NASA Contractor Report 3378 

Case Studies of Clear Air-. 

Using the Diagnostic Richardson 

Number Tendency Formulation 

John L. Keller and Patrick A. Haines 

CONTRACT NASs-32 111 

JANUARY 198 1 
TECH LIBRARY KAFB, NM 

NASA Contractor Report 3378 

Case Studies of Clear Air Turbulence 

Using the Diagnostic Richardson 

Number Tendency Formulation 

John L. Keller and Patrick A. Haines 

University of Dayton 

Dayton, Ohio 

Prepared for 

Marshall Space Flight Center 

under Contract NASS-32 111 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Branch 

1981 

- 

TABLE OF CONTENTS 

INTRODUCTION 

THE DRT FORMULATION 

GENERAL NATURE OF INPUT DATA 

RESULTS 

SUMMARY AND DISCUSSION 

REFERENCES 

iii 

PAGE 

1 

5 

12 

21 

46 

48 

-- 

- - -----_ ‘. ------- 

., ..: 

. ‘. -7’. ‘~.. ,’ ” >: : 

7, ,..,, (‘., 

, / ” ., -; :, , ~ .‘. 
._ _. -.-J .-.L - - -..-- ~~---. _ simula-cion moael. aeveloDea iw t511cK. a 

FIGURE 

1 

2 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

LIST OF ILLUSTRATIONS 

The CAT Mechanism. 

DRT Horizontal Grid as Subset of the NMC Octagonal 

Hemispheric Grid. 

DRT Case Study Methodology. 

DRT Module Formulation. 

Scatter Diagram Comparing Perturbed and Unperturbed 

Eastward Wind Components for the 500 mb Pressure 

Level. 

Vertical Motion Field for the Unperturbed and 

Perturbed Wind Fields at 500 mb for OZ 20 February 

1970. 

Time to Reach Ri,, for the Unperturbed and 

Perturbed Wind Fields at 500 mb for OZ 20 February 

1970. 

Time to Reach Ri,, and Pilot Encounters with 

Moderate to Severe Turbulence at 300 mb for 

oz 9 March 1976. 

Same as Figure 8, except for 122. 

Hypothetical Flight Path from Rome to New York 

at the 300 mb Surface. 

Same as Figure 

Same as Figure 

Same as Figure 

Pressure-Level 

Same as Figure 

Pressure-Level 

Same as Figure 

Same as Figure 

Pressure-Level 

Vertical Cross 

Same as Figure 

8, Except for Map Time Indicated. 

9, Except for Map Time Indicated 

8, Except for Map Time and 

Indicated. 

9, Except for Map Time and 

Indicated. 

8, Except for Map Time Indicated. 

9, Except for Map Time and 

Indicated. 

Section of tcr Along 50°N Parallel. 

17, Except 12 Hours Later. 

Pressure Heights and Isotachs at 250 mb. 

CAT Potential Index for the 4000 Foot Interval 

centered at 250 mb, and Pilot Encounters with 

Moderate to Severe Turbulence. 

Same as Figure 20, Except for Map Time and 

Pressure-Level Indicated. 

PAGE 

1 

13 

15 

16 

17 

18 

19 

22 

23 

25 

26 

27 

28 

29 

30 

31 

32 

33 

35 

36 

37 
FIGURE PAGE 

22 

23 

24 

25 

26 

27 

28 

LIST OF ILLUSTRATIONS (Continued) 

CAT Potential Index, I 

Encounters with Modera E eT;o 

and Pilot 

Severe Turbulence 

at 300 mb for 02 9 March 1976. 

39 

Same as Figure 22, Except for 122. 

Same as Figure 22, Except for Map Time Indicated. 

Same as Figure 23, Except for Map Time Indicated. 42 

Same as Figure 22, Except for Map Time Indicated 43 

Same as Figure 23, Except for Map Time Indicated 44 

Fraction of Reported Encounters as a Function of 45 

CAT Potential Parameters tcr and ICAT for all 

cases. 

V 

..- 

,, _-- __- / 
. . I 

__. _- ---.-cI- .-i _Iu_ -__-. a__. -_.- .- ___-e - -- - 

cP 

E 

f 

H 

x 

*CAT 

3 

P 

R 

Ri 

Ricr 

t . . 

t cr 

U 

V 

-f 

V 

8 

K 

P 

(5 

GS 

a 

X 

VJ 

w 

wn 

WN 

%g 

LIST OF SYMBOLS 

Specific heat at constant pressure 

Perturbation energy density 

Coriolis parameter 

Height of unstable shear layer 

Unit vector (positive = eastward) 

Index of CAT potential 

Unit vector (positive = northward) 

Pressure 

Gas constant for dry air 

Richardson number 

Critical Richardson number 

Time 

Time to reach critical Richardson number (Ricr) 

Eastward component of wind 

Northward component of wind 

Horizontal wind velocity 

Vertical (height) coordinate 

Magnitude of the vertical difference of the 

horizontal velocity through a given layer 

Potential temperature 

Wp 

Density of air 

Standard atmospheric static stability 

Constant related to the Diagnostic Ri tendency 

Diagnostic Ri tendency 

Velocity potential 

Stream function 

Vertical wind component (DP/Dt) 

Iterated value of the vertical wind component 

Final value of the vertical wind component 

Quasi-geostrophic value of the vertical wind component 

vi 

-_-_-- .._ .- - _ .-.__-~-- . . ~- --- 

. . 1. . . 
QL JUU 111. 

FORFWORD 

Merely interjecting the word "turbulence" into a conversation 

with anyone in aviation will elicit an unanticipatable response 

that depends on the person's experience, education, and role within the aviation community: So many preconceptions and misconceptions are associated with clear air turbulence or "CAT" that 

it has entered the realm of weather folklore. This research effort 

was initiated in order to apply recent advances in theoretical 

research into the CAT mechanism for ultimate operational uses. 

The initial motivation was, of course, passenger comfort and 

safety. As is true of many contemporary endeavors, the economics 

of petroleum changed the perspective. Conversations with some 

commercial pilots have revealed that flying in turbulent conditions 

increases fuel consumption "significantly." A decade ago, a 

change of several percent in aggregate fuel consumption due to 

essentially random encounters with CAT may not have been significant. Now, the effort needed to detect and forecast CAT and 

favorably change fuel consumption by just one percent is almost 

certainly cost-effective. We hope that this report will support 

the contention that CAT detection and forecasting can be made 

practical. 

This research was conducted by the University of Dayton 

Research Institute under the auspices of the National Aeronautics 

and Space Administration, George C. Marshall Space Flight Center, 

under the technical.direction of Mr. Dennis W. Camp of the Space 

Sciences Laboratory. The support for this effort was provided by 

Mr. A. Richard Tobiason of Transport Aircraft, Aeronautical 

Systems Division, Office of Aeronautics and Space Technology. 

The authors wish to thank Mr. M. J. Oard, currently of the 

National Weather Service, for his personal notes and prototype 

computer source code, and Mr. M. J. 0. Dutton of the British 

Meteorological Office for the detailed records of pilot encounters 

with CAT. We also wish to thank Mr. Dennis W. Camp for his valuable 

comments and suggestions about the report manuscript. 

vii 

. -__--.- -- 
_ 

_____-.--._- 

’ 
QL JUU 111. 

INTRODUCTION 

Encounters with atmospheric turbulence are an important 

and persistent problem for the aviation community. Though 

progress has been made in the communication of turbulence 

reports to pilots, both commercial and general aviation would 

benefit considerably by improved forecasting and monitoring of 

significant regions of moderate to severe turbulence. 

When addressing any problem of atmospheric turbulence, 

it is necessary to distinguish between the two.fundamental types 

of turbulence. The type of turbulence which occurs in the 

planetary boundary layer (generally in the lower kilometer of 

the atmosphere) is most important for smaller aircraft but can 

be significant during takeoffs and landings by high-performance 

aircraft. This more intense turbulence is generally associated 

with nonhydrostatic cumulus convection, the most severe being 

created by thunderstorms. The dangers posed by such storms are 

due to both the extreme updrafts and downdrafts within the cloud 

itself and to the outflows at its base. The former problem is 

generally encountered during in-flight periods,while the latter 

can present a particularly serious threat during landings. 

Although strong thunderstorms can often reach to the high levels 

of flight paths used by high-performance aircraft, their turbulent nature still fits within the framework of nonhydrostatic 

planetary boundary layer turbulence in which the individual 

cells can be regarded as a local intrusion of the boundary layer 

through the free atmosphere. 

A most troublesome type of turbulence is associated with 

statically stable shear layers. While turbulence generated by 

thunderstorms can usually be avoided by steering clear of the 

cloud area, turbulence embedded within statically stable shear 

1 
layers can often be cloud-free, giving rise to clear air 

turbulence (CAT). Aircraft observations of CAT indicate that 

the mean horizontal extent of turbulence is on the order of 

100 km in width and approximately 200 km in the direction of 

the wind. The vertical depth of a layer can be as much as 

1500 m. Nearly 5.3 percent of the time, a patch of CAT can be 

expected to exist for more than six hours (Pinus, 1964). The 

commercial and military aviation communities would especially 

benefit from accurate quantitative monitoring and prediction 

of CAT. This is due to the fact that commercial airlines 

tend to fly near the tropopause level where statically stable 

shear layers are more common. 

What is the motivation for developing and implementing 

improved turbulence detection and forecasting techniques? 

The reason which initially formed the basis for earlier work 

remains: passenger comfort and safety. However, a more 

contemporary motivation has been added. Pilots report that 

fuel consumption rates increase perceptibly while flying in 

turbulent conditions (Camp and Frost, 1980). Avoiding turbulence can save money. 

With the development of the FPS-16 Jimpshere/Jimsonde (J/J) 

balloon system came the capability for obtaining high resolution 

profiles of wind, temperature, and rise-rate (e.g., Fichtl, 1971; 

Camp and Caplan, 1969; Range Commander's Council Document, 

110-77, 1977). The resolution appears sufficient to accurately 

sense the vertical structure of mesoscale disturbances which 

support the CAT producing mechanism, i.e., unstable KelvinHelmholtz gravity waves created within statically stable shear 

layers (e.g., Lalas, et al., 1976; Lalas and Einaude, 1976; 

Lindzen and Rosenthal, 1976). Unfortunately, the existence 

and intensity of the turbulence can only be inferred from 

statistical considerations of a given stratified shear layer, 

under an assumption of Kelvin-Helmholtz (K-H) instabilities, as 

resolved by a J/J profile (e.g., Essenwanger and Reiter, 1969; 

2 

-- _____ ---.-.. ---_I_ .~ - 

. . ._ .: ” -: , -__ .:: ’ 
QL JUU 111. 

Keller, 1978). The expense of implementing an operational J/J 

system does not appear to be justified by its potential usefulness 

as a CAT detector. 

Currently, operational turbulence forecast products supplied 

to the av' ration community are generally considered to be highly 

imprecise and often inaccurate. In effect, the synoptic scale 

information obtained by the Rawinsonde (RW) observations acts 

merely to supplement airline pilot reports of turbulence encounters. 

There would appear to be more practical merit in using the existing j 

operational RW more effectively. Such an approach, the Diagnostic 

Richardson number Tendency (DRT) technique, involves the calculation of vertical wind shear, static stability (and hence the 

Richardson number), and their tendencies at the synoptic scale. 

The Richardson number is defined as 

-1 Zln8 Ri=-- 

P ap (1) 

with 

P = density, 

8 = potential temperature, 

P = atmosphere, and 

2 = horizontal wind velocity. 

The most recent formulation of the DRT technique is that of 

Oard (1974). Oard demonstrated the potential usefulness of the 

DRT technique by applying the technique to a previously welldocumented CAT event (Reed and Hardy, 1972). The indices produced 

by the Oard formulation of the DRT technique corresponded well 

to regions where documented aircraft encounters had occurred. i 

r 

In a more recent investigation (Dutton, 1979) indices 

corresponding to the DRT predictors were evaluated for their 

correlation to documented moderate to severe CAT encounters 

experienced during trans-Atlantic flights. Dutton found 
_ 

. ._. . f-he wind- The VI 

these indices correlated poorly to the encountered regions 

of CAT. However, Oard contended that very accurate values for 

the vertical wind velocity (~1 were necessary for the calculation of the DRT indices. Thus, Oard used vertical velocities 

determined from an iteration of the "Balance w equation" to 

obtain highly refined diagnostic w's. The vertical velocities 

used by Dutton (1979) were a forecast product of the lo-level 

model currently used by the British Meteorological Office 

(Burridge and Gadd, 1977). Prognosticated w's are generally 

considered to be inaccurate. Thus .' since accurate values of 

w are required to give reasonable DRT indices, the results of 

Dutton',s study are not too surprising. Dutton also calculated, 

from the prognostic model output, an index which he found to 

have forecast skill with respect to CAT. These regions will 

be compared with the DRT indices in the results of this report. 
THE DRT FORMULATION 

An index for quantitatively measuring the potential for 

CAT has been defined (Roach, 1970) using the time rate of change 

of the Richardson number (Ri) following the air motion. This 

is the so-called Diagnostic Ri Tendency (DRT). Ignoring small 

terms, it is 

where 

D 

Dtw = 

The initial 

encountered 

calculating 

application of Eq. (2) in a case study (Roach, 1970) 

problems which arose primarily due to the method of 

the first term on the 

shear tendency. This shortcoming 

using the thermal wind to specify 

(Oard, 1974). 

(2) 

a + a 

at + v-V +wap and 

DP 

Dt' 

right, the vertical wind 

was greatly remedied by 

the vertical shear tendency 

I 

5 

j 

._-. --. _ 

Following the method of Haltiner and Martin (1957), the 

vertical wind shear tendency is expressed in terms of isotherm 

packing related to frontogenesis, i.e., 

(3) 

The second term, the static stability tendency, is expressed as 

- (q-qg) = (i!q(g2&+ IEE+?22) (4) 
where 

R= 

C = F 

K= 

f = 

u= 

v= 

w = 

v= 

gas constant for dry air, 

specific heat at constant pressure, 

R/cp I 

coriolis parameter, 

eastward component of wind, 

northward component of wind, 

vertical component of wind, and 

"del" operator. 

The meteorological variables of u, v, and 8 can be obtained 

more-or-less directly from the synoptic data network. Details 

of the input data procedure will be discussed later. One 

possible weakness of this formulation, as was pointed out by 

Oard, is its reliance exclusively on the thermal wind approximation for the vertical wind shear. The vertical velocity, w, 

must be obtained indirectly. Since accurate values are required, 

the field is obtained through an iterative solution of the full 

balance w-equation using quasi-geostrophic w's for the initial 

guess field. 

The primary index for CAT used by Oard [1974), obtained 

from Eq. (21, is defined as 

t = !LnRi/(Q,) (5) cr 

which is an idealized measure of the time required by an 

atmospheric layer to reach the critical value of the Richardson 

number (Ricr) assuming the atmosphere continues the trend at 

the instance of observation. The value of Ricr for an infinitesimal layer has been found theoretically to be 0.25 (Miles 

and Howard, 1964). Since such a condition would never be 

resolved due to the synoptic scale nature of the Rawinsonde 

network input data, Oard arbitrarily chose Ricr = 1. Given 

the idealistic nature of tcr, the primary usefulness of tcr 

as a relative measure of CAT development probability remains 

intact. 

6 

-_-~. 

;-‘ 

-- --. _. 

_I , ‘. 
It can readily be seen from Eq. (5) that regions where 

Q << 0 and Ri is relatively small are regions which would be 

prime candidates for supporting CAT (i.e., give small to). 

Such a conclusion is supported by the results of the case study 

performed by Oard (19.74) and by the results of this work, which 

will be discussed later. While Eq. (5) takes into account 

vertical shear with respect to the time required for the atmosphere to reach a state favorable for the initiation of turbulence, 

a given value of t cr will not uniquely imply what its potential, 

intensity will be. Thus, based upon what is now known about the 

nature of the CAT mechanism, a modification of this index is 

also considered. Both the index used by Oard and this latter 

index are calculated and compared to documented CAT encounter 

reports. 

In the many empirical, theoretical, and numerical studies 

of CAT if any one element were to be considered to play a 

dominant role for both the initiation of turbulence and determining its intensity, it is the variation of the horizontal wind 

velocity with height, the vertical wind shear. It has been 

shown (Roach, 1970; Lindzen, 1974) that the vertical shear plays 

a dominant role as the energy source for K-H instabilities and 

that for a hyperbolic tangent shear profile, the rate of increase 

of K-H perturbation energy is proportional to the cube of the 

shear amplitude (Lalas and Einaudi, 1976; Haines, 1980), that is 

(6) 

where 

E = perturbation energy density, and 

H = height of unstable shear layer. 

The energy being fed into the amplifying K-H instabilities 

ultimately cascades into the turbulent subrange. An additional 

consideration is based upon the theoretical revelation that 

the unstable K-Hwave is carriedalong by the larger scale wind 

7 
field. Such behavior seems to be substantiated by recent 

observations (Gary, 1980). Hence, one is confronted with the 

situation where patches of CAT produced in some source region 

are propagated away and may be maintained as long as the largerscale shear is sufficiently large. Figure 1 schematically 

illustrates the CAT producing mechanism as it is currently 

believed to operate in the free atmosphere. 

From left to right, Figure 1 shows the creation of larger, 

synoptic-scale statically stable shear layers within which smallerscale discontinuities can exist. These shallower layers can 

then absorb gravity wave energy due to cumulus convection, 

terrain, or other random sources below through critical layer 

absorption (e.g., Hines, 1968; Bretherton, 1969; Jordan, 1972). 

Amplifying K-H instabilities may then form at these levels, 

ultimately "breaking" to form turbulence. That the spectra 

breaking K-H instabilities are consistent with the observed 

turbulent spectra was shown by Reiter (1969). Despite the 

consistent results shown by Haines (1980) in his K-H wave 

of 

energetical analyses of detailed J/J profiles, the fact that it 

is the RW system and not the J/J system which is currently 

operational compels the development of an index which is valid 

for the RW resolution. 

An index has been devised accounting for the role of 

vertical shear, as determined from RW observations, with 

respect to the potentially available energy for K-H instabilities. 

The rate of turbulent energy dissipation in an unstable shear 

layer was determined, based upon the synoptic-scale origin of 

the energy destined to become turbulent motion by Roach (1970), as 

(7) 

8 
W 

t 

I 

Wave 

Absorption 

by 

Shear Layer 

Net Wave 

t 

Energy Loss 

Rate 

Wave Production 

Rate 

J 

Waveave 

Overreflection 

v 

Amplifying 

+ Gravity Waves 

time (or space) ----) 

Ground 

Figure 1. The CAT mechanism. 
The value of E has units of energy per unit time. Upon inspectior 

it can be seen that Eqs. (6) and (7) are differing functions of 

IAfl. This is because they are actually measures of different 

quantities. Equation (6), in its complete form, includes factors 

involving the growth rate of a given unstable K-H wave obtained 

from a normal-mode instability analysis with respect to a given 

highly idealized vertical wind profile. Thus, Eq. (6) represents 

the amplification rate of a single K-H mode. Alternatively, 

Eq- (7) is essentially a measure of the "rate of accumulation" 

of the energy potentially available to K-H instabilities within 

a given synoptic-scale volume of atmosphere. As well, since 

Eq- (7) takes into account the contribution of the surrounding 

atmosphere (via a), it is a much more physically realistic model 

than the highly idealized model which is the source of Eq. (6). 

+2 In practice a value of /Avl is calculated by 

where 

AP = 100 mb. 

A modified form of Eq. (7) was used as an index of CAT potential, 

*CAT = lAv'12(@ + @) (9) 

The constant @ is included to take into account potentially 

turbulent situations where I, is small and oscillating around 

zero. Thus r ICAT could be large (due to large ;Av'j), locally 

supporting turbulent layers where Ri -+ l/4. The value of Q 

is chosen to be -2.5 x 10m6s -1 which is small enough not to 

affect significantly large positive values of @. ICAT is set 

equal to zero for all I CAT < 0. It might just as well have 

been kept negative and used as an indicator of "low-risk" 

areas. However, to provide clearer graphical display, the 

former choice is made. 

10 
The turbulent energy dissipation rate, c, defined by 

Eq- (7) is strictly valid only within the relatively shallow 

statically-stable shear layer itself where Ri approaches l/4 

and unstable K-H waves can exist. Thus, the use of Eq. (7) 

would require the same vertically detailed data as does Eq. (6) 

which is available only via special data systems such as the 

J/J system. The modified expression Eq. (9) is applied to 

fixed synoptic-scale vertical increments (AP = 100 mb) resolved 

by the RW system which may contain these shallow layers of 

unstable K-H waves. The situation of relatively shallow CAT 

layers existing withiLl synoptic scale layers of the atmosphere 

was discussed by Roach (1970) who suggested an attempt should 

be made to statistically relate mesoscale Richardson numbers 

for which Eq. (7) is valid to synoptic-scale Richardson numbers. 

This analysis was ultimately performed using conjunctive RW 

and J/J balloon profiles (Keller, 1978). Unfortunately, neither 

the Richardson numbers nor the vertical shears proved to be 

consistently related. Hence, the factor l/24 in Eq. (7) is 

dropped and Eq. (9) becomes a relative measure of CAT potential, 

with an implicit understanding that it should be interpreted 

in terms of the probability for the existence of one or more 

CAT layers embedded somewhere within the given synoptic layer. 

The units remain as energy per unit time, which can be interpreted 

physically as a rate of generation (or as stated earlier, 

"rate of accumulation") of the energy potentially available 

to K-H instabilities. 

An assumption whose effects remain as in the Oard study 

is that of the adiabatic assumption 

DRn@/Dt = 0 (10) 

used in the derivation of Eq. (4) the static stability tendency. 

This assumption is valid in the absence of radiation effects and 

in relatively dry air. Since the main interest is with 

respect to turbulence in clear air, Eq. (10) would seem to 

be a reasonable approximation. 

11 
GENERAL NATURE OF INPUT DATA 

In this case study Oard graphically generated from the 

raw RW ascent data a grid consisting of 15,X 15 points in the 

horizontal at 10 constant pressure levels. This corresponded 

to a horizontal resolution of 250 km and 100 mb vertical 

resolution. Each data-set required about two to three months 

of effort to construct. It is not surprising, then, that Oard 

investigated only one case of CAT. 

Such a methodology for generating input data-sets is 

obviously impractical for case study applications of the DRT 

approach. In this study, a DRT data preprocessor was developed 

which uses the archived National Meteorological Center (NMC) 

formatted data. A 15 X 15 point grid, consisted.of a subset 

of the NMC 1977 point octagonal grid, can be specified for 

virtually any area of interest in the Northern hemisphere 

(see Figure 2). This corresponds to a horizontal resolution 

of approximately 360 km for cases that are considered prior to 

1 July 1978. Starting at this date, the horizontal resolution 

has doubled to approximately 180 km. 

The DRT data preprocessor provides to the DRT model, for 

an arbitrarily specified geographic region, a "data-net" of 

horizontal wind velocity, 2, pressure height, z, and temperature, 

T, at 19 constant pressure surfaces from 100 to 1000 mb. This 

corresponds to vertical resolution of 50 mb--twice that of the 

input data-sets of Oard. The NMC gridded data, obtained from 

archived NMC data tapes, exists at the standard pressure levels: 

100, 150, 200, 250, 300, 400, 500, 700, and 850 mb. Only 2 and 

z are available at 1000 mb. This fact requires interpolated 

data fields to be created at 350, 450, 550, 600, 650, 750, 800, 

900, and 950 mb as well as extrapolation for T at 1000 mb. 

This task is performed by simple linear interpolation (Bevington, 

1969). 

12 
. 1 

. 

. 

0” . . . . . . . . . 

. . . . . . . . . . . . . . . . 

Figure 2. DRT horizontal grid as subset of the NMC octagonal 

hemispheric grid. 

input data grid. 

The outer square encompasses the 

output grid. 

The inner square encompasses the 

13 
The scalar T and z fields can be interpolated in this 

fashion with relative impunity. However, strictly speaking, 

the horizontal wind velocity, v', should not be handled simply 

by interpolating its components u and v (e.g., Luers and 

MacArthur, 1975; Schaefer and Doswell, III, 1979). Accurate 

interpolation would require the decomposition of v' into its 

stream-function and velocity potential which could then be 

interpolated and finally used to obtain 2 at the necessary 

levels (Shukla and Saha, 1974). The current version of the 

DRT data preprocessor uses linearly interpolated values of G 

based upon the contention that the results obtained would be 

changed only in minor detail. The reasons are as follows: 

(a) the vertical motion (w) field can be obtained iteratively, 

requiring only a rough initial guess field, (b) the region 

of greatest interest is generally above 350 mb where interpolation of G is not required, and (c) a sensitivity analysis 

was performed (Keller, et al., 1979) which indicates that the 

important features of areas of high CAT potential are essentially 

maintained even for significantly disturbed wind fields. 

Figure 3 shows the DRT case study methodology. The NMC 

formatted data fields for the 

documented CAT encounters are 

As detailed earlier, the data 

15 x 15 x 19 "data-net" of $, 

map time corresponding to pilot 

addressed by a data preprocessor. 

preprocessor supplies the required 

Z‘ and T to the DRT module which 

DRT module, formulated in Section 2, 

primary task performed by the DRT 

calculates t cr and ICAT. The 

is outlined in Figure 4. The 

module is the calculation of 6, the Ri tendency. The values 

of 1 a$/aPI and Ri, which are factors in the indices t and cr 

I CAT' are calculated directly from the input data. The Ri 

tendency, 0, is a function of G, T, and w N’ the balance vertical 

velocity. The Balance-omega, tiN, is obtained iteratively 

using the quasi-geostrophic omega (cl, qg) for the initial guess 

(w,) . 

Consider Figures 5, 6, and 7. Figure 5 compares the 

magnitude of unperturbed and the randomly perturbed eastward 

14 
INPUT DATA: 

NMC 1977 PT. GRID 

DOCUMENTED OR 

SUSPECTED 

CAT EVENT 

* 

I I Horizontal Resolution: 75 x 15 

DATA PROCESSOR (~3.20 km) 

Vertical Resolution: 19 levels 

(AP = 50 mb) 

‘I 

ARBIT;;tW;L;E~RIDDED -+ 

V = Horizontal velocity 

1 

z = Pressure-height 

T= Temperature 

DRT MODULE 

"r' 

DIAGNOSTIC CAT 

PARAMETERS 

t cr"CAT 

7 

SUBJECTIVE 

ANALYSIS 

OBJECTIVE, STATISTICAL 

ANALYSIS 

SYNTHESIS AND 

EVALUATION 

Figure 3. DRT case study methodology. 

15 
Q = WC, T, w,) 

Wn+l = Wn+l (T, $1 x w n' n' w/at) 

w,- = wqg = olGg, z, as) 

s 4 = YQZ) 

0 

S 

= constant reference atmosphere static stability 

$ = vJ(3, 

xn = xnhn) 

w/at - fbN, Q, x,) 

t cr = RnRi(@)-' 

'CAT = I&l2 (@+$I) 

(I = 0.25 x 10 -5s-1 

Figure 4. DRT Module Formulation 

16 
Perturbed Eastward wind component (m*s -1 ) 

Figure 5. Scatter diagram comparing perturbed and unperturbed 

eastward wind components for the 500 mb pressure 

level. 

17 
Figure 6. Vertical motion field for the (A) unperturbed 

(B) perturbed wind fields at 500 mb for OZ 20 

ruary 1970. Units are in 10-1Nt=m'2as'1. 

and 

Feb18 
Figure 7. Time to reach Ri,r for the (A) unperturbed and (B) 

perturbed wind fields at 500 mb for 02 20 February 

1970 (x10%. 

19 
wind component that was used to assess the sensitivity of the 

DRT technique.to errors in the horizontal wind field for the 

500 mb surface of OZ 20 February 1970. It can be seen that 

this component was perturbed as much as 15 percent at the 

,500 mb level. Other pressure-levels were similarly perturbed. 

Figures 6 and 7 show, respectively, the corresponding comparisons 

for the derived vertical motion field and tcr. The hatching 

highlights the relative maxima of these fields. Note that 

except for differences in detail (notably over New England 

for to) the general patterns remain the same. The significance 

of these results is that the DRT algorithm is relatively insensitive to reasonably small errors or inconsistencies in the 

input horizontal wind field. Since 0 is the primary factor 

calculated by the DRT algorithm, ICAT must be similarly stable. 

One final constraint was with respect to the vertical 

stratification. The vertical interpolation of temperature in 

the lower part of the atmosphere occasionally would result in 

an artificial superadiabatic lapse rate. This relatively 

infrequent event (generally occuring at no more than 10 percent 

of the points below 500 mb) results in a computational instability in the DRT algorithm. Physically, this corresponds to 

a negative Ri and thus a convectively unstable stratification-- 

regardless of the degree of vertical wind shear. Turbulence 

associated with such layers is more characteristic of the 

planetary boundary layer. Since CAT of significant intensity 

occurs exclusively within statically stable shear layers, 

the main focus of consideration will be restricted to atmospheric layers which possess vertical stratification far 

greater than the adiabatic lapse rate. Points at which this 

problem occurred were simply adjusted to the adiabatic lapse 

rate. 

20 
RESULTS 

In this study the DRT technique was applied to four case 

studies of pilot encounters with CAT. The results of the DRT 

calculations are compared to the distribution of actual encounters 

documented as part of a study performed by the British Central 

Forecast Office (CFO) published by Dutton (1979). The dates are 

9 March 1976, 18 March 1976, and 24 March 1976 over the European 

sector of the trans-Atlantic route. 

Figure 8 shows, for OZ 9 March 1976 at the 300 mb level, 

lines of constant tCr, in units of lo4 seconds. Also shown 

are the pilot encounters (described as moderate to severe) 

reported near this level. The levels were chosen to show the 

fields of tcr intersecting regions of relative maxima of CAT 

encounters. Because there is considerable subjectivity about 

the reported intensity, only those incidents described as at 

least "moderate" are included. The encounters are illustrated 

as follows: The location where CAT was initially reported is 

indicated by an "Encounter Letter." The location where turbulence ended is indicated by the head of an arrow. The distance 

over which turbulence occurred is indicated by the length of the 

arrow. The corresponding temporal extent is-listed in the 

table to the right of the figure. Figure 9 illustrates the same 

information for 122. Additionally, the prognosticated regions 

of high CAT probability calculated by Dutton (1979) are shown 

by a dot-dash line and the channel of highest aircraft density 

by a dotted line. Dutton performed calculations for only the 

122 map times. 

Figures 8 and 9 reveal that most of the moderate to severe 

turbulence encounters documented generally possess the characteristics usually attributed to CAT; that is, a spatial scale 

usually less than 100 km and temporal scales usually less than 

one hour. The number of encounters at the OZ map time is small 

and no skill can be ascertained from Figure 8. Figure 9, however, 

21 
OZ 9 March 1976 

t cro104 seconds 

300 mb (27 to 35,000 ft.) 

ENCOUNTER/TIME 

A 0315-03322 

B 0411-0424 

C 0552-0554 

D 0610-0621 

Figure 8. Time to reach Ri and pilot encounters with moderate 

to severe turbulggce at 300 mb for OZ 9 March 1976. 

22 
t cr.1O4 seconds 

300 mb (27 to 31,000 ft.) 

ENCOUNTER/TIME .--_____---__ 

A 

c" 

D 

E 

F 

G 

H 

: 

K 

L 

M 

1033-10512 

1244-1246 

1309-1310 

1317-1324 

1331-1340 

1334-1350 

1359-1403 

1439- 1454 

1446- 1459 

1509-1528 

1634- 1637 

1704-1715 

1705-1733 

122 9 March 1976 

Figure 9. Same as Figure 8, except for 122. The dot-dash line 

shows the forecast regions of CAT (Dutton, 1979); 

The dotted line shows the "channel" of highest aircraft density. 

23 
seems to show better results. This trend generally continues 

for subsequent cases, as will be shown. An alternative way to 

view the potential for CAT can be seen in Figure 10. A vertical 

cross-section of tcr intersecting a hypothetical flight path at 

300 mb from Rome to New York is shown. It can be seen that a 

pilot assigned this flight-path can expect to pass through two 

regions of relatively high CAT potential, over western Europe 

and west of Ireland. 

It becomes immediately evident that the turbulence 

probability index supplied by the DRT formulation is much more 

specific and detailed than the current CAT forecast products 

supplied by the National Weather Service (NWS). For example, 

from an operational viewpoint the information shown in Figures 8 

and 9 would be of even greater value. While it may not be 

practical to circumvent the regions of higher CAT potential, 

a picture of the vertical structure, as shown in Figure 10, 

would suggest to the pilot that the turbulence potential should 

be lessened by dropping only several thousand feet in altitude. 

Figures 11 through 16 show the fields of to, and CAT 

encounter records as well as the Dutton CAT forecast regions 

for the remaining three cases. In addition to information 

shown in previous figures, encounter letters which are circled 

correspond to reports of "significantly greater than moderate" 

turbulence. 

Some of the discrepancy can perhaps be explained in 

considering Figures 17 and 18. These figures consist of vertic;,l 

cross sections of t cr along the 50"N meridian for OZ and 122 

18 March 1976. The CAT encounters which occurred along this 

corridor two to seven hours after the OZ map time are indicated. 

The general eastward and downward progression of the turbulence 

reports appears consistent with the similar propagation in the 

minimum in tcr initially at 11.5 km. One other feature,.which 

is important with respect to application of the DRT technique 

to CAT forecasting, is the considerable degree of conservatism 

24 
Feet L”, 

45,000 

40,000 

35,000 

30,000 

25,000 

a 15,000 

10,000 

5,000 

'14 

,13 

*12 

11 

,lO 

9 

8 

5 

4 

’ 3 

* 2 

- NEW YORK IRELAND PARIS ROME 

1 L J 4 3 b 7 8 9 

t cr’1O4 SECONDS OZ 9 March 1976 

Figure 10. Hypothetical flight-path from Rome to New York at the 300 mb surface. The 

flight-path is shown against a vertical cross section of to. 
. . . . . . - -.......- _ ..__. _- .____ ~ -- 

t cr*104 seconds 

300 mb (27 to 31,000 ft.) 

ENCOUNTER/TIME 

A 0530-05452 

B 0717-0729 

C 0739-0800 

D 0807-0821 

E 0821-0840 

F 0838-0846 

G 0840-0850 

H 0947- 1004 

I 1010-1013 

J 1021-1037 

K 1024-1038 

L 1102-1105 

OZ 12 March 1976 

Figure 11. Same as Figure 8, except for map time indicated. 

26 
t c,a1O4 seconds 

300 mb (27 to 31,000 ft.) 

ENCOUNTER/TIME 

A 

B 

C 

D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

! 

; 

R 

S 

1132-11352 

1141-1152 

1154-1210 

1157-1201 

1310-1314 

1331-1351 

1437-1519 

1455-1509 

1502-i510 

1536-1548 

1555-1605 

1610-1612 

1710-1727 

1757-1811 

1828-1830 

1838-1850 

1840-1845 

1856-1902 

1935-1936 

122 12 March 1976 

Figure 12. Same as Figure 9, except for map time indicated. 

27 
t cr'1O4 seconds 

250 mb (31 to 35,000 ft. 

ENCOUNTER/TIME 

A 0220-02402 

B 0300-0320 

C 0435-0504 

D 0448-0500 

E 0448-0521 

F 0511-0520 

G 0522-0550 

H 0537-0544 

I 0640-0705 

J 0650-0654 

K 0753-0755 

L 0829-0844 

M 0917-0920 

Fi 0935-0940 1009-1014 

; 1044-1047 1031-1041 

OZ 18 March 1976 

Figure 13. Same as Figure 8, except for map time and pressurelevel indicated. 

28 
t cr*1O4 seconds 

250 mb (31 to 35,000 ft.) 

_---________ ENCOUNTER/TIME 

A 1044-10472 

B 1107-1131 

C 1237- 1306 

D 1325-1335 

E 1351-1434 

F 1405-1419 

G 1428- 1458 

H 1444-1502 

I 1456-1521 

J 1509-1511 

K 1513-1530 

L 1537-1545 

M 1545-1621 

i 1559-1659 1640-1650 

: 1708-1720 1648-1703 

122 18 March 1976 

Figure 14. Same as Figure 9, except for map time and pressurelevel indicated. 

29 
t crs1O4 seconds 

350 mb (23 to 28,000 

ENCOUNTER/TIME 

A 0725-07302 

B 0745-0754 

C 0845-0855 

D 0852-0858 

ft.) 

OZ 24 March 1976 

Figure 15. Same as Figure 8, except for map time indicated. 
t c,S104 seconds 

350 mb (23 to 28,000 ft.) 

ENCOUNTER/TIME _---I 

A 0955-10182 

F 1000-1010 1000-1005 

E 1100-1105 1100-1111 

G' 

i114-1124 

1159-1205 

H 1206-1216 

: 1334-1335 1357-1404 

r 1437-1442 1534-1540 

M 1626-1629 

N 1753-1756 

N 1805-1809 

P 1815-1826 

Figure 16. Same as for Figure 9, except for map time and 

pressure -level indicated. 

31 
Feet km 

'6 

* 5 

'4 

3 

. 1 

I I I I I 1 I . I I I L I I I 

I I I I I I 1 I 

35w 28W 21w 

Longitude 

t cr*104 seconds OZ 18 March 1976 

ENCOUNTER/TIME 

A 0220-02402 

B 0300-0320 

D 0448-0521 

I4 0537-0544 

: 0650-0654 0640-0705 

Figure 17. Vertical cross section of tcr along 50°N parallel. 
w 

W 

Feet km 

45,000 

40,000 

35,000 

11 

10 

30,000 

25,000 

44 

LO,000 

J *Ic, 

z 15,000 

10,000 

5,000 

5 

3 

2 

’ 1 

L 

I 1 I I a I I 

I I I I I I I 

35w 28W 21w 

Longitude 

t c,.1O4 seconds 121. 18 March 1976 

Figure 18. Same as Figure 17, except 12 hours later. 
in the fields of tcr. Indeed, it may be possible, for relatively 

short forecast periods, to construct a prognostic scheme, analogous 

to the absolute vorticity conserving barotropic models used as 

early weather prediction models, where tcr would simply be 

advected by the three-dimensional wind field. 

As discussed in the second section, there is reason to 

believe that including the magnitude of the vertical shear in 

a DRT-based CAT potential index would improve its performance. 

Hence, yet another way to improve the agreement between CAT 

encounters and their proximity to source regions might be 

use of a more physically realistic index. 

Consider Figure 19. This figure shows the pressureheights (solid lines) and isotachs (dashed lines) for 02 

18 March 1976 at 250 mb. The isotachs are in units of m-s -1 

and the heights in decameters. Shaded areas delineate regions 

of highest wind velocity. The direction will generally parallel 

the lines of constant height. Comparison with Figure 13 reveals 

that a minimum in tcr exists just west of 25 W longitude. This 

center is mainly north and west of a very strong wind maximum 

(60 m-s-1). How this translates into the CAT potential index 

defined by Eq. (9) is seen in Figures 20 and 21. The index ICAT 

is constructed so that larger values correspond to higher 

potential. Although a source region, related to the minimum in 

t cr' is shown positioned closer to the reported encounters, 

Figure 21 reveals an even more interesting event. It can be 

seen that between OZ and 122, the area of I CAT intensified at 

this level. Observational confirmation of this intensification 

is strongly suggested by the report of "considerably greater 

than moderate turbulence" designated as encounter "J" which had 

the latest time for the initiation of its encounter at this 

latitude. 

Further analysis is needed to determine whether this 

apparent intensification in I CAT implies that this index is not 

as conservative as is t cr. Indeed, the degree of conservatism 

34 
OZ 18 March 1976 

Figure 19. Pressure heights, in decameters (solid lines), and 

Isotachs (m-s-l ) (dashed lines), at 250 mb. 

35 
'CAT( cm**s -2),-l .' 

250 mb (31 to 35,000 ft.) 

ENCOUNTER/TIME 

A 0220-02402 

B 0300-0320 

C 0435-0504 

D 0448-0500 

E 0448-0521 

F 0511-0520 

G 0522-0550 

H 0537-0544 

: 0640-0705 0650-0654 

K 0753-0755 

L 0829-0844 

M 0917-0920 

i 0935-0940 1009-1014 

; 1044-1047 1031-1041 

OZ 18 March 1976 

Figure 20. CAT potential index for the 4000 foot interval 

centered at 250 mb, and pilot encounters with 

moderate to severe turbulence. Maximum values 

of I CAT are indicated by underlining. 

36 
ICAT (cm2as -2)s-1 

250 mb (27 to 31,000 ft.) 

ENCOUNTER/TIME 

1044-10472 

1107-1131 

1237-1306 

1325-1335 

1351-1434 

1405-1419 

1428- 1458 

1444-1502 

1456-1521 

1509-1511 

1513-1530 

1537-1545 

1545-1621 

1559-1659 

1640- 1650 

1640-1650 

1648-1703 

1708- 1720 

122 18 March 1976 

Figure 21. Same as Figure 20, except for map time and pressurelevel indicated. 

37 
of t cr itself is speculative at 

to be considered in this regard 

this time. Important elements 

include the relative roles 

played by advective versus the actual amplification processes. 

Figures 22 through 27 show the fields of ICAT and CAT 

encounters for the other three cases. In general this index 

appears to be superior to t cr for specifying potential regions 

of CAT. In particular, that for 02 9 March 1976 (Figure 22) 

reveals clearly why there were few encounters of CAT during 

this time. While Figure 8 indicates very small values of tcr 

over Europe for this time, the corresponding I CAT field shows 

generally small values of ICAT. Hence it can be concluded that 

although CAT may have occurred associated with the small tcr 

values, it was of weak intensity. 

A more quantitative comparison is shown in Figure 28, 

which summarizes the CAT encounters in relation to the "centers 

of action" responsible for their existence. For example, 85 

percent of the encounters were reported in regions where ICAT > 0, 

57 percent for ICAT > 2(cm2.s -2) s-l. Similarly, 73 percent of 

the encounters occurred when tcr < 32 x 104's . It can be seen 

that ICAT is apparently a more accurate indicator of CAT 

potential. 

38 
'CAT (cm23 )S -2 -1 

300 mb (27 to 31,000 ft.) 

ENCOUNTER/TIME 

A 0315-03322 

B 0411-0424 

k 

0552-0554 

0610-0621 

OZ 9 March 1976 

Figure 22. CAT potential index, ICA 

with moderate to severe T' 

and pilot encounters 

0.Z 9 March 1976. 

urbulence at 300 mb for 

39 
300 mb (27 to 31,000 ft.) 

ENCOUNTER/TIME 

A 1033-10512 

B 1244-1246 

C 1309-1310 

F 1317-1324 1331- 1340 

F 1334-1350 

G 1359-1403 

H 1439- 1454 

I 1446- 1459 

J 1509-1528 

K 1634-1637 

k 1705-41715 1733 

122 9 March 1976 

Figure 23. Same as Figure 22, except for 122 

40 
'CAT (cm2s5-2)s-1 

300 mb (27 to 31,000 ft.) 

ENCOUNTER/TIME 

A 

F 

E 

F 

G 

H 

I 

J 

K 

L 

0530-05452 

0717-0729 

0739-0800 

0807-0821 

0821-0840 

0838-0846 

0840-0850 

0947- 1004 

1010-1013 

1021-1037 

1024-1038 

1102-1105 

OZ 12 March 1976 

Figure 24. Same as Figure 22, except for map time indicated. 

41 
'CAT (CJA )s -2 -1 

350 ml: (27 to 37,000 ft.) 

ENCOUNTER/TIME 

A 1132-11352 

B 1141-1152 

C 1154-1210 

E" 1310-1314 1157-1201 

F 1331-1351 

G 1437-1519 

H 1455- 1509 

I 1502-1510 

J 1536-1548 

K 1555-1605 

L 1610-1612 

M 1710-1727 

i 1828-1830 1757-1811 

p 1856-1902 1840-1845 1838-1850 

S 1935- 1936 

122 12 March 1976 

Figure 25. Same as Figure 23, except for map time indicated. 

42 
'CAT (cm'.* )s -2 -1 

350 mb (23 to 28,000 ft.) 

ENCOUNTER/TIME --P 

A 0725-07302 

B 0745-0754 

C 0845-0855 

D 0852-0858 

OZ 24 March 1976 

Figure 26. Same as Figure 22, except for map time indicated. 

43 
'CAT ( cm'*5 )s 12 -1 

350 mb (23 to 28 ,000 ft.) 

ENCOUNTER/TIME 

A 0955-10182 

B 1000-l 005 

C 1000-1010 

II 1100-1105 

E 1100-1111 

F 1114-1124 

G 1159-1205 

H 1206-1216 

I 1334-1335 

J 1357-l 404 

K 1437-1442 

L 1534-l 540 

M 1626-1629 

N 1753-1756 

0 1805-1809 

P 1815-1826 

Figure 27. Same as for Figure 23, except for map time indicated. 

44 
- 

0.80 

0.20 

0 

/ 

/ 

/ 

/ 

/ 

/ 

'CAT '16 ‘8 >4 / ‘2 >O 2 -2 -1 

I (cm 4 )s 

--- I 

'cr < 4 ~8 (16 < 32 x10 4 s 

DRT PARAMETER 

Figure 28. Fraction of reported encounters as a function of CAT 

potential parameters tcr and ICAT for all cases. 
SUMMARY AND DISCUSSION 

The results of several case studies of clear air.turbulence 

(CAT) using the Diagnostic Richardson number Tendency (DRT) 

formulation have been highlighted. The performance of this 

technique in resolving regions of documented CAT encounters was 

encouraging. An attractive aspect of its operational adaptability 

is that the input data can be supplied by the currently 

Rawinsonde system. 

operational 

Two indices are calculated deterministically from the 

synoptic-scale changes in static stability and vertical wind 

shear conducive for supporting meso-scale CAT layers. The 

products generated by the DRT formulation highlight specific 

regions of the atmosphere which could be interpreted operationally in terms of a CAT-encounter probability. The apparent 

improvement inherent in the index I CAT can be argued in the course 

of comparing the tcr fields for the 18 March 1976 case with the 

'CAT fields for this case. The low tcr region under consideration 

shows an eastward and southward progression at this level, but 

the values of tcr actually increased (Figures 13 and 14). This 

might lead to an interpretation of decreasing CAT activity. 

The fields of ICAT illustrated in Figures 20 and 21 imply both 

a south-eastward propagation and general increase in CAT activity 

and intensity. Pilot reports of CAT suggest that the patches 

of CAT generated by this region increased in number and intensity. 

Of course, further detailed investigations are required to 

establish the definite advantage of ICAT. 

The availability of archived data of twice the horizontal 

resolution used in this study suggests that the DRT technique 

be tested on post-July 1978 encounters. This could be done in 

conjunction with an improved method for vertical interpolation 

of the horizontal wind field. A further refinement would be 

to include a parameterization for the effects of significant 

terrain at the lower boundary. Thus, the generally higher 

46 
I 

frequency ,of turbulence observed above and downwind of 

mountainous regions could be accounted for. 

Pursuing the DRT technique further through .more detailed 

quantitative analyses of CAT case studies and refinements in 

the algorithm seems justifiable. Such a conclusion is based 

upon the encouraging results presented here as well as the 

motivations related to passenger comfort and safety, and 

reduction in fuel consumption. 

47 
REFERENCES 

Bevington, P. R., 1969: Data Reduction and Error Analysis for 

the Physical Sciences, New York, McGraw-Hill, 259-265. 

Bretherton, F. P., 1969: "Momentum Transport by Gravity Waves," 

Quart. J. Roy. Meteor. Sot., 95(404), 213-243. - 

Burridge, D. M., and A. J. Gadd, 1977: "The Meteorological Office 

Operational lo-Level Numerical Weather Prediction Model," 

Sci. paperr Meteor. Off., No. 34. 

Camp, D. W., and F. E. Caplan, 1969: "High Resolution 

Balloonborne Temperature Sensor," J. Appl. Meteor.; g(l), 

259-166. 

Camp, D. W., and W. Frost, 1980: "Fourth Annual Workshop on 

Meteorological and Environmental Inputs to Aviation 

Systems," Rebecca A. Durocher, editor. FAA report in 

preparation. 

Dutton, M. J. O., 1979: "Performance of Conventional Operational 

Forecasts of Clear-Air Turbulence During the 1976 Turbulence 

Survey," The Meteorological Magazine, 108, 61-76. 

Essenwanger, 0. M., and E. R. Reiter, 1969: "Power Spectrum, 

Structure Function, Vertical Wind Shear, and Turbulence in 

Troposphere and Stratosphere," Arch. Met. Geoph. Biokl., 

Ser. A., 18, 17-24. 

Fichtl, G. H., 1971: "The Responses of Rising or Falling 

Spherical Wind Sensors to Atmospheric Wind Perturbations," 

J. Appl. Meteor., 10(6), 1275-1284. - 

Gary, B. L., 1980: JPL-NASA, personal COmmUniCatiOn. 

Haines, P., 1980: Report in progress. 

Haltiner, G. L., and F. L. Martin, 1957: Dynamical and Physical ._._- 

Meteorology, New York, McGraw-Hill, 287-289. 

Hines, C. O., 1968: "Some Consequences of Gravity-Wave Critical 

Layers in the Upper Atmosphere," J. Atmos. and Terres. 

Phys., 30, 837-843. 

Jordan, A. R., 1972: "Atmospheric Gravity Waves from Winds and 

Storms," J. Atm. Sci. 29(3), 445-456. 

48 
Keller, J. L., 1978: "Prediction and Monitoring of Clear Air 

Turbulence: An Evaluation of the Applicability of the 

Rawinsonde System," NASA CR-3072. 

Keller, J. L., P. A. Haines, and J. K. Luers, November 1979: "The 

Diagnostic Richardson Number Tendency as an Index of 

Clear Air Turbulence Potential," WMO Technical 

Conference on Aviation Meteorology, TECAM-32, Geneva, 

Switzerland. 

Lalas, D. P., F. Einaudi, and D. Fua, 1976: "The Destabilizing 

Effect of the Ground on Kelvin-Helmholtz Waves in the 

Atmosphere," J. Atm. Sci., 33(l), 59-69. - 

Lalas, D. P., and F. Einaudi, 1976: "On the Characteristics of 

Gravity Waves Generated by Atmospheric Shear Layers," J. 

Atm. Sci., 33(7), 1248-1259. - 

- 

Lindzen, R. S., 1974: "Stability of a Helmholtz Velocity Profile 

in a Continuously Stratified, Infinite Boussinesq of 

Fluid-Applications to Clear Air Turbulence," J. Atm. Sci., - 

31(6), 1507-1514. 

Lindzen, R. S., and A. J. Rosenthal, 1976: "On the Instability 

of Helmholtz Velocity Profiles in Stably Stratified Fluids 

When a Lower Boundary is Present," J. Geophys. Res., 81(g), - 

1561-1571. 

Luers, J. K., and C. D. MacArthur, 1975: "Optimum Radars and 

Filters for the Passive Sphere System," NASA-CR-111952. 

Miles, J. W., and L. N. Howard, 1964: "Note on a Heteorogeneous 

Shear Flow," J. Fluid Mech., 20, 331-336. 

Oard, M. J., 1974: "Application of a Diagnostic Richardson 

Number Tendency to a Case Study of CAT," J. Appl. Meteor., 

13(7), 771-777. 

Pinus, N. Z., 1964: "Atmospheric Turbulence," NASA TT F-246. 

Range Commanders Council Document 110-77, 1977: Committee of the 

Meteorological Group, Secretariat Range Commanders Council, 

White Sands Missile Range, New Mexico, 19pp. 

Reed, R. J., and K. R. Hardy, 1972: "A Case Study of Persistent 

Intense Clear Air Turbulence in an Upper Level Frontal Zone," 

J. APP~. Meteor., 11(3), 541-549. - 

Reiter, E. R., 1969: "The Nature of Clear Air Turbulence: A 

Review;" in Clear Air Turbulence and Its Detection, Y. Pao 

and A. Goldburg, editors, New York, Plenum Press, 7-33. 

49 
Roach, W. T., 197.0: "On the Influence of Synoptic Development on 

the Production of High Level Turbulence," Quart. J. Roy. 

Meteor. Sot., 96, 413-429. - 

Schaeffer, J. T., and C. A. Doswell III, 1979: "On the 

Interpolation of a Vector Field," Mon. Wea. Rev., 107(4), 

458-476. 

Shukla, J., and K. R. Saha, 1974: "Computation of Non-divergent 

Streamfunction and Irrotational Velocity Potential from the 

Observed Winds," Mon. Wea. Rev., 102(6), 419-425. 

50 
1. REPORT NO. 2. GOVERNMENT ACCESSION NO. 3. RECIPIENT’S CiTiGO.-- 

NASA CR-3378 

4. TITLE AND SUBTITLE 5. REPoRT DATE 

Case Studies of Clear Air Turbulence Using the January 1981 

Diagnostic Richardson Number Tendency Formulation 6. PERFORMlNG ORGANIZATION CODE 

7. AUTHOR(S) 6. PERFORMING ORGANIZATION REPOR I’ 

11- John L. Keller and Patrick A. Haines UDR TR .:.. --- 80-97 

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. WORK UNIT, NO. 

University of Dayton 

M-324 

11. CONTRACT OR GRANT NO. 

Research Institute NAS8-32111 

Dayton, Ohio 45469 ,S. TYPE OF REPORT & PERIOD COVEREI 

12. SPONSORING AGENCY NAME AND ADDRESS 

Contractor Report 

National Aeronautics and Space Administration 

Washington, D.C. 20546 14. SPONSORING AGENCY CODE 

IS. SUPPLEMENTARY NOTES 

Marshall Technical Monitor: Dennis W. Camp 

Final Report 

16. ABSTRACT 

The results of four case studies of Clear Air Turbulence (CAT) using the 

Diagnostic Richardson Number Tendency (DRT) formulation are highlighted. The performance of this technique in resolving regions of documented CAT encounters is 

encouraging. Its operational adaptability appears particularly attractive in that 

input data can be supplied by the currently operational Rawinsonde system. 

Two CAT indices are calculated deterministically, sensing synoptic-scale changes 

in static stability and vertical wind,shear conducive for supporting mesoscale CAT 

layers. These two indices reveal volumes of the troposphere which act as "source 

regions" for patches of CAT. The first, the time to reach the critical Richardson 

number necessary for the initiation of turbulent conditions, was first used by 

Oard (1974). A second index is devised which is an adaptation of Roach's (1970) 

work relating synoptic-scale and mesoscale energetical coupling and gives more 

information on the relative intensity of these "source regions." The output from 

the DRT computer module highlights specific regions of the atmosphere which can be 

interpreted operationally in terms of a CAT encounter probability. 

7. KEY WORDS 

Clear Air Turbulence 

__.~... 

16. DISTRIBUTION ;TATEMENT 

Unclassified - Unlimited 

3. SECURITY CLASSIF. (of thla ce~oct, 

Unclassified 

Subject Category 47 

20. SECURITY CLASSIF. (of this pegs) 

Unclassified 

For de by National Technical Information Service. Springfield. ViIgbli.4 22161 

NASA-Langley, 1981