Mechanistic View Of The Circulation

To what extent can we explain the main features of the observed general circulation on the basis of the fluid dynamics of a simple representation of the atmosphere driven by latitudinal gradients in solar forcing? The emphasis here is on "simple." In reality, the Earth's surface is very inhomogeneous: there are large mountain ranges that disturb the flow and large contrasts (e.g., in temperature and in surface roughness) between oceans and continents. In the interests of simplicity, however, we shall neglect such variations in surface conditions. We also neglect seasonal and diurnal1 variations, and assume maximum solar input at the equator, even though the subsolar point (Fig. 5.3) migrates between the two tropics through the course of a year. Thus we shall consider the response of an atmosphere on a longitudinally uniform, rotating planet (that is otherwise like the Earth) to a latitudinal gradient of heating. We shall see that the gross features of the observed circulation can indeed be understood on this basis; later, in Section 8.5, the shortcomings introduced by this approach will be briefly discussed.

Thus we ask how an axisymmetric atmosphere responds to an axisymmetric forcing, with stronger solar heating at the equator than at the poles. It seems reasonable to assume that the response (i.e., the induced circulation) will be similarly axisymmetric. Indeed, this assumption turns out to give us a qualitatively reasonable description of the circulation in the tropics, but it will become evident that it fails in the extratropical atmosphere, where the symmetry is broken by hydrodynamical instability.

8.2.1. The tropical Hadley circulation

If the Earth was not rotating, the circulation driven by the pole-equator temperature difference would be straightforward, with warm air rising in low latitudes and cold air sinking at high latitudes, as first suggested by Hadley and sketched in Fig. 5.19. But, as seen in Figs. 5.20 and 5.21, this is not quite what happens. We do indeed see a meridional circulation in the tropics, but the sinking motion is located in the sub-tropics around latitudes of ± 30°. In fact, the following considerations tell us that one giant axisymmetric meridional cell extending from equator to pole is not possible on a rapidly rotating planet, such as the Earth.

Consider a ring of air encircling the globe as shown in Fig. 8.3, lying within and being advected by the upper level poleward flow of the Hadley circulation. Because this flow is by assumption axisymmetric, and also because friction is negligible in this upper level flow, well above the near-surface boundary layer, absolute angular momentum will be conserved by the ring as it moves around. The absolute angular momentum per unit mass is (recall our discussion of angular momentum in the radial inflow experiment in Section 6.6.1)

A = Qr2 + ur, the first term being the contribution from planetary rotation, and the second from the eastward wind u relative to the Earth, where r is the distance from the Earth's rotation axis. Since r = a cos p,

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

FIGURE 8.3. Schematic of a ring of air blowing west —> east at speed u at latitude p. The ring is assumed to be advected by the poleward flow of the Hadley circulation conserving angular momentum.

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

momentum at the equator is simply A0 = Qa2. As the ring of air moves poleward, it retains this value, and so when it arrives at latitude p, its absolute angular momentum is

Therefore the ring will acquire an eastward velocity

Q (a2 - a2 cos2p) sin2p , , u (p) = _______1 = Qa. (8-2)

Note that this is directly analogous to Eq. 6-23 of our radial inflow experiment, when we realize that r = a cos p. Equation 8-2 implies unrealistically large winds far from the equator, at latitudes of (10°, 20°, 30°), u(p) = (14, 58,130) m s-1, and of course the wind becomes infinite as p ^ 90°. On the grounds of physical plausibility, it is clear that such an axisymmetric circulation cannot extend all the way to the pole in the way envisaged by Hadley (and sketched in Fig. 5.19); it must terminate 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 circulation, as depicted in Fig. 8.4. Near the equator, where f is small and the Coriolis effect is weak, angular momentum constraints are not so severe and the equatorial atmosphere acts as if the Earth were rotating slowly. As air moves away from the equator, however, the Coriolis parameter becomes increasingly large and in the northern hemisphere turns the wind to the right, resulting in a westerly component to the flow. At the poleward extent of the Hadley cell, then, we expect to find a strong westerly flow, as indeed we do (see Fig. 5.20). This subtropical jet is

FIGURE 8.4. Schematic of the Hadley circulation (showing only the northern hemispheric part of the circulation; there is a mirror image circulation south of the equator). Upper level poleward flow induces westerlies; low level equatorward flow induces easterlies.

driven in large part by the advection from the equator of large values of absolute angular momentum by the Hadley circulation, as is evident from Eq. 8-2 and depicted in Fig. 8.4. Flow subsides on the subtropical edge of the Hadley cell, sinking into the subtropical highs (very evident in Fig. 7.27), before returning to the equator at low levels. At these low levels, the Coriolis acceleration, again turning the equatorward flow to its right in the northern hemisphere, produces the trade winds, northeasterly in the northern hemisphere (southeasterly in the southern hemisphere) (see Fig. 7.28). These winds are not nearly as strong as in the upper troposphere, because they are moderated by friction acting on the near-surface flow. In fact, as discussed above, there must be low-level westerlies somewhere. In equilibrium, the net frictional drag (strictly, torque) on the entire atmosphere must be zero, or the total angular momentum of the atmosphere could not be steady. (There would be a compensating change in the 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 circulation would extend farther (or less far) poleward.

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

desert ie a r-equ atonal regions belt

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

desert ie a r-equ atonal regions belt

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 in our axisymmetric model would be westerly at the poleward edge of the circulation cell, and eastward near the equator. This is similar to the observed pattern (see Fig. 7.28, middle panel), but not quite the same. In reality the surface westerlies maximize near 50° N, S, significantly poleward of the subtropical jet, a point to which we return in Section 8.4.2.

As sketched in Fig. 8.5, the subtropical region of subsidence is warm (because of adiabatic compression) and dry (since the air aloft is much drier than surface air); the boundary formed between this subsiding air and the cooler, moister near-surface air is the ''trade inversion'' noted in Chapter 4, and within which the trade winds are located. Aloft, horizontal temperature gradients within the Hadley circulation are very weak (see Figs. 5.7 and 5.8), a consequence of very efficient meridional heat transport by the circulation.

Experiment on the Hadley circulation: GFD Lab VIII revisited

A number of aspects of the Hadley circulation are revealed in Expt VIII, whose experimental arrangement has already been described in Fig. 7.12 in the context of the thermal wind equation. The apparatus is just a cylindrical tank containing plain water, at the center of which is a metal can filled with ice. The consequent temperature gradient (decreasing ''poleward'') drives motions in the tank, the nature of which depends on the rotation rate of the apparatus. When slowly rotating, as in this experiment (Q < 1 rpm—yes, only 1 rotation of the table per minute: very slow!), we see the development of the thermal wind in the form of a strong ''eastward'' (i.e., super-rotating) flow in the upper part of the fluid, which can be revealed by paper dots floating on the surface and dye injected into the fluid as seen in Fig. 7.13.

The azimuthal current observed in the experiment, which is formed in a manner analogous to that of the subtropical jet by the Hadley circulation discussed previously, is maintained by angular momentum advec-tion by the meridional circulation sketched in Fig. 7.12 (right). Water rises in the outer regions, moves inward in the upper layers, conserving angular momentum as it does, thus generating strong ''westerly'' flow, and rubs against the cold inner wall, becoming cold and descending. Potassium permanganate crystals, dropped into the fluid (not too many!), settle on the bottom and give an indication of the flow in the bottom boundary layer. In Fig. 8.6 we see flow moving radially outwards at the bottom and being

FIGURE 8.6. The Hadley regime studied in GFD Lab VIII, Section 7.3.1. Bottom flow is revealed by the two outward spiralling purple streaks showing anticyclonic (clockwise) flow sketched schematically in Fig. 7.12 (right); the black paper dots and green collar of dye mark the upper level flow and circulate cyclonically (anticlockwise).

deflected to the right, creating easterly flow opposite to that of the rotating table; note the two purple streamers moving outward and clockwise (opposite to the sense of rotation and the upper level flow). This bottom flow is directly analogous to the easterly and equatorward trade winds of the lower atmosphere sketched in Fig. 8.4 and plotted from observations in Fig. 7.28.

Quantitative study of the experiment (by tracking, for example, floating paper dots) shows that the upper level azimuthal flow does indeed conserve angular momentum, satisfying Eq. 6-23 quite accurately, just as in the radial inflow experiment. With Q = 0.1 s-1 (corresponding to a rotation rate of 0.95 rpm; see Table A.1, Appendix A.4.1) Eq. 6-23 implies a hefty 10cms-1 if angular momentum were conserved by a particle moving from the outer radius, t\ = 10 cm, to an inner radius of 10 cm. Note, however, that if Q were set 10 times higher, to 10 rpm, then angular momentum conservation would imply a speed of 1 m s-1, a very swift current. We will see in the next section that such swift currents are not observed if we turn up the rotation rate of the table. Instead the azimuthal current breaks down into eddying motions, inducing azimuthal pressure gradients and breaking angular momentum conservation.

8.2.2. The extratropical circulation and GFD Lab XI: Baroclinic instability

Although the simple Hadley cell model depicted in Fig. 8.5 describes the tropical regions quite well, it predicts that little happens in middle and high latitudes. There, where the Coriolis parameter is much larger, the powerful constraints of rotation are dominant and meridional flow is impeded. We expect, however, (and have seen in Figs. 5.8 and 7.20), that there are strong gradients of temperature in middle latitudes, particularly in the vicinity of the subtropical jet. So, although there is little meridional circulation, there is a zonal flow in thermal wind balance with the temperature gradient. Since T decreases poleward, Eq. 7-24 implies westerly winds increasing with height. This state of a zonal flow with no meridional motion is a perfectly valid equilibrium state. Even though the existence of a horizontal temperature gradient implies horizontal pressure gradients (tilts of pressure surfaces), the associated force is entirely balanced by the Coriolis force acting on the thermal wind, as we discussed in Chapter 7.

Our deduction that the mean meridional circulation is weak outside the tropics is qualitatively in accord with observations (see Fig. 5.21) but leaves us with two problems:

1. Poleward heat transport is required to balance the energy budget. In the tropics, the overturning Hadley circulation transports heat poleward, but no further than the subtropics. How is heat transported further poleward if there is no meridional circulation?

2. Everyday observation tells us that a picture of the midlatitude atmosphere as one with purely zonal winds is very wrong. If it were true, weather would be very predictable (and very dull).

Our axisymmetric model is, therefore, only partly correct. The prediction of purely zonal flow outside the tropics is quite wrong. As we have seen, the midlatitude atmosphere is full of eddies, which manifest themselves as traveling weather systems.4 Where do they come from? In fact, as we shall now discuss and demonstrate through a laboratory experiment, the extratropical atmosphere is hydrodynamically unstable, the flow spontaneously breaking down into eddies through a mechanism known as baroclinic instability.5 These eddies readily generate meridional motion and, as we shall see, affect a meridional transport of heat.

Baroclinic instability

GFD Lab XI: Baroclinic instability in a dishpan To introduce our discussion of the breakdown of the thermal wind through baroclinic instability, we describe a laboratory experiment of the phenomenon. The apparatus, sketched in Fig. 7.12, is identical to that of Lab VIII used to study the thermal wind and the Hadley circulation. In the former experiments the table was rotated very slowly, at a rate of Q < 1 rpm. This time, however, the table is rotated much more rapidly, at Q ~ 10 rpm, representing the considerably greater Cori-olis parameter found in middle latitudes. At this higher rotation rate something remarkable happens. Rather than a steady axisymmetric flow, as in the Hadley regime shown in Fig. 8.6, a strongly eddying flow is set up. The thermal wind remains, but breaks down through instability, as shown in Fig. 8.7. We see the development of eddies that sweep (relatively) warm fluid from the periphery to the cold can in one sector of the tank (e.g., A in Fig. 8.7), and simultaneously carry cold fluid from the can to the periphery at another (e.g., B in Fig. 8.7). In this way a radially-inward heat transport is achieved, offsetting the cooling at the center caused by the melting ice.

For the experiment shown we observe three complete wavelengths around the tank in Fig. 8.7. By repeating the experiment but at different values of Q, we observe that the scale of the eddies decreases, and the flow becomes increasingly irregular as Q is increased. These eddies are produced by the same mechanism underlying the creation of atmospheric weather systems shown in Fig. 7.20 and discussed later.

Before going on we should emphasize that the flow in Fig. 8.7 does not conserve angular momentum. Indeed, if it did, as estimated at the end of Section 8.2.1, we would observe very strong horizontal currents near the ice bucket, of order 1 m s-1 rather than a few cm s-1, as seen in the experiment. This, of course, is exactly what Eq. 8-2 is saying for the atmosphere: if rings of air conserved angular momentum as they move in axisym-metric motion from equator to pole, then we obtain unrealistically large winds. Angular momentum is not conserved because of the presence of zonal (or, in our tank experiment, azimuthal) pressure gradient associated with eddying motion (see Problem 6 of Chapter 6).

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

5In a "baroclinic" fluid, p = p (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 [p = p (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 the centre at point A and vice versa at point B. The eddies are created by the instability of the thermal wind induced by the radial temperature gradient due to the presence of the ice bucket at the center of the tank. The diameter of the ice bucket is 15 cm.

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 the centre at point A and vice versa at point B. The eddies are created by the instability of the thermal wind induced by the radial temperature gradient due to the presence of the ice bucket at the center of the tank. The diameter of the ice bucket is 15 cm.

Middle latitude weather systems The process of baroclinic instability studied in the laboratory experiment just described is responsible for the ubiquitous waviness of the midlatitude flow in the atmosphere. As can be seen in the observations shown on Figs. 5.22, 7.4, and 7.20, these waves often form closed eddies, especially near the surface, where they are familiar as the high- and low-pressure systems associated with day-to-day weather. In the process, they also affect the poleward heat transport required to balance the global energy budget (see Fig. 8.1). The manner in which this is achieved on the planetary scale is sketched in Fig. 8.8. Eddies ''stir'' the atmosphere, carrying cold air equator-ward and warm air poleward, thereby reducing the equator-to-pole temperature contrast. To the west of the low (marked L) in Fig. 8.8, cold air is carried into the tropics. To the east, warm air is carried toward the pole, but since poleward flowing air is ascending (remember the large-scale slope of the 0 surfaces shown in Fig. 5.8) it tends to leave the surface. Thus we get a concentrated gradient of temperature near point 1, where the cold air ''pushes into'' the warm air, and a second, less marked concentration where the warm air butts into cold at point 2. We can thus identify cold and warm fronts, respectively, as marked in the center panel of Fig. 8.8. Note that a triangle is used to represent the ''sharp'' cold front and a semicircle to represent the ''gentler'' warm front. In the bottom panel we present sections through cold fronts and warm fronts respectively.

Timescales and length scales We have demonstrated by laboratory experiment that a current in thermal wind balance can, and almost always does, become hydrodynam-ically unstable, spawning meanders and eddies. More detailed theoretical analysis6 shows that the lateral scale of the eddies that form, Leddy (as measured by, for example,

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

1. The Eady growth rate of the disturbance eat is given by a = 0.31 U/Lp = 0.31 f-- using Eq. (7-23) and U = -j-H

(see Gill, 1982). Inserting typical numbers for the troposphere, N = 10-2 s-1, = 2 x 10-3 s-1, f = 10-4 s-1, we find that a ~ 10-5 s-1, an e-folding timescale of 1 day.

2. The wavelength of the fastest-growing disturbance is 4Lp where Lp is given by Eq. 7-23. This yields a wavelength of 2800 km if Lp = 700 km. The circumference of the Earth at 45° N is 21,000 km, and so about 7 synoptic waves can fit 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-pole temperature contrast. Middle: To the west of the "L," cold air is carried in to the tropics. To the east, warm air is carried 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.

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-pole temperature contrast. Middle: To the west of the "L," cold air is carried in to the tropics. To the east, warm air is carried 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.

the typical lateral scale of a low pressure system in the surface analysis shown in Fig. 7.25, or the swirls of dye seen in our tank experiment, Fig. 8.7) is proportional to the Rossby radius of deformation discussed in Section 7.3.4:

eddy

where Lp is given by Eq. 7-23.

The timescale of the disturbance is given by:

r LP

where U = du/dz H is the strength of the upper level flow, du/dz is the thermal wind given by Eq. 7-17, and H is the vertical scale of the flow. Eq. 8-4 is readily interpretable as the time it takes a flow moving at speed U to travel a distance Lp and is proportional to the (inverse of) the Eady growth rate, after Eric Eady7, a pioneer of the theory of baroclinic instability.

In our baroclinic instability experiment, we estimate a deformation radius of some 10 cm, roughly in accord with the observed scale of the swirls in Fig. 8.7 (to determine a scale, note that the diameter of the ice can in the figure is 15 cm). Typical flow speeds observed by the tracking of paper dots floating at the surface of the fluid are about 1cms-1. Thus Eq. 8-4 suggests a timescale of Teddy ~ 7.0 s, or roughly one rotation period. Consistent with Eq. 8-3, the eddy scale decreases with increased rotation rate.

Applying the above formulae to the middle troposphere where f ~ 10-4 s-1, N ~ 10-2s-1 (see Section 4.4), H ~ 7 km (see Section 3.3), and U ~ 10 ms-1, we find that Lp = NH/f ~ 700 km and Teddy ~ 700 km/10 m s-1 ~ 1 d. These estimates are roughly in accord with the scales and growth rates of observed weather systems.

We have not yet discussed the underlying baroclinic instability mechanism and its energy source. To do so we now consider the energetics of and the release of potential energy from a fluid.

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