14
1 General Problem Formulation
and Pa of the boundary layer in the atmosphere. But the pressure Pw (at
z = TJ) can be directly expressed by the potential velocity derivatives (1.9).
The complete system of equations (1.3), (1.9)-(1.14) for determining the
surface evolution TJ with the initial conditions (1.8) presents considerable
difficulty for analysis. Unlike in the usual classic theory of potential waves
with the given pressure distribution Pa at the surface ry, either the surface
itself or the pressure are not determined in wind wave theory. These two
unknown functions are not independent, so a joint solution of both equations
(1.9)-(1.12) for wave disturbances at z < TJ and complicated equations of
vortex current above the wave boundary at z > TJ are required.
1.2 Geometric Optics Approximation
The problem of mathematical wind wave modelling is aggravated by the fact
that there are different horizontal and vertical non-uniformities significantly
influencing the propagation and generation of surface gravity waves in the
ocean under natural conditions. The most typical non-uniformities include
spatial and temporal variability of current, turbulent motion and uneven
bottom relief in oceanic areas. That is why it is interesting to consider the
effect of non-uniformity on wave propagation and generation.
This is quite a complicated problem to be solved in the general formulation. First of all, it seems reasonable to consider wind wave propagation,
with their length and period of the wave much less than the typical spatial
and temporal medium variation scales, i.e. 1-100 km and 1-10 h. If the latter
are assumed to be typical for a wide class of oceanic motions, it is possible
to consider this problem using the method of geometric optics.
The geometric optics method is based on the assumption of the existence
of plane waves. Plane waves are assumed to be described by the same propagation direction, length and amplitude everywhere. Natural waves obviously
do not have such properties. But they can often be considered as plane in
each local space area. In this case it is necessary for the wave amplitude a, the
wave vector k and the frequency w to remain almost unchanged at a distance
of about the wavelength and over the time of about the wave period. Changes
of these parameters are connected with variations of the wave propagation
media, requiring small changes of these parameters in the media variability scale. Large-scale currents and the uneven bottom are considered to be
non-uniform media. Thus, if the typical horizontal scale of the bottom relief
variation is M 1 , the spatial current scale is M2 and T is its temporal scale
variation, the necessary condition for the applicability of the geometric optics
method: conditions:
(1.15)
If these conditions are satisfied, the so-called wave surfaces may be introduced in which the wave phase is the same at every point at a given moment.
1 General Problem Formulation
and Pa of the boundary layer in the atmosphere. But the pressure Pw (at
z = TJ) can be directly expressed by the potential velocity derivatives (1.9).
The complete system of equations (1.3), (1.9)-(1.14) for determining the
surface evolution TJ with the initial conditions (1.8) presents considerable
difficulty for analysis. Unlike in the usual classic theory of potential waves
with the given pressure distribution Pa at the surface ry, either the surface
itself or the pressure are not determined in wind wave theory. These two
unknown functions are not independent, so a joint solution of both equations
(1.9)-(1.12) for wave disturbances at z < TJ and complicated equations of
vortex current above the wave boundary at z > TJ are required.
1.2 Geometric Optics Approximation
The problem of mathematical wind wave modelling is aggravated by the fact
that there are different horizontal and vertical non-uniformities significantly
influencing the propagation and generation of surface gravity waves in the
ocean under natural conditions. The most typical non-uniformities include
spatial and temporal variability of current, turbulent motion and uneven
bottom relief in oceanic areas. That is why it is interesting to consider the
effect of non-uniformity on wave propagation and generation.
This is quite a complicated problem to be solved in the general formulation. First of all, it seems reasonable to consider wind wave propagation,
with their length and period of the wave much less than the typical spatial
and temporal medium variation scales, i.e. 1-100 km and 1-10 h. If the latter
are assumed to be typical for a wide class of oceanic motions, it is possible
to consider this problem using the method of geometric optics.
The geometric optics method is based on the assumption of the existence
of plane waves. Plane waves are assumed to be described by the same propagation direction, length and amplitude everywhere. Natural waves obviously
do not have such properties. But they can often be considered as plane in
each local space area. In this case it is necessary for the wave amplitude a, the
wave vector k and the frequency w to remain almost unchanged at a distance
of about the wavelength and over the time of about the wave period. Changes
of these parameters are connected with variations of the wave propagation
media, requiring small changes of these parameters in the media variability scale. Large-scale currents and the uneven bottom are considered to be
non-uniform media. Thus, if the typical horizontal scale of the bottom relief
variation is M 1 , the spatial current scale is M2 and T is its temporal scale
variation, the necessary condition for the applicability of the geometric optics
method: conditions:
(1.15)
If these conditions are satisfied, the so-called wave surfaces may be introduced in which the wave phase is the same at every point at a given moment.