Scaling the Urban Boundary Layer S. E. Belcher & O. Coceal Department of Meteorology, University of Reading

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1 Scaling the Urban Boundary Layer S. E. Belcher & O. Coceal Department of Meteorology, University of Reading This work is motivated by current issues in urban meteorology and urban air quality, such as How do urban areas affect weather? How is weather felt in urban areas? How is mixing between street air and boundary-layer air mediated? To address these questions requires a model for mixing and transport within urban areas. The dynamical effects of urban areas are usually represented in mesoscale models as a region of high roughness, with a roughness length of say 1m. But the roughness length is not a physical length scale. Instead it is defined by taking the logarithmic wind profile measured in the inertial sublayer of the boundary layer and extrapolating down to the level where the wind speed is zero. This level is then defined to be the roughness length (plus the displacement height). Hence the roughness length can only be even defined when there is a logarithmic velocity profile. In an urban area, the building density typically changes continuously from a suburban district, with low buildings, to a central district, with high buildings. In such a situation, where the boundary layer constantly evolves to the local building density, it is not obvious that there even exists a logarithmic region within the roughness sublayer. Hence it is not obvious that a local roughness length can be defined. In addition, there are applications of urban meteorology, such as modelling urban air quality, that require estimates of the transport and mixing actually within the roughness sublayer. For these reasons a more complete model is required for the dynamical effect of urban areas on the atmospheric boundary layer, and one that models the winds actually within the roughness sublayer and within the building canopy. Hence we are developing an urban canopy model. The model builds on developments in modelling flow in vegetation canopies (reviewed by Finnigan 2000; Raupach & Thom 1980). This approach has the advantages that: (i) no logarithmic layer is assumed, (ii) some measures of the winds are resolved within the urban canopy, and finally (iii) the model yields values for the effective roughness length of the surface in terms of measurable parameters of the building layout and density. Here the urban canopy model is briefly summarised: more complete details can be found in Belcher, Jerram & Hunt (2001). The model equations are then scaled to identify the key parameters that control the transport in urban canopies. Finally, the question of how a rural boundary layer adjusts to an urban canopy is addressed. The key theme to emerge is a new length scale, L c, which depends upon the aerodynamical resistance of the buildings and which characterises the length taken for winds within the canopy to adjust to the urban canopy.

2 2 Inertial sublayer u w Roughness sublayer u w u~ w~ u w u~ w~ + D Figure 1 Schematic showing how the different components of the stress act in different layers. 2. A canopy model for the roughness sublayer It is neither practical nor desirable to resolve the winds around every building within a large urban area. Even if we did have sufficicently large computers, from where would be get all the required data to define the geometry of the model and what would we do with all the output? Instead we aim to forecast average measures of the wind vector. The average is defined over both time, as in any boundary layer flow, and also over space. The spatially-averaged wind speed is then defined by where W ( x) { ( x + y ) L } δ ( z) u ( x x) u( x ) = W dx = 2 A 4πL exp is a weighting for the spatial average. Notice that A the spatial averaging introduces an averaging length L A. Prognostic variables, such as the wind components, are then separated into 3 components: u = U + u ~ + u. These three components are defined thus: U = u the Mean wind, the result of averaging a series of u~ = u U measurements over both space and time the spatial fluctuation, which is the variation of timeaveraged wind about the spatial mean = u U u ~ the turbulent wind, which fluctuates in both time and space u The aim here is to develop a model for the mean wind U(x,z).

3 3 2.1 Momentum equations in the urban canopy model Dynamical effects of the canopy on the mean flow are found by considering the momentum equations (cf. Finnigan 2000). Substitution of the triple velocity decomposition into the momentum equation and averaging over space and time yields an equation for the mean velocity components. When the flow is stationary, the streamwise momentum equation becomes DU P ρ = ρ u w ρ uw ~ ~ D. Dt x Notice how three new terms appear on the right hand side as a result of the two averaging procedures. They are ρ u w Spatial average of the turbulent stress (Reynolds stress): u ~ w ~ This term represents the transport of momentum by turbulent eddies and occurs in all turbulent flows ρ Dispersive stress: This term represents the transport of momentum by spatial fluctuations Has been quantified for flow over hills and ocean waves (Belcher & Hunt 1998) D Distributed aerodynamic drag: Represents the pressure and viscous forces exerted on roughness elements: It has been studied in flow through vegetation canopies (Finnigan 2000) Figure 1 shows a schematic of the role played by these new terms in different regions of the boundary layer. Firstly, the Reynolds stress is present throughout the boundary layer where there is turbulent transport of momentum. Lower down towards the surface, the dispersive stress is non-zero. This can be taken as a dynamical definition of the depth of the roughness sublayer. Finally, within the building canopy itself, the pressure difference across individual buildings gives rise to an aerodynamic drag on the air flow. Each of these terms needs to be parameterised to complete the model 2.2 Modelling the turbulent Reynolds stress The Reynolds stress is modelled here using the mixing length model, namely u w = l 2 ( du dz) 2, where l is the mixing length. Two models are used for the mixing length. Firstly, we use the standard mixing length model with l = κz. Secondly, we use a displaced mixing length model, with l = l m constant within the building canopy l = κ(z-d) above the canopy.

4 4 l l z l l (z-d) d Figure 2 Schematics showing eddies represented by (a) the standard mixing length model and (b) the displaced mixing length model. The standard mixing length model is the form used in standard boundary layer models and might be regarded as a limit for very sparse canopies. The displaced mixing length model represents the blocking of turbulent transport from the boundary layer into the building canopy. The mixing length within the canopy is constant to reflect either the wake turbulence shed off the buildings, or the mixing layer turbulence that develops at the top of a fine canopy, such as a vegetation canopy (see Finnigan 2000). It is of real interest to know when these models are expected to be valid. 2.3 Modelling the dispersive stress There is some evidence from wind tunnel studies that the dispersive stress is much smaller than the Reynolds stress, at least above the building canopy (Bohm & Finnigan 2000; Chen & Castro 2001). These studies, however, used uniform roughness elements. We might expect the dispersive stress to be larger when the roughness elements have non-uniform size. Then the vorticity associated with the mean wind shear a wrapped around the tallest buildings, presumably yielding large downward momentum transfer. However, in the light of the data that is currently available, we assume here that the dispersive stress can be neglected altogether. (And note that we compare the model only with data from experiments with uniform roughness elements!) 2.3 Modelling the distributed drag The distributed drag represents the effects of the aerodynamic drag of the buildings on the mean wind. Hence we represent the building canopy as a porous block with a resistance to the flow.

5 5 The distributed drag can be parameterised by considering the drag on an individual building, which is 1/2 ρ U 2 C D A f,, where C D is the drag coefficient, and A f is the frontal area of the building. Now assume that the drag coefficient is unaffected by the neighbouring buildings, so that the total drag on N buildings is N 1/2 ρ U 2 C D A f. Finally this drag must be distributed through the volume of air, which is h βa t, where h is the height of the buildings, β is the fraction of floor area covered by buildings and A t is the total floor area covered by the N buildings. The force per unit volume acting on the mean wind is then 2 2 N (1/ 2) ρu CD Af CDλ 2 U D = = U =. h β A βh L t Here L c is the canopy length scale: it is a length scale that characterises the distributed drag. For typical urban areas, L c ranges from a few tens to a few hundreds of metres. The formula for the distributed drag contains much information about the characteristics of the buildings C D characterizes the form drag depends on shape λ characterizes arrangement of roughness elements 1-β characterizes volume occupied by buildings c 3. Adjustment of a rural boundary layer to an urban canopy: x h x = 0 x Lc Figure 3 Adjustment of a rural boundary layer to an urban canopy. Three regions can be distringuished: (1)an impact region; (2) an adjustment region; (3) a roughness change region. Belcher, Jerram & Hunt (2001) performed a linear analysis of the adjustment of a rural boundary layer to an urban canopy. They show that the adjustment occurs over three distinct stages (see figure 3):

6 6 1. Impact region: -h > z > 0 U U x p x U 2 L c Incident flow decelerated almost inviscidly Large shear develops at top of canopy 2. Canopy adjustment region: 0 > z > L c U U x p x + τ U 2 Flow within canopy adjusts to a local balance between downwards transport of momentum and canopy drag. Flux of air upwards out of canopy L c 3. Roughness change region: z > L c τ U 2 L c Flow in canopy adjusted to a local balance. The canopy acts as a roughness to the flow above. Flow above the canopy develops an internal boundary layer as in flow over a roughness change. The fully-nonlinear equations developed above for the urban canopy were also solved numerically. The urban area is represented as a rectangular area (in 2-d) or cuboid (in 3-d) within which there is an additional drag force U 2 /L c in the momentum equation Numerical experiments were performed to demonstrate further that L c characterises the adjustment of the boundary layer within the urban canopy. The parameters for this series of runs were based on the array configuration used in the Davidson et al (1995, 1996) experiments (which are described in more detail below). Three different values of the drag coefficient, C D, were chosen and they corresponding to values of L c of 13.1m, 7.8m and 5.0m. Figure 4 shows a sequence of vertical profiles of the horizontal wind component for a range of values of x/l c. The three plots correspond to the three values of L c. The wind speed, U, is nondimensionalised by its value at the top of the canopy U h and vertical distance z by the canopy height h. These plots show that the flow adjusts by a distance of about 3L c downstream of the leading edge of the canopy. Hence, these fully-nonlinear simulations confirm the scaling obtained by Belcher, Jerram & Hunt (2001). Figure 5 shows the variation of the mean wind at the top of the canopy, U h, with distance downstream of the leading edge of the canopy. This figure shows that the wind speed at the canopy top itself continues to adjust beyond 3L c. Now, U h is proportional to the local friction velocity once the flow in the canopy has adjusted. Hence this variation of U h is associated with the adjustment of the boundary layer above the canopy, which by this stage is adjusting as in a classical roughness change (e.g. Kaimal & Finnigan 1990). Figure 5 shows that although the flow within the canopy is now adjusted to the canopy, it is an adjustment to the local friction turbulent stress at the top of the canopy, which is itself evolving slowly downwind.

7 7 4. Comparisons with the measurements The numerical and analytical models are now compared with the measurements of Davidson et al. (1995,1996), who performed two sets of experiments using the array shown in figure 6. The first was a field experiment with obstacle dimensions w b h = 2.2m 2.45m 2.3m, while the second was a wind tunnel boundary layer experiment with w = b = h = 0.12m. In the simulations the canopy parameters were chosen to correspond to the array dimensions used in the field experiment of Davidson et al. (1995). Corresponding to the prevailing experimental conditions, the simulations were carried out for neutral stratification. The numerical model here was run with the simple mixing length profile l = κz. Davidson et al. measured the time-mean streamwise velocities at half canopy height at several locations in the cross-stream direction. These were then averaged to obtain spatially-averaged winds. Figure 7 shows the resulting streamwise variation of the mean velocity, made dimensionless using the value far upstream of the array. Also shown for comparison are the results from the numerical simulation as well as the wind profile predicted on the basis of the 2-d quasi-linear theory (Belcher, Jerram and Hunt 2001). The following points may be noted from this comparison: 1. Both sets of data collapse onto each other when non-dimensionalised as in figure 7. This is because the relevant dimensionless parameters (h/l, L/L c and z 0 /h) are approximately the same in both experiments. 2. There is good agreement overall between the numerical simulations and the experimental data, assuming a drag coefficient of $C_d=5.$ With the obstacle dimensions and layout given above this corresponds to $L_c=7.8m.$ 3. The simulation agrees very well with linear theory within the canopy. However there are significant departures in the recovery region, where the linear theory underpredicts the mean velocity. This indicates that nonlinear processes are important in that region. 4. The results for 3-d simulations (not shown) are substantially the same as the 2d simulations, indicating that the flow is essentially 2-d. 5. The wind profile appears to be adjusted at x 10b, which corresponds to 3L c, as explained above.

8 8 5 Conclusions and Future Work We have outlined an urban canopy model for the space and time averaged winds in an urban area. Further details are available in Belcher, Jerram & Hunt (2001). Formal development of the model shows that the averaging procedure introduces three new terms into the momentum equations. They are (i) the turbulent Reynolds stress, which appears in all turbulent flows, (ii) a dispersive stress which has been set to zero here and (iii) a distributed drag that represents the aerodynamic drag of the buildings on the wind. We have used the model to investigate adjustment of a rural boundary layer to an urban canopy. The key finding is that L c characterises the distance necessary for the winds within the urban canopy to adjust to a local equilibrium. Numerical simulations indicate that 3L c is a good quantitative estimate. This work suggests further questions. Firstly, we have assumed following evidence in the literature that the dispersive stress is zero where the buildings are of uniform size. Where the buildings have variable height we expect the dispersive stress to be larger. Further measurements are needed to establish whether or not the dispersive stress can still be neglected in this case. Secondly, it is of great interest to know the pattern of the turbulence in urban areas. Is it similar to the turbulence generated in a collection of wakes? Or is it the mixing layer type turbulence seen to characterise turbulence in vegetation canopies? The answer to this question will be of value in developing models for turbulent mixing in urban areas. Acknowledgements This work forms part of the UWERN Urban Meteorology Programme. It is a pleasure to acknowledge the helpful conversations we have had with Julian Hunt and Neil Jerram during the course of this work. References Belcher, Jerram & Hunt (2001) Adjustment of the atmospheric boundary layer to a group of canopy elements. Submitted to Quart J. Roy Meteorol. Soc. Belcher & Hunt (1998) Turbulent flow over hills and waves. Ann. Rev. Fluid Mech. 30 Bohm, Finnigan & Raupach (2000) Dispersive fluxes and canopy flows: Just how important are they? Proc 24 th AMS conf. On Agricultural and Forest Met. pp106. Chen & Castro (2001) Submitted to Boundary-Layer Met. Finnigan, (2000) Turbulence in plant canopies. Ann. Rev. Fluid. Mech. 32 Kaimal & Finnigan (1990) Atmospheric boundary layer flows. Oxford. Raupach & Thom (1980) Turbulence in plant canopies. Ann. Rev. Fluid Mech. 13,

9 9 Figure 4 Vertical profiles of mean horizontal wind speed for increasing values of x/l c. Wind speed is made dimensionless using its value at the canopy top U h, and height is normalised by the canopy height h. (a) L c = 13.1m; (b) L c = 7.8m; (a) L c = 5.0m. Figure 5 Mean wind speed at top of canopy normalised by its upstream value, as a function of dimensionless distance x/l c, for the three different values of L c indicated in figure 4.

10 10 Figure 6. Plan of the geometry of the buildings in the experiments of Davidson et al (1995, 1996) Figure 7 Normalised mean wind speed measured at half canopy height as a function of dimensionless distance from leading edge of the array. The wind speed is normalised by the upstream value, U 0, and distance is normalised by obstacle breadth b.