The Improvement of Turbulent Heat Flux Parameterization for Use in the Tropical Regions Using Low Wind Speed Excess Resistance Parameter

Reliable simulation of turbulent heat fluxes needed for modeling land‐atmosphere interactions remains a challenge over the humid tropical region. This may be connected with the inadequate parameterization of the roughness lengths for momentum (z0m) and heat (z0h) transfer usually expressed in terms of excess resistance factor (κB−1). This paper assesses the performance of existing κB−1 schemes developed for high wind speed conditions over the humid tropical region. Thereafter, a more appropriate κB−1 suitable for low wind speed condition is developed for use in the aerodynamic resistance parameterization. Based on observed surface heat fluxes and profile measurements of wind speed and temperature from Nigeria Micrometeorological Experimental site, new κB−1 parameterization was derived through the application of the Monin‐Obukhov similarity theory and Brutsaert theoretical model for heat transfer. The derived κB−1=6.66Re*0.02−5.47 , where Re* is the Reynolds number. Turbulent flux parameterization with this new formula provides better estimates of heat fluxes with reference to results from existing κB−1 schemes. The R2 increased by about 85%, while mean bias error and root‐mean‐square error in the parameterized QH based on the derived κB−1 reduced by about 63% and 66.7%, respectively. Similarly, the R2 increased by about 38%, while mean bias error and root‐mean‐square error in the parameterized QE based on the derived κB−1 reduced by about 47.8% and 52.6%, respectively. The derived κB−1 gave better estimates of QH than QE during the daytime. The derived κB−1 scheme corrects a well‐documented, large overestimation of turbulent heat fluxes, and it is therefore recommended for use in regions where low wind speed is prevalent.


Introduction
Realistic surface representation is required in climate models (CMs) to define the lower boundary condition (Pope et al., 2000;Williams et al., 2003). Diurnal variations of sensible and latent heat fluxes obtained from partitioning of available energy have a significant influence on climate simulation (Gao et al., 2004(Gao et al., , 2009. Reliable turbulent heat fluxes near the Earth's surface are necessary for improved weather and climate simulations (Chou et al., 2003). However, current operational CMs tend to systematically overestimate the diurnal range of turbulent heat fluxes for the tropical region, which might be related to inappropriate parameterizations for bare-soil heat transfer (Su et al., 1998;Trenberth, 2011;Zeng et al., 1997Zeng et al., , 2005. The turbulent heat fluxes can be described by means of the roughness length concept in Aerodynamic Resistance Approach (ARA) used in CMs. The aerodynamic (momentum, z 0m ) and thermal (z 0h ) roughness lengths are two crucial parameters for bulk transfer equation (Adeniyi, 2013). And both parameters are not physically based and thus cannot be measured directly. Their values usually are inversely derived from observations in field experiments or empirically estimated for practical applications. The logarithmic ratio of the two roughness lengths is called the excess resistance factor, κB −1 , where κ is the von Karman constant and B −1 is the Stanton number. The latter is related to the excess resistance to heat transfer. The behavior of the turbulent fluxes in the microlayer of the atmospheric surface layer (ASL) is controlled by this factor, and it is needed in most climate and hydrological models (Su et al., 1998;Zeng et al., 1997Zeng et al., , 2005. Several studies have developed different κB −1 schemes based on the synoptic observations of the field experiments for both vegetated and bare soil surfaces for relatively high wind speed conditions, though field studies over bare soil surfaces are still limited (Zeng et al., 1997(Zeng et al., , 2005. Despite the efforts of most climatic modelers, varied κB −1 algorithms are used by the various research groups and forecast centers. For this reason, the World Climate Research programme (WCRP) workshop on air-land interaction recommended the experimental verification of the widely accepted Brutsaert algorithm and intercomparison of various κB −1 algorithms (Su et al., 1998;Zeng et al., 1997Zeng et al., , 2005. However, different authors have carried out WCRP mandate in various regions especially in high-latitude regions where high wind speed is prevalent (Mölder & Lindroth, 2001;Overgaard, 2005;Zeng et al., 1997). However, studies on this are still very scarce for low wind speed conditions which are prevalent in the tropics. Stewart et al. (1994) determined κB −1 for eight semiarid areas and obtained values ranging between 3.5 and 12.5. The κB −1 calculated by Voogt and Grimmond (2000) for an urban site (London) using Brutsaert expression ranged between 13 and 27. Yang et al. (2007) evaluated several κB −1 schemes using the same data set in Japan. They obtained values ranging between 3 and 10. Zeng et al. (1997) formulated a new κB −1 scheme using a well-established algorithm in Arizona and compared with different κB −1 schemes. Liu et al. (2010) estimated κB −1 using seven schemes. They found that κB −1 has obvious diurnal variation, but no physical explanation was given for the large variability. Mölder and Lindroth (2001) and Molder and Kellner (2002) formulated κB −1 schemes for bare and vegetated surfaces using Brutsaert theoretical model in Sweden. Furthermore, some international experiments under the WCRP have verified the Brutsaert's formula for different κB −1 schemes. For instance, the European Center for Medium range Weather Forecast (ECMWF) retained the function form of the Brutsaert's formula. The Tropical Ocean Global Atmosphere (TOGA) also retained n = 0.25, while the n value was adjusted for the National Center for Environmental Prediction (NCEP) and Goddard Earth Observing System (GEOS); their excess resistance parameters were developed under moderate to high wind speed (wind up to 18 m/s; Brutsaert, 2005). Developed a comprehensive bulk algorithm using the data from the TOGA Coupled Ocean-Atmosphere Response Experiment under weak to moderate wind conditions (e.g., less than 12 m/s).
The above mentioned works formulated kB −1 schemes under high wind speed conditions and compared different κB −1 schemes for high-latitude regions. An expression for the κB −1 factor is yet to be formulated based on the tropical synoptic observations, and no intercomparision of κB −1 schemes of different global climate model (GCMs) is yet to be carried out in this region. Such an intercomparison is now possible because of the availability of several observational data sets taken under low wind conditions. In particular, Nigeria Micrometeorological Experiment (NIMEX) provides simultaneous hourly data of fluxes and bulk environmental variables over a tropical station under weak wind conditions (i.e., less than 3 m/s). The purpose of this study is to develop a more appropriate excess resistance factor ( κB −1 ) suitable for low wind speed condition and incorporate it into the aerodynamic resistance approach in the Regional Climate Models (RCM)s to simulate the diurnal pattern of turbulent heat fluxes. The performance of various existing κB −1 schemes developed for high wind speed conditions in simulating the turbulent fluxes will be assessed. radiation, and the soil (subsurface) parameters using low-response sensors. The EC system consists of an ultrasonic anemometer and a Krypton hygrometer. The sensors were mounted at 2-m height to capture the turbulent wind, acoustic temperature, and humidity components. The equipment was sampled at a frequency of 16 Hz in order to distinguish turbulent structures adequately (Foken, 2003). However, higher sampling frequencies enable a better distinction in moments of high and low kinematic stress, which results in a shortening of the estimated duration of a turbulent structure. The infrared thermometer was used to remotely measure surface temperature at a height of 1.8 m above the ground level. The water vapor and carbon dioxide analyzers were added to the setup to measure CO 2 and H 2 O vapor fluxes. Jegede et al. (2004) documented an overview of the NIMEX experiment.

Data Analysis and Quality Assessment
Several quality tests were regularly applied to the data, including controls of steady state flow conditions and intermittent turbulence. Simple visual test according to Foken (2003) was used on daily basis to check the quality of the basic meteorological variables (slow response), while TK2, a software package written by Mauder and Foken (2006), was used for the quality control and analysis of the EC data. The following processes were incorporated into TK2: 1. Spike detection method of Vickers and Mahrt (1997) based on Højstrup (1993) was used to remove the values that were not physically possible, before the calculation of the variances and covariances. 2. To determine the time delay between the sonic anemometer and Krypton hygrometer that was sampled at different frequencies, cross-correlation analysis was done for each averaging interval of the sensors. 3. Crosswind correction was done for the sonic temperature following Liu et al. (2005). 4. The planar fit method of Wilczak et al. (2001) was applied for coordinate transformation. Spectral corrections were done using the spectral models of Kaimal et al. (1972) and Højstrup (1981). 5. Conversion of buoyancy flux into sensible heat flux was done following Schotanus et al. (1983). 6. The latent heat flux was corrected for fluctuations in density and mean vertical mass flow according to Webb et al. (1980). 7. A test for steady state conditions and well-developed turbulence was done by applying the methods of Foken and Wichura (1996) and Foken et al. (2004). Details are in Mauder et al. (2007) and Adeniyi and Ogunsola (2012).

Bulk Aerodynamic Excess Resistance Algorithm (κB −1 )
Bulk aerodynamic κB −1 algorithm for the computation of turbulent heat fluxes is based on the Brutsaert theoretical framework (Brutsaert, 1979;1982), and it has been used by various authors over natural and artificial surfaces (Mölder & Lindroth, 2001). The algorithm is both applicable to vegetated and bare soil surfaces. Meanwhile, it has been documented that the turbulent structure over sparse and dense canopies varies significantly; thus, vegetation or lack-thereof can play a key role in determining κB −1 (Belcher et al., 2012;Finnigan, 2000;Raupach & Thom, 1981). Furthermore, the estimation of roughness sublayer height as related to canopy height might be a mere simplification but does not seem to be crucial for κB −1 estimation (Brutsaert, 1979;1982). The algorithm involves the estimations of the mass/heat transfer coefficients and roughness lengths for wind and temperature in the roughness sublayer. In addition, the algorithm also assumed neutral atmospheric condition at the top of interfacial roughness sublayer (Molder & Kellner, 2002;Brutsaert, 1982).
The atmosphere surface layer profile of the wind speed u and air temperature T a are given as (Foken, 2003) u where ψ m and ψ h are integrated stability function for heat transfer and momentum, respectively. κ is the von Karman constant, L is the Monin-Obukov length, T s is surface temperature, and T * is the scaling temperature (Molder & Kellner, 2002).
The κB −1 factor can be expressed as a difference of two resistance terms.
where st −1 o is the sublayer Stanton number, c do is the drag coefficient, and κ is the von Karman constant.
where T * is the scaling temperature From equations (1)-(3) Assuming neutral condition at the top of the interfacial sublayer of the surface layer where the zero-plane displacement d over a bare soil surface is taken as 0. Also, the value of ln z−d zoh or sublayer Stanton number has been known to be Reynolds number dependent (Kustas et al., 1989;Molder & Kellner, 2002) After simplification, we have the final expression for κB −1 as The functional form of Brutsaert expression took the form The Brutsaert theoretical framework scaled the exponent (n) of roughness Reynolds as Re 0:25 * (Brutsaert, 1965(Brutsaert, , 1975a(Brutsaert, , 1975b. Earlier proposed scaling results gave the value of n as 0.5 and 1 Zilitinkevich et al., 2001). The power laws "1/4" and "1/2" of Re both have strong theoretical foundations which have been widely used in various parameterizations, but 1/4 scaling is still in use to date Yang et al., 2003). The Brutsaert drag (b 2 = 2.75) and interfacial heat (b 1 = 2.46) coefficients were obtained from laboratory measurements using the surface renewal algorithm. The algorithm was based on the notion that at the interface the heat transfer is controlled by molecular diffusion which is translated into internal Kolmogorov scale eddies, and these eddies are renewed intermittently after random times of contact with the evaporating surface. This algorithm could not couple the near-surface molecular diffusion with the turbulent dynamic layer, and renewal rate of the eddies were not time dependent; the distribution would not be exponential, in which case the value of 1/4 would not be perfectly accurate. And turbulence parameters were not taken into consideration by this algorithm. In view of these limitations of the Brutsaert theoretical algorithm using surface renewal method, the EC technique gave room for field verifications. Moreover, experimental verification of the Brutsaert algorithm scaled the value of n as 0.30 and 0.50 by Mölder and Lindroth (2001) using Cartesian analysis. They varied the value of n from 0 to 1 at 0.1 steps; the n value corresponding to the minimum deviation was chosen (no theoretical augment. Furthermore, international experiments conducted by Global Energy and Water Cycle Experiment (GEWEX), GEOS, and NCEP obtained n values, ranging from 0.45 to 0.75 (Zeng et al., 1997).
The two scaling laws exponents (n = 0.25 and n = 0.50), drag, and Stanton number coefficients inferred from the Brutsaert theoretical algorithm were used to estimate the turbulent heat fluxes using ARA, and expected preliminary results showed strong overestimation during early and late hours of the morning (>40 W/m 2 ), the midday (>120 W/m 2 ), and nighttime hours (>10 W/m 2 ; see section 3.4). The observations from the results showed that the functional form of Brutsaert (1982) cannot be retained for this region. Motivated by this, we inferred the Brutsaert exponent and coefficients from NIMEX experimental data.

Estimating the Heat Transfer Coefficient (Stanton Number)
To obtain the Stanton number, the values of k 1 and n are determined from the value of (T s − T a) and T * using experimental data and Cartesian analyses (Mölder et al., 1999). Since the relationship between κB −1 and Re * has been established (Molder & Kellner, 2002;Mölder & Lindroth, 2001), we obtained the value of n by plotting the kB −1 value estimated using EC data against Re * value using power function regression Because kB −1 cannot be directly measured, we inferred the κB −1 from the measured friction velocity (u * ), stability parameter, and measured kinematic heat flux using Monin-Obukhov similarity theory (Brutsaert, 2005;Foken, 2003;Garratt, 1994;Voogt & Grimmond, 2000;Rigden et al., 2017) k 1 is the slope of the temperature difference between T s and T a plotted against the scale T* = κ À Á Re n * .

Estimating the Drag Coefficient
The drag coefficient in equation (10) is estimated using wind speed measurement, and z 0m is estimated using the EC technique. Zero place displacement is taken as 0 (bare surface). According to the Monin-Obukhov similarity theory (Monin & Obukhov, 1954), the gradient of nondimensional wind speed is written as Equation (13) is integrated to obtain the average wind speed u at height z.
where φ m z L À Á is the similarity universal function and ψ m z L À Á is the stability function of the wind profile which becomes 0 under neutral conditions. The aerodynamic roughness length z 0m was defined (Ma et al., 2002) where u, u * , and ψ m z L À Á are measured wind speed, friction velocity, and stability function from sonic the Spain Zeng et al. (1997) GEWEX (Global Energy and Water Cycle Experiment) China Yang et al. (2008) GEOS (Goddard Earth Observing system) United States Wang et al. (2004) NCEP (National centre for Environment prediction)

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anemometer. Since EC data are not always available, we proposed a new z 0m polynomial equation of order 2 using z 0m estimated from equation (15).

New Excess Resistance Parameter Estimated Using NIMEX Data
The new scheme for κB −1 takes the original form of Brutsaert theoretical model, but all the coefficients were obtained using the NIMEX synoptic data (see sections 3.2 and 3.3). The validity of the Brutsaert exponent was tested for performance in the equatorial atmosphere at the NIMEX site to ascertain the nature of atmospheric surface layer turbulence in this region.

Comparison With Other Tropical Overland GCMs κB −1 Algorithms
The different Tropical Overland κB −1 schemes used in CMs are presented in Table 1. Using observations reported by Zeng et al., 1997, Smith et al. (1990 and Garratt (1992), the Charnock (1955) and Roll (1948) roughness models used in the estimation of z 0m are appropriate for strong wind speed condition; in other words, they are inappropriate under weak wind speed condition. Therefore, the proposed z 0m was used in this work. The coefficients in GEOs are determined by interpolation between the relations of Large and Pond (1981) for moderate to large wind speed and relation of Kondo (1975).

Research Data Analysis 2.4.1. Coefficient of Determination (r2)
The ratio of explained variation, (X obser − X m ) 2 , to the total variation, (X obser − X m ) 2 , is called the coefficient of determination. X m is the mean of the observed X values. The ratio lies between 0 and 1. A high value of r 2 is desirable as this shows a lower unexplained variation.

Root-Mean-Square Error and Mean Bias Error
The root-mean-square error (RMSE) gives the information on the short-term performance of the correlations by allowing a term-by-term comparison of the actual deviation between the estimated and measured values. The lower the RMSE, the more accurate is the estimate. A positive value of mean bias error (MBE) shows an overestimate, while a negative value shows an underestimate by the model (Igbai, 1993).
The MBE is defined as the sum of the absolute value of the difference between the estimated and observed variables for the 24-hr period and is given as The RMSE is calculated to reflect the overall accuracy of the shape of the predicted curved and is defined as where n is the number of observations. The closer the estimated temperature is to the observed temperature, the smaller the RMSE. The RMSE tends to penalize large individual error so heavily and such as may be the better criterion of the performance (Evans et al., 1993).

Estimating the Drag Coefficients Using NIMEX Data
The measured friction velocity and wind speed from the EC system were used to obtain the mean values of cd − 1 2 0 for the low wind speed condition at

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the NIMEX site. The drag coefficients ( cd − 1 2 0 ) of bare soil surface were obtained using quation (10). Since measurements from EC system are not always available and roughness length models used in aerodynamic resistance approach in GCMs are only appropriate for high wind speed conditions, a new aerodynamic roughness length was proposed using the polynomial fit of order 2 from z om estimated from EC data and wind speed measurements. The proposed z om can be expressed as a function of wind speed (Fairall et al., 2003), Equation (18) is valid for weak wind speed condition (magnitude up to 3 m/s), with a coefficient of determination of 0.85 and standard error of 3.23×10 −4 .
The daily cd − 1 2 0 of the tested bare soil surface ranged from 5.21 to 5.81 for both periods. The mean cd − 1 2 0 was almost the same for the two experimental transitional periods, and this can be attributed to the reference height adopted for wind speed measurements using the EC technique (Foken, 2003). The diurnal variation of cd under low wind speed conditions and explained that the extensive wave breaking associated with high wind speed is not always common under low wind speed conditions. This factor has been known to reduced effective surface roughness and drag coefficient (Hwang, 2005). The mean surface roughness (z 0m ) ranged from 0.023 to 0.034 m for 2004, and 0.023 to 0.035 m for 2010. The correspondent mean z 0m for both periods were 0.026 and 0.025 m, respectively, with the average standard error of 0.0003 m. Since all periods have a fairly smooth aerodynamic condition, these values were used for z 0m in all subsequent analysis.

Estimating the Heat Transfer Coefficient (Stanton Number) Using NIMEX Data
The values of n and k were obtained from experimental data to determine the Stanton number using Brutsaert algorithm. The exponent (n) of the roughness Reynolds number was obtained from power function regression analysis of the κB −1 value estimated using EC data plotted against Re * value for 2004 and 2010, Ile-Ife site (Figures 2 and 3). The n value obtained for the experimental period (2004) was 0.020 under observed wind speed condition, ranging from 0 to 3 m/s. The approximated κB −1 factor obtained for stable and unstable conditions is expressed as The coefficient of determination is 0.99 with a minimum standard deviation of 0.05.
In addition, the n value obtained for the experimental period (2010) was 0.019 under observed wind speed condition, ranging from 0 to 3.1 m/s.

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The approximated κB −1 factor obtained for stable and unstable conditions is expressed as The coefficient of determination is 0.96 with a minimum standard deviation of 0.05. The mean value obtained for n was 0.02 for the two periods. It could be inferred that the n value obtained was much smaller than the ones obtained under high wind speed conditions (Molder & Kellner, 2002). Figure 2 also shows that the κB −1 factor increases as Re * increases, though lower Re * value was observed for the experimental sites with a range of 10-120. The higher range has been reported for pasture, vegetable surface, and forest (over Re * > 1000) under high wind speed conditions (Molder & Kellner, 2002;Zeng et al., 1997).
The temperature difference between T s and T a was plotted against the scale T * =κ ð ÞRe 0:02 * , where the power of 0.02 was obtained from experimental data. The average slopes of the data were 6.66 ( Figure 2) and 6.65 (Figure 4), both with a minimum deviation of 0.01 (The regression line was forced through the origin). The mean slope of the two experimental periods was 6.66 as shown in Figure 4 and 5 for both stable and unstable conditions. The Stanton number coefficient obtained is of the form The estimated Stanton number obtained ranged from 7.03 to 7.11. The same value was obtained for the two experimental periods (7.06).

Derived κB −1 Factor for Low Wind Speed Condition
The exponent fixed at 0.25 by Brutsaert functional form of the κB −1 factor was replaced by n = 0.02 from experimental data, though the Brutsaert exponent has only been found valid during the neutral condition in stable conditions for the equatorial region whenever drag and Stanton number coefficients are replaced with NIMEX field-based coefficients ( Figure 6). Substituting equation (19) and estimated mean value of drag coefficient in equation (10) where Re * is the roughness Reynolds number, Re * ¼ z0mu* ν . The kinematics molecular viscosity of air was assumed constant as 1.461 × 10 −5 m 2 /s. The empirical constants in equation (21) are roughness sublayer Stanton number and drag coefficient, respectively. The new κB −1 scheme had n value much more lower than n values obtained for high wind speed conditions. This implies that atmosphere surface layer turbulence associated with low wind speeds can be different from that at moderate to high wind speed. So, it can be said that low atmosphere turbulence is prevalent within the ASL of an equatorial atmosphere. The value of Re * decreases as the z 0m decreases for stable condition, while Re * increases as z 0m increases for the unstable condition.
The Brutsaert temperature roughness length for an equatorial atmosphere is therefore expressed as z 0h ¼ z 0m exp κ ð Þ 6:66Re 0:02 * −5:47 The new scheme κB −1 is good because we studied the general trend in surface-air temperature versus a relevant scale (here T * Re 0:02 * ). And it is  κu * respectively, where ρC p , r a , γ, e s , and e a are volumetric heat capacity, aerodynamic resistance to heat transfer, psychometric constant, saturated vapor pressure for soil, and saturated vapor pressure for air, respectively. Values of Q H and Q E were simulated using derived κB −1 and other six existing overland κB −1 schemes. Figures 6 and 7 showed the intercomparison of the composite diurnal variations of the modeled sensible heat fluxes obtained from different κB −1 schemes with the measured data for wet and dry days, respectively. The Q H fluxes estimated by GEOS and ECMWF were overestimated by 60% (>200 W/m 2 ) during the wet days and 50% during the dry days for the daytime period. The Q H estimated by NCEP showed a strong overestimation of about 45% (>180 W/m 2 ) and 47% (>198 W/m 2 ) for wet and dry days, respectively. The GEWEX κB −1 scheme shows closeness to the measured value during the early hours of the morning, while a strong overestimation of about 100 W/m 2 was still observed during the midday. The GEWEX κB −1 scheme seems to show better performance during the nighttime stable atmospheric condition. Most of the time, the NCEP κB −1 scheme overestimated Q H relative to the others, whereas the GEWEX scheme overestimated measured Q H in the morning hours when Q H is negative. The TOGA scheme showed closeness to the observed data, especially during the early hours of the morning, though an underestimation of about 30 W/m 2 was still observed during the midday. The reason is the fact that TOGA was developed under a moderate wind speed condition; the diurnal wind speed range is about 5 m/s higher than value obtained for this region. The derived NIMEX κB −1 scheme showed better performance for daytime and nighttime atmospheric conditions. The scheme gave almost zero bias in κB −1 with respect to observed κB −1 around 1100 hr LT and 1500 hr LT. This is not surprising because most of the field based coefficients were estimated from the NIMEX site. The large difference in wind drag coefficient may be primarily responsible for the overestimation by some schemes like GEOS, GEWEX, and NCEP. Also, the diurnal variation of Q H is similar for the different κB −1 schemes; however systematic differences exist in heat flux values. In Figure 7, the performance of some schemes like GEOS, NCEP, and ECMWF still showed overestimation of Q H greater than 80 W/m 2 for daytime and nighttime atmospheric conditions, respectively. The performance of the NIMEX κB −1 scheme was also very good for dry days, although slight overestimation of about 10 to 30 W/m 2 was observed during the early hours of the morning and late hours of the nighttime. The NIMEX κB −1 scheme gave reliable Q H estimation in the ARA. The TOGA scheme showed a slight overestimation (>15 W/m 2 ) during the morning, strong agreement with measured data during the midday, and strong overestimation during the night. This implies that the TOGA scheme is reliable for daytime Q H estimation. Figures 8 and 9 showed the intercomparison of the composite diurnal variations of the modeled latent heat fluxes from different κB −1 schemes with the measured data for wet and dry days, respectively. The Q E fluxes estimated by GEOS, NCEP, and ECMWF κB −1 schemes were strongly overestimated (>150 W/m 2 ) during the daytime for wet and dry days. The Q E  computed based on the NIMEX κB −1 scheme has almost zero bias with respect to the measured value during the early hours of the morning but a slight underestimation of about 25 W/m 2 during the midday. The TOGA scheme also showed closeness to the measured data during the early hours of the morning and late hours of the night, though strong underestimation of about 80 W/m 2 exists during the midday. During the dry days, all the schemes, except NIMEX and TOGA, simulated Q E with little or no bias compared to the measured data during the early hours of the morning and late hours of the night. And slight underestimations of about of 10 and 30 W/m 2 were obtained during the midday from NIMEX and TOGA, respectively. The validity of Brutsaert exponent for an equatorial atmosphere was tested in this work. The field-based exponent was replaced by 0.25, while other coefficients were retained in NIMEX κB −1 scheme. The Brutsaert exponent seems to be valid for both wet and dry days during the early hours of the morning (u > 0.02 m/s), while it gives strong overestimation during the midday and nighttime conditions (>120 W/m 2 ; Figures 6 and 7). It can therefore be concluded that the Brutsaert exponent is valid for neutral conditions during the stable atmospheric condition, but the exponent is not valid for unstable atmospheric conditions. The atmospheric condition is in line with the assumption of Brutsaert theoretical model for the neutral stable condition Table 2.

Evaluation of the Performance of the Different κB −1 Schemes
The daytime and nighttime MBEs, RMSEs, and r 2 values calculated for Q H and Q E are presented in Table 3. The NIMEX κB −1 scheme has the highest r 2 values for stable (0.82) and unstable (0.90) conditions, with  . The ECMWF, GEWEX, and GEOS showed strong overestimations with high MBE and RMSE for stable period during the wet days. The performances of the GEOS and ECMWF were also poor during unstable period for wet days. The performance of TOGA κB −1 was also good for stable and unstable condition. Generally, NIMEX κB −1 scheme has the best performance during the stable and unstable conditions for both wet and dry days. This is as a result of the following: first, the field-based coefficients are estimated from the observed NIMEX data. Second, there was a thorough study of the general trends in surface-air temperature difference versus a relevant scale (T * Re 0:02 * Þ; and third, the inclusion of the Re 0.0.2 term improved the r 2 value.

Conclusion
A derived excess resistance factor (κB −1 ) needed in the ARA was formulated using synoptic data from the humid tropical region, and Brutsaert theoretical algorithm was verified using NIMEX surface layer observations under low wind speed condition. The derived κB −1 scheme had n value much lower than n values obtained for high-latitude climatic region. This implies that atmosphere surface layer turbulence associated with low wind speeds can be different from that at moderate to high wind speed. Weak atmosphere turbulence is prevalent within the ASL of an equatorial atmosphere. The new κB −1 scheme was derived based on the general trend in surface-air temperature versus a relevant scale (here T * Re 0:02 * ). It is very simple to apply, especially in most synoptic stations where the estimation of interfacial heat transfer coefficients is not possible. However, the new κB −1 scheme has some limitations. First, it has empirical constant values for Stanton number and drag coefficients which can only be the same under similar experimental conditions at other sites. Second, it is limited to air flow over fairly smooth surfaces with low roughness Reynolds number range, 5 < Re * < 200. The inclusion of the new κB −1 factor improved the performance of ARA for the humid tropical region. The RMSE reduced from 56 to 25 W/m 2 for Q H ; and 66 to 16 W/m 2 for Q E . Therefore, the aerodynamic resistance due to heat transfer and momentum resistance cannot be assumed equal within the equatorial atmosphere. Turbulent fluxes generally reduce aerodynamic 1982aresistance during the daytime and increase it during the nighttime.
Using the conceptual framework developed by Brutsaert (1982), the micrometeorological measurements at NIMEX provided coefficients for the algorithm that adequately describe the nature of surface layer atmospheric turbulence at the sites concerned. The field-based empirical coefficients for the relationship between the Stanton number and the roughness Reynolds number were different from the ones based on laboratory measurements that Brutsaert initially derived. However, the Brutsaert exponent (0.25) seems to be valid for the neutral condition of stable conditions in the equatorial atmosphere. All of the κB −1 paraameterization schemes considered here showed different skills for turbulent fluxes simulation. In general, the new κB −1 factor, which corrects a well-documented overestimation of mean κB −1 by Brutsaert formulation (Trenberth, 2011;Zeng et al., 2005;Liu et al., 2010), performed better for these land sites and may be a better choice to be incorporated into current land surface models, especially for areas where low wind speed is prevalent. Lastly, the aerodynamic resistance approach has been tested on a period covering the transitional period from dry to wet weather conditions, but only for bare soil surface. Other types of vegetation need to be investigated, but it is believed that, due to the use of new κB −1 factor (and since vegetation is not a dominant factor in the estimation of κB −1 or z 0h ), the model will perform well regardless of the types of vegetation in a low wind speed condition. In case of snow cover, however, special treatment would be required. The result obtained encourages the use of aerodynamic resistance approach for evaluating turbulent heat fluxes as a means of determining the stability of the atmospheric surface layer. Also, the outcome of this work is most needed for verification purposes in most CMs, in order to ensure proper climate change prediction.