The coupling of wave drift and wind velocity profiles
The Stokes drift velocity profile due to Toba's equilibrium wave spectrum is shown to consist of a surface constant shear layer, an intermediate logarithmic layer and a deep exponentially decaying tail. On identifying the logarithmic layer with a wall boundary layer (which is justified a posteriori by showing that the major part of the energy dissipation by wave breaking occurs in the roughness sublayer), for a range of directionality (p) of the wave spectrum ½–2, Toba's constant (α) lies in the range 0.12–0.10 in good agreement with data. The roughness length for water (z0) of this profile has the Charnock form, z0 = au2*g–1 in which u* is the friction velocity in air, g is the acceleration of gravity and a is a constant of order unity determined by the condition that momentum transfer by wave breaking just supports the wind stress, and using this formula the transition from smooth to intermediate flow at which rippling commences is quite well estimated. The velocity profiles in air and water with respect to z0, are predicted from a similarity hypothesis to have the forms
u′ = u* ( + 1/ ln z/z0)
u = w* ( − 1/ ln z/z0) z > z0
where z is the distance from the sea surface, u′ and u are respectively the velocities in air and water, is Von Karman's constant, w* is the friction velocity in water, w* is the Stokes surface drift velocity, and u* is a wave speed centered in the equilibrium range. Observations in the open sea indicate that ∼ 12.
An alternate pair of profiles, also predicted from the similarity hypothesis, is,
u′ = us + u*/ ln z/z′0
u = us − w*/ ln z/z′0 z > z′0
where z′0 is the roughness length for air, and us ∼ 2w* is the surface current. Observations suggest that small surface drifters travel at speeds intermediate between w* and 2w*.
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