6 The Role of Eddies in the General Circu-lation

6.1 The axisymmetric state

At the beginning of the class, we discussed the nonlinear, inviscid, axisym-metric theory of the meridional structure of the atmosphere. The importantaxisymmetric equations can be written as

@m

@t+ u � rm =

1

�

@�

@z; (1)

@T

@t+ v

@T

@y+ wS = Q ; (2)

where m = a2 cos2 '+ ua cos' is the speci�c absolute angular momentum,� is the frictional stress, S = (p=p0)

� @�=@z, and Q = J=�cp, where J is thediabatic heating rate. We discussed how, in the inviscid free atmosphere, (1)implies that in steady state u � rm = 0, and in turn this led us to concludethat a nonzero meridional circulation can exist only in a tropical region wherem is uniform; in the extratropics, there can be no meridional circulation, asdepicted in Fig. 6.1. As we also discussed, in the tropics this produces a �at

v=w =0-> Q=0-> τ =0s

Q<0Q>0

EQ POLE

-> T=T-> u =0s

temperature distribution, with diabatic heating in the upwelling region and

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diabatic cooling in the subtropical downwelling, with a westerly subtropicaljet in the upper troposphere, at the outer wings of the Hadley cell. Outsidethe Hadley cell, the absence of a meridional circulation implies radiativeequilibrium withQ = 0 in steady state, and T = Te, the radiative equilibriumtemperature. Further, since there is no angular momentum transport insteady state, the vertical integral of (1) gives � s = 0: there can be no surfacestress whence the zonal velocity us at the surface must vanish.Especially in the extratropics, this picture is not right. For one thing, tem-

peratures are not in radiative (or radiative-convective) equilibrium; e.g., thepoles are much warmer than they would then be. For another, the meridionalcirculation is shown in Fig. 1; while it is indeed strongest in the tropics, there

Figure 1: Annual and zonal mean meridional streamfunction [Oort]

are noticeable reverse (thermally indirect) circulations in middle latitudes�

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the �Ferrel�cells1. Further, as Fig. 2 shows, the distribution of zonal winds

Figure 2: Annual and zonal mean zonal wind [Peixoto & Oort].

is also inconsistent with the axisymmetric picture; the well-known surfaceextratropical westerlies are at odds with the prediction of zero surface stress,and the upper tropospheric jets, while being in more-or-less the right place,are much weaker than they would be if there were no gradient of absoluteangular momentum across the tropical upper troposphere. Missing from ourpicture are the ubiquitous extratropical eddies.

6.2 The extratropical mean state in the presence ofeddies

Let�s consider the impact of extratropical eddies. To be consistent with ourtreatment of eddies, we�ll assume QG theory is valid and (for the moment)

1The Ferrel cells, unlike the Hadley cells, are not robust with respect to changes in howone does the zonal averaging. E.g., averages along isentropes produce an extratropicalcirculation in the same sense as, but weaker than, the Hadley circulation (the same is trueof the �residual�circulation.)

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we�ll ignore spherical geometry. Separate everything into its zonal mean andeddy component:

�a(y; z; t) =1

L

Z L

0

a(x; y; z; t) dx ;

a0(x; y; z; t) = a(x; y; z; t)� �a(y; z; t) :

The zonally averaged QG equations are then

@�u

@t� f�v =

1

�

@�

@z� @

@y

�u0v0

�@ �T

@t+ S �w = �Q� @

@y

�v0T 0

�(3)

@�v

@y+1

�

@

@z(� �w) = 0

and

f@�u

@z+R

H

@ �T

@y= 0 : (4)

(Note that �v; �w are ageostrophic, because �vg = @ =@x = 0, and so terms like�v@�u=@y and �v@ �T=@y are formally negligible.) Thus, the impact of the eddieson the mean state is felt through only two terms2 ;3: an e¤ective zonal force(per unit mass) and an e¤ective diabatic heating, which are represented bythe convergence of the eddy �uxes of momentum and of heat, respectively. Itis tempting, then, to regard the eddy momentum �ux as impacting the meanzonal wind, and the eddy heat �ux as impacting mean T , but that would bea mistake, since thermal wind shear balance (4) ensures that the mean windand temperature impacts are linked (the linkage occurs via the meridionalcirculation). So one needs to be careful deciding what does what.First, let�s look at the eddy �uxes in a climatology. The eddy heat �uxes,

shown in Fig. 3(a) and (b) for the transient and stationary waves respectively,are generally poleward except in the subtropics, where they are weak. Theeddy momentum�uxes, Fig. 4(b) and (c), are also generally poleward, except

2There could be less direct impacts, e.g., if rain is produced within the eddies, theywill have an impact on �Q.

3In fact, it is possible to reorganize (3) such that the eddy term disappears from theheat budget, in which case the eddy term in the momentum budget becomes just ��1r�F,the divergence of the EP �ux (but that is another story).

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Figure 3: Stationary and eddy heat �uxes (annual and zonal means). [Peixoto& Oort]

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Figure 4: Eddy angular momentum �uxes for (b) transient eddies and(c)stationary waves. [Peixoto & Oort]

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poleward of 50� 60o latitude, where they become equatorward.Why do the �uxes look like this? The eddy heat �uxes are relatively

straightforward. Recall our energetic argument that for baroclinic eddiesv0T 0 must be directed generally poleward; we saw that we expect the same tobe true for upward-propagating stationary Rossby waves. The momentum�uxes are a little less obvious. We can actually understand them in simpleterms, by invoking some of the principles we discussed earlier in the class,including our �nding that the EP �ux,

F = (Fy; Fz) =

���u0v0; �f v

0T 0

S

�; (5)

indicates the direction of wave activity propagation. That this is so is evidentfrom the Eliassen-Palm law for conservation of wave activity, which makesit clear that, e.g., a convergence of F indicates a local accumulation of waveactivity. It is con�rmed, for the case of �almost-plane�waves for which wecan de�ne group velocity, by the result that F = cgA, where A is the waveactivity density (P Set 4, Question 2). This means that if we know F we candeduce the direction of propagation and, conversely, if we know the directionof propagation (because, e.g., we know where the sources are) we can deducea qualitative picture of what F is, and hence of the momentum and heat�uxes.Now, as noted above, we expect the EP �uxes to be generally upward

(v0T 0 poleward) for both types of eddy, as is con�rmed by observations (Fig.5). The �uxes are generally upward except in summer, when the stationarywave �ux is actually downward, but the �uxes are actually weak then (notethe arrow scale on the upper right of the �gure) and so are of little con-sequence. Note, especially for the transients, the �uxes �splay out� in theupper troposphere, producing a predominantly equatorward component, butpoleward in higher latitudes. Through (5), we can see that this is consistentwith the behavior of the momentum �ux.To understand the gross aspects of the �uxes, consider Fig. 6. On a

plane with a spatially homogeneous baroclinic zone [Fig. 6(a)] the eddy heat�ux is poleward whence Fz > 0: there is a source of wave activity at thelower boundary, where the surface temperature gradient allows the stabilitycriterion to be violated. But spatial homogeneity requires Fy = 0 whenceu0v0 = 0 everywhere in this case4. Suppose now, however [Fig. 6(b)], that

4Hence, e.g., the momentum �ux is zero in the classic Eady problem.

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Figure 5: Northern hemisphere EP �uxes for transient and stationary waves.[Peixoto & Oort]

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EQ POLE

v'T' > 0

u' v' > 0 u' v' < 0

EQ POLE

v'T' > 0

u' v' > 0 u' v' < 0

v'T' > 0

u' v' = 0z

TROPOPAUSE

SURFACE

(a)

(b)

(c)

Figure 6: Fluxes associated with baroclinic eddies (schematic). Arrows showEP �ux vector, �uxes of momenum and heat represented by light and darkshading, repectively. (a) homogeneous case, (b) localized baroclinic zone ona �-plane, (c) localized baroclinic zone on a sphere.

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the baroclinic zone is localized in latitude. Then the source of wave activityis similarly localized; the EP �uxes point upward (v0T 0 poleward) out of thesurface baroclinic zone, but spread out laterally away from the source. Thus,Fy points away from the baroclinic zone (and thus away from the jet), whencethe eddy momentum �ux �u0v0 = �Fy, on both sides, is directed into the jet.From the cautionary note above, we cannot immediately conclude from thisthat the eddies strengthen the jet; we�ll see below that it does mean.On a sphere, we saw that for simple geometric reasons there is a bias

toward equatorward propagation. This means [Fig. 6(c)] that , compared tocase (b), the region of equatorward EP �uxes and poleward eddy momen-tum �ux is enhanced, while that of poleward EP �ux and equatorward eddymomentum �ux is weakened. The scenario depicted in Fig. 6(c) is indeedsimilar to what is observed, both for EP �uxes, and for heat and momentum�uxes.What does the eddy transport do? In steady state (such as we expect to

encounter near the solstices) the momentum budget is straightforward:

�f�v = �1�

@�

@z� @

@y

�u0v0

�: (6)

If we integrate this in the vertical, given that mass balance demands no netnorthward mass �ux, i.e. Z 1

0

��v dz = 0 ;

and the impossibility of any frictional stress at in�nity, we �nd that thesurface stress must be balanced by the eddy momentum transport:

� s =@

@y

Z 1

0

� u0v0 dz : (7)

This column balance, illustrated in Fig. 6.2(a), simply states that the onlycontributions to the column are from eddy momentum �uxes and surfacestress. The mean circulation makes no contribution because the absence of anet mass �ux means no net advection of planetary angular momentum, andbecause the QG assumption renders advection of relative angular momentumnegligible here5. Hence, in the vicinity of the generation of eddy activity�the midlatitude baroclinic zone� where the momentum �uxes are convergent

5Unlike in the tropical Hadley cell.

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u' v'

τs

W

(a)

(u'v') balanced by fvy

fv balanced by surface stress

WARMING COOLING

EQ POLE

(b)

WARMINGCOOLING

POLE

(c)

EQ v'T'

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in the upper troposphere, the most direct impact is a positive surface stress,and hence surface westerlies.If we go beyond the vertical integral, and if we make the reasonable

assumption that frictional stresses are negligible outside the boundary layer,then in the free atmosphere (6) tells us that

f�v =@

@y

�u0v0

�(8)

so the region of momentum �ux convergence in the upper troposphere mustbe balanced by mean equatorward �ow, as depicted in Fig. 6.2(b). Applyingcontinuity, this must become vertical motion at the edges of the region; whichway does it go? We could integrate the continuity equation upward ordownward to get �w, but going downward involves the boundary layer, where(8) is not valid. If instead we integrate from in�nity and apply the constraintthat � �w ! 0 at in�nity, we have

� �w(z) =@

@y

Z 1

z

��v dz

So the vertical motion must be as shown, upward in low latitudes and down-ward at higher latitudes. The return �ow then occurs in the boundary layer,where the Coriolis term is balanced by surface stress. Thus, we see thatthe thermally indirect Ferrel cell must be driven by eddy momentum �uxes:heat �uxes have not entered the argument.There is a thermal impact of the momentum �uxes, however: adiabatic

warming/cooling associated with the Ferrel cell will produce warming in lowlatitudes and cooling in high latitudes! But this is where the eddy heat�uxes come into play, as depicted in Fig. 6.2(c). The poleward transport ofheat represented by the eddy �ux v0T 0 dominates over the Ferrel cell e¤ect(as it must, in a gross sense) and the net e¤ect of the eddies is to reducethe pole-to-equator temperature gradient overall, and correspondingly to re-duce the baroclinic shear of the mean �ow. How the strength of the uppertropospheric jet is impacted depends on the competing e¤ects of barotropicacceleration/baroclinic deceleration, which depends on many factors. [Theseissues were addressed by Robinson, J. Atmos. Sci., 1991).]

6.3 Role of eddies in the tropics (some brief remarks)

The e¤ects of eddies are not con�ned to the extratropical regions; as we haveseen, the tropical �ow is not axisymmetric. There are eddies produced within

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the tropics (some of which are the stationary eddies that make up the Walkercirculation, and monsoon �ows). In the upper troposphere the tropicalatmosphere is also bombarded by eddies of extratropical origin which, as wesaw, inevitably propagate into the tropics because of the e¤ects of sphericalgeometry. These latter eddies provide a wave drag (EP �ux convergence)on the upper tropospheric westerlies. How eddies of tropical origin impactthe zonal mean state is less obvious.That eddy transport is at least quantitatively important is clear from

climatology: the subtropical jets, for example, are much weaker than theywould be if angular momentum were conserved in the upper troposphericout�ow, as we earlier assumed it was in our 2D theory. Further, it appearsthat eddies may in fact be in control of the key parameter dependencies ofthe Hadley circulation (Walker & Schneider, J. Atmos. Sci., 2006) and of itsseasonal variations (Schneider & Bordoni, J. Atmos. Sci., 2008). Furtherstill, the width of the cell may be controlled by extratropical eddies: if oneregards that edge as being the boundary between the �direct�Hadley cell andthe �indirect�Ferrel cell, then since the Ferrel cell (as we have just discussed)is driven by baroclinic eddies, the location of the edge may be determinedby extratropical baroclinicity (e.g., Korty & Schneider, Geophys. Res. Lett.,2008).

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