274 Gerbrand J. Komen
2) One does not go for a deterministic description, but merely seeks to make statements about average quantities. So one does not want to predict the height of
the sea surf ace at a particular position and a particular time, but one wants to
predict the probability that the waves have a particular height at a particular
place and time. Statistics is introduced by assuming that the different components are - and remain - independent:
(8)
3) It can now be shown that the spectrum F(k) satisfies the folIowing equation:
(9)
The form given here applies to deep water. A more general version with shallow water effects and wave-current interactions is given in Komen et al. (1994).
The left-hand side describes propagation; the source terms on the right describe
the interaction ofwave components with the atmosphere (Sin; the input), among
themselves (SnD and with the ocean (Sds; dissipation). Thanks to major research
efforts made over the last 30 years we now have suitable parametrisations for
the different source terms. In general, these are based on a mixture oftheoretical arguments and empirical observations. AlI source terms are functionals of
the wave spectrum, F. Dissipation and wave-wave interactions are nonlinear in
F. The wind input is linear in F, but it depends also on the local wind.
It seems appropriate to repeat the main assumptions that went into the derivation
of equation 9 beeause they presumably speeity its validity range. These assumptions were: 1) the geometrical optics approximations; 2) weak nonlinearity (in the
mean); and 3) (local) statistical homogeneity. These conditions seem to be well met
at sea, but they surely break down in situations in which the waves ehange on a
scale which is short with respect to their wavelength or period, sueh as may happen
close to the shore.
Equation 9 can be solved numericalIy by forward integration in time. To do this,
one needs to specify forcing wind fields at each grid point and for each time, as
well as initial and boundary conditions. Wind fields are usualIy taken from NWP
(numeric al weather predietion) models. Coasts are normalIy treated as fulIy
absorbing. Open boundaries may be avoided by a suitable ehoice of the integration
area. If this is not possible one may nest a limited area model within a larger scale
model.
The initial conditions need some discussion. Physically, the instability of air flow
over aflat sea oecurs at centimetre wave lengths. At these short length scales surface tension still plays a role. Aiso the physics of the interaetion between the
boundary layer and these short waves is rather different from the physics goveming
2) One does not go for a deterministic description, but merely seeks to make statements about average quantities. So one does not want to predict the height of
the sea surf ace at a particular position and a particular time, but one wants to
predict the probability that the waves have a particular height at a particular
place and time. Statistics is introduced by assuming that the different components are - and remain - independent:
(8)
3) It can now be shown that the spectrum F(k) satisfies the folIowing equation:
(9)
The form given here applies to deep water. A more general version with shallow water effects and wave-current interactions is given in Komen et al. (1994).
The left-hand side describes propagation; the source terms on the right describe
the interaction ofwave components with the atmosphere (Sin; the input), among
themselves (SnD and with the ocean (Sds; dissipation). Thanks to major research
efforts made over the last 30 years we now have suitable parametrisations for
the different source terms. In general, these are based on a mixture oftheoretical arguments and empirical observations. AlI source terms are functionals of
the wave spectrum, F. Dissipation and wave-wave interactions are nonlinear in
F. The wind input is linear in F, but it depends also on the local wind.
It seems appropriate to repeat the main assumptions that went into the derivation
of equation 9 beeause they presumably speeity its validity range. These assumptions were: 1) the geometrical optics approximations; 2) weak nonlinearity (in the
mean); and 3) (local) statistical homogeneity. These conditions seem to be well met
at sea, but they surely break down in situations in which the waves ehange on a
scale which is short with respect to their wavelength or period, sueh as may happen
close to the shore.
Equation 9 can be solved numericalIy by forward integration in time. To do this,
one needs to specify forcing wind fields at each grid point and for each time, as
well as initial and boundary conditions. Wind fields are usualIy taken from NWP
(numeric al weather predietion) models. Coasts are normalIy treated as fulIy
absorbing. Open boundaries may be avoided by a suitable ehoice of the integration
area. If this is not possible one may nest a limited area model within a larger scale
model.
The initial conditions need some discussion. Physically, the instability of air flow
over aflat sea oecurs at centimetre wave lengths. At these short length scales surface tension still plays a role. Aiso the physics of the interaetion between the
boundary layer and these short waves is rather different from the physics goveming