The urban topography model, used in describing the bottom wall boundary of the LES domain, is prepared from a detailed 2 m resolution laser-scanned dataset of the Helsinki area . The data are conveniently available in raster map format and, in addition to the height distribution h(x,y), also include a distribution of land cover types LC(x,y)∈{1,2,…,NLC} where NLC is the number of land cover classes in the dataset. Both raster maps are shown in Fig. . Access to similar surface data source is a critical prerequisite for the presented methodology.

The horizontal domain for the LES analysis extends Lx=4096 m in the mean wind direction and Ly=2048 m in the crosswind direction and is spatially oriented such that x axis is coincident with the geostrophic wind direction of the case study. The EC measurement site at Hotel Torni is pivotally located in the LES domain to facilitate the determination of its footprint. However, the extracted raster map has to be first purposefully preprocessed to attain a form that complies with the LES analysis-specific requirements. The following manipulations were applied to obtain the final topography model depicted in Fig. .

The first half of the topography model (where x<Lx/2) is flattened for the purpose of generating physically realistic ABL conditions at the inlet through turbulence recycling technique (see below).

The meteorological conditions for the simulation are adopted from 9 September in 2012 when near-neutral ABL conditions were recorded with the EC measurements made on top of the Torni building. Lidar measurements from the chosen time frame yielded |ug|=10 m s-1 for the geostrophic wind in a southwesterly direction (α=218∘) and δ≈300 m for the boundary layer height. The Coriolis force (corresponding to latitude 60∘ N) is included to account for the turning of the flow within the boundary layer. The meteorological conditions are conveyed to the simulation by means of a precomputed ABL solution over flat surface, which in this context represents the surface of the Baltic Sea bordering Helsinki from the south. The boundary conditions for the velocity solution in this precursor simulation were set such that a fixed value is applied at the top and a no-slip condition at the bottom boundary of the domain while setting all the lateral boundaries as periodic.

For the precursor simulation the solver was run with an option that explicitly conserves the initial mass flow rate across the system, which was specified by initializing the velocity field with a constant value u|t=0=0.95 ug. This initialization value was determined by trial and error with the objective that the precursor solution would ultimately yield the desired geostrophic wind value at z=δ for the horizontally averaged velocity field u¯pre. The boundary layer growth was controlled by initializing the potential temperature field with a vertical profile θ0(z) that features a strong inversion layer at 300<z<350 m. This θ0(z) profile is defined according to the following lapse rates: ∂θ0∂z=0Kkm-10m≤z<300m50Kkm-1300m≤z<350m3Kkm-1350m≤z.

The precursor LES solution was computed on a grid that has the same resolution and vertical dimension as the principal urban LES grid, but its lateral dimensions are smaller by an integer division. Table  summarizes the respective grid characteristics. The study features a spatial resolution of 1 m, which is unprecedented at this scale. found the same resolution to be sufficient to capture the relevant turbulence physics within a real urban roughness sublayer. However, the effect of grid resolution on the final result is not investigated in this work. The influence of the structural details of the urban surface (balconies, chimneys, ventilation ducts, stationary cars, small-scale vegetation, etc.) not included in the urban topography model are taken into account by specifying a uniform roughness length z0=0.05 m on the bottom boundary surfaces .

The precursor simulation generates a highly resolved ABL solution that will be utilized, first, in a recursive manner to initialize the entire urban LES flow field with turbulence and, second, to aid construction of appropriate inlet boundary conditions though a technique labeled turbulence recycling, which is based on the method by with modifications by . The implementation of this boundary condition in PALM is presented in , but to aid discussion the description is also covered here with modified notation.

Denoting prognostic field variables by ψ=ψ(x,t) where ψ∈u,v,w,θ,e, the precursor solution is used to extract horizontally averaged vertical profiles ψ¯pre(z) for the turbulence recycling boundary condition. These stationary profiles are utilized at the inlet boundary in the urban simulation to conserve the global state of the mean flow, but in a manner that also incorporates physically sound turbulent fluctuations that occur in an ABL flow. This is achieved by specifying a recycling plane, that is, a yz plane at a windwise coordinate xrc, placed sufficiently far downstream from the inlet to prevent feedback of disturbances between the two planes. The fluctuations are obtained from the recycling plane through the following technique: ψ′|x=xrc=ψ|x=xrc-ψy|x=xrc, where the spatial mean (in the crosswind direction) ψy=ψy(z,t) at the recycling plane is computed as a time dependent vertical profile ψy|x=xrc=1Ny∑i=1Nyψ(xrc,yi,z,t) that carries a dependence on Ny. Finally, utilizing the precursor generated mean profiles, the turbulence recycling inlet boundary condition becomes ψ|x=xin=ψ¯pre+ψ′|x=xrc. In association with the turbulence recycling, the top boundary condition in the main simulation is specified as a slip-wall.

In this study, the recycling plane is situated, as shown in Fig. , in accordance with the precursor domain length such that (xrc-xin)=1024 m ≈ 3.4 δ and the same distance is allocated from the recycling plane to the edge of the urban topography to ensure that disturbances originating from the urban terrain are not conveyed back to the inlet. The chosen turbulent inlet arrangement generated no observable feedback effect on the incoming turbulence field.

The embedded LS particle model is employed such that, after the initial transients in the LES solution have subdued (after approximately 5 min of simulation), the release of particles is activated within the region outlined in Fig. . The release area extends 3030 m (≈41zm) in the upwind direction and 780 m (≈10.5zm) in both lateral directions from the Hotel Torni's EC site. The release area has been trimmed according to preliminary trial simulations to reduce the number of redundant particles in the domain.

Denoting the Lagrangian coordinate vector of the lth particle by Xl(t)=Xl(t),Yl(t),Zl(t), the release locations Xol=Xl|t=0 are uniformly distributed 2 m apart in the x and y directions while the vertical coordinate is set ΔZo=1 m above the topography: Zol=h(Xol,Yol)+ΔZo. The release height of one grid spacing at 1 m resolution is inferred to be a justifiably close to the surface to represent both the traffic emissions as well as the surface atmosphere exchanges. It also lowers the risk of accumulating a large number of particles within the first grid cell where the velocity values are dictated by the logarithmic wall function and the vertical advection of particles solely by the stochastic model due to . Thus, the underlying assumption is that, at 1 m resolution, the release height of 1 m above solid surfaces does not significantly influence the footprint distribution, which is evaluated at 2 m horizontal resolution.

The raw particle data for constructing footprints through LES–LS modeling in an arbitrarily heterogeneous environment are obtained by setting a target volume around the specified sensor location xm and recording which particles hit this target. Although this approach appears natural and straight-forward at first sight, a closer scrutiny reveals a number of problematic issues which arise with this setup, particularly when the flux sensor is mounted on a building (or close to one) instead of a tower. Purely from the perspective of particle data acquisition in the LES–LS simulation, setting a larger target volume would directly alleviate the computational effort required to gather a large enough dataset of particle hits, but this would clearly violate the formal premise that the footprint should be evaluated for the coordinate xm of the sensor. However, it turns out that the discrete setting of the LES–LS approach calls into question the relevance of seeking an urban footprint for a precise point near the surface of a solid structure.

Consider the problem of strictly concentrating on the exact location xm of the sensor. This effort becomes immediately futile as the spatial resolution with which the buildings are described in the topography model (which contains information on elevation changes only) cannot account for structural details that, in reality, influence the flow conditions at the precise location of the sensor. The same reasoning also extends to the LES flow analysis where the computational cost would become prohibitively expensive if the resolution would be set according to the ∼ 10-1 m scale of structural detail of building facades and rooftops in the hypothetical situation that such datasets were available. Therefore, it is important that the methodology for evaluating footprints in urban environments comes with a prerequisite that the resolution demands of the LES–LS model are purely dictated by the turbulent structures within the urban canopy and not the fine details of the sensor site. On these grounds, the method to collect particle data in the LES–LS simulation is based on setting a finite target volume around the sensor location xm without strictly dictating the appropriate size. This is done understanding the fact that the flow around the sensor mounting building strongly interacts with the flow, resulting in strong gradients in the mean velocity field in the vicinity of the sensor. This is bound to further complicate the subsequent postprocessing of the flux footprint because the eddy-covariance approach necessitates that the effect of the mean flow should be eliminated through the process of coordinate rotation , which is presented in the context of this study in Sect. . Clearly, the discrete LES–LS approach in an arbitrarily complex urban environment is endowed with pronounced uncertainties. For this reason, the postprocessing procedure has to encompass a capability to conduct spatial sensitivity analysis on the intermediate footprint results and, according to its outcome, selectively exploit the particle dataset in the final processing of the result.

Adopting this strategy reduces the level of rigor required at the setup stage of the LES–LS analysis and simplifies the guidelines for the particle acquisition: the target volume should be centered at xm and its dimensions chosen to represent the sensor site proportionately (vagueness intended) to the dimensions of the building geometry. In all cases, it is important to acknowledge that, as a rule of thumb, more than 107 particle hits need to be recorded at the target volume during the course of the LES–LS simulation to gather a large enough dataset for flexible postprocessing. In general, it pays off to specify an oversized target and gather a large dataset, accepting that it contains certain percentage of particle hits whose contribution will be discarded. In this study, using Lt≈4 m to represent the horizontal length scale of Hotel Torni's apex structure on which the sensor is mounted, the target for monitoring particle hits is specified as a box of volume Vt=Δxt×Δyt×Δzt=2Lt×5Lt×3Lt. The box is centered at the apex (which closely coincides with the actual sensor location xm) such that it extends 2.5Lt in the crosswind and upward directions, 1Lt in both streamwise directions and 0.5Lt downward, entering partly into the building structure. And, to reiterate, these dimensions were chosen under the guiding principle that the target box reasonably represents the sensor site and enables particle hits to be gathered at higher rate. Figure  provides an illustration of the size and placement of the target box in relation to the surrounding urban topography. The monitoring is performed at 0.5 s intervals, which corresponds to approximately eight LES time steps. This allows the same particle to be recorded multiple times at different locations within the target box. This feature is intentional and desirable because of the chosen postprocessing strategy.

The precursor simulation is run for 1.5 h physical time to develop the desired ABL profile. The initialization of the primary LES–LS computation with this precursor solution expectedly results in short-lived unphysical fluctuations around the urban topography, but after 3 min of simulation these overshoots have been advected away from the domain. The release of LS particles is initiated after 5 min of simulation, and from there on particles are released simultaneously in puffs at 10 s intervals such that two particles are seeded from each location at every instance. This translates into releasing approximately 2.36×106 particles every interval. The release schedule was determined by trial and error to best utilize the computational capacity of the supercomputer. Each particle is assigned a maximum lifetime Tmax⁡l=1200 s, which is long enough to guarantee that even the particles that are advected by the slowest ∼ 0.2ug velocity scales manage to travel over 2 km during this time frame. The total number of particles in the whole domain converged to approximately 68×106. Particles reaching any of the lateral boundaries or the top boundary are “absorbed”, that is, deleted and deallocated from the computer's memory while the wall boundary below functions as an ideally smooth reflective surface for the particles. The simulation was run for 3 h physical time during which ca. 19×106 particle hits were recorded at the target volume. The computation cost of this simulation is comparable to running 3–4 urban flow simulations with the objective of studying turbulence. In absolute terms, the simulation took ca. 10 days on the Cray XC40 supercomputer “Sisu” (CSC – IT Center for Science, Finland) with 2048 CPUs which amounts to ca. 5.3×105 CPU hours. The LS model constituted merely 20 % of the total CPU time of the LES–LS simulation, which is an appreciably moderate value considering the high number of particles handled by the solver.

The piecewise processing of the footprint carries an inherent difficulty that arises in situations where the mean flow displays strong gradients within the target volume. This is evidently present in the considered case study featuring an EC sensor mounted close to the top of a building. The difficulty relates to the evaluation of w¯i,j,k which is used in the footprint evaluation to extract the fluctuating velocity components about the mean value within its corresponding subvolume (as shown above). The initially implemented piecewise processing approach naturally involved utilizing LES to obtain the (45 min time-averaged) mean velocity distribution w¯ from within the target box volume and evaluating the spatial average w¯i,j,k for each subvolume Vi,j,k. However, with the Hotel Torni case study it became evident that this approach gave rise to a systematic negative bias in the footprints, which becomes immediately apparent in the far-field distributions. This outcome persisted until the discretization of the target volume was refined according to nx=8, ny=20, and nz=12 such that the subvolumes corresponded with the uniform 1 m resolution of the LES grid. This involved generating nx×ny×nz=1920 independent fi,j,k contributions via Eq. (), where the w¯i,j,k values were now directly obtained from their corresponding LES grid cells. Figure  illustrates the effect of target volume discretization by comparing crosswind integrated footprints f¯y at different levels of refinement. The comparison reveals how the negative bias in the far field and the reduction in near-field magnitude immediately emerge with coarser discretizations. It should also be noted that a targeted refinement in the z direction, while using coarser horizontal resolution in effort to generate thin subvolumes that approximate planes, does not remedy the situation because the mean flow gradients around the sensor site are significant in all directions. Such finite planes or one-cell-high grid layers are conventionally used as targets when Lagrangian-particle-based methods are utilized to evaluate footprints under heterogeneous conditions with undisturbed sensor sites e.g.,.

Unfortunately, at the required level of target volume discretization, the excessive number of individual fi,j,k contributions causes the postprocessing to become highly tedious. Since the proposed LES–LS methodology is founded on the premise that the size of the original target box around the sensor site can be chosen arbitrarily, the postprocessing effort must entail a procedure that enables the exclusion of those fi,j,k contributions that are deemed unfit for the final assembly. However, this selection operation becomes overly laborsome to manage when the number of subvolumes becomes large (viz. values exceeding 102) and particularly when the individual sectional footprints inadequately converge and thereby become uninformative when examined independently. For instance, in this case study, the required level of discretization gives rise to fi,j,k contributions that are generated from ca. 104 particle entries, which is a decidedly insufficient amount even for generating informative approximations for the near-field distributions. For these reasons, it is deemed unacceptable that the evaluation of urban footprints solely relies on the established piecewise postprocessing method. In response, this paper introduces an augmented coordinate rotation technique, labeled far-field correction, which incorporates well into the proposed piecewise postprocessing strategy and brings significant savings in the associated data manipulation efforts. This alternative technique allows much coarser target volume discretization to be employed in the assembly of the final footprint without unacceptably compromising the result.

The method has a prerequisite that the deficiently obtained footprint (for instance, obtained via insufficient target box discretization) must exhibit a properly leveled off far field, because the approach fundamentally relies on the following simple assertion: if the footprint distribution plateaus in the far field, this asymptote can be amended to become the zero reference level, which deviates from the “correct” asymptote by a negligibly small offset. Accepting this assertion and the associated approximation paves the way for a corrective coordinate rotation scheme which can be laid out by first classifying the data contributing to the far-field footprint via subsets ri,j,k⊂si,j,k which are defined as the sets of particle entries whose Xo fall into the outermost portion of the domain ri,j,k=si,j,k|0≤Xol-Xomin⁡<β100xm-Xomin⁡. Here Xomin⁡=min⁡l∈{1,…,Nrp}(Xol) is the farthest upstream coordinate where particles are seeded (thus, farthest away from xm) and β specifies the remotest percentage of the footprint across which the mean value of f¯y no longer changes, that is, ∂f¯y∂x≈0 when averaging over the length of the far field. (The β value is case-specific, but a typical range is expected to fall between 10 and 20.) With the help of the far-field datasets ri,j,k, the fluctuating vertical velocity component, used in Eq. () and previously defined by Eq. (), can now be evaluated as W′tl=Wtl-w¯i,j,k*, where w¯i,j,k*=ci,j,kw¯i,j,k defines the far-field-corrected mean vertical velocity, which is obtained by scaling the initially obtained value by a coefficient ci,j,k to satisfy the criterion that the particle entries in each ri,j,k do not contribute to the corresponding fi,j,k. This becomes a simple one-dimensional optimization problem in which the objective is to minimize J=∫Ωβfi,j,kdx, where Ωβ represents the far-field domain, by the means of controlling ci,j,k. Thus, this technique bears resemblance to a planar fit method . Because the control variable here is a single scalar, a rudimentary implementation of an iterative gradient decent search algorithm suffices (see, for instance, ).

Table  displays selected diagnostic data obtained from an application of this far-field correction technique to the Hotel Torni footprint case study. The data indicate that, when the mean vertical velocity values are initially obtained from the LES solution, the ci,j,k scaling coefficients concentrate near the mean value of 0.9. The range of individual values naturally depends on the magnitude of the starting value w¯i,j,k in Eq. () which, in turn, depends on the chosen discretization of the target volume. But it is important to emphasize that, although the far-field correction method is guaranteed to yield a physically justifiable asymptotic behavior for the footprint, the combined effect of the correction method and the target volume discretization on the final footprint result cannot be inferred from Table .

This realignment of the coordinate rotation plane within a larger subvolume, when examined in contrast with the reference technique where the coordinate rotation is performed at full LES resolution, alters how some of the individual particles contribute to the footprint. However, this discrepancy gives rise to an error that is distributed throughout the footprint domain. Therefore, the validity of the far-field correction approach hinges upon the magnitude of this distributed error and its sensitivity to the target box discretization. The sensitivity can be established by carrying out the selective assembly of the footprint result for different levels of target box discretizations.

Since its conception it has been clear that the piecewise postprocessing approach must be endowed with the capacity to incorporate a sensitivity analysis phase into the final assembly of the footprint result. One of the driving motivators for developing the piecewise approach arose from the need to reduce the computational cost of collecting a large number of particle hits by an arbitrarily sized target volume around xm. However, the reduction can only be achieved by the piecewise postprocessing approach if the sectional footprint results are allowed to be inspected and combined in a partially converged state. This is an important stipulation without which the proposed postprocessing strategy fails to offer considerable computational savings.

Thus, the process of selectively assembling the final footprint result begins by first defining an inadequately converged initial footprint, which represents the desired preform at xm. This reference footprint, labeled fref, should be constructed from at least 106 particle entries to facilitate a sufficiently informative evaluation of sensitivities. The selection process proceeds by iteratively introducing partial contributions f(l) that are independent from fref and evaluating the sensitivity of the footprint distribution with respect to the selection of target box indices in K (see Eq. ). The objective is to obtain a sufficiently converged footprint while minimizing the discrepancy between the constituent fi,j,k included in the final result. Thus, the selection process is quantitatively guided by the evaluation of “deltas” between fref and f(l), constructed from a partial set K(l) of target box indices (i.e., Δf(l)=fref-f(l)) and utilizing a norm over a subdomain Ω⋆⊂Ω encompassing only the near field (a fraction of the total LES footprint domain nearest to xm) as a measure for the associated discrepancy. In this connection, it has been found most effective to define the extent of Ω⋆ such that the integral over the near-field domain constitutes approximately half of the total integral of the footprint: ∫Ω⋆fdx≈12∫Ωfdx. The near-field norm is computed as ||Δf(l)||2,Ω⋆=∫Ω⋆Δf(l)(xf)2dxf1/2, utilizing identically normalized footprints for this evaluation. In this study the footprints are normalized to yield ∫Ωfdx=1. The exclusion of the outer portion of the footprint domain allows the relevant deviations in the near field to be reflected in ||Δf(l)||2,Ω⋆ while avoiding the contamination due to poorly defined “deltas” in the weakly converged outer region. In this study, the nearest 30 % of the total length of the LES footprint domain is used to represent the near field as this yields for the normalized reference footprint ∫Ω⋆frefdx=0.51. The search for the fitting contributions entails an iterative procedure, which is described herein for the case study utilizing target volume discretization nx×ny×nz=3×5×3. The relevant intermediate results and ||Δf(l)||2,Ω⋆ values are depicted in Fig. .

The process begins by setting at the 0th iteration K(0)={im,jm,km}⊂K, where the indices correspond to the subvolume containing xm. The obtained footprint f(0)=fim,jm,km is composed of ca. 4×105 particle entries, which does not meet the desired level of convergence to act as fref. Thus, through a qualitative inspection, the original set is augmented K(1)=K(0)+{im,jm±1,km} to yield f(1), which is chosen as the reference footprint.

The iterative process continues such that new candidate contributions f(l) are introduced incrementally in a radially outward progressing manner. This process is demonstrated in Fig.  where intermediate entries f(2)-f(6) introduce differently combined additions in y, x and z directions. For the sake of brevity, the example contributions combine a relatively large number of fi,j,k entries. The decision to include a candidate contribution in the final assembly is done according to a criteria ||Δf(l)||2,Ω⋆≤||Δf||max, where the maximum allowable discrepancy ||Δf||max must be determined according to the case-specific requirements. In this case study, the threshold was set to include f(3) such that ||Δf||max=||Δf(3)||2,Ω⋆. Naturally this threshold level can be varied to generate alternative footprint assemblies (with different levels of convergence), which allow, in the context of the considered footprint applications, the impact and uncertainty associated with these choices to be transparently monitored.

The obtained final result, which combines the earlier accepted additions, features 20/45 of all subvolume contributions. The obtained footprint also exhibits adequate convergence in the far field, having been constructed from ca. 8×106 particle entries. Subsequently, the lowest vertical (k=1) plane and the farthest (i=3) plane were completely excluded from K in the final assembly. This outcome indicates that the contributions with the largest deviations arise from Vi,j,k that are either in contact with the tower structure or in its wake region. Therefore, this suggests that it is not advantageous to set up Vt such that the building structure cuts into the volume.

As long as the individual subsets contain a sufficient number of particle data entries (>105), as is required by the far-field correction approach, it is beneficial to discretize the target volume as finely as possible (by increasing nx, ny, and nz) as it enables a more flexible and fine-tuned assembly and permits a more accurate coordinate rotation treatment. Depending on the total number of particles gathered during the simulation and the size of the target box, the maximum number of admissible subvolumes is expected to be ∼ 102. At this scale, when the postprocessing techniques are implemented with appropriate automations, the labor cost is not significantly affected by the total number of subvolumes. However, when the standard coordinate rotation is applied and the target volume discretization is carried out in accordance with the LES grid resolution, the total number of subvolumes readily exceeds 103 (as in this example study nx×ny×nz=1920) the selective assembly phase becomes prohibitively laborsome. But, given sufficient computational capacity, the far-field correction approach can be exploited to perform the selective assembly process to provide a description for an effective target volume Vt,eff=∑i,j,k∈KVi,j,k, which can subsequently be reassembled from the finely resolved Vi,j,k contributions. Such a result is depicted in Fig.  together with two footprints that are obtained through an identically guided selection process utilizing far-field correction (ffc) with different combinations of nx, ny, and nz. The comparison reveals that the differences between the three results are remarkably insignificant, indicating, first, that a significant part of the error contributions, introduced by the far-field correction method, have a compensating effect and, second, that the obtained footprint is not highly sensitive to the sensor placement despite the variable flow conditions around the sensor. This demonstrates the utility and robustness of the selective piecewise postprocessing approach. From here on the presented results correspond to the nx=3, ny=5, nz=3 target volume discretization level.

Taking into account the far-field correction procedure, the postprocessing procedure for evaluating a footprint from a LES–LS obtained dataset can be described in the following steps.

Split the original dataset S into nx×ny×nz number of subsets labeled si,j,k according to a Cartesian division of the target volume Vt.

It is noteworthy that in step 2 for the approximate footprint evaluation and in step 4a for the initialization of the optimization loop, the values for the mean vertical velocities w¯i,j,k do not have to be accurate. Therefore, the use of vertical velocity data from LES can be omitted altogether, which simplifies the case setup and data handling considerably. The approximate values can be obtained more simply, for instance, by evaluating the mean of incident vertical velocity value from particle data in each si,j,k.