Second-order trajectory in a sinusoidal velocity field ($\overline{u_L}=k{u}_0^2/(2\omega )$)

For prescribed one-dimensional velocity $u=u_0\sin (kx-\omega t)$, let $\alpha =k{X}_0$, $\theta =\alpha -\omega t$ and expand the Lagrangian trajectory as $x=X_0+u_{0}a+u_0^{2}b$. The Taylor expansion of $\dot{x}=u(x,t)$ gives $\dot{a}=\sin \theta$ and $\dot{b}=ka\cos \theta$. Initial position $X_0$ requires $a=(\cos \theta -\cos \alpha )/\omega$ andThe secular velocity $k{u}_0^2/(2\omega )$ is Stokes drift, even though the Eulerian time average vanishes. This fixed-time expansion requires $|k{u}_0/\omega |\ll 1$ and is not uniform once the secular displacement changes the phase by order unity.