## 1. Introduction

In many micro- to regional-scale air quality models, atmospheric stability is considered a dominant meteorological parameter that impacts model performance and results. Therefore, the method used to quantify atmospheric stability could have major implications for air pollution simulations (Zoras et al. 2006; Behrendt et al. 2011; Valdebenito et al. 2011). Historically, atmospheric stability has been calculated through the Pasquill stability classification [e.g., the Industrial Source Complex (ISC3) model; Seinfeld 1986], which only considers surface wind speed, daytime incoming solar radiation, and nighttime cloud cover. More recently, atmospheric stability has been alternatively estimated using the surface energy balance method (Holtslag and van Ulden 1983), which incorporates the thermal and mechanical characteristics of different land-use types in several air quality models [e.g., the American Meteorological Society/Environmental Protection Agency Regulatory Model (AERMOD; USEPA 2004) and the California Puff modeling system (CALPUFF; Scire et al. 2000)], as well as other photochemical models, such as Comprehensive Air Quality Model with Extensions (CAMx; ENVIRON 2015) and the Community Multiscale Air Quality (CMAQ; Byun and Ching 1999) model. In this approach, the accuracy of the land-use classification is critical for producing accurate simulation results. For example, Cheng and Byun (2008) and Cheng et al. (2008) used the fifth-generation Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model (MM5) for simulations across the Houston–Galveston, Texas, metropolitan area. These authors suggested that using land-use and land-cover (LULC) datasets from the Texas Forest Service, which are much closer to the actual LULC types than the previously recommended LULC dataset from the U.S. Geological Survey (USGS), could improve meteorological simulation results (e.g., boundary layer heights and local circulation patterns for large emission sources). In the simulations by Cheng and Byun (2008) and Cheng et al. (2008), the improved LULC dataset resolved several significant issues, such as the overestimation of daily maximum temperatures and the delay in simulated extreme wind speed. The improved model also further enhanced the performance of the air quality model.

In an air quality model that considers LULC, atmospheric stability is calculated by 1) determining the surface net radiation flux, 2) partitioning the amounts of latent heat *Q*_{e} and sensible heat fluxes *Q*_{h} based on the Bowen ratio (*B* = *Q*_{h}/*Q*_{e}) for different seasons and land-use types, and 3) quantifying the vertical diffusion parameters that may influence air pollutant concentrations through the determination of the Monin–Obukhov length scale *L*, which is estimated from friction velocity *u*_{*}, potential temperature, and *Q*_{h}, or some experimental approaches (Pal et al. 2013, 2015). Although calculations of atmospheric stability using air quality models have improved with the use of Bowen ratios, CALPUFF employed the 14-category USGS land-use classification system (Scire et al. 2000) but used only a single default value for the Bowen ratio to describe the entire year (and, thus, did not include seasonal dynamics). AERMOD adopted the approach used by Paine (1987), which specifies the seasonal Bowen ratio by land-use type (USEPA 2004). In 2013, the U.S. Environmental Protection Agency (USEPA) suggested the use of a 21-class land-cover classification (i.e., the National Land Cover Dataset 1992; USEPA 2013). Although AERMOD did consider possible variations in Bowen ratios across different seasons and humidity conditions, changing the ratios by season did not account for the small hourly changes in Bowen ratios that were more relevant to the temporal resolution considered. The meteorological models used in other photochemical models include MM5 and the Weather Research and Forecasting Model. Previous studies have also explored the influences of different land-use classifications in Bowen ratio estimates (Chen et al. 2001; Lee et al. 2004; Quintanar et al. 2008).

Daytime Bowen ratios are also often different from nighttime values. Daytime Bowen ratios usually range between 0 and 2, while nighttime values often vary more widely from −50 to large positive values (Cook 2007). Bowen ratios vary through time because of the strong diurnal fluctuations in net radiation and their associated effects on latent and sensible heat fluxes. Latent heat fluxes are orders of magnitude higher during the day, leading to more stable Bowen ratios. On the other hand, during the night, net radiation fluxes are usually small or negative, so latent and sensible heat flux magnitudes are much smaller, leading to larger variations in Bowen ratios. Precisely because of the relatively smaller sensible heat flux, the atmospheric is usually stable during the nighttime. Several previous studies related to vapor circulation and energy-exchange processes concluded that significant variations in nighttime Bowen ratios could possibly affect the simulation results. Therefore, it was suggested that small or negative values for nighttime Bowen ratios should not to be utilized in the calculations (Ortega-Farias et al. 1996; Unland et al. 1996). Thus, the methods used to calculate daytime Bowen ratios are more important than those used for setting nighttime ratios in air quality models. Consequently, the objective of this study was to improve methods for determining daytime and hourly Bowen ratios for applications in air quality models used to calculate atmospheric stability.

## 2. Model description and validation

### a. Model description

According to the fundamental principles of thermodynamics and micrometeorological approaches, it was assumed that, when the surface air maintains contact with the ground surface for a long enough time (i.e., nearly balanced conditions), the amount of vapor (humidity) in the air can be considered the “index for land moisture content” (Hanel 1976). In addition, if plants cover a large land surface area, the vapor flux resulting from evapotranspiration would be reflected in the observed humidity levels (Hargreaves and Samani 1982). These were considered the basic assumptions in the development of our approach.

*B*is a widely used surface parameter that represents the land-use-specific characteristics of surface energy composition and is defined as the ratio of the sensible heat flux to the latent heat flux at the ground surface (Bowen 1926):In Eq. (2.1),

*Q*

_{h}at the ground surface can be calculated using the vertical profiles of the air temperature:where

*ρ*is the air density (kg m

^{−3}),

*C*

_{p}is the specific heat of the air at constant pressure (J kg

^{−1}°C

^{−1}), ∂

*T*/∂

*z*is the vertical temperature gradient near the ground (°C m

^{−1}), and

*K*

_{e}is the turbulent diffusivity (eddy viscosity).

*Q*

_{e}in Eq. (2.1) can be quantified using the vertical gradient of humidity:where

*L*

_{e}is the latent heat for the evaporation of water and

^{−3}).

*w*can be calculated as the ratio of vapor mass to dry air mass in a certain volume of air, as shown:where

^{−3}),

*M*

_{a}is the molecular weight of air,

*M*

_{υ}is the molecular weight of water vapor,

*e*is the vapor pressure (mb or hPa),

*p*is the atmospheric pressure (mb or hPa),

^{−1}mole

^{−1}),

*T*is the air temperature (absolute temperature; K), and ε is a constant (equal to

*M*

_{υ}

*/M*

_{a}= 0.625). Furthermore, the approximation of the rightmost part of Eq. (2.4) can be made because the order of the magnitude for atmospheric pressure is much greater than that of vapor pressure under realistic atmospheric conditions. Equation (2.4) could then be expressed according to the following relationship:

*Q*

_{e}, the latent heat flux can be converted to the following:where

*L*

_{e}is the latent heat of water. Finally, by substituting the equations for

*Q*

_{h}[Eq. (2.2)] and

*Q*

_{e}[Eq. (2.6)] into the equation for the Bowen ratio [Eq. (2.1)], we can calculate the Bowen ratio according to the following equation:To further simplify Eq. (2.7), we substituted the equation for relative humidity (RH), whereand

*e*

_{s}is the saturated vapor pressure at a given temperature. It was further assumed that vapor resulting from evapotranspiration would be present as a result of the large gradients in the boundary resistance layer, which is only a few centimeters thick. As a result, the relative humidity should change substantially only in the viscous sublayer of the ground surface or a leaf surface, while the relative humidity should remain constant outside of these viscous sublayers (i.e., the boundary resistance layer). Also for the thermodynamic aspect, all of the deduction considered the vertical direction, and so combined Eqs. (2.7) and (2.8) to obtain the following relationship:

*T*is the absolute temperature (K). By using Eq. (2.12), the hourly Bowen ratio can be determined from relative humidity and air temperature observations acquired from local weather stations. We can then substitute the calculated Bowen ratio into the surface energy balance equation to calculate the sensible heat flux and final daytime atmospheric stability.

### b. Model validation

Over the past decade, a variety of studies have been conducted to estimate Bowen ratios as functions of land cover; this research has been made possible, in part, by large improvements in remote sensing technology, computing power, and storage speed. Such studies have been applied to measurements of evapotranspiration (Kosugi et al. 2007), the transmission of latent and sensible heat (Prueger et al. 1998), and CO_{2} sequestration (Dugas et al. 1999). Other methods used to estimate Bowen ratios typically calculate ratios based on estimates of latent heat flux and assumptions regarding ground surface heat flux. Among these methods, the Penman–Monteith equation has been evaluated at 11 dry–wet areas around the world using 20 internationally famous evapotranspiration empirical formulas. These formulas were recommended by the American Society of Civil Engineers in 1990; published by the International Committee of Irrigation and Drainage in 1994; continuously applied and verified by the Food and Agriculture Organization (FAO); recommended and updated by the FAO in 1977 and 1984, respectively; applied to the concepts of crop canopy resistance and aerodynamic resistance (Monteith 1981); and further revised and recommended in 1998 (Allen et al. 1998). This approach is the only model for estimating evapotranspiration that has been recognized by professional and academic institutions around the world. This model includes a radiation term for the source of evapotranspirated heat and an aerodynamic force term as the driver of evapotranspiration. Although the Penman–Monteith model is considered a strong approach for estimating evapotranspiration, it requires a large number of parameters that can be difficult to estimate. In particular, wind speed can easily be influenced by local environments and is not available at many weather stations. The method was developed by Priestley and Taylor (1972) and combines aerodynamics and the energy balance by simplifying the Penman–Monteith model.To verify the reliability of the equation proposed in this study, the Penman–Monteith (PM) method and the simplified Priestley–Taylor (PT) method were used to verify the estimates of latent heat flux that were calculated by our proposed equation.

## 3. Results and discussion

### a. Sensitivity analysis for the Bowen equation

An analytical expression for calculating the Bowen ratio was derived in this study using parameters for absolute temperature and relative humidity [Eq. (2.12)]. The equation’s sensitivities to changes in temperature (assuming a fixed relative humidity) and relative humidity (assuming a fixed temperature) are shown in Figs. 1 and 2, respectively. When the relative humidity was maintained at a fixed value, the Bowen ratio decreased as the absolute temperature increased (Fig. 1). In this situation, the saturated water vapor pressure increased with the rising temperature, increasing the amount of water vapor contained in the air. In other words, when the water content at the ground surface was sufficient, more energy was consumed for evapotranspiration, and ground surface water was transformed into vapor. As this occurred, the latent heat flux (*Q*_{e}) increased, causing the corresponding Bowen ratio to decrease. In addition, when the temperature was under 290 K, changes in the Bowen ratio were substantial.

When the absolute temperature was maintained at a fixed value, the Bowen ratio decreased as the relative humidity increased (Fig. 2). Because relative humidity was defined as the ratio of the air’s vapor pressure to the saturated vapor pressure at a given temperature, the air’s vapor pressure increased with relative humidity. In this case, a larger volume of surface water evaporated and was transformed into vapor, leading to a higher net available energy and latent heat flux. This caused the Bowen ratio to decrease as a result. Furthermore, when the relative humidity was under 0.2, the changes in the Bowen ratio were substantial.

In Taiwan, the temperature ranges from 293 to 311 K, and relative humidity ranges between 0.6 and 0.9. According to Figs. 1 and 2, the Bowen ratio would be >1 if the relative humidity was <0.44 at 293 K and >1 if the relative humidity was <0.25 at 303 K. At a relative humidity between 0.6 and 0.9, the Bowen ratio would be >1 if the temperature was <287 K. Table 1 shows the Bowen ratio at different combinations of these two parameters in Taiwan. Therefore, under common weather conditions in Taiwan, changes in the theoretical Bowen ratio are predicted to be small, and it would be rare to find a Bowen ratio > 1. The Bowen ratio was predicted to be >1 only at low temperatures, such as in mountainous regions. This result also indicates that evapotranspiration is very large in Taiwan.

Bowen ratios as a function of the parameter values used for temperature and RH.

### b. Case study and monitoring data

To validate our model performance and to compare the results with the PM and PT methods, we selected study cases from Central Weather Bureau weather stations in Taiwan. According to the basic assumptions and applications of the PM and PT approaches, these two methods are more adequate for quantification of latent heat flux in vegetated landscapes like grasslands, wetlands, or farmlands. Therefore, two representative agriculture-landscape weather stations, Yilan station (24°45′56″N, 121°44′53″E, and 7.2 m AGL) at the center of Lanyang Plain in northeastern Taiwan and Chiayi station (23°29′52″N, 120°25′28″E, and 26.9 m AGL) at Chianan Plain in southwestern Taiwan, were chosen for model validation and comparison. According to the long-term climatic statistics, the annual mean temperature is 23.1°C at Chiayi and 22.5°C at Yilan and the seasonal variations of the temperature at both stations are very mild, so there is no frost season at either station. To explore how the latent heat fluxes and Bowen ratios vary over different seasons under distinct radiative environments, the simulation periods were chosen from 19 to 21 January for wintertime and from 22 to 24 July for the summer season during 2006 for both weather stations.

Figures 3a and 4a show the variations of the latent heat fluxes obtained using the Bowen ratios estimated by the three methods and data from the Yilan and Chiayi weather stations during the winter. For all three days of simulation, the latent heat fluxes gradually increased starting at 0800 local time (LT). At noon, the latent heat fluxes reached the peak value and then declined. On the first and second days, the latent heat fluxes were all above 100 W m^{−2}. These trends were similar for both weather stations and for all three methods of estimation. The fluxes for the first day were slightly higher than those of the second day for all methods of estimation at the Yilan weather station and for the PT and Bowen ratio methods at the Chiayi weather station. However, the PM estimation method generated slightly higher fluxes for the second day compared to the first at the Chiayi weather station because of the enhanced winter monsoon, where a higher wind speed led to the differences in the latent heat fluxes in the methods. On the third day, the latent heat fluxes were much lower than those of the first two days for all methods of estimation and at both weather stations as a result of the weather conditions.

Figure 3b shows the simulation results for summer at the Yilan weather station. On the first day, the latent heat fluxes gradually increased, starting at 0700 LT and peaking at noon. After noon, the fluxes declined for all methods of estimation. On the second day, the maximum values for the latent heat fluxes occurred later, from 1300 to 1500 LT. Trends on the third day were similar to those on the first day, with lower maximum values at 1200 LT on the third compared to the first day. At 1100 LT, the latent heat fluxes suddenly dropped as the cloud cover increased and then began to gradually decrease starting at 1200 LT. These trends were similar for all three methods of estimation.

Results also showed that the latent heat fluxes measured from the PM method were the highest, while the fluxes quantified by the proposed Bowen ratio method were the lowest. The latent heat fluxes obtained through the PM method were particularly different from those estimated by the other two methods on the third day of measurement during the study period in the winter. Results differed less between the estimation methods in the summer season. Latent heat fluxes obtained using the Bowen ratio and PT methods were particularly more consistent (Fig. 3).

The simulation results for summer at the Chiayi weather station are shown in Fig. 4b. Latent heat fluxes began to increase at 0600 LT on the first two days, reaching maximum values at noon before declining. On the first two days, the latent heat fluxes were all above 400 W m^{−2}. On the third day, because of the weather conditions, the maximum values for latent heat fluxes were lower than those observed on the first two days, with peaks between 1000 and 1200 LT. At 1100 LT, following an increase in cloud cover, the latent heat fluxes suddenly dropped and then gradually continued to decrease after noon. These trends were similar for all three methods.

In addition, the values for the latent heat fluxes obtained using the PM method were the highest, while those estimated through the Bowen ratio method were the lowest. At the Chiayi station, the results among the three methods were closest during the summertime study period, while the fluxes quantified from the Bowen ratio and PT methods were relatively more consistent for both the winter and summer study periods.

In general, all three of the methods have very consistent patterns that characterize the dynamics of latent heat fluxes in the simulations. However, the fluxes estimated by the PT and the proposed Bowen ratio methods showed a much closer relationship. This is likely because the PT method is a simplification of the PM method and does not consider the aerodynamic mechanism (wind speed and aerodynamic conductance), which is similar to the Bowen ratio method derived in this study. However, the advantage of this proposed Bowen ratio method is that it required only the parameters of relative humidity and atmospheric temperature, which are more applicable and easier to obtain from the routine measurement data at the weather station. However, on the other hand, the traditional PT and PM methods are more complicated and require more data such as radiation and energy fluxes to estimate the Bowen ratios. Thus, the proposed Bowen ratio method could allow users from a variety of applications to more easily estimate Bowen ratios without the use of more complicated parameters that may not be available from the ordinary weather stations. In addition, the proposed methodology would allow for the more realistic calculation of hourly atmospheric stability levels in air quality models, increasing the spatiotemporal resolution of the predictions.

## 4. Conclusions

In many air quality models, atmospheric stability is determined using sensible heat fluxes obtained from the ground surface energy balance equation, where the Bowen ratio is a critical parameter. However, the Bowen ratios used in air quality models are typically assumed as predefined values based on land-cover classifications despite the fact that the dynamics of Bowen ratios can possibly change over different hours. In 2013, the USEPA suggested the use of Bowen ratio reference values with high spatiotemporal resolution in the AERMOD model, but the model still does not consider temporal variations in Bowen ratios. To improve calculations of atmospheric stability, this study developed an equation for calculating hourly Bowen ratios using only parameters for relative humidity and absolute temperature. To test the validity of the proposed approach, the value for latent heat flux calculated from the developed equation was compared to the values determined through the PM and PT methods. The results showed that the estimated latent heat flux values were consistent among the three methods, supporting the strength of the proposed approach. The major advantage of this method is to easily quantify the partitions of surface net radiation by simply using the data of relative humidity and temperature from the in-field measurement. Therefore, this proposed method is relatively more convenient compared to the PM and PT methods. In addition, this proposed method considers the temporal variations of the Bowen ratio and the atmospheric stability on the hourly scale, which makes it more realistic and applicable for improving the quantification of atmospheric stability in the air quality models. However, there are still some minor limitations associated with this method. First, if the surface is covered with ice or snow, part of the net radiation will be used for melting, but this mechanism is not considered in this method. Second, this method may not apply to the unsteady condition. Although there are minor limitations, the model accuracy and the data accessibility ensure this proposed method will be applicable for use in air quality models. Therefore, Bowen ratios calculated according to the proposed method could have important implications for improving air quality models by allowing users to incorporate realistic changes in ratios over short time scales.

The authors thank all colleagues who contributed to this study. This study was supported in part by the Ministry of Science and Technology (MOST) of Taiwan (MOST 101-2221-E-002-106, 99-2621-M-002-029, 100-2111-M-002-006, 101-2621-M-002-022, 101-2111-M-002-003, 103-2621-M-002-012, and 104-2621-M-002 -026) and by National Taiwan University (EcoNTU: NTU-CESRP- 104R7604-2). The authors have no conflicts of interest to declare.

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