&EPA
United States
Environmental Protection
Agency
Environmental Sciences Research EPA 600/4-79-034
Laboratory May 1 979
Research Triangle Park NC 27711
Research and Development
Diffusion
Coefficients from
Metrac System
Turbulence
Measurements
-------
RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into nine series. These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The nine series are.
1. Environmental Health Effects Research
2 Environmental Protection Technology
3 Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
6. Scientific and Technical Assessment Reports (STAR)
7 Interagency Energy-Environment Research and Development
8. "Special" Reports
9. Miscellaneous Reports
This report has been assigned to the ENVIRONMENTAL MONITORING series.
This series describes research conducted to develop new or improved methods
and instrumentation for the identification and quantification of environmental
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studies to determine the ambient concentrations of pollutants in the environment
and/or the variance of pollutants as a function of time or meteorological factors.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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EPA-600/4-79-034
May 1979
DIFFUSION COEFFICIENTS FROM
METRAC SYSTEM TURBULENCE MEASUREMENTS
by
W. H. Jasperson
Control Data Corporation
Minneapolis, MN 55440
Contract No. 68-02-2444
Project Officer
R. E. Eskridge
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, NC 27711
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NC 27711
-------
DISCLAIMER
This report has been reviewed by the Environmental Sciences Research
Laboratory, U.S. Environmental Protection Agency, and approved for publica-
tion. Approval does not signify that the contents necessarily reflect the
views and policies of the U.S. Environmental Protection Agency nor does men-
tion of trade names or commercial products constitute endorsement or recommen-
dation for use.
ii
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ABSTRACT
The results from 34 "constant level" tetroon flights made near St. Cloud,
Minnesota, and tracked with the METRAC positioning system are presented. These
flights were made throughout the year and primarily at heights between 700 and
1400 meters above the surface. Flight times ranged in length from 2100 to
6000 seconds.
Three-dimensional velocity variances, autocorrelation functions, power
spectra and diffusion coefficients are presented. Relationships showing an
increase of vertical velocity variance with decreasing atmospheric stability
and with increased wind speed are illustrated. The autocorrelation functions
of the velocities tend to be oscillatory, and problems of estimating Lagrangian
integral time scales from them are discussed. Diffusion coefficients estimated
from the power spectra ranged over more than two orders of magnitude with the
horizontal diffusion coefficients generally larger than the vertical diffusion
coefficient. While the vertical diffusion coefficient increases with decreasing
atmospheric stability, the two horizontal diffusion coefficients show no clear
relationship to the atmospheric stability, wind speed or time of day.
This report was submitted in fulfillment of Contract No. 68-02-2444 by
Control Data Corporation under the sponsorship of the U.S. Environmental Pro-
tection Agency. This report covers a period from September 23, 1976 to Decem-
ber 22, 1977, and work was completed as of May 12, 1978.
iii
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CONTENTS
j
Abstract iii
Figures vi
1. Introduction 1
2. Summary and Conclusions 3
3. Recommendations 4
4. Flight Summary 5
5. Computations and Discussion 8
Variances 9
Autocorrelations 11
Spectra 16
Diffusion Coefficients 16
References. 26
Appendix 27
-------
FIGURES
Number Page
1 Tetroon trajectories. Flight number is given near the end of
each trajectory. The basic four tracking system receivers
are denoted by X's. Tetroons were launched from near the
central receiver 7
2 RMS vertical velocity versus environmental lapse rate 12
3 RMS vertical velocity versus horizontal wind speed for day
and nighttime launches. Environmental lapse rate estimates
are printed near each data point 13
4 The longitudinal, lateral and vertical velocity autocorrela-
tion functions for tetroon flight TF3 14
5 The longitudinal, lateral and vertical velocity autocorrela-
tion functions for tetroon flight TF9 15
6 Example of a correlation function before and after digitally
filtering out the dominant periodic component. This example
is the vertical velocity component of tetroon flight TF7. ... 18
7 Energy spectra for the vertical velocity component of TF22
and the longitudinal velocity component of TF39 22
8 Vertical diffusion coefficient, K , versus environmental
z
lapse rate 24
9 Lateral diffusion coefficient, K , versus the longitudinal
diffusion coefficient, K . The solid line indicates equality
Jv
between K and K while the dashed lines indicate one order
x y
of magnitude differences 25
VI
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SECTION 1
INTRODUCTION
The demands of modern society require an increased understanding of the
diffusive capabilities of the atmosphere. No longer is it possible to assume
that the atmosphere can digest all we put into it without there being some eco-
logical consequence. Predictions must be made concerning the environmental
impact or effect of effluent sources such as stacks, highways or cities. The
development of numerical transport and diffusion models is a necessary step if
we are to realistically and safely utilize the atmosphere to disperse our
wastes.
Diffusion occurs on all scales. If one assumes the appropriate time
scale, the atmosphere can be considered turbulent, and eddy diffusion coeffi-
cients computed, from space scales of centimeters to hundreds of kilometers.
Indeed, Hage, et al. (1966), have looked at stratospheric diffusion over time
periods of many months by tracing radioactive debris from nuclear explosions.
Classically, however, most studies have centered on time scales of
seconds to minutes and several hours to a few days. Small-scale Eulerian
measurements can be made from towers, and quasi-Eulerian measurements can be
made from aircraft. Tracking of smoke puffs allow a measure of diffusion in
a Lagrangian reference frame. Larger scale diffusion has been investigated
using upper air sounding data directly or by constructing geostrophic trajec-
tories to estimate Lagrangian diffusion coefficients. Many of these methods
and results have been surveyed and summarized by Bauer (1974), and Gage and
Jasperson (1975).
The diffusive properties of the atmosphere on the mesoscale are
the least well documented. These scales of order tens of minutes to hours
and kilometers to tens of kilometers are the fundamental scales for models
designed for cities. Quasi-horizontally floating super-pressured balloons
-------
(tetroons) have been shown to provide first approximations to the three-
dimensional atmospheric motion (e.g., Angell and Pack, 1961, and Hass, et al.,
1967). Some experimental turbulence data on the mesoscale have been collected
utilizing tetroons tracked by radar transponder systems (e.g., Angell and Pack,
1962, Angell, et al., 1961, and Sato, 1975). In these studies, tetroon
height was determined with the aid of helicopters.
This report presents the results of turbulence data collected with the
use of tetroons and the METRAC positioning system installed near St. Cloud,
Minnesota. The METRAC system described by Gage and Jasperson (1974) allows
accurate three-dimensional tracking of a lightweight expendable transmitter
attached to the tetroons.
-------
SECTION 2
SUMMARY AND CONCLUSIONS
A total of 45 tetroon flights were tracked by the METRAC positioning
system. Time series of three-dimensional velocity were computed for 34 of
these flights. The data, collected at many times of day and throughout a
year, show a considerable range of variance. The magnitude of the RMS
vertical velocities are dominated by the local tetroon height lapse rate
for stable atmospheres. When the local lapse rate is near adiabatic, the
magnitude of the RMS vertical velocities are apparently determined by other
factors. For instance, a day-night stratification of the data produces two
different relationships between the RMS vertical velocities and wind speed.
In both cases, the vertical velocity variances increase with increased wind
speed. However, for the daytime flights, when heat is being added at the
surface and the mixed layer is better developed, the variance increases much
more rapidly with wind speed than for nighttime flights.
Autocorrelation functions, spectra and diffusion coefficients were
computed for the 34 successful flights, and they illustrate the complicated
atmospheric structure. The autocorrelation functions tended to be oscillatory,
thereby complicating the evaluation of the Lagrangian integral time scale.
Therefore, the diffusion coefficients were estimated directly from the
spectra.
The diffusion coefficients computed for each of the three velocity com-
ponents range over more than two orders of magnitude. The horizontal diffusion
coefficients are generally larger than the vertical diffusion coefficient. The
vertical diffusion coefficient, like the vertical velocity variance, shows a
relationship to the local lapse rate. The two horizontal diffusion coefficients
show no clear relationship to the local atmospheric stability, wind speed,
or time of day.
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SECTION 3
RECOMMENDATIONS
The 34 flights presented in this report show one fact very clearly:
the structure of the atmosphere is very complex. The scale depicted by these
flights is one in which waves and turbulent eddies coexist. Even if one
could compute a diffusion coefficient exactly, the data from a single flight
would be of questionable utility in developing a useful model to explain and
predict the diffusive quality of the atmosphere at any given time. One needs
to work with averages and find parameters that group and separate data in
meaningful ways.
With data already collected, more emphasis should be placed on data
stratification. The day-night stratification presented in this report was a
first attempt at data separation and showed systematic results. Classification
of flights by whether or not they are in the mixed layer and surface layer
observations which could quantify atmospheric heat input and the strength of
convective activity would be useful. An evaluation of the stationarity of the
variance statistics should be made to determine the desired length of data
series. Shorter flights could be made at lower levels. A detailed case study
look at several flights should be made to fully evaluate the influence of the
periodic motion on the diffusion calculations.
Furthermore, additional tetroon turbulence measurements should be collec-
ted to provide a larger data base for stratification. With these flights, an
effort should be made to obtain additional surface layer measurements and
boundary layer profiles so that the turbulence data could be better
classified.
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SECTION 4
FLIGHT SUMMARY
A total of 45 super-pressured tetroon flights were made between
November 18, 1976, and December 7, 1977. Of these, 34 flights were considered :
successful and were fully processed. The 11 remaining flights suffered from
"f
one or more of the following problems:
1. balloon failure,
2. data recording failure,
and/or 3. transmitter failure.
Table 1 presents the basic launch and flight statistics. The average
heights of the flights ranged from 700 m to 1350 m with one flight at 2350 m.
Flights were made throughout the year and at many different times of day.
Wind direction varied over three quadrants for these 34 flights and flight
level wind speed ranged from 1.5 m/s to 21.8 m/s. Flight duration over which
turbulent statistics were computed ran from 2100 to 6000 seconds.
Figure 1 presents the two-dimensional flight trajectories with lines
connecting the initial and final tetroon positions for which statistics were
computed. Actual flight deviations from these lines were small. The four
basic receivers used by the METRAC positioning system are denoted by X's, and
the launch point was very near the central receiver. The flight numbers are
noted near the end of each trajectory. Tracking distances ranged from approxi-
mately 10 km to 70 km.
-------
TABLE 1. TETROON FLIGHT DATA
Mean Flight Level Wind
Flight
TF2
3
4
5
6
7
8
9
11
12
14
16
17
19
20
21
22
23
24
26
28
29
30
31
35
36
37
38
39
41
42
43
44
45
Date
11/23/76
11/23/76
12/ 1/76
12/ 2/76
12/ 3/76
1/12/77
1/18/77
1/25/77
2/18/77
2/27/77
8/11/77
8/16/77
8/17/77
9/20/77
9/20/77
9/20/77
9/21/77
9/21/77
9/27/77
9/27/77
9/28/77
9/28/77
10/ 3/77
10/ 4/77
ll/ 4/77
11/15/77
11/15/77
11/15/77
11/15/77
11/16/77
11/16/77
12/ 7/77
12/ 7/77
12/ 7/77
Launch Time
1153 GST
1500
1339
1002
1213
1203
1253
1950
1542
1323
1755
1700
0454
1012
1457
2005
0455
0847
1017
1610
0418
0752
2047
0357
1247
0627
0956
1358
1634
0354
0655
0727
1033
1240
Flight Level
1010 m
930
1080
940
970
810
1040
900
880
980
900
1070
1350
2350
1050
800
1000
1300
1150
1000
825
1050
900
700
1250
1050
1150
1150
1050
1200
1050
950
975
850
Dir/Speed
257/ 5.7m/s
290/ 7.4
315/11.6
24/ 2.2
316/11.4
273/ 6.9
336/ 7.2
249/13.8
334/ 9.3
7/13.2
268/12.3
294/ 7.1
319/14.2
255/ 3.6
133/ 6.9
152/11.0
171/ 6.5
176/ 9.6
332/12.5
317/10.1
303/ 4.8
241/ 5.3
209/10.6
238/ 8.5
165/ 9.6
266/ 5.7
285/16.2
246/16.7
259/18.9
298/21.8
305/17.0
281/ 1.5
193/ 4.4
164/ 6.6
Data Sample
Length
2800 s
2720
2800
3600
2720
2800
3040
2720
2800
2720
2100
5120
3520
6000
4000
2720
3600
3400
3680
3600
4000
5040
3000
3200
3600
3900
2720
2720
2420
2720
2720
3200
3900
2800
-------
21
38
12
Figure 1. Tetroon trajectories. Flight number is given near the end of each
trajectory. The basic four tracking system receivers are denoted
by X's. Tetroons were launched from near the central receiver.
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SECTION 5
COMPUTATIONS AND DISCUSSION
The data collected from the tetroon flights consist of time series of
three-dimensional velocity components evaluated at one second intervals.
Some smoothing was necessary in order to separate motions of a few centi-
meters per second from any system induced variance. This was particularly
true of the vertical velocity component for flights that traveled a long
distance.
Several smoothing schemes were tested. The data presented in this report
were smoothed by passing a symmetric 20-point filter over the one second data.
The normalized weights, C , for this filter were generated from the binomial
coefficients, b , given by Equation 1.
b
C = m . . _ N! N=20 QX
m~ 2Q,b ; bm m!(N-m)! m=0,1, 2,... , 20
Subsequently, each 20 smoothed points were averaged to give one estimate of
each wind component every 20 seconds. For each flight processed, between 35
and 100 minutes of data were used resulting in a time series of between 105
and 300 points for each velocity component. The coordinate system was then
rotated in the direction of the mean wind, and all velocities were transformed
into longitudinal, lateral and vertical components. Linear trends were removed
from each of the velocity components.
Later in the analysis, it was found that even after the removal of a
linear trend, there was still considerable energy remaining in the low fre-
quency, long period components of the velocity time series. This energy was
removed by utilizing time-frequency convolution theory (e.g., see Jenkins
-------
and Watts, 1968; or Lathi, 1968) and generating a new filtered time series y(t)
given by
y(t) = / Y(f)eJ"""df. (2)
Y(f) is derived by multiplication of the frequency filter H(f) and the Fourier
transform of the original data series:
Y(f) = H(f)X(f). (3)
X(f) is given by
X(f) = "" X(t)e"j2*ftdt (4)
where X(t) is the original data series. H(f) was defined to be zero for the
mean and fundamental frequency and to be unity for all other frequencies.
VARIANCES
Perhaps the easiest variables to compute from the component velocity
fluctuations are the variances. The longitudinal, lateral and vertical compo-
nent variances are presented in Table 2 along with the estimated environmental
lapse rate at the tetroon float level and the average environmental lapse rate
from the surface to the float level. The lapse rates were taken from the
temperature profiles of the St. Cloud National Weather Service soundings.
These soundings are taken daily at approximately 0500 and 1700 CST. In some
cases the lapse rates were difficult or impossible to reliably estimate due to
the proximity of a significant level, and hence a change in lapse rate at the
tetroon flight level, or due to time differences between the soundings and the
tetroon flights. In these instances, no lapse rate was computed. Positive
lapse rates given in Table 2 represent a decrease of temperature with height.
The dry adiabatic lapse rate is 1.00 ฐC/100m.
The variances of each of the velocity components vary over approximately
two orders of magnitude. The variances of the longitudinal and lateral compo-
nents often tend to be comparable in size though the longitudinal component is
larger on the average. The vertical variance tends to be smaller than the
horizontal variances.
9
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TABLE 2. STABILITY AND VARIANCES
Lapse Rate
TF2
3
4
5
6
7
8
9
11
12
14
16
17
19
20
21
22
23
24
26
28
29
30
31
35
36
37
38
39
41
42
43
44
45
.60 C/ 100m
.60
.46
-.11
-.10
1.11
-.87
.89
1.02
.71
1.16
1.17
.33
.32
.67
.62
-.40
-.15
1.04
1.10
-.85
-.67
1.00
-.86
.08
-.10
.00
.26
.48
.33
.28
-.63
-.19
-.24
0.1 C/ 100m
0.1
0.6
-.5
-1.8
-.7
0.3
0.8
0.7
0.6
0.9
0.9
0.6
0.1
-.2
-.6
0.5
1.0
0.9
0.7
0.9
0.9
0.7
-1.0
0.4
0.9
0.9
-.3
-.3
-.2
y
Longitudinal
.1562m2/s2
.1335
.4118
.0283
.2572
.0064
.1336
.2079
.1547
.0538
.0349
.1424
.0393
.0323
.0297
.0188
.0231
.0628
1.2902
.0531
.0052
.0423
.0057
.0318
.0406
.0932
.1247
.1974
.1903
.1102
.0116
.0248
.0339
.0705
Lateral
.1604m2/s2
.1211
.1989
.0440
.1297
.0715
.1355
.0564
.2200
.0905
.0130
.0660
.0318
.0041
.0434
.0403
.0175
.0395
.3060
.0636
.0060
.0270
.0125
.0920
.0459
.0079
.0402
.1320
.0447
.0612
.0482
.0415
.0210
.0277
\
Vertical
.0241m2/s2
.0517
.1531
.0037
.0204
.0153
.0231
.3795
.0594
.3278
.0411
.0553
.0277
.0045
.0210
.0284
.0091
.0369
.5396
.1320
.0106
.0066
.0308
.0522
.0407
.0498
.0353
.0531
.1527
.0620
.0566
.0038
.0072
.0366
a Average lapse rate between the surface and tetroon float level.
b Lapse rate at the tetroon float level.
10
-------
One would expect the intensity of the vertical variance to be dependent
upon the atmospheric stability. Figure 2 presents the RMS vertical velocity
deviation as a function of the lapse rate at the tetroon float altitude. It
can be seen that there is, in fact, a tendency for increased variance and
scatter with increased float altitude lapse rate. This implies that, while
the local lapse rate dominates the vertical turbulent intensity in a stable
layer, other factors such as the presence of convection can act to enhance the
turbulent intensity in a less stable layer. The curve is similar to one
presented by Hass, et al. (1967), and Sato (1975).
One might also expect that there would be an increase in the vertical
velocity variance associated with an increase in wind speed. A plot of the
RMS vertical velocities versus horizontal wind speed did not, at first glance,
show any consistent pattern. Including environmental lapse rates only confused
the picture more. However, if one stratifies the data by time of day (day
versus night), a pattern does seem to emerge.
Figure 3 presents a plot of horizontal wind speed versus the RMS vertical
velocity deviations. The data has been stratified into day (1000 - 1700 CST)
and night (2000 - 0700 CST), and the estimated environmental lapse rates have
been printed near the appropriate points. Two curves have been sketched to
indicate the pattern of the day-night stratification. This figure indicates
that, at least for wind speeds greater than 7 m/s, the time of day is of
greater importance in determining the RMS fluctuations than is the float level
lapse rate. This effect is likely due to the effectiveness of convection in
transferring energy from the mean wind to turbulent eddies.
AUTOCORRELATIONS
Lagrangian autocorrelation functions were computed for each of the three
velocity components of the 34 fully processed flights. These data are presented
in the Appendix, and two examples are plotted in Figures 4 and 5.
One noteworthy feature of virtually all the autocorrelation functions is
that they contain strong periodicities. These periodicities have time scales
ranging from a few minutes to nearly half an hour. The dominant vertical
periods tend to be shorter than the horizontal component periods. For some of
the flights, the vertical component frequency compared favorably with the
11
-------
CO
ll
H
U
o
s
H
44
fc
0)
.6
.5
.4-
.3
.2-
.1
-2 -1 0
Stable Unstable
Lapse Rate (ฐC/100m)
Figure 2. RMS vertical velocity versus environmental lapse rate.
12
-------
W
"a
4J
H
U
O
at
y
H
1/1
Launch between 1000 and
1700 GST
Launch between 2000 and
0700 CST
25
Wind Speed (m/s)
Figure 3. RMS vertical velocity versus horizontal wind speed for day and night
time launches. Environmental lapse rate estimates are printed near
each data point.
13
-------
1
o
\J
+1
g
I-l
u
a}
r- 1
0)
u n
M ฐ
O
u
o
w
3
<
71
o
VJ
_1
-ซ 1 I ' '
x TF3
X
* x* * * k ป *
% longitudinal x* ***
X x
* *
* If
x
X *
*x x"
^ -
X
X
X
X
If
X
*
* lateral
> * ^ * x
*ป 4. *
x ซ*
*>ปปป X *
f-
X
X
*** X*
y * k * X
X ^ X Jf <
* * ^ vertical ป
ป * * * ซ
X w * X *
< < * ***
* * v u
X ป ป *
X ป
X ซ
1 1 1 1
200
400 600
Lag (sec) >
800
1000
Figure 4. The longitudinal, lateral and vertical velocity autocorrelation
functions for tetroon flight TF3.
14
-------
ion
**
V.
Vx
200
********
lateral
vertical
****
*X*x>*
1
1
400 600
Lag (sec) >
800
1000
Figure 5. The longitudinal, lateral and vertical velocity autocorrelation
functions for tetroon flight TF9.
15
-------
Brunt-Vaisala frequency, 17, which can be computed from the environmental
temperature and stability by
Since ป7 is strongly dependent upon the lapse rate which was often difficult
to estimate, no explicit comparisons are presented in this report.
The fact that the velocity component periodicities tend to remain so
consistent within a flight for long periods of time indicate that they are
atmospheric in origin and not solely a balloon response to a transitory velo-
city eddy. These periodicities, however, give rise to problems in computing
diffusion coefficients as will be discussed in a subsequent section.
SPECTRA
Spectral density estimates were also computed for each of the velocity
components, and they, too, are presented in the Appendix. These spectral
estimates were computed using a Fast Fourier transform (e.g., see Jenkins and
Watts, 1969, pp. 313-317). Utilizing the Fast Fourier transform allowed
efficient computation and maximum frequency resolution (equivalent to a harmonic
analysis). The spectra in the Appendix were smoothed by a Hanning (quarter,
half, quarter) filter, and values from the second harmonic through approximately
the fiftieth harmonic are presented.
DIFFUSION COEFFICIENTS
As has been shown elsewhere (e.g., Pasquill, 1962), diffusion coefficients
can be formulated as the product of the mean square velocity perturbations and
the Lagrangian integral time scale of that velocity component. That is, for
the x or longitudinal diffusion coefficient,
Kx = ~* I/ '(6)
*
where u is the longitudinal perturbation and I is the limiting value of
T1
KT) = f R (r)dr ' (7)
Jo u
16
-------
where R (T) is the Lagrangian autocorrelation function defined by
RU(T) = u(r)ji(t + r) (8)
u2
Physically, u2 represents the energy of the velocity fluctuations and I
X
represents the Lagrangian scale size of the fluctuating eddies.
Unfortunately, these correlation functions discussed earlier in this
section and presented in the Appendix do not tend to converge to a value I
over the sample interval. One solution to the convergence problem which has
been used successfully is to assume that the autocorrelation function can be
approximated by a function which can be integrated from zero to infinity. Two
functions used by Murgatroyd (1968) were
R! = e-at (9)
and R = e~^ cos 7t.
In experiments with our data, a function of the form Rป = R.. + Rซ fit some of
the correlation functions out to lag 30 to 50. In these cases, the integral
was dominated by R.. as the periodic component damped very slowly. In other
cases, R, and R? did a poor job of fitting the correlation function.
Other experiments were made to digitally filter out the dominant periodic
component of the autocorrelation functions. If one assumes that the strong
periodic components were well defined atmospheric waves, the tetroon data can
be separated into variance associated with waves and variance associated with
"turbulence". Over the time scale of the data (say one to three hours), the
wave component can be assumed to produce the meander of a plume while the
turbulent component distributes or diffuses the material within the plume.
It was found that by filtering out the dominant wave, the shape of the auto-
correlation functions were significantly altered and tendencies for the
functions to converge were improved. An example is presented in Figure 6.
However, the somewhat arbitrary question of deciding how much of which waves
to filter out of the data created a certain amount of uneasiness with this
procedure.
17
-------
0
Autocorrelation
i
i-1 i-1
-1
(
1 i 1 1 l
x TF7
unfiltered
*"'ซ ***
* ซ , * *
X*
filtered
*
* * <"* v*<
K, X >-lt^
^ V k
X
1 1 1 1
3 200 400 600 800 10(
Lag (sec) ป>
Figure 6. Example of a correlation function before and after digitally
filtering out the dominant periodic component. This example
is the vertical velocity component of tetroon flight TF7.
18
-------
An alternative formulation for estimating the diffusion coefficient has
been discussed by Hinze (1959). Because of the mathematical duality between
spectra and correlation functions, the integral time scale in the x direction
with u2 the associated velocity variance can be formulated as
where f is the frequency and E is the spectral energy density, i.e.,
/oc
E(f)df = u2. (11)
Therefore,
K = 7- lim E (f) (12)
x 4 ,. u
f-ปo
This method was used in the computation of diffusion coefficients in this
report. It is similar to the last method in that the presence of waves or
isolated strong periodicities are recognized but not included as a part of the
diffusion process. That is, the application of Equation 12 tends to ignore
isolated spikes in the power spectrum.
Two examples of E(f) are presented in Figure 7. For most of the flight
data, there appeared to be a limiting value of E(f). The working plots were
made on a linear frequency scale and E(f) was extrapolated to zero frequency.
The numbers are, therefore, representative for a time scale of up to one to
three hours depending upon the data sample length. Local maxima such as that
illustrated in the vertical component of TF22 were not considered in the
estimate of the limit. In several flights, the slope of E(f) versus f changed.
This is illustrated in the longitudinal data of TF39 where the change in slope
is very near the Brunt-Vaisala frequency. This slope change may reflect the
inclusion of waves in the diffusion process for periods greater than several
minutes.
19
-------
The Hinze formulation illustrates the importance of the low frequency
eddies (i.e., as f-K>) in determining diffusion. This also, of course, can be
seen in the autocorrelation function formulation. The first few lags of the
autocorrelation function are the most important in determining the integral
time scale, and these lags are determined largely by the higher energy, low
frequency components. For flights TF2 - TF12, diffusion coefficients were
computed from both the power spectrum and the autocorrelation function and
similar results were obtained. All of the computed diffusion coefficients
are presented in Table 3. In a few instances, a limiting value for E(f)
could not be determined. The values in parentheses for flights TF2 - TF12
are diffusion coefficients computed from the autocorrelation function.
Values of the integral time scale are not presented explicitly in this
report because the diffusion coefficients were computed directly from the
power spectra by Equation 12. The variances given by Equation 11 where E(f)
is assumed to be a smooth function were not computed but could be approximated
by the flight variances tabulated in Table 2.
The characteristics of the diffusion coefficients are, of course, very
similar to the characteristics of the velocity componenent variances. All
three components, the longitudinal, lateral and vertical, range over more
than two orders of magnitude. The vertical diffusion coefficient, K , is
z
inhibited by the local thermal stability as is shown in Figure 8. Six flights
were determined to have taken place in the daytime connective boundary layer.
The Pasquill stability index which relates stability to cloud cover, surface
wind speed and sun angle was estimated for each of these flights. In this
scheme A represents extremely unstable conditions, C represents slightly un-
stable and F represents moderately stable conditions (Gifford, 1968). Table 4
presents these results along with the vertical diffusion coefficients.
TABLE 4. PASQUILL STABILITY CLASSIFICATION
Pasquill Vertical Diffusion
Flight Index Coefficient
TF11 D 7.9 m2/5
14 D 3.5
16 C 6.3
24 B 20
26 C 37
30 E 4.2
20
-------
For these flights within the daytime convective boundary layer, there appears
to be a dramatic break in the value of the vertical diffusion coefficient near
the C classification.
As shown in Figure 9, the lateral and longitudinal diffusion coefficients
are approximately equal in only about one-third of the cases. In seven cases,
the two horizontal diffusion coefficients differ by more than one order of mag-
nitude (as shown by the dashed lines of Figure 9). The longitudinal diffusion
coefficient tends to be larger than the lateral diffusion coefficient. Unlike
the vertical diffusion coefficient, there is no obvious relationship between
the values of either of the horizontal diffusion coefficients and either the
float level or the average stability between the surface and the float level.
21
-------
10
o
QJ
CO
CO
C
a)
Q
00
t-l
3
.1
100
10
.1
lII MM)
I I I i i I I
I I I III
I l l i l l i
,1
TF22
Vertical
Mill
TF39
Longitudinal ~
l l i I I
I 111 i l I l l i i I
.0001
.001
.01
Frequency (Cy/sec)
Figure 7 . Energy spectra for the vertical velocity component of TF22 and the
longitudinal velocity component of TF39.
22
-------
TABLE 3. DIFFUSION COEFFICIENTS
Flight Longitudinal Lateral Vertical
TF2
3
4
5
6
7
8
9
11
12
14
16
17
19
20
21
22
23
24
26
28
29
30
31
35
36
37
38
39
41
42
43
44
45
7
60 in /s
49
100
8.9
44
1.1
28
30
56
7.9
5.6
112
6.6
11.4
2.9
6.3
5.7
14
337
22
1.6
13
0.4
3.3
2.9
20
10
70
40
15
0.7
11
12
13
(23)
(20)
(82)
(6.8)
(51)
(0.8)
(35)
(42)
(28)
(4.8)
7
80 m Is
25
25
5.2
44
10
22
2.1
56
2.4
0.5
8.1
14
1.8
10
10
2.6
13
16
0.7
__-_
1.9
28
4.2
0.8
14
26
3.6
9.7
8.9
13
3.1
__-_
(24)
(18)
(24)
(13)
(26)
(ID
(34)
(2.8)
(55)
(18)
7
0.8 m la
0.9
14
0.1
0.7
0.4
2.0
50
7.9
1.6
3.5
6.3
1.7
0.5
0.5
2.5
0.3
1.3
20
37
0.1
4.2
4.1
0.6
____
1.5
2.2
1.7
1.0
1.1
0.3
0.4
2.6
(1.0)
(2.1)
(ID
(0.2)
(1.0)
(0.6)
(2.3)
(30)
(5.9)
(1.6)
23
-------
100 c
10
o
0)
ซn
1 0
Lapse Rate (ฐC/100m)
Figure 8. Vertical diffusion coefficient, Kz, versus environmental lapse
rate.
24
-------
100
10
o
-------
REFERENCES
1. Angell, J. K. and D.H. Pack. Estimates of vertical air motions in desert
terrain from tetroon flights. Mon. Wea. Rev., 89, 273-283, 1961.
2. Angell, J. K. and D. H. Pack. Analysis of low-level constant volume
balloon (tetroon) flights from Wallops Island. J. Atmos. Sci., 19, 87-98,
1962.
3. Angell, J. K., P. W. Allen and E. A Jessup. Mesoscale relative diffusion
estimates from tetroon flights. J. Appl. Meteor., 10, 43-46, 1971.
4. Bauer, E. Dispersion of tracers in the atmosphere and ocean. J, Geophys.
Res., 79, 789-796, 1974.
5. Gage, K. S. and W. H. Jasperson. Prototype METRAC balloon-tracking system
yields accurate, high-resolution winds in Minneapolis field test.
Bull. Am. Meteor. Soc., 55, 1107-1114, 1974.
6. Gage, K. S. and W. H. Jasperson. A feasibility study for measuring
stratospheric turbulence using METRAC positioning system. Final Report,
Contract NAS-2-8400, Control Data Corp., Minneapolis, MN, 1974, 84pp.
7. Gifford, F. A. Chapter 3. Meteorology and Atomic Energy, 1968, D. H.
Slade, Ed. U. S. Atomic Energy Commission, 1968.
8. Hage, K. D., G. Arnason, N. F. Bowne, P. S. H. D. Entrekin, M. Levitz
and J. S. Sekorski. Particle fallout and dispersion in the atmosphere.
Report SC-CR-66-2031, Nat. Tech. Inform. Serv., Springfield, Virginia,
1966.
9. Hass, W. A., W. H. Hoecker, D. H. Pack and J. K. Angell. Analysis of low-
level, constant volume balloon (tetroon) flights over New York City.
Quart. J. Roy. Met. Soc., 93, 483-493, 1967.
10. Hinze, J. 0. Turbulence, McGraw Hill, New York, 1959, 586pp.
11. Jenkins, G. M. and D. G. Watts. Spectral Analysis and its Applications.
Holden-Day, San Francisco, 525pp., 1968.
12. Lathi, B. P. Communications Systems. Wiley, New York, 431pp., 1968.
13. Murgatroyd, R. J. Estimations from geostrophic trajectories of horizontal
diffusivity in the mid-latitude troposphere and lower stratosphere.
Quart. J. Roy. Met. Soc., 95, 40-62, 1969.
14. Pasquill, F. Atmospheric Diffusion. Van Nastrand, London, 1962, 297pp.
15. Sato, J. Estimation of turbulent diffusivity over Tokyo metropolitan
area from constant-volume balloons. Pap. Meteor, and Geophys., 26, 35-46
1975.
26
-------
APPENDIX
FLIGHT DATA STATISTICS
This appendix contains autocorrelations and spectral estimates for the
longitudinal, lateral and vertical velocity perturbations. The dimensions
2
of the spectral estimates are in m /sec.
27
-------
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing}
1. REPORT NO.
EPA-600/4-79-034
3. RECIPIENT'S ACCESSION-NO.
4. TITLE AND SUBTITLE
DIFFUSION COEFFICIENTS FROM METRAC SYSTEM
TURBULENCE MEASUREMENTS
5. REPORT DATE
May 1979
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
W. H. Jasperson
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Control Data Corporation
Minneapolis, MN 55440
10. PROGRAM ELEMENT NO.
1AA603A (FY-76)
11. CONTRACT/GRANT NO.
68-02-2444
12. SPONSORING AGENCY NAME AND ADDRESS
13. TYPE OF REPORT AND PERIOD COVERED
Environmental Sciences Research Laboratory-RTP, NC
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park. N.C. 27711
Final
14. SPONSORING AGENCY CODE
EPA/600/09
15. SUPPLEMENTARY NOTES
16. ABSTRACT
ResuUs from 34 "constant level" tetroon flights made near St. Cloud, MN and
tracked with the METRAC positioning system are presented. These flights were made
throughout the year and primarily at heights between 700 and 1400 meters above the
surface. Flight times ranged in length from 2100 to 6000 seconds.
Three-dimensional velocity variances, autocorrelation functions, power spectra
and diffusion coefficients are presented. Relationships showing an increase of
vertical variance with decreasing atmospheric stability and with increased wind speed
are illustrated. The autocorrelation functions of the velocities tend to be oscilla-
tory, and problems of estimating Lagrangian integral time scales from them are
discussed. Diffusion coefficients estimated from the power spectra ranged over more
than two orders of magnitude with the horizontal diffusion coefficients generally
larger than the vertical diffusion coefficients. While the vertical diffusion
coefficient increases with decreasing atmospheric stability, the two horizontal
diffusion coefficients show no clear relationship to atmospheric stability, wind
speed or time-of-day.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS
COS AT I Field/Group
* Meteorology
Meteorological balloons
* Turbulence
* Diffusion coefficient
METRAC System
04 B
20 D
20 M
18. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (ThisReport)
IINri ASSTFTFR
21. NO. OF PAGES
68
20. SECURITY CLASS (Thispage)
UNCLASSIFIED
22. PRICE
EPA Form 2220-1 (9-73)
62
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