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Figure 2. Schematic of wave breaking, with the resultant convergence of gravity wave momentum flux. From Geller [1983].

the gravity wave energy propagates. This is shown in

Figure 1, which is Figure 2 in Hines’ [1960] paper.

In the previous discussion of the noninteraction theorems,

one of the conditions for noninteraction was u0 ≠ c; that is, the

mean zonal flow is unequal to the wave phase velocity.

Bretherton [1966] examined the case of a gravity wave in a

shear flow where u0 = c (the critical level) and the Richardson

number (to be defined shortly) is very large. He found that

in this case the gravity wave vertical group velocity → 0 as

ˉ vanishes on

u0 → c. Thus, the gravity wave energy flux p′w′

the far side of the critical level, in which case the momentum

ˉ is also zero. Since there is no wave interaction

flux ρ0 u′w′

below the critical level, this implies a convergence (or

divergence) of the wave momentum flux at the critical level.

Booker and Bretherton [1967] generalized this result to the

case of finite Richardson number, in which case they derived

the results that in passing through the critical level,



ffi wave


momentum flux is attenuated by a factor of e−2π Ri−4 , where

Ri, the Richardson number, is given by


Ri ¼  2 :



Thus, at a gravity wave critical level, the absorption of

the wave will tend to bring the mean flow toward the wave

phase velocity (by Eliassen and Palm’s first theorem).

There followed a period of very active research into the

nature of gravity wave critical levels. Hazel [1967] showed

that the Booker and Bretherton [1967] result was essentially

correct in the case of a fluid with viscosity and heat

conduction. Breeding [1971] suggested that nonlinear effects

might lead to some wave reflection in addition to absorption,

but Geller et al. [1975] suggested that as the wave approached

a critical level, it produces turbulence that would likely lead to

wave absorption before nonlinear effects would lead to wave


In an isothermal atmosphere, the densityz decreases

exponentially with increasing altitude z as e H , H being

the pressure scale height. Without dissipation or critical

levels, the gravity wave kinetic energy per unit volume ρ0v′2

should remain constant, in which case the amplitude of the

wave’s horizontal velocity (and as it turns out temperature)


uctuations should grow as e 2H . This being the case, the

wave eventually becomes unstable. Hodges [1967] was

the first to point out that this will be a source of turbulence in

the middle and upper atmosphere.

This provided the starting point for Lindzen’s [1981]

seminal paper that suggested a self-consistent way of

parameterizing the effects of unresolved gravity waves in

climate models. The principle for this parameterization is

illustrated in Figure 2. On the right is illustrated a gravity wave

whose wind and temperature amplitude are exponentially

increasing with height, and as pictured, the wave momentum

ˉ is constant with height. Since the vertical

flux ρ0 u′w′

wavelength of this wave is fixed, ∂v′/∂z and ∂T′/∂z also

increase with height exponentially, as illustrated by the outer

envelope. Eventually, the wave becomes either convectively

unstable or shear unstable and breaks down. Lindzen [1981]

made the assumption that above the level where the wave

breaks down, it loses just enough energy to turbulence to keep

the wave amplitude constant above that level, as illustrated.

ˉ decreases with height above the

This means that ρ0 u′w′

breaking level so that there is a divergence of wave

momentum flux above the level where the gravity wave

begins to break, also as pictured in Figure 2. Of course, one

could make different assumptions of what occurs above the

breaking level. For instance, Alexander and Dunkerton

[1999] assume that gravity waves deposit all of their

momentum at the breaking level.

This gravity wave breaking and the subsequent drag on

mesospheric winds (by Eliassen and Palm’s first theorem)

gave physical justification to the Rayleigh drag used by Leovy

Figure 3. Time-height section of the monthly mean zonal winds (in m sÀ1 ) over equatorial stations from Geller et al. [1997], which was an

update from Naujokat [1986]. Copyright American Meteorological Society.


Figure 3. (continued)



[1964] in his modeling of the mesospheric wind structure,

since many gravity waves have their source in the troposphere

where their source phase velocity is small. Developing and

implementing ways of parameterizing the effects of unresolved gravity waves in climate models is a research topic of

great current interest, but one might say that this had its

intellectual roots in the papers of Eliassen and Palm [1961],

Hodges [1967], and Booker and Bretherton [1967], since

critical levels are also of great importance in this.


The quasi-biennial oscillation (QBO) was discovered

independently by Reed et al. [1961] and by Veryard and

Ebdon [1961]. Figure 3 shows its structure over the equator. A

quasiperiodic pattern of descending easterlies (unshaded)

followed by descending westerlies (shaded) is evident. The

average period of a complete cycle is about 28 months, but the

period varies considerably, being about 21 months in 1972–

1974 and about 35 months in 1983–1986. Moreover, the

westerlies descend more quickly than the easterlies. The

maximum amplitude of the QBO is about 20 m sÀ1, occurring

in the middle stratosphere.

Following the discovery of the QBO, there were many

attempts to explain why this phenomenon occurred, but the

key papers that led to today’s generally accepted explanation

for the QBO were those of Wallace and Holton [1968],

Lindzen and Holton [1968], and Holton and Lindzen [1972].

There were many efforts that tried to explain the QBO in

terms of a hypothesized periodic radiative forcing, but

Wallace and Holton [1968] constructed a diagnostic model to

Table 1. Characteristics of the Dominant Observed Planetary-Scale

Waves in Equatorial Lower Stratospherea

Theoretical Description


Period (ground-based)

Zonal wave number

Vertical wavelength

Average phase speed

relative to ground

Observed when mean

zonal flow is

Average phase speed

relative to maximum

zonal flow

Average phase speed

relative to maximum

zonal flow




Approximate inferred





meridional scales




Figure 4. Schematic illustration of the geopotential and wind fields

for the equatorial trapped (top) Kelvin and (bottom) mixed Rossbygravity waves. Adapted from Andrews et al. [1987], who, in turn,

adapted it from Matsuno [1966].

Kelvin Wave



Wallace and

Kousky [1968]

15 days


6–10 km

+25 m sÀ1

Yanai and

Maruyama [1966]

4–5 days


4–5 km

À23 m sÀ1


(maximum ≈

À25 m sÀ1

+50 m sÀ1


(maximum ≈

+7 m sÀ1)

À30 m sÀ1

8 m sÀ1



2–3 m sÀ1

2–3 m sÀ1


30 m


1.5 10À3 m sÀ1

1300–1700 km



1.5 10À3 m sÀ1

1000–1500 km

1 =2

From Andrews et al. [1987].

see what kind of radiative and momentum forcings would be

necessary to explain the observed characteristics of the QBO.

They found that only an unrealistic radiative forcing could

explain the observed features. On the other hand, they found

that momentum forcings could explain the observations but

only if the momentum forcing itself had a downward propagation. Lindzen and Holton [1968], noting the results of

Wallace and Holton [1968], published their famous paper that

gave essentially today’s accepted explanation for the QBO

only 8 months after the appearance of the Wallace and Holton

[1968] paper. They noted that there were reasons to believe

that there were strong gravity waves in the equatorial region.

They noted that Matsuno [1966] had predicted the existence

of equatorially trapped eastward propagating Kelvin waves

and westward propagating mixed Rossby-gravity waves (see


Figure 5. Schematic representation of the Lindzen and Holton [1968]/Holton and Lindzen [1972] theory for the QBO: (a)

initial state and (b) initial state (curve 1) and evolutionary progression. Curves 2 and 3 show successive stages of evolution,

as explained in the text. After Plumb [1984].

Figure 4). These waves had subsequently been observed by

Yanai and Maruyama [1966] and Wallace and Kousky [1968]

(see Table 1). They noted that these equatorial gravity waves

would encounter critical levels and that the theory of Booker

and Bretherton [1967] implied the needed downward propagating momentum flux to explain the QBO. Their theory

was updated by Holton and Lindzen [1972] so that the gravity

wave momentum absorption now occurred through radiative

damping together with critical levels to produce the QBO.

Since the pioneering work of J. M. Wallace, J. R. Holton,

and R. S. Lindzen, much more work on the theory of the QBO

has taken place, and Alan’s work on this topic has been

seminal. An interesting laboratory analogue to the QBO was

demonstrated by Plumb and McEwan [1978]: a standing wave

pattern was forced by pistons oscillating a membrane at the

bottom of a cylinder filled with a stratified fluid. A descending

pattern of alternating angular velocities was observed to

result. This has been nicely interpreted by Plumb [1977], as is

Figure 6. Schematic sketch of the winter and summer mean zonal wind patterns. Also shown are the planetary wave raypaths

for the weak westerly wind waveguides in the winter hemisphere. From Dickinson [1968]. Copyright American

Meteorological Society.


Figure 7. (left) Mean zonal wind state, (middle) m = 0 refractive index, (right) m = 1 stationary planetary wave energy

propagation. From Matsuno [1970]. Copyright American Meteorological Society.

illustrated in Figure 5. This clearly showed that the essence

of the mechanism for the QBO was to have both eastward

and westward momentum fluxes that would be preferentially

absorbed in regions of small Doppler-shifted intrinsic wave

frequencies. Thus, in Figure 5a, positive phase speed waves

are preferentially absorbed, leading to a downward propagating westerly shear zone as shown in curve 1 of Figure 5b.

The negative phase speed waves, having high intrinsic

frequencies, propagated to higher altitudes, but they were

ultimately absorbed as indicated by the arrows at the top of

Figure 5a and of curve 1 in Figure 5b. As time passes, the

absorption of the two waves leads to curve 2 and then to

curve 3 in Figure 5b. Ultimately, the bottom shear zone gets

so extreme, it is subject to diffusive smoothing, which effectively leads to the mirror image of Figure 5a, so that the oscillation continues. While equatorially trapped waves no doubt

play a role in forcing the QBO, Haynes [1998] has demonstrated that a geographically uniform source of gravity waves

gives rise to the QBO through the different manner in which

the equatorial atmosphere reacts to momentum and heat

fluxes, which is distinct from the situation in the extratropics.

Thus, both “garden variety” gravity waves and equatorially

trapped waves play an important role in forcing the QBO.



The importance of the Charney-Drazin [1961] results on

planetary wave propagation was quickly appreciated. For

instance, Dickinson [1968] considered stationary planetary

wave propagation through a basic state that had its mean

zonal winds varying both in altitude and latitude. He

concluded that the strong mean zonal winds of the polar night

jet would be a “barrier” to planetary wave propagation,

resulting in the picture shown in Figure 6. In Figure 6,

note that the planetary wave rays refracted toward the equator

are hypothesized to be absorbed at the u0 = 0 critical line.

(Later work showed that there is also reflection.) Also pictured

is the polar waveguide ray where the waves are refracted

poleward by the strongest winds of the polar night jet and

reflected by the polar geometry. Some of the waves in this polar

waveguide are pictured as propagating to very high altitudes.

Matsuno [1970] advanced an alternative picture for the

propagation of stationary planetary waves in winter. His

formulation differed from that of Dickinson [1968] in that

Matsuno [1970] formulated his quasi-geostrophic equations

on a sphere so that they conserved energy. His picture is

shown in Figure 7. Figure 7 (left) shows Matsuno’s winter

mean zonal wind state. Figure 7 (middle) shows the effective

refractive index for the m = 0 planetary wave (m here being

the zonal wave number). The refractive index minimum results from the minimum in the latitudinal gradient of the zonally averaged quasi-geostrophic potential vorticity (not

shown here). Figure 7 (right) shows the energy propagation

vectors. Note that, unlike in the work of Dickinson [1968], the

minimum in the refractive index acts as a “barrier” to planetary

wave propagation rather than the “barrier” necessarily being the

region of strongest westerly winds. The planetary wave propagation shows a bifurcation around this “barrier” with propagation toward the equator but also upward propagation through

the lower portion of the strong polar night jet. Matsuno’s [1970]

picture is confirmed by observations [e.g., Geller, 1993].

Stationary planetary waves propagate both vertically and

latitudinally, so they encounter a critical line where u0 = 0.


Figure 8. Matsuno’s [1971] model results. (a) His imposed planetary wave forcing at his lower boundary at 10 km

compared to an observation by I. Hirota. (b) Time-height section of the evolution of his modeled wind at 608N. (c) Timeheight section of his modeled changes in temperature at 608N. (d) Changes in the zonal mean temperature as a function

of latitude at t = 0, 10, 20, and 21 days at an altitude of 30 km. Figures 8b–8d are from Matsuno [1971] for his experiment

C2. Copyright American Meteorological Society.

This situation was analyzed by Dickinson [1970]. His timedependent linear analysis indicated that the planetary wave

perturbation zonal velocities u′ → ∞ as the planetary wave

energy approaches the critical level but that the time scale for

this process is long compared with the times scale on which u0

varies. He concluded from his analysis that in realistic

situations, planetary wave–mean flow interactions take place

over a region hundreds of kilometers in width rather than at a

singular line. Later nonlinear analyses by Stewartson [1977],

Warn and Warn [1976], Killworth and McIntyre [1985], and

Haynes [1985] indicate that it is likely that planetary waves

are partially reflected by these critical regions. Observations

indicate that these critical regions now correspond to what we

now call the subtropical “surf zone” [e.g., McIntyre and

Palmer, 1983, 1984]. These later works were published while

Alan was already a leading researcher in middle atmosphere

dynamics and transport, and Alan would go on to clarify

many aspects of this “surf zone” on stratospheric transport.

A sudden stratospheric warming is an event in which lower

stratospheric temperatures increase dramatically by several

tens of degrees Celsius, and the winter westerly vortex actually reverses to easterlies in a period of only a few days. The

sudden stratospheric warming was discovered by Scherhag

[1952], and what is now the accepted explanation for these

warmings was given by Matsuno [1971]. The basis for

Matsuno’s [1971] treatment is found in results derived by

Eliassen and Palm [1961]. Their equations (10.11) and

(10.12) give the following results for quasi-geostrophic

planetary waves:

ˉ ¼ −u0ˉ




ˉ ¼ −u0 fR v′T





where φ′ is the wave geopotential perturbation, ω′ is the

pressure vertical velocity perturbation, the overbars represent


Figure 9. Evolution of Matsuno’s [1971] modeled stratospheric warming at 30 km (about 10 hPa) for the case shown in

Figure 8. Thick lines show the isobaric height (with 500 m contours), and the thin lines show temperature deviations (in 8C)

from its value at the pole before the warming. Modeled from Matsuno [1971]. Copyright American Meteorological Society.

zonal averaging, R is the gas constant, and σ is the static

stability. Equation (16) implies that extratropical planetary

waves with upward energy flux must have an associated

northward heat flux. Furthermore, one can show from this

that such waves must have their phase lines sloping to the

west with height. Matsuno [1971] noted that an intensifying

planetary wave propagating energy upward in winter will be

accompanied by a northward heat flux and a thermally indirect

circulation with ascending motion at the cold pole and

descending motion at low latitudes. This transient wave

intensification implies that not all of the polar heating caused

by the convergence of the meridional heat flux is canceled by

the ascending motion there. This acts to diminish the

meridional temperature gradient, thus leading to a decreasing

mean zonal westerly flow. In time, a critical level can develop,

in which case, there is even a greater convergence of the

planetary wave heat flux so that very rapid heating occurs.

Eventually, radiation reestablishes the cold winter stratospheric pole, and the easterly polar vortex is also reestablished.

This sequence of events is illustrated in Figure 8 from

Matsuno [1971]. Matsuno [1971] imposed a sharp increase

in the amplitude of the planetary waves at his lower

boundary at 10 km altitude. The results shown in Figure 8

are for a spherical domain with wave number 2 forcing.

Note that a very large increase in planetary wave forcing

occurs between day 7 and 11. A corresponding decrease in the

strength of the westerlies is seen between about day 7 and

day 18, after which time a critical level exists where u0 = 0

(Figure 8b). Large temperature increases are then seen

below about 50 km, with somewhat smaller temperature

decreases above this altitude (Figure 8c). The polar temperature rises by some 508K over a period of 10 days. Notice that

although the transient rise in planetary wave forcing causes

significant changes, much more rapid changes occur after the

establishment of the critical layer. What we see then in the

work of Matsuno [1971] is that the planetary wave transience

initially breaks the noninteraction, leading to changes in the

mean zonal state, but the much larger zonal mean changes

occur after the u0 = 0 condition occurs.

Sudden stratospheric warmings can occur because of

planetary wave number 1 or 2 forcing. The results above are

shown for Matsuno’s [1971] wave number 2 case. The change

in the polar vortex is seen in the 30 km maps in Figure 9,

corresponding to the case shown in Figure 8. This wave number


Figure 10. Barometric variations (on two different scales) at Batavia

(the present Jakarta at 68S) and Potsdam (528N) in November 1919.

From Chapman and Lindzen [1970], who, in turn, took this image

from Bartels [1928]. Reprinted with kind permission of Springer

Science and Business Media.

2 warming proceeds by splitting the polar westerly vortex (day

10–20) after which a polar easterly vortex is set up (day 22). A

number of subsequent papers followed Matsuno’s [1971]

work, modeling stratospheric warmings, but they all followed

his basic prescription, with relatively minor variations.


The theory for atmospheric tides goes back to the work

of Laplace [1799], who derived the idealized equations for

the free and forced oscillations for a thin atmosphere on

a spherical planet. While the Moon’s gravitation forces the

oceanic tide, which is semidiurnal, it is the Sun’s heating

that forces the Earth’s atmospheric tides. Under simplifying

assumptions, Laplace used the traditional method of separation of variables to obtain an ordinary differential equation

for the latitudinal structure of the tide as well as one for its

vertical structure. The equation for the latitude structure is

called Laplace’s tidal equation. Its eigenvalues are often referred to in terms of equivalent depths (an analogy to the ocean),

and its eigenfunctions are called Hough functions. The equivalent depth then occurs as a parameter in the vertical structure

equation. One can then expand the latitudinal structure of the

solar heating in terms of these Hough functions.

Early observations contradicted simple intuition, in that the

surface pressure showed tidal variations that were predominantly semidiurnal, while the solar forcing is predominantly

diurnal. Figure 10, from Chapman and Lindzen [1970],

illustrates two things. One is that surface pressure variations

in connection with extratropical systems are much greater

than surface pressure variations in the tropics (except when

tropical cyclones occur). The other is that tidal variations are

larger in the tropics and are predominantly semidiurnal

rather than diurnal, as expected.

Initial suggestions to explain the dominance of the

semidiurnal tide in surface pressure over the expected diurnal

tide were that the semidiurnal solar forcing excited a resonance in the atmosphere, while the diurnal forcing was off

resonance. As more and more was learned about atmospheric

tides and their forcing, this did not prove to be the case. It was

not until Kato [1966] and Lindzen [1966] independently

discovered the existence of negative equivalent depth eigenmode solutions to Laplace’s tidal equation that this problem

was solved. The solution is best explained with the aid of

Figure 11, from Chapman and Lindzen [1970]. In Figure 11,

V the vertical variation of the solar heating of atmospheric

water vapor (V1) and ozone (V2) are shown, along with the

latitudinal variations of these heating functions. Note the

different scales for the diurnal and semidiurnal heating functions. When these latitudinal heating functions are expanded

in the diurnal and semidiurnal Hough function solutions of

Laplace’s tidal equation, the semidiurnal heating only gives

rise to positive equivalent depth modes, the principal one of

which has a very long vertical wavelength of about 100 km,

while the diurnal heating gives rise to negative equivalent

depth modes, which cannot propagate vertically, and a principal positive equivalent depth mode of rather short vertical

wavelength (about 25 km). For the diurnal tide, the ozone

heating forces a negative equivalent depth mode that cannot

propagate down to the surface plus a short vertical wavelength

mode. In contrast, the semidiurnal ozone heating principally

forces a long wavelength mode that reaches the surface. For

the semidiurnal solar tide, the vertical wavelength is longer

than the depth of the ozone heating, and it propagates down to

the surface, so the tidal pressure variations forced by ozone

and water vapor add to produce a sizable response in surface

Figure 11. Vertical and horizontal variations of solar heating of

atmospheric water vapor and ozone. From Chapman and Lindzen

[1970], who, in turn, took this image from Lindzen [1968]. Reprinted

with kind permission of Springer Science and Business Media.


pressure. For the diurnal ozone heating, not only does the

negative equivalent depth mode not propagate down to the

surface, but there is also destructive interference over the

depth of the ozone heating for the short vertical wavelength

solution. Therefore, the sizable semidiurnal variation in

surface pressure is a consequence of the additive solutions

from water vapor and ozone heating, while for the diurnal

tide there is very little surface response to the ozone heating.

Not only did Lindzen’s [1966] and Kato’s [1966] papers

lead to a resolution of the long-standing questions about the

solar atmospheric tides, their work also led to an appreciation

of the completeness of the Hough functions. This, in turn,

led to Longuet-Higgins [1968] paper that examined the family

of oscillations on a fluid envelope on a rotating sphere, which

gave a theoretical framework for understanding global

atmospheric wave motions.


Alan Plumb’s entrance on the middle atmosphere scene

occurred just before a fundamental change occurred in the

field with the famous publication by Molina and Rowland

[1974]. While the study of stratospheric ozone already had a

rich history, it was the Molina and Rowland [1974]

publication that thrust stratospheric ozone research into the

prominent position it occupies today.

The Chapman [1930] reactions

O2 þ hνðλ < 240 nmÞ → O þ O,

O þ O2 ỵ M O3 ỵ M,

O3 ỵ h < 320 nmị O ỵ O2 ,

O ỵ O3 2O2





are the simplest set of chemical reactions to account for the

production and destruction of stratospheric ozone. In these

reactions, ozone is produced by the dissociation of

molecular oxygen by ultraviolet radiation (reaction (17))

followed by the attachment of one of the freed oxygen

atoms to an oxygen molecule (reaction (18)). It is then

destroyed by reaction (20), where an oxygen atom combines

with an ozone molecule to make two stable oxygen

molecules. Note that reaction (19) does not really destroy

Figure 12. A supply of dry air is maintained by a slow circulation from the equatorial tropopause. Figure and caption from

Brewer [1949]. Reprinted with permission.


ozone since the resulting oxygen atom quickly combines

with an oxygen molecule to remake ozone.

A problem arose with these simple Chapman [1930]

reactions when the rate of reaction (20) was measured by

Benson and Axworthy [1957] and was found to proceed too

slowly to account for measured ozone concentrations. This

led Hampson [1964] to suggest the following reactions as

being important. Note that the net result of reactions

OH ỵ O3 HO2 ỵ O2


HO2 ỵ OOH ỵ O2


programs, extensive international assessments of stratospheric ozone, the implementation of the Montreal Protocol,

and the award of the 1995 Nobel Prize in Chemistry to P. J.

Crutzen, M. J. Molina, and F. S. Rowland.


Given the fact that ozone is mainly produced in the tropical

stratosphere, where the UV radiation is sufficiently intense,

is reaction (20) since the OH radical acts as a catalyst. Hunt

[1966] suggested a set of rate constants for reactions (21)

and (22) that accounted for observed ozone concentrations.

This was followed by papers by Crutzen [1970] and

Johnston [1971] that pointed out the importance of the

catalytic cycle involving nitrogen oxides,

NO ỵ O3 NO2 ỵ O2


NO2 þ O→NO þ O2 .


Crutzen [1970] suggested that reactions (23) and (24)

dominate ozone loss at altitudes between 25 and 40 km,

while reactions (21) and (22) increase in importance at

higher altitudes. Johnston [1971] suggested that reactions

(23) and (24) would lead to decreased stratospheric ozone

loss if a large fleet of supersonic transport planes were to be

implemented, emitting large amounts of nitrogen oxides.

In 1974, just as Alan was entering the scene, the paper by

Molina and Rowland [1974] appeared, suggesting that manmade chlorofluorocarbon atmospheric concentrations were

rapidly increasing because of their increasing use as aerosol

propellants in refrigeration and other industrial sources.

These chlorofluorocarbons are very stable in the troposphere

but are dissociated in the stratosphere where they encounter

the energetic solar ultraviolet radiation. This released chlorine

to the stratosphere that participates in the very fast catalytic


Cl ỵ O3 ClO þ O2


ClO þ O→Cl þ O2 .


The Molina and Rowland [1974] paper changed the face

of stratospheric research, as did the discovery of the

Antarctic ozone hole by Farman et al. [1985]. Large

national and international programs were implemented to

study stratospheric ozone. These included new satellite

Figure 13. Contributions from the mean meridional circulation, the

large-scale eddies, and horizontal diffusion to the continuity equation for an ozone-like tracer at two levels. The net rate of change of

this ozone-like tracer is also shown. From Hunt and Manabe [1968].

Copyright American Meteorological Society.

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