04/22/23 Hadley cell lec 02, AFCB 1

The Hadley Cell continued…

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Overview…• What does it look like?

– Observations were reviewed last week– Three-cell pattern in each hemisphere– Winter Hadley cell stronger than summer

cell (N-S temperature gradients?)– Strong connection with eddy fields

• being established in class at the moment and we will come back to this

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Overview…• How do we know this?

– How do we take observations and construct the time- and/or zonally-averaged fields shown?

– What observations do we use?• Problem – many fields have considerable

uncertainty in measurements, e.g., – We’ll look at this later

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Overview…• Why is the overturning circulation the

way it is? What “maintains” it?– We’re working on this!

• Is it changing or expected to change?– Yes – according to some.– We’ll get to this soon.

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The overturning circulation dynamics continued…

• Back to Holton…

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The angular momentum story…

• From Holton 10.3• The TOTAL angular momentum for the

earth-ocean-atmosphere system is conserved in the absence of torques.

• http://www.scienceagogo.com/news/20030209203254data_trunc_sys.shtml

• Angular momentum M is defined by: M = r ×V

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• Naturally there is a component associated with the rotating planet (through ) and a component associated with motions relative to the rotating earth – through the zonal wind component, u.

• We can show:

• which is conserved in the absence of torques.

cos cos aM a u a M

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• It is observed that earth’s Me is reduced at times when the atmosphere’s Ma is increased (stronger-than-average westerlies for a period).

• In this case, the length of the day is increased as the planet spins more slowly.

• For our discussions, we’ll ignore this and assume Ma is conserved (if no torques).

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• The observed pattern of westerlies and easterlies implies the following:

tropics

midlatitudes

easterlies

earth

westerlies

earth

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• So the atmosphere loses Ma in mid-latitudes and gains Ma in tropical latitudes.

• This implies that there MUST be a poleward flux of Ma to maintain the balance!

• How is this accomplished – and how does this relate to the Hadley cell?

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Some notes on averaging…

• So far we have met zonal and time averaging. • In studying the general circulation of the

atmosphere, we take it a step further and write:

• where the overbar is a time average and the prime is a departure

• where the brackets give a zonal average and the asterisk is a departure

'u u u

*u u u

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Some notes on averaging…

• So more generally we have:

* *' 'u u u u u

Time and zonal average, e.g., trade winds

Stationary eddies e.g., forced by flow over mountains

Transient (e.g., baroclininc) eddies

Seasonally-varying

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• And for two variables…* *

' 'vT v T v T v T

Transport by time- and zonally-averaged motions (mean meridional flow)

Transport by stationary eddies Transport by transient

eddies

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• So what motions account for the poleward flux of Ma required to maintain balance?

• Fig. 11.7 in Peixoto & Oort shows the answer:

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• This shows that:

– Overall transport is dominated by transient eddies!

– Stationary eddy transport is smaller, and smaller still in the southern hemisphere!

– Mean meridional circulation transport is also small, and shows the 3-cell structure.

– Again – eddies are important!!

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• Back to Holton and Ma…suppose for now that it is conserved.

• Consider a zonal ring of air at the equator and then displaced poleward.

• Since Ma is conserved, we have [assuming u(equator)=0]:

• This would give u(30N) 130 m/s !!!

2sin( )cosau

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• From this we conclude:– Angular momentum conservation does

NOT (quite) explain the observed westerly jets at about 30N or S (and Ma is not conserved).

– Also, since dMa/dt 0, there must be torques acting to remove angular momentum from the atmosphere.

– It is of interest to see what these are…

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• Holton’s Eq (10.27) is the usual momentum equation written in angular momentum form.

– On the RHS,torques involve either eddy stresses or zonal pressure gradients.

• Holton’s Eq (10.42) is the same equation expanded and written in sigma-coordinates.

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• Holton’s Eq (10.43) is (10.42) vertically integrated (summed).

• This equation is used to understand how momentum is removed as a hypothetical zonal ring of air moves poleward.

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• There are 3 terms on the RHS:– Angular momentum flux– Small-scale turbulent eddies– Surface pressure torque

• Holton shows (culminating in Eq 10.46) that the first term can be written as the vertically-integrated meridional Ma flux is 1

0

cos ' 'sa p u v d

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• So angular momentum flux is related to mom flux.

• When we examine the “average” structure of mid-latitude eddies, we find the SW-NE tilt, which implies a northward momentum flux.

• This therefore gives a northward Ma flux – as required by the conservation of M.

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• What is the “surface pressure torque”?• In -coordinates it is written as:

• Where ps is surface pressure (=p/ps) and h=h(x,y) is terrain height.

shpx

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• This term is especially effective at removing Ma in middle latitudes of the northern hemisphere – why?

– Obs suggest ps > 0 where h/x >0 and vice versa, so the term gives a net negative effect (“ Ma/ t < 0”) – reducing Ma.

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• Summary…– The assumption of conservation of Ma does not explain

the observed westerly jets at 30 latitude.– Thus – torques must exist to remove momentum.– At the same time, there must be a poleward flux of

momentum.– Accomplished largely by eddies.

– Despite the observations, Held & Hou successfully used the principle of angular momentum conservation to deduce properties of the Hadley cell.

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Questions regarding the Hadley cell:

• Why does the Hadley cell have the observed latitudinal extent?

– What factors control this? – Could it extend pole-equator?

• What controls its strength?

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This was first looked at in the paper by Held & Hou – download it and start reading! We will go thru it in class.

Held, I.M. & A. Y. Hou, 1980: Nonlinear axially symmetric circulations in a near inviscid atmosphere. J. Atmos. Sci., 37, 515-533.

Find – download – print – read – read again!Also, check out http://www.meteo.physik.uni-muenchen.de/~roger/Tropical_Meteorology/Tropical_05.pdf

We will have a graded group discussion in class on the paper.