the hadley cell continued…
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04/22/23 Hadley cell lec 02, AFCB 1
The Hadley Cell continued…
04/22/23 Hadley cell lec 02, AFCB 2
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
04/22/23 Hadley cell lec 02, AFCB 3
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
04/22/23 Hadley cell lec 02, AFCB 4
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.
04/22/23 Hadley cell lec 02, AFCB 5
The overturning circulation dynamics continued…
• Back to Holton…
04/22/23 Hadley cell lec 02, AFCB 6
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).
04/22/23 Hadley cell lec 02, AFCB 9
• The observed pattern of westerlies and easterlies implies the following:
tropics
midlatitudes
easterlies
earth
westerlies
earth
04/22/23 Hadley cell lec 02, AFCB 10
• 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?
04/22/23 Hadley cell lec 02, AFCB 11
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
04/22/23 Hadley cell lec 02, AFCB 12
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
04/22/23 Hadley cell lec 02, AFCB 13
• 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
04/22/23 Hadley cell lec 02, AFCB 14
• So what motions account for the poleward flux of Ma required to maintain balance?
• Fig. 11.7 in Peixoto & Oort shows the answer:
04/22/23 Hadley cell lec 02, AFCB 15
04/22/23 Hadley cell lec 02, AFCB 16
• 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!!
04/22/23 Hadley cell lec 02, AFCB 17
• 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
04/22/23 Hadley cell lec 02, AFCB 18
• 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…
04/22/23 Hadley cell lec 02, AFCB 19
• 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.
04/22/23 Hadley cell lec 02, AFCB 20
• 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.
04/22/23 Hadley cell lec 02, AFCB 21
• 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
04/22/23 Hadley cell lec 02, AFCB 22
• 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.
04/22/23 Hadley cell lec 02, AFCB 23
• 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
04/22/23 Hadley cell lec 02, AFCB 24
• 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.
04/22/23 Hadley cell lec 02, AFCB 25
• 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.
04/22/23 Hadley cell lec 02, AFCB 26
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?
04/22/23 Hadley cell lec 02, AFCB 27
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.
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