Wave breaking and the surface velocity field for three-dimensional water waves

Abstract

It is shown that there is an exact reduction from the full three-dimensional inviscid water wave problem with vorticity to a set of hyperbolic equations for the free surface horizontal velocity field. The key term in the reduced equations is g + (Dw/Dt)|S where g > 0 is the gravitational constant and (Dw/Dt)|S is the Lagrangian vertical acceleration at the surface. Vertical accelerations—both Eulerian and Lagrangian—at the surface are widely used as a diagnostic for wave breaking and this reduction gives a precise theoretical argument for how accelerations drive the surface dynamics. When the acceleration term is small, the surface equations reduce to the model equation proposed in Pomeau et al (2008 Nonlinearity 21 T61–T79, Proc. R. Soc. Lond. A 464 1851–66) which provides a universal law for spreading of the crest during wave breaking, and when the acceleration term is negative the equations can switch type (hyperbolic to elliptic) and solutions can fail to exist. Addition of surface tension leads to dispersive regularization of the surface equations.