NCAR MANUSCRIPT 70-182
PROGRESS IN RESEARCH ON ATMOSPHERIC TURBULENCE
D. K. LillyNational Center for Atmospheric Research
Boulder, Colorado 80302
December 1970
PROGRESS IN RESEARCH ON ATMOSPHERIC TURBULENCE
Instrumentation
In ground- and tower-based instrumentation for direct sensing few if any
new concepts have appeared, but steady improvements in technique and reli-
ability have been reported (Dyer, et al., 1967; Goddard, 1970; B. B. Hicks,
1969; Kaimal, 1968; Kaimal, et al., 1968; Thurtell, et al., 1970; Wesely, et
al., 1970) and inter-comparisons have been conducted between different kinds
of instruments (Businger, et al., 1967; Miyake, et al., 1970) leading generally
to increased knowledge and confidence in the characteristics of each. More
rapid advances have occurred in aircraft-based instrumentation. In particular
the utilization of inertial navigation systems by Axford (1968), the Air Force
HICAT program (Crooks, et al., 1968) and the National Center for Atmospheric
Research-Desert Research Institute program (1970), is leading to the ability
to measure the larger turbulence scales and mesoscales where much of the im-
portant energy generation resides in levels above the surface boundary layer.
Sheih (1971) has effectively utilized hot wire anemonetry to measure the
micro-scales of turbulence from an aircraft. Balloons of either the constant
level type (Angell, et al., 1968) or with a roughened surface to improve
stability (DeMandel & Scoggins, 1967) tracked by precision radar, have been
used to reveal details of the near-scale wind field.
Another rapidly developing field is that of remote sensing by electro-
magnetic or sonic signals. Observations taken by high resolution radar
(Atlas, et al., 1970; Glover, et al., 1969; Gossard, et al., 1970; Hardy &
Ottersten, 1969; Hicks & Angell, 1968; J. J. Hicks, 1969; Konrad, 1970;
Richter, 1969) have shown the ability of this sensor to detect the outlines
of non-condensing thermals in the planetary boundary layer and turbulence
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elements in shear layers at levels up to the tropopause. Doppler radar has
been used to obtain complete fields of motion (Lhermitte, 1969) although it
has thus far been mainly limited to areas of precipitating cloud environ-
ments or large amounts of artificially introduced reflectors. Acoustic sound-
ing methods can be designed to duplicate most of the abilities of radar
systems, with a considerably higher power return efficiency (Little, 1969;
McAllister, 1968). Early results are now becoming available (Beran, personal
communication).
Surface and planetary boundary layer
The improved instrumentation of recent years and better understanding of
proper site selection criteria have led to much improved data for evaluating
the current statistical theories of boundary layer similarity and turbulence
spectra, although some advances have been made from further analysis of older
data. Fairly good agreement appears to have been reached on the empirical
universal functional relationships between profiles and fluxes in the surface
boundary layer in steady homogeneous conditions. For both the stable and the
unstable cases Businger, et al. (1970) have produced new formulations, based
on recent measurements, which are only rather small changes from the "KEYPS"
formulation for the unstable case and essentially agree with the results of
Oke (1970) and Webb (1970) for the stable case. It is found that the eddy
Prandtl number (KM/KH) is constant in the stable case with the constant equal
to 1.0 according to Oke and Webb and 1.5 according to Businger, et al. This
difference is associated with a proposed alteration of the von Karman constant
from 0.40 to 0.35 in the latter paper. Considerable uncertainty still exists
regarding the existence and/or value of a critical Richardson number. There
also remains the not inconsiderable problem of the physical meaning and
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rationale of the various universal functions and constants, a problem which
may only be resolved by their abstraction from the results of numerical simu-
lation experiments. There is some hope, however, that some of the newer
generalized turbulence closure theories (discussed subsequently) can close
the understanding gap more effectively.
The problem of the transient adjustment of the surface boundary layer
to changes in roughness has received attention (Blom and Wartena, 1969;
Bradley, 1968; Nickerson, 1968; Taylor, 1969a,b) but apparently only for the
neutral stability case. The most useful work seems to be the observational
study by Bradley, with the principal practical conclusion being that a "good"
micrometeorological site should have a height-fetch ratio of 1:200 or less.
The problem of wind-generation of surface water waves and the associated
drag on the wind and water has been both attractive and resistant to attacks
on several fronts. For a review of the field up to 1967 see Stewart (1967).
Several groups have recently developed wind-water tunnel facilities and con-
ducted experiments intended to illustrate the process of wave generation.
The results (Hidy & Plate, 1965, 1966; Plate & Hidy, 1967; Plate, Chang, &
Hidy, 1969: Shemdin & Hsu, 1967; Stewart, 1970; Wu, 1968) cannot be summarized
briefly, except to say that none of the existing theories of wave generation
has been confirmed adequately and few have been totally rejected. Among the
generally agreed-upon features: Phillips' (1958) prediction of an inertial
range in the wave height spectrum proportional to (frequency)-5 has been
verified, and the mean velocity profile over waves is found to be only subtly
different, if at all, from that over a solid surface. In a series of papers
Wu (1969a,b; 1970) has produced substantial evidence, in general agreement
with Stewart (1967), that the smallest wavelets, those with phase speed less
than u , are the principal absorbers of wind energy and momentum.
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The planetary boundary layer (turbulent Ekman layer) is now receiving
greatly increased attention, perhaps because of its importance in pollution
problems and in numerical simulation of the large scale atmosphere. There
is substantial agreement (Blackadar and Tennekes, 1968; Csanady, 1967) that
the correct scaling of the planetary boundary layer thickness in neutral
conditionsgoes as u*/f, although the similarity theories invoked to produce
this result are not wholly convincing and observational evidence is unclear.
However, the statistical conclusions of Deardorff's (1970b) numerical simula-
tion experiments essentially verify the validity of that scaling. Deardorff's
results also bear strongly on the work in Ekman layer stability and large eddy
structure conducted by Brown (1970), Faller and Kaylor (1966), Lilly (1966),
and Tatro and Mollo-Christensen (1967). The principal burden of those studies
has been that the Ekman layer is normally unstable to downwind-oriented roll
vortices, and that these secondary flow systems should be considered as
essential elements in the planetary boundary layer. Deardorff's results for
neutral stability show the existence of large eddies without doubt, but they
seem to be much too irregular and transitory to be easily identified as Ekman
layer rolls. In fact they differ little from the eddies obtained in
Deardorff's earlier (1970a) simulation of non-rotating Couette flow. None-
theless, empirical evidence continues to pile up (Angell, Pack & Dickson,
1968; Hanna, 1969) that some sort of relatively regular and steady-state
downstream roll elements do commonly exist, especially in unstable condi-
tions. Deardorff's results for the unstable case (1970a) show somewhat
stronger indications of persistent downstream rolls.
Not unrelated to the above are the observational and theoretical
studies of convective elements in the sub-cloud mixed layer. On the theore-
tical side models of plumes have been developed, following the lead of Turner
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(1963) which incorporate eddy energy of the plume (Fox, 1970) and its environ-
ment (Telford, 1966, 1970; see also Morton, 1968 for a critique of Telford's
model). The observations that need to be explained include the fact that the
thermal elements apparently remain of about the same radius throughout their
ascent through the mixed layer (Warner and Telford, 1967), have sharp edges
at the back (upwind) side (Kaimal and Businger, 1970; Lenschow, 1970; Warner
and Telford, 1967) and have an elliptical cross-section at any given altitude,
being considerably elongated in the downwind direction (Lenschow, 1970).
Lilly (1971) has suggested that most of these features are consistent with
the bent over plume model of Scorer (1959). Another type of thermal element
that has come under increasing interest is the dust devil (Kaimal and
Businger, 1970; Sinclair, 1969) but few new or unexpected conclusions have
yet been reached.
A large field program, the Barbados Oceanographic and Meteorological
Experiment (BOMEX) was conducted in the eastern tropical Atlantic during the
early summer of 1969. About 100 separate observational sub-programs were
incorporated into the total program, with a major goal to obtain as many
measurements as possible, by different methods, of the turbulent fluxes of
heat, moisture, and momentum across the sea-air interface and to construct
an integrated budget of these parameters for a substantial period over an
observational region of order 106 km2. A pilot project of this type was
conducted by Fleagle, et al. (1967). Descriptions of the total program,
including most of the sub-programs, have been published (BOMEX Bulletins
1-8, 1969-70) but due to data processing delays most of the "core" data from
the principal ship and aircraft stations has not yet been fully analyzed.
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Clear air turbulence and gravity waves
Progress in the area of clear air turbulence (CAT) has been particularly
significant in two areas. First, in the study of turbulence for itself, con-
siderable evidence has been amassed now that the Richardson number is the
most significant environmental parameter associated with turbulence generation
and maintenance and that the unstable Kelvin-Helmholtz wave is frequently the
proximate cause of turbulence development. Second, the studies of Kung and
others now indicate an important role for upper tropospheric and tropopause
turbulence in the atmospheric general circulation and perhaps in the evolution
of individual large scale meteorological events.
Looking at the latter subject first, the papers by Kung (1966a,b; 1967;
1969a,b) show rather clearly that energy dissipation in the atmosphere is
confined principally to the planetary boundary layer and to a somewhat deeper
layer centered near the tropopause, and that the total dissipation in the
2upper layer is about 1 watt/m , of the same magnitude but somewhat
smaller than that near the lower boundary. Kung's values were determined
strictly from residuals of the large scale energy budget. From estimates of
turbulence amplitudes and frequencies Trout and Panofsky (1969) and
Vinnichenko (1970) arrived at very similar values. From a more specific
synoptic study Reed (1969) showed that CAT can interact significantly with
the development of a baroclinic motion field.
In a well-known paper with later extensions by both authors Miles and
Howard (1964) showed that a necessary condition for small amplitude instabi-
lity of an inviscid, steady, parallel incompressible flow is that the gradient
Richardson number is less than 1/4 somewhere in the flow. In spite of the
limitations contained in the underlined words, the Richardson number criterion,
which is associated with unstable Kelvin-Helmholtz waves, is the leading
candidate as a criterion for CAT formation. Laboratory experiments have been
conducted with stratified shearing flows (Stoeffler, 1970; Thorpe, 1968, 1969)
tending to verify the criterion, and the theory has also been extended to the
finite amplitude case (Drazin, 1970). Observational studies generally show
a good correlation of CAT with strong wind shear and low Richardson number
(Boucher, 1970; Browning, et al., 1970; Ludlam, 1967; Mancuso & Endlich, 1969;
Mather, 1969; Panofsky, et al., 1968; Reed, 1969; Reiter & Foltz, 1967; Waco,
1970), with Mather and Reed providing some of the most convincing data. CAT
can apparently occur within a wide range of synoptic conditions (Collis,
Endlich & Mancuso, 1969; Reiter, 1969) but the existence of strong mesoscale
circulations, including mountain waves (Crooks, et al., 1968; Kuettner &
Lilly, 1968; Lilly & Toutenhoofd, 1969; Reiter & Foltz, 1967) and nearby
thunderstorms (Burnham, 1970; Prophet, 1970) adds greatly to the occurrence
probabilities, especially in the stratosphere. There is a feeling, backed
up by rather sparse observational evidence (Reed, 1969; Reiter, 1969) that
when CAT is sustained for some time it tends to form along inversion sur-
faces representing the outer boundaries of old CAT regions which have become
vertically well mixed. Recent work by Long (1970) gives a theoretical frame-
work for this view.
Gravity waves, especially those induced by topography, are believed to
be strongly implicated in many cases of turbulent development and in addi-
tion are of considerable interest in themselves. In a series of papers
Miles and Huppert (Huppert and Miles, 1969; Miles, 1968, 1969, 1970;
Miles and Huppert, 1968, 1969) extended the work of Drazin and Moore (1967)
in theoretical modelling of finite amplitude lee wave flow over various
shaped obstacles and in non-rotating and rotating flow. Related work was
reported by Pao (1969). Booker and Bretherton (1967), Bretherton (1969a,b),
and Jones (1968) considered the interactions between lee waves and their environ-
ment. These two papers are particularly relevant to the general circulation
and large scale prediction of the atmosphere because of the conclusion that
rather large amounts of momentum are likely to be transported to the strato-
sphere or beyond by gravity waves.
Two theoretical models have been developed (Danielson & Bleck, 1970;
Vergeiner, 1971) for predicting lee waves in the real atmosphere. Although
both of these are based on linearized two-dimensional equations, they are de-
signed to incorporate arbitrary upstream wind and temperature stratification and
real topography. Some degree of skill is shown by both models when compared
with real data. Houghton & Kasahara (1968) and Houghton & Isaacson (1969) de-
veloped models describing hydraulic jump-type flows in two and three layered
atmospheres. The models, which are based on the shallow water approximation,
are believed to be relevant to the strong downslope windstorms frequently ex-
perienced on the eastern slopes of the Rocky Mountains (Julian & Julian, 1969).
Observational data on mountain waves have been exhibited by Axford (1970)
and Reynolds, et al. (1968) and from a continuing observational program in the
central Colorado area (Kuettner & Lilly, 1968; Lilly & Toutenhoofd, 1969;
Vergeiner & Lilly, 1970) using multiple aircraft flights and constant volume
balloons. The most recent and complete set of observations in this series,
from the winter of 1969-1970, have not yet been reported.
Cloud convection
By comparison with most other fields reviewed here, work in this impor-
tant area of atmospheric dynamics seems somewhat disjointed, not being domi-
nated by an acknowledged main line of attack, and with relatively poor
contact between the theoretical and observational viewpoints. The most
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coherent sub-area is that where cloud dynamics interacts with cloud physics,
especially with respect to cloud modification. Here a series of one-dimensiona
plume types models incorporating microphysics processes to some degree
(Simpson & Wiggert, 1969; Srivastava, 1967; Weinstein, 1970) have been develope
and used with reported success in comparison with observed convective clouds.
In a recent paper, based partly on his observational work on cloud updrafts
(Warner, 1969, 1970a) Warner (1970b) has found the plume models seriously de-
fective in some respects. In particular he shows that the models cannot
simultaneously predict correct liquid water content and cloud depth, possibly
because of the lack of a downdraft regime in most of these models.
Two-dimensional numerical simulation models of clouds or thermals appear
rather frequently (Arnason, et al., 1968; Liu & Orville, 1969; Murray, 1970;
Orville, 1968; Orville & Sloan, 1970; Takeda, 1969). The most notable of
these are the results of Orville and Sloan, which have very high resolution
and interesting microphysical-dynamic interactions and Takeda's rather
sophisticated model, which also shows important microphysical interactions,
including a downdraft development. It is known, however, that the limita-
tion to two dimensions not only removes the possibility of simulating im-
portant three-dimensional phenomena such as those related to vertical shear,
but also makes the proper simulation of the turbulent energy cascade diffi-
cult or impossible. The successes obtained by Deardorff in three-dimensional
simulation of the planetary boundary layer suggests the potential usefulness
of three-dimensional cloud simulation models.
Only a little progress has been made toward developing a statistical
model of a convective cloud ensemble above the planetary boundary layer.
Asai's (1967) simple (compared even to plume models) and somewhat attrac-
tive cellular model has been further analyzed by Schlesinger and Young (1970)
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but has not been developed to the point of being suitable for comparison
with observations. Quite possibly no suitable observations exist, at least
until final processing of the BOMEX data has been completed. An imaginative
paper by Fraser (1968) represents a possible alternative, in principle, to
Asai's approach but also requires further development. In the case of a
shallow cloud-topped mixed layer capped by an inversion a layer model has
been developed (Lilly, 1968) which is probably suitable for comparison with
observations. Again, however, it is difficult to obtain fully adequate
observational data, especially with respect to the critical divergence and
subsidence parameters.
On the other hand observational data and semi-empirical models continue
to accumulate on the intense convective cloud systems with which dynamicists
have been relatively helpless, on account of the apparent great complexity
of the three-dimensional time-dependent dynamic and microphysical interac-
tions. Zipser (1969) has presented evidence that some wet-season tropical
disturbances (now commonly called "cloud clusters", from their appearance in
satellite photographs) are intimately associated, and in effect produced,
by cloud systems similar to those of the middle latitude squall line. The
squall line thunderstorm circulation itself has been further investigated by
Alberty (1969), Bates (1970, compiled by the Severe Storms Research Group of
St. Louis University), Carlson and Ludlam (1968), Fankhauser (1971), Fujita
and Grandoso (1968), Newton (1967, 1969), and Roach (1967).
Turbulence structure and closure theory
With the nearly universal acceptance of the Kolmogorov-Obukhov inertial
range concepts, some additional attention has been focused on the higher and
lower ends of the spectrum. Tower measurements discussed by Busch and
In efforts to improve the concepts of the universal equilibrium theory
Kolmogorov (1962) and other Soviet scientists suggested the possibility of
a log-normal frequency distribution for the dissipation rate. Observational
tests of this hypothesis have been made in the laboratory (Wyngaard & Tennekes,
1970) and in atmospheric boundary layers (Gibson, Stegen, & Williams, 1970;
Sheih, 1971; Stewart, Wilson & Burling, 1970). The results of these measure-
ments are not fully conclusive, since they disagree with each other in some
important points, but all show a substantial degree of conformity with the
hypothesis of log-normality. Wyngaard and Tennekes show that the hypothesis
also requires that the skewness and kurtosis of the velocity derivatives must
be increasing functions of Reynolds number, and produce verifying evidence
of these predictions. Sheih, however, finds that skewness is small for his
large Reynolds number planetary boundary layer observations. Orszag (1970b)
has shown that the log-normal distribution is theoretically awkward in the
respect that the distribution is not uniquely determined by its moments and
therefore that theories of turbulence based on the usual velocity moments
may become unworkable. The deviations from log-normality reported by
Stewart, et al. though not by the other authors, may be sufficient to remove
the difficulty.
Substantial progress appears to have been made in the formulation of
useful and general closures to the moment problem of turbulence theory,
bearing in mind the possible reservations implied by the above paragraph.
The most promising concept in the reviewer's opinion is the eddy-damped
quasi-normal and eddy-damped Markovian models, introduced by Orszag (1970a)
and extended by Kraichnan (1971). The basic concept is extraordinarily
simple for this rather esoteric field. In the eddy-damped quasi-normal
equation for rate of change of third order moments, the transformation of
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fourth order moments to products of second order moments by means of the
quasi-normal assumption is accepted but with the addition of a damping term,
that is a term negatively proportional (by a local turbulent time scale) to
the third order correlations. For the eddy-damped Markovian model the equa-
tion is simply assumed to be steady state, and thus becomes a diagnostic
equation relating third to second order correlations. The damping term is
highly analogous to the "non-linear scramblin" term of Crow's viscoelastic
model (1968) and the eddy-damped Markovian model itself has strong analogies
with Smagorinsky's (1963) and Lilly's (1967) eddy viscosity hypothesis for
mesh scale damping of numerically simulated turbulence. Early tests sug-
gest that the model may have comparable accuracy with Kraichnan's Lagrangian
History Direct Interaction Approximation (1965) but is much simpler to
utilize. No catastrophic failures of the type obtained from solution of
the unaltered quasi-normal equations are observed.
Two other, much simpler closures to homogeneous isotropic turbulence
problems were pesented by Leith (1967) and Pao (1965, 1968). These are of
the same nature as the Heisenberg-Obukhov-Kovasznay closures and should
have similar usage as semi-quantitative conceptual models. Both authors
made interesting comparisons of the decay spectra predicted by their and
other models.
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Konrad, T. G., 1970: The dynamics of the convective process in clear air
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Mancuso, R. L., and R. M. Endlich, 1969: Analyzing and forecasting clear-
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Murray, F. W., 1970: Numerical models of a tropical cumulus cloud with bi-
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Srivastava, R. C., 1967: A study of the effects of precipitation on cumulus
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-27-
Thorpe, S. A., 1969: Experiments on the instability of stratified shear
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U.S. National Oceanic and Atmospheric Administration, 1969-1970: Bomex Bulle-
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Vergeiner, I., 1971: An operational linear lee wave model for arbitrary
basic flow and two-dimensional topography. Accepted for publication
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Vinnichenko, N. K., and J. A. Dutton, 1969: Empirical studies of atmospheric
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-30-
Zipser, E. J., 1969: The role of organized unsaturated convective downdrafts
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ADDITIONAL BIBLIOGRAPHY
INSTRUMENTATION:
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SURFACE AND PLANETARY BOUNDARY LAYER:
Barger, W. R., W. D. Garrett, E. L. Mollo-Christensen, and K. W. Ruggles,
1970: Effects of an artificial sea slick upon the atmosphere and the
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Chi, S. W., S. J. Ying, and C. C. Chang, 1969: The ground turbulent boundary
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Davis, R. E., 1970: On the turbulent flow over a wavy boundary. J. Fluid
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Fichtl, G. H., 1968: Characteristics of turbulence observed at the NASA
150-m meteorological tower. J. Appl. Meteor., 7, 838-844.
Fichtl, G. H., and G. E. McVehil, 1970: Longitudinal and lateral spectra
of turbulence in the atmospheric boundary layer at the Kennedy Space
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Garstang, M., 1967: Sensible and latent heat exchange in low latitude synop-
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Kaimal, J. C., 1969: Measurement of momentum and heat flux variations in the
surface boundary layer. Radio Sci., 4, 1147-1153.
Krishna, K., 1968: A numerical study of the diurnal variation of meteorologi-
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Leslie, L. M., and R. K. Smith, 1970: The surface boundary layer of a hurri-
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-34-
Swinbank, W. C., 1968: A comparison between predictions of dimensional analy-
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CLEAR AIR TURBULENCE AND GRAVITY WAVES:
Bretherton, F. P., 1969: Waves and turbulence in stably stratified fluids.
Radio Sci., 4, 1279-1287.
Dutton, J. A., 1969: An energy budget for a layer of stratospheric CAT.
Radio Sci., 4, 1137-1142.
Hall, J. M., and Y.-H. Pao, 1969: Spectra of internal waves and turbulence
in stratified fluids. Part 2. Experiments on the breaking of internal
waves in a two-fluid system. Radio Sci., 4, 1321-1325.
Hodge, Mary W., 1967: Large irregularities of rawinsonde ascensional rates
within 100 nautical miles and three hours of reported clear air turbu-
lence. Mon. Weather Rev., 95, 99-106.
Lane, J. A., 1969: Some aspects of the fine structure of elevated layers
in the troposphere. Radio Sci., 4, 1111-1114.
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over the equatorial Pacific during the Line Islands Experiment. J.
Atmos. Sci., 27, 336-342.
Miller, A. J., H. M. Woolf, and F. G. Finger, 1968: Small-scale wind and
temperature structure as evidenced by meteorological rocket systems.
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Pao, Y.-H., 1969: Spectra of internal waves and turbulence in stratified
fluids. Part 1. General discussion and indications from measurements
in stably stratified atmosphere and ocean. Radio Sci., 4, 1315-1320.
Scorer, R. S., 1969: Billow mechanics. Radio Sci., 4, 1299-1308.
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Scotti, R. S., and G. M. Corcos, 1969: Measurements on the growth of small
disturbances in a stratified shear layer. Radio Sci., 4, 1309-1313.
Stewart, R. W., 1969: Turbulence and waves in a stratified atmosphere.
Radio Sci., 4, 1269-1278.
Thorpe, S. A., 1969: Experiments on the stability of stratified shear flows.
Radio Sci., 4, 1327-1331.
Waco, D. E., 1970: Temperatures and turbulence at tropopause levels over
Hurricane Beulah (1967). Mon. Weather Rev., 98, 749-755.
Woods, J. D., 1969: On Richardson's number as a criterion for laminar-
turbulent-laminar transition in the ocean and atmosphere. Radio Sci.,
4, 1289-1298.
CLOUD CONVECTION:
Chernikov, A. A., Yu. V. Mel'nichuk, N. Z. Pinus, S. M. Shmeter, and N. K.
Vinnichenko, 1969: Investigations of the turbulence in convective atmo-
sphere using radar and aircraft. Radio Sci., 4, 1257-1259.
Das, Phanindramohan, 1969: The thermodynamic equation in cumulus dynamics.
J. Atmos. Sci., 26, 399-407.
Fosberg, M. A., 1967: Numerical analysis of convective motions over a
mountain ridge. J. Appl. Meteor., 6, 889-904.
Holle, R. L., 1968: Some aspects of tropical oceanic cloud populations.
J. Appl. Meteor., 7, 173-183.
Krishnamurti, T. N., 1968: A calculation of percentage area covered by con-
vective clouds from moisture convergence. J. Appl. Meteor., 7, 184-195.
Plank, V. G., 1969: The size distribution of cumulus clouds in representa-
tive Florida populations. J. Appl. Meteor., 8, 46-67.
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Warner, J., 1967: Some observations on the orographic cloud of the island
of Hawaii and on trade wind cumuli nearby. Tellus, 19, 456-461.
TURBULENCE STRUCTURE AND CLOSURE THEORY:
Baines, W. D., and J. S. Turner, 1969: Turbulent buoyant convection from a
source in a confined region. J. Fluid Mech., 37, 51-80.
Barcilon, A. I., 1968: Phase space solution of buoyant jets. J. Atmos. Sci.,
25, 796-807.
Betchov, Robert, 1967: Review of Kraichnan's theory of turbulence. Phys.
Fluids Suppl., 17-24.
Borkowski, Janusz, 1969: Spectra of anisotropic turbulence in the atmo-
sphere. Radio Sci., 4, 1351-1355.
Bradshaw, P., 1967: The turbulence structure of equilibrium boundary layers.
J. Fluid Mech., 29, 625-645.
Canavan, G. H., 1970: Some properties of a Lagrangian Wiener-Hermite expan-
sion. J. Fluid Mech., 41, 405-412.
Crow, S. C., and G. H. Canavan, 1970: Relationship between a Wiener-Hermite
expansion and an energy cascade. J. Fluid Mech., 41, 387-403.
Deardorff, J. W., and G. E. Willis, 1967: The free-convection temperature
profile. Quart. J. Roy. Meteor. Soc., 93, 166-175.
Deardorff, J. W., and R. L. Peskin, 1970: Lagrangian statistics from numeri-
cally integrated turbulent shear flow. Phys. Fluids, 13, 584-595.
Dutton, J. A., and D. G. Deaven, 1969: A self-similar view of atmospheric
turbulence. Radio Sci., 4, 1341-1349.
Frenkiel, F. N., and P. S. Klebanoff, 1967: Higher-order correlations in a
turbulent field. Phys. Fluids, 10, 507-520.
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Gibson, C. H., 1968: Fine structure of scalar fields mixed by turbulence.
I. Zero-gradient points and minimal gradient surfaces. Phys. Fluids,
11, 2305-2315.
Gibson, C. H., 1968: Fine structure of scalar fields mixed by turbulence.
II. Spectral theory. Phys. Fluids, 11, 2316-2327.
Harlow, F. H., and P. I. Nakayama, 1967: Turbulence transport equations.
Phys. Fluids, 10, 2323-2332.
Justus, C. G., 1969: A theory for the energy spectrum of shear-dependent
turbulence. J. Atmos. Sci., 26, 1238-1244.
Kahng, Woo-Hyung, 1970: Spectral analysis of inviscid Burgers' model of
turbulence using Cameron-Martin-Wiener exact expansion. Phys. Fluids,
13, 1970-1977.
Kato, H., and 0. M. Phillips, 1969: On the penetration of a turbulent
layer into stratified fluid. J. Fluid Mech., 37, 643-655.
Kirwan, A. D., Jr., 1968: Constitutive equations for a fluid containing non-
rigid structures. Phys. Fluids, 11, 1440-1446.
Kovasznay, L. S. G., 1967: Structure of the turbulent boundary layer.
Phys. Fluids Suppl., 25-30.
Kovasznay, L. S. G., Valdis Kibens, and R. F. Blackwelder, 1970: Large-scale
motion in the intermittent region of a turbulent boundary layer. J.
Fluid Mech., 41, 283-325.
Kraichnan, R. H., 1968: Lagrangian-history statistical theory for Burgers'
equation. Phys. Fluids, 11, 265-277.
Kraichnan, R. H., 1968: Small-scale structure of a scalar field convected
by turbulence. Phys. Fluids, 11, 945-953.
Kraichnan, R. H., 1970: Convergents to turbulent functions. J. Fluid Mech.,
41, 189-217.
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Kraichnan, R. H., 1970: Instability in fully developed turbulence. Phys.
Fluids, 13, 569-575.
Lin, J.-T., S. Panchev, and J. E. Cermak, 1969: A modified hypothesis on
turbulence spectra in the buoyancy subrange of stably stratified shear
flow. Radio Sci., 4, 1333-1337.
Lumley, J. L., 1967: Rational approach to relations between motions of
differing scales in turbulent flows. Phys. Fluids, 10, 1405-1408.
Meecham, W. C., and D.-T. Jeng, 1968: Use of the Wiener-Hermite expansion
for nearly normal turbulence. J. Fluid Mech., 32, 225-249.
Meecham, W. C., 1970: Equilibrium characteristics of nearly normal turbu-
lence. J. Fluid Mech., 41, 179-188.
Morton, B. R., 1967: Entrainment models for laminar jets, plumes, and wakes.
Phys. Fluids, 10, 2120-2127.
Morton, B. R., 1969: The strength of vortex and swirling core flows. J.
Fluid Mech., 38, 315-333.
Orszag, S. A., and M. D. Kruskal, 1968: Formulation of the theory of turbu-
lence. Phys. Fluids, 11, 43-60.
Panchev, S., 1968: Coefficient of horizontal macroturbulent exchange in the
atmosphere. J. Atmos. Sci., 25, 933-935.
Phillips, 0. M., 1967: The maintenance of Reynolds stress in turbulent
shear flow. J. Fluid Mech., 27, 131-144.
Plate, E. J., and S. P. Arya, 1969: Turbulence spectra in a stably strati-
fied boundary layer. Radio Sci., 4, 1163-1168.
Reiter, E. R., 1969: Structure of vertical wind profiles. Radio Sci., 4,
1133-1136.
Van Atta, C. W., and T. T. Yeh, 1970: Some measurements of multi-point time
correlations in grid turbulence. J. Fluid Mech., 41, 169-178.
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Walton, J. J., 1968: Turbulent spectra from the Kraichnan-Spiegal approxima-
tion. Phys. Fluids, 11, 435-437.
Wyngaard, J. C., H. Tennekes, J. L. Lumley, and D. P. Margolis, 1968: Struc-
ture of turbulence in a curved mixing layer. Phys. Fluids, 11, 1251-1253.
MISCELLANEOUS:
Murgatroyd, R. J., 1969: Estimations from geostrophic trajectories of hori-
zontal diffusivity in the mid-latitude troposphere and lower stratosphere.
Quart. J. Roy. Meteor. Soc., 95, 40-62.
Zimmerman, L. I., 1969: Atmospheric wake phenomena near the Canary Islands.
J. Appl. Meteor., 8, 896-907.
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