236
5 Wave Evolution in Non-uniform Currents in Deep Water
It is easy to find a solution for (5.135) in analytical form as:
V = 2qFz + z canst
(z:::; 0) ,
(5.136)
where "canst" is the constant of integration, prescribed by the problem formulation.
The profile of the current velocity (5.136) is shown in Fig. 5.37. It should
be noted that the value V(z) is not dependent on the wave number k. In
the case of waves propagating in increasing countercurrent and its value with
the vertical profile (5.136) being achieved, the blocking of all wave spectrum
components will take place at one spatial point. This is the point where the
location of caustics of all spectral components coincides. The spectral solution
does not eliminate these singularities, as was done in case of a vertically
uniform current. A more intensive increase of the wave height and total energy
is observed in the vicinity of this point.
It should be noted that the solution (5.136) is formally obtained using
the approximate dispersion relation (5.127), (5.128), and its correct use is
limited by the condition of the small vorticity parameter D 2 / gk :::; 1 (Skop,
1987). This means that the vorticity of the current velocity profile should be
less than the wave frequency. Thus, the applicability condition of the solution
(5.136) can be written in the form:
02
= J_ ( 8 V)
2
= J_ ( {29 +canst)
2
:::; 1.
gk
gk az
gk v ru
(5.137)
In the case with canst = 0, the condition (5.137) can be written as
Q 2 jgk = 2/(zkn) :::; 1. Due to the value z becoming zero this condition
becomes unfulfilled in the uppermost layer of the surface current.
It can be expected that the considered solution is realized locally, i.e. neither in entire surface layer, nor for all wave spectrum components. Even in the
case when the blocking is simultaneously observed within some range of wave
numbers, a more intensive increase of the total wave height and, correspondingly, wave breaking could be expected. This situation (see Fig. 5.37) might
be observed in the presence of a background current entraining a significant
water layer with the surface countercurrent, probably of drift origin, being
superimposed on it. The described wave blocking appears in the case with
the wave propagation direction coinciding with the drift current direction
and being opposite to the background flow.
In conclusion it should be noted that the vertical non-uniform current
influences the waves in a sufficiently diverse way. Another aspect of wave interaction with a vertical non-uniform current, generated by wind, is discussed
in the paper by Zakharov & Shrira (1990), explaining the narrowness of the
angular spectrum. The evolution of a random field of weak non-linear surface
waves is investigated with consideration of the induced dispersion mechanism in the wind-driven near-surface current. This makes a contribution to
5 Wave Evolution in Non-uniform Currents in Deep Water
It is easy to find a solution for (5.135) in analytical form as:
V = 2qFz + z canst
(z:::; 0) ,
(5.136)
where "canst" is the constant of integration, prescribed by the problem formulation.
The profile of the current velocity (5.136) is shown in Fig. 5.37. It should
be noted that the value V(z) is not dependent on the wave number k. In
the case of waves propagating in increasing countercurrent and its value with
the vertical profile (5.136) being achieved, the blocking of all wave spectrum
components will take place at one spatial point. This is the point where the
location of caustics of all spectral components coincides. The spectral solution
does not eliminate these singularities, as was done in case of a vertically
uniform current. A more intensive increase of the wave height and total energy
is observed in the vicinity of this point.
It should be noted that the solution (5.136) is formally obtained using
the approximate dispersion relation (5.127), (5.128), and its correct use is
limited by the condition of the small vorticity parameter D 2 / gk :::; 1 (Skop,
1987). This means that the vorticity of the current velocity profile should be
less than the wave frequency. Thus, the applicability condition of the solution
(5.136) can be written in the form:
02
= J_ ( 8 V)
2
= J_ ( {29 +canst)
2
:::; 1.
gk
gk az
gk v ru
(5.137)
In the case with canst = 0, the condition (5.137) can be written as
Q 2 jgk = 2/(zkn) :::; 1. Due to the value z becoming zero this condition
becomes unfulfilled in the uppermost layer of the surface current.
It can be expected that the considered solution is realized locally, i.e. neither in entire surface layer, nor for all wave spectrum components. Even in the
case when the blocking is simultaneously observed within some range of wave
numbers, a more intensive increase of the total wave height and, correspondingly, wave breaking could be expected. This situation (see Fig. 5.37) might
be observed in the presence of a background current entraining a significant
water layer with the surface countercurrent, probably of drift origin, being
superimposed on it. The described wave blocking appears in the case with
the wave propagation direction coinciding with the drift current direction
and being opposite to the background flow.
In conclusion it should be noted that the vertical non-uniform current
influences the waves in a sufficiently diverse way. Another aspect of wave interaction with a vertical non-uniform current, generated by wind, is discussed
in the paper by Zakharov & Shrira (1990), explaining the narrowness of the
angular spectrum. The evolution of a random field of weak non-linear surface
waves is investigated with consideration of the induced dispersion mechanism in the wind-driven near-surface current. This makes a contribution to