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5 Wave Evolution in Non-uniform Currents in Deep Water
In this section an attempt is undertaken to describe the evolution of wave
parameters along the fetch in the presence of a spatially uniform current.
The wave development in a current is assumed to occur according to the
same laws as in its absence. The role of a current is attributed to a change
of the relative wind speed generating the waves, i.e. the speed measured in
the coordinate system connected with the current velocity. Furthermore, the
current velocity is added to the wave propagation velocity relative to a fixed
fetch changing the time of the wave interaction with the air flow.
In accordance with the general formulation of the problem (5.1), (5.2),
a constant wind speed U directed along the Ox axis is assumed to be prescribed at the horizon: z = 10m. The current velocity is equal to the value V.
Wind wave evolution along the fetch x in the established regime will be
investigated with the initial energy spectral density valueS= S (k, /3), being
prescribed at the boundary x = x 0 (see Fig. 5.38).
In order to describe wind wave development a coordinate system moving
with the current V will be used. The coordinates of a mobile system x'
and y' are connected with the initial coordinate system x' = x- Vt; y' = y.
The current is absent in the new coordinate system and the waves develop
according to similar laws as in calm water with the exception that the wind
speed relative to the water surface is equal to the value U - V.
The evolution of spectral density of the wind wave energy is assumed to
be described by (5.1) which can be written as follows:
dS _aS
aS dx'
aS dy' aS dk aS d/3 _ G'
dt- at+ ax'dt + ay'dt + ak dt + af3dt- '
X
y
-~(5.142)
Fig. 5.38. Schematic presentation of wind wave generation in current: U - wind
speed; V - current speed
5 Wave Evolution in Non-uniform Currents in Deep Water
In this section an attempt is undertaken to describe the evolution of wave
parameters along the fetch in the presence of a spatially uniform current.
The wave development in a current is assumed to occur according to the
same laws as in its absence. The role of a current is attributed to a change
of the relative wind speed generating the waves, i.e. the speed measured in
the coordinate system connected with the current velocity. Furthermore, the
current velocity is added to the wave propagation velocity relative to a fixed
fetch changing the time of the wave interaction with the air flow.
In accordance with the general formulation of the problem (5.1), (5.2),
a constant wind speed U directed along the Ox axis is assumed to be prescribed at the horizon: z = 10m. The current velocity is equal to the value V.
Wind wave evolution along the fetch x in the established regime will be
investigated with the initial energy spectral density valueS= S (k, /3), being
prescribed at the boundary x = x 0 (see Fig. 5.38).
In order to describe wind wave development a coordinate system moving
with the current V will be used. The coordinates of a mobile system x'
and y' are connected with the initial coordinate system x' = x- Vt; y' = y.
The current is absent in the new coordinate system and the waves develop
according to similar laws as in calm water with the exception that the wind
speed relative to the water surface is equal to the value U - V.
The evolution of spectral density of the wind wave energy is assumed to
be described by (5.1) which can be written as follows:
dS _aS
aS dx'
aS dy' aS dk aS d/3 _ G'
dt- at+ ax'dt + ay'dt + ak dt + af3dt- '
X
y
-~(5.142)
Fig. 5.38. Schematic presentation of wind wave generation in current: U - wind
speed; V - current speed