Introduction
Temperature fluctuations and their vertical organization inherently govern
the energy budget in the convective planetary boundary layer (CBL) by
determining the vertical heat flux and modifying the interaction of vertical
mean temperature gradient and turbulent transport (Wyngaard and Cote, 1971;
Wyngaard, 2010). Thus, the measurement of turbulent temperature fluctuations
and characterizations of their statistics are essential for solving the
turbulent energy budget closure (Stull, 1988). In situ measurements (near the
ground, on towers, or on airborne platforms) sample certain regions of the
CBL within certain periods and have been used for a long time for turbulence
studies. But to the best of our knowledge, there are
no previous observations based on a remote-sensing technique suitable for
this important task, i.e., resolving temperature fluctuations in high
resolution and covering simultaneously the CBL up to the interfacial layer
(IL). In this work, it is demonstrated that rotational Raman lidar (RRL)
(Cooney, 1972; Behrendt, 2005) can fill this gap.
By simultaneous measurements of turbulence at the land surface and in the IL,
the flux divergence and other key scaling variables for sensible and latent
heat entrainment fluxes can be determined, which is key for the evolution of
temperature and humidity in the CBL and thus for verifying turbulence
parameterizations in mesoscale models (Sorbjan, 1996, 2001, 2005).
Traditionally, studies of turbulent temperature fluctuations in the
atmospheric CBL were performed with in situ instrumentation operated on tethered balloons, helicopters, and
aircraft (e.g., Clarke et al., 1971; Muschinski et al., 2001) as well as
recently with unmanned aerial vehicles (UAVs, e.g., Martin et al., 2011).
However, it is not possible to obtain instantaneous profiles of turbulent
fluctuations with in situ sensors and it is difficult to identify the exact
location and characteristics of the IL. Recently, it was demonstrated that
the combination of remote-sensing instruments (for guiding) and a UAV also
allows for the study of entrainment processes at the CBL top (Martin
et al., 2014). However, the UAV cannot continuously examine the processes due
to its short endurance.
For studying turbulent processes and their parameterizations, however, it is
essential that the turbulent transport and the temperature gradient are
measured simultaneously in the same volume. Therefore, the shortcomings of in
situ observations call for new
remote-sensing technologies. These instruments can be operated on different
platforms and can provide excellent long-term statistics, if applied from
ground-based platforms. Passive remote-sensing techniques, however, show
difficulties in contributing to turbulence studies because of their inherent
limitation in range resolution which flattens turbulent fluctuations.
Nevertheless, Kadygrov et al. (2003) published a study on turbulent
temperature fluctuations based on passive remote-sensing techniques. The
authors used a scanning microwave temperature profiler to investigate thermal
turbulence and concluded that the spectral density of brightness temperature
fluctuations at 75 m above ground indeed followed the expected -5/3-power law of Kolmogorov (1991).
Kadygrov et al. (2003) concluded that “measurements can be provided in all
weather conditions, but the technique has limitations in altitude range” as
their turbulence studies could only reach up to a maximum height of 200 m.
In recent years, new insights in CBL turbulence were provided by studies
based on active remote sensing with different types of radar and lidar
systems. Radar wind profilers were used to study the vertical CBL wind
profile and its variance (e.g., Angevine et al., 1994; Eng et al., 2000;
Campistron et al., 2002). A radio-acoustic sounding system (RASS)
provides profiles of virtual temperature which can
be used as a scaling parameter for turbulence studies also in higher
altitudes (e.g., Hermawan and Tsuda, 1999; Furomoto and Tsuda, 2001). But
temperature and moisture fluctuations cannot be separated with RASS.
Furthermore, the RASS profiles have typical resolutions of a few minutes
which is too large to resolve the inertial subrange. In addition to radar,
lidar techniques have also been used for turbulence studies: elastic
backscatter lidar (Pal et al., 2010, 2013), ozone differential absorption
lidar (ozone DIAL) (Senff et al., 1996), Doppler lidar (e.g., Lenschow
et al., 2000, 2012; Wulfmeyer and Janjic, 2005; O'Connor et al., 2010;
Träumner et al., 2015), water vapor differential absorption lidar (WV
DIAL) (e.g., Senff et al., 1994; Kiemle et al., 1997; Wulfmeyer, 1999a;
Lenschow et al., 2000; Muppa et al., 2015), and water vapor Raman lidar
(e.g., Wulfmeyer et al., 2010; Turner et al., 2014a, b) have been employed or
a combination of these techniques (e.g., Giez et al., 1999; Wulfmeyer, 1999b;
Kiemle et al., 2007, 2011; Behrendt et al., 2011a; Kalthoff et al., 2013).
However, so far, profiling of turbulent temperature fluctuations with active
remote sensing was missing.
In general, daytime measurements are more challenging than nighttime
measurements for lidar because of the higher solar background which increases
the signal noise and even prohibits measurements for most Raman lidar
instruments. In order to address the measurement needs, the University of Hohenheim (UHOH) RRL was
optimized for high temperature measurement performance in daytime in the CBL
(Radlach et al., 2008). The data of the UHOH RRL have already been used for
studies on the characterization of transport and optical properties of
aerosol particles near their sources (Behrendt et al., 2011b; Valdebenito
et al., 2011), on the initiation of convection (Groenemeijer et al., 2009;
Corsmeier et al., 2011), and on atmospheric stability indices (Behrendt
et al., 2011; Corsmeier et al., 2011). Here, the formalism introduced by
Lenschow et al. (2000) is applied for the first time to the data of an RRL to
study the extension of the variable set of lidar turbulence studies within
the CBL to temperature.
The measurements discussed here were carried out at around local noon
(11:33 UTC) on 24 April 2013 during the Intensive Observations Period (IOP)
6 of the HD(CP)2 (High-Definition Clouds and Precipitation for advancing
Climate Prediction) Observational Prototype Experiment (HOPE), which is
embedded in the project HD(CP)2 of the German Research Ministry. The
UHOH RRL was positioned during this study at
50∘53′50.56′′ N,
6∘27′50.39′′ E, 110 m a.s.l. near the
village of Hambach in western Germany where it performed measurements between
1 April and 31 May 2013.
This paper is organized as follows. In Sect. 2, the setup of the UHOH RRL is
described briefly; more details can be found in Hammann et al. (2015). The
meteorological background and turbulence measurements are presented in
Sect. 3. Finally, conclusions are drawn in Sect. 4.
Setup of the UHOH RRL
The RRL technique is based on the fact that different portions of the pure
rotational Raman backscatter spectrum show different temperature dependence.
By extracting signals out of these two portions and forming the signal ratio,
one obtains a profile which, after calibration, yields a temperature profile
of the atmosphere (see, e.g., Behrendt, 2005, for details).
Overview of key parameters of the rotational Raman lidar of
University of Hohenheim (UHOH RRL) during the measurements discussed here.
Transmitter
Flash-lamp-pumped injection-seeded frequency-tripled Nd:YAG laser
Pulse energy: ∼200 mJ at 354.8 nm
Repetition rate: 50 Hz
Pulse duration: ∼5 ns
Receiver
Diameter of primary mirror: 40 cm
Focal length: 4 m
Field of view: 0.75 mrad (selectable)
Scanner
Manufactured by the NCAR, Boulder, CO, USA
Mirror coating: protected aluminum
Scan speed: up to 10∘ s-1
Detectors
Photomultiplier tubes, Hamamatsu R7400-U02 (Elastic), R1924P (RR1+2)
Data acquisition system
3-channel transient-recorder, LICEL GmbH, Germany
Range resolution
3.75 m in analog mode up to 30 km range
3.75 m in photon-counting mode up to 30 km range
37.5 m in photon-counting mode up to 75 km range
Scheme of the UHOH RRL. The beam-steering unit (BSU) consists
of two plane mirrors which scan the laser beam and receiving
telescope field-of-view. LM: laser mirror; PBP: Pellin–Broca prism;
BE: beam expander; BD: beam dump; L1 to L4: lenses; IF0 to IF3:
interference filters; PMT1 to PMT3:
photomultiplier tubes; RR1 and RR2: rotational Raman channel 1 and
2, respectively. The beam splitter for the water vapor Raman channel
between L1 and IF0 has been omitted for clarity here.
A scheme of the UHOH RRL during HOPE is shown in Fig. 1. Key system
parameters are summarized in Table 1. As laser source, an injection-seeded
frequency-tripled Nd:YAG laser (354.8 nm, 50 Hz,
10 W), model GCR 290-50 of Newport Spectra-Physics GmbH, is used. The
UV laser radiation is separated from the fundamental and frequency-doubled
radiation near 532 and 1064 nm, respectively, with a Pellin–Broca
prism (PBP), so that only the UV radiation is sent to the atmosphere. This
improves eye safety significantly compared to systems which use harmonic beam
splitters because there is definitely no potentially hazardous green laser
light present in the outgoing laser beam. But the main reason for using UV
laser radiation for the transmitter of the UHOH RRL is that the backscatter
cross section is proportional to the inverse wavelength to the fourth power.
This yields significantly stronger signals and thus lower statistical
uncertainties of the measurements in the lower troposphere (see also Di
Girolamo et al., 2004, 2006; Behrendt, 2005) when using the third harmonic
instead of the second harmonic of Nd:YAG laser radiation. Behind the PBP, the
laser beam is expanded 6.5-fold in order to reduce the beam divergence to
< 0.2 mrad. The laser beam is then guided by three mirrors
parallel to the optical axis of the receiving telescope (coaxial design) and
reflected up into the atmosphere by two scanner mirrors inside of a so-called
beam-steering unit (BSU). The same two mirrors reflect the atmospheric
backscatter signals down to the receiving telescope which has a primary
mirror diameter of 40 cm. The scanner allows for full hemispherical
scans with a scan speed of up to 10∘ s-1. In the present
case study, the scanner was pointing constantly in vertical direction. In the
focus of the telescope, a field-stop iris defines the field of view. For the
data shown here, an iris diameter of 3 mm was selected which yielded
a telescope field of view of 0.75 mrad. The light is collimated
behind the iris with a convex lens and enters a polychromator which contained
three channels during the discussed measurements: one channel for collecting
atmospheric backscatter signals around the laser wavelength (elastic channel)
and two channels for two signals from different portions of the pure
rotational Raman backscatter spectrum. During the HOPE campaign, the
polychromator was later extended with a water vapor Raman channel; the beam
splitter for this channel was already installed during the measurements
discussed here. Within the polychromator, narrow-band multi-cavity
interference filters extract in a sequence the elastic backscatter signal and
the two rotational Raman signals with high efficiency. The filters are
mounted at angles of incidence of about 5∘. This setting allows for
high reflectivity of the signals of the channels following in the chain
(Behrendt and Reichardt, 2000; Behrendt et al., 2002, 2004). The filter
passbands were optimized within detailed performance simulations for
measurements in the CBL in daytime (Behrendt, 2005; Radlach et al., 2008;
Hammann et al., 2015). The new daytime/nighttime switch for the second
rotational Raman channels (Hammann et al., 2015) was set to daytime
optimizing the signal-to-noise ratio of the RR2 channel for high-background
conditions. Further details on the receiver setup and the filter passbands
can be found in Hammann et al. (2015).
Time–height cross section of particle backscatter coefficient
βpar at 354.8 nm measured with the UHOH RRL
on 24 April 2013 between 11:00 and 12:00 UTC. The temporal and
spatial resolution of the data is Δt=10 s and
Δz=3.75 m with a gliding average of 109 m.
The instantaneous CBL heights determined with the Haar-wavelet
analysis of βpar profiles are marked. a.g.l. (above
ground level).
Turbulence case study
Data set
The synoptic condition on 24 April 2013 was characterized by a large
high-pressure system over central Europe. Because no clouds were forecasted
for the HOPE region, this day was announced as Intensive Observation Period
(IOP) 6 with the goal to study CBL development under clear-sky conditions.
Indeed, undisturbed solar irradiance resulted in the development of a CBL
which was not affected by clouds. A radiosonde launched at the lidar site at
11:00 UTC showed moderate westerly winds throughout the CBL and also in the
lower free troposphere. The horizontal speeds were < 2 m s-1 near
the ground increasing to about 5 ms-1 in the CBL between about
100 and 1000 m a.g.l. (above ground level). Between 1000 and
1300 m a.g.l., the horizontal wind increased further to about
10 ms-1 while ranging between this value and 8 ms-1
in the lower free troposphere; 3 m temperatures at the lidar site increased
between 09:00 and 11:00 UTC from 280 to 294 K. The sensible heat
flux at noon was about 170 Wm-2 at the lidar site.
The time–height plot of the particle backscatter coefficient
βpar (Fig. 2) between 11:00 and 12:00 UTC shows the
CBL height around local noon (11:33 UTC with a maximum solar elevation of
54∘ on this day). βpar was measured with the
rotational Raman lidar technique by use of a temperature-independent
reference signal (Behrendt et al., 2002). Data below 400 m were affected by incomplete
geometrical overlap of the outgoing laser beam and the receiving
telescope and have been excluded from this study.
As seen in Fig. 2, indeed no clouds were present in this period. The CBL is
clearly marked by higher values of βpar which result from
aerosol particles which are lifted up from the ground into the CBL. The
instantaneous CBL height was determined with the Haar-wavelet technique which
detects the strongest gradient of the aerosol backscatter signal as tracer
(Pal et al., 2010, 2012; Behrendt et al., 2011a) (Fig. 2). The mean of the
instantaneous CBL heights zi in the observation period was
1230 m a.g.l. This value is used in the following for the normalized height
scale z/zi. The standard deviation of the instantaneous CBL heights was
33 m; the absolute minimum and maximum were 1125 and 1323 m a.g.l.,
i.e., the instantaneous CBL heights were within 200 m. Besides its
vertical structure, the βpar field in the CBL also shows a
temporal trend in this case which may be explained by changing aerosol number
density or size distribution in the advected air over the lidar.
The temperature profile, which is the primary data product of the UHOH RRL,
for the period of 11:00–11:20 UTC, is shown in Fig. 3 together with zi
and the data of a local radiosonde launched at the lidar site at 11:00 UTC.
Calibration of the RRL temperature data used in this study was made with
these radiosonde data in the CBL between 400 and 1000 m a.g.l.; the RRL
data above result from extrapolation of the calibration function. For the
calibration, we used a 20 min average of the RRL data in order to reduce
sampling effects between the two data sets. Longer averaging periods for the
RRL reduce the statistical uncertainty of the measurements but increase the
sampling differences; shorter averaging results in larger statistical errors
and additionally in sampling of fewer air masses which makes the comparison
with the snapshot data of the radiosonde more difficult. It would be optimum,
of course, to track the sonde with the RRL but such a synchronization of the
lidar scanner with the sonde is not yet possible with the UHOH RRL.
Upper panel: average temperature profile measured with the
UHOH RRL on 24 April 2013 between 11:00 and 11:20 UTC and
temperature profiles measured with a local radiosonde launched at
the lidar site at 11:00 UTC. Lower panel: same but potential
temperature profiles. The dashed line shows zi for
comparison. Error bars show the uncertainties derived with Poisson
statistics from the intensities of the rotational Raman signals.
The uncertainty of the calibration depends mainly on the calibration of the
radiosonde; their uncertainty is < 0.2 K (see
http://www.graw.de/home/products2/radiosondes0/radiosondedfm-090/ and
Nash et al., 2011). It is noteworthy that the accuracy of the measured
temperature fluctuations do not depend on the absolute accuracy of the
temperature measurements but on their relative accuracy. Even with an error
of 1 K, the relative accuracy of the measured temperature fluctuations would
be better than (1 K) / (250 K) = 0.4 %. For the statistical
analysis of the turbulent temperature fluctuations, we then used this
calibration for the 1 h RRL data set between 11:00 and 12:00 UTC. This
1 h period seems here to be a good compromise: for much longer periods, the CBL
characteristics may change considerably while shorter periods would reduce
the number of sampled thermals and thus increase the sampling errors.
The temperature profiles of RRL and radiosonde shown in Fig. 3 agree
within fractions of 1 K in the CBL. Larger differences occur in
the IL due to the different sampling methods: the mean lidar profile
shows an average over 20 min, while the radiosonde data sample an
instantaneous profile along the sonde's path which was determined by
the drift of the sonde with the horizontal wind. In this case, the
sonde needed about 5 min to reach the top of the boundary layer and
drifted by about 1.6 km away from a vertical column above
the site. Depending on the part of the thermal eddies in the CBL and
the IL that are sampled, the radiosonde data thus represent different
CBL features and are not representative for a mean profile (Weckwerth
et al., 1996) which is a crucial point to be considered when using
radiosonde data for scaling of turbulent properties in the
CBL. Furthermore, averaged lidar temperature data are also more representative
for a certain site for model validations.
Inside the CBL, the potential temperature (derived from the RRL temperature
data with the radiosonde pressure profile) is nearly constant indicating
a well-mixed CBL (Fig. 3, lower panel); zi lies approximately in the
middle of the temperature inversion in the IL (Fig. 3). Figure 4 shows the
temperature gradients of the radiosonde and the RRL profiles, the latter for
two averaging periods, namely, 11:00 to 11:20 UTC and 11:00 to 12:00 UTC.
The maximum temperature gradient is in this case very similar for all three
profiles, i.e., between 0.6 and 0.7 K / (100 m). It is
interesting to note furthermore that the height of maximum temperature
gradient agrees with zi for both RRL profiles as determined with the
Haar-wavelet technique. In contrast to this, the height of the maximum
temperature gradient in the radiosonde profile is about 60 m lower.
But, as already mentioned, the radiosonde data are not representative for
a mean profile.
Average temperature gradients measured with the UHOH RRL on
24 April 2013 between 11:00 and 11:20 UTC, between 11:00 and 12:00 UTC and temperature gradient measured with a local
radiosondes launched at the lidar site at 11:00 UTC. The horizontal dashed
line shows zi, the mean CBL top height for the period between
11:00 and 12:00 UTC, which agrees with the maximum temperature
gradients of both RRL profiles. The vertical dashed line shows the
dry-adiabatic temperature gradient. Error bars show the uncertainties
derived with Poisson statistics from the intensities of the
rotational Raman signals.
Same as Fig. 2 but for temperature, potential temperature,
and detrended temperature fluctuations: time–height cross sections
measured with the UHOH RRL on 24 April 2013 between 11:00 and
12:00 UTC. The temporal and spatial resolution of the data is
Δt=10 s and Δz=3.75 m with
a gliding average of 109 m. The instantaneous CBL heights
determined with the Haar-wavelet analysis are marked (same as shown
in Fig. 2). a.g.l. (above ground level). (Black vertical lines are
gaps which result from gridding the data to exact
10 s intervals; these artifacts do not influence the
turbulence analysis.)
Turbulent temperature fluctuations
For CBL turbulence analyses, the instantaneous value of temperature T(z) at
height z is separated in a slowly varying component T(z)‾
derived from applying a linear fit to the data typically over 30 to 60 min
and the temperature fluctuation T′(z) according to, e.g.,
Wyngaard (2010)
T(z)=T(z)‾+T′(z).
Figure 5 shows the time–height cross sections
of temperature, potential temperature, and detrended temperature fluctuations
T′(z) in the discussed period. For detrending, the same linear
regression was applied to the temperature time series of all heights.
Furthermore, the data set with the temperature fluctuations was gridded to
exact 10 s time steps in order to ensure that all derived parameters are
correct. (The vertical black lines in the lower panel of Fig. 5 are artifacts
from this procedure.) One can see the positive and negative temperature
fluctuations inside the CBL. In the IL, the fluctuations in the measured data
become larger than in heights below. Above the CBL in the free troposphere,
one finds fewer structures in the temperature fluctuations and mostly
uncorrelated instrumental noise.
Lidar data contain significant stochastic instrumental noise, which has to be
determined and for which has to be corrected in order to obtain the
atmospheric fluctuation of a variable of interest. In general, the
signal-to-noise ratio can be improved by averaging the signal in time and/or
range but this in turn would of course reduce the ability to resolve
turbulent structures. In principle, very high time resolution, i.e., the
maximum allowed by the data acquisition system, is preferred in order to keep
most frequencies of the turbulent fluctuations. But this is only possible as
long as the derivation of temperature does not result in a non-linear
increase of the noise errors; this noise regime should be avoided. A temporal
resolution of 10 s turned out to be a good compromise for the
temporal resolution of our data as explained below.
(a) Autocovariance functions (ACF) around the zero lag
obtained at different heights from the temperature measurements
shown in Fig. 5, i.e., with the data of 24 April 2013 between 11:00
and 12:00 UTC. (b) Zoom of (a) for lower heights only. (c) ACF with power-law fit for 600 m a.g.l.
The variance of the atmosphere xa′(z)2‾ and the noise variance
xn′(z)2‾ of a variable
x are uncorrelated. Thus, we can write (Lenschow et al., 2000)
xm′(z)2‾=xa′(z)2‾+xn′(z)2‾
with xm′(z)2‾ for the
measured total variance. Overbars denote here and in the following temporal
averages over the analysis period. The separation of the atmospheric variance
from the noise contribution can be realized by different techniques. Most
straightforward is the autocovariance method, which makes use of the fact
that atmospheric fluctuations are correlated in time while instrumental noise
fluctuations are uncorrelated. Further details were introduced by Lenschow
et al. (2000) so that only a brief overview is given here. The atmospheric
variance can be obtained from the autocovariance function (ACF) of a variable
by extrapolating the tails (non-zero lags) to zero lag with a power-law fit
(see Eq. 32 of Lenschow et al., 2000). As the ACF at zero lag is the total
variance, the instrumental noise variance is the difference of the two.
Alternatively, one may calculate the power spectrum of the fluctuations and
use Kolmogorov's (1991) -5/3 law within the
inertial subrange in order to determine the noise level (e.g., O'Connor et
al., 2010). We prefer the ACF method to the spectral analysis because the ACF
method is less prone to errors since the statistical noise does not show up
at the non-zero lags which are used for the fit; the determination of the
statistical noise level from the power spectra is more prone to errors.
Figure 6 shows the ACF obtained from the measured temperature fluctuations
for heights between 400 and 1230 m a.g.l., i.e., 0.3 to 1.0zi for
time lags from -200 to 200 s. The increase of the values at zero
lag with height shows mainly the increase of the statistical noise with
height. Different values of the ACF close to the zero lag show differences in
the atmospheric variance at different heights.
The following question arises: what it the most suitable number of lags for the
extrapolation of the structure function to lag zero? This has been discussed
in Wulfmeyer et al. (2010) and Turner et al. (2014b) but here we are
providing more details. We have applied the following procedure to the
measured temperature fluctuations for the determination of the integral
scale, all higher-order moments, and for the separation of noise and
atmospheric variances: first of all, the profile of the integral scale is
derived using a standard number of lags. Usually, we are taking 20 time lags
of 10 s covering thus 200 s, as this turned out from previous measurements
to be a value which is typically appropriate. The resulting integral scale is
a measure of the mean size of an eddy in time. If the resulting integral
scale is larger than the averaging time of the measured data, which is in
this case 10 s, one can state that the most important part of the turbulent
fluctuations is resolved. It can be theoretically shown that the zero
crossing of the ACF appears at 2.5 times the integral scale (Wulfmeyer et
al., 2015). Thus, we are choosing ≤ 2.5 times the mean value of the
integral scale throughout the CBL as a reasonable number of fit lags. Please
note that this refinement was not discussed in the literature before except
only very recently by Turner et al. (2014b) and Wulfmeyer et al. (2015).
Previously, very simple approaches were used such as just the value of the
first lag as an approximation for the extrapolation to lag zero. Our approach
is more appropriate and may further be refined by applying an iteration
between the determination of the integral scale and the derivation of the
optimal number of fit lags at each height. As the integral timescale has a
mean of about 80 s in the CBL corresponding to a mean zero crossing of the
ACF at 200 s, we finally decided to use 15 fit lags in this study (see
Fig. 6c) which is on the safe side. We found that we can interrupt the
iteration procedure in the first step because all resulting profiles are
within the range of the noise error bars in this case regardless of whether
we use 10, 15, or 20 fit lags. As a result, 15 fit lags finally seemed for us
to be the best selection. For the higher-order moments, the same number of 15
fit lags was used as for the variance but here linear extrapolations to lag
zero was applied (Lenschow et al., 2000). We consider this as best approach,
as the shape of the higher-order structure function is still unknown to date.
Noise errors
The resulting profiles of the noise error of the temperature measurements
ΔT(z)=Tn′(z)2‾
are shown in Fig. 7 together with profiles of the errors due to shot noise
derived with Poisson statistics from the signal intensities (as detailed
below). Both profiles are similar but it should be noted that the
autocovariance technique specifies the total statistical error, while the
shot-noise error is a part of the total statistical error.
For calculating the shot-noise errors from the signal intensities, the
following approach was made: the lidar signals are detected simultaneously in
analog and in photon-counting mode. As the intensities of our rotational
Raman signals are too strong, the photon-counting signals are affected by
dead-time effects in lower heights
of about 6 km in daytime. Correction of these dead-time
effects (Behrendt et al., 2004) is
possible down to about 1.5 km. As this height limit is still too high
for CBL studies, the analog signals have been used for the measurements of
this study. In order to derive the shot-noise errors of the measurements with
Poisson statistics, the analog signals of each 10 s profile were fitted to
the photon-counting signals in heights between about 1.5 and 3 km,
where both detection techniques were providing reliable data after dead-time correction of the
photon-counting data. By this scaling, photon-counting rates could then be
attributed to the analog signal intensities in lower altitudes. These
attributed count rates were consequently used. The background photon-counting
numbers were derived from the photon-counting signals detected from high
altitudes.
The ratio of the two background-corrected photon-count numbers
NRR1 and NRR2 of lower and higher rotational
quantum number transition channels
Q=NRR2NRR1
is the measurement parameter which yields the atmospheric temperature profile
after calibration of the system.
Statistical uncertainties of 10 s, 1 min, and 20 min
temperature profiles at noontime determined with a 2/3-power-law
fit of the ACF data (see Fig. 6). Shot-noise errors calculated by use of Poisson statistics from the
detected signal intensities in each height are shown for comparison. It
can be seen that the statistical uncertainty of the RRL temperature
measurements is mainly governed by shot noise. The range
resolution of the data was 109 m.
The shot-noise error of a signal with N photon counts according to Poisson
statistics is
ΔN(z)=N(z).
Error propagation for the RRL temperature data then yields (Behrendt et al.,
2002)
ΔT(z)=∂T∂QNRR2(z)NRR1(z)NRR1∗(z)+ΔB‾RR12NRR1(z)2+NRR2∗(z)+ΔB‾RR22NRR2(z)2,
with NRR1∗(z) and NRR2∗(z) for the
photon counts in the two rotational Raman channels before background
correction. NRRi(z)=NRRi∗(z)-B‾RRi with i=1,2 are the signals which are
corrected for background noise per range bin B‾RRi.
∂T/∂Q is provided by the temperature calibration
function. As outlined already above (see Sect. 3.1), the uncertainty of this
calibration for the analysis of turbulent temperature fluctuations is
negligible.
Since the background is determined over many range bins, the statistical
uncertainty of the background can be neglected (Behrendt et al., 2004) so
that one finally gets
ΔT(z)=∂T∂QNRR2(z)NRR1(z)NRR1(z)+B‾RR1NRR1(z)2+NRR2(z)+B‾RR2NRR2(z)2.
The data in Fig. 7 show that the shot-noise errors
calculated with Poisson statistics provide lower estimates for the total
errors. But the comparison also confirms that the photon shot noise gives the
major contribution (about 75 %) and that other statistical error sources
(like the electric noise of the analog signals) are comparatively small. A
similar result, also for analog signals which were glued to photon-counting
signals, has already been obtained before for water vapor Raman lidar by
Whiteman et al. (2006).
The background-corrected rotational Raman signals scale according to
NRRi(z)∝PΔtΔzηtηrA,
where i=1,2, P is laser power, Δt is measurement time, Δz is range
resolution, ηt and ηr are transmitter and receiver efficiency, respectively, and A is receiving telescope area. The background counts in each signal range bin scale in a similar way
but without being influenced by power P and ηt, so that we
get
B‾RRi(z)∝ΔtΔzηrA.
One can see from Eqs. (7) to (9) that the statistical measurement uncertainty
scales consequently with the parameters which are found in both previous
equations according to
ΔT∝1ΔtΔzηrA.
It is noteworthy, that increases of the laser power P and transmitter
efficiency ηt are even more effective in reducing ΔT
than increases of Δt, Δz, ηr, or A because
the former improve only the backscatter signals and do not increase the
background simultaneously like the latter. The value of the improvement
obtained from increases of P or ηt, however, depends on
the intensity of the background and thus on height and background-light
conditions (see also Radlach et al., 2008; Hammann et al., 2015).
The statistical uncertainties for the RRL temperature measurements at
noontime shown in Fig. 7 were determined with 10 s temporal
resolution and for range averaging of 109 m. The resulting error
profiles for other temporal resolutions were then derived from the 10 s
error profile by use of Eq. (10). The errors for other range resolutions can
be easily obtained from Eq. (10) in a similar way.
The results of the error analysis show the very high performance of the UHOH
RRL temperature data: with 10 s resolution, the total statistical
uncertainty ΔT at noontime determined from the variance analysis of
the temperature fluctuations is below 1 K up to 1020 m a.g.l. With
1 min resolution, ΔT is below 0.4 K up to 1000 m a.g.l.
and below 1 K up to 1510 m a.g.l. With 20 min averaging, ΔT is below 0.1 and 0.3 K up to 1050 and 1710 m a.g.l.,
respectively.
Integral scale
Figure 8 shows the profile of the integral scale of the temperature
fluctuations. It was obtained with the 2/3-power-law fit of the structure
function to the ACF (Lenschow et al., 2000; Wulfmeyer et al., 2010). The
integral scale is about 80 s in the mixed layer decreasing towards smaller
values in the IL. At zi, the integral scale was (56±17) s. The
integral scale is significantly larger than the temporal resolution of the
UHOH RRL data of 10 s. This confirms that the resolution of our data
is high enough to resolve the turbulent temperature fluctuations including
the major part of the inertial subrange throughout the CBL. The integral timescale, which can be related to a length scale provided that the mean
horizontal wind speed is known, is considered as a measure of the mean size
of the turbulent eddies involved in the boundary layer mixing processes.
Integral scale of the temperature fluctuations shown in Fig. 5 (1 h
period between 11:00 and 12:00 UTC, 24 April 2013). Error bars show the
noise errors. The mean CBL height zi of 1230 m (dashed line) was
determined with the Haar-wavelet analysis of βpar and was
used for the relative height scale z/zi.
Temperature variance
To the best of our knowledge, the first profile of the temperature variance
of the atmosphere Ta′(z)2‾ measured with a lidar system is shown in Fig. 9; the profile
starts at about 0.3 zi and covers the whole CBL. We found that between
0.3 and 0.9 zi, i.e., the major part of the CBL, the atmospheric
variance was much smaller than in the IL. Here the values were only up to
0.1 K2 (at 1100 m = 0.9 zi with 0.01 and
0.06 K2 for the sampling and noise error, respectively). We also
used the methods of Lenschow et al. (2000) for deriving these errors. While
the noise errors denote the 1σ statistical uncertainties of the data
product due to uncorrelated noise in the time series of the input data, the
sampling errors describe those uncertainties resulting from the limited
number of atmospheric eddies in the analysis period. Taking the error bars
into account, one finds that the apparent minimum of the temperature variance
profile at 0.6 zi is only weakly significant. What remains is a profile
with slightly increasing variance with height in the CBL and a clear maximum
in the IL close to zi. This maximum of the variance profile was
0.39 K2 with a sampling error of 0.07 and 0.11 K2 for
the noise error (root-mean-square variability). Above, the variance decreased
again. One expects such a structure for the variance profile: except at the
surface, the temperature variance in the CBL is largest in the IL, since the
temporal variability is driven by entrainment caused by turbulent
buoyancy-driven motions acting against the temperature inversion at the top
of the CBL (e.g., Deardorff, 1974; André et al., 1978; Stull, 1988; Moeng
and Wyngaard, 1989).
Profile of temperature variance (1 h period between 11:00 and
12:00 UTC, 24 April 2013). Error bars show the noise errors (thin error
bars) and the sampling errors (thick error bars). The mean CBL height zi
of 1230 m (dashed line) was used for the relative height scale
z/zi.
For quantitative comparisons, often normalization of the temperature variance
profile with T* is used (Deardorff, 1970). But in the real world with
its heterogeneous land use and soil properties and thus corresponding flux
variability such scaling becomes difficult. Instead of a single scaling
value, one could employ several flux stations and try to find a more
representative scaling parameter by weighted averaging of the measurements
made over different land-use types. But even then one expects that the scaled
temperature variance profile depends on the ratio of the mean entrainment and
surface flux (e.g., Moeng and Wyngaard, 1989). Thus, we decided not to scale
the variance profile here and leave further generalizations to future studies
based on more cases.
Third-order moment and skewness
The third-order moment (TOM) of a fluctuation is a measure of
the asymmetry of the distribution. The skewness S is the
TOM normalized by the variance to a dimensionless parameter
defined for temperature as
S(z)=T′(z)3‾T′(z)2‾3/2.
The normal distribution (Gaussian curve) has zero TOM and
S. Positive values for TOM and S show a right-skewed
distribution where the mode is smaller than the mean. If the mode is
larger than the mean, TOM and S become negative
(left-skewed distribution).
TOM and S profiles for the atmospheric temperature fluctuations of our case
were derived with the technique of Lenschow et al. (2000), as explained in
Sect. 3.2. The results are shown in Fig. 10. Up to about 0.9 zi, the
TOM was not different to zero (taking the 1σ statistical uncertainties
into account). In the IL, i.e., between 0.9 and 1.1 zi, a negative peak
is found with values down to -0.93 K3 with 0.05 and
0.16 K3 for the sampling and noise errors, respectively. The
skewness profile shows the same characteristics. Only data around
0.6 zi had to be omitted from the skewness profile because the measured
variance values are close to zero here and thus dividing by these values
yields too large relative errors. At zi, we found a skewness of -4.1
with 1.1 and 1.9 for the sampling and noise errors, respectively.
Same as Fig. 9 but profiles of the third-order moment (TOM) and the
skewness S. Error bars show the noise errors (thin error bars) and the
sampling errors (thick error bars). The mean CBL height zi of
1230 m (dashed line) was used for the relative height scale
z/zi. The dotted vertical line marks zero skewness. Skewness data around
0.6 and above 1.1 z/zi were omitted because the data were too noisy
here due to variances close to zero.
TOM and S profiles reveal interesting characteristics of the thermal plumes
which were present in the CBL in this case. As rising plumes of warmer air
are typically narrow and surrounded by larger areas of air close to the
average temperature, one expects slightly positive temperature skewness in
the major part of the CBL; e.g., Mironov et al. (1999) show values between 0
and 2 (see their Fig. 1b); they did not show negative values which would
indicate narrow cold plumes. In the CBL up to about 0.9 zi, the
measured values in our case agree with these data taking the uncertainties
into account.
The negative minima of TOM and S in the IL above show a clear difference
between the IL and the CBL below. Between 0.9 and 1.1 zi, negative and
positive fluctuations were not symmetric but fewer very cold fluctuations
were balanced by many warm fluctuations with less difference to the mean.
Because turbulent mixing occurs in the IL in a region of positive vertical
temperature gradient, the air present in the free
troposphere is warmer than the air in the CBL below. Consequently, the
negative peak indicates that the cold overshooting updrafts in the IL
were narrower in time than the downdrafts of warmer air.
Similar characteristics of the temperature TOM and skewness profiles in the
IL were discussed, e.g., by Mironov et al. (1999), Canuto et al. (2001), and
Cheng et al. (2005) who compare experimental data (tank, wind tunnel,
airborne in situ), large eddy simulation (LES) data, and analytical expressions. Now, more comparisons
can be performed between real atmospheric measurements and models.
Interestingly, an inverse structure of the TOM profile is found with respect
to humidity fluctuations (Wulfmeyer, 1999b; Wulfmeyer et al., 2010; Turner et
al., 2014b). Combining these results, it should be possible to perform very
detailed comparisons with LES and to refine turbulence parameterizations.
This concerns particularly the TKE (turbulent kinetic energy) 3.0 order schemes that are using the
closure of the variance budget for determining the turbulent exchange
coefficients.
Fourth-order moment and kurtosis
The fourth-order moment (FOM) is a measure of the steepness at the sides of
the distribution and the corresponding flatness of the peak. The kurtosis is
the FOM normalized by the variance to a dimensionless parameter according to
Kurtosis(z)=T′(z)4‾T′(z)2‾2.
With this definition, the normal distribution (Gauss curve) has a kurtosis of
3. Equation (12) is also used by Lenschow et al. (2000); we follow this
definition here. Please note that sometimes kurtosis is defined differently
including a subtraction of 3 which then results in a kurtosis of 0 for the
normal distribution, but mostly Kurtosis – 3 is called “excess kurtosis”.
Figure 11 shows FOM and kurtosis profiles of the measured temperature
fluctuations of our case which have also been obtained with the method of
Lenschow et al. (2000) for noise correction. For both FOM and kurtosis, the
noise errors of the data are quite large; the importance of an error analysis
becomes once more obvious. Throughout the CBL, no significant differences to
the normal distribution are found. While the values for the FOM are close to
zero in the CBL (< 0.5 K4 up to 0.9 zi), they appear
larger in the IL, but the noise error does not allow for determining exact
values, zero is still within the 1σ noise error bars. At zi, FOM
was 3.0 K4 with 0.1 and 4.2 K4 for the sampling and
noise errors, respectively. The kurtosis at zi was 23 with 8 and 35 for
the sampling and noise errors, respectively. We conclude that the
distribution of atmospheric temperature fluctuations was not significantly
different to a Gaussian distribution (quasi-normal) regarding its fourth-order
moment and kurtosis in our case.
Same as Fig. 9 but profiles of the fourth-order moment (FOM) and
kurtosis. Only kurtosis data below 0.55 and around 1.0 z/zi are shown
because other data are too noisy. The dotted vertical line in the lower panel
marks a value of 3 which is the kurtosis of the normal distribution. Error
bars show the noise errors (thin error bars) and the sampling errors (thick
error bars). The mean CBL height zi of 1230 m (dashed line in
the upper panel) was used for the relative height scale z/zi.
Even if the data here are too noisy to identify non-zero FOM or kurtosis in
the IL, it is interesting to note that higher values of kurtosis in the IL
would reflect a situation for which a large fraction of the temperature
fluctuations occurring in this region would exist due to infrequent, very
large deviations in temperature; the related most vigorous thermals would
then be capable to yield quite extreme temperature fluctuations, while mixing
intensively in the IL with the air of the lower free troposphere. In contrast
to this, the temperature fluctuations would be more moderate (Gaussian) in
the CBL below.
Conclusions
We have shown that rotational Raman lidar provides a remote-sensing technique
for the analysis of the turbulent temperature fluctuations within the
well-developed CBL during noontime – even though the background-light
conditions at noon are least favorable for the measurements. The required
high temporal and spatial resolution combined with low-enough statistical
noise of the measured data is reached by the UHOH RRL which is to the best of our
knowledge for the first time. The data can thus be evaluated during all
time periods of the day for studying the structure of the atmospheric
boundary layer – of course also at night.
A case of the HOPE campaign was analyzed. The data were collected between
11:00 and 12:00 UTC on IOP 6, 24 April 2013, i.e., exactly around local noon
(11:33 UTC). The UHOH RRL was located near the village of Hambach in western
Germany (50∘53′50.56′′ N,
6∘27′50.39′′ E; 110 m a.s.l.).
A profile of the noise variance was used to estimate the statistical
uncertainty ΔT of the temperature data with a structure function fit
to the ACF. A comparison with a ΔT profile
derived with Poisson statistics demonstrated that the statistical error is
mainly due to shot noise. The Haar-wavelet technique was applied to 10 s
profiles of βpar and provided the mean CBL height over the
observation period of zi=1230 m a.g.l. This value was used for
normalizing the height scale. The integral scale had a mean of about
80 s in the CBL confirming that the temporal resolution of the RRL
data of 10 s was sufficient for resolving the majority of
turbulence down to the inertial subrange.
The results of this study give further information on turbulent temperature
fluctuations and their statistics in the CBL and within the IL.
The atmospheric variance profile showed clearly the largest values close to
zi. A maximum of the variance of the atmospheric temperature
fluctuations was found in the IL: 0.39 K2 with a sampling and
noise error of 0.07 and 0.11 K2, respectively.
Subsequently, also profiles of the third- and fourth-order moments were
derived:
TOM and skewness were not significantly different to zero within the CBL up
to about 0.9 zi. In the IL between 0.9 and 1.1 zi, a negative
minimum was found with values down to -0.93 K3 with 0.05 and
0.16 K3 for the sampling and noise errors, respectively. Skewness
at zi was -4.1 and with 1.1 and 1.9 for the sampling and noise errors,
respectively. We conclude that the
turbulent temperature fluctuations were not significantly skewed in the CBL.
In contrast to this, the atmospheric temperature fluctuations in the IL were
clearly skewed to the left (negative skewness). This finding is related to
narrower cold overshooting updrafts and broader downward mixing of warmer air
from the free troposphere in the IL.
Throughout the CBL, no significant differences to the normal distribution
were found for FOM and the kurtosis. For all moments but especially the FOM,
the importance of an error analysis became once more obvious.
A quasi-normal FOM even when TOM is non-zero, agrees with the hypothesis of
Millionshchikov (1941) which forms the basis for a large number of closure
models (see Gryanik et al., 2005, for an overview). However, some recent
theoretical studies, measurement data, and LES data suggest that this
hypothesis would not be valid for temperature in the CBL
(also see Gryanik et al., 2005, for an overview).
Gryanik and Hartmann (2002) suggested furthermore a parameterization between
the FOM, skewness, and variance of turbulent temperature fluctuations which
can be tested as soon as a larger number of measurement cases on turbulent
temperature fluctuations with rotational Raman lidar have become available.
It is planned to extend the investigation of CBL characteristics in future
studies also by combining the UHOH RRL data with humidity and wind
observations from water vapor DIAL (Behrendt et al., 2009; Wagner et al.,
2013; Muppa et al., 2015) and Doppler lidar. Furthermore, also the scanning
capability of the UHOH RRL will be used in the future to collect data closer
to the ground and even the surface layer (Behrendt et al., 2012) in order to
investigate heterogeneities over different terrain.
The combination of different turbulent parameters measured by lidar –
preferably, at the same atmospheric coordinates simultaneously – promises to
provide further understanding on the important processes taking place in the
CBL including the IL. For instance, up until now, the key physical processes
governing the IL and their relationships with other CBL properties unfortunately remain
only poorly understood: they are oversimplified in empirical
studies and poorly represented in the models. In consequence, more data
should be evaluated to get the statistics of the turbulent temperature
fluctuations under a variety of atmospheric conditions. We believe that
corresponding measurements with RRL will contribute significantly to better
understanding of boundary layer meteorology in the future – not only in
daytime but also at night so that the entire diurnal cycle is covered and the
characteristics of turbulent temperature fluctuations in different stability
regimes can be observed.