On the Dependence of the Equilibrium Range of Sea Wave Spectra with Wave Age

Donelan and Pierson have proposed a semiphysical model of the equilibrium sea wave spectrum, based upon a parameterization of wave growth and dissipation terms. Their model is applicable for fully developed seas only. In the framework of Donelan and Pierson's approach, this paper explores the dependence of the equilibrium spectrum upon wave age. To this end, we examine how the dissipation through wave breaking is expected to vary with wave age, according to the approach proposed by Longuet-Higgins in 1969. The constraint imposed by Longuet-Higgins' theory requires an increase of the equilibrium spectrum F(k,0) in the wind direction with increasing inverse wave age U/C_p. This is in accordance with Banner's empirically deduced statement that F(k,0) is proportional to (U/C_p)^0.5 in the equilibrium range. Our inferred F(k,0) tends to increase more or less linearly with U/C_p (we find F(k,0) proportional to 1 + 0.25(U/C_p - 0.83), rather than through a power law. If a power law is fitted we obtain F(k,0) approximately proportional to (U/C_p)^0.35 for the range 0.83 < U/C_p > 5. Finally, the roughness length of the air-sea interface is inferred from our modelled spectrum through integration of the form drag over wave number under rough conditions. This shows a wave age dependence that is compatible with measurements of wind stress performed in the field at various wave ages.