This paper was written by P. B. Smit, T. T. Janssen, and T. H. C. Herbers.
Estimation of second-order, near-surface wave kinematics is important for interpretation of ocean surface remote sensing and surface-following instruments, determining loading on offshore structures, and understanding of upper-ocean transport processes. Unfortunately, conventional wave theories based on Stokes-type expansions do not consider fluid motions at levels above the unperturbed fluid level. The usual practice of extrapolating the fluid kinematics from the unperturbed free surface to higher points in the fluid is generally reasonable for narrowband waves, but for broadband ocean waves this results in dramatic (and nonphysical) overestimation of surface velocities. Consequently, practical approximations for random waves are at best empirical and are often only loosely constrained by physical principles. In the present work, the authors formulate the governing equations for water waves in an incompressible and inviscid fluid, using a boundary-fitted coordinate system (i.e., sigma or s coordinates) to derive expressions for near-surface kinematics in nonlinear random waves from first principles. Comparison to a numerical model valid for highly nonlinear waves shows that the new results 1) are consistent with second-order Stokes theory, 2) are similar to extrapolation methods in narrowband waves, and 3) greatly improve estimates of surface kinematics in random seas.
