60
3 Numerical Implementation of the Wave Energy Balance Equation
along the characteristics can be found as:
cos f3t cos 'Pi = cos f3? cos 'P? 0
(3.28)
And again, the initial angle f3? does not coincide with the angles prescribed by
spectral representation in the grid point. The spectral density corresponding
to f3? may be approximated by a polynomial. The initial propagation angle
at the grid point ( '13i, r.pj) is assumed to be {31• It follows from (2.8) that the
angle {3 is dependent only on the latitude r.p, but neither on the frequency w,
nor on the longitude '13. That is why for all grid points, located at latitude r.pj,
the value of the propagation angle is known and equal to {3j. The appropriate
spectrum value is also defined. For the grid points located at another latitude,
say 'Pj- 1 , an equivalent propagation angle can be derived using the relation
(2.8).
In order to determine the spectrum value for f3? the following interpolation
is used:
1
si-l,j-l(f3?) = L amSi-l,j-l(f3t+m)'
(3.29)
m=-1
where am are the interpolation coefficients.
The wave spectrum component is assumed to be zero in the wave shadow
zone, which can take place in a spherical surface (in our case formally
I cosf3?('Pj-l)l > 1).
The smoothing operator (3.25) is applied at the next step of the interpolation-ray method.
The boundary conditions are numerically implemented as follows. If a regular grid point corresponds to land, the appropriate spectral component is
assumed to be equal to zero in (3.27).
3.6 Comparison of Results
of Numerical Wave Propagation Schemes
Numerical grid and determination of main value results.
For
numerical simulations of swell propagation over a spherical surface a grid
stretching in longitude from -12° to 12° and in latitude from 51 to 75° is
chosen. The grid includes 25 x 49 points, with a spacing of 0.5° in latitude
and 1.0° in longitude, comprising approximately 55 km between neighbouring points at the grid area centre. This can be considered as a simplified area
of the Norwegian and North Seas. A simplified form of the area allows us to
obtain an accurate analytical solution of the problem.
The problem of wave propagation starting from the initial condition (3.5)(3.8) is solved analytically, and after that using in turn the numerical upwind
scheme (3.9) and the INTERPOL method. The numerical results of different
methods is compared by their integral values enumerated below.
3 Numerical Implementation of the Wave Energy Balance Equation
along the characteristics can be found as:
cos f3t cos 'Pi = cos f3? cos 'P? 0
(3.28)
And again, the initial angle f3? does not coincide with the angles prescribed by
spectral representation in the grid point. The spectral density corresponding
to f3? may be approximated by a polynomial. The initial propagation angle
at the grid point ( '13i, r.pj) is assumed to be {31• It follows from (2.8) that the
angle {3 is dependent only on the latitude r.p, but neither on the frequency w,
nor on the longitude '13. That is why for all grid points, located at latitude r.pj,
the value of the propagation angle is known and equal to {3j. The appropriate
spectrum value is also defined. For the grid points located at another latitude,
say 'Pj- 1 , an equivalent propagation angle can be derived using the relation
(2.8).
In order to determine the spectrum value for f3? the following interpolation
is used:
1
si-l,j-l(f3?) = L amSi-l,j-l(f3t+m)'
(3.29)
m=-1
where am are the interpolation coefficients.
The wave spectrum component is assumed to be zero in the wave shadow
zone, which can take place in a spherical surface (in our case formally
I cosf3?('Pj-l)l > 1).
The smoothing operator (3.25) is applied at the next step of the interpolation-ray method.
The boundary conditions are numerically implemented as follows. If a regular grid point corresponds to land, the appropriate spectral component is
assumed to be equal to zero in (3.27).
3.6 Comparison of Results
of Numerical Wave Propagation Schemes
Numerical grid and determination of main value results.
For
numerical simulations of swell propagation over a spherical surface a grid
stretching in longitude from -12° to 12° and in latitude from 51 to 75° is
chosen. The grid includes 25 x 49 points, with a spacing of 0.5° in latitude
and 1.0° in longitude, comprising approximately 55 km between neighbouring points at the grid area centre. This can be considered as a simplified area
of the Norwegian and North Seas. A simplified form of the area allows us to
obtain an accurate analytical solution of the problem.
The problem of wave propagation starting from the initial condition (3.5)(3.8) is solved analytically, and after that using in turn the numerical upwind
scheme (3.9) and the INTERPOL method. The numerical results of different
methods is compared by their integral values enumerated below.