Stokes drift is the difference between the mean velocity of fluid particles and the Eulerian mean velocity at fixed positions. It appears at second order because particles sample the oscillatory velocity at displaced positions.
A fluid displacement measured from an initial position must vanish at the initial time. Integrating a harmonic velocity therefore introduces phase-dependent constants. Mean-orbit labels absorb those constants and leave purely oscillatory displacements. The instantaneous displacement correction to velocity can differ under the two reference conventions, but its period mean gives the same leading Stokes drift.
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Stokes drift refers to the phenomenon in fluid dynamics where particles or fluid elements move in a direction slightly different from that of the wave propagation itself due to the non-linear nature of surface waves. This effect is most commonly observed in ocean waves. In a simplified explanation, as a wave travels through the water, it creates circular or elliptical orbital motions of water particles.