Abstract: A
generalization of the quasi-laminar model is constructed by averaging the
linearized equations of motion for a turbulent shear ow in the direction
parallel to the crest of the Stokes wave. It is shown that the resulting mean
momentum transfer comprises (i) a singular part, which is proportional to
product of the velocity-profile curvature and the mean square of the
wave-induced vertical velocity in the critical layer, where the mean wind speed
is equal to the wave speed; (ii) a vertical integral of mean product of the
vertical velocity and the vorticity w
where w is the wave-induced perturbation in the total
velocity along a streamline of theaveraged motion; (iii)
the perturbation in the mean turbulent shear stress at the air-water interface.
A closure model, based on Townsend [38], is constructed for the specification of
turbulent Reynolds stresses. The resulting equation together with its
corresponding boundary conditions is solved asymptotically. The growth rate of
Stokes wave is then calculated from the derived expressions for the momentum
flux for slow wind-wave regime . The result of
calculations for the energy transfer parameter agrees well with the numerical
integration of the Reynolds-stress transport equations over a Stokes wave (Sajjadi
[28]), also with the numerical calculations of Ierley and Miles [12], and
provides further evidence to support the earlier postulation of Belcher and Hunt
[1] for rapid distortion theory of turbulence over water waves. Furthermore, it
is demonstrated that (1) for turbulent flow over surface waves there is no
evidence that critical-layer mechanism is responsible for growth of surface
waves, and (2) in comparing the results with field data there is clearly evidence
of presence of higher harmonics in surface waves. An expression is derived for
the energy transfer parameter from wind to surface wave, valid for strong wind such as those due
to tropical cyclone (up to category 2), which can be incorporated in spectral
wave models.
Keywords and phrases: air-sea interactions, stokes wave, turbulence, tropical cyclone.
