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

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1 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 zonal wind structure) departs signi cantly from radiative equilibrium. For January, these di erences are summarized in Fig 1. The major features Figure 1: Circulation schematic. are: 1. winter (northern) polar stratosphere K warmer than radiative equilibrium 2. summer (southern) polar stratosphere 0-20K warmer than radiative equilibrium 3. winter mesosphere 100K warmer than radiative equilibrium in high latitudes, 4. summer mesosphere 0 50K colder than radiative equilibrium in high latitudes, and 5. equatorial lower stratosphere a little colder (by about 5K) than radiative equilibrium. The di erences for July (July radiative equilibrium is very similar to January not quite the same, but the di erence is only a degree or two) are broadly similar to those shown in Fig. 1; the major di erence is that the winter (southern) lower stratosphere is much closer to radiative equilibrium than is northern winter; departures in southern winter range from about 10K in the lower stratosphere up to 100K near the stratospause. 1

2 Since, by de nition, there would be no departures from radiative equilibrium in the absence of dynamical heat transport, it is clear that dynamics must be responsible for these di erences. In fact, as we shall see, we can be more speci c about what type of motions must be responsible. Suppose that we neglect departures from zonal symmetry in the atmosphere. Since we are considering the climatological state of the solstice atmosphere, the atmosphere is in quasi-steady state at this time; therefore the heat budget gives + ws = Q (1) where S + g=c p. If we restrict attention to the polar regions where v vanish, then ws = Q. (2) [In fact, on the large scales we are interested in, scaling analysis would tell us that this equation is quite a good approximation everywhere and not just near the poles]. Therefore the diabatic heating/cooling (corresponding to departures from radiative equilibrium) must be balanced by adiabatic cooling/heating. Where the atmosphere is warmer/cooler than radiative equilibrium (and therefore Q < 0=Q > 0) the vertical motion must be downward/upward: We can use a Newtonian cooling approximation (not too bad for the stratosphere) Q = (T T e ), where T e is the radiative equilibrium temperature, to quantify this motion. Typical values of the radiative adjustment time 1 are up to 100d. in the lower stratosphere, 10 20d. in the middle stratosphere, and a few days around the stratopause. Using these numbers for, and the values S 10K km 1 (stratosphere), 5K km 1 (mesosphere) we can estimate w as shown schematically for January as shown in Fig 1. Vertical velocities near the north pole range from m s 1 (lower stratosphere) to m s 1 (mesosphere). Weaker downward motion is implied in the summer lower stratosphere and weak upwelling in the tropics. Strong ascent ( 10 2 ms 1 ) is implied in the summer mesophere. This is not, of course, the whole story. Continuity demands that there be latitudinal motion. From the continuity equation, we may estimate v as follows whence jvj L ' jwj H, since the vertical scale of w is greater than or comparable with the scale height H (7km). For L = 5000km (hemispheric 2 ) this gives approximately the values shown in Fig 1. Note the following features: 1 Here we are using log-pressure coordinates, with z = H ln (p=p 0 ), where H is a constant scale height and p 0 = 1000hPa. = p=gh is density in these coordinates. 2 Assuming that the circulation is hemispheric in scale, jvj ranges from 0 to its maximum value over half the distance from equator to pole, i.e., 5000 km. 2

3 1. strong ( 0:5 2:5ms 1 ) equator-to-pole ow in the winter stratosphere 2. very strong ( 10ms 1 ) pole-to-pole (summer-to-winter) ow in the upper mesosphere 3. weak (< 0:5ms 1 ) poleward ow in the summer lower stratosphere. [Note that (i) these are overestimates, since we have not allowed for the spherical geometry the fastest vertical motion is con ned to the relatively small polar caps and (ii) the implied vertical motion in the tropics will be much weaker than the polar velocities for the same reason]. Very large time-mean mesospheric velocities are indeed observed by radar at and around the solstices. In July, the picture we get is similar, but implied velocities in the lower stratosphere in southern winter are smaller than in northern winter. Qualitatively, at least, the pattern of circulation that we have deduced here, in the absence of eddy motions, looks rather like what is suggested by the global distribution of stratospheric trace gases. Does this man that the eddies, which we know to be present, do not matter much? Well, no, as we can soon discover. We must now turn to consider the implications of this circulation for the angular momentum budget. From the zonal momentum equation + f) = G x. (3) Scaling arguments convince us that w@u=@z is a negligible contribution outside the tropics and analysis of the wind climatology shows is small (but not everywhere negligible) compared with f in midlatitudes. Now, under our assumption of zonal symmetry, the frictional term must represent molecular friction, which is tiny over the region of interest (though it does become larger at high levels). Therefore, in the absence of eddies, the mean zonal winds would have to accelerate in response to the Coriolis e ects associated with the meridional motion these accelerations are substantial, ranging from about 5ms 1 =day in the winter stratosphere to around 100ms 1 = day in the mesosphere. Clearly, over the 100 days or so of the winter season, such accelerations are not occurring. Clearly some other process must be maintaining the observed (almost) steady state. Since we cannot appeal to frictional e ects (note that we assume no eddies, so there is no turbulent eddy viscosity in G x ) we are therefore led to conclude that our assumption of no eddies is inconsistent with the observed climatology the observed thermal structure (together with a radiative equilibrium calculation, the heat budget and continuity) demands the presence of eddy transports of angular momentum to balance the angular momentum budget. In principle, any eddies (small or large scale) might do the job. However, note that the angular momentum requirements are for a coherent input of (easterly) angular momentum into the winter stratosphere and mesophere and (westerly) angular momentum in the summer mesosphere. So small-scale mixing which merely redistributes angular momentum in a 3

4 localized region will not do we need momentum transport over large distances in the vertical and/or latitude (and of course eddies can only move angular momentum around they cannot create or destroy it). It is worth noting (though we do not consider it in detail) that the requirement for eddy transport under the observed circumstances is a fundamental one. If there were no eddies, then the angular momentum balance (3) tells us that inviscid steady state solutions must have either (i) constant angular momentum density along streamlines, or (ii) v = w = 0. Possibility (i) is the basis of the Held-Hou theory of the tropospheric Hadley cell; destruction of the angular momentum gradient can only occur in the tropics where it is already weak. Such solutions are possible in the stratosphere, as we shall discuss later. More than around 20 o o the equator, however, where Rossby number U=fL is everywhere less than 1 and so is de nitely nonzero, we must have v = 0 (again relying on scaling analysis to neglect w@u=@z). If v = 0, then w = 0 by continuity and then the steady heat budget (2) tells us that Q = 0: the extratropical atmosphere must be in radiative equilibrium (with no meridional circulation). So, on a fundamental level, sustained departures of the extratropical atmosphere from radiative equilibrium must be driven by eddy transports. We note in passing that the picture of the meridional circulation shown in Fig. 1 that we deduced from the heat budget is consistent with the very earliest deductions based on tracer distributions (and with what we have seen of the distributions of CH 4 and HF). In 1929 Dobson et al. noted that the winter/spring ozone maxmum implies an equator-to-winter pole circulation in the lower stratosphere. (But they went on to dismiss this suggestion on grounds that, in retrospect, seem spurious.) In a classic paper Brewer (QJRMS, 1949) discussed the extreme dryness of the stratosphere; its typical minimum mixing ratios of around 3ppmv are far short of saturation to get such low moisture via condensation he argued that all air entering the stratosphere must do so through the tropical tropopause which is the only place cold enough to produce such low humidity (in fact at tropopause pressures T needs to be as low as about 192K which is even colder than we see in the zonal means plots we looked at earlier, though it does get cold enough locally). The circulation he postulated is shown in Fig. 2. We will come to appreciate that the circulation does indeed look that which we have deduced here, despite the presence of eddies, but to see this we will rst have to learn to look at the mean circulation, and the role of eddies, in the right way. 4

5 Figure 2: The (lower) stratospherc circulation deduced by Brewer from water vapor (and other) observations. 5