1. Introduction
In recent decades, the number of urban meteorological and dispersion modeling publications has grown rapidly as a result of increased computational speed. These studies help us to understand better the atmospheric and environmental effects of urbanization (urban heat islands, pollution, etc.), and they aid in the search for mitigation strategies. However, contemporary increases in computational speed are no panacea [see a recent review of urban modeling by Martilli (2007)]. A mesoscale model needs a horizontal domain of tens/hundreds of kilometers to simulate mesoscale circulations. Yet, for computational reasons the resolution cannot be better than a few kilometers, and much of the heterogeneity in the urban landscape cannot be explicitly represented in the model.
Several urban parameterizations (e.g., Masson 2000; Kusaka et al. 2001; Martilli et al. 2002; Kanda et al. 2005) have been developed to communicate the mean thermal and dynamic effects of the city to the mesoscale model. Urban parameterizations have notably improved numerical results as they have developed, though their complexity and their computational demands have also increased. A key problem is the assignment or derivation of averaged physical properties that optimally represent or define each local-scale urban neighborhood, “urban climate zone” (Oke 2006), or urban model grid cell. These averaged physical properties are very important because they feed the urban parameterization; as a consequence, they are directly responsible for the quality of the numerical results. An important issue, then, is how we can obtain averaged material properties that represent the same physical interaction with the atmosphere as a given heterogeneous ensemble of materials in a city (note that we use the word “average” throughout this paper to mean “weighted” or “geometrically combined”). In this work, we will focus on the modeled accuracy of the sensible heat flux, because it is the most relevant interaction between (dry) urban surfaces and the atmosphere. This work is a first exploration of this question.
In section 2, three different formulations for averaging thermal properties of materials are presented. The numerical framework is explained in section 3. The comparison between the different proposals is carried out in section 4. Conclusions and future directions are discussed in section 5.
2. Theoretical framework







The first average [Eq. (5a)] is the standard approach. It weights the parameters with the area fractions αj of the different materials present in the urban zone. The second average [Eq. (5b)] is similar to the first one but weights layer-integrated thermal conductivity (djλj; W K−1) and heat capacity (djcj; J K−1 m−2) as opposed to weighting λj and cj alone without accounting for the variation in dj. By including the dj in the numerator, this last approach assumes that material thicknesses are small relative to the damping depth associated with the period of the dominant forcing (i.e., it assumes that the time scale for thermal adjustment across the layer, which depends on λj, cj, and dj, is small relative to the period of the forcing).




3. Simulation development
First, nine base-case simulations are carried out using three typical materials (metal, softwood, and concrete; see Table 1), with three different thicknesses (d1 = 0.025 m, d2 = 0.05 m, and d3 = 0.10 m) for each material. In the simulations, Eqs. (3) and (9) are solved numerically. The nomenclature used to describe a base case is κi_dl, meaning that the simulation was carried out for the material i (κi = λi/ci, i = 1, 2, 3) with thickness l (dl, l = 1, 2, 3). These nine simulations, when appropriately weighted by their area fraction, provide the correct results with which the averaged material simulation results are evaluated. Second, 162 different simulations were carried out for each proposed average; each one of these simulations is represented by κijk_dlmn. Each of the six subscripts can take any of the values 1, 2, and 3, but the first three cannot be repeated, for a total of 6 × 27 = 162 different scenarios. In other words, a given simulated average material κijk_dlmn may be “composed” of materials of identical depth (the idea is to consider all of the possible combinations of thickness) but not identical material type. The case κijk_dlmn represents the averaged material that attempts to describe an urban zone formed by 20% of the κi material with dl thickness, 30% of the κj material with dm thickness, and 50% of the κk material with dn thickness (these percentages have been chosen arbitrarily and are fixed in all of the comparisons so as to keep the number of scenarios reasonable). To assess the impact of including one “outlier” material in terms of its thermal properties (metal), the above simulations are repeated with brick (Table 1) instead of metal.



4. Simulation results
To compare the averaging methods, the sensible heat obtained in each case (i.e., for each κijk_dlmn) is compared with the sensible heat representative of the urban zone: αQhi_l + βQhj_m + γQhk_n (where Qhi_l is the sensible heat obtained with the κi material of depth dl, and so forth, and α = 0.2, β = 0.3, and γ = 1 − α − β are the area fractions fixed previously). Sensible heat flux is used for the comparison, because equivalent results are obtained for the surface temperature since the outdoor temperature Ta and convective heat transfer coefficient h do not vary between simulations. The root-mean-square error (RMSE) is computed using results at all 5760 time steps (2 days with 30-s time steps).
The results for the different averages can be seen in Fig. 1 for the first boundary condition (internal temperature fixed), and in Fig. 2 for the second (internal temperature free) for different area fractions and thicknesses of metal, softwood, and concrete. It is difficult to distinguish which of the first two (approach a or approach b) averages is superior; however the approach-c average notably improves the results—in particular, for the case of the fixed internal temperature. This is not unexpected given that the assumption in Eq. (8) that the ΔTj are equal for all materials is better satisfied with this boundary condition. RMSE for each combination of the three different materials does not exhibit coherent behavior dependent on the thickness of the materials [the thinnest is represented on the left (d111) and the thickest is represented on the right (d333) in all of the figure panels]. Only with the free internal boundary condition, large κj (=λj/cj) (relative to the mean κj), and small dj does approach b clearly outperform approach a and demonstrate RMSE equal to that of approach c (not shown). Thus, average b only yields good results for a relatively small subset of the total number of simulations (about 20) when certain conditions are met. It seems clear that the assumption of small damping depth relative to layer thickness (which underpins approach b) does not hold for most of the combinations of depths and materials modeled here.
To obtain a better overall picture of the differences, the mean and standard deviation of the RMSE values are calculated for each averaging approach. The results show that overall the approach-b average is similar to the approach-a average (Table 3). On the other hand, the best results are obtained on average for the weighting approach c. However, when brick replaces metal (simulations with the internal boundary condition ∂T/∂z = 0), RMSE magnitude decreases substantially and the three averaging methods converge to a significant extent (Table 3), but the relative decrease in RMSE with averaging method c still remains significant (≈30% decrease vs ≈40% with metal). Thus, the chosen averaging method is less important in an absolute sense when materials with similar thermal behavior are averaged, as evidenced by the overall RMSE decrease. In essence, the greater the variability in a material thermal property is, the more poorly any average value may be expected to represent the distribution of material thermal properties. Nevertheless, method c appears, on average, to yield a significantly smaller RMSE in a relative sense.
Last, the downward radiation used so far in the simulations is more representative of horizontal than vertical surfaces (which may be shaded for portions of the day). To take into account the effects of vertical surface (wall) orientation, the metal, softwood, and concrete simulations are repeated with a different variation of incident shortwave radiation. The K↓ is set to zero before midday (P/2), at which point it jumps to the value prescribed in Eq. (10) where it remains for the duration of the day. This rapid variation in solar forcing is typical of a west-facing wall in the Northern Hemisphere, for example. The results are similar to those obtained previously (see Table 4), and again the approach-c average is superior.
5. Conclusions
This work is a first step toward the determination of physical properties that represent the behavior (in terms of its interaction with the atmosphere) of an ensemble of different materials in an urban zone or neighborhood. A new averaging method for thermal parameters has been proposed [Eqs. (5c) and (8)], and better results have been obtained—in particular, for an ensemble of materials with large variability in thermal behavior. To use the new averaging scheme, information on the area occupied by each material and the thickness of each material is needed. This information can be obtained from building-construction databases. Similar formulations can be used for other surface descriptors necessary in (urban) mesoscale atmospheric modeling. That is, the equation that describes the physics of the problem for an ideal situation is solved, and subsequently new values of physical parameters that better represent the net effect of the subgrid-scale heterogeneity are obtained.
The authors thank CIEMAT for the doctoral fellowships held by Francisco Salamanca and NSERC and UBC for the doctoral scholarships held by Scott Krayenhoff. We also thank the reviewers for important comments on the manuscript. This work was funded by the Ministry of Environment of Spain.
REFERENCES
Clarke, J. A., , P. P. Yaneske, , and A. A. Pinney, 1991: The harmonisation of thermal properties of building materials. Tech. Note 91/6, BEPAC, 87 pp.
Kanda, M., , T. Kawai, , M. Kanega, , R. Moriwaki, , K. Narita, , and A. Hagishima, 2005: A simple energy balance model for regular building arrays. Bound.-Layer Meteor., 116 , 423–443.
Kusaka, H., , H. Kondo, , Y. Kikegawa, , and F. Kimura, 2001: A simple single-layer urban canopy model for atmospheric models: Comparison with multi-layer and slab models. Bound.-Layer Meteor., 101 , 329–358.
Martilli, A., 2007: Current research and future challenges in urban mesoscale modelling. Int. J. Climatol., 27 , 1909–1918.
Martilli, A., , A. Clappier, , and M. W. Rotach, 2002: An urban surface exchange parameterization for mesoscale models. Bound.-Layer Meteor., 104 , 261–304.
Masson, V., 2000: A physically-based scheme for the urban energy budget in atmospheric models. Bound.-Layer Meteor., 94 , 357–397.
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APPENDIX
List of Symbols Used in Solving Energy Budget Equations
- Q* Net radiative flux density (W m−2)
- Qh Sensible heat flux density (W m−2)
- K↓ Downward shortwave radiative flux density (W m−2)
- K0 Maximum value of the shortwave radiation (W m−2)
- L↓ Downward longwave radiative flux density (W m−2)
- Tsfc Surface temperature (K)
- Ta Air temperature (K)
- Tmax Maximum value of the air temperature (K)
- Tmin Minimum value of the air temperature (K)
- α Shortwave albedo
- ε Longwave emissivity
- σ Stefan–Boltzmann constant (W m−2 K−4)
- h Convective heat transfer coefficient (W m−2 K−1)

RMSE of the sensible heat Qh for fixed internal temperature obtained with the three different averages (approaches a, b, and c) and for the combination formed by (a) 20% of the κ1 material, 30% of the κ2 material, and 50% of the κ3 material with 27 combinations of thicknesses dlmn (represented by the notation κ123_dlmn); (b) 20% κ1, 30% κ3, and 50% κ2 (κ132_dlmn); (c) 20% κ2, 30% κ1, and 50% κ3 (κ213_dlmn); (d) 20% κ2, 30% κ3, and 50% κ1 (κ231_dlmn); (e) 20% κ3, 30% κ1, and 50% κ2 (κ312_dlmn); and (f) 20% κ3, 30% κ2, and 50% κ1 (κ321_dlmn). In (a)–(f), at the extreme left there is d111 and at the extreme right there is d333. From left to right, the n index permutes the fastest and the l index permutes the slowest (i.e., the sequence is d111, d112, d113, etc.).
Citation: Journal of Applied Meteorology and Climatology 48, 8; 10.1175/2009JAMC2176.1

As in Fig. 1, but for free internal temperature.
Citation: Journal of Applied Meteorology and Climatology 48, 8; 10.1175/2009JAMC2176.1
Inputs and parameters used in the numerical simulations.

Mean and standard deviation of the RMSE values for each of the three thermal property averaging approaches.

As in Table 3, but for the case in which the downward shortwave radiation is zero before midday.

We want to stress here that we are not proposing to average together materials from roofs with those from walls. Rather, the averages are among roof materials and wall materials, to arrive at average values for each of these two surfaces. Urban canopy parameterizations, in fact, resolve different budgets for walls and roofs, because the radiation and dynamics behave differently for vertical and horizontal surfaces.