the general circulation of the atmosphereaczaja/pg2008/marshallplumb_atmosphere.pdf · energetics...

C H A P T E R

8

The general circulation of theatmosphere

8.1. Understanding the observed circulation8.2. A mechanistic view of the circulation

8.2.1. The tropical Hadley circulation8.2.2. The extra tropical circulation and GFD Lab XI: baroclinic instability

8.3. Energetics of the thermal wind equation8.3.1. Potential energy for a fluid system8.3.2. Available potential energy8.3.3. Release of available potential energy in baroclinic instability8.3.4. Energetics in a compressible atmosphere

8.4. Large-scale atmospheric heat and momentum budget8.4.1. Heat transport8.4.2. Momentum transport

8.5. Latitudinal variations of climate8.6. Further reading8.7. Problems

In this chapter we return to our dis-cussion of the large-scale circulation of theatmosphere and make use of the dynamicalideas developed in the previous two chap-ters to enquire into its cause. We begin byreviewing the demands on the atmosphericcirculation imposed by global energy andangular momentum budgets as depicted inFig. 8.1. The atmosphere must transportenergy from equator to pole to main-tain the observed pole-equator tempera-ture gradient. In addition, because thereare westerly winds at the surface in middlelatitudes and easterly winds in the trop-ics (see, e.g., Fig. 7.28), angular momentum

must be transported from the tropics tohigher latitudes. We discuss how this trans-fer is achieved by the Hadley circulation inthe tropics and weather systems in middlelatitudes. We will discover that weathersystems arise from a hydrodynamic insta-bility of the thermal wind shear associatedwith the pole-equator temperature gradient.Produced through a mechanism knownas baroclinic instability, the underlyingdynamics are illustrated through laboratoryexperiment and discussed from the per-spective of the available potential energy‘‘stored’’ by the horizontal temperaturegradients.

135

136 8. THE GENERAL CIRCULATION OF THE ATMOSPHERE

8.1. UNDERSTANDING THEOBSERVED CIRCULATION

The simplest observed global characteris-tic of the atmosphere is that the tropics aremuch warmer than the poles. As discussedin Chapter 5, this is a straightforward con-sequence of the geometry of the earth: theannually averaged incoming solar radiationper unit area of the earth’s surface is muchgreater at the equator than at the poles, adifference that is compounded by the factthat the polar regions are covered in iceand snow and therefore reflect much of theincoming radiation back to space. A little lessobvious is the fact that the tropical regionsactually receive more energy from the sunthan they emit back to space, whereas theconverse is true in high latitudes. Since bothregions are, on an annual average, in equilib-rium, there must be a process acting to trans-port excess energy from the tropics to makeup the deficit in high latitudes, as depictedschematically in Fig. 8.1 (left).

The implied transport of some 6 × 1015W(see Fig. 5.6) must be effected by the atmo-spheric circulation, carrying warm air pole-ward and cold air equatorward. (In fact,

the ocean circulation also contributes, asdiscussedinChapter11.)Asaresult, thetrop-icsarecooler,andpolarregionswarmer, thanthey would be in the absence of such trans-port. Thus, in this as in other respects, theatmospheric general circulation plays a keyrole in climate.

What are the motions that deliver therequired transport? The gross featuresof the observed atmospheric circulationdiscussed in Chapter 5 are depicted inFig. 8.2. The zonal flow is strongly west-erly aloft in middle latitudes, a fact thatwe can now, following Section 7.3, under-stand as a straightforward consequence ofthe decrease of temperature with latitude.Surface winds are constrained to be weakby the action of friction near the ground,and thermal wind balance, Eq. 7-24, impliesthat a poleward decrease of temperatureis necessarily accompanied by increas-ing westerly winds with height. Takentogether, these two facts require a zonalflow in middle latitudes that increases fromnear zero at the ground to strong westerliesat altitude, as schematized in Fig. 7.19.

Although the near-surface winds areweak—a few m s−1—they nevertheless

FIGURE 8.1. Latitudinal transport of (left) heat and (right) angular momentum (mtm) implied by the observedstate of the atmosphere. In the energy budget there is a net radiative gain in the tropics and a net loss at highlatitudes; to balance the energy budget at each-latitude, a poleward heat flux is implied. In the angular momentumbudget the atmosphere gains angular momentum in low latitudes (where the surface winds are easterly) and losesit in middle latitudes (where the surface winds are westerly). A poleward atmospheric flux of angular momentumis thus implied.

8.2. A MECHANISTIC VIEW OF THE CIRCULATION 137

FIGURE 8.2. Schematic of the observed atmospheric general circulation for annual-averaged conditions. Theupper level westerlies are shaded to reveal the core of the subtropical jet stream on the poleward flank of theHadley circulation. The surface westerlies and surface trade winds are also marked, as are the highs and lows ofmiddle latitudes. Only the northern hemisphere is shown. The vertical scale is greatly exaggerated.

exhibit a distinct spatial distribution, withthe zonal component being easterly in thetropics and westerly in middle latitudes(Fig. 7.28). In the presence of surface friction,the atmosphere must therefore be losingangular momentum to the ground in middlelatitudes and gaining angular momentum inthe tropics. As Fig. 8.1 (right) illustrates, theangular momentum balance of the atmo-sphere thus requires a transport of westerlyangular momentum from low to middlelatitudes to maintain equilibrium.

Even though the west-to-east circulationin the upper troposphere is the dominantcomponent of the large-scale atmosphericflow, it cannot be responsible for therequired poleward transports of heat andangular momentum, for which north-southflow is needed. As we saw in Chapter 5(Fig. 5.21) there is indeed a mean circulationin the meridional plane, which is dominatedby the Hadley circulation of the tropicalatmosphere with, on an annual average,mean upwelling near the equator, polewardflow aloft, subsidence in the subtropics,and equatorward return flow near the sur-face. This circulation transports heat andangular momentum poleward, as required,

within the tropics; however, the meridionalcirculation becomes much weaker in mid-dle latitudes and so cannot produce muchtransport there. We saw in Section 5.4.1 thatmost of the north-south flow in the extrat-ropical atmosphere takes the form of eddies,rather than a mean overturning as in thetropics. As we will shall see, it is these eddiesthat produce the poleward transports in theextratropics.

8.2. A MECHANISTIC VIEW OFTHE CIRCULATION

To what extent can we explain the mainfeatures of the observed general circulationon the basis of the fluid dynamics of a simplerepresentation of the atmosphere driven bylatitudinal gradients in solar forcing? Theemphasis here is on ‘‘simple.’’ In reality,the Earth’s surface is very inhomogeneous:there are large mountain ranges that disturbthe flow and large contrasts (e.g., in tem-perature and in surface roughness) betweenoceans and continents. In the interests ofsimplicity, however, we shall neglect suchvariations in surface conditions. We also


neglect seasonal and diurnal1 variations,and assume maximum solar input at theequator, even though the subsolar point(Fig. 5.3) migrates between the two Tropicsthrough the course of a year. Thus we shallconsider the response of an atmosphere ona longitudinally uniform, rotating, planet(that is otherwise like the Earth) to a lat-itudinal gradient of heating. We shall seethat the gross features of the observed cir-culation can indeed be understood on thisbasis; later, in Section 8.5, the shortcomingsintroduced by this approach will be brieflydiscussed.

Thus we ask how an axisymmetric atmo-sphere responds to an axisymmetric forcing,with stronger solar heating at the equatorthan at the poles. It seems reasonable toassume that the response (i.e., the inducedcirculation) will be similarly axisymmetric.Indeed, this assumption turns out to give usa qualitatively reasonable description of thecirculation in the tropics, but it will becomeevident that it fails in the extratropical atmo-sphere, where the symmetry is broken byhydrodynamical instability.

8.2.1. The tropical Hadley circulation

If the Earth was not rotating, the circulationdriven by the pole-equator temperaturedifference would be straightforward, withwarm air rising in low latitudes and coldair sinking at high latitudes, as first sug-gested by Hadley and sketched in Fig. 5.19.But, as seen in Figs. 5.20 and 5.21, this isnot quite what happens. We do indeed seea meridional circulation in the tropics, butthe sinking motion is located in the sub-tropics around latitudes of ± 30. In fact,the following considerations tell us that onegiant axisymmetric meridional cell extend-ing from equator to pole is not possible on arapidly rotating planet, such as the Earth.

Consider a ring of air encircling the globeas shown in Fig. 8.3, lying within andbeing advected by the upper level polewardflow of the Hadley circulation. Becausethis flow is by assumption axisymmetric,and also because friction is negligible inthis upper level flow, well above the near-surface boundary layer, absolute angularmomentum will be conserved by the ringas it moves around. The absolute angularmomentum per unit mass is (recall our dis-cussion of angular momentum in the radialinflow experiment in Section 6.6.1)

A = Ωr2 + ur,

the first term being the contribution fromplanetary rotation, and the second from theeastward wind u relative to the Earth, wherer is the distance from the Earth’s rotationaxis. Since r = a cosϕ,

A = Ωa2 cos 2ϕ + ua cosϕ. (8-1)

Now, suppose that u = 0 at the equator(Fig. 5.20 shows that this is a reasonableassumption in the equatorial uppertroposphere). Then the absolute angular

FIGURE 8.3. Schematic of a ring of air blowing west−→ east at speed u and latitude ϕ. The ring is assumedto be advected by the poleward flow of the Hadleycirculation conserving angular momentum.

1In reality, of course, there are strong instantaneous longitudinal variations in solar forcing between those regionsaround local noon and those in nighttime. However, except very near the surface, the thermal inertia of the atmospheredamps out these fluctuations and temperature varies little between day and night. Hence neglect of diurnal variations is areasonable approximation.


momentum at the equator is simply A0 =Ωa2. As the ring of air moves poleward, itretains this value, and so when it arrives atlatitude ϕ, its absolute angular momentumis

A = Ωa2 cos 2ϕ + ua cosϕ = A0 = Ωa2 .

Therefore the ring will acquire an eastwardvelocity

u (ϕ) =Ω(a2 − a2 cos 2ϕ

)

a cosϕ= Ωa

sin2 ϕ

cosϕ. (8-2)

Note that this is directly analogous toEq. 6-23 of our radial inflow experiment,when we realize that r = a cosϕ.Equation 8-2 implies unrealistically largewinds far from the equator, at latitudesof (10, 20, 30), u(ϕ) = (14, 58, 130) m s−1,and of course the wind becomes infinite asϕ → 90. On the grounds of physical plausi-bility, it is clear that such an axisymmetriccirculation cannot extend all the way tothe pole in the way envisaged by Hadley(and sketched in Fig. 5.19); it must termi-nate somewhere before it reaches the pole.Just how far the circulation extends depends(according to theory) on many factors.2

Consider the upper branch of the circula-tion, as depicted in Fig. 8.4. Near the equa-tor, where f is small and the Coriolis effectis weak, angular momentum constraints arenot so severe and the equatorial atmosphereacts as if the Earth were rotating slowly. Asair moves away from the equator, however,the Coriolis parameter becomes increasinglylarge and in the northern hemisphere turnsthe wind to the right, resulting in a westerlycomponent to the flow. At the polewardextent of the Hadley cell, then, we expectto find a strong westerly flow, as indeedwe do (see Fig. 5.20). This subtropical jet is

FIGURE 8.4. Schematic of the Hadley circulation(showing only the Northern Hemispheric part ofthe circulation; there is a mirror image circulationsouth of the equator). Upper level poleward flowinduces westerlies; low level equatorward flow induceseasterlies.

driven in large part by the advection fromthe equator of large values of absolute angu-lar momentum by the Hadley circulation,as is evident from Eq. 8-2 and depicted inFig. 8.4. Flow subsides on the subtropicaledge of the Hadley cell, sinking into thesubtropical highs (very evident in Fig. 7.27),before returning to the equator at low levels.At these low levels, the Coriolis accelera-tion, again turning the equatorward flowto its right in the northern hemisphere,produces the trade winds, northeasterly inthe northern hemisphere (southeasterly inthe southern hemisphere) (see Fig. 7.28).These winds are not nearly as strong asin the upper troposphere, because they aremoderated by friction acting on the near-surface flow. In fact, as discussed above,there must be low-level westerlies some-where. In equilibrium, the net frictional drag(strictly, torque) on the entire atmospheremust be zero, or the total angular momen-tum of the atmosphere could not be steady.(There would be a compensating change inthe angular momentum of the solid Earth,and the length of the day would drift.3)

2For example, if the Earth were rotating less (or more) rapidly, and other things being equal, the Hadley circulationwould extend further (or less far) poleward.

3In fact the length of the day does vary, ever so slightly, because of angular momentum transfer between theatmosphere and underlying surface. For example, on seasonal timescales there are changes in the length of the day of abouta millisecond. The length of the day changes from day to day by about 0.1 ms! Moreover these changes can be attributedto exchanges of angular momentum between the earth and the atmosphere.


FIGURE 8.5. A schematic diagram of the Hadley circulation and its associated zonal flows and surface circulation.

So as shown in Fig. 8.5 the surface winds inour axisymmetric model would be westerlyat the poleward edge of the circulation cell,and eastward near the equator. This is sim-ilar to the observed pattern (see Fig. 7.28,middle panel), but not quite the same. Inreality the surface westerlies maximize near50 N, S, significantly poleward of the sub-tropical jet, a point to which we return inSection 8.4.2.

As sketched in Fig. 8.5, the subtropicalregion of subsidence is warm (because ofadiabatic compression) and dry (since theair aloft is much drier than surface air; theboundary formed between this subsidingair and the cooler, moister near-surface airis the ‘‘trade inversion’’ noted in Chapter 4,and within which the trade winds arelocated. Aloft, horizontal temperature gra-dients within the Hadley circulation are veryweak (see Figs. 5.7 and 5.8), a consequenceof very efficient meridional heat transportby the circulation.

Experiment on the Hadley circulation: GFDLab VIII revisited

A number of aspects of the Hadley cir-culation are revealed in Expt VIII, whoseexperimental arrangement has already beendescribed in Fig. 7.12 in the context of thethermal wind equation. The apparatus is just

a cylindrical tank containing plain water,at the center of which is a metal can filledwith ice. The consequent temperature gradi-ent (decreasing ‘‘poleward’’) drives motionsin the tank, the nature of which dependson the rotation rate of the apparatus.When slowly rotating, as in this experiment(Ω 1 rpm—yes, only 1 rotation of the tableper minute: very slow!), we see the devel-opment of the thermal wind in the formof a strong ‘‘eastward’’ (i.e., super-rotating)flow in the upper part of the fluid, whichcan be revealed by paper dots floating onthe surface and dye injected into the fluid asseen in Fig. 7.13.

The azimuthal current observed in theexperiment, which is formed in a manneranalogous to that of the subtropical jet by theHadley circulation discussed previously, ismaintained by angular momentum advec-tion by the meridional circulation sketchedin Fig. 7.12 (right). Water rises in the outerregions, moves inward in the upper layers,conserving angular momentum as it does,thus generating strong ‘‘westerly’’ flow, andrubs against the cold inner wall, becomingcold and descending. Potassium perman-ganate crystals, dropped into the fluid (nottoo many!), settle on the bottom and give anindication of the flow in the bottom bound-ary layer. In Fig. 8.6 we see flow moving


FIGURE 8.6. The Hadley regime studied in GFD LabVIII, Section 7.3.1. Bottom flow is revealed by the twooutward spiralling streaks showing anticyclonic (clock-wise) flow sketched schematically in Fig. 7.12 (right);the black paper dots and collar of dye mark the upperlevel flow and circulate cyclonically (anticlockwise).

radially outwards at the bottom and beingdeflected to the right, creating easterly flowopposite to that of the rotating table; notethe two dark streamers moving outwardand clockwise (opposite to the sense of rota-tion and the upper level flow). This bottomflow is directly analogous to the easterlyand equatorward trade winds of the loweratmosphere sketched in Fig. 8.4 and plottedfrom observations in Fig. 7.28.

Quantitative study of the experiment (bytracking, for example, floating paper dots)shows that the upper level azimuthal flowdoes indeed conserve angular momentum,satisfying Eq. 6-23 quite accurately, justas in the radial inflow experiment. WithΩ = 0.1 s−1 (corresponding to a rotation rateof 0.95 rpm; see Table 13.1, Appendix 13.4.1)Eq. 6-23 implies a hefty 10 cm s−1 if angularmomentum were conserved by a particlemoving from the outer radius, r1 = 30 cm,to an inner radius of 30 cm. Note, how-ever, that if Ω were set 10 times higher, to10 rpm, then angular momentum conserva-tion would imply a speed of 1 m s−1, a veryswift current. We will see in the next sec-tion that such swift currents are notobserved if we turn up the rotation rateof the table. Instead the azimuthal cur-rent breaks down into eddying motions,

inducing azimuthal pressure gradients andbreaking angular momentum conservation.

8.2.2. The extratropical circulation andGFD Lab XI: baroclinic instability

Although the simple Hadley cell modeldepicted in Fig. 8.5 describes the tropicalregions quite well, it predicts that little hap-pens in middle and high latitudes. There,where the Coriolis parameter is much larger,the powerful constraints of rotation aredominant and meridional flow is impeded.We expect, however, (and have seen inFigs. 5.8 and 7.20), that there are stronggradients of temperature in middle lati-tudes, particularly in the vicinity of thesubtropical jet. So, although there is littlemeridional circulation, there is a zonal flowin thermal wind balance with the tempera-ture gradient. Since T decreases poleward,Eq. 7-24 implies westerly winds increasingwith height. This state of a zonal flow withno meridional motion is a perfectly validequilibrium state. Even though the existenceof a horizontal temperature gradient implieshorizontal pressure gradients (tilts of pres-sure surfaces), the associated force is entirelybalanced by the Coriolis force acting on thethermal wind, as we discussed in Chapter 7.

Our deduction that the mean meridionalcirculation is weak outside the tropics isqualitatively in accord with observations(see Fig. 5.21) but leaves us with twoproblems:

1. Poleward heat transport is requiredto balance the energy budget. In thetropics, the overturning Hadleycirculation transports heat poleward,but no further than the subtropics.How is heat transported furtherpoleward if there is no meridionalcirculation?

2. Everyday observation tells us that apicture of the midlatitude atmosphereas one with purely zonal winds is verywrong. If it were true, weather wouldbe very predictable (and very dull).


Our axisymmetric model is, therefore,only partly correct. The prediction of purelyzonal flow outside the tropics is quitewrong. As we have seen, the midlatitudeatmosphere is full of eddies, which manifestthemselves as traveling weather systems.4

Where do they come from? In fact, as weshall now discuss and demonstrate througha laboratory experiment, the extratropicalatmosphere is hydrodynamically unstable,the flow spontaneously breaking down intoeddies through a mechanism known asbaroclinic instability.5 These eddies readilygenerate meridional motion and, as we shallsee, effect a meridional transport of heat.

Baroclinic instability

GFD Lab XI: baroclinic instability in adishpan To introduce our discussion ofthe breakdown of the thermal wind throughbaroclinic instability, we describe a labora-tory experiment of the phenomenon. Theapparatus, sketched in Fig. 7.12, is identicalto that of Lab VIII used to study the thermalwind and the Hadley circulation. In the for-mer experiments the table was rotated veryslowly, at a rate ofΩ 1 rpm. This time, how-ever, the table is rotated much more rapidly,atΩ ∼ 10 rpm, representing the considerablygreater Coriolis parameter found in middlelatitudes. At this higher rotation rate some-thing remarkable happens. Rather than asteady axisymmetric flow, as in the Hadleyregime shown in Fig. 8.6, a strongly eddyingflow is set up. The thermal wind remains,but breaks down through instability, as

shown in Fig. 8.7. We see the developmentof eddies that sweep (relatively) warm fluidfrom the periphery to the cold can in onesector of the tank (e.g., A in Fig. 8.7), andsimultaneously carry cold fluid from thecan to the periphery at another (e.g., B inFig. 8.7). In this way a radially-inward heattransport is achieved, offsetting the coolingat the center caused by the melting ice.

For the experiment shown we observethree complete wavelengths around thetank in Fig. 8.7. By repeating the experimentbut at different values of Ω, we observethat the scale of the eddies decreases, andthe flow becomes increasingly irregular asΩ is increased. These eddies are producedby the same mechanism underlying creationof atmospheric weather systems shown inFig. 7.20 and discussed later.

Before going on we should emphasizethat the flow in Fig. 8.7 does not conserveangular momentum. Indeed, if it did, asestimated at the end of Section 8.2.1, wewould observe very strong horizontal cur-rents near the ice bucket, of order 1 m s−1

rather than a few cm s−1, as seen in theexperiment. This, of course, is exactly whatEq. 8-2 is saying for the atmosphere: if ringsof air conserved angular momentum asthey move in axisymmetric motion fromequator to pole, then we obtain unrealisti-cally large winds. Angular momentum is notconserved because of the presence of zonal(or, in our tank experiment, azimuthal)pressure gradient associated with eddyingmotion (see Problem 6 of Chapter 6).

4 Albert Defant (1884–1974) German Professor of Meteorology and Oceanography, who madeimportant contributions to the theory of the general circulation of the atmosphere. He was the first to liken the meridionaltransfer of energy between the subtropics and pole to a turbulent exchange by large-scale eddies.

5In a ‘‘baroclinic’’ fluid, ρ = ρ(p,T

), and so there can be gradients of density (and therefore of temperature) along

pressure surfaces. This should be contrasted to a ‘‘barotropic’’ fluid [ρ = ρ(p)] in which no such gradients exist.


FIGURE 8.7. Top: Baroclinic eddies in the ‘‘eddy’’regime viewed from the side. Bottom: View from above.Eddies draw fluid from the periphery in toward thecentre at point A and vice versa at point B. The eddiesare created by the instability of the thermal windinduced by the radial temperature gradient due tothe presence of the ice bucket at the center of the tank.The diameter of the ice bucket is 15 cm.

Middle latitude weather systems Theprocess of baroclinic instability studied inthe laboratory experiment just describedis responsible for the ubiquitous wavinessof the midlatitude flow in the atmosphere.As can be seen in the observations shownon Figs. 5.22, 7.4, and 7.20, these waves

often form closed eddies, especially nearthe surface, where they are familiar asthe high and low pressure systems asso-ciated with day-to-day weather. In theprocess, they also affect the poleward heattransport required to balance the globalenergy budget (see Fig. 8.1). The mannerin which this is achieved on the planetaryscale is sketched in Fig. 8.8. Eddies ‘‘stir’’the atmosphere, carrying cold air equator-ward and warm air poleward, therebyreducing the equator-to-pole temperaturecontrast. To the west of the low (marked L)in Fig. 8.8, cold air is carried into the trop-ics. To the east, warm air is carried towardthe pole, but since poleward flowing air isascending (remember the large-scale slopeof the θ surfaces shown in Fig. 5.8) it tendsto leave the surface. Thus we get a concen-trated gradient of temperature near point 1,where the cold air ‘‘pushes into’’ the warmair, and a second, less marked concentra-tion where the warm air butts into cold atpoint 2. We can thus identify cold and warmfronts, respectively, as marked in the centerpanel of Fig. 8.8. Note that a triangle is usedto represent the ‘‘sharp’’ cold front and asemicircle to represent the ‘‘gentler’’ warmfront. In the bottom panel we present sec-tions through cold fronts and warm frontsrespectively.

Timescales and length scales We havedemonstrated by laboratory experiment thata current in thermal wind balance can, andalmost always does, become hydrodynam-ically unstable, spawning meanders andeddies. More detailed theoretical analysis6

shows that the lateral scale of the eddies that

6Detailed analysis of the space-scales and growth rates of the instabilities that spontaneously arise on an initially zonaljet in thermal wind balance with a meridional temperature gradient show that:

1. The Eady growth rate of the disturbance eσt is given by σ = 0.31U/Lρ = 0.31fN

dudz

using Eq. (7-23) and U =dudz

H

(see Gill, 1982). Inserting typical numbers for the troposphere, N = 10−2 s−1,dudz

= 2 × 10−3 s−1, f = 10−4 s−1, we

find that σ 10−5 s−1, an e-folding timescale of 1 day.

2. The wavelength of the fastest growing disturbance is 4Lρ where Lρ is given by Eq. 7-23. This yields a wavelength of2800 km if Lρ = 700 km. The circumference of the earth at 45 N is 21,000 km, and so about 7 synoptic waves canfit around the earth at any one time. This is roughly in accord with observations; see, for example, Fig. 5.22.


FIGURE 8.8. Top: In middle latitudes eddies transport warm air poleward and upward and cold air equator-ward and downward. Thus the eddies tend to ‘‘stir’’ the atmosphere laterally, reducing the equator-to-poletemperature contrast. Middle: To the west of the ‘‘L,’’ cold air is carried in to the tropics. To the east, warm air iscarried toward the pole. The resulting cold fronts (marked by triangles) and warm fronts (marked by semicircles)are indicated. Bottom: Sections through the cold front, a −→ a′, and the warm front, b −→ b′, respectively.

form, Leddy (as measured by, for example,the typical lateral scale of a low pressuresystem in the surface analysis shown inFig. 7.25, or the swirls of dye seen in ourtank experiment, Fig. 8.7 is proportional tothe Rossby radius of deformation discussedin Section 7.3.4:

Leddy ∼ Lρ, (8-3)

where Lρ is given by Eq. 7-23.The timescale of the disturbance is

given by:

Teddy ∼LρU

, (8-4)

where U = du/dz H is the strength of theupper level flow, du/dz is the thermal windgiven by Eq. 7-17 and H is the vertical scale

8.3. ENERGETICS OF THE THERMAL WIND EQUATION 145

of the flow. Eq. 8-4 is readily interpretable asthe time it takes a flow moving at speed Uto travel a distance Lρ and is proportional tothe (inverse of) the Eady growth rate, after EricEady, a pioneer of the theory of baroclinicinstability.

In our baroclinic instability experiment,we estimate a deformation radius of some10 cm, roughly in accord with the observedscale of the swirls in Fig. 8.7 (to determinea scale, note that the diameter of the icecan in the figure is 15 cm). Typical flowspeeds observed by the tracking of paperdots floating at the surface of the fluidare about 1 cm s−1. Thus Eq. 8-4 suggestsa timescale of Teddy ∼ 7.0 s, or roughly onerotation period. Consistent with Eq. 8-3, theeddy scale decreases with increased rotationrate.

Applying the above formulae to themiddle troposphere where f ∼ 10−4 s−1,N ∼ 10−2 s−1 (see Section 4.4), H ∼ 7 km(see Section 3.3), and U ∼ 10 m s−1, wefind that Lρ = NH/f ∼ 700 km and Teddy ∼700 km/10 m s−1 ∼ 1 d. These estimates areroughly in accord with the scales andgrowth rates of observed weather systems.

We have not yet discussed the underly-ing baroclinic instability mechanism and itsenergy source. To do so we now considerthe energetics of and the release of potentialenergy from a fluid.

8.3. ENERGETICS OF THETHERMAL WIND EQUATION

The immediate source of kinetic energyfor the eddying circulation observed in ourbaroclinic instability experiment and in themiddle-latitude atmosphere, is the poten-tial energy of the fluid. In the spirit of theenergetic discussion of convection devel-oped in Section 4.2.3, we now computethe potential energy available for conver-sion to motion. However, rather than, asthere, considering the energy of isolatedfluid parcels, here we focus on the potential

energy of the whole fluid. It will becomeapparent that not all the potential energy ofa fluid is available for conversion to kineticenergy. We need to identify that componentof the potential energy—known as availablepotential energy—that can be released by aredistributions of mass of the system.

To keep things as simple as possible, wewill first focus on an incompressible fluid,such as the water in our tank experiment. Wewill then go on to address the compressibleatmosphere.

8.3.1. Potential energy for a fluid system

We assign to a fluid parcel of volumedV = dx dy dz and density ρ, a potentialenergy of gz ×

(parcel mass

)= gzρ dV. Then

the potential energy of the entire fluid is

PE = g∫

zρ dV, (8-5)

where the integral is over the whole sys-tem. Clearly, PE is a measure of the verticaldistribution of mass.

Energy can be released and converted tokinetic energy only if some rearrangementof the fluid results in a lower total poten-tial energy. Such rearrangement is subjectto certain constraints, the most obvious ofwhich is that the total mass

M =∫ρ dV, (8-6)

cannot change. In fact, we may use Eq. 8-6to rewrite Eq. 8-5 as

PE = gM 〈z〉, (8-7)

where

〈z〉 =∫

z ρ dV∫ρ dV

=

∫z ρ dV

M(8-8)

is the mass-weighted mean height of thefluid. We can think of 〈z〉 as the ‘‘height ofthe center of mass.’’ It is evident from Eq. 8-7that potential energy can be released onlyby lowering the center of mass of the fluid.


8.3.2. Available potential energy

Consider again the stratified incom-pressible fluid discussed in Section 4.2.2.Suppose, first, that ∂ρ/∂z < 0 (stably strat-ified) and that there are no horizontalgradients of density, so ρ is a functionof height only, as depicted in Fig. 4.5. InSection 4.2.3 we considered the energeticimplications of adiabatically switching theposition of two parcels, initially at heights z1

and z2, with densities ρ1 and ρ2, respectively,with z2 > z1 and ρ2 < ρ1. Since ρ is conservedunder the displacement (remember we areconsidering an incompressible fluid here),the final state has a parcel with density ρ1 atz2, and one with ρ2 at z1. Thus density hasincreased at z2 and decreased at z1. The cen-ter of mass has therefore been raised, and soPE has been increased by the rearrangement;none can be released into kinetic energy.This of course is one way of understandingwhy the stratification is stable. Even thoughsuch a fluid has nonzero potential energy (asdefined by Eq. 8-7), this energy is unavailablefor conversion to kinetic energy. It cannot bereduced by any adiabatic rearrangement offluid parcels. Therefore the fluid has avail-able potential energy only if it has nonzerohorizontal gradients of density.7

Consider now the density distributionsketched in Fig. 8.9 (left); a two-layer fluid isshown with light fluid over heavy with theinterface between the two sloping at angle γ.This can be considered to be a highly idea-lized representation of the radial densitydistribution in our tank experiment GFDLab XI with heavier fluid (shaded) to the

right (adjacent to the ice bucket) and lightfluid to the left. Let us rearrange the fluidsuch that the interface becomes horizontalas shown. Now all fluid below the dashedhorizontal line is dense, and all fluid aboveis light. The net effect of the rearrangementhas been to exchange heavy fluid down-ward and light fluid upward such that, inthe wedge B, heavy fluid has been replacedby light fluid, while the opposite hashappened in the wedge A. (Such a rear-rangement can be achieved in more thanone way, as we shall see.) The center ofmass has thus been lowered, and the finalpotential energy is therefore less than thatin the initial state: potential energy has beenreleased. Since no further energy can bereleased once the interface is horizontal, thedifference in the potential energy betweenthese two states defines the available potentialenergy (APE) of the initial state.

The important conclusions here are thatavailable potential energy is associated withhorizontal gradients of density and thatrelease of that energy (and, by implication,conversion to kinetic energy of the motion)is effected by a reduction in those gradients.

The previous discussion can be madequantitative as follows. Consider againFig. 8.9, in which the height of the inter-face between the two fluids is given byh(y) = 1/2H + γy, and to keep things sim-ple we restrict the slope of the interfaceγ < H/2L to ensure that it does not inter-sect the lower and upper boundaries atz = 0, z = H respectively. Direct integrationof Eq. 8-5 for the density distribution shownin Fig. 8.9, shows that the potential energy

7 Eric Eady (1918–1967). A brilliant theorist who was a forecaster in the early part of his career,Eady expounded the theory of baroclinic instability in a wonderfully lucid paper—Eady (1949)—which attempted toexplain the scale, growth rate, and structure of atmospheric weather systems.


FIGURE 8.9. Reduction in the available potential energy of a two-layer fluid moving from an initial state inwhich the interface is tilted (left) to the final state in which the interface is horizontal (right). Dense fluid is shaded.The net effect of the rearrangement is to exchange heavy fluid downward and light fluid upward such that, in thewedge B, heavy fluid is replaced by light fluid, whereas the opposite occurs in the wedge A.

per unit length in the x−direction is givenby (as derived in Appendix 13.1.2)

P =∫H

0

∫L

−Lgρz dy dz

= H2Lρ1

(g +

g ′

4

)+

13ρ1g ′γ2L3,

where g′ = g(ρ1 − ρ2)/ρ1 is the reduced grav-ity of the system. P is at its absoluteminimum, Pmin, when γ = 0, and given bythe first term on the lhs of the above. Whenγ = 0, no rearrangement of fluid can lowerthe center of mass, since all dense fluid liesbeneath all light fluid. When γ = 0, however,P exceeds Pmin, and the fluid has availablepotential energy in the amount

APE = P − Pmin =13ρ1g ′γ 2L3 . (8-9)

Note that APE is independent of depthH, because all the APE is associated withthe interfacial slope; only fluid motion inthe vicinity of the interface changes poten-tial energy, as is obvious by inspection ofFig. 8.9.

It is interesting to compare the APE asso-ciated with the tilted interface in Fig. 8.9with the kinetic energy associated with thethermal wind current, which is in balancewith it. Making use of the Margules formula,Eq. 7-21, the geostrophic flow of the upper

layer is given by u2 = g ′γ/2Ω if the lowerlayer is at rest. Thus the kinetic energy ofthe flow per unit length is, on evaluation:

KE =∫H

h

∫L

−L

12ρ2u2

2 dy dz =12ρ2

g ′2γ 2

(2Ω)2HL

(8-10)

The ratio of APE to KE is thus:

APEKE

=23

(L

Lρ

)2

(8-11)

where we have set ρ1/ρ2 −→ 1 and usedEq. 7-23 to express the result in terms ofthe deformation radius Lρ.

Equation 8-11 is a rather general resultfor balanced flows—the ratio of APE to KEis proportional to the square of the ratioof L, the lateral scale over which the flowchanges, to the deformation radius Lρ. Typ-ically, on the largescale, L is considerablylarger than Lρ, and so the potential energystored in the sloping density surfaces greatlyexceeds the kinetic energy associated withthe thermal wind current in balance with it.For example, T varies on horizontal scales ofL ∼ 5000 km in Fig. 5.7. In Section 8.2.2 weestimated that Lρ ∼ 700 km; thus

(L/Lρ

)2 50. This has important implications foratmospheric and (as we shall see in Chapter10) oceanic circulation.8

8In fact in the ocean(L/Lρ

)2 400, telling us that the available potential energy stored in the main thermoclineexceeds the kinetic energy associated with ocean currents by a factor of 400. (See Section 10.5.)


8.3.3. Release of available potentialenergy in baroclinic instability

In a nonrotating fluid, the release of APEis straightforward and intuitive. If, at someinstant in time, the interface of our two-component fluid were tilted as shown inFig. 8.10 (left), we would expect the inter-face to collapse in the manner depicted bythe heavy arrows in the figure. In reality theinterface would overshoot the horizontalposition, and the fluid would slosh arounduntil friction brought about a motionlesssteady state with the interface horizontal.The APE released by this process wouldfirst have been converted to kinetic energyof the motion, and ultimately lost to fric-tional dissipation. In this case there wouldbe no circulation in the long term, but ifexternal factors such as heating at the leftof Fig. 8.10 (left) and cooling at the rightwere to sustain the tilt in the interface, therewould be a steady circulation (shown bythe light arrows in the figure) in which theinterface-flattening effects of the circulationbalance the heating and cooling.

In a rotating fluid, things are not soobvious. There is no necessity for a circula-tion of the type shown in Fig. 8.10 (left) todevelop since, as we saw in Section 7.3.5, anequilibrium state can be achieved in whichforces associated with the horizontal den-sity gradient are in thermal wind balance

with a vertical shear of the horizontal flow.In fact, rotation actively suppresses suchcirculations. We have seen that if Ω is suf-ficiently small, as in the Hadley circulationexperiment GFD Lab VIII (Section 8.2.1),such a circulation is indeed established.At high rotation rates, however, a Hadleyregime is not observed; nevertheless, APEcan still be released, albeit by a very differentmechanism. As we discussed in Section 8.2.2,the zonal state we have described is unstableto baroclinic instability. Through this insta-bility, azimuthally asymmetric motions aregenerated, within which fluid parcels areexchanged along sloping surfaces as theyoscillate from side to side, as sketched inFig. 8.8. The azimuthal current is thereforenot purely zonal, but wavy, as observed inthe laboratory experiment shown in Fig. 8.7.The only way light fluid can be movedupward in exchange for downward motionof heavy fluid is for the exchange to takeplace at an angle that is steeper than thehorizontal, but less steep than the densitysurfaces, or within the wedge of instability,as shown in Fig. 8.10 (right). Thus APE isreduced, not by overturning in the radialplane, but by fluid parcels oscillating backand forth within this wedge; light (warm)fluid moves upward and radially inwardat one azimuth, while, simultaneouslyheavy (cold) fluid moves downward and

FIGURE 8.10. Release of available potential energy in a two-component fluid. Left: Nonrotating (or very slowlyrotating) case: azimuthally uniform overturning. Right: Rapidly rotating case: sloping exchange in the wedge ofinstability by baroclinic eddies.


outward at another, as sketched in Fig. 8.8.This is the mechanism at work in GFDLab XI.9

8.3.4. Energetics in a compressibleatmosphere

So far we have discussed energetics inthe context of an incompressible fluid, suchas water. However, it is not immediatelyclear how we can apply the concept ofavailable potential energy to a compress-ible fluid, such as the atmosphere. Parcelsof incompressible fluid can do no workon their surroundings and T is conservedin adiabatic rearrangement of the fluid;therefore potential energy is all one needsto consider. However, air parcels expandand contract during the redistribution ofmass, doing work on their surroundingsand changing their T and hence internalenergy. Thus we must also consider changesin internal energy. It turns out that for anatmosphere in hydrostatic balance and witha flat lower boundary it is very easy toextend the definition Eq. 8-5 to include theinternal energy. The internal energy (IE)of the entire atmosphere (assuming it tobe a perfect gas) is, if we neglect surfacetopography (see Appendix 13.1.3)

IE = cv

∫ρT dV =

cv

Rg∫

zρ dV . (8-12)

Note the similarity to Eq. 8-5. In fact, for adiatomic gas like the atmosphere, cv/R =(1 − κ) /κ = 5/2, so its internal energy

(neglecting topographic effects) is 2.5 timesgreater than its potential energy. The sumof internal and potential energy is conven-tionally (if somewhat loosely) referred to astotal potential energy, and given by

TPE = PE + IE =gκ

∫zρ dV =

gκ

M 〈z〉 ,

(8-13)

where, as before, 〈z〉 is defined by Eq. 8-8and M by Eq. 8-6.

We can apply the ideas discussed in theprevious section to a compressible fluid if,as discussed in Chapter 4, we think in termsof the distribution of potential tempera-ture, rather than density, since the former isconserved under adiabatic displacement.

Consider Fig. 8.11, in which the distri-bution of θ in the atmosphere, θ increasingupward and equatorward as in Fig. 5.8, isschematically shown. Again, we supposethat two air parcels are exchanged adia-batically. Suppose first that we exchangeparcel 1, with θ = θ1, with a second parcelwith θ = θ2 > θ1, vertically along path A. Itis clear that the final potential energy willbe greater than the initial value because theparcel moving upward is colder than theparcel it replaces. This is just the problemconsidered in our study of stability condi-tions for dry convection in Section 4.3. Ifparcels are exchanged along a horizontalpath or along a path in a surface of con-stant θ (dotted lines in the figure), there willbe no change in potential energy. But con-sider exchanging parcels along path B in

9 Jule Gregory Charney (1917–1981), American meteorologist and MIT Professor. He madewide-ranging contributions to the modern mathematical theory of atmospheric and oceanic dynamics, including thetheory of large-scale waves, and of baroclinic instability. He was also one of the pioneers of numerical weather prediction,producing the first computer forecast in 1950.


FIGURE 8.11. Air parcels 1 and 2 are exchangedalong paths marked A and B, conserving potentialtemperature θ. The continuous lines are observed θsurfaces (see Fig. 5.8). The tilted dotted line is parallelto the local θ.

Fig. 8.11, which has a slope between that ofthe θ surface and the horizontal. The parcelmoving upward is now warmer than theparcel moving downward, and so poten-tial energy has been reduced by the parcelrearrangement. In other words the center ofgravity of the fluid is lowered. In the process,a lateral transport of heat is achieved fromcold to warm. It is this release of APE thatpowers the eddying motion. As discussedin Section 8.3.3, fluid parcels are exchangedin this wedge of instability as they oscillatefrom side to side, as sketched in Fig. 8.8.

8.4. LARGE-SCALE ATMOSPHERICHEAT AND MOMENTUM BUDGET

8.4.1. Heat transport

We have seen two basic facts about theatmospheric energy budget:

1. There must be a conversion of energyinto kinetic energy, from the availablepotential energy gained from solarheating, which requires that motionsdevelop that transport heat upward

(thereby tending to lower theatmospheric center of mass).

2. Accompanying this upward transport,there must be a poleward transport ofheat from low latitudes; this transportcools the tropics and heats the polarregions, thus balancing the observedradiative budget.

It is fairly straightforward to estimate therate of poleward heat transport by atmo-spheric motions. Consider the northwardflow at latitude ϕ across an elemental areadA, of height dz and longitudinal width dλ,so dA = a cosϕ dλ dz. The rate at which massis flowing northward across dA is ρv dA,and the associated flux of energy is ρvE dA,where

E = cpT + gz + Lq +12

u · u (8-14)

is the moist static energy, Eq. 4-27 (madeup of the ‘‘dry static energy’’ cpT + gz, andthe latent heat content Lq) and the kineticenergy density 1

2 u · u. If we now integrateacross the entire atmospheric surface at thatlatitude, the net energy flux is

Hλatmos =

∫∫ρvE dA

= a cosϕ∫ 2π

0

∫∞

0ρvE dz dλ

=ag

cosϕ∫ 2π

0

∫ ps

0vE dp dλ, (8-15)

(using hydrostatic balance to replace ρdzby −dp/ρ) where ps is surface pressure. Itturns out that the kinetic energy componentof the net transport is negligible (typicallyless than 1% of the total), so we only need toconsider the first three terms in Eq. 8-14.

Let us first consider energy transport inthe tropics by the Hadley circulation which(in the annual mean) has equatorward flowin the lower troposphere and polewardflow in the upper troposphere. We maywrite

Hλtropics =

2πag

cosϕ∫ ps

0v(cpT + gz + Lq

)dp,

(8-16)

8.4. LARGE-SCALE ATMOSPHERIC HEAT AND MOMENTUM BUDGET 151

where we have assumed that the tropicalatmosphere is independent of longitude λ.

Since temperature decreases with alti-tude, the heat carried poleward by the uppertropospheric flow is less than that carriedequatorward at lower altitudes. Thereforethe net heat transport—the first term inEq. 8-16—is equatorward. The Hadley cir-culation carries heat toward the hot equatorfrom the cooler subtropics! This does notseem very sensible. However, now let usadd in the second term in Eq. 8-16, thepotential energy density gz, to obtain theflux of dry static energy v

(cpT + gz

). Note

that the vertical gradient of dry static energyis ∂

(cpT + gz

)/∂z = cp∂T/∂z + g. For an atmo-

sphere (like ours) that is generally stableto dry convection, ∂T/∂z + g/cp > 0 (as dis-cussed in Section 4.3.1) and so the dry staticenergy increases with height. Thereforethe poleward flow at high altitude car-ries more dry static energy poleward thanthe low-level flow takes equatorward, andthe net transport is poleward, as we mightexpect.

To compute the total energy flux wemust include all the terms in Eq. 8-16. Thelatent heat flux contribution is equatorward,since (see Fig. 5.15) q decreases rapidly withheight, and so the equatorward flowing airin the lower troposphere carries more mois-ture than the poleward moving branch aloft.In fact, this term almost cancels the dry staticenergy flux. This is because the gross verticalgradient of moist static energy cpT + gz + Lq,a first integral of Eq. 4-26 under hydro-static motion, is weak in the tropics. This isjust another way of saying that the tropicalatmosphere is almost neutral to moist con-vection, a state that moist convection itselfhelps to bring about. This is clearly seen inthe profiles of moist potential temperatureshown in Fig. 5.9. Thus the equatorward andpoleward branches of the circulation carryalmost the same amount in each direction. Inthe net, then, the annually averaged energyflux by the Hadley cell is poleward, butweakly so. In fact, as can be seen in Fig. 8.13,the heat transport by the ocean exceeds that

of the atmosphere in the tropics up to 15 orso, particularly in the northern hemisphere.

The relative contributions of the variouscomponents of the energy flux are differ-ent in the extratropics, where the meancirculation is weak the greater part of thetransport is effected by midlatitude eddies.In these motions, the poleward and equa-torward flows occur at almost the samealtitude, and so the strong vertical gradi-ents of gz and Lq are not so important.Although these components of the energyflux are not entirely negligible (and must beaccounted for in any detailed calculation),we can use the heat flux alone to obtainan order of magnitude estimate of the netflux. So in middle latitudes we can representEq. 8-15 by

Hλmid−lat ∼ 2π

acp

gcosϕ ps [v][T],

where [v] is a typical northward wind veloc-ity (dominated by the eddy component inmiddle latitudes), and [T] is the typicalmagnitude of the temperature fluctuationsin the presence of the eddies—the temper-ature difference between equatorward andpoleward flowing air. Given a = 6371 km,cp = 1005 J kg−1 K−1, g = 9.81 m s−2, ps 105 Pa, then if we take typical values of[v] 10 m s−1 and [T] 3 K at a latitude of

45, we estimate Hλmid−lat ∼ 8 PW. As dis-

cussed earlier in Section 5.1.3 and in moredetail later, this is of the same order asimplied by the radiative imbalance.

In cartoon form, our picture of the low-and high-latitude energy balance is there-fore as shown in Fig. 8.12 (left). In thetropics energy is transported poleward bythe Hadley circulation; in higher latitudes,eddies are the principal agency of heattransport.

The results of more complete calcula-tions, making use of top of the atmosphereradiation measurements and analyzedatmospheric fields, of heat fluxes in theatmosphere and ocean are shown inFig. 8.13. The bulk of the required transport


FIGURE 8.12. Schematic of the transport of (left) energy and (right) momentum by the atmospheric generalcirculation. Transport occurs through the agency of the Hadley circulation in the tropics, and baroclinic eddies inmiddle latitudes (see also Fig. 8.1).

FIGURE 8.13. The ocean (thin) and atmospheric(dotted) contributions to the total northwards heat flux(thick) based on the NCEP reanalysis (in PW = 1015 W)by (i) estimating the net surface heat flux over the ocean,(ii) the associated oceanic contribution, correcting forheat storage associated with global warming and con-straining the ocean heat transport to be −0.1 PW at68 S, and (iii) deducing the atmospheric contributionas a residual. The total meridional heat flux, as inFig. 5.6, is also plotted (thick). From Trenberth andCaron (2001).

is carried by the atmosphere in middleand high latitudes, but the ocean makesup a considerable fraction, particularly inthe tropics where (as we have seen) atmo-spheric energy transport is weak. The roleof the ocean in meridional heat transport,and the partition of heat transport betweenthe atmosphere and ocean, are discussed insome detail in Section 11.5.2.

8.4.2. Momentum transport

In addition to transporting heat, thegeneral circulation also transports angularmomentum. In the Hadley circulation, aswe have seen, the upper, poleward-flowing,branch is associated with strong westerlywinds. Thus westerly momentum is carriedpoleward. The lower branch, on the otherhand, is associated with easterlies that areweakened by surface friction; the equator-ward flow carries weak easterly momentumequatorward. The net effect is a polewardtransport of westerly angular momentum.Midlatitude eddies also transport angularmomentum (albeit for less obvious and, infact, quite subtle reasons, as sketched inFig. 8.14), again mostly transporting east-ward angular momentum poleward. Thishas the effect of modifying the surfacewinds from that sketched in Fig. 8.5, byshifting the low latitude westerlies intomiddle latitudes.

The atmospheric angular momentumbudget may therefore be depicted as inFig. 8.12 (right). Because there is a net exportof momentum out of low latitudes, theremust be a supply of momentum into thisregion; the only place it can come from isthe surface, through friction acting on thelow level winds. For the friction to supplyangular momentum to the atmosphere (andyet act as a brake on the low level winds),the low level winds in the tropics must

8.5. LATITUDINAL VARIATIONS OF CLIMATE 153

FIGURE 8.14. Circular eddies on the left are unable to affect a meridional transfer of momentum. The eddieson the right, however, by virtue of their ‘‘banana shape,’’ transfer westerly momentum northwards south of, andsouthwards north of, the midline. Weather systems transport westerly momentum from the tropics

(uv > 0

)toward

higher latitudes, as required in Fig. 8.12 (right), by ‘‘trailing’’ their troughs down in to the tropics (see, e.g., the H2Odistribution shown in Fig. 3), as in the southern half of the ‘‘banana-shaped’’ eddy sketched on the right.

be easterly, consistent with that deducedfor the Hadley circulation here. The bal-ancing loss of westerly momentum fromthe atmosphere—which must be associatedwith drag on the near-surface westerlies—islocated in middle latitudes and balancesthe supply of westerly momentum via eddytransport.

8.5. LATITUDINAL VARIATIONSOF CLIMATE

Putting everything together, we candepict the atmospheric wind systems in theupper and lower troposphere schematicallyas in Fig. 8.15.

As we have remarked at various pointsin this chapter, the structure of the circula-tion dictates more than just the pattern ofwinds. In the near-equatorial regions, theconvergence of the trade winds is associ-ated with frequent and intense rainfall, asis characteristic of the deep tropics. It ishere, for example, that the world’s greattropical rainforests are located. Around thesubtropics (20–30 latitude), in the warm,dry, descending branch of the Hadley cir-culation, the climate is hot and dry: thisis the subtropical desert belt. Poleward ofabout 30, where baroclinic eddies domi-nate the meteorology, local winds vary from

day to day, though they are predominantlywesterly, and the weather is at the mercyof the passing eddies: usually calm and finein anticyclonic high pressure systems, andfrequently wet and stormy in cyclonic, lowpressure systems.

At least from a ‘‘big picture,’’ we havethus been able to deduce the observedpattern of atmospheric winds, and the divi-sion of the world into major climate zones,through straightforward consideration ofthe properties of a dynamic atmosphereon a longitudinally uniform rotating planetsubject to longitudinally independent dif-ferential heating. We cannot proceed muchfurther without taking account of seasonalvariations, and of longitudinal asymmetryof the Earth, which we do not have the spaceto develop here. However, we will brieflynote some of the major differences betweenour simple picture and observed circulation.

During the course of a year, the patternof solar forcing migrates, north in northernsummer, south in southern summer, andthis has a significant impact on the atmo-spheric response. Our picture of the ‘‘typi-cal’’ Hadley circulation (Fig. 8.5) inevitablyshows symmetry about the equator. Inreality, the whole circulation tends to shiftseasonally such that the upwelling branchand associated rainfall are found on thesummer side of the equator (see Fig. 5.21).


Longitute

FIGURE 8.15. Schematic of the global distributions of atmospheric winds in the upper (top) and lower (bottom)troposphere together with the primary latitudinal climate zones.

The degree to which this happens is stronglycontrolled by local geography. Seasonalvariations over the oceans, whose tempera-tures vary relatively little through the year,are weak, while they are much strongerover land. The migration of the main areaof rainfall is most dramatic in the region ofthe Indian Ocean, where intense rain movesonto the Asian continent during the summermonsoon.

Although prevailing westerlies and mig-rating storms characterize most of the extra-tropical region throughout the year, thereare seasonal variations (generally with morestorms in winter, when the temperature gra-dients are greater, than in summer) andstrong longitudinal variations, mostly asso-ciated with the distribution of land andsea. Longitudinal variations in climate canbe dramatic: consider the contrast acrossthe Atlantic Ocean, where the Januarymean temperature at Goose Bay, New-

foundland (at about 53 N) is about 30 Ccolder than that at Galway, at almost thesame latitude on the west coast of Ireland.We cannot explain this with our longitudi-nally symmetric model of the world, but theexplanation is nevertheless straightforward.Prevailing winds bring cold continental airto the coast of Newfoundland, but mildoceanic air to the west coast of Ireland. TheNorth Atlantic Ocean is much warmer thanthe continents at the same latitude in winter,partly because of the large heat capacity ofthe ocean, and partly because of heat trans-port by the ocean circulation. The oceancirculation and its role in climate is thesubject of the following chapters.

8.6. FURTHER READING

An interesting historical account of evolv-ing perspectives on the general circulation

8.7. PROBLEMS 155

of the atmosphere is given in Lorenz (1967).Chapter 6 of Hartmann (1994) presentsa thorough discussion of the energy andangular budgets of the general circulation;a more advanced treatment can be found inChapter 10 of Holton (2004).

8.7. PROBLEMS

1. Consider a zonally symmetriccirculation (i.e., one with nolongitudinal variations) in theatmosphere. In the inviscid uppertroposphere, one expects such a flowto conserve absolute angularmomentum, so that DA/Dt = 0, whereA = Ωa2 cos 2ϕ + ua cosϕ is the absoluteangular momentum per unit mass(see Eq. 8-1) where Ω is the Earthrotation rate, u the eastward windcomponent, a the Earth’s radius, and ϕ

latitude.

(a) Show for inviscid zonallysymmetric flow that the relationDA/Dt = 0 is consistent with thezonal component of the equation ofmotion (using our standardnotation, with Fx the x-componentof the friction force per unit mass)

DuDt

− fv = −1ρ

∂p∂x

+ Fx ,

in (x, y, z) coordinates, wherey = aϕ (see Fig. 6.19).

(b) Use angular momentumconservation to describe how theexistence of the Hadley circulationexplains the existence of both thesubtropical jet stream in the uppertroposphere and the near-surfacetrade winds.

(c) If the Hadley circulation issymmetric about the equator, andits edge is at 20 latitude,determine the strength of thesubtropical jet stream.

FIGURE 8.16. A schematic diagram of the Hadleycirculation rising at 10 S, crossing the equator andsinking at 20 N.

2. Consider the tropical Hadleycirculation in northern winter, asshown in Fig. 8.16. The circulationrises at 10 S, moves northward acrossthe equator in the upper troposphere,and sinks at 20 N. Assuming that thecirculation outside the near-surfaceboundary layer is zonally symmetric(independent of x) and inviscid (andthus conserves absolute angularmomentum about the Earth’s rotationaxis), and that it leaves the boundarylayer at 10 S with zonal velocity u = 0,calculate the zonal wind in the uppertroposphere at (a) the equator,(b) 10 N, and (c) 20 N.

3. Consider what would happen if a forcetoward the pole was applied to thering of air considered in Problem 1 andit conserved its absolute angularmomentum, A. Calculate the impliedrelationship between a smalldisplacement δϕ and the change in thespeed of the ring δu. How manykilometers northwards does the ringhave to be displaced to change itsrelative velocity 10 m s−1? How doesyour answer depend on theequilibrium latitude? Comment onyour result.

4. An open dish of water is rotatingabout a vertical axis at 1 revolution per


minute. Given that the water is 1Cwarmer at the edges than at the centerat all depths, estimate, under statedassumptions and using the followingdata, typical azimuthal flow speeds atthe free surface relative to the dish.Comment on, and give a physicalexplanation for, the sign of the flow.

How much does the free surfacedeviate from its solid body rotationform?

Briefly discuss ways in which thisrotating dish experiment is a usefulanalogue of the general circulation ofthe Earth’s atmosphere.

Assume the equation of state given byEq. 4-4 with ρref = 1000 kg m−3,α = 2 × 10−4 K−1, and Tref = 15 C, themean temperature of the water in thedish. The dish has a radius of 10 cmand is filled to a depth of 5 cm.

5. Consider the incompressible,baroclinic fluid [ρ = ρ(T)] sketched inFig. 8.17 in which temperature surfacesslope upward toward the pole at anangle s1. Describe the attendant zonalwind field assuming it is in thermalwind balance.

By computing the potential energybefore and after interchange of two

FIGURE 8.17. Schematic for energetic analysis of thethermal wind considered in Problem 5. Temperaturesurfaces have slope s1: parcels of fluid are exchangedalong surfaces which have a slope s.

rings of fluid (coincident with latitudecircles y at height z) along a surface ofslope s, show that the change inpotential energy PE = PEfinal− PEinitialis given by

PE = ρref N2 (y2 − y1

)2 s (s − s1) ,

where N 2 = −g/ρref ∂ρ/∂z is thebuoyancy frequency (see Section 4.4),ρref is the reference density of the fluid,and y1, y2 are the latitudes of theinterchanged rings. You will find ituseful to review Section 4.2.3.

Hence show that for a givenmeridional exchange distance(y2 − y1

):

(a) energy is released if s < s1;

(b) the energy released is a maximumwhen the exchange occurs alongsurfaces inclined at half the slopeof the temperature surfaces.

This is the wedge of instabilitydiscussed in Section 8.3.3 andillustrated in Fig. 8.10.

6. Discuss qualitatively, but from basicprinciples, why most of the Earth’sdesert regions are found at latitudes of± (20–30).

7. Given that the heat content of anelementary mass dm of air attemperature T is cpT dm (where cp isthe specific heat of air at constantpressure), and that its northwardvelocity is v:

(a) show that the northward flux ofheat crossing unit area (in the x − zplane) per unit time is ρcpvT;

(b) hence, using the hydrostaticrelationship, show that the netnorthward heat flux H in theatmosphere, at any given latitude,can be written

8.7. PROBLEMS 157

FIGURE 8.18. The contribution by eddies ( K m s−1) to the meridional flux vT (Eq. 8-17, plotted as a function oflatitude and pressure for (top) DJF and (bottom) JJA.

H = cp

∫ x2

x1

∫∞

0ρ vT dx dz

=cp

g

∫ x2

x1

∫ ps

0vT dx dp ,

where the first integral (in x) iscompletely around a latitude circleand ps is surface pressure.

(c) Fig. 8.18 shows the contribution ofeddies to the atmospheric heat flux.

What is actually shown is thecontribution of eddies to thequantity vT, which in the abovenotation is

vT =1

x2 − x1

∫ x2

x1

vT dx (8-17)

(i.e., the zonal average of vT),where x2 − x1 = 2πa cosϕ, where ais the Earth radius and ϕ latitude.Use the figure to estimate the netnorthward heat flux by eddiesacross 45 N. Compare this withthe requirement (from the Earth’sradiation budget and Fig.8.13 thatthe net (atmosphere and ocean)heat transport must be about5 × 1015 W.

the general circulation of the atmosphereaczaja/pg2008/marshallplumb_atmosphere.pdf · energetics...

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