The Green Roof 2 Layer (GR2L) model implements an interactive dual-layer water balance bucket model for green roofs with a substrate layer and an optional retention layer underneath on daily time steps. Key compartments and parameters are depicted in Figure 1. Both layers are hydraulically connected. In practice, this is often achieved by using high capillary distribution fleeces for horizontal water distribution at the base of the substrate layer in combination with glass wool or fabric wicks that penetrate into the retention layer for vertical upward movement of water from the retention layer to the substrate layer. Vertically downward movements of excess water from the substrate layer into the retention layer is achieved through direct seepage. The substrate layer is exposed to environmental conditions and experiences rainfall (water input) and evapotranspiration (water loss). Potential irrigation (water input) during operation can be applied both directly to the substrate layer or indirectly via the retention layer as rainfall.

The substrate layer can exchange water with the top atmospheric boundary and the retention layer beneath. Water exchange between the layers is assumed to be instantaneous in modeling terms, i.e., local equilibria are reached within < 1 day. Thus, given available retention water the substrate is assumed to be at a homogenous water holding capacity that allows the planted vegetation to thrive (see also Gößner et al., 2021).

If the substrate is parameterized sufficiently, e.g., known van Genuchten (1980) parameters, the water storage could also be expressed as soil moisture or volumetric water content. However, due to the relatively coarse and thin substrate layer in green roofs (order of cm to dm, with considerable fractions of gravel), the presence of a shallow water reservoir directly beneath it (retention layer), and a general lack of experimentally-derived soil-moisture release curves for green roof substrates, GR2L does not consider vertical soil moisture gradients in the substrate, but expresses the water content in the substrate layer simply in terms of water head, i.e., water stored, at any given time.

Surplus rainfall or irrigation onto the substrate layer above a maximum water storage capacity will seep into the retention layer while a temporary water deficit in the substrate layer, caused by evapotranspiration will initiate upward movement of water from the retention layer to the substrate layer, if available (Figure 1). Excess water in the retention layer beyond its storage capacity which is usually determined by the position of an outlet or overflow pipe (with or without throttle orifice) is considered as overflow. GR2L does not consider conditions where parts of the water in the system are immobile, e.g., freezing conditions or snow cover.

Both the substrate and retention layer have upper and lower boundaries for water storage. In the substrate layer minimum substrate layer storage capacity S_{sub; min} and maximum substrate layer storage capacity S_{sub; max} correspond to the substrate's saturated and residual water content, respectively, and are generally estimated experimentally from weighing lysimeter data. In the retention layer, these water storage parameters would be a direct result of the retention space design. In most practical cases, the minimum retention layer storage capacity S_{ret; min} would be zero. When the maximum retention layer storage capacity S_{ret; max} is set to zero, the model acts as a single-layer bucket model, representing a green roof without a retention layer (Equation 1).

Hence, the total water storage S_tot in the system is given as the sum of water stored in the substrate layer (S_sub) and water stored in the retention layer (S_ret):

with

and

Expressed as a sequential model, storage in the substrate layer on day i depends on the previous day's storage plus incoming water from precipitation P and/or upward pull Q_up minus evapotranspiration and/or downward seepage Q_down. Similarly, the retention layer storage on day i depends on the previous day's storage plus downward seepage from the substrate layer minus potential upward transport and/or loss of excess water through the outlet.

Precipitation (i.e., rainfall and/or surface irrigation) and potential evapotranspiration ET_p data can be obtained from on-site measurements or publicly available weather time series data for a given location. For this study precipitation and potential evapotranspiration (Penman Monteith, FAO formula) were taken from the German Weather Service (DWD Climate Data Center—CDC, 2019, 2021). The model implements an interception loss term I that reduces the effective precipitation that infiltrates into the substrate layer during each time step. Interception is assumed to remain constant on the scale of one vegetation period, but could be derived from experimental data (if available), e.g., leaf area index (LAI) and a canopy storage function (Liu et al., 2021). The capability of vegetated green roofs to evaporate initial precipitation without transfer into the substrate layer is mainly dependent of vegetation type and overall exposed surface area (Nagase and Dunnett, 2012). A constant value of 1 mm per day was chosen in line with urban roof runoff models (such as Gash et al., 2008). Evapotranspirative loss is either measured directly in lysimeter set-ups or calculated dynamically from ET_p data using a moisture extraction function for determining actual evapotranspiration ET_a. The moisture extraction function is the ratio of the actual water storage S_sub and the maximum effective water storage S_{sub; max} in the substrate layer, scaled by a green roof factor k_g which is conceptually not dissimilar to a crop factor k_c (Allen et al., 1998). Classically, the crop factor is used to convert reference evaporation into actual evapotranspiration for different crop types. By using k_g as calibration parameter we do not only consider vegetation type but also aggregate green roof design and operational conditions into k_g:

A variety of different moisture extraction function shapes were tested following the parametrizations by Zhao et al. (2013). While the group of non-quadratic functions provided comparable and generally good results, quadratic functions did not perform better in preliminary model evaluations (results not shown). Thus the function of least complexity was chosen (Equation 4). Alternative, step-wise approaches to calculate evapotranspiration on daily time steps can be used, based on available data; similar to Liu et al. (2021) who used a Hargreaves evapotranspiration approach with linear moisture extraction function and a Dalton-type transfer between reservoirs.

The full model parameterization can, hence, be expressed within the two layers as

and

with all parameters as defined above and the respective time step index i or i−1 in the subscript. OUT_i is the overflow from the retention layer that occurs when S_{ret, max} (Figure 1; Table 1) is reached. Fluxes such as P, Et_a, Et_p, Q_up, Q_down, I, and OUT are in [mm day⁻¹], the states S_sub, S_ret, and S_tot are in [mm] and k_g is [–].

Data for calibrating the GR2L model were courteously provided by Optigrün International AG (Krauchenwies-Göggingen, Germany). The dataset consists of time series data spanning 187 days from April 1 to October 4, 2021, gathered from four experimental plots of pilot-scale dual-layer green roof types: an Economy Roof, a Garden Roof, a Nature Roof, and a Retention Roof (Gößner et al., 2021). Each green roof type consists of an upper substrate layer and a lower retention layer separated by a distribution fleece, but differs when it comes to layer thickness, substrate specifications, vegetation, overall water storage capacity, and drainage type. Each roof type was constructed as a weighting-scale lysimeter with additional green roof area surrounding it to limit edge effects. Details about the experimental design including information on location and weather, irrigation schedule, design parameters, sensor specifications, and data acquisition system have previously been presented in Gößner et al. (2021). Key operational and design parameters relevant for this study are summarized in Table 1.

The lysimeters were located at the Optigrün premises in Göggingen (Germany) and instrumented to record weight and outflow at 5-min intervals from April 1 to October 5, 2021. Precipitation and temperature data were collected on site. The exact study set-up, data collection, and a comparative study of evapotranspiration rates are explained in Gößner et al. (2021).

The data provided by Optigrün included recorded weight and outflow data on 5-min intervals as well as weather data. The data had to be pre-processed before being suitable for GR2L model calibration. Hereby, we followed four consecutive steps of data pre-processing. First, the raw data (weight, outflow, precipitation) were screened for outliers and physically insensible data. Then, the data were scaled to a common reference area, i.e., 1 m², and converted into units of water head (mm) to allow for a meaningful analysis of the water balance. Third, missing data (predominantly due to temporary sensor failure) was gap-filled using (i) linear regression for temperature data (0.1% of data); (ii) a bias-corrected replacement from publicly available data for precipitation data (DWD Climate Data Center—CDC, 2021, Station Pfullendorf, ID 3927) (60.1% of data); and (iii) a bias-corrected replacement of lysimeter weight and outflow data across green roof types based on correlation during respective non-zero outflow or non-zero weight change day (7.0% of data). Finally, data were aggregated to daily values, i.e., sums for outflow and precipitation, and differences between midnight observations for lysimeter weight changes.

Geometry and time series data provided by Gößner et al. (2021) were used to parametrize four green roof types (Table 1). Water storage capacity in the retention layer is given by green roof design and geometry. While the retention layer minimum water storage capacity for all green roof types is zero, representing an empty retention layer, the retention layer maximum water storage capacity is related to the position of the outlet.

Substrate layer minimum and maximum water storage capacity depend on substrate type and composition, as well as green roof specific parameters such as vegetation type, root depth, and density. These parameters were extracted empirically from the lysimeter and soil moisture time series data. Assuming an empty retention layer, the minimum effective water storage in the substrate layer S_{sub; min} was approximated by the residual water content θ_res in the substrate, i.e., at the time of minimum recorded soil moisture. The minimum effective substrate water storage capacity can, then, be derived by scaling θ_res with the substrate layer thickness d_sub. Assuming a filled retention layer, the maximum effective water storage in the substrate layer S_{sub; max} was approximated by using the median lysimeter weight at times of soil moisture near (i.e., >95% percentile) an apparent saturated water content θ_sat and subtracting the weight of a filled retention layer. Cleaned and gap-filled substrate water content time series data with derived, corresponding minimum and maximum water storage capacities for all four roof types are shown in Supplementary Figure S1.

Initial substrate and retention layer storage must be supplied to the model as initial conditions. An allocation of available water in the system at t = 0 between substrate and retention layer storage was done based on lysimeter weight and the conception that water accumulates in the substrate layer first, before seeping into the retention layer.

A key indicator of the progressing climate crisis is the higher frequency and intensity of precipitation extremes (wet and dry periods) globally (Thackeray et al., 2022). Using the calibrated models for each green roof type, we tested their respective performance on historical meteo ensembles. Publicly available rainfall and ET_p data from 1991 to 2021 for the weather station located closest to the experimental plot (DWD station 3927 in Pfullendorf) (DWD Climate Data Center—CDC, 2019, 2021) were used to assess the influence of green roof design on overall water balance for three reference years: a dry extreme, a wet extreme, and an average year. The year 2018 was selected as the extreme dry reference year based on having the lowest overall total recorded precipitation (603 mm). 2001 was selected as the extreme wet reference year based on having the highest overall total recorded precipitation (1,041 mm). In addition, an ensemble of 10 randomly selected years within the period 1991 to 2021 was selected to represent climate variability; these were 1993, 1998, 2000, 2005, 2006, 2007, 2009, 2012, 2013, and 2017.

The model water balance prediction for all four green roof types over the course of an entire calendar year is shown for both the extreme dry and extreme wet reference year in Figure 4. All green roof types express similar patterns when it comes to predicted storage and outflow over time, albeit with varying degrees of severity when it comes to gradual filling and emptying. In the extreme dry reference year, the Retention Roof was predicted to have run dry (i.e., reached < 10% of storage capacity) on 76 days throughout the year, despite having the overall largest retention layer capacity. Increased evapotranspirative loss during periods of extended drought through the relatively thinner substrate layer, led to sharper decreases in overall storage as compared to the Garden Roof, which was predicted to risk running dry on 29 days throughout the year. Conversely, the green roofs were predicted to be filled > 90% of their storage capacity on 204, 190, 201, and 140 days for the Economy Roof, Garden Roof, Nature Roof, and Retention Roof, respectively. The Retention Roof, thus, makes most effective use of its retention storage and provides the overall largest scope for evaporative cooling and for retaining heavy rainfall events without outflow in summer months. Both the Garden Roof and Retention Roof provided no considerable outflow between March and November. During the extreme wet year, the Retention Roof again proves to be the roof design that provides the largest scope for additional temporary water retention, because of its relatively high evapotranspirative water loss as the green roofs were predicted to be filled >90% of their storage capacity on 307, 310, 314, and 285 days for the Economy Roof, Garden Roof, Nature Roof, and Retention Roof, respectively. Both the Garden Roof and Retention Roof provided considerable retention of two heavy summer rainfall events in July and August as compared to the Economy Roof and Nature Roof.

When examining the overall water balance over the course of the extreme dry, average, and extreme wet reference years, the Retention Roof and Garden Roof tend to lose the majority of their incoming water through evapotranspiration, while the Economy Roof and Nature Roof lose the majority of their incoming water through direct outflow (Figure 5). While in average years, evapotranspiration losses account for 27, 43, 33, and 46% of the total water balance for the Economy Roof, Garden Roof, Nature Roof, and Retention Roof, respectively, the losses to outflow make up 54, 36, 47, and 34%, respectively. The remainder is attributed to the interception term, which is assumed constant across all green roof types due to lack of reliable experimental data. Thus, the Garden Roof and Retention Roof types tend to have the ability to be more favorable design options when evaporative cooling is preferred over the potential for water reuse through irrigation or groundwater recharge through targeted infiltration.

The distribution of the overall predicted relative storage capacity for each of the two green roof layers—i.e., the available water stored in the retention and substrate layer normalized to the respective layer's total water storage capacity, respectively—for all four green roof types across the reference dry, average, and wet years is shown in Figure 6. All four roof types express similar patterns regarding storage behavior between the three reference years, with dry years having the highest spread and lowest overall median relative storage, average years having the smallest spread and wet years having the highest overall relative storage and similar spread to wet years over the year.

For the retention layer, the median water storage over the course of the reference dry, average, and wet year was 48, 64, and 90% for the Economy Roof, 48, 74, and 94% for the Garden Roof, 51, 70, and 91% for the Nature Roof, and 35, 71, and 92% for the Retention Roof, respectively. Retention layers are used to temporally retain stormwater or to irrigate the vegetation and buffer dry periods. It is thus expected that the water level inside the retention layer fluctuates considerably during the course of a year. In dry years, the retention layer of all four roof types runs empty at least once (Figure 6), resulting in a total of 173, 181, 166, and 181 days where the retention layer may run empty in the Economy Roof, Garden Roof, Nature Roof, and Retention Roof, respectively, thus not being able to perform the desired function of providing water during extended dry conditions to the substrate layer. These periods fall predominantly into the summer months of June, July, and August for all four roof types. During these periods, however, the retention layer has its largest capacity to store excess rainfall, such as from localized heavy summer rain events.

Even during the reference wet year, the retention layer falls dry for all green roof types at least once, albeit with two distinct patterns emerging; while the green roof types with considerable retention capacity only show extremely short periods of zero-water storage in the retention layer (5 and 7 days in July for the Garden Roof and the Retention Roof, respectively), the green roof types with lower retention storage capacity dry out more frequently and over longer periods of time (41 and 73 days predominantly in May and July for the Nature Roof and Economy Roof, respectively). During the reference wet year, two major rainfall events in August and September, accounting for a combined approximate 12% of annual rainfall, replenished the retention layer completely after several weeks of extended dry conditions (Figure 4). This is in contrast to the reference average year, where the retention layer never completely emptied in any of the four green roof types. This is a function of both more regular rainfall patterns and lower evapotranspirative pull from the superjacent substrate layer. The Nature Roof and Economy Roof, however, occasionally fell to < 25% storage capacity in the retention layer during average years, thus emphasizing their limited capacity to serve as multifunctional BGI within a larger BGI cascade.

Most pairwise comparisons of the predicted retention layer relative storage capacity time series during the reference years showed no statistically significant differences (Supplementary Table S2). Exceptions are the comparisons between (i) Retention Roof and Economy Roof, and (ii) Garden Roof and Nature Roof during average years which again exemplifies the influence of the presence/absence of a meaningful retention layer across the tested roof types on green roof performance.

The substrate layer in the current model parameterization tries to remain at full water storage capacity. This is clearly reflected in Figure 6, where the median relative water storage for all four green roof types and all three reference years reaches 100%, indicating that the substrate layer is sufficiently served with water throughput a majority of days during the year. Similar to the patterns observed in the retention layer, the Retention Roof made best use of the full range of water storage available throughout the reference dry year and falling only occasionally to relative storage capacities of < 50% following the relatively high evapotranspirative upward water movement in summer months. The Nature Roof and Economy Roof, on the other hand, tend to have the lowest overall median relative storage capacities, which is especially pronounced in the dry year, thus indicating that these green roof types are more susceptible to changing climatic patterns and do not provide sufficient buffering capacity for even their low-transpiration sedum vegetation and mixed vegetation during extended extreme dry conditions. It has to be noted that none of the roofs, however, experienced severe water stress for the substrate layer vegetation over prolonged periods of time (as all tested green roof types have at least some type of retention layer). Due to their generally high relative elevation, wind and solar exposure, and thin and coarse substrate layer, green roofs are extreme environments when compared to natural ecosystems. Adding even small water retention capacity to green roofs appears to reduce water stress considerably.

Unlike the retention layer data, most pairwise comparisons of the predicted substrate layer relative storage capacity time series during the dry, average, and wet reference years showed statistically significant differences (Supplementary Table S2). This is likely to be a result of the fact that the substrate used in the experimental plots was of similar composition across all roof types (lawn substrate for the Garden Roof and extensive multi-layer substrate for all other roof types), resulting in similar total water retention capacities and behavior in their experimental design. Improved parameterization and extended test data time series might improve substrate layer processes modeling, however.