Orbital Velocity in Spatial Ocean Wave Elevation Measurement: Nonlinear Computation and Approximation

A data base of the recent Gulf of Tehuantepec experiment, of measured elevation of ocean surface waves in strong wind conditions, over swaths that are 6 km long by 200 m wide, is used as input for computation of the orbital velocity of the waves. Linear, quadratic and cubic approximations are compared to computations using a fully nonlinear-dispersive method in three dimensions, where the effect of a current is accounted for. A cancellation among the leading order quadratic terms is observed, and explains why a linear approximation when corrected by evaluation at the actual free surface provides a good estimate of the orbital velocity. Highly accurate results are obtained by including terms up to the cubic nonlinearity. The waves examined here have an estimated nonlinearity Kˆη_m up to 0.4, Kˆ local wavenumber, η_m = 5 m maximal elevation, while historically high waves like those in Hurricane Camille and the Draupner wave have Kˆη_m = 0.49 and η_m = 18.5 m (Draupner wave).