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

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

ρ

f

×

u

S

,

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

to be added to the common Coriolis forcing

ρ

f

×

u

.

{\displaystyle \rho {\boldsymbol {f}}\times {\boldsymbol {u}}.}

Here

u

{\displaystyle {\boldsymbol {u}}}

is the mean flow velocity in an Eulerian reference frame and

u

S

{\displaystyle {\boldsymbol {u}}_{S}}

is the Stokes drift velocity – provided both are horizontal velocities (perpendicular to

z

^

{\displaystyle {\hat {\boldsymbol {z}}}}

). Further

ρ

{\displaystyle \rho }

is the fluid density,

×

{\displaystyle \times }

is the cross product operator,

f

=

f

z

^

{\displaystyle {\boldsymbol {f}}=f{\hat {\boldsymbol {z}}}}

where

f

=

2

Ω

sin

⁡

ϕ

{\displaystyle f=2\Omega \sin \phi }

is the Coriolis parameter (with

Ω

{\displaystyle \Omega }

the Earth's rotation angular speed and

sin

⁡

ϕ

{\displaystyle \sin \phi }

the sine of the latitude) and

z

^

{\displaystyle {\hat {\boldsymbol {z}}}}

is the unit vector in the vertical upward direction (opposing the Earth's gravity).

Since the Stokes drift velocity

u

S

{\displaystyle {\boldsymbol {u}}_{S}}

is in the wave propagation direction, and

f

{\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

u

S

=

c

(

k

a

)

2

exp

⁡

(

2

k

z

)

{\displaystyle {\boldsymbol {u}}_{S}={\boldsymbol {c}}\,(ka)^{2}\exp(2kz)}

with

c

{\displaystyle {\boldsymbol {c}}}

the wave's phase velocity,

k

{\displaystyle k}

the wavenumber,

a

{\displaystyle a}

the wave amplitude and

z

{\displaystyle z}

the vertical coordinate (positive in the upward direction opposing the gravitational acceleration).