FRACTAL PROPERTIES OF TEMPERATlTRE ... - Semantic Scholar

FRACTAL PROPERTIES OF TEMPERATlTRE FLUCTUATIONS IN THE CONVECTIVE SURFACE LAYER SZYMON P. MALINOWSKI 1,2 and MONIQUE Y. LECLERC 1 ,3

IDepartment of Physics, University of Quebec at Montreal, P.O. Box 8888, Station A, Montreal,

Quebec, CANADA, H3C 3P8; 20 n leave from University of Warsaw, Institute of Geophysics ul.

Pasteura 7, 02-093 Warsaw, Poland; 3 Centre for Global and Climate Change, McGill

University, Montreal, Quebec, Canada

(Received in final foml 11 April, 1994)

Abstract. Temperature fluctuations in a convective surface layer were investigated. Box counting analysis was performed to investigate fractal properties of surfaces of constant temperature and was performed on sets of points obtained by setting thresholds on detrended records. Results indicate that surfaces of constant temperature have fractal properties for thresholds far from the mean. Estimated fractal dinlensions of one-dimensional cuts through these surfaces varied between 0.23 and 0.66, increasing with threshold value approaching the nlean temperature. For thresholds close to the mean, no fractal behavior was found. Asymmetry in results for thresholds above and below the mean temperature was attributed to the asymmetry between updrafts and downdrafts in the convective surface layer. The tenlperature dissipation rate (TD) was also investigated. It was found to be strongly inter mittent with large fluctuations of the intermittency exponent. Moments were analyzed in order to investigate multifractal properties of TD. Results indicate scaling in the range of 50TJ-l OOOTJ (where TJ is the Kolmogorov scale) and multifractal properties resembling those observed for passive scalar dissipation in laboratory flows.

1. Introduction In turbulence research, the fractal approach has brought a new perspective to understanding the energy cascade and its influence on the intermittence of tur bulent energy dissipation (Mandelbrot, 1974; Meneveau and Sreenivasan, 1991). The diffusion in turbulent fluids has been described in terms of fractals (Hentschel and Procaccia, 1984). Fractal geometry allows a concise description of surfaces and interfaces in turbulent flows (Sreenivasan et al., 1989). The fractal dimension of passive scalar isoconcentration surfaces together with their scaling range con stitute in1portant indicators of the effectiveness of turbulent n1ixing (Sreenivasan et al., 1989; Constantin et al., 1991). In atmospheric physics, fractals have been used to describe cloud properties (Lovejoy, 1982; Cahalan and Joseph, 1988; Lovejoy and Schertzer, 1990; Malinowski and Zawadzki, 1993; Malinowski et al., 1994). Attempts have also been undertaken to construct a universal descrip tion of atn10spheric properties encompassing the whole range of atmospheric motions in the framework of multifractal formalism (Schertzer and Lovejoy, 1989). So far, however, few attempts have been made to use these approaches in studies of atmospheric boundary layer (ABL) turbulence. In this paper, tech niques developed by researchers studying passive scalars in laboratory turbulent Boundary-Layer Meteorology 71: 169-187, 1994.

© 1994 Kluwer Academic Publishers. Printed in the Netherlands.

170

SZYMON P. MALINOWSKI AND MONIQUE Y. LECLERC

flows will be used to investigate temperature fluctuations in the lower boundary layer in convective conditions. The following techniques are applied to temperature records: 1) box counting on points obtained by setting thresholds on time series and 2) scale analyses of moments due to the temperature dissipation rate by determining T(q) and f(a). The first technique is aimed at determining the fractal dimension of surfaces of constant temperature. Assun1ing that the temperature field is advected by the mean wind, points at the threshold level correspond to the surface of constant temperature passing through the sensor. Fractal dimension of a set of such points can be estimated by means of box counting and compared with the results of Sreenivasan et ale (1989), Constantin et ale (1991) and Miller and Din10takis (1991), obtained for passive scalars in laboratory flows. The second technique is aimed at determining whether the temperature dissipa tion rate in the turbulent ABL has multifractal properties. Again, the temperature field is assumed to be advected by the mean wind and then the temperature dissi pation is estimated using the temporal derivative of the temperature fluctuations squared. These results are then investigated by means of multifractal analysis and results are compared to those obtained by Prasad et ale (1988) and Prasad and Sreenivasan (1990) for passive scalars in laboratory flows.

2. Literature Review

Sreenivasan et ale (1989) visualized laser-sheet sections through surfaces of iso concentration of a passive scalar in well developed entraining turbulent flows (e.g., turbulent jets, shear layers and boundary-layer flows) from which they derived a fractal dimension of the surface. The spatial resolution of the investi gated 1-D and 2-D sections through the flow was close to the Kolmogorov scale, so that twists and convolutions of the surface due to all acting dynamic scales of turbulent eddies could be observed. The fractal dimension D of surfaces in these experiments assuming isotropic turbulence was found to be 2.35 + 0.05. This is in agreement with their theoretical predictions, achieved assun1ing Reynolds number similarity, constant turbulent energy flux down to the Kolmogorov scale and considering the action of n10lecular dissipation in the smallest eddies. This theoretical prediction gives D = 7/3 = 2.33 when the intermittency of the turbu lent energy dissipation is not considered and D = 2.36 when it is. Sreenivasan et ale (1989) estimated the area through which molecular mixing takes place con sidering the range of scales in which surfaces are convoluted (from the integral scale of turbulence to the Kolmogorov scale), and then, accounting for molecular fluxes, they estimated the entrainment rate of turbulent flows. Their calculations agree well with the one based on Taylor's entrainment hypothesis widely used in geophysical flows (see review by Tun1er, 1986). Constantin et al. (1991) expand ed findings of Sreenivasan et ale (1989) showing that isoconcentration surfaces

TEMPERATURE FLUCTUATIONS IN THE CONVECTIVE SURFACE LAYER

171

completely embedded in turbulent regions are characterized by a fractal dimen sion D = 8/3 = 2.67. On the other hand, Miller and Dimotakis (1991), performing experiments sin1ilar to those of Sreenivasan et ale (1989), did not find any evidence of constant fractal dimension of isoconcentration surfaces in turbulent flows. They argued that the behavior of isoscalar surfaces is better described by a stochastic lognormal model, and they attributed Sreenivasan et ale 's (1989) findings to the particular choice of isoconcentration surfaces characterized by concentrations far from the mean. Prasad and Sreenivasan (1990) investigated three-dimensional fields of con centration of passive scalars in turbulent wakes and jets. Their method consisted of mapping closely spaced two-dimensional cross-sections (slice after slice) by means of laser-induced fluorescence. The resolution of these measurements was of the order of 3 Kolmogorov scales TJ. In addition to investigating isoconcentra tion surfaces of turbulent dye (which resulted in the confirmation of Sreenivasan et ale 's (1989) findings), they analyzed properties of passive scalar dissipation rate r defined as: (1)

They found extreme temporal and spatial intermittency of r. Despite the observed anisotropy of the r field, they reported that singularity spectra f(a) obtained for three components of r : [(8X/8x)2, (8X/8y)2, (8X/8z)2] and for r itself, agreed well within experin1ental uncertainties, indicating that multifractal properties can be retrieved from only one component. Furthermore, based on comparison of their results with previous one-dimensional measuren1ents resolv ing Koln10gorov and Batchelor scales, Prasad and Sreenivasan (1990) suggested that a resolution of the order of several Kolmogorov scales is sufficient to inves tigate multifractal properties of the passive scalar dissipation rate. In this paper, we shall investigate which approach, Sreenivasan et ale (1989) or Miller and Dimotakis (1991), holds better for surfaces of constant temperature in the convective surface layer. We shall also examine the temperature dissipation rate and compare our findings with those of Prasad and Sreenivasan (1990). Since atmospheric measurements cover a broad range of scales, they hold a distinct advantage over laboratory observations. However, they are not performed in controlled conditions. These factors will be discussed below. 3. The Experimental Data This experiment was performed in Ste-Anne-de-Bellevue, Quebec, Canada, dur ing September, 1991. Temperature fluctuations in the atmospheric surface layer were measured at high resolution (0.01 °C) and with fast response (beyond 200 Hz, 200 Hz low-pass filtered output) fine wire (2.5 /-lm diameter, 4.5 mm length)

172


TABLE I

Experimental data sets

No.

Date

Time

f

U

U/ 21 / 2

L

-z/L

1 2 3 4 5

91.09.04 91.09.16 91.09.18 91.09.18 91.09.18

1505-1525 1410-1420 1150-1200 1458-1508 1300-1450

400 400 400 400 20

3.3 3.9 5.2 5.8 5.6

0.78 0.87 1.08 0.99 0.98

-35 -707 -145 -197 -189

-0.17 -0.01 -0.04 -0.03 -0.03

Date: yy.mm.dd Time: EST; f - sampling frequency [Hz]. V - mean windspeed at 6 m [nl/s]. -1/2 U/2 -

root mean square windspeed fluctuations at 6 m [m/s].

L - Obukhov length at 6 m [m].

- z / L - Richardson flux nUlnber.

-

-1/2

For 400 Hz files U, U /2 ,L, z / L are estinlated fronl 30 min segnlents of 20 Hz data from the ultrasonic anemometer, collected immediately before or after 400 Hz data sets.

resistance thennometers mounted at 1 and 2 m above the ground. The data were digitized with a 12 bit AID converter and stored on a personal computer hard disk. The resolution of the digitizer was set to the resolution of the temperature sensor, so that the noise level of the them10meter was below the resolution of the AID converter. 10 min (256,000 samples in each channel) data sets were collected at 400 Hz. This data collection followed the rv 1 hour san1pling (rv 68,000 samples in each channel) of 20 Hz observations of turbulent velocity and temperature from a 3-D ultrasonic anemometer/thennometer (Kaijo Denki, Japan) mounted at 6 m from which atmospheric stability was calculated. Table I contains basic infor mation: mean windspeed at 6 m, wind fluctuations for comparison with Taylor's hypothesis and the surface-layer stability. Even though temperature is a scalar, it is not a passive scalar such as those investigated by Sreenivasan et ale (1989), Miller and Dimotakis (1991) and Prasad and Sreenivasan (1990). Fortunately in the experimental conditions (Table I), the Obukhov length was significantly greater than the position of the sensors above the ground, meaning that the wind shear was the main source of turbulence and buoyancy effects due to temperature fluctuations were small. Secondly, mean temperature increased due to solar heating of the ABL during the day. In order to investigate temperature fluctuations with respect to slowly increasing mean temperature, records have been detrended by subtracting the best linear fit. Typ ical mean temperature fluctuations in a 10 min experimental record were of the order of 0.1-0.2 °C, while the maximum amplitude of the observed tempera

173


FI091805.400 detrended T at 1m. 0.5 0.0

Q ~

-0.5

-1.0 0

0.5

1.5

2

2.5

3

3.5

4

4.5

5

9

9.5

10

time [s] FI091805.400 detrended T at 1m. 0.5 0.0

Q E-

-0.5 -1.0 5

5.5

6

6.5

7

7.5

8

8.5

time [s] Fig. 1.

An exanlple of detrended 400 Hz temperature fluctuations at 1 nl.

ture fluctuations was usually over 3 ac. Examples of such detrended records are shown in Figure 1. In this study, it is assumed that the high frequency response and small sam pling volume of the sensing element are such that resolved scales are in the dissipation range. The Kolmogorov scale TJ is typically of the order of 1 mm, and the dissipating range begins at scales of several tens of TJ. The Corrsin scale, suitable for scalar fields, is slightly larger than TJ for temperature (TJ = hPr for Pr = 0.71 in the boundary layer), lending support to our assumption. Anoth er experimental confirmation of its validity can be found by analyzing power spectra of detrended temperature records. A typical power spectrunl, obtained by averaging 30 Haar-windowed segments of 8192 data points is shown in Figure 2. The constant slope in the frequency range 0.1-80 Hz is close to - 5/3 (from -1.6 to -1.65 in various cases). The slight departure from the -5/3 slope at higher frequencies suggests that observations resolve turbulence scales within the dissipation range. This, together with the frozen flow hypothesis and the Prasad and Sreenivasan (1990) findings quoted above, allow us to estimate the local temperature dissipation TD along the record using:

(2)

174


FI091802.400, T2m, PS.

FI09I802.400, TIm, PS. 7

7

6

6

5

5

4

4

0:

bI)

.9

L...-

L..--

o

"

o

-

2

-

2

-

3

~

3

o

l..---_ _- - - J

o

2

..L-----I

2

log(f[Hz])

log(f[Hz])

Fig. 2. Examples of power spectra of temperature fluctuations observed at I and 2 n1 from the surface.

where aT/at is the temporal derivative of temperature estimated from experi mental data by finite differences. Examples of estimated TD of temperature data are given in Figure 3.

4. Results 4.1.

BOX COUNTING ON TEMPERATURE RECORDS

Detrended 400 and 20 Hz observations of temperature fluctuations (Table I) were thresholded every 0.1 °C, resulting in sets of points corresponding to intersec tions of the threshold with the record. Assun1ing that surfaces of constant tem perature can be found in a turbulent flow, the above points may be interpreted as the passage of such a surface through the sensor. Thus, assuming the frozen flow hypothesis, the set of points represents a one-dimensional Eulerian section through a convoluted surface of constant ten1perature advected with the flow. The fractal nature of such sets of points was investigated using box counting analysis. The length (in time) occupied by the set is divided into a nun1ber of

175


150

Q

C

100

~

0

E

0

50

E

O

a

0.5

1.5

2

2.5

3

3.5

4

4.5

5

9

9.5

10

time [s] FI091805.400 TO/mean TO at 1m. 100

0' E § 0

E

0'

50

E

O

5

5.5

6

6.5

7

7.5

8

8.5

time [s] Fig. 3.

Estimate of temperature dissipation rate, corresponding to first lOs of data in Figure 1.

boxes of size d. The number of boxes N(d) containing elements of the set of points follows a power law: (3)

where the negative value of the exponent D is an estimate of the fractal dimension of the set (for details consult, e.g. Falconer, 1990). In practice, results of box counting are plotted in log-log coordinates. When plots within a certain range of box sizes can be reasonably approximated by a straight line, the data exhibit a fractal dimension equal to the negative value of the slope of this approximation. Examples of box counting for sets of points obtained from setting a threshold on temperature records are presented in log-log coordinates in Figure 4. At scales smaller than 0.02 s for 400 Hz records and 0.4 s for 20 Hz record, most of the plots bend horizontally; this is typical when approaching the limit of data resolution (see Malinowski and Zawadzki, 1993; Malinowski et al., 1994). At scales larger than 6 s for 400 Hz records and 100 s for 20 Hz records, most of the slope values are close to -1, suggesting the space-filling behavior of convolutions at these scales (i.e., each large box is occupied). Thus the linear best fit was calculated by means of the least squares method for a range of scales between 0.02-6 s for 400 Hz data and 0.4-100 s for 400 Hz data. The negative slope of S associated with the best fit is plotted vs. the T threshold in Figure 5.

176

SZYMON P. MALINOWSKI AND MONIQlTE Y. LECLERC

Fl 091802.400, T2m, FBCr (-1.2)-(0.)]. 4

FI091802.400, T2n1, FBC[(0.)-(1.9)]. 4

3.5

3.5

3

3

2.5

.........

2.5

+.

.+

'+-,

'+

+. '~

2

2

+ ~,

" "'

1.5

1.5 '-to ~

.+. +.~

0.5

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OL..-----l--------L--------' -2

o

o

L--_--L..

2

-2

log(L[s])

.L.

o

--'

2

log(L[s])

Fl 090403.400, Tl nl, FBC[(-1.1)-(0.)]. 4.5 4

FI090403.400, TIm, FBC[(0.):(2.3)]. 4.5 4

3.5 3 2.5 2

1.5

0.5

0.5

OL----l--------L--------' -2

o

2

-2

o log(L[s])

log(L[s])

Fig.4a-d.

2

177


FI091603.400, Tl n1, FBC[(-0.2):(0.)]. 4.5 .------y------r--------, +

4

FI091603.400, TIm, FBC[(0.):(0.5)]. 4.5.--------r-----r--------, 4

'+.

+'+

"'+

+"+,

''"'', "+.,

+"'\ '~....

3.5

Z

bi} ~

'.

+,+,

'+, :t:,.

\'+. , ,

3

.,

3.5

3

" +

"

'.

'\ +.

\.,\.

Z

\'.

2.5

'+

~

\. \~~

\

'+ '+

•+

"\ '\

+.

.+

+

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'+.

'bi)

\"

2

+.....,

2.5

~,

\

'+. ~

2

.,

1.5 \ \ \

1.5

"\ \

+- ........'\

\

0.5

'., ~-+,

\-+-+-\

0.5 '---_ _ -4

~

__a..__ _~

o

-2

log(L[s])

2

o

\ ..L.-.-+-+~--+'

l . - -_ _- L

-4

-2

0

2

log(L[s])

Fig. 4. Exan1ples of box counting on detrended and thresholded temperature records. d - box size, N - amount of non-empty boxes. Curves from bottom to top correspond to thresholds approaching the mean temperature value in 0.1 °C increments at two levels (2 and 1 m). (a, c, e) Fluctuations below the mean value. b, d, f) Fluctuations above the mean value.

As one can see, S decreases wilh increasing absolute threshold value. For the same reason, the statistical reliability of S as an estimate of fractal dimension D decreases with increasing absolute threshold level. In the limit of large absolute thresholds, the resulting fractal dimension is 0 with zero reliability because of the lack of points. For large absolute thresholds, even in cases of reasonable statistics (say 10g(N) > 10), results are often scattered. When the bcst fit is applied on a log-log plot with correlation 0.998 were assumed to be reasonable estimates of D (indicated by boxed areas in Figure 4 and thick lines in Figure 5). Summarizing, fractal dimensions obtained for the investigated series at two observation levels (1 and 2 m) range between 0.31 and 0.66 for 400 Hz data and between 0.23 and 0.66 for 20 Hz data. Data sets are insufficient to decide whether there is a possible dependence of the results on Obukhov length and on observation level.

178


S vs. T threshold, 1m

rn

S vs. T threshold, 2m

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6 rn

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 -2

0 T threshold

2

0 -2

o

2

T threshold

Fig. 5. Negative of the slope of the linear best fit to the results of box counting analysis within the scaling range vs. temperature threshold at two levels. Crosses correspond to data set 1, circles to data set 2, stars to set 3, x' s to set 4 and dots to set 5.

For thresholds close to the most positive fluctuations (Figures 4b, d, f, h), results differ from those close to the largest negative fluctuations (Figures 4a, c, e, g). The most positive threshold yields a small number of data points, thus providing few clues on the fractal nature of those extreme but rare events. In contrast with the 20 Hz series, even for threshold values differing by as little as 0.1 °C from the most negative of the observed fluctuations, a much larger set of hundreds of points with well defined fractal properties is obtained. This asymmetry will be elaborated in Section 5. 4.2. lVI0MENT ANALYSIS OF TEMPERATl;RE DISSIPATION

The extren1e intermittency of the TD record (Figure 3) is striking. The maximum instantaneous TD observed (not shown on the plot) was over 600 times greater than the mean TD value and fluctuations 100 times greater than the mean value of TD were common. Such intermittent records of passive scalar dissipation rate or turbulent energy dissipation have often been successfully described in terms of multifractal formalism (Prasad et al., 1988; Prasad and Sreenivasan, 1990; Meneveau and Sreenivasan, 1991). Here, methods similar to those described in Mandelbrot (1989) and Feder (1988) will be adopted to study multifractal


179

properties of temperature dissipation in the ABL. Monlent analysis can be applied to the measure associated with a set. In order to perfonn this analysis, the measure is integrated (coarse-grained) on boxes of size d, then raised to the q-th power (moment order q) and averaged over a set. By analogy with box counting, if the averaged moments follow power laws with exponents T(q) as functions of box SIze:

(4) the measure is multifractal. Thus, q and T(q) constitute a pair of variables describ ing multifractal properties of the multifractal measure. One can also define, after Hentschel and Procaccia (1983), the generalized dimensions D q as: D q = T(q)/(q - 1).

(5)

The latter showed that Do corresponds to the fractal dimension support of the measure, D 1 to the entropy (infornlation) dimension and D 2 to the correlation dimension. Another equivalent description could be given by another pair of variables: the Lipschitz-Holder exponent a and singularity spectrum f(a) (see e.g. Feder, 1988). They can be calculated from a Legendre transfonn of T(q): T(q) = dT(q)/dq;

f(a)

= qa(q) -

T(q).

(6)

The temperature dissipation estimate may be analyzed in this way. Figure 6 shows the scaling properties of moments of this measure vs. segment size. Despite some variability in scaling ranges inherent in the data, a scaling range common to all investigated data can be found. In this scaling range (0.16-3 s), the sequence of moment scaling exponents T(q) (slope in the scaling range vs. moment order q from Figure 6) and singularity spectrum f(a) were retrieved. Centered differences were used to estinlate dT / dq. Results are presented in Figure 7. When retrieving T(q) from moments analysis, several uncertainties arise. The first concerns the scaling range; a quantitative method to bypass a subjective detennination of the scaling range has yet to be proposed. Correlation coefficient of the best fit is not a good indicator; that will become clearer following the discussion of the box counting analysis. In fact, scaling ranges obtained above are the result of subjective evaluation. To minimize the uncertainty, scaling ranges were evaluated subjectively in plots with slopes approaching 45°, where a visual evaluation of those ranges is most accurate (during analysis, the aspect ratio was modified in order to obtain a slope of 45 deg). The second uncertainty is in detennining which moments of q are statistically significant, similarly as in Saucier (1991). The T(q) functions for 1/2 and 1/4 of the data set were compared on the same plot (exanlple given in Figure 8). Regions in which T(q) curves agree suggest that results are significant for q corresponding to these regions. An

180


FI090403.400, T02m, mts[-5:-0.5]. 10 r - - - , . . - - - - - - - , . - - - - - - - . -

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-15

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+ - + - + -+ -+ -+ -+ -... -+- -+-.

~~~~~~+

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0'------'------....1-----....L---l

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-2

log(X[s])

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5

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-15

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+ ... +.+.. +. .' + .. +' +,+'

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5

-30

-35

-2

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log(X[s))

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181


TO, tau(q). 4 r-----------r---------,

TO, f(alfa).

1.2 r - - - - - - - - , - - - - - - , . . . - - - - - .

+ FI091603.400 1m

,

* FI091603.4002m

2

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o

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0.4

-4 0.2

-6

.

I,

I,

o

-8 '--

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-5

--1

-0.2 '---_ _

-...L.

---I

o

5

--'

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q

1.5 alfa

TO, tau(q).

4o.....-------r-------,

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1.4 .---------.,-----.------.------.

1.2

o -2

0.8 I.

-4

. ,.

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;

-8

+ FI091802.400 1m

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t,

--'

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---.l-_ _---L.._ _---J

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1.5

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alfa

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182


analysis of Figure 8 indicates that for -5 > q > 3.5, moments converge to the sanle T(q) curve. However, it does not imply that higher negative nloments give us important physical information. In most cases, for q < -1.5, the estimated T(q) curve is not concave, as the theory suggests. On a f(a) plot, points corresponding to q < -1.5 do not form the right branch of a parabola and are omitted in plots of f(a). The reason lies in the resolution of the sensor and digitizer. Negative moments emphasize weak singularities (or small values) of TD, corresponding to small temperature fluctuations. Fluctuations smaller than the sensor and the AID converter resolution cannot be detected; thus weak singularities are not measured and moments of q smaller than -1.5 represent digital noise rather than weak singularities in the temperature dissipation. Curves in Figure 7 for -1.5 < q < 3.5 resemble similar curves presented by Prasad et ale (1988) and by Sreenivasan (1991) for the dissipation rate of passive scalars in laboratory flows suggesting that TD has similar multifractal properties. In all of the investigated cases, Do (dimension of the support of tem perature dissipation) was equal to 1. In order to nlake quantitative comparisons, the information dimension DI, the correlation dimension D2 and an intermittency exponent f.-L, calculated by Meneveau and Sreenivasan (1991) from the relation:

(7) are included in Table II. Both dimensions were well established in all the investi gated data sets. The last parameter, f.-L, shows strong variability from case to case, suggesting weak intermittency in data sets 1-2 and very strong intermittency in sets 3-4. Despite this, the mean value of f.-L = 0.31 + 0.13 corresponds well to the result for passive scalars in laboratory flows reported by Sreenivasan (1991) (f.-L = 0.36 + 0.05).

5. Discussion While results of box counting analysis on detrended and thresholded temperature records show no fractal behavior for thresholds close to the mean, the picture is quite different farther away where scaling is between 0.02-100 s and the fractal dimension of the cut through surfaces of constant tenlperature, depending on the chosen threshold, can be defined. Findings of Miller and Dimotakis (1991) indicate that the scalar interface cannot be described in terms of a constant frac tal dimension. However, findings of Sreenivasan et ale (1989) and Prasad and Sreenivasan (1990) argue that isoconcentration surfaces of passive scalars in tur bulent flows should have constant fractal dimension of 2.36 (D = 0.36 on a one-dimensional cut, similar to our case). On the other hand, Sreenivasan et ale (1989) approach seems to work well (except for the value of the fractal dimen sion) for cloud-clear air interfaces (Malinowski and Zawadzki, 1993; Malinowski et al., 1994). We believe that differences in those results may have a physical basis.

183


Fl 090403.400, TDl m, tau(q).

FI 090403.400, TDt m, f(aIfa).

3

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0 0.6

-1 ::s

;g

-2

~

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0

+ 256000

* 512000 o 128000

-0.2 ,

0

-0.4

0

5

0.5

q

1.5 alfa

F1090403.400, TD2m, tau(q).

F1090403.4oo, TD2m, f(a1fa).

3

1.2

2

0, I

~

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,

0.8

4.

0 0.6

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-2

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-4

-5 -6

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+ 256000

* 512000 o 128000 0 q

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Fig. 8. Multifractal characteristics of tenlperature dissipation at two levels. Example of sensitivity to the length of data series. (a) Sequences of moment exponents T(q). (b) Singularity spectra f(a).

184


TABLE II Multifractal properties of the temperature dissipation estinlate Data

2 3 4 Mean value

Sensor

D

D

/-L

1m 2m 1m 2m 1m 2m 1m 2m

0.82 0.81 0.87 0.83 0.80 0.81 0.81 0.81 0.82 +0.02

0.67 0.66 0.69 0.72 0.66 0.68 0.65 0.68 0.67 +0.03

0.18 0.25 0.24 0.13 0.46 0.40 0.41 0.42 0.31 +0.13

D - entropy (information) dimension;

D - correlation dimension;

/-L - intermittency exponent.

The box counting n1ethod is aimed at investigating the fractal dimension of the set. In thresholding, points of a given value of the function are picked. This procedure, however, does not guarantee picking up points of the same physi cal significance, i.e., box counting is perfonned on points representing various stages of a mixing process: primary and secondary mixing. For example, when initial stages of mixing are examined (usually in the laboratory experiments), the interface of mean concentration of a passive scalar outlines two volumes of fluid of different origin with minimum and maximum concentrations observed. Then mixing of these two previously unmixed volumes, the primary n1ixing event, takes place. When continuous turbulence is observed (as is the case here), the passive scalar field becomes more complex and isoconcentration surfaces created due to secondary mixing events (mixing of previously mixed fluid parcels with concentrations closer to the mean) are quite common. Thus, interfaces between both previously unmixed parcels as well as already n1ixed parcels are detected by thresholding values close to the mean. For thresholds closer to large concentra tions (extreme temperature fluctuations in our case), the probability of crossing an interface due to a secondary mixing event decreases; thus relatively more points in the set have a similar physical origin. These correspond to the primary mixing events. Previous arguments become more apparent when analyzing thresholds closer to the small fluctuations observed. This is well illustrated by the first 2 s of the record in Figure 1. In the convective surface layer, cooler parcels originate from the adiabatic region above the surface layer. Regardless of their original level,


185

such parcels arrive unmixed at the surface. Their temperature equals that of the potential temperature in the adiabatic mixed layer 8. Thus, by setting thresh olds close to 8, we pick up points on the surfaces that delineate parcels which descended from the mixed layer. Such points form fractal sets (which can be seen in Figure 4, with lines marked with arrows) of fractal dimensions D in the range of 0.23-0.66. This corresponds well to the findings of Sreenivasan et ale (1989), although we observe a wider range of fractal dimensions. However, the situation is more complicated for positive temperature fluctuations. The most positive fluc tuations, advected past the temperature sensors (the buoyant updrafts) originated from the hot surface. The surface nlay have a non-uniform temperature, resulting in extreme variability of positive fluctuations. Thus, results of functional box counting for positive thresholds differ from those for negative thresholds. This is attributed to the asymmetry between updrafts and downdrafts. A convective boundary layer exhibits a small positive skewness in the vertical velocity and temperature fields. It is this property, resulting from surface heating, which is responsible for the asynlmetric distribution of the fractal dinlensions between positive and negative thresholds. The above discussion suggests that the approaches of Sreenivasan et ale (1989) and Miller and Dimotakis (1991) may be complementary. While interfaces corre sponding to the primary mixing events exhibit a fractal behavior, the inability of detemlining the physical origin of parcels involved in secondary mixing events by thresholding alone results in sets of points which cannot be described in terms of fractals. However, it should be pointed out that evidence of constant fractal dimension of interfaces as found in Sreenivasan et ale (1989) was not found here. There are other possible reasons for this discrepancy. First of all, the use of the frozen flow hypothesis may cause improper division of the investigated records into boxes since box width depends on the instantaneous velocity value rather than on the mean value. Secondly, the tenlperature fluctuations may be interpreted as some passive scalar concentration fluctuations. This, in the con vective surface layer with dynamical influences of temperature fluctuations due to buoyancy of air parcels, is questionable and nlay influence the measurenlent results. Thirdly, it is possible that the convective surface layer conditions are sim ilar to the assumptions made by Constantin et al. (1991) rather than to those used by Sreenivasan et ale (1989). In this case, the value D = 0.66 is consistent with results of Constantin et ale (1991). Finally, the discrepancy may be explained by using detrended data due to changes of mean temperature in the surface layer. The temperature dissipation rate TD has a multifractal nature, and results resembling those of the dissipation rate of passive scalars in laboratory flows (Prasad et al., 1988; Prasad and Sreenivasan, 1990), which varies highly with the observed intermittency exponent, have been obtained. There are also other differences. The first and most important one is the scaling range. While Prasad and Sreenivasan (1990) observed scaling of moments in the range of 3 'r/ (pix el size, the smallest resolvable scale) to rv250 'r/, we find evidence of scaling

186

SZYMON P. MALINOWSKI AND MONIQlTE Y. LECLERC

between 0.16 and 3 s which, for typical measurement conditions (windspeed 5 mls at 6 m, assuming roughness length z = 2 cm, logarithmic wind profile and TJ rv 1 mm), corresponds to the range 50-1000 TJ, while the estimated smallest resolvable scale was smaller than 10 TJ. While data of Miller and Dimotakis (1991) did not cover large scales, the difference in the upper end of the scaling range is due to different experimental conditions; the discrepancy in the lower end of the scaling range can be attributed to one or several possibilities. For example, it can result from the measurement method (point Eulerian n1easurements), sensor asymmetry or small-scale convective effects driven by temperature fluctuations. Unfortunately, we do not have information which could suggest the most like ly cause. In the observed scaling range, the multifractal properties of TD can be stated for the small range of moments (-1.5 < q < 3.5). The extension of results to more negative moments is limited by the resolution of the temperature sensor while short records cause divergence of high positive moments. However, it seems interesting that despite a clear divergence of T(q) for q > 3.5, the left branch of f(a) for higher moments does not diverge (Figure 8), suggesting that f(a) is less sensitive to the divergence of moments than T(q).

6. Conclusions Results suggest a potential for the use of fractal techniques in investigations of turbulence in the atmospheric boundary layer. In contrast with most laboratory flows, turbulence in this layer obeys a wide range of scales and is a physical system ideally suited for studies using an analysis based on scale invariance. The above study indicates that surfaces of constant temperature in the convec tive surface layer closely resemble fluctuations of isoconcentration surfaces of passive scalars in turbulent laboratory flows. The fact that the temperature is adiabatic above the unstable surface layer enables one to trace back the origin of some of those freshly descended parcels to the observation point. Surfaces characterizing the primary mixing events appear to be fractal. The dimension D, estimated from one-dimensional sections through such a surface, is in the range of 0.23-0.66. Thresholding and box counting were incapable of determining the origin of other surfaces of constant temperature, especially for temperature val ues close to the mean which resulted in non-fractal sets, corroborating findings by Miller and Din10takis (1991). The temperature dissipation rate was found to be multifractal. However, the lower limit of the scaling range was significant ly larger than observed by Miller and Din10takis (1991). Further investigations should be performed to explain this. We think that further pursuit of investigations sin1ilar to those presented above could be fruitful. In particular, temperature fluctuations in the nocturnal boundary layer should be particularly revealing since it is characterized by high intermitten cy and sporadic vertical transport. Further work examining interfaces of virtual potential temperature should be pursued to isolate the contribution to buoyancy


187

by ten1perature fluctuations. Analyses of primary and secondary mixing events can potentially lead to unusual and new insights on the physics of mixing in the ABL.

7. Acknowledgments The authors wish to thank the Fonds pour la Formation de Chercheurs et I' Aide a la Recherche (FCAR), the National Science and Engineering Council (NSERC) and the Atmospheric Environment Service (AES) for the funding provided to develop the fast response temperature sensor used in this research, to perform field observations of turbulence and to support the analysis presented here.

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