# 21: Appendix J- The Stokes Drift

In the limit of small amplitude waves, fluid parcels oscillate in place (section 8.2.5), i.e., there is no overall current associated with the wave. But at finite amplitude, waves drive a nonzero mean motion called the Stokes drift. Floating objects such as driftwood are carried ashore by the Stokes drift. The essential reason for the Stokes drift is that the amplitude of the particle ellipses (Figure 8.4) increases toward the surface. As a result, a particle moves slightly faster at the top of its ellipse than at the bottom.

Here we will estimate the speed of the Stokes drift using the small-amplitude theory developed in chapter 8. To begin with, write the horizontal velocity as a small perturbation $$\{x^\prime ,z^\prime \}$$ from its value at $$\{x_0,z_0\}$$:

\begin{aligned} u(x, z, t) &=u^{0}+u^{\prime} \\ &=u^{0}+u_{x}^{0} x^{\prime}+u_{z}^{0} z^{\prime}, \end{aligned}

where the superscript 0 denotes evaluation at $$\{x_0,z_0\}$$. Now substitute from Equations 8.2.37, 8.2.25, and 8.2.44:

$u^{\prime}=k \frac{\left(U^{0}\right)^{2}}{\omega} \sin ^{2}\left(k x_{0}-\omega t\right)+U_{z}^{0} \frac{W^{0}}{\omega} \cos ^{2}\left(k x_{0}-\omega t\right).$

We now average this over one wave period, $$2\pi/\omega$$. The averages of the squared sine and cosine functions over a period are both equal to 1/2. Noting also that $$U^0_z = kW^0$$, we have

\begin{aligned} u^{S}=\frac{\omega}{2 \pi} \int_{0}^{2 \pi / \omega} u^{\prime} d t &=\frac{1}{2} \frac{k}{\omega}\left[\left(U^{0}\right)^{2}+\left(W^{0}\right)^{2}\right] \\ &=\omega k \eta_{0}^{2} \frac{\cosh ^{2} k\left(z_{0}+H\right)+\sinh ^{2} k\left(z_{0}+H\right)}{2 \sinh ^{2} k H} \end{aligned}

The reader may check that $$w^\prime$$, calculated in similar fashion, averages to zero.

• In the short wave (deep water) limit $$kH \rightarrow \inf$$ this becomes $u^{S}=c\left(k \eta_{0}\right)^{2} e^{2 k z_{0}}.$ he Stokes drift is therefore proportional to the phase speed $$c$$, the square of the wave steepness $$k\eta_0$$, and a rapidly-decreasing function of depth.
• In the long wave (shallow water) limit $$kH \rightarrow 0$$, the Stokes drift is independent of depth: $u^{S}=\frac{c}{2} \frac{\eta_{0}^{2}}{H^{2}}.$