CURRENTS WITH FRICTION

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CURRENTS WITH FRICTION
Nansens qualitative argument on effects of
friction
?
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EKMAN SOLUTION (1905)
Assumptions homogeneous fluid no horizontal
pressure gradients infinitely deep and wide
ocean no horizontal friction constant eddy
viscosity steady wind northward
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Boundary conditions
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V0 is 45 to the right of the wind (in the
northern hemisphere) V0 decreases exponentially
with depth as it turns clockwise (NH) At depth z
-DE the flow speed falls to e-p 0.04 that at
the surface and in opposite direction
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Indicates that surface currents are 1 of the
wind speed at the poles 2.5 of the wind
speed at 450 11 of the wind speed at 200
Empirically, it is seen that V0 / W oscillates
between 1 and 5
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Ekman Transport
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Ekman Equations
Integrating over two Ekman layers, from depth z
2DE to z 0
Ekman Transport
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Ekman Transport
m2/s
Ekman transport is inversely proportional to f
Water is replaced from the side but what
happens at the coast?
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Equatorward winds on ocean eastern
boundaries Poleward wind on ocean western
boundaries
Poleward winds on ocean eastern
boundaries Equatorward wind on ocean western
boundaries
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Consequence of Upwelling
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Bottom Friction and Shallow Water Effects
Assumptions Bottom at z 0 u, v 0 at z 0
(no flow at the bottom) u ug, v 0 at the top
of the bottom Ekman layer (z DB) ug is
geostrophic flow
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BOTTOM EKMAN LAYER
ug
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Flow at interior?
Flow at bottom?
z
-x
Overlap of bottom and surface Ekman layers
Importance of shelf break depth
Problems with Ekman theory constant Az,
constant wind, linear flow, steady
state, infinite ocean, no pressure gradients
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SVERDRUP SOLUTION
Assumed gradients in the wind field -- in
contrast to Ekmans spatially uniform wind
Ekman Transport
Convergence
Divergence
Differentiating Ekman equations with respect to y
and x, to look at gradients in wind field
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Adding the two yields
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Meridional transport of water given by the curl
of the wind
How about zonal (in x) transport?
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Integrating from Eastern Boundary (x0) to the
west -x
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NORTH EQUATORIAL CURRENT
NORTH EQUATORIAL COUNTERCURRENT
SOUTH EQUATORIAL CURRENT
Streamlines of mass transport from mean wind
stress (Reid, 1948)
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Conservation of mass is forced by including
north-south currents confined to a thin,
horizontal boundary layer. From Tomczak and
Godfrey (1994).
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SVERDRUP SOLUTION
Limited to the east side of the oceans because Qx
grows with x. Neglects friction which would
eventually balance the wind-driven flow
Solutions may be used for describing the global
system of surface currents. Conservation of mass
is forced at the western boundaries by including
north-south currents confined to a thin,
horizontal boundary layer Only one boundary
condition can be satisfied, no flow through the
eastern boundary. More complete descriptions of
the flow require more complete equations.
Solutions give no information on the vertical
distribution of the current.
What happens on the western part of the oceans?
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STOMMEL SOLUTION (1948)
Currents are fast and narrow on the western
part of ocean basins slow and broad on the
eastern part of basins
Change of f with latitude is the main responsible
for western intensification of ocean currents
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Western Intensification can also be understood
with vorticity arguments
Vorticity tendency for portions of fluid to
rotate
We will consider RELATIVE, PLANETARY, ABSOLUTE,
AND POTENTIAL
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Relative vorticity
Planetary vorticity Equals f. A stationary
object on the earth has planetary vorticity that
varies with latitude
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Absolute vorticity Planetary plus relative
vorticities
Changes in absolute vorticity are useful to help
understand tendencies for fluids to rotate
To describe changes in absolute vorticity, take
Changes of absolute vorticity in time are related
to divergences
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Side View
Convergence Gain of Absolute Vorticity Column
Stretching
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Side View
Divergence Loss of Absolute Vorticity Column
Squashing
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Now consider a layer of thickness D, whose
equation of continuity is
Changes of layer thickness are given by
divergences/convergences Convergences increase
in D Divergences decrease in D
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D constant and f changing
Consider
D changing and f constant
Use concept of Potential Vorticity Conservation
to explain Western Intensification
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MUNKS SOLUTION
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