A breaking wave carries a record of the bottom beneath it. New modeling by Scripps Institution of Oceanography researchers turns that familiar coastal relationship into measurements, predicting how the slope of the seafloor and the height of an incoming wave determine the size and angle of its overturning crest.
UC San Diego detailed the work Sept. 2 after its publication in the Journal of Fluid Mechanics. Doctoral student Kanoa Pick and oceanographer Falk Feddersen used a two-dimensional, fully nonlinear potential-flow model to simulate solitary waves as they entered shallower water, steepened and pitched forward.
The model produced a consistent geometry. Steeper bottom slopes generated larger overturns that projected more horizontally, with thicker jets extending from the crest. Gentler slopes produced smaller overturns inclined more sharply toward the surface, and their jets were thinner. Offshore wave height changed the pattern alongside the slope.
That description goes beyond the usual division between spilling and plunging breakers. Those labels identify broad behavior. The new relationships estimate the area of the overturn, the size of the projecting jet, its angle and its proportions. In effect, the equations describe not only whether a wave breaks forcefully but what shape the breaking portion takes before impact.
The distinction matters after the crest falls. A thicker jet starts higher and carries more potential energy. When it strikes the water, that energy becomes turbulence and bubbles. Pick and Feddersen found jet potential energy was strongly related to jet area, tying the visible shape above the surface to mixing below and around it.
Turbulence is one of the engines of sand suspension in the surf zone. Grains lifted from the bed can then be carried shoreward, seaward or along the coast by waves and currents. A better description of overturning geometry therefore adds a physical link between seabed slope, an individual breaker and the repeated motion that reshapes beaches.
The calculation is controlled by design. Solitary waves simplify the more complicated trains of swell and wind waves arriving at a real shoreline, and a two-dimensional model leaves out alongshore variation. The study also did not test wind effects, which earlier research has shown can alter where and how waves break. Its value is isolating relationships that can be examined before more processes are added. Nor does one set of equations turn a beach profile into a permanent forecast. Nearshore slopes change as storms remove sand, calmer seasons return it and engineered structures redirect transport. The same incoming wave can encounter a different bottom weeks later. That moving boundary is precisely why geometry grounded in both wave height and bathymetry is useful.
The paper was published online July 27 and appears in the journal's Aug. 10 volume. The Sept. 2 university release brought the findings into a local coastal frame. Funding came from the Mark Walk Wolfinger Foundation and the Mark Walk Wolfinger Surfzone Research Fund, created in memory of a La Jolla surfer. The work leaves a measured chain from slope to crest, crest to turbulence and turbulence to the sand moving under the next wave.