Role of wave breaking in the air–sea interaction - Breaking and Dissipation of Ocean Surface Waves - page 380

τ c = ρ w g ω r
0

π

k F (ω, θ)γ (ω, θ) d θ d ω (9.21)

− π

where is the fractional wave dissipation rate.

In (9.20) - (9.21) , the upper-limit of integration frequency

γ(ω,θ)

ω r is such that at the spectrum

tail ω>ω r the momentum flux goes straight to the current, bypassing the wave-growth

stage. This flux is given by (9.19) , and therefore the total flux transferred to the currents is
τ c + T r which provides the upper boundary condition.

Total energy exchange in Figure 9.8 is

g ω r
0

π

E = ρ w ω F (ω m , θ)β(ω, θ) d θ d ω. (9.22)

− π

Thus, the energy flux from waves to currents and from wind to currents is

E c = τ c · U c (9.23)

and

E

v =

T r · U c , (9.24)

respectively. In these expressions, k and U c are vectors of wavenumber and surface current
(see also Section 2.10 ).
Wave breaking and the corresponding energy dissipation produce turbulent energy flux
E T to the upper ocean:

g ω r
0

π

E T = ρ w ω F (ω m , θ)γ (ω, θ) d θ d ω − E c . (9.25)

− π

The last term E cT in the upper panel of Figure 9.8 is due to the wave-induced turbulence
(see Sections 7.5 and 9.2.2 ).
The bottom panel of Figure 9.8 provides a general scheme of the system of atmosphere-
ocean interactions, subdivided into eight layers with different dynamics. Zero level in
the figure corresponds to the wavy surface, and the wave boundary layer in the atmo-
sphere includes regions IV and V . The outer WBL extends to the height h λ p .Itis

here that the dominant-wave-induced pressure oscillations and therefore the wave-induced
momentum flux are felt. The 'dominant wave' scale in this scenario is determined by
ω<ω r .
The meaning of the

w ∼

0 . 1

ω>ω r play a role in
the internal WBL sublayer V . These short waves form the local roughness parameter and
the tangential stress (9.19) . The height of this layer

ω r scale now becomes clearer as frequencies

ζ t is linked to the oscillating surface

rather than to the mean water level.
Layer III is the part of WBL where stratification effects become important as described
by the Monin-Obukhov theory (see (3.19) and Section 3.1 ). In the case of neutral stratifi-
cation, the profile here is logarithmic, and the roughness parameter is defined by the waves
and the friction drag. Since h w is usually much smaller than the Monin-Obukhov scale L ,
stratification effects can be neglected in layers I and II . At the upper-boundary height h s
of the Monin-Obukhov layer III , Coriolis force has to be introduced.