4.3 Wave Energy Dissipation in Deep Water
135
The wave energy dissipation function is significantly different in various
frequency bands. Thus, it can be presented as a linear or quasi-linear dependence on the spectral density in the case of wave interaction with turbulent movement of water masses in the low-frequency band. As for the
high-frequency band with dissipation defined by crest wave breaking (being a strongly non-linear mechanism), its spectral density dependence should
also be non-linear.
The high-frequency spectral area determines, mainly, sea surface roughness, with the main impulse and energy flows from wind to waves. The highfrequency spectral components reach their maximum development within
a short period of time. As shown by numerical experiments, the presence
of an equilibrium interval determines the essential weak non-linear energy
flows to the low-frequency spectral area. For example, in order to create the
low-frequency spectrum development it is enough to solve the problem of
non-linear spectrum evolution with a constant equilibrium interval (4.47) as
a condition and without taking into consideration the influence of the wind.
The Phillips parameter, equal too:= 7.8 · w- 3 , provides more than 30 per
cent of energy supply, necessary for typical development of wind waves. The
same value, equal to o: = 20.0 · w- 3 , provides the usual wave development
without wind, due only to the action of non-linear energy transfer from the
equilibrium interval to the low spectral area.
Parametrization of high-frequency dissipation. Since there is no theory of wave energy dissipation at present, an attempt is made to obtain its
value, proceeding from general considerations. In order to do so, estimations
of the known components of the source function at large frequency values
can be used, i.e. when a» amax· It can also be assumed that the frequency
spectrum value should asymptotically approach the value of the isotropic
equilibrium interval: S00 (a) = o:g 2 a-5.
The existence of the asymptotic stationary spectrum range means that
physical mechanisms should be balanced properly. The sum of source functions should be equal to zero within the stationary spectrum interval. Integrating the source function (3.1) over directions one can obtain:
J Gcts(a,/3) d/3 ~- J Gin(a,/3) d/3- J Gni(a,/3) d/3. (4.55)
The estimation of the non-dimensional relaxation time (normalized by
dynamic velocity) of the wave process connected with energy dissipation is
as follows:
(4.56)
where the non-dimensional relaxation time is determined as:
j- 1 = ~* j G(a, {3) d/3 / j S(a, {3) d/3.
(4.57)
135
The wave energy dissipation function is significantly different in various
frequency bands. Thus, it can be presented as a linear or quasi-linear dependence on the spectral density in the case of wave interaction with turbulent movement of water masses in the low-frequency band. As for the
high-frequency band with dissipation defined by crest wave breaking (being a strongly non-linear mechanism), its spectral density dependence should
also be non-linear.
The high-frequency spectral area determines, mainly, sea surface roughness, with the main impulse and energy flows from wind to waves. The highfrequency spectral components reach their maximum development within
a short period of time. As shown by numerical experiments, the presence
of an equilibrium interval determines the essential weak non-linear energy
flows to the low-frequency spectral area. For example, in order to create the
low-frequency spectrum development it is enough to solve the problem of
non-linear spectrum evolution with a constant equilibrium interval (4.47) as
a condition and without taking into consideration the influence of the wind.
The Phillips parameter, equal too:= 7.8 · w- 3 , provides more than 30 per
cent of energy supply, necessary for typical development of wind waves. The
same value, equal to o: = 20.0 · w- 3 , provides the usual wave development
without wind, due only to the action of non-linear energy transfer from the
equilibrium interval to the low spectral area.
Parametrization of high-frequency dissipation. Since there is no theory of wave energy dissipation at present, an attempt is made to obtain its
value, proceeding from general considerations. In order to do so, estimations
of the known components of the source function at large frequency values
can be used, i.e. when a» amax· It can also be assumed that the frequency
spectrum value should asymptotically approach the value of the isotropic
equilibrium interval: S00 (a) = o:g 2 a-5.
The existence of the asymptotic stationary spectrum range means that
physical mechanisms should be balanced properly. The sum of source functions should be equal to zero within the stationary spectrum interval. Integrating the source function (3.1) over directions one can obtain:
J Gcts(a,/3) d/3 ~- J Gin(a,/3) d/3- J Gni(a,/3) d/3. (4.55)
The estimation of the non-dimensional relaxation time (normalized by
dynamic velocity) of the wave process connected with energy dissipation is
as follows:
(4.56)
where the non-dimensional relaxation time is determined as:
j- 1 = ~* j G(a, {3) d/3 / j S(a, {3) d/3.
(4.57)