= Solution
The dyed particle satisfies the <Lagrangian trajectory> equations
$$
\dot x=\varepsilon y\cos(x-t),
\qquad
\dot y=\varepsilon\sin(x-t),
\qquad
x(0)=y(0)=0.
$$
For the proposed approximation, $x=O(\varepsilon^2)$ and
$$
\dot y=-\varepsilon\sin t
=\varepsilon\sin(x-t)+O(\varepsilon^3).
$$
Also
$$
\dot x
=\varepsilon^2(\cos^2t-\cos t)
=\varepsilon y\cos(x-t)+O(\varepsilon^3).
$$
The initial conditions hold, verifying
$$
x=\varepsilon^2\left(\frac14\sin2t+\frac t2-\sin t\right),
\qquad
y=\varepsilon(\cos t-1)
$$
through order $\varepsilon^2$.
Over one period, the periodic terms return to their initial values while the secular term changes $x$ by $\varepsilon^2\pi$. Hence the dyed particle has <Stokes drift>
$$
\boxed{\overline v_{\rm particle}
=\left(\frac{\varepsilon^2}{2},0\right)}
$$
to this order, despite the zero Eulerian mean velocity.
Back to article page
