Theoretical and Applied Climatology

, Volume 103, Issue 3, pp 501–517

Energy balance in urban Mexico City: observation and parameterization during the MILAGRO/MCMA-2006 field campaign

Authors

    • Department of Geography, Faculty of Arts and Social SciencesNational University of Singapore
    • Molina Center for Energy and the Environment (MCE2)
  • Shelley Pressley
    • Laboratory for Atmospheric Research, Department of Civil and Environmental EngineeringWashington State University
  • Rasa Grivicke
    • Laboratory for Atmospheric Research, Department of Civil and Environmental EngineeringWashington State University
  • Eugene Allwine
    • Laboratory for Atmospheric Research, Department of Civil and Environmental EngineeringWashington State University
  • Luisa T. Molina
    • Molina Center for Energy and the Environment (MCE2)
  • Brian Lamb
    • Laboratory for Atmospheric Research, Department of Civil and Environmental EngineeringWashington State University
Original Paper

DOI: 10.1007/s00704-010-0314-7

Cite this article as:
Velasco, E., Pressley, S., Grivicke, R. et al. Theor Appl Climatol (2011) 103: 501. doi:10.1007/s00704-010-0314-7

Abstract

The parameterization of the energy balance from a residential and commercial neighborhood of Mexico City was investigated using direct measurements of radiative and heat fluxes carried out during the MILAGRO/MCMA-2006 field campaign as a reference. The measured fluxes were used to evaluate different models of the energy balance based on parameterizations that require standard meteorological observations: ambient temperature, relative humidity, atmospheric pressure and cloudiness. It was found that these models reproduce with reasonable accuracy the diurnal features of the radiative and heat fluxes. The largest differences between modeled and observed fluxes correspond to the incoming longwave radiation, mainly due to errors in the cloudiness data. This paper contributes to the understanding of the energy partitioning in (sub)tropical urban environments, particularly in the developing world, where energy balance models have not been evaluated.

1 Introduction

A clear description of the surface energy balance provides the framework for understanding many of the chemical and physical processes occurring in the atmosphere. For instance, the evolution of the boundary layer is strongly dependent upon the way in which the net all-wave radiation reaching the surface is partitioned among the different heat fluxes. In urban environments, urban structures, land cover, and human activities have a significant effect on the energy balance and, in turn, on many boundary layer meteorological parameters. Urbanization has led to profound alterations in the local environment; for example, the reduction of vegetation evapotranspiration alters the water balance; the low albedo of urban surfaces enhances the absorption of solar radiation during daytime. This extra heat stored in the buildings increases the ambient temperature, enhances buoyant production of turbulence, and reduces nighttime cooling. Vehicles, industries, and residential heating represent additional sources of heat. These climatic alterations affect the production and chemical characteristics of pollutant gases and particles, cause changes in the physiological comfort of humans, increase public health risks, modify cooling and heating requirements, and perturb weather patterns with increases in cloud coverage, precipitation, and fog.

Following the scheme proposed by Oke (1988), the energy balance for an urban ecosystem can be described by placing the whole urban canopy within a volume box with its top above the blending height or roughness sublayer, in the inertial sublayer where the heat fluxes merge into a net exchange flux. Its base is placed at the depth at which there is no net heat exchange over the time period of interest. The lower boundary of the inertial sublayer is predicted to be 15–40 m for residential areas and larger for central city locations. In practice, this works out to be two to four times the mean height of buildings (Grimmond and Oke 1999a; Roth 2000). At this height, the energy fluxes are fully representative of the underlying surface in a radius of similar size to the size of a typical urban neighborhood (102–104 m). In this way, the energy balance at the top of the box can be expressed as:
$$ Q* + {Q_{\rm{F}}} = {Q_{\rm{H}}} + {Q_{\rm{E}}} + \Delta {Q_{\rm{S}}} $$
(1)
where Q* is the net all-wave radiation; QH and QE are the turbulent fluxes of sensible and latent heat, respectively; QF is the anthropogenic heat flux released from human activities, mainly from combustion processes; and ΔQS is the change in the energy storage per unit of time over the whole urban canopy.

In principle, the different components of the energy balance can be measured. However, such measurements are usually not available, even though they are prerequisites of weather forecasting and air quality models. This lack of atmospheric monitoring is accentuated in the developing world, including low latitude cities with tropical or subtropical climates. This is unfortunate because much of the projected future urban growth will take place in these cities. To improve our understanding of the atmosphere/surface exchange of mass and energy in this type of urban environment, we deployed an eddy covariance system to measure fluxes of selected pollutant gases, aerosols, and energy in a typical residential and commercial district of Mexico City as part of the Mexico City Metropolitan Area (MCMA-2006) field campaign, a component of the Megacity Initiative: Local and Global Research Observations (MILAGRO) project conducted during March, 2006 (Molina et al. 2010). This paper presents results of the energy flux measurements including Q*, QH and QE which were measured directly, and ΔQS which was calculated as the residual in the energy balance equation. In this approach, it was assumed that the contribution from QF is incorporated in the other components of the energy balance. Results from the fluxes of pollutant gases and aerosols are presented in separate papers (Velasco et al. 2009; Grivicke et al. 2010, in preparation).

The measurements of Q* included individual measurements of the incoming and outgoing short- (0.3–4 μm) and longwave (4–100 μm) radiations (K↓, K↑, L↓, and L↑, K, and L indicate short- and longwave radiation, respectively, while the arrows give the direction of the flux). The measured components of Q* were used to test different simple models of the radiative budget and energy balance based on parameterizations that require standard meteorological observations (ambient temperature, relative humidity, atmospheric pressure, cloudiness fraction, and cloud ceiling height). The models tested here calculate Q*, QH, QE, and ΔQS using parameterizations developed for rural environments, and so far, they have only been evaluated with measurements from mid-latitudes cities. Only three data sets from studies conducted in Miami, FL; Tucson, AZ; and Mexico City have been used to assess the utility of these models in subtropical cities (Oke et al. 1999; Grimmond and Oke 2002; Newton et al. 2007).

The work by Oke et al. (1999) in Mexico City was conducted in the historic core of the city, which consists of large areas of old massive stone buildings built during the colonial period. Those measurements were carried out in 1993 and included the first direct observations of Q*, QH, and QE in a subtropical city. In 1985, Oke et al. (1992) investigated the energy balance from a mixed residential, commercial, and industrial area of Mexico City through direct measurements of Q* and QH, parameterization of ΔQS, leaving QE as the residual in the energy balance. The energy balance in other three locations of Mexico City has been investigated, two in semirural ecosystems and one within the urban core (Barradas et al. 1999; Tejeda-Martinez and Jauregui 2005). The measurements presented here are the first set of energy fluxes obtained by direct methods including the four components of the radiative budget in urban Mexico City as part of a comprehensive air quality research project.

2 Flux measurements

2.1 Study site and observation period

Mexico City is located within a basin on the central Mexican Plateau at an elevation of 2,240 m above sea level. The climate is highland subtropical, with a well-defined rainy season (May–October). Mexico City is one of the largest metropolitan areas in the world with a population of nearly 20 million. The local meteorological conditions and climate are strongly influenced by the surrounding terrain (Whiteman et al. 2000) and by the extensive and heavily developed urbanization (Jauregui 1997). The energy balance measurements were conducted in a busy district (Escandon district: 19°24′12.63′′ N, 99°10′34.18″ W) surrounded by congested avenues and close to the center of the city. Following the local climate zone classification proposed by Steward and Oke (2009), the monitored urban surface corresponds to “compact housing”. The topography was completely flat and relatively homogeneous in terms of building material, density, and height. The mean height of the surrounding buildings was zh = 12 m. The aerodynamic surface roughness was estimated to be zo = 1 m, and the zero displacement plane zd was calculated to be 8.4 m height following the rule-of-thumb estimate zd = 0.7zh (Grimmond and Oke 1999a). The predominant land use of the monitored district was residential and commercial, with 57% of the surface covered by buildings of three and four stories high (most of them built of concrete with flat roofs). Roadways and other impervious surfaces accounted for 37%, and vegetation covered the remaining 6%. This means that the biomass was scarce, and consequently, the potential for evapotranspiration from vegetation was small. The walls of the buildings represented an additional surface for the heat exchange, and the ratio of the total 3D area to the plan area, known as active surface area ratio (A3D/A2D), was 1.36, with roofs accounting 42%, walls 27%, roads and other impervious surfaces 27%, and vegetation 4% to the complete 3D active surface area.

The fetch of the observed fluxes was calculated using a hybrid Lagrangian model (Hsieh et al. 2000). If the footprint is defined as the upwind area that contributes 80% of the total measured flux, the calculated footprint extent ranged from 650 to 6,800 m, averaging 1,150 m during the entire period of observations. For these distances, we can assume that the measured fluxes are representative of the heat fluxes for a typical neighborhood of Mexico City. Further details of the footprint analysis are provided in Velasco et al. (2009).

The turbulent fluxes of sensible and latent heat and the four components of Q* were measured continuously during 13 days in March, 2006. March is one of the warmest months of the year in Mexico City with mean minimum and maximum temperatures of 7.7°C and 24°C, respectively, and average monthly precipitation rate of 9.3 mm. The synoptic conditions over the plateau during the first 6 days of observations led to mostly clear sky during the morning and partly cloudy conditions during the afternoon. The rest of the days were affected by the passage of two cold fronts, with increased humidity and cloudiness, a reduction in the ambient temperature, and scattered showers during the afternoons. For the analysis of the energy budget, the first days are referred as “warm days” and the second as “cool days”. Time series of ambient temperature, relative humidity, and fractional cloudiness during the period of observations are shown in Fig. 1. The daytime cloudiness data were measured by a multi-band rotating shadow band radiometer (MERSR) located 44 km north of the measurement site, while at night the cloudiness data were obtained from routine meteorological observations at the Mexico City airport located 10 km northeast. A detailed description of the meteorological conditions during the MILAGRO/MCMA-2006 field campaign is provided by Fast et al. (2007).
https://static-content.springer.com/image/art%3A10.1007%2Fs00704-010-0314-7/MediaObjects/704_2010_314_Fig1_HTML.gif
Fig. 1

Ambient temperature (black curve), relative humidity (blue curve with open circles), fractional cloudiness (gray shadow), and rain presence (red dots at the bottom, their size indicates the relative intensity of the shower) during the period of observations

2.2 Instrumentation and measurements

The energy flux measurements were conducted from a 25-m walk-up tower mounted on the rooftop of a building, with a total height of 42 m, more than three times the mean height of the surrounding buildings, and of sufficient height to be in the constant flux layer. The net all-wave radiation was measured by a Kipp and Zonen net radiometer CNR1, which is a four-component system containing upward- and downward-facing pyranometers and pyrgeometers. The turbulent fluxes of sensible and latent heat were measured using the eddy covariance method. The wind speed, virtual temperature, and humidity fluctuations were sampled at 10 Hz with a 3D sonic anemometer (Applied Technologies, Inc., model SATI-3K) and an open-path infrared gas analyzer (OP-2 IRGA, ADC BioScientific). The fluxes were calculated over 30-min periods and were corrected for the effects of air density using the Webb corrections. A coordinate rotation on 3D velocity components was performed to eliminate errors due to sensors tilt relative to the surface, and a low-pass filter was applied to eliminate the presence of a possible trend in the 30-min series. A detailed description of the instrumentation and methodology used in the eddy covariance system is provided by Velasco et al. (2009).

2.3 Observed fluxes

2.3.1 Radiation budget

The net all-wave radiation flux is expressed as:
$$ Q* = K* + L* = K \downarrow - K \uparrow + L \downarrow - L \uparrow $$
(2)
where K* is the sum of the two shortwave radiation components and L* is the sum of the two longwave radiation components. Figure 2 shows the observed components of the radiation budget during both types of days, warm and cool days. In the morning, no significant differences were observed, but the larger cloudiness observed on cool days during the rest of the diurnal course reduced Q* by about 28% to 39% when compared to the warm days. In the afternoon, this reduction was driven mainly by the reduction of the shortwave radiation, which diminished the heat sequestered within the urban canopy and the intensity of L↑ during nighttime. In contrast, the thicker layer of clouds during cool days increased the intensity of L↓ from 7% to 11% during the afternoons and nights. At night, the net radiative loss was larger during warm days. The average loss during warm days was −114 W m−2, but only −68 W m−2 during cool days (see Table 1). The lower release of heat during cool nights may reduce the depth of the nocturnal mixing height and therefore enhance the accumulation of emissions of pollutant gases and aerosols in a smaller layer, driving important implications to the local air quality.
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Fig. 2

Average diurnal pattern of the radiation budget during warm (a) and cool (b) days. By convention, the downward radiation fluxes are positive and the upward negative. The colored shadows represent ±1 standard deviation of the total averages of measured fluxes and give an indication of the day-to-day variability in each phase of the daily cycle. In the same context, the dashed lines indicate ±1 standard deviation of Q*. The time scale corresponds to the local standard time

Table 1

Mean observed fluxes and flux ratios for different periods of the diurnal course and both types of days

 

Q*

QH

QE

ΔQS

QH/Q*

QE/Q*

ΔQS /Q*

β

QHQS

(QH + ΔQS)/QE

Daytime (Q* > 0)

Cool days

337 ± 30

128 ± 20

44 ± 17

193 ± 24

0.37 ± 0.05

0.13 ± 0.06

0.60 ± 0.07

3.48 ± 2.00

0.62 ± 0.09

8.89 ± 4.44

Warm days

380 ± 60

137 ± 17

51 ± 16

191 ± 34

0.38 ± 0.05

0.13 ± 0.02

0.50 ± 0.07

2.91 ± 0.58

0.77 ± 0.17

6.82 ± 1.40

Nighttime (Q* < 0)

Cool days

−68 ± 12

17 ± 13

9 ± 6

−97 ± 15

−0.21 ± 0.15

−0.16 ± 0.11

1.43 ± 0.18

11.35 ± 25.82

−0.15 ± 0.10

−8.12 ± 3.40

Warm days

−114 ± 11

7 ± 2

7 ± 3

−129 ± 9

−0.11 ± 0.13

−0.06 ± 0.03

1.19 ± 0.13

2.32 ± 2.89

−0.09 ± 0.08

−23.19 ± 11.32

Morning (8–12 h)

Cool days

406 ± 28

132 ± 21

43 ± 17

233 ± 20

0.32 ± 0.05

0.10 ± 0.05

0.58 ± 0.02

4.57 ± 4.56

0.55 ± 0.09

12.22 ± 10.63

Warm days

397 ± 49

136 ± 18

39 ± 13

216 ± 32

0.35 ± 0.03

0.11 ± 0.02

0.54 ± 0.03

3.62 ± 1.07

0.65 ± 0.09

8.57 ± 2.17

Afternoon (12–18 h)

Cool days

286 ± 55

128 ± 35

46 ± 21

157 ± 41

0.45 ± 0.13

0.17 ± 0.10

0.59 ± 0.11

3.21 ± 1.42

0.76 ± 0.21

7.78 ± 3.74

Warm days

366 ± 101

139 ± 40

58 ± 25

160 ± 41

0.42 ± 0.06

0.16 ± 0.04

0.47 ± 0.11

2.72 ± 0.75

0.98 ± 0.30

5.65 ± 1.66

The fluxes are in units of watts per square meter, while the ratios are non-dimensional

At regional level, the atmospheric pollution emitted by urban centers may lead to reductions of the solar radiation reaching the surface by as much as 5% to 10% (Ramanathan and Carmichael 2008). For Mexico City, Jauregui and Luyando (1999) found a maximum reduction close to 20%. In general, atmospheric pollution produces similar alterations in the radiation budget as cloudiness does: The intensity of K* is reduced and L↓ is increased. Pollutants such as black carbon aerosols absorb solar radiation, while other pollutants such as sulfates and nitrates reflect solar radiation.

Based upon observations of K↑ and K↓ from 8 a.m. to 5 p.m., when the solar elevation angle was >20° and the solar radiation intense, the observed albedo \( \left( {\alpha = K \uparrow /K \downarrow } \right) \) was 0.115 ± 0.027. This low albedo is a direct consequence of the scarce vegetation and represents an average of those albedos typically reported for roofs, roads, walls, and other typical urban surfaces.

2.3.2 Energy budget

Figure 3 shows the ensemble diurnal energy budget for both types of days, and Table 1 presents the statistics and the energy partitioning. Contrary to most of the previous observations from urban areas, QH was not the most important heat sink in the energy balance during warm and cool days. During daytime, QH represented 37% of Q* in both types of days. The larger cloud cover during cool days reduced the magnitude of QH by 8% during the afternoon, but at nighttime, QH increased by a factor of 2 or more when compared to warm days. This increment of up to 25 W m−2 during cool days was due to the lower loss of L*.
https://static-content.springer.com/image/art%3A10.1007%2Fs00704-010-0314-7/MediaObjects/704_2010_314_Fig3_HTML.gif
Fig. 3

Average diurnal pattern of the energy budget during warm (a) and cool (b) days. The colored shadows represent ±1 standard deviation of the total averages of measured fluxes and give an indication of the day-to-day variability in each phase of the daily cycle. The time scale corresponds to the local standard time

As expected, the smallest term in the energy balance was QE. In general, it was usually positive throughout the diurnal course with peaks above 100 W m−2 at noon. In both types of days, 13% of the net radiation was expended in evaporation during daytime. At night, the evaporation was almost null; cool nights showed slightly higher values of QE than warm nights as a consequence of the afternoon and evening rains during those days. Because of the relative small values of QE, the Bowen ratio (β = QH/QE) was high during the whole diurnal course, and even though no significant differences were observed in the magnitude and diurnal pattern of QE in both types of days, β was higher and more variable during cool days, mainly during the evenings and early mornings after the rain events.

The heat sequestered by the urban surface during daytime represented at least 50% of Q*. Although the increased cloudiness during cool days did not affect ΔQS during daytime, the nocturnal release of ΔQS produced a lower upward-directed flux than during warm days. In both types of days, this nocturnal upward flux was initially larger than the net radiation; in warm nights, it became within ±5% of the net radiation loss 2 or 3 h after sunset, as observed typically in urban environments, but during cool days, this occurred until 3 hours before sunrise.

The large fraction of heat sequestered by the urban canopy is due to the nature of the integrated landscape and climate of Mexico City, including the dry and warm weather, intense solar radiation, scarce vegetation, extended impervious cover, densely built up, thermal properties of the constituent building materials, large active surface area for absorption provided by walls and roofs, and shelter provided by the urban canyon morphology. The daytime ΔQS/Q* ratio observed in this site varied from 0.5 to 0.6 and was consistent with ratios reported for urban sites with surface vegetation <5%, such as a light-industrial site in Vancouver, Canada, downtown Mexico City, and a residential district of Ouagadougou, Burkina Faso (Grimmond and Oke 1999b; Oke et al. 1999; Offerle et al. 2005). In suburban districts with surface vegetation fractions >25%, ΔQS is typically 20–30% of Q*.

3 Parameterization

Each one of the components in the radiation and energy budgets was modeled with parameterizations using standard meteorological observations as input data. The radiation and energy modeling schemes are similar to the net all-wave radiation parameterization (NARP) scheme developed by Offerle et al. (2003) and the local-scale urban meteorological parameterization scheme (LUMPS) developed by Grimmond and Oke (2002). Table 2 shows the structure of the scheme applied in this study, including the data requirements and values of the parameterization coefficients. These coefficients were obtained via regression analyses from the observed flux data and therefore are specific for the monitored neighborhood. In the following sections, we describe how these coefficients compare to those for other sites and to coefficients suggested as default values for modeling purposes. This comparison is one way to evaluate how well the simple parameterizations of energy balance components applies to Mexico City. A second approach is to predict fluxes using the parameterizations and compare these predictions to the observed fluxes. However, this approach is not a completely independent test of the models since the predictions employ the best-fit coefficients. This evaluation can be made more rigorous by using standard meteorological observations from the Mexico City airport and local meteorological service, and we include results obtained in this way as part of our evaluation. Overall, this modeling approach is expected to provide further insight of the energy partitioning in a subtropical city of a developing country, as well as use as a reference for evaluating the micrometeorological schemes used to model the evolution of the convective boundary layer and the dispersion of pollutants in Mexico City.
Table 2

Overview of the scheme proposed in this study for modeling the components of the radiative and energy budgets from an urban surface

Step

Radiative or energy component

Formulation

Constants and parameterization coefficients

Input data: meteorological observations and modeled variables

Alternative formulation

1

K

Direct observations of K

a1 = 1,100 W m−2

ϕ and N

Kasten and Czeplak (1980). Equation 3a

a2 = −30 W m−2

b1 = −0.75

b2 = 3.4

2

K

K↑ = α K

α = 0.115

K

 

3

K*

K* = K↓(1 − α)

α = 0.115

K

 

4

L

Stefan Bolzmann’s law. Equation 5. εsky by Prata (1996) including altitude (Prata 1996) and cloudiness (Oke 1978) corrections. Equations 68

σ = 5.67 × 10−8 W m−2 K−4

Ta, ea, pa, and N

Swinbank (1963), Brutsaert (1975), Satterlund (1979), and Idso (1981) for εsky

p0 = 1,013 mbar

c = see Table 3

5

L

L↑-NARP, Offerle et al. (2003). Equation 10

σ = 5.67 × 10−8 W m−2 K−4

Ta, K↓ and L↓ (modeled)

L↑-Mex. Equation 9 for L↑ and Eqs. 1112 for Ts

εs = 0.92

α = 0.115

6

L*

L* = L↓ − L

 

L↓ and L↑ (modeled)

 

7

Q*

Q* = K* + L*. Equation 2

 

K* and L* (modeled)

 

8

ΔQS

Camuffo and Bernadi (1982). Equation 13

d1 = 0.671

Q* (modeled)

 

d2 = 0.450 h

d3 = −52 W m−2

9

QH

Simplified Penman–Monteith approach. Holtslag and Van Ulden (1983). Equation 14

αPM = 0.29

s = f(Ta, pa)

 

β = 7.33 W m−2

Q* and ΔQS (modeled)

10

QE

Simplified Penman–Monteith approach. Holtslag and Van Ulden (1983). Equation 15

αPM = 0.29

s = f(Ta, pa)

 

β = 7.33 W m−2

Q* and ΔQS (modeled)

 

The parameterization coefficients are specific for the monitored district during the MILAGRO/MCMA-2006 field campaign. See text for more detailed explanations

aConstants, parameterization coefficients, and input data correspond to the alternative formulation of K

3.1 Radiation budget

3.1.1 Incoming and outgoing shortwave radiation

The incoming shortwave radiation was calculated using the approach proposed by Kasten and Czeplak (1980). This approach has been widely used and depends only on the solar elevation angle (ϕ) and the fractional cloudiness (N):
$$ K \downarrow = \left( {{a_1}\sin \varphi + {a_2}} \right)\left( {1 + {b_1}{N^{{b_2}}}} \right) $$
(3)
where the first factor represents K↓ under clear sky conditions and the second factor is a correction for the presence of clouds. a1 and a2 are empirical coefficients that describe the average atmospheric attenuation of K↓ by water vapor, ozone, other gases, and aerosols, and b1 and b2 are empirical coefficients which may depend on the climate of the specific site. For 10 years of observations at Hamburg, Kasten and Czeplak (1980) obtained b1 = −0.75 and b2 = 3.4. From a number of measurements at mid-latitude locations, it has been found that a1 varies from 900 to 1,100 W m−2 and a2 from −30 to −69 W m−2. For sites where these coefficients are unknown, Holtslag and Van Ulden (1983) suggest to consider a1 = 990 W m−2 and a2 = −30 W m−2 for ϕ > 20°. For the observed period and latitude and conditions of Mexico City, the best results were obtained by considering a1 = 1,100 W m−2 and a2 = −30 W m−2 with a coefficient of determination r2 = 0.89.

There are more sophisticated radiation transfer schemes to calculate K↓ that require more specific information such as meteorological data from upper air soundings, as well as intensive numerical efforts. However, it does not appear that more complex approaches improve substantially the approximations obtained from simple parameterizations (e.g., Niemelä et al. 2001b). Fortunately, the monitoring of K↓ has become part of the routine meteorological measurements in airports and meteorological agencies. This is the case of Mexico City, where the National Meteorological Service has five automatic meteorological stations measuring K↓. In this context, K↓ can be considered as input data instead of a variable within the radiation and energy schemes, with the advantage that the errors from its parameterization are not propagated to the subsequent computations of the other fluxes.

The outgoing shortwave radiation is the fraction of K↓ reflected by the urban surface defined by α, and therefore, the net shortwave radiation is expressed as:
$$ K* = K \downarrow \left( {1 - \alpha } \right) $$
(4)

The albedo varies in response to the solar elevation angle; however, this variation was small during the diurnal course of our observations, at midday α averaged 0.111 ± 0.002, and increased to 0.140 ± 0.024 1 h before sunset. In terms of general applications, it is not necessary to include the diurnal variation of α.

3.1.2 Incoming longwave radiation

The incoming longwave radiation is not commonly measured, and therefore, its estimation is needed to complete the radiation budget scheme. L↓ is modeled by the Stefan Bolzmann’s law as:
$$ L \downarrow = {\varepsilon_{\rm{sky}}}\sigma T_{\rm{a}}^4 $$
(5)
where Ta is the ambient temperature near the surface and σ is the Stefan Boltzmann’s constant (5.67 × 10−8 W m−2 K−4). Different approaches to calculate εsky have been proposed in terms of Ta and humidity. They have been evaluated in detail elsewhere (Prata 1996; Crawford and Duchon 1999; Niemelä et al. 2001a; Newton et al. 2007); it appears that the approach of Prata (1996) offers the most reliable estimation. Using the observed data, we evaluated five models (Swinbank 1963; Brutsaert 1975; Satterlund 1979; Idso 1981; Prata 1996) including the cloudiness and altitude corrections described below. The Prata (1996) formulation produced the smallest deviations when compared to the observed values of εsky. At night, the deviations were small, but during the day, εsky was consistently underestimated by up to 10% and 25% at midday during warm and cool days, respectively. For clear sky conditions, Prata’s approach is given by:
$$ \begin{array}{*{20}{c}} {{\varepsilon_{{\rm{sky - clear}}}} = 1 - \left( {1 + \xi } \right)\exp \left( { - {{\left( {1.2 + 3.0\xi } \right)}^{0.5}}} \right)} \\{\xi = 46.5\frac{{{e_{\rm{a}}}}}{{{T_{\rm{a}}}}}} \\\end{array} $$
(6)
where ea is the vapor pressure in millibar and Ta in kelvin. At high altitude sites, the emissivity decreases by the proportionally smaller water vapor path that can emit downward radiation; for this effect, Prata (1996) established an altitude correction (Δεheight) for εsky-clear in terms of the atmospheric pressure:
$$ \begin{array}{*{20}{c}} {{\varepsilon_{{\rm{sky - clear - height}}}} = {\varepsilon_{{\rm{sky - clear}}}} + \Delta {\varepsilon_{\rm{height}}}} \\{\Delta {\varepsilon_{\rm{height}}} = - 0.05\left( {\frac{{{p_0} - {p_{\rm{a}}}}}{{{p_0} - 710}}} \right)} \\\end{array} $$
(7)
where p0 is the standard atmospheric pressure (1,013.25 mbar) and pa is the ambient pressure at the studied site. The emissivity needs to be adjusted also by the effects of cloudiness. We used the approach suggested by Oke (1978), where εsky-clear-height is modified by a non-linear cloud term in terms of the fractional cloudiness and a coefficient (c) that accounts for the decrease of cloud-base temperature with increasing cloud height (see Table 3):
$$ {\varepsilon_{\rm{sky}}} = {\varepsilon_{{\rm{sky - clear - height}}}}\left( {1 + c{N^2}} \right) $$
(8)
Table 3

Values of the coefficients used to account for the cloudiness effect in εsky-clear in terms of type and height of cloud

Cloud type

Typical cloud height (km)

Coefficient c

Cirrus

12.20

0.04

Cirrostratus

8.39

0.08

Altocumulus

3.66

0.17

Altostratus

2.14

0.20

Stratocumulus

1.22

0.22

Stratus

0.46

0.24

Fog

0

0.25

Taken from Oke (1978)

Usually airports report the cloud ceiling height as a routine meteorological variable. If observations of this variable are not available, cloudiness corrections as those suggested by Offerle et al. (2003), Niemelä et al. (2001a), and Crawford and Duchon (1999) can be applied, instead of the correction proposed by Oke (1978).

3.1.3 Outgoing longwave radiation

The outgoing longwave radiation is driven primarily by the black body radiation from the urban surface, and only a small fraction is due to the reflection of L↓ from the surface,
$$ L \uparrow = \sigma {\varepsilon_{\rm{s}}}T_{\rm{s}}^4 + \left( {1 - {\varepsilon_{\rm{s}}}} \right)L \downarrow $$
(9)
εs and Ts are the weighted emissivity and temperature of the 3D active surface areas within the urban canopy (roofs, walls, roads, other impervious surfaces, and vegetation), respectively. Using individual values of εs for each type of surface (εroof = 0.94, εroads = 0.95, εwalls = 0.85, εveg. = 0.97; Oleson et al. 2008), a weighted εs = 0.92 was determined for the monitored district. In contrast, Ts is difficult to determine directly because of the highly heterogeneous urban surface, and it is almost never available from routine observations. Ts can be obtained from numerical solutions of the heat conduction equation for each urban surface accounting for materials, geometries, and thermal characteristics (e.g., Oleson et al. 2008). A simple approach is to approximate Ts by Ta including a correction term based on Q* or K↓ as proposed by van Ulden and Holtslag (1985). Offerle et al. (2003) used this approach in NARP applying a correction term based on K↓,
$$ L \uparrow = {\varepsilon_{\rm{s}}}\sigma T_{\rm{a}}^4 + 0.08K \downarrow \left( {1 - \alpha } \right) + \left( {1 - {\varepsilon_{\rm{s}}}} \right)L \downarrow $$
(10)
the correction term (second term on the right-hand side) does not account for any hysteresis between Ts and Ta (Ts tends to be greater than Ta during daytime and lower during nighttime), but accordingly to its authors and confirmed in this study, it has a little impact relative to errors associated with cloudiness and εsky determinations.
The energy balance study of Oke et al. (1999) in downtown Mexico City (in the following referred as Mex93) included surface temperature measurements of roofs (Tr) and roads (Tig). Later, Oleson et al. (2008) simulated those measurements including surface temperatures of walls (Tw). As an alternative method to estimate Ts, we considered the hourly temperature difference between each type of surface and the ambient \( \left( {\Delta {T_i} = {T_i} - {T_{\rm{a}}}} \right) \) measured or modeled in downtown Mexico City. ΔTi was normalized by the difference between the maximum and minimum diurnal ambient temperatures \( \left( {\Delta {T_{\rm{a}}} = {T_{{ \max }}}-{T_{{ \min }}}} \right) \) observed in Mex93, and using the maximum diurnal temperature difference observed in the present study (refereed as Mex06), the hourly temperature for each surface type was estimated on a daily base,
$$ {T_i} = {\left( {{T_{\rm{a}}}} \right)_{\rm{Mex06}}} + {\left( {\frac{{\Delta {T_i}}}{{\Delta {T_{\rm{a}}}}}} \right)_{\rm{Mex93}}}{\left( {\Delta {T_{\rm{a}}}} \right)_{\rm{Mex06}}} $$
(11)
a weighted Ts was subsequently calculated using the 3D fractional (λi) active surface areas of the monitored district in Mex06,
$$ {T_{\rm{s}}} = {\lambda_{\rm{r}}}\Delta {T_{\rm{r}}} + {\lambda_{\rm{ig}}}\Delta {T_{\rm{ig}}} + {\lambda_{\rm{w}}}\Delta {T_{\rm{w}}} + {\lambda_{\rm{v}}}\Delta {T_{\rm{v}}} $$
(12)
where impervious grounds such as sidewalks and other asphalted areas (λig) are accounted as roads (λr), as well as vegetated surfaces (λv), since no surface temperatures of them were available. λw accounts for walls. The application of this parameterization of Ts to estimate L↑ through Eq. 9, which we will refer as L↑-Mex in the following, is only possible because of the similitude in the spatial distribution of buildings, scarce vegetation, and climatology in both locations. For this reason, L↑-Mex may be valid only for some districts of Mexico City, opposite to the parameterization in Eq. 10 proposed by Offerle et al. (2003), which is not restricted to any location. We will refer to this last approach as L↑-NARP hereafter.

3.2 Energy budget

As indicated in Eq. 2, Q* is the sum of K* defined by Eq. 4 considering either the modeled or observed K↓, L↓ modeled following the approach of Prata (1996) including the altitude and cloudiness corrections and L↑ modeled by L↑-NARP or L↑-Mex. With the assumption that QF can be neglected from Eq. 1 because is small magnitude relative to Q* and in the observations is incorporated in the other components of the energy balance, QF is not included in the parameterization of Q*.

3.2.1 Storage heat flux (ΔQS)

ΔQS is calculated as a function of Q* and the temporal variation of Q* (∂Q*/∂t) using the parameterization proposed by Camuffo and Bernadi (1982). This parameterization captures the hysteresis behavior of ΔQS and has been widely used to investigate the uptake and loss of heat in buildings and urban surfaces (e.g., Anandakumar 1999; Grimmond and Oke 1999b),
$$ \Delta {Q_{\rm{S}}} = {d_1}Q* + {d_2}\frac{{\partial Q*}}{{\partial t}} + {d_3} $$
(13)

The parameter d1 indicates the overall strength of the dependence of the storage heat flux on Q*. The parameter d2 describes the degree and direction of the phase relations between ΔQS and Q*. When d2 is positive, ΔQS precedes the peak in Q*; when it is zero, the two fluxes are exactly in phase, i.e., there is no hysteresis. The parameter d3 is an intercept term that indicates the relative timing when ΔQS and Q* become negative. A large d3 coefficient indicates that ΔQS becomes negative much earlier than Q*. These three coefficients were fitted statistically to the values of ΔQS obtained from the 30-min observations. The temporal variation of Q* was calculated as the average of the previous and subsequent 30-min values of Q*.

Using the entire set of flux measurements, we found d1 = 0.671, d2 = 0.450 h, and d3 = −52 W m−2 with an r2 = 0.95. These values are within the observed ranges reported for other urban locations The large value of d1 indicates the importance of ΔQS in the energy balance of Mexico City; similar values have been observed only in an industrial site of Vancouver, Canada and downtown Mexico City (Grimmond and Oke 1999b). The observed value of d2 is much higher than the value observed by Oke et al. (1999) in downtown Mexico City (0.069) and similar to those observed in Tucson and Miami (Grimmond and Oke 1999b). With regards to d3, the observed coefficient was very high, only lower than the coefficient reported for a suburban sector of Tucson, and indicates that storage turns negative well before Q*.

3.2.2 Sensible and latent heat fluxes (QH and QE)

The parameterization of QH and QE presented in this study uses the simplified Penman–Monteith approach proposed by Holtslag and Van Ulden (1983),
$$ {Q_{\rm{H}}} = \frac{{\left( {1 - {\alpha_{\rm{PM}}}} \right) + \left( { \frac{\gamma }{s} } \right)}}{{1 + \left( { \frac{\gamma }{s} } \right)}}\left( {Q* - \Delta {Q_{\rm{S}}}} \right) - \beta $$
(14)
$$ {Q_{\rm{E}}} = \frac{{{\alpha_{\rm{PM}}}}}{{1 + \left( { \frac{\gamma }{s} } \right)}}\left( {Q* - \Delta {Q_{\rm{S}}}} \right) + \beta $$
(15)
where s is the slope of the saturation vapor pressure versus temperature (∂qs/∂T), γ is the psychometric “constant”, and αPM and β are empirical parameters. αPM accounts for the correlation of QH and QE with the available energy \( \left( {{Q_{\rm{A}}} = Q*{ } - { }\Delta {Q_{\rm{S}}}} \right) \) depending on the moisture availability, whereas β accounts for the uncorrelated portion. αPM = 0.29 and β = 7.33 W m−2 were derived from a linear regression (r2 = 0.58) between QE and QA using the entire set of 30-min observations. These values are between those calculated by Grimmond and Oke (2002) for a number of urban locations, including downtown Mexico City.

4 Parameterization evaluation

In a first evaluation, the parameterizations described above were tested for both types of days using the observed data. The time series of the modeled and observed components of the radiative and energy budgets are shown in Figs. 4 and 5, respectively. Figures 6 and 7 show the correlations between observations and parameterizations. The statistical results of the model, including the mean bias error (MBE), root mean square error (rmse), r2, and fractional error (FE) are summarized in Table 4. The evaluation was performed per type of day and different periods of the diurnal course. The fractional error is defined by:
$$ {\hbox{FE}} = \frac{1}{N}\sum\limits_{i = 1}^N {\frac{{\left| {\left( {{F_{{\rm{m}}i}} - {F_{{\rm{o}}i}}} \right)} \right|}}{{\tfrac{1}{2}\left( {{F_{{\rm{m}}i}} + {F_{{\rm{o}}i}}} \right)}}\left( {100\% } \right)} $$
(16)
where Fm and F0 represent the modeled and observed fluxes, respectively.
https://static-content.springer.com/image/art%3A10.1007%2Fs00704-010-0314-7/MediaObjects/704_2010_314_Fig4_HTML.gif
Fig. 4

Time series of modeled and observed fluxes of solar radiation. The gray areas represent the observations. The time series in black were modeled using the observations of K↓, while in e and f represent L↑ and L* calculated by L↑-NARP. The red dashed curves in the same panels represent L↑ and L* calculated by L↑-Mex, while in a represents K↓ modeled by Eq. 3. Note that by convention the magnitude of outgoing radiation (K↑, L↑, and L*) is negative; here is presented as positive for practical purposes

https://static-content.springer.com/image/art%3A10.1007%2Fs00704-010-0314-7/MediaObjects/704_2010_314_Fig5_HTML.gif
Fig. 5

Time series of modeled and observed energy fluxes. The gray areas represent the observations. The time series in black were modeled using the observations of K↓ and L↑-NARP to calculate Q*

https://static-content.springer.com/image/art%3A10.1007%2Fs00704-010-0314-7/MediaObjects/704_2010_314_Fig6_HTML.gif
Fig. 6

Correlations between observations and parameterizations of the radiation budget components. In a the parameterization of K↓ corresponds to the approach of Kasten and Czeplak (1980). In b and c the parameterizations of K↑ and K* considers the observations of K↓. While the parameterizations of L↑ and L* in e and f correspond to L↑-NARP

https://static-content.springer.com/image/art%3A10.1007%2Fs00704-010-0314-7/MediaObjects/704_2010_314_Fig7_HTML.gif
Fig. 7

Correlations between observations and parameterizations of the energy budget components. The parameterizations consider the observations of K↓ and L↑ modeled by L↑-NARP

Table 4

Statistical results of the modeled components of the radiative and energy budgets for different periods of the diurnal course and both types of days: cool and warm days

 

 

K

K*

L

L

L*

Q*

QH

QE

ΔQS

Warm days

All day

MBE

1.1

1.0

−7.0

−9.0

2.0

5.6

−6.0

−2.8

11.0

rmse

4.2

4.2

19.1

11.8

15.2

16.7

32.2

25.2

45.6

FE

16.6

4.4

4.1

2.1

9.0

13.4

67.5

73.5

23.4

r2

1.00

1.00

0.61

0.98

0.85

1.00

0.86

0.50

0.94

Daylight (8–16 h)

MBE

1.0

−1.0

−13.3

−14.2

0.8

−0.1

−13.1

−5.1

18.5

rmse

2.7

2.7

23.7

17.2

16.9

15.9

48.7

33.2

67.9

FE

3.1

0.4

5.1

3.1

8.6

9.0

27.7

48.5

29.2

r2

1.00

1.00

0.45

0.96

0.90

1.00

0.61

0.26

0.79

Night (20–6 h)

MBE

0.2

−5.4

5.6

12.4

0.3

3.7

4.2

rmse

7.1

6.0

9.5

14.8

7.5

10.5

15.6

FE

2.0

1.4

6.9

11.5

99.5

93.4

9.3

r2

0.49

0.96

0.42

0.41

0.10

0.00

0.13

Morning (8–12 h)

MBE

0.8

−0.8

−6.1

−6.7

0.7

−0.1

−27.1

−5.6

31.5

rmse

2.5

2.5

13.8

9.9

10.1

9.9

46.2

28.9

58.7

FE

2.9

0.4

3.5

1.7

5.4

2.5

28.6

54.4

19.2

r2

0.99

1.00

0.18

0.97

0.88

1.00

0.66

0.36

0.82

Afternoon (12–16 h)

MBE

1.1

−1.1

−19.7

−20.6

1.0

−0.2

1.5

−4.6

4.3

rmse

2.8

2.8

29.7

21.6

21.1

19.7

51.3

37.1

76.6

FE

3.3

0.4

6.5

4.2

11.5

14.6

26.9

42.4

40.1

r2

1.00

1.00

0.34

0.93

0.97

1.00

0.60

0.15

0.80

Evening (16–19 h)

MBE

−4.0

4.0

−17.5

−11.0

−6.5

−3.0

0.51

−16.3

4.8

rmse

5.1

5.1

31.8

11.7

25.1

25.2

28.9

37.1

56.9

FE

42.2

13.8

8.5

2.5

17.0

30.2

39.0

57.8

52.1

r2

0.97

1.00

0.67

0.95

0.55

0.96

0.64

0.06

0.69

Cool days

All day

MBE

−0.7

0.7

−32.2

−8.3

−23.9

−21.3

−4.9

0.8

−15.2

rmse

5.0

5.0

40.6

11.4

31.9

29.7

37.4

21.0

48.5

FE

24.0

6.6

10.3

2.3

31.0

31.3

78.4

85.6

32.9

r2

0.99

1.00

0.37

0.96

0.71

0.99

0.76

0.43

0.93

Daylight (8–16 h)

MBE

1.7

−1.7

−25.2

−9.3

−15.9

−17.5

0.4

0.4

−18.4

rmse

3.8

3.8

31.7

13.8

20.6

21.1

50.4

30.6

69.8

FE

4.7

0.6

7.9

2.5

12.9

8.2

33.0

57.2

34.3

r2

0.99

1.00

0.57

0.91

0.80

1.00

0.50

0.14

0.80

Night (20–6 h)

MBE

−32.5

−7.8

−24.6

−20.6

−9.0

3.0

−13.5

rmse

40.5

9.2

32.9

30.9

25.2

10.8

27.6

Fractional error

10.7

2.1

37.1

34.2

101.7

107.5

22.2

r2

0.19

0.78

0.03

0.03

0.01

0.01

0.11

Morning (8–12 h)

MBE

2.2

−2.2

−9.8

−1.7

−8.2

−10.4

−13.9

−3.7

7.7

rmse

4.6

4.6

12.9

7.1

10.0

11.9

47.9

27.2

54.5

FE

5.4

0.7

3.3

1.3

6.3

2.6

28.5

46.9

19.0

r2

0.99

1.00

0.62

0.97

0.94

1.00

0.48

0.25

0.78

Afternoon (12–16 h)

MBE

1.2

−1.2

−40.6

−17.0

−23.6

−24.7

15.3

4.5

−47.5

rmse

3.0

3.0

42.9

18.3

27.3

27.5

52.7

33.8

83.7

FE

4.0

0.5

12.6

3.7

19.6

13.9

37.7

67.9

51.4

r2

1.00

1.00

0.01

0.92

0.88

1.00

0.56

0.08

0.84

Evening (16–19 h)

MBE

−4.0

4.0

−58.8

−12.5

−46.3

−41.2

2.8

−7.5

−35.8

RMSE

5.2

5.2

61.1

13.8

49.0

44.5

41.1

20.4

55.1

FE

64.4

20.6

18.0

3.1

59.6

80.2

86.4

65.7

74.0

r2

0.94

1.00

0.00

0.90

0.60

0.97

0.03

0.01

0.74

The parameterization considers the observations of K↓ and L↑ modeled by L↑-NARP. MBE and rmse are in watts per square meter and FE in percent

To conduct a more independent evaluation, ambient data including cloudiness observations from the airport and MERSR and K↓ from the closest station of the National Meteorological Service located at 2.1 km from the monitored site were used to model QH and QE from March 7–17, 2006. These fluxes were compared to fluxes measured by the eddy covariance system as shown in Fig. 8.
https://static-content.springer.com/image/art%3A10.1007%2Fs00704-010-0314-7/MediaObjects/704_2010_314_Fig8_HTML.gif
Fig. 8

Time series and correlations between modeled and observed QH (a) and QE (b) from March 7 to 17, 2006. In the time series, the gray areas represent the observations and the black curves the parameterizations considering observations of K↓ and L↑-NARP. The color gradient in the correlations’ bins indicates the hour of day like in previous figures

4.1 Radiation budget

4.1.1 Shortwave radiation

For comparison purposes, Figs. 4a and 6a show the performance of the parameterization of K↓ defined by Eq. 3. The subsequent parameterizations of K↑, K*, and Q* employ the measured data of K↓. As expected, the parameterization of K↓ showed better results during periods of low cloudiness (mornings and warm days). Excluding the 30 min after sunrise and a couple of hours before sunset, when the fluxes were small due to the low ϕ and/or the intense cloudiness, the modeled K↓ showed mean fractional errors of 10% and 18% during warm and cool days, respectively. During warm days, this deviation did not change significantly during the entire day, but during cool days, it passed from 12% in the morning to 33% in the afternoon. Holtslag and Van Ulden (1983) determined an accuracy of 10% under clear sky for this parameterization. The empirical coefficients a1 and a2 used to describe the attenuation of K↓ in Eq. 3 might not account properly the turbidity caused by the atmospheric pollution of Mexico City, particularly by organic aerosols, which are known to be significant absorbers of shortwave radiation (Barnard et al. 2008). However, it is more likely that the observed deviations were due to an under-prediction of the cloudiness data. The parameterization consistently overestimated K↓ during the afternoons of cool days, when the cloud cover was always high, as shown in Fig. 6a.

The parameterization errors of K↑ and K* were negligible during daytime. The relative large deviations (>50%, >4 W m−2) of K↑ after the sunrise and before the sunset were due to the small magnitude of K↑ rather than to a problem in the parameterization. The negligible errors suggest that the albedo obtained from the observations of K↑ and K↓ represents properly the urban surface of the studied area of Mexico City.

4.1.2 Incoming longwave radiation

Figure 4d shows the time series of L↓ modeled using the parameterization of εsky proposed by Prata (1996) including the altitude and cloudiness corrections (Eqs. 68). As indicated before, εsky was consistently underestimated by the model during daytime, leading underestimations of L↓ by about 5.1% (13.3 W m−2) and 7.9% (25.2 W m−2) during warm and cool days, respectively. The largest underestimations (18.0%, 58.8 W m−2) were observed in evenings of cool days, when the cloud cover over-passed 70%. In general, periods reporting significant cloudiness were poorly reproduced, in contrast to periods with scarce cloudiness during the 4 days at the beginning of the study (March 17–20). The model underestimation of L↓ during daytime suggests an over-prediction of the cloud cover. In a place like Mexico City, where the aerosol pollution is high, in particular of carbonaceous aerosols, which are known to reduce effectively the L↓ that reaches the ground (Bond and Bergstrom 2006), it would be expected to observe a consistent modeling over-prediction during the entire day, in particular during the morning, when the aerosols level is high.

4.1.3 Outgoing longwave radiation

In a first test, both schemes, L↑-NARP and L↑-Mex, were compared to the observations of L↑, with the finding that both methods reproduce accurately L↑ in terms of magnitude and diurnal evolution. Both schemes presented deviations no larger than 6% from the observations. Only at early morning, when the minimum ambient temperature is reached, L↑-NARP performed clearly better than L↑-Mex as shown in Fig. 4e. L↑-Mex over-predicted consistently L↑ (3.3%, 13.1 W m−2) during nighttime due to overestimations of Ts. Even though the indirect formulation proposed here to calculate Ts using the surface temperature measurements conducted in Mex93 (Eqs. 11 and 12) correlates strongly (r2 = 0.94) with Ts extracted from the direct observations of L↑ and Eq. 9, the modeled Ts by L↑-Mex was consistently over-predicted between 2 and 3 K during early morning (5:30–6:30 a.m.). The opposite was true during early afternoon (12:30–13:30 p.m.) when underestimations from 5 to 6 K in Ts led to underestimations of up 3.4% (12.8 W m−2) in L↑. The higher ambient temperatures during warm days accentuated only slightly the deviations of L↑ compared to cool days (e.g., 12.8 and 10.1 W m−2 during warm and cool days, respectively, in the early afternoons). The deviations in Ts are due to the physical differences between downtown Mexico City and the neighborhood studied here (e.g., zh = 18 m and A3D/A2D = 1.78 in Mex93 and zh = 12 and A3D/A2D = 1.36 in Mex06).

With the premise that L↑-NARP performs slightly better than L↑-Mex and is not restricted to any location, the radiative and energy approach presented here uses L↑-NARP as primary parameterization. The deviations of L↑-NARP during daylight were essentially equal to those of L↑-Mex, mean underestimations of 14.2 (3.1%) and 9.3 W m−2 (2.5%) during warm and cool days, respectively. During nighttime, L↑-NARP under-predicted L↑ by less than 8 W m−2 in both types of days. These deviations represent less than 2.5% errors from the observations.

L* integrates the modeling errors of L↓ and L↑, prevailing errors of L↓, in particular during the cool days. Even though the absolute magnitude of L↑ was larger than L↓ (from 1.5 to 1.7 times during daytime and 1.1 to 1.4 times during nighttime), the modeling errors of L↓ led to underestimations in L* of up to 46.3 W m−2 (59.6%) during evenings with cloudiness >70%. During warm days, the deviations of both L↓ and L↑ were of similar magnitude, but not during cool days, when the deviations of L↓ were clearly larger than the deviations of L↑ (e.g., −40.6 and −17.0 W m−2 for L↓ and L↑, respectively, during the afternoons).

4.2 Energy budget

4.2.1 Net radiation

The parameterization of Q* integrates the modeling errors of L↓, L↑, and K↑, in addition to the errors of K↓ if observations are not available. K↓ is the main radiative component of Q* during daytime, while at nighttime, when K* is null, both components of the longwave radiation are important. In this context, if K↓ is obtained from observations, the modeling errors of Q* must be small during daytime and sometimes larger during nighttime.

During warm days, the major deviations in the parameterization of Q* occurred at nighttime, with mean overestimations of 12.4 W m−2 (11.5%). During daytime, the modeling deviations were in general small and for some periods negligible. The largest deviations in the parameterization of Q* occurred during afternoons and nights of cool days (−41.2 W m−2 in average before sunset), as a direct consequence of the poor modeling of L↓ during periods of intense cloudiness.

The accuracy in the parameterization of Q* is fundamental for the good modeling of the other components of the energy balance since all of them relies directly on Q*. Because the quality of the longwave radiation components, especially L↓, depends strongly on the accuracy of the cloudiness data, accurate and continuous observations of the cloud cover are essential. In our case, the uncertainty in the cloudiness data was due to the relative long distance between the monitored neighborhood and the two locations where the cloudiness observations were conducted, besides that the nocturnal observations at the airport were human-made and limited to only four different fractions of cloud cover.

4.2.2 Storage heat flux

Overall, the modeling of ΔQS using the parameterization of Camuffo and Bernadi (1982) with the empirical coefficients fitted statistically from the measurements reproduced correctly the observations of ΔQS, including its hysteresis behavior in both types of days. The model was capable of reproducing the exact moment when ΔQS turns negative, which occurs between 2 and 3 h before the sunset as shown in Fig. 5d. As expected, the uncertainties in the modeling of Q* were propagated to ΔQS. The periods with large uncertainties in Q* were also for ΔQS. In general, the parameterization tends to overestimate ΔQS in periods of low cloudiness (e.g., warm days and mornings of cool days), contrary to periods of intense cloudiness, such as afternoons and evenings of cool days when mean underestimations of up to 47.5 and 35.8 W m−2 were observed, respectively.

4.2.3 Sensible and latent heat fluxes

The simplified Penman–Monteith approach to calculate QH and QE produced relatively good results for both types of days. The cloud cover produced many irregular features and spikes in the time series of QH and QE. During daytime, this approach predicted almost all these irregular features and spikes but failed to predict the magnitude of the observed spikes of QE. At nighttime, the fractional errors were large (~100%) because of the small magnitude of these two fluxes; however, QE was consistently overestimated by about 3.5 W m−2 during both types of days. The lower scattering of QH relative to QE in the correlations between observations and parameterizations shown in Fig. 7 indicates the difficulty of modeling QE under cloudy and rainy conditions. The modeling deviations of L* produced larger or similar fractional errors in the modeling of QH and QE than in the modeling of Q* and ΔQS, but not in terms of magnitude, for example during cool days their mean daily FE were 78.4%, 85.6%, 31.3%, and 32.9%, respectively, while their MBE were −4.9, 0.8, 5.6, and 15.2 W m−2, respectively.

As indicated before, a second evaluation to the complete modeling approach was performed through an estimation of the energy fluxes during 10 days in which only measurements of QH and QE were available and whose values were not used to determine any parameterization coefficient. Figure 8 shows the time series and correlations between modeled and observed fluxes. The statistical results of this test presented slightly higher deviations than those observed in the previous evaluation, in particular QH presented a larger underestimation during daytime. This underestimation was very consistent and might be due to the different measurement location of K↓, but a calibration or installation failure in the instrument is more probable since a comparison of its readings from later days, with simultaneous measurements of K↓ at the studied site, showed a consistent underestimation of ~10% during the mornings and early afternoons. Nevertheless, this second test demonstrated also the capability of this modeling scheme to simulate the energy fluxes of a neighborhood of Mexico City with reasonable accuracy.

5 Conclusions

The parameterization of the energy balance from a residential and commercial neighborhood of Mexico City was investigated using direct measurements of radiative and heat fluxes carried out during the MILAGRO/MCMA-2006 field campaign. The measured fluxes were used to evaluate different models of the radiative and energy balances based on parameterizations that require standard meteorological observations. These models have the advantage of integrating the heterogeneous urban canopy in terms of morphology, surface cover, and sources and sinks of heat. However, these models require a priori knowledge of many physical and empirical parameters, which in many cases depend on the characteristics of the site and prevailing atmospheric conditions. These parameters can be obtained from the literature, by analysis of the urban canopy composition, or from measured flux data, as they were obtained in this study.

Overall, we found that these models reproduce correctly the diurnal features of the observed fluxes, regardless of the additional difficulties imposed by the prevailing weather conditions characterized by cloudy afternoons and nights and scattered showers during the latter part of the study. Better predictions were obtained for periods of low cloudiness during mornings and warm days. The energy balance scheme proposed here captures the hysteresis of Q* and ΔQS with relatively high precision. This is important for the modeling of the chemical and physical processes occurring near the surface. In the same context, the accurate simulation of QH during daytime is important for the accurate modeling of the convective boundary layer. As expected, the uncertainties in the modeling of Q* were transmitted to the modeling of QH, QE, and ΔQS. Periods with large uncertainties in Q* presented also large uncertainties in the other energy fluxes. The largest deviations were observed during afternoons and evenings of cool days. These deviations were directly associated with the quality of the cloudiness data.

Since K↓ can be obtained from meteorological observations, the major source of error in the modeling of the energy balance depends upon the parameterization of L↓, particularly at nighttime when K* is null and after sunrise and before sunset when K* is small. The accurate modeling of L↓ depends strongly on the parameterization of εsky and, in turn, on the accuracy of the cloudiness data. Also, the overlaying pollution layer in Mexico City may impose an additional cause of error. This needs further investigation and is an important aspect of coupled meteorological and chemical transport modeling.

Overall, the evaluation results of the parameterization approach presented here are consistent with those results observed in evaluations of the modeling schemes NARP and LUMPS with flux observations from urban locations in the USA, Canada, Europe, and downtown Mexico City. As expected, better results were obtained with the evaluation in which the modeled fluxes were compared to the measured fluxes using the best-fit coefficients compared to the more independent evaluation using meteorological observations from the local meteorological agency and airport. The larger errors in the latter evaluation were probably due to the distance between the flux system and the meteorological observations, in addition to possible errors in the measurements of K↓. However, both evaluations demonstrated the capability of the proposed scheme to predict with reasonable accuracy the energy fluxes from an urban neighborhood of a subtropical city, as well as the need of reliable standard meteorological observations.

This study is only the second evaluation of these relatively simple schemes and the first set of energy flux measurements including the four components of the radiative budget in a subtropical city in a developing country. In future work, the flux measurements presented here may be used as an alternative data set to evaluate more comprehensive parameterizations of the energy balance, such as the town energy balance model (Masson 2000; Masson et al. 2002), which considers the urban canopy geometry.

Acknowledgments

This study was supported by the National Science Foundation (ATM-0528227) and the Metropolitan Environmental Commission of Mexico. E. Velasco acknowledges the postdoctoral funding provided by the Faculty of Arts and Social Sciences of the National University of Singapore for the data analysis. The assistance and logistical support provided by the Atmospheric Monitoring System of the Federal District Government (SIMAT) was fundamental for the satisfactory development of this study. The cloudiness data were provided by J. Fast and J. Barnard from the Pacific Northwest National Laboratory and B. de Foy from Saint Louis University. The K↓ data used in the second evaluation of the modeling approach was provided by E. Alvarez from the National Meteorological Service. The authors acknowledge the comprehensive comments made by two anonymous reviewers that helped to improve substantially the original manuscript.

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© Springer-Verlag 2010