The coupling between WRF and SUEWS occurs via the biophysical interactions between the land surface – with SUEWS introduced as a new land surface module option – and other physics modules in WRF (Fig. ).

The radiation module provides radiative forcing variables, incoming short- K↓ and long-wave radiation L↓, to the land surface module. The land surface module returns outgoing short- and long-wave radiation K↑ and L↑.

SUEWS simulates both the energy balance , 1Q*+QF=QH+QE+ΔQS, and water balance , 2P+Ie=E+R+ΔS. The two are linked through the latent heat (QE) or evaporative E fluxes (by the latent heat of vaporisation). The water balance is driven by precipitation (P) and external water use (Ie). Whereas the surface energy balance is driven by the net all-wave radiation (Q*=K↓-K↑+L↓-L↑, where K and L denote the short- and long-wave components, respectively, and arrows ↓ and ↑ in the subscript denote the incoming and outgoing directions, respectively) in all environments but additionally in cities, the human activities result in anthropogenic heat flux emissions (QF). The turbulent sensible (QH), latent (QE) and net storage heat flux (ΔQS); runoff (R); and change in water storage (ΔS) each have distinct responses that differ with land use and land cover. Traditionally, SUEWS has been mostly used for urban areas , but for WRF–SUEWS it has been extended to non-urban contexts . Each model grid cell has up to seven land cover types (paved, buildings, deciduous trees, evergreen trees, grass/crops, bare soil and water) whose fractions and properties (e.g. height, albedo, leaf area index) can each vary between grid cells.

In WRF–SUEWS, QF is calculated using heating and cooling degree days (HDDs and CDDs) following the approach: 3QF=ρpop,taF0+aF1CDD+aF2HDD, where aF0,aF1 and aF2 are grid-specific coefficients. The grid population densities ρpop,t could be a daily mean (i.e. day and night) value e.g. or capture the diurnal variations e.g.. Typically, for cities with strong commuting flows (e.g. London), QF and ρpop,t are larger in the central business districts (CBDs) during the day due to the commuting but are higher in residential areas at night. As using daily mean population density may bias QF and lose intra-daily variability (e.g. difference in large city centres between work-intensive and non-work periods), following other models e.g. we divide the day into four periods (morning transition, day, afternoon transition and night). Day and night population densities are used in their respective periods, and their averages are used in both transition periods. The other parameter values can be derived from more detailed models that are not rapid enough for NWP (e.g. GQF, ; DASH, ). ΔQS for each grid cell is calculated using the Objective Hysteresis Model (OHM) : 4ΔQS=∑ifia1,iQ*+a2,i∂Q*∂t+a3,i, where t is the time, fi is the fraction of the area of each land cover type (i) and a1-3,i values are material-related coefficients for each land cover type that can vary with the grid cell (cf. Sect. ). This approach allows for much more rapid calculation of this flux than other methods. QE is introduced to the Penman–Monteith equation : 5QE=sQ*+QF-ΔQS+ρcpV/ras+γ1+rs/ra, where s is the slope of saturation vapour pressure curve, ρ is the density of air, cp is the specific heat capacity of air at constant pressure, V is the vapour pressure deficit, γ is the psychrometric “constant”, ra is the aerodynamic resistance for heat or water vapour, and rs is the surface or canopy resistance. Following , assuming energy balance closure, QH is calculated as 6QH=Q*+QF-ΔQS-QE. With the latent heat of vaporisation Lv and Eq. () we obtain E=QELv to link surface energy (Eq. ) and water (Eq. ) balance. R includes the runoff from individual surfaces, in channels and to groundwater. External water use Ie is estimated based on the automatic and/or manual irrigation or external application (e.g. street cleaning) as follows : 7Ie=Airrfautb0,a+b1,aTd+b2,atr+1-fautb0,m+b1,mTd+b2,mtr, where Airr is the area irrigated, faut is the fraction of Airr that is automatically irrigated, b0-2,a,m values are site-specific coefficients, Td is the daily mean temperature and tr is the number of days since the rain. The net change in the water storage ΔS (e.g. in soil, in waterbodies, on the surface) is determined at each time step as the change in each surface water state compared to the previous time step.

The aerodynamic resistance ra is calculated at first at the atmospheric level in WRF–SUEWS, where the wind speed (U) is determined (Fig. ): 8ra=ln⁡Zm-zdz0m-ψmln⁡Zm-zdz0v-ψvκ2u, where zd is the zero plane displacement height (m), z0m (and z0v) is the roughness length for the momentum (and heat/water vapour), u is wind speed at height Zm, κ is the von Kármán constant (0.4) and ψm (and ψv) is the atmospheric stability functions for momentum (and water vapour). Stability is determined iteratively using the Obukhov length and initiated with a LUMPS-calculated sensible heat flux taking the grid land cover fractions into account.

To compute the grid-integrated surface resistance rs, its inverse, the surface conductance gs, is used : 9rs-1=gs=∑iGPFTgmax⁡,ifiLAIiLAImax⁡,i⋅gK↓g(Δq)gTagΔθsoil. The mix of vegetation within the grid is taken into account by considering each vegetation type i with land cover fraction fi, the maximum conductance gmax⁡,i, leaf area index (LAIi), maximum LAI (LAImax,i) and the surface conductance parameter GPFT determined by plant functional type. Functions gK↓, g(Δq), gTa and gΔθ are related to how the environmental variables – downwelling short-wave K↓ radiation, specific humidity deficit (Δq), air temperature Ta and soil moisture deficit Δθsoil  – control the surface resistance. These functions have the following forms : 10gK↓=K↓GK+K↓K↓,maxGK+K↓,max,11g(Δq)=Gq, base +1-Gq, baseGq, shapeΔq,12gTa=Ta-TLTH-TaTcGT-TLTH-GTTc,13gΔθsoil =1-exp⁡GθΔθsoil-ΔθWP1-exp⁡-GθΔθWP, where the G parameters are related to environmental controls indicated by subscripts K for solar radiation, q for the specific humidity deficit (“base” and “shape” for base value and curve shape, respectively), Ta for air temperature and θ for the soil moisture deficit. Table  gives the values used in the evaluation of the coupled WRF–SUEWS system (detailed in Sect. ). K↓,max⁡ is the maximum incoming short-wave radiation (1200 W m-2 used in this work); Tc=TH-GTGT-TL with TL and TH being the lower and upper limits for switching off evaporation, respectively (TL=-10 ∘C and TH=55 ∘C); and ΔθWP is the wilting point (120 mm).

LAI varies with growing degree days (GDDs) and senescence degree days (SDDs) via 14LAId,i=min⁡LAImax⁡,i,(LAId-1,i)0.03GDD×5×10-4+LAId-1,i,TBase,SDD<Ta<TBase,GDDmax⁡LAImin⁡,i,(LAId-1,i)0.03SDD×5×10-4+LAId-1,i,TBase,GDD<Ta<TBase,SDD, where the previous-day (subscript d-1) LAI is used with the base temperature corresponding to the initiation of leaf-off TBase,SDD  and leaf-on periods TBase,GDD. The model also requires LAImax⁡,i and LAImax⁡,i for each vegetation type i.

SUEWS accounts for the running water balance of the multiple surface types. The water amount on the canopy of each surface Ci determines the surface resistance between dry and wet rs=0sm-1 by replacing rs in Eq. () with rss : 15rss=Wrb(s/γ+1)+(1-W)rs+rb(s/γ+1)-1-rb(s/γ+1), where W is a function of the relative amount of water present on each surface to its water storage capacity Si: 16W=1Ci≥SiKr-1Kr-Si/CiCi<Si. Kr depends on the aerodynamic and surface resistances, 17Kr=rs/ra/ra-rbrs+rb(s/γ+1), where rb, the boundary layer resistance, is a function of friction velocity u* : 18rb=1.1u*-1+5.6u*13. Equations ()–() ensure that the surface resistance rss has a smooth transition from 0 s m-1 (a completely wet surface) to rs (a dry surface).

This work presents a coupling framework and its evaluation using SUEWS v2018, ensuring consistency in internal physics with the offline version for the comparison later in Sect. . However we note that the coupling structure designed for WRF–SUEWS enables seamless upgrades to more recent SUEWS versions.

Current offline versions of SUEWS have options not in the coupled WRF–SUEWS system, including

CO2 fluxes for local-scale anthropogenic and biogenic urban–atmosphere exchanges

The following are considered in the design of coupled WRF–SUEWS system.

Performance. Given coupling with file-based IO (input–output) exchanges has unacceptable computational performance, we use the SuMin module under the WRF framework (Figs.  and ).

Incoming short-wave radiation K↓ is known to be overestimated by WRF because of unresolved clouds and/or aerosols . However, if the forcing radiation is too large, the other surface fluxes and variables will be impacted. Thus, they should not compare well to observations, or alternatively if they do compare well, the variables are correct for the wrong reason. In urban areas, even on clear-sky days, there is often a large presence of aerosols that impact bulk transmissivities (e.g. Shanghai, ; Beijing, ; London, ). A bulk atmospheric transmissivity (τ) can be specified in the namelist.suews to partially correct the overestimation of K↓ by WRF, which can be determined using K↓ at both the top of atmosphere K↓,TOA and the surface K↓,S are used : 19τ=K↓,sK↓,TOA. As τ can vary seasonally (e.g. the cases in London and Swindon as shown in Table ) we determine the median clear-sky difference (Δτc) between τWRF and τobs from the analysis of clear-sky days observations around the peak K↓ (which occurs between 40 % and 60 % of the daylight hours). The K↓ forcing (Fig. ) for SUEWS in WRF (K↓,W-S) is then corrected using the original one produced by WRF (K↓,WRF) as 20K↓,W-S=K↓,WRF-1-ΔτcK↓,TOA. Given the empirical nature of the parameter values, this correction can only be applied where observations are available. Here, we apply the correction to all time periods but separate the evaluation (Sect. ) by sky conditions to assess effectiveness. Obviously, this simple correction is not a complete solution but rather an attempt to obtain more accurate K↓ forcing for the coupled SUEWS and, hence, better surface feedbacks for the WRF atmospheric modules (Fig. ).

WRF–SUEWS is evaluated at the same two UK sites as in a previous SUEWS evaluation study (; W16 hereafter) for consistency – these sites exhibit distinct urban characteristics:

KCL. The King's College London Strand Campus (51∘30′ N, 0∘07′ W) is a dense central business district area in London (d03 of Fig. ).

WRF–SUEWS is run with two-way nesting mode of 33 vertical levels (top at 5 kPa 11 layers in the boundary layer below 2000 m with lowest levels in d03 and d04 being ∼40 m a.g.l.; above ground level) for all four domains (Fig. ). We note more vertical levels may be needed in detailed investigations of atmospheric features (e.g. temperature, precipitable water, etc.); here a moderate number of vertical levels are used as a balance between the computational cost and necessary representation of atmospheric profiles considering the focus of this work on the model development and evaluation of essential urban–atmosphere interactions. The atmospheric boundary conditions used are the 1∘ × 1∘ (latitude × longitude) National Centre for Environmental Prediction FNL (final) data . The well-tested WRF “CONUS” (contiguous United States) physics suite (configuration since v3.9) is used with the land surface scheme changed to SUEWS (Table ). The SUEWS physics schemes (Table , details provided in ) are selected for simplicity including using the building and tree heights with a rule-of-thumb method for momentum roughness length and displacement height; additionally, snow and irrigation modules are turned off (following W16's KCL and SWD configuration). We use the WRF adaptive time step option to reduce the total run time while being numerically stable considering both the horizontal and vertical extent . For us, the adopted time steps for domain 1 to 4 are around 72, 9, 3 and 3 s.

We evaluate WRF–SUEWS during the four seasons using 2-week periods in 2012 (Table ). To generate appropriate initial conditions (i.e. model spin-up), we conduct offline SUEWS runs driven by observations collected at KCL and SWD (refer to the purple boxes in Fig.  and related notations in Sect.  for details about the atmospheric forcing variables as well as Table  for surface property settings) for 2012 until the soil moisture converges (<0.1 % difference in last time step between consecutive years): a period of 15 years is needed for KCL, while a period of 5 years is needed for SWD. For fully vegetated grids (i.e. fDCT/EVT/GRA=1), the observations at SWD are used as forcing conditions to spin-up SUEWS for 4 years before convergence. The required initial states (i.e. soil moisture, leaf area index) are used for each WRF–SUEWS period (Table ). For grid cells with fPAV+BDG>0.6, the KCL-based initial states are prescribed to represent areas dominated by impervious surfaces, while SWD is used for the rest that are not completely vegetated. The appropriate complete pervious cover type values are assigned to the pervious cells based on their dominant land cover type.

The W16 evaluation of SUEWS (v2016a) uses 60 min radiation and turbulent fluxes observed at KCL and SWD . Both flux towers are located close to the centre (within a 200–300 m radius circle) of a 1 km2 model grid cell. The source areas of the observed turbulent fluxes have a probable 50 % contribution from within ∼400 m of the flux tower at KCL and a probable 80 % contribution from within ∼700 m at SWD . Here, we average the 30 min sample output from land surface scheme (i.e. SUEWS) to 60 min to compare with the observations.

The mixed-layer height (MLH), derived from continuous high-resolution (15 s and 10 m) attenuated backscatter observed with a Vaisala CL31 ceilometer at Marylebone Road (MR) in London , is used to evaluate the model's ability to predict atmospheric boundary layer (ABL) dynamics. The MLH values have been compared to AMDAR, or Aircraft Meteorological Data Relay (the median difference between inversion heights and MLH is 346 m based on all time periods; for more evaluation results, refer to ). Various observations can be used to obtain the height of the boundary layer, but the results depend on the variable used . For example, differences occur between using temperature inversion and MLH (e.g. at night, ) or between MLH and the turbulence-derived mixing height (MH; ), the height where the vertical velocity variations falls below a threshold . On the other hand, when using the Mellor–Yamada–Janjić scheme, the WRF output PBLH (planetary boundary layer height) is derived from the height where turbulent kinetic energy falls below 0.2 m2 s-2 .

Although the comparison of the aerosol-derived MLH from observations and the turbulence-based mixing height diagnosed from the model output (WRF PBL, hereafter referred to as WRF MH) may be affected to systematic differences (e.g. those associated with vertical resolution as suggested by ), the comparable nature between MLH and MH enables the former to be a proxy to examine the latter modelled by WRF.

The evaluation metrics used with the number of data points (N) available from the model output Ymod and observation Yobs time series are the following.

Hit rate (HR).21HR=∑j=1NHδY,j-Ymod,j-Yobs,jNwith Heaviside step function H defined by22H(x)=0,x<01,x≥0and the threshold δY,j being a value dependent on evaluation variable Y. We use the HR to evaluate the surface energy fluxes with δYj=50 W m-2 for radiative K↓,K↑,L↓,L↑,Q* and δY,j=0.1Q*+50Wm-2 for turbulent (QE and QH) fluxes , respectively. If HR=0, it suggests none of model predictions are within the acceptable threshold set, while HR=1 indicates all fall into the acceptance range.

The WRF–SUEWS model demonstrates satisfactory performance across various seasons and for two distinct urban areas (Sect. ). Therefore, we employ it to examine the influence of QF on the atmospheric boundary layer – a unique characteristic of the coupled WRF–SUEWS system compared to the standalone SUEWS – in d03 (Fig. ) during April 2012, before the Olympics disrupted typical patterns in July. As daylight saving time has begun by this time of the year, people's activities are shifted an hour earlier (e.g. typical workday is 08:00–16:00 UTC or 09:00–17:00 BST local time; British summer time). Peak QF emissions occur in central London (Fig. ) during the daytime (>300Wm-2, Fig. c). By the late afternoon and through the night (20:00–05:00 UTC) the values in these areas (Fig. b) are smaller (<130Wm-2, Fig. d). The areas where large values occur differ between night and day.

At night the peaks occur across a larger area beyond central London, with some larger values towards the outskirts (Fig. b; cf. Fig. a). The nocturnal QF mean in d03 is smaller 17.1Wm-2 than the daytime 19.8Wm-2 but is more spatially consistent (Fig. d; cf. Fig. c).

These spatiotemporal patterns (Fig. a, b) are largely consistent with previous studies in London. Daily peak values differ between studies because of model grid cell size and efforts over time to reduce carbon emissions and therefore energy use . Peak values (grid size) vary between 120Wm-2 (3.2km2, ), ∼150Wm-2 (1km2, ; 1km2, ) and 210Wm-2 (3.2km2, ). Besides, the WRF–SUEWS-predicted average QF values for London (i.e. ∼18Wm-2) are comparable with those estimated in other mega-cities (e.g. peak values of ∼50Wm-2 in the city of Osaka, Japan, by ; annual average of ∼20Wm-2 in the Beijing–Tianjin–Hebei agglomeration by ; annual average of ∼30Wm-2 in Beijing by ).

To assess the feedback, we focus on two example grids with contrasting population density profiles:

a central business district (CBD) area, within the City of Westminster (grid C, Fig. ), with tall buildings (roughness length z0=2.0m, zero plane displacement zd=13.9m; calculated using the rule-of-thumb approach as in ), a low fraction of vegetation fVEG=13.1% and large daytime population density 763inhha-1 but small nocturnal density 111inhha-1