Breaking wave
In physics, a breaking wave is a wave whose amplitude reaches a critical level at which some process can suddenly start to occur that causes large amounts of wave energy to be dissipated. At this point, simple physical models describing the dynamics of the wave will often become invalid, particularly those which assume linear behavior.
The most generally familiar sort of breaking wave is the breaking of water surface waves on a coastline. Because of the horizontal component of the fluid velocity associated with the wave motion, wave crests steepen as the amplitude increases; wave breaking generally occurs where the amplitude reaches the point that the crest of the wave actually overturns — though the types of breaking water surface waves are discussed in more detail below. Certain other effects in fluid dynamics have also been termed "breaking waves", partly by analogy with water surface waves. In meteorology, gravity waves are said to break when the wave produces regions where the potential temperature decreases with height, leading to energy dissipation through convective instability; likewise Rossby waves are said to break when the potential vorticity gradient is overturned. Wave breaking also occurs in plasmas, when the particle velocities exceed the wave's phase speed.
An analogous effect in electromagnetic waves, although not commonly termed wave breaking, is when the electric field amplitude associated with the wave is sufficient to cause dielectric breakdown and sparking (such as when sharp metal objects are placed in a microwave oven).
Breaking water surface waves
Breaking of water surface waves may occur anywhere that the amplitude is sufficient, including in mid-ocean. However, it is particularly common on beaches because waves are refracted towards the region of shallower water (because the phase velocity is lower there), and the shallow water also means that the same wave energy density gives rise to a greater surface amplitude.
There are four basic types of breaking water waves. They are spilling, plunging, collapsing, and surging.
Spilling Breakers
In this type of wave, the crest undergoes deformation and destabilizes, resulting in it spilling over the front of the wave. This wave tends to create a frothy appearance. It occurs most often on gentle beaches.
Plunging
Purportedly a rather dramatic form of break. The crest of the wave curls over and crashes into the base of the wave, creating a sizable splash.
It tends to happen most often on steep beaches.
Surging
On steeper beaches, a wave might advance up without breaking at all. It deforms and flattens from the bottom. The front of the wave advances up towards the crest, creating reflection.
Collapsing
Collapsing waves are a cross between plunging and surging, in which the crest never fully breaks, yet the bottom face of the wave gets steeper and collapses, resulting in foam.
Mathematics
During breaking, a deformation (usually a bulge) forms at the wave crest, either leading side of which is known as the "toe". Parasitic capillary waves are formed, with short wavelengths. Those above the "toe" tend to have much longer wavelengths. This theory is anything but perfect, however, as it's linear. There have been a couple non-linear theories of motion (regarding waves). One put forth uses a perturbation method to expand the description all the way to the third order, and better solutions have been found since then. As for wave deformation, methods much like the boundary integral method and the Boussinesq model have been created.
It has been accounted for, that the high-frequencies detail present in a breaking wave play a part in crest deformation and destabilzation. The same theory expands on this, stating that the valleys of the capillary waves create a source for vorticity. It is said that surface tension (and viscosity) are significant for waves up to $2m$ in wavelength.
These models are flawed, however, as they can't take into account what happens to the water after the wave breaks. Post-break eddy forms and the turbulence created via the breaking is mostly unresearched. Understandably, it might be difficult to glean predictable results from the ocean.
After the tip of the wave overturns and the jet collapses, it creates a very coherent and defined horizontal vertex. The plunging breakers create secondary eddies down the face of the wave. Small horizontal random eddides that form on the sides of the wave suggest that, perhaps, prior to breaking, the water's velocity is more or less two dimensional. This becomes three dimensional upon breaking.
The main vortex along the front of the wave diffuses rapidly into the interior of the wave after breaking, as the eddies on the surface become more viscous. Advection and molecular diffusion play a part in stretching the vortex and redistributing the vorticity, as well as the formation turbulence cascades. The energy of the large vortices are, by this method, is transferred to much smaller isotropic vortices.
Experiments have been conducted to deduce the evolution of turbulence after break, both in deep water and on a beach.