[Reprinted from Bulletin of the American Meteorological Society, Vol. 50, No. 10, October, 1969, pp. 792-799]
Printed in U. S. A.
An occasional series reporting on U. S. and international GARP activities and scientific concerns presented as
a public service to the meteorological community by the American Meteorological Society through arrangements
with the U. S. Committee on the Global Atmospheric Research Program of the National Academy of SciencesNational Research Council. Opinions expressed in GARP TOPICS do not necessarily reflect the point of view
of the U. S. Committee.
>
Editor’s note: The following is the third in a series of articles designed to call atten­
tion to scientific problems, new analysis techniques, and recent findings in tropical
meteorology which are of special interest and relevance to the Global Atmospheric
Research Program.
Some Recent Developments in the Study of Tropical Wave Disturbances^
John M. Wallace
University of Washington, Seattle
Abstract
Since 1966, two types of wave motions have been dis­
covered in the tropical stratosphere. These have been
identified with the two gravest modes of a family of
equatorial waves. These waves are characterized by
downward phase propagation, which renders them im­
portant in the vertical transport of energy and zonal
momentum. In the tropical lower troposphere there
exists a separate class containing wave modes which do
not propagate vertically, one of these being the familiar
easterly wave.
The role of these two classes of waves in the tropical
general circulation is discussed and the possible energy
sources for the waves are enumerated.
1. Introduction
The discovery of the easterly wave during the 1940’s
gave a strong impetus to the study of tropical wave dis­
turbances. By the early 1950’s the main characteristics of
the easterly wave and the closely related equatorial wave
had been well documented, and considerable progress
had'been made in interpreting these phenomena from a
1 Contribution No. 208, Department of Atmospiheric Sci­
ences, University of Washington, Seattle, Wash.
792
dynamical point of view. Riehl (1954) gives a detailed
summary of these early research efforts.
Attempts to extend and generalize these early wave
models were seriously hampered by the paucity of
tropical upper air observations and by the lack of a
sound theoretical basis for interpreting synoptic results.
Consequently, in the years which followed, the emphasis
in tropical research tended to shift to other problems
such as hurricane formation, convection, and the tropi­
cal general circulation.
Within the past few years, the inadequacies in data
and theoretical support for synoptic investigations have
been alleviated to some extent. Application of spectrum
analysis techniques to long time series of tropical data
has provided some compensation for the poor spatial
distribution of tropical stations. Theoretical contribu­
tions by Matsuno (1966) and Lindzen (1967) have pro­
vided invaluable guidance in the planning of synoptic
investigations and the interpretation of results. These
developments have contributed to an expanding re­
search effort in the field of tropical disturbances. This
activity has already produced some significant results,
particularly with regard to disturbances at the higher
levels.
Vol. 50, No. 10, October 1969
Bulletin American Meteorological Society
2. Stratospheric wave disturbances
The strong, steady zonal currents which encircle the
Earth have been the subject of numerous observational
and theoretical studies. The “quasi-biennial” or “26
month” oscillation in zonal winds in the equatorial
stratosphere, first noted by Reed (1960), has com­
manded a particularly large amount of attention be­
cause of the perplexing problem of explaining the
large zonal accelerations taking place within a band of
Fig. 1. Time-height section of zonal wind at 8° latitude
with annual cycle removed. Solid isotachs are placed at
intervals of 10 m secrh Shaded areas indicate westerlies.
Below 35 km monthly mean rawinsonde data for the Canal
Zone (9N) and Ascension Island (8S) were averaged together
to remove all fluctuations with odd symmetry about the
equator. Above 34 km, this procedure could not be used
because rocket data were available for Ascension Island
only. At these levels the annual cycle was removed by
harmonic analysis. Some minor smoothing was done to make
the analyses computable at 35 km. (After Lindzen and
Holton, 1968.)
latitude straddling the equator. The time-height section
shown in Fig. 1 illustrates this phenomenon, as well as
the equally remarkable semi-annual oscillation at higher
levels.
Almost without exception, the early observational
studies of these phenomena employed mean monthly
wind statistics as raw data because of the convenience
which they afforded. This automatically eliminated the
possibility of identifying fluctuations with periods
shorter than a month. As early as 1963 Ebdon (1963)
noted that there have been instances where, during the
midst of an easterly regime of the quasi-biennial oscilla­
tion, westerly winds had appeared for a day or two and
vice versa, but until recently there has been no attempt
to analyze these fluctuations on a systematic basis.
This recent revival of interest in tropical wave dis­
turbances began when Yanai and Maruyama (1966)
noticed regular, wavelike fluctuations in the meridional
wind component at stratospheric levels. These were
observed to propagate westward and downward, with a
horizontal wavelength of about 10,000 km, a vertical
wavelength of about 6 km, and a period of 4-5 days.
The #aves are evident in' Fig. 2 which shows a timeheight section of wind. Subsequent studies (Maruyama
and Yanai, 1967; Maruyama, 1967) have revealed the
existence of associated fluctuations in the zonal wind
component and in temperature. The observed phase
relationships between these parameters are such that
the wind vector at any given point in the Northern
Hemisphere rotates clockwise with time as a wave
passes, and the maximum temperature occurs at the time
of the maximum southerly wind. The reverse is true for
a point in the Southern Hemisphere. The waves have
odd symmetry about the equator and appear to be
confined within about 12° of the equator. A horizontal
depiction of the pressure field and streamline patterns
is given in Matsuno (1966).
KAPINGAMARANGi (Noi'oe, e i54*46')
lAt
MARCH-APRiL 1958
iiiiiiuiLt.
tiiliiilii
ii
V
’k
V,
'-^1
i\%,
IkkxiL^
k TvSi;
k.
pi:.
kkkk <
1
25
1
1
28
1
1
31
Fig.
I
1
“\>nV
__ 1_ _ 1_ _6__ !_ _ 1__9__ 1_ _ 1__12
3
15
t
1
18
1
t
21
t
1
24
\
1
1
27
1
tT"
30
2. Time-height section of wind in knots. Shaded regions denote southerly components.
(After Maruyama and Yanai, 1967.)
793
Vol. 50, No. 10, October 1969
1963
Fig. 4. Time-height section of temperature. Contours are
Fig. 3. Time-height section of zonal wind. Contours are
placed at increments of 5 m sec"'. Westerlies are shaded.
(After Wallace and Kousky, 1968b.)
placed at increments of 2C. Dashed linfs indicate easterly
maxima transcribed from zonal wind analyses. (After Wallace
and Kousky, 1968b.)
HEIGHT
Soon after the discovery of 4-5 day stratospheric
waves, Wallace and Kousky (1968a) noted a large 10-15
day period oscillation in the zonal winds in the same
region (Fig. 3). There was no evidence of any related
fluctuations in the meridional wind component. In
retrospect it is easily seen why these waves, despite their
large amplitudes, were not discovered earlier. Their
period is too long for them to be noticed in a short
sequence of daily data, but too short for them to appear
in a monthly average. Moreover, they cause no change
in wind direction. It is quite likely that they were re­
sponsible for Ebdon’s observation that westerlies oc­
casionally intruded into easterly regimes and vice
versa.
These waves also produce distinct temperature fluc­
tuations (Fig. 4). Warmest temperatures precede the
maximum westerly winds by 1/4 cycle, with both oscilla­
tions propagating downward. The wave amplitude is
largest at the equator and decays' to about half the
maximum value at 10° latitude.' This observational
evidence suggests the physical interpretation shown
schematically in Fig. 5.
a) The absence of meridional wind fluctuations re­
quires that the zonal wind component must be in
geostrophic equilibrium with the meridional pressure
gradient. It follows-that the levels of maximum zonal
winds and maximum pressure should coincide.
b) Hydrostatic equilibrium requires warmest tem­
peratures 1/4 wavelength below (in advance of) the
level of maximum pressure. This is verified observationally.
c) The zonal momentum equation requires that maxi­
mum westerlies follow 1/4 cycle behind the maximum
east to west pressure gradient. Given the above re­
strictions, this can only be satisfied for an eastward
propagating wave.
d) Given the fact that radiative processes in this
region operate on a time scale considerably longer
than that of the waves, the first law requires that the
warmest temperatures lag the maximum subsidence
by 1/4 cycle.
794
:---------------
Fig. 5. Idealized cross section along a latitude circle show­
ing phases of the zonal wind, temperature, pressure and
vertical motion oscillations associated with Kelvin waves, as
deduced from theoretical considerations. (After Wallace and
Kousky, 1968a.)
e) In the absence of the meridional wind component,
continuity requires that the zonal wind and the
vertical motion oscillations be in phase.
The arrangement shown in Fig. 5 is the only one
which satisfies all these requirements. Similar but some­
what more involved reasoning leads to a corresponding
arrangement for the waves discovered by Yanai and
Maruyama.
Matsuno (1966) and Lindzen (1967) had predicted
these same phase relationships theoretically by obtaining
the zonally propagating wave solutions to the linearized
equations of motion, continuity and thermodynamic
energy on an equatorial beta-plane. The resulting family
of wave modes contains gravity waves and Rossby
waves. The gravest mode with even symmetry about the
Bulletin American Meteorological Society
equator is an eastward propagating gravity wave with
no meridional wind component, and a structure identi­
cal to ithat shown in Fig. 5. It has been called the
atmospheric “Kelvin wave” because of its resemblance to
a type of shallow water gravity wave which propagates
along a coastal boundary and has no velocity compo­
nent normal to the boundary. In the atmospheric case,
the equator plays the same role as the coast line
(Lindzen and Holton, 1968).
In the theoretical solutions, the gravest mode with
odd symmetry about the equator has a structure simi­
lar to that indicated for the observed 4-5 day wave.
This solution is unique in that it is a mixed Rossbygravity wave mode.
There are also higher modes which represent solu­
tions to the equations. Theoretical considerations sug­
gest that these should be characterized by disturbances
with very short vertical wavelengths (Lindzen and Matsuno, 1968). As yet these modes have no precise observa­
tional counterparts, but there is some indication that
a number of them may be associated with the fine struc­
ture which sometimes appears in soundings with high
vertical resolution. An example, taken from the Line
Island Experiment, is shown in Fig. 6. In this sequence
of soundings we see evidence of persistent features
with vertical wavelengths as small as 1-2 km which tend
to propagate downward. The fact that these same fea­
tures appear at the same levels at neighboring stations,
suggests that they are rather large in horizontal extent.
km or more to disturbances with 1 km wavelengths or
less. In general, these waves are characterized by down­
ward phase propagation, long periods (several days or
more) and large wavelengths (up to 40,000 km for the
Kelvin waves). A summary of the properties of the two
modes which have been identified observationally is
given in Table 1.
Table
1. A description of the vertically propagating wave modes.
Theoretical description
Mixed Rossby—
gravity wave mode
n = 0 mode
Region of occurrence
Stratosphere,
upper troposphere
Period
4-S days
Vertical wavelength
4-8 km
Horizontal wavelength
10,000 km
Direction of propagation
Westward
downward
Amplitudes E-W component 2-3 m sec“'
N-S component 2-3 m sec”'
Kelvin wave
n = — 1 mode
Stratosphere
12-18 days
> 6 km
> 20,000 km
Eastward
downward
8-12 m see”*
0
Holton (1969) has shown that Charney’s (1963) scale
analysis, in which he assumed that tropical distur­
bances have a vertical scale on the order of the atmo­
spheric scale height, does not apply to these vertically
propagating disturbances. In the next section, we dis­
cuss another type of wave disturbances to which
Charney’s scaling apparently does apply.
3. Tropospheric wave disturbances
Fig. 6. Vertical profiles of meridional wind component at
Christmas Island (2N, 158E) at 6 hour intervals over a 2
day period. Units are m seer'.
The foregoing observational evidence strongly sug­
gests that there is a whole spectrum of wave modes
present in the equatorial atmosphere. These range
from the Kelvin wave with vertical wavelengths of 8
There appears to be another class of tropical wave
disturbances which does not propagate vertically. Evi­
dence of such waves is presented in a recent study of
lower tropospheric data by Wallace and Chang (1969).
Fig. 7, which is taken from that study, shows vertically
averaged time series of wind, temperature and relative
humidity for the layer between the surface and 500 mb
at Truk. Large fluctuations are evident in both wind
components. Fig. 8 shows the. power spectra for the
vertically averaged wind at Truk for 4 successive 6
month periods, including the one for which time series
data are shown. There is considerable power at low
frequencies (corresponding to periods longer than 10
days) and also in the 4-5 day period range which is
particularly pronounced during the last 6 month
period in the meridional wind component. Both types
of waves appear to be capable of existing without verti­
cal phase propagation, which suggests that they are
equivalent barotropic in nature.
The 4-5 day disturbances propagate westward, with
a speed slightly in excess of the mean easterly flow.
They have a longitudinal wavelength on the order of
3000 km, and thus are clearly distinct from the 4-5 day
disturbances described above, which have 10,000 km
795
Vol. 50, No. 10, October 1969
to AUG
31 AUG
30 SEPT
TIME
31 OCT
21 NOV
(DAYS)
Fig. 7. Time series of vertically averaged zonal and meridional wind components, temperature and relative humidity
(averaged for ten levels at 50 mb intervals between the surface and 550 mb) and surface pressure for Truk (7N, 151E).
Missing observations are interpolated. (After Wallace and Chang, 1969.)
wavelengths. At times when they are active (e.g., late
1964) they produce fluctuations in relative humidity,
with peak values occurring near or just to the east of the
trough in the streamline field. Thus, in many respects
diey resemble the classical model of easterly or “equa­
torial” waves. There are indications that these same
waves may be characterized by an eastward tilt with
height (upward phase propagation) when they occur
at stations further east in the Pacific, such as Christmas
Island (Yanai et al., 1968).
It has been pointed out ^ that the observed speed of
propagation of these waves is about the same as would
be expected for a free Rossby mode with the appro­
priate zonal wave number.
The longer period disturbances appear to be of
considerably larger horizontal scale; perhaps on the
2 J. R. Holton, personal communication.
796
order of 10,000 km. They are characterized by a strong
in-phase relationship between the zonal and meridional
wind components. It may not be appropriate to view
these disturbances as zonally propagating waves since
they appear to be confined to limited regions of
longitude.
We note that the 'vertical scale of the disturbances
discussed in this section is comparable with Charney’s
(1963) scaling assumption and the result of his analysis
(i.e., that the disturbances must be equivalent' barotropic) seems to be applicable to them.
4. Role of tropical wave disturbances in
the general circulation
The vertically propagating wave disturbances in the
equatorial region influence the atmospheric general cir­
culation in several ways:
VARIANCE PER UfJIT FREQUENCY INTERVAL (M^SEC"^PER2% oDAY ')
Bulletin American Meteorological Society
(DAYS)
Fig. 8. Power spectra for the vertically averaged (surface—550 mb) zonal and meridional wind
components at Truk. (After Wallace and Chaiig, 1969.)
797
Vol. 50, No. JO, October 1969
1) They transport wave energy upward. Because of
this property they may serve as the primary mecha­
nism for the leakage of wave energy from the tropo­
sphere. (Lindzen, 1967).
2) They are an important mechanism for the vertical
transport of momentum. Under certain conditions
this momentum may be absorbed by the mean zonal
flow. Lindzen and Holton (1968) have proposed a
mechanism whereby interactions between the wave
disturbances and the mean zonal flow could provide
the momentum source for the quasi-biennial oscilla­
tion. There is already substantial observational evi­
dence which suggests that such an interaction actually
does exist (Maruyama, 1968a, 1969; Wallace and
Kousky, 1968b).
3) They transport heat away from the equator (with
the exception of the Kelvin wave). According to
calculations by Maruyama (1968b) the divergence of
heat flux out of the equatorial zone may represent a
significant factor in the heat budget at the tropopause
level.
In order to better understand the role of these wave
disturbances in the general circulation, we must review
their structure in more detail. In the discussion of Fig.
5 we noted that the pressure, zonal wind and vertical
motion oscillations are in phase. This means that at a
given level, ascending air is marked by higher pressure
and larger zonal velocity than descending air. As a
result of the pressure difference, air below the level in
question is doing work on the air above, and, in effect,
mechanical energy is imparted to the air at higher
levels. It is a common property of all the members of
this wave family that downward phase propagation is
indicative of upward energy propagation. Yanai and
Hayashi (1969) have made estimates of the vertical flux
of energy associated with the mixed Rossby-gravity
wave mode.
As a result of the difference in westerly wind compo­
nent between rising and sinking air, it is readily seen
that westerly momentum is being transported upward in
the Kelvin wave. The interpretation in the case of the
other wave modes is somewhat more complicated. In
general, the other modes transport westerly or easterly
momentum upward, depending upon whether their
phase propagation is eastward or westward, respectively.
In like manner, the coincidence of warm tempera­
tures with poleward moving air and vice versa is re­
sponsible for a transport of heat away from the equator.
This is strongest at or just above the tropopause level
where the temperature fluctuations are largest.
The role of the equivalent barotropic wave disturb­
ances in the general circulation is somewhat less clear.
Vertical transports of momentum or energy by the large
scale motion field would appear to be insignificant.
(This does not preclude the possibility that convective
activity associated with these waves may be an im­
portant vertical transport mechanism.) Meridional heat
798
transports should also be small in comparison to those
in the vertically propagating waves. The strong positive
correlation between the zonal and poleward wind com­
ponents in the low frequency waves suggests that these
may be effective in the poleward transport of zonal
momentum. The 4-5 day waves also exhibit a positive
correlation between the wind components, but to a
considerably lesser extent (Wallace and Chang, 1969).
5. Energy source of the waves
The vertical propagating waves are continuously car­
rying energy upward from the lower atmosphere to a
sink at higher levels. The energy sink may be the mean
zonal flow, or it may be eddy viscosity or radiative
damping if the waves propagate to very high levels
without being absorbed by the mean zonal flow. In
either case, a source of wave energy in the lower atmo­
sphere is implied. For this reason these waves are
referred to as “forced” modes.
On the other hand, the modes we have designated
as "equivalent barotropic” do not transport energy or
momentum in the vertical. Thus, were it not for fric­
tion they would have no energy sink, and could persist
indefinitely in the absence of external forcing. For this
reason they are sometimes referred to as “free modes.”
In reality, the presence of friction does require an
energy source for these waves as well.
At present, the source of excitation of these waves is
not thoroughly understood. Mak (1969) has proposed
that they might be driven by unstable baroclinic dis­
turbances at higher latitudes through lateral coupling..
He demonstrated this mechanism by means of a two
layer model of the tropical atmosphere which was
driven by stochastic forcing at the middle latitude
boundaries. The forcing function was derived from actual
data. The tropical disturbances which developed in the
model have frequencies and wave numbers which are
strongly suggestive of the vertically propagating modes.
However, their vertical structure more closely resem­
bles that of the equivalent barotropic modes described
above. These provocative results encourage future re­
search along these lines.
Another possible energy source is the release of
latent heat. In Uieir moist general circulation model
Manabe and Smagorinsky (1967) obtain small ampli­
tude disturbances which resemble the equivalent baro­
tropic modes described above. These result from convec­
tive adjustment in regions where the model atmosphere
becomes statistically unstable with respect to moist adia­
batic processes. Convective adjustment warms the upper
troposphere- and cools the lower troposphere, creating
disturbances which are warm core at high levels and
cold core at low levels. The response of the mass field
to the resulting horizontal pressure gradients produces
equivalent barotropic vorticlty perturbations with maxi­
mum amplitude in the middle troposphere. Whether
this process takes place in the actual atmosphere remains
to be seen.
Bulletin American Meteorological Society
A third possible energy source for the waves is the
kinetic energy of the mean zonal flow. The necessary
conditions for barotropic instability .may be met when
strong shear zones are located close to the equator.
The ITCZ in the western Pacific during the Northern
Hemisphere summer is often characterized by strong
cyclonic shear in the zone .between the equatorial
westerlies and the northeast trades. At times, it is pos­
sible that the zonal flow may become barotropically
unstable in regions such as these, in which case,
disturbances of the equivalent barotropic type would
result.
Besides the question of energy sources there are a
number of related problems which deserve attention. Of
particular interest is the relation between the various
types of wave disturbances and cloud and precipitation
patterns. Will it be necessary to understand and be
able to predict the movements of all tropical waye
modes in order to make accurate weather forecasts in
the tropics? One would hope that it would not be
necessary to define in detail the structure and move­
ment of the vertically propagating disturbances; particu­
larly those with short vertical wavelengths. Of equal
interest is the question of how, and on what time scale,
tropical wave disturbances influence the middle latitude
circulation patterns. This will ultimately determine the
density of trdpical observations required for extended
prediction in middle latitudes. It is hoped that the
various subprograms within GARP will provide defini­
tive answers to some of these questions.
Acknowledgments. The author is grateful to Mr.
Vernon E. Kousky and Dr. M. Yanai who furnished
several of the figures. The work was supported by the
National Science Foundation under grant number GA
629X.
References
Chamey, J. G., 1963; A note on large scale motions in the
tropics. J. Atmos. Sci., 20, 607-609.
Ebdon, R. A., 196^: The tropical stratospheric wind fluctua­
tion: evidence of its permanency from earlier data.
Weather, 18, 2-7.
Holton, J. R., 1969: A note on the scale analysis of tropical
-motions. J. Atmos. Sci., 26, HO-lTt.
------, and R. S. Lindzen, 1968: A note on "Kelvin” waves
in the atmosphere. Mon. Wea. Rev., 96, 385-386.
Lindzen, R. S., 1967: Planetary waves on beta-planes. Mon.
Wea. Rev., 95, 441-451.
—^—•, and J. R. Holton, 1968: A theory of the quasi-biennial
oscillation. J. Atmos. Sci., 25, 1095-1107.
------, and T. Matsuno, 1968: On the nature of large scale
wave disturbances in the equatorial lower stratosphere.
J. Meteor. Soc. Japan, 46, 215-221.
Mak, M.-K., 1969; Laterally driven stochastic motions in the
tropics. J. Atmos. Sci., 26, 41-63.
Manabe, S., and J. Smagorinsky, 1967: Simulated climatol­
ogy of a general circulation model with a hydrologic cycle
II. Mon. Wea. Rev., 95, 151-169.
Maruyama, T., 1967: Large scale disturbances in the equa­
torial lower stratosphere. J. Meteor. Soc. Japan, 45, 391408.
------, 1968a: Time series of power spectra of disturbances
in the equatorial lower stratosphere in relation to the
quasi-biennial oscillation. J. Meteor. Soc. Japan, 44, 327341.
------, 1968b: Upward transport of westerly momentum due
to large scale disturbances in the equatorial lower
stratosphere. J. Meteor. Soc. Japan, 45, 404-417.
------, 1969: Long-term behavior of Kelvin waves and mixed
Rossby-gravity waveg. Submitted for publication.
------, and M. Yanai, 1967: Evidence of large scale wave
disturbances in the equatorial lower stratosphere. J.
Meteor. Soc. Japan, 45, 195-196.
Matsuno, T., 1966: Quasi-geostrophic motions in the equa­
torial area. J. Meteor. Soc. Japan, 46, 215-222.
Reed, R. J., 1960: The circulation of the stratosphere. Paper
presented at the 40th Anniversary meeting of the Amer.
Meteor. Soc., Boston, January 1960.
Riehl, H., 1954: Tropical Meteorology. McGraw-Hill Book
Company, New York, 392 pp.
Wallace, J. M., and C. P. Chang, 1969: Spectrum analysis of
large scale wave disturbances in the tropical lower
troposphere. J. Atmos. Sci., 26, 1010-1025.
------, and V. E. Kousky, 1968a; Observation evidence of
Kelvin waves in the tropical stratosphere. J. Atmos. Sci.,
25, 900-907.
------, and------, 1968b: On the relation between Kelvin waves
and the quasi-biennial oscillation. J. Meteor. Soc. Japan,
47, 496-502.
Yanai, M., and T, Hayashi, 1969: Large-scale equatorial
waves penetrating from the upper troposphere into the
lower stratosphere. Submitted for publication.
------, and T. Maruyama, 1966: Stratospheric wave distur­
bances propagating over the equatorial Pacific. J. Meteor.
Soc. Japan, 44, 291-294.
------, ------, T. Nitta, and Y. Hayashi, 1968: Power spectra
of large scale disturbances over the tropical Pacific. J.
Meteor.
Soc.
Japan,
46, 308-323.
n
799