# Scalar turbulent fluxes and variances in the interfacial layer from lidar observations and assessment of Lagrangian Stochastic Models

Fabien Gibert, Dimitri Edouart, Paul Monnier, Julie Collignan, Julio Lopez, Claire Cénac

<figure><img src="image_1.png" /></figure>

## To cite this version:

Fabien Gibert, Dimitri Edouart, Paul Monnier, Julie Collignan, Julio Lopez, et al.. Scalar turbulent fluxes and variances in the interfacial layer from lidar observations and assessment of Lagrangian Stochastic Models. Quarterly Journal of the Royal Meteorological Society, 2025, pp.e70016. 10.1002/qj.70016 . hal-05379361

HAL Id: hal-05379361

https://hal.science/hal-05379361v1

Submitted on 24 Nov 2025

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Scalar turbulent fluxes and variances in the interfacial layer from lidar
observations and assessment of Lagrangian Stochastic Models

Fabien Gibert,*a Dimitri Edouart,ª Paul Monnier,ª Julie Collignan,ª Julio Lopez,ª Claire
Cénac,ª

ª Laboratoire de Météorologie Dynamique (LMD/IPSL), École polytechnique, Institut polytechnique de Paris, Sorbonne Université, École normale supérieure, PSL Research University, CNRS, École des Ponts, 91128 Palaiseau, France

*Corresponding author: Fabien Gibert, gibert@lmd.polytechnique.fr
The link between scalar second order moments and gradient in the interfacial layer (IL) capping the convective boundary layer (CBL) is analyzed using new lidar turbulent-scale observations. Temperature and moisture variances as well as sensible and latent heat fluxes have been measured during one hundred hours in CBL temperate and arid regions with various free-cloud meteorological conditions. The IL fluxes and variances have been confronted with Lagrangian Stochastic Model (LSM) theories using scalar gradients, vertical velocity variance, kinetic energy dissipation rate and integral scales of turbulence with the purpose to build a local parametrization. The results show that Pearson diffusion model in a stratified flow, that relies on IL vertical velocity variance, $\sigma_w^2$, and buoyant oscillation period, $1/N$ as the critical scale of turbulent diffusion, gives universal relevant parameters for IL temperature and moisture flux and variance. Our observations enable to infer for the first time IL local scaling for temperature, $K_\theta = (0.35 \pm 0.03) \sigma_w^2/N$ and moisture $K_q = (0.27 \pm 0.03) \sigma_w^2/N$ eddy-diffusivities. The LSM local scaling parameters have been compared successfully for temperature with previous Large-Eddy Simulations and with in situ observations made in the nocturnal stratified atmospheric surface layer. Other LSM tested in this work that uses different scales to characterize IL scalar statistics failed especially for moisture statistics.

SIGNIFICANCE STATEMENT

The purpose of this study is to analyze the link between turbulent fluxes and gradient of scalars such as temperature and humidity at the top of the atmospheric convective boundary layer in the interfacial layer. This is critical to improve future transport models in the atmosphere. Our results based on lidar observations assess the Pearson LSM theoretical formulations of scalar turbulent flux and diffusivity in the interfacial layer and LES results concerning IL temperature flux and variance.
28

# 1. Introduction

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

The turbulent transport of heat, matter and momentum in the convective boundary layer (CBL) is a key process to understand the 3D distribution of scalars and their budget in the atmosphere (Wulfmeyer et al. 2015). The surface and the interfacial layer (IL) that encapsulate the CBL and where scalar exchanges occur with the soil and the free troposphere, respectively, are critical zones of interest. The IL is defined between the height of maximum negative sensible heat flux and the height where the flux vanishes (Wyngaard, 2010). The IL has many similarities with the nocturnal stably stratified layer (ASL), i.e. there exists large temperature inversions and gradients of scalars that may produce large fluxes and variances depending on the size and strength of turbulent eddies. The IL is also referred as CBL capping “inversion layer” that prevents the dynamically active layer to deepen, also with an enhanced horizontal wind shear. Nevertheless, the source of turbulence in the IL results from the transport of turbulent kinetic energy created by surface positive buoyancy rather than from the interfacial layer wind shear (Le Mone 2019). Different conditions above the IL may exist due to the CBL diurnal cycle. Initially the interfacial layer is part of the residual layer (usually less stratified) and later in the day it becomes part of the free troposphere with synoptic scale characteristics. Last but not least, the diurnal cycle of surface sensible heat flux creates variable diurnal dynamical forcing conditions to the CBL that are expected to play a role in IL properties.

46

47

48

49

50

51

52

53

54

55

56

57

58

Although ASL exchanges are usually well documented by in situ observations and Monin-Obukhov similarity theory (Stull 1988; Wyngaard 2010), some recent issues have arisen (Wulfmeyer et al. 2023), and the IL processes remain a largely unknown field for both observations and models. As a result, parametrization of turbulent transport in the atmospheric boundary layer remains challenging. Parametrization of the dry CBL usually suffer from problems to represent IL linked counter-gradient fluxes (Deardorff 1966, Troen and Mahrt, 1986; Lock et al. 2000; Bretherton and Park 2009). A now classical approach of scalar CBL transport is to combine a mass flux representation (Arakawa and Schubert, 1974; Siebesma and Cuypers, 1995) and a K-diffusion representation of small-scale turbulence (Hourdin et al. 2002, Soares et al. 2004; Siebesma et al. 2007), in a so-called “eddy diffusion mass flux” (EDMF) scheme. The scalar K-profile, especially in the IL, has to be parametrized either with standard CBL non-local scales (Siebesma et al. 2007) or local scales such as the local gradient Richardson number and the turbulent kinetic energy (TKE) in Hourdin et al. (2019).

59
Concerning local scaling, turbulent dispersion in the atmosphere has been studied in the past using Lagrangian Stochastic Model (LSM) (Taylor 1921; Pearson 1983). Taylor introduced such a model for homogeneous turbulence. Several improvements were proposed over time to take care of heterogenous velocity variance (Legg and Raupach 1982) and non-perfect mixing (Thomson 1987). Wilson and Sawford (1996) show that such model could be not relevant in strong stably stratified flow as thermal stratification is not specifically considered there. Pearson et al. (1983) corrected this missing aspect considering both equations for the momentum of fluid elements and for the fluctuating density. He introduced a parameter gamma that was described as a measure of molecular exchange between fluid elements. Although experiments could not support the role of molecular mixing, this idea agrees to the fact that vertical turbulent transport in stable atmosphere requires irreversible mixing to take place. Note that a parallel between LSMs and Eulerian second-moment closure models has been subsequently addressed in Das and Durbin (2005). Following Pearson et al., Lindborg and Brethouwer (2008) proposed a modified form of the density exchange term that involves the mean Brunt-Vaisala frequency and dissipation of turbulent potential energy per unit mass. Their irreversible mixing term supports former Osborn and Cox eddy diffusion model (1972) that was commonly used for stable stratification assuming a balance between dissipation and production in the Reynolds-averaged equation for mean-square buoyancy fluctuations.

Later to these theoretical considerations, Large-Eddy Simulations (LES) have been widely used to assess IL CBL parametrizations (Sullivan et al. 1998, Otte and Wyngaard 2001, Federovitch et al. 2004, Siebesma et al. 2007, Hourdin et al. 2019, among others). Sorbjan built non-local scaling laws and IL parametrization of turbulent fluxes and second order moments (Sorbjan 2005, 2006). In addition to LES, direct numerical simulations (DNS) have also been used (Garcia and Mellado 2014, Haghshenas and Mellado 2019). These recent studies suggest an original view of the IL as a superposition of two sub-layers centered at maximum buoyancy gradient (upper one) and minimum buoyancy flux (lower one) with specific properties and characteristic scales. Both LES and DNS studies have a huge statistical advantage to usual observations, nevertheless one may wonder the significance and representativity of such simulations in the real world.

To do so, new observations are needed with turbulence-scale temporal and spatial resolutions to investigate flux-gradient relationships and estimate higher-order moments. These observations should be made in different climate zones to test LSM theories and study the robustness of IL advanced parametrizations. As detailed by Wulfmeyer et al. (2016, 2018) lidar
measurements should play a role as they have the potential to make continuous, precise, and simultaneous measurements of wind speed, temperature, water vapor profiles as well as eddy-covariance turbulent flux profiles up to the top of the CBL. Such measurements have been successfully achieved in the past (Senff et al. 1994; Giez et al. 1999, Kiemle et al. 1997, 2007, 2011) for latent heat flux and (Behrendt et al. 2015, Di Girolamo et al. 2017) for sensible heat flux and have been operated routinely using commercially available Doppler and Raman lidars (Behrendt et al. 2020). Such mobile ground-based 3-D lidar observatory has been built at Laboratoire de Météorologie Dynamique, Palaiseau, France (Gibert et al. 2022). It combines simultaneous measurements of wind speed, temperature, water vapour ($H_2O$) and carbon dioxide ($CO_2$) in the lower troposphere.

This work presents a dataset collected over several days (a hundred hours) including Lidar measurements and in situ sensors. The dataset is used to analyze the scalar quantities, such as temperature and moisture, in IL stratified turbulence. Particularly, the relevance of LSM in inferring local scaling turbulent fluxes and variances in the IL is analyzed. The paper is organized as follows. First, section 2 contains a summary of the main LSM theories and results. Then in section 3, the Lidar dataset characteristics are provided and discussed with respect to uncertainties. Typical case studies in temperate and arid regions are analyzed. Section 4 analyses the similarities and differences of scalar statistics. Next, the application of LSM theories and local parametrizations of IL scalar turbulent fluxes and variances are presented and discussed. Finally, a conclusion section summarized main results.

## 2. LSM review concerning scalar turbulent flux, eddy-diffusivity and variance in the interfacial layer

We consider here the diffusion of a scalar at the top of the CBL in the interfacial layer with the free troposphere. For the IL, we consider a persistent movement of homogeneous-turbulent flow with mean and constant wind velocity $U$. We further assume a two-dimensional problem ($x, z$), $x$ being the mean horizontal wind direction, and $z$ the vertical one. Upon consideration of space and time ergodicity, $x$ means either the distance motion of an air mass during a time $t$ (Lagrangian view) or the side distance of a box taken at a given time $t$ over which one operates an ensemble averaging (LES view). In this view, $x$ may be also replaced by $t$ with $x=Ut$ (Eulerian view and practical case for the Lidar observations).
a. Turbulent diffusion

Using a standard decomposition of velocity and concentration of a given scalar in their mean (U, W, C) and fluctuating part (u, w, c) and then time-averaging the results, the conservation equation for the turbulent flow may be written as:

$$U \frac{\partial c}{\partial x} = - \frac{\partial \overline{wc}}{\partial z} \qquad (1)$$

with the following assumptions (Wyngaard 2010): (i) negligible vertical advection (negligible change in CBL height in our case) (ii) negligible turbulent diffusion in the streamwise direction with respect to the one in the vertical (two-dimensional problem) and (iii) transport due to molecular diffusion is negligible compared to the one due to turbulent flux. Note however that molecular diffusion is still an important mechanism for mixing at the smallest scales. To solve Eq. (1), it is common to use a standard first order closure which assumes that the turbulent flux is proportional to the mean concentration gradient,

$$\overline{wc}(x, z) = -K_c(x, z) \frac{\partial c}{\partial z} \qquad (2)$$

where $K_c$ is the eddy-diffusivity coefficient which is commonly derived as a function of the flow properties. Nevertheless, assuming $K_c$ is constant over the vertical distance that is used to calculate the vertical gradient of C and using Eq. (1) and (2), one obtains

$$U \frac{\partial c}{\partial x} = K_c(x) \frac{\partial^2 c}{\partial z^2} \qquad (3)$$

and the mean scalar concentration C at any x location has the following Gaussian solution:

$$C(x, z) = \frac{a}{\sqrt{2\pi U \sigma_z(x)}} e^{\frac{-z^2}{2\sigma_z^2(x)}}, \text{ with } \sigma_z^2 \approx \langle z'^2 \rangle \qquad (4)$$

Here, a is a constant and $\sigma_z$ is the characteristic vertical length of scalar dispersion. $\sigma_z$ is usually interpreted to the rms of Lagrangian vertical displacement of fluid particles $\sqrt{\langle z'^2 \rangle}$ (Wyngaard 2010). Using Equations (3) and (4), the eddy diffusivity $K_c$ can be expressed as:

$$K_c = \frac{U}{2} \frac{d\sigma_z^2}{dx} = \frac{1}{2} \frac{d\sigma_z^2}{dt} \approx \frac{1}{2} \frac{d\langle z'^2 \rangle}{dt} \qquad (5)$$

Following Pearson et al. (1983) and Hunt (1985), the scalar concentration fluctuations in a gradient $\partial C/\partial z$ can also be related to the fluctuations of vertical fluid particles displacement:

$$\sigma_c \sim \sqrt{\langle z'^2 \rangle} \left( \frac{\partial c}{\partial z} \right) \qquad (6)$$
The gradient-diffusion hypothesis (Eq. 2) supposes that interfacial layer turbulence is regarded as a microscopic process in analogy to molecular diffusion. This is only the case if the characteristic IL turbulent length and time scales are much smaller than boundary layer macroscopic scales.

155

## b. LSM formalism to link scalar gradient and turbulent diffusion statistics

The specificity of LSM is to link turbulent scalar dispersion, gradient and relevant scalar second order moments like variance and flux to the rms of fluid parcel vertical displacement $\sqrt{\langle z'^2 \rangle}$. As mentioned in the introduction, several LSM variants were built, with important parameters being the strength of vertical stratification in the atmosphere and relevant integral scales characterizing turbulent diffusivity.

## Taylor approach

Taylor approach ignores the IL vertical stratification (i.e. neutral condition). It consists in relating the vertical dispersion of fluid particles to the correlation of the vertical velocity field. The growth of the mean square of vertical fluid parcel displacements is (Taylor 1921):

$$ \langle z'^2 \rangle_{Taylor} = 2\sigma_{wL}^2 \int_0^t (t-\tau)R(\tau)d\tau \quad (7) $$

where $\sigma_{wL}^2$ is the variance and $R$ is the normalized autocorrelation of the random walk fluid particle Lagrangian vertical velocities. A Lagrangian turbulent vertical velocity integral time scale, $I_{wL}$, can be defined from $R$:

$$ I_{wL} = \int_{0}^{+\infty} R(\tau)d\tau \quad (8) $$

Assuming $t > I_{wL}$, the vertical mean spread of fluid particles increases as the square root of time:

$$ \langle z'^2 \rangle_{Taylor} = 2\sigma_{wL}^2 I_{wL}t \quad (9) $$

In homogeneous and stationary turbulence, Lagrangian and Eulerian statistics and especially variance are equivalent: $\sigma_{wL}^2 = \sigma_w^2$. Lagrangian and Eulerian integral scales may also be considered as roughly equal, $I_{wL} = I_w$ (Corrsin, 1963).

Thus, using Eq. (5) one obtains a formulation for the Taylor scalar eddy-diffusivity:

$$ K_{Taylor} = \sigma_w^2 \cdot I_w \quad (10) $$
In addition, using Eqs. (6) and (9) and assuming that the characteristic width of Taylor dispersion is defined for a characteristic time scale $t = I_w$, one obtains the associated Taylor scalar variance:

$$ \sigma_{Taylor}^2 \sim \sigma_w^2 I_w^2 \left(\frac{\partial C}{\partial z}\right)^2 \qquad (11) $$

## Pearson approach

In stable condition, a vertical displacement of fluid parcels from their equilibrium position produces buoyancy fluctuations that increase the mean potential energy. If the process is adiabatic, Equation (9) will produce an increase of potential and then turbulent kinetic energy which is not possible if the turbulence is stationary. A finite vertical dispersion is indeed observed in such condition (van Aartrijk and Clercx 2008).

The stratification of the IL introduces naturally a new time scale in the turbulent flow: $1/N$ where $N$ is the Brunt-Väisälä frequency:

$$ N = \left( \frac{g}{\bar{\theta}_v} \cdot \frac{\partial \bar{\theta}_v}{\partial z} \right)^{1/2} \qquad (12) $$

Pearson et al. (1983) used this time scale to propose a new form for vertical mean spread of fluid particles taking into account stratification:

$$ \langle z'^2 \rangle_{Pearson} \sim \frac{\sigma_w^2}{N^2} (1 + O(\gamma^2)Nt) \qquad (13) $$

where the first term accounts for the finite rms displacement of fluid parcels in stationary turbulence and the second term adds a diffusive interchange of density between fluid elements by the coefficient $\gamma$. Note that Eq. (13) is given for $\gamma \ll 1$, i.e. strong stable stratification.

Using Eq. (13) and Eqs. (5,6) one obtains a formulation for the Pearson scalar diabatic eddy diffusivity and variance in stable flow:

$$ K_{Pearson} \sim \gamma^2 \frac{\sigma_w^2}{N} \qquad (14) $$

$$ \sigma_{Pearson}^2 \sim \frac{\sigma_w^2}{N^2} \left(\frac{\partial C}{\partial z}\right)^2 \qquad (15) $$

Note that Eqs. (14) and (15) have been tested successfully for potential temperature with LES in stably stratified interfacial layer (Otte and Wyngaard 2001).
205 Lindborg-Osborn approach

206 Using Boussinesq equations in a constant background stratification, Lindborg and Brethouwer

207 (2008) arrived to a similar form of Eq. (13) in stationary turbulence:

$$
208 \qquad \langle z'^2 \rangle_{\text{Lindborg}} \sim \frac{\langle b'^2 \rangle}{N^2} + \frac{2}{N^2} \epsilon_p t \qquad (16)
$$

209 where $\epsilon_p$ is the mean dissipation of turbulent potential energy per unit of mass and $\langle b'^2 \rangle$ is the

210 Lagrangian structure function of normalized buoyancy fluctuations, $b = g\theta_v/(N\theta_{v,0})$. To

211 obtain this derivation, the authors assumes that the interaction time scale for exchange of

212 density between fluid elements is equal to the Kolmogorov time scale ($\nu/\epsilon$)$^{1/2}$ where $\nu$ is the

213 molecular viscosity and $\epsilon$ is the mean dissipation rate of kinetic energy. Considering the inertial

214 subrange where the motion of fluid parcels is dominated by a turbulent cascade with $t > 1/N$,

215 Lindborg and Brethouwer (2008) proposes a specific development of Eq. (16):

$$
216 \qquad \langle z'^2 \rangle_{\text{Lindborg}} \sim (\pi C_{PL} + 1) \frac{2}{N^2} \epsilon_p t, \text{ with } C_{PL} \sim 3 \qquad (17)
$$

217 This yields the following form for eddy diffusivity:

$$
218 \qquad K_{\text{Lindborg}} \sim (\pi C_{\text{PL}} + 1) \frac{\epsilon_p}{N^2} \qquad (18)
$$

219 Actually, this form of the vertical eddy diffusivity is similar to the one derived from Osborn

220 (1980) assuming a balance between production and dissipation in the equation of conservation

221 of the buoyancy fluctuations:

$$
222 \qquad K_{\text{Osborn}} \sim \Gamma \frac{\epsilon}{N^2} \qquad (19)
$$

223 where $\Gamma = \epsilon_p/\epsilon$ is a coefficient which refers to a “mixing efficiency” and which varies with

224 the type of flow considered and with the Froud number (Maffioli and Davidson 2016).

225 Following Wulfmeyer et al. (2016), $\epsilon$ can be determined experimentally using horizontal wind

226 speed and vertical velocity measurements: $\epsilon = 2\sigma_w^3/(5U.I_w)$. Using Eqs. (6) and (18), $\epsilon_p =$

227 $\Gamma\epsilon$ and assuming that the characteristic width of dispersion is defined for a characteristic time

228 scale $t = 1/N$, one obtains the Lindborg-Osborn scalar eddy-diffusivity and variance:

$$
229 \qquad K_{\text{Lindborg-Osborn}} \sim (\pi C_{\text{PL}} + 1) \frac{\Gamma \epsilon}{N^2} \qquad (20)
$$

$$
230 \qquad \sigma_{\text{Lindborg-Osborn}}^2 \sim (\pi C_{\text{PL}} + 1) \frac{2\Gamma\epsilon}{N^3} \left(\frac{\partial C}{\partial z}\right)^2 \qquad (21)
$$

231 Two turbulent diffusion time scales are indeed involved in Lindborg-Osborn LSM: 1/N and

232 $I_w$.
233

In this work, we follow Driedonks and Tennekes (1984) and neglect the role of IL shear (and its characteristic scale) in scalar variance and flux, making the assumption that IL shear production creates small-scale turbulence that is rapidly dissipated locally and does not contribute much to the entrainment processes. Such assumption could be investigated in the future.

234

235

236

237

238

## 3. Turbulent scalar fluxes and other observations in the convective boundary layer

239

240

### a. Instrumentation, experimental sites and dataset

241

A new wind, temperature, H₂O and CO₂ scanning lidars mobile observatory has been developed at Laboratoire de Météorologie Dynamique, Ecole Polytechnique, France (Gibert et al. 2022) (Fig. 1).

242

243

<figure><img src="image_3.png" /></figure>

244

245

Fig. 1. 3-D lidar observatory and other instrumentation used in this study during two field experiments in a temperate region at SIRTA observatory, France (left panels) and in a semi-arid region with irrigated fields during LIAISE experiment, Spain (right panels). EC: eddy-covariance station, RS: radiosondes.

246

247

248

249

250

As part of this system, two different scanning lidars are involved: 1) a temperature and water vapor Raman lidar at 355 nm; 2) a prototype DIAL and Doppler lidar at 2051 nm with a coherent and direct detection for radial wind speed and CO₂ absorption measurements (Gibert et al. 2015). The mobile lidar observatory was operated in a temperate region at SIRTA observatory, a heterogeneous suburban site in Palaiseau, France (https://sirta.ipsl.fr) in June

251

252

253

254
2021 and then in the homogeneous arid area of Lleida, Spain during the LIAISE experiment (Land surface interactions with the atmosphere over the Iberian semi-arid environment) in July 2021 (Boone et al. 2025). The lidar observations were completed by surface layer measurements of pressure, temperature and specific humidity, wind speed and direction at 30-m (SIRTA) and 50-m (LIAISE). Daily (SIRTA) and hourly (LIAISE) soundings were also made to provide lidar-independent profiles for temperature, humidity and wind. These were used for calibration and comparison. Several eddy-covariance stations with sonic anemometers and gas analyzers were spatially distributed at different locations and heights close to the lidar observatory to provide turbulence-linked surface layer data like sensible and latent heat fluxes, friction and free-convection scaling velocities. In addition, a radar UHF and a Doppler lidar were used to get continuous profiles of horizontal wind speed, direction and shear.

The two scanning lidars work on a one-hour-based cycle making horizontal and vertical scanning measurements during the first 20 min of the hour for other studies. The rest of the time, almost 40 min, which concerns the present study, the lidars made vertical measurements. Although the two lidar acquisition systems were referenced with the same internet reference time, a potential time lag between the lidars data was checked and corrected. The standard space and time resolution of the data are 48 m and 8 s respectively. The dataset consists in 7 days of measurements in the arid area and 3 days in the temperate one with almost 10 hours of measurements per day collected in the convective boundary layer. Table 1 reports the median characteristics of each day. Note that for simplification of the notation the variable with no index is calculated at the height of minimum sensible heat flux, $z_{if}$. Index « 0 » is used for the surface variable.

Definitions, calculations and discussion of these data will appear later in the paper but one can already notice the various cases studied with especially different conditions for surface fluxes and IL stratification and wind shear. The arid dataset (A-) corresponds to very large Bowen ratio (> 100), dry unstable to windy dry conditions (Mahrt, 1991). The temperate dataset (T-) corresponds to small Bowen ratio from 0.3 to 0.8, moist unstable to moist windy conditions.

283
284
285
286
<table><caption>Table 1. Observations characteristics during several days of measurements in the arid region (A) during July 2021 and in the temperate one (T) during June 2021. In the first column, [11;7] represents the numbers of lidar and radiosondes measurements during the day. Median variables for each day: z<sub>if</sub> CBL height from minimum sensible heat flux; H<sub>0</sub>, Q<sub>0</sub> surface (in situ) and H, Q interfacial layer (lidar) sensible and latent heat fluxes; IL vertical gradient of potential temperature ∂θ̃<sub>v</sub>/∂z and specific humidity -∂q̃/∂z, s horizontal wind shear (radar/ lidar), R<sub>i</sub>, gradient Richardson number, vertical velocity variance, 1/N, Brunt-Väisälä oscillation period and I<sub>w</sub>, vertical velocity integral scale. Gradient, shear, Brunt-Väisälä pulsation and gradient Richardson number are calculated at z<sub>if</sub>. Radiosondes measurements, when available, are provided between parenthesis. Data and uncertainties are detailed in Appendix A and Fig. A2.</caption><thead><tr><th rowspan="2">Day</th><th rowspan="2">z<sub>if</sub><br>(m)</th><th rowspan="2">H<sub>0</sub><br>(K.m/s)</th><th rowspan="2">Q<sub>0</sub><br>(g/kg.m/s)</th><th rowspan="2">∂θ̃<sub>v</sub>/∂z<br>(K/km)</th><th rowspan="2">-∂q̃/∂z<br>(g/kg/km)</th><th rowspan="2">s<br>(s<sup>-1</sup>)</th><th rowspan="2">Ri</th><th rowspan="2">σ<sup>2</sup><sub>w</sub><br>(m<sup>2</sup>.s<sup>-2</sup>)</th><th rowspan="2">1/N<br>(s)</th><th rowspan="2">I<sub>w</sub><br>(s)</th></tr><tr></tr></thead><tbody><tr><td>A-0716<br>[10;7]</td><td>698</td><td>0.23<br>0.027</td><td>0.000<br>0.050</td><td>11<br>(9.9)</td><td>5.7<br>(6.9)</td><td>0.008<br>(0.008)</td><td>4.6<br>(3.5)</td><td>0.21</td><td>57<br>(43)</td><td>21</td></tr><tr><td>A-0717<br>[11;5]</td><td>728</td><td>0.22<br>0.060</td><td>0.000<br>0.062</td><td>17<br>(13)</td><td>10<br>(10)</td><td>0.011<br>(0.013)</td><td>4.5<br>(2.9)</td><td>0.26</td><td>43<br>(48)</td><td>26</td></tr><tr><td>A-0718<br>[9;0]</td><td>582</td><td>0.24<br>0.034</td><td>-0.002<br>0.028</td><td>7.4</td><td>2.8</td><td>0.013</td><td>1.2</td><td>0.31</td><td>64</td><td>24</td></tr><tr><td>A-0719<br>[6;0]</td><td>660</td><td>0.25<br>0.053</td><td>-0.002<br>0.037</td><td>7.5</td><td>3.4</td><td>0.012</td><td>2.6</td><td>0.54</td><td>64</td><td>26</td></tr><tr><td>A-0720<br>[6;6]</td><td>631</td><td>0.28<br>0.092</td><td>0.000<br>0.057</td><td>8.8<br>(4.7)</td><td>2.5<br>(5.6)</td><td>0.011<br>(0.012)</td><td>1.5<br>(0.5)</td><td>0.35</td><td>93<br>(97)</td><td>29</td></tr><tr><td>A-0721<br>[9;6]</td><td>395</td><td>0.19<br>0.017</td><td>0.011<br>0.014</td><td>7.9<br>(11)</td><td>8.1<br>(13)</td><td>0.020<br>(0.010)</td><td>0.7<br>(1.7)</td><td>0.33</td><td>63<br>(53)</td><td>42</td></tr><tr><td>A-0727<br>[12;3]</td><td>784</td><td>0.23<br>0.11</td><td>0.012<br>0.026</td><td>3.2<br>(2.2)</td><td>0.8<br>(1.7)</td><td>0.013<br>(0.018)</td><td>0.3<br>(0.3)</td><td>0.69</td><td>77<br>(99)</td><td>43</td></tr><tr><td>T-0613<br>[12;1]</td><td>890</td><td>0.045<br>0.194</td><td>0.027<br>0.160</td><td>11</td><td>7.8</td><td>0.009</td><td>5.3</td><td>0.43</td><td>47</td><td>31</td></tr><tr><td>T-0614<br>[8;1]</td><td>788</td><td>0.078<br>0.066</td><td>0.106<br>0.052</td><td>1.3</td><td>4.5</td><td>0.007</td><td>3.3</td><td>0.22</td><td>73</td><td>52</td></tr><tr><td>T-0615<br>[9;1]</td><td>832</td><td>0.11<br>0.065</td><td>0.101<br>0.045</td><td>4.3</td><td>3.4</td><td>0.007</td><td>13</td><td>0.29</td><td>88</td><td>43</td></tr></tbody></table>

287

288

289

290

291

292

293

294

295

296

297
298
299

b. Scalar profiles, gradient, variances, fluxes observations and uncertainties. Two cases study.

300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326

One 40-min time serie of the A-0716 case, which is characterized by a strong IL stratification, is detailed in Fig. 2 with vertical velocity, potential temperature and specific humidity data. UHF radar measurements brought the mean components of the horizontal wind speed (space resolution 75 m) during the same period. A radiosonde was launched at 14h and provided simultaneous meteorological profiles. The lidar scalar vertical gradient estimate uses high space resolution lidar with 7.5 m - 2 min space and time resolution that are later averaged over 40 min. Note that below z = 400 m a different overlap function in the two Raman lidar temperature channels creates a bias in potential temperature profile (that we have not been able to fix so far) but this does not affect the present study located in the IL. Moreover, this bias has negligible effect on statistical measurements (variance and flux). The statistical measurements like variance, eddy-covariance flux, integral scale use the standard lidar space and time resolution of 48 m and 8 s as a compromise to obtain a sufficient precision and to respect integral scales of turbulence and avoid biases (Gibert et al. 2011a). The natural and instrumental noise contributions in the variance budget are estimated using the autocovariance functions and a 2/3 power law fit from inertial subrange predictions (Lenschow et al. 2000, Wulfmeyer et al. 2016). This method is also used to estimate integral scale values and uncertainties. Eddy-covariance fluxes are calculated with removed trend data (Lenschow et al. 2000, Behrendt et al. 2020). Instrumental and sampling errors have been estimated using variance and integral scale estimates (Senff et al. 1994, Giez et al. 1999). Note that specific humidity must be considered for virtual potential temperature gradient and flux. This significantly modifies the buoyancy flux profile and may entail wrong $z_{if}$ and minimum buoyancy flux estimates (LeMone 2019). The IL scalar gradients are calculated with a regression over 10 points (~ 80 m) centered at the height $z_{if}$. This was chosen first to take into account the 48-m statistical measurements vertical resolution (variance, flux) and second to have a sufficient precision on the derivative. The same signal processing is made for wind shear estimates using Doppler lidar or UHF radar measurements. These local estimates are then used to calculate Brunt-Väisälä frequency and gradient Richardson number.

327
328

The full dataset and relevant estimates and uncertainties presented in this study are detailed in Appendix A.
<figure><img src="image_4.png" /><figcaption>Fig. 2. A-0716 case study at 14h local time. (a) Vertical velocity; horizontal wind speed profiles (radar with thick solid lines and radiosondes with thin solid lines); vertical velocity variance (markers). The similarity law is indicated with the dotted line; integral scale (markers) and buoyant period oscillation (magenta solid line). (b) Virtual potential temperature; 40-min mean lidar profile (red line) and standard deviation and radiosonde profile (black solid line) at 14h; virtual potential temperature variance (red square); integral scale; sensible heat flux profile with bars from instrumental (cap) and sampling (no cap) errors. (c) Specific humidity; 40-min mean lidar profile (blue line) and standard deviation and radiosonde profile (black solid line) at 14h; specific humidity variance (blue triangle); integral scale; latent heat flux profile with bars from instrumental (cap) and sampling (no cap) errors. The height at minimum sensible heat flux, $z_{if}$, and at maximum specific humidity variance, $z_{iv}$, are indicated with dashed and dashed and dotted lines respectively.</figcaption></figure>

Lidar sensible heat flux profile is nearly linear with height up to $z_{if}$ as expected by mixed-layer similarity theory (Deardorff 1972). Latent heat flux profile shows a large increase just below the interfacial layer, which is explained by a decrease of specific humidity (drying) in the afternoon (Deardorff 1974). Surface sensible and latent heat fluxes provided by a co-located instrumented tower with eddy-covariance stations are reported in Fig. 2b and 2c fifth panels with a cross marker. These surface data agree with the lidar flux profiles although one must keep in mind that surface flux might spatially varies with surface heterogeneity. To
350

prevent such mismatch the surface flux is calculated at the top level of the towers (50 m here)
to have the best surface representativity.

351

352

Fig. 3 displays a different case in the temperate region with low stratification and the
following characteristics: (i) an unexpected enhancement of IL vertical velocity variance with
respect to CBL similarity law (Fig. 3a third panel) (ii) a large IL sensible heat flux with respect
to the surface one (Fig. 3b fourth panel) (iii) a large IL thickness given by the large difference
between the height of minimum sensible heat flux z<sub>if</sub> and of maximum water vapor variance
z<sub>iv</sub> (iv) a IL virtual potential temperature gradient similar to the one in the free troposphere.

353

354

355

356

357

358

Similar wind shear conditions are measured as for the arid case provided in Fig. 2. Then the
observed enhancement of the second maximum of vertical velocity variance in the IL is unlikely
to be only due to the wind shear (Haghshenas and Mellado, 2019). An important feature (and
a main difference with Fig. 2) is that the vertical velocity integral scale increases in the IL and
is larger than the Brunt-Väisälä oscillation period (Fig. 3a fourth panel). This case is then likely
to be characterized by gravity waves that break in the IL stratified layer (Gibert et al. 2011b).

359

360

361

362

363

364

The top of the CBL is usually characterized by a decrease of vertical velocity variance and
integral scale (Fig. 2a third and fourth panels), showing the destruction of large-scale turbulent
eddies by the capping temperature inversion and resulting negative buoyancy (A-0716 case).

365

366

367

The vertical velocity variance also follows well the free convection similarity profile (Stull
1988) for this case. These usual features do not work for the T-0615 case study (Fig. 3).

368

369

As shown by radiosondes profiles (Fig. 2b, 3b and 2c, 3c second panels), the instantaneous top
of the CBL may be very sharp with steep increase of potential temperature and/or decrease of
specific humidity. Ensemble or time-average profiles as monitored by the lidar instrument
provide a smoother interface that can be related to the eddy-covariance fluxes calculated over
the same period of time. However, as shown by the two examples (Fig. 2 and 3), the smoothness
of the profiles may be different for the different scalars. This unexpected difference with respect
to traditional LES results (Sorbjan 2005, Cancelli et al. 2014) is likely to be linked to different
integral scales of turbulence for the different scalars, temperature and specific humidity.

370

371

372

373

374

375

376

377
<figure><img src="image_5.png" /><figcaption>Fig. 3. T-0615 case study at 15h local time. (a) Vertical velocity; horizontal wind speed profiles (radar with thick solid lines and radiosondes with thin solid lines); vertical velocity variance (markers). The similarity law is indicated with the dotted line; integral scale (markers) and buoyant period oscillation (magenta solid line). (b) Virtual potential temperature; 40-min mean lidar profile (red line) and standard deviation and radiosonde profile (black solid line) at 14h; virtual potential temperature variance (red square); integral scale; sensible heat flux profile with bars from instrumental (cap) and sampling (no cap) errors. (c) Specific humidity; 40-min mean lidar profile (blue line) and standard deviation and radiosonde profile (black solid line) at 14h; specific humidity variance (blue triangle); integral scale; latent heat flux profile with bars from instrumental (cap) and sampling (no cap) errors. The height at minimum sensible heat flux, $z_{if}$, and at maximum scalar variance $z_{iv}$, are indicated with dashed and dotted lines respectively.</figcaption></figure>

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395
# 4. Assessment of LSM theory and local scaling of IL scalar turbulent fluxes and variances

## a. Comparing scalar characteristics

Can fluid parcel temperature and water vapor mixing ratio be considered identically to apply the same scaling laws for their second order moment in the IL? To answer to that question, we compare their main characteristics in the IL: gradient, integral scales, standard deviation and eddy-covariance fluxes (Fig. 4). In order to reduce the instrumental noise of Lidar estimates (Fig. A2) and infer a significant geophysical information, the data are averaged over ten samples. The mean, associated with its standard deviation, and the median values are analyzed.

<figure><img src="image_6.png" /><figcaption>Fig. 4. (a) Gradient, (b) standard deviation, (c) integral scales (black: 1/N; green: vertical velocity; red: virtual potential temperature) and (d) turbulent flux of temperature as a function of moisture's ones in the IL. Open markers and error bars are for the ten-samples mean and its standard deviation. Filled markers are for the ten-samples median values.</figcaption></figure>

Temperature and moisture gradient are well correlated in the IL ($\rho$ = 0.98 for median values) although some dispersion still remains especially for the highest values (standard deviation of the mean values in Fig. 4a). Fig. 2 gives an example of this dispersion where a strong stratification $\partial\bar{\theta}_v/\partial z = (0.035 \pm 0.010)$ K/m is associated with a weaker moisture gradient, $-\partial\bar{q}/\partial z = (0.009 \pm 0.004)$ g/kg/m, than expected but the contrary is also possible. Fig. 4c shows that temperature and moisture integral scales are also well correlated ($\rho$ = 0.98
for median values). Instead, no correlation is observed with vertical velocity integral scale and buoyancy scale 1/N. Temperature integral scales are lower than moisture and vertical velocity ones (by a round factor of 2) which confirms the results obtained with former observations (Lenschow and Stankov, 1986; Gibert et al. 2011a) or LES (Couvreux et al. 2005). Fig. 4c suggests that: $I_{\theta v} < I_w \le I_q < 1/N$. The LSM scaling laws (section 2) use different integral scales, i.e. $I_w$ and $1/N$, that compare differently with temperature and moisture ones. If the characteristic scale of scalar turbulent fluctuations is larger than the characteristic scale of turbulent diffusion, then another physical process than diffusion (for example fluctuations due to advection) must be considered and LSM diffusion theory is expected to fail.

There is also a significant correlation between temperature and moisture standard deviations ($\rho = 0.92$ for median values) although there is a large dispersion of the data for the largest values. Fig. 2 shows that temperature and moisture standard deviations at minimum sensible heat flux height are often located on a slope, their maximum being at higher height ($z_{if} \neq z_{iv}$). When the maximum values at $z_{iv}$ are large, the values at $z_{if}$ are more sensitive to the vertical resolution of lidar measurements and the range gate to calculate gradients. This may explain a part of the variability of the highest values. The correlation between eddy-covariance fluxes is slightly lower than for the standard deviations, although remaining significant ($\rho = 0.72$). This loss of correlation is due to large $H$ that corresponds to lower $Q$ values than expected by the linear trend seen in Fig. 4d. Fig. 3 gives an explanation for this. At $z_{if}$, $H = - (0.08 \pm 0.04)$ K.m/s whereas $Q = (0.04 \pm 0.02)$ g/kg.m/s. In addition, $-\partial \bar{q}/\partial z = (0.004 \pm 0.001)$ g/kg/m whereas $\partial \bar{\theta}_v/\partial z = (0.011 \pm 0.002)$ K/m which is also a larger value than expected (see the associated anomaly in linear trend of Fig. 4a. These anomalies in scalar gradient and fluxes linear trends are indeed linked to the observed large difference between $z_{if}$ and $z_{iv}$. Such difference is due to an increasing stratification of the upper part of the CBL. This stratification is the cause of significant $H$ and $\partial \bar{\theta}_v/\partial z$ values while the changes in the corresponding passive scalar (like moisture) gradient and flux remain negligible in the middle of the CBL. LeMone (2019) noted that this usually happens in the afternoon when the surface heat flux starts to decrease.

In conclusion, gradients, integral scales, standard deviations and fluxes of temperature and moisture in the IL are significantly linearly correlated. Therefore, general scaling laws are likely to be applied identically for both scalars. However, the LSM characteristic scale of diffusion, $I_w$ is usually lower than the moisture integral scale and is therefore expected not to be adapted to explain moisture fluctuation and eddy-diffusivity in the IL.
Fig. 5 shows the test of the different LSM for sensible (panels a-c) and latent (panels d-f) heat fluxes: Taylor with Eq. (10), Pearson with Eq. (14) and Lindborg-Osborn with Eq. (20). For Taylor LSM (Fig. 4a), a linear trend on median values is observed for sensible heat flux for small values of $\sigma_w^2 \cdot I_w \cdot \partial\bar{\theta}_v/\partial z$ (< 0.1 K.m/s). Within this limit, a bi-square linear regression through the origin (on the full dataset with 95% confidence) gives $K_{\theta,Taylor} = (1.02 \pm 0.15) \sigma_w^2 \cdot I_w$ which is fully in agreement with Eq. (10). One might think that a low N-pass filter would improve the linear regression given that Taylor LSM was not built for stratified layer but we checked with our dataset that this is actually not the case. The failure of Taylor LSM local scaling, which is even more obvious for latent heat flux (Fig. 5d), results more from the choice of $I_w$ as the turbulent diffusion scale as mentioned before.

<figure><img src="image_7.png" /></figure>

Fig. 5: IL sensible (H) and latent (Q) heat flux as a function of local parametrized flux from Taylor (a, d), Pearson (b, e) and Lindborg-Osborn (c, f) LSM. Open and filled markers are for mean and median values respectively (over 10 samples). (a) and (b): dashed line is for $y = x$ predicted by Taylor (1921); (b) and (e): dashed line is for $y = 0.26x$ (LES) and dashed and dotted line for $y = 0.31x$ (Kansas data) obtained by Otte and Wyngaard (2001); (c) and (f) dashed line is for $y = 3.1x$ from Lindborg and Brethouwer (2008). Colored solid lines are for bi-square regressions through the origin on the full dataset with 95% confidence.
468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

Using $1/N$ as the characteristic time scale of IL scalar eddy diffusion (Pearson LSM) improves LSM flux scaling as a similar linear trend is observed for the two scalars, temperature and moisture. Linear regression through the origin gives the eddy-diffusivity for temperature $K_{\theta,Pearson} = (0.35 \pm 0.03) \sigma_w^2/N$ (Fig. 5b) and moisture $K_{q,Pearson} = (0.27 \pm 0.03) \sigma_w^2/N$ (Fig. 5e) with significant $R^2$ ($> 0.6$), mainly limited by the dataset instrumental errors (confirmed by Monte Carlo simulations). Our result for temperature gives actually a slightly larger value than the LES study made by Otte and Wyngaard (2001) in the interfacial layer $K_{\theta} = 0.26\sigma_w^2/N$. However, observations in the stable surface layer in Kansas, USA showed a similar result as our lidar experiment: $K_{\theta} \approx 0.31\sigma_w^2/N$ (Businger et al. 1971). This supports the analogy between CBL interfacial layer and stable nocturnal layers as already mentioned by Otte and Wyngaard. Other measurements by Nieuwstad (1984) in the nocturnal boundary layers at the Cabau tower in the Netherlands presents also the same relationship: $K_{\theta} \approx 0.25\sigma_w^2/N$. In addition to these former observations that concerns only temperature, our lidar experiment enables to prove for the first time that a similar diffusivity coefficient may also be used for moisture. Pearson formalism and Eq. (14) is given for $\gamma \ll 1$ and strong stratification. Fig. 5b and 5e show that Eq. (14) is still valid from weak to strong stratification $3 \ 10^{-3} < N < 5 \ 10^{-2} \text{ s}^{-1}$ and a mean $\gamma \approx 0.5$.

485

486

487

488

489

490

491

492

493

Lindborg formalism gives similar results as Taylor's one with a failure of the scaling for moisture. In contrary to sensible heat flux where a linear trend is still observed for low values as for Taylor scaling, no trend can be seen for latent heat flux. This may also be explained by the fact that Lindborg-Osborn scaling involves two different scales, $1/N$ and $I_w$ through $\epsilon$ (Eq. 20), $I_w$ being expected not to be representative of moisture fluctuations in the IL. Nevertheless, for small values, i.e. $\epsilon/N^2 \partial \bar{\theta}_v/\partial z < 0.04$ K.m/s, a bi-square linear regression gives $K_{\theta,Lindborg-Osborn} \sim (3.31 \pm 0.53) \epsilon/N^2$. Assuming $\Gamma \sim 0.3$ (Maffioli 2016) in Eq. (20), one obtains the theoretical Lindborg-Osborn LSM eddy-diffusivity $K_{\text{Lindborg-Osborn}} \sim 3.1 \epsilon/N^2$ which agrees with our experimental eddy-diffusivity coefficient value and uncertainty.

494

495

496

497

In conclusion, Pearson LSM gives the best scaling law for sensible and latent heat fluxes in the IL. This local scaling has been applied to the full dataset (Table 1) to analyze how such parametrization matches IL turbulent flux diurnal cycles in temperate and semi-arid regions (Fig. 6).

498

499
<figure><img src="image_8.png" /><figcaption>Fig. 6. Comparison of sensible and latent heat flux diurnal variations from direct Lidar eddy-covariance measurements and Pearson LSM local scaling law for the ten different days in temperate and arid regions (Table 1). Open markers are for Pearson LSM using radiosondes data (to calculate gradients and buoyancy period).</figcaption></figure>

The local scaling with the lidar dataset follows well the diurnal variations of IL fluxes within the uncertainties: the increases (A-0718, A-0727, T_0613), decreases (T-0614, T-0615) as well as complex evolutions (A-0717, A-0615) along the day are well returned by LSM scaling, a little better for sensible than for latent heat fluxes though. Significant underestimates of the latent heat flux largest values (A-0716, A-0717, T-0614) by LSM scaling are not explained by the uncertainties. LSM with radiosondes dataset (used to calculate the scalar gradient) gives better agreement for the A-0716 case as there is a sharper decrease of specific humidity at $z_{if}$ than for the Lidar smooth profile (Fig. 2). LSM with radiosondes dataset present usually a higher dispersion of LSM scaling (A-0721 case) given that they provide instantaneous profiles that are not representative of the mean IL statistics (flux and variance). However, for most of the radiosonde data used in this study, the agreement between LSM scaling (markers in Fig. 6) and IL Lidar fluxes is still very good which means that a good estimate of CBL IL scalar fluxes with the current radiosonde worldwide network is feasible, provided that the variance of IL vertical velocity could be estimated accurately through another way.
c. Variance of temperature and water vapor local scaling in the IL

Figure 7 shows the test of the different LSM for rms of virtual potential temperature (panels a-c) and specific humidity (panels d-f) fluctuations in the interfacial layer at $z_{if}$: Taylor with Eq. (11), Pearson with Eq. (15) and Lindborg-Osborn with Eq. (21). Unlike flux estimates that decrease to the origin in Fig. 5, the temperature and moisture rms present here an offset value, $\sigma_{\theta v,0} \sim 0.3$ K and $\sigma_{q,0} \sim 0.2$ g/kg. Note that these offsets are unlikely to be due to the instrumental noises (corrected, see Appendix A and Fig. A1), especially for specific humidity ($\sigma_{q,inst} < 0.1$ g/kg).

<figure><img src="image_9.png" /></figure>

Fig. 7: IL virtual potential temperature and specific humidity standard deviation as a function of local parametrized standard deviation from Taylor (a, d), Pearson (b, e) and Lindborg-Osborn (c, f) LSM. Open and filled markers are for mean and median values respectively (over 10 samples). (b) and (e): dashed line is for $y = 1.15x$ (LES) and dashed and dotted line for $y = 0.7x$ (Kansas data) obtained by Otte and Wyngaard (2001); (c) dashed line is for $y = 2.49x$ from Lindborg and Brethouwer (2008). Colored solid lines are for bi-square regressions through the origin on the full dataset with 95% confidence.

For Taylor LSM (Fig. 7a and b), no linear trend is observed for temperature as well as for moisture which shows that this scaling is, once again, not adapted to represent scalar turbulent fluctuations in the IL. For Pearson LSM (Fig. 7a and b), the choice of $1/N$ as the diffusion scale improves the alignment of the data for both scalars. A linear regression through the origin gives $\sigma_{\theta,Pearson} \cong 0.66$. $\sigma_w / N \cdot \partial\bar{\theta}_v / \partial z + 0.34$ (Fig. 7b) and moisture $\sigma_{q,Pearson} \cong 0.98$. $\sigma_w /$
$N \cdot \partial \bar{q} / \partial z + 0.22$ (Fig. 7e) with limited $R^2 (> 0.5)$ though, once again fully explained by the data uncertainties. The result for temperature agrees (within the uncertainties) with the one obtained with in situ observations in the stable nocturnal layer $\sigma_{\theta}/(\partial \bar{\theta}_{v}/\partial z) \approx 0.8 \sigma_{w}/N$ (Hunt 1985). The Kansas data used in Otte and Wyngaard (2001) suggested a similar proportionality constant of about 0.7 while Nieuwstadt's data proposed 0.96. LES results in the interfacial layer gave a similar constant of about 1.15 (Otte and Wyngaard 2001). This former constant corresponds rather for moisture than for temperature rms in the present study while still being in the uncertainty range of both scalars. Lindborg formalism failed for the scaling of moisture fluctuations. However, for temperature, a linear trend is observed and a linear regression through the origin gives $\sigma_{\theta,Lindborg-Osborn} \approx 2.11 \cdot \epsilon^{1/2}/N^{3/2} \cdot \partial \bar{\theta}_{v}/\partial z + 0.32$. Assuming $\Gamma \sim 0.3$ (Maffioli 2016) in Eq. (21), one obtains the theoretical Lindborg-Osborn LSM temperature rms $\sigma_{\theta,Lindborg-Osborn} \approx 2.49 \cdot \epsilon^{1/2}/N^{3/2} \cdot \partial \bar{\theta}_{v}/\partial z$ which fully agrees with our result within the uncertainty.

<figure><img src="image_10.png" /><figcaption>Fig. 8: Comparison of virtual potential temperature and specific humidity standard deviation diurnal variations from direct Lidar measurements and Pearson LSM local scaling law for the ten different days in temperate and arid regions (Table 1). Open markers are for Pearson LSM using radiosondes data (to calculate gradients and buoyancy period).</figcaption></figure>
In conclusion, Pearson LSM and Lindborg-Osborn LSM gives a suitable scaling law for potential temperature standard deviation in the IL although the offset at the origin remains to be determined. For moisture fluctuations, only Pearson LSM proved to be efficient and suitable to be applied almost identically for both scalars. The Pearson local scaling has been applied, as for eddy-covariance fluxes, to the full dataset (Table 1) to analyze how such parametrization matches IL scalar standard deviation diurnal cycles in temperate and semi-arid regions. Fig. 8 shows that the complex evolutions of rms scalar fluctuations are well modelized by Pearson LSM for both scalars. In the same way as for fluxes, the use of radiosonde data gives mostly similar agreement between scalar rms fluctuations and LSM scaling. If the IL vertical velocity standard deviation is known, the parametrization of IL scalar second order moments could be done with the current observation network and meteorological models.

The precedent relations for flux and variance contain variables such as $\sigma_w^2$ for Pearson LSM or $\epsilon$ for Osborn-Lindborg, that are turbulent and need to be parametrized too. Although a network of Doppler lidar is likely to be widespread in the coming years, one may wonder how these variables could be parametrized with current in situ instrumentation. For vertical velocity variance, Fig. 2a shows that CBL similarity profile (Stull p375, 1988) could be used to do so. However, Fig. 3a also shows that CBL similarity is likely to fail in the afternoon when $z_{if} \neq z_{iv}$. Such parametrization needs for further investigation and dedicated work that will be considered in the future.

## 6. Conclusion

Simultaneous lidar measurements of scalar turbulent fluxes, variances and gradients in the CBL interfacial layer enabled to address for the first time the important issue of IL scalar turbulent diffusion with a direct observational point of view. The dataset consists in almost one hundred 40-min time gate of measurements in the CBL with different conditions for surface sensible heat fluxes (temperate and arid region, morning and afternoon transitions included), IL wind shear and scalar gradients that could be used in further numerical studies. When we analyzed the differences between the two scalars considered here, i.e. temperature and moisture, the observations show that gradients, integral scales, standard deviations and fluxes of temperature and moisture in the IL are significantly linearly correlated and may be considered identically for local scaling in the IL. Using LSM theories, a local scaling of scalar rms fluctuations, flux and eddy-diffusivity has been reminded implying different dynamics parameters and critical scales of diffusion. The test of different LSM with our dataset show that
the vertical velocity variance and the buoyancy period that are used in Pearson LSM prove to be efficient and universal to scale locally temperature and moisture standard deviations, fluxes and eddy-diffusivities. Our results for IL temperature, $K_{\theta} = (0.35 \pm 0.03)\sigma_w^2/N$ and moisture $K_q = (0.27 \pm 0.03)\sigma_w^2/N$ eddy-diffusivities and fluxes agree with LES results and in situ observations made in the ASL. This supports the analogy between IL and ASL processes. For the scalar standard deviations, the results agree as well with Pearson LSM although they present an offset value, $\sigma_{\theta v,0} \sim 0.3$ K and $\sigma_{q,0} \sim 0.2$ g/kg that was not mentioned in LES studies. Note that these offsets are unlikely to be due to instrumental noises and so need for further investigation. The use of the vertical velocity integral scale as a scalar diffusion scale entails a failure of LSM local scaling that relies on it, i.e. Taylor and Osborn-Lindborg, especially for moisture. Indeed, our dataset show that $I_w$ is usually lower than the moisture integral scale and is therefore expected not to be adapted to explain moisture fluctuation and then diffusivity in the IL. To improve IL turbulence flux parametrization in transport models, vertical velocity variance parametrization as well as non-local scaling will be addressed in a future work. This will be made with the same dataset used in the present study.

608

## Acknowledgments.

This work is supported by Centre National de Recherche Scientifique (CNRS) with the program HyMeX and by Agence Nationale de la Recherche (ANR) with the project LIAISE (ANR-19-CE01-0017). We want to thank SIRTA observatory team for the in-situ data and Ludovic Thobois from Vaisala factory who provides additional Doppler lidar measurements during the temperate region experiment. We also thank the UK Met Office team (Jeremy Price, Jennifer Brooke) for the in-situ data and the Laboratoire d'Aérologie (LA) for the UHF radar measurements (Marie Lothon, Yannick Bezombes) during LIAISE field experiment (arid region experiment).

618

## Data Availability Statement.

The native lidar data collected during LIAISE field experiment are available on AERIS database (https://liaise.aeris-data.fr/products/).

622

623
Dataset and uncertainties

Each variable of interest (gradient, variance, integral scale, flux) is calculated in an almost 626
40 min period of time. The data time series are detrended at each level by subtracting a least 627
square linear fit. Uncertainties are estimated following Lenschow et al. (2000) formalism. The 628
temperature and humidity profiles from the Raman lidar are calibrated using radiosondes 629
profiles so that systematic errors on gradients and statistical estimates due to the calibration 630
constants are estimated to be negligible for the signal processing used in this paper. 631

a. Atmospheric, instrumental variances and uncertainties

In order to estimate the uncertainties on scalar and velocity profiles but also to discriminate
natural and instrumental standard deviation in the measurements we use the autocovariance
functions *ACV*. Atmospheric and noise variances are discriminated using a 2/3 power law fit
and looking at the extrapolated value to lag 0. This gives the atmospheric variance (Fig. A1).
The 2/3 power law fit is made using ACV data with lags until ACV first zero crossing. To limit
the influence of mesoscale structures, the 2/3 power law was made using a 1-minute maximum
lag (here we use the first 6 lag points, i.e. 48 s).

<figure><img src="image_11.png" /><figcaption>Fig. A1. Autocovariance functions for (a) wind speed, (b) temperature and (c) humidity for times series of A-0716 day at z = 0.552 m. 2/3 power law fits are indicated which leads to instrumental and atmospheric variance estimates, integral scale estimates as well as uncertainties.</figcaption></figure>
The instrumental variance is calculated using the total variance and the fit-estimated atmospheric variance (Fig. A1), assuming no correlation between noise and atmospheric fluctuations:

$$ \sigma_d^2 = \sigma_{d, \text{atm}}^2 + \sigma_{d, \text{inst}}^2 \qquad (\text{A1}) $$

where d is for the data used, i.e. vertical wind speed, virtual potential temperature and specific humidity.

## b. Integral scale and uncertainty

Following Wulfmeyer et al. (2016), an inertial range integral scale is estimated assuming that all autocovariance function has the form:

$$ \mathrm{ACV}_d(\tau) = \sigma_{d, \text{atm}}^2 - k_d \tau^{2/3} \qquad (\text{A2}) $$

The integral scale is defined by:

$$ I_d(z) = \int_0^\infty ACR_d(z, \tau)d\tau \qquad (\text{A3}) $$

where $ACR_d(z, \tau) = ACV_d(z, \tau)/\sigma_{d, \text{atm}}^2(z)$ is the autocorrelation function. Using Eq. (A2) and limiting the integration to the first zero crossing of $ACR_d$, $\tau_0$:

$$ I_d(z) \approx \int_0^{\tau_0} ACR_d(z, \tau)d\tau = \int_0^{\tau_0} \left(1 - \frac{k_d}{\sigma_{d, \text{atm}}^2}\tau^{2/3}\right)d\tau \qquad (\text{A4}) $$

$$ I_d(z) \approx \frac{2}{5} \left( \frac{\sigma_{d, \text{atm}}^2}{k_d} \right)^{3/2} \qquad (\text{A5}) $$

The integral scale uncertainty is then provided by the uncertainties on the 2/3 power fit law coefficients (Fig. A1) using Eq. (A5).

## c. Instrumental and sampling error in eddy-covariance flux estimates

The sensible or latent heat flux is calculated using the correlation between the fluctuations of vertical velocity and those of virtual potential temperature or specific humidity due to turbulence only:

$$ H = \overline{w'\theta'_{v}} \qquad (\text{A6}) $$

$$ Q = \overline{w'q'} \qquad (\text{A7}) $$

From Eq. (A6) and (A7) one infers the random instrumental error in eddy flux measurements (Giez et al. 1999):
$$672 \qquad \sigma_{H,inst}^2 = \frac{\Delta t}{T} \frac{\Delta z}{Z} (\sigma_{w,atm}^2 \sigma_{\theta v,inst}^2 + \sigma_{w,inst}^2 \sigma_{\theta v,atm}^2 + \sigma_{w,inst}^2 \sigma_{\theta v,inst}^2) \qquad (A8)$$

$$673 \qquad \sigma_{Q,inst}^2 = \frac{\Delta t}{T} \frac{\Delta z}{Z} (\sigma_{w,atm}^2 \sigma_{q,inst}^2 + \sigma_{w,inst}^2 \sigma_{q,atm}^2 + \sigma_{w,inst}^2 \sigma_{q,inst}^2) \qquad (A9)$$

where $\Delta t$ and $\Delta z$ corresponds to the time and space resolution of lidar measurements and $T$ and $Z$ are respectively the time and space resolution of eddy-covariance flux measurements.

As shown in Fig. 2, this error is weak for latent heat flux. The reason is that instrumental error in vertical velocity and specific humidity is usually very small (Fig. A1). Even for sensible heat flux instrumental error is not the main source of uncertainty. The main source of error in ground-based lidar eddy-flux measurement is due to the limited number of dominant eddies or integral scales that passes over the instrument in the time interval $T$. It definitively depends on the horizontal wind speed. This statistic uncertainty is defined as follows (Lenschow and Stankov 1986, Senff et al. 1994, Giez et al. 1999):

$$683 \qquad \sigma_{H,samp}^2 = 2 \frac{I_H}{T} (H^2 + \sigma_{w,atm}^2 \sigma_{\theta v,atm}^2) \qquad (A10)$$

$$684 \qquad \sigma_{Q,samp}^2 = 2 \frac{I_Q}{T} (Q^2 + \sigma_{w,atm}^2 \sigma_{q,atm}^2) \qquad (A11)$$

where $I_H$ and $I_Q$ are the sensible and latent heat flux integral scales respectively. Given that vertical velocity and eddy-covariance fluxes integral scales are usually similar in the convective boundary-layer (Gibert et al. 2011a) we used $I_w$ in practice to calculate the sampling errors. Even if both instrumental and sampling errors make sense in eddy-covariance flux error estimate, Fig. 2 and Fig. 3 show that the sampling error calculated with Eq. (A10) and (A11) is no doubt over-estimated given that the error bars are far larger than the random part of lidar flux estimates. Then a reasonable estimate of lidar flux uncertainty is likely to be in the same order of magnitude than the instrumental uncertainty.

### d. Dataset and uncertainties

The full dataset with relevant variables used in this paper are gathered in Fig. A2 as well as associated relative errors. A median value of uncertainty is indicated. The relative error on CBL turbulent velocity is calculated assuming a negligible error on the in situ surface flux (sampling error is considered using the highest level of the tower) and a relative error on CBL height estimate $z_{if}$ which is due to the lidar vertical resolution, i.e. 48 m. Relative error on gradient estimates are provided by the linear least square fit coefficient errors on scalar high-resolution time-averaged profiles (i.e. with 7.5 m vertical resolution). This yields the error on Brunt-
Väisälä frequency using Eq. (12). Relative errors on variance, integral scale and flux have been detailed in the former sections of this Appendix. The relative errors on wind shear from radar UHF and Doppler lidar measurements are estimated as the difference between the two nearest points around $z_{if}$. The gradient Richardson number relative error is calculated using Brunt-Väisälä frequency and wind shear errors.

<figure><img src="image_12.png" /><figcaption>Fig. A2: Relevant variable and relative error: (a) CBL velocity turbulent scale (b) vertical wind speed integral scale (c) vertical wind speed variance (d) sensible heat flux with instrumental (square) and sampling (cross) errors (e) gradient of virtual potential temperature (f) variance of virtual potential temperature (g) latent heat flux with instrumental (triangle) and sampling (cross) errors (h) gradient of specific humidity (i) variance of specific humidity (j) horizontal wind shear (k) Brunt-Väisälä frequency (l) gradient Richardson number. All the variables except $w_*$ are calculated at $z_{if}$. Median relative errors are indicated with solid and dashed lines (only for flux sampling error).</figcaption></figure>

706
REFERENCES

Arakawa, R.A. and W.H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the large-scale environment. Part I. *J. Atmos. Sci.*, **31**, 674–701.

Behrendt, A., Wulfmeyer, V., Hammann, E., Muppa, S. K., and Pal, S.: Profiles of second- to fourth-order moments of turbulent temperature fluctuations in the convective boundary layer: first measurements with rotational Raman lidar, *Atmos. Chem. Phys.*, **15**, 5485–5500, https://doi.org/10.5194/acp-15-5485-2015, 2015.

Behrendt A., V. Wulfmeyer, C. Senff, S. Kumar Muppa, F. Späth, D; Lange, N. Kalthoff and A. Wieser, 2020: Observation of sensible and latent heat flux profiles with lidar. *Atmos. Meas. Tech.*, **13**, 3221-3233, https://doi.org/10.5194/amt-13-3221-2020

Boone, A., J. Bellvert, M. Best, J. Brooke, G. Canut-Rocafort, J. Cuxart, O. Hartogensis, P. Le Moigne, J. R. Miró, J. Polcher, J. Price, P. Quintana Seguí, J. Bech, Y. Bezombes, O. Branch, J. Cristobal, K. Dassas, P. Fanise, F. Gibert, Y. Goulas, J. Groh, J. Hanus, G. Hmimina, L. Jarlan, E. Kim, V. Le Dantec, M. Le Page, F. Lohou, M. Lothon, M. R. Mangan, B. Marti, D. Martinez-Villagrasa, J. McGregor, A. Kerr-Munslow, N. Ouaadi, A. Philibert, J. Quiros-Vargas, U. Rasher, B. Siegmann, M. Udina, A. Vial, B. Wrenger, V. Wulfmeyer, M. Zribi, 2025: The Land Surface Interactions with the Atmosphere over the Iberian Semi-Arid Environment (LIAISE) Field Campaign, Journal of the European Meteorological Society, **2**, 100007, ISSN 2950-6301, https://doi.org/10.1016/j.jemets.2025.100007.

Bretherton, C. S., and S. Park, 2009: A new moist turbulence parameterization in the Community Atmosphere Model. *Journal of Climate*, **22**(12), 3422–3448, https://doi.org/10.1175/2008JCLI2556.1

Businger, J.A., J.C. Wyngaard, Y. Izumi and E.F. Bradley, 1971: Flux-profile relationships in the atmospheric surface layer, *J. Atmos. Sci.*, **28**, 181-189. https://doi.org/10.1175/1520-0469(1971)028%3C0181:FPRITA%3E2.0.CO;2

Cancelli D.M., M. Chamecki and N.L. Dias, 2014: A large-eddy simulation study of scalar dissimilarity in the convective atmospheric boundary layer, *J. Atmos. Sci.*, **71**, 3-15, https://doi.org/10.1175/JAS-D-13-0113.1.
746 Corsin S., 1963: Estimates of the relations between Eulerian and Lagrangian scales in large
747 Reynolds number turbulence, *J. Atmos. Sci.*, **20**, 115-119. https://doi.org/10.1175/1520-0469(1963)020%3C0115:EOTRBE%3E2.0.CO;2

749 Couvreux F., F. Guichard, J.-L. Redelsperger, C. Kiemle, V. Masson, J.-P. Lafore, and C. Flamant, 2005: Water-vapour variability within a convective boundary-layer assessed by large-eddy simulations and IHOP 2002 observations. *Q. J. R. Meteorol. Soc.*, **131**, 2665-2693.

753 Das S.K. and Durbin P.A., 2005: A Lagrangian stochastic model for dispersion in stratified turbulence, *Physics of fluids*, **17**, 025109, https://doi.org/10.1063/1.1849184

755 Deardorff, J. W., 1966: The counter-gradient heat-flux in the lower atmosphere and in the laboratory. *J. Atmos. Sci.*, **23**, 503-506.

757 Deardorff, J.W., 1972: Numerical investigation of neutral and unstable planetary boundary layers, *J. Atmos. Sci.*, **29**, 91-115.

759 Deardorff, J.W., 1974: Three-dimensional numerical study of the height and mean structure of a heated planetary boundary-layer, *Boundary-Layer Meteorol.* **7**, 81-106.

761 Di Girolamo, P., Cacciani, M., Summa, D., Scoccione, A., De Rosa, B., Behrendt, A., and Wulfmeyer, V., 2017: Characterisation of boundary layer turbulent processes by the Raman lidar BASIL in the frame of HD(CP)2 Observational Prototype Experiment, *Atmos. Chem. Phys.*, **17**, 745-767, https://doi.org/10.5194/acp-17-745-2017.

765 Driedonks, A.G.M. and H. Tennekes, 1984: effects in the well-mixed atmospheric boundary layer. *Boundary-Layer Meteorol* **30**, 75-105, https://doi.org/10.1007/BF00121950

767 Federovich, E., R. Conzemius and D. Mironov, 2004: Convective entrainment into a shear-free, linearly stratified atmosphere: Bulk models reevaluated through large eddy simulations, *J. Atmos. Sci.*, **61**, 281-295. https://doi.org/10.1175/1520-0469(2004)061%3C0281:CEIASL%3E2.0.CO;2

771 Garcia J.R. and J.P. Mellado, 2014: The two-layer structure of the entrainment zone in the convective boundary-layer, *J. Atmos. Sci.*, **71**, 1935-1955. https://doi.org/10.1175/JAS-D-13-0148.1

774 Gibert F., G.J. Koch, J. Y. Beyon, T.W. Hilton, K.J. Davis, A. Andrews, P.H. Flamant and U. N. Singh, 2011a: Can CO₂ turbulent flux be measured by lidar? A preliminary study., *J. Atmos. and Oceanic Tech.*, **28**, 365-377, https://doi.org/10.1175/2010JTECHA1446.1
777 Gibert F., N. Arnault, J. Cuesta, R. Plougonven and P.H. Flamant, 2011b: Internal gravity
778 waves convectively forced in the atmospheric residual layer during the morning transition,
779 *Q.J.R. Meteorol. Soc.*, **137**, 1610-1624, https://doi.org/10.1002/qj.836

780 Gibert F., D. Edouart, C. Cénac, F. Le Mounier, and A. Dumas, 2015: 2-µm Ho emitter-based
781 coherent DIAL for CO₂ profiling in the atmosphere, *Opt. Lett.* **40**, 3093-3096,
782 https://doi.org/10.1364/OL.40.003093

783 Gibert F., D. Edouart, P. Monnier, C. Cénac, V. Gauthier and H. Salvador, 2022: A wind,
784 temperature, H₂O, CO₂ scanning lidar mobile observatory for a 3D thermodynamic view
785 of the atmosphere, Proceedings of the 30th International Laser Radar Conference
786 (ILRC30), Big Sky Conference Center, Montana, USA

787 Giez, A., G. Ehret, R.L. Schwiesow, K.J. davis and D.H. Lenschow, 1999: Water vapor flux
788 measurements from ground-based vertically pointed water vapor differential absorption
789 and Doppler lidars. *J. Atmos. Oceanic Technol.*, **16**, 237-250,
790 http://dx.doi.org/10.1175/1520-0426(1999)016%3C0237:WVFMFG%3E2.0.CO;2

791 Haghshenas A. and J. P. Mellado. Characterization of wind-shear effects on entrainment
792 in a convective boundary layer. *J. Fluid Mech.*, 858:145-183, 2019.

793 Hourdin, F., F. Couvreux and L. Menut, 2002: Parameterisation of the dry convective
794 boundary layer based on a mass flux representation of thermals. *J. Atmos. Sci.*, **59**,
795 1105-1123.

796 Hourdin, F., A. Jam, C. Rio, F. Couvreux, I. Sandu, M.-P. Lefebvre, et al., 2019: Unified
797 parameterization of convective boundary layer transport and clouds with the
798 thermal plume model. *Journal of Advances in Modeling Earth Systems*, **11**, 2910-2933, https://doi.org/10.1029/2019MS001666

800 Hunt J.C.R., J.C. Kaimal and J.E. Gaynor, 1985: Some observations of turbulence structure in
801 stable layers, *Quart. J. R. Met. Soc.*, **111**, 793-815,
802 https://doi.org/10.1002/qj.49711146908

803 Kiemle, C., Ehret, G., Giez, A., Davis, K., Lenschow, D., and Oncley, S. P.: Estimation of
804 boundary layer humidity fluxes and statistics from airborne differential absorption
805 lidar (DIAL), J. Geophys. Res., 102, 29189-29203,
806 https://doi.org/10.1029/97JD01112, 1997.
807 Kiemle, C., Ehret, G., Fix, A., Wirth, M., Poberaj, G., Brewer, W. A., Hardesty, R. M., Senff, C. J., and LeMone, M. A.: Latent heat flux profiles from collocated airborne water vapor and wind lidars during IHOP_2002, J. Atmos. Ocean. Tech., 24, 627–639, https://doi.org/10.1175/JTECH1997.1, 2007.

811 Kiemle, C., Wirth, M., Fix, A., Rahm, S., Corsmeier, U., and Di Girolamo, P.: Latent heat flux measurements over complex terrain by airborne water vapour and wind lidars, *Q. J. Roy. Meteor. Soc.*, 137, 190–203, https://doi.org/10.1002/qj.757, 2011.

814 Legg, B. J. and Raupach, M. R. 1982: Markov-chain simulation of particle dispersion in inhomogeneous flows: the mean drift velocity induced by a gradient in Eulerian velocity variance. *Boundary-Layer Meteorology*. 24 (1), 3-13. https://doi.org/10.1007/BF00121796

817 LeMone M.A., W.M. Angevine, C.S. Bretherton, F. Chen, J. Dudhia, E. Federovič, K.B. Katsaros, D.H. Lenschow, L. Mahrt, E.G. Patton, J. Sun, M. Tjernström and J. Weil, 2019: 100 years of progress in boundary layer meteorology, *Meteorological Monographs*, 59, 9.1-9.85, https://dx.doi.org/10.1175/AMSMONOGRAPHS-D-18-0013.1

821 Lenschow D.H. and B.B. Stankov, 1986: Length scales in the convective boundary layer, *J. Atmos. Sci.*, **43**, 1198-1209, https://doi.org/10.1175/1520-0469(1986)043%3C1198:LSITCB%3E2.0.CO;2

824 Lenschow, D. H., V. Wulfmeyer, and C. Senff, 2000: Measuring second-through fourth-order moments in noisy data, *J. Atmos. Ocean. Tech.*, **17**, 1330–1347, https://doi.org/10.1175/1520-0426(2000)017<1330:MSTFOM>2.0.CO;2

827 Lindborg E. and Brethouwer G., 2008: Vertical dispersion by stratified turbulence, *J. Fluid Mech.*, **614**, 303–314, doi:10.1017/S0022112008003595

829 Lock, A.P., A.R. Brown, M.R. Bush, G.M. Martin, and R.N.B. Smith, 2000: A new boundary layer mixing scheme. Part I: Scheme description and single-column model tests. *Monthly Weather Review*, **128**(9), 3187. https://doi.org/10.1175/15200493(2000)128h3187:ANBLMSi2.0.CO;2

833 Maffioli A. and Davidson P.A., 2016: Dynamics of stratified turbulence decaying from a high buoyancy Reynolds number. *Journal of Fluid Mechanics*, **786**, 210-233, doi:10.1017/jfm.2015.667

836 Mahrt L., 1991: Boundary-layer moisture regimes. *Boundary-Layer Meteorol.*, **152**,151-176.
838 Nieuwstadt F.T.M., 1984: The turbulent structure of the stable nocturnal boundary layer, *J. Atmos. Sci.*, **41**, 2202-2216, https://doi.org/10.1175/1520-0469(1984)041%3C2202:TTSOTS%3E2.0.CO;2

841 Osborn T.R. and Cox C.S., 1972: Oceanic Fine Structure. *Geophysical & Astrophysical Fluid Dynamics*, **3**, 321-345, https://doi.org/10.1080/03091927208236085

843 Osborn T.R., 1980: Estimates of the Local Rate of Vertical Diffusion from Dissipation Measurements. *J. Phys. Oceanogr.*, **10**, 83-89, https://doi.org/10.1175/1520-0485(1980)010<0083:EOTLRO>2.0.CO;2.

846 Otte M.J. and J. Wyngaard, 2001: Stably stratified interfacial-layer turbulence from large-eddy simulation, *J. Atmos. Sci.*, **58**, 3424-3442, https://doi.org/10.1175/1520-0469(2001)058%3C3424:SSILTF%3E2.0.CO;2

849 Pearson H.J., Puttock J.S., Hunt J.C.R., 1983: A statistical model of fluid-element motions and vertical diffusion in a homogeneous stratified turbulent flow. *Journal of Fluid Mechanics*, **129**, 219-249, doi:10.1017/S0022112083000749

852 Senff, C., J. Bösenberg, and G. Peters, 1994: Measurement of water vapor flux profiles in the convective boundary layer with lidar and radar-RASS, *J. Atmos. Ocean. Tech.*, **11**, 85-93.

855 Siebesma, A.P., and J.W.M. Cuypers, J. W. M., 1995: Evaluation of parametric assumptions for shallow cumulus convection. *J. Atmos. Sci.*, **52**, 650-666.
https://doi.org/10.1175/1520-0469(1995)052<0650:EOPAFS>2.0.CO;2

858 Siebesma, A.P., P.M.M. Soares, and J. Teixeira, 2007: A combined eddy-diffusivity mass-flux approach for the convective boundary layer. *J. Atmos. Sci.*, **64**, 1230.
https://doi.org/10.1175/JAS3888.1

861 Soares, P.M.M., P.M.A. Miranda, A.P. Siebesma and J. Teixeira, 2004: An eddy-diffusivity/mass-flux parametrization for dry and shallow cumulus convection. *Q. J. R. Meteorol. Soc.*, **130**, 3365-3383.

864 Sorbjan, Z., 2005: Statistics of scalar fields in the atmospheric boundary layer based on large-eddy simulations. Part I: Free convection. *Boundary-Layer Meteorol.*, **116**, 467-486, https://doi.org/10.1007/s10546-005-0907-3
867 Sorbjan, Z., 2006: Statistics of scalar fields in the atmospheric boundary layer based on large-eddy simulations. Part II: Forced convection. *Boundary-Layer Meteorol.*, **119**, 57-79,
https://doi.org/10.1007/s10546-005-9014-8

870 Sullivan, P.P., C.-H. Moeng, B. Stevens, D.H. lenschow and S.D. Mayor, 1998: Structure of
871 the entrainment zone capping the convective atmospheric boundary layer, *J. Atmos. Sci.*,
872 55, 3042-3064,
873 https://doi.org/10.1175/15200469(1998)055%3C3042:SOTEZC%3E2.0.CO;2

874 Stull, R.B., 1988: *An introduction to boundary-layer meteorology*, new edition, 1997. Kluwer,
875 Dordrecht, 670 pp.

876 Taylor, G.I. 1921: Diffusion by continuous movements, *Proc. London Math. Soc.*, Sec 2, **20**,
877 196-211.

878 Troen, I. and L. Mahrt, 1986: A simple model of the atmospheric boundary layer: Sensitivity
879 to surface evaporation. *Boundary-Layer Meteorol.* **37**, 129-148.

880 Thomson D. J., 1987: Criteria for the selection of stochastic models of particle trajectories in
turbulent flows, J. *Fluid Mech.* **180**, 529-556, doi:10.1017/S0022112087001940

882 van Aartrijk, M. and Clercx H.J.H., 2008: Preferential concentration of heavy particles in
883 stably stratified turbulence, *Phys. Rev. Lett.*, **100**, 254501,
884 https://doi.org/10.1103/PhysRevLett.100.254501

885 Wilson, J. D. and Sawford, B. L., 1996: Review of Lagrangian stochastic models for
886 trajectories in the turbulent atmosphere, *Boundary-Layer Meteorol.*, **78**, 191-210,
887 https://doi.org/10.1007/BF00122492

888 Wulfmeyer et al., 2015: A review of the remote sensing of lower tropospheric thermodynamic
889 profiles and its indispensable role for the understanding and simulation of water and
890 energy cycles, Rev. Geophys., 53, 819-895, doi:10.1002/2014RG000476

891 Wulfmeyer V., S.K. Mua, A. Behrendt, E. Hamann, F. Späth, Z. Sorbian, D.D. Turner and
892 R.M. Hardesty, 2016: Determination of convective boundary layer entrainment fluxes,
893 dissipation rates and the molecular destruction of variances: theoretical description and a
894 strategy for its confirmation with a novel lidar system synergy, *J. Atmos. Sci.*, 73, 667 -
895 692, https://doi.org/10.1175/JAS-D-14-0392.1
896 Wulfmeyer, V., and Coauthors, 2018: A New Research Approach for Observing and

897 Characterizing Land-Atmosphere Feedback. *Bull. Amer. Meteor. Soc.*, **99**, 1639–

898 1667, https://doi.org/10.1175/BAMS-D-17-0009.1.

899 Wulfmeyer, V., Pineda, J.M.V., Otte, S. *et al.*, 2023: Estimation of the Surface Fluxes for

900 Heat and Momentum in Unstable Conditions with Machine Learning and Similarity

901 Approaches for the LAFE Data Set. *Boundary-Layer Meteorol* **186**, 337–371.

902 https://doi.org/10.1007/s10546-022-00761-2

903 Wyngaard J. C., 2010: Turbulence in the Atmosphere. Cambridge University Press, 393

904 pp.

905

906

907

908