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.

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 :


 * $$\rho\boldsymbol{f}\times\boldsymbol{u}_S,$$

to be added to the common Coriolis forcing $$\rho\boldsymbol{f}\times\boldsymbol{u}.$$ 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, $$\times$$ 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 $$\boldsymbol{u}_S=\boldsymbol{c}\,(ka)^2\exp(2kz)$$ 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).