Journal of Geophysical Research: Atmospheres
RESEARCH ARTICLE
10.1002/2013JD021008
Key Points:
• Observation of stratospheric
turbulence layers with
balloon-borne instrument
• Energy dissipation rate profiles
with unprecedented precision
of millimeters
• No clear correlation between the
occurrence of turbulence and
Richardson number
Correspondence to:
A. Haack,
[email protected]
Citation:
Haack, A., M. Gerding, and F.-J. Lübken
(2014), Characteristics of stratospheric
turbulent layers measured by LITOS
and their relation to the Richardson
number, J. Geophys. Res.
Atmos., 119, 10,605–10,618,
doi:10.1002/2013JD021008.
Characteristics of stratospheric turbulent layers measured
by LITOS and their relation to the Richardson number
A. Haack1 , M. Gerding1 , and F.-J. Lübken1
1 Leibniz Institute of Atmospheric Physics at the Rostock University, Kühlungsborn, Germany
Abstract
Based on high-resolution turbulence measurements performed with the newly established
balloon-borne instrument Leibniz Institute Turbulence Observations in the Stratosphere (LITOS) during
the Balloon Experiments for University Students (BEXUS) 6 and BEXUS 8 campaigns from Kiruna, we
derived characteristics of stratospheric turbulence layers, like their thickness and distance in between.
Typically, the layers are ∼15–130 m thick and have a distance of ∼60–270 m, and their number increases
with altitude. Due to the very high measurement resolution of LITOS in the range of millimeters, we
obtain energy dissipation rate profiles with unprecedented precision. Within the turbulent layers we
get a mean dissipation rate of 3.4×10−2 W/kg (BEXUS 6) and 1.1×10−2 W/kg (BEXUS 8) corresponding to
a heating rate of 1 to ∼3 K/d. The profiles show an increase of the energy dissipation with altitude.
Comparisons with the Richardson number Ri preclude a clear correlation between the occurrence of
turbulence and Ri <1∕4. Despite the expected occurrence of turbulence at Ri < 1/4, we also observed
turbulent layers where Ri was >1∕4 and far beyond, independent of the scale over which Ri has
been determined.
1. Introduction
Received 10 OCT 2013
Accepted 19 AUG 2014
Accepted article online 23 AUG 2014
Published online 18 SEP 2014
Although turbulence and associated processes play an important role within many different aspects of
the atmosphere, they are not fully quantified or understood [e.g., Wyngaard, 1992; Fritts et al., 2003]. Even
within strongly stratified environments like the stratosphere turbulence occurs in layers of limited depth
induced by wave breaking or shear instabilities [e.g., Barat, 1982a; Sato and Woodman, 1982; Fritts et al.,
2003; Theuerkauf et al., 2011]. Those intermittent layers of enhanced turbulence are embedded in regions of
weak or no turbulence contributing to mixing and transport within this altitude range [e.g., Fritts et al., 2003;
Clayson and Kantha, 2008]. However, the generation of stratospheric turbulence and its consequences on,
e.g., the energy balance of the middle atmosphere as well as on the mixing of trace gases are not completely
understood so far. But almost all theoretical and numerical models dealing with atmospheric circulation,
dynamics, energetics, and composition must include the effects of turbulence [Gavrilov et al., 2005]. Furthermore, experimental and theoretical studies over the past two decades have shown that the stratosphere
plays an important role in the natural variability and forced response of the Earth system [e.g., Gerber et al.,
2012]. However, the existing observational uncertainties regarding stratospheric turbulence lead to, e.g., a
reduced quality of turbulence parameterization within models.
An important parameter for the atmospheric energy budget is the energy dissipation rate 𝜀, which estimates the strength of turbulence, i.e., the amount of energy dissipated into heat at the small scales of
the energy cascade. High-resolution balloon measurements offer a nice possibility to determine 𝜀 within
the stratosphere. For example, radiosonde data are able to achieve a vertical resolution of roughly 5 m
and thus permit observations of small-scale atmospheric structures and processes [Love and Geller, 2012].
Based on these data, the energy dissipation rate is determined using Thorpe scale analysis, which was previously developed in studies of oceanic turbulence [e.g., Thorpe, 1977; Gavrilov et al., 2005; Clayson and
Kantha, 2008; Wilson et al., 2011]. It turns out that a higher measurement resolution significantly improves
the accuracy of the energy dissipation rate. But those soundings are technically challenging, and therefore, stratospheric soundings quantifying turbulence at small scales are rare. In this paper we report the first
subcentimeter resolution observations of turbulence in the stratosphere.
One method to measure atmospheric turbulence is to deduce the power spectra of winds. The transition from the inertial subrange of turbulence to the viscous subrange, also known as the inner-scale of
turbulence (typically a few centimeters in the stratosphere), is determined by the energy dissipation. At the
HAACK ET AL.
©2014. American Geophysical Union. All Rights Reserved.
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Journal of Geophysical Research: Atmospheres
Table 1. Flight Parameters of BEXUS 6 and BEXUS 8
Balloon size/type
Gondola weight
Distance balloon gondola
Sensor type
Mean ascent rate
Maximum altitude
Sampling rate
Rotation and acceleration sensor
Tropopause height
a From
BEXUS 6
8 October 2008
BEXUS 8
10 October 2009
10,000 m3
121 kg
50 m
CTA
4.45 m/s
29 km
2 kHz
+a
9.7 km
12,000 m3
140 kg
50 m
CTA + CCA
4.7 m/s
27.8 km
2 + 8 kHz
+
8.0 km
LowCoINS experiment.
10.1002/2013JD021008
Leibniz Institute of Atmospheric
Physics we have developed the
balloon-borne in situ instrument LITOS
(Leibniz Institute Turbulence Observations in the Stratosphere) which
allows to measure temperature and
wind fluctuations down to scales of a
few millimeters and to directly determine the inner scale of turbulence by
analyzing the measured power spectra [Theuerkauf et al., 2011]. To the best
of our knowledge, LITOS is the only
instrument applied in atmospheric
physics that resolves these scales.
In section 2 we will shortly describe LITOS and the performed measurement campaigns. Section 3 presents
the characteristics of the observed turbulent layers. A statistical method for the exact determination of
the energy dissipation rate based on the measured spectra of velocity fluctuations will be outlined in
section 4, followed by a presentation of altitude profiles of 𝜀. Finally, in section 5 we discuss the relation of
the observed turbulent layers with the Richardson number Ri, which has been questioned recently.
2. Measurement Method, Database
2.1. Instrument
The new lightweight, compact balloon-borne instrument called LITOS (Leibniz Institute Turbulence Observations in the Stratosphere) has been designed for investigations of fine-scale turbulent fluctuations in the
temperature and wind field. Achieving a measurement resolution of typically 2.5 mm, the entire turbulence
spectrum down to the viscous subrange in the stratosphere is studied. So far, LITOS has been launched successfully several times from the institute site at Kühlungsborn (54◦ N, 12◦ E) as a stand-alone payload. During
two BEXUS campaigns (Balloon Experiments for University Students) in 2008 and 2009 at Kiruna (67◦ N, 21◦ E),
LITOS has been integrated into a larger gondola for stratospheric balloons. Generally, due to its large diameter, the balloon follows the ambient wind field during the ascent phase and the payload is also following
the wind at balloon height. Consequently, with our instrument we obtain the wind differences between the
wind at balloon height and at payload height caused by variations of the wind velocity. To observe wind
fluctuations we use a constant temperature anemometer (CTA) also called hot wire, where the measurement principle is based on convective cooling of a heated wire caused by the air flow passing the wire. For
our cases we use a commercial CTA system from Dantec Dynamics consisting of a Wheatstone bridge and a
tungsten wire with a diameter of 5 μm and a length of 1.25 mm. The convective heat loss of the wire caused
by changes of the ambient flow velocity generates differences in the bridge voltages. This signal is sampled
with a rate of 2 kHz (BEXUS 6) or with 2 kHz and 8 kHz (BEXUS 8). Hence, a spatial resolution of 2.5 mm or
0.6 mm is achieved, assuming that the balloon ascends with an average speed of 5 m/s. It should be pointed
out that in order to obtain absolute wind velocities the wire has to be calibrated during ambient conditions
similar to the conditions during the experiments. Extensive laboratory measurements have shown that it
is not necessary to derive absolute wind velocity values from the CTA signal to obtain turbulence parameters [Theuerkauf et al., 2011]. In fact, a spectral analysis of the unscaled voltage signal also yields the spectral
slope of the observed wind variations. Further details concerning the measurement principle and the payload design as well as laboratory results can be found in Gerding et al. [2009] and Theuerkauf et al. [2011].
In order to study the atmospheric background conditions during flight, a radiosonde has been integrated
into the BEXUS 6 gondola. The radiosonde measurements provide altitude profiles of temperature with a
resolution of ∼ 2 s, i.e., ∼ 10 m at ∼ 5 m/s ascent speed.
2.2. Campaigns and Database
LITOS has been launched from Kiruna as part of the BEXUS campaigns on 8 October 2008 (07 UT) and
10 October 2009 (08 UT). The most important launch parameters of both flights are summarized in Table 1.
During BEXUS 6 (2008) only wind fluctuations have been observed, while during BEXUS 8 (2009) wind and
HAACK ET AL.
©2014. American Geophysical Union. All Rights Reserved.
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altitude [m]
Journal of Geophysical Research: Atmospheres
19.5
19.5
19.4
19.4
19.3
19.3
19.2
19.2
19.1
19.1
19.0
19.0
10.1002/2013JD021008
L3
L2
1.34
1.36
1.38
1.4
L1
-0.03 -0.02 -0.01
voltage [V]
0
0.01 0.02 0.03
voltage fluctuations [V]
Figure 1. Example of the observed profiles of velocity fluctuations between 18.950 km and 19.585 km during the BEXUS
8 flight. Regions with strong turbulent fluctuations (marked by red points) can be clearly distinguished from calm regions
which are solely characterized by instrumental noise.
temperature fluctuations have been measured simultaneously. Additionally, during this flight two CTA sensors have been installed at two different corners of the BEXUS 8 gondola simultaneously measuring wind
fluctuations with a sampling rate of 2 kHz and 8 kHz. For further analyses of the BEXUS 8 flight, we considered only the turbulent layers which have been observed by both CTA sensors. In the following we will
concentrate on the results obtained within the stratospheric wind field, i.e., within the altitude range of
∼7 km and ∼29 km (BEXUS 6) or ∼27 km (BEXUS 8), respectively. As a typical example, Figure 1 shows
the measured voltage fluctuations which are directly related to wind fluctuations for the altitude region
between 18.95 km and 19.585 km during the BEXUS 8 flight. The left figure presents the raw data signal
(blue) and fitted spline (black line) used to subtract larger-scale variations from the raw data signal. As a
result, we obtain the profile of fluctuations as shown in the right part of the figure. The profile is characterized by distinct regions of strong voltage fluctuations followed by calm regions with few or no fluctuations.
Three turbulent layers are clearly visible, i.e., the first layer L1 between ∼19.0 and 19.24 km, the second
turbulent layer L2 between 19.3 and ∼19.41 km, and a third layer L3 between 19.43 and ∼19.54 km. The
turbulent regions can easily be identified by dense occurrences of high signal amplitudes up to ∼20 mV. In
between the turbulent layers, no wind fluctuations at all are detected and the signal basically shows instrumental noise of ∼3 mV which is twice the standard deviation. The boundaries of the turbulent layers have
been determined with a cluster algorithm which detects the previously mentioned regions of accumulated
high signal amplitudes within the given data profile. Of course, the results of this algorithm depend directly
on predefined parameters, in our case the distance d and turbulent neighbors n. More details and the results
of the cluster algorithm are presented in section 3.
Figure 2 shows the radiosonde profiles of temperature and the meridional and zonal wind components for
both BEXUS flights. During BEXUS 6 (Figure 2, left) the temperature decreases down to a minimum of −64◦ C
around 27 km. A slight tendency to small negative temperature gradients is prevailing, and therefore, more
or less instable regions inducing turbulence can be expected. Within the profiles of u and v the jet stream is
clearly visible at an altitude of ∼10 km, i.e., at the tropopause. Above 10 km the zonal wind remains positive
(i.e., in eastward direction) and the meridional wind component stays negative (i.e., in southward direction).
Compared to BEXUS 6, the temperature profile of BEXUS 8 (Figure 2, right) shows a similar behavior with
altitude. Again, above the tropopause (i.e., ∼8 km) the profile is characterized by regions with increasing
and decreasing temperature values. Consequently, these multiple changes in the temperature gradient provide the necessary conditions for turbulence. The profiles of the wind components reveal a meridional flow
in southward directions up to 20 km and a change to northward directions above 20 km. The zonal wind
components remain positive, i.e., an eastward wind flow dominates in the entire profile.
3. Properties of Stratospheric Turbulent Layers
One major point of interest is the number and vertical expansion of turbulent layers within the stratosphere. Since the obtained data set of LITOS is quite heterogeneous and comprehensive, an autonomous
and effective method is required to analyze the given data profiles.
HAACK ET AL.
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Journal of Geophysical Research: Atmospheres
-40
-20
0
20
40
25
-40
10.1002/2013JD021008
-20
0
20
40
25
20
20
T
v
u
15
10
10
5
5
-60
-40
-20
0
T
v
u
15
-60
-40
-20
0
Figure 2. Profiles of the radiosonde temperature (red) and meridional (black) and zonal (blue) wind components of (left)
BEXUS 6 and (right) BEXUS 8. The black line marks the tropopause height.
As outlined before, the observed turbulent layers are characterized by a high variability of the measured
wind values within a certain altitude distance. Hence, such concentrations or clusters of “turbulent values”
within the wind profile have to be detected by an algorithm in order to identify the beginning and ending
of a turbulent layer. Before such a cluster algorithm can be applied, a preprocessing of the given data set
has to be done in order to distinguish turbulent data points from nonturbulent data points. Based on the
noise level (3 mV) which is constant with altitude, each data point has been classified as turbulent (above
the noise level) or nonturbulent (below the noise level). The obtained profile of turbulent and nonturbulent points provides the basis for the cluster algorithm. Now the algorithm assigns turbulent data points to
one turbulent layer, i.e., cluster, depending on a certain number of adjacent turbulent points within a certain altitude distance. In other words, if a turbulent point has at least n turbulent neighbors within distance
d, it is associated with a cluster. We have chosen a distance parameter d of 5 m and a minimum number of
turbulent data points n = 300 (sampling rate of 2 kHz) and 800 (sampling rate of 8 kHz) for our study. That
means, if the next neighbor to an identified turbulent layer has itself at least 300 (800) neighbors above the
noise level within the next 5 m, this data point is added to the turbulent layer. If not, the layer ends and a
new layer is identified as soon as there are again 300 (800) turbulent data points within 5 m. With a larger
distance parameter of, e.g., d = 100 m different turbulent layers separated by 5–100 m would be attributed
as a single, broad layer. Therefore, a statistical analysis would result in less but broader layers compared to
our choice. Vice versa, a smaller distance parameter would reveal more but thinner turbulent layers. Obviously, the selection of d and n biases the results of the analyses, and we will therefore focus on qualitative
results. The choice of n has a smaller influence on the results, probably because the number of data points
within an even thin turbulent layer easily exceeds n, while the statistical noise is low enough not to produce
many data points above the lower signal limit. Turbulent layer thickness, distance between the layers, and
the number of layers have been counted in 1000 m steps for the complete profile of wind fluctuations of
BEXUS 6 and BEXUS 8. The profiles of the mean number of layers in Figure 3a show a high similarity between
BEXUS 6 and BEXUS 8. Generally, the values range between ∼10 and ∼27 turbulent layers per 1000 m with
a typical layer thickness of ∼40 m. Both profiles show a maximum around 13 km and a second maximum
around 25 km, while the latter reaches much higher values than the first one. That means that more turbulent layers have been observed within the middle stratosphere (i.e., above ∼16 km) compared to the lower
stratosphere. An explanation could be the vertical expansion of the turbulent layers, i.e., thinner turbulent
layers within the middle stratosphere possibly lead to a higher number of layers. Indeed, the profiles of the
mean layer thickness for BEXUS 6 and BEXUS 8 (see Figure 3b) show a general decrease with altitude. Only
around ∼18 km a maximum of the mean thickness is found. Again, both profiles are very similar and the
values vary between ∼15 m and ∼130 m. During BEXUS 8 the turbulent layers are a bit thinner compared
to BEXUS 6. However, during both flights more but thinner layers have been observed with increasing altitude. Figure 3c shows the mean distance between the turbulent layers ranging from ∼60 m to 270 m. The
HAACK ET AL.
©2014. American Geophysical Union. All Rights Reserved.
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Journal of Geophysical Research: Atmospheres
altitude[km]
30
a
30
b
30
10.1002/2013JD021008
c
30
25
25
25
25
20
20
20
20
15
15
15
15
10
10
10
10
5
5
0
10
20
30
0
50
100
5
0
100 200 300
5
d
0
50
100
Figure 3. (a) Mean number of turbulent layers within 1 km steps smoothed over 4 km for BEXUS 6 (red) and BEXUS 8
(blue). (b and c) Turbulent layer thickness and the mean distance between turbulent layers. (d) Turbulent fraction within
1 km steps.
data reveal a maximum distance around ∼18 km, where the thickest turbulent layers (see Figure 3b) and
the lowest number of layers (see Figure 3a) have been obtained. That means that this region must have
been quite turbulent resulting in thick turbulent layers with only thin nonturbulent regions in between. Furthermore, the distance between the layers is smaller for BEXUS 8 which corresponds to a higher number of
thinner layers compared to BEXUS 6. The lowest distance has been identified around 23 km (BEXUS 8) and
25 km (BEXUS 6), where on the other hand the highest numbers of turbulent layers have been obtained (see
Figure 3a). Finally, the turbulent fraction within 1 km steps has been determined and is shown in Figure 3d
for BEXUS 6 and BEXUS 8. Both profiles show values of nearly 100% in the tropopause region and decrease
with altitude. In accordance with the higher values for the thickness of the turbulent layers for BEXUS 6, a
higher turbulent fraction has been obtained compared to BEXUS 8. Or vice versa, the higher number of thinner turbulent layers during BEXUS 8 leads to a smaller turbulent fraction, even so the distance between the
layers is smaller compared to BEXUS 6. In summary, during both flights thinner turbulent layers have been
observed in the middle stratosphere and they are closer together resulting in a higher number of turbulent
layers than in the lower stratosphere. Furthermore, the turbulent fraction decreases with altitude. However,
the most remarkable point is the general similarity between the profiles of both flights.
In the literature only little information is given concerning the vertical thickness of turbulent layers in the
stratosphere or the distance between the layers. During the 1980s pioneering balloon measurements studying stratospheric turbulence were performed by, e.g., Barat [1982a, 1982b], Barat and Genie [1982], and Barat
and Bertin [1984]. Their observations reveal vertical depths of the turbulent layers ranging from less than
50 m up to 800 m. Gavrilov et al. [2005] obtained turbulent layer thicknesses of some 10 m from temperature
balloon measurements using Thorpe displacement profiles. Another paper by Sato and Woodman [1982]
shows radar measurements of thin stratospheric turbulent layers. The thickness was usually less than the
altitude resolution of the radar so that they could only estimate the vertical depth to be less than 150 m.
But from a zenith-swinging experiment they got an average value of about 50 m. Additionally, they determined a vertical separation between the turbulent layers from a few to several hundred meters. We have
yield similar results with LITOS for the vertical thickness as well as for the distance between the layers. Based
on the cluster results, we found that between 7 and 29 km, nearly 78% of the BEXUS 6 flight has been turbulent, while for BEXUS 8 the value is a bit lower, namely, 63% between 7 and 26.5 km. These surprisingly high
values probably result from the high measuring sensitivity of LITOS. Due to this sensitivity, even very small
turbulent layers are detected with LITOS leading to an unexpected high amount of turbulent regions in the
stratosphere. These findings emphasize the importance of stratospheric turbulence observations and the
analysis of its consequences.
4. Energy Dissipation Rates for BEXUS 6 and BEXUS 8
Turbulent spectra of wind fluctuations are expected to have a specific gradient, namely, a −5∕3 slope in
the inertial subrange and a −7 slope in the viscous subrange [e.g., Heisenberg, 1948; von Weizsäcker, 1948;
Tennekes and Lumley, 1985]. Due to the very high measurement resolution of LITOS, we are able to resolve
the turbulent spectrum down to the viscous subrange for the first time in the stratosphere [Theuerkauf et al.,
HAACK ET AL.
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Journal of Geophysical Research: Atmospheres
10.1002/2013JD021008
2011]. Figure 4 shows a typical spectrum of turbulent wind fluctuations
between 19.330 km and 19.355 km of
the BEXUS 8 flight. The spatial scale
L is derived from L = 2𝜋∕k = vb ∕f
where k is the wave number, f the
frequency, and vb the balloon ascent
velocity. As expected from theory,
the characteristic −5∕3 slope of the
inertial subrange is well identified
between spatial scales of 2 m and
0.06 m followed by a transition to
Figure 4. Turbulent spectrum of velocity fluctuations for a 25 m altitude
interval obtained during BEXUS 8 (2 kHz). The mean balloon ascent rate for the −7 slope in the viscous subrange,
i.e., < 0.045 m. Below 0.03 m the
this interval is 6.2 m/s. The red line shows the theoretical fit based on the
Heisenberg model. An inner scale of 4.5 cm and an energy dissipation rate spectrum shows instrumental noise.
of 0.9 mW/kg have been determined.
At scales larger than 2 m only a few
points are given due to the length of
the data section of only 25 m. Additionally, the first part of the spectrum is influenced by the spline fitting, i.e., the background conditions at
larger scales. Consequently, for a representation of the typical −3 slope of the buoyancy subrange, a wider
altitude section has to be chosen. However, this is not necessary for the determination of the energy dissipation rate. In order to retrieve the energy dissipation rate from the measured spectrum, a theoretical model
based on Heisenberg [1948] is fitted to the data similar to a technique developed for sounding rocket flights
[Lübken, 1992; Lübken et al., 1993]. The one-dimensional Heisenberg model W exhibits a k−5∕3 power law in
the inertial subrange and a smooth transition to the k−7 slope in the viscous subrange and is given by
W(𝜔) =
(𝜔∕vb )−5∕3
Γ(5∕3) sin(𝜋∕3) 2
C [
]2
2𝜋vb
1 + (𝜔∕vb ∕k0 )8∕3
(1)
where Γ is the gamma function (Γ(5∕3) = 0.90167), vb the balloon velocity, 𝜔 = 2𝜋f the cyclic frequency,
and C 2 the structure function constant. The model “breaks” at k0 , which basically constitutes the intersection
of the asymptotic form of W(𝜔) in the inertial and viscous subrange. Tatarskii [1971] has shown that k0 is
determined from the behavior of the structure function D at origin, i.e.,
d2
8𝜋
D(0) =
3 ∫0
dr2
∞
Φ(k)k4 dk.
(2)
Based on Taylor’s frozen field hypothesis [Taylor, 1935], the three-dimensional function Φ(k) is related to
W(𝜔) via
Φ(k) = −
vb2
2𝜋k
⋅
d
W(𝜔)
d𝜔
(3)
where vb is the balloon velocity, i.e., speed by which instrument is moved. With l0 = 2𝜋∕k0 we finally obtain
the so called inner scale:
3
⎡
⎤4
C2 ⎥
⎢
l0 = 2.5 ⋅ 2
.
⎢ d D(0) ⎥
⎣ dr2
⎦
(4)
The structure function D for velocity fluctuations within the viscous subrange is given by
Dtot (r) =
1𝜀 2
r
3𝜈
(5)
where 𝜀 is the energy dissipation rate and 𝜈 the kinematic viscosity. After inserting in equation (4) and with
2
the structure function constant for velocity CV2 = 4𝛼𝜀 3 [Barat and Bertin, 1984] and the empirical constant
HAACK ET AL.
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heating rate [K/d]
8.6x10-5
30
8.6x10-4
8.6x10-3
8.6x10-2
8.6x10-1
8.6x100
8.6x101
8.6x102
10-5
10-4
10-3
10-2
10-1
100
101
altitude [km]
25
20
15
10
10-6
epsilon [W/kg]
Figure 5. The energy dissipation rate (blue) of BEXUS 6 plotted in logarithmic scale against the altitude. The red line
presents the linear regression of log 𝜀.
𝛼 = 0.5 [Antonia et al., 1981; Bertin et al., 1997], a relation between the inner scale l0 and the energy
dissipation rate 𝜀 is obtained:
(
l0 = 5.7 ⋅
𝜈3
𝜀
)1
4
.
(6)
To retrieve the energy dissipation rate 𝜀 from the measured wind fluctuations, the Heisenberg model is
fitted to the spectrum. From the best fit l0 is determined and therewith 𝜀 via equation (6). The kinematic
viscosity has been calculated based on the in situ measurements of the radiosonde on board of the BEXUS
gondola. As an example, the best fit has been plotted as a red line to the spectrum in Figure 4. The theoretical fit nicely agrees with the measured spectrum and yields an inner scale l0 of 4.5 cm and an energy
dissipation rate 𝜀 of 9.4 mW/kg. The statistical uncertainty of 𝜀 due to instrument noise and discretization
error is very small. Potential errors arise due to spurious spectral signals by movements of the gondola.
These may affect the fit and the determination of the inner scale l0 . But as mentioned before, the BEXUS
gondola shows only very small pendulum motions that do not influence the spectra. Another potential error
source are the fit limits. We apply an automatic noise level detection that is also used to fix the lower limit
for the fitted spectral range. We have made different sensitivity studies for the influence of the choice of fit
parameters on the results and estimate an uncertainty in 𝜀 by about a factor of 3.
4.1. Altitude Profiles of 𝜺
In order to obtain a complete profile of 𝜀, the data set is divided into segments by a moving window of
5 s, i.e., ∼25 m (assuming a balloon ascent rate of 5 m/s) with an overlap of 2 s, i.e., 10 m. For each segment,
the spectrum is calculated based on Welch’s method [Welch, 1967]. After evaluating whether the spectrum
is turbulent or not by means of the individual noise characteristic, the Heisenberg model is fitted to the turbulent spectrum. The resulting 𝜀 value is assigned to the mean altitude of the segment. For segments where
the Heisenberg model does not fit the data (e.g., due to a vanishing inertial subrange), 𝜀 is set to zero. Thus,
a profile of the energy dissipation rate with a step size of 10 m for the complete altitude profile is obtained.
Figure 5 shows the energy dissipation rate 𝜀 from 7 km up to 29 km for the BEXUS 6 flight. It can easily be
seen that the energy dissipation rate increases with altitude, at least above 16 km. Accordingly, the lowest
value of 1.45 × 10−6 W/kg has been measured below 16 km and the highest value of 1.81 W/kg above 25 km,
respectively. Hence, the values of the energy dissipation rate cover several orders of magnitude within a
height range of ∼22 km. To emphasize the increase of 𝜀 with altitude we plotted a straight line to the profile. In order to get a more detailed picture of the individual turbulent regions, Figure 6 shows again the
profile of the energy dissipation rate but this time in linear scale and divided into three altitude ranges. It
should be emphasized that due to the height variation of the 𝜀 values, the 𝜀 axes have been scaled differently for the three plots to visualize individual turbulent layers within the complete altitude region. The
intermittency of the turbulent regions is clearly recognizable by the alternation between regions with high
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Journal of Geophysical Research: Atmospheres
10.1002/2013JD021008
𝜀 values, i.e., turbulent layers, and regions
with lower values of 𝜀, i.e., calm regions.
13
Up to 10.3 km (Figure 6, left) the pre20
28
12
dominant part is characterized by very
19
27
11
low energy dissipation rates and only a
18
26
few smaller turbulent layers stand out
10
17
25
of the calm background. Above 10.3 km
9
16
24
up to 14 km hardly no altitude region
8
15
23
can be found, where there is no turbu14
7
22
lent layer. In Figure 6 (middle), i.e., from
0
0.01
0.02
0
0.1
0.2
0
1
2
14 km up to 22 km, the further increase
epsilon [W/kg]
of the energy dissipation rate is clearly
Figure 6. (left, middle, and right) The profile of 𝜀 obtained during
visible. Hence, higher-energy dissipaBEXUS 6 in linear scale divided into three altitude segments. Due to
tion rates can be found above 19.6 km,
the high variation of 𝜀, the 𝜖 axes have been scaled different for the
while below only a few turbulent laythree plots.
ers with 𝜀 larger than ∼0.02 W/kg are
detected. The increase of the energy
dissipation rate proceeds also in the altitude region from 22 km up to 29 km (Figure 6, right). Most conspicuous is the turbulent region above 27.5 km, where the highest 𝜀 values of the complete altitude profile
can be found, namely, up to 1.8 W/kg. In summary, the BEXUS 6 profile of energy dissipation rates displays
nicely the intermittency of the stratospheric turbulence. Moreover, a significant increase of the dissipation
rate with altitude is found. Figure 7 shows the resulting profile of the energy dissipation rate for the wind
fluctuations of BEXUS 8. Similar to the BEXUS 6 data, the energy dissipation rate increases with altitude.
Thus, the lowest value of 5 × 10−6 W/kg has been measured at 7.1 km and the highest value of 0.7 W/kg at
25.8 km. Again, a straight line has been plotted to emphasize the dissipation increase with altitude. Figure 8
shows the energy dissipation rate profile in a linear scale divided into three altitude regions. Up to 12.6 km
(Figure 8, left) several turbulent layers have been observed with varying vertical thicknesses. A section with
barely no turbulence has been observed between 12.8 km and 13.8 km. A tendency to higher 𝜀 values can
also be noticed within the altitude region from 14 to 21 km (Figure 8, middle). Furthermore, the spatial intermittency of the detected turbulence can easily be recognized again. Regions with considerably less or even
no turbulence dissipation at all (e.g., 16.2–16.8 km) alternate with turbulent regions (e.g., 18–19 km). The
region between 21 and 27 km (Figure 8, right) is characterized by turbulent layers with much higher-energy
dissipation rates than in the altitude regions below. In addition, there exist much larger regions without turbulence (e.g., between 21.7 km and 23.2 km). The turbulent layer with the highest energy dissipation rate
is located just below 26 km. Summing up, an increase of 𝜀 with altitude is clearly visible. Furthermore, the
distance between the individual layers also increases with altitude.
22
30
21
29
altitude [km]
14
heating rate [K/d]
8.6x10-5
30
8.6x10-4
8.6x10-3
8.6x10-2
8.6x10-1
8.6x100
8.6x101
8.6x102
10-5
10-4
10-3
10-2
10-1
100
101
altitude [km]
25
20
15
10
10-6
epsilon [W/kg]
Figure 7. Energy dissipation rate (blue) of BEXUS 8 together with the linear regression (red line).
HAACK ET AL.
©2014. American Geophysical Union. All Rights Reserved.
10,612
altitude [km]
Journal of Geophysical Research: Atmospheres
10.1002/2013JD021008
14
21
27
13
20
26
12
19
11
18
10
17
9
16
8
15
25
24
23
22
21
14
7
0
0.025
0.05
0.075
0
0.05
0.1
0
0.25
0.5
0.75
epsilon [W/kg]
Figure 8. (left, middle, and right) Profile of 𝜀 of BEXUS 8 in linear scale showing the intermittent structure of stratospheric
turbulence. (Note the scale change.)
Based on the energy dissipation rate profiles shown above, we determined mean 𝜀 values and the corresponding heating rates for the complete altitude profiles. Generally, higher-energy dissipation rates have
been obtained during the BEXUS 6 flight. In order to emphasize the large amount of energy which is dissipated into heat within the turbulent layers, the mean values have been determined for the turbulent regions
only. Additionally, for comparison, we also calculated mean values for the complete altitude profile including turbulent and nonturbulent regions. Table 2 contains the mean values for both BEXUS flights divided
into tropopause region (7–15 km) and stratospheric region (15–26.5 km/29 km). Clear differences appear
between 𝜀mean for the turbulent layers and 𝜀mean when also the nonturbulent regions are taken into account.
For almost all altitude regions, the energy dissipation rate is up to 1 order of magnitude higher within the
turbulent layers compared to the sum of turbulent and nonturbulent regions. The higher 𝜀 values within
turbulent layers and the resulting heating rate of up to 3.3 K/d point out that if turbulence occurs in the
stratosphere, it has indeed the potential to influence, e.g., the mixing of trace species.
Similar to the altitude plots of the energy dissipation rate shown above for the BEXUS 6 (Figure 5) and
BEXUS 8 (Figure 7) flights, all values of 𝜀 are higher in the stratosphere than in the tropopause region. The
difference between both regions occurs regardless of whether only the turbulent regions or the complete
altitude regions are considered. The observed increase of 𝜀 is partly caused by the fact that kinematic viscosity increases with altitude. Equation (6) shows that even with unchanged wind fluctuations (i.e., unchanged
l0 ), the energy dissipation rate would increase with altitude due to increasing kinematic viscosity. However,
the differences between both flights show (even in our limited data set) that additional factors contribute to
the change of epsilon with altitude, e.g., wave activity and wind shear.
However, energy dissipation rates obtained during earlier experiments differ up to 2 orders of magnitude
from 1 × 10−5 W/kg, i.e., 4 × 10−3 K/d [Barat, 1982a; Barat and Bertin, 1984] up to 1.7 × 10−3 W/kg, i.e.,
∼1.5×10−3 K/d [Lilly et al., 1974]. This illustrates the large variance of 𝜀 in the stratosphere which is clearly
verified by our measurements. Both our epsilon profiles revealed larger dissipation rates in the stratosphere compared to the tropopause region, even if the increase itself is different. An increasing dissipation
has also been found in previous studies [e.g., Jumper et al., 2003; Alexander and Tsuda, 2008; Kantha and
Table 2. Mean Energy Dissipation Rates for BEXUS 6 and BEXUS 8
BEXUS 6
BEXUS 8
HAACK ET AL.
7–15 km
15–29 km
7–29 km
7–15 km
15–26.5 km
7–26.5 km
Only Turbulent Layers
(W/kg)
(K/d)
Turbulent and Nonturbulent Regions
(W/kg)
(K/d)
1.3 × 10−3
3.8 × 10−2
3.4 × 10−2
4.6 × 10−3
1.9 × 10−2
1.1 × 10−2
9.9 × 10−4
2.9 × 10−2
1.8 × 10−2
2.7 × 10−3
6.9 × 10−3
5.0 × 10−3
0.1
3.3
2.9
0.4
1.6
1.0
©2014. American Geophysical Union. All Rights Reserved.
0.1
2.5
1.6
0.2
0.6
0.4
10,613
Journal of Geophysical Research: Atmospheres
b)
altitude [km]
a)
10.1002/2013JD021008
c)
d)
24.7
24.7
24.7
24.7
24.6
24.6
24.6
24.6
24.5
24.5
24.5
24.5
24.4
24.4
24.4
24.4
24.3
24.3
24.3
24.3
24.2
24.2
24.2
24.2
24.1
24.1
24.1
24.1
24.0
24.0
24.0
24.0
23.9
23.9
23.9
23.9
23.8
0.1
0.2
epsilon [W/kg]
23.8
23.8
23.8
0
0
50
100
Richardson number
0
0.01
0.02
wind shear [1/s]
0
0.002
N2 [1/s2]
Figure 9. (a) Energy dissipation rate profile between 23.8 km and 24.7 km for BEXUS 6. (b) The Richardson number for
the same altitude region. The red line marks Ric = 1∕4. (c and d) The wind shear and Brunt-Väisälä frequency.
Hocking, 2011]. Depending on the maximum altitude of the measurements presented there, the increase
of 𝜀 is more or less pronounced. On the other hand, an increase as well as a decrease of 𝜀 in the stable
stratified stratosphere is described in the publication of Clayson and Kantha [2008]. We cannot give a final
answer on this from our limited data set, but we like to point out that the increase of dissipation is consistent with the increase of kinematic viscosity. Nevertheless, further experiments are needed to determine
representative or even seasonal mean energy dissipation rates within the stratosphere.
5. Relation to Richardson Number
A classical approach to analyze the relation of turbulence to the atmospheric background conditions is the
determination of the Richardson number Ri, i.e., the ratio of the squared Brunt-Väisälä frequency to the
squared wind shear. Additionally, the critical Richardson number Ric defines the threshold where the atmosphere changes from stability to turbulence, and from linear theory it is suggested to be Ric = 1∕4. However,
an ongoing discussion questions the existence of such a critical Richardson number or rather the relation of
the Richardson number to the occurrence of turbulence regions. On the one hand, many observations reveal
a correlation between low Richardson numbers and turbulence [e.g., Barat, 1982a; Mauritsen and Svensson,
2007; Clayson and Kantha, 2008]. But an increasing number of observations and also theoretical simulations reveal a more complex and not straightforward relation of Ric to turbulent flows [Achatz, 2005, 2007;
Mauritsen and Svensson, 2007; Galperin et al., 2007; Balsley et al., 2008; Zilitinkevich et al., 2008]. The different
analyses lead to three main assumptions:
1. Suggestion of hysteresis. Laminar air flow must drop below Ric = 1∕4 before turbulence may occur, but
turbulent flow can exist up to Ri = 1.0 before becoming laminar [Galperin et al., 2007; Balsley et al., 2008].
2. Scale-dependent problem. The distribution of Ri depends strongly on scale size, and Ric may exist if
sufficiently small scales are examined [Balsley et al., 2008].
3. Ric does not exist. Extensive body of experimental, observational, and theoretical results points to the fact
that a single-valued Ric at which turbulence is suppressed totally and laminarized simply does not exist
and turbulence can survive in flows with Ri far exceeding unity [Achatz, 2005, 2007; Galperin et al., 2007;
Zilitinkevich et al., 2008].
In the following it is examined whether the turbulence results of LITOS show a correlation with the
Richardson number or not. Based on our radiosonde measurements, Ri has been calculated with the
method described by Balsley et al. [2008]. They calculated the vertical gradients of stepwise linear fits for N2
(Brunt-Väisälä frequency) and S2 (wind shear) and using the relation Ri = N2 ∕S2 to determine the Richardson
number. In order to investigate the scale dependence of the Richardson number, different height increments are used. Figure 9 shows an altitude section of the energy dissipation rate profile and the Richardson
number Ri of the BEXUS 6 flight. Wind shear and Brunt-Väisälä frequency are calculated across 90 m vertical distance. Around 23.88 km, Ri is smaller than 1∕4 and the theoretical condition for the development
of turbulence is fulfilled. Accordingly, a turbulent layer, i.e., an increase of the energy dissipation rate 𝜀 is
HAACK ET AL.
©2014. American Geophysical Union. All Rights Reserved.
10,614
Journal of Geophysical Research: Atmospheres
altitude [km]
a)
b)
10.1002/2013JD021008
c)
d)
24.21
24.21
24.21
24.21
24.20
24.20
24.20
24.20
24.19
24.19
24.19
24.19
24.18
24.18
24.18
24.18
24.17
24.17
24.17
24.17
24.16
24.16
24.16
24.16
24.15
24.15
24.15
24.15
24.14
24.14
24.14
24.14
24.13
0.05
0.1
epsilon [W/kg]
24.13
24.13
24.13
0
0
20
40
Ri (20 m)
0
20
40
Ri (70 m)
0
20
40
Ri (200 m)
Figure 10. Example of a ∼43 m thick turbulent layer observed during BEXUS 6 and the corresponding scale-dependent
Richardson number Ri. (a) The 𝜀 profile. Ri has been calculated over (b) 20 m, (c) 70 m, and (d) 200 m. The red line
presents the critical Richardson number Ric = 1∕4.
epsilon [W/kg]
observed. However, this clear correlation between Ric and 𝜀 is only observed at this altitude. Instead, even
though Ri tends to 1∕4 at, e.g., 24.06 km, no distinct turbulent layer can be identified; i.e., no significant
increase of 𝜀 has been detected. Similarly, at 24.66 km Ri is clearly below Ric , but absolutely no increase of 𝜀
can be noticed. On the other hand, turbulent layers have been observed, where Ri is far beyond any critical
number. For instance, at 24.0 km a turbulent layer is well identified, while the values for Ri increase up to 50.
Similarly, the distinctive turbulent layer shortly below 24.2 km occurs in a region where Ri decreases from
75 to 13 but remains higher than Ric . This altitude section from 23.8 km to 24.7 km illustrates that increased
energy dissipation rates have been measured while the Richardson number is higher than 1. Therefore, the
first assumption of hysteresis as described above has to be rejected based on our case study. The second
assumption implies a scale-dependent problem, i.e., the relation between Ri and 𝜀 depends on the scale
at which both profiles have been obtained. That means that in order to resolve turbulent layers with vertical depths of only some 10 m, Ri has to be determined at approximately the same scale. Hence, the lower
limit of the Ri scale is given by the thickness of the turbulent layers. Based on the radiosonde observations
which provide data at ∼10 m steps, Ri has been calculated for different altitude increments. In Figure 10
an example of a turbulent layer with a thickness of ∼43 m observed during the BEXUS 6 flight is plotted.
The Richardson number has been calculated over 20 m, 70 m, and 200 m, and the profiles are shown in
Figures 10b–10d, respectively. Obviously, if Ri is calculated over 20 m, the occurrence of the turbulent layer
corresponds to a region where Ri is smaller
than Ric . Whereas, if the Richardson numBEXUS
ber has been determined with a scale
102
of 70 m or 200 m, its values do not drop
below 1∕4 but remain larger than 10. Even
100
though Ri tends to small values between
24.16 km and 24.20 km, this region would
-2
have been declared as nonturbulent which
10
is contradictory to the observed turbulent layer within the wind field. However,
10-4
the relation between Ri and 𝜀 seems to
depend on the scale over which Ri has
10-6
been calculated. For further investigations,
0
10
-200
-100
100 1000 10000
the complete data set of Ri has been plotRichardson number
ted as a function of the obtained energy
Figure 11. Energy dissipation rates of the BEXUS 6 flight against the
dissipation rates of BEXUS 6 and BEXUS 8,
Richardson number Ri scaled over 10 m. All values of Ri below 1 are
i.e., only for all these altitude bins where
marked red. For a better presentation the x axis scale is split into a
4
turbulence is evident. Figure 11 shows the
linear part up to 1 and a logarithmic scale up to 10 . The green line
Richardson number of the BEXUS 6 flight
denotes Ri = 1.
HAACK ET AL.
©2014. American Geophysical Union. All Rights Reserved.
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Journal of Geophysical Research: Atmospheres
10.1002/2013JD021008
determined for 10 m resolution. Surprisingly, even for a rather small scale like 10 m,
the majority of Ri values are larger than 1.
That means that turbulence has been
100
observed although Ri indicates stability.
A similar result has been obtained for the
-2
10
BEXUS 8 flight. Figure 12 shows the result
for the wind fluctuations of BEXUS 8 for a Ri
10-4
scale of 10 m. Again, a strikingly low number of turbulent events occur at Ri smaller
10-6
than 1. Instead, the Richardson number
-20
-10
0
10
100 1000 10000
Richardson number
is highly variable. Ri values of up to ∼103
Figure 12. Energy dissipation rates of the BEXUS 8 flight against the have been obtained during both BEXUS
Richardson number Ri scaled over 10 m. All values of Ri below 1 are flights within turbulent layers. There is a
marked red. For a better presentation the x axis scale is split into a
chance that the calculation of the Richardlinear part up to 1 and a logarithmic scale up to 104 . The green line son number at this very high resolution is
denotes Ri = 1.
affected by pendulum motions of the gondola that may perturb the observed wind
shear. But the pendulum of the BEXUS 6 and BEXUS 8 gondolas was very small due to the large weight of the
payload (not shown here). The remaining pendulum motion may result in an underestimation of the true
Richardson number or a larger scatter, but typically not in an overestimation. The radiosonde provides a horizontal wind measurement that is some kind of average across the whole flight train (payloads and balloon),
resulting in an altitude difference between wind and temperature data. This results in a larger uncertainty
of the radiosonde data, but from the best of our knowledge not in a systematic error, i.e., an overestimation of Ri. Therefore, the general result of the observation of turbulence at large Richardson numbers
remains unaffected.
BEXUS 8
epsilon [W/kg]
102
For further analysis, the percentage of altitudes where the Richardson number is below 1 or 1/4 has been
determined depending on the scale of Ri calculation for both BEXUS flights. The results are summarized in
Table 3. During BEXUS 6 the Richardson number (at 10 m scale) was < 1 for 42.3% and < 1∕4 for only 29.9%
of the altitude profile. Hence, for only 42% of the altitude region between 7 and 29 km the theoretical conditions for turbulence existence are given. Furthermore, if the Richardson number is calculated for larger
scales than 10 m, those values decrease dramatically and go even down to 3% for a Ri scale of 200 m. Similar results have been obtained for BEXUS 8. But the percentage of altitude with Ri < 1 is smaller than for
BEXUS 6. Only 34.1% of the altitude is characterized by Ri values < 1, and the values decrease rapidly again
with increasing Ri scale.
Additionally, mean values of energy dissipation rates have been determined for altitude regions where
Ri < 1 and also for Ri > 1. For almost all scales over which Ri has been calculated, higher 𝜀mean values have
been obtained for Ri < 1 than for Ri > 1. Even though these results suggest a relation between Ri and the
occurrence of turbulence, i.e., higher-energy dissipation rates in regions with smaller Ri, the analyses shown
above are contradictory. Figures 11 and 12 show a large variation of 𝜀 for all values of Ri, i.e., also for Ri > 1.
Several turbulent layers have been measured where Ri is > 1 and far beyond. Additionally, the comparison of the 𝜀 profiles also reveals regions where Ri is smaller than 1, but no turbulent layer has been found.
Apparently, Ri cannot be used as a reliable indicator for the occurrence of turbulence.
Table 3. Percentage of Altitude for Ri < 1∕4 and Ri < 1 and 𝜀mean for Regions With Ri < 1 and Ri > 1
Scale of Ri
Ri < 1∕4
Ri < 1
𝜀mean (Ri < 1)
𝜀mean (Ri > 1)
HAACK ET AL.
10 m
BEXUS 6
70 m
200 m
10 m
BEXUS 8
70 m
200 m
29.9%
42.3%
24 mW/kg
13 mW/kg
17.7%
36.3%
23 mW/kg
15 mW/kg
3.4%
21.5%
28 mW/kg
15 mW/kg
11.1%
34.1%
66 mW/kg
66 mW/kg
3.2%
29.6%
67 mW/kg
66 mW/kg
0.7%
19.8%
71 mW/kg
66 mW/kg
©2014. American Geophysical Union. All Rights Reserved.
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Journal of Geophysical Research: Atmospheres
10.1002/2013JD021008
6. Summary
During two BEXUS campaigns in 2008 and 2009 from Kiruna, fine-scale turbulent fluctuations in the
stratospheric wind field between ∼7 km and ∼29 km have been studied with LITOS. In contrast to earlier
measurements, we detected a higher number of thinner turbulent layers. Furthermore, our measurements
reveal that thinner turbulent layers occur in the middle stratosphere and the distance between the turbulent layers is smaller compared to the lower stratosphere. We found that the number of turbulent layers
increases with altitude. Due to a measurement resolution of at least 2.5 mm, the entire turbulence spectrum down to the viscous subrange has been obtained, which enables the determination of the energy
dissipation rate with unprecedented precision. During both flights, we observed higher-energy dissipation
rates in the stratosphere compared to the tropopause region (7–15 km). Additionally, 𝜀mean is up to a factor of ∼2 higher within the turbulent layers compared to the sum of turbulent and nonturbulent regions.
Averaging across turbulent and nonturbulent regions of the stratosphere (15–29 km), we found mean
energy dissipations rates of 7 and 29 mW/kg (corresponding to heating rates of 0.6 and 2.5 K/d) from
BEXUS 8 and BEXUS 6, respectively. This fact emphasizes the potential importance of stratospheric turbulent
layers for, e.g., the mixing of trace gases or the energy budget of the middle atmosphere.
The comparison between the Richardson number Ri and the energy dissipation rate shows contradictory
results. We observed a large variation of 𝜀 for all values of Ri, i.e., also for Ri > 1. In regions where Ri was
> 1 and far beyond, turbulent layers have been detected. On the other hand, we found cases where Ri < 1
and no increase of 𝜀 has been observed. From our measurements we therefore preclude a clear correlation
between Ri and the occurrence of turbulence.
Acknowledgments
We acknowledge the support by the
Leibniz graduate school ILWAO, jointly
funded by the MBWK (for the government of Mecklenburg-Vorpommern)
and the BMBF (for the German federal
government). Additionally, we gratefully thank DLR (German Aerospace
Center) and SNSB (Swedish National
Space Board) for the opportunity to
participate in the BEXUS campaigns.
We thank the reviewers for their helpful comments, especially for finding a
major mistake in the manuscript.
HAACK ET AL.
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