Coriolis–Stokes force explained

In fluid dynamics, the Coriolis–Stokes force is a forcing of the mean flow in a rotating fluid due to interaction of the Coriolis effect and wave-induced Stokes drift. This force acts on water independently of the wind stress.

This force is named after Gaspard-Gustave Coriolis and George Gabriel Stokes, two nineteenth-century scientists. Important initial studies into the effects of the Earth's rotation on the wave motion – and the resulting forcing effects on the mean ocean circulation – were done by, and .

The Coriolis–Stokes forcing on the mean circulation in an Eulerian reference frame was first given by :

\rho\boldsymbol{f} x \boldsymbol{u} _{S,}

to be added to the common Coriolis forcing

\rho\boldsymbol{f} x \boldsymbol{u}. Here \boldsymbol{u} is the mean flow velocity in an Eulerian reference frame and \boldsymbol{u} _{S} is the Stokes drift velocity – provided both are horizontal velocities (perpendicular to \hat{\boldsymbol{z}} ). Further \rho is the fluid density, x is the cross product operator, \boldsymbol{f} = f\hat{\boldsymbol{z}} where f =2 \Omega\sin\phi is the Coriolis parameter (with \Omega the Earth's rotation angular speed and \sin\phi the sine of the latitude) and \hat{\boldsymbol{z}} is the unit vector in the vertical upward direction (opposing the Earth's gravity).

Since the Stokes drift velocity

\boldsymbol{u} _{S} is in the wave propagation direction, and \boldsymbol{f} is in the vertical direction, the Coriolis–Stokes forcing is perpendicular to the wave propagation direction (i.e. in the direction parallel to the wave crests). In deep water the Stokes drift velocity is \boldsymbol{u} ^{2 \exp( 2 kz)} _{S = \boldsymbol{c}(ka)} with \boldsymbol{c} the wave's phase velocity, k the wavenumber, a the wave amplitude and z the vertical coordinate (positive in the upward direction opposing the gravitational acceleration).

See also

• Ekman layer
• Ekman transport