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

In an urban neighborhood of interest, the walls and roofs of the buildings are generally built with a selection of different materials. Because we are interested in interactions with the atmosphere, the averaged material properties for walls and roofs used in urban parameterizations should give the same total sensible heat flux as the weighted sum of the sensible heat fluxes from the individual materials considered.¹ So, in a general sense, indicating with α_j the area fraction of the material j in the zone and indicating with H_j the sensible heat flux from the material j to the atmosphere, we are seeking a set of averaged material thermal properties that would yield a sensible heat flux H such that

Assuming that the convective heat exchange coefficients h are equal for each different material, this translates into

where T_A is the air temperature, T_j is the surface temperature of the jth material, and T is the surface temperature of the averaged material. So, clearly the thermal properties of the averaged material should be such that the previous relation [Eq. (2)] is satisfied. Therefore, the surface temperature of the averaged material is equal to the weighted sum of the individual surface temperatures.

Within a layer of homogeneous material, the temperature T satisfies the heat diffusion equation:

where λ (W m⁻¹ K⁻¹) is the thermal conductivity and c (J m⁻³ K⁻¹) is the volumetric heat capacity of the material. So, the question is how to determine the physical properties λ, c, and the thickness d of the new (averaged) material such that

where d_j represents the thickness of the jth material present in the urban zone. The physical properties (d, c, and λ) of the averaged material could be determined in many ways, but three approaches will be analyzed here—

—where λ_j is the thermal conductivity of the jth material and c_j is its volumetric heat capacity. In the third approach [Eq. (5c)], P is a time period of 1 day (because in this context the diurnal cycle typically dominates) and Z_D is a characteristic depth of the averaged material.

The first average [Eq. (5a)] is the standard approach. It weights the parameters with the area fractions α_j 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 (d_jλ_j; W K⁻¹) and heat capacity (d_jc_j; J K⁻¹ m⁻²) as opposed to weighting λ_j and c_j alone without accounting for the variation in d_j. By including the d_j 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 λ_j, c_j, and d_j, is small relative to the period of the forcing).

The third average [Eq. (5c)] is significantly different. A justification for the last average proposed is as follows. Assuming that at the internal side of a material the temperature exhibits a periodic signal with period P and that the amplitude increases as the outdoor side (where heat exchanges with the atmosphere occur) is approached, a particular solution of the diffusion equation [Eq. (3)] can be written as

where T₀ is a reference temperature, ΔT_j is the amplitude of the sinusoidal (within the material), z is the distance from the interior boundary of the material, and Z_Dj = [(λ_jP)/(c_jπ)]^1/2 is a characteristic depth of the material (when the amplitude of a signal is damped, Z_Dj is known as the “damping depth”). Using the same reasoning for the averaged material, Eq. (4) can be written as

Considering the case in which each of the different materials has the same amplitude ΔT_j, Eq. (7) can be reduced to

which is independent of the amplitude ΔT. With this approach, the physical parameters d and c of the averaged material can be chosen freely but the thermal conductivity [λ = (cπZ_D²)/P] is fixed through Eq. (8) and the consideration that the averaged temperature is described by the left-hand side of Eq. (7). The assumptions and simplifications used in the above derivation are never completely fulfilled in any real situation. However, the goal is to find an averaged set of material properties (d, c, and λ) that is able to satisfy Eq. (2), and we conduct numerical simulations solving Eq. (3) (because no analytical solutions of the diffusion equation exist for most real situations) to evaluate the physical appropriateness of our approach. If the numerical results indicate that the third average [Eq. (5c)] better describes the interaction between urban surfaces and the atmosphere, the above hypotheses are supported. In essence, the new approach consists of calculating an average from an analytical solution of the heat conduction equation and then comparing the results with those obtained numerically, with less stringent and more realistic assumptions. Thus, a large number of simulations are carried out, and for each proposed average in Eq. (5) the surface temperature T is compared with the “correct” average temperature

(i.e., that temperature that will result in the correct sensible heat flux to the atmosphere).

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 κ_i_d_l, meaning that the simulation was carried out for the material i (κ_i = λ_i/c_i, i = 1, 2, 3) with thickness l (d_l, 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 κ_ijk_d_lmn. 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 κ_ijk_d_lmn 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 κ_ijk_d_lmn represents the averaged material that attempts to describe an urban zone formed by 20% of the κ_i material with d_l thickness, 30% of the κ_j material with d_m thickness, and 50% of the κ_k material with d_n 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.

In numerically solving Eq. (3), two different boundary conditions are considered here. With the first, the internal boundary condition has been fixed (i.e., the temperature at the deepest level inside the material is constant for the whole simulation period); with the second, the internal surface temperature can change freely such that ∂T/∂z = 0 at the lower boundary throughout the simulation. One might expect the former boundary condition to more closely approximate homogeneous walls and roofs close to the relatively constant internal building temperature, whereas the latter is expected to better represent external layers closer to ambient forcing and interior layers in contact with insulation materials. At the surface, the boundary condition is defined by solving an energy budget equation in both cases (an explanation of the symbols used can be found in the appendix):

where and Q_h = h(T_sfc − T_a). The term Q_G (storage heat flux density) is the net heat flowing into the material. The sensible heat flux from the surface is a function of the difference between the air temperature and the surface temperature, and of the wind speed through the convective heat transfer coefficient h. A constant value for h is considered in the simulations, representing a day with little wind variability, for simplicity. The air temperature T_a and the downward shortwave radiation K↓ are taken as follows (values of the parameters can be found in Table 2):

Note that the radiation reaches a maximum π/4 or one-eighth of a cycle before the air temperature (i.e., 3 h for diurnal cycles). The heat diffusion equation is solved by an implicit finite difference approach at each time step by inverting the corresponding tridiagonal matrix.

4. Simulation results

To compare the averaging methods, the sensible heat obtained in each case (i.e., for each κ_ijk_d_lmn) is compared with the sensible heat representative of the urban zone: αQ_{hi_l} + βQ_{hj_m} + γQ_{hk_n} (where Q_{hi_l} is the sensible heat obtained with the κ_i material of depth d_l, 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 T_a 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 ΔT_j 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 κ_j (=λ_j/c_j) (relative to the mean κ_j), and small d_j 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.