Physics:Coriolis–Stokes force

Short description: Concept in fluid dynamics

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), (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]