44
2 Mathematical Simulation of Wave Propagation at Global Distances
wave dispersion, i.e. the difference in propagation velocity of various spectral
components, should result in "blurring" of some surface sphericity effects.
One of the first basic papers (Snodgrass et al., 1966) investigating swell
propagation over global distances should be mentioned. Six wave stations located along the great circle arc between New Zealand and Alaska had been
making swell observations in the Pacific Ocean for two and a half months.
The observation results had large scatter, although the average values of the
wave height revealed a noticeable decrease of swell travelling over relatively
small distances compared to the area of the storm. The wave height decrease
coefficient 8 (according to the formula h0 "' e- 61 , where l is the travelled
distance) is estimated as 8"' 2.1 x 10- 7 m- 1 for the frequency 0.07 Hz. The
height decrease actually ceases with further swell propagation, and the average decrease coefficient makes up no more than 0.2 x w- 7 m- 1 .
These data can be compared with our calculation. The generation area
size is assumed to be 2200 km. This corresponds approximately to the storm
horizontal scale estimated by Snodgrass et al. (1966). According to the data,
the wave height makes up 0. 79 of its initial value at a distance of 1100 km
from the storm. The wave height is 0.63 of the initial value at a distance of
12,000 km, being approximately equal to the distance between Alaska and
New Zealand. According to our wave calculations for a spherical surface, the
wave heights are 0.82 and 0.65 of the initial value, while the wave calculations
for the plane surface are 0.95 and 0.38, respectively. As can be seen from the
comparison of the aforementioned estimates, the wave heights on a sphere are
generally in satisfactory agreement with full-scale observations. The present
study points out the importance of taking into account the Earth's spherical
surface in the numerical simulation of swell propagation in the ocean.
2.4 The Influence of Current on the Evolution
of Waves at the Global Scale
The influence of the Earth's spherical surface on wave propagation in the
ocean are considered in this chapter. The influence of current and shallow
water was not taken into account previously. This is quite understandable as
in most cases the influence of these factors on waves is of a more local character. These problems are considered in more detail in the following chapters.
However, the influence of global-scale currents on waves probably with no
large gradients, but with some contribution comparable to spherical effects,
is considered in this section. For example, the circumpolar current (or the socalled west wind current) is a typical current influencing wave propagation in
the Southern ocean, located approximately at latitude 40-60° S. Its current
distribution is zonally directed westward.
It should be noted that it was already mentioned in some papers (Snodgrass et al., 1966) that waves due to Pacific Ocean storms near Antarctic
shores changed their direction from that calculated with the help of weather
2 Mathematical Simulation of Wave Propagation at Global Distances
wave dispersion, i.e. the difference in propagation velocity of various spectral
components, should result in "blurring" of some surface sphericity effects.
One of the first basic papers (Snodgrass et al., 1966) investigating swell
propagation over global distances should be mentioned. Six wave stations located along the great circle arc between New Zealand and Alaska had been
making swell observations in the Pacific Ocean for two and a half months.
The observation results had large scatter, although the average values of the
wave height revealed a noticeable decrease of swell travelling over relatively
small distances compared to the area of the storm. The wave height decrease
coefficient 8 (according to the formula h0 "' e- 61 , where l is the travelled
distance) is estimated as 8"' 2.1 x 10- 7 m- 1 for the frequency 0.07 Hz. The
height decrease actually ceases with further swell propagation, and the average decrease coefficient makes up no more than 0.2 x w- 7 m- 1 .
These data can be compared with our calculation. The generation area
size is assumed to be 2200 km. This corresponds approximately to the storm
horizontal scale estimated by Snodgrass et al. (1966). According to the data,
the wave height makes up 0. 79 of its initial value at a distance of 1100 km
from the storm. The wave height is 0.63 of the initial value at a distance of
12,000 km, being approximately equal to the distance between Alaska and
New Zealand. According to our wave calculations for a spherical surface, the
wave heights are 0.82 and 0.65 of the initial value, while the wave calculations
for the plane surface are 0.95 and 0.38, respectively. As can be seen from the
comparison of the aforementioned estimates, the wave heights on a sphere are
generally in satisfactory agreement with full-scale observations. The present
study points out the importance of taking into account the Earth's spherical
surface in the numerical simulation of swell propagation in the ocean.
2.4 The Influence of Current on the Evolution
of Waves at the Global Scale
The influence of the Earth's spherical surface on wave propagation in the
ocean are considered in this chapter. The influence of current and shallow
water was not taken into account previously. This is quite understandable as
in most cases the influence of these factors on waves is of a more local character. These problems are considered in more detail in the following chapters.
However, the influence of global-scale currents on waves probably with no
large gradients, but with some contribution comparable to spherical effects,
is considered in this section. For example, the circumpolar current (or the socalled west wind current) is a typical current influencing wave propagation in
the Southern ocean, located approximately at latitude 40-60° S. Its current
distribution is zonally directed westward.
It should be noted that it was already mentioned in some papers (Snodgrass et al., 1966) that waves due to Pacific Ocean storms near Antarctic
shores changed their direction from that calculated with the help of weather