7 The General Circulation

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Jennifer Parks
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1 7 The General Circulation 7.1 The axisymmetric state At the beginning of the class, we discussed the nonlinear, inviscid, axisymmetric theory of the meridional structure of the atmosphere. The important axisymmetric equations can be + u rm ws = Q ; where m = a 2 cos 2 ' + ua cos ' is the speci c absolute angular momentum, is the frictional stress, S = (p=p 0 and Q = J=c p, where J is the diabatic 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 conclude that a nonzero meridional circulation can exist only in a tropical region where m is uniform; in the extratropics, there can be no meridional circulation, as depicted in Fig As we also discussed, in the tropics this produces a at Q>0 Q<0 v=w=0 -> Q=0 -> τ =0 s -> T=T -> u =0 s EQ POLE temperature distribution, with diabatic heating in the upwelling region and diabatic cooling in the subtropical downwelling, with a westerly subtropical jet in the upper troposphere, at the outer wings of the Hadley cell. Outside 1

2 the Hadley cell, the absence of a meridional circulation implies radiative equilibrium with Q = 0 in steady state, and T = T e, the radiative equilibrium temperature. Further, since there is no angular momentum transport in steady state, the vertical integral of (1) gives s = 0: there can be no surface stress whence the zonal velocity u s at the surface must vanish. Especially in the extratropics, this picture is not quite right. For one thing, temperatures are not in radiative (or radiative-convective) equilibrium; e.g., the poles are much warmer than they would then be. For another, the meridional circulation is shown in Fig. 1; while it is indeed strongest Figure 1: Annual and zonal mean meridional streamfunction [Oort] in the tropics, there are noticeable reverse (thermally indirect) circulations 2

3 in middle latitudes the Ferrel cells 1. Further, as Fig. 2 shows, the dis- Figure 2: Annual and zonal mean zonal wind [Peixoto & Oort]. tribution of zonal winds is also inconsistent with the axisymmetric picture; the well-known surface extratropical 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 absolute angular momentum across the tropical upper 1 The Ferrel cells, unlike the Hadley cells, are not robust with respect to changes in how one does the zonal averaging. E.g., averages along isentropes produce an extratropical circulation in the same sense as, but weaker than, the Hadley circulation (the same is true of the residual circulation.) 3

4 Missing from our picture are the ubiquitous extratropical ed- troposphere. dies. 7.2 The extratropical mean state in the presence of eddies Let s consider the impact of extratropical eddies. To be consistent with our treatment of eddies, we ll assume QG theory is valid and (for the moment) we ll ignore spherical geometry. Separate everything into its zonal mean and eddy component: Z L a(y; z; t) = 1 a(x; y; z; t) dx ; L 0 a 0 (x; y; z; t) = a(x; y; z; t) a(y; z; t) : The zonally averaged QG equations are then f v + S w + ( u0 v0 T 0 + T = 0 : (4) (Note that v; w are ageostrophic, because v g =@x = 0, and so terms like v@u=@y and v@ T =@y are formally negligible.) Thus, the impact of the eddies on the mean state is felt through only 2 terms 23 : an e ective zonal force (per unit mass) and an e ective diabatic heating, which are represented by the convergence of the eddy uxes of momentum and of heat, respectively. It is tempting, then, to regard the eddy momentum ux as impacting the mean 2 There could be less direct impacts, e.g., if rain is produced within the eddies, they will have an impact on Q. 3 In fact, it is possible to reorganize (3) such that the eddy term disappears from the heat budget, in which case the eddy term in the momentum budget becomes just 1 rf, the divergence of the EP ux. 4

5 zonal wind, and the eddy heat ux as impacting mean T, but that would be a mistake, since thermal wind shear balance (4) ensures that the mean wind and temperature impacts are linked (the linkage occurs via the meridional circulation). 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. The eddy momentum uxes, Fig. 4(b) and (c), are also generally poleward, except poleward of o 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 eddies v 0 T 0 must be directed generally poleward; we saw that we expect the same to be true for upward-propagating stationary Rossby waves. The momentum uxes are a little less obvious. We can actually understand them in simple terms, by invoking some of the principles we discussed earlier in the class, including our nding that the EP ux, F = (F y ; F z ) = u 0 v 0 ; f v0 T 0 ; (5) S indicates the direction of wave activity propagation. As just noted, we expect the EP uxes to be generally upward (v 0 T 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 stationary wave ux is actually downward, but the uxes are actually weak then (note the arrow scale on the upper right of the gure) and so are of little consequence. Note, especially for the transients, the uxes splay out in the upper troposphere, producing a predominantly equatorward component, but poleward in higher latitudes. Through (5), we can see that this is consistent with 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 F z > 0: there is a source of wave activity at the lower boundary, where the surface temperature gradient allows the stability criterion to be violated. But spatial homogeneity requires F y = 0 whence u 0 v 0 = 0 everywhere in this case 4. Suppose now, however [Fig. 6(b)], that the baroclinic zone is localized in latitude. Then the source of wave activity is similarly localized; the EP uxes point upward (v 0 T 0 poleward) out of the 4 Hence, e.g., the momentum ux is zero in the classic Eady problem. 5

6 Figure 3: Stationary and eddy heat uxes (annual and zonal means). [Peixoto & Oort] 6

7 Figure 4: Eddy angular momentum uxes for (b) stationary waves and (c) transient eddies. [Peixoto & Oort] 7

8 8 Figure 5: Northern hemisphere EP uxes for transient and stationary waves. [Peixoto & Oort]

9 TROPOPAUSE (a) z v't' > 0 u' v' = 0 SURFACE (b) u' v' > 0 u' v' < 0 v't' > 0 EQ POLE (c) u' v' > 0 u' v' < 0 v't' > 0 EQ POLE Figure 6: Fluxes associated with baroclinic eddies (schematic). Arrows show EP ux vector, uxes of momenum and heat represented by light and dark shading, repectively. (a) homogeneous case, (b) localized baroclinic zone on a -plane, (c) localized baroclinic zone on a sphere. 9

10 surface baroclinic zone, but spread out laterally away from the source. Thus, F y points away from the baroclinic zone (and thus away from the jet), whence the eddy momentum ux u 0 v 0 = F y, on both sides, is directed into the jet. From the cautionary note above, we cannot immediately conclude from this that 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 to case (b), the region of equatorward EP uxes and poleward eddy momentum ux is enhanced, while that of poleward EP ux and equatorward eddy momentum ux is weakened. The scenario depicted in Fig. 6(c) is indeed similar 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 u0 v 0 : (6) If we integrate this in the vertical, given that mass balance demands no net northward mass ux, i.e. Z 1 v dz = 0 ; 0 and the impossibility of any frictional stress at in nity, we nd that the surface stress must be balanced by the eddy momentum transport: Z 1 0 u 0 v 0 dz : (7) This column balance, illustrated in Fig. 7.2(a), simply states that the only contributions to the column are from eddy momentum uxes and surface stress. The mean circulation makes no contribution because the absence of a net mass ux means no net advection of planetary angular momentum, and because the QG assumption renders advection of relative angular momentum negligible here 5. Hence, in the vicinity of the generation of eddy activity the midlatitude baroclinic zone where the momentum uxes are convergent in the upper troposphere, the most direct impact is a positive surface stress, and hence surface westerlies. 5 Unlike in the tropical Hadley cell. 10

11 (a) u' v' W τ s (b) (u'v') balanced by fv y WARMING COOLING EQ POLE fv balanced by surface stress (c) COOLING WARMING EQ v't' POLE 11

12 If we go beyond the vertical integral, 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 u0 v 0 (8) so the region of momentum ux convergence in the upper troposphere must be balanced by mean equatorward ow, as depicted in Fig. 7.2(b). Applying continuity, this must become vertical motion at the edges of the region; which way does it go. We could integrate the continuity equation upward or downward 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 constraint that w! 0 at in nity, we have Z 1 z v dz So the vertical motion must be as shown, upward in low latitudes and downward at higher latitudes. The return ow then occurs in the boundary layer, where the Coriolis term is balanced by surface stress. Thus, we see that the 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 low latitudes and cooling in high latitudes! But this is where the eddy heat uxes come into play, as depicted in Fig. 7.2(c). The poleward transport of heat represented by the eddy ux v 0 T 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 reduce the pole-to-equator temperature gradient overall, and correspondingly to reduce the baroclinic shear of the mean ow. How the strength of the upper tropospheric jet is impacted depends on the competing e ects of barotropic acceleration/baroclinic deceleration, which depends on many factors. [These issues were addressed by Robinson, J. Atmos. Sci., 51, 2553 (1994).] 12

1 Climatological balances of heat, mass, and angular momentum (and the role of eddies)

1 Climatological balances of heat, mass, and angular momentum (and the role of eddies) We saw that the middle atmospheric temperature structure (which, through thermal wind balance, determines the mean