A simultaneous deployment of Doppler, temperature, and water-vapor lidars is able to provide profiles of molecular destruction rates and turbulent kinetic energy (TKE) dissipation in the convective boundary layer (CBL). Horizontal wind profiles and profiles of vertical wind, temperature, and moisture fluctuations are combined, and transversal temporal autocovariance functions (ACFs) are determined for deriving the dissipation and molecular destruction rates. These are fundamental loss terms in the TKE as well as the potential temperature and mixing ratio variance equations. These ACFs are fitted to their theoretical shapes and coefficients in the inertial subrange. Error bars are estimated by a propagation of noise errors. Sophisticated analyses of the ACFs are performed in order to choose the correct range of lags of the fits for fitting their theoretical shapes in the inertial subrange as well as for minimizing systematic errors due to temporal and spatial averaging and micro- and mesoscale circulations. We demonstrate that we achieve very consistent results of the derived profiles of turbulent variables regardless of whether 1 or 10 s time resolutions are used. We also show that the temporal and spatial length scales of the fluctuations in vertical wind, moisture, and potential temperature are similar with a spatial integral scale of

Improved understanding and modeling of turbulent transport processes in the convective boundary layer (CBL) require in-depth studies of the budgets of second- and other higher-order moments of atmospheric variables. Key variables include turbulent kinetic energy (TKE) and water-vapor and temperature variances; the latter is in terms of absolute humidity, mixing ratio, or specific humidity variances as well as temperature or potential temperature variances. The analysis of TKE and variance budgets and their components is important for the verification of weather forecast, climate, and Earth system models. Furthermore, the representation of budgets of second-order moments of atmospheric variables is essential for the parameterization of turbulent transport processes in mesoscale models, in which the turbulence is not explicitly resolved, or for the parameterization of subgrid-scale cloud processes.

In a turbulent flow, eddy diameters span a range of length scales

Large-eddy and direct numerical simulation (LES and DNS) models predict turbulence fields by explicitly resolving turbulent processes into the inertial subrange. Resolving these processes avoids the need for TPs at their spatial grid increments (

This verification should be performed under a range of different meteorological conditions, from the surface layer (SL), through the mixed layer (ML) and the interfacial layer (IL) at the CBL top to the lower troposphere. Previous studies, as early as the 1970s, mainly used in situ measurements

A powerful alternative is the operation and application of ground-based active remote systems, which are considered in this work. In the SL, scanning lidar systems were successfully combined to derive high-resolution profiles for studying Monin–Obukhov similarity theory

For the measurement of scalar variances and their destruction rates, high-resolution vertical profiling of temperature and moisture is also possible with active remote sensing. Sufficient spatial–temporal resolutions can be reached with water-vapor differential absorption lidar (WVDIAL) and water-vapor Raman lidar (WVRL). For WVDIAL, this was demonstrated in

With respect to temperature measurements, a breakthrough has recently been achieved using the temperature rotational Raman lidar (TRRL) technique. The first demonstration of high-resolution temperature profiling during daytime in the CBL, from which higher-order moments could be derived, was presented in

DL measurements are available from observatories such as the Atmospheric Radiation Measurement (ARM) program's Southern Great Plains (SGP) site

In this work, we present the derivation of TKE dissipation and molecular destruction rate profiles from the HOPE campaign. This study is organized as follows: in Sect. 2, we present the field campaign and the data set used in this study. We revisit the TKE, water-vapor, and temperature budget equations and discuss the terms containing the dissipation rates in Sect. 3. Some examples of their parameterizations are presented. In Sect. 4, we show how we derived and evaluated the transverse temporal autocovariance functions (ACFs) and the power spectra of the lidar time series in order to derive profiles of variances, the ACF coefficients, and the integral timescales dependent on temporal and spatial resolutions. Examples of the profiles of TKE dissipation and molecular destruction rates using our method for a case during HOPE are presented in Sect. 5. These results contain detailed error analyses. In Sect. 6, the results are discussed and compared with previous methods and corresponding results. A summary and an outlook are given in Sect. 7.

We present analyses of turbulence profiles from intensive observation period (IOP) 5 of HOPE, which was performed in spring 2013, close to the city of Jülich in Germany. IOP5 was executed on 20 April 2013. The data set was collected with the WVDIAL and TRRL of the Institute of Physics and Meteorology (IPM) at the University of Hohenheim (UHOH) and with a DL operated by the Karlsruhe Institute of Technology (KIT) between 11:30–12:30 UTC. The lidar systems were located at a site close to the village of Hambach near the Jülich Research Centre at 50

The UHOH WVDIAL is based on a Ti:sapphire laser transmitter tuned to

The UHOH TRRL is a combined water-vapor and temperature rotational Raman lidar. The laser transmitter is a frequency-tripled, injection-seeded Nd:YAG laser, which delivers up to 15 W average power at 355 nm. The heart of this system is a very efficient high-transmission series of interference filters in front of four sensitive photomultipliers, which collect four signals: the elastic backscatter, two channels sensitive to the rotational Raman scattering by nitrogen and oxygen, and the vibrational–rotational Raman channel of water vapor. In this work, we focus on the temperature profiles measured with the TRRL.

At the Hambach site, KIT operated a coherent DL based on an Er:YAG 1.6

The WVDIAL, TRRL, and DL data were processed consistently with temporal resolutions of either 1 s or 10 s from 11:30–12:30 UTC. All measurements started at 350 m above ground level (a.g.l.). We maintained a vertical resolution of 50 m in the DL vertical wind profiles, as the resulting SNR was good enough for accurate measurements up to the IL. The WVDIAL and TRRL data were processed with vertical resolutions of 70 and 100 m, respectively, in order to maintain an acceptable SNR up to the IL. For the WVDIAL, we evaluated data with 1 and 10 s resolutions to study changes in the ACFs and the power spectra, whereas the TRRL data could only be evaluated with 10 s due to the lower SNR. The TRRL data were overlap corrected up to 800 m

The TKE dissipation and the molecular destruction rates are the sink or loss terms in the TKE and mixing ratio and potential temperature variance budget equations. TKE is defined as

The TKE dissipation can be parameterized in multiple ways; see, e.g.,

The budgets for the water-vapor mixing ratio variance

If the turbulence is stationary and isotropic and the temporal and vertical resolutions of the lidar observations are high enough to resolve the inertial subrange, then the resulting transversal temporal ACF

Generally, lidar-based measurements of the ACFs are influenced by system noise and spatial and temporal filtering effects. Therefore, it is reasonable to study the ACFs using the following transformations:

Alternatively, the transformations

The ACFs should also be studied with respect to their temporal and spatial integral scales

Solving Eqs. (

As demonstrated in Eqs. (

The results are presented in Fig.

Horizontal wind profiles determined from a radiosonde launch at 13:00 UTC and three conical DL scans. For the soundings (black bullets), an error of 1 m s

The determination of the ACF and its coefficients with respect to its theoretical shape in the inertial subrange is more demanding than the estimation of variances for two reasons. (1) The difference in the total and the atmospheric variances around lag 0 has to be determined with high accuracy. (2) A suitable range of lags for the ACF must be evaluated for deriving its coefficient by a fit to the data. The determination of this coefficient is influenced by system noise, filter effects of temporal and vertical averaging, and atmospheric effects, e.g., the influence of regional meso- or microscale circulations that may influence the fluctuations on the corresponding temporal and spectral scales.

For the derivation of turbulence profiles we applied Taylor's hypothesis of frozen turbulence. We assumed that this was valid, as

As described above, the basic data sets were time–height cross sections of

gridding of all data to vertical resolutions of 15 m and temporal resolutions of 1 or 10 s;

despiking, detrending, and high-pass filtering of the data at each height

derivation of the ACFs

application of the functions defined in Eqs. (

fit of the ACFs over this range of lags to determine atmospheric variances, the ACF coefficients, and their errors as a function of height;

characterization of vertical and temporal filter effects and its influence on the shape of the ACFs;

determination of the profiles of the integral timescales;

corresponding derivation and evaluation of the power spectra and detection of the inertial subranges for consistency with respect to filter effects and system noise in the frequency domain;

combination of the profiles of the ACF coefficients and the horizontal wind profile for deriving profiles of TKE dissipation and molecular destruction rates including error propagation.

Figure

Time–height cross section of

After the determination of the ACFs of the time series, we studied their shapes for all three variables of interest.

We started with the examination of

3D plot of

The suitable ranges for the fits of the ACFs can be evaluated in more detail using Eqs. (

Example of the evaluation of the range of suitable lags according to Eq. (

Figure

Power spectrum

It is essential to investigate the sensitivity of the ACFs for both 1 s and 10 s resolutions in order to find out whether the coarser time resolutions can be used as well.

Figure

Comparison of

Using these lag ranges for fitting the ACFs, we determined the profiles of the integral timescale

Small dark green and large green circles: profiles of the variance of the vertical velocity fluctuations for 1 and 10 s data, respectively, including noise error bars. Small pink and large red squares: the corresponding results for the integral timescale

A similar conclusion can be derived for the profiles of the vertical wind variance

For the mixing ratio fluctuations, we are challenged by the trade-off between the accuracy and resolution of the WVDIAL measurements. The nonlinear reduction in system noise as a function of the spatial resolutions is explained in

The ACFs are presented in Fig.

The comparison between the ACFs computed using the 1 and 10 s resolutions is presented in Fig.

Comparison of

Power spectra of the WVMR fluctuations using 1 s data. The 10 s spectra are very consistent (not shown). The fits of the Kolmogorov spectrum with the noise floor are also shown.

Using these lag ranges for fitting the ACFs, we determined the profiles of the integral timescale

Small cyan and large blue circles: profiles of the variance of the WVMR fluctuations for 1 and 10 s data, respectively. Small pink and large red squares: the corresponding results for the integral timescale

The profiles of

For the potential temperature fluctuations, it is particularly difficult to derive structures in the ACF and the power spectra due to the large noise level in the TRRL observations. Although the improvement of TRRL towards the resolution of temperature fluctuations was substantial in recent years (see, e.g.,

The corresponding ACFs are presented in Fig.

Power spectrum

Using lags 1–3 for fitting the ACFs, we determined the profiles of the integral timescale

Pink circles: profiles of the variance of the potential temperature fluctuations for 10 s data. Red squares: the corresponding results for the integral timescale.

Using the fits of the ACFs, we are able to detect the expected small

A comparison of the integral length scales is presented in Fig.

Comparison of the integral length scales

The derivation of the profiles of the ACF coefficient

Small dark green and large light green circles: profiles of TKE dissipation

For parameterizations of

Small dark green circles: TKE dissipation

Further refinements are possible using the observed dependence of

Dark green diamonds:

Using high-resolution profile observations of the water-vapor mixing ratio

The derivation of the profiles of the ACF coefficient

Small pink and large cyan circles: profiles of

For a parameterization of

Small cyan circles:

The profiles of the ACF coefficient

Small blue and large red circles: profiles of

For a parameterization of

Small cyan circles:

In this work, we used high-resolution time series of

According to its definition,

We determined

With respect to

Our profile of vertical velocity variance reaches a maximum at

Fundamental for the derivation of TKE dissipation or molecular destruction rates is the relationship between the variances and the coefficient

TKE dissipation

The determination of

The magnitude of the

In contrast to TKE dissipation measurements, we are not aware of the use of any remote sensing efforts to determine values or even profiles of the molecular destruction rates for temperature and water-vapor variances. Both

It is interesting to relate the ACF coefficients, variances, and molecular destruction rates in more detail. We have demonstrated how this can be done using a combination of spectral and ACF analyses, as done above for

Since our method allows for the measurement of vertical profiles of

In the future, our WVDIAL measurements will be improved with respect to SNR to achieve better performance. This is now possible because recent updates to this lidar system have resulted in an average power of the laser transmitter of up to 10 W

In this work, transverse temporal ACFs were used to derive vertical profiles of TKE dissipation

We applied the methodology proposed in

A weakly convective case from the HOPE data set was selected and profiles of temporal and integral scales as well as of variances were determined. Several relationships between

We found a maximum of

We also showed that

This combination of measurements has been realized during the HOPE and LAFE field campaigns

The long-term goal should be to provide routine analyses of diurnal cycles of turbulence profiles in different climate regions for confirming the universality of scaling or to refine them by characterizations of wind shear, the strength of the inversion layer, and other potential scaling variables. A corresponding setup of instrumentation was proposed for the GEWEX Land–Atmosphere Feedback Observatories (GLAFOs;

The HOPE data used in this work are available from the Institute of Physics and Meteorology (IPM), University of Hohenheim. The IDL codes used to derive the variables can also be provided by IPM per email by contacting the main author of this publication.

VW: writing, data analyses, turbulence theory, methodology, discussion; CS: turbulence methodology, data analysis, review, editing; FS: WVDIAL analysis and methodology; AB, DL: TRRL data analysis and methodology; RMB: turbulence theory, methodology, editing; WAB, AW: DL lidar data analysis and methodology; DDT: methodology, review, editing.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This study was supported by the Federal Ministry of Education and Research (BMBF) High Definition Clouds and Precipitation (HD(CP)

This research has been supported by the Bundesministerium für Bildung und Forschung (grant no. 89243019SSC000034); the Department of Energy, Labor and Economic Growth (grant no. 89243019SSC000034); NOAA; and NCAR.

This paper was edited by Markus Rapp and reviewed by two anonymous referees.