Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 299

effective in the observed attenuation of swell, provided the relevant Reynolds numbers are
above their critical value. This is turbulence induced by the mean wave orbital motion in
the water, rather than produced by breaking or by friction against the air interface. Below,
we will follow Babanin ( 2006 ) and Babanin & Haus ( 2009 ) to introduce the phenomenon
of this turbulence.
Turbulence can be generated by a sheared fluid motion if a certain inertia-to-viscosity
ratio limit is overcome (e.g. Reynolds , 1883 ). While this should be true in a general case
(e.g. Kinsman , 1965 ), the motion due to surface water waves is generally regarded to be
irrotational (e.g. Young , 1999 ) and thus does not produce shear stresses and turbulence
directly. The conjecture of irrotationality in such theories is a consequence of the initial
assumption that the waves are free, that is have no viscosity and surface tension, and there-
fore cannot cause shear stresses (e.g. Komen et al. , 1994 ). Albeit small and negligible
from the point of view of many applications, water viscosity, however, is not zero. In the
presence of a strong exponential vertical gradient of the wave orbital velocity, and this
velocity being one or two orders of magnitude larger than the other velocities in the water
column usually attributed with the shear stresses, wave-caused shear stresses are unavoid-
able. Thus, the existence of wave-induced turbulence has been suggested ( Babanin , 2006 ).
In the ocean, which is always turbulent, the instability of pre-existing, even if negligibly
small vortices can also take energy from potential waves ( Benilov et al. , 1993 ).
Babanin ( 2006 ) proposed the concept of a wave-amplitude-based Reynolds number that
indicates a transition from laminarity to turbulence for the mean wave orbital motion. Esti-
mates of the critical wave Reynolds number (see (7.70) below for the definition) provided
an approximate value of

Re wave critical ≈

.

3000 (7.67)

Babanin ( 2006 ) tested this number on mechanically-generated laboratory waves and it was
confirmed. Once this number is used for ocean conditions when mixing due to heating and
cooling is less important than that due to the waves, quantitative and qualitative characteris-
tics of the ocean mixed layer depth (MLD) were shown to be predicted with a good degree
of agreement with observations. Testing the hypothesis against other known results in tur-
bulence generation and wave attenuation, including swell propagation across the Pacific,
was also conducted. Later, in further laboratory experiments it was demonstrated that, inter-
mittently, the wave-induced turbulence can appear at even lower Reynolds numbers, i.e.
Re wave ∼ 1000 ( Babanin & Haus , 2009 ; Dai et al. , 2010 ).

In the discussion below, the linear wave theory will be both employed for estimates and
criticised for not including viscosity. Therefore, some clarifications should be mentioned
at this stage to avoid what may seem a contradiction.
The wave motion considered is that due to the wind-generated waves at the ocean sur-
face. Nonlinear corrections to the mean orbital velocities and amplitudes are unimportant
from the point of view of turbulence generation, and the linear wave theory is used here
to scale the mean wave motion as a function of water depth. Although such an approach
is routinely used and is regarded to be reliable on average, it is necessary to comment that