Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 298

role of the normal stresses is in this regard, it does not discard of course the swell-induced
shear stresses.
Measurements of air flow over swell are quite rare (see e.g. a recent study and review by
Soloviev & Kudryavtsev , 2010 ), and Ardhuin et al. ( 2009a ) relied on analogy with the wall
boundary layer. They argued that the air-sea boundary layer should have the same proper-
ties if the waves are treated as surface undulations, but magnitudes of the orbital velocity
u orb and displacement a orb should be doubled in the respective estimates of Reynolds num-
bers, compared to the fixed boundary ( Collard et al. , 2009 ). For the boundary-layer flow to
be turbulent then, the Reynolds numbers have to be:

4 u orb a orb
ν a

10 5

Re = > (7.63)

where ν a is the air viscosity (e.g. Jensen et al. , 1989 ).

If the flow is laminar, strong shear above the interface makes the air viscosity important,
and the dissipation coefficient is expected to be (e.g. Dore , 1978 ; Collard et al. , 2009 ):

2 5 / 2

2

2 ρ a
ρ w

T

1
gc g

α ν = ν a .

(7.64)

10 − 5 m 2 s − 1 ), At ambient conditions ( ν a = 1 . 4 · α ν is a function of T only, and for observed

10 − 8 m − 1 down swell periods of T = 13-19 s it is expected to vary in the range from 2 . 2 ·

10 − 9 m − 1 , respectively.
For the turbulent boundary layer, the energy decay can be estimated based on knowledge
of the temporal dissipation coefficient

to 5 . 8 ·

2
gT 2

ρ a
ρ w

4 π

α time = c g α = f e u orb . (7.65)

Here, f e is an analogy of the smooth-wall friction factor f w and is expected to be of the
order of f e =

008 (e.g. Jensen et al. , 1989 ).
Thus, Ardhuin et al. ( 2009a ) interpreted their observations of swell decay in terms of
such a mechanism of swell friction against the air. The ratio of observed values of

0 . 002 − 0 .

over
α ν (7.64) ranged from 1 to 28, indicating that both laminar and turbulent behaviours were
present. For the latter, curve fitting needed the friction constant in the range

α

− 0 . 001 ≤

019 which is therefore, with the median value of 0.007, close to the expectations
for a smooth surface. Transition occurs at

f e ≤ 0 .

10 4 Re ≈ 5 · (7.66)

which is another swell-attenuation threshold to keep in mind, in addition to the steepness
(7.61) - (7.62) .
There is, however, another phenomenon which also has laminar-to-turbulent transition
and also leads to dissipation of energy of waves, including swell. That is, it may also be