Stokes drift

Remarks

For a small oscillation, expand $u(X(t), t) \approx u(x_0, t) + (\xi(t) \cdot \nabla) u(x_0, t)$ with $\xi(t) = \int_0^t u(x_0, s)\,ds$; this gives the standard approximation $u_S \approx \overline{(\xi \cdot \nabla) u}$. Example: take the linear deep-water wave with surface $\eta = a\cos(kx - \omega t)$ and $\omega^2 = gk$. Its Eulerian mean velocity below the troughs is zero, while the approximation gives the horizontal drift $u_S = \omega k a^2 e^{2kz}$ at depth $z < 0$. So the drift is of second order in the amplitude and decays twice as fast with depth as the wave motion. A spatially uniform oscillation $u(x, t) = U_0\cos(\omega t)\,e_1$ has zero Stokes drift, since the particle sees the same velocity as the fixed point; drift needs a velocity that varies in space. Non-example: the field $u(x, t) = (t, 0, \dots, 0)$ is not periodic in $t$, so the averages above, and the Stokes drift, are not defined for it.