Mass transport induced by 5-th order nonlinear water waves

Linear theory of water waves is reasonable to use only in the case of small waves. As the waves’ steepness increases, nonlinear effects start to play a role too significant to be neglected. One of the widely used and commonly accepted methods of calculating water wave kinematics is Fenton’s method (Fenton 1985). It allows one to find free surface elevation and velocity potential up to 5^th order in wave steepness.

In case of wave-related phenomena, among quantities of interest is wave-induced mass transport as its knowledge is necessary to find tracer transport, like oil pollution, algae bloom, or plastic. Cieślikiewicz & Gudmestad (1994) introduced a method of calculating mass transport induced by harmonic and random water waves. A key contribution in this study was taking into account the emergence effect: in the Eulerian frame of reference, a fixed point in space in the vicinity of the free surface emerges and submerges under the water.

 

The primary objective of the present study was to integrate Fenton’s kinematics into the Cieślikiewicz & Gudmestad methodology for more accurately calculating the wave-induced mass transport. I used the perturbation scheme and the technique of transformation of random variables (Huang et al. 1983). The results demonstrate strong agreement with previous approaches.

 

Cieślikiewicz, W. & Gudmestad, O. T. (1994). Mass transport within the free surface zone of water waves. Wave Motion, 19(2), 145–158. https://doi.org/10.1016/0165-2125(94)90063-9

Fenton, J. D. (1985). A Fifth-Order Stokes Theory for Steady Waves. Journal of Waterway, Port, Coastal, and Ocean Engineering, 2(111), 216–234

Huang, N. E., Long, S. R., Tung, C.-C., Yuan, Y., & Bliven, L. F. (1983). A Non-Gaussian Statistical Model for Surface Elevation of Nonlinear Random Wave Fields. Journal of Geophysical Research, 88(C12), 7597–7606; Papoulis, A., & Pillai, S. U. (2002). Probability, Random Variables and Stochastic Processes. McGraw-Hill