Advances in Water Resources 28 (2005) 1230–1239
www.elsevier.com/locate/advwatres
Testing TaylorÕs hypothesis in Amazonian rainfall fields
during the WETAMC/LBA experiment
Germán Poveda *, Manuel D. Zuluaga
Posgrado en Recursos Hidráulicos, Escuela de Geociencias y Medio Ambiente, Universidad Nacional de Colombia, Cra 80 x Calle 65,
Bloque M2-315, Medellı́n, Colombia
Received 9 July 2004; received in revised form 8 March 2005; accepted 22 March 2005
Available online 9 June 2005
Abstract
TaylorÕs hypothesis (TH) for rainfall fields states that the spatial correlation of rainfall intensity at two points at the same instant
of time can be equated with the temporal correlation at two instants of time at some fixed location. The validity of TH is tested in a
set of 12 storms developed in Rondonia, southwestern Amazonia, Brazil, during the January–February 1999 Wet Season Atmospheric Meso-scale Campaign. The time Eulerian and Lagrangian Autocorrelation Functions (ACF) are estimated, as well as the
time-averaged space ACF, using radar rainfall rates of storms spanning between 3.2 and 23 h, measured at 7–10-min time resolution,
over a circle of 100 km radius, at 2 km spatial resolution. TH does not hold in 9 out of the 12 studied storms, due to their erratic
trajectories and very low values of zonal wind velocity at 700 hPa, independently from underlying atmospheric stability conditions.
TH was shown to hold for 3 storms, up to a cutoff time scale of 10–15 min, which is closely related to observed features of the life
cycle of convective cells in the region. Such cutoff time scale in Amazonian storms is much shorter than the 40 min identified in midlatitude convective storms, due to much higher values of CAPE and smaller values of storm speed in Amazonian storms as compared to mid-latitude ones, which in turn contribute to a faster destruction of the rainfall field isotropy. Storms satisfying TH
undergo smooth linear trajectories over space, and exhibit the highest negative values of maximum, mean and minimum zonal wind
velocity at 700 hPa, within narrow ranges of atmospheric stability conditions. Non-dimensional parameters involving CAPE (maximum, mean and minimum) and CINE (mean) are identified during the storms life cycle, for which TH holds: CAPE mean/CINE
mean = [30–35], CAPE max/CINE mean = [32–40], and CAPE min/CINE mean = [22–28]. These findings are independent upon the
timing of storms within the diurnal cycle. Also, the estimated Eulerian time ACFÕs decay faster than the time-averaged space and the
Lagrangian time ACFÕs, irrespectively of TH validity. The Eulerian ACFÕs exhibit shorter e-folding times, reflecting smaller correlations over short time scales, but also shorter scale of fluctuation, reflecting less persistence in time than over space. No significant
associations (linear, exponential or power law) were found between estimated e-folding times and scale of fluctuation, with all estimates of CAPE and CINE. Secondary correlation maxima appear between 50 and 70 min in the Lagrangian time ACFÕs for storms
satisfying TH. No differences were found in the behavior of each of the three ACFÕs for storms developed during either the Easterly
or Westerly zonal wind regimes which characterize the development of meso-scale convective systems over the region. These results
have important implications for modelling and downscaling rainfall fields over tropical land areas.
2005 Elsevier Ltd. All rights reserved.
Keywords: Taylor hypothesis; Amazonia; Rainfall fields; CAPE; CINE; Space-time correlation
*
Corresponding author. Tel.: +57 4 4255122; fax: +57 4 4255103.
E-mail address: [email protected] (G. Poveda).
0309-1708/$ - see front matter 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.advwatres.2005.03.012
G. Poveda, M.D. Zuluaga / Advances in Water Resources 28 (2005) 1230–1239
1. Introduction
1.1. The significance of TaylorÕs hypothesis
Understanding the space–time dynamics of rainfall
fields constitutes one the most challenging topics of geophysics. An important observation that has guided the
research refers to the validity of ‘‘TaylorÕs hypothesis
of the frozen field’’, in the space–time correlations of
rainfall fields [45,5], which was originally stated in the
study of turbulence of fluid flows [41,24]. To understand
the hypothesis, let n(t, x) denote the space–time rainfall
intensity at some time t > 0, and some location x in a
two dimensional region, and assume that rainfall is
homogeneous in space and stationary in time. Let the
space–time correlation be denoted by, q(s, r) =
Corr{n(t, x), n(t + s, x + r)}. It depends only on the time
lag s and spatial distance r, due to stationarity and
homogeneity of the random field. Then TaylorÕs hypothesis states that
qðs; 0Þ ¼ qð0; UsÞ;
ð1Þ
where U is some characteristic speed vector. It connects
the spatial correlation at two points at the same instant
of time with temporal correlation at two instants of time
at some fixed location. The hypothesis implies that spatial fluctuations are carried past a fixed point by the
mean flow speed without undergoing any essential
change. Zawadzki [45] studied radar rainfall data of
mid-latitude convective storms, and concluded that TaylorÕs hypothesis holds in rainfall intensity fields for time
periods shorter than 40 min, and then it breaks down.
ZawadzkiÕs results showed that at time lag s > 40 min,
the temporal correlation falls below the spatial correlation. This time scale roughly corresponds to the life cycle
of mid-latitude convective rainfall cells. His results also
indicated that correlation in space exhibits more memory than correlation in time, since the tails of the temporal correlations fit well to exponential functions, whereas
the spatial correlation tails fit well to power laws. Subsequently, Crane [5] using spectral analysis of rapid-response gages located in the Rhine valley (Germany),
observed that TaylorÕs hypothesis holds in rainfall data
up to a cutoff time of 30 min and for spatial scales less
than 20 km, and then it breaks down. At the microscale,
Lovejoy and Scherzer [22] reports lidar observations of
raindrops in which TaylorÕs hypothesis does not hold,
within a range of spatial scales 3–500 m, and timescales
from 0.1 s to 500 s.
The validity of TaylorÕs hypothesis and its cutoff time
reflects important dynamical features of rainfall fields,
and therefore it provides a test for models of the
space–time dynamics of rainfall fields, including those
based on the stochastic theory of point random fields
[19,43,9,11,17,4,14]. The breakdown of TaylorÕs hypothesis is caused by the presence of anisotropy between
1231
space and time in the advection of rainfall cells across
the EarthÕs surface, because the life cycle of rainfall cells
themselves destroy the space–time isotropy. This explains why TaylorÕs hypothesis does hold infinitely in
isotropic space–time theories of rainfall, [43,13,32].
The role of dissipation in the cutoff in TaylorÕs hypothesis for rainfall correlation was generalized to a wide
class of possible stochastic short-memory rainfall models [11]. In fact, lack of evidence for scaling in time,
which translate into a finite scale of fluctuation [42],
does not preclude scaling properties in space. For time
lags for which TaylorÕs hypothesis holds, one can expect
that scaling in space is reflected in time scaling [46].
The validity of TaylorÕs hypothesis is also used as a
test for scale-independent models of rainfall fields,
including fractal, multi-scaling, and random cascade
models [20–22,12,31,33,8,15,29]. For instance, [32] indicate that TaylorÕs hypothesis is valid up to a correlation
time of the generator process Wt of a random cascade
model of meso-scale rainfall. Other efforts within the
context of cascade theories, based on generalizations
of TaylorÕs hypothesis are discussed in Lovejoy and
Scherzer [23].
1.2. Amazonian rainfall and the WETAMC/LBA
campaign
The Amazon River basin constitutes a paradigm of
complexity in the land surface-atmosphere system, due
to its areal extent of more than 6.4 million km2 (the largest river basin in the world), its tropical setting, and
complex eco-hydro-climatological dynamics, which exert global influence. The hydro-climatic and ecological
functioning of the Amazon is currently being investigated by the Large Scale Atmosphere–Biosphere Experiment in Amazonia (LBA) [1,35,28,7]. Modelling results
suggest that strong changes may occur in local, regional
and global atmospheric circulation patterns associated
with deforestation over the Amazon [39,48,49,26,44].
The annual cycle of precipitation exhibits a wet season
during November–March and a dry season during
May–September, due to the latitudinal migration of
the Intertropical Convergence Zone [30], which interacts
with the seasonal cycle of moisture advection by the low
level winds from the Atlantic Ocean, and through complex interactions of the land surface-atmosphere system,
including important precipitation recycling feedback
from evapotranspiration [36,10].
At smaller space–time scales, convective rainfall over
the Amazon has been investigated during the Wet Season Atmospheric Meso-scale Campaign/LBA (WETAMC/LBA) experiment in Amazonia, developed
during the period January–February 1999. The WETAMC/LBA campaign focused on the dynamical,
microphysical, electrical, and diabatic heating characteristics of tropical convection in this part of Amazonia.
1232
G. Poveda, M.D. Zuluaga / Advances in Water Resources 28 (2005) 1230–1239
For a detailed review and results of the experiment, see
Silva Dias et al. [37,38] and [25]. During the WETAM/
LBA campaign, two main regimes in the convective
activity were observed [34,18,3]. These two regimes were
closely associated with the zonal wind component in the
low tropospheric layer at 850–700 hPa; hence they were
coined as Westerly and Easterly regimes.
Using information form a set of 12 storms recorded
during the 1999 WETAMC/LBA field campaign, in this
paper we intend: (i) to test for the validity of TaylorÕs
hypothesis and the possible existence of a cutoff timescale in land-based convection-dominated tropical rainfall, and how the results compare with those obtained by
Zawadzki [45] for mid-latitude convective rainfall; (ii) to
quantify differences in the decay of ACFÕs for Amazonian storms, and (iii) to understand how the results in
(i) and (ii) are linked to dynamical and thermodynamical characteristics of the atmosphere during the storms,
which include their regime, average zonal wind velocity
at 700 hPa, and atmospheric stability indices including
Convective Available Potential Energy (CAPE), and
Convection Inhibition Energy (CINE). Required definitions and theoretical considerations are shown in Section 2, while data sets are described in Section 3,
results are provided in Section 4, and conclusions are
summarized in Section 5.
2. Definitions and theory
This section uses Zawadzki [45] definitions for the
time Eulerian, Lagrangian, and Spatial autocorrelation
functions (ACF). Let R(x, y, t) represents the rainfall
rate as a function of space and time, whose scales are defined for an isolated storm in terms of an area S and a
time T, such that
Rðx; y; tÞ P 0;
for ðx; yÞ 2 S and t 2 T ;
Rðx; y; tÞ ¼ 0;
outside this region.
ð2Þ
A time–space autocorrelation function of R is therefore
given as
AS;T ða; b; sÞ
Z Z
1
Rðx; y; tÞRðx þ a; y þ b; t þ sÞ dx dy dt;
¼
ST T S
ð3Þ
where a, b and s represent the lag variables, and the normalized ACF defined as,
a0 ða; b; sÞ ¼
AS;T ða; b; sÞ
AS;T ð0; 0; 0Þ
ð4Þ
will be independent of S and T, provided these parameters are large enough to contain the entire storm system.
Note that the normalized ACF is an even function varying between 0 and 1, and that the definition of the ACF
does not require the assumption of stationarity and
homogeneity of R(x, y, t). Henceforth, we denote time
averages with an overbar, and space averages by h.i,
and thus Eq. (3) becomes
AS;T ða; b; sÞ ¼ hRðx; y; tÞRðx þ a; y þ b; t þ sÞi.
ð5Þ
The instantaneous space ACF is defined by
AS ða; bÞ ¼ hRðx; y; tÞRðx þ a; y þ b; tÞi.
ð6Þ
The time-averaged space ACF is defined for s = 0 in
Eq. (5), as
AS;T ða; b; 0Þ ¼ hRðx; y; tÞRðx þ a; y þ b; tÞi.
ð7Þ
And the time Eulerian ACF is obtained from (5),
with a = b = 0. Thus,
ES;T ðsÞ ¼ AS;T ð0; 0; sÞ ¼ hRðx; y; tÞRðx; y; t þ sÞi.
ð8Þ
The cross-correlation function of rain rates at two
different times is defined as
hRðx; y; t1 ÞRðx þ a0 ; y þ b0 ; t2 Þi.
ð9Þ
There exist values of a0 and b0, which will maximize
this expression. These lags represent the linear displacement of the pattern R(x, y, t1) with respect to a second
pattern R(x, y, t2), which gives the optimal matching between them. Thus, a vector displacement of the storm is
defined during the interval (t2 t1) by the pair (a0, b0).
In general, a0 and b0 depend on t1 and s = t2 t1.
The velocity components of the storm system are defined by
a0
b
; U y ðtÞ ¼ lim 0 .
ð10Þ
s!0 s
s
Assuming a constant velocity U for the storm system,
we get
U x ðtÞ ¼ lim
s!0
a0 ¼ U x s;
b0 ¼ U y s.
ð11Þ
Now we define a Lagrangian ACF, by making a = a0
and b = b0 in Eq. (5), as
LS;T ðsÞ ¼ AS;T ða0 ; b0 ; sÞ
¼ hRðx; y; tÞRðx þ a0 ; y þ b0 ; t þ sÞi.
ð12Þ
It is worth noting that the Eulerian ACF is dependent
on the motion of the storm, while the time Lagrangian
ACF is defined relative to the storm local coordinates,
and therefore it is independent of the stormÕs motion.
For clarity,
AS;T ð0; 0; 0Þ ¼ AS ð0; 0Þ ¼ ES;T ð0Þ ¼ LS;T ð0Þ ¼ hR2 i;
ð13Þ
2
where hR i is the mean square value over S and T, of
R(x, y, t). The normalized ACFÕs are obtained by dividing Eqs. (7), (8) and (12) by hR2 i, and using Eqs. (4)
and (13):
aða; bÞ ¼ a0 ða; b; 0Þ;
eðsÞ ¼ a0 ð0; 0; sÞ;
lðsÞ ¼ a0 ða0 ; b0 ; sÞ.
ð14Þ
G. Poveda, M.D. Zuluaga / Advances in Water Resources 28 (2005) 1230–1239
The instantaneous space ACF is normalized by dividing Eq. (6) by AS(0, 0) = hR2i to obtain
aða; bÞ ¼
AS ða; bÞ
.
hR2 i
ð15Þ
1233
tically stationary in the sense that the properties of space
variations can be obtained from time variations at a
point by converting time into space through the velocity
of the storm system, [45].
2.1. TaylorÕs hypothesis in rainfall fields
3. Datasets
For the case in which rainfall rates are expressed in a
time-independent coordinate system, i.e. (x 0 , y 0 ) are the
coordinates of the storm, and R(x 0 , y 0 , t) is the rainfall
rate in such coordinates, then
The state of Rondonia (Brazil) is located in the southwestern part of Amazonia. Its vegetation is dominated
mainly by tropical forests, deforested pastures, and spatially forested savannas. The relief averages 300 m, ranging between 50 and 1000 m. Average temperature is
24–25 C, with maxima between 28 and 29 C and minimum around 22 C. Mean annual precipitation ranges
between 2000 and 2500 mm, with a monthly average
of 250 mm/month during the wet season (October–
April), and 50 mm/month during the dry season.
Our data set consisted of radar rainfall rates of
storms developed in Rondonia during the WETAMC/
LBA experiment, which were recorded by the S-POL
(S-band, dual polarimetric) radar located at 61.9982W,
11.2213S. Original data consisted of microwave band
reflectivity, which is directly related to rainfall intensity at a spatial scale of 2 km · 2 km, over a circular
area of 31,000 km2. Surveillance scans were available
between 7 and 10 min. The resulting processed rainfall
maps were produced by the Colorado State University
Radar Meteorology Group, and made available from
the URL http://tornado.atmos.colostate.edu/lbadata/radar/radar_main.html. Our data set consisted of 12 storms
developed during the WETAMC/LBA campaign, whose
main characteristics are shown in Table 1. Fig. 1 shows a
sequence of the January 11th 1999 storm, from 13:53 to
17:45, local time.
To estimate the Eulerian time ACF, radar data are
used at each time interval, s, without spatial displacement (a = b = 0), while the Lagrangian ACF requires
the search of a pair of parameters a and b that maximize
Eq. (12). Such values of a and b represent the linear
Rðx; y; tÞ ¼ Rðx0 ; y 0 Þ ¼ R0 ðx U x t; y U y tÞ.
ð16Þ
Therefore the Eulerian ACF, from Eq. (8) becomes
ES;T ðsÞ
¼ hR0 ðx U x t; y U y tÞR0 ðx U x t U x s; y U y t U y sÞi
ð17Þ
and the time-averaged space ACF, from Eq. (7),
becomes
AS;T ða; b; 0Þ
¼ hR0 ðx U x t; y U y tÞR0 ðx U x t þ a; y U x t þ bÞi.
ð18Þ
Taking a = Uxs and b = Uys in Eq. (18), then by
comparing (17) and (18), in the direction of motion we
get under the assumption of isotropy,
ES;T ðsÞ ¼ AS ðU sÞ ¼ AS ðU sÞ;
ð19Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
U ¼ 2 U 2x þ U 2y .
ð20Þ
Therefore, using Eq. (13), we obtain
aðU sÞ ¼ eðsÞ.
ð21Þ
TaylorÕs hypothesis is applicable to precipitation processes if Eq. (21) does hold. That is, the storm is statisTable 1
Physical and statistical characteristics of the 12 analyzed storms
Storm
date
Start
time (LTC)
End
time (LTC)
Duration
(h)
Time
interval (min)
Maximum
intensity (mm/h)
1/e Eulerian
(min)
1/e Lagrangian
(min)
990111
990112
990113
990114
990115
990116
990126a
990201
990220
990228
990126b
990225
13:53
14:31
12:39
09:02
19:06
00:02
00:09
00:07
00:06
00:09
19:06
21:07
18:46
18:45
23:46
14:34
23:53
05:53
09:57
23:10
23:56
22:06
22:18
01:50
3.8
4.2
11.1
5.5
4.7
5.9
9.8
23.7
23.1
21.9
3.2
4.7
7
11
11
7
10
11
10
10
10
10
10
10
183
150
163
117
392
180
160
128
163
146
214
165
8
11
26
12
16
11
11
15
14
21
17
17
37
23
47
14
23
32
32
25
29
33
34
36
1234
G. Poveda, M.D. Zuluaga / Advances in Water Resources 28 (2005) 1230–1239
Fig. 1. A sequence of precipitation patterns of the storm of January 11, 1999 from 13:53 to 17:45 local time. Higher intensity rainfall rates
correspond to darker colors, and the central point indicates the radar location. Spatial scale is shown on the top left graph.
displacement of the storm system. The time-averaged
spatial ACF is estimated using the averaged values of
a and b for all lags, and using Eq. (7), as input values.
Thus, there exists an estimated Spatial ACF for each
lag, s. As we already mentioned, all the estimated ACFÕs
are divided by hR2 i, to obtain normalized estimates.
Information of zonal wind velocity at 700 hPa during
the studied storms was obtained from the Climate
Reanalysis project from the National Center for Atmospheric Research-National Center for Environmental
Prediction (NCEP/NCAR) [16], for the region between
61W to 62.8W and 10.4S to 12.1S, corresponding to
the S-POL radar coverage region. Data were obtained
four times per day at 00, 06, 12 and 18 h. Estimation
of thermodynamic indices (CAPE and CINE) was
performed using data recorded from radiosondes deployed during the WETAMC campaign inside the ABRACOS project region. ABRACOS is the acronym of the
Anglo-Brazilian Amazonian Climate Observation Study;
http://www.nwl.ac.uk/ih/www/research/babracos.html),
which is located inside the S-POL radar coverage region.
Values of CAPE and CINE were estimated according to
definitions given by Curry and Webster [6, p. 199].
4. Results and discussion
Analysis of estimated ACFÕs indicates that 9 out of 12
storms exhibit significantly different decaying autocorrelation functions, meaning that TaylorÕs hypothesis does
not hold. Those storms are identified in the first column
of Table 1 as 990113, 990115, 990114, 990116, 990201,
990220, 990228, 990126b, and 990225. Fig. 2 shows
the normalized Lagrangian, Eulerian and Spatial ACFÕs,
and the space–time trajectories of two selected storms.
The remaining 3 storms (990111, 990112 and 990126a)
do satisfy TaylorÕs hypothesis. Fig. 3 shows the ACFÕs
for two of the storms which exhibit similar decay for
the time Eulerian and the time-averaged space (Spatial)
ACFÕs. It turns out that TaylorÕs hypothesis is valid up
to a certain cutoff time lag of approximately s =
10–15 min. Results (not shown) indicate that the timing
of the storms during the diurnal cycle does not affect the
validity of these conclusions.
A detailed analysis of the storm paths suggests that
TaylorÕs hypothesis is valid depending upon the type
of storm trajectory. TaylorÕs hypothesis is not valid for
storms showing erratic (space-filling) trajectories. One
of TaylorÕs hypothesis fundamental assumptions is that
the velocity field advects the storm properties, thus making it possible to interchange time for space in the
ACFÕs. On the other hand, storms undergoing smooth,
linear and constant velocity trajectories, as shown for
the second set of storms (Fig. 3), do satisfy the underlying assumptions and therefore TaylorÕs hypothesis,
within the aforementioned cutoff time scales.
For storms satisfying TaylorÕs hypothesis, the identified cutoff timescale of 10–15 min could be associated
with some observed physical characteristics of rainfall
cells. Silva Dias et al. [38] describe the organization of
convection into precipitating lines, whose convective
cells exhibit fairly erect vertical structures (see their Figures 15 and 16), except during the mature phase when
the convective vertical structure become tilted as the
cells extend to above 9 km. In any case, storm cells exhibit horizontal extents of approximately 4000–4500 m,
and horizontal velocities on the order of 6–8 ms1,
which means that the timescale of such rainfall cells is
G. Poveda, M.D. Zuluaga / Advances in Water Resources 28 (2005) 1230–1239
1.0
60
_____
Eulerian
. . . . . Lagrangian
___
Spatial
0.6
T= 0
Latitude (km)
ACF value
0.8
1235
0.4
40
x
T= 287 T= 197
x T=
xx
T= 49 x x x
T= 98 T= 148
247
20
0
xxx Xxx x
-20
0.2
-40
0
10
20
30
40
50
60
70
-40 -20
Lag (minutes)
1.0
Latitude (km)
ACF value
. . . . . Lagrangian
_ _ _ Spatial
0.6
20
60
_____ Eulerian
0.8
0
40
Longitude (km)
0.4
x
x
40
x
xX
T= 1317
x
x
x
x
xxx x
x
x
T= 1025
x T= 917
x
x
T= 971
20
0
x
x T= 809
T= 863
T= 755
x
x
x T= 593
x x T= 377
x
T= 215
-20
0.2
x
xx
x T= 1187
T= 1133
T= 1079
x
T= 0
x T= 161
x x
x
T= 53
T= 107
-40
0
10
20
30
40
50
60
70
-40 -20
Lag (minutes)
0
20
40
Longitude (km)
Fig. 2. Time-averaged space ACF, with the scale of space lag converted to time scale, together with the Eulerian and Lagrangian time ACFÕs, of two
selected storms for which TaylorÕs hypothesis does not hold. Storm 990115 (top), and storm 990228 (bottom). Trajectories of the storm centroids are
shown on the right, with time (T, min) evolution shown at the inset.
1.0
_____
Eulerian
. . . . . Lagrangian
_ _ _ Spatial
0.6
50
Latitude (km)
ACF value
0.8
0.4
T= 33
T= 66
T= 99
0
x x
x
T= 198
x
0.2
x
x
0
10
20
30
40
50
60
70
-50
0
50
Longitude (km)
Lag (minutes)
1.0
T= 0
_____
T= 57
Eulerian
. . . . . Lagrangian
_ _ _ Spatial
0.6
0.4
0
10
20
30
40
50
60
Lag (minutes)
70
T= 115
T= 173
x
x
xx
T= 230
0
x
x
T= 254
x
-50
0.2
x
x
50
Latitude (km)
0.8
ACF value
x
x
x
x T= 232
x
x
-50
x
T= 132
T= 165
T= 0
x x
x
x
xx
-50
0
50
Longitude (km)
Fig. 3. Similar to Fig. 2 for two selected storms satisfying TaylorÕs hypothesis. Storm 990111 (top), and storm 990112 (bottom).
on the order of 10–15 min. On the other hand, convective activity over the region exhibits pulses of maximum
vertical wind velocity, with an average periodicity of
approximately 30 min; see Fig. 12 of Silva Dias et al.
[38]. One can conjecture that the cutoff time of TaylorÕs
hypothesis corresponds to half the time interval between
such convective activity pulses, during which rainfall
cells regenerate in association with the updrafts, and dissipate in association with download precipitation.
Also, our results point out that in Amazonian convective storms, either TaylorÕs hypothesis does not hold
or its validity is restricted to a much shorter time scales
than the 40-min threshold identified for mid-latitude
convective storms, [45]. Differences between cutoff time
1236
G. Poveda, M.D. Zuluaga / Advances in Water Resources 28 (2005) 1230–1239
scales between Amazonian and mid-latitude storms may
be explained in terms of the physical mechanisms
governing both tropical and extra-tropical land-based
convective storms, including the interaction between
highly localized deep convection mechanisms with the
large-scale flow, strength of land surface-atmosphere
interactions, evapotranspiration and surface moisture
convergence rates, diurnal cycle of surface heating, vertical wind speed rates, and atmospheric stability characteristics. Indeed, relevant differences are observed
between values of CAPE and wind speed for Amazonian
and mid-latitude convective storms. Our results (Table
2) indicate that average CAPE range between 1177 J/
kg and 3197 J/kg, whereas for the mid-latitude storms
studied by Zawadzki et al. [47], CAPE hardly exceeds
the value of 850 J/kg. Such differences in CAPE reflect
a much more unstable atmosphere in tropical storms,
in comparison to mid-latitude storms. A stronger atmospheric instability contributes to destroy isotropy in
space, thus explaining the shorter cutoff time in Amazonian storms. Also, wind speed does not surpass 9 m/s in
Amazonian storms, whereas they get 15–20 m/s in midlatitude storms, as reported by Zawadzki [45, see his
Table 2]. Higher wind speeds for mid-latitude storms
contribute to maintain the isotropy of the field for a
longer time scale, and the validity of TaylorÕs hypothesis
up to a longer cutoff time.
For the entire set of analyzed storms, the Eulerian
time ACFÕs exhibit the fastest decaying rates, as they
combine both space and time variability. This conclusion is evidenced by a shorter decorrelation or e-folding
time, defined as the time lag for which the ACFÕs reach
the value 1/e of its maximum. Our results indicate that
the Eulerian time ACFÕs exhibit e-folding times that
are nearly half of those for the Lagrangian time ACFÕs
(see last two columns of Table 1). These results can be
explained by the very definition of the time Eulerian
and Lagrangian ACFÕs, due to their dependence with
respect to the stormÕs motion, as stated before. Another
property characterizing the rate of decay of the three
ACFÕs is the scale of fluctuation, defined as [40,42]
Z 1
h¼
qðs; 0Þ ds;
ð22Þ
1
where q(s, 0) has already been defined in (1). The scale of
fluctuation was introduced by Taylor himself to estimate
the time interval between independent observations in
the study of turbulent flows, but it also quantifies the degree of persistence and the type of memory of stochastic
processes [27]. The scale of fluctuation of the time-averaged space ACFÕs is longer than the scale of fluctuation
of the Eulerian ACFÕs, irrespectively of TaylorÕs hypothesis validity. This result implies that, for Amazonian
storms, the time-averaged space ACFÕs are more persistent than the time Eulerian ACFÕs, in agreement with
findings of Zawadzki [45, see his Fig. 15] for mid-latitude convective storms. Another noticeable feature of
the estimated ACFÕs is the secondary correlation maxima appearing in the Lagrangian time ACFÕs for storms
satisfying TaylorÕs hypothesis. For storm 990111 (Fig. 3,
top), the secondary correlation maximum (around 0.4)
appears around 50–55 min, whereas for storm 990112
(Fig. 3, bottom), it appears approximately at 70 min.
The physics behind this observation requires further
investigation.
TaylorÕs hypothesis was also tested for storms developed during the Easterly and Westerly zonal wind regimes identified in Amazonian convection by Cifelli et
al. [3]. The first one occurred on January 26th
(990226b in Table 1), during an Easterly phase, and
the second one on February 25th (990225 in Table 1),
during a westerly phase. A relevant question refers to
statistical differences emerging from physical and
dynamical features of both storm generating mechanisms. Fig. 4 shows that differences between the estimated Eulerian, Lagrangian and Spatial ACFÕs are
Table 2
Values of the zonal wind velocity at 700 hPa, and estimated CAPE (maximum, minimum, mean) and CINE (mean) during the analyzed storms
Storm data
TH
Regime
Uwind
max
(m/s)
Uwind
min
(m/s)
Uwind
mean
(m/s)
CAPE
max
(J/kg)
CAPE
min
(J/kg)
CAPE
mean
(J/kg)
CINE
mean
(J/kg)
CAPEmean/
CINEmean
CAPEmax/
CINEmean
CAPEmin/
CINEmean
990111
990112
990113
990114
990115
990116
990126a
990201
990220
990228
990126b
990225
Yes
Yes
No
No
No
No
Yes
No
No
No
No
No
W
W
W
W
W
W
E
W
E
E
E
W
8.5
5.0
2.5
1.5
7.0
3.5
9.5
4.0
6.5
7.0
5.5
8.0
6.0
3.0
0.5
0.5
4.5
2.0
8.5
1.0
5.0
5.0
4.5
6.0
7.5
4.0
1.5
0.5
5.5
2.5
9.0
2.5
5.5
6.0
5.0
7.3
1460
3064
2167
2564
2107
1541
3846
2844
3562
3212
–
–
869
2631
1740
645
870
830
2197
817
1572
2279
–
–
1177
2847
1897
1579
1488
1212
3261
1717
2408
2745
2671
3791
39
95
200
31
95
90
95
177
150
52
129
85
30.2
30.0
9.5
50.9
15.7
13.5
34.3
9.7
16.1
52.8
20.7
44.6
37.4
32.3
10.8
82.7
22.2
17.1
40.5
16.1
23.7
61.8
–
–
22.3
27.7
8.7
20.8
9.2
9.2
23.1
4.6
10.5
43.8
–
–
TH in the second column refers to whether TaylorÕs hypothesis holds. Third column refers to Westerly (W), or Easterly (E) zonal wind regimes.
G. Poveda, M.D. Zuluaga / Advances in Water Resources 28 (2005) 1230–1239
-0.20
ACF Differences
-0.10
0.00
_____ Eulerian
0.10
. . . . . Lagrangian
_ _ _
Spatial
0.20
0
2
4
6
8
Number of lag
Fig. 4. Differences between the three ACFÕs for two storms pertaining
to the easterly and westerly zonal wind regimes identified in Amazonian convection by Cifelli et al. [3]. The first one developed on January
26th (990226b in Table 1), during an easterly phase, and the second
one on February 25th (990225 in Table 1), during a westerly phase.
negligible for these storms pertaining to both zonal wind
regimes. This result may be explained because of the
number of rain cells per day is very similar during the
two regimes, and also because the structural characteristics of the convective systems (size and lifetime) are quite
similar between the two regimes, although differences
have been found in the organization of the convection
[18]. Differences between the space–time variability of
storms during both regimes also arise in estimates of
higher statistical moments, and in measures of nonlinear
dependence [2].
Another objective of this study aims to investigate the
validity of TaylorÕs hypothesis with respect to dynamical
and thermodynamical characteristics of the atmosphere
during the studied storms. Table 2 shows the values of
zonal wind velocity at 700 hPa (maximum, mean, minimum), obtained from the NCEP/NCAR Reanalysis
Project, and values of CAPE (maximum, minimum,
and mean) and CINE (mean), during the entire duration
of the storms, which have been estimated from radiosondes during the WETAMC field campaign. Results
shown in Table 2 allow to draw the following general
conclusions:
• TaylorÕs hypothesis holds for storms exhibiting the
highest negative values of maximum, minimum and
mean zonal wind velocity at 700 hPa. High negative
zonal wind velocity values are consistently associated
with more straight and smoother storm trajectories,
and conversely small zonal wind velocity values are
associated with more erratic and space-filling storm
paths.
• Atmospheric stability conditions during the studied
storms are associated with the validity of TaylorÕs
hypothesis. Non-dimensional parameters were formed
1237
between CAPE (maximum, mean, and minimum) and
CINE (mean), for which TaylorÕs hypothesis was
identified to hold, as follows: CAPE mean/CINE
mean = [30–35], CAPE max/CINE mean = [32–40],
and CAPE min/CINE mean = [22–28]. The physical
significance of these non-dimensional ratios with
regard to TaylorÕs hypothesis remains to be investigated.
• Besides these two previous conclusions, there appears
to be no clear-cut differentiated values of the studied
dynamical and thermodynamical variables during
Amazonian storms, with respect to TaylorÕs hypothesis validity. Such variables exhibit a large range of
values, including maximum zonal wind velocity at
700 hPa [9.5 m s1 to 7 m s1], minimum wind
velocity at 700 hPa [9 m s1 to 7.25 m s1], maximum storm CAPE [1460 J kg1 to 3846 J kg1], minimum storm CAPE [645 J kg1 to 2631 J kg1], and
average CINE [31 J kg1 to 200 J kg1], without
any significant correlation among them (not shown).
Such disparity of atmospheric environments during
the studied Amazonian storms may explain their
(mostly) erratic trajectories, which in turn contribute
to explain why most of the storms do not satisfy TaylorÕs hypothesis, but also the short cutoff time of
those storms satisfying it.
• No significant association (linear, exponential or
power law) was found between estimated values of
e-folding time or scale of fluctuation, with respect to
all estimates of CAPE and CINE, for the set of studied storms.
5. Conclusions
The validity of TaylorÕs hypothesis in rainfall fields is
tested using information of radar rainfall rates of storms
spanning between 3 and 23 h, at 7–10-min time resolution. Storms were recorded in southwestern Amazonia,
at Rondonia, Brazil, during the wet season WETAMC/
LBA experiment, during January–February 1999. Our
findings indicate that TaylorÕs hypothesis does not hold
in most studied storms (9 out of 12), due to their erratic
(more space-filling) trajectories, characterized by exhibiting the lowest values of zonal wind velocity at 700 hPa,
irrespectively of the values of atmospheric stability indices such as CAPE and CINE, during the studied storms.
The identified erratic trajectories of those storms invalidate the underlying assumptions of TaylorÕs hypothesis.
On the other hand, TaylorÕs hypothesis was shown to
hold for a small set of 3 storms undergoing smooth,
linear trajectories over space, characterized with the
highest negative values of zonal wind velocity at
700 hPa, and at the same time depending upon atmospheric stability indices. We found narrow ranges of
1238
G. Poveda, M.D. Zuluaga / Advances in Water Resources 28 (2005) 1230–1239
non-dimensional parameters formed between CAPE
(maximum, mean and minimum) and CINE (mean),
whose values are associated with the validity of TaylorÕs
hypothesis, namely: (i) CAPE mean/CINE mean = [30–
35], (ii) CAPE max/CINE mean = [32–40], and (iii)
CAPE min/CINE mean = [22–28]. Otherwise, have have
found no more significant correlations between dynamical and thermodynamical values of the studied storms
with regard to the validity (or lack thereof) of TaylorÕs
hypothesis. The identified disparity of atmospheric environments pertaining to these Amazonian storms can be
a reflect of their (for the most part) erratic trajectories,
which contribute to explain why most of them do not
satisfy TaylorÕs hypothesis, but also the short time scale
for its break down (10–15 min) in those storms satisfying it. Such brief timescale may be explained by
observed physical characteristics (size and velocity) during the life cycle of convective rainfall cells in this Amazonian region [38]. For those storms satisfying TaylorÕs
hypothesis, such cutoff time scale is much shorter than
the one found by Zawadzki [45] for mid-latitude convective storms (40 min), due to characteristic life cycles of
the tropical storms, but also due to much larger values
of CAPE and smaller wind speeds in tropical storms
as compared with mid-latitude storms, which contributes to destroy the isotropy of the field. Up to such time
lag (10–15 min), one could expect that scaling in space
is reflected in time scaling [46]. The timing during
the diurnal cycle do not affect the validity of these
conclusions.
For our data set, Eulerian time ACFÕs decay much
faster than the Lagrangian time and the time-averaged
space ACFÕs, thus indicating that the Eulerian ACFÕs
exhibit shorter e-folding times, as well as less persistence.
Secondary correlation maxima appeared in the
Lagrangian time ACFÕs for storms which satisfy TaylorÕs hypothesis. The secondary correlation maximum
(around 0.4) appeared between 50 and 70 min. No significant correlations were found between e-folding time
and scale of fluctuation of the storms with thermodynamic stability indices like CAPE and CINE. Also, no
differences were found among each of the three ACFÕs
for storms developed during the Easterly and Westerly
zonal wind regimes, in spite of the differences found in
the kinematic and microphysical vertical structure characteristics [34,18,3].
Testing TaylorÕs hypothesis in rainfall fields is based
upon the type decaying features of the ACFÕs, which
are in turn associated with second-order statistical moments of the probability distribution function of the relevant stochastic processes. Analysis of higher–higher
order moments of the space–time scaling of Amazonian
storms indeed exhibit significant differences for
storms pertaining to both Easterly and Westerly regimes
[2]. As it was previously mentioned in Section 1, these
results shed light towards linking the physics and
statistical characteristics of Amazonian convection, but
also they serve as ground truth for model testing and
downscaling the results of meso-scale models over tropical land regions.
Acknowledgements
G.P. is grateful to the Cooperative Institute for Research in Environmental Sciences (CIRES), University
of Colorado, Boulder, USA, for their support as a Visiting Fellow, which allowed fruitful interaction with
V.K. Gupta and H. Diaz. G.P. thanks the LBA project
for insightful meetings of the scientific steering committee over the years. Thanks to W.F. Krajewski and two
anonymous reviewers for their valuable comments. This
research was funded by Dirección de Investigaciones de
Medellı́n (DIME), of Universidad Nacional de Colombia at Medellı́n, Colombia. M.D.Z. was also supported
by the ‘‘Académicos en Formación’’ Program from Universidad Nacional de Colombia.
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