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]
\( {\displaystyle \rho {\boldsymbol {f}}\times {\boldsymbol {u}}_{S},} \)
to be added to the common Coriolis forcing \( {\displaystyle \rho {\boldsymbol {f}}\times {\boldsymbol {u}}.} \) Here \( \boldsymbol{u} \) 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 \( {\hat {{\boldsymbol {z}}}}) \). Further \( \rho \) is the fluid density, \( \times \) is the cross product operator, \( {\displaystyle {\boldsymbol {f}}=f{\hat {\boldsymbol {z}}}} \) where \( {\displaystyle f=2\Omega \sin \phi } \) is the Coriolis parameter (with \( \Omega \) the Earth's rotation angular speed and \( {\displaystyle \sin \phi } \) 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 \( {\displaystyle {\boldsymbol {u}}_{S}} \) 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 \( {\displaystyle {\boldsymbol {u}}_{S}={\boldsymbol {c}}\,(ka)^{2}\exp(2kz)} \) with \( {\displaystyle {\boldsymbol {c}}} \) 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]
See also
Ekman layer
Ekman transport
Notes
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
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
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
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