2 Mathematical Simulation
of Wave Propagation at Global Distances
2.1 Problem Formulation of Ocean Wind Wave
Modelling Using Spherical Variables
Chapter 1 presents a general problem formulation of the evolution of the wind
wave spectrum considering various determining factors. Due to the complexity of formulating the general problem, sphericity solution effects are considered in this chapter. The investigation is limited to the deep-water case,
taking into consideration that the average ocean depth essentially exceeds
the characteristic lengths of wind waves and swell.
It should be noted that traditionally most wind wave models are based
on the wave energy balance equation, written in the usual rectangular coordinates r = {x,y} (it is not used in the WAM model (Komen et. al., 1994)
and some of its modifications (Tolman, 1992)). The problems are solved in
the horizontal plane. This is justified by considering relatively limited water areas, for example, lakes. However, it is doubtful that we can use this
approach for waves in oceans with their sizes comparable to the Earth's radius. A specific correction for wave estimation can be introduced taking into
account the Earth's curvature, first of all describing swell propagation over
large distances that can exceed 90° along a great circle arc (Snodgrass et al.,
1966).
The spherical shape of the oceanic surface results in another geometrical
divergence of wave energy compared to that on a plane surface. This leads
to a smaller (or greater) wave height decrease with waves propagating from
the generation area. The Earth's rotation may have some influence on the
waves resulting in deviation of their propagation trajectory from a geodesic
line. However, these effects are so insignificant that they can be completely
neglected (Backus, 1962).
Generally speaking, waves travelling directly from the source and waves
having curled around the Earth and returning back from the other side can
be observed due to the Earth's round shape. This situation is observed, for
example, in seismology in the case of Rayleigh wave propagation. It is excluded for hydrodynamic waves in the ocean, as the water surface on the
Earth is actually completely overlapped by continents. Therefore, only waves
moving directly from the source are considered in this problem (see Fig. 2.1).
of Wave Propagation at Global Distances
2.1 Problem Formulation of Ocean Wind Wave
Modelling Using Spherical Variables
Chapter 1 presents a general problem formulation of the evolution of the wind
wave spectrum considering various determining factors. Due to the complexity of formulating the general problem, sphericity solution effects are considered in this chapter. The investigation is limited to the deep-water case,
taking into consideration that the average ocean depth essentially exceeds
the characteristic lengths of wind waves and swell.
It should be noted that traditionally most wind wave models are based
on the wave energy balance equation, written in the usual rectangular coordinates r = {x,y} (it is not used in the WAM model (Komen et. al., 1994)
and some of its modifications (Tolman, 1992)). The problems are solved in
the horizontal plane. This is justified by considering relatively limited water areas, for example, lakes. However, it is doubtful that we can use this
approach for waves in oceans with their sizes comparable to the Earth's radius. A specific correction for wave estimation can be introduced taking into
account the Earth's curvature, first of all describing swell propagation over
large distances that can exceed 90° along a great circle arc (Snodgrass et al.,
1966).
The spherical shape of the oceanic surface results in another geometrical
divergence of wave energy compared to that on a plane surface. This leads
to a smaller (or greater) wave height decrease with waves propagating from
the generation area. The Earth's rotation may have some influence on the
waves resulting in deviation of their propagation trajectory from a geodesic
line. However, these effects are so insignificant that they can be completely
neglected (Backus, 1962).
Generally speaking, waves travelling directly from the source and waves
having curled around the Earth and returning back from the other side can
be observed due to the Earth's round shape. This situation is observed, for
example, in seismology in the case of Rayleigh wave propagation. It is excluded for hydrodynamic waves in the ocean, as the water surface on the
Earth is actually completely overlapped by continents. Therefore, only waves
moving directly from the source are considered in this problem (see Fig. 2.1).