Rotational Raman lidar makes use of Raman signals of atmospheric molecules (mainly N2 and O2) of different temperature dependencies (Cooney, 1972; Behrendt, 2005). From the ratio of the atmospheric backscatter signals, a measurement signal is obtained which depends on atmospheric temperature. The rotational Raman lidar of the University of Hohenheim was designed with the focus on high-resolution measurements during the daytime (Radlach et al., 2008; Hammann et al., 2015; Hammann and Behrendt, 2015). As a laser transmitter, a frequency-tripled injection-seeded Nd:YAG laser is used (see also Di Girolamo et al., 2004), which emits 200 mJ laser pulses at 355 nm with a repetition rate of 50 Hz. A Pellin–Broca prism refracts the other laser wavelengths out of the optical path, which makes the system eye-safe from short distances onward. The whole lidar is housed in a truck so that it can be moved easily to field campaigns. A two-mirror scanner allows for three-dimensional observations. For the measurements discussed here, this scanner was pointing vertically. The backscatter signals of the atmosphere are collected with a telescope with a 40 cm primary mirror and then separated with interference filters into four channels: the elastic channel, two rotational Raman channels, and a water-vapor Raman channel. The first three of these channels are mounted in a cascade, which makes the signal separation very efficient (Behrendt and Reichardt, 2000; Behrendt et al., 2002, 2004). The water-vapor Raman channel was not yet in operation for the measurements discussed here, but the beam splitter for this channel was already in place. Photomultipliers collect the four signals and give electric signals, which are analyzed with a combined photon-counting and analog transient recorder (LICEL GmbH). A temporal resolution of 10 s and a range resolution of 3.75 m were selected for the raw data of all channels. For the temperature measurements, the two rotational Raman signals were first smoothed with a gliding average of 108.75 m; then the ratio of the signals was calculated. For the temperature calibration, data of a radiosonde launched at the lidar site were used. The second-order logarithmical function suggested by Behrendt and Reichart (2000) was used as calibration function. Since the strong daytime photon-counting signals were influenced by dead-time effects in the ranges discussed in this study, we used only analog signals.

Doppler lidar measures the radial wind velocity via the Doppler shift of laser radiation scattered in the atmosphere (e.g., Werner, 2005). In this study, we used data of the heterodyne Doppler lidar Wind-Tracer WTX of Lockheed Martin Coherent Technologies, USA, operated by the Karlsruhe Institute of Technology (KIT) (Träumner et al., 2014). The lidar transmitter is a Er:YAG laser that emits laser pulses at a wavelength of 1.6 µm using a pulse repetition frequency of 750 Hz with 2.7 mJ pulse energy. The lidar can be operated in different scan patterns. Vertical stare mode yields vertical velocity w with a time resolution of typically 1 s from about 375 m a.g.l. to the top of the boundary layer and partly above, depending mainly on the aerosol concentration. The effective range-gate resolution was about 60 m.

In the following, we also introduce briefly the WVDIAL of the University of Hohenheim (UHOH) because we show latent heat flux profiles for comparison. WVDIAL provides absolute humidity profiles with high temporal and spatial resolution in the lower troposphere (Wulfmeyer and Bösenberg, 1998). During the HOPE campaign (see Sect. 4.1 below for definition; Macke et al., 2017), the scanning water-vapor WVDIAL of the University of Hohenheim (Wagner et al., 2013) was operated in vertical mode during clear-sky conditions and in scanning mode during cloudy periods (Späth et al., 2016). The operational wavelength of the UHOH WVDIAL is near 818 nm. The laser transmitter was switched shot by shot between the online and offline frequencies. The backscatter signals were recorded for each laser shot (250 Hz) with a range resolution of 15 m. The measured absolute humidity has typical temporal and spatial resolutions of 1 s to 1 min and 15 to 300 m, respectively, depending on the range of interest. Due to the instrument's high laser power (about 2 W) in combination with a very efficient receiver (0.8 m telescope), the data have low noise uncertainties up to the CBL top. When deriving absolute humidity from the UHOH WVDIAL data used in this study, 10 s averages and a gliding window length of 135 m for the Savitzky–Golay algorithm (Savitzky and Golay, 1964) were used, resulting in a triangular weighting function with a full width at half maximum of about 60 m.

Lenschow et al. (2000) introduced a procedure for the estimation of higher-order moments of turbulent fluctuations that accounts for random instrumental noise. We follow this method for resolving the turbulent moments of temperature, vertical wind, and humidity for estimating instrument noise uncertainties. Further, important refinements were presented in Wulfmeyer et al. (2016) such as automated spike detection and the proper choice of lags used in the autocovariance analyses.

The flux profiles in the CBL were calculated using the eddy covariance method (see, e.g., Senff et al., 1994; Wulfmeyer, 1999b). The data processing procedure is described in detail in Wulfmeyer et al. (2016). In the following, we explain the method with the sensible heat flux as an example; the measurement of the latent heat flux works in an analogous way. First, temperature and wind data measured by the TRRL and Doppler lidar, respectively, were despiked. This means that histograms of the data at each height were calculated for the selected period, and then all data outside of 4 standard deviations from the median were removed. Other authors (Turner et al., 2014a, b) refined the despiking of noisy lidar data by also considering non-Gaussian distributions and asymmetric despiking thresholds on either side of the histogram, but we found that for the case shown here such further refinements do not change the results significantly. A despiking procedure is required because the lidar data analysis algorithms are non-linear and noise in the data may result in large (non-linear) outliers in some cases.

In a second step, the despiked temperature data were detrended using a linear fit at each height level. This procedure is required in order to focus on the turbulent fluctuations by removing influences of large-scale advection, synoptic processes, and the diurnal cycle. Detrending means that the time series of the scalar observations s(z,t), with s being, e.g., temperature T or humidity q, z being height above ground, and t being time, are used over the selected period to obtain mean linear trends s*(z,t) and that these trends were then subtracted from the instantaneous values s(z,t) at each height to obtain the fluctuations 1s′(z,t)=s(z,t)-s*(z,t). Consequently, the mean of these fluctuations becomes zero.

In case of the wind data w(z,t), we decided to subtract only time-independent means at each height level of the vertical wind data in order to ensure fluctuations with perfect zero mean. Detrending alters the real atmospheric fluctuations quite significantly when the updrafts and downdrafts are not perfectly evenly distributed in the analysis period (which is in practice never the case due to the statistical nature of the thermals in the CBL).

In practice, both despiking and detrending has to be performed with caution in order not to eliminate real atmospheric features. This means that the time series of data should be investigated first and only quasi-stationary time series with small trends in the scalar (T or q), small biases in w, and a sufficient number of thermals should be used in order to obtain fluxes with low systematic errors. Typically, the time period for the analysis thus need to be at least 30 min.

Before the scalar and wind time series can be combined, one must ensure in a third step that the time and height for each data point are as close as possible. For this, we gridded the data to closest neighbors.

The correlation of the temperature fluctuations T′ and the vertical wind fluctuations w′ provides profiles of the eddy sensible heat flux according to 2H(z)=ρa(z)cpw′(z)T′(z)‾, with ρa(z) for the air density at height z and cp for the specific heat capacity of air.

For the lidar data, which are discrete in time, Eq. (2) results in 3H(z)=ρa(z)cpN(z)∑i=1N(z)w′i(z)T′i(z), where i is the number of each data point and N(z) is the total number of common data points at height z of T′ and w′. N(z) is not the same for all heights z because the despiking routine removes a few data points.

The instrumental noise uncertainty and the representativeness uncertainty play a major role when deriving the statistics of turbulent fluctuations in the atmosphere with noisy data (Lenschow et al., 1994, 2000). The noise uncertainty is due to the instrumental noise of the lidar data. The uncertainty due to sampling only a limited period of time covering a limited number of turbulent eddies is referred to as representativeness uncertainty or sampling uncertainty.

Fortunately, when H(z) is calculated with the lidar data with Eq. (3), the noise contributions of the vertical wind and temperature fluctuations, w′i,n(z) and T′i,n(z), do not cause a bias in the covariance and – unlike for the variances – no subtraction of an instrumental noise term is required.

This can be understood by splitting the fluctuations into an atmospheric part and into a noise part according to 4w′i(z)=w′i,a(z)+w′i,n(z) and 5T′i(z)=T′i,a(z)+T′i,n(z).

Inserting Eqs. (4) and (5) in Eq. (3) yields 6H(z)=ρa(z)cpN(z)∑i=1N(z)(w′i,a(z)T′i,a(z)+w′i,a(z)T′i,n(z)+w′i,n(z)T′i,a(z)+w′i,n(z)T′i,n(z)), and hence 7H(z)=ρa(z)cpN(z)∑i=1N(z)w′i,a(z)T′i,a(z) because w′i,n(z) and T′i,n(z) are uncorrelated with each other as well as with the atmospheric fluctuations.

But even though there is no noise term to be subtracted when determining fluxes from noisy data, of course, there still remains an uncertainty of the flux value due to noise: because the real atmospheric data set is always of a finite length, the noise terms do not cancel fully. This noise uncertainty of the covariance of the fluctuations of a scalar s (e.g., T or q) and the vertical wind w can be estimated by applying Gaussian error propagation to Eq. (7) giving (Wulfmeyer et al., 2016) 8σw′s′‾(z)≅s′a(z)2‾σw′(z)2+w′a(z)2‾σs′(z)2N(z), with σw′(z)2 and σs′(z)2 for the noise variances of the scalar and the vertical wind at height z, respectively. Furthermore, the noise uncertainties of the fluctuations are equal to the noise uncertainties of the original (non-detrended) data sets – σw′(z)2=σw(z)2 and σs′(z)2=σs(z)2 (assuming that the noise uncertainty in the trend determination can be neglected); nevertheless, in the following we keep the notation with primes for clarity as we are dealing with fluctuations here.

Equation (8) can be further approximated and rearranged so that the relative noise of the covariance is expressed with the relative noise variances according to 9σw′s′‾w′s′‾≅1Nr2σs′2s′a2‾+σw′2w′a2‾, where r is the correlation coefficient between atmospheric vertical wind fluctuations and atmospheric scalar fluctuations given by 10r=w′s′‾w′a2‾s′a2‾. Please note that we omit the height dependence for simplicity in Eq. (9) and the following equations. Consequently, it follows for the noise uncertainty of the sensible heat flux that 11σHH≅1Nr2σT′2T′a2‾+σw′2w′a2‾. It is important to note that we find the variances of the atmospheric fluctuations of temperature and vertical wind in Eq. (11) but not the variances of the total fluctuations (which include both the atmospheric and noise fluctuations).

In order to identify these atmospheric variances of the temperature and vertical wind fluctuations, we use the method of Lenschow et al. (2000) to separate the noise variance from the total variance: the atmospheric variance σa2 is obtained from the total variance σtot2 by subtracting the noise variance σn2: 12σa2=σtot2-σn2. σn2 is determined from an autocovariance analysis of the high-resolution time series of the lidar data. The autocovariance at zero lag is the sum of the atmospheric and noise variances. While the atmospheric fluctuations are correlated in time, the random instrumental noise fluctuations are not. Consequently, one can separate the atmospheric variance from the noise variance by extrapolating the fit of the autocovariance function (also called “structure function”) to lag zero. The most suitable choice for the lag is a factor of 2.5 larger than the integral timescale (Wulfmeyer et al., 2016).

After the noise uncertainty profiles have been determined from the variance analysis for both the temperature and vertical wind measurements, σT′2 and σw′2 are used for calculating the noise uncertainty of the fluxes with Eq. (11).

The representativeness uncertainty or sampling uncertainty of the flux expressed as a variance is the square of the difference of the mean flux 〈F〉 measured in the sampled time period and the real mean flux F‾ which would be determined in an infinitely long measurement period: 13σF2=〈(F‾-〈F〉)2〉. This uncertainty can be estimated for τ≫τws with (Lenschow et al., 1994) 14σF=21+1r22τwsτF‾2, where τws is the integral timescale of the vertical flux of the scalar s, τ is the length of the measurement period, and r is again the correlation coefficient of s′ and w′. (Note that Lenschow et al., 1994, call this type of uncertainty “random error” which must not be confused with the noise uncertainty due to random noise.)

An upper limit of the representativeness uncertainty can be obtained from the minimum of the integral timescales of vertical wind τw and the integral timescale of the scalar τs via (Lenschow and Kristensen, 1985) 15σF=4r2min⁡τw,τsτF‾2. Thus, it follows for the sensible heat flux that 16σH,sampling=2Hrmin⁡τw,τsτ.

The vertical divergence of the sensible heat flux can be related to a temperature tendency via 171ρacp∂H∂z=∂w′T′‾∂z=-∂θ‾∂tFluxDiv. The temperature tendency term on the right-hand side of Eq. (17) is just a fraction of the total tendency, namely the contribution of the vertical flux divergence to the total tendency. We indicate this relation by the subscript FluxDiv. The tendency contributions of advection, radiative cooling, etc., are not included and need to be determined separately.

With the lidar data, we obtain the fluxes in the lower part of the CBL and at the CBL top simultaneously. Thus, we get the temperature trend due to flux divergence in the observed part of the CBL with 18∂θ‾∂tFluxDiv=-1ρacpH(zi)-H(zbottom)zi-zbottom, where zbottom stands for the lowest observed height of the CBL.

In a similar way, the divergence of the latent heat flux is related to the moisture tendency in the CBL via 191Lv∂L∂z=∂w′q′‾∂z=-∂q‾∂tFluxDiv. with Lv for the latent heat of vaporization and thus for the lidar data we use 20∂q‾∂tFluxDiv=-1LvL(zi)-L(zbottom)zi-zbottom.

The data used in this study were collected during the HOPE campaign (Macke et al., 2017). HOPE stands for the HD(CP)2 Observational Prototype Experiment. HD(CP)2 (High Definition Clouds and Precipitation for advancing Climate Prediction) was a German research initiative aiming at a reduction in the uncertainty of climate-change predictions by means of a better understanding and simulation of cloud and precipitation processes. The HOPE domain was located near the Research Center Jülich in western Germany. During the HOPE period in April and May 2013, three so-called supersites were set up forming a triangle with side lengths of about 4 km. At the site near the village of Hambach, the UHOH set up its scanning TRRL and its scanning WVDIAL. The KIT brought its so-called KITcube (Kalthoff et al., 2013) with – among a suite of other instruments – a scanning Doppler wind lidar and a surface energy balance station.

In this study, we use data collected on 24 April 2013, the HOPE intensive observation period (IOP) 6. The HOPE domain was under the influence of an anticyclone located over central Europe on this day (see also Behrendt et al., 2015; Muppa et al., 2016). At the lidar site, the CBL was well developed by 10:00 UTC (Muppa et al., 2016). We selected the period from 11:05 to 11:50 UTC around solar noon at 11:32 UTC for the analysis of fluxes. Similar periods have been used regarding separate analyses of higher-order moments of the turbulent wind, temperature, and humidity fluctuations (Behrendt et al., 2015; Maurer et al., 2016; Muppa et al., 2016; Wulfmeyer et al., 2016).

As discussed in Behrendt et al. (2015), the mean of the instantaneous CBL heights zi in the observation period was 1230 m above ground level (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. At the ground, sensible heat flux H0 was 192 W m-2 and thus lower than the latent heat flux L0 of 255 W m-2 measured with the KITcube energy balance station at the lidar site.

Figure 1 shows the fluctuations of temperature, humidity, and vertical wind T′(z,t), q′(z,t), and w′(z,t), respectively, measured with the three lidar systems on 24 April 2013 in the 45 min period between 11:05 and 11:50 UTC. We found that this period is long enough to provide results with low uncertainties. It should be noted that these data include both atmospheric fluctuations and instrumental noise (compare Eqs. 4 and 5). While the first are correlated in time, the latter are not. It can already be seen here that the noise in the temperature data is higher than in the humidity and vertical wind data. Nevertheless, the correlated atmospheric fluctuations stand out from the noise in all three plots.

It is interesting to now compare the simultaneous fluctuations of temperature, humidity, and wind with each other. Updrafts (w′>0) are generally, but not always, related to warmer and moister air, while downdrafts (w′<0) are generally, but not always, cooler and dryer. This illustrates the CBL dynamics, which are mainly driven by buoyancy, but also their complexity.

The products of vertical wind fluctuations with temperature and with humidity fluctuations are shown in Fig. 2. Positive instantaneous latent heat flux values are dominant throughout the CBL, while the instantaneous sensible heat flux values are dominantly positive in the lower half of the CBL but negative in the upper half.

For completeness, Fig. 2 also shows the product of temperature and humidity fluctuations. Apart from noise, the data show partly positive and partly negative values. Positive values indicate that warmer air was moister, while cooler air was drier in the CBL. The fact that there are also negative data points reveals that cooler and moister as well as warmer and dryer fluctuations also appeared simultaneously here.

For the variance data and noise uncertainties presented here, 20 data points of the structure function were used for the fit, giving a period of 200 s with 10 s resolution of the data. This is a reasonable number because the first zero crossing of the autocovariance function τ0 is found at 2.5 times the integral scale (Behrendt et al., 2015; Wulfmeyer et al., 2016), which is mostly between 40 and 120 s in this case (see Fig. 3a). The integral scales were determined according to Eq. (42) of Wulfmeyer et al. (2016). We found that it is not a problem to use a few more lags (even beyond the zero crossing) in order to get a stable fit. The integral scale is a measure of the mean horizontal size of the eddies in the temporal domain during the measurement period (see, e.g., Lenschow et al., 2000, for details). Interestingly, this scale is usually different for different variables. In addition, by comparing the temporal resolution of the lidar measurements with the profile of the integral scale, we can make sure that the temporal resolution is high enough throughout the profile to resolve the major part of the turbulent fluctuations.

The temperature variance profile in this case shows the typical peak near the CBL top (Fig. 3b). The value at zi was (0.46±0.08±0.13) K2, with the first error for the sampling uncertainty and the second for the noise uncertainty. The humidity variance has a more complex structure here with a double peak near zi (see also Muppa et al., 2016). The peak at 0.9 zi was (0.17±0.04±0.01) (g m-3)2, while the upper peak at 1.1 zi was (0.58±0.11±0.10) (g m-3)2. The vertical wind variance also shows a secondary peak at zi in this case, while the maximum is found in the middle CBL, as is typical. The maximum vertical wind variance appeared at 0.6 zi and was (1.67±0.40±0.03) (m s-1)2.

The correlation coefficients (Eq. 10) of the lidar data (Fig. 1) are shown in Fig. 4. Figure 5 shows the sensible heat flux profile H(z) derived with the data shown in Fig. 1 via Eq. (3) together with the latent heat flux profile L(z). While positive values are found for H(z) in the lower and middle CBL, negative values are found in the upper CBL as well as in the lower free troposphere. The upward sensible heat flux in the lower and middle CBL is related to upward energy flux from the surface into the CBL, while the negative values above show the downward energy flux into the CBL by entrainment. The sign of the correlation coefficient r of w′ and T′ is just the same as the sign of H at the same height. The values of r lie between -1 and 1 with the exception of two data points between 1000 and 1100 m. These outliers are due to the statistical uncertainty of the data since the temperature variance is close to zero here. r is larger than 0.5 from 400 to about 600 m with a decreasing tendency with height. The maximum H was (182±112±32) W m-2 and appears at 0.5 zi, again with the first error for the sampling uncertainty and the second for the noise uncertainty. At about 0.7 zi, H and r change signs to negative values above. We estimated the entrainment flux near zi by averaging the measurements between 0.95 and 1.05 zi and obtained a value of (-62±27±42) W m-2. At our lowest measurement points between 400 and 500 m (corresponding to 0.3 to 0.4 zi) we found a mean sensible heat flux of (156±34±8) W m-2. Taking these representative values at the lower and upper parts of our measurement range in the CBL, we obtain -0.28 W m-3 for the vertical divergence of the sensible heat flux. This corresponds (Eq. 18) to a temperature tendency term of 2.3×10-4 K s-1 or 0.83 K h-1.

Figure 5 also shows the latent heat flux profile L(z) derived with the lidar data for comparison. The values are positive throughout the CBL as being typical. So are also the values of the correlation coefficients of moisture and vertical wind fluctuations (Fig. 4). This is because both upward moisture transport from the surface into the CBL and downward entrainment of dry air from the lower free troposphere into the CBL are related to positive values. Taking again representative values for the latent heat flux at the lower and upper parts of the boundary layer, we can determine the latent heat flux divergence and furthermore the water-vapor tendency (Wulfmeyer, 1999b). Between 400 and 500 m (corresponding to 0.3 to 0.4 zi), we found a latent heat flux of (95±37±5) W m-2. The entrainment flux near zi was (190±63±39) W m-2 (obtained again from 0.95 and 1.05 zi. but with the outlier at 1275 m excluded). Over the range of the measurements, we thus get a latent heat flux divergence of 0.12 W m-3, which corresponds to a tendency of the absolute humidity due to the interplay of evapotranspiration and entrainment of -5.3×10-5 (g m-3) s-1 or -0.19 (g m-3) h-1.