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 Kˆηm up to 0.4, local wavenumber, ηm = 5 m maximal elevation, while historically high waves like those in Hurricane Camille and the Draupner wave have Kˆηm = 0.49 and ηm = 18.5 m (Draupner wave).

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