Coriolis–Stokes force

In

This force is named after

The Coriolis–Stokes forcing on the mean circulation in an

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

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

(with ${\displaystyle \Omega }$ the Earth's rotation and ${\displaystyle \sin \phi }$ the ) 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

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]

See also

• Ekman layer
• Ekman transport

Notes

References