AOKI S.

ASANUMA J.

CHOI T. J.

GAO Z.

HAYAKAWA N.

KIM J.

LEE H.

WANG J.

Tsukuba Repository

Scale Analysis of the Turbrulence Heat
Transfer over the Tibetan Plateau with Wavelet
Transform (Session 4: Problems for Flux
Measurements studies)
著者 AOKI S., ASANUMA J., HAYAKAWA N., KIM J., CHOI
T. J., LEE H., GAO Z., WANG J.
journal or
publication title
Bulletin of the Terrestrial Environment
Research Center, the University of Tsukuba,
Proceedings of the International Workshop on
GAME-AAN/Radiation
volume 1
number 別冊
page range 79-80
year 2001-03
URL http://doi.org/10.15068/00147189
Scale Analysis of the Turbrulence Heat Transfer over the Tibetan Plateau 
with Wavelet Transform 
S. Aoki (Grad. School, Nagaoka Univ. of Tech.), J. Asanuma* (Tsukuba Univ.), N. Hayakawa (Nagaoka Umv. 
of Tech.), J. Kim, T.J. Choi, H. Lee {Yonsei Univ.), Z. Gao, J. Wang {Chinese Acad. Sci.) 
1. Introduction 
Turbulent transfer of heat, momentum and trace 
gases is a phenomenon whose characteristics scale 
spans over several decades. As a rough sketch, Kol-
mogoroff's law of 5/3 power well describes obser-
vation at length scales between the integral scale 
and the micro scale, while an established theory is 
lacked at the scales larger than the integral scale. 
This region of the scale resides between the spec-
trum peak of turbulence and that corresponds to 
daily cycle of meteorological variables, and inte-
faces the two different scales, i.e. turbulence and 
larger scale phenomena. 
From a practical point of view, "energy closure 
problem", which currently draws much attention 
from the micrometeorological community[e.g. 8], 
necessitates the investigation of this scale region, 
邸 theaveraging time/length for the Reynolds av-
eraging is usually set within or around this scale re-
gion and as its inadequate selection could cause flux 
underestimation with the eddy correlation tech-
nique[e.g. 6].
Purpose of this paper is to identify the scales rel-
evant to turbulence heat transfer by applying the 
wavelet transform to the data acquired over the Ti-
betan Plateau, and to discuss possible causes of the 
energy imbalance. 
2. Data Analyzed 
Data subject to the analysis is a turbulence data 
acquired with an eddy correlation system over a flat 
and sparse grassland near Naqu city during GAME-
Tibet'98 IOP as a cooperated work between Ko-
rean, Chinese, and Japanese scientists. ・Detailed 
description of the observation site and the mea-
surement system are given elsewhere[!]. Energy 
budget study [1] with this data set has already re-
vealed that there is a significant energy imbalance 
of the order of 20% of the available energy when the 
turbulent flux are evaluated with eddy correlation 
technique. From this data set, 9 consecutive data 
•corresponding address: Terrestrial Environmental Re-
search Center, Tsukuba University, Tsukuba, lbaraki, 305-
8577, Japan. e-mail: asanumaCerc2.suiri.tsukuba.ac.jp 
Table 1: Analyzed runs. Time in BST 
??? ????Date 
07/28 
08/03 
08/16 
08/22 
08/30 
09/06 
09/12 
09/13 
09/14 
start time 
10:00:10 
10:06:59 
10:08:31 
10:08:30 
09:59:17 
10:00:50 
08:00:38 
10:08:31 
10:06:51 
end time 
17:17:04 
17:23:54 
17:25:26 
17:25:24 
17:16:11 
17:17:44 
15:17:32 
17:25:25 
17:23:45 
were selected for the subsequent analysis (Table 1). 
3. Wavelet Transform 
Wavelet transform came into the micrometeo-
rological community in the late'80s [e.g., 3] and 
quickly became a common tool with an anticipa-
tion that it could better describe the turbulence 
eddy than the Fourier transform do. Mathematical 
basis of the wavelet transform and its application 
to the turbulence are fully described elsewhere[e.g., 
2; 5] and wil be briefly explained here. Wavelet 
transform of the time series J(t) can be written as, 
閏 (a,b) =嘉i:J(t) ゅ(~) dt (1) 
whereゆ(t)is caled mother wavelet, which usually 
have finite support unlike its Fourier counterpart, 
i.e. trigonometric functions. Parameters a and b 
are called "scale" and "translate" respectively. As 
shown in equation (1), wavelet transform decom-
poses original time series into a wavelet packet spec-
ified by the scale a and the location b. 
Multiplication of the wavelet transform of two 
time series J(t) and g(t), i.e., W1(a,b)W9(a,b) in-
dicates the component of the covariance between 
J(t) and g(t) with the scale a observed at the time 
location b. Integration of this over the time, i.e., 
知 (a)=Jo助 (a,b)W9(a, b)db (2) -o 
is called wavelet covariance, and means the covari-
ance component with the scale a. This wavelet co-
variance, often called wavelet cospectrum, is the 
wavelet counterpart of Fourier cospectrum. 
30m1n 
? ?
?
0.03 
0.025 
0.02 
0.015 
0,01 
0.005 
orthogonal o 
non-orthogonal 
-0.05 l...l2_ 
10000 
゜
゜
1000 100 10 
a, (sec) 
orthogonal o 
non-orthogonal -
0.1 
30mm 
? ?
?
1.6e-05 
1.4e-05 
1.2e-05 
1e-05 
Se-06 
6e-06 
4e-06 
2e-06 
〇 ←
-2e-06 
-4e-06 
-6e-06 
10000 
4. 
??
?
Q 
100 10 
l1,, (se・c) 
0.1 
Figure 1:Calculated wavelet covariance for run 6. 
Upper panel: sensible heat -flux, lower: latent heat 
flux 
In this study, wavelet covariance for the heat 
transfer, Ewe ,Ewq is calculated for each of the Runs 
listed above. Haar wavelet is selected as a mother 
wavelet due to its simplicity and an computational 
methodology described in [4) was used to imple-
ment the orthogonal wavelet transform. 
Results and discussion 
Figure 1 shows the calculated wavelet covariance 
of the sensible and latent heat flux for Run 6. The 
length scale corresponds to the 30 minutes averag-
ing time that is used to evaluate Reynolds average 
Table 2: Contribution of the heat transfer at the 
scale larger than 30 minutes relative to the total 
heat transfer(in %). 
Run 
1 
2 
3 
4 
5 
6 
7 
8 ， 
In 
w-0 
1.4 
-1.8 
11.6 
-9.2 
-13.0 
3.4 
-17.0 
-3.9 
-6.1 
is also indicated in the figure. It is apparent that 
while the most of the heat istransferred at the scale 
of the order of 1 to 100 seconds, some portion of the 
transfer occurred by the eddy larger than the size 
corresponds to the 30 minutes; this suggests that 
the averaging time of 30 minutes is not long enough 
to capture alof the heat transfer. It should be also 
noted, however, that the large amount of the sen-
sible heat is also transferred at the scale around 7 
hours where the negative value of the latent heat 
flux is observed. This behavior of the wavelet co-
variance at the larger scale was found to differ with 
each Run: some Runs have heat transfer with op-
posite direction to those shown in Figure 1 within 
thes scale range beyond 30 minutes. 
In order to identify the contribution of the heat 
transfer at the larger scale, calculated covariance 
are integrated from 30 minutes to the total length 
of the data and for whole range of the scale, and the 
ratio of the two are calculated. The results are list 
in Table 2. These numbers indicates that the con-
tribution of the eddy larger than 30 minutes is not 
negligibly smal, but not large enough to explain al 
of the energy imbalance found in the surface energy 
budget. 
When the large scale components around 7 hrs 
were reconstructed into a time scale using inverse 
wavelet transform, it isidentified as a trend in the 
temperature and humidity with time scale of one 
day. However, lack of the data longer than one day 
in the data set prohibitted the further investigation. 
References 
w-q 
0.7 
2.3 
-5.5 
4.1 
3.4 
1.3 
11.7 
3.4 
2.4 
[1] Choi T, et al. Boundary Layer Meteorol, sub-
mitted, 2001. 
[2] Daubechies I. Ten Lectures on Wavelets. SIAM, 
1992. 
[3] Foufoula-Georgiou E and Kumar P, eds. 
Wavelets in Geophysics, vol. 4 of Wavelet Anal-
ysis and Its Application. Academic Press, 1994. 
[4] Howell J and Mahrt L. In Foufoula-Georgiou 
and Kumar [3], 107-128. 
[5] Kumar P and Foufoula-Georgiou E. 
Foufoula-Georgiou and Kumar [3], 1-44. 
[6] Mahrt L. J Atmos Oceanic Tech, 15(2), 416, 
1998. 
[7] Tennekes H and Lumley JL. A First Course in 
Turbulence. The MIT Press, Cambridge, MA, 
1972. 300pp. 
[8] Twine T, et al. Agri and Forest Met, 103, 279, 
2000. 


Scale Analysis of the Turbrulence Heat Transfer over the Tibetan Plateau with Wavelet Transform (Session 4: Problems for Flux Measurements studies)

Scale Analysis of the Turbrulence Heat Transfer over the Tibetan Plateau with Wavelet Transform (Session 4: Problems for Flux Measurements studies)

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