168
5 Wave Evolution in Non-uniform Currents in Deep Water
in the limit No » N 0 , , N ~ {/§N00 • As shown below, this situation can be
fulfilled with waves propagating at increasing countercurrent. The spectrum
increase is limited by wave breaking. There is the opposite case - the wave
energy density is decreased due to interaction with a fair current. The solution
is not influenced by the equilibrium interval (5.23).
One can pass from the action density spectrum N(k) to the energy frequency spectrum S(a) measured in the readout system connected with the
current S(a) = ak8kj8aN(a)lk=k(u)· It should be noted that the wave spectrum measurements at the current (Barenblatt et al., 1985) were made precisely in that coordinate system. As shown below, it is significantly different
from the spectrum S(w) measured in a fixed coordinate system, described in
Sect. 5.2. Based on the ratio (5.23), the solution of the spectrum S(a) can
be written in the form:
S(a) =So(ao)~ 8k (8ko)-1
aoko 8a 8ao
x { 1 + ~ [ So(ao~ ]q [(.!!...) 9 q _ 1 ] }l/q ,
9 apg 2 a 0
5
ao
where S0 (a0 ) is the initial wave spectrum in the absence of current.
(5.24)
The expression for initial wave frequency spectrum in the absence of current (i.e. w =a) is taken in the form (5.16), where n is the parameter determining the spectrum form. It is assumed to be equal to 5.5.
It is necessary to solve the equation system (5.20)-(5.22) for the given
current speed profile V(x) in order to determine the value ao as a function of
the variables a, V, r in (5.24). In its exact form, this system is not integrated
even for the one-dimensional case. That is why the following approaches are
taken into consideration.
It follows from (5.22) that the frequency w is preserved for the stationary
current speed V along the wave packet propagation trajectory. It can be
written as a- Vk = a0 in the one-dimensional case. This ratio determines the
wave number k0 depending on k and the current speed V. However, the same
speed value V can correspond to two different wave numbers. Both straight
(Cgx > 0) and reverse waves (Cgx < 0) can exist in the countercurrent at
the same point (with x < Xm)· It is a result of straight waves reflected from
the horizontal non-uniform current. The straight waves can only exist after
having passed the maximum current speed (in the segment x > Xm)· Due to
this reason, the motion integral is insufficient for an unambiguous solution.
It is necessary to apply kinematic schemes in this case.
Thus, a wave packet is reflected from the current and rolled down propagating in the increasing countercurrent until it reaches the point V = V*,
where the blocking condition Cgx = Cg - V* = 0 is fulfilled. It is necessary
that the conditions V* < Vmax should be formally met. Otherwise, the wave
packet passes through the "barrier", i.e. the area with maximum current
5 Wave Evolution in Non-uniform Currents in Deep Water
in the limit No » N 0 , , N ~ {/§N00 • As shown below, this situation can be
fulfilled with waves propagating at increasing countercurrent. The spectrum
increase is limited by wave breaking. There is the opposite case - the wave
energy density is decreased due to interaction with a fair current. The solution
is not influenced by the equilibrium interval (5.23).
One can pass from the action density spectrum N(k) to the energy frequency spectrum S(a) measured in the readout system connected with the
current S(a) = ak8kj8aN(a)lk=k(u)· It should be noted that the wave spectrum measurements at the current (Barenblatt et al., 1985) were made precisely in that coordinate system. As shown below, it is significantly different
from the spectrum S(w) measured in a fixed coordinate system, described in
Sect. 5.2. Based on the ratio (5.23), the solution of the spectrum S(a) can
be written in the form:
S(a) =So(ao)~ 8k (8ko)-1
aoko 8a 8ao
x { 1 + ~ [ So(ao~ ]q [(.!!...) 9 q _ 1 ] }l/q ,
9 apg 2 a 0
5
ao
where S0 (a0 ) is the initial wave spectrum in the absence of current.
(5.24)
The expression for initial wave frequency spectrum in the absence of current (i.e. w =a) is taken in the form (5.16), where n is the parameter determining the spectrum form. It is assumed to be equal to 5.5.
It is necessary to solve the equation system (5.20)-(5.22) for the given
current speed profile V(x) in order to determine the value ao as a function of
the variables a, V, r in (5.24). In its exact form, this system is not integrated
even for the one-dimensional case. That is why the following approaches are
taken into consideration.
It follows from (5.22) that the frequency w is preserved for the stationary
current speed V along the wave packet propagation trajectory. It can be
written as a- Vk = a0 in the one-dimensional case. This ratio determines the
wave number k0 depending on k and the current speed V. However, the same
speed value V can correspond to two different wave numbers. Both straight
(Cgx > 0) and reverse waves (Cgx < 0) can exist in the countercurrent at
the same point (with x < Xm)· It is a result of straight waves reflected from
the horizontal non-uniform current. The straight waves can only exist after
having passed the maximum current speed (in the segment x > Xm)· Due to
this reason, the motion integral is insufficient for an unambiguous solution.
It is necessary to apply kinematic schemes in this case.
Thus, a wave packet is reflected from the current and rolled down propagating in the increasing countercurrent until it reaches the point V = V*,
where the blocking condition Cgx = Cg - V* = 0 is fulfilled. It is necessary
that the conditions V* < Vmax should be formally met. Otherwise, the wave
packet passes through the "barrier", i.e. the area with maximum current