Hadley circulation extent

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Hadley circulation extent
Halley (1686) first identified the dynamic forcing of air heating
by differential heating between various latitudes on the planet.
Halley envisioned a vertical circulation whereby the
northeasterly trades rose at the equator, turned southwesterly
aloft, then sank.
No explanation of why there is any eastward component, but
a remarkable insight that predated direct measurement of the
upper atmosphere by ~250 years.
Hadley (1735): “the causes of the general trade-winds have
not been fully explained by any of those who wrote on that
subject.”
Essentially understood the conservation of angular
momentum to be critical to the circulation.
Hadley circulation (first published in 1735)
Why can’t an equator to pole cell exist on Earth when it
does on Venus?
Consider angular momentum conservation. A parcel of air
has absolute angular momentum M:
M  r  ur
2
where the first term comes from the planetary rotation, and
the second is from eastward motion relative to the moving
Earth.


r
a

Why can’t an equator to pole cell exist on Earth when it
does on Venus?
Consider angular momentum conservation. A parcel of air
has absolute angular momentum M:
M  r  ur
2
where the first term comes from the planetary rotation, and
the second is from eastward motion relative to the moving
Earth.

r  acos
 M  a cos   uacos
2
2
Why can’t an equator to pole cell exist on Earth when it
does on Venus?
Now consider a ring of air at the equator. If u = 0 here, then
its absolute angular momentum is:
M0  a
2
As the ring moves poleward, it keeps this value.
M  a cos   uacos  M 0  a
2
2
sin 
 u
  a
cos
2
2
Why can’t an equator to pole cell exist on Earth when it
does on Venus?
This produces untenable wind speeds:
m
u10 ,20 ,30 ,90  14,58,130,
s
As an air parcel moves poleward, its distance to Earth’s axis
of rotation decreases (a cos ), so to conserve angular
momentum, the eastward component of the velocity must
increase.
By the time this reaches the subtropics, this circulation has
become unstable, and the large-scale waves in middle
latitudes develop.
90°N
60°N
Polar easterlies
Westerlies
30°N
Trade winds
0°
30°S
60°S
90°S
Held and Hou (1980) derived a scaling relation for the
meridional extent of the Hadley circulation based on
angular momentum conservation and Hide’s theorem.
Suppose that, above a planetary boundary layer, angular
momentum diffusion and frictional forces are weak,
hydrostatic balance holds, and meridional winds are much
weaker than zonal winds, as is generally the case on Earth.
The meridional momentum equation can then be
approximated by gradient-wind balance:
tan  2
1 
fu 
u 
a
a 
If we substitute Hide’s result, that the actual zonal wind cannot
exceed the value from a few slides back:
2
sin 
u  uM  a
cos
and use the small angle approximation (good in the Tropics),
we can derive an expression for the meridional extent of the
Hadley circulation.

gz*  h
M 
2 2
 a T0
This actually predicts a circulation to extend about as far as it
does on Earth: ~25o-30o latitude.
An alternate view of the Hadley cell--one actually that is more
traditional--has been gaining in popularity this decade.
In this view, the Hadley cell continues poleward until the
resulting vertical shears become baroclinically unstable.
Using Phillips’ two layer model to define this latitude, one can
derive an alternate expression for the poleward extent:
gz*  v
c ~ 4 2 2
 a To
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