Material derivative ($D/Dt$)

The material derivative follows a moving continuum parcel:

Convective acceleration is the spatial part $(\mathbf{u}\cdot \nabla )\mathbf{u}$ of the material derivative of fluid velocity. It can remain nonzero in steady flow, because a moving particle passes through positions with different velocities. The identity $(\mathbf{u}\cdot \nabla )\mathbf{u}=\nabla (|\mathbf{u}{|}^2/2)-\mathbf{u}\times (\nabla \times \mathbf{u})$ relates it to vorticity.

The material acceleration is the rate of change of velocity following a continuum parcel:

An Eulerian coordinate labels a fixed position in space at which continuum fields are observed.

A Lagrangian coordinate labels a material particle and remains attached to it as the continuum moves.

An Eulerian time average holds spatial position fixed while averaging a time-dependent field:

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.