Role of wave breaking in the air–sea interaction - Breaking and Dissipation of Ocean Surface Waves - page 391

taken into account (that is the wave stays symmetric over its period/length). Such are the
various approaches which led, for example, to the concept of the wave-caused turbulent
diffusion ( Jacobs , 1978 ; Qiao et al. , 2004 , 2010 ; Gayer et al. , 2006 ).
The asymmetry of the wave-induced water-particle motion, as a residual effect of the
gradual (i.e. non-breaking) dissipation, can be included in the mean flow, and this possi-
bility has also been considered before (e.g. Ardhuin & Jenkins ( 2006 ) obtained it by using
interaction of the Stokes drift and wind-induced currents). Pleskachevsky et al. ( 2011 )
argued that both symmetric and asymmetric effects are important and must be taken into
account. In the models, the overall effect on turbulent diffusivity can be obtained based
on an idealised solution for symmetric wave profiles (e.g. linear wave theory), if the dis-
sipation of the wave is known and can be taken into account independently (e.g. from
measurements or wave modelling which includes the dissipation term).
In fluid flows, the turbulence is described by the random fluctuations of velocity u i (i.e.
see (9.2) above) for i

z directions. The instantaneous velocity u i is generally (also
due to wave motion) presented as

= x , y ,

u i u i = u i + (9.28)

where u i is the mean fluid velocity (for wave motion, the integration increment should be
dt

T in order to capture variations of u i ). By definition, the time-averaged fluctuations
u i are zero, and therefore at the time scale of wave period T this turbulence is 'invisible',
but it does influence turbulent viscosity ν t and turbulent diffusion.

If the turbulence intensity (that is ratio u i /

u i ) is less than 1% of the mean flow, this
is usually classified as a low-level turbulence, 1-5% corresponds to a medium-level tur-
bulence intensity and greater than 5% is a high-level turbulence case. Since wave orbital
velocities can exceed 3-5m s, even the low-turbulence condition will signify mean-orbit

fluctuations of the order of several cm/s and can produce turbulent mixing orders of mag-
nitude higher than the mixing due to molecular viscosity or due to background circulation
currents (for depths of 30-50m, these currents are about 0-0

/

. 2m / s in the lower water

layers of the North Sea).
The mean orbital ve loc ity u orb , i is a solution of linear theo ry, a f unction of H s and T and
is a particular case of u i in (9.28) . Input of H s and T into u orb , i (which is how the wave
information is transferred in the circulation models) presents static values for the wave
height, period and length, whereas these properties change dynamically: such input does
not include information about wave energy dissipation of any kind. Therefore, turbulent
fluctuatio ns an d damping effects due to them (loss of energy) are not presented in the
idealised u orb , i which is typically used for the techni cal ap plications.
In reality, even at laminar motion the idealised u orb , i will be slowing down due to
molecular water viscosity

ν

. The difference can be denoted as u Ti :

u Ti = u orb , i − u i (9.29)

and characterises deviation of the real mean-orbital wave motion in viscous fluid from
ideal non-viscid fluid. While WRN (7.70) is below critical value (7.67) , the wave-induced