EULER-LAGRANGE TRANSFORMATION AND THE STOKES DRIFT IN THE PRESENCE OF OSCILLATORY AND RESIDUAL CURRENTS

It is shown that in an Eulerian velocity field consisting of a spatially varying component on which a uniform oscillatory component is superimposed, the residual mass transport velocity at a certain place is in general different from the residual Lagrangian velocity of a parcel labeled at that place. The approximation of the rigorous Euler-Lagrange transformation leading to the familiar derivation of the Stokes' drift is examined. The latter is a two-fold truncation of expansions of the former in a series of equations and a Taylor series of derivatives. The truncation of the latter series breaks down in cases of considerable spatial variability of the current velocity field, whereas the truncation of the former series may still be used if the uniform current velocity component energetically dominates the spatially varying component. It is then shown that the error introduced by substituting the local mass transport velocity for the residual Lagrangean velocity depends on the ratio of the amplitude of horizontal displacement, due to the uniform oscillatory component, to the integral length scale of the spatially varying component of the velocity field. Errors are largest if this ratio is of order 1. Such a situation may occur particularly in shallow tidal areas of complicated geometry. © 1979.