1. 7 Spatial-Temporal Scale Considerations
33
In order to obtain quantitative estimates of various factors, the functions in the right-hand side of (1.86)-(1.90) are transformed into their nondimensional form:
,_
tj;R
'P = (lcgl) ;
. : .
kR
k = (lcgl) ;
where (lcgl) is a mean group velocity estimate. In this case the right-hand
sides of the equations also become non-dimensional with the new parameters:
a: = (lVI) I (leg I) is the ratio of the mean current velocity to the wave propagation speed; X = ( J6. VI) I (I k VI) is the ratio defining the effectiveness of
non-uniform current on the wave kinematics, where (lkl) is the mean estimate
of the wave number; and J = (lki)(6.H)e- 2 (ki)(H) is a parameter characterizing the refraction effect in shallow water. It is clear that a comparison of the
non-dimensional parameters a:, x, J determines the quantitative significance
of the different mechanisms. Thus, for waves with a period T = 6 s in shallow
water at kH = 1 and depth gradient 6.H I 6.L ~ 10- 3 the parameter J is of
order 10 3 -10 4 . The parameter (3 is of a smaller value. For 6. VI 6.L"' 10- 4 s- 1
the value (3 is estimated as 10 2 . This is significantly greater than the effects
due to sphericity of the ocean surface (in this case they are about 1).
Thus, even the roughest estimates indicate that the shallow-water and the
large-gradient current effects are of the greatest influence on the change of
the wave elements at comparatively small distances. The sphericity is practically of no importance for describing such effects. It is manifested at global
distances. In this case the small-gradient currents with global scales (Kenyon,
1981) can play their role.
It is reasonable to carry out a study of the influences of different effects
on the solution of the problem, isolating the spatial-temporal scales. This
allows the simplification of the analysis of the problem. It reveals the most
effective mechanisms forming the wind wave spectrum, taking into account
the spatial-temporal scale of wave development in a specific geographical
area. It should be noted that the geometric aspect of the problem is analysed
by a description of wave packet propagation in phase space. This is described
not only by the right-hand side, but also by the left-hand side of the kinetic
equation.
Thus, it is possible to investigate the problem at the following spatialtemporal scales:
1. The global scale (for the spatial scale L1 "' 10 6 -10 7 m and the temporal
scale T1 "' 10 6 s) taking into account the Earth's surface curvature and global
currents for modelling wind waves. In this case the typical spectrum-forming
mechanisms are effective under deep water conditions (G2 , G5 , G8 , etc.). The
spatial scale of wave field non-uniformity is determined by a typical scale of
atmospheric perturbations (cyclones). An example of such a model is given
in Chap. 2.
2. The regional scale I (L2 "' 10 5 -10 6 m, T2 "' 10 5 s) with the wind waves
modelled under deep water conditions in seas, large lakes, basins, etc; the
33
In order to obtain quantitative estimates of various factors, the functions in the right-hand side of (1.86)-(1.90) are transformed into their nondimensional form:
,_
tj;R
'P = (lcgl) ;
. : .
kR
k = (lcgl) ;
where (lcgl) is a mean group velocity estimate. In this case the right-hand
sides of the equations also become non-dimensional with the new parameters:
a: = (lVI) I (leg I) is the ratio of the mean current velocity to the wave propagation speed; X = ( J6. VI) I (I k VI) is the ratio defining the effectiveness of
non-uniform current on the wave kinematics, where (lkl) is the mean estimate
of the wave number; and J = (lki)(6.H)e- 2 (ki)(H) is a parameter characterizing the refraction effect in shallow water. It is clear that a comparison of the
non-dimensional parameters a:, x, J determines the quantitative significance
of the different mechanisms. Thus, for waves with a period T = 6 s in shallow
water at kH = 1 and depth gradient 6.H I 6.L ~ 10- 3 the parameter J is of
order 10 3 -10 4 . The parameter (3 is of a smaller value. For 6. VI 6.L"' 10- 4 s- 1
the value (3 is estimated as 10 2 . This is significantly greater than the effects
due to sphericity of the ocean surface (in this case they are about 1).
Thus, even the roughest estimates indicate that the shallow-water and the
large-gradient current effects are of the greatest influence on the change of
the wave elements at comparatively small distances. The sphericity is practically of no importance for describing such effects. It is manifested at global
distances. In this case the small-gradient currents with global scales (Kenyon,
1981) can play their role.
It is reasonable to carry out a study of the influences of different effects
on the solution of the problem, isolating the spatial-temporal scales. This
allows the simplification of the analysis of the problem. It reveals the most
effective mechanisms forming the wind wave spectrum, taking into account
the spatial-temporal scale of wave development in a specific geographical
area. It should be noted that the geometric aspect of the problem is analysed
by a description of wave packet propagation in phase space. This is described
not only by the right-hand side, but also by the left-hand side of the kinetic
equation.
Thus, it is possible to investigate the problem at the following spatialtemporal scales:
1. The global scale (for the spatial scale L1 "' 10 6 -10 7 m and the temporal
scale T1 "' 10 6 s) taking into account the Earth's surface curvature and global
currents for modelling wind waves. In this case the typical spectrum-forming
mechanisms are effective under deep water conditions (G2 , G5 , G8 , etc.). The
spatial scale of wave field non-uniformity is determined by a typical scale of
atmospheric perturbations (cyclones). An example of such a model is given
in Chap. 2.
2. The regional scale I (L2 "' 10 5 -10 6 m, T2 "' 10 5 s) with the wind waves
modelled under deep water conditions in seas, large lakes, basins, etc; the