242
5 Wave Evolution in Non-uniform Currents in Deep Water
S ( k, {3, U, x - x0 ), then the expression ( 5.145) indicates that in the presence of a constant current the spectrum expression is transformed according
to the substituted variables as follows:
X -Xo
X- Xo -t
;
1 - 2V Jk79 cos- 1 ({3)
U-tU-V.
(5.147)
In should be noted that after replacing the expression (5.147), the
fetch value can be formally extremely large and it becomes negative at
1 - 2V Jk!Y/ cos ({3) ~ 0. This is due to the fact that the group velocity
projection Cgx = Cgx - V can attain zero or negative values for some spectral
components. These waves cannot propagate along the positive Ox axis due
to large current velocities, i.e. from the initial boundary x 0 , with the initial
spectral density 80 being set, to the calculated point x. However, these spectral components can still arrive at the calculated point as they are driven
backwards by the current (i.e. from right to left, see Fig. 5.38).
Since the fetch value is not limited to the right of the calculated point (see
Fig. 5.38), the spectral components drifting backwards can be considered to
travel an infinitely large distance. The time period of their interaction with
the air flow is not limited either. That is why the fetch value can be assumed
to be extremely large at Cgx < 0. In fact this means that the energy of these
spectral components should have reached the saturation condition.
The dependence (5.16) is used as the spectrum approximation S in the
absence of current. There are the following values: m 0 is the zero moment, and
u max is the spectral maximum frequency. They are functions of the fetch x and
the wind speed U. The value n is assumed to be equal to 5.5, corresponding to
the mean parameter value at the main stage of wave development (Davidan
et al., 1985).
The following relationship between the non-dimensional values m0 =
m 0 g 2 /U 4 and CTmax = Umax9/U 2 in the absence of a current is used (Davidan
et al., 1985):
-
0 11 - -0.34
O"max = . mo
.
(5.148a)
The dependence of the spectral zero moment m 0 of the non-dimensional
fetch X= (x- x0 ) gfU 2 is assumed to be equal to:
0 84
mo = m= { th ( 2. 727 X 10- 5 X)
8 } ---r ,
(5.148b)
where m= = 3.03 x 10 3 . This is a non-dimensional zero moment of the spectrum for completely developed waves, with the spectral maximum frequency
being equal to CTmax = 0.8.
The expression (5.148b) is a known generalized power dependence (Davidan et al., 1985) for X > 3.6 x 10 4 . The value m0 is asymptotically smoothly
transferred into m= for large values of the argument to the hyperbolic
5 Wave Evolution in Non-uniform Currents in Deep Water
S ( k, {3, U, x - x0 ), then the expression ( 5.145) indicates that in the presence of a constant current the spectrum expression is transformed according
to the substituted variables as follows:
X -Xo
X- Xo -t
;
1 - 2V Jk79 cos- 1 ({3)
U-tU-V.
(5.147)
In should be noted that after replacing the expression (5.147), the
fetch value can be formally extremely large and it becomes negative at
1 - 2V Jk!Y/ cos ({3) ~ 0. This is due to the fact that the group velocity
projection Cgx = Cgx - V can attain zero or negative values for some spectral
components. These waves cannot propagate along the positive Ox axis due
to large current velocities, i.e. from the initial boundary x 0 , with the initial
spectral density 80 being set, to the calculated point x. However, these spectral components can still arrive at the calculated point as they are driven
backwards by the current (i.e. from right to left, see Fig. 5.38).
Since the fetch value is not limited to the right of the calculated point (see
Fig. 5.38), the spectral components drifting backwards can be considered to
travel an infinitely large distance. The time period of their interaction with
the air flow is not limited either. That is why the fetch value can be assumed
to be extremely large at Cgx < 0. In fact this means that the energy of these
spectral components should have reached the saturation condition.
The dependence (5.16) is used as the spectrum approximation S in the
absence of current. There are the following values: m 0 is the zero moment, and
u max is the spectral maximum frequency. They are functions of the fetch x and
the wind speed U. The value n is assumed to be equal to 5.5, corresponding to
the mean parameter value at the main stage of wave development (Davidan
et al., 1985).
The following relationship between the non-dimensional values m0 =
m 0 g 2 /U 4 and CTmax = Umax9/U 2 in the absence of a current is used (Davidan
et al., 1985):
-
0 11 - -0.34
O"max = . mo
.
(5.148a)
The dependence of the spectral zero moment m 0 of the non-dimensional
fetch X= (x- x0 ) gfU 2 is assumed to be equal to:
0 84
mo = m= { th ( 2. 727 X 10- 5 X)
8 } ---r ,
(5.148b)
where m= = 3.03 x 10 3 . This is a non-dimensional zero moment of the spectrum for completely developed waves, with the spectral maximum frequency
being equal to CTmax = 0.8.
The expression (5.148b) is a known generalized power dependence (Davidan et al., 1985) for X > 3.6 x 10 4 . The value m0 is asymptotically smoothly
transferred into m= for large values of the argument to the hyperbolic