Stokes Drift - Example: Deep Water Waves

Example: Deep Water Waves

See also: Airy wave theory and Stokes wave

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:

\eta\, =\, a\, \cos\, \left( k x - \omega t \right),

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 ūS(z) of the Stokes drift velocity for deep-water waves is approximately:

\overline{u}_S\, \approx\, \omega\, k\, a^2\, \text{e}^{2 k z}\, =\, \frac{4\pi^2\, a^2}{\lambda\, T}\, \text{e}^{4\pi\, z / \lambda}.

As can be seen, the Stokes drift velocity ūS 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.

Read more about this topic:  Stokes Drift

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