Coriolis–Stokes force

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 (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]

\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).^[1]

Notes

^ ^a ^b ^c ^d Polton, J.A.; Lewis, D.M.; Belcher, S.E. (2005), "The role of wave-induced Coriolis–Stokes forcing on the wind-driven mixed layer" (PDF), Journal of Physical Oceanography, 35 (4): 444–457, Bibcode:2005JPO....35..444P, CiteSeerX 10.1.1.482.7543, doi:10.1175/JPO2701.1, archived from the original (PDF) on 2017-08-08, retrieved 2009-03-31

References

Hasselmann, K. (1970), "Wave‐driven inertial oscillations", Geophysical Fluid Dynamics, 1 (3–4): 463–502, Bibcode:1970GApFD...1..463H, doi:10.1080/03091927009365783
Leibovich, S. (1980), "On wave–current interaction theories of Langmuir circulations", Journal of Fluid Mechanics, 99 (4): 715–724, Bibcode:1980JFM....99..715L, doi:10.1017/S0022112080000857, S2CID 14996095
Pollard, R.T. (1970), "Surface waves with rotation: An exact solution", Journal of Geophysical Research, 75 (30): 5895–5898, Bibcode:1970JGR....75.5895P, doi:10.1029/JC075i030p05895
Ursell, F.; Deacon, G.E.R. (1950), "On the theoretical form of ocean swell on a rotating Earth", Monthly Notices of the Royal Astronomical Society, 6 (Geophysical Supplement): 1–8, Bibcode:1950GeoJ....6....1U, doi:10.1111/j.1365-246X.1950.tb02968.x