## 1. Introduction

All general circulation models (GCMs) analyzed and interpreted by Solomon et al. (2007) [Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report, chapter 10] project an increasing global air temperature for the future. As one consequence, the number of days with heat stress will increase in many regions around the globe. The expected changes in strength and frequency of extreme daytime or nighttime temperatures will provoke enhanced discomfort and even an increase in mortality (Souch and Grimmond 2004).

The additional heat load associated with global warming will especially affect cities since it adds to the well-known urban heat island (UHI) effect. The higher temperatures in cities are caused by the high density of buildings, the nonpermeability of the ground surface, and the surplus of energy due to anthropogenic activities. During daytime, land-use and surface characteristics have a greater influence on the UHI intensity than does anthropogenic release of heat (Hart and Sailor 2009). Furthermore, UHI intensity is found to be strongly controlled by the amount of evaporation and shading from buildings and trees in the urban environment (Spronken-Smith and Oke 1998). In districts with many high buildings, the daily maximum temperature can even be lower than in surrounding areas. During nighttime, heat release from anthropogenic activities plays a more important role. In addition, the emission of heat, which has been stored in the urban fabric during daytime, the reduced ventilation, and the sky-view factor of the urban canopy contribute to higher nocturnal temperatures in cities (Oke 1987). The warmer urban boundary layer can be advected to the surrounding areas and thereby can influence temperature records taken at rural sites (Brandsma et al. 2003).

Mitigation concepts for the urban heat load address geometry and spacing of buildings (height-to-width ratio of street canyons) to promote ventilation and building materials and color to reduce the heat storage capacity and absorption of solar radiation. In this context, urban vegetation is important for shading and evaporative cooling to moderate the thermal and moisture climate (Roth 2007). Stone and Norman (2006) investigated the influence of the size and material composition of built-up areas on UHI intensity. They found that the contribution of individual land patches to regional heat island formation could be reduced by 40% if suitable land-use planning policies are adopted.

Observations already provide evidence of the impact of global climate change and land-use changes on the heat load in cities. In Tokyo, the rise in the annual mean air temperature during the past 100 years is found to be much higher (3°C per century) than that of global air temperature (0.8°C per century) (Fujibe 2009). Projected trends for the city of London until 2080 show a surplus of urban warming of about 0.04°C per decade in addition to a regional warming of about 0.5°C per decade for midrange emission scenarios (Wilby 2003). It is pointed out that this projected urban warming is probably a conservative estimate since changes in urban population, building density, energy consumption, and so on are neglected in this study.

To maintain or improve the quality of living in cities, urban planners need detailed information on future urban climate on the residential scale. This study introduces an approximate but computationally inexpensive procedure, which we call the cuboid method, to provide the data needed to quantify the impact of climate change on the urban heat load with the required spatial and temporal resolution. We utilize global climate projections from the “ECHAM5” model downscaled to the regional scale using four different regional climate models (RCM). The regional climate time series provided by the RCM simulations and a limited set of diurnal simulations with a high-resolution urban climate model are used to estimate the expected changes in the frequency of air temperature threshold exceedances. The applicability of the method is demonstrated for the city of Frankfurt am Main, Germany.

This paper is structured as follows. The regional climate models used as input for the cuboid method and the basic features of the Microscale Urban Climate Model in 3 dimensions (MUKLIMO_3) are introduced in section 2. The cuboid method is described in section 3, and its application to Frankfurt is discussed in section 4. An evaluation of the cuboid method and its results is presented in section 5. In section 6 we discuss the results of the cuboid method. We close the paper with a summary and some conclusions in section 7.

## 2. Model description

Because of the coarse resolution of GCMs, climate-change signals projected by GCMs need to be scaled down to higher spatial resolutions before they can be used for climate impact studies. The first downscaling step is accomplished by regional climate models and statistical downscaling approaches. Based on these results, the second downscaling step uses MUKLIMO_3 at urban scale. In this section all models and simulations used are introduced.

### a. Regional climate projections

To represent the regional climate we use an ensemble of regional climate projections generated by four approaches to assess the uncertainty inherent in these projections. Two of them are derived from numerical RCMs, and two use statistical downscaling techniques.

The numerical regional climate projections utilized in this study are the simulations with the hydrostatic Regional Model (REMO) (Jacob 2001) commissioned by the German Federal Environmental Agency (Jacob et al. 2008) (grid width: 0.088° ≈ 10 km) and the so-called consortium runs (Hollweg et al. 2008) (grid width: 0.165° ≈ 18 km) performed with the nonhydrostatic Climate Limited-Area Model (CLM) RCM (Böhm et al. 2006). Both models were derived from routine weather prediction models that were adapted for climate applications. A brief overview of the models and the simulation setup for data stream 2 (data on a rotated grid) is given in Table 1. For our analysis we use the derotated data (data stream 3) with a horizontal grid width of 0.2° for CLM and 0.1° for REMO.

In addition, we make use of the projections from the two statistical downscaling approaches Wetterlagen-basierte Regionalisierungsmethode (WETTREG) and Statistical Regional Model (STAR). The WETTREG model (Enke et al. 2005) is a circulation-pattern-based weather generator. The projection time series are generated by randomly rearranging the time slices of recent climate. However, the process of resampling is performed to conserve the previously derived frequency distribution of circulation patterns from the simulations of GCMs as closely as possible. This conditioning allows creation of new time series of characteristics that are significantly different from current climate. WETTREG data for all relevant climate parameters are available for 283 stations across Germany. In distinction from the numerical models, the WETTREG model only yields data points on the daily basis (e.g., daily mean temperature).

Similar to WETTREG, STAR model simulations (Orlowsky et al. 2008) are also generated by resampling observational data from weather stations. However, the coupling to global climate projections is more straightforward than within WETTREG. Here, temperature change projections for the region of interest are derived from the global models and are simply prescribed as linear trend for the recombination of weather episodes without regard to the simulated large-scale circulation patterns. STAR provides data for 2342 weather stations of Germany on a daily time resolution.

All regional climate projections are forced by data from the global climate model ECHAM5 (Roeckner et al. 2006; Hagemann et al. 2006; see also online at http://cera-www.dkrz.de/WDCC/ui/Compact.jsp?acronym=EH5-T63L31_OM-GR1.5L40_20C_1_MM). The evaluation period 1971–2000 is chosen to get a sufficiently long time span for the comparison of observations and simulation results. In addition, we take the ECHAM5 time slice 2021–50 resulting from the IPCC Special Report on Emission Scenarios (SRES) A1B scenario (Nakicenovic and Swart 2000) to estimate the future change. We investigate the A1B scenario because it is the only scenario available for all four regional climate projections in Germany.

The ECHAM5 simulation (twentieth-century simulation, realization 1) uses observed anthropogenic forcing for carbon dioxide, methane, nitrous oxide, chlorofluorocarbons, ozone, and sulfate initialized by a preindustrial control simulation. It neglects the natural forcing from volcanoes and changes of solar activity. The grid resolution is T63 (1.88° ≈ 200 km) with 31 vertical layers. ECHAM5 is run in a coupled mode with the Max Planck Institute Ocean Model (MPI-OM). The CLM simulations were nested directly into the ECHAM5 fields. For REMO a two-step nesting was applied. A coarser REMO simulation at 0.44° grid width is driven by ECHAM5 and provides the boundary values for the high-resolution REMO simulation at 0.088°.

### b. MUKLIMO_3

To downscale further the regional climate projections to a residential scale, enabling a simulation of a city and its surroundings, MUKLIMO_3 (Sievers 1995) is employed on the urban scale. The basic version of MUKLIMO_3 solves the Navier–Stokes equations to simulate the atmospheric flow fields. It is based on the two-dimensional atmospheric model MUKLIMO (Sievers and Zdunkowski 1986) with a generalization of the streamfunction–vorticity method to three dimensions.

The current version of MUKLIMO_3 is intended to simulate the atmospheric temperature fields in urban environments. For this purpose, MUKLIMO_3 is augmented with prognostic equations for atmospheric temperature and humidity, balanced heat and moisture budgets in the soil, and a sophisticated vegetation model. Cloud processes and precipitation are not considered. MUKLIMO_3 simulates idealized atmospheric conditions with pronounced influence of local land-use properties. Typical integration times reach from several hours up to one day.

In MUKLIMO_3 special attention is dedicated to interactions between the buildings and the atmosphere. A fundamental feature of MUKLIMO_3 is, thus, the parameterization of unresolved buildings. The need for such parameterization arises for scales in which buildings are resolved vertically but not horizontally. To account for the interaction between the atmosphere and the buildings, the structural properties of the buildings within a grid cell are described by three statistical parameters: the building density, their wall area per grid volume, and their mean height. Based on a technical description of nine different building structures [Bundesministerium für Raumordnung, Bauwesen und Städtebau (BMBau) 1980] we deduced characteristic values of those parameters (Table 2) to define building classes for use in MUKLIMO_3. Because in reality the building structures vary distinctly within a grid cell of ∼50–100-m horizontal resolution, MUKLIMO_3 allows a secondary building structure within one building class (e.g., class 10 and 15).

To obtain the fraction of buildings *γ _{b}*, the mean site occupancy ratio in BMBau (1980) is reduced by 10% to account for the roads and the city squares. From the given number of floors we calculate the mean building height

*h*(5 m for the first floor and 3.5 m for others). The wall-area index

_{b}*w*is computed as the product of the surface-to-volume ratio, the wall area–to-surface ratio, and the mean building height. The characteristic values for these parameters are given in BMBau (1980). The fraction of pavement

_{b}*ν*refers to the building-free part of the grid cell. Their values are essentially based on expert knowledge. The roughness length of all building classes is

*z*

_{0}= 0.2 m since the friction at the walls and roofs is calculated separately. For all building classes in Table 2, a lower-level vegetation is assumed (Table 3).

The consequences of the parameterization of the unresolved buildings for the model equations are as follows: In grid cells with unresolved buildings, 1) the volume of the atmosphere is reduced relative to that in the building-free grid cells, 2) heat and momentum are exchanged between the walls and roofs of the buildings and the atmosphere, and 3) subgrid-scale perturbations of the wind field modifying the turbulent exchange in the atmosphere are induced.

The transport equations for the heat and the moisture in the soil are derived from Sievers et al. (1983). The treatment of vegetation in the canopy model is based on Siebert et al. (1992). The latter is ameliorated using three vertical levels for the vegetation. The topmost level contains the trees with the mean tree height *h _{t}*, the leaf area density for the tree top

*ρ*, and the fraction of tree cover

_{t}*σ*. The stem room differs from that by a lower leaf area density

_{t}*ρ*. All lower-level vegetation is characterized by the height of the canopy layer

_{s}*h*, the vegetation cover (

_{c}*σ*not including

_{υ}*σ*), and the leaf area index for the canopy layer (LAI

_{t}*). LAI*

_{c}*is defined as the ratio of the upper leaf surface divided by the surface area on which the vegetation grows. The impact of leaves in the model is threefold. 1) Leaves act as obstacles for the airflow. They are sources and sinks of 2) radiative energy as well as 3) water vapor. The values for these parameters are listed for different land-use classes in Table 3.*

_{c}On top of the land-use classes in Tables 2 and 3, MUKLIMO_3 provides the opportunity to consider broader streets by turning a part of the affected grid cell into a paved surface. Here we distinguish between the highways (six lanes; width = 25 m) and the national roads (four lanes; width = 15 m). In addition we consider the runways (width = 65 m) and taxiways (width = 30 m) at the airport. The additional fraction of pavement due to these streets is calculated by the length of the street in the grid cell times its width and is merged with the land-use-specific *ν* (Tables 2 and 3). Vegetation cover and buildings are, thus, restricted to the pavement-free part of the grid cell.

Radiation above the buildings is computed separately for short and long wavelengths. For the shortwave radiation, we compute the direct and the diffuse irradiance using an empirical approach [Verein Deutscher Ingenieure (VDI) 1994]. The parameterization for the longwave net radiative flux is based on the approach of Möller (1954) and Zdunkowski et al. (1975) with an additional parameterization to consider the effect of clouds. In layers with buildings, shortwave radiation is reflected and absorbed by their walls and roofs depending on the building density and height as well as by the soil. The model also includes a scheme for the emission and absorption of longwave radiation by the buildings and the soil.

Since MUKLIMO_3 simulates a limited area of the atmosphere, the model requires initial and boundary specifications. From a given set of initial air temperature *T*, relative humidity rh, and horizontal wind vector ** υ** characterizing the environmental conditions, the model first computes a one-dimensional profile up to 1100-m altitude. This profile represents the background meteorological conditions with low influence from elevation and land use. It is, in the beginning, stamped to the complete three-dimensional model domain. In the course of the three-dimensional integration the one-dimensional profile values at 750-m altitude are transferred to the top of the three-dimensional domain and taken as time-dependent upper boundary values. This transfer is done for the wind velocity and the air temperature as well as the exchange coefficients.

The lateral boundary conditions are derived from a sequence starting with one-dimensional simulations at the corners and subsequent two-dimensional simulations at the edges using the corner results as boundaries. Only the boundary conditions of the inflow edges are then considered for the integration.

## 3. Cuboid method

To estimate the impact of climate change on the urban heat load we need microscale simulations of the urban environment covering climatological time scales (30 yr) for both past and future periods. To conduct these urban-scale simulations for several 30-yr time periods would lead to an enormous computational effort. For the reduction of this effort we simulate a set of characteristic meteorological conditions and derive the actual conditions of a specific day by means of interpolation.

To facilitate the description of the procedure, we first assume that all conditions potentially leading to heat stress can be characterized by air temperature only. The temperature range enabling these conditions can be estimated to be limited by a daily mean air temperature of minimum *T*_{c,min} and maximum *T*_{c,max}. Both values are taken as initial temperatures, representing the ambient meteorological conditions, for two MUKLIMO_3 simulations. They result from the daily average of the one-dimensional initialization procedure of MUKLIMO_3. As a result, from the subsequent three-dimensional MUKLIMO_3 simulations we receive hourly fields providing two 24-h idealized diurnal cycles of the urban-scale air temperature *T*, the daily values of *T*_{max} and *T*_{min}, and other meteorological parameters. This first step is schematically displayed in Fig. 1 with solid arrows.

The MUKLIMO_3 simulations can subsequently be linearly interpolated to an ambient air temperature *T _{i}* (dashed arrows). The interpolation weight

*w*is computed from the distance of

_{i}*T*to the two fixed points

_{i}*T*

_{c,min}and

*T*

_{c,max}and applied to the simulated urban-scale MUKLIMO_3 fields to yield the interpolated urban-scale fields of

*T*

_{int},

*T*

_{int,max}or

*T*

_{int,min}. The ambient air temperature

*T*can be derived either from the observations (OBS) or from the projected time series resulting from the RCMs.

_{i}From the assumption of 1 degree of freedom only we receive the above described one-dimensional problem. In general, meteorological conditions have a higher degree of freedom and depend on multiple (*n*_{dim}) parameters. Then, the interpolation can be extended to an *n*_{dim}-dimensional linear interpolation with 2^{ndim} fixed points.

For the study of potential heat stress in an urban environment we reduce this *n*_{dim}-dimensional problem and assume 3 degrees of freedom: the 2-m air temperature *T _{c}*, the 2-m relative humidity rh

*, and the 10-m wind speed*

_{c}*υ*. We place these three parameters in a three-dimensional cuboid with the first dimension referring to

_{c}*T*, the second dimension referring to rh

_{c}*, and the third dimension referring to*

_{c}*υ*. These 3 degrees of freedom need eight fixed points and, thus, eight MUKLIMO_3 simulations for idealized diurnal cycles initialized with the permutation of the minimum and maximum values of the triplet

_{c}*T*, rh

_{c}*, and*

_{c}*υ*. These eight fixed points can be assigned to the eight corners of the cuboid. A schematic of this cuboid is displayed in Fig. 2.

_{c}The limits of the cuboid depend on the background climate of the environment. They should be as small as possible to minimize the errors due to the linear interpolation. However, they must be large enough to encompass most of the relevant conditions to avoid extrapolations.

The method to provide MUKLIMO_3 simulations for idealized diurnal cycles at the eight fixed points of the trilinear interpolation as well as the input time series representing the regional climate conditions either from observations or regional climate projections is called the cuboid method in the following. The results of the cuboid method are the interpolated MUKLIMO_3 fields.

Ecologists use a similar approach, which is called a climate envelope model, to establish a multidimensional space defined by a set of climate variables (Farber and Kadmon 2003). The application of the two methods is different. Whereas the climate envelope models are used to convert point information of species distribution into predictive maps (Farber and Kadmon 2003), we use the cuboid method to reduce the computational effort induced by climatological analysis. Climate envelope models are not able to cope with correlations and interactions among the climate factors (Farber and Kadmon 2003), whereas for the cuboid method this relation is given implicitly by the simulation results at the fixed points of the interpolation, which are the cuboid’s corners. Furthermore, it is assumed that all combinations of climate variables within the boundaries of the climate envelope are equally suitable. This assumption is not necessary for the cuboid method, because the relation between the input climate factors is provided by the background climate of the observations or the regional climate projections.

## 4. Application to Frankfurt

We applied the cuboid method to the city of Frankfurt am Main (50.1°N, 8.7°E). To characterize the urban heat load in this area the interpolated MUKLIMO_3 fields of *T*_{int}, *T*_{int,max}, or *T*_{int,min} are analyzed by counting the threshold exceedances *N*_{T>25}, defined as the number of summer days with a daily maximum temperature *T*_{max} ≥ 25°C, *N*_{T>30} defined as the number of hot days with *T*_{max} ≥ 30°C, and *N*_{t>20} referring to the number of beer-garden days with a 2000 central European summertime (CEST) temperature *T*_{20CEST} ≥ 20°C.

### a. Cuboid configuration

The cuboid established for the Frankfurt area covers the temperature range between *T*_{c,min} = 15°C and *T*_{c,max} = 25°C. Using this range for the ambient daily mean air temperature *T _{i}*, we cover all days with potential heat stress in the area around Frankfurt. If

*T*is lower than

_{i}*T*

_{c,min}(e.g., in winter months), then the probability for a threshold exceedance in the urban-scale temperature field is very low, such that the number of

*T*that need to be analyzed is also very low. If

_{i}*T*is higher than

_{i}*T*

_{c,max}, the probability for a threshold exceedance in the urban-scale temperature field is everywhere almost 1. Thus, the error due to the extrapolation is negligible. The cuboid dimension for the relative humidity is restricted by rh

_{c,min}= 42% and rh

_{c,max}= 80%. This covers the range of daily average conditions in Frankfurt in summer. The wind speed dimension ranges between

*υ*

_{c,min}= 0.7 m s

^{−1}and

*υ*

_{c,max}= 3.0 m s

^{−1}. The lower bound of the wind speed is justified because a further reduction of the wind speed would not yield significantly different results for the temperature field. With a wind speed of 3 m s

^{−1}or higher, the development of a local flow system and, thus, the typical urban temperature characteristics would be suppressed. In this case the synoptic-scale wind system would prevail and lead to an almost horizontally homogeneous temperature field.

The ambient conditions used as input for the cuboid method are derived from both measured and simulated 30-yr time series representative for the Frankfurt environment. For the evaluation of the climate indices we use past 30-yr time series observed at Rhein-Main airport (southwest of the city of Frankfurt) as “perfect boundary conditions” resembling the real weather conditions as closely as possible. To estimate the climate-change impact on the city, we employ time series from regional climate projections and statistical downscaling approaches (section 2) to dynamically downscale regional climate to the local scale.

From the numerical RCM simulations REMO and CLM we extract the time series of four grid cells close to Frankfurt to account for the effective model resolution, which is coarser than the numerical grid size. These grid cells are situated in the southwest of the city of Frankfurt (Fig. 4, described below). They are chosen as input for the cuboid method because 1) they are situated close to the city but 2) are not dominated by the buildings and because 3) the elevation is in a good agreement with that of the city (REMO: 103–129 m MSL; CLM: 124–150 m MSL; MUKLIMO_3: 87–281 m MSL).

From the daily average of the 2-m air temperature *T _{i}* and 2-m dewpoint we calculate the relative humidity rh

*. From the 10-m horizontal wind components the wind speed*

_{i}*υ*and direction are derived. Last, the time series of the four grid cells are spatially averaged to serve as input for the cuboid method.

_{i}From the statistical downscaling approaches WETTREG and STAR we selected the time series of the stations Geisenheim (110 m MSL) and Kahl am Main (107 m MSL) (Fig. 4, described below). Both simulations supply the daily mean *T _{i}*, rh

*, and*

_{i}*υ*. The average of both stations is used as input for the cuboid method.

_{i}To consider the wind direction dependence of the temperature fields the cuboid has to be established for each dominant wind direction separately. The wind direction distribution of the hourly measurements at Rhein-Main airport during 1971–2000 is displayed in Fig. 3. It is clear that the observed wind direction is either northeast (NE) or southwest (SW). Thus, the set of eight MUKLIMO_3 simulations for the 24-h idealized diurnal cycles is carried out for the two dominant wind directions 45° (NE) and 225° (SW) to yield an NE and an SW cuboid. For all days with daily mean wind directions between 315° and 135° the NE cuboid is used. For days with daily mean wind directions between 135° and 315° the SW cuboid is applied.

Because neither WETTREG nor STAR provides information on the wind direction, we generate a random time series for this parameter adjusted to the observed directions at Rhein-Main airport. To build this time series, first of all the observed time series of daily *T _{i}*, rh

*, and*

_{i}*υ*at Rhein-Main airport are used as input for the cuboid method. Then, the resulting interpolated

_{i}*T*

_{int,max}and

*T*

_{int,20CEST}fields are analyzed by counting the number of threshold exceedances for

*N*

_{T>25},

*N*

_{T>30}and

*N*

_{t>20}at the MUKLIMO_3 grid cell of Rhein-Main airport for NE and SW separately. The observed frequencies are listed in Table 4. Afterward, for each day of the time series of WETTREG and STAR, the wind direction is randomly chosen to fit the fraction observed (Table 4) for each climate index separately. This estimation relies on the assumption that the wind direction partitioning of WETTREG and STAR used as input for the cuboid method can be replaced by the observed partitioning and that it is constant throughout the projection period.

### b. Model configuration

Figure 4a displays an overview of the Frankfurt environment (Rhein-Main region). The green-shaded area depicts the model domain of MUKLIMO_3. It is rotated by 30° to the north. Figure 4b depicts the elevation of the MUKLIMO_3 domain in Gauss–Krüger coordinates. The river Main flows from NE to SW. In the northwest of the domain the foothills of the Taunus with a model elevation of up to 281 m arise. The Berger ridge has a model height of up to 205 m.

Figure 5 shows a map of the land use. The physical parameters of the building and land-use classes are listed in Tables 2 and 3. The higher fraction of buildings with a higher degree of inhomogeneity is situated in the city center. The two longish commercial and industrial zones (SW of the domain) show the Rhein-Main airport with the runways in between the two building structures.

The horizontal resolution of the simulation domain ranges from 500 m at the outskirts to 50 m at the city center. In total we use 177 grid cells (25 km) in the *x* direction and 149 grid cells (17 km) in the *y* direction. The vertical resolution ranges from 10 m in the lowest 100 m to 50 m from altitudes of 200 up to 750 m at 25 grid levels. The position of the sun is set according to the latitude of Frankfurt on 16 July. This date results in solar angles for an idealized diurnal cycle for typical summer conditions in that region.

With this cuboid and model configuration, the set of MUKLIMO_3 simulations for the idealized 24-h diurnal cycle initialized with the eight fixed points of the trilinear interpolation is performed for the two dominant wind directions NE (45°) and SW (225°). The one-dimensional MUKLIMO_3 initialization was started at 0900 CEST 15 July. The daily average of this one-dimensional initialization until 0900 CEST 16 July matches the 2-m air temperature, the 2-m relative humidity, and the 10-m wind speed of one of the eight cuboid’s corners. The three-dimensional computation proceeds with the initial profile on 0900 CEST 16 July and performs a 24-h simulation until 0900 CEST 17 July.

As an example, Fig. 6 shows the air temperature and horizontal wind vectors resulting from the MUKLIMO_3 simulation for the NE flow initialized with *T*_{c,max} = 25°C, rh_{c,min} = 42%, and *υ*_{c,min} = 0.7 m s^{−1}, that is, the lower-right front corner of the cuboid (Fig. 2).

Figure 6a refers to the 5-m air temperature *T* and horizontal wind vectors at 1800 CEST. At that time *T* is close to the daily maximum temperature. A higher *T* occurs in regions with a high building density. However, the horizontal variability is low, but it can only partly be attributed to the elevation differences (between 87 and 281 m) in the domain. It ranges between a minimum *T* of 31.7°C and a maximum of 35.9°C. Because of the very low regional wind speed of about 1 m s^{−1} the local influence dominates. The wind speed at 5-m height in the forest (e.g., south of the river Main) is almost zero and convective convergence zones develop.

Figure 6b displays the nocturnal *T* at 0200 CEST. During the night, a slight wind from the Taunus ridge develops transporting clean air into the city. Overall, the wind speed is very low, with a spatial average wind speed of 0.5 m s^{−1}. In the city center the nocturnal air temperature is higher than in its surroundings. The horizontal variability of *T* at 0200 CEST, which ranges between a minimum *T* of 13.8°C and a maximum of 29.4°C, is higher than at 1800 CEST. Again *T* is higher in regions with a high building density than in its environment. Averaging the domain (not weighting the grid cell size) yields 33.8°C at 1800 CEST and 26.2°C at 0200 CEST.

Figure 7 shows an overview of the complete procedure of the cuboid method beginning from the MUKLIMO_3 simulations of the diurnal cycles and the time series of the regional background climate through the cuboid method to the statistical analysis of the results.

## 5. Evaluation

### a. Evaluation of the cuboid method

To evaluate the technical functionality of the cuboid method for use in urban-heat-load studies, we compare the temperature field resulting from the cuboid method for the central point of the cuboid (*T*_{c,cp} = 20°C, rh_{c,cp} = 61%, and *υ*_{c,cp} = 1.85 m s^{−1}) with a direct MUKLIMO_3 simulation initialized with the values of this central point.

The following nomenclature is used in this section. The MUKLIMO_3 simulation initialized with the central point of the cuboid *T*_{c,cp}, rh_{c,cp}, and *υ*_{c,cp} is called MUK3_{cp}. The result of the cuboid method with an interpolation of the eight corner simulations to the central point of the cuboid (*T*_{c,cp}, rh_{c,cp}, *υ*_{c,cp}) is called CUB_{int}.

Figure 8a shows the map of the daily maximum temperature *T*_{max} for the NE flow resulting from CUB_{int}. Figure 8b displays the difference between *T*_{max} from CUB_{int} and that from MUK3_{cp}, that is, Δ*T*_{max}(CUB_{int} − MUK3_{cp}). It is apparent that CUB_{int} underestimates MUK3_{cp} in almost every grid cell by about 0.6°C. The spatial pattern of the deviation suggests that it is slightly higher in built-up areas (≈−1.5°C) when compared with the forest in the southwest of the domain (≈−0.5°C). For the maximum and minimum deviation see Table 5. The wavelike pattern in Δ*T*_{max}(CUB_{int} − MUK3_{cp}) can be explained by the nonlinear small-scale compensating flow that is not fully captured by the linearly interpolated CUB_{int}.

For the daily maximum (*T*_{max}; °C) daily minimum (*T*_{min}; °C) and 2000 CEST (*T*_{20CEST}; °C) 5-m air temperature of MUK3_{cp}, CUB_{int}, and the difference between MUK3_{cp} and CUB_{int}, the spatial average, minimum, and maximum of the domain are listed in Table 5 for the NE and SW flow. Note that to compute the domain average the grid values are not weighted with the grid area. In this way, the more-highly-resolved city center has intentionally a higher influence on the spatial average than does the outskirt area.

The Δ*T*_{max}(CUB_{int} − MUK3_{cp}) has a comparable magnitude and a similar pattern for NE (Fig. 8b) and SW (not shown). The magnitude of Δ*T*_{20CEST}(CUB_{int} − MUK3_{cp}) for the 2000 CEST temperature field *T*_{20CEST} is very similar to that of Δ*T*_{max}(CUB_{int} − MUK3_{cp}) (Table 5). Again CUB_{int} underestimates the directly simulated MUK3_{cp} in almost every grid cell, with a similar pattern (not shown) like Δ*T*_{max}(CUB_{int} − MUK3_{cp}).

The magnitude of Δ*T*_{min}(CUB_{int} − MUK3_{cp}) for the daily minimum temperature *T*_{min} is slightly larger (see Table 5) than for Δ*T*_{max}(CUB_{int} − MUK3_{cp}) and Δ*T*_{20CEST}(CUB_{int} − MUK3_{cp}). The presented configuration of the cuboid method together with the setting for *T*_{c,min}, *T*_{c,max}, rh_{c,min}, rh_{c,max}, *υ*_{c,min}, and *υ*_{c,max} appears to be less appropriate for threshold exceedances with respect to the minimum temperature like, for example, sultry nights. This might be caused by the lower nocturnal boundary layer height and the higher spatial temperature gradients due to the weaker mixing during the night. A change of the values at the cuboid’s corner to better cover the range of nocturnal conditions might improve the result for *T*_{min}.

Since Δ*T*_{max}(CUB_{int} − MUK3_{cp}) and Δ*T*_{20CEST}(CUB_{int} − MUK3_{cp}) are smaller than 1°C at more than 90% of the grid cells and are smaller than 0.5°C at more than 50% of the grid cells, we conclude that the cuboid method results in an acceptable accuracy and, thus, represents an appropriate approximation.

### b. Evaluation of the simulated climate indices

To evaluate the results of the cuboid method with respect to the climate indices we use the observed time series at Rhein-Main airport as input for the cuboid method for the time period 1971–2000. We call this application the evaluation run (EVAL).

A map of the resulting annual mean number of summer days (*N*_{T>25}) for EVAL is displayed in Fig. 9a. The *N*_{T>25} is highest in areas with a high building density such as in the center of the city or at the Rhein-Main airport. In the financial district (Fig. 5; class 10: dark purple) *N*_{T>25} is lower than in the neighboring areas. This is caused by the shadows from high towers, which reduce the daily maximum temperature. During the night, the daily minimum temperature in the financial district is higher than in the neighborhood (not shown). This can be explained by the higher heat capacity of the large buildings (Oke 1987; Hart and Sailor 2009) as compared with the environment and the reduced ventilation and sky-view factor due to the tall buildings. Therefore, the nocturnal UHI can be interpreted as a consequence of the daytime heat storage (Giridharan et al. 2005).

Table 6 lists the climate indices derived from station measurements (OBS) together with the cuboid method results for EVAL analyzed at the grid cells of the measurement stations Rhein-Main airport and Offenbach. The deviation of *N*_{T>25} is less than 20% for both Rhein-Main airport and Offenbach and is even less than 10% for 1971–2000. This is an acceptable range for the simulations. The deviation of *N*_{t>20} and *N*_{T>30} is higher. The higher deviation for *N*_{t>20} can be partly explained by the small difference between the fixed sunset time of our idealized MUKLIMO_3 simulation and the varying true sunset times during the course of the summer. The higher deviation for *N*_{T>30} is mainly due to its very small absolute number (see Table 6).

## 6. Results

The results of the cuboid method using a small ensemble of regional climate projections as input are discussed for the number of summer days (*N*_{T>25}). For this purpose we apply the regional climate projections provided by REMO, CLM, WETTREG, and STAR (section 4). We call these applications using the names of the generating regional climate models. We analyze the past 1971–2000 (C20) and the future 2021–50 (A1B) time period. The future development is assumed to follow the SRES A1B scenario (Nakicenovic and Swart 2000).

### a. Number of summer days for 1971–2000

Figure 9 shows the maps of the number of summer days (*N*_{T>25}) resulting from the cuboid method for EVAL (Fig. 9a), REMO (Fig. 9b), CLM (Fig. 9c), WETTREG (Fig. 9d), and STAR (Fig. 9e) for the time period 1971–2000. The visual verification shows that the climate projections of REMO, WETTREG, and STAR agree very well with EVAL. However, the simulation using CLM results in a much lower *N*_{T>25}.

To estimate the sampling uncertainty of the climate indices with respect to the interannual variability we employ a practical approach using a nonparametric bootstrap sampling algorithm. For this purpose we draw 30 random values with replacement from each model’s set of annual *N*_{T>25}, *N*_{T>30}, and *N*_{t>20} to create a 30-yr synthetic sample at each grid cell. This drawing is repeated 1000 times to generate a sufficiently high number of possible realizations. The 5th and 95th percentiles of the resulting collection of estimates are then used as lower and upper 90% confidence bounds for the true climate indices. These confidence intervals account for the uncertainty due to the short-term natural variability (Kendon et al. 2008).

Table 7 lists the spatial average, minimum, and maximum *N*_{T>25} resulting from the cuboid method using EVAL, REMO, CLM, WETTREG, and STAR as input. Again, we see the good agreement of the spatial average between REMO, WETTREG, and STAR with EVAL but a poor one between CLM and EVAL. The horizontal variability is slightly overestimated by REMO, WETTREG, and STAR but is underestimated by CLM in comparison with EVAL.

In addition, Table 7 lists the domain-averaged width of the confidence interval at the 90% significance level (difference between the 95th and 5th percentile) for the time period 1971–2000. For CLM and STAR, the magnitude of this value is very similar to that of EVAL. However, REMO overestimates and WETTREG underestimates the interannual variability in comparison with EVAL.

Figure 10 shows the box-and-whisker plot for the domain-averaged climate indices resulting from the cuboid method with EVAL, REMO, CLM, WETTREG, and STAR as input for the time period 1971–2000. As customary, the bottom and top of the box depict the 25th and 75th (lower and upper quartile) percentile. The band near the middle of the box refers to the 50th percentile (median). The lower and upper ends of the whiskers represent the 5th and 95th percentile, respectively. The range of the whiskers can be interpreted as width of the confidence interval at the 90% significance level.

For *N*_{T>25} the confidence intervals of REMO, WETTREG, and STAR coincide with that of EVAL. Thus, we conclude that those samples originate from the same population and that the climate projections resemble the observations very well. However, this is not true for CLM.

For the number of hot days (*N*_{T>30}), the confidence interval of CLM coincides with that of EVAL. For all others, the conformity with the confidence interval of EVAL is small. However, the absolute number of *N*_{T>30} is very low so that reliable conclusions are difficult to draw. For the number of beer-garden days (*N*_{t>20}) the conformity of all regional climate projections with the confidence interval of EVAL is small. For *N*_{t>20} we must therefore conclude that the regional climate projections do not resemble the observations well.

### b. Future change in number of summer days for 2021–50

To estimate the change of climate indices due to global temperature rise we compare the results from regional climate projections in the future (2021–50) resulting from the SRES A1B scenario (Nakicenovic and Swart 2000) with those of the control period (C20) in the past (1971–2000).

To estimate the sampling uncertainty with respect to the change of climate indices we use 100 000 synthetic bootstrap samples. The higher sample size relative to that of the control period is necessary because both the past and the future climate indices add variability to the difference.

Table 8 lists the spatial average, minimum, and maximum change in annual mean number of summer days (Δ*N*_{T>25}) for the future time period 2021–50 with respect to 1971–2000 using REMO, CLM, WETTREG, and STAR as input for the cuboid method. The horizontal variability (differences between maximum and minimum values in Table 8) of Δ*N*_{T>25} amounts to about 5 days per year. This value is much smaller than the horizontal variability of *N*_{T>25} for 1971–2000 (Table 7), which amounts to about 40 days per year. Because the horizontal variability of Δ*N*_{T>25} is also lower than the domain-averaged width of the confidence interval at the 90% significance level, the respective maps are not displayed here. Therefore, we conclude that the surplus of days with heat stress in the city is not significantly different from the surplus in the city’s environment.

The spatially averaged magnitude of Δ*N*_{T>25} is very similar for REMO, CLM, and WETTREG projecting a future increase of about 30%. The increase of Δ*N*_{T>25} using STAR is much higher.

Figure 11 shows the box-and-whisker plot for the domain-averaged change in climate indices between the time periods 1971–2000 and 2021–50. For Δ*N*_{T>25}, the confidence intervals of REMO, CLM, and WETTREG coincide well. However, STAR simulates a much higher increase of Δ*N*_{T>25}, Δ*N*_{T>30}, and Δ*N*_{t>20} than the others. However, the results of all four RCMs are in the range of the three existing ECHAM5 realizations, reflecting the internal variability of the climate system. For Δ*N*_{T>30} and Δ*N*_{t>20}, the agreement of the confidence intervals of REMO, CLM, and WETTREG is not very high.

The length of the whiskers is larger, and therefore the confidence interval is broader, for the future change (Fig. 11 and Table 8) than for the evaluation time slice 1971–2000 (Fig. 10 and Table 7). This is due to the added variability of the past and future interannual spreading but also to the higher variability of future air temperature (Schär et al. 2004) itself.

The range embraced by all boxes for a certain climate index shows the uncertainty of the projection due to different RCM input. From the results of our four-member ensemble we conclude that the annual mean number of summer days will increase, at the 90% significance level, by 5–32 days in Frankfurt by 2021–50. The annual mean number of hot and beer-garden days will increase (at the 90% significance level) by 3–16 and 4–29 days, respectively. However, the results for Δ*N*_{T>30} and Δ*N*_{t>20} are difficult to interpret because the RCM input of the cuboid method for the evaluation time slice 1971–2000 does not resemble EVAL very well.

## 7. Summary and conclusions

We presented a pragmatic approach to estimate the impact of climate change on the urban environment using the cuboid method. This method allows one to simulate the urban heat load on a climatological time scale using eight representative simulations for an idealized diurnal cycle with MUKLIMO_3 on the urban scale for each relevant wind direction. From these simulations all potential heat stress conditions leading to air temperature threshold exceedances can be derived.

The cuboid method is based on the assumption that all meteorological situations potentially leading to heat load can be characterized by 3 degrees of freedom, which are the 2-m air temperature, 2-m relative humidity, and 10-m wind speed. Since only certain ranges of these parameters potentially lead to heat load conditions, a three-dimensional cuboid can be constructed for which the minimum and maximum values of the three parameters are assigned to the cuboid’s corners, respectively. The MUKLIMO_3 simulations initialized with these corner values form the basis of the trilinear interpolation, which results in meteorological fields for specific daily average conditions derived from representative regional observations or regional climate projections. Because of the combined use of only a few computationally expensive MUKLIMO_3 simulations together with a simple and computationally inexpensive trilinear interpolation procedure, the cuboid method represents an innovative dynamical–statistical downscaling technique to quantify the impact of climate change on urban heat load with the high spatial resolution needed by urban planers for the development of suitable adaptation strategies.

We applied the cuboid method to the Frankfurt region and technically evaluated the cuboid method by a comparison between a MUKLIMO_3 simulation initialized with the cuboid’s central point for the air temperature, relative humidity, and wind speed and the cuboid method with the central point as a target point of the interpolation. When counting air temperature threshold exceedances of the daily maximum temperature, an acceptable accuracy of the cuboid method was found.

In addition, the results of the cuboid method were evaluated with respect to the annual mean number of summer, hot, and beer-garden days by using an observed time series of daily average air temperature, relative humidity, and wind as input for the cuboid method (EVAL). These results were compared with in situ measurements. A good accuracy was found for the number of summer days, but the number of beer-garden days did not agree equally well. The percentage deviation for the number of hot days turned out to be large because of the low absolute number.

Using the cuboid method, we investigated the ability of the regional climate projections to correctly represent the past climate indices, applying an ensemble of four different regional climate projections. Using REMO, WETTREG, and STAR as input for the cuboid method the annual mean number of summer days agreed well with EVAL. However, CLM was found to underestimate the number of summer days significantly. Also the RCM input for the cuboid method did not resemble the annual mean number of beer-garden days of EVAL well, and conclusions for the number of hot days were difficult because of the low absolute number.

In addition, we estimated the impact of climate change on the climate indices. According to the analysis of the regional climate projections from four ensemble members, we found that the annual mean number of summer days will increase by 5–32 days at the 90% significance level in Frankfurt for 2021–50 relative to 1971–2000. For the number of hot and beer-garden days, we do not interpret the resulting change because the conformity of the confidence intervals for the evaluation time period between EVAL and the regional climate projections is small.

Another important point to follow is that the outcome of climate research has to be linked more explicitly to the objectives of sustainable settlement, since settlement planning is a key aspect of sustainability (Mills 2006). In this context we will use MUKLIMO_3 as a numerical simulation laboratory to investigate the effect of land-use changes on the urban heat load under changing climate conditions. The combined study of climate change in cities due to global climate change and local climate change caused by urban development shall facilitate the development of precautionary measures and adaptation strategies for decision makers in the city administration to be prepared for the expected intensification of the urban heat stress. We will use these results as a basis to communicate the impact of climate change on the city of Frankfurt to urban planners and the city administration.

The promising results of the cuboid method for the heat load in Frankfurt indicate that this method might represent a useful downscaling technique for a wide range of applications. In our study, three meteorological parameters (regional air temperature, humidity, and wind speed) were identified as dominating factors for possible urban-heat-load conditions. This required 2^{3} MUKLIMO_3 simulations and a trilinear interpolation procedure. If for a different application only two dominating factors are sufficient to describe most of the expected variability (e.g., wind speed and mixing layer depth for estimating urban ventilation), then only 2^{2} MUKLIMO_3 simulations and a bilinear interpolation procedure are required; that is, the cuboid method could be reduced to an even faster rectangle method. By analogy, 2^{4} MUKLIMO_3 simulations and a four-dimensional linear interpolation are necessary if four dominating meteorological parameters can be identified, and so on. The approximate validity of linear interpolation has to be evaluated for all applications. Furthermore, for nonurban climate-change impact studies the urban climate model can be replaced by other computationally expensive impact models, such as air-quality models or hydrological models.

The authors thank the Umweltbundesamt, Dessau, Model and Data Group at the Max Planck Institute for Meteorology, Hamburg; the Potsdam Institute for Climate Impact Research, Potsdam; the Climate and Environment Consulting GmbH, Potsdam; and the CLM-Community for providing the regional climate projections. We also appreciate the support of the administration of the city of Frankfurt am Main. The work was done within the cooperation between the “Umweltamt der Stadt Frankfurt am Main” and the “Deutscher Wetterdienst” (DWD: German Meteorological Service). We thank the three anonymous reviewers for their comments, which really helped to improve the manuscript.

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Description of the CLM and REMO model setups for data stream 2 (data on rotated grid).

Characteristic parameters for the building classes relevant for the simulations: fraction of buildings (*γ _{b}*; 1), mean building height (

*h*; m), wall-area index (

_{b}*w*; 1), and fraction of pavement (

_{b}*ν*; 1).

Characteristic parameters for the land-use classes relevant for the simulations: roughness length (*z*_{0}; m), mean tree height (*h _{t}*; m), leaf area density for the tree top (

*ρ*; m

_{t}^{2}m

^{−3}), mean stem height (

*h*; m), leaf area density in stem height (

_{s}*ρ*; m

_{s}^{2}m

^{−3}), height of the canopy layer (

*h*; m), leaf area index for the canopy layer (LAI

_{c}*; 1), fraction of tree cover (*

_{c}*σ*; 1), fraction of vegetation cover (

_{t}*σ*; 1), and fraction of pavement (

_{υ}*ν*; 1).

Number (frequency) of temperature threshold exceedances separately for NE and SW, and the climate indices *N*_{T>25}, *N*_{T>30}, and *N*_{t>20} at Rhein-Main airport in 1971–2000.

Daily maximum (*T*_{max}; °C), daily minimum (*T*_{min}; °C), and 2000 CEST (*T*_{20CEST}; °C) temperature of CUB_{int}, MUK3_{cp}, and the difference between CUB_{int} and MUK3_{cp} for the NE and SW flow: spatial average, minimum, and maximum value of the domain.

Comparison of observed and simulated climate indices (days yr^{−1}). OBS is derived from station measurements at Rhein-Main airport and Offenbach (plus sign and open circle, respectively, in Fig. 9a). EVAL results from the cuboid method using the measurement as input analyzed at the respective grid cells of the plus sign and the open circle. Pct dev denotes the percentage deviation of EVAL from OBS. An asterisk indicates missing observations in 1978–79 and 1996–2000.

Spatial average, minimum, and maximum annual mean number of summer days (*N*_{T>25}; days yr^{−1}) and the spatial average of the width of the confidence interval at the 90% significance level resulting from the cuboid method using EVAL, REMO, CLM, WETTREG, and STAR as input for 1971–2000.

Spatial average, minimum, and maximum change in the annual mean number of summer days (Δ*N*_{T>25}; days yr^{−1}) and the spatial average of the width of the confidence interval at the 90% significance level resulting from the cuboid method using REMO, CLM, WETTREG, and STAR as input. The values refer to the difference between 2021–50 and 1971–2000.