In PALM, the transport of passive scalar quantities, including trace gases, is governed by the filtered scalar conservation equation : 1∂s‾∂t=-1ρ∂ρu‾js‾∂xj-1ρ∂∂xjρu‾j′′s′′+χs, where s is the concentration of the trace gas in kg m⁻³, t is time, ρ is the density of dry air, uj is the jth component of velocity, xj is the spatial coordinate, and χs is the source or sink term of the passive scalar. Variables with double prime (^′′) refer to the subgrid-scale (SGS) component of the corresponding variables, while the over-bar indicates the filtered quantities. The advection and SGS turbulent flux divergence terms are discretized using finite-difference schemes implemented in PALM, and SGS fluxes are parametrised via an eddy-diffusivity approach consistent with the LES formulation.

PALM-CO₂ leverages these existing scalar transport codes and augments them with dedicated modules to represent anthropogenic and biogenic CO₂ fluxes. These fluxes enter the scalar transport equation exclusively through the source or sink term χs, ensuring a clean separation between atmospheric transport processes and surface exchange parametrisations.

Figure  illustrates the overall structure of PALM-CO₂, including the coupling between the flow solver, the biogenic flux module, and the prescribed anthropogenic emission inputs.

Biogenic CO₂ exchange is represented using the Vegetation Photosynthesis and Respiration Model (VPRM; ), which parametrises the ecosystem-scale carbon uptake and release based on vegetation state and meteorological drivers. VPRM has been widely applied in regional and urban-scale carbon studies . The parameters in the VPRM model are typically fitted using measurements from local eddy-covariance flux towers. In this study, we select a set of parameters (see Appendix ) from the WRF-Chem module that has been calibrated using flux tower measurement in Europe .

In PALM-CO₂, VPRM computes gross primary production (GPP) and ecosystem respiration (RESP) using the following equations: 2GPP=λ×Tscale×Pscale×Wscale×EVI×11+PAR/PAR0×PAR, where λ is a parameter fitted with the measurement of biogenic fluxes. Vegetation status and water stress are represented by the Enhanced Vegetation Index (EVI) and the Land Surface Water Index (LSWI). Pscale is a coefficient accounting for the degree of leafiness which is computed from LSWI and also depends on EVI to determine whether the vegetation has reached full leaf expansion, 3Pscale=1,EVI>THleaf full expansion,1+LSWI2,EVI≤THleaf full expansion. THleaf full expansion is a threshold for vegetation growing cycle. EVI above the threshold means the growing period between leaf full expansion and senescence and below the threshold is during bud burst to leaf full expansion and during senescence. The threshold is computed from the annual maximum and minimum EVI for the computation grid, 4THleaf full expansion=EVImin+0.55×(EVImax-EVImin). Wscale accounts for the water content in the vegetation and is computed from LSWI, 5Wscale=LSWI-LSWIminLSWImax-LSWImin,grassland and savanna,,1+LSWI1+LSWImax,all other classes. PAR is photosynthetically active radiation which is correlated with short-wave (SW) radiation: PAR≈SW/0.505; PAR0 is a parameter for half photosynthetic saturation. Tscale is a coefficient computed from air temperature T, 6Tscale=(Tair-Tmin)(Tair-Tmax)(Tair-Tmin)(Tair-Tmax)-(Tair-Topt)2, with Tmin, Topt, and Tmax denoting minimum, optimal, and maximum temperatures for photosynthesis. Tair is the air temperature.

The respiration flux is mainly determined by the air temperature Tair and is computed from 7RESP=αTair+β, where α and β are coefficients that vary among different vegetation types. The net ecosystem exchange (biogenic flux) is then calculated 8NEE=-GPP+RESP, where a negative value means net uptake and a positive value means net emission.

To facilitate comparison with eddy-covariance observations and to analyse vertical CO₂ exchange, PALM-CO₂ includes a diagnostic computation of turbulent scalar fluxes. The vertical turbulent flux (F) is defined as 9F=〈w′s′〉, where angle brackets denote the temporal average, 10〈ϕ〉=1τ∫t0t0+τϕ(t)dt, where τ is the time window for averaging. w′ and s′ are the deviations from the mean of the vertical velocity component and tracer gas concentration (e.g. CO₂) respectively, 11aw′=w-〈w〉,11bs′=s-〈s〉. Then the flux can be computed from the following temporal averaged quantities, 12〈w′s′〉=〈(w-〈w〉)(s-〈s〉)〉=〈ws-〈w〉s-w〈s〉+〈w〉〈s〉〉=〈ws〉-〈w〉〈s〉=〈w‾s‾〉-〈w‾〉〈s‾〉. In the last line, the spatially filtered quantities w‾ and s‾ and their product w‾s‾ from the Large-eddy simulation (LES) code are used to compute the vertical turbulent flux for the CO₂ concentration. A major assumption is the omission of the sub-grid scale turbulent flux, which requires the grid size to be small enough to resolve the total turbulent flux. It is shown in the grid sensitivity study (Sect. ) that the resolved flux at medium height boundary layer (higher than three roughness-layer heights) takes over 96 % of the total turbulent flux. Thus, the assumption is valid. A minor assumption is that the spatial filtering and time averaging commute and that the filter preserves the mean, such as 〈⋅〉=〈⋅‾〉. This assumption requires the averaging window is long enough to converge the statistics at all spatial locations. A computation of eddy turnover time (about 450 s for the grid sensitivity study case) shows that a half-hour average window (1800 s) covers about four eddy turnover periods, validating this minor assumption. The temporal averaging of w‾s‾ is added to the code via a user-defined procedure. Its rolling sum is computed during time-stepping in the code, and its temporal average 〈ws〉 is computed and reported after each averaging time window τ. 〈w〉 and 〈s〉 can be easily recorded using the default output settings in PALM. The time window for averaging is set as 0.5 h.

Anthropogenic CO₂ emissions for urban areas are commonly provided as coarse-resolution inventories, typically with a kilometre-scale spatial resolution, and as annual totals. For use in high-resolution urban transport models, such inventories must be disaggregated both spatially and temporally to match the computational grid and time step. A direct interpolation of coarse grid values onto a fine-resolution model grid is straightforward but insufficient to represent realistic urban emission patterns, which are often dominated by localised sources such as building heating systems, traffic corridors, and industrial point sources. The method for processing emission inputs for chemistry transport models has been implemented in several existing tools, including the online emission module in COSMO  and the stand-alone processors HERMES , EMIPS , and FUME . However, these tools are primarily designed for global and regional applications and directly use emission inventories at those scales; therefore, they are not readily applicable to our study.

To address this, a sector-based disaggregation strategy is adopted. Disaggregation is applied on the sub-subsection levels defined in the LAEI dataset listed in Appendix . Emission sub-subsectors contributing only a small fraction of total emissions are distributed uniformly in space and time after interpolation to the fine grid. In contrast, dominant sub-subsectors are spatially disaggregated using additional geospatial surrogate data that represent the underlying emission-generating activities. This approach balances accuracy with data availability, as comprehensive surrogate datasets are not uniformly available for all emission sub-subsectors. The selection of sub-subsectors for detailed disaggregation is therefore based on their relative contribution to total emissions.

For sub-subsectors selected for fine-scale treatment, spatial disaggregation relies on auxiliary geospatial datasets such as building-level energy information, industrial point-source inventories, and road network data. These datasets are used as spatial proxies to redistribute emissions from coarse administrative or grid-based units to individual buildings, road segments, or point sources, which are then aggregated to the target model grid resolution.

Temporal disaggregation is applied to convert annual emissions into hourly fluxes. Sub-subsector-specific temporal profiles are used where available, capturing diurnal, weekly, and seasonal variability in anthropogenic activities. Sub-subsectors with negligible contributions are assigned constant temporal profiles, while dominant sub-subsectors employ time-varying profiles derived from established emission databases.

PALM provides an interface for user-defined chemical mechanisms where one can set up the chemical components . In PALM-CO₂, CO₂ is represented as a single passive scalar without chemical reactions. Both anthropogenic and biogenic fluxes are treated as surface source terms and are incorporated into the scalar transport equation through χs in Eq. ().

Biogenic CO₂ fluxes are computed by VPRM, whereas anthropogenic emissions are provided as externally prepared, time-resolved gridded datasets. VPRM (described in Sect. ) is implemented in PALM as part of an online chemistry module, where biogenic fluxes are computed prior to solving the scalar transport equation using near-surface air temperature and short-wave radiation retrieved from PALM runtime variables. Anthropogenic emission sources (described in Sect. ) are supplied as hourly NetCDF files, requiring activation of the level-of-detail (LOD) 2 option in PALM.

Both biogenic and anthropogenic fluxes are prescribed as source terms in the lowest atmospheric grid cell adjacent to the surface. In locations where complex terrain or building structures elevate the first fluid grid cell above ground level, the surface fluxes are applied at that elevated cell. This approach maintains a physically consistent coupling between surface emissions and the resolved airflow, ensuring that fluxes are introduced directly into the urban canopy layer where turbulent exchange and transport are explicitly represented.

A study area in London was selected for model validation and analysis, as shown in Fig. . London is selected for its high population density, high level of urbanisation, and its vast amount of green infrastructures. In Camden, there is a CO₂ flux measurement station located at 190 m height on the BT tower , providing an independent dataset for evaluating the performance of the model developed in this study.

The Greater London Authority (GLA) area (Fig. a) is characterised by a highly heterogeneous land-use and urban landscape. At its core lies a dense, multifunctional urban structure dominated by commercial, financial, and high-density residential uses, with extensive transport infrastructure and limited open space. Moving outward, land use transitions to predominantly residential neighbourhoods of medium to low density, interspersed with local centres, industrial estates, and major road and rail corridors. Despite its metropolitan character, the GLA area contains a substantial proportion of green and blue spaces, including large public parks, river corridors – most notably the River Thames – and protected areas .

Sitting at the core of the GLA, the borough of Camden (Fig. b) reflects this heterogeneous landscape. From north to south, there is a mixing of green spaces (Hampstead Heath and the Regent's Park), residential area, and commercial and business centres around three main train stations (Euston, St Pancras and King's Cross). These are also shown in Fig.  which depicts a bird's-eye view of dense building areas with parks and stations. Overall, land use in the GLA and Camden shows a complex mosaic in which intense urban development coexists with natural and semi-natural landscapes, resulting in strong spatial contrasts across relatively short distances.

A nested domain is used, which consists of a coarse-resolution domain (parent domain, denoted as N01 domain) and a fine-resolution domain (child domain, denoted as N02 domain). The parent domain (GLA)'s range is 60 km by 60 km in the horizontal and 3 km in the vertical, while the grid resolution in x,y,z is 100, 100, and 50 m respectively. The child domain (Camden)'s range is 8 km by 8 km in the horizontal and 1 km in the vertical, while the grid resolution is 10 m in all directions.

For the London case study, anthropogenic CO₂ emissions are derived from the London Atmospheric Emissions Inventory (LAEI) , published by the Greater London Authority. LAEI employs a bottom-up methodology, combining activity data with sector-specific emission factors to estimate emissions across multiple sectors. The most recent dataset available during this study is LAEI2019, released in 2022, which reports annual total emissions for the year 2019 on a 1 km² grid covering the region within the M25 motorway. The inventory grid conforms to borough boundaries, so that grid cells intersecting multiple boroughs are subdivided accordingly.

Since LAEI only reports annual total emission on 1 km² tiles, it is essential to further disaggregate the emission data spatially and temporarily for use in an urban model. A direct approach would be to interpolate the LAEI 1 km² tile data on the finite difference grid defined in the urban model. However, this approach cannot capture the real emission scenario where emission within a neighbourhood can originated from several large localised emission sources such as gas boilers within households or industrial chimney stacks. While it is ideal to disaggregate all sectors into a fine grid map, there is a lack of available geo-spatial surrogate data for the task. For example, one might be able to find domestic building information in a city to disaggregate domestic energy usage sector, but there rarely exists a comprehensive list of construction site for non-road mobile machinery sector. Therefore, one needs to decide which sectors should be disaggregated with additional surrogate data and which should be simply interpolated. Here, we examine the proportions of each sub-subsector in the LAEI dataset. A list of the sector categories is included in Appendix . Note that some sub-subsectors have been merged due to their small contribution. For sub-subsectors whose proportions are less than 10 %, their emissions would be simply distributed evenly in the finer grid and in time (annual total amount spread to hourly emission). Otherwise, we consider them to be significant and require fine-scale geospatial data as spatial proxies to disaggregate. In LAEI, CO₂ emission is reported for four major sectors (domestic, commercial and industrial, transport, and others) with sub-sectors and sub-subsectors. Figure  shows the significant emission sub-subsectors, which are domestic gas combustion, non-domestic (commercial and industrial) gas combustion, road transport, and industrial processes A1 (A1 refers to large industrial point sources).

It is noted here that although the word “disaggregation” is used to describe the operation of processing emission inventory data to the desired grid resolution (10 m), data in LAEI for the following sub-subsectors are not used: domestic gas combustion, non-domestic gas combustion, and industrial processes part A1. Instead, the source data for activities that LAEI has used to generate their 1 km² grid map are retrieved. Together with additional geospatial data as spatial proxies, we processed the 10 by 10 m² gird map for these sub-subsectors. Road transport was provided in LAEI as shapefile of the major road network and is convenient to interpolate to a grid of 10 by 10 m².

For the London case study, vegetation state variables required by the biogenic CO₂ flux model are derived from satellite-based surface reflectance products. Due to frequent cloud cover over the United Kingdom, EVI and LSWI are processed as monthly composites rather than at higher temporal resolution. For each calendar month in 2019, all available cloud-free satellite observations from the corresponding month throughout the 2018–2020 period are selected and temporally mosaicked. The median values are used to generate spatially complete monthly EVI and LSWI fields, reducing data gaps caused by cloud contamination.

The resulting monthly EVI and LSWI maps are provided to PALM as static input variables and are used by the biogenic flux module to represent vegetation phenology and water stress. This approach ensures spatial continuity of vegetation indices while preserving seasonal variability relevant for photosynthesis and respiration.

The parameters of the biogenic model are selected from the VPRM chemistry (WRF-Chem) weather research and forecast model , using parameter sets calibrated against measurements of the flux tower of eddy-covariance in Europe . Parameters are listed in Appendix . These parameters are applied uniformly across the study domain, acknowledging that vegetation-specific calibration at the neighbourhood scale is not feasible given the available observations.

To obtain simulated eddy-covariance (EC) fluxes from the LES, two approaches are employed. The primary approach computes EC fluxes from temporally averaged three-dimensional model fields using Eq. (). An independent approach uses virtual measurement points that output high-frequency time-series data, from which EC fluxes are calculated following the standard eddy-covariance procedure. The agreement between these two approaches is used to verify the temporal-averaging method.

For the temporal-averaging approach, resolved vertical scalar fluxes are averaged over a 30 min window, consistent with the averaging period used in flux-tower observations. The mean vertical velocity and CO₂ concentration are obtained from the standard PALM output, while the covariance term is accumulated online during model integration. These correspond to the 〈w‾〉, 〈s‾〉, and 〈w‾s‾〉 terms in Eq. (). All are 3D output quantities. These fields are then substituted into Eq. () in post-processing to obtain a 3D eddy-covariance flux field on all grid points. The 3D EC flux at the exact grid point where the BT tower measurement is located is extracted to compare the simulated EC flux with the site measurement described in the Sect. .

Besides the EC flux computed via Eq. (), we also set up virtual measurement points in the domain to show the validity of this temporal averaging method. This is an independent calculation of the EC flux using time-series data. The virtual measurements output velocity component and CO₂ concentration at a frequency of 1 Hz, the same as that used in BT measurement . After the simulation, 30 min means are computed from the 1 Hz time series and subtracted from instantaneous values to obtain the fluctuations w′ and s′. The EC flux is then calculated as the covariance 〈w′s′〉 over each half-hour period. The comparison of the EC fluxes obtained via virtual measurement and via temporal averaging (Eq. ) is shown in Fig. . The two approaches yield almost the same flux profiles. In the main text, we use the EC flux computed from temporal averaging 3D fields as the simulated flux to compare with BT tower measurement.

A 30 min averaging window is used throughout this study to match the averaging period of the BT measurements and standard eddy-covariance processing. While EC fluxes are computed using 30 min averaging intervals, a 2 h moving average is applied to both simulated and measured flux time series when calculating RMSE and correlation coefficients to reduce high-frequency variability. The moving average is computed by taking four half-hourly EC fluxes and compute their mean. The timestamp for the moving average is at the end of the 2 h window.

The measurement site is in Camden, London. An eddy-covariance flux tower was set up on the BT tower in Camden by previous researchers . It collected the vertical CO₂ flux , turbulent flow quantities and other meteorological and greenhouse gas variables. We used the measurements in 2019 to validate our CO₂ model. Specifically, the wind speed and the eddy-covariance CO₂ flux were used in the validation. The temporal interval of measurements was half an hour. The CO₂ flux is computed via the procedure described in Sect. . In the following, we provide some essential information on the measurement site and its surrounding areas. For more details, the reader is referred to .

The eddy-covariance (EC) measurements of CO₂ were conducted from the BT Tower in central London (51°31′17.4′′ N, 0°08′20.0′′ W). The EC system was installed on a lattice structure mounted on the roof of the telecommunications tower, providing an effective measurement height of approximately 190 m above street level. The system consisted of a three-dimensional sonic anemometer and a closed-path cavity ring-down spectrometer measuring CO₂, CH₄ and H₂O, with air sampled through a 45 m inlet line. Owing to the unusually large measurement height, the footprint of the observations extended over several kilometres and integrated emissions from a substantial portion of central London.

The BT Tower is located within a densely urbanized area of central London characterized by a mixture of commercial, residential and transport infrastructure. Within a radius of approximately 10 km, the mean building height is 8.8 ± 3.0 m, while typical suburban building heights are 5.6 ± 1.8 m. Consequently, the 190 m measurement height is more than an order of magnitude greater than the average surrounding building height, placing the EC sensors well above the urban roughness sublayer and enabling observation of integrated fluxes from a large urban source area.

Footprint analyses indicate that the source area varies seasonally from a few kilometres under convective conditions to several tens of kilometres during wintertime stable conditions. The footprint encompasses a diverse range of land uses. To the southwest and northwest, the source area includes two major urban parks, Hyde Park (142 ha) and Regent’s Park (197 ha), respectively. Northern sectors are dominated by suburban residential neighbourhoods, whereas eastern and southern sectors contain a mixture of densely built residential and commercial districts. The southeastern footprint also includes part of the River Thames. As a result, the measured CO₂ fluxes represent the integrated influence of emissions from road traffic, residential and commercial energy use, and human activities across a heterogeneous urban landscape.

Because CO₂ fluxes measured at the BT tower are dominated by anthropogenic emissions, they provide limited information on the realism of the biogenic component of the model. Therefore, an additional qualitative evaluations of the implemented VPRM scheme were made.

First, we conducted using eddy-covariance measurements from the ICOS site CZ-Lnz, a deciduous forest ecosystem in the Czech Republic . The objective of this comparison is not to perform a site-specific validation of vegetation fluxes in Regent's Park, but rather to assess whether the model reproduces realistic magnitudes, seasonal variations, and diurnal patterns of net ecosystem exchange (NEE) for deciduous vegetation. Given the climatic and ecological differences between the two sites, the comparison is interpreted as an assessment of the VPRM implementation rather than as a direct validation of local biogenic fluxes.

The CZ-Lnz site is located in a deciduous floodplain forest, and its eddy-covariance measurements are primarily influenced by local biogenic carbon exchange. Although the climatic and ecological conditions differ from those in London, CZ-Lnz provides a long-term, high-quality record of net ecosystem exchange (NEE) from a deciduous forest ecosystem, allowing a qualitative assessment of the VPRM representation of forest carbon uptake and respiration.

Monthly ensemble averages of tower observations were used to derive representative half-hourly diurnal NEE profiles for each month. For the model, NEE was extracted directly from the VPRM output for a grid cell in Regent's Park classified as deciduous forest. The comparison therefore focuses on the temporal characteristics of NEE rather than on direct site-to-site agreement.

Second, the total NEE for the parent domain (roughly Greater London Authority area) was compared with the ICOS-VPRM product . This product runs offline VPRM module with MODIS-derived satelite indices and meteorological inputs from ECMWF IFS analysis data. The product has a spatial resolution of ∼ 0.1°. The daily sum of NEE for the twelve simulated days from PALM-CO₂ and from ICOS-VPRM were compared. Using the simulated days as representative values for their corresponding months, annual NEE rate (both total amount and monthly correlation) were reported.

In this subsection, we show comparison of the modelled flux and the measured flux by the eddy covariance method. This comparison validates: (1) the input anthropogenic emissions and online-computed biogenic fluxes, and (2) the CO₂ transport process calculated by PALM.

In previous studies, CO₂ transport has been approximated using simplified two-dimensional advection–dispersion approaches, in which turbulent source areas or flux footprints were parameterized rather than explicitly resolved; see for an urban carbon emission model that leverages the 2D model for validation. In contrast, PALM resolves 3D turbulence using large-eddy simulation, enabling a more detailed representation of turbulent dispersion and vertical transportation processes, and thereby providing a more physically consistent characterization of source-receptor relationships.

To reduce the influence of high-frequency variability and facilitate comparison between simulated and measured fluxes, both time series were additionally smoothed using a 2 h moving average prior to the calculation of RMSE and correlation coefficients. The 30 min EC fluxes were retained for all other qualitative analyses. Wind speed and wind direction have also been smoothed using the same method.

Figure  shows the validation results for the CO₂ flux for selected months representing each season of a year. Overall, good agreement between the simulated CO₂ flux and the measurement at the BT tower has been reached. When July is excluded, the average RMSE and correlation coefficient are 16.74 µmol m⁻² s⁻¹ and 0.74, respectively. The trend of the diurnal cycle has been captured by the model and matches well with observations. Specifically, the CO₂ flux remains low in the early morning before sunrise, increases during the morning hours (08:00 a.m. to 12:00 p.m.), reaches a peak in the afternoon (04:00 p.m. to 06:00 p.m.), and gradually decreases toward midnight (00:00 a.m.). As the measurement site is located in a commercial-residential mixed area with limited green spaces, the flux pattern is dominated by anthropogenic emission. Therefore, such a diurnal cycle meets the expectation of human activity patterns within a day. Quantitatively, the magnitudes of fluxes in January, April, and October show good agreement with observations. Statistics show relatively low error and high correlation between simulation and measurement.

Modelled fluxes in July show a consistent pattern as in other seasons, while discrepancy from measurement exists. The site measurement in 16 July has poor quality which already shows a non-typical diurnal pattern in the figure. Besides, an overestimation of flux on 16 July persists. As one would expect that anthropogenic emission in July (summer time) is smaller than in October and January, yet the simulated flux in July is higher. This overestimation originates from extremely weakly forced atmosphere during daytime peak hours. As can be seen in Fig. c, wind speed at 190 m height is close to 1 m s⁻¹ between 10:00 to 16:00. Poor advection caused build-up of CO₂ in the domain and an over-estimation of EC flux. The quality control flag and additional analysis of July data can be found in Appendix .

Figures  and show validation of the wind speed and direction. The average RMSEs for wind speed and wind direction in the figures were 2.30 m s⁻¹ and 37.50°, respectively, while the corresponding average correlation coefficients were 0.50 and 0.68. The wind velocity on the 10 by 10 m² grid at 190 m height corresponding to the measurement location is extracted from the simulation. In the figures, wind speed and direction at a half-hourly interval and its 2 h moving average daily profile are shown. The general trend of daily variation of wind (the 2 h moving average profile) matches that in the measurement. For all days simulated in 2019, statistics show good agreement between simulation and measurement for both wind speed and direction, with average RSME around 2.4 m s⁻¹ and 28.65° and correlation coefficient 0.55 and 0.75 (see Appendix ). The validation results demonstrate that ERA5 provides suitable dynamic boundary conditions for the parent domain and confirm the effectiveness of the nesting strategy in representing processes within the nested domain.

Results for all months can be found in Appendix .

Comparison with the BT tower measurements provides the primary validation of the modelling framework in the study area. However, CO₂ fluxes measured at the BT tower are dominated by anthropogenic emissions, making it difficult to directly assess the realism of the vegetation component of the model. Here, we compare biogenic flux with two independent sources in the literature to evaluate the biogenic fluxes in PALM-CO₂.

To provide an independent evaluation of the implemented VPRM scheme, we first compare the simulated biogenic fluxes against eddy-covariance observations from a European forest site, Lanzhot in Czech (CZ-Lnz) . The purpose of this comparison is not to perform a site-specific validation of vegetation in Regent's Park, but rather to assess whether the model produces realistic magnitudes, seasonal variations, and diurnal cycles of ecosystem carbon exchange.

Figure  compares the simulated and observed NEE profile. Both datasets exhibit the expected seasonal behaviour of temperate deciduous forests. During winter (December, January, February), NEE remains relatively small and positive, reflecting limited photosynthetic activity and continued ecosystem respiration. During summer (May, June, July), stronger daytime carbon uptake produces larger negative NEE values and a pronounced diurnal cycle. The model reproduces these broad seasonal and diurnal characteristics, indicating that the implemented VPRM module captures the dominant controls on biogenic carbon exchange.

Differences are evident during the spring green-up period and autumn senescence. The simulated NEE shows stronger carbon uptake in March and April and weaker uptake in August, September and October compared with CZ-Lnz observations. These discrepancies are likely related to differences in phenology between the two sites since they correspond to the budburst and senescence stages of deciduous forests . The timing of budburst and senescence is influenced by local climate, species composition, soil conditions, and management practices, all of which differ between the urban park environment of Regent's Park and the rural floodplain forest represented by CZ-Lnz. As shown in Fig. , EVI and LSWI increase rapidly in March at the Regent's Park location, leading to enhanced simulated GPP through Eq. ().

The stronger uptake simulated during spring is consistent with the rapid increase in EVI and LSWI at the Regent's Park location. However, the differences observed during late summer and early autumn cannot be explained by EVI alone. Although EVI remains relatively high during August and September, NEE is influenced by additional environmental controls in VPRM, including temperature and moisture stress, as well as site-specific ecosystem characteristics. Furthermore, the comparison involves two ecologically distinct systems: an urban park in London and a rural floodplain forest in the Czech Republic. Differences in species composition, soil moisture conditions, forest structure, and phenological timing may therefore lead to different seasonal trajectories of carbon uptake despite similar vegetation index values. Consequently, the late-summer discrepancy is likely attributable to site-specific ecological differences rather than changes in canopy greenness alone.

Second, the sum of the total biogenic flux in the N01 domain (60 by 60 km²) is compared with the ICOS-VPRM product . Figure  shows the comparison. The annual trend of biogenic flux computed in PALM agrees with that in ICOS-VPRM in general. A positive net flux during winter, late summer and autumn seasons appeared in both computations, while spring and early summer show negative net flux. The values are listed in Table . Using the NEE on the 16th day of each month to represent monthly flux, average biogenic flux in 2019 is -3.95 kton d⁻¹ in our model while ICOS-VPRM reports average NEE -4.92 kton d⁻¹. Correlation coefficient between them is 0.97. To show the ratio of biogenic fluxes and anthropogenic emissions, Table  also lists the total biogenic fluxes in the N02 domain (8 by 8 km²) as well as the N01 domain. For the N02 domain, biogenic flux to anthropogenic flux ratio is between -4.72 % to 2.29 %; for the N01 domain the ratio is between -78.2 % and 8.6 %. This shows that in the densely populated Camden borough biogenic fluxes have very limited offset to anthropogenic emissions.

In this subsection, we show results on carbon dioxide concentration and its relation to the height of the Atmospheric Boundary Layer (ABL). Three snapshots of CO₂ concentration in the morning 08:30 a.m., noon 12:30 a.m., and evening 08:30 p.m. are selected in the following analysis. The ABL heights and Obukhov lengths for the snapshots are listed in Table . Figure  shows the CO₂ concentration at the surface level at selected times during a day in January and July. Due to terrain height and building geometries, the surface level at different horizontal places are at different levels. This has been processed and in the figure only the lowest level in the air is plotted. In all three snapshots in both January and July, individual large anthropogenic emission locations such as major roads and buildings in the commercial region (lower right in each sub-figures) can be identified. They contribute substantially to the variation of local concentrations. In January, due to a relatively higher wind speed (about 10 m s⁻¹, see Fig. a), advection near the ground level is prominent throughout the day. However, its effect weakens around noon (12:30 p.m.), due to an increase in upward turbulent flow driven by ground heating in the air. At night, reduced heat turbulence causes a build-up of CO₂ in the region. In July, the wind speed is lower (about 4 m s⁻¹, see Fig. c). In the morning, only limited advection occurs due to weak lateral wind and minimal heat turbulence. Therefore, a large concentration region appears in the dense commercial region (lower right in the sub-figure). At noon, due to increased heat turbulence and upward flow, concentration is reduced in the region. At night, mixed effects from suppressed upward turbulence and increased lateral advection (due to increased wind speed) produce a moderate concentration. 24 h surface concentration profiles for the two days discussed here can be found in Appendix .

The January cases are characterized by a relatively shallow boundary layer (411–827 m) and near-neutral to weakly unstable/stable conditions (|L|≈ 100–300 m). In contrast, the July cases exhibit substantially stronger instability during daytime (L=−1.9 to −7.4 m), leading to enhanced turbulent mixing and the development of a deep convective boundary layer reaching 1.7 km at midday. Following sunset, both periods transition to stable stratification (L>0), accompanied by a reduction in boundary-layer height, although the summer boundary layer remains considerably deeper than the winter boundary layer due to stronger daytime heating and residual turbulence. The ABL height and stability are reflected in the vertical CO₂ concentration profile. Figure  shows the concentration on a x=4000 m vertical cross-section over the domain at different times in January and July. The location of the vertical cross-section is shown in Fig. b. In general, during morning and night, the shallow boundary layer traps CO₂ below approximately ABL height, while during the day the ABL is thicker and the unstable ABL can mix CO₂ well. An exception occurs in the morning (08:30 a.m.) in January when advective transport lifts some plumes above the boundary layer, reducing near-surface accumulation.

Figure  also shows the impact of spatial patterns of anthropogenic emissions on CO₂ concentrations within the lower atmospheric layer. As shown in the geography tags, urban land-use types are interwoven from south to north (+Y direction in the figure), alternating commercial districts and green spaces: beginning with parkland (Hyde Park), followed by dense commercial areas (Marylebone), then another parkland and commercial areas (Regent's Park and Camden Town), mixed natural land and residential zones (Hampstead Heath and Highgate). Concentration hotspots correspond to densely populated areas with higher human activities. Throughout the day and in both months, the effect of turbulent dispersion (plumes) can be clearly observed in the figure.

The above pattern is revealed through LES that explicitly represents urban topography and spatially heterogeneous anthropogenic and biogenic fluxes at a 10 m resolution. The resulting complex CO₂ dispersion pattern highlights the importance of resolving urban-scale flow structures and emission heterogeneity when simulating urban carbon transport. On this basis, the model can help constrain emission patterns in an inversion problem. This conclusion echoes the argument in that vertical distribution of carbon fluxes greatly affects the transport process.

The grid resolution in the case study was selected as 10 m in the child domain. This is a balance between resolution, availability of static and dynamic driver data, and the purpose of the carbon dioxide case study in general. From the perspective of numerical discretisation, one wishes to have a high resolution to resolve turbulent transport at as small scales as possible. However, this can be prohibitive for computational cost. It also suggests a new requirement for static and dynamic drivers. For static driver, it requires finer topography data (buildings and terrain height), land use data, and vegetation data which could be scarce. Yet from the perspective of the user end, a study on carbon dioxide does not require as fine a resolution as those used in air pollution study where the individual exposure can benefit from a finer concentration map. Eventually, we selected the 10 m resolution in the main case study.

Nevertheless, it is worth exploring the turbulent transport process at different grid resolution. Here, we include two additional simulations of an ideal convective boundary layer (CBL) case at 10 and 5 m resolution and compare the results. The domain is 2 km by 2 km by 3.5 km, i.e. a smaller area surrounding the BT tower in the child domain in the main case study shown in Fig. . The 2 km domain is created using the same Ordnance Survey data for terrain height and buildings at 10 and 5 m resolution while the flow and emission conditions are controlled. The flow is driven by both the geostrophic wind, ug=5 m s⁻¹ and vg=5 m s⁻¹ and the bottom heating at 0.1 K m s⁻¹. These values are typical in the study area. The side boundaries for momentum and temperature are cyclic. The top boundary for momentum is the free slip, while for energy it is a constant gradient at 0.01 K m⁻¹. The emission of passive scalar CO₂ is set as a constant rate in time but spatially heterogeneous, whose pattern is determined by a typical winter day in the study area. The side boundaries for CO₂ are set as 0, while the top boundary is zero gradient Neumann. The initial flow field is geostrophic wind. The initial potential temperature is constant below 800 m and increasing at 0.01 K m⁻¹ until the top boundary to form an inversion layer. The initial concentration for CO₂ is set as 0. Both cases have been run for 48 h. Horizontally, the grid resolution is 10 and 5 m respectively. Vertically, below 400 m the grid resolution is 10 and 5 m respectively, while above 400 m the grid space increases at a ratio of 1.08 until reaches approximately 3.5 km.

Figure  shows the vertical flux profile at the end of simulation (48 h) for two grid sizes. Note that the temporal mean is used to compute the turbulent fluctuation and turbulent flux for subplot (a) and (e), while spatial (horizontal) mean is used for the purpose in the rest subplots. In Fig. b and f, resolved fluctuations are defined using horizontal averaging. As a result, the resolved flux may include contributions from persistent spatial heterogeneity arising from terrain-induced flow structures and spatially varying emission patterns. Therefore, the resolved flux fraction should be interpreted as the portion of transport resolved by the LES grid, rather than as a strict decomposition between turbulent and dispersive transport processes. This interpretation becomes more reliable above the roughness sublayer, where turbulence is increasingly horizontally homogeneous and the influence of dispersive motions associated with surface heterogeneity is diminished. In the current case, average building height or roughness height is about 37.7 m for 10 m grid resolution and 33.2 m for 5 m grid resolution.

Overall, the flux profiles obtained with the two grid resolutions show good agreement at intermediate and higher elevations, particularly above approximately 75 m (about two roughness heights). Below this level, however, notable differences emerge. As shown in Fig. a and e, the turbulent flux at 5 m resolution exhibits a larger standard deviation, reflecting the stronger influence of surface heterogeneity captured by the finer representation of urban morphology, including street canyons, building geometry, and spatially varying emission patterns.

The extent to which turbulent transport is resolved also differs within the roughness sublayer, as illustrated by the profiles of resolved and under-resolved turbulent fluxes in panels b–d and f–g. Below 75 m, the 5 m simulation resolves a greater fraction of the turbulent flux than the 10 m simulation. At higher elevations (> 100 m, or roughly three roughness heights), both resolutions resolve the vast majority of the turbulent transport, with resolved flux fractions exceeding 96.9 % and 98.5 % for the 10 and 5 m simulations, respectively. At 190 m, corresponding to the height of the BT Tower measurements, the resolved flux fractions are 98.9 % for both grid resolutions.

These results indicate that, under convective boundary-layer conditions, a 5 m grid resolution provides a more accurate representation of surface heterogeneity and turbulent transport within the lower urban canopy and roughness sublayer (up to approximately three roughness heights). Above this region, however, turbulent transport of passive scalars is already well resolved at a 10 m grid resolution. Additional details of the grid-sensitivity experiment and model configuration are provided in the Appendix D.

While the validation in previous subsections show promising agreement, it should be noted that the LES presented there represents only one realisation of turbulent flow field for a given day. Because atmospheric turbulence is chaotic, even under identical large-scale forcing and boundary conditions, small perturbations in the turbulent field can lead to different instantaneous evolutions. Therefore, a perfect match between a single LES realization and the measured half-hourly eddy-covariance fluxes should not be expected. To better quantify the variability associated with turbulent realization uncertainty, a 10-member ensemble simulation was performed for 16 October 2019.

The ensemble members were generated by using different random number seeds in the PALM synthetic turbulence generator, while keeping all other model settings unchanged. In this way, the ensemble isolates the sensitivity of the simulated EC flux to differences in the turbulent realization, rather than to changes in forcing, domain configuration, or model physics.

Figure  shows the half-hourly EC flux from all 10 ensemble members together with the ensemble mean. At the native 30 min resolution, both the model output and the measurements exhibit strong short-term fluctuations. Such variability is expected for turbulent fluxes and makes a direct point-by-point comparison between simulation and observation difficult, especially when the purpose is to assess whether the LES reproduces the overall temporal evolution and magnitude of the fluxes. For this reason, the comparison was additionally carried out using a 2 h moving average, applied consistently to both the LES output and the measurements. This smoothing reduces the influence of high-frequency fluctuations while retaining the diurnal structure of the flux evolution.

The smoothed ensemble results indicate that, despite substantial spread among individual members at half-hourly resolution, the realizations show a broadly consistent diurnal pattern after applying the 2 h moving average. The ensemble mean captures the main daytime increase in EC flux, with values rising from low nighttime and early-morning levels to a midday/afternoon maximum, followed by a decline toward evening. This suggests that the large-scale temporal development of the flux is robust across different turbulent realisations, whereas part of the short-term mismatch between a single simulation and the observations can be attributed to intrinsic flow variability.

Figure  compares the measured EC flux and the ensemble mean using the 2 h moving average. The black line represents the ensemble mean, and the shaded band indicates the uncertainty range of the ensemble mean given by the standard deviation. The red line shows the smoothed measurements. Overall, the ensemble mean reproduces the broad diurnal evolution seen in the observations, including the increase during the morning, elevated daytime fluxes, and the decrease in the evening. Quantitatively, the comparison yields an RMSE of 16.60 µmol m⁻² s⁻¹, and a Pearson correlation coefficient of 0.77 for the 2 h moving-mean series, indicating moderate-to-good agreement in the temporal evolution.

At the same time, differences remain between the modelled and observed fluxes, particularly during periods when the measured half-hourly values show sharp excursions. Some of these discrepancies are reduced after temporal smoothing, which supports the interpretation that part of the disagreement in the single-realisation comparison arises from the stochastic nature of turbulence rather than from a systematic model error alone. However, the remaining differences also indicate that ensemble variability does not explain all mismatches between LES and observations. The residual mismatches can be due to uncertainties in the emission profiles of the particular day.