Lagrangian Motions in Simple Kinematic Oscillatory Flow Field

A kinematic approach is taken to examine analytically the Euler‐Lagrange transformation and the basic behavior of fluid parcel trajectories associated with a monochromatic oscillatory current field having progressive wave characteristics. This study examines (1) The dependency of the Lagrangian motions on the initial release conditions; and (2) the distinction between the Eulerian and Lagrangian time scales and its implications on the residual Lagrangian flow field. While the instantaneous trajectory of a labeled parcel depends strongly on the initial conditions, the appropriate period in the Lagrangian framework $T_{\mathrm{Lag}}$ and the residual Lagrangian drift over such a period are independent of the initial conditions. Because $T_{\mathrm{lag}}$ and the Eulerian time scale $T_{\mathrm{eu}}\text{,}$ defined as the period of the oscillatory Eulerian current, may be quite different, the residual Lagrangian drift over a complete $T_{\mathrm{eu}}$ may show strong dependency with the initial release conditions. It was also found that while the Eulerian current is monochromatic, the Lagrangian trajectory exhibits multiple harmonics, with components at both higher and lower frequencies relative to the fundamental frequency.