1.5 Kinetic Equation of the Evolution of the Wind Wave Spectrum
27
Substituting the values q 1 and p 1 from the Hamiltonian equation (1.55)
gives
df = aj {Hf}
dt
at +
'
(1.61)
where the designation:
(1.62)
is introduced.
The expression (1.62) is called the Poisson brackets for the values Hand f.
Thus, the kinetic equation (1.54) can also be considered as a sum of a nonstationary term aN I at with the corresponding Poisson brackets for N and w.
The functions of dynamic variables that remain constant in the motion
of the system are usually called motion integrals. It can be seen from (1.61)
that the condition for f to be a motion integral ( df I dt = 0) can be written
in the form:
aj
at+ {Hf} = o.
(1.63)
If the motion integral is not dependent explicitly on time, then {H!} = 0,
i.e. the Poisson brackets with the Hamiltonian function must be equal to zero.
An important feature of the Poisson brackets is that iff and g are two motion
integrals, then the brackets composed of them are also the motion integrals
{fg} (Poisson's theorem).
For the geometric interpretation of the dynamical system the notion of
phase space is often used, as the space of 2s dimensions where coordinate
axes are the values of the s coordinates and s momentum of the system.
Point motion in the phase space depicts a corresponding line, called the phase
trajectory. The entire area is moved, assuming that every point of the area
of the given phase space propagates with time according to the equations
of motion of the considered dynamical system. Its volume can be proved to
be invariant J dF = const. (Lantsosh, 1965). This statement (the Liouville
theorem) follows directly from the phase volume invariants under canonical
transformations and from the fact that variations under motion themselves
can be considered as canonical transformations.
In the mathematical modelling of wind waves the traditional transfer
from the hydrodynamic equations (1.5)-(1.13) to the kinetic equation (1.54)
is as follows (Hasselmann, 1979; Yefimov&Polnikov, 1991). The hydrodynamic fields are assumed to be random functions, written in the form of the
Fourier (or Fourier-Stieltjes) integral. The equations of motion are presented
for spectral components of water surface elevation according to the hydrodynamic equations of the homogeneous field approximation. Using the statistical formulas for closing the higher moments of the Fourier components the
solution leads to the evolution equation of the wind wave field spectrum S.
It should be noted that the initial formulation of the hydrodynamic problem
27
Substituting the values q 1 and p 1 from the Hamiltonian equation (1.55)
gives
df = aj {Hf}
dt
at +
'
(1.61)
where the designation:
(1.62)
is introduced.
The expression (1.62) is called the Poisson brackets for the values Hand f.
Thus, the kinetic equation (1.54) can also be considered as a sum of a nonstationary term aN I at with the corresponding Poisson brackets for N and w.
The functions of dynamic variables that remain constant in the motion
of the system are usually called motion integrals. It can be seen from (1.61)
that the condition for f to be a motion integral ( df I dt = 0) can be written
in the form:
aj
at+ {Hf} = o.
(1.63)
If the motion integral is not dependent explicitly on time, then {H!} = 0,
i.e. the Poisson brackets with the Hamiltonian function must be equal to zero.
An important feature of the Poisson brackets is that iff and g are two motion
integrals, then the brackets composed of them are also the motion integrals
{fg} (Poisson's theorem).
For the geometric interpretation of the dynamical system the notion of
phase space is often used, as the space of 2s dimensions where coordinate
axes are the values of the s coordinates and s momentum of the system.
Point motion in the phase space depicts a corresponding line, called the phase
trajectory. The entire area is moved, assuming that every point of the area
of the given phase space propagates with time according to the equations
of motion of the considered dynamical system. Its volume can be proved to
be invariant J dF = const. (Lantsosh, 1965). This statement (the Liouville
theorem) follows directly from the phase volume invariants under canonical
transformations and from the fact that variations under motion themselves
can be considered as canonical transformations.
In the mathematical modelling of wind waves the traditional transfer
from the hydrodynamic equations (1.5)-(1.13) to the kinetic equation (1.54)
is as follows (Hasselmann, 1979; Yefimov&Polnikov, 1991). The hydrodynamic fields are assumed to be random functions, written in the form of the
Fourier (or Fourier-Stieltjes) integral. The equations of motion are presented
for spectral components of water surface elevation according to the hydrodynamic equations of the homogeneous field approximation. Using the statistical formulas for closing the higher moments of the Fourier components the
solution leads to the evolution equation of the wind wave field spectrum S.
It should be noted that the initial formulation of the hydrodynamic problem