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Conservation of mass

Revised March 22, 2000 3:22 pm

Copyright 2000 David A. Randall

This chapter is intended to provide a “quick look” at the observed seasonally varying general circulation of the atmosphere. A few basic ?elds will be shown and described. Lots of questions will be raised, but for the most part the answers will be deferred until later chapters. Most of the ?elds shown in this chapter are based on analyses performed at the European Centre for Medium Range Weather Forecasts (ECMWF; see the 1997 ECMWF report in the reference list at the end of this chapter). The observations discussed in this chapter show many things, but among the most important phenomena that can be glimpsed here are the tropical Hadley and Walker circulations, the monsoons, stationary planetary waves, and some aspects of the hydrologic cycle. We need some equations for use in interpreting the observations. The table below gives the ones that we will use in this chapter; you should already be familiar with them. In writing these equations, pressure has been used as the vertical coordinate.

Conservation of horizontal momentum (“primitive” form) Hydrostatic equation

?F V DV ? u tan ?? -------- + ? f + --------------? k × V = – ? p φ + g ?p Dt a ?φ = –α ?p ? -ω = 0 ? p ? V + ----?p ?F DT - = ωα + gc p T + Q rad + Q lat dc p ------Dt ?p


(2.2) (2.3)

Conservation of mass Conservation of thermodynamic energy


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Conservation of moisture

?F q Q lat Dq ------=g – -------?p Dt L p α = RT

(2.5) (2.6)

Equation of state

The symbols used in these equations are de?ned as follows:

V D ----Dt

The horizontal wind vector The three-dimensional Lagrangian time derivative, i.e. the time derivative following a particle, which satisfies D( ) ? ? ----------- ≡ ? ? + V ? ?p + ω ( ) ? ? t? p Dt ?p The Coriolis parameter A unit vector pointing upwards. Note that the direction denoted by “upwards” varies from place to place on the Earth. The zonal and meridional wind components (the meridional component is not actually used in the equations given above, but is defined here for completeness) Longitude and latitude (longitude is not actually used above) Geopotential The radius of the Earth The upward turbulent flux of quantity x Pressure, used here as a vertical coordinate Acceleration of gravity Latent heat of condensation The specific “gas constant” of dry air Temperature The vertical “pressure velocity” The specific heat of air at constant pressure

f k u, v

λ, ? φ a Fx p g L R T ω cp

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q α Q rad Q lat

The mixing ratio of water vapor The specific volume of the air, the inverse of the density, equal to RT/p The radiative heating rate The latent heating rate
With the approximation of geostrophic balance, the momentum equations reduce to

1 ? ?φ ? - ----- , 0 = f v g – -------------a cos ? ? ?λ? p 1 ? ?φ ? - ----- . 0 = – f u g – -a ? ??? p



Here u g and v g are the zonal and meridional components of the geostrophic wind, respectively. If (2.7) and (2.8) are combined with the hydrostatic equation, (2.1), and the equation of state,(2.6), we can derive the thermal wind equations:

?u g 1 R ? ?T ? = -------, fpa ? ? ?? p ?p ?v g – R ? ?T ? = --------------------. fpa cos ? ? ? λ? p ?p



These relationships between the vertical derivatives of the winds and the horizontal derivatives of the temperature will be used in the following discussion of the observations.

Fig. 2.1 shows the January and July monthly mean maps of sea level pressure, respectively. These maps are much smoother than weather maps plotted for particular observation times, because the moving highs and lows that represent individual weather systems have been averaged out. Fig. 2.2 shows the corresponding zonally averaged distributions of the sea level pressure for January and July.
Certain features are seen in both January and July, including a tendency for high pressure to occur in the subtropics. The highs typically appear as “cells,” e.g. in the North Atlantic and North Paci?c in July, or off the west coast of South America in January. In many cases, the subtropical highs are found over the eastern parts of the oceans. In the Northern Hemisphere, they are particularly strong in the northern summer (Hoskins, 1996). Strong highs are also apparent in middle latitudes during winter, e.g. in Siberia and western North America. Both regions are quite mountainous.

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Figure 2.1:

Throughout the year, there is a belt of low pressure in the tropics. The decrease of sea level pressure from the subtropics towards the equator implies tropical easterlies near the surface, from geostrophic balance. In the Northern Hemisphere during northern winter, prominent low-pressure cells also appear, most notably near the Aleutian Islands and Iceland. These are regions where low-pressure centers are often found on individual days. There is a tendency for a minimum of the sea level pressure near 60 ∞N, especially in January but also to some extent in July. A very obvious belt of low pressure is found over the ocean north of Antarctica, throughout the year, although it is more intense in July (winter) than January (summer). Because the sea level pressure generally decreases from the subtropics to middle
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Figure 2.2:

latitudes, the geostrophic relation leads us to expect a tendency for surface westerlies on the poleward side of the subtropical highs. Similarly, the relatively high pressure over the poles suggests surface easterlies in high latitudes. Generally there is less seasonal change in the Southern Hemisphere than in the Northern Hemisphere. Also, the departures from the zonal means, which represent “stationary eddies,” are much more prominent in the Northern Hemisphere than the Southern Hemisphere. Especially in the Northern Hemisphere, there is a tendency for low pressure over the oceans in winter, and high pressure over the continents in winter, and vice versa in summer. The tropical sea level pressure distribution is generally very smooth and featureless, compared to that of middle latitudes. A simple explanation for this was given by Charney (1963), in terms of the differences in dynamical balance between the tropics; speci?cally, the effects of the Earth’s rotation are much more important than particle accelerations in middle latitudes, and the opposite is true in the tropics. Jule Charney, one of the giants of twentiethcentury meteorology, is cited here for the ?rst time in Chapter 2; his name will come up again and again throughout the remainder of this course; a photo of Charney is shown in Fig. 2.3 . Charney began his analysis with the equation of motion, (2.1), which can be written in simpli?ed form as

DV -------- + f k × V = – ?φ . Dt


Here we have omitted the tan ? term and the friction term, for simplicity, so that there are only three terms left. We have also dropped the subscript p on the del operator, to make the

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Figure 2.3:

notation simpler. The three terms shown in (2.11)represent most of the “action” throughout most of the atmosphere. Their orders of magnitude can be estimated as follows:

DV V -------- ? ----Dt L f k × V ? fV δφ ?φ ? ----L


(2.12) (2.13) (2.14)

Here V is a “velocity scale,” which might be on the order of 10 m s-1, L is a length scale, which might be on the order of 106 m, and δφ is a typical ?uctuation of the geopotential height. The numerical values of these scales have been chosen to be representative of “large-scale” motions on the Earth; if we wanted to analyze small-scale motions, we would choose different numerical values. The same numerical values of V and L can be used for both the tropics and middle latitudes because we use the term “large-scale” in the same way for both regions.

DV - , is typically negligible In middle latitudes, the acceleration following a particle, -------Dt

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DV - might be 10 m s-1 over a compared to the rotation term. For example, a typical value of -------Dt 10 –4 DV ? V - = ------time in which the air travels 1000 km, so that so that -------- = 10 m s-2, while - ----6 L Dt 10
a representative midlatitude value of the Coriolis parameter is f midlat ? 10
–3 –4 2 2

s-1, and a

typical wind speed is 10 m s-1, so that f midlat V ? 10 m s-2, about one order of magnitude larger than the acceleration. Geostrophic balance is, therefore, approximately satis?ed in midlatitudes, i.e.

( δφ ) midlat -, f midlat V ? ----------------------L ( δφ ) midlat ? f midlat VL .

(2.15) (2.16)

Here we have added the “midlat” subscript to δφ , just for clarity. According to (2.16), rotation strongly determines the magnitude of midlatitude height ?uctuations. On the Equator, however, the Coriolis parameter vanishes, so we expect that in the deep tropics geostrophic balance breaks down (a point discussed further in a later chapter), and particle accelerations tend to balance the pressure gradient force, much as they do on small scales almost everywhere in the atmosphere, and in many engineering problems:
2 ( δφ ) V tropics ----- ? ------------------------ , L L



( δφ ) tropics ? V .
Comparing (2.16) and (2.18), we see that



( δφ ) tropics V ------------------------ ? ------------------- ≡ Ro midlat , ( δφ ) midlat f midlat L


where Ro midlat is a midlatitude Rossby number. By substituting the numerical values given above, we ?nd that Ro midlat ? 0.1 . Eq. (2.19) therefore tells us that geopotential height ?uctuations in the tropics are much smaller that those in middle latitudes. At this point, we can bring in the hydrostatic equation [Eq. (2.1)] to show that large-scale ?uctuations of temperature and surface pressure are also much smaller in the tropics than in middle latitudes. Later in this chapter we will return to Charney’s analysis to interpret the relationships between vertical motion and heating in the tropics.

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Fig. 2.4 shows the latitude-height distribution of the zonally averaged zonal wind for January and July, respectively.

Figure 2.4:

The plot extends from the surface to the middle stratosphere. We plot the wind components and other variables against height, rather than pressure, because the pressure coordinate tends to “squash” the stratosphere into a thin region at the top of the plot,

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concealing its structure. Although the stratosphere contains only a small fraction of the mass of the whole atmosphere, its dynamical in?uence extends downward into the troposphere, and so the stratosphere should be of interest even to students who are mainly interested in the general circulation of the troposphere. In both January and July, easterlies extend through the entire depth of the tropical troposphere. They are somewhat stronger in July. They are concentrated in the Northern Hemisphere in July and the Southern Hemisphere in January. The near-surface easterlies are stronger in the winter hemisphere. Westerly jets (“jet streams”) are quite prominent, especially in the winter hemisphere. The main tropospheric winter jet tends to occur at about 30 ° latitude. The summer jet, which is weaker, tends to be at about 45 ° latitude. The midlatitude jet maxima are consistently found near the 200 mb level. In the Southern Hemisphere in July, there is a clear minimum in the westerlies near 150 mb, at about 40 °S. Above and poleward of this minimum is a stratospheric westerly jet called the polar night jet. A similar but weaker jet occurs in the Northern Hemisphere winter, although it is not evident in the ?gure. Weak surface easterlies occur near the poles. The most striking things about the stratospheric zonal winds are the strong westerly vortex in the winter hemisphere, often called the “polar night jet,” and the easterlies ?lling the summer hemisphere. Note that the polar night jet is separated from the westerly jet at the tropopause level by a local minimum in the westerlies; nevertheless there is a band of strong westerlies which extends upwards from the midlatitude troposphere into the high-latitude stratosphere. As discussed later, the summer stratosphere is radiatively controlled, while the winter stratosphere is strongly in?uenced by dynamics, including upward wave propagation from the troposphere.

Fig. 2.5 shows maps of the 850 mb zonal wind for January and July, respectively. Again, keep in mind that many intense small-scale (~ 1000 km) features would appear in daily maps, but have been smoothed out here by time averaging.
The monthly mean maps show very obvious alternating bands of easterlies and westerlies, which qualitatively remind us of Jupiter (see Fig. 2.6 ), although of course Jupiter has more bands; generally speaking, the Earth’s atmosphere features easterlies in the tropics, westerlies in middle latitudes, and easterlies again near the poles. We can also see features associated with the strong cells in the sea level pressure maps, e.g. the easterlies in the extreme North Paci?c in January, associated with the Aleutian Low. Again, stationary eddies are much more apparent in the Northern Hemisphere than the Southern Hemisphere. Generally speaking, however, the features seen in the maps have a very zonal orientation, with strong north-south gradients and relatively weak east-west gradients. Note the intensi?cation of both the zonal mean ?ow and the eddies of the midlatitude westerlies in winter, in each hemisphere. The westerlies in the Northern Hemisphere in winter are particularly strong over the oceans. The strong positive maximum in the Arabian Sea in July is associated with the Indian Summer Monsoon. The westerlies north of Australia but (just south of the Equator) in January are indicative of the Winter Monsoon. In both regions, the sign of the zonal wind reverses seasonally.
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Figure 2.5:

Fig. 2.7 shows the corresponding maps for 200 mb. The winds are generally stronger aloft than near the surface; this is true of both the zonal mean and the eddies. Note the very prominent January westerly jet maxima off the east coasts of North America and, especially, Asia. There is also a westerly jet maximum at about 30° S, near the Date Line.

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Figure 2.6:

In the Northern winter, there are “dipoles” consisting of easterly-westerly pairs, straddling the equator, near the Americas and also near the longitude of Australia. The midlatitude westerlies come pretty close to the equator in January. In July, there are equatorial easterlies at all longitudes, and the westerlies have intensi?ed in the midlatitudes of the Southern Hemisphere, and weakened in the midlatitudes of the Northern Hemisphere. Two intrusions of westerlies are seen in the Northern Hemisphere tropics: one just east of the Date Line, and another over the Atlantic Ocean. As will be discussed later, these regions of mean westerlies allow waves to propagate from middle latitudes into the tropics.

Fig. 2.8 shows the latitude-height distribution of the zonally averaged meridional wind for January and July, respectively. The zonal means reach about 2 m s-1 in absolute value; the strongest values occur in the tropics. In the winter-hemisphere tropics, in both months, there is an obvious dipole structure, with poleward ?ow aloft and equator ward ?ow near the surface. Evidently there is convergence near the equator at low levels, and divergence aloft. The convergence zone near the surface shifts from the Southern Hemisphere in January to the Northern Hemisphere in July. These features are of course associated with the Hadley circulation. Poleward ?ow is also found near the surface in middle latitudes, with weak quaternary ?ow above. We can also see weaker regions of low-level convergence near 60 °S and 60 °N.

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Figure 2.7:

As with most other ?elds, the seasonal changes of the meridional wind in the midlatitudes of the Southern Hemisphere are quite weak. The mass-weighted vertical mean of the time- and zonally averaged meridional wind must be very close to zero at all latitudes. This can be understood from the surface pressuretendency equation, which is

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Figure 2.8:

? pS ? ? = – ??? ∫ V dp ? . ?t ? ?



Problem 3 at the end of this chapter challenges you to derive (2.20). Averaging (2.20) around a latitude circle gives

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1 ?? ? - ----[ p S ] = – -------------a cos ? ?? ? ?t ?



? dp cos ? ? . ?


Here subscript S denotes a surface value, and square brackets represent zonal averages:

1 ? - ∫ ( ) d λ . In an average over a suf?ciently long time, [ p S ] must become small at [ ] ≡ ----2π ?t

each latitude, because p S is bounded within a fairly narrow range. It follows that

?? ----?? ? ?



? dp cos ? ? = 0 , ?


where the overbar denotes a time average. This means that


dp cos ? is independent of

latitude. Since cos ? = 0 at both poles, we can conclude that, in a time average,

0 pS


= 0


at all latitudes. The physical meaning of (2.23) is very simple. Suppose, for instance, that


dp was positive at the Equator. This would mean that, in a zonal and vertical average,

air would be ?owing out of the Southern Hemisphere and into the Northern Hemisphere; of course this can really happen at a given instant, but if it continued the surface pressure in the Southern Hemisphere would decrease to zero and the surface pressure in the Northern Hemisphere would increase to roughly double its normally observed values. Obviously the pressure-gradient force would resist such a scenario. Recall that in the derivation of (2.23) we assumed that the surface pressure does not change much on suf?ciently long time scales. In addition, it follows immediately from (2.7) that the zonally averaged meridional component of the geostrophic wind vanishes on each pressure surface:

0 = [ vg ] .


This shows that all of the zonally averaged meridional circulations, { [ v ], [ ω ] } , including for example the Hadley circulation, are completely ageostrophic. The implication is that the important large-scale circulations are not necessarily quasigeostrophic. We note, however, that the strongest features in Fig. 2.7 do occur in the tropics, where geostrophy would be expected to lose its grip.

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Fig. 2.9 shows maps of the 850 mb meridional wind for January and July, respectively. Fig. 2.10 shows the corresponding 200 mb maps. Unlike the zonal wind, the


Figure 2.9:

time-averaged meridional wind does not show a banded, east-west structure; the east-west gradients are at least as strong as the north-south gradients. In the Northern Hemisphere, there is a tendency for alternating southerly and northerly ?ows, with a structure that resembles zonal wave number 3 or 4. The time-averaged meridional currents in the Southern Hemisphere are generally weaker than those in the Northern Hemisphere. The intensities of the meridional currents are stronger at 200 mb than at 850 mb. In the Northern Hemisphere especially, there is a tendency for stronger features in winter than in summer. As can be seen in Fig. 2.7 , the zonally averaged meridional ?ow in the tropics of the winter hemisphere is generally toward the summer pole at low levels, and toward the winter pole aloft. This is not very apparent in Fig. 2.9 or Fig. 2.10, however. At 850 mb we can see a

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Figure 2.10:

strong southerly ?ow just north of the Equator in July, associated with the Indian summer monsoon. The northerly ?ow near 120 ° E in January is associated with the winter monsoon, but is relatively inconspicuous. There are many parts of the world in which the mean meridional wind reverses seasonally. Examples include the Arabian Sea, most of the North Paci?c, and the southern Great Plains of North America.

Fig. 2.11 shows the streamlines at 850 mb, for January and July, and Fig. 2.12

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Figure 2.11:

shows the corresponding streamlines at 200 mb. Streamlines indicate direction but not magnitude. Such features as the subtropical highs and midlatitude lows are clearly evident at 850 mb. There are strong cross-equatorial ?ows at 850 mb in both the Paci?c and Indian Oceans, in July. The 200 mb streamlines show the planetary wave patterns in the midlatitude winter, and also tropical phenomena including a strong anticyclone over the Indian subcontinent in July. In the Northern Hemisphere, the height ?elds show pronounced seasonal changes; the gradients are tight in the winter and loose in the summer. The subtropical highs are easier to

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Figure 2.12:

pick out in summer than in winter. At 850 mb, the winter ?elds in the Northern Hemisphere are decidedly “wavy,” while the summer ?elds are “lumpy.” The seasonal changes in the Southern Hemisphere are much weaker.

Fig. 2.13 shows maps of the geopotential height on the 850 mb surface, and Fig. 2.14 shows the corresponding maps for 200 mb. The geopotential thickness between two pressure

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Figure 2.13:

surfaces measures the average temperature of the intervening layer. The meridional temperature gradients are clearly evident in the plots. We notice the weak gradients in the tropics, as discussed already in connection with Charney’s (1963) analysis. In addition, the geostrophic wind is proportional to the gradient of the height ?eld. The relationships between the zonal and meridional wind maps and the geopotential maps are clearly evident. A “stream function,” ψ, can be de?ned for the mean meridional circulation. Plots of
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Figure 2.14:

this stream function effectively show the zonally averaged vertical velocity and the zonally averaged meridional velocity, together in one diagram.The de?nition of the stream function is:

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?ψ -, [ v ] 2 π acos ? = g -----?p
2 ?ψ -. [ ω ] 2 π a cos ? = – g -----??


Here again the square brackets denote zonal averages. You should con?rm that for, any given distribution of ψ , the continuity equation (2.3) is automatically satis?ed. Note that ψ is, by virtue of its de?nition, independent of longitude. It is easy to see that lines of constant ψ cannot intersect the Earth’s surface; if they did, that would imply a ?ow of air across the Earth’s surface. Similarly, lines of constant ψ cannot extend upward into space; if they did, that would imply that the Earth’s atmosphere was being accumulated from or lost into space. Note from (2.25) that

δ p? L M L M ----- ? -- ? L ? ---- ? ---- = ----, ψ?? 2 ? g? t t t L



i.e. ψ has dimensions of mass per unit time. Typically ψ is expressed in units of 10 12 g s -1, or 109 kg s-1. This unit is sometimes called a “Sverdrup,” especially in the oceanographic literature. The sign of ψ is arbitrary [i.e. the signs could be reversed in (2.25)], and also an arbitrary constant can be added to ψ without changing [ v ] or [ ω ] . In view of (2.25), we can compute ψ from either [ v ] or [ ω ] ; here [ v ] has been used, with the boundary condition

ψ = 0 at the Earth’s surface.


The physical content of (2.31) is that ψ is a constant along the Earth’s surface; the particular constant chosen (i.e. zero) has no physical signi?cance.

Fig. 2.15 shows the latitude-height distribution of the stream function of the mean meridional circulation, for January and July, respectively. A banded structure is clearly evident. Again, this is somewhat reminiscent of the banded circulation systems that are readily visible in images of Jupiter and Saturn. Deep rising motion occurs in the summer-hemisphere tropics, with sinking motion on either side. The strongest tropical rising motion is near 300 mb, but notice that weak rising motion continues into the tropical stratosphere. The strongest subsidence is in the winter hemisphere subtropics, again near 300 mb. Rising motion occurs in middle latitudes, and is strongest in the winter. Maximum values tend to occur near 500 mb. Sinking motion is found near the poles, mainly in the lower troposphere.
The dominant cellular structures in the tropics are called “Hadley Cells.” There is a “large” Hadley circulation at each solstice, with its rising branch in the summer-hemisphere tropics and its body extending into the winter-hemisphere subtropics. Its peak magnitude is about 160 sverdups. A weaker Hadley circulation occurs in the summer hemisphere. Both Hadley cells are “direct” circulations, which means that their rising branches are warm and their sinking branches are cold. As discussed later, such circulations convert potential energy into kinetic energy. There are also indirect circulations in the middle latitudes, most clearly in the

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Figure 2.15:

Southern Hemisphere in both seasons. These are called “Ferrel cells.” Their physical interpretation is explained later. Finally, the polar regions play host to weak direct circulations. The correspondence between the zonally averaged vertical motion and the zonally averaged meridional motion is fairly obvious. The meridional currents can be interpreted as

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out?ows from or in?ows to the vertical currents.

Fig. 2.16 shows maps of the 500 mb vertical velocity for January and July,


Figure 2.16:


respectively. The units are nanobars per second. The strongest maxima and minima have absolute values of roughly 1000 nanobars per second, which is about 100 millibars per day.

Fig. 2.15 gives the impression that there are regular bands of rising and sinking motion, arranged along latitude circles. Zonal bands of rising and sinking motion are not easy
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to pick out in Fig. 2.16 , however; at ?rst glance, the patterns look almost random. There is some tendency, however, for rising motion in the tropics; sinking motion in the subtropics, especially in the winter hemisphere; rising motion in middle latitudes; and sinking motion over the poles. Sinking motion tends to be associated with surface pressure maxima, and rising motion with surface pressure minima. For example, the subtropical highs are clearly regions of large-scale sinking motion in the middle troposphere. The seasonal change of the large-scale vertical motion is very spectacular in the region of the Tibetan plateau. Rising motion occurs in summer, and sinking motion in winter. These changes are associated with the Indian monsoon. There are some cases of rising motion upstream of mountain ranges, and sinking motion downstream; examples are the Rocky Mountains and the Himalayas, in winter. Looking at the other major mountain ranges of the world, however, it is hard to see a clear pattern of orographically forced vertical motions. Such motions do of course exist, but a more re?ned analysis is needed to detect them. Note the rising motion over southern Africa and tropical South America in January, in the same regions where we will see later that there are water vapor maxima 850 mb in January. There is a tendency for large water vapor mixing ratios in regions of rising motion, and small water vapor mixing ratios in regions of sinking motion. In particular, deserts, like the Sahara, are regions of sinking motion

As discussed in introductory physics textbooks (e.g. Feynman et al. 1962), the angular momentum per unit volume of a particle, L , with respect to some origin, is a vector, given by the cross product of the particle’s linear momentum, ρ V , and the displacement vector separating the particle from the origin, r :

L = r × ρV .


The angular momentum vector of the Earth’s atmosphere, with respect to an “origin” at the center of the Earth, could be computed by applying (2.28) to each air parcel, and integrating over the entire atmosphere. The angular momentum of the air is mostly due to the rotation of the Earth. In addition, the motion of the air relative to the Earth includes strong jets which are oriented very nearly along latitude circles. For these reasons, the angular momentum vector for the atmosphere as a whole is very nearly parallel to the axis of the Earth’s rotation, and in practice when we discuss the angular momentum of the atmosphere we are almost always concerned with the component of the angular momentum vector that is parallel to the axis of the Earth’s rotation. This component of the angular momentum vector, per unit mass, is denoted by M in the discussion below, and is given by

M ≡ ( ? a cos ? + u ) a cos ? .


The ? term of (2.29) represents the angular momentum due to the Earth’s rotation. The u term represents the relative angular momentum due to the rotation of the atmosphere relative to the Earth’s surface. Fig. 2.17 shows the zonally averaged total angular momentum and

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relative angular momentum, both per unit mass, for January and July, as analyzed by ECMWF. Note that the absolute angular momentum is generally much larger than the relative angular momentum. The absolute angular momentum varies mainly with latitude, and only slightly with height. It is positive everywhere, and takes its largest values near the Equator. The plots of the zonally averaged relative angular momentum per unit mass resemble those of the zonally averaged zonal wind (c.f. Fig. 2.4). In the tropical upper troposphere, the absolute angular momentum contours “tilt” slightly towards the winter pole, in both seasons. This is an indication that the air ?owing poleward in the main Hadley circulation is conserving its absolute angular momentum as it goes. The Lagrangian derivative of (2.29) can be written as

D? DM -------- = – ( ? a cos ? + u ) a sin ? ------Dt Dt D ?? Du ------+ a cos ? ? - . ? – ? a sin ? Dt ? + -----Dt D? - , we can rewrite (2.30) as Collecting terms, and using v ≡ a ------Dt Du DM - – ( 2 ? a cos ? + u ) v sin ? . -------- = a cos ? -----Dt Dt (2.31)


Du - , which appears on the right-hand side of (2.31), we To derive an expression for a cos ? -----Dt use the zonal component of (2.1) which can be written as ?F 1 ?φ Du uv tan ? - ----- – α u. ------ = fv + ----------------- – -------------a cos ? ?λ ?z Dt a
Multiply (2.32) by a cos ? , to obtain:


?F Du ?φ - = fva cos ? + uv sin ? – ----a cos ? ------ – a cos ? ? α u? ? ?z ? Dt ?λ ?F u? ?φ = ( 2 ? a cos ? + u ) v sin ? – ----- – a cos ? ? α ? ?z ? . ?λ
Substituting (2.34) into (2.31), we ?nd that


?F DM ?φ -------- = – ----- – a cos ? ? α u? . ? ?z ? Dt ?λ


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Figure 2.17:

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This shows that M is conserved in the absence of (zonal) pressure torques and frictional torques. Note that (2.34) is considerably simpler than (2.32), indicating that angular momentum is more nearly conserved than linear momentum is. Now use the continuity equation to go over to ?ux form:

?F u ? ?M ?φ - ( ω M ) = – ----- ( a cos ? ) . ------- + ? ? ( V M ) + ----- + g -------?p ?p ?t ?λ
Integrate (2.35), term by term, through the entire column. For example,


p S ?M ------- dp 0 ?t

? pS ? pS - ( ∫0 M dp ) – M S ------- . = --?t ?t


Here p S is the surface pressure. In this way, we ?nd that

--- ( ∫0 S M dp ) + ?t


? pS p ------? ( ∫0 S V M dp ) – M S ? ? ?t + V S ?

pS –



? pS ? - ∫ φ dp + φ S ------- + ga cos ? ( F u ) S = – ----?λ ?λ

?φ S ? pS - ( ∫0 φ dp – φ S p S ) – p S ------- + ga cos ? ( F u ) S . = – ----?λ ?λ (2.37)
The last term on the left-hand side of (2.37) drops out because the condition that no mass can cross the Earth's surface can be expressed as

? pS ------- + VS ? ?t

pS –


= 0.


Finally, integrate (2.37) around a latitude circle, to obtain:

? 2 π pS 1 ? 2π ---( - ----M dp d λ ) + -------------?t ∫0 ∫0 a cos ? ?? ∫0



v cos ? M dp d λ

?φ S 2π π - d λ + ga cos ? ∫2 = – ∫0 p S ------( F u )S d λ . 0 ?λ


The ?rst term on the right hand side of (2.39) represents the effects of “mountain torque.” It vanishes if either pS or φS is independent of λ. See Fig. 2.18. The mountain torque also

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vanishes if is pS is a function of φS only. According to (2.39), in the absence of mountain torque and surface friction, M is conserved. Note that both torques in (2.39) involve angular momentum exchange between the atmosphere and the underlying surface.
westerly flow

high pS Mountain low pS

?φ S ------->0 ?λ

?φ S -------<0 ?λ λ


?φ S ?M ------- ? – ∫ p s ------- dλ < 0 ?t ?λ
Figure 2.18:

The Earth-atmosphere system can exchange angular momentum with other bodies through tidal forces. Such forces are causing the length of the day and the radius of the Moon's orbit to increase slowly. To the extent that we can neglect these small effects, the angular momentum of the Earth-atmosphere system must remain constant with time. Evidence that this is really true is shown in Fig. 2.19. The data plotted in the ?gure indicate that changes in the rate of rotation of the solid Earth, i.e., the length of day, are highly correlated with changes in the angular momentum of the atmosphere. Because the tendency of the angular momentum of the atmosphere is negligible in a long time average, the angular momentum exchanged between the Earth (i.e., the solid Earth and oceans together) and the atmosphere must average out to zero. This can be seen from the time average of (2.37):

1 ? 2π -------------- ----a cos ? ?? ∫0

pS cos ? 0

?φ S? π? π ------- d λ + ga cos ? ∫2 vM dp d λ = – ∫2 p ( F u ) S d λ . (2.40) S 0 ? 0 ?λ ?

The left-hand side of this equation represents the meridional transport of vertically and zonally integrated atmospheric angular momentum. The two terms on the right-hand side represent the sources and sinks due to mountain torque and surface friction torque, respectively. The surface frictional stress transfers positive angular momentum from the atmosphere to the ocean-solid Earth system where the surface winds are westerly, and positive angular momentum ?ows back into the atmosphere where the surface winds are easterly. The mountain torques generally have the same sign as the frictional torques. It follows that for the Earth as a whole there must be a mixture of easterly and westerly winds at the Earth’s surface. Observations discussed above demonstrate that this is in fact the case.

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Figure 2.19:

Fig. 2.20 shows the observed zonally averaged zonal wind stress for the oceans only, as compiled by Han and Lee (1983). In this ?gure, positive zonal wind stresses indicate that positive (westerly) atmospheric momentum is being transferred to the ocean; such positive values naturally occur in regions where the surface wind is westerly, i.e. in the middle latitudes of both hemispheres. Negative zonal surface stresses, somewhat weaker in magnitude, occur in the tropics and subtropics, in association with the trade winds. Fig. 2.21 shows estimates of the zonally averaged friction and mountain torques. The frictional torque dominates overall, but mountain torque is appreciable in the Northern Hemisphere. Fig. 2.22 shows the ?ow of angular momentum from the tropics, where the surface easterlies extract it from the oceans and solid earth, to middle latitudes, where the surface westerlies deposit it in the oceans and the solid earth. Oort (1989) argues that the return ?ow occurs largely through the movement of the continents, rather than through the circulation of the oceans.
Conservation of angular momentum in the air ?owing poleward in the upper branch of the Hadley cell leads to the formation of westerlies, which take their maximum values in the jet streams near the latitudes where the Hadley circulations stop. The westerly jets can thus be interpreted as consequences of the poleward ?ow of air in the Hadley circulations. Further discussion is given in later. Suppose that the zonal surface wind is nearly geostrophic. Then, in the absence of mountains (e.g. over the oceans) the surface pressure must have a meridional maximum at a
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Figure 2.20:

latitude where u passes through zero, i.e. where surface easterlies meet surface westerlies. If compare Fig. 2.2 with Fig. 2.4, you will see that in fact the subtropical surface pressure maximuma occur at the latitudes where the zonal surface wind passes through zero. This partially explains why the surface pressure tends to have a maximum in the subtropics.

Fig. 2.23 shows the (somewhat idealized) vertical distribution of temperature from the surface to the 100 km level1. In the lowest 10 to 15 km, the temperature decreases monotonically upward; this is the troposphere. Above the upper boundary of the troposphere, which is called the tropopause, the temperature becomes uniform with height and then begins to increase upward. This region is the stratosphere. The upward increase of temperature in the stratosphere is due to the absorption of solar radiation by ozone, which is created in the stratosphere by photochemical processes. Without ozone there would be no stratosphere. The vertical distribution of ozone is also shown in Fig. 2.23. The stratopause occurs near the 1 mb (~50 km) level, which is above the ozone layer. Above the stratosphere is the mesosphere, within which the temperature decreases upwards again. The mesopause is near the 0.01 mb level. Above the mesopause is the “thermosphere,” within which the temperature increases upwards again. Within the thermosphere, the composition of “dry air” begins to change signi?cantly, the air loses the qualities of a perfect gas, electromagnetic forces become important for the dynamics of the atmosphere, and the molecular viscosity (per unit mass) becomes very large. In this course, we consider the troposphere and the stratosphere, but we will not discuss the mesosphere or the thermosphere. Fig. 2.24 shows the latitude-height distributions of the zonally averaged temperature for January and July, respectively. At low levels, the warmest air is near the Equator, but near
1. The

100 km level is approximately the height at which objects entering the Earth’s atmosphere begin to experience significant frictional heating.

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Figure 2.21:

100 mb the coldest air is over the Equator. In fact, the lowest temperatures in the ?gures are found near the tropical tropopause. It is also true that the tropopause is highest in the tropics and lowest near the poles. The tropopause height is nearly discontinuous in the mid-latitudes, particularly in the winter hemisphere. In the stratosphere, extremely cold temperatures are found above the winter pole, especially in the Southern Hemisphere. The summer pole is much warmer due to the absorption of solar radiation by ozone. The midlatitude low-level temperature gradients are quite strong in the winter hemisphere. Above 200 mb, the summer pole is considerably warmer than the winter pole; in fact, the zonally averaged temperature increases monotonically from the Equator to the summer pole near 100 mb. This suggests easterlies in the summer stratosphere, which do in fact occur. In the winter hemisphere, the warmest 100 mb temperatures occur in middle latitudes. The strong decrease of temperature between midlatitudes and the poles is consistent with the polar night jets mentioned earlier. In January, a temperature “inversion” (i.e. temperature increasing upward) appears ?T over the North Pole. Generally speaking, the lapse rate, – ----- , is largest in the tropics, and ?z smaller (or even negative) near the poles. Thermal wind balance between the meridional temperature gradients and the vertical

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Figure 2.22:

shear of the zonal wind is well satis?ed, as can be seen by comparison of Fig. 2.4 and Fig. 2.24. It is interesting to compare the latitude-height section of zonally averaged temperature, shown in Fig. 2.24, with the corresponding latitude-height section of zonally p 0? κ averaged potential temperature, which is shown in Fig. 2.25. Recall that θ ≡ T ? ---? p ? , where R - . The key p 0 is a constant reference pressure, usually taken to be 1000 mb, and κ ≡ ---cp physical facts about potential temperature are that it is conserved under dry adiabatic processes and that it increases upward in a statically stable atmosphere. This upward increase is evident in the ?gure. In the troposphere, potential temperature surfaces slope downwards

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Figure 2.23:


from the polar regions to the tropics, while in the stratosphere the opposite occurs. The stratosphere is easily identi?ed because the potential temperature increases upward very sharply there, showing that the static stability is very strong. On the other hand, the static stability is particularly weak in the tropical upper troposphere.

Fig. 2.26 shows maps of the 850 mb temperature for January and July, respectively. Fig. 2.27 shows the corresponding results for 200 mb. The expected winter-to-summer warming at 850 mb is very obvious in the Northern Hemisphere, but less so in the Southern Hemisphere, except over land. Temperature gradients in the tropics are very small. Monthly mean temperatures over the high Antarctic terrain reach about -50 °C in July, while those over the Arctic ocean in January do not fall below -35 °C. In the tropics there is very little seasonal change, and the temperature distribution is very uniform.
Naturally, the temperature gradient points mainly from the poles toward the tropics at 850 mb, but stationary eddies are plainly visible in the Northern Hemisphere in January. There are some regions in which the mean temperature actually increases poleward, at 850 mb, e.g. from Northern Africa across to India in July, and near Australia in January. From thermal wind considerations, we might expect easterlies aloft in these regions. This expectation is borne out in Fig. 2.5. In winter, there is a tendency for the eastern sides of the Northern Hemisphere

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Figure 2.24:

continents to be colder than the western sides. This leads to particularly strong meridional temperature gradients on the east coasts. We know that such strong temperature gradients favor a rapid upward increase of the westerlies; also, such highly baroclinic regions are preferred centers of cyclogenesis. The strongest temperature gradients at 200 mb are found in high latitudes, especially in winter. The eddy pattern is much stronger at 200 mb than at 850 mb. Particularly noticeable are the maxima over the North Paci?c in January, over eastern North America in January, and over southern Asia in July. Note than in each of these regions the 200 mb temperature increases as we move from the tropics towards middle latitudes. This implies a

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Figure 2.25:

tendency for the westerlies to weaken above this level.

Fig. 2.28 shows zonal and time-averages, plotted as functions of latitude and potential 1?p temperature of pressure, the pseudodensity – -- , zonal wind, and meridional wind, g?θ according to Edouard et al. (1997). The pseudodensity measures the amount of mass between potential temperature surfaces. For a variety of reasons (Hoskins et al., 1985), the use of
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Figure 2.26:

potential temperature as a vertical coordinate is becoming increasingly common in both observational and modeling studies. It is logically possible to use potential temperature as a vertical coordinate because potential temperature increases upward (with just a few minor exceptions), so that there is a one-to-one correspondence between height and potential temperature. In the absence of heating, parcels remain on surfaces of constant potential temperature. Fig. 2.29 shows the zonally averaged potential vorticity (PV), which is de?ned by

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Figure 2.27:

PV = – ( ζ θ + f )

?θ g, ?p


where ζ θ is the vertical component of the vorticity in which the horizontal derivatives are taken along isentropic surfaces. The PV is a highly conservative variable which strongly controls the large-scale dynamical ?elds (e.g. Hoskins et al., 1985). The ?gure shows that in middle latitudes the tropopause roughly coincides with a constant-PV surface, namely the

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Figure 2.28:

± 2 PVU surface. Note from Fig. 2.28 that in the tropics the tropopause roughly coincides with a constant- θ surface, namely the 390 K surface. The stratosphere is a region of high PV, and intrusions of stratospheric air into the troposphere are characterized by anomalously large values of the PV. The troposphere has a relatively uniform PV, suggesting that PV is being mixed there. Stratospheric air entering the troposphere can be recognized by its large (absolute) PV.

Representative vertical distributions of relative humidity and mixing ratio are shown in Fig. 2.30. There is a nearly monotonic upward decrease of the relative humidity in the climatology, although of course this is not necessarily true at any particular time and place.

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Figure 2.29:

The mixing ratio pro?le is representative of middle latitudes, in the annual mean. Note that the horizontal scale is logarithmic. The mixing ratio decreases by four orders of magnitude between the surface and the lower stratosphere. It actually increases upward in the stratosphere, because there is a chemical source of water vapor in the stratosphere due to the oxidation of methane

Fig. 2.31 shows the latitude-height distributions of the zonally averaged water vapor mixing ratio for January and July, respectively. Because the data presented in these ?gures have been fed through the analysis/forecast system of a numerical weather prediction center (namely ECMWF), which can easily distort the distribution of water vapor, they should be taken with a grain of salt.

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Figure 2.30:

The ?gures show the most humid air near the equator, and the driest air near the winter pole. The seasonal change in the Northern Hemisphere is quite dramatic. There is an extremely rapid upward decrease of the mixing ratio at all latitudes. The largest average values, near the surface in the tropics, are close to 18 g kg-1, which means that about 2% of the air is water vapor. Despite the extreme dryness of the air aloft (a few parts per million by volume in the stratosphere), upper tropospheric water vapor is very important radiatively. Since the mixing ratio is greatest near the surface, regions of low-level mass convergence, such as those apparent in the zonally averaged meridional wind (Fig. 2.8) tend to be regions of vertically integrated moisture convergence as well. Although as much mass diverges at upper levels as converges at lower levels, the diverging air aloft is dry, while the converging air near the surface is moist.

Fig. 2.32 shows maps of the 850 mb water vapor mixing ratio for January and July, respectively. As would be expected, the largest values occur in the tropics and the summer hemisphere. A very clear maximum extends around the circumference of the Earth in the tropics, mainly somewhat north of the Equator. Meridional moisture gradients are often quite sharp.
There are also strong east-west variations, however. For example, in January there are strong maxima over southern Africa and in the Amazon basin. Minima are found in the subtropical highs. There are very dramatic seasonal changes over the midlatitude continents, with larger values in summer. Major desert regions like the Sahara and western North America are clearly associated with water vapor minima. The distributions of temperature and moisture can be combined in a variable called

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Figure 2.31:

the “moist static energy,” which is de?ned by

h ≡ c p T + gz + Lq ,


The moist static energy is of interest in part because it is conserved under both moist and dry

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Figure 2.32:

adiabatic processes, and even under pseudoadiabatic processes in which precipitation occurs; these conservation properties are proven later. The latitude-height distribution of moist static energy, as analyzed by ECMWF, is shown in Fig. 2.33 . In the upper levels, the moist static energy increases upwards; in the tropics and in the midlatitude summer, the moist static energy decreases upwards in the lower troposphere; and especially in the tropics the moist static energy has a minimum in the middle troposphere. The reasons for this distribution are as follows: Above the middle troposphere, or at any altitude near the poles, the water vapor mixing ratio is negligible. In that case, the moist static energy reduces (approximately) to the

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Figure 2.33:

dry static energy, which is given by s ≡ c p T + gz . It can be shown that s (like the potential temperature) increases upward in a dry statically stable atmosphere. At lower levels, where water vapor is plentiful, especially in the tropics, the upward decrease of the water vapor mixing ratio overwhelms the upward increase of s, so that the moist static energy decreases upwards. It follows that the moist static energy has a minimum in the middle troposphere in the tropics.

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The zonally averaged precipitation rates for January and July are shown in Fig. 2.34. Fig. 2.35 shows maps of the precipitation rate for January and July, respectively. Because

Figure 2.34:

precipitation tends to be very “spotty” in both space and time, average values are dif?cult to measure accurately even under ideal conditions. As a result, these values are uncertain by at least 25%. Certainly the data for such remote regions as the South Paci?c Ocean cannot be strongly defended. The zonally averaged precipitation has its strongest maximum in the tropics, with secondary maxima in the middle latitudes. The tropical maximum is associated with the intertropical convergence zone and the monsoons, while the middle latitude maxima are associated with baroclinic wave activity in the winter, and monsoon circulations in the summer. There are minima in the subtropics, where the major deserts occur. The global mean of the precipitation rate is around 3 mm day-1; again, this value is uncertain by perhaps 25%.

Fig. 2.35 clearly shows that the rainiest regions of the world are in the tropics. There are major seasonal shifts in the locations of the tropical precipitation. In January, heavy rain falls over the Amazon basin, over southern Africa, the Indian Ocean, the maritime continent north of Australia, in the South Paci?c Convergence Zone that extends southeastward from the intersection of the Date Line with the Equator, and across most of the tropical Paci?c and Atlantic Oceans north of the Equator. In July, the tropical rains have generally shifted to the north. We ?nd heavy rainfall in the extreme northern part of South America, the neighboring Caribbean Sea and tropical North Atlantic Ocean, over India and neighboring regions of Southeast Asia, and to the north of the maritime continent, off the east coast of tropical Asia. The seasonal shifts of tropical precipitation are quite spectacular, and are most clearly seen in the longitudes of South America, Africa, India, and Southeast Asia.

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Figure 2.35:

As is clear from Fig. 2.34 , minima of the zonally averaged precipitation occur in the subtropics, and secondary maxima occur in the middle latitudes. Midlatitude precipitation is also highly variable. For example, the precipitation over northern Asia occurs mainly in the summer. The warm currents off the east coasts of North America and Asia receive heavy precipitation mainly in January. The northwestern portion of the United States receives heavy precipitation in January but not in July. Regions that receive plentiful precipitation throughout the year include eastern North

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America, the extreme southern tip of South America, and England. As can be seen by comparing Fig. 2.16 and Fig. 2.35 , there is a strong correlation between vertical motion and precipitation. Precipitation maxima correspond to regions of latent heat release inside the column. Regions of latent heat release also tend to be regions in which the net radiative heating of the atmosphere is positive, because of the preponderance of high cold clouds that block the emission of longwave radiation to space. (This will be discussed further later.) In short, rising motion tends to occur where there is heating, and this is particularly true in the tropics. Here we return to the analysis of Charney (1963). As already discussed, Charney concluded that horizontal temperature gradients tend to be weak in the tropics. Referring to the thermodynamic energy equation, (2.4), we re-write it in expanded form, as

?F T ?T ? ?T c p? -----+ V ? ? T + ω = ωα + gc p ? p + Q rad + Q lat . ? ?t ? p?


As already explained in Chapter 1, the time average of the time-rate-of-change term can be made as small as desired if the averaging interval is chosen to be long enough. For the case of atmospheric temperature, an averaging interval of one month is suf?cient to make the time-rate-of-change term of (2.43) negligible. Then the time average of (2.43) can be written as

V ? ?c pT + ω

? ? ( c p T + φ ) = g F c p T + Q rad + Q lat . ?p ?p


Here the overbar is used to denote a time average, and the hydrostatic equation has been used. The terms on the left-hand side of (2.44) represent dry adiabatic processes, while those on the left-had side represent “heating” of various kinds. The quantity c p T + φ is sometimes called the dry static energy. We show later that the dry static energy increases ? upwards when the atmosphere is stably strati?ed in the dry sense, i.e. ( c T + φ ) < 0 . The ?p p ? vertical motion term of (2.44) therefore represents cooling ( – ω ( c p T + φ ) < 0 ) when the ?p ? air is rising ( ω < 0 ) and warming ( – ω ( c p T + φ ) > 0 ) when the air is sinking ( ω > 0 ). ?p Eq. (2.)(2.44) simply says that in a time average, heating has to be balanced by a combination of horizontal and vertical advection. Now recall Charney’s conclusion that horizontal temperature gradients are negligible in the tropics. This means that the ?rst term on the left-hand side of (2.)(2.44) is relatively small in the tropics. It follows that the only way to balance tropical heating or cooling is through vertical motion, and so clearly there should be a very strong correspondence between the pattern of vertical motion and the pattern of heating. This is what we see when we compare Fig. 2.16 and Fig. 2.35. Tropical rising motion occurs almost exclusively where ?F c p T latent and radiative heating are active, i.e. where g + Q rad + Q lat > 0 , and tropical ?p

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sinking motion occurs almost exclusively where radiative cooling is dominant, i.e. where ?F c p T g + Q rad + Q lat < 0 . ?p These conclusions apply fairly well even in the middle latitudes in summer, simply because horizontal temperature gradients are weak there as well. They de?nitely do not apply in the middle latitude winter, where the horizontal advection term of (2.44) is critically important.

We do not yet have a satisfactory theory of the general circulation of the atmosphere. The main dif?culties are:

? While there is no doubt that the general circulation is thermally forced, the heating mechanisms are very complicated, and motion-dependent. ? Even if we pretend that the heating is prescribed, the dynamical response to the heating is complicated because of the existence of eddies and their interactions with the zonalmean flow. The eddies are neither purely random nor purely regular. We do have a qualitative “view” of the general circulation, based on many varied studies. This view is much more than a mere speculation. It can be summarized as follows:
As will be discussed later, a steady, zonally symmetric circulation is mathematically possible, or at least it would be if the Earth’s surface were zonally uniform. Such a symmetric circulation could be driven by tropical heating and cooling at higher latitudes. The associated “thermal wind” shear of the zonal component would be very large, however. Correspondingly, a steady symmetric circulation would have a very large equator-to-pole temperature gradient, indicating that the poleward energy transport is inef?cient in this symmetric regime. This inef?ciency arises from the Earth’s rapid rotation, which inhibits meridional motions. If the energy transport was ef?cient, the temperature gradient would be much weaker Nature chooses a more ef?cient way of transporting energy poleward, which results in a weaker meridional temperature gradient. This is possible because the symmetric circulation is unstable. The relevant mechanism is baroclinic instability, which causes the growth of quasi-horizontal, quasi-geostrophic eddies of cyclone scale. These eddies transport heat poleward, as indicated schematically in Fig. 2.36. The ?ux is greatest in middle latitudes because that is where the vertical shear is strongest; the eddies are inactive in the tropics partly because f is small there. Eddies can also be forced by zonal inhomogeneities in the boundary conditions, e.g. mountains and heating due in part to air-sea contrast. Because the mechanisms which produce them are “anchored” to the Earth’s surface, such forced eddies tend to be stationary. Observations (presented later) show that such forced eddies do transport some energy poleward, mainly in the Northern Hemisphere. Although the meridional circulation is not symmetric, a mean meridional circulation (MMC) can, of course, be de?ned; that is precisely what is plotted in Fig. 2.15 . The MMC “feels” the eddy energy transport as a cooling in the subtropics and a warming in higher latitudes. In response, the MMC tends to produce sinking in the subtropics and rising motion in higher latitudes. This is a crude interpretation of why there is a three-cell circulation in the winter hemisphere (where baroclinic waves are particularly active), with a Ferrel Cell in
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Pole warming Figure 2.36: cooling


middle latitudes, as shown in Fig. 2.15 and indicated schematically in Fig. 2.37 . Note that

Eddy warming

Eddy cooling

Eddy Heat Flux

Ferrel Cell Pole Figure 2.37:

Hadley Cell Equator

the MMC resists the efforts of the eddies. That turns out to be true quite generally, as we will see later.
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This chapter is intended to impart some familiarity with basic features of the observed general circulation, without much attempt to explain why the circulation appears as it does. The observations presented in this chapter raise many questions. For example:

? Why does the sea level pressure tend to attain its maximum values in the subtropics? ? What determines the intensity and latitudes of the jet streams? Why do the winter jet maxima occur at particular longitudes? ? Why is the upper-level circulation “wavy,” and the lower-level circulation “lumpy?” ? What determines the number of “bands” seen, for example, in Fig. 2.15? ? What mechanisms generate the observed stationary waves? ? Why are there surface easterlies near the poles? ? Why are seasonal changes generally weaker in the Southern Hemisphere than in the Northern Hemisphere? ? What determines the magnitude of the pole-to-equator gradient of the surface temperature? What determines the observed lapse rate of temperature? Why does the lapse rate change as we move from the tropics to the middle latitudes to the poles? ? What determines the height of the tropopause, as a function of latitude? Why is the tropical tropopause so cold? ? Why do the subtropical highs tend to occur in the eastern portions of the ocean basins? ? Why is there such a strong belt of low pressure around Antarctica? What determines the locations and intensities of the Aleutian and Icelandic Lows? ? What causes the summer and winter monsoons? ? Why is there a strong Siberian winter high? ? How much do individual Januaries and Julys, for particular years, differ from the “average” January and July conditions shown here? What causes such year-to-year variations? ? What determines the vertical distribution of water vapor? ? How are the observed patterns of large-scale rising and sinking motion produced and maintained? ? What are the geographical patterns of the day-to-day weather fluctuations that accompany the monthly mean maps shown here, and how do these fluctuations affect the time means? ? Why does the general circulation appear “smooth,” rather than “noisy?”
These and many other questions will be discussed in the remainder of this course.

References and Bibliography
Ambaum, M., 1997: Isentropic formation of the tropopause. J. Atmos. Sci., 54, 555-568. Bengtsson, L., M. Kanamitsu, P. Kallberg, and S. Uppala, 1982: FGGE 4-dimensional data assimilation at ECMWF. Bull. Amer. Meteor. Soc., 63, 29-43. Charney, J. G., 1963: A note on large-scale motions in the tropics. J. Atmos. Sci., 20, 607Randall 605 Book Title

609. Dickey, J. O., S. L. Marcus, J. A. Steppe, and R. Hide, 1992: The Earth's angular momentum budget on subseasonal time scales. Science, 255, 321-324. Dong, D., R. S. Gross, and J. O. Dickey, 1996: Seasonal variations of the Earth's gravitational field: an analysis of atmospheric pressure, ocean tidal, and surface water excitation. Geophys. Res. Lett., 23, 725-728. Dutton, J. A., 1976: The Ceaseless Wind. McGraw-Hill, New York, 579 pp. ECMWF 1997: ERA Description. ECMWF Re-analysis Project Report Series 1. Available from ECMWF.4 Edouard, S., R. Vautard, and G. Brunet, 1997: On the maintenance of potential vorticity in isentropic coordinates. Quart. J. Roy. Meteor. Soc., 123, 2069-2094. Eliassen, A., and E. Raustein, 1968: A numerical integration experiment with a model atmosphere based on isentropic coordinates. Meteorologiske Annaler, 5, 45-63. Eliassen, A., and E. Raustein, 1970: A numerical integration experiment with a six-level atmospheric model with isentropic information surface. Meteorologiske Annaler , 5, 429-449.0 Feynman, R. P., R. B. Leighton, and M. Sands, 1963: The Feynman lectures on physics. Vol. 1: Mainly mechanics, radiation, and heat. Addison-Wesley Pub. Co., Han, Y.-J., and S.-W. Lee, 1983: An analysis of monthly mean wind stress over the global ocean. Mon. Wea. Rev., 111, 1554-1566. Hide, R, and J. O. Dickey, 1991: Earth's variable rotation. Science, 253, 629-637. Hoskins, B. J., M. E. McIntyre, and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877-946. Hoskins, B., 1996: On the existence and strength of the summer subtropical anticyclones. Bull. Amer. Meteor. Soc., 77, 1287-1292. Hsu, Y.-J., and A. Arakawa, 1990: Numerical modeling of the atmosphere with an isentropic vertical coordinate. Mon. Wea. Rev., 118, 1933-1959. Huffman. G. J., R. F. Adler, P. Arkin, A. Chang, R. Ferraro, A. Gruber, J. Janowiak, A. McNab, B. Rudolf, and U. Schneider, 1997: The Global Precipitation Climatology Project (GPCP) Combined Precipitation Dataset. Bull. Amer. Meteor. Soc ., 78, 520.

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Kasahara, A., 1974: Various vertical coordinate systems used for numerical weather prediction. Mon. Wea. Rev., 102, 509-522. Konor, C. S., and A. Arakawa, 1997: Design of an atmospheric model based on a generalized vertical coordinate. Mon Wea. Rev., 125, 1649-1673. Mo, K., J. O. Dickey, and S. L. Marcus, 1997: Interannual fluctuations in atmospheric angular momentum simulated by the National Centers for Environmental Prediction medium range forecast model. J. Geophys. Res., 102, 6703-6713. Manabe, S., and R. T. Wetherald, 1967: Thermal equilibrium of the atmosphere with a given distribution of relative humidity. J. Atmos.Sci., 24, 241-259. Newton, C. W., 1971: Mountain torques in the global angular momentum balance. J. Atmos. Sci., 28, 623-628. Oort, A. H., 1985: Balance conditions in the Earth's climate system. Adv. in Geophys., 28A, 75-98. Oort, A. H., 1989: Angular momentum cycle in the atmosphere-ocean-solid earth system. Bull. Amer. Meteor. Soc., 70, 1231-1242. Ponte, R. M., D. Stammer, and J. Marshall, 1998: Oceanic signals in observed motions of the Earth’s pole of rotation. Nature, 391, 476-479. Rosen, R. D., D. A. Salstein, T. M. Eubanks, J. O. Dickey, and J. A. Steppe, 1984: An El Ni?o Signal in Atmospheric Angular Momentum and Earth Rotation. Science, 225, 411-414. Trenberth, K. E., and J. G. Olson, 1988: An evaluation and intercomparison of global analyses from the National Meteorological Center and the European Centre for Medium Range Weather Forecasts. Bull. Amer. Meteor. Soc., 69, 1047-1057. Wahr, J. M., and A. H. Oort, 1984: Friction- and mountain-torque estimates from global atmospheric data. J. Atmos. Sci., 41, 190-204. Xie, P., and P. A. Arkin, 1996: Analyses of Global Monthly Precipitation Using Gauge Observations, Satellite Estimates, and Numerical Model Predictions. J. Climate, 9, 840-858.

1. a) Estimate the total water vapor content of the atmosphere, in kg. b) Estimate the total mass of liquid and ice (combined) in the global atmosphere, in kg. This number is not actually known to better than an order of magnitude. You
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will have to be creative to come up with a credible estimate of your own. State any assumptions that you make. 2. Make a rough estimate of the total kinetic energy of the atmosphere, in joules. If all of the solar radiation absorbed by the Earth were used to supply this kinetic energy, how long would it take to accumulate the observed amount? Derive (2.20) by starting from (2.3). Show that the mountain torque vanishes if p S depends only on φ S .

3. 4.

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