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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]
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).
[1]
See also
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