1. Introduction

Atmospheric turbulence and the related fluctuations of the refractive index of air affect the propagation of electromagnetic waves. These so-called scintillations are a challenge for communication and imaging systems (e.g., ground-based telescopes) that use radio waves or visible or infrared radiation [1]. Fluctuations in the refractive index of air are mainly related to temperature and humidity. Therefore, turbulent fluctuations of temperature and humidity determine the intensity of turbulence-induced refraction. Various instruments and calculation methods have been developed to obtain the structure parameter of the refractive index of air ($C_{n^2}$) for a certain wavelength, to qualify signals of imaging or communication systems that use electromagnetic radiation. However, these instruments (sonic anemometer, scintillometer, air refractometer) are not easy to operate and are rather expensive, such that frequent observations are rare. A robust method to quantitatively estimate $C_{n^2}$ based on readily available data is therefore needed. Our method could moreover be used for gap filling in case of failures of a sonic anemometer, scintillometer, or an air refractometer.

Forty years ago, Wyngaard et al. [2] developed a semi-empirical theory which relates the temperature structure parameter $C_{T^2}$ (which is related to $C_{n^2}$) to atmospheric stability and the vertical temperature gradient at a certain height in the atmospheric surface layer (the lowest meters of the atmosphere). In the years after, several studies showed that $C_{n^2}$ does not only depend on $C_{T^2}$, but also on the humidity structure parameter $C_{q^2}$ and on the joint structure parameter $C_{Tq}$ [3]. For wavelengths in the visible and near-infrared, $C_{n^2}$ mainly depends on $C_{T^2}$, while for radio wavelengths, $C_{q^2}$ mainly determines $C_{n^2}$. A summary of past research on models and measurements for optical turbulence is presented in [1]. Friehe [4] and Davidson et al. [5] estimated $C_{n^2}$ from dual-level meteorological data over the ocean for visible light. Andreas [6] performed a sensitivity study for a $C_{n^2}$ derivation method based on observed surface fluxes of heat, moisture, and momentum using Monin–Obukhov similarity theory [7], and for a method based on dual-level meteorological data. He tested both methods above snow for a broad wavelength range (visible, infrared, millimeter, and radio).

However, fluxes or vertical gradients are usually not available from regular meteorological observations. Therefore, Sadot and Kopeika [8] developed a regression-based method based on single-level standard weather-station data to estimate $C_{n^2}$ in the atmospheric surface layer, restricted to one height and one wavelength. Also, weather forecasts can be used to predict $C_{n^2}$ by their method. However, the method needs several empirical parameters which were only derived for a desert region, and which need to be first determined for other regions. Rachele and Tunick [9] and Bendersky et al. [10] developed methods to similarly obtain $C_{n^2}$ from single-level weather data for prairie grasslands and coastal environments, which are, however, also not applicable for other regions. Leclerc et al. [11] evaluated three models that are all based on the work of [8] and [10]. They compared model output with scintillo-meter data of two days atop a mountain in Florida, and found moderate results. Besides the regional limitation, these models also only hold for optical wavelengths. Cheinet et al. [12] used large-eddy simulation on weather forecasts, to characterize near-surface optical turbulence at four sites in western Europe. However, new, robust, and easy-to-use methods for estimating $C_{n^2}$ from single-level weather data are lacking, which motivates our study.

The daily course of $C_{n^2}$ depends on the atmospheric conditions which depend on the time of the day, the day of the year (DOY), the weather type of that specific day, and the vegetation activity. Observations of $C_{n^2}$ above grass in The Netherlands are given in Fig. 1 for cloudy and clear sky conditions, for a summer and an autumn day, for an optical and a millimeter wavelength. With our scheme, we aim to capture both the daily and seasonal changes of $C_{n^2}$ for different locations.

 

Fig. 1. Daily course of observed values of $C_{n^2}$ (from an eddy-covariance station) for $\lambda =670\mathrm{nm}$ [(a) and (c)] and $\lambda =2\mathrm{mm}$ [(b) and (d)] for different weather types (blue asterisk is clear sky, red circle is cloudy) and seasons: (a) and (b) two summer days; (c) and (d) two autumn days above grass in The Netherlands (Haarweg Wageningen, 2005).

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We use an adaptation of the scheme introduced in [13] and [14], later referred to as dRH99, that estimates surface heat and moisture fluxes from air temperature, humidity, pressure, wind speed, and incoming shortwave radiation, all measured at one height only. This scheme is based on the Penman–Monteith equation, which estimates evapotranspiration rates from atmospheric conditions and vegetation-specific parameters (e.g., canopy resistance, surface albedo, and roughness) that are all easy to estimate. We provide lookup tables for these parameters in Section 2.A and in the Appendices, where we describe our implementation of the scheme. In Section 2, we furthermore explain how $C_{T^2}$, $C_{q^2}$, and $C_{Tq}$ are derived from surface fluxes (using Monin–Obukhov similarity theory), and how $C_{n^2}$ is derived from the meteorological structure parameters. Our method is easily applicable, and can be used to provide insight into optical turbulence at a specific location for a certain moment, but also for a period of several years to derive its climatology. Estimates of optical turbulence effects on electro-optic or laser systems could be derived, mainly for horizontal atmospheric paths, as a forecast or during or after a specific operation. The complete scheme that estimates $C_{n^2}$ from standard single-level weather data follows from the Appendix and Sections 2.B and 2.C.

We validate estimated fluxes and structure parameters with summertime measurements from different locations and vegetation types: grass in The Netherlands, grass in the South of France, and wheat in the West of Germany. Each dataset covers at least one month of data. In Section 3, we describe the datasets, and in Section 4 we explain the evaluation of our scheme. We restrict ourselves to daytime data, because in our current implementation, we lack information to estimate incoming long-wave radiation at nighttime. In Section 5, we compare fluxes and structure parameters derived using our method with high-frequency measurements (eddy covariance). In Section 6, we present the weaknesses of previous methods and we show the sensitivity of our own scheme. In contrast to other methods, our scheme only weakly depends on empirical assumptions and is still computationally cheap.

2. Framework

In this section, we describe how we derive $C_{n^2}$ from the atmospheric variables temperature, humidity, pressure, wind speed, and radiation, and from the surface characteristics albedo, roughness, leaf area index, vegetation height, and the minimal stomatal resistance of the canopy. Our scheme follows three steps. First (Section 2.A), fluxes are estimated from the single-level weather data using an adaptation of an existing scheme based on a linearized version of the surface energy balance and the Penman–Monteith equation. Then (Section 2.B), the obtained fluxes are used to estimate the structure parameters $C_{T^2}$, $C_{q^2}$, and $C_{T{q}^2}$, following Monin–Obukhov similarity theory. $C_{n^2}$ is finally estimated from the latter (Section 2.C) using the theory described by [15].

A. Estimation of Surface Fluxes from Single-Level Weather Data

1. Basic Method

dRH99 presented a scheme that relates surface fluxes of momentum and sensible and latent heat to only a few weather variables. They successfully tested their scheme for a full year of observations above grass in Cabauw, The Netherlands. The scheme is based on a linearized version of the surface energy balance equation, where every dependence of a flux on the surface temperature is replaced by a linearized dependence on the temperature difference between surface and atmosphere.

For the evapotranspiration flux ($L_{\mathrm{v}}E$) this leads to the Penman–Monteith equation

Table 1. Vegetation Types and Parameter Values after the ERA40 Surface Scheme (TESSEL) Used in the ECMWF Model

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The sensible heat flux $H$ can be calculated as a residual of the surface energy balance

To obtain $L_{\mathrm{v}}E$, the estimation of the surface skin temperature is crucial because of its role in both $Q_{\mathrm{n}}$ and $G$. The estimation of the surface skin temperature depends on $r_{\mathrm{a}}$; however, $H$ and $L_{\mathrm{v}}E$ depend on $r_{\mathrm{a}}$ as well. Therefore, the scheme consists of an iteration loop. $r_{\mathrm{a}}$ depends on atmospheric stability, such that the surface skin temperature is estimated via an iteration that starts with a neutral atmosphere (further explained in Section 4). The net radiation and thus the available energy for heat fluxes further depends on the downward shortwave radiation ($S_{\text{in}}$), the surface albedo ($a$), cloud cover, surface long-wave emissivity, and atmospheric long-wave emissivity.

2. Modifications to dRH99

Our scheme to solve Eq. (1) to estimate heat fluxes from single-level weather data is given in Appendix A, where it is shown how all terms of the surface energy balance are determined based on $S_{\mathrm{in}}$, $T_{\mathrm{a}}$, $q$, $u$ (wind speed), and $p$. In this section, we discuss the adaptations that we made to the scheme of dRH99 regarding $r_{\mathrm{c}}$ and radiation calculations.

Because the scheme of dRH99 was only tested for one specific location in The Netherlands (grass-covered), its performance for other regions or vegetation types is uncertain. Here, we focus on the parameterization of the dependence of the canopy resistance $r_{\mathrm{c}}$ on water vapor content of the air. dRH99 uses the formulation

Here, we show a comparison of the approximations for the dependence of $r_{\mathrm{c}}$ on water vapor deficit given by dRH99 and [16], referred to as BB97, and three approximations used in the land surface schemes of the weather and climate models of the National Center for Meteorological Research in France (CNRM, the ISBA-Ags scheme), the United States National Centers for Environmental Prediction (NCEP, the Noah scheme), and the European Center for Medium-Range Weather Forecasts (ECMWF, the TESSEL scheme). The Noah and TESSEL schemes are based on the Jarvis–Stewart approach [17,18], where

 

Fig. 2. Reaction of $r_{\mathrm{c}}$ to water vapor deficit $\mathrm{\Delta }q$ according to different schemes for the observed range of $\mathrm{\Delta }q$. The reactions by dRH99 and BB97 were defined for the same grass area (Cabauw, The Netherlands), and are therefore both calculated by $r_{\mathrm{c}}\mathrm{LAI}/r_{\mathrm{s},\min }$ ($\mathrm{LAI}=2$, $r_{\mathrm{s},\min }=110\mathrm{s}{\mathrm{m}}^{-1}$, being representative for Cabauw).

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The approach of dRH99 significantly deviates from the other approximations (Fig. 2). Their response function (and thus the canopy resistance) at zero water vapor deficit (relative humidity of 100%) is zero. Herewith dRH99 empirically takes into account that, at low water vapor deficits, the evapotranspiration term is dominated by evaporation whereas it is assumed in the Penman–Monteith equation that transpiration is the only contribution to evapotranspiration. For the three land surface schemes, the response function is 1 ($r_{\mathrm{c}}=r_{s,\min }/\mathrm{LAI}$) for a saturated atmosphere ($\mathrm{\Delta }q=0$). The function of BB97 leads to a response value of 0.24 for a saturated atmosphere (see also Eq. 6).

The other difference is that the function of dRH99 shows a much stronger increase of $r_{\mathrm{c}}$ with $\mathrm{\Delta }q$ than the other four functions, which leads to large deviations for dry air conditions ($\mathrm{\Delta }q>10\mathrm{g}{\mathrm{kg}}^{-1}$). This is probably caused by the fact that they tested their scheme only for a Dutch grassland (Cabauw), where $\mathrm{\Delta }q$ typically ranges between 5 and $11\mathrm{g}{\mathrm{kg}}^{-1}$.

Given these limitations, we replace the representation by dRH99 by a function that is also applicable for other locations and land use types (e.g., a function with a small $r_{\mathrm{c}}$ for very humid conditions and a weak dependence on $\mathrm{\Delta }q$). However, we found that in our data, $r_{\mathrm{c}}$ calculated from the water vapor deficit dependence as in Noah, TESSEL, and Ags is very different from $r_{\mathrm{c}}$ derived from measurements via the inverted Penman–Monteith equation. Namely, the values obtained with Noah, TESSEL, and Ags all stay within a very short range around $r_{s,\min }/\mathrm{LAI}$. The function of BB97 is a nice integration between a small offset and a weak slope of the response functions. In our scheme, we use Eq. (5), omitting $f_{\theta }$ (see Appendix A), and we replace the scaling factor 25.9 by ${\mathrm{f}}_r{r}_{s,\min }/\mathrm{LAI}$.

To estimate incoming long-wave radiation, dRH99 uses an expression given by [24], where $L_{\mathrm{in}}$ depends on the apparent emissivity of the atmosphere, the fractions of low and high clouds, and two empirical cloud coefficients. However, as information on cloud cover for high and low clouds is not available in our datasets, we use relations described in [25] for clear skies and [26] for cloudy skies, which depend on the total cloud cover only (as detailed in Appendix A). We also slightly adapted the albedo calculation of dRH99, such that it does not depend on the fraction of diffuse radiation, but on the total cloud cover fraction.

B. Estimation of Structure Parameters of Temperature and Humidity from Surface Fluxes

The relations between structure parameters and surface fluxes, based on Monin–Obukhov similarity theory, have been studied, among others, by [2,6,27–30]. Maronga [31] found, from a large-eddy simulation resolving surface layer turbulence, that the functions derived for $C_{T^2}$ follow Monin–Obukhov similarity theory while the functions for $C_{q^2}$ are sensitive and not universal. Deviations from Monin–Obukhov similarity theory for measured variances were quantified as well, by [32]. Here we use the universal functions for the structure parameters of temperature and humidity, for unstable conditions presented by [30]. They describe the relation between the structure parameters and surface fluxes via the temperature and humidity scales $T_{*}$ and $q_{*}$ [Eq. (A4d) in Appendix A] as

For homogeneous turbulence, the joint structure parameter $C_{Tq}$ can be estimated from the temperature and humidity structure parameters following [34]

The step from surface fluxes to $C_{T^2}$ and $C_{q^2}$ introduces uncertainty in our $C_{n^2}$ scheme, because Monin–Obukhov similarity theory is not always valid [32]. Also, the step from $C_{T^2}$ and $C_{q^2}$ to $C_{Tq}$ introduces uncertainty, because an estimate of $r_{Tq}$ has to be made (both issues are discussed in Section 6).

C. Estimation of ${\mathsf{C}}_{{\mathsf{n}}^{\mathsf{2}}}$ from Temperature, Humidity, and Joint Structure Parameters

The relationship between the structure parameter of the refractive index of air $n$, and the temperature, humidity, and joint structure parameter is given by [35] as

3. Data

In the first part of this section, we describe the three measurement sites, specifications on instrumentation and vegetation, and the available data that we used to test our scheme. An overview of the measurements used for input or validation is given per site in Table 2. We test our scheme for two vegetation types, grass (G) and wheat (W), and two climatic regions, central (C) and southern Europe (S). The names of the sites are Haarweg (GC), Lannemezan (GS), and Merken (WC). In the second part of this section, we explain our data processing and selection.

Table 2. Instrumentation Specifications Per Input or Validation Variable for the Three Measurement Sites

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A. Datasets

1. Haarweg: GC

The weather station at the Haarweg (http://www.met.wau.nl/haarwegdata) in Wageningen [The Netherlands, 7 m above sea level (a.s.l.)], an agrometeorological station provided all four radiation components and the soil heat flux (7.5 cm below ground level). Additionally, in 2005, fluxes of sensible and latent heat were measured using an eddy-covariance (EC) station [3.2 m above ground level (a.g.l.)]. Over the year, the grassland was mowed frequently to keep the vegetation height ($z_{\mathrm{c}}$) $\approx 10\mathrm{cm}$. The soil at the Haarweg site contains clay and is rich in organic matter. We used data from the period of April 1 to September 30, 2005.

2. Merken: WC

Surface energy components were measured during the FLUXPAT campaign near Merken (Germany, 114 m a.s.l.) in the summer of 2009. This campaign was organized to study the soil-vegetation-atmosphere system (see, e.g., [36] and [37] for details). A station with an EC system (see Table 2 for details) at 2.4 m a.g.l. was installed in the middle of a flat winter wheat field. The four radiation components were measured at 1.7 m a.g.l., and the soil heat flux at 7.5 cm depth. The site contains a bouldered silt-loam soil. From June 12 on, the wheat height remained 0.85 m, which we specified as the start of the ripening process. Data from April 15 to June 12 (DOY 104–163) were considered as the growing phase of the wheat, and data from June 13 to July 26 (DOY 164–207) as the ripening phase. Canopy height and LAI measurements were performed biweekly, such that we could derive a linear growth and LAI increase for the growing phase (see Table 3).

Table 3. Land-Surface Input Variables per Dataset^d

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3. Lannemezan: GS

The BLLAST (Boundary Layer Late Afternoon and Sunset Turbulence) campaign took place near the Pyrenees in southern France (582 m a.s.l.) from June 14 to July 8, 2011 [38]. The main objective of this campaign was to better understand the physical processes that control and follow from the transition of a convective boundary layer towards a stratified nocturnal boundary layer. Surface and boundary-layer measurements were performed around Lannemezan, a village located on a large plateau ($200{\mathrm{km}}^2$) with several mainly agricultural land use types on a sandy loamy soil. An EC station was installed in a grass field, equipped with sensors for the main surface energy components: sensible and latent heat flux (2.55 m a.g.l.), incoming and outgoing shortwave and long-wave radiation (1.68 m a.g.l.), and soil heat flux (3 cm below ground level); see Table 2 for details. The grass was cut before the installation, and grew throughout the campaign, reaching 35 cm at the end of the campaign (see Table 3; we took $z_{\mathrm{c}}=0.25\mathrm{m}$).

B. Data Processing and Selection

For the input of our scheme, we averaged 1-min values of $p$ and $S_{\mathrm{in}}$ to 30-min data. Half-hourly wind speed, temperature, and humidity values were obtained from the eddy covariance data (as described below).

The long-wave radiation components and soil heat flux values were averaged for validation from 1-min to 30-min data as well. A storage term was added to the soil heat flux using the calorimetric method [39], applying a volumetric soil heat capacity of $2.2\times {10}^6\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$ for all three sites (assuming that the contributions of sand, clay, organic matter, water, and air are similar at the sites).

Eddy covariance data has been processed to provide both mean variables (wind speed, temperature, and humidity) needed to drive the model, and fluxes and other turbulent quantities (see Table 2) that are used to validate the model results. For the validation values, we determined averages of atmospheric stability, friction velocity, and sensible and latent heat fluxes from the 20 Hz EC data (10 Hz in the case of GC) using the software EC-pack-2.5.23-1.3 [40] for every measurement site. The averaging time was 30 min, which adequately captures surface-layer turbulence and excludes mesoscale processes. For the heat fluxes, we performed a planar fit rotation [41] where the rotation angles were determined over a period of 7 days to adjust the coordinate system. Linear trends were removed and the Webb correction [42] was applied to correct humidity fluctuations for density fluctuations induced by temperature fluctuations. The sonic temperature (used as input) was corrected for humidity effects using the Schotanus correction [43]. 95% confidence intervals were estimated by quantifying the sampling error for each scalar average and flux, following [40], used for the data selection. We used fixed upper and lower plausible limits for the radiation and flux components. Because the sum of observed $H$ and $L_{\mathrm{v}}E$ is generally lower than the observed available energy, we adjusted the sensible and latent heat fluxes to the total available energy ($Q_{\mathrm{n}}-G$), while maintaining the value of the Bowen ratio ($H/L_{\mathrm{v}}E$) following [44].

We also used the EC-pack software to obtain validation values for the structure parameters of $T_{\mathrm{a}}$, $q$, and their joint structure parameter based on the structure function. We applied path-averaging factors to correct the structure parameters that are derived from measurements that originate from a finite path instead of one point. The correction for $C_{T^2}$ related to a measurement height of 2.5 m, a wind speed of $2\mathrm{m}{\mathrm{s}}^{-1}$, the path length in the sonic anemometer of 0.12 m, and a lag of 0.9 m is a factor of 1.12 following the Kaimal spectrum. For $C_{q^2}$, this factor is 1.13 (with a path length in the gas analyzer of 0.13 m), and for $C_{Tq}$ this factor is 1.49, which is higher because it additionally depends on the distance between the anemometer and the gas analyzer (0.15 m).

Furthermore, we applied a correction to the structure parameters obtained by EC-pack {Eq. (5) in [45]} to improve the conversion from time to space (for a nonconstant wind speed) that is needed for the structure function following [46]. They derived this correction factor for the structure parameters from Wyngaard and Clifford [47], who only determined it for $({\sigma }_U^2/U^2)<0.1$. If our data exceeded this limit, we applied a correction of $95/90$ [Eq. (5) in [45] for $({\sigma }_U^2/U^2)=0.1$]. To obtain $C_{n^2}$ from the calculated structure parameters, we used the expressions of [15] described in Section 2.C. We calculated $C_{n^2}$ for an optical wavelength ($\lambda =670\mathrm{nm}$) and for a millimeter wavelength ($\lambda =2\mathrm{mm}$).

We selected daytime data between 7 and 17 UTC and eliminated stable situations by excluding data with $z/L>-0.02$. Furthermore, we excluded data when less than 35,980 raw data points were counted per half hour of 20 Hz data (17,990 in the case of GC). We also excluded rainy days from the data, because wet sonic anemometers and gas analyzers introduce errors in our variables. Furthermore, turbulence in the lowest part of the atmosphere is very weak on rainy days. For GC, we removed 36 days such that 148 days remained. We also excluded rainy days for WC, such that 44 days were left for the growing period and 24 days for the ripening period. For the grass data in Southern France, we excluded 6 days (DOY 167, 169, 173, 174, 180, and 185) at which rain events were observed (17 days remained).

4. Methods

Here, we first describe how we use the scheme with our data, and then how we validate the scheme with other data and derivations.

From the 30-min daytime data of $T_{\mathrm{a}}$, $q$, $p$, $u$, and $S_{\mathrm{in}}$, we calculated the sensible, latent, and soil heat fluxes, and the radiation components from the first step in our scheme as discussed in Section 2.A, following the parameterizations given in Appendix A. The required land surface characteristics that differ per measurement site are given in Table 3. Some constants are given in Appendix B.

From the estimated values for $z/L$, $H$, $L_{\mathrm{v}}E$, and $u_{*}$ that follow from the first part of the scheme, we calculated the structure parameters $C_{T^2}$, $C_{q^2}$, and $C_{Tq}$ following the theory described in Section 2.B, applying a fixed $r_{Tq}$ of 0.8. We derived $C_{n^2}$ for $\lambda =670\mathrm{nm}$ and $\lambda =2\mathrm{mm}$ via Eq. (8).

We validated our estimates of the radiation components, fluxes, and structure parameters with the values directly obtained by radiation and EC measurements. We also validate the key parameters and variables of the first part of our scheme: $T_{\mathrm{s}}$, $r_{\mathrm{a}}$, and $r_{\mathrm{c}}$. However, these are not directly measured. To compare the estimated surface temperature $T_{\mathrm{s}}$ with measurements, we calculated the measured $T_{\mathrm{s}}$ using

We calculated regression fits per variable for the four datasets, using orthogonal distance regression. The offset was set to zero, and the slope was calculated, including its standard deviation. We normalized the root mean squared error (RMSE) by dividing by the mean observed value of the specific variable to obtain the normalized RMSE (NRMSE) to compare regressions of different variables. Because the derivation of the measured resistances is sensitive for low fluxes and low available energy, we additionally eliminated data for the regression fit for $r_{\mathrm{c}}$ and $r_{\mathrm{a}}$ when $H$, $L_{\mathrm{v}}E$, or $Q_{\mathrm{n}}-G<20\mathrm{W}{\mathrm{m}}^{-2}$.

We tested the agreement of the datasets with Monin–Obukhov similarity theory by calculating the parameters in the similarity functions for the structure parameters as follows: we kept the second parameter $c_2$ in Eqs. (7) and (8) constant (14.9 for $T$, 4.5 for $q$), and fitted the data to a regression with the prescribed shape of $y=c_1{(1-c_{2}x)}^{-2/3}$. The structure parameter and stability data were weighted using the inverse of the confidence intervals for $C_{T^2}$, $C_{q^2}$, and $z/L$ (via the intervals of $u_{*}$, $T_{\mathrm{a}}$, $q$, $\overline{w^{\prime }{T}^{\prime }}$, and $\overline{w^{\prime }{q}^{\prime }}$) that were calculated by EC-pack.

5. Results

Here, we discuss the results in the same steps and order as we described our scheme; first the estimation of fluxes from weather data (Section 5.A), then the estimation of temperature and humidity structure parameters from fluxes, and finally the estimation of $C_{n^2}$ (Section 5.B).

A. Fluxes

The difference between the surface temperature $T_{\mathrm{s}}$ and air temperature $T_{\mathrm{a}}$ is very important for the calculation of the energy that is available for $H$ and $L_{\mathrm{v}}E$. This difference, however, is overestimated for all four datasets [see the regression fits and errors for all datasets in Tables 4 and 5, and for GC in Fig. 3(c)]. Despite the overestimation of this difference, and thus of $T_{\mathrm{s}}$, the long-wave radiation components are estimated fairly well for all datasets. Thereby, $Q_{\mathrm{n}}$ estimates fit the observations with a general underestimation of 2%–8% [see also Fig. 3(f) for GC]. For GS, this is caused by an underestimated incoming long-wave radiation. The surface reflectance of shortwave radiation (albedo), however, is underestimated for all sites (not shown). $G$ is underestimated for all datasets. However, its value is very low compared to $Q_{\mathrm{n}}$ such that this underestimation does not produce large errors for the energy that is available for $H$ and $L_{\mathrm{v}}E$.

Table 4. Regression Results of Estimated Versus Observed Fluxes, Structure Parameters, and Variables at the Haarweg (GC left) and in Lannemezan (GS right)^a

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Table 5. Regression Results of Estimated Versus Observed Fluxes, Structure Parameters, and Variables During the Growing Season (Left) and Ripening Season (Right) in Merken (WC)^a

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Fig. 3. Validation of estimates (model) of $r_{\mathrm{a}}$, $r_{\mathrm{c}}$, $T_{\mathrm{s}}$, $H$, $L_{\mathrm{v}}E$, and $Q_{\mathrm{n}}-G$ with measurements (obs) at a grass field in The Netherlands (GC) in 2005. Dark markers indicate clear-sky conditions, whereas light markers indicate very cloudy conditions.

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$r_{\mathrm{c}}$ estimates are much lower than the values we derived from observations using the inverted Penman–Monteith approach [Eq. (11)] for all datasets. However, we did not validate our estimated $r_{\mathrm{c}}$ with direct observations, and thus the presented underestimation of $r_{\mathrm{c}}$ is uncertain. The NRMSE for $r_{\mathrm{c}}$ was still quite high despite the extra data selection. Nevertheless, the partitioning of the available energy into $H$ and $L_{\mathrm{v}}E$ is appropriate for GC and GS. This indicates that $r_{\mathrm{c}}$ is of the correct order of magnitude. For WC, $H$ is generally overestimated and $L_{\mathrm{v}}E$ is slightly underestimated. However, the estimations of all variables are quite scattered for the datasets (see the standard deviations and NRMSE values given in the first columns in Table 5). The WC grow dataset is by far the smallest we used, partly because of the stability criterion. $r_{\mathrm{a}}$ estimates are quite scattered; however, most of the data used here is in a $r_{\mathrm{c}}$ regime in which the fluxes are not very sensitive to $r_{\mathrm{a}}$, see [48].

B. Structure Parameters

Although $H$ and $L_{\mathrm{v}}E$ are predicted well for GC, $C_{T^2}$ and $C_{q^2}$, and thereby $C_{Tq}$, are overestimated. For the growing wheat (WC), $H$ is largely overestimated, which leads to a clear overestimation of $C_{T^2}$ as expected from the relation between $H^2$ (or $T_{*}^2$) and $C_{T^2}$ given in Eq. (7a). For GS, the regression slope for $L_{\mathrm{v}}E$ is satisfying, whereas the slope for $C_{q^2}$ is more than twice as large. This is not expected from the relation between the slopes of $L_{\mathrm{v}}{E}^2$ and $C_{q^2}$ as mentioned before. We think that this is partly related to the Monin–Obukhov parameters used for this relation. To check this hypothesis, we determined the similarity relationships [Eqs. (7a) and (7b)] where we kept $c_2$ to its literature values of 14.9 for temperature and 4.5 for humidity.

The obtained values for $c_1$ deviate the most from values given by [30] for the GS data, for both temperature and humidity (5.2 and 2.5 respectively, instead of 6.7 and 3.5). The coefficients for GC agree better with [30], with 6.7 and 3.9. For WC, we found 4.7 and 2.8 for the growing phase of the wheat (WC), and 4.7 and 2.6 for the ripening phase (WC). If we use the newly obtained coefficients for GS (5.2 and 2.5 for temperature and humidity, respectively), we obtain improved regression slopes (indicated with MO in Table 4). However, $C_{q^2}$ is still overestimated (albeit with a large NRMSE) while the estimation of $L_{\mathrm{v}}E$ is close to perfect with a very low NRMSE. The large NRMSE for $C_{q^2}$ is probably caused by the fact that at GS, humidity fluctuations were smaller than at GC and WC, and therefore more sensitive to instrumental errors.

The estimates of $C_{T^2}$, $C_{q^2}$, and $C_{Tq}$ lead to reasonable estimates of $C_{n^2}$ (see Tables 4 and 5 and Figs. 4 and 5). Note that the structure parameters occur in a large range (mind the logarithmic scale in the figures, and hence a slope of 2 should be interpreted as an error of much less than an order of magnitude). For GS, we see the overestimation of $C_{q^2}$ again in the estimate for $C_{n^2}$ for $\lambda =2\mathrm{mm}$ because, at this wavelength, scintillations are more sensitive to humidity fluctuations than at nanometer wavelengths. For the growing season in Merken, we find the opposite. The large uncertainty of $C_{T^2}$ for the data of both the growing and ripening phase leads to an overestimation of $C_{n^2}$ for $\lambda =670\mathrm{nm}$, a wavelength at which $C_{n^2}$ is more sensitive to temperature fluctuations.

 

Fig. 4. Validation of estimates (model) of the structure parameters of $n$ for (a) an optical and (b) a millimeter wavelength, with derivations from measurements (obs) at a grass field in The Netherlands in 2005. Dark markers indicate clear-sky conditions, whereas light markers indicate very cloudy conditions.

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Fig. 5. Validation of estimates (model) of the structure parameters of $n$ for an optical [(a) and (c)] and a millimeter wavelength [(b) and (d)], with derivations from measurements (obs). Above, a grass field in southern France in 2011 for adjusted Monin–Obukhov parameters. Below, a wheat field in the ripening phase in western Germany in 2009. Dark markers indicate clear-sky conditions, whereas light markers indicate very cloudy conditions.

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As an example of the possible application of our scheme, we show in Fig. 6 the performance of our scheme for a sunny and cloudy day at the Haarweg (GC). The scheme estimates the turbulence at the cloudy day very well, and for the clear sky day, there is a slight overestimation. It can be seen that for the optical wavelength, scintillation values are highest around noon, decrease until sunset, and increase again towards midnight. On the other hand, for the millimeter wavelength, the scintillations decrease to zero in the night. Cloudy conditions weaken the scintillations. For these cases, our method performs better for $\lambda =2\mathrm{mm}$ than for $\lambda =670\mathrm{nm}$, because humidity fluctuations are larger and better defined.

 

Fig. 6. $C_{n^2}$ measured (markers) and estimated (lines) for a clear (circles and dashed) and cloudy (asterisks) summer day (June 19, 12 UTC, 2005) at the Haarweg (similar to Fig. 1).

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6. Discussion

Here, we compare our method with previous methods, and discuss the sensitivity of an existing method that is often used. Furthermore, complementary to the validation of our method presented here, we show the sensitivity of our method.

A. Sensitivity of Previous Methods

Existing methods for determining $C_{n^2}$ that are based on single-level data only were developed for daytime and nighttime data, for desert and mountain areas. Probably, that is because one important application that suffers from poor electromagnetic wave propagation is the ground-based telescope. We tested the algorithm that is often used, which is presented and compared with other empirical algorithms in [11], Eq. (6).

As mentioned in the introduction, other methods for the determination of $C_{n^2}$ from single-level meteorological data are available in the literature. These methods are generally less complex in construction, being based on nonlinear regressions to $T_{\mathrm{a}}$, $u$, relative humidity (RH), and occasionally $S_{\mathrm{in}}$. The methods are ignorant of the properties of the underlying land, but tracing back their origin reveals that they were derived for desert and mountain areas, combining daytime and nighttime data. We have tried to assess the skill of these methods for our datasets but discovered a number of issues. First, the methods are only applicable to optical wavelengths (exact wavelength unspecified) and for a single height (not clearly specified, but turns out to be 15 m). But above all, for a large part of our data, the methods give negative $C_{n^2}$ values. Here, we will more closely look at a method developed by Bendersky et al. [10], based on work by Sadot and Kopeika [8]. This method is presented in [49] as Eq. (15.4.34) {also tested in [11], Eq. (6)}. In Fig. 7, we show the sensitivity of that method to RH and $T_{\mathrm{a}}$ (the sensitivity to $u$ and $S_{\mathrm{in}}$ is much weaker).

 

Fig. 7. Sensitivity of derivations of $C_{n^2}$ using the algorithm given in [49], to RH and $T_{\mathrm{a}}$. Diamond markers indicate the 25th and 75th percentile of RH and $T_{\mathrm{a}}$ for the grass data of the Haarweg in 2005.

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For a dry and warm atmosphere, the method results in positive $C_{n^2}$ values that are of the order of magnitude of our measurements. However, for humid or colder areas, the method results in a negative value, which is physically impossible. We used the method with RH and $T_{\mathrm{a}}$ taken from our GC data: 25th and 75th percentiles, represented by the diamond markers in Fig. 7. These values lead to a value just around the zero line, indicated in the figure. The 25th percentile of temperature combined with the 75th percentile of humidity lead to a negative value (the upper left diamond). This shows that the method does not work for other regions than desert-like areas.

Therefore, we recommend our method over previous methods, especially when using optical or microwave systems above other land-use types than a desert or mountain peak. Our method, however, is not yet suitable for nighttime data. It is beyond the scope of this work, but for nighttime applications, we suggest replacing the cloud-fraction estimate [Eq. (A2c)] or the long-wave radiation estimates by parts of the semiempirical nocturnal surface fluxes scheme by [50]. Furthermore, nighttime data often corresponds to stable conditions, for which the Monin–Obukhov equations described in Section 2.B should be replaced by functions that are valid for a stable atmosphere. Once such a nighttime scheme has been implemented, it would be good to additionally test the model for desert and mountain regions, and for high elevations.

B. Sensitivity of Our Method

We developed a method to estimate $C_{n^2}$ from single-level weather data that consists of three steps:

For step (a), we applied an existing scheme of dRH99 that was developed for grass in The Netherlands (Cabauw). We adjusted the scheme for other field conditions, and tested it for another grass field in The Netherlands, a grass field in southern France, and a wheat field in western Germany. This scheme is based on the Penman–Monteith equation, which is an appropriate description of the process of transpiration. But this only holds provided that it is used with observations from the location of interest; a different surface results in different temperature, humidity, and wind observations.

In our scheme, we replaced the water vapor deficit dependence of $r_{\mathrm{c}}$, described by dRH99 as $r_{\mathrm{c}}={10}^4\mathrm{\Delta }q$, with the approach of BB97. In Fig. 8, we show the performance of the scheme for $L_{\mathrm{v}}E$ and $r_{\mathrm{c}}$ if we use the water vapor deficit dependence described by dRH99. When comparing these results with the results presented in Figs. 3(b) and 3(e), we see that the estimation of the latent heat flux deteriorates when using the original dRH99 formulation, especially for dry conditions. This result agrees with Fig. 2, in which we see that the approach of dRH99 differs from the other approaches mainly for dry air conditions.

 

Fig. 8. $r_{\mathrm{c}}$ and $L_{\mathrm{v}}E$ estimates (model) versus measurements (obs) for the Haarweg (GC) using the approach for $r_{\mathrm{c}}$ from dRH99, colored by $\mathrm{\Delta }q$ (light is 0, dark is $10\mathrm{g}{\mathrm{kg}}^{-1}$).

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To investigate which parameters have the largest influence on the derived $C_{n^2}$, we performed a sensitivity analysis. In Table 6, we list the parameters of step (a), and show for one midday summer situation the relative increase of $C_{n^2}$ for a realistic increase of the parameters. We see that the dependence on the parameters in $f_{\mathrm{dq}}$ are very strong. Furthermore, our scheme is very sensitive to the estimated LAI, canopy height, and albedo. This indicates that for the first step of our method, a good estimate of the vegetation characteristics is the most crucial part. The increase of the parameters that lead to an increase or decrease of $C_{n^2}$ for both wavelengths (e.g., $N$, $a$, and $A_g$) are related to the available energy. If a certain increment leads to a decrease of $C_{n^2}$ for one wavelength and an increase for the other wavelength, then the parameter is mainly related to the partitioning of energy between latent and sensible heat (e.g., LAI, $r_{s,\min }$, and ${\mathrm{f}}_r$).

Table 6. Sensitivity Estimates of $C_{n^2}$ to the Main Input Parameters and Constants (or Parameters Directly Derived from $S_{\mathrm{i}\mathrm{n}}$) for June 19, 2005, 12 UTC at the Haarweg^a

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Next to the values given for the TESSEL land surface scheme, we tested our method with vegetation parameters from the Noah land surface model (not shown). However, the flux and structure parameter validation results are much worse using values from Noah. It seems that the definition of LAI and $r_{s,\min }$ cannot be separated from their appearance in the land surface schemes. We did not implement the dependence of $r_{\mathrm{c}}$ on the type of carbon fixation the vegetation uses (C3 or C4). Adding such a plant-physiological process requires a clear separation between transpiration and evaporation processes. If $C_{n^2}$ estimates are needed for partly wet vegetated surfaces, our method is empirically solid. However, we actually need a more realistic treatment to capture interception and soil evaporation. For that, a soil vegetation atmosphere could be used.

In step (b), the uncertainties in the scheme are the validity of Monin–Obukhov similarity theory, and the value for $r_{Tq}$. Furthermore, the derivations of structure parameters using EC-pack might introduce validation errors. Therefore, we also tested our scheme while bypassing step (a): we determined structure parameters from observed fluxes (scaled as $T_{*}$ and $q_{*}$) rather than modeled fluxes. In Fig. 9, we show that if measured fluxes are used, an almost $1:1$ fit is visible (for both wavelengths), with some scatter for the lowest values. This means that even if the flux estimation in the first step would be perfect, $C_{n^2}$ results will still be scattered.

 

Fig. 9. Validation of derivations of $C_{n^2}$ from measured values for $C_{T^2}$, $C_{q^2}$, $C_{Tq}$, and $r_{Tq}$, with derivations from flux measurements ($T_{*}$ and $q_{*}$) at a grass field in The Netherlands in 2005. Dark markers indicate an optical wavelength; light markers indicate a millimeter wavelength.

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Values for $C_{n^2}$ calculated from measured fluxes using Monin–Obukhov similarity theory are somewhat smaller than $C_{n^2}$ values derived from the observed structure parameters of $T_{\mathrm{a}}$ and $q$ (see the slopes of the regression in the legend). This holds for both wavelengths for this data (GC), which indicates that here, the Monin–Obukhov similarity functions for both temperature and humidity should have a slightly larger $c_1$ or smaller $c_2$ [see Eqs. (7a) and (7b)]. In the previous section we wrote that, indeed, we found a slightly larger value for $c_1$ for $q$ for the GC data.

We found that for the GS dataset, we should certainly change the coefficients in the Monin–Obukhov relations. In the previous section, we calculated $c_1$ that fitted the data with a $c_2$ that was kept constant. However, we would like to point out that if we fit the GS data to the function without keeping $c_2$, we obtain for temperature $c_1=4.2$ and $c_2=10.8$, and for humidity $c_1=2.5$ and $c_2=4.7$. These values are very unusual compared to reported values for several datasets [54], especially for humidity. However, more data in the neutral range would be needed for a good fit of $c_2$.

Considering a fixed value for $r_{Tq}$, we cannot omit this assumption when only weather data is available, because $r_{Tq}$ cannot be derived from low-frequency temperature and humidity data. One would expect a value close to 1 because temperature and humidity are assumed to behave similar during daytime. We found in our datasets that the daytime values differ, with a median of $\approx 0.6$ taking all scales together, which indicates that our datasets suffered from nonlocal effects. However, this does not necessarily affect the correlation in the inertial subrange. Therefore, the introduced error in the calculation of $C_{Tq}$ (and therefore also in $C_{n^2}$) by using a fixed $r_{Tq}$ of 0.8 is unknown.

Step (c) is the most straightforward part of our method, because that part is based on established physics theories. Only the dependence on the wavelength of interest influences the performance of our method. For the tested datasets, our method generally performs better for $\lambda =2\mathrm{mm}$ than for $\lambda =670\mathrm{nm}$, due to a better-defined humidity flux (microwaves are more sensitive to humidity fluctuations than to temperature fluctuations). Note that a more elaborated $C_{n^2}$ scheme would also include the scattering effect of aerosols on the wave propagation equation and thus on $C_{n^2}$.

7. Conclusions

A good estimate of the optical turbulence provides knowledge of the performance of communication and imaging systems. Often, weather data is available at only one level, precluding the use of methods based on vertical gradients [55]. Previous methods are rather empirical, and thereby only applicable to a specific environment. Therefore, we present an approach to estimate the structure parameter of the refractive index of air ($C_{n^2}$) based on single-level weather station data. Our estimates of $C_{n^2}$ are accurate enough to help in the development of systems based on the propagation of electromagnetic waves (e.g., radiowave communication and ground-based telescopy). Based on a long-time series of weather station data, the climatology of $C_{n^2}$ for arbitrary wavelengths could be determined. For existing weather and imaging data, our estimates provide information on the quality of the images.

For a grass field in The Netherlands, the estimated values for $C_{n^2}$ are in agreement with values derived directly from eddy covariance measurements. The results of wheat in western Germany contained more scatter than validation results from the Dutch grass, especially for the optical wavelength. The scheme is most sensitive to the first part of the scheme, the estimation of surface fluxes from single-level weather data. The parametrization of the canopy resistance is especially important, which differs per vegetation type and growing state. It determines the partitioning of energy between sensible and latent heat fluxes.

However, the scatter introduced by flux estimation uncertainties is only slightly larger than the scatter that is obtained when calculating $C_{n^2}$ from observed surface fluxes. For the second step, in which the structure parameters of temperature and humidity are estimated from surface fluxes, validity of Monin–Obukhov similarity theory is of importance. For the grass field in southern France, we had to fit the Monin–Obukhov parameters to the data to obtain good validation results. In the third step, $C_{n^2}$ is calculated from the structure parameters for one specific wavelength. For our grassland datasets, estimated values for $C_{n^2}$ agree slightly better with eddy-covariance data for the millimeter wavelength than for the optical wavelength. The code for this application (MATLAB) can be obtained from the second author.

Appendix A: Scheme to Estimate Fluxes from Weather Data

The scheme to compute the surface fluxes consists of four blocks. In the first block, all variables that can be determined directly from the meteorological input data are computed [Eqs. (A1a)–(A2g)]. The second block defines the dependence of the surface energy balance on the surface temperature [Eqs. (A3a)–(A3i)]. The third block describes the iteration that is needed to deal with the stability dependence (and hence surface-flux dependence) of the aerodynamic resistance [Eqs. (A4a)–(A4e)]. The final block describes how, after convergence of the iteration, the various energy fluxes can be determined [Eqs. (A5a)–(A5e)].

First, our scheme that is adapted from dRH99 directly estimates air density $\rho$, air heat capacity $c_p(=1004(1+0.84q))$, water vapor pressure $e_{\mathrm{a}}$, and saturated water vapor pressure $e_{\mathrm{sat}}$ (both in Pa) from air temperature ($T_{\mathrm{a}}$), specific humidity ($q$), and pressure ($p$). The slope of the saturated vapor pressure curve is $s(T)=\mathit{d}{e}_{\mathrm{sat}}/dT$, and the psychrometric constant is $\gamma =c_{p}p{R}_{\mathrm{v}}/(L_{\mathrm{v}}{R}_{\mathrm{d}})$, where $R_{\mathrm{d}}$ is the specific gas constant for dry air ($287{\mathrm{J}\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$) and $R_{\mathrm{v}}$ is the specific gas constant for water vapor ($462{\mathrm{J}\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$). The temperature dependence of $e_{\mathrm{sat}}$, $s$, and $L_{\mathrm{v}}$ can be found in [56], page 355.

After that, the surface resistance can be calculated following

(A1a)$$r_{\mathrm{c}}={\mathrm{f}}_r\frac{r_{\mathrm{s},\min }}{\mathrm{LAI}}{f}_{\mathrm{dq}}{f}_{\mathrm{rad}},$$

where the scaling factor ${\mathrm{f}}_r=0.47$, and $f_{\mathrm{dq}}$ is calculated following BB97:

(A1b)$$f_{\mathrm{dq}}=1+h_s(\mathrm{\Delta }q-\mathrm{\Delta }{q}_0),$$

in which $\mathrm{\Delta }q$ is the specific humidity deficit in $\mathrm{kg}{\mathrm{kg}}^{-1}$ (calculated as $q_{\mathrm{sat}}-q$), $h_s=160{\mathrm{kg}}^{-1}\mathrm{kg}$, $\mathrm{\Delta }{q}_0=0.003\mathrm{kg}{\mathrm{kg}}^{-1}$, and where

(A1c)$$f_{\mathrm{rad}}=\frac{1000S_{\mathrm{in}}+230(1000-2S_{\mathrm{in}})}{S_{\mathrm{in}}(1000-230)}.$$

Then, most of the radiation components can be calculated. Outgoing shortwave radiation ($S_{\mathrm{out}}$) is calculated following

(A2a)$$S_{\mathrm{out}}=a{S}_{\mathrm{in}},$$

where $a$ is the surface albedo that depends on the solar elevation angle $\alpha$ and the effective cloud cover fraction $N$ as

(A2b)$$\begin{gathered} a=a_{\max }-(1-N)\sin (\alpha )(a_{\max }-a_{\min }) \\ -N(a_{\max }-a_{\text{cloud}}). \end{gathered}$$

$a_{\max }$, $a_{\min }$, and $a_{\text{cloud}}$ are the respective surface albedos for a minimum solar elevation and a clear sky (the highest $a$), a maximum solar elevation and a clear sky (the lowest $a$), and a very cloudy sky (see Table 7 for the constants). We calculated $N$ as

(A2c)$$N=\frac{\frac{S_{\mathrm{in},0}-S_{\mathrm{in}}}{S_{\mathrm{in},0}}-0.2}{0.8},$$

where $S_{\mathrm{in},0}$ is the solar constant $I_0$ ($\approx 1365{\mathrm{W}}^2{\mathrm{m}}^{-2}$) multiplied with the cosine of the zenith angle ${\theta }_z$ of the specific location and time and corrected for the orbital eccentricity via the difference in distance ($d$) to the sun,

(A2d)$$S_{\mathrm{in},0}={\overline{I}}_0{\left(\frac{{\overline{d}}_{\mathrm{sun}}}{d_{\mathrm{sun}}}\right)}^2\cos {\theta }_z.$$

Incoming long-wave ($L$) radiation is calculated as

(A2e)$$L_{\mathrm{in}}={ϵ}_{\mathrm{a}}\sigma T_{\mathrm{a}}^4,$$

where ${ϵ}_{\mathrm{a}}$ is the air emissivity and $\sigma$ is the Stefan–Boltzmann constant ($5.67\times {10}^{-8}$). We used the following expressions for the emissivity of air, adapted from [25]:

(A2f)$${ϵ}_{a,\text{clear}}=0.63+5.95\times {10}^{-7}{e}_{\mathrm{a}}{e}^{1500/T_{\mathrm{a}}},$$

and [26]

(A2g)$${ϵ}_{\mathrm{a}}=N+(1-N){ϵ}_{\mathrm{a},\text{clear}}.$$

Second, $H$ is calculated from the energy balance equation, which we can write as

(A3a)$$\begin{gathered} H=Q_{n,0}+Q_{n,T}(T_{\mathrm{s}}-T_{\mathrm{a}})-G_0-G_T(T_{\mathrm{s}}-T_{\mathrm{a}}) \\ -L_{\mathrm{v}}{E}_0-L_{\mathrm{v}}{E}_T(T_{\mathrm{s}}-T_{\mathrm{a}}), \end{gathered}$$

where all energy terms consist of an isothermal term that is calculated using $T_{\mathrm{a}}$ instead of $T_{\mathrm{s}}$ (indicated with the subscript 0), and a correction term that corrects for that assumption of an isothermal atmosphere. This correction term therefore depends on $(T_{\mathrm{s}}-T_{\mathrm{a}})$ (indicated with the subscript $T$). Thereby, the correction terms depend on $H$, via

(A3b)$$T_{\mathrm{s}}-T_{\mathrm{a}}=\frac{H{r}_{\mathrm{a}}}{\rho c_p}.$$

Because $r_{\mathrm{a}}$ depends on the atmospheric stability, which is unknown, an iteration loop is needed to calculate $H$. Inserting the $H$-dependent correction terms in Eq. (A3a) results in

(A3c)$$H=\frac{Q_{n,0}-G_0-L_{\mathrm{v}}{E}_0}{1+Q_{n,T}+G_T+L_{\mathrm{v}}{E}_T},$$

where the isothermal terms are calculated following

(A3d)$$Q_{n,0}=(1-a)S_{\mathrm{in}}+({ϵ}_{\mathrm{a}}-1){ϵ}_{\mathrm{e}}\sigma T_{\mathrm{a}}^4,$$

(A3e)$$G_0=Ag(T_{\mathrm{a}}-T_{24\mathrm{h}}).$$

Here, $A_g$ is the soil heat transfer coefficient (see Table 7), and $T_{24\mathrm{h}}$ is the temperature history of the last 24 h that represents the deep-soil temperature, and

(A3f)$$L_{\mathrm{v}}{E}_0=\frac{\rho c_p}{\gamma }\frac{e_{\mathrm{sat}}(T_a)-e}{r_{\mathrm{a}}+r_{\mathrm{c}}}.$$

Note that $L_{\mathrm{v}}{E}_0$ depends on $r_{\mathrm{a}}$ too, which means that the calculation of $L_{\mathrm{v}}{E}_0$ is part of the iteration loop that is used for the calculation of $H$.

The correction terms are all part of the iteration loop, because via $H$, we introduced an $r_{\mathrm{a}}$ dependence. The iteration loop for the energy balance components [Eqs. (A3c)–(A4e)] is initiated with neutral conditions, and is stopped when the sensible heat flux converged within $1{\mathrm{Wm}}^{-2}$ (three iterations are usually enough):

(A3g)$$Q_{n,T}=\frac{r_{\mathrm{a}}}{\rho c_p}{ϵ}_{\mathrm{s}}\sigma 4T_{\mathrm{a}}^3,$$

where ${ϵ}_{\mathrm{s}}$ is the surface emissivity (see Table 7),

(A3h)$$G_T=\frac{r_{\mathrm{a}}}{\rho c_p}{A}_g,$$

and

(A3i)$$L_{\mathrm{v}}{E}_T=\frac{r_{\mathrm{a}}}{\gamma }\frac{s(T_{\mathrm{a}})}{r_{\mathrm{a}}+r_{\mathrm{c}}}.$$

The loop starts with neutral conditions (${\mathrm{\Psi }}_T={\mathrm{\Psi }}_u=0$) for a first estimate of the atmospheric-stability-dependent $r_{\mathrm{a}}$ and $u_{*}$. The aerodynamic resistance is calculated as

(A4a)$$r_{\mathrm{a}}=\frac{1}{\kappa u_{*}}(\ln \frac{z}{z_{0h}}-{\mathrm{\Psi }}_T\left(\frac{z}{L}\right)+{\mathrm{\Psi }}_T\left(\frac{z_{0h}}{L}\right)).$$

Here, $\kappa$ is the Von Kàrmàn constant (0.4), $z$ is the height ($z=z_{\mathrm{m}}-d$, $d=\frac{2}{3}{z}_{\mathrm{c}}$ is the displacement height), and $z_{0h}$ is the roughness length for heat, calculated as $0.1z_{0m}$ [$z_{0m}=0.4(z_{\mathrm{c}}-d)$, the roughness length for momentum]. ${\mathrm{\Psi }}_T$ is the Businger–Dyer integrated Monin–Obukhov flux profile relation function for temperature gradients as described in [57]. $L$ is the Obukhov length calculated following Eq. (A4c), and the friction velocity $u_{*}$ is calculated as

(A4b)$$u_{*}=\frac{u\kappa }{\ln \left(\frac{z}{z_{0m}}\right)-{\mathrm{\Psi }}_u\left(\frac{z}{L}\right)+{\mathrm{\Psi }}_u\left(\frac{z_{0m}}{L}\right)},$$

where $u$ is the wind speed and ${\mathrm{\Psi }}_u$ is the Businger–Dyer integrated Monin–Obukhov flux profile relation function for wind speed gradients. The Obukhov length $L$ is calculated within the loop as

(A4c)$$L=\frac{T_{\mathrm{a}}{u}_{*}^2}{\kappa g{T}_{\mathrm{v}*}},$$

where the temperature scale $T_{\mathrm{v}*}$ is a scaled buoyancy flux. The scalar scales are calculated as

(A4d)$$T_{\mathrm{v}*}=\frac{-H_{\mathrm{v}}}{c_p\rho u_{*}},T_{*}=\frac{-H}{c_p\rho u_{*}},q_{*}=\frac{-L_{\mathrm{v}}E}{L_{\mathrm{v}}\rho u_{*}},$$

where the buoyancy flux

(A4e)$$H_{\mathrm{v}}=H(1+0.61q)+0.61c_p{T}_{\mathrm{a}}\frac{L_{\mathrm{v}}{E}_0}{L_{\mathrm{v}}},$$

and $g$ is the acceleration due to gravity ($9.81\mathrm{m}{\mathrm{s}}^{-2}$).

After $H$ converges to one value, the following variables are extracted explicitly for diagnosis:

(A5a)$$T_{\mathrm{s}}=T_{\mathrm{a}}+\frac{H{r}_{\mathrm{a}}}{\rho c_p}+z{\mathrm{\Gamma }}_{\mathrm{d}},$$

where ${\mathrm{\Gamma }}_{\mathrm{d}}$ is the dry adiabatic lapse rate of $0.01{\mathrm{Km}}^{-1}$. If $T_{\mathrm{s}}$ is known, the long-wave outgoing radiation, net radiation, and soil and latent heat flux can be calculated following

(A5b)$$L_{\mathrm{out}}={ϵ}_{\mathrm{s}}\sigma T_{\mathrm{s}}^4+(1-{ϵ}_{\mathrm{s}})L_{\mathrm{in}},$$

(A5c)$$Q_{\mathrm{n}}=S_{\mathrm{in}}-S_{\mathrm{out}}+L_{\mathrm{in}}-L_{\mathrm{out}}=Q_0-{ϵ}_{\mathrm{s}}\sigma 4T_{\mathrm{a}}^3(T_{\mathrm{s}}-T_{\mathrm{a}}),$$

(A5d)$$G=-A_g(T_{24\mathrm{h}}-T_{\mathrm{s}})=G_0+A_g(T_{\mathrm{s}}-T_{\mathrm{a}}),$$

and

(A5e)$$\begin{gathered} L_{\mathrm{v}}E=\frac{s(Q_{\mathrm{n}}-G)+\frac{\rho c_p}{r_{\mathrm{a}}}(e_{\mathrm{sat}}-e_{\mathrm{a}})}{s+\gamma (1+\frac{r_{\mathrm{c}}}{r_{\mathrm{a}}})} \\ =L_{\mathrm{v}}{E}_0+\frac{\rho c_p}{\gamma }\frac{s(T_{\mathrm{a}})(T_{\mathrm{s}}-T_{\mathrm{a}})}{r_{\mathrm{a}}+r_{\mathrm{c}}}. \end{gathered}$$

Appendix B: Land Surface Input

Table 7. Land Surface Input Variables Used for All Sites^a

View Table | View all tables in this article

Appendix C: Coefficients Capturing the Wavelength Dependence of $C_{n^2}$

For $\lambda <1\mathrm{mm}$,

(C1a)$$b_1={10}^{-6}(\frac{0.237134+68.39397}{130-{\lambda }^{-2}}+\frac{0.45473}{38.9-{\lambda }^{-2}}),$$

(C1b)$$\begin{gathered} b_2={10}^{-6}(0.648731+5.8058\times {10}^{-3}{\lambda }^{-2} \\ -7.115\times {10}^{-5}{\lambda }^{-4}+8.851\times {10}^{-6}{\lambda }^{-6})-b_1, \end{gathered}$$

(C1c)$$b_{q2}=b_2,$$

and for $\lambda >1\mathrm{mm}$,

(C1d)$$b_1=0.776\times {10}^{-6},$$

(C1e)$$b_2={10}^{-6}(\frac{7500}{T_{\mathrm{a}}}-0.056),$$

(C1f)$$b_{q2}={10}^{-6}(\frac{3750}{T_{\mathrm{a}}}-0.056).$$

This work was financed by the DFG (Deutsche Forschungsgemeinschaft), projects SI606/26-1 and GR2687/4-1, based on project SCHU2350/2-1; links between local-scale measurements and catchment-scale measurements and modeling of gas-exchange processes over land surfaces. The Haarweg site was maintained by the Meteorology and Air Quality Group Wageningen, The Netherlands. Data from the EC station at the Merken site was provided by the DFG Transregional Collaborative Research Center 32, TR32, Germany. The BLLAST field experiment was hosted by the instrumented site of Centre de Recherches Atmosphériques, Lannemezan, France (Observatoire Midi-Pyrénées, Laboratoire d’Aérologie). We thank all people involved in these campaigns. Furthermore, we thank the anonymous reviewers for their positive feedback.

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