The application of the method of characteristics for large areas is not justified, since all
rays must be collected from the entire area at each calculating point at every
time step (there are about 400 points in the North Atlantic for a numerical
grid with 2.5° X 2.5°).
For numerically solving (2.1), the widely-used finite-difference method
can be employed. A review of these methods for the plane case is given by
Rogers et al. (1999). However, numerical implementation of (2.1) on a sphere
is rather difficult. Unlike the solution of the similar problem in a plane, it
is necessary to approximate the additional term as 1 8f3 . df3 1 dt, increasing
the dimension of the equation. The problem becomes more complicated when
it is implemented in full form, taking into account a spatially non-uniform
current and an uneven bottom.