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

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

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

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

The overturning circulation dynamics continued… Back to Holton… 1/1/2019 Hadley cell lec 02, AFCB

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: 1/1/2019 Hadley cell lec 02, AFCB

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

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

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

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

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

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

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

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

1/1/2019 Hadley cell lec 02, AFCB

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

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

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

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

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

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

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

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

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

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

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

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