1.5 Kinetic Equation of the Evolution of the Wind Wave Spectrum
25
wind wave theory is to determine this relation with a help of the dynamic
equations describing the "water-air" system. Sea waves and the wind speed
field are of a random character. That is why the main problem of wind wave
theory (on the level of second moments) can be formulated specifically as the
problem of determining the surface wave spectrum by the statistical characteristics of a random wind speed field of the atmospheric turbulent boundary
layer.
1.5 Kinetic Equation of the Evolution
of the Wind Wave Spectrum
The description of the formulation of the hydrodynamic problem of wind
waves is given in Sect. 1.1. Apart from the complexity of solving this problem, there is an additional difficulty in wind wave modelling, connected with
the random character of wind waves. That is why any attempt to solve the
problem of the numerical simulation of wind waves on real ocean scales with
the help of the deterministic hydrodynamic formulation is impractical. The
number of degrees of freedom of the system is practically unlimited.
Significant achievements in the numerical simulation of wind waves are
connected with the kinetic equation describing the evolution of the wave
spectrum under the external field action with the wind field being an external field action. The phenomenological description of the equation is as
follows. The statistically space-homogeneous, stationary, random water-air
interface ry(r, t) is considered in the previous Sect. 1.4. For describing the
random field evolution "slow" coordinates and time are introduced. Field
variability scales essentially exceed the typical lengths and periods of waves.
The corresponding generalization of the statistically homogeneous and
stationary field can be achieved by considering the local spectra, depending
on the slow coordinates r e and the time te (the index "e" is omitted below):
S = S(k,w,r,t).
(1.52)
Similarly, the wave action spectrum can be written as:
N = N(k,w,r,t) = Sja-.
(1.53)
Now the general phenomenological equation for the spectral density evolution
of the wave action can be formally written as the transfer equation:
aN aN
.
a
.
a
.
8t + a:;:(Nr) + ak (Nk) +ow (Nw) =G.
(1.54)
If the derivatives r, k, w are written in the form of Hamiltonian equations:
dr
81{
dk
81{
dw
dt
ak '
dt
or'
dt
81{
at '
(1.55)
25
wind wave theory is to determine this relation with a help of the dynamic
equations describing the "water-air" system. Sea waves and the wind speed
field are of a random character. That is why the main problem of wind wave
theory (on the level of second moments) can be formulated specifically as the
problem of determining the surface wave spectrum by the statistical characteristics of a random wind speed field of the atmospheric turbulent boundary
layer.
1.5 Kinetic Equation of the Evolution
of the Wind Wave Spectrum
The description of the formulation of the hydrodynamic problem of wind
waves is given in Sect. 1.1. Apart from the complexity of solving this problem, there is an additional difficulty in wind wave modelling, connected with
the random character of wind waves. That is why any attempt to solve the
problem of the numerical simulation of wind waves on real ocean scales with
the help of the deterministic hydrodynamic formulation is impractical. The
number of degrees of freedom of the system is practically unlimited.
Significant achievements in the numerical simulation of wind waves are
connected with the kinetic equation describing the evolution of the wave
spectrum under the external field action with the wind field being an external field action. The phenomenological description of the equation is as
follows. The statistically space-homogeneous, stationary, random water-air
interface ry(r, t) is considered in the previous Sect. 1.4. For describing the
random field evolution "slow" coordinates and time are introduced. Field
variability scales essentially exceed the typical lengths and periods of waves.
The corresponding generalization of the statistically homogeneous and
stationary field can be achieved by considering the local spectra, depending
on the slow coordinates r e and the time te (the index "e" is omitted below):
S = S(k,w,r,t).
(1.52)
Similarly, the wave action spectrum can be written as:
N = N(k,w,r,t) = Sja-.
(1.53)
Now the general phenomenological equation for the spectral density evolution
of the wave action can be formally written as the transfer equation:
aN aN
.
a
.
a
.
8t + a:;:(Nr) + ak (Nk) +ow (Nw) =G.
(1.54)
If the derivatives r, k, w are written in the form of Hamiltonian equations:
dr
81{
dk
81{
dw
dt
ak '
dt
or'
dt
81{
at '
(1.55)