Definitions for wave breaking - Breaking and Dissipation of Ocean Surface Waves - page 18

f = f p ± 0 . 3 f p of a Black Sea record with peak frequency f p = 0 . 25 Hz. Frequency

f (inverse period f = 1 / T ) of each wave in the record is plotted versus steepness

08, squared boxes
correspond to waves exhibiting whitecapping. Solid lines indicate the peak frequency
(horizontal line) and the breaking threshold

of the wave; + indicates all waves, ∗ are those waves with > 0 .

08 (vertical) below which even two-
dimensional waves are not expected to break ( Babanin et al. , 2007a , 2009a , 2010a , see also
Chapter 5 ).
A significant number of waves, however, detected visually as breaking, are below the
threshold. Some of them have a steepness as low as

= 0 .

03 and still exhibit whitecapping.
This is the subsiding phase, still detected as breaking if relying on whitecap observations.
At the other end, out of two waves with

= 0 .

27 which is very high for field waves by any
standards, one wave does not exhibit whitecaps and another does, that is, the first one is on
its way up to limiting steepness (incipient breaker) and another is on its way down while
collapsing (developing breaker). In Figure 8.6 , out of 6347 individual waves considered
54 were in this high-steepness range (

≈ 0 .

3), half were breaking and half were
not. These observations highlight uncertainties and ambiguities in existing definitions of
breaking rates and the need for classification of the wave-breaking phases.

= 0 . 25-0 .

2.3 Residual breaking

The last, residual stage is formally introduced here following Rapp & Melville ( 1990 )as
the phase of the breaking process when the whitecap is already left behind, but the under-
water turbulent front is still moving downstream. During this fourth stage of breaking,
whitecaps are decreasing in size as entrained bubbles rise to the surface, but spatial evolu-
tion of mixing continues as the turbulent front continues to move ( Rapp & Melville , 1990 ;
Melville & Matusov , 2002 ).
This stage will not be detectable by either wavelet or similar analytical methods based
on interpretation of surface elevation, or by means of whitecapping-oriented measurement
approaches. Rapp & Melville ( 1990 ) used dye to investigate propagation of the induced
turbulence following a breaking event. They found that propagation of the turbulent front
continues for many wave periods, although at a much slower speed once the breaking
wave has passed by. The horizontal extent L of the turbulent mixing reaches kL ≈ 5 and

the vertical extent D reaches kD 1, where k is the wavenumber of the breaking wave

and positive values of the turbulent plume mean propagation forward and upward.
Rapp & Melville 's ( 1990 ) experiments were conducted with wave breaking caused by
superposition of linear waves. Post-breaking dynamic effects and outcomes of such break-
ing can be very different to those resulting from other breaking mechanisms, for example,
breaking brought about as a result of nonlinear modulation in the wave train or even super-
position of nonlinear waves. Therefore, it is actually not clear whether the residual stage
is a general feature of wave breaking and will persist in cases of breaking other than those
caused by linear superposition.

≈−