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.

60°N

30°N

0°

30°S

60°S

90°S

90°N

Equator

Dry air sinking

Dry air sinking

Moistairrising

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:

where the first term comes from the planetary rotation, and the second is from eastward motion relative to the moving Earth.

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M =Ωr2 + ur

a

r

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:

where the first term comes from the planetary rotation, and the second is from eastward motion relative to the moving Earth.

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M =Ωr2 + ur

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r = acosφ

∴M =Ωa2 cos2 φ + uacosφ

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:

As the ring moves poleward, it keeps this value.

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M0 =Ωa2

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M =Ωa2 cos2 φ + uacosφ = M0 =Ωa2

∴ u φ( ) = Ωasin2 φ

cosφ

Why can’t an equator to pole cell exist on Earth when it does on Venus?

This produces untenable wind speeds:

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.

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u 10o,20o,30o,90o( ) = 14,58,130,∞( )m

s

60°N

30°N

0°

30°S

60°S

90°S

90°N

Polar easterlies

Westerlies

Trade winds

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:

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fu+tanφ

au2 ≈ −

1

a

∂Φ

∂φ

If we substitute Hide’s result, that the actual zonal wind cannot exceed the value from a few slides back:

and use the small angle approximation (good in the Tropics), we can derive an expression for the meridional extent of the Hadley circulation.

This actually predicts a circulation to extend about as far as it does on Earth: ~25o-30o latitude.

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u ≤ uM =Ωasin2 φ

cosφ

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φM ≈gz*Ω2a2

ΔhT0

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:

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φc ~gz*Ω2a2

ΔvTo

4