Drag of the sea surface

It is shown how the drag of the sea surface can be computed from
the wind speed and the sea state. The approach, applicable both
for fully developed and for developing seas, is based on
conservation of momentum in the boundary layer above the sea,
which allows one to relate the drag to the properties of the
momentum exchange between the sea waves and the atmosphere.

The total stress is split into two parts: a turbulent part and
a wave-induced part. The former is parameterized in terms of
mixing-length theory. The latter is calculated by integration of
the wave-induced stress over all wave numbers. Usually, the
effective roughness is given in terms of the empirical Charnock
relation. Here, it is shown how this relation can be derived from
the dynamical balance between turbulent and wave-induced stress.
To this end, the non-slip boundary condition is assigned to the
wave surface, and the local roughness parameter is determined by
the scale of the molecular sublayer.

The formation of the sea drag is then described for fully
developed and developing seas and for light to high winds.

For the Charnock constant, a value of about $0.018 - 0.030$ is
obtained, depending on the wind input, which is well within the
range of experimental data.

It is shown that gravity-capillary waves with a wavelength less
than 5 cm play a minor role in the momentum transfer from wind
to waves. Most of the momentum is transferred to decimeter and
meter waves, so that the drag of developing seas depends
crucially on the form of the wave spectrum in the corresponding
high wavenumber range.

The dependence of the drag on wave age depends sensitively on
the dependence of this high wavenumber tail on wave age. If the
tail is wave-age independent, the sea drag appears to be
virtually independent of wave age. If the tail depends on
wave age, the drag also does. There is contradictory evidence as
to the actual dependence. Therefore, additional experiments are
needed.