1 Introduction

The lower part of the troposphere in contact with the Earth's surface represents the atmospheric boundary layer (ABL). Being an essential and fundamental component of the Earth's climate, the investigation of the land surface–atmosphere exchange processes plays a prominent role in the ABL research including many related disciplines, e.g., weather forecasting, climate science, environmental impact studies, hydrology, and agricultural science. Quantifying air–land turbulent fluxes is of obvious relevance for modelling coupled atmosphere–land surface systems. Its study is critical for understanding pollution episodes because the pollutants emitted on the ground will be mixed and diluted in the ABL. Also, a stably stratified boundary layer (SSBL) remains a matter of keen interest in fluid dynamics as the airflow generates a wide range of processes encompassing all scales of the atmospheric circulation.

Traditionally and within the limits of a homogeneous underlying surface, the MOST describes the surface fluxes (i.e., the flux-variance and flux-gradient) in the surface layer (Monin and Obukhov 1954), including thermal and aerodynamic roughnesses (Sorbjan 1989; Kaimal and Finnigan 1994; Wyngaard 2010). In addition, the sources of turbulent mixing as wind shear production and buoyancy production (Deardorff 1972; Moeng and Sullivan 1994) are variable in space and time; hence, accurate measurements and their parameterization are a challenge (Baklanov et al. 2011; Holtslag et al. 2013). Even as these assumptions are observable in many situations for one-dimensional processes, they remain violated on the coastal footprint where horizontal gradients are significant. The relatively dry inland footprints (compared to the coastal zone) are almost entirely aerodynamically rough with a low heat storage capacity, reflecting a low nightly cycle of sensible heat. The ocean surface is relatively smooth (compared to the Earth's) and has a very high heat storage capacity, making it able to store heat for a protracted period, thus fostering turbulent mixing at great depths with impacts on the diurnal cycle over the water. In contrast, the less watery (relatively drier) inland areas are rough and have low heat storage capacity, conveying a more pronounced diurnal cycle of sensible heat flux. Hence, the upland footprints are aerodynamically rough surfaces (higher sensible heat flux relative to the coast), whilst the coastal (water/land interface) surfaces are relatively smooth (aerodynamically) owed to their vicinity to the ocean.

The interaction sought is based on the transfer of turbulent coefficients in terms of heat and drag that affects the wind between the coast and the mainland, with its impact on the application of MOST.

Notwithstanding consistent literature outlining multitudes of experimental investigations of MOST for a horizontal and relatively homogeneous surface-atmosphere (Högström 1988; Stull 1988; Garratt 1992; Kaimal and Finnigan 1994; Grachev et al. 2005; Wyngaard 2010), very few studies are performed on coastal footprints and then compared to those on onshore. For the most part, observational studies in the coastal zone are primarily associated with measurements of turbulent transfer coefficients such as the drag coefficient, with a focus on deviations from MOST, the trend of the surface wave field over the water (Geernaert 1988; Mahrt and Geernaert 1999; Grachev et al. 2017). In addition, some investigations have also been carried out on the African continent (e.g., HAPEX-Sahel in 1992, DACCIWA in 2016), which have contributed to the development and applicability of MOST (Businger et al. 1971; Merry and Panofsky 1976; Nieuwstadt 1984; Mahrt et al. 1998). The HAPEX-Sahel (cf. Goutorbe et al. 1994) and DACCIWA (cf. Flamant et al. 2017) campaigns were conducted on a small scale (one or two sites), while the AMMA campaign was expanded (eight locations as part of this work). The complexity of the climatic and living conditions (famine, water scarcity, high temperature, deregulated rainfall cycle, the atmosphere–surface interaction (continental and oceanic)-climate) regulated by the West African monsoon (WAM), has allowed the scientific community to focus on this region of West Africa through the AMMA project. The lack of data in this region makes this project unique in terms of its scope and the means used (financial, human resources, and instrumentation). The AMMA project intensified in 2006, called Special Observing Periods (SOPs), is subdivided as follows: the pre-monsoon (SOP1), peak monsoon (SOP2), and the post-peak monsoon (SOP3) (Redelsperger et al. 2006).

During SOP3, air masses move from the sea to the land. These air masses, which interact with those of the continent, are at the source of several processes, especially MCS (Mesoscale Convective systems), LLJ (Low-Level Jet), advection (surface inhomogeneity), LLCs (low-level stratus clouds), gravity waves (Guichard et al. 2010). Penide et al. (2010) examined the processes (MCSs coupled with other variables) causing precipitation (not shown) in West Africa and revealed their strong presence and lifetime in this area during SOP3. Calm nights can suddenly become unstable due to their occurrence.

Variations in the structure of ABL, as well as local meteorological conditions near the surface, considerably affect the propagation of electromagnetic waves. Moreover, the usual meteorological characteristics and atmospheric refractivity are directly linked. Actually, the increase (or decrease) in atmospheric refractivity varies depending on whether a radio ray is bending upwards (or downwards). In a shallow layer near the surface with a negative refractive gradient still called a duct, any ray bending toward that surface can be ‘trapped’ (Brooks et al. 1999; Kaissassou et al. 2015, 2020, 2021; Grachev et al. 2017). As a result, electromagnetic waves can travel through the duct. The present contribution, performed within the framework of the AMMA program and specifically at SOP3 (August 15–September 15, 2006), aims to better understand ducting in the coastal and inland regions. This knowledge requires the controlled air-coastal/onshore exchanges, as well as the flow and turbulence properties in the ABL. The two classes of areas referred above are the coastal areas (Dakar, Douala, and Nouakchott) bordered by the Atlantic Ocean, and the onshore areas (Abuja, Agadez, Bamako, Niamey, and Tombouctou). This article is structured as follows: investigation area and data are outlined in the next section, numerical tools and methodology in section 3, results and discussion in section 4, and lastly, section 5 presents the conclusion.

2 Study area and data

The AMMA field program helped to generate the coastal and inland observational data, available in the AMMA database, collected through the Vaisala RS-80 and RS-92 sondes, which sampled every 2 s and provided data with vertical resolution ranging from 5 to 12 m, depending on the characteristics of each site and the meteorological conditions. Each sensor has a different relative humidity bias, which varies according to time of day and temperature. Indeed, Vaisala RS-80 models are known to have a high dry bias in humidity during the day and a lower one at night. Meanwhile, the Vaisala RS-92 models have a moderate dry bias during the day and low humidity at night. Given all these uncertainties about the moisture bias of radiosondes, assessment studies were performed using estimates of the integrated water vapour content derived from Global Positioning System (GPS) ground station measurements during the AMMA special observation period 2006. Correction functions (not shown) were applied to all sensor types (Vaisala RS-80, Vaisala RS-92) to correct for radiosonde bias, with the aim of significantly reducing vertical moisture and temperature measurement uncertainties (Karbou et al. 2012). According to Steinbrecht et al. (2008), Vaisala RS-80 radiosonde systems reveal lower temperatures in the daytime than Vaisala RS-92 systems. They also report some specifications for each type of radiosonde at 1000 hPa (surface layer). For RS-80, temperature (K) bias: \(<\pm 0.2\), temperature accuracy: \(0.3\), pressure (hPa) bias: \(<\pm 0.5\), pressure accuracy: \(1.2\). For RS-92, temperature (K) bias: \(<\pm 0.2\), temperature accuracy: \(0.15\), pressure (hPa) bias: \(<\pm 0.5\), pressure accuracy: \(0.6\). This allowed us to be in the surface layer depending on the three levels (level 1, about 3 m above the surface of each site, then level 2, ~10 m and finally level 3, ~20 m) chosen, and to better characterize and understand the ducting in this layer of the ABL. The AMMA SOP3 is peculiar in that it occurs after the turbulent period of the WAM, and so conditions from ABL to SOP3 (August 15–September 15, 2006) can be considered stable at night-time. Measurements at each site highlight the multi-scale spatial inhomogeneities of the ABL. The development of ABL over heterogeneous regions is of particular interest because of the non-stationary conditions, which conflict with the underlying hypotheses of MOST and consequently with the coupled environmental prediction and evaporation duct designs. Of the 20 or so active sites (as shown in figure 1) during the campaign, only eight were selected based on the low percentage of missing data and the type of data to be used in this work. The data collected are separated into two subgroups of surveys per day, at 12:00 UTC and 00:00 UTC, during the investigated period. Those of the subset of concern to us are those taken at 00:00 UTC, i.e., under stable conditions.

Figure 1
figure 1

Summary map of AMMA’s meteorological stations over West Africa, where horizontal axis represents longitude coordinates (from left (west) to right (east)), vertical axis is latitude coordinates (from down (south) to top (north)). The coloured locations (blue for coastal (Dakar, Douala, Nouakchott) and orange for onshore (Abuja, Agadez, Bamako, Niamey, Tombouctou)) represent those in our investigation.

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Mentes and Kaymaz (2007) examined the surface ducting conditions over Istanbul, Turkey, using radiosonde measurements recorded at the Göztepe meteorological station located on the Asian coast. Steiner and Smith (2002) and Mesnard and Sauvageot (2010) show the unsuitability of radiosonde data in the study of ducts due to their poor resolution (about 100 m). Meanwhile, Bech et al. (1998) in Barcelona report that the occurrence of surface ducts using low-resolution radiosonde data is 83% of those captured using high-resolution radiosonde measurements. Agusti et al. (2010) add that new equipment was deployed during the 2006 AMMA field season, enabling high vertical resolution (ranging from 5 to 12 m, between two successive levels) observations. All data collected during the AMMA campaigns have been recorded and stored in raw form and available in a common database (information online at http://database.amma-international.org). The radiosonde data are stocked in their original form, i.e., with a vertical resolution of an average 10 times more refined than a standard World Meteorological Organization (WMO) sounding (Faccani et al. 2009). The raw radiosonde data have a sampling rate of 2 seconds during the balloon ascent. As a result, radiosonde data are being used for this investigation. Table 1 summarizes the details of the data used for each location.

Table 1 Geographical coordinates and station altitudes in meters above sea level (m a.s.l.) including rates of nightly averaged data used per station during AMMA SOP3 campaign.
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3 Numerical tools and methodology

Turbulent atmospheric data gathered during the 2006 AMMA SOP3 field campaign in West Africa are used to study the air-coast/inland coupling. Firstly, this work consists of outlining the daily averages of the nightly time series of some measured variables (wind speed and direction, air temperature, relative humidity) as well as other statistics (drag coefficient), including turbulent fluxes (sensible and latent heat) of the various sites, aiming to show the state of the meteorological conditions and the ruggedness of the surfaces, and the air-coastal/onshore footprints interaction. The drag coefficient is expressed as follows:

$${C}_{D}={\left(\frac{{u}_{*}}{U}\right)}^{2}$$
(1)

where \({u}_{*}={({\langle {u}^{\prime}{w}^{\prime}\rangle }^{2}+{\langle {v}^{\prime}{w}^{\prime}\rangle }^{2})}^{1/4}\) (in m/s) and U (m/s) represent the friction and wind velocity, respectively. u′, v′ and w′ are the wind turbulent fluctuations of longitudinal, lateral and vertical components of wind velocity.

The turbulent fluxes of latent heat \({H}_{L}\) and sensible heat \({H}_{S}\) can be written due to eddy-correlation method by:

$${H}_{L}={L}_{e}\rho \overline{{w }^{\prime}{q}^{\prime}}$$
(2)

and

$${H}_{S}={c}_{p}\rho \overline{{w }^{\prime}{\theta }^{\prime}}$$
(3)

\({L}_{e}\) is the latent heat of evaporation of water, \({c}_{p}\) is the specific heat capacity of air at constant pressure, \(q\) is the air-specific humidity, \(\rho\) is the mean air density, \(\theta =T{\left(\frac{1000}{P}\right)}^{\gamma }\) is the potential temperature, with \(\gamma = 0.286\), P is the pressure in hPa and T the temperature in Kelvin. The overbar represents an averaging operator.

Additionally, we compare and verify the validation of the universal Phi (\(\varPhi\)) functions whose empirical expressions were formulated by Kaimal and Finnigan (1994) in the 1968 Kansas field experiment, with the data collected from AMMA SOP3 under stable conditions (night-time) for the surface layer, over West Africa. The classical \(\varPhi\) functions from this landmark Kansas experiment were computed on flat, homogeneous surface footprints (Businger et al. 1971; Kaimal and Finnigan 1994). Thus, according to MOST, the non-dimensional vertical gradients of mean potential temperature (equation 4) and wind speed (equation 5), properly scaled, are universal functions of the Monin–Obukhov stability parameter (MOSP), scaled as follows:

$${\varPhi }_{h}\left(\zeta \right)=\left(\frac{\kappa z}{{\theta }_{*}}\right) \left(\frac{d\theta }{dz}\right)$$
(4)

and

$${\varPhi }_{m}\left(\zeta \right)=\left(\frac{\kappa z}{{u}_{*}}\right) \left(\frac{dU}{dz}\right)$$
(5)

with \(\kappa =0.4\) is the von Kármán constant, \({\theta }_{*}= -\overline{{w }^{\prime}{\theta }^{\prime}}/{u}_{*}\) is the scaling temperature.

Similarly, the normalized standard deviation of the velocity components \({\sigma }_{\alpha }\) (for \(\alpha =u\) or \(\alpha =v\)), air temperature \({\sigma }_{\theta }\) and vertical velocity component \({\sigma }_{w}\) are expressed as:

$${\varPhi }_{\alpha }(\zeta )=\frac{{\sigma }_{\alpha }}{{u}_{*}}$$
(6)
$${\varPhi }_{\theta }(\zeta )=\frac{{\sigma }_{\theta }}{\left|{\theta }_{*}\right|}$$
(7)
$${\varPhi }_{w}=\frac{{\sigma }_{w}}{{u}_{*}}$$
(8)

where \(\zeta =z/L\) is the MOSP (\(z\) is level and \(L=-({{\theta }_{00}{u}_{*}^{2}}/{\kappa g{\theta }_{*}})\) is the Obukhov length). \({\theta }_{00}\) here is the reference temperature near the ground.

The coupling between the different computations and the necessary settings made on each zone enables us to discuss this in the next section.

4 Results and discussions

4.1 Measured atmospheric turbulence parameters

The aim of the traditional time series is to present an overview of the distribution and sampling of relevant variables, the meteorological conditions of the region, and additional information not usually derived from the scattered plots. Figure 2 outlines the time series of wind speed and direction, air temperature, and relative humidity, averaged daily at each coastal site and over three levels, as shown in table 2. The time distributions of wind speed (all typically <6 m/s) and direction over the coastal footprints are nearly similar, with wind speeds over Douala near neutral. Douala also has cooler temperatures than Dakar and Nouakchott. Douala's surface humidity decreases from level 1 to level 3, Dakar's is deeper in level 3 than in the others, while in Nouakchott, the gaps are negligible. In figure 3, the Tombouctou site shows higher wind speed intensities on all three levels and low for the Abuja and Bamako stations. The wind speeds at the onshore parcels are almost similar to those at the coastal sites. The nocturnal time series of wind directions are patchy and typically range from 90° to 315°. Tombouctou and Agadez are warmer, with air temperatures around 30°C, while Abuja is colder (air temperature < 25°C). The air is more humid in Abuja and Bamako, but more pronounced in Abuja, reaching practically 100% on all three levels. This contrasts with the air in Tombouctou and Agadez, which is drier (RH < 75%), justifying their status as Sahelian cities. Based on the comparison of the two categories of sites in a global way, the studied parameters (wind speed, wind direction, air temperature and relative humidity) are quasi-similar in both types, in spite of some specificities.

Figure 2
figure 2

Time series of nightly averages (00.00 UTC) of wind speed (1st row), wind direction (2nd row), air temperature (3rd row), and relative humidity (4th row) over the three levels (see table 2) for the coastal footprints (Dakar, Douala, Nouakchott) from August 15 to September 15, 2006.

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Table 2 Instrumentation, parameters and averaged measurement height in meters above ground level (m a.g.l.) for each site during AMMA SOP3 campaign.
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Figure 3
figure 3

Time series of nightly averages (00.00 UTC) of wind speed (1st row), wind direction (2nd row), air temperature (3rd row), and relative humidity (4th row) over the three levels (see table 2) for the mainland footprints (Abuja, Agadez, Bamako, Niamey, Tombouctou) from August 15 to September 15, 2006.

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4.2 Turbulent fluxes and the MOSP

The magnitude of the averaged night-time values of friction velocity is approximately similar on both coastal and inland surfaces (see figure 4). Douala (figure 4a, b, c) outlines a relatively smoother surface than all the others; in contrast, Nouakchott and Tombouctou have the roughest surfaces. Friction velocity is thus disparately distributed and can therefore, fluctuate greatly (increasing or decreasing) from one night to the next. In view of these findings, friction velocity is not an accurate indicator to typify the two types of surfaces (coastal and inland).

Figure 4
figure 4

Time series of nightly averages (00.00 UTC) of friction velocity \({u}_{*}\) over the three levels (see table 2) for the coastal footprints (a, b, c) and mainland footprints (d, e, f) from August 15 to September 15, 2006. The symbols are the same as those in figures 2 (for coastal areas) and 3 (for mainland areas).

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Figure 5 explores the nightly averages of the drag coefficient plotted versus wind direction. This figure reveals a wind speed slope ranging from 100 to about 350° at coastal sites, while this slope ranges from 45 to 340° at inland areas. Despite the low values of the drag coefficient observed at the coastal sites of Dakar and Nouakchott, its amplitude remains close to that of the inland areas. The non-existence of \({C}_{D}\) values (figure 5a) for level 1 at the Douala site reflects the neutrality of the wind's velocity. The order of magnitude of \({C}_{D}\) over the investigated areas is \({C}_{D}=1.{10}^{-1}\), with the majority of the values below 0.4. This order contrasts with \({C}_{D}\approx 1.{10}^{-3}\) obtained by Grachev et al. (2017) over the North Carolina coastal area during the CASPER-East field experiment between October and November 2015, conveying relatively smoother surfaces. The influence of obstacles on the different sites, in general, is very inconspicuous. The turbulent air shear flow in the African coastal areas becomes as complex as that in the inland. Because of the discontinuity (inhomogeneity) of the coastal footprint and a similar order of magnitude, the drag coefficient cannot be an indicator between coastal and inland sites in West Africa.

Figure 5
figure 5

Nightly averages of the drag coefficient plotted versus wind direction for data recorded during the AMMA SOP3 field campaign (August 15–September 15, 2006) on the three levels (see table 2), for the coastal (a, b, c) and mainland (d, e, f) areas. The symbols are the same as those in figures 2 (for coastal areas) and 3 (for mainland areas).

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Figures 6 and 7 highlight the time series of turbulent fluxes (\({H}_{S}\) and \({H}_{L}\)) at each site. Traditionally, the sign convention specifies stable stratifications when \({H}_{S} < 0\) and unstable for \({H}_{S}>0\). In view of these features, we depict stable and unstable stratifications on the coast and inland (figures 6 and 7). Note also that the magnitudes of \({H}_{S}\) are larger inland than on the coast, values linked to the different specific heat capacities and aerodynamic properties of the underlying surfaces. In contrast, the amplitude of \({H}_{L}\) (figure 7) is virtually identical both on the coast and inland, reflecting the similarity of the two types of underlying surfaces in terms of latent heat flux. Additionally, figure 8 reports the temporal patterns of MOSP (\(\zeta =z/L\)); this figure shows the type, number, and degree (weak or strong) of stratification periods observed. According to the sign convention, \(\zeta >0\) indicates a stable stratification or a stable boundary layer (SBL) and unstable or a convective boundary layer (CBL), when \(\zeta <0\). The amplitude of \(\zeta\) is substantially more pronounced inland than on coastal surfaces, and its values in the surface layer as one moves away from the surface, i.e., the level. Except for the Douala location, the values at all coastal sites (at all levels), as well as those at level 1 of all inland sites, tend towards neutrality, i.e., \(\zeta \to 0\) (see figure 8a, b, c, d). These values are more prominent at levels 2 and 3 of the inland surfaces (figure 8e, f).

Figure 6
figure 6

Time series of nightly averages (00.00 UTC) of sensible heat flux \({H}_{S}\) over the three levels (see table 2) for the coastal footprints (a, b, c) and mainland footprints (d, e, f) from August 15 to September 15, 2006. The horizontal gray line indicates the boundary between a stable boundary layer (\({H}_{S}<0)\) and a convective boundary layer (\({H}_{S}>0)\). The symbols are the same as those in figures 2 (for coastal areas) and 3 (for mainland areas).

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Figure 7
figure 7

Time series of nightly averages (00.00 UTC) of latent heat flux \({H}_{L}\) over the three levels (see table 2) for the coastal footprints (a, b, c) and mainland footprints (d, e, f) from August 15 to September 15, 2006 of AMMA campaign. The symbols are the same as those in figures 2 (for coastal areas) and 3 (for mainland areas).

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Figure 8
figure 8

Time series of nightly averages (00.00 UTC) of MOSP \(\zeta =z/L\) on the three levels (see table 2) for the coastal footprints (a, b, c) and mainland footprints (d, e, f) from August 15 to September 15, 2006. The horizontal gray line indicates the boundary between a stable boundary layer (\(\zeta >0)\) and a convective boundary layer (\(\zeta <0)\). The symbols are the same as those in figures 2 (for coastal areas) and 3 (for inland areas).

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In the down part of the ABL (surface layer), the MOST describes the average temperature of a fluid in the layer, the aerodynamic and thermal roughnesses, and the flow as a function of height, under the conditions of a stable non-neutral atmosphere. Because of the jaggedness of the various surfaces, the results generated (e.g., the variance of the various fluxes) sparsely point to a neutral atmosphere, hence the violation of MOST.

The description and interpretation of the notion of atmospheric turbulence rely strongly on the upwind ‘flux markers’ on which the different statistics have been sampled. Typically, flux markers are locations that lie at some distance along with the upwind fetch and actually contribute to the initiation and sinks of a signal (Leclerc and Foken 2014). Kljun et al. (2004), Klaassen and Sogachev (2006), and Glazunov et al. (2016) indicate that existing marker models attempt to describe the position and extent of the surface contributing to the sampling of a turbulent flux as a function of windward distances, thermal stratification, ground roughness, and measurement height. Thus, for heterogeneous surfaces such as those in our study, measurements collected at different levels may have very different flux markers that correspond to either an aerodynamically rough or relatively smooth surface. This steep change in surface characteristics due to the coast-inland transition causes the development of an internal boundary layer (IBL). So, according to the observations of the measurements sampled during the AMMA SOP3 campaign, the sensible heat flux could be qualified as an indicator of the different surface footprints.

4.3 Applicability of MOST in the inland and coastal zones

The dataset collected during the AMMA SOP3 campaign permits us, in this section, to examine the classical MOST universal functions over heterogeneous terrain by comparing them to those established during the Kansas 1968 field campaign over uniform flat surfaces.

All correctly scaled turbulence statistics, especially the dimensionless vertical gradients of potential temperature (equation 4) and mean wind speed (equation 5), are universal MOSP functions (\(\zeta\)). Similarly, the normalized standard deviations of the wind speed and air temperature components are defined by equations (6 and 7), respectively. Although traditionally, field experiments establish the exact forms of universal functions, there is nevertheless asymptotic modelling in the prediction of these functions for the cases of strong stable stratification (\(\zeta >1\)) and extreme instability (\(\zeta <-1\)), through self-similarity assumptions (Stull 1988; Kaimal and Finnigan 1994; Wyngaard 2010). Studying similarity functions (equations 4–7) requires filtering the data to extract spurious ones, as described by Grachev et al. (2016). Figure 9 exposes the plot of the normalized standard deviation of the vertical velocity component \({\varPhi }_{w}\) as a function of the local MOSP (\(\zeta\)) of the collected turbulent data. These data were sectioned into stable (\(\zeta > 0\), figure 9a, b) and unstable (\(\zeta < 0\), figure 9c, d) conditions to evaluate the MOSP at different levels of the local scale. Other similar graphs of the normalized standard deviation of the longitudinal velocity component \({\varPhi }_{u}\) versus \(\zeta\) are shown in figure 10. From figures 9 and 10, the two universal functions \({\varPhi }_{u}(\zeta )\) and \({\varPhi }_{w}(\zeta )\) are consistent with the local-scale MOST predictions. Moreover, the averaged data from the different levels converge reasonably well to a unique universal curve; in particular, \({\varPhi }_{w}(\zeta )\) and \({\varPhi }_{u}(\zeta )\) are approximately uniform for \(\zeta > 0\). Grachev et al. (2017) prove that the same results are obtained for the normalized standard deviation of the lateral component of the velocity \({\varPhi }_{v}={\sigma }_{v}/{u}_{*}\), not shown (with \({\varPhi }_{v}\approx 1.80\) when \(\left|\zeta \right|\to 0\)). As expected in the near-neutral regime, the asymptotic boundaries predict that \({\varPhi }_{u}(0)>{\varPhi }_{v}(0)>{\varPhi }_{w}(0)\), underscoring the anisotropy of the airflow. Our results agree with the classical results produced by Panofsky and Dutton (1984), who argue that \({\varPhi }_{w}\left(0\right)=1.25\), \({\varPhi }_{v}\left(0\right)=1.91\) and \({\varPhi }_{u}\left(0\right)=2.39\), which are derived from data gathered from the flat surface overland under near-neutral stability conditions.

Figure 9
figure 9

Plots of the non-dimensional standard deviation of the vertical velocity component (\({\varPhi }_{w}\)) versus local MOSP (\(\zeta\)) for the nightly averages data recorded during the AMMA SOP3 field campaign (August 15–September 15, 2006) over the coastal (a, c) and mainland (b, d) footprints. Figures (a) and (b) correspond to stable conditions (ζ > 0), or the stable boundary layer (SBL); figures (c) and (d) reflect the unstable conditions (\(\zeta\)< 0), or the convective boundary layer (CBL). The dashed lines represent the \({\varPhi }_{w}=1.25(1+0.2\zeta )\) for ζ > 0 and \({\varPhi }_{w}=1.25{(1-3\zeta )}^{1/3}\) for ζ < 0.

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Figure 10
figure 10

As for figure 9, but for the normalized standard deviation of the longitudinal velocity component (\({\varPhi }_{u}\)). The dashed lines correspond to \({\varPhi }_{u}=2.3(1 + 0.2\zeta )\) for \(\zeta > 0\) (a, b) and \({\varPhi }_{u}=2.3{(1-3\zeta )}^{1/3}\) for ζ < 0 (c, d).

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Figure 11 shows the plot of the dimensionless standard deviation of potential temperature versus \(\zeta\). Grachev et al. (2003, 2008) show the ambiguity of \({\varPhi }_{\theta }\) for near-neutral conditions because when \(\zeta\) tends to zero, the temperature scale \({\theta }_{*}\) also tends to zero asymptotically. They add that under these near-neutral conditions, \({\sigma }_{\theta }\) has finite values associated with thermally heterogeneous surfaces. Regardless of the behaviours of these different quantities, we nevertheless find that the averaged data asymptotically follow the classical MOST curves (figure 11) except for the highly stable and near-neutral cases; indeed, \({\varPhi }_{\theta }\) is approximately constant for \(\zeta > 0\) (figure 11a, b), but decreasing \(\zeta\) leads to decreasing \({\varPhi }_{\theta }\) when \(\zeta < 0\) (figure 11c, d). Furthermore, the bulk of our computed \({\varPhi }_{\theta }\) values consistently lie above the plotted curves for our two classes of surfaces.

Figure 11
figure 11

As for figure 9, but for the normalized standard deviation of the temperature (\({\varPhi }_{\theta }\)). The dashed lines correspond to \({\varPhi }_{\theta }=2(1 + 0.5\zeta {)}^{-1}\) for \(\zeta > 0\) (a, b) and \({\varPhi }_{\theta }=2{(1-9.5\zeta )}^{-1/3}\) for ζ < 0 (c, d).

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So far, the results of the evaluation of normalized standard deviations of vertical velocity (\({\varPhi }_{w}\)), longitudinal velocity (\({\varPhi }_{u}\)) and temperature (\({\varPhi }_{\theta }\)) for our different surfaces using the data collected (nightly averaged) during the AMMA SOP3 campaign at multi-level in the surface layer (about 10%) of the ABL, qualitatively and closely corroborate with those from the flat and homogeneous areas where MOST is applied. Our results are found in the limit of low stability, i.e., \(0<\zeta <1\), and low convection (instability), i.e., \(- 1 < \zeta < 0\), on all surfaces of the eight locations studied in West Africa.

Equations (4 and 5) define the multi-level flux-profile relationships, or simply the link between the dimensionless vertical gradients of potential temperature (\({\varPhi }_{h}\)) and mean wind speed (\({\varPhi }_{m}\)) with the turbulent fluxes. \({\varPhi }_{h}\) and \({\varPhi }_{m}\) are estimated in a layer that is several meters deep. Indeed, their evaluation is based on the finite difference method such that to estimate the vertical gradients at an intermediate level \(x\), linear interpolations of the potential temperature and wind speed from the adjacent levels \(x-1\) and \(x+1\) are performed.

The dimensionless vertical gradient of mean wind speed \({\varPhi }_{m}\) is plotted as a function of \(\zeta\) and then plotted in figure 12 according to equation (5) for both stable and unstable conditions. It can be seen in figure 12 that the averaged data disagree with the canonical Businger–Dyer expression in the convective case (see figure 12c, d); this observation contrasts with that made in the stable case where the data appear to agree poorly with the classical expression (figure 12a, b). Similarly, analogous plots are made for the dimensionless vertical gradient of the mean potential temperature \({\varPhi }_{h}\) versus \(\zeta\) according to equation (4) and shown in figure 13. The scatter in the averaged data does not support the classical Businger–Dyer formulations. Thus, the flux-profile and more precisely flux-gradient similarity relations of \({\varPhi }_{m}\) and \({\varPhi }_{h}\) are not consistent. Exploiting eddy covariance data collected over a pine forest, and data obtained over an alpine slope, Rannik (1998) and Nadeau et al. (2013) respectively concluded that flux-gradient similarity performs worse than flux-variance similarity.

Figure 12
figure 12

As for figure 9, but for the non-dimensional vertical gradient of mean wind speed (\({\varPhi }_{m}\)). The dashed lines refer to the Businger–Dyer relationships \({\varPhi }_{m}=(1 + 5\zeta )\) for \(\zeta > 0\) (a, b) and \({\varPhi }_{m}={(1-16\zeta )}^{-1/4}\) for ζ < 0 (c, d).

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Figure 13
figure 13

As for figure 9, but for the non-dimensional vertical gradient of mean potential temperature (\({\varPhi }_{h}\)). The dashed lines refer to the Businger–Dyer relationships \({\varPhi }_{h}=(1 + 5\zeta )\) for \(\zeta > 0\) (a, b) and \({\varPhi }_{h}={(1-16\zeta)}^{-1/2}\) for ζ < 0 (c, d).

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The reason for the inconsistency of MOST in the plots of \({\varPhi }_{m}\) and \({\varPhi }_{h}\) versus ζ (figures 12, 13) is due to self-correlation. Indeed, the identical variables (e.g., the friction velocity \({u}_{*}\)) are involved in both the definition of MOSP and the universal functions, reflecting a weak trend in the data. However, this self-correlation problem can be adjusted by creating and plotting, for example, a ‘mixed’ universal function or a stability parameter by removing \({u}_{*}\). This mixing becomes the combination (multiplication) of any universal function with the inverse of another function such that we have terms \({\varPhi }_{m}{\varPhi }_{u}^{-1}\), \({\varPhi }_{h}{\varPhi }_{w}^{-1}\), \({\varPhi }_{u}{\varPhi }_{w}^{-1}\), and so on. Self-correlation does not affect these new hybrid functions because apart from the height \(z\), they do not share any variables with \(\zeta\). To prune this sensitivity from the self-correlation dependence, we define the following function:

$${\varPhi }_{m}{\varPhi }_{w}^{-1}=\left(\frac{\kappa z}{{\sigma }_{w}}\right)\frac{dU}{dz}.$$
(9)

This universal function of equation (9) yields the plot in figure 14. Figure 14 shows a dispersion of data that does not change significantly around the curves from the canonical formulations developed in Kansas, so that autocorrelation does not affect our results overall.

Figure 14
figure 14

As for figure 9, but for the function \({\varPhi }_{m}{\varPhi }_{w}^{-1}=(\kappa z/{\sigma }_{w})dU/dz\), which is a composite of the universal functions (5) and (6), for \(\alpha =w\), and is not impacted by auto-correlation. The dashed lines refer to \({\varPhi }_{m}{\varPhi }_{w}^{-1}=0.8(1+ 5\zeta )(1 + 0.2\zeta {)}^{-1}\) for \(\zeta > 0\) (a, b) and \({\varPhi }_{m}{\varPhi }_{w}^{-1}={0.8(1 - 16\zeta )}^{-1/4}(1-3\zeta {)}^{-1/3}\) for ζ < 0 (c, d).

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Since universal functions are a means of assessing local degrees of dispersion as well as general trends (figures 9–14), the bin-averaged data can produce additional information for us. Figures 15, 16, and 17 display the bin-averaged relationships of the non-dimensional functions \({\varPhi }_{m}\), \({\varPhi }_{h}\), and \({\varPhi }_{w}\) evaluated at different levels as a function of \(\zeta\) using data gathered for both stable and unstable conditions, and separately for coastal (Dakar and Nouakchott) and inland (Bamako and Niamey) flows. The two examples chosen per category (coastal and onshore) are based on the density of data collected, as presented in table 1.

Figure 15
figure 15

The bin-averaged non-dimensional universal functions of the vertical velocity component (\({\varPhi }_{w}\)) plotted versus the local MOSP (\(\zeta\)) for the nightly averages data recorded during the AMMA SOP3 field experiment (August 15–September 15, 2006) over two examples (Dakar and Nouakchott) of the coastal (a, b) and two examples (Bamako and Niamey) of the mainland (c, d) footprints. (a and c) correspond to stable conditions (ζ > 0); (b and d) reflect the unstable conditions (\(\zeta\)< 0). The black dashed lines represent the Businger–Dyer relationships reported by Kaimal and Finnigan (1994).

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Figure 16
figure 16

As for figure 15, but for the non-dimensional vertical gradient of mean wind speed (\({\varPhi }_{m}\)). The dashed lines refer to the Kansas experiment type-relationships output by Kaimal and Finnigan (1994). The horizontal gray line corresponds to \({\varPhi }_{m}=0\).

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Figure 17
figure 17

As for figure 15, but for the non-dimensional vertical gradient of mean potential temperature (\({\varPhi }_{h}\)). The dashed lines correspond to the Kansas experiment type-relationships written by Kaimal and Finnigan (1994). The horizontal gray line corresponds to \({\varPhi }_{h}=0\).

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By exploiting the bin-averaged values of \({\varPhi }_{w}\) observed at different levels during the coastal and inland flows in figure 15, we find that they collapse quite well to a single universal curve while remaining systematically above it; this means that the variance of the non-dimensional vertical velocity is a universal function of \(\zeta\). It can, therefore, be assumed that \({\varPhi }_{w}\) asymptotically follows the empirical Monin–Obukhov predictions, predicted for flat and homogeneous surfaces. The AMMA SOP3 data for \({\varPhi }_{w}\) tend to overestimate the canonical Kansas-type expressions for \(-1 < \zeta < 0\) (figure 15b, d). However, our \({\varPhi }_{w}\) values better corroborate (for \(0.01 < \zeta < 5\), figure 15c) the Kansas-type formulations under stable conditions while remaining slightly above them for several levels over the inland area; this can be explained by the fact that the stable stratification absorbs the vertical motion and the surface–turbulence interaction becomes nearly neutral. Similar results were achieved by Kaimal and Finnigan (1994) and Grachev et al. (2017), who proved that during onshore flow, the average data for \({\varPhi }_{w}\) melt quite well and closely follow the canonical Monin–Obukhov predictions, valid for homogeneous and flat terrain, in the CASPER-East campaign. The bin-averaged \({\varPhi }_{w}\) data for the coastal flow show less scatter between different collection levels (figure 15a) in a smaller range of the stability parameter \(\zeta\) compared to the \({\varPhi }_{w}\) values for the inland flow (figure 15c) under stable conditions. The reason for this statement is that the \(\zeta\) values are essentially closer around a mean value on the coast (too complex terrain), while the scatter is more pronounced in the mainland.

Figures 16 and 17 correspond to plots of bin-averaged mean wind speed \({\varPhi }_{m}\) (figure 16) and mean temperature \({\varPhi }_{h}\) (figure 17) as a function of \(\zeta\) for our two surface types. A weak match emerges in these two figures with the canonical Monin–Obukhov Kansas-type predictions (plotted as dashed lines), contrasting with the \({\varPhi }_{w}\) data. The bin-averaged \({\varPhi }_{m}\) and \({\varPhi }_{h}\) data obtained at different surface layer levels do not collapse entirely to a single curve, both in the coastal and inland areas, with a tendency for our observations to underestimate the MOST predictions. The dependence of \({\varPhi }_{m}\) and \({\varPhi }_{h}\) on \(\zeta\) appears to be very weak. However, the existing gap between the bin-averaged values versus MOST expressions for \({\varPhi }_{m}\) (figure 16) is less pronounced than that for \({\varPhi }_{h}\). Thus, we can note that the curves for \({\varPhi }_{m}\) versus \(\zeta\) are qualitatively better behaved as they collapse relatively well with the classical functions than the curves for \({\varPhi }_{h}\).

The representations of our different universal functions follow various behaviours depending on the case. This diversification of behaviours can be attributed to the local (haze, sinuous motion, small eddies) and non-local (momentum and heat dynamics) turbulent effects at small scales in the ABL. At these scales, these effects adapt instantaneously to changes in local conditions (due to surface heterogeneity) at large scales, allowing to maintain a dynamic thermal equilibrium caused by the large eddies. The standard deviations obtained in this work were reasonably expected due to the fact that the MOST were established for local scale turbulence and non-neutral conditions and, thus, for inhomogeneous terrains such as those studied here. Similarly, the derivatives of the expressions for the vertical gradients of mean wind speed and potential temperature in \({\varPhi }_{m}\) and \({\varPhi }_{h}\), whose estimates are based on a deep layer \(\Delta z\), are mass parameters. Overall, large motions (e.g., due to irregular surface heat and pressure variations) as well as eddy dispersion control the mean vertical gradients. Large eddies enhanced by the surface texture are responsible for transporting heat and momentum throughout the depth of the ABL. Under these conditions, vertical and local gradients can have opposite directions. For this reason, classical MOST models are said to overlook the dynamics of non-local effects in the ABL.

The statement made in the above paragraph can be considered as a specific approach to the study of classical MOST. Indeed, the study of the generalized classical MOST implies the integration of additional influences on the flux-gradient relations, namely the boundary layer height, the thermal conductivity, the Coriolis parameter and several other statistics.

5 Conclusion

Using radiosonde data collected during the AMMA SOP3 field experiment conducted from August 15 to September 15, 2006 (summer monsoon) in West Africa, we examined local (small-scale) atmospheric turbulence data. These data were observed on three levels of the ABL surface layer over two types of terrain, including coastal and inland footprints. Based on the good resolution of these radiosonde data, we performed a spatio-temporal analysis of the ABL structure under non-stationary conditions to understand the coastal/inland air coupling. They enabled a comparative study of turbulent fluxes with other statistics over a range of aerodynamically rough and dry footprints, and relatively smooth and less dry footprints.

  • A time-series analysis of the nightly averages of turbulent fluxes at each site was first performed. Unstable (\(\zeta < 0\)) and stable (\(\zeta > 0\)) stratifications were observed on each footprint type and site. Several time series of the different turbulent fluxes are similar from one surface to another. However, there are significant differences related to the thermal and aerodynamic properties of the lower surfaces. Although the differences between the drag coefficient, friction velocity as well as latent heat flux are not very noticeable, i.e., having the same order of magnitude for our range of surfaces, the nightly averages of the sensible heat flux are, however, more pronounced over the inland surfaces than over the coastal surfaces, due to the discontinuity of the heat capacity between the areas. Thus, sensible heat flux can be viewed as an indicator of different surface types.

  • We basically focused on testing and approving MOST on the surface types using the dimensionless empirical functions according to MOST plotted as a function of \(\zeta\) for the stable and unstable conditions, as well as the bin-averaged values scaled versus \(\zeta\). The scatter plots of the nightly averages of the individual values of the scaled classical functions of \({\varPhi }_{w}\), \({\varPhi }_{u}\), and \({\varPhi }_{\theta }\) prove that the dimensionless standard deviations globally adhere to MOST within the stability conditions and the uncertainty of the experiment. Furthermore, the scatter of the \({\varPhi }_{\theta }\) data compared to the classical MOST curves is likely to be related to heterogeneous field temperature. Also, statistically, \({\varPhi }_{w}\) has a better MOSP dependence than \({\varPhi }_{u}\). A careful study of the bin-averaged plotted data shows that \({\varPhi }_{w}\) converges better at multi-level to a single universal Kansas-type curve for inland flow than for coastal flow while remaining consistently above it in the ranges \(0.01 <\zeta < 5\) and \(-1 <\zeta < 0\).

  • The dimensionless vertical gradient of mean wind speed \({\varPhi }_{m}\) and potential temperature \({\varPhi }_{\theta }\) for our observations on the coast show a poor match with the Businger–Dyer and Monin–Obukhov empirical predictions than inland. Compared to \({\varPhi }_{w}\), \({\varPhi }_{h}\) and \({\varPhi }_{m}\) show, statistically, very little or no dependence on \(\zeta\). This suggests that coastal flow requires coastal-specific coding.

  • In view of the above, we can say that the conjunction of local and non-local turbulent effects is responsible for the divergent behaviour of the diverse universal functions in the ABL. The turbulent measurements observed at a single level enabled us to establish the scaled deviations of MOST associated with uncodified small- and large-scale turbulent motions (small and large eddies, for example). However, local features of the airflow (local small-scale fluxes and gradients) are strongly responsible for small-scale turbulence, while large-scale forcing has less influence on stratification. Since the flux–profile relationships are based on measurements inside the thick layers and controlled by motions and structures (small eddies, small structures, and other local motions) considered as ‘noises’, the classical expressions become inadequate to describe \({\varPhi }_{h}\) and \({\varPhi }_{m}\) in the coastal environment. Thus, the coastal zone, which is the sea/land interface, becomes more complex and deserves further investigations to obtain a better match of \({\varPhi }_{h}\) and \({\varPhi }_{m}\) with the classical MOST functions.