Coriolis–Stokes force

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.[1]

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 Ursell & Deacon (1950), Hasselmann (1970) and Pollard (1970).[1]

The Coriolis–Stokes forcing on the mean circulation in an Eulerian reference frame was first given by Hasselmann (1970):[1]

${\displaystyle \rho {\boldsymbol {f}}\times {\boldsymbol {u}}_{S},}$

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

Since the Stokes drift velocity ${\displaystyle {\boldsymbol {u}}_{S}}$ is in the wave propagation direction, and ${\displaystyle {\boldsymbol {f}}}$ 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 ${\displaystyle {\boldsymbol {u}}_{S}={\boldsymbol {c}}\,(ka)^{2}\exp(2kz)}$ with ${\displaystyle {\boldsymbol {c}}}$ the wave's phase velocity, ${\displaystyle k}$ the wavenumber, ${\displaystyle a}$ the wave amplitude and ${\displaystyle z}$ the vertical coordinate (positive in the upward direction opposing the gravitational acceleration).[1]