5.2 Frequency-Angular Spectrum Evolution in a Current
163
~
-----------.,
/~~~\
~
Fig. 5.3. Angular energy distribution function in countercurrent with V =
-1 ms- 1 (a) and in fair current with V = 3 ms- 1 (b) for different frequencies w:
1- 0.85 rads- 1 ; 2 - 0.65 rads-\ 3 - 1.5 rads- 1 . Dot-and-dash lines denote the
initial angular energy distribution
a fair current. At the same time, the frequency of the spectrum maximum is
almost unchanged.
The range of spectral directions for different frequencies is shown in
Fig. 5.3a,b. It is normalized by the spectral maximum at a given current
velocity. As can be seen, the direction range is narrowed by the countercurrent. It is narrowed to a greater extent at high frequencies. The process is
vice versa in a fair current.
If waves propagate to the area with a current opposite to the wave direction the wave energy per unit area is increased due to interaction with
the countercurrent. This can be limited by dissipation connected with wave
breaking. There is an equilibrium interval appearing in the spectrum, i.e.
the range of frequencies w and the angles {3, in which the energy influx from
a current is balanced by breaking.
Generally speaking, there is no reason to assume that the equilibrium
interval of the wave frequency spectrum in a current is approximated by the
well-known Phillips formula S00 (w) "'w- 5 (1980). The frequency-angular interval spectrum in a current can be found by recalculating the corresponding
dispersion relation using the idea of the universally applied spatial spectrum within the equilibrium interval (Kitaigorodskii et al., 1975). Thus, the
frequency-angular spectrum of the equilibrium interval can be written in the
following form:
163
~
-----------.,
/~~~\
~
Fig. 5.3. Angular energy distribution function in countercurrent with V =
-1 ms- 1 (a) and in fair current with V = 3 ms- 1 (b) for different frequencies w:
1- 0.85 rads- 1 ; 2 - 0.65 rads-\ 3 - 1.5 rads- 1 . Dot-and-dash lines denote the
initial angular energy distribution
a fair current. At the same time, the frequency of the spectrum maximum is
almost unchanged.
The range of spectral directions for different frequencies is shown in
Fig. 5.3a,b. It is normalized by the spectral maximum at a given current
velocity. As can be seen, the direction range is narrowed by the countercurrent. It is narrowed to a greater extent at high frequencies. The process is
vice versa in a fair current.
If waves propagate to the area with a current opposite to the wave direction the wave energy per unit area is increased due to interaction with
the countercurrent. This can be limited by dissipation connected with wave
breaking. There is an equilibrium interval appearing in the spectrum, i.e.
the range of frequencies w and the angles {3, in which the energy influx from
a current is balanced by breaking.
Generally speaking, there is no reason to assume that the equilibrium
interval of the wave frequency spectrum in a current is approximated by the
well-known Phillips formula S00 (w) "'w- 5 (1980). The frequency-angular interval spectrum in a current can be found by recalculating the corresponding
dispersion relation using the idea of the universally applied spatial spectrum within the equilibrium interval (Kitaigorodskii et al., 1975). Thus, the
frequency-angular spectrum of the equilibrium interval can be written in the
following form: