Abstract

Atmospheric scintillations cause difficulties for applications where an undistorted propagation of electromagnetic radiation is essential. These scintillations are related to turbulent fluctuations of temperature and humidity that are in turn related to surface heat fluxes. We developed an approach that quantifies these scintillations by estimating Cn2 from surface fluxes that are derived from single-level routine weather data. In contrast to previous methods that are biased to dry and warm air, our method is directly applicable to several land surface types, environmental conditions, wavelengths, and measurement heights (lookup tables for a limited number of site-specific parameters are provided). The approach allows for an efficient evaluation of the performance of, e.g., infrared imaging systems, laser geodetic systems, and ground-to-satellite optical communication systems. We tested our approach for two grass fields in central and southern Europe, and for a wheat field in central Europe. Although there are uncertainties in the flux estimates, the impact on Cn2 is shown to be rather small. The Cn2 daytime estimates agree well with values determined from eddy covariance measurements for the application to the three fields. However, some adjustments were needed for the approach for the grass in southern Europe because of non-negligible boundary-layer processes that occur in addition to surface-layer processes.

© 2014 Optical Society of America

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 (Cn2) 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 Cn2 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 CT2 (which is related to Cn2) 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 Cn2 does not only depend on CT2, but also on the humidity structure parameter Cq2 and on the joint structure parameter CTq [3]. For wavelengths in the visible and near-infrared, Cn2 mainly depends on CT2, while for radio wavelengths, Cq2 mainly determines Cn2. A summary of past research on models and measurements for optical turbulence is presented in [1]. Friehe [4] and Davidson et al. [5] estimated Cn2 from dual-level meteorological data over the ocean for visible light. Andreas [6] performed a sensitivity study for a Cn2 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 Cn2 in the atmospheric surface layer, restricted to one height and one wavelength. Also, weather forecasts can be used to predict Cn2 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 Cn2 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 Cn2 from single-level weather data are lacking, which motivates our study.

The daily course of Cn2 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 Cn2 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 Cn2 for different locations.

 

Fig. 1. Daily course of observed values of Cn2 (from an eddy-covariance station) for λ=670nm [(a) and (c)] and λ=2mm [(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 CT2, Cq2, and CTq are derived from surface fluxes (using Monin–Obukhov similarity theory), and how Cn2 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 Cn2 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 Cn2 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 CT2, Cq2, and CTq2, following Monin–Obukhov similarity theory. Cn2 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 (LvE) this leads to the Penman–Monteith equation

LvE=s(QnG)+ρcpra(esatea)s+γ(1+rcra),
where Qn is the net radiation, G the soil heat flux, and rc and ra are the canopy and aerodynamic resistances, respectively, which together determine the partitioning between sensible and latent heat flux. For the canopy resistance, a few site-specific input parameters are needed, for which we provide a lookup table (Tables 1 and 7). esatea is the water vapor deficit in Pa, where ea is the actual water vapor pressure and esat is the saturation water vapor pressure (which only varies with temperature). s is the slope of the saturated vapor pressure curve, ρ is the air density, cp is the specific heat capacity of air at constant pressure (p) that depends on humidity, and γ is the psychrometric constant (all given in Appendix A). Note that Eq. (1) describes transpiration only. However, if rc would be set to zero, evaporation of intercepted water is simulated.

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

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

H=QnGLvE.

To obtain LvE, the estimation of the surface skin temperature is crucial because of its role in both Qn and G. The estimation of the surface skin temperature depends on ra; however, H and LvE depend on ra as well. Therefore, the scheme consists of an iteration loop. ra 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 (Sin), 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 Sin, Ta, q, u (wind speed), and p. In this section, we discuss the adaptations that we made to the scheme of dRH99 regarding rc 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 rc on water vapor content of the air. dRH99 uses the formulation

rc=104Δq=104esateapRdRv,
where Δq is the specific humidity deficit in kgkg1, calculated as qsatq, and Rd and Rv are the specific gas constants for dry air and water vapor, respectively (287 and 462Jkg1K1).

Here, we show a comparison of the approximations for the dependence of rc 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

rc=rs,minLAIfdqfradfTfθ.
This approach is based on rs,min, a minimum stomatal resistance for optimal conditions, which is vegetation dependent (see Table 1), and that is scaled from a square meter of leaf surface to a canopy with a specific leaf area index (LAI, see Table 1 as well). The four functions represent the reaction of vegetation to environmental factors: water vapor deficit, global radiation, air temperature, and soil moisture. In this discussion, we only address the water vapor deficit dependence. In Noah [19], fdq is calculated as 1+hsΔq [20], where hs is an empirical coefficient that describes the reaction of stomatal resistance on humidity deficit (45kgkg1, the inverse of K3 in [21]). The Ags scheme [22] contains a similar type of response, but with hs=58kgkg1. In TESSEL [23], fdq=egD8/5pΔq is used, which has a similar effect as the Noah implementation (see Fig. 2) for gD=0.02mb1 (in TESSEL, gD=0.03mb1 for trees and zero for crops). Based on Cabauw data, BB97 calculates the canopy resistance as
rc=25.9fdqfradfθ,
where fdq=1+hs(ΔqΔq0). hs=160kg1kg, Δq0=0.003kg1kg, and the factor 25.9 represents a specific rs,min/LAI. For comparison, we need the approach of BB97 in the Jarvis–Stewart format [Eq. (4)], leading to a scaling factor fr=0.47, such that
frrs,minLAICabauwfdq=25.9(1+hs(ΔqΔq0)),
for LAI=2 and rs,min=110sm1, being representative for Cabauw.

 

Fig. 2. Reaction of rc to water vapor deficit Δq according to different schemes for the observed range of Δq. The reactions by dRH99 and BB97 were defined for the same grass area (Cabauw, The Netherlands), and are therefore both calculated by rcLAI/rs,min (LAI=2, rs,min=110sm1, 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 (rc=rs,min/LAI) for a saturated atmosphere (Δq=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 rc with Δq than the other four functions, which leads to large deviations for dry air conditions (Δq>10gkg1). This is probably caused by the fact that they tested their scheme only for a Dutch grassland (Cabauw), where Δq typically ranges between 5 and 11gkg1.

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 rc for very humid conditions and a weak dependence on Δq). However, we found that in our data, rc calculated from the water vapor deficit dependence as in Noah, TESSEL, and Ags is very different from rc 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 rs,min/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 fθ (see Appendix A), and we replace the scaling factor 25.9 by frrs,min/LAI.

To estimate incoming long-wave radiation, dRH99 uses an expression given by [24], where Lin 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 [2,6,2730]. Maronga [31] found, from a large-eddy simulation resolving surface layer turbulence, that the functions derived for CT2 follow Monin–Obukhov similarity theory while the functions for Cq2 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

(κz)2/3CT2T*2=6.7κ2/3(114.9zL)2/3,
(κz)2/3Cq2q*2=3.5κ2/3(14.5zL)2/3.
In these equations, κ is the Von Kármán constant, with a value of 0.4. z is the height of interest above the canopy, assuming that the atmospheric surface layer is a constant-flux layer: T* and q* do not change with height. L is the Obukhov length defined by [33] as a function of surface friction and buoyancy (see Appendix A).

For homogeneous turbulence, the joint structure parameter CTq can be estimated from the temperature and humidity structure parameters following [34]

CTq=rTqCT2Cq2,
where rTq is the correlation coefficient between temperature and humidity. CTq has the value of the joint structure function (scaled with a function of the separation distance) in the inertial subrange, assuming that rTq is similar for all scales within the inertial subrange. Usually, Ta and q fluctuations are better correlated in the inertial subrange than at larger scales. For daytime data without disturbances from nonlocal processes, an rTq of 0.8 is often observed close to the surface.

The step from surface fluxes to CT2 and Cq2 introduces uncertainty in our Cn2 scheme, because Monin–Obukhov similarity theory is not always valid [32]. Also, the step from CT2 and Cq2 to CTq introduces uncertainty, because an estimate of rTq has to be made (both issues are discussed in Section 6).

C. Estimation of Cn2 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

Cn2=AT2T¯a2CT2+Aq2q¯2Cq2+2ATAqT¯aq¯CTq,
where AT and Aq depend mainly on pressure, air temperature, specific humidity, and the wavelength. Note that, although historically absolute humidity is used in the definitions of AT and Aq, here we will follow [15] and use specific humidity. Functions for AT and Aq are given by [15] as
AT=p¯T¯a(b1+b2q¯RvR¯),
and
Aq=p¯Ta¯bq2q¯RvR¯(1q¯(RvRd)R¯),
where R is the universal gas constant (8.314JK1mol1). The wavelength dependence of AT and Aq is captured by the coefficients b1 and bq2 (see Appendix C).

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.

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

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 (zc) 10cm. 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).

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Table 3. Land-Surface Input Variables per Datasetd

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 (200km2) 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 zc=0.25m).

B. Data Processing and Selection

For the input of our scheme, we averaged 1-min values of p and Sin 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 2.2×106Jm3K1 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 LvE is generally lower than the observed available energy, we adjusted the sensible and latent heat fluxes to the total available energy (QnG), while maintaining the value of the Bowen ratio (H/LvE) following [44].

We also used the EC-pack software to obtain validation values for the structure parameters of Ta, 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 CT2 related to a measurement height of 2.5 m, a wind speed of 2ms1, 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 Cq2, this factor is 1.13 (with a path length in the gas analyzer of 0.13 m), and for CTq 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 (σU2/U2)<0.1. If our data exceeded this limit, we applied a correction of 95/90 [Eq. (5) in [45] for (σU2/U2)=0.1]. To obtain Cn2 from the calculated structure parameters, we used the expressions of [15] described in Section 2.C. We calculated Cn2 for an optical wavelength (λ=670nm) and for a millimeter wavelength (λ=2mm).

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 Ta, q, p, u, and Sin, 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, LvE, and u* that follow from the first part of the scheme, we calculated the structure parameters CT2, Cq2, and CTq following the theory described in Section 2.B, applying a fixed rTq of 0.8. We derived Cn2 for λ=670nm and λ=2mm 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: Ts, ra, and rc. However, these are not directly measured. To compare the estimated surface temperature Ts with measurements, we calculated the measured Ts using

Ts=(Lout(1ϵs)Linϵsσ)1/4,
in which we also used 0.96 for the surface emissivity ϵs, and 5.67×108Wm2K4 for the Stefan–Boltzman constant σ. The measured resistances were calculated following
ra=ρcpTaTsH,
and
rc=ra(s(QnG)+ρcp(esatea)raγLvEsγ1),
which is an inverted form of the Penman–Monteith equation [Eq. (1)].

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 rc and ra when H, LvE, or QnG<20Wm2.

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 c2 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=c1(1c2x)2/3. The structure parameter and stability data were weighted using the inverse of the confidence intervals for CT2, Cq2, and z/L (via the intervals of u*, Ta, q, wT¯, and wq¯) 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 Cn2 (Section 5.B).

A. Fluxes

The difference between the surface temperature Ts and air temperature Ta is very important for the calculation of the energy that is available for H and LvE. 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 Ts, the long-wave radiation components are estimated fairly well for all datasets. Thereby, Qn 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 Qn such that this underestimation does not produce large errors for the energy that is available for H and LvE.

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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

 

Fig. 3. Validation of estimates (model) of ra, rc, Ts, H, LvE, and QnG 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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rc 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 rc with direct observations, and thus the presented underestimation of rc is uncertain. The NRMSE for rc was still quite high despite the extra data selection. Nevertheless, the partitioning of the available energy into H and LvE is appropriate for GC and GS. This indicates that rc is of the correct order of magnitude. For WC, H is generally overestimated and LvE 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. ra estimates are quite scattered; however, most of the data used here is in a rc regime in which the fluxes are not very sensitive to ra, see [48].

B. Structure Parameters

Although H and LvE are predicted well for GC, CT2 and Cq2, and thereby CTq, are overestimated. For the growing wheat (WC), H is largely overestimated, which leads to a clear overestimation of CT2 as expected from the relation between H2 (or T*2) and CT2 given in Eq. (7a). For GS, the regression slope for LvE is satisfying, whereas the slope for Cq2 is more than twice as large. This is not expected from the relation between the slopes of LvE2 and Cq2 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 c2 to its literature values of 14.9 for temperature and 4.5 for humidity.

The obtained values for c1 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, Cq2 is still overestimated (albeit with a large NRMSE) while the estimation of LvE is close to perfect with a very low NRMSE. The large NRMSE for Cq2 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 CT2, Cq2, and CTq lead to reasonable estimates of Cn2 (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 Cq2 again in the estimate for Cn2 for λ=2mm 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 CT2 for the data of both the growing and ripening phase leads to an overestimation of Cn2 for λ=670nm, a wavelength at which Cn2 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 λ=2mm than for λ=670nm, because humidity fluctuations are larger and better defined.

 

Fig. 6. Cn2 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 Cn2 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 Cn2 from single-level meteorological data are available in the literature. These methods are generally less complex in construction, being based on nonlinear regressions to Ta, u, relative humidity (RH), and occasionally Sin. 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 Cn2 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 Ta (the sensitivity to u and Sin is much weaker).

 

Fig. 7. Sensitivity of derivations of Cn2 using the algorithm given in [49], to RH and Ta. Diamond markers indicate the 25th and 75th percentile of RH and Ta for the grass data of the Haarweg in 2005.

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For a dry and warm atmosphere, the method results in positive Cn2 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 Ta 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 Cn2 from single-level weather data that consists of three steps:

  • (a) From single-level weather data to surface fluxes;
  • (b) From surface fluxes to structure parameters of temperature and humidity; and
  • (c) From structure parameters to Cn2.

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 rc, described by dRH99 as rc=104Δq, with the approach of BB97. In Fig. 8, we show the performance of the scheme for LvE and rc 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. rc and LvE estimates (model) versus measurements (obs) for the Haarweg (GC) using the approach for rc from dRH99, colored by Δq (light is 0, dark is 10gkg1).

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To investigate which parameters have the largest influence on the derived Cn2, 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 Cn2 for a realistic increase of the parameters. We see that the dependence on the parameters in fdq 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 Cn2 for both wavelengths (e.g., N, a, and Ag) are related to the available energy. If a certain increment leads to a decrease of Cn2 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, rs,min, and fr).

Tables Icon

Table 6. Sensitivity Estimates of Cn2 to the Main Input Parameters and Constants (or Parameters Directly Derived from Sin) for June 19, 2005, 12 UTC at the Haarwega

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 rs,min cannot be separated from their appearance in the land surface schemes. We did not implement the dependence of rc 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 Cn2 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 rTq. 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 11 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, Cn2 results will still be scattered.

 

Fig. 9. Validation of derivations of Cn2 from measured values for CT2, Cq2, CTq, and rTq, 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 Cn2 calculated from measured fluxes using Monin–Obukhov similarity theory are somewhat smaller than Cn2 values derived from the observed structure parameters of Ta 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 c1 or smaller c2 [see Eqs. (7a) and (7b)]. In the previous section we wrote that, indeed, we found a slightly larger value for c1 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 c1 that fitted the data with a c2 that was kept constant. However, we would like to point out that if we fit the GS data to the function without keeping c2, we obtain for temperature c1=4.2 and c2=10.8, and for humidity c1=2.5 and c2=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 c2.

Considering a fixed value for rTq, we cannot omit this assumption when only weather data is available, because rTq 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 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 CTq (and therefore also in Cn2) by using a fixed rTq 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 λ=2mm than for λ=670nm, due to a better-defined humidity flux (microwaves are more sensitive to humidity fluctuations than to temperature fluctuations). Note that a more elaborated Cn2 scheme would also include the scattering effect of aerosols on the wave propagation equation and thus on Cn2.

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 (Cn2) based on single-level weather station data. Our estimates of Cn2 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 Cn2 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 Cn2 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 Cn2 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, Cn2 is calculated from the structure parameters for one specific wavelength. For our grassland datasets, estimated values for Cn2 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 ρ, air heat capacity cp(=1004(1+0.84q)), water vapor pressure ea, and saturated water vapor pressure esat (both in Pa) from air temperature (Ta), specific humidity (q), and pressure (p). The slope of the saturated vapor pressure curve is s(T)=desat/dT, and the psychrometric constant is γ=cppRv/(LvRd), where Rd is the specific gas constant for dry air (287Jkg1K1) and Rv is the specific gas constant for water vapor (462Jkg1K1). The temperature dependence of esat, s, and Lv can be found in [56], page 355.

After that, the surface resistance can be calculated following

rc=frrs,minLAIfdqfrad,
where the scaling factor fr=0.47, and fdq is calculated following BB97:
fdq=1+hs(ΔqΔq0),
in which Δq is the specific humidity deficit in kgkg1 (calculated as qsatq), hs=160kg1kg, Δq0=0.003kgkg1, and where
frad=1000Sin+230(10002Sin)Sin(1000230).

Then, most of the radiation components can be calculated. Outgoing shortwave radiation (Sout) is calculated following

Sout=aSin,
where a is the surface albedo that depends on the solar elevation angle α and the effective cloud cover fraction N as
a=amax(1N)sin(α)(amaxamin)N(amaxacloud).
amax, amin, and acloud 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
N=Sin,0SinSin,00.20.8,
where Sin,0 is the solar constant I0 (1365W2m2) multiplied with the cosine of the zenith angle θz of the specific location and time and corrected for the orbital eccentricity via the difference in distance (d) to the sun,
Sin,0=I¯0(d¯sundsun)2cosθz.
Incoming long-wave (L) radiation is calculated as
Lin=ϵaσTa4,
where ϵa is the air emissivity and σ is the Stefan–Boltzmann constant (5.67×108). We used the following expressions for the emissivity of air, adapted from [25]:
ϵa,clear=0.63+5.95×107eae1500/Ta,
and [26]
ϵa=N+(1N)ϵa,clear.

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

H=Qn,0+Qn,T(TsTa)G0GT(TsTa)LvE0LvET(TsTa),
where all energy terms consist of an isothermal term that is calculated using Ta instead of Ts (indicated with the subscript 0), and a correction term that corrects for that assumption of an isothermal atmosphere. This correction term therefore depends on (TsTa) (indicated with the subscript T). Thereby, the correction terms depend on H, via
TsTa=Hraρcp.
Because ra 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
H=Qn,0G0LvE01+Qn,T+GT+LvET,
where the isothermal terms are calculated following
Qn,0=(1a)Sin+(ϵa1)ϵeσTa4,
G0=Ag(TaT24h).
Here, Ag is the soil heat transfer coefficient (see Table 7), and T24h is the temperature history of the last 24 h that represents the deep-soil temperature, and
LvE0=ρcpγesat(Ta)era+rc.
Note that LvE0 depends on ra too, which means that the calculation of LvE0 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 ra 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 1Wm2 (three iterations are usually enough):

Qn,T=raρcpϵsσ4Ta3,
where ϵs is the surface emissivity (see Table 7),
GT=raρcpAg,
and
LvET=raγs(Ta)ra+rc.

The loop starts with neutral conditions (ΨT=Ψu=0) for a first estimate of the atmospheric-stability-dependent ra and u*. The aerodynamic resistance is calculated as

ra=1κu*(lnzz0hΨT(zL)+ΨT(z0hL)).
Here, κ is the Von Kàrmàn constant (0.4), z is the height (z=zmd, d=23zc is the displacement height), and z0h is the roughness length for heat, calculated as 0.1z0m [z0m=0.4(zcd), the roughness length for momentum]. ΨT 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
u*=uκln(zz0m)Ψu(zL)+Ψu(z0mL),
where u is the wind speed and Ψu 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
L=Tau*2κgTv*,
where the temperature scale Tv* is a scaled buoyancy flux. The scalar scales are calculated as
Tv*=Hvcpρu*,T*=Hcpρu*,q*=LvELvρu*,
where the buoyancy flux
Hv=H(1+0.61q)+0.61cpTaLvE0Lv,
and g is the acceleration due to gravity (9.81ms2).

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

Ts=Ta+Hraρcp+zΓd,
where Γd is the dry adiabatic lapse rate of 0.01Km1. If Ts is known, the long-wave outgoing radiation, net radiation, and soil and latent heat flux can be calculated following
Lout=ϵsσTs4+(1ϵs)Lin,
Qn=SinSout+LinLout=Q0ϵsσ4Ta3(TsTa),
G=Ag(T24hTs)=G0+Ag(TsTa),
and
LvE=s(QnG)+ρcpra(esatea)s+γ(1+rcra)=LvE0+ρcpγs(Ta)(TsTa)ra+rc.

Appendix B: Land Surface Input

Tables Icon

Table 7. Land Surface Input Variables Used for All Sitesa

Appendix C: Coefficients Capturing the Wavelength Dependence of Cn2

For λ<1mm,

b1=106(0.237134+68.39397130λ2+0.4547338.9λ2),
b2=106(0.648731+5.8058×103λ27.115×105λ4+8.851×106λ6)b1,
bq2=b2,
and for λ>1mm,
b1=0.776×106,
b2=106(7500Ta0.056),
bq2=106(3750Ta0.056).

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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30. D. Li, E. Bou-Zeid, and H. A. R. De Bruin, “Monin–Obukhov similarity functions for the structure parameters of temperature and humidity,” Bound-Layer Meteor. 145, 45–67 (2012).

31. B. Maronga, “Monin-Obukhov similarity functions for the structure parameters of temperature and humidity in the unstable surface layer: results from high-resolution large-eddy simulations,” J. Atmos. Sci. 71, 716–733 (2014). [CrossRef]  

32. A. van de Boer, A. F. Moene, A. Graf, D. Schüttemeyer, and C. Simmer, “Detection of entrainment influences on surface-layer measurements and extension of Monin-Obukhov similarity theory,” Bound-Layer Meteor. 152, 19–44 (2014).

33. A. M. Obukhov, “Turbulence in an atmosphere with a non-uniform temperature,” Tr Geofiz. Inst. Akad. Nauk. SSSR 1, 95–115 (1946).

34. A. F. Moene, “Effects of water vapour on the structure parameter of the refractive index for near-infrared radiation,” Bound-Layer Meteor. 107, 635–653 (2003).

35. R. J. Hill and S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. 68, 892–899 (1978).

36. A. Graf, D. Schüttemeyer, H. Geiß, A. Knaps, M. Möllmann-Coers, J. H. Schween, S. Kollet, B. Neininger, M. Herbst, and H. Vereecken, “Boundedness of turbulent temperature probability distributions, and their relation to the vertical profile in the convective boundary layer,” Bound-Layer Meteor. 134, 459–486 (2010).

37. A. van de Boer, A. F. Moene, and D. Schuettemeyer, “Sensitivity and uncertainty of analytical footprint models according to a combined natural tracer and ensemble approach,” Agric. Forest. Meteor. 169, 1–11 (2013).

38. M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

39. W. R. Rouse, “Microclimate at arctic tree line 3. The effects of regional advection on the surface energy balance of upland tundra,” Water Resour. Res. 20, 74–78 (1984). [CrossRef]  

40. A. van Dijk, A. F. Moene, and H. A. R. de Bruin, “The principles of surface flux physics: theory, practice and description of the ECPACK library,” Internal Report 2004/1 (Meteorology and Air Quality Group, 2004).

41. J. M. Wilczak, S. P. Oncley, and S. A. Stage, “Sonic anemometer tilt correction algorithms,” Bound-Layer Meteor. 99, 127–150 (2001).

42. E. Webb, G. Pearman, and R. Leuning, “Correction of flux measurements for density effects due to heat and water vapour transfer,” Quart. J. Roy. Meteor. Soc. 106, 85–100 (1980).

43. P. Schotanus, F. Nieuwstadt, and H. de Bruin, “Temperature measurement with a sonic anemometer and its application to heat and moisture fluxes,” Bound-Layer Meteor. 26, 81–93 (1983).

44. T. E. Twine, W. P. Kustas, J. M. Norman, D. R. Cook, P. Houser, T. P. Meyers, J. H. Prueger, P. J. Starks, and M. L. Wesely, “Correcting eddy-covariance flux underestimates over a grassland,” Agric. Forest. Meteor. 103, 279–300 (2000).

45. M. Braam, F. C. Bosveld, and A. F. Moene, “On Monin Obukhov scaling in and above the atmospheric surface layer: the complexities of elevated scintillometer measurements,” Bound-Layer Meteor. 144, 157–177 (2012).

46. F. C. Bosveld, “The KNMI Garderen experiment: micro-meteorological observations 1988-1989,” Technical Report TR-208 (De Bilt, 1999).

47. J. C. Wyngaard and S. F. Clifford, “Taylor’s hypothesis and high-frequency turbulence spectra,” J. Atmos. Sci. 34, 922–929 (1977). [CrossRef]  

48. C. M. J. Jacobs and H. A. R. De Bruin, “The sensitivity of regional transpiration to land-surface characteristics: significance of feedback,” J. Climate 5, 683–698 (1992).

49. N. S. Kopeika, A System Engineering Approach to Imaging (SPIE Optical Engineering, 1998).

50. A. A. M. Holtslag and H. A. R. De Bruin, “Applied modeling of the nighttime surface energy balance over land,” J. Appl. Meteor. 27, 689704 (1988).

51. A. K. Betts and A. C. Beljaars, “Estimation of effective roughness length for heat and momentum from FIFE data,” Atmos. Res. 30, 251–261 (1993). [CrossRef]  

52. F. Kasten and G. Czeplak, “Solar and terrestrial radiation dependent on the amount and type of cloud,” Sol. Energy 24, 177–189 (1980). [CrossRef]  

53. A. Spena, G. D’Angiolini, and C. Strati, “First correlations for solar radiation on cloudy days in Italy,” in ASME-ATI-UIT 2010 Conference on Thermal and Environmental Issues in Energy Systems, Sorrento, Italy, May 2010.

54. L. M. J. Kooijmans, “Testing the universality of Monin-Obukhov similarity functions,” Master’s thesis (Wagenigen University, 2013).

55. M. Sivasligil, C. B. Erol, O. M. Polat, and H. Sari, “Validation of refractive index structure parameter estimation for certain infrared bands,” Appl. Opt. 52, 3127–3133 (2013). [CrossRef]  

56. A. F. Moene and J. C. van Dam, Transport in the Atmosphere-Vegetation-Soil Continuum (Cambridge University, 2014).

57. C. A. Paulson, “The mathematical representation of wind speed and temperature profiles in the unstable atmospheric surface layer,” J. Appl. Meteor. 9, 857–861 (1970).

References

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  1. A. Tunick, “A critical assessment of selected past research on optical turbulence information in diverse microclimates,” Technical Report (Army Research Laboratory, 2002).
  2. J. C. Wyngaard, O. R. Coté, and Y. Izumi, “Local free convection, similarity, and the budgets of shear stress and heat flux,” J. Atmos. Sci. 28, 1171–1182 (1971).
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  3. M. L. Wesely, “The combined effect of temperature and humidity fluctuations on refractive index,” J. Appl. Meteor. 15, 43–49 (1976).
  4. C. A. Friehe, “Estimation of the refractive-index temperature structure parameter over the ocean,” Appl. Opt. 16, 334–340 (1977).
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  5. K. L. Davidson, T. M. Houlihan, C. W. Fairall, and G. E. Schacher, “Observation of the temperature structure function parameter, C T 2, over the ocean,” Bound-Layer Meteor. 15, 507–523 (1978).
  6. E. L. Andreas, “Estimating Cn2 over snow and sea ice from meteorological data,” J. Opt. Soc. Am. A 5, 481–495 (1988).
    [Crossref]
  7. A. S. Monin and A. M. Obukhov, “Basic laws of turbulent mixing in the surface layer of the atmosphere,” Tr. Geofiz. Inst. Akad. Nauk. SSSR 24, 163–187 (1954).
  8. D. Sadot and N. S. Kopeika, “Forecasting optical turbulence strength on the basis of macroscale meteorology and aerosols: models and validation,” Opt. Eng. 31, 200–212 (1992).
    [Crossref]
  9. H. Rachele and A. Tunick, “Energy balance model for imagery and electromagnetic propagation,” J. Appl. Meteor. 33, 964–976 (1994).
  10. S. Bendersky, N. S. Kopeika, and N. Blaunstein, “Atmospheric optical turbulence over land in middle east coastal environments: prediction modeling and measurements,” Appl. Opt. 43, 4070–4079 (2004).
    [Crossref]
  11. T. T. Leclerc, R. L. Phillips, L. C. Andrews, D. T. Wayne, P. Sauer, and R. Crabbs, “Prediction of the ground-level refractive index structure parameter from the measurement of atmospheric conditions,” Proc. SPIE 7685, 76850A (2010).
    [Crossref]
  12. S. Cheinet, A. Beljaars, K. Weiss-Wrana, and Y. Hurthaud, “The use of weather forecasts to characterise near-surface optical turbulence,” Bound-Layer Meteor. 138, 453–473 (2011).
  13. A. A. M. Holtslag and A. P. Van Ulden, “A simple scheme for daytime estimates of the surface fluxes from routine weather data,” J. Climate Appl. Meteor. 22, 517–529 (1983).
  14. W. C. de Rooy and A. A. M. Holtslag, “Estimation of surface radiation and energy flux densities from single-level weather data,” J. Appl. Meteor. 38, 526–540 (1999).
  15. H. C. Ward, J. G. Evans, O. K. Hartogensis, A. F. Moene, H. A. R. De Bruin, and C. S. B. Grimmond, “A critical revision of the estimation of the latent heat flux from two-wavelength scintillometry,” Q. J. R. Meteorol. Soc. 139, 1912–1922 (2013).
  16. A. Beljaars and F. C. Bosveld, “Cabauw data for the validation of land surface parameterization schemes,” J. Climate 10, 1172–1193 (1997).
  17. P. G. Jarvis, “The interpretation of the variations in leaf water potential and stomatal conductance found in canopies in the field,” Phil. Trans. 273, 593–610 (1976).
  18. J. B. Stewart, “Modelling surface conductance of pine forest,” Agric. Forest Meteor. 43, 19–35 (1988).
  19. F. Chen, K. Mitchell, J. Schaake, Y. Xue, H. L. Pan, V. Koren, Q. Y. Duan, M. Ek, and A. Betts, “Modeling of land surface evaporation by four schemes and comparison with FIFE observations,” J. Geophys. Res. 101, 7251–7268 (1996).
    [Crossref]
  20. F. Chen and J. Dudhia, “Coupling an advanced land surface-hydrology model with the Penn State-NCAR MM5 modeling system. Part I: Model implementation and sensitivity,” Mon. Weather Rev. 129, 569–585 (2001).
    [Crossref]
  21. J. Stewart and L. Gay, “Preliminary modelling of transpiration from the fife site in Kansas,” Agric. Forest Meteor. 48, 305–315 (1989).
  22. R. J. Ronda, H. A. R. de Bruin, and A. A. M. Holtslag, “Representation of the canopy conductance in modeling the surface energy budget for low vegetation,” J. Appl. Meteor. 40, 1431–1444 (2001).
  23. B. J. J. M. van den Hurk, P. Viterbo, A. C. M. Beljaars, and A. K. Betts, “Offline validation of the ERA40 surface scheme,” (2000).
  24. G. W. Paltridge and C. M. R. Platt, Radiative Processes in Meteorology and Climatology (Elsevier, 1976).
  25. S. B. Idso, “A set of equations for full spectrum and 8 to 14  μm and 10.5 to 12.5  μm thermal radiation from cloudless skies,” Water Resour. Res. 17, 295–304 (1981).
    [Crossref]
  26. T. M. Crawford and C. E. Duchon, “An improved parameterization for estimating effective atmospheric emissivity for use in calculating daytime downwelling longwave radiation,” J. Appl. Meteor 38, 474–480 (1999).
  27. R. J. Hill, “Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes,” Waves Random Media 2, 179–201 (1992).
    [Crossref]
  28. V. Thiermann and H. Grassl, “The measurement of turbulent surface-layer fluxes by use of bichromatic scintillation,” Bound-Layer Meteor. 58, 367–389 (1992).
  29. H. A. R. De Bruin, W. Kohsiek, and J. J. M. van den Hurk, “A verification of some methods to determine the fluxes of momentum, sensible heat, and water vapour using standard deviation and structure parameter of scalar meteorological quantities,” Bound-Layer Meteor. 63, 231–257 (1993).
  30. D. Li, E. Bou-Zeid, and H. A. R. De Bruin, “Monin–Obukhov similarity functions for the structure parameters of temperature and humidity,” Bound-Layer Meteor. 145, 45–67 (2012).
  31. B. Maronga, “Monin-Obukhov similarity functions for the structure parameters of temperature and humidity in the unstable surface layer: results from high-resolution large-eddy simulations,” J. Atmos. Sci. 71, 716–733 (2014).
    [Crossref]
  32. A. van de Boer, A. F. Moene, A. Graf, D. Schüttemeyer, and C. Simmer, “Detection of entrainment influences on surface-layer measurements and extension of Monin-Obukhov similarity theory,” Bound-Layer Meteor. 152, 19–44 (2014).
  33. A. M. Obukhov, “Turbulence in an atmosphere with a non-uniform temperature,” Tr Geofiz. Inst. Akad. Nauk. SSSR 1, 95–115 (1946).
  34. A. F. Moene, “Effects of water vapour on the structure parameter of the refractive index for near-infrared radiation,” Bound-Layer Meteor. 107, 635–653 (2003).
  35. R. J. Hill and S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. 68, 892–899 (1978).
  36. A. Graf, D. Schüttemeyer, H. Geiß, A. Knaps, M. Möllmann-Coers, J. H. Schween, S. Kollet, B. Neininger, M. Herbst, and H. Vereecken, “Boundedness of turbulent temperature probability distributions, and their relation to the vertical profile in the convective boundary layer,” Bound-Layer Meteor. 134, 459–486 (2010).
  37. A. van de Boer, A. F. Moene, and D. Schuettemeyer, “Sensitivity and uncertainty of analytical footprint models according to a combined natural tracer and ensemble approach,” Agric. Forest. Meteor. 169, 1–11 (2013).
  38. M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).
  39. W. R. Rouse, “Microclimate at arctic tree line 3. The effects of regional advection on the surface energy balance of upland tundra,” Water Resour. Res. 20, 74–78 (1984).
    [Crossref]
  40. A. van Dijk, A. F. Moene, and H. A. R. de Bruin, “The principles of surface flux physics: theory, practice and description of the ECPACK library,” (Meteorology and Air Quality Group, 2004).
  41. J. M. Wilczak, S. P. Oncley, and S. A. Stage, “Sonic anemometer tilt correction algorithms,” Bound-Layer Meteor. 99, 127–150 (2001).
  42. E. Webb, G. Pearman, and R. Leuning, “Correction of flux measurements for density effects due to heat and water vapour transfer,” Quart. J. Roy. Meteor. Soc. 106, 85–100 (1980).
  43. P. Schotanus, F. Nieuwstadt, and H. de Bruin, “Temperature measurement with a sonic anemometer and its application to heat and moisture fluxes,” Bound-Layer Meteor. 26, 81–93 (1983).
  44. T. E. Twine, W. P. Kustas, J. M. Norman, D. R. Cook, P. Houser, T. P. Meyers, J. H. Prueger, P. J. Starks, and M. L. Wesely, “Correcting eddy-covariance flux underestimates over a grassland,” Agric. Forest. Meteor. 103, 279–300 (2000).
  45. M. Braam, F. C. Bosveld, and A. F. Moene, “On Monin Obukhov scaling in and above the atmospheric surface layer: the complexities of elevated scintillometer measurements,” Bound-Layer Meteor. 144, 157–177 (2012).
  46. F. C. Bosveld, “The KNMI Garderen experiment: micro-meteorological observations 1988-1989,” (De Bilt, 1999).
  47. J. C. Wyngaard and S. F. Clifford, “Taylor’s hypothesis and high-frequency turbulence spectra,” J. Atmos. Sci. 34, 922–929 (1977).
    [Crossref]
  48. C. M. J. Jacobs and H. A. R. De Bruin, “The sensitivity of regional transpiration to land-surface characteristics: significance of feedback,” J. Climate 5, 683–698 (1992).
  49. N. S. Kopeika, A System Engineering Approach to Imaging (SPIE Optical Engineering, 1998).
  50. A. A. M. Holtslag and H. A. R. De Bruin, “Applied modeling of the nighttime surface energy balance over land,” J. Appl. Meteor. 27, 689704 (1988).
  51. A. K. Betts and A. C. Beljaars, “Estimation of effective roughness length for heat and momentum from FIFE data,” Atmos. Res. 30, 251–261 (1993).
    [Crossref]
  52. F. Kasten and G. Czeplak, “Solar and terrestrial radiation dependent on the amount and type of cloud,” Sol. Energy 24, 177–189 (1980).
    [Crossref]
  53. A. Spena, G. D’Angiolini, and C. Strati, “First correlations for solar radiation on cloudy days in Italy,” in ASME-ATI-UIT 2010 Conference on Thermal and Environmental Issues in Energy Systems, Sorrento, Italy, May2010.
  54. L. M. J. Kooijmans, “Testing the universality of Monin-Obukhov similarity functions,” Master’s thesis (Wagenigen University, 2013).
  55. M. Sivasligil, C. B. Erol, O. M. Polat, and H. Sari, “Validation of refractive index structure parameter estimation for certain infrared bands,” Appl. Opt. 52, 3127–3133 (2013).
    [Crossref]
  56. A. F. Moene and J. C. van Dam, Transport in the Atmosphere-Vegetation-Soil Continuum (Cambridge University, 2014).
  57. C. A. Paulson, “The mathematical representation of wind speed and temperature profiles in the unstable atmospheric surface layer,” J. Appl. Meteor. 9, 857–861 (1970).

2014 (3)

B. Maronga, “Monin-Obukhov similarity functions for the structure parameters of temperature and humidity in the unstable surface layer: results from high-resolution large-eddy simulations,” J. Atmos. Sci. 71, 716–733 (2014).
[Crossref]

A. van de Boer, A. F. Moene, A. Graf, D. Schüttemeyer, and C. Simmer, “Detection of entrainment influences on surface-layer measurements and extension of Monin-Obukhov similarity theory,” Bound-Layer Meteor. 152, 19–44 (2014).

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

2013 (3)

H. C. Ward, J. G. Evans, O. K. Hartogensis, A. F. Moene, H. A. R. De Bruin, and C. S. B. Grimmond, “A critical revision of the estimation of the latent heat flux from two-wavelength scintillometry,” Q. J. R. Meteorol. Soc. 139, 1912–1922 (2013).

A. van de Boer, A. F. Moene, and D. Schuettemeyer, “Sensitivity and uncertainty of analytical footprint models according to a combined natural tracer and ensemble approach,” Agric. Forest. Meteor. 169, 1–11 (2013).

M. Sivasligil, C. B. Erol, O. M. Polat, and H. Sari, “Validation of refractive index structure parameter estimation for certain infrared bands,” Appl. Opt. 52, 3127–3133 (2013).
[Crossref]

2012 (2)

M. Braam, F. C. Bosveld, and A. F. Moene, “On Monin Obukhov scaling in and above the atmospheric surface layer: the complexities of elevated scintillometer measurements,” Bound-Layer Meteor. 144, 157–177 (2012).

D. Li, E. Bou-Zeid, and H. A. R. De Bruin, “Monin–Obukhov similarity functions for the structure parameters of temperature and humidity,” Bound-Layer Meteor. 145, 45–67 (2012).

2011 (1)

S. Cheinet, A. Beljaars, K. Weiss-Wrana, and Y. Hurthaud, “The use of weather forecasts to characterise near-surface optical turbulence,” Bound-Layer Meteor. 138, 453–473 (2011).

2010 (2)

T. T. Leclerc, R. L. Phillips, L. C. Andrews, D. T. Wayne, P. Sauer, and R. Crabbs, “Prediction of the ground-level refractive index structure parameter from the measurement of atmospheric conditions,” Proc. SPIE 7685, 76850A (2010).
[Crossref]

A. Graf, D. Schüttemeyer, H. Geiß, A. Knaps, M. Möllmann-Coers, J. H. Schween, S. Kollet, B. Neininger, M. Herbst, and H. Vereecken, “Boundedness of turbulent temperature probability distributions, and their relation to the vertical profile in the convective boundary layer,” Bound-Layer Meteor. 134, 459–486 (2010).

2004 (1)

2003 (1)

A. F. Moene, “Effects of water vapour on the structure parameter of the refractive index for near-infrared radiation,” Bound-Layer Meteor. 107, 635–653 (2003).

2001 (3)

F. Chen and J. Dudhia, “Coupling an advanced land surface-hydrology model with the Penn State-NCAR MM5 modeling system. Part I: Model implementation and sensitivity,” Mon. Weather Rev. 129, 569–585 (2001).
[Crossref]

R. J. Ronda, H. A. R. de Bruin, and A. A. M. Holtslag, “Representation of the canopy conductance in modeling the surface energy budget for low vegetation,” J. Appl. Meteor. 40, 1431–1444 (2001).

J. M. Wilczak, S. P. Oncley, and S. A. Stage, “Sonic anemometer tilt correction algorithms,” Bound-Layer Meteor. 99, 127–150 (2001).

2000 (1)

T. E. Twine, W. P. Kustas, J. M. Norman, D. R. Cook, P. Houser, T. P. Meyers, J. H. Prueger, P. J. Starks, and M. L. Wesely, “Correcting eddy-covariance flux underestimates over a grassland,” Agric. Forest. Meteor. 103, 279–300 (2000).

1999 (2)

T. M. Crawford and C. E. Duchon, “An improved parameterization for estimating effective atmospheric emissivity for use in calculating daytime downwelling longwave radiation,” J. Appl. Meteor 38, 474–480 (1999).

W. C. de Rooy and A. A. M. Holtslag, “Estimation of surface radiation and energy flux densities from single-level weather data,” J. Appl. Meteor. 38, 526–540 (1999).

1997 (1)

A. Beljaars and F. C. Bosveld, “Cabauw data for the validation of land surface parameterization schemes,” J. Climate 10, 1172–1193 (1997).

1996 (1)

F. Chen, K. Mitchell, J. Schaake, Y. Xue, H. L. Pan, V. Koren, Q. Y. Duan, M. Ek, and A. Betts, “Modeling of land surface evaporation by four schemes and comparison with FIFE observations,” J. Geophys. Res. 101, 7251–7268 (1996).
[Crossref]

1994 (1)

H. Rachele and A. Tunick, “Energy balance model for imagery and electromagnetic propagation,” J. Appl. Meteor. 33, 964–976 (1994).

1993 (2)

H. A. R. De Bruin, W. Kohsiek, and J. J. M. van den Hurk, “A verification of some methods to determine the fluxes of momentum, sensible heat, and water vapour using standard deviation and structure parameter of scalar meteorological quantities,” Bound-Layer Meteor. 63, 231–257 (1993).

A. K. Betts and A. C. Beljaars, “Estimation of effective roughness length for heat and momentum from FIFE data,” Atmos. Res. 30, 251–261 (1993).
[Crossref]

1992 (4)

C. M. J. Jacobs and H. A. R. De Bruin, “The sensitivity of regional transpiration to land-surface characteristics: significance of feedback,” J. Climate 5, 683–698 (1992).

R. J. Hill, “Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes,” Waves Random Media 2, 179–201 (1992).
[Crossref]

V. Thiermann and H. Grassl, “The measurement of turbulent surface-layer fluxes by use of bichromatic scintillation,” Bound-Layer Meteor. 58, 367–389 (1992).

D. Sadot and N. S. Kopeika, “Forecasting optical turbulence strength on the basis of macroscale meteorology and aerosols: models and validation,” Opt. Eng. 31, 200–212 (1992).
[Crossref]

1989 (1)

J. Stewart and L. Gay, “Preliminary modelling of transpiration from the fife site in Kansas,” Agric. Forest Meteor. 48, 305–315 (1989).

1988 (3)

E. L. Andreas, “Estimating Cn2 over snow and sea ice from meteorological data,” J. Opt. Soc. Am. A 5, 481–495 (1988).
[Crossref]

J. B. Stewart, “Modelling surface conductance of pine forest,” Agric. Forest Meteor. 43, 19–35 (1988).

A. A. M. Holtslag and H. A. R. De Bruin, “Applied modeling of the nighttime surface energy balance over land,” J. Appl. Meteor. 27, 689704 (1988).

1984 (1)

W. R. Rouse, “Microclimate at arctic tree line 3. The effects of regional advection on the surface energy balance of upland tundra,” Water Resour. Res. 20, 74–78 (1984).
[Crossref]

1983 (2)

A. A. M. Holtslag and A. P. Van Ulden, “A simple scheme for daytime estimates of the surface fluxes from routine weather data,” J. Climate Appl. Meteor. 22, 517–529 (1983).

P. Schotanus, F. Nieuwstadt, and H. de Bruin, “Temperature measurement with a sonic anemometer and its application to heat and moisture fluxes,” Bound-Layer Meteor. 26, 81–93 (1983).

1981 (1)

S. B. Idso, “A set of equations for full spectrum and 8 to 14  μm and 10.5 to 12.5  μm thermal radiation from cloudless skies,” Water Resour. Res. 17, 295–304 (1981).
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1980 (2)

E. Webb, G. Pearman, and R. Leuning, “Correction of flux measurements for density effects due to heat and water vapour transfer,” Quart. J. Roy. Meteor. Soc. 106, 85–100 (1980).

F. Kasten and G. Czeplak, “Solar and terrestrial radiation dependent on the amount and type of cloud,” Sol. Energy 24, 177–189 (1980).
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1978 (2)

R. J. Hill and S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. 68, 892–899 (1978).

K. L. Davidson, T. M. Houlihan, C. W. Fairall, and G. E. Schacher, “Observation of the temperature structure function parameter, C T 2, over the ocean,” Bound-Layer Meteor. 15, 507–523 (1978).

1977 (2)

C. A. Friehe, “Estimation of the refractive-index temperature structure parameter over the ocean,” Appl. Opt. 16, 334–340 (1977).
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J. C. Wyngaard and S. F. Clifford, “Taylor’s hypothesis and high-frequency turbulence spectra,” J. Atmos. Sci. 34, 922–929 (1977).
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1976 (2)

M. L. Wesely, “The combined effect of temperature and humidity fluctuations on refractive index,” J. Appl. Meteor. 15, 43–49 (1976).

P. G. Jarvis, “The interpretation of the variations in leaf water potential and stomatal conductance found in canopies in the field,” Phil. Trans. 273, 593–610 (1976).

1971 (1)

J. C. Wyngaard, O. R. Coté, and Y. Izumi, “Local free convection, similarity, and the budgets of shear stress and heat flux,” J. Atmos. Sci. 28, 1171–1182 (1971).
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1970 (1)

C. A. Paulson, “The mathematical representation of wind speed and temperature profiles in the unstable atmospheric surface layer,” J. Appl. Meteor. 9, 857–861 (1970).

1954 (1)

A. S. Monin and A. M. Obukhov, “Basic laws of turbulent mixing in the surface layer of the atmosphere,” Tr. Geofiz. Inst. Akad. Nauk. SSSR 24, 163–187 (1954).

1946 (1)

A. M. Obukhov, “Turbulence in an atmosphere with a non-uniform temperature,” Tr Geofiz. Inst. Akad. Nauk. SSSR 1, 95–115 (1946).

Alexander, D. C.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

Andreas, E. L.

Andrews, L. C.

T. T. Leclerc, R. L. Phillips, L. C. Andrews, D. T. Wayne, P. Sauer, and R. Crabbs, “Prediction of the ground-level refractive index structure parameter from the measurement of atmospheric conditions,” Proc. SPIE 7685, 76850A (2010).
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Angevine, W. M.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

Augustin, P.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

Bargain, E.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

Barrié, J.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

Bazile, E.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

Beljaars, A.

S. Cheinet, A. Beljaars, K. Weiss-Wrana, and Y. Hurthaud, “The use of weather forecasts to characterise near-surface optical turbulence,” Bound-Layer Meteor. 138, 453–473 (2011).

A. Beljaars and F. C. Bosveld, “Cabauw data for the validation of land surface parameterization schemes,” J. Climate 10, 1172–1193 (1997).

Beljaars, A. C.

A. K. Betts and A. C. Beljaars, “Estimation of effective roughness length for heat and momentum from FIFE data,” Atmos. Res. 30, 251–261 (1993).
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Beljaars, A. C. M.

B. J. J. M. van den Hurk, P. Viterbo, A. C. M. Beljaars, and A. K. Betts, “Offline validation of the ERA40 surface scheme,” (2000).

Bendersky, S.

Betts, A.

F. Chen, K. Mitchell, J. Schaake, Y. Xue, H. L. Pan, V. Koren, Q. Y. Duan, M. Ek, and A. Betts, “Modeling of land surface evaporation by four schemes and comparison with FIFE observations,” J. Geophys. Res. 101, 7251–7268 (1996).
[Crossref]

Betts, A. K.

A. K. Betts and A. C. Beljaars, “Estimation of effective roughness length for heat and momentum from FIFE data,” Atmos. Res. 30, 251–261 (1993).
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B. J. J. M. van den Hurk, P. Viterbo, A. C. M. Beljaars, and A. K. Betts, “Offline validation of the ERA40 surface scheme,” (2000).

Bezombes, Y.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

Blaunstein, N.

Blay-Carreras, E.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

Boichard, J. L.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

Bosveld, F. C.

M. Braam, F. C. Bosveld, and A. F. Moene, “On Monin Obukhov scaling in and above the atmospheric surface layer: the complexities of elevated scintillometer measurements,” Bound-Layer Meteor. 144, 157–177 (2012).

A. Beljaars and F. C. Bosveld, “Cabauw data for the validation of land surface parameterization schemes,” J. Climate 10, 1172–1193 (1997).

F. C. Bosveld, “The KNMI Garderen experiment: micro-meteorological observations 1988-1989,” (De Bilt, 1999).

Bourdon, A.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

Bou-Zeid, E.

D. Li, E. Bou-Zeid, and H. A. R. De Bruin, “Monin–Obukhov similarity functions for the structure parameters of temperature and humidity,” Bound-Layer Meteor. 145, 45–67 (2012).

Braam, M.

M. Braam, F. C. Bosveld, and A. F. Moene, “On Monin Obukhov scaling in and above the atmospheric surface layer: the complexities of elevated scintillometer measurements,” Bound-Layer Meteor. 144, 157–177 (2012).

Butet, A.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

Campistron, B.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

Cheinet, S.

S. Cheinet, A. Beljaars, K. Weiss-Wrana, and Y. Hurthaud, “The use of weather forecasts to characterise near-surface optical turbulence,” Bound-Layer Meteor. 138, 453–473 (2011).

Chen, F.

F. Chen and J. Dudhia, “Coupling an advanced land surface-hydrology model with the Penn State-NCAR MM5 modeling system. Part I: Model implementation and sensitivity,” Mon. Weather Rev. 129, 569–585 (2001).
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F. Chen, K. Mitchell, J. Schaake, Y. Xue, H. L. Pan, V. Koren, Q. Y. Duan, M. Ek, and A. Betts, “Modeling of land surface evaporation by four schemes and comparison with FIFE observations,” J. Geophys. Res. 101, 7251–7268 (1996).
[Crossref]

Clifford, S. F.

R. J. Hill and S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. 68, 892–899 (1978).

J. C. Wyngaard and S. F. Clifford, “Taylor’s hypothesis and high-frequency turbulence spectra,” J. Atmos. Sci. 34, 922–929 (1977).
[Crossref]

Cook, D. R.

T. E. Twine, W. P. Kustas, J. M. Norman, D. R. Cook, P. Houser, T. P. Meyers, J. H. Prueger, P. J. Starks, and M. L. Wesely, “Correcting eddy-covariance flux underestimates over a grassland,” Agric. Forest. Meteor. 103, 279–300 (2000).

Coté, O. R.

J. C. Wyngaard, O. R. Coté, and Y. Izumi, “Local free convection, similarity, and the budgets of shear stress and heat flux,” J. Atmos. Sci. 28, 1171–1182 (1971).
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Couvreux, F.

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M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

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Evans, J. G.

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M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

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M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

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M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

Viterbo, P.

B. J. J. M. van den Hurk, P. Viterbo, A. C. M. Beljaars, and A. K. Betts, “Offline validation of the ERA40 surface scheme,” (2000).

Wacker, S.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

Ward, H. C.

H. C. Ward, J. G. Evans, O. K. Hartogensis, A. F. Moene, H. A. R. De Bruin, and C. S. B. Grimmond, “A critical revision of the estimation of the latent heat flux from two-wavelength scintillometry,” Q. J. R. Meteorol. Soc. 139, 1912–1922 (2013).

Wayne, D. T.

T. T. Leclerc, R. L. Phillips, L. C. Andrews, D. T. Wayne, P. Sauer, and R. Crabbs, “Prediction of the ground-level refractive index structure parameter from the measurement of atmospheric conditions,” Proc. SPIE 7685, 76850A (2010).
[Crossref]

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E. Webb, G. Pearman, and R. Leuning, “Correction of flux measurements for density effects due to heat and water vapour transfer,” Quart. J. Roy. Meteor. Soc. 106, 85–100 (1980).

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S. Cheinet, A. Beljaars, K. Weiss-Wrana, and Y. Hurthaud, “The use of weather forecasts to characterise near-surface optical turbulence,” Bound-Layer Meteor. 138, 453–473 (2011).

Wesely, M. L.

T. E. Twine, W. P. Kustas, J. M. Norman, D. R. Cook, P. Houser, T. P. Meyers, J. H. Prueger, P. J. Starks, and M. L. Wesely, “Correcting eddy-covariance flux underestimates over a grassland,” Agric. Forest. Meteor. 103, 279–300 (2000).

M. L. Wesely, “The combined effect of temperature and humidity fluctuations on refractive index,” J. Appl. Meteor. 15, 43–49 (1976).

Wilczak, J. M.

J. M. Wilczak, S. P. Oncley, and S. A. Stage, “Sonic anemometer tilt correction algorithms,” Bound-Layer Meteor. 99, 127–150 (2001).

Wildmann, N.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

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J. C. Wyngaard and S. F. Clifford, “Taylor’s hypothesis and high-frequency turbulence spectra,” J. Atmos. Sci. 34, 922–929 (1977).
[Crossref]

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[Crossref]

Xue, Y.

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[Crossref]

Yagüe, C.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

Zaldei, A.

M. Lothon, F. Lohou, D. Pino, F. Couvreux, E. R. Pardyjak, J. Reuder, J. Vilà-Guerau de Arellano, P. Durand, O. Hartogensis, D. Legain, P. Augustin, B. Gioli, I. Faloona, C. Yagüe, D. C. Alexander, W. M. Angevine, E. Bargain, J. Barrié, E. Bazile, Y. Bezombes, E. Blay-Carreras, A. van de Boer, J. L. Boichard, A. Bourdon, A. Butet, B. Campistron, O. de Coster, J. Cuxart, A. Dabas, C. Darbieu, K. Deboudt, H. Delbarre, S. Derrien, P. Flament, M. Fourmentin, A. Garai, F. Gibert, A. Graf, J. Groebner, F. Guichard, M. A. Jimenez Cortes, M. Jonassen, A. van den Kroonenberg, D. H. Lenschow, V. Magliulo, S. Martin, D. Martinez, L. Mastrorillo, A. F. Moene, F. Molinos, E. Moulin, H. P. Pietersen, B. Piguet, E. Pique, C. Román-Cascón, C. Rufin-Soler, F. Saïd, M. Sastre-Marugán, Y. Seity, G. J. Steeneveld, P. Toscano, O. Traullé, D. Tzanos, S. Wacker, N. Wildmann, and A. Zaldei, “The BLLAST field experiment: Boundary-layer late afternoon and sunset turbulence,” Atmos. Chem. Phys. Discuss. 14, 10789–10852 (2014).

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A. van de Boer, A. F. Moene, A. Graf, D. Schüttemeyer, and C. Simmer, “Detection of entrainment influences on surface-layer measurements and extension of Monin-Obukhov similarity theory,” Bound-Layer Meteor. 152, 19–44 (2014).

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

Q. J. R. Meteorol. Soc. (1)

H. C. Ward, J. G. Evans, O. K. Hartogensis, A. F. Moene, H. A. R. De Bruin, and C. S. B. Grimmond, “A critical revision of the estimation of the latent heat flux from two-wavelength scintillometry,” Q. J. R. Meteorol. Soc. 139, 1912–1922 (2013).

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Figures (9)

Fig. 1.
Fig. 1. Daily course of observed values of Cn2 (from an eddy-covariance station) for λ=670nm [(a) and (c)] and λ=2mm [(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).
Fig. 2.
Fig. 2. Reaction of rc to water vapor deficit Δq according to different schemes for the observed range of Δq. The reactions by dRH99 and BB97 were defined for the same grass area (Cabauw, The Netherlands), and are therefore both calculated by rcLAI/rs,min (LAI=2, rs,min=110sm1, being representative for Cabauw).
Fig. 3.
Fig. 3. Validation of estimates (model) of ra, rc, Ts, H, LvE, and QnG 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.
Fig. 4.
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.
Fig. 5.
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.
Fig. 6.
Fig. 6. Cn2 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).
Fig. 7.
Fig. 7. Sensitivity of derivations of Cn2 using the algorithm given in [49], to RH and Ta. Diamond markers indicate the 25th and 75th percentile of RH and Ta for the grass data of the Haarweg in 2005.
Fig. 8.
Fig. 8. rc and LvE estimates (model) versus measurements (obs) for the Haarweg (GC) using the approach for rc from dRH99, colored by Δq (light is 0, dark is 10gkg1).
Fig. 9.
Fig. 9. Validation of derivations of Cn2 from measured values for CT2, Cq2, CTq, and rTq, 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.

Tables (7)

Tables Icon

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

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

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Table 3. Land-Surface Input Variables per Datasetd

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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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Table 6. Sensitivity Estimates of Cn2 to the Main Input Parameters and Constants (or Parameters Directly Derived from Sin) for June 19, 2005, 12 UTC at the Haarwega

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Table 7. Land Surface Input Variables Used for All Sitesa

Equations (50)

Equations on this page are rendered with MathJax. Learn more.

LvE=s(QnG)+ρcpra(esatea)s+γ(1+rcra),
H=QnGLvE.
rc=104Δq=104esateapRdRv,
rc=rs,minLAIfdqfradfTfθ.
rc=25.9fdqfradfθ,
frrs,minLAICabauwfdq=25.9(1+hs(ΔqΔq0)),
(κz)2/3CT2T*2=6.7κ2/3(114.9zL)2/3,
(κz)2/3Cq2q*2=3.5κ2/3(14.5zL)2/3.
CTq=rTqCT2Cq2,
Cn2=AT2T¯a2CT2+Aq2q¯2Cq2+2ATAqT¯aq¯CTq,
AT=p¯T¯a(b1+b2q¯RvR¯),
Aq=p¯Ta¯bq2q¯RvR¯(1q¯(RvRd)R¯),
Ts=(Lout(1ϵs)Linϵsσ)1/4,
ra=ρcpTaTsH,
rc=ra(s(QnG)+ρcp(esatea)raγLvEsγ1),
rc=frrs,minLAIfdqfrad,
fdq=1+hs(ΔqΔq0),
frad=1000Sin+230(10002Sin)Sin(1000230).
Sout=aSin,
a=amax(1N)sin(α)(amaxamin)N(amaxacloud).
N=Sin,0SinSin,00.20.8,
Sin,0=I¯0(d¯sundsun)2cosθz.
Lin=ϵaσTa4,
ϵa,clear=0.63+5.95×107eae1500/Ta,
ϵa=N+(1N)ϵa,clear.
H=Qn,0+Qn,T(TsTa)G0GT(TsTa)LvE0LvET(TsTa),
TsTa=Hraρcp.
H=Qn,0G0LvE01+Qn,T+GT+LvET,
Qn,0=(1a)Sin+(ϵa1)ϵeσTa4,
G0=Ag(TaT24h).
LvE0=ρcpγesat(Ta)era+rc.
Qn,T=raρcpϵsσ4Ta3,
GT=raρcpAg,
LvET=raγs(Ta)ra+rc.
ra=1κu*(lnzz0hΨT(zL)+ΨT(z0hL)).
u*=uκln(zz0m)Ψu(zL)+Ψu(z0mL),
L=Tau*2κgTv*,
Tv*=Hvcpρu*,T*=Hcpρu*,q*=LvELvρu*,
Hv=H(1+0.61q)+0.61cpTaLvE0Lv,
Ts=Ta+Hraρcp+zΓd,
Lout=ϵsσTs4+(1ϵs)Lin,
Qn=SinSout+LinLout=Q0ϵsσ4Ta3(TsTa),
G=Ag(T24hTs)=G0+Ag(TsTa),
LvE=s(QnG)+ρcpra(esatea)s+γ(1+rcra)=LvE0+ρcpγs(Ta)(TsTa)ra+rc.
b1=106(0.237134+68.39397130λ2+0.4547338.9λ2),
b2=106(0.648731+5.8058×103λ27.115×105λ4+8.851×106λ6)b1,
bq2=b2,
b1=0.776×106,
b2=106(7500Ta0.056),
bq2=106(3750Ta0.056).

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