Stokes Drift - Example: Deep Water Waves

Example: Deep Water Waves

The Stokes drift was formulated for water waves by George Gabriel Stokes in 1847. For simplicity, the case of infinite-deep water is considered, with linear wave propagation of a sinusoidal wave on the free surface of a fluid layer:

where

η is the elevation of the free surface in the z-direction (meters),

a is the wave amplitude (meters),

k is the wave number: k = 2π / λ (radians per meter),

ω is the angular frequency: ω = 2π / T (radians per second),

x is the horizontal coordinate and the wave propagation direction (meters),

z is the vertical coordinate, with the positive z direction pointing out of the fluid layer (meters),

λ is the wave length (meters), and

T is the wave period (seconds).

As derived below, the horizontal component ū_S(z) of the Stokes drift velocity for deep-water waves is approximately:

As can be seen, the Stokes drift velocity ū_S is a nonlinear quantity in terms of the wave amplitude a. Further, the Stokes drift velocity decays exponentially with depth: at a depth of a quart wavelength, z = -¼ λ, it is about 4% of its value at the mean free surface, z = 0.