On wave breaking and the equilibrium spectrum of wind-generated waves

A theoretical calculation is made of the loss of energy by wave breaking in a random sea state in terms of the spectral density function. In the special case of the equilibrium spectrum F(σ) = αg2σ-5 the proportion ɷ of energy lost per mean wave cycle is found to be given by ω ≑ e-1/8α irrespective of the low-frequency cut-off in the spectrum. Assuming that in the equilibrium state the loss of energy by breaking is comparable to that supplied by the wind, one can estimate the constant α in terms of the drag coefficient of the wind on the sea surface. It is found that α≑ -1/8/ln[1600C3/2(ρair/ρwater)]. Taking a representative value of C one finds α ≑ 1.3 x 10-2, which falls within the range of observed values of α. The above equation for α is rather insensitive to the various assumptions made in the analysis. There is some evidence, derived from observation, that α may not in fact be quite constant, but may decrease slightly as the wave age (gt/U) or the non-dimensional fetch (gx/U2) is increased. It is suggested that the drag coefficient may behave similarly.