Lagrangian trajectory

A Lagrangian trajectory follows one marked fluid particle and solves

A fluid element is an infinitesimal material parcel followed along a Lagrangian trajectory.

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.