Intercomparison studies of models simulating the partitioning of energy over urban land surfaces have shown that the heat storage term is often poorly represented. In this study, two implicit discrete schemes representing heat conduction through urban materials are compared. We show that a well-established method of representing conduction systematically underestimates the magnitude of heat storage compared with exact solutions of one-dimensional heat transfer. We propose an alternative method of similar complexity that is better able to match exact solutions at typically employed resolutions. The proposed interface conduction scheme is implemented in an urban land surface model and its impact assessed over a 15-month observation period for a site in Melbourne, Australia, resulting in improved overall model performance for a variety of common material parameter choices and aerodynamic heat transfer parameterisations. The proposed scheme has the potential to benefit land surface models where computational constraints require a high level of discretisation in time and space, for example at neighbourhood/city scales, and where realistic material properties are preferred, for example in studies investigating impacts of urban planning changes.

The climate of cities differ from surrounding natural landscapes because
urban structures change the terms of the surface energy balance (Oke, 1982).
One important term is storage heat flux density (

The paper is structured as follows. Section 1 introduces key issues in measuring and modelling urban heat storage. Section 2 describes the conduction schemes and evaluation methods. Section 3 presents scheme results for a simple idealised environment, and within an urban model and subjected to observational forcing. Section 4 discusses and concludes findings.

In simple environments it is possible to measure

The residual approach has the inherent problem of accumulating all
observational errors, which can be significant for turbulent fluxes (Wilson
et al., 2002). Estimates of

At scales of individual buildings,

A potential weakness in evaluating urban model performance is the wide
variety of parameters that could describe urban materials. Urban models are
sensitive to material thermal parameter variation, particularly the magnitude
and phase of storage heat flux and sensible heat flux (Oleson et al., 2008b).
Selecting the best thermal parameters is not necessarily straightforward; as
land surface models are a simplified representation of reality, model
material parameters should only be viewed as abstract representations of
observed physical quantities (Gupta et al., 1999). Researching and inputting
realistic values based on local material parameters requires considerable
effort, and can result in little improvement to performance, or even degrade
it (Loridan and Grimmond, 2012). In PILPS-Urban, average model performance in

Three calculation methods of heat storage are compared: two discrete schemes
and an exact solution. The two discrete schemes lump a material's heat
capacitance at a temperature node and calculate solutions numerically at
each time step. The exact method calculates continuous harmonic solutions to
a periodic forcing. All three solve Fourier's law

A common discretised approach is to locate the temperature node centrally within a homogeneous layer – a half-layer scheme (Fig. 1a). This approach is used in many urban land surface schemes, for example Town Energy Budget (TEB) (Masson, 2000), Single-Layer Urban Canopy Model (SLUCM) (Kusaka et al., 2001), Building Effect Parameterization (BEP) (Martilli et al., 2002), Community Land Model – Urban (CLMU) (Oleson et al., 2008a), Vegetated Urban Canopy Model (VUCM) (Lee and Park, 2008), and the Australian Town Energy Budget (aTEB) (Thatcher and Hurley, 2012). Models utilising this method vary in their spatial and temporal resolution, but typically resolve between 1 and 10 substrate temperature nodes, at between 5 and 60 min time steps (Grimmond et al., 2009, 2010). The half-layer method is based on well-established land surface models representation of thermal conduction through soil (e.g. Oleson et al., 2010), and is also used in multi-layer snow and sea ice models (e.g. West et al., 2016).

A discretised form of the conduction equation (Eq. 3) is

Conceptual diagram for two methods of discretising heat transfer
through (three) homogeneous layers.

Energy conservation is demonstrated by the equivalency of Eqs. (7) and (8)
for layers 1 to

While the half-layer scheme lumps capacitance at the centre of layers, an
alternative approach is to lump capacitance at the interface between layers
(Fig. 1b). Since the paths of conduction between nodes are now completely
within homogeneous layers, Eq. (6) simplifies to

Total heat storage flux density for the interface scheme is

Again, energy conservation is demonstrated through the equivalency of
Eqs. (11) and (12), which after cancelling inner conduction terms leaves heat
storage flux density equal to outer conduction terms:

Here we use the admittance procedure (Butcher, 2006; Davies, 1973), which
calculates exact solutions to planar heat transfer through a series of
homogeneous layers when subject to a steady sinusoidal forcing on one side,
with a fixed temperature on the other. The international standard ISO
13786:2007 documents the method. The exact solution to the heat storage flux
density

Periodic areal heat capacity (ISO, 2007a) is a useful measure of a
composite materials ability to store heat over a sinusoidal cycle. It is a
better measure than overall heat capacity or surface thermal admittance as
it accounts for the periodic penetration depth of each material layer (for
thick composites) as well as heat lost through transmittance (for thin
composites). It can be calculated exactly as

The half-layer and interface discrete schemes are compared with exact
solutions of heat transfer. The external boundary is forced by

For both discrete schemes, a linear system of equations describing the temperature evolution of each node is generated and solved by decomposition and back-substitution of a tridiagonal matrix (Thomas' algorithm). Discrete schemes are run for six forcing periods of 24 h, with the first five periods discarded as spin-up, and the last compared with the exact solution. For the exact solution, components of the heat transfer matrix are calculated numerically and harmonic solutions computed. Performance statistics are based those used in PILPS-Urban Phase 2, as described in Phase 1 (Grimmond et al., 2010).

The discrete schemes are also compared within an urban land surface model (aTEB) forced by observations using the methodology of the PILPS-Urban Phase 2 (Grimmond et al., 2011).

The proposed interface scheme is implemented in the Australian Town Energy Budget (aTEB) urban land surface model (Thatcher and Hurley, 2012). aTEB was developed to act as the urban component of a regional or global climate model, so takes the highly efficient building-averaged approach where the generic urban unit is an infinite street canyon (Nunez and Oke, 1977). Canyon surfaces (walls, road, snow and vegetation) are connected to a bulk canyon air layer via an aerodynamic resistance network. Roofs and canyon air are then connected in parallel to the overlying atmosphere.

Although written from the ground up, aTEB is conceptually based on the
influential Town Energy Budget (TEB) urban canopy model (Masson, 2000) with
some modifications for Australian conditions. Modifications include the following:

In-canyon vegetation for suburban areas represented by a big-leaf model, adapted from Kowalczyk et al. (1994) but with a largely reduced set of prognostic variables.

An air-conditioning component which pumps waste heat into canyons and prevents buildings acting as energy sinks during high-temperature periods, allowing energy closure at each time step.

A two-wall canyon allowing radiative interactions between a sunlit and shaded
walls, and a canyon airflow parameterisation with venting and recirculating
regions, each integrated through 180

aTEB would be categorised as a “complex” urban model following the methodology of Grimmond et al. (2010, 2011), primarily because of its canyon-based approach. Conceptually similar models include TEB (Masson, 2000), SLUCM (Kusaka et al., 2001) and CLMU (Oleson et al., 2008a). A significant differentiator amongst conceptually similar models is the parameterisation of heat exchange between canyon surfaces and turbulent air. By default, the SLUCM uses a form of the Jürges formula (1924), while TEB and CLMU use a form of Rowley et al. (1930). An alternative approach was developed by Harman et al. (2004b) where aerodynamic conductance is separately calculated for each canyon surface based on the airflow in different regions of the canyon for different canyon geometries. Urban canopy models that utilise forms of the Harman circulation scheme include the Single Column Reading Urban Model (SCRUM) (Harman and Belcher, 2006), the Met Office Reading Urban Surface Exchange Scheme (MORUSES) (Porson et al., 2010) and aTEB (Thatcher and Hurley, 2012). In order to assess how the proposed conduction scheme is affected by different aerodynamic heat transfer parameterisations, the Jürges, Rowley and Harman methods have been implemented in aTEB as described in Appendix C. Further details on aTEB is available in Thatcher and Hurley (2012) and Luhar et al. (2014).

Observational data were obtained from flux tower measurements in a suburban
site in Melbourne, Australia (Coutts et al., 2007a, b). Data include
up-welling and down-welling long and shortwave radiation (

The data used here are identical to that used in the First International Urban
Land Surface Model Comparison Project (PILPS-Urban) Phase 2 (Grimmond et al.,
2011), from which our evaluation methodology follows, that is:

Observed downwelling radiation, air temperature, pressure, wind, humidity and rainfall data were used as forcing data to run the urban model offline, i.e. without the need to be coupled to an atmospheric/earth system model.

Observed upwelling radiation, turbulent heat and residual heat storage flux observations were compared with model output to evaluate the performance of the model.

The initial 108.4 days of observation were treated as spin-up and excluded from analysis.

The remaining 366 days were analysed, but if any flux were missing or gap filled in a time interval, all data in that interval were ignored, resulting in 8520 usable half-hour time intervals.

The site at Preston, Melbourne, is typical of low-to-medium density suburban housing in Australia, with detached one- to two-storey brick, timber and steel framed buildings, separated by roads, lawn and large trees. The site is classified in Best and Grimmond (2014b) as a local climate zone (LCZ) 6 (Stewart and Oke, 2012) or as an urban zone for energy exchange (UZE) medium density (Loridan and Grimmond, 2011, 2012).

Conduction schemes are evaluated through two independent methods: (1) in a highly idealised environment to allow comparison to exact solutions to heat transfer and (2) within an urban land surface model and compared with observations.

Discrete schemes are run at very high resolution (200 layers, each 1 mm
deep) in order to test code validity. Table 1 shows the normalised standard
deviation (SD) of high spatial resolution simulations over various time steps,
where an SD of 1.0 means amplitudes of discrete and exact solutions match. At
the higher time and space resolutions, both schemes converge towards the
exact solution as expected. As the temporal resolution is decreased to
30 min time step amplitudes of

Normalised standard deviation (SD) of

We now test the performances of the discrete schemes at space and time
resolutions more practical and typical of urban land surface models. The
20 wall and 12 roof configurations described in the CLMU database
were tested at 30 min time steps over various levels of complexity. In
Fig. 2, errors for 2–10 layer configurations are plotted individually (288
total), along with layer means. Of the two discrete schemes, the interface
scheme has an average normalised SD closer to the exact solution for each
layer up to ten (Fig. 2a). The mean normalised SD of

Heat storage errors of the half-layer and the proposed interface
schemes vs. exact solutions over various levels of complexity. Thirty-two roof
and wall types from the CLMU database (Jackson et al., 2010) were modelled
with nine levels of complexity, for a total of 288 representations (dots).
Mean of layer errors indicate scheme performance at different levels of
complexity (lines). For both

Heat storage errors of the half-layer and the proposed interface
schemes vs. exact solution over increasing periodic areal heat capacity (or
thermal mass), with lightweight framed construction on the left, and
heavyweight stone/ rubble construction on the right. For both

Figure 2b shows the layer mean of the interface scheme mean absolute errors
(MAE) are smaller than the half-layer scheme. The normalised MAE for the
half-layer scheme monotonically decreases as the number of layers increases.
Time and space discretisations both lead to negative bias in

Heat storage response of half-layer and proposed interface schemes
vs exact solutions for four wall types:

Having evaluated the performance of the two schemes over increasing material complexity, we evaluate performance for different composite thermal characteristics. Figure 3 shows the normalised SD and the MAE for each scheme at 30 min time steps for the 288 CLMU walls and roofs, plotted against the composite material's periodic areal heat capacity (Eq. 20), herein called thermal mass. Materials with low thermal mass (e.g. lightweight framed walls or sheet metal) do not absorb and store as much heat over the diurnal cycle as materials with high thermal mass (e.g. concrete or brick). In contrast to discrete layers, thermal mass varies continuously, so the response of each scheme is represented as a locally weighted linear regression (LOWESS) (Cleveland, 1979). The thicker lines represent all 288 material configurations, while the thin dashed lines represent a subset of walls and roofs with four layers (the number of layers used later in the urban model analyses).

Figure 3a shows the interface scheme LOWESS of normalised SD is closer to the
exact solution for all values of thermal mass represented in the CLMU
database. The half-layer scheme's LOWESS of normalised SD shows an increasing
negative bias for larger thermal mass, while the interface scheme LOWESS is
less steep. Figure 3b plots MAE (non-normalised) and shows schemes have
greater difference in absolute errors for assemblies with higher thermal
mass. Heat storage becomes a larger proportion of the energy balance in
neighbourhoods with more heat capacity, so a scheme that is better able to
represent

We now evaluate the performance of the discrete schemes using wall and roof
characteristics that are typical of previous urban modelling studies. Figure
4 displays the results of four walls, one realistic (SITE) and three
optimised (WRF, UZE and aTEB). All walls are made up of four layers and later
analysed in the aTEB urban model (Sect. 3.2). The upper panels show

Heat storage response of half-layer and proposed interface schemes vs exact solutions. Material datasets per Fig. 4 and Appendix A. Upper panels show storage heat flux density over a periodic cycle, lower panels show normalised error from exact solutions. The interface scheme reduces errors for these roof representations, but improvements are less pronounced than for walls (Fig. 4).

Hourly averaged diurnal heat storage response of an urban land surface model to four wall/roof datasets over 12 months. Material datasets per Fig. 4 and Appendix A. In each case, the interface scheme reduces errors by increasing heat storage amplitude. Only Harman aerodynamic heat transfer parameterisation shown (others qualitatively similar). When evaluation is limited to 3-month seasons, differences between schemes are qualitatively similar to annual results, although overall errors are higher in summer where radiation flux magnitudes are larger (not shown). The MAE for hourly averaged fluxes of a mean day shown here is different to MAE of each half-hour time step throughout the evaluation period (Table 2).

The same analysis is undertaken for roofs and shown in Fig. 5. Overall the interface scheme improves performance of these four-layer representations; however, the degree of improvement is smaller than the walls analysed above. Errors for both schemes are more pronounced in the UZE and aTEB roofs, which have greater total depths and therefore higher spatial discretisation errors (roof and wall characteristics listed in Appendix A).

As a consequence of the method of discretisation, the interface scheme has
one additional temperature node over the half-layer scheme for any given
number of layers (see Fig. 1). In Fig. 2, the interface average errors are
lower than half-layer average errors for assemblies with

aTEB half-hourly performance statistics for

Figure 6 compares the net storage heat flux density (

Half-hourly

The interface scheme in general improves the urban simulation performance in

A Taylor diagram of energy fluxes for each simulation with interface scheme (filled markers) and half-layer scheme (unfilled markers) connected by a line. Change toward a normalised standard deviation of 1 indicates improved variance (radial distance) and toward the bottom axis indicates improved pattern correlation (radial angle). Where both are improved, the centred root mean square (cRMSE) error decreases towards zero (interior arcs). Different colours indicate different material parameter datasets (SITE, WRF, UZE, aTEB; see Appendix A). Different marker shapes indicate different aerodynamic heat transfer parameterisations (Rowley, Harman, Jüges; see Appendix C). Anonymous results of participants in the PILPS-urban Phase 2 intercomparison are also plotted (Grimmond et al., 2011). Overall the interface scheme improves storage heat, sensible heat and longwave radiation fluxes. Latent heat flux response is less clear, but on average the interface scheme degrades performance. See Table 3 for summary statistics.

Average performance statistics: change from half-layer to interface scheme
for 12 pairs of experiments (improvement in bold). The significance value (

In Fig. 7, a Taylor diagram (Taylor, 2001) extends the statistical evaluation
of the two conduction schemes to include sensible and latent heat, and
upwelling longwave radiation fluxes. Shortwave radiation flux is omitted
because it is unaffected by the conduction schemes, and downwelling radiation
fluxes are prescribed. A Taylor diagram is useful in determining whether the
difference error (measured in centred, or bias corrected, root mean square
error: cRMSE) occurs from a change in variance (standard deviation: SD) or a
change in pattern correlation (Pearson's correlation coefficient:

For net storage heat flux (

For upwelling longwave radiation flux (

For sensible heat flux (

For latent heat flux (

The mean change in performance from half-layer to interface schemes for the
12 sets of experiments are presented in Table 3, with improvements in bold. A
paired, two-sided

We evaluated the performance of two implicit discrete schemes that represent heat conduction through urban materials. The half-layer scheme is well established and widely used in land surface models for urban structures, soil, snow and ice. It lumps heat capacitance at nodes centred within discrete layers. We proposed an alternative scheme of similar complexity that lumps heat capacitance at the interface between layers. We used two independent methods to evaluate the schemes: comparison with exact solutions to heat transfer in an idealised environment, and comparison with long time-series observations for an urban site with heat storage calculated as a residual of the urban energy balance.

In the idealised evaluation, a series of multi-layered assemblies of various
complexities were subjected to a sinusoidal temperature forcing representing
the diurnal cycle. The half-layer scheme was found to systematically
under-estimate

In the urban model evaluation, we assessed the impact of implementing the
interface scheme on performance of an urban land surface model (aTEB) over a
15-month observation period for a site in Melbourne, Australia. We evaluated
four material parameter datasets and three common aerodynamic heat transfer
parameterisations. In the urban model, the interface scheme tended to
increase the diurnal magnitude of

By physical reasoning, the interface scheme increases storage available to
the transient external environment by representing heat capacity at skin
surfaces, resulting in larger diurnal amplitudes of

Other than affecting flux predictions, the interface scheme can provide structural benefits to urban land surface models. The skin temperatures of urban surfaces are used in balancing energy budgets and determining radiant and turbulent fluxes. The interface scheme calculates skin temperatures prognostically, while models using the half-layer schemes diagnose skin temperatures as an additional calculation, or assume the first layer bulk temperature is representative of the skin temperature. For aTEB, moving from a half-layer to an interface conduction scheme avoided the additional calculations required to diagnose skin temperatures, and resulted in a 5 % reduction in average runtime for offline simulations.

In conclusion, the interface conduction scheme has the potential to benefit urban land surface models simulating environmental phenomena at scales that require a high level of discretisation in time and/or space for reasons of efficiency. Examples include numerical weather prediction, where many simulations are required in short timeframes, or climate studies that require simulation over long timescales. The interface scheme also improves performance in assemblies using realistic material thermal parameters, so may benefit large-scale studies investigating future impacts of urban design or climate mitigation measures. Results presented here are based on a single urban model with multiple configurations, and on a single observation site, so future work may extend evaluation to other sites and other urban models.

Source, evaluation and plotting code is included in supplementary material,
available through the online Git repository at

alpha01.dat (used as forcing data for the urban model, includes observation quality flags),

observation_preston.csv (flux observations to assess model performance),

QF.txt (estimates of anthropogenic heat sources).

Urban land surface model material dataset parameters. References per text in Appendix A.

From a dataset of global urban characteristics (Jackson et al., 2010) intended for use in the Community Land Model – Urban (CLMU) (Oleson et al., 2008a), or other global climate models. The database catalogues the properties of 32 common walls and roofs from around the world. Attempts are made by Jackson et al. (2010) to reconcile the whole wall/roof thermal conductivity with real-world values by estimating the effects of thermal bridging, air leakage and poor construction. For our analysis, all 10-layer composite walls and roofs were collapsed through each iteration down to two layers using depth-weighted averages of conductivity and heat capacity, resulting in 288 representations of walls and roofs of varying thermal characteristics and complexity.

As presented in PILPS-Urban Phase 2 (Grimmond et al., 2011). Characteristics were derived from an area and depth weighted average of material thermal properties at the observation site. Roofs and walls were an aggregate of metal, terracotta, concrete and asbestos, insulation, lightweight framing and plasterboard, separated into four layers: external skin, structure, insulation and internal lining.

From the WRF/urban integrated urban modelling system v3.2 (Chen et al., 2011), which includes the SLUCM and BEP urban schemes. In WRF, three default sets of parameters are available for various densities of urban land cover, here we use the low-intensity residential set following the observation site's classification by Loridan and Grimmond (2012). The WRF default parameters represent a generic homogeneous material with a heat capacity and conductivity similar to lightweight concrete throughout.

From the updated WRF/urban parameters described in Loridan and Grimmond (2012) based on results of a multi-objective optimisation algorithm to minimise root mean square error over 15 urban locations. New parameter values were recommended for three categories of urban areas based on Urban Zone for Energy exchange (UZE: Loridan and Grimmond, 2011) and were subsequently included in releases of WRF/urban as an optional dataset. We use the medium urban category following the observation site's classification by Loridan and Grimmond (2012).

From the ECOCLIMAP database (Masson et al., 2003) on which TEB (Masson, 2000) defaults are based, but with increased layer depths per Thatcher and Hurley (2012). The walls and roofs are not representative of typical building methods in Australia, but nonetheless give reasonable results for generic Australian cities. The roof is a layered composite with thermal conductivity and heat capacity of dense concrete, aerated concrete and insulation; the walls a composite of concrete and insulation and the road/soil of asphalt and dry soil.

Urban land surface model parameters: constant in each experiment. Based on observation site characteristics (Grimmond et al., 2011) and aTEB defaults.

Three parameterisations of heat exchange between surfaces and turbulent air
are evaluated within aTEB: the Jürges, Rowley and Harman methods.
Sensible heat flux between a surface (

The Jürges method (Jürges, 1924) is implemented in aTEB as
described by Kusaka et al. (2001):

The Rowley method (Rowley et al., 1930) is implemented in aTEB as
described by Masson (2000):

The Harman method (Harman et al., 2004b) is implemented in aTEB as described
by Thatcher and Hurley (2012):

Mean heat storage errors of the half-layer and the proposed
interface schemes vs exact solutions over 2–10 layer representations and for
1, 15, 30 and 60 min time steps (per Fig. 2). For both

In Fig. D1 the analysis of Sect. 3.1.2 is repeated for 1, 15, 30 and 60 min
time steps. For the half-layer scheme, the optimum combination of time step
length and layer number is achieved by increasing the number of layers to
10

M. Thatcher developed the original aTEB urban model. M. Lipson proposed and developed the interface scheme, wrote the analysis code and prepared the manuscript, with assistance from M. Hart and M. Thatcher.

The authors declare that they have no conflict of interest.

This study was supported by the Australian Research Council (ARC) Centre of Excellence for Climate System Science (CE110001028). Mathew Lipson was supported by an Australian Postgraduate Award. We thank Andrew Coutts for sharing original observational datasets and anthropogenic heat flux estimates; Martin Best, Sue Grimmond and Maggie Hendry for answering questions and providing additional information regarding the PILPS-Urban intercomparison analysis methods; Yannick Copin for the base Taylor diagram code; and Andrew Pitman for useful feedback on the manuscript. Finally, we thank the two anonymous reviewers, whose comments improved the clarity and robustness of the study. Edited by: J. Kala Reviewed by: two anonymous referees