1. The Hadley Cell continued…
1/1/2019
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
1/1/2019
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
1/1/2019
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.
1/1/2019
Hadley cell lec 02, AFCB

5. The overturning circulation dynamics continued…
Back to Holton…
1/1/2019
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.
Angular momentum M is defined by:
1/1/2019
Hadley cell lec 02, AFCB

7. which is conserved in the absence of torques.
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.
1/1/2019
Hadley cell lec 02, AFCB

8. 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).
1/1/2019
Hadley cell lec 02, AFCB

9. The observed pattern of westerlies and easterlies implies the following:
tropics
earth
westerlies
midlatitudes
earth
1/1/2019
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?
1/1/2019
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
1/1/2019
Hadley cell lec 02, AFCB

12. Some notes on averaging…
So more generally we have:
Transient (e.g., baroclininc) eddies
Time and zonal average, e.g., trade winds
Stationary eddies e.g., forced by flow over mountains
Seasonally-varying
1/1/2019
Hadley cell lec 02, AFCB

13. And for two variables…
Transport by time- and zonally-averaged motions (mean meridional flow)
Transport by stationary eddies
Transport by transient eddies
1/1/2019
Hadley cell lec 02, AFCB

14. Fig. 11.7 in Peixoto & Oort shows the answer:
So what motions account for the poleward flux of Ma required to maintain balance?
Fig in Peixoto & Oort shows the answer:
1/1/2019
Hadley cell lec 02, AFCB

15. 1/1/2019
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!!
1/1/2019
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(30N)  130 m/s !!!
1/1/2019
Hadley cell lec 02, AFCB

18. From this we conclude:
Angular momentum conservation does NOT (quite) explain the observed westerly jets at about 30N 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…
1/1/2019
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.
1/1/2019
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.
1/1/2019
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 
1/1/2019
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.
1/1/2019
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.
1/1/2019
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.
1/1/2019
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.
1/1/2019
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?
1/1/2019
Hadley cell lec 02, AFCB

27. Find – download – print – read – read again!
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,
Find – download – print – read – read again!
Also, check out
We will have a graded group discussion in class on the paper.
1/1/2019
Hadley cell lec 02, AFCB