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5 Wave Evolution in Non-uniform Currents in Deep Water
From the theoretical point of view, the rips are suggested to be due to
either wind wave transformation in non-uniform currents or wave formation
in the water surface with a current flowing around a submarine obstacle
(Barenblatt et al., 1985). It is also believed (Leykin & Monin, 1985) that
a suitable wave field model is a "soliton gas", i.e. a system of solitons with
different direction propagation and amplitudes. The waves do not interact,
but simply superimpose each other.
The first of these hypotheses is discussed in this section. The rip model can
be considered as a result of wave transformation in a large-scale horizontal
non-uniform current. The indirect role of different underwater obstacles is not
excluded. They have a specific influence on surface waves, creating a strong
non-uniform current.
Problem formulation.
The problem is considered in the geometrical
optics approximation. It is based on the wave action density balance equation
in the spectral form (5.1). The main difficulty is insufficient knowledge of the
source function G. There are theoretical ratios for the wind input generating
mechanism Gin and the non-linear energy transfer Gn!, but unfortunately
there is still no generally accepted expression for the wave dissipation Gds·
The limitation of the wave spectrum increase is often achieved in wind wave
models as follows: the source function (without dissipation) is multiplied by
the term 1-11 = 1- J(B/B00 ). The correcting function 1-11 establishes the
ultimate possible value of the spectral density B00 • Thus, it takes the wave
energy dissipation into account indirectly. The Pierson-Moskowitz spectrum
is used as an ultimate spectrum Boo, formed by the constant wind, to calculate
the wave in deep water without currents in wind wave models of the first
and second generations. The approximation of such an ultimate spectrum is
unknown in the case of a horizontal non-uniform current. Thus, Basovich et
al. (1982) take the source function in the action density balance equation
in the form G = f3uN(1- N/N0 ), where N0 is the action density in the
absence of a current and f3u is the wave increase increment. However, the
wave spectrum increase is insufficiently limited by the source function. For
example, in the case of the wind action being neglected (f3u = 0), the spectral
energy density B = N a is changed significantly. It can greatly exceed the
equilibrium interval spectrum (Kitaigorodskii et al., 1975; Phillips, 1958).
An attempt should be made to determine the term limiting the increase
of the wave action density spectrum in the kinetic equation (5.1) with waves
in a horizontal non-uniform current and wind input. For this purpose the
hypothesis of the equilibrium interval can be used as some ultimate state
of the spatial energy spectrum (Kitaigorodskii et al., 1975). The equilibrium
interval is considered to be invariant and its approximation can be written as
Boo(k) = (1/2)apk- 4 Q(f3). Proceeding from the wave energy density balance
equation (which takes into account the wind and current influence on waves)
to the wave action density equation with the correcting function in the form
1-11 = 1- (B/Boo)q, the following equation is obtained:
5 Wave Evolution in Non-uniform Currents in Deep Water
From the theoretical point of view, the rips are suggested to be due to
either wind wave transformation in non-uniform currents or wave formation
in the water surface with a current flowing around a submarine obstacle
(Barenblatt et al., 1985). It is also believed (Leykin & Monin, 1985) that
a suitable wave field model is a "soliton gas", i.e. a system of solitons with
different direction propagation and amplitudes. The waves do not interact,
but simply superimpose each other.
The first of these hypotheses is discussed in this section. The rip model can
be considered as a result of wave transformation in a large-scale horizontal
non-uniform current. The indirect role of different underwater obstacles is not
excluded. They have a specific influence on surface waves, creating a strong
non-uniform current.
Problem formulation.
The problem is considered in the geometrical
optics approximation. It is based on the wave action density balance equation
in the spectral form (5.1). The main difficulty is insufficient knowledge of the
source function G. There are theoretical ratios for the wind input generating
mechanism Gin and the non-linear energy transfer Gn!, but unfortunately
there is still no generally accepted expression for the wave dissipation Gds·
The limitation of the wave spectrum increase is often achieved in wind wave
models as follows: the source function (without dissipation) is multiplied by
the term 1-11 = 1- J(B/B00 ). The correcting function 1-11 establishes the
ultimate possible value of the spectral density B00 • Thus, it takes the wave
energy dissipation into account indirectly. The Pierson-Moskowitz spectrum
is used as an ultimate spectrum Boo, formed by the constant wind, to calculate
the wave in deep water without currents in wind wave models of the first
and second generations. The approximation of such an ultimate spectrum is
unknown in the case of a horizontal non-uniform current. Thus, Basovich et
al. (1982) take the source function in the action density balance equation
in the form G = f3uN(1- N/N0 ), where N0 is the action density in the
absence of a current and f3u is the wave increase increment. However, the
wave spectrum increase is insufficiently limited by the source function. For
example, in the case of the wind action being neglected (f3u = 0), the spectral
energy density B = N a is changed significantly. It can greatly exceed the
equilibrium interval spectrum (Kitaigorodskii et al., 1975; Phillips, 1958).
An attempt should be made to determine the term limiting the increase
of the wave action density spectrum in the kinetic equation (5.1) with waves
in a horizontal non-uniform current and wind input. For this purpose the
hypothesis of the equilibrium interval can be used as some ultimate state
of the spatial energy spectrum (Kitaigorodskii et al., 1975). The equilibrium
interval is considered to be invariant and its approximation can be written as
Boo(k) = (1/2)apk- 4 Q(f3). Proceeding from the wave energy density balance
equation (which takes into account the wind and current influence on waves)
to the wave action density equation with the correcting function in the form
1-11 = 1- (B/Boo)q, the following equation is obtained: