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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-19-

Hsueh, Y., 1968: Mesoscale turbulence spectra over the Indian Ocean. J.

Atmos. Sci., 25, 1052-1057.

Huppert, H. E., and J. W. Miles, 1969: Lee waves in a stratified flow.

Part 3. Semi-elliptical obstacle. J. Fluid Mech., 35, 481-496.

Hwang, H. J., 1970: Power density spectrum of surface wind speed on

Palmyra Island. Mon. Weather Rev., 98, 70-74.

Jones, W. L., 1968: Reflexion and stability of waves in stably stratified

fluids with shear flow: A numerical study. J. Fluid Mech., 34, 609-624.

Julian, L. T., and P. R. Julian, 1969: Boulder's winds. Weatherwise, 22,

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Julian, P. R., W. M. Washington, L. Hembree, and C. Ridley, 1970: On the

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Kaimal, J. C., and D. A. Haugen, 1967: Characteristics of vertical velocity

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93, 305-317.

Kaimal, J. C., 1968: The effect of vertical line averaging on the spectra

of temperature and heat-flux. Quart. J. Roy. Meteor. Soc., 94, 149-155.

Kaimal, J.C., J. C. Wyngaard, and D. A. Haugen, 1968: Deriving power spec-

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827-837.

Kaimal, J. C., and J. A. Businger, 1970: Case studies of a convective plume

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Kao, S.-K., 1968: Large-scale atmospheric motion and transports in frequency

wave-number space. J. Atmos. Sci., 25, 32-38.

Kolmogorov, A. N., 1962: A refinement of previous hypothesis concerning

the local structure of turbulence in a viscous incompressible fluid at

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-20-

Konrad, T. G., 1970: The dynamics of the convective process in clear air

as seen by radar. J. Atmos. Sci., 27, 1138-1147.

Kraichnan, R. H., 1965: Lagrangian-history closure approximation for tur-

bulence. Phys. Fluids, 8, 575-598.

Kraichnan, R. H., 1966: Isotropic turbulence and inertial-range structure.

Phys. Fluids, 9, 1728-1752.

Kraichnan, R. H., 1967: Inertial ranges in two-dimensional turbulence.

Phys. Fluids, 10, 1417-1423.

Kraichnan, R. H., 1971: An almost-Markovian Galilean-invariant turbulence

model. Accepted for publication in J. Fluid Mech. (February or March).

Kuettner, J. P., and D. K. Lilly, 1968: Lee waves in the Colorado Rockies.

Weatherwise, 21, 180-186;193;196-197.

Kung, E. C., 1966: Kinetic energy generation and dissipation in the large-

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Kung, E. C., 1969: Further study on the kinetic energy balance. Mon.

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Kung, E. C., 1969: Seasonal variation of kinetic energy in the atmosphere.

Quart. J. Roy. Meteor. Soc., 95, 501-512.

Leith, C. E., 1967: Diffusion approximation to inertial energy transfer in

isotropic turbulence. Phys. Fluids, 10, 1409-1416.

Lenschow, D. H., 1970: Airplane measurements of planetary boundary layer

structure. J. Appl. Meteor., 9.

-21-

Lhermitte, R. M., 1969: Note on the observation of small-scale atmospheric

turbulence by doppler radar techniques. Radio Sci., 4, 1241-1246.

Lilly, D. K., 1966: On the instability of Ekman boundary flow. J. Atmos.

Sci. , 23, 481-494.

Lilly, D. K., 1967: The representation of small-scale turbulence in numeri-

cal simulation experiments. In Proceedings of the IBM Scientific Com-

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White Plains, N.Y., 195-210.

Lilly, D. K., 1968: Models of cloud-topped mixed layers under a strong in-

version. Quart. J. Roy. Meteor. Soc., 94, 292-309.

Lilly, D. K., 1969: Numerical simulation of two-dimensional turbulence.

Phys. Fluids Suppl. II, 240-249.

Lilly, D. K., and W. Toutenhoofd, 1969: The Colorado lee wave program. In

Clear Air Turbulence and Its Detection. New York, Plenum Press, 232-245.

Lilly, D. K., 1971: Comments on "Case studies of a convective plume and a

dust devil". J. Appl. Meteor.

Lilly, D. K., 1971: Numerical simulation of developing and decaying two-

dimensional turbulence. Accepted for publication in J. Fluid Mechs.

Little, C. G., 1969: Acoustic methods for the remote probing of the lower

atmosphere. Proc. IEEE, 57, 571-578.

Liu, J. Y., and H. D. Orville, 1969: Numerical modeling of precipitation

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Long, R. R., 1970: A theory of turbulence in stratified fluids. J. Fluid

Mech., 42, 349-365.

Ludham, F. H., 1967: Characteristics of billow clouds and their relation

to clear-air turbulence. Quart. J. Roy. Meteor. Soc., 93, 419-435.

-22-

Mancuso, R. L., and R. M. Endlich, 1969: Analyzing and forecasting clear-

air turbulence probabilities over the United States. Mon. Weather Rev.,

97, 527-533.

Mather, G. K., 1969: Clear air turbulence research activities at the

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Miles, J. W., and L. N. Howard, 1964: Note on a heterogeneous shear flow.

J. Fluid Mech., 20, 331-336.

Miles, J. W., 1968: Lee waves in a stratified flow. Part 1. Thin barrier.

J. Fluid Mech., 32, 549-567.

Miles, J. W., and H. E. Huppert, 1968: Lee waves in a stratified flow.

Part 2. Semi-circular obstacle. J. Fluid Mech., 33, 803-814.

Miles, J. W., and H. E. Huppert, 1969: Lee waves in a stratified flow.

Part 4. Perturbation approximations. J. Fluid Mech., 35, 497-525.

Miles, J. W., 1969: The lee-wave regime for a slender body in a rotating

flow. J. Fluid Mech., 36, 265-288.

Miles, J. W., 1970: The lee-wave regime for a slender body in a rotating

flow. Part 2. J. Fluid Mech., 42, 201-206.

Miyake, M., M. Donelan, G. McBean, C. Paulson, F. Badgley, and E. Leavitt,

1970: Comparison of turbulent fluxes over water determined by profile

and eddy correlation techniques. Quart. J. Roy. Meteor. Soc., 96, 132-137.

Miyake, M., R. W. Stewart, and R. W. Burling, 1970: Spectra and cospectra

of turbulence over water. Quart.J. Roy. Meteor. Soc., 96, 138-143.

Morton, B. R., 1968: On Telford's model for clear air convection. J. Atmos.

Sci., 25, 135-138.

-23-

Murray, F. W., 1970: Numerical models of a tropical cumulus cloud with bi-

lateral and axial symmetry. Mon. Weather Rev., 98, 14-28.

Myrup, L. 0., 1969: Turbulence spectra in stable and convective layers in

the free atmosphere. Tellus, 21, 341-354.

National Center for Atmospheric Research, 1969: Instrumenting NCAR's

Buffalo Aircraft. Facilities for Atmospheric Res., 8, 9-12.

Newton, C. W., 1967: Severe convective storms. Adv. Geophys., 12, 257-308.

Newton, C. W., 1969: The role of extratropical disturbances in the global

atmosphere. In The Global Circulation of the Atmosphere, ed. G. A.

Corby, London, Salisbury Press, 137-158.

Nickerson, E. C., 1968: Boundary layer adjustment as an initial value pro-

blem. J. Atmos. Sci., 25, 207-213.

Oke, T. R., 1970: Turbulent transport near the ground in stable conditions.

J. Appl. Meteor., 9, 778-786.

Oort, A. H., and Albion Taylor, 1969: On the kinetic energy spectrum near

the ground. Mon. Weather Rev., 97, 623-636.

Orszag, S. A., 1967: Approximate calculation of the Kolmogorov-Obukhov con-

stant. Phys. Fluids, 10, 454-455.

Orszag, S. A., 1970a: Analytical theories of turbulence. J. Fluid Mech., 41,

363-386.

Orszag, S. A., 1970b: Indeterminacy of the moment problem for intermittent

turbulence. Phys. Fluids, 13, 2211-2212.

Orville, H. D., 1968: Ambient wind effects on the initiation and develop-

ment of cumulus clouds over mountains. J. Atmos. Sci., 25, 385-403.

Orville, H. D., and L. J. Sloan, 1970: A numerical simulation of the life

history of a rainstorm. J. Atmos. Sci., 27, 1148-1159.

Panofsky, H. A., J. A. Dutton, K. H. Hemmerich, G. McCreary, and N. V. Lovin

1968: Case studies of the distribution of CAT in the troposphere and

stratosphere. J. Appl. Meteor., 7, 384-389.

Panofsky, H. A., and E. Mares, 1968: Recent measurements of cospectra for

heat-flux and stress. Quart. J. Roy. Meteor. Soc., 94, 581-585.

Panofsky, H. A., 1969a: Spectra of atmospheric variables in the boundary

layer. Radio Sci., 4, 1101-1109.

Panofsky, H. A., 1969b: The spectrum of temperature. Radio Sci., 4, 1143-

1146.

Pao, Y.-H., 1965: Structure of turbulent velocity and scalar fields at

long wavenumbers. Phys. Fluids, 8, 1063-1075.

Pao, Y.-H., 1968: Transfer of turbulent energy and scalar quantities at

large wavenumbers. Phys. Fluids, 11, 1371-1372.

Pao, Y.-H., 1969: Inviscid flows of stably stratified fluids over barriers.

Quart. J. Roy. Meteor. Soc., 95, 104-119.

Phillips, 0. M., 1958: The equilibrium range in the spectrum of wind-

generated waves. J. Fluid Mech., 4, 426-434.

Plate, E. J., and G. M. Hidy, 1967: Laboratory study of air flowing over

a smooth surface onto small water waves. J. Geophys. Res., 72, 4627-

4641.

Plate, E. J., P. C. Chang, and G. M. Hidy, 1969: Experiments on the genera-

tion of small water waves by wind. J. Fluid Mech., 35, 625-656.

Prophet, D. T., 1970: Vertical extent of turbulence in clear air above the

tops of thunderstorms. J. Appl. Meteor., 9, 320-321.

Reed, R. J., 1969: A study of the relation of clear-air turbulence to the

mesoscale structure of the jet stream region. In Proceedings of the

Boeing Conference on Clear-Air Turbulence, New York, Plenum.

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0

-25-

Reiter, E. R., and H. P. Foltz, 1967: The prediction of clear air turbulence

over mountainous terrain. J. Appl. Meteor., 6, 549-556.

Reiter, E., 1969: The nature of clear-air turbulence. In Clear Air Turbulence

and Its Detection, New York, Plenum.

Reynolds, R. D., R. L. Lamberth, and M. G. Wurtele, 1968: Investigation of a

complex mountain wave situation. J. Appl. Meteor., 7, 353-358.

Richter, J. H., 1969: High resolution tropospheric radar sounding. Radio

Sci., 4, 1261-1268.

Roach, W. T., 1967: On the nature of the summit areas of severe storms in

Oklahoma. Quart. J. Roy. Meteor. Soc., 93, 318-336.

Schlesinger, R. E., and J. A. Young, 1970: On the transient behavior of

Asai's model of moist convection. Mon. Weather Rev., 98, 375-384.

Scorer, R. S., 1959: The behavior of chimney plumes. Int. J. Air Poll., 1,

198-220.

Sheih, Ching-Ming, 1971: Airborne hot-wire measurements of the small-scale

structure of the atmospheric turbulence. Accepted for publication in

Phys. Fluids (February).

Shemdin, 0. H., and E. Y. Hsu, 1967: Direct measurements of aerodynamic

pressure above a simple progressive gravity wave. J. Fluid Mech., 30,

403-416.

Simpson, Joanne, and Victor Wiggert, 1969: Models of precipitating cumulus

towers. Mon. Weather Rev., 97, 471-489.

Sinclair, P. C., 1969: General characteristics of dust devils. J. Appl.

Meteor., 8, 32-45.

Smagorinsky, J., 1963: General circulation experiments with the primitive

equations. I. The basic experiment. Mon. Weather Rev., 91, 99-164.

-26-

Srivastava, R. C., 1967: A study of the effects of precipitation on cumulus

dynamics. J. Atmos. Sci., 24, 36-45.

Stewart, R. H., 1970: Laboratory studies of the velocity field over deep-

water waves. J. Fluid Mech., 42, 733-754.

Stewart, R. W., 1967: Mechanics of the air-sea interface. Phys. Fluids Suppl.,

47-55.

Stewart, R. W., J. R. Wilson, and R. W. Burling, 1970: Some statistical pro-

perties of small scale turbulence in an atmospheric boundary layer. J.

Fluid Mech., 41, 141-152.

Stoeffler, R. C., 1970: Further Research on Instabilities in Atmospheric Flow

Systems Associated with Clear Air Turbulence. United Aircraft Research

Laboratories, Rept. J910563-14 (NASA Contract NASW-1582), East Hartford,

Connecticut, 63 pp.

Takeda, T., 1969: Numerical simulation of large convective clouds. Stormy

Weather Group Scientific Report MW-64. McGill University, Montreal,

Canada.

Tatro, P. R., and E. L. Mollo-Christensen, 1967: Experiments on Ekman layer

instability. J. Fluid Mech., 28, 531-543.

Taylor, P. A., 1969: On the planetary boundary layer flow under conditions

of neutral thermal stability. J. Atmos. Sci., 26, 427-431.

Taylor, P. A., 1969: The planetary boundary layer above a change in surface

roughness. J. Atmos. Sci., 26, 432-440.

Telford, J. W., 1966: The convective mechanism in clear air. J. Atmos. Sci.,

23, 652-666.

Telford, J. W., 1970: Convective plumes in a convective field. J. Atmos.

Sci., 27, 347-358.

Thorpe, S. A., 1968: A method of producing a shear flow in a stratified

fluid. J. Fluid Mech., 32, 693-704.

-27-

Thorpe, S. A., 1969: Experiments on the instability of stratified shear

flows: Immiscible fluids. J. Fluid Mech., 39, 25-48.

Thurtell, G. W., C. B. Tanner, and M. L. Wesely, 1970: Three-dimensional

pressure-sphere anemometer system. J. Appl. Meteor., 9, 379-385.

Trout, Dennis, and H. A. Panofsky, 1969: Energy dissipation near the tropo-

pause. Tellus, 21, 355-358.

Tsvang, L. R., 1969: Microstructure of temperature fields in the free atmo-

sphere. Radio Sci., 4, 1175-1177.

Turner, J. S., 1963: The motion of buoyant elements in turbulent surroundings.

J. Fluid Mech., 16, 1-16.

U.S. National Oceanic and Atmospheric Administration, 1969-1970: Bomex Bulle-

tin No. 1-8. The BOMAP Office, National Oceanic and Atmospheric Admini-

stration, Rockville, Md.

Vergeiner, I., and D. K. Lilly, 1970: The dynamic structure of lee wave flow

as obtained from balloon and airplane observations. Mon. Weather Rev.,

98, 220-232.

Vergeiner, I., 1971: An operational linear lee wave model for arbitrary

basic flow and two-dimensional topography. Accepted for publication

in Quart. J. Roy. Meteor. Soc.

Vinnichenko, N. K., and J. A. Dutton, 1969: Empirical studies of atmospheric

structure and spectra in the free atmosphere. Radio Sci., 4, 1115-1126.

Vinnichenko, N. K., 1970: The kinetic energy spectrum in the free atmosphere -

1 second to 5 years. Tellus, 22, 158-166.

Waco, D. E., 1970: A statistical analysis of wind and temperature variables

associated with high altitude clear air turbulence (HICAT). J. Appl.

Meteor., 9, 300-309.

Warner, J., and J. W. Telford, 1967: Convection below cloud base. J. Atmos.

Sci., 24, 374-382.

Warner, J., 1969: The microstructure of cumulus cloud. Part II. The effect

on droplet size distribution of the cloud nucleus spectrum and updraft

velocity. J. Atmos. Sci., 26, 1272-1282.

Warner, J., 1970a: The microstructure of cumulus cloud. Part III. The

nature of the updraft. J. Atmos. Sci., 27, 682-688.

Warner, J., 1970b: On steady-state one-dimensional models of cumulus convec-

tion. J. Atmos. Sci., 27, 1035-1040.

Webb, E. K., 1970: Profile relationships: The log-linear range, and exten-

sion to strong stability. Quart. J. Roy. Meteor. Soc., 96, 67-90.

Weinstein, A. I., 1970: A numerical model of cumulus dynamics and micro-

physics. J. Atmos. Sci., 27, 246-255.

Wendell, L. L., 1969: A study of large-scale atmospheric turbulent kinetic

energy in wave-number frequency space. Tellus, 21, 760-788.

Wesely, M. L., G. W. Thurtell, and C. B. Tanner, 1970: Eddy correlation

measurements of sensible heat flux near the earth's surface. J. Appl.

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Winn-Nielsen, A., 1967: On the annual variation and spectral distribution

of atmospheric energy. Tellus, 19, 540-559.

Wu, Jin, 1968: Laboratory studies of wind-wave interactions. J. Fluid Mech.,

34, 91-111.

Wu, Jin, 1969: Froude number scaling of wind-stress coefficients. J.

Atmos. Sci., 26, 408-413.

Wu, Jin, 1969: Wind stress and surface roughness at air-sea interface.

Geophys. Res., 74, 444-455.

Wu, Jin, 1970: Wind-wave interactions. Phys. Fluids, 13, 1926-1930.

Wyngaard, J. C., and H. Tennekes, 1970: Measurements of the small-scale

structure of turbulence at moderate Reynolds numbers. Phys. Fluids,

13, 1962-1969.

-30-

Zipser, E. J., 1969: The role of organized unsaturated convective downdrafts

in the structure and rapid decay of an equatorial disturbance. J. Appl.

Meteor., 8, 799-814.

-30-

ADDITIONAL BIBLIOGRAPHY

INSTRUMENTATION:

Atlas, David, 1969: The measurement of crosswind by non-coherent dual arm

bistatic radio tropo-scatter techniques. J. Atmos. Sci., 26, 1122-1127.

Atlas, David, R. C. Srivastava, R. E. Carbone, and D. H. Sargeant, 1969:

Doppler crosswind relations in radio tropo-scatter beam swinging for a

thin scattered layer. J. Atmos. Sci., 26, 1104-1117.

Atlas, David, R. C. Srivastava, and W. S. Marker, Jr., 1969: The influence

of specular reflections on bistatic tropospheric radio scatter from tur-

bulent strata. J. Atmos. Sci., 26, 1118-1121.

Atlas, David, R. C. Srivastava, and P. W. Sloss, 1969: Wind shear and reflec-

tivity gradient effects on doppler radar spectra: II. J. Appl. Meteor.,

8, 384-388.

Battan, Louis J., and J. B. Theiss, 1970: Measurement of vertical velocities

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radio refractivity and water vapor in the atmosphere. Radio Sci., 4,

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Bean, B. R., and R. 0. Gilmer, 1969: Comparison of barium fluoride humidity

element with the microwave refractometer for studies of rapid fluctua-

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Coantic, M., and D. Leducq, 1969: Turbulent fluctuations of humidity and

their measurement. Radio Sci., 4, 1169-1174.

Danielsen, E. F., and R. T. Duquet, 1967: A comparison of FPS-16 and GMD-1

measurements and methods for processing wind data. J. Appl. Meteor.,

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Endlich, R. M., and J. W. Davies, 1967: The feasibility of measuring turbu-

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Lee, R. W., 1969: A review of line-of-sight propagation studies of the small-

scale structure of the atmosphere. Radio Sci., 4, 1211-1233.

Ottersten, Hans, 1969: Atmospheric structure and radar backscattering in

clear air. Radio Sci., 4, 1179-1193.

Ottersten, Hans, 1969: Mean vertical gradient of potential refractive index

in turbulent mixing and radar detection of CAT. Radio Sci., 4, 1247-1249.

Ottersten, Hans, 1969: Radar backscattering from the turbulent clear atmo-

sphere. Radio Sci., 4, 1251-1255.

Stegen, G. R., and C. W. Van Atta, 1970: A technique for phase speed measure-

ments in turbulent flow. J. Fluid Mech., 42, 689-699.

Suomi, V. E., K. J. Hanson, and R. J. Parent, 1967: The "chirp" digital

radiosonde. J. Appl. Meteor., 6, 195-198.

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

ocean. J. Appl. Meteor., 9, 396-400.

Bonner, W. D., S. Esbensen, and R. Greenberg, 1968: Kinematics of the low-

level jet. J. Appl. Meteor., 7, 339-347.

Busch, N. E., R. M. Brown, and J. A. Frizzola, 1970: Vertical velocity

variances and Reynolds stresses at Brookhaven. J. Appl. Meteor., 9,

583-587.

-32-

Chi, S. W., S. J. Ying, and C. C. Chang, 1969: The ground turbulent boundary

layer of a stationary tornado-like vortex. Tellus, 21, 693-700.

Clarke, R. H., 1970: Observational studies in the atmospheric boundary layer.

Quart. J. Roy. Meteor. Soc., 96, 91-114.

Csanady, G. T., 1967: Concentration fluctuations in turbulent diffusion.

J. Atmos. Sci., 24, 21-28.

Csanady, G. T., 1969: Diffusion in an Ekman layer. J. Atmos. Sci., 26, 414-

426.

Davis, R. E., 1970: On the turbulent flow over a wavy boundary. J. Fluid

Mech., 42, 721-731.

Dickson, C. R., and J. K. Angell, 1968: Eddy velocities in the planetary

boundary layer as obtained from Tetroon flights at Idaho Falls. J.

Appl. Meteor., 7, 986-993.

Dyer, A. J., 1968: An evaluation of eddy flux variation in the atmospheric

boundary layer. J. Appl. Meteor., 7, 845-850.

Estoque, M. A., 1968: Vertical mixing due to penetrative convection. J.

Atmos. Sci., 25, 1046-1051.

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

Center. J. Appl. Meteor., 9, 51-63.

Garstang, M., 1967: Sensible and latent heat exchange in low latitude synop-

tic scale systems. Tellus, 19, 492-508.

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-

cal parameters in the planetary boundary layer. Mon. Weather Rev., 96,

269-276.

-33-

Leslie, L. M., and R. K. Smith, 1970: The surface boundary layer of a hurri-

cane. II. Tellus, 22, 288-297.

Lettau, H., and J. Zabransky, 1968: Interrelated changes of wind profile

structure and Richardson number in air flow from land to inland lakes.

J. Atmos. Sci., 25, 718-728.

Lettau, H., 1969: Note on aerodynamic roughness-parameter estimation on the

basis of roughness-element description. J. Appl. Meteor., 8, 828-832.

Monin, A. S., 1967: Turbulence in the atmospheric boundary layer. Phys.

Fluids Suppl., 31-37.

Myrup, L. 0., 1969: A numerical model of the urban heat island. J. Appl.

Meteor., 8, 908-918.

Peterson, K. R., 1968: Continuous point source plume behavior out to 160

miles. J. Appl. Meteor., 7, 217-226.

Priestley, C. H. B., 1967: Handover in scale of the fluxes of momentum,

heat, etc. in the atmospheric boundary layer. Phys. Fluids Suppl.,

38-46.

Readings, C. J., and D. R. Rayment, 1969: The high-frequency fluctuation of

the wind in the first kilometer of the atmosphere. Radio Sci., 4,

1127-1131.

Rider, L. J., and M. Armendariz, 1970: Vertical wind component estimates

up to 1.2 km above ground. J. Appl. Meteor., 9, 64-71.

Ruggles, K. W., 1970: The vertical mean wind profile over the ocean for

light to moderate winds. J. Appl. Meteor., 9, 389-395.

Slade, D. H., 1969: Wind measurement on a tall tower in rough and inhomo-

geneous terrain. J. Appl. Meteor., 8, 293-297.

Stewart, R. E., 1968: Atmospheric diffusion of particulate matter released

from an elevated continuous source. J. Appl. Meteor., 7, 425-432.

-34-

Swinbank, W. C., 1968: A comparison between predictions of dimensional analy-

sis for the constant-flux layer and observations in unstable conditions.

Quart. J. Roy. Meteor. Soc., 94, 460-467.

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.

Madden, R. A., and E. J. Zipser, 1970: Multi-layered structure of the wind

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.

J. Appl. Meteor., 7, 390-399.

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.