Mass transport induced by nonlinear surface gravity waves

As waves pass, fluid elements experience not only periodic motion but also a movement in a direction of wave propagation (Stokes, G.G. (1847) On the Theory of Oscillatory Waves. Transactions of the Cambridge Philosophical Society, 8, 441-455). Defined as a difference between the average Lagrangian flow velocity of a particle and the average Eulerian flow velocity of the fluid, the Stokes drift entails, amongst others, the existence of wave-induced mass transport (van den Bremer TS, Breivik Ø. 2017 Stokes drift. Phil. Trans. R. Soc. A 376:20170104. http://dx.doi.org/10.1098/rsta.2017.0104). Knowledge of it is of high significance since it allows one to calculate tracer transport, for instance, plastic or oil pollution.
While operating in the Eulerian frame of reference, one should recognize that a fixed point in space in the vicinity of a free surface emerges and submerges under the water during wave motion. This phenomenon is called the emergence effect and it does impact the particle kinematics properties. Cieślikiewicz & Gudmestad developed a method of calculating the wave-induced mass transport for deterministic and random waves taking into account the emergence effect (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).
The goal of the study was to introduce numerical examples and verification of both deterministic and random wave cases presented by Cieślikiewicz & Gudmestad (1994) depending on wind wave parameters. Wolfram Mathematica software was used to carry out the calculations and draw figures. The wave energy spectrum was determined using the JONSWAP formula (Hasselmann, K., Barnett, T. P., Bouws, E., Carlson, H., Hasselmann, D. E., Kruseman, P., Meerburg, A., Mûller, P., Olbers, D. J., Richter, K., Sell, W., & Walden, H. (1973). Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Ergaenzungsheft Zur Deutschen Hydrographischen Zeitschrift, Reihe A., 12(A8), 1–95). The results show that the mass transport values for a representative deterministic wave agree with values for random waves. Therefore, the deterministic wave formulas may be used to initial estimate mass transport induced by random water wave field.