Physics: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.[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]
- [math]\displaystyle{ \rho\boldsymbol{f}\times\boldsymbol{u}_S, }[/math]
to be added to the common Coriolis forcing [math]\displaystyle{ \rho\boldsymbol{f}\times\boldsymbol{u}. }[/math] Here [math]\displaystyle{ \boldsymbol{u} }[/math] is the mean flow velocity in an Eulerian reference frame and [math]\displaystyle{ \boldsymbol{u}_S }[/math] is the Stokes drift velocity – provided both are horizontal velocities (perpendicular to [math]\displaystyle{ \hat{\boldsymbol{z}} }[/math]). Further [math]\displaystyle{ \rho }[/math] is the fluid density, [math]\displaystyle{ \times }[/math] is the cross product operator, [math]\displaystyle{ \boldsymbol{f}=f\hat{\boldsymbol{z}} }[/math] where [math]\displaystyle{ f=2\Omega\sin\phi }[/math] is the Coriolis parameter (with [math]\displaystyle{ \Omega }[/math] the Earth's rotation angular speed and [math]\displaystyle{ \sin\phi }[/math] the sine of the latitude) and [math]\displaystyle{ \hat{\boldsymbol{z}} }[/math] is the unit vector in the vertical upward direction (opposing the Earth's gravity).
Since the Stokes drift velocity [math]\displaystyle{ \boldsymbol{u}_S }[/math] is in the wave propagation direction, and [math]\displaystyle{ \boldsymbol{f} }[/math] 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 [math]\displaystyle{ \boldsymbol{u}_S=\boldsymbol{c}\,(ka)^2\exp(2kz) }[/math] with [math]\displaystyle{ \boldsymbol{c} }[/math] the wave's phase velocity, [math]\displaystyle{ k }[/math] the wavenumber, [math]\displaystyle{ a }[/math] the wave amplitude and [math]\displaystyle{ z }[/math] the vertical coordinate (positive in the upward direction opposing the gravitational acceleration).[1]
See also
Notes
- ↑ 1.0 1.1 1.2 1.3 Polton, J.A.; Lewis, D.M.; Belcher, S.E. (2005), "The role of wave-induced Coriolis–Stokes forcing on the wind-driven mixed layer", Journal of Physical Oceanography 35 (4): 444–457, doi:10.1175/JPO2701.1, Bibcode: 2005JPO....35..444P, http://www.met.rdg.ac.uk/bl_met/papers/Polton05.pdf, retrieved 2009-03-31
References
- Hasselmann, K. (1970), "Wave‐driven inertial oscillations", Geophysical Fluid Dynamics 1 (3–4): 463–502, doi:10.1080/03091927009365783, Bibcode: 1970GApFD...1..463H
- Leibovich, S. (1980), "On wave–current interaction theories of Langmuir circulations", Journal of Fluid Mechanics 99 (4): 715–724, doi:10.1017/S0022112080000857, Bibcode: 1980JFM....99..715L
- Pollard, R.T. (1970), "Surface waves with rotation: An exact solution", Journal of Geophysical Research 75 (30): 5895–5898, doi:10.1029/JC075i030p05895, Bibcode: 1970JGR....75.5895P
- 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, doi:10.1111/j.1365-246X.1950.tb02968.x, Bibcode: 1950GeoJ....6....1U
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