ABSTRACT: Using an energy balance model we find a linear relationship between surface layer temperature reduction DTs and domain averaged latent heat flux from vegetated green roofs E, i.e. DTs = ECP * E. We call the slope of this relation Evaporative Cooling Potential (ECP) with a value of ∼0.014Km^{2}W^{1} or 0.35Kday mm^{1}. We explain this linear dependence using a Taylor series expansion and derive a simple and fast method to determine ECP.
INTRODUCTION
The fast growth of global population, mainly in urban areas and the concurrent increase in likelihood of extreme weather events forces cities to prepare for water scarcity and heat spells. The fact that cities are several degrees warmer than the surrounding rural environment, known as the urban heat island effect, amplifies this risk (Rosenzweig et al., 2015).
Green roofs offer the possibility to mitigate multiple environmental issues in an urban environment. A common benefit attributed to green roofs is the temperature reduction through evaporation. Yet, the water availability depends on the local climate as well as green roof design. Fast tools to assess the benefits of such installations are thus needed for architects and urban planners to make better decisions in urban development.
This study focuses on evaluating the effect that evaporative cooling has on outdoor air temperatures in an urban environment. An established urban energy balance model (Masson, 2000, TEB) was modified to quantify the cooling potential of green roofs and study the scalability of this mitigation strategy.
SIMULATIONS
In the energy balance model used the representation of the urban surface is simplified as a generic urban canyon. This approach thus cannot be used to assess individual buildings, but is suitable to study the local outdoor climate on a neighborhood scale. The model comprises of a set of differential equations describing the time evolution of energy stored in the urban surfaces, i.e. to determine the temperature of the urban surfaces sensible, latent and ground heat flux are considered along with short and long wave radiation.
Simulations were performed for different climates (daily mean short wave radiation S and atmospheric temperature T_{atm}) and urban geometries (building height h, building width b and road width l), with varying soil moisture con tent, green roof fraction and urban surface layer thickness. The forcings were held constant and the simulations thus reached a steady state.
Case
 S [Wm^{1}]
 T_{atm} [°C]
 h [m]
 b [m]
 l [m]

CN
 480
 22
 20
 26
 12

CW
 480
 22
 20
 20
 20

HN
 520
 32
 20
 26
 12

HW
 520
 32
 20
 20
 20

To better understand what processes influence the cooling potential of green roofs and retrieve a fast but accurate estimate without having to solve the set of differential equations numerically, we also derive an alternative method to determine the effect of evaporative cooling by studying a Taylor series expansion of the models set of ODEs.
RESULTS
Running the full model all simulations show a linear relationship between surface layer temperature reduction DTs and domain averaged latent heat flux from vegetation E, i.e. DTs = ECP * E. Figure 1 shows this collapse of all data on a single line. The slope of this relation is approximately
0.0135Km^{2}W^{1 }and we call it the Evaporative Cooling Potential (ECP). On the top axis the latent heat flux is expressed as daily evaporated water column and we can better compare this to the available precipitation. This allows to assess the maximal cooling that can be achieved with given precipitation or we can estimate the effect additional watering would have on green roof performance, with an upper limit given by green roof area and climate. Previous studies have already suggested such a link. Li et al. (2014) describe a linear decrease of air temperature with green roof fraction, Mackey et al. (2012) compare temperature reduction to the normalised difference vegetation index (NDVI) from satellite measurements and also find a linear relationship. However, this dependency is not well understood.
FIGURE1
ECP appears to be universal and does not depend on the urban geometry or climate. We can examine this linear relation by studying the Jacobian of the Taylor series expansion. Indeed ECP can be estimated quite accurately by the equation:
ECPs=a1J56J55J66=a1∂g5∂TG∂g5∂Ts∂g6∂TG
Where in this case a is the green roof percentage of the total urban surface area, g5 describes the evolution of surface layer temperature with time and g6 describes the evolution of the green roof temperature with time. We calculate ECPs = 0.0126Km^{2}W^{1} which is close to the value of 0.0135Km^{2}W^{1} we obtain by solving the ODEs numerically. The evaporative cooling of green roofs in an urban environment can thus be predicted quickly without having to solve differential equations.
Furthermore this approach can also be expanded to diurnal cases and predict the temperature reduction of daily mean and maximum temperatures. For such general cases ECP becomes a function of the stability of the surface air layer. We can compare the steady state results to an unsteady case with a diurnal atmospheric temperature and radiation cycles. The average forcing corresponds to the HW case with a surface layer thickness of 300m and a green roof fraction of 50%. The resulting surface layer temperatures are shown in the top panel of Figure 2. By changing the soil moisture content we can reproduce Figure 1 for daily mean temperatures and maximum temperatures. The bottom panel of Figure 2 reveals a linear behavior for both cases even under unsteady conditions. Note that the effect on mean temperature in the unsteady case is very close to the findings for a steady state.
FIGURE2