Mean winds through an inhomogeneous urban canopy

Mean winds through an inhomogeneous urban canopy O. Coceal∗ and S. E. Belcher Department of Meteorology, University of Reading, P.O. Box 243, Reading,...
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Mean winds through an inhomogeneous urban canopy O. Coceal∗ and S. E. Belcher Department of Meteorology, University of Reading, P.O. Box 243, Reading, RG6 6BB, U.K. Abstract. The mean flow within inhomogeneous urban areas is investigated using an urban canopy model. The urban canopy model provides a conceptual and computational tool for representing urban areas in a way suitable for parameterisation within numerical weather prediction and urban air quality models. Average aerodynamic properties of groups of buildings on a neighbourhood scale can be obtained in terms of the geometry and layout of the buildings. These canopy parameters then determine the spatially averaged mean wind speeds within the canopy as a whole. Using morphological data for real cities, computations are performed for representative sections of cities. Simulations are performed to study transitions between different urban neighbourhoods, such as residential areas and city centres. Such transitions are accompanied by changes in mean building density and building height. These are considered first in isolation, then in combination, and the generic effects of each type of change are identified. The simulation of winds through a selection of downtown Los Angeles is considered as an example. An increase in canopy density is usually associated with a decrease in the mean wind speed. The largest difference between mean winds in canopies of different densities occurs near ground level. Winds generally decrease upon encountering a taller canopy of the same density, but this effect may be reversed very near the ground, with possible speed-ups if the canopy is especially tall. In the vicinity of a transition there is an overshoot in the mean wind speed in the bottom part of the canopy. Mechanisms for these effects are discussed. Keywords: Atmospheric boundary layer, Urban Meteorology, Urban canopy model

1. Introduction An ongoing challenge in the modelling of flow and dispersion in urban areas, as well as in the design and interpretation of field experiments, is the heterogeneity of urban areas. Towns and cities consist of large buildings with random shapes, sizes and distributions. Further complications include changes in mean building density and building height between different parts of a city, such as residential compared to commercial areas. Yet, many applications such as the representation of urban areas in numerical weather prediction and urban air quality models, require that complex urban effects be parameterised in a simple, operationally viable way. ∗

E-mail: [email protected] c 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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A significant simplification results if one can assume some measure of statistical homogeneity in the underlying urban morphology. This is usually possible over a spatial scale of order 1 km, known as the neighbourhood scale (Britter and Hanna, 2003). One may then consider spatial averages of flow quantities over a suitable averaging area. The urban area is represented aerodynamically as a porous medium. Each building exerts a resistive force, or drag, on the local air flow, whose effect on the spatially-averaged wind is represented as a body force spread smoothly through the urban canopy region. Such an approach originates from modelling of flow over plant canopies (see the excellent review by Finnigan, 2000). Macdonald (2000) applied the canopy approach to extensive regular arrays of urban-like cubical obstacles, and showed that the predicted exponential velocity profile matched well the spatially averaged velocity profiles computed from measurements. Simple ‘distributed drag’ models of urban areas have also been proposed in the literature to parameterise the effects of urban areas within mesoscale meteorological models (Brown, 2000; Martilli et al., 2002). Recently, Belcher et al. (2003) developed a quasi-linear analytical canopy model for the adjustment of the mean flow of a turbulent boundary layer to a canopy of large roughness elements. Coceal and Belcher (2004) further developed an urban canopy model which solves numerically the fully nonlinear dynamical equations, based on parameterisations suitable for large-scale urban roughness elements. They show that the urban canopy model compares well with wind tunnel measurements of the mean wind profile through a homogeneous canopy of roughness elements and with measurements of the effective roughness length of such a canopy. They also show that the urban canopy model predicts the deceleration of the mean wind associated with the adjustment of a rural boundary layer to a canopy of cubical roughness elements. These comparisons give confidence that the canopy approach can be extended from fine-scale vegetation canopies to canopies of large roughness elements that characterise urban areas. If the building characteristics over a given urban area are uniformly distributed, then we may speak of the overall urban area as a homogeneous urban canopy. The spatially-averaged wind within the canopy can then be related to average morphological parameters within the area as a whole. If, however, there is a variation in one or more of the morphological characteristics (such as local average building density or building height), then the canopy is inhomogeneous. A whole city is therefore an inhomogeneous canopy; it could be composed of many smaller patches, each being reasonably homogeneous. Coceal and Belcher (2004) considered idealised canopies which were spatially homogeneous. They did not

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consider changes of canopy characteristics with fetch. Here we apply the urban canopy model to realistic cities, by making use of actual urban morphological data, and by explicitly taking into account variations in morphological characteristics from one neighbourhood to another. Hence, we model flow through an inhomogeneous urban canopy. The following questions are specifically addressed in the present paper: − What are the relevant urban canopy parameters, and how can they be computed from the building morphology? − How does the mean wind within an urban canopy depend on these parameters? − How does the mean wind change from an urban patch to another one with different canopy parameters? These questions are here investigated under adiabatic conditions and strong winds. The effect of buoyancy, which may be important under weak wind conditions, is not considered. The paper is organised as follows. In section 2, the urban canopy model is described, and relevant canopy parameters are identified. Typical values of these canopy parameters are computed from morphological data for some North American cities. Section 3 investigates the effects of large-scale inhomogeneity, namely changes in building density and in building height, which arise between different sections of cities. In section 4, a simulation of mean winds through an inhomogeneous section of a real city, namely Los Angeles, is described. Finally, section 5 sums up the conclusions from these studies.

2. Canopy parameters derived from urban morphological data This section reviews the formulation of the urban canopy model and some pertinent results from Coceal and Belcher (2004). The reader is referred to that paper for further details about the urban canopy model, including its numerical implementation. In subsection 2.3, some relevant canopy parameters are then computed for real cities. 2.1. Canopy model formulation In the canopy formulation, the governing equations are formally averaged, both over time and over a volume (Raupach and Shaw, 1982; Finnigan, 2000). The averaging volume is taken to be very thin in

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the vertical, and large enough in the horizontal to include a number of canopy elements, but not so large as to lose any spatial variation in the density of canopy elements. The horizontal averaging area has dimensions LA × LA , where LA is an averaging lengthscale. The factors constraining LA will be discussed in Section 2.2 below. Under the two operations of time and space averages, prognostic variables then have three components, which for the streamwise velocity, u, are u =U +u ˜ + u0 . (1) Here U = hui is the time and space averaged velocity, referred to here as the mean velocity, u ˜ = u − U is the spatial variation of the time mean flow around individual roughness elements and u 0 = u − U − u ˜ is the turbulent fluctuation, and overbar denotes time average and angle brackets denote spatial average. The aim is to calculate the mean wind vector, U i (x, y, z), which is obtained by solving the time and space averaged momentum equations, which following Raupach and Shaw (1982) and Finnigan (2000) are ∂ ∂ DUi 1 ∂P + =− hu0i u0j i − h˜ ui u ˜ j i − Di . Dt ρ ∂xi ∂xj ∂xj

(2)

The averaging procedures thus produce three new terms in the momentum equation. There is a spatially-averaged Reynolds stress, hu 0i u0j i, which represents spatially-averaged momentum transport due to turbulent velocity fluctuations. There is a dispersive stress, h˜ uiu ˜j i, due to momentum transport by the spatial deviations from the spatially-averaged wind. And finally, within the canopy volume, there is a smoothlyvarying canopy element drag, Di , which arises from spatially averaging the localised drag due to individual roughness elements. The source terms of Equation (2) need to be parameterised in a way suitable for urban areas . The dispersive stress is neglected, on the strength of experimental evidence that near the top of the canopy the dispersive stress is very small compared to the Reynolds stress (Finnigan, 1985; Cheng & Castro, 2002a). The Reynolds stress is represented using a mixing length profile lm which is the harmonic mean between a surface-controlled part κz and a constant part l c . 1 1 1 = + lm κz lc

(3)

The reasoning behind Equation (3) is that for small z turbulent eddies are blocked by the ground, whereas near the top of the canopy the turbulence is dominated by shear layer instability which gives a mixing length lc that is constant with height (Raupach et al., 1996). The value of lc depends on the canopy density.

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2.2. Canopy model parameters The canopy element drag Di , which arises in the spatially averaged momentum Equation (2) is a body force per unit volume acting on the spatially averaged flow and can be parameterised in the following way. Consider an array of N obstacles each with frontal area A f , height h and sectional drag coefficient (Macdonald, 2000) c d (z), distributed over a total averaging area At . The force acting at height z on each element is 21 ρU 2 (z)cd (z)Af dz/h, and the thin averaging volume at height z is given by (1 − β) At dz, where (1 − β) is the fractional volume occupied by air in the canopy. Hence, the total force per unit volume acting on the air at height z is P

1 cd (z) Af |U |Ui , ρDi = ρ 2 hAt (1 − β)

(4)

The roughness density, λf = Af /At , is the total frontal area per unit ground area, and hence expresses a measure of the packing density of the obstacles (e.g. Wooding et al., 1973). The canopy element drag can then be expressed as P

Di =

|U |Ui 1 cd (z)λf |U |Ui = , 2 h(1 − β) Lc

(5)

where the canopy drag length scale, L c , is defined by Lc =

2h (1 − β) . cd (z) λf

(6)

For urban areas it is appropriate to think of canopy elements, i.e. buildings that have a horizontal cross-section that is uniform with height. The canopy volume may then be defined as hA , where h is the plan Pt P area weighted average building height h ≡ hAp / Ap , and Ap is the plan area of an individual building. Then simple geometry shows that the volume fraction β occupied by the buildings is equal to the plan area density λp ,Pdefined as the total building plan area per unit ground area, λp = Ap /At . Hence, the canopy drag length scale, Lc , can be calculated from the morphological parameters λ f , λp and h together with the drag coefficient cd (z). These parameters also suffice to completely specify the turbulence closure employed in the urban canopy model. Note that the parameters λ f , λp , and λs (defined as the mean building height to street width aspect ratio) are defined in the same way as in Grimmond and Oke (1999). The parameter λ s is used in subsection 2.3 below. The drag coefficient cd (z) has a dependence on the shape of the obstacles. If different shapes of obstacles are present in an averaging

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volume then the cd (z) that appears above represents a mean value. Using wind tunnel data from Cheng and Castro (2002a), Coceal and Belcher (2004) computed the profile of c d (z) for flow over a cubical array. They recommended a height-averaged, mean value of the sectional drag coefficient, denoted by cd , of 2. Using this value of cd Equation (6) reduces to 1 − λp Lc ≈ h (7) λf and therefore Lc is here modelled as a lengthscale that is constant with height. The dynamical significance of Lc is two-fold. First Belcher et al. (2003) show, using scaling analysis of the momentum equation for canopy flow, that mean winds within the canopy depend on the parameter Lc /h. This parameter determines the penetration of winds into the canopy, and hence provides a dynamical measure of canopy density. Canopies with lower values of L c /h are denser. Secondly, using scaling arguments motivated by a linear analysis, they also argue that Lc represents a lengthscale for an incident wind profile to adjust to the canopy. They specifically show that the wind adjusts on a lengthscale of order x0 given by x0 ∼ Lc ln K, (8) where the factor ln K depends on upwind conditions, and varies between roughly 0.5 and 2 for typical urban settings. Coceal and Belcher (2004) show by numerical simulations that the scaling coefficient in Equation (8) is approximately equal to 3, so that x 0 is given by x0 = 3 Lc ln K,

(9)

This adjustment pertains to the wind profile within the canopy normalised by the wind speed at the top of the canopy U h . The wind speed Uh is itself evolving much more slowly, in adjustment to the local turbulent stress at the top of the canopy. One may view the significance of x0 as follows. Within the canopy, only buildings within a radius of order x0 affect local winds - this may be viewed as a dynamical definition of a neighbourhood. It also gives the fetch needed for the wind to be representative of the upstream urban morphology. Hence, information about x0 can guide the siting of measuring instruments in field campaigns, so that these measurements may be interpreted appropriately. The issue is more complicated in the case of instruments that measure point data, since there is a great deal of local small-scale inhomogeneity. What is needed then is an array of such instruments deployed over a wide enough area. Spatial averaging would then yield data that are truly representative of the urban neighbourhood.

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Note that the lengthscale LA over which spatial averaging is performed must be larger than the building spacing but less than the lengthscale x0 over which the flow is evolving. For real urban areas, with variations in building density and building height, L A must also be smaller than the spatial scale over which such variations occur. Hence, the model is applicable if there is a scale separation between typical building spacings and the scale of spatial inhomogeneities. The validity of this assumption for real urban areas is considered in section 4. 2.3. Computation of canopy parameters from urban data The values of the parameters Lc /h and x0 are here computed using values of frontal area density λf , plan area density λp , and average building height h compiled by Burian et al. (2002) for Los Angeles and Salt Lake City. Table I gives the morphological parameters given by Burian et al. and the values of Lc /h and x0 computed from these parameters using Equations (7) and (9). The value of ln K is taken to be 1 in these calculations. The computed data shows a consistent trend, with the downtown core of the cities having the smallest values of Lc /h. The value of Lc /h for these cities decreases monotonically with frontal area density λf . This makes sense, given that the reciprocal of this parameter represents the ‘density’ of the canopy. Note that in all cases Lc /h > 1, whereas generally Lc /h  1 for plant canopies. This means that urban canopies are ‘shallow’ whereas plant canopies are generally ‘deep’ (Belcher et al., 2003). The values of the adjustment lengthscale x 0 range from a few tens to a few hundreds of metres, except for inordinately large values of several kilometres in the first two categories. This is because these areas are exceptionally sparse, too sparse perhaps for them to be usefully thought of as comprising a canopy, since the individual wakes of the urban structures comprising them do not interact. A dimensionless parameter, the adjustment number N c , may be defined as being proportional to the number of rows the wind takes to adjust in an equivalent regular array of cuboids with the same average morphological parameters. The adjustment number N c is defined to be equal to Lc /w, where w is the separation of the cuboids. Using Equation (6) then gives # " 1 − λp 2 , (10) Nc = cd (z) λf /λs + λp where λs is the mean building height to street width aspect ratio. Values of Nc computed from the present urban data indicate that winds would adjust after about three rows of buildings in a uniform environment. This is true of nearly all the areas listed in Table I,

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except for the two exceptionally sparse categories (‘transportation’ and ‘other urban/built-up’). The land use category ‘transportation’ includes airports, roads, railways as well as areas devoted to communications and utilities. ‘Other urban/built-up’ includes urban land uses that are predominantly vegetation, such as parks or cemeteries, as well as predominantly built-up areas that do not fall into any other category. These estimates are well supported by the observations of Macdonald et al. (2000) who observed little change in vertical profiles of mean horizontal velocity within regular arrays of cubes after the second row. This is in contrast to the much longer fetches needed for equilibrium in the roughness sublayer above the canopy (Macdonald et al. 2000, Cheng and Castro 2002b).

3. Effects of large-scale inhomogeneity: changes in building density and building height Here the effects of changes in building density and building height on winds in an urban canopy are analysed by means of idealised numerical experiments and a simple analytical model. Two canonical cases are considered: first, two successive canopies of same height but different density, and secondly two canopies of same density but different height. 3.1. Relative equilibrium mean wind speeds To guide our thinking, we invoke a simple analytical model of equilibrium mean winds in a canopy following Cionco (1965) and more recently Macdonald (2000). Within the canopy there is a dynamical balance between the vertical stress gradient and drag force, namely ∂ 0 0 hu w i = Dx , ∂z

(11)

2 (∂U/∂z)2 and D = U 2 /L . For a deep homogeneous where hu0 w0 i = lm x c canopy, the mixing length is constant with height, l m = lc . Solution of Equation (11) shows that the equilibrium wind profile within such a deep homogeneous canopy is exponential (Cionco 1965)

U = Uh ea(z/h−1) ,

(12)

where Uh is the value of the wind speed at the top of the canopy, z = h. The quantity a is known as the attenuation coefficient and is given by a=

h (2lc2 Lc )1/3

,

(13)

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If one assumes that the mixing length is controlled by the vortices shed in the obstacle wakes, and therefore that l c ∝ h (Belcher et al. 2003), then Equation (13) gives a ∝ (h/Lc )1/3 .

(14)

Therefore, Lc /h gives a measure of the penetration of winds into the canopy, as mentioned before. Coceal and Belcher (2004) show that the urban canopy model gives velocity profiles which are significantly different from the simple exponential solutions above. This arises because urban canopies are generally not deep, since L c /h > 1. Nevertheless, the exponential model qualitatively captures several effects also predicted by the urban canopy model, as will be shown below. As a first approximation therefore, the exponential model gives a simple and convenient way in which to compare mean winds in different canopies, outside of any adjustment regions. Suppose that air flows from one canopy to a second canopy (see Figure 1) . How do the equilibrium wind speeds at a given height compare? Referring to Figure 1, the equilibrium mean wind profile in canopies 1 and 2 are given respectively by U1 (z) for z U2 (z) for z

= ≤ = ≤

Uh1 ea1 (z/h1 −1) , h1 Uh2 ea2 (z/h2 −1) , h2

(15) (16)

At a particular height z, the ratio of the equilibrium wind speed in canopy 2 to that in canopy 1 is then U2 (z) Uh2 = e(a2 /h2 −a1 /h1 )z+(a1 −a2 ) , U1 (z) Uh1 for z ≤ min(h1 , h2 ). 



(17)

This ratio is significant because it gives a dimensionless measure of the relative wind speed between different regions, and hence quantifies the effect of any morphological changes between these regions on the wind in a generic way. Case I: (canopies of same height but different densities) Suppose h1 = h2 = h, so that Equation (17) simplifies to U2 (z) = U1 (z)



Uh2 Uh1



e(a1 −a2 )(1−z/h) .

(18)

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Suppose now that the wind blows from a less dense to a denser canopy, i.e. Lc2 /h2 < Lc1 /h1 , and hence from Equation (14) a1 < a2 . Since z/h ≤ 1, the exponential factor is less than unity. We suppose here that the canopies are not so dense that skimming flow predominates and mutual sheltering of buildings causes a reduction in roughness (Raupach et al., 1991). This then implies that the denser canopy is also rougher. In that case the denser canopy extracts more momentum from the air and hence the ratio of windspeeds at the respective canopy tops Uh2 /Uh1 is less than unity. This implies that U 2 (z)/U1 (z) < 1. Hence, at any height the equilibrium windspeed decreases in going from a less dense to a denser canopy, as would be expected. There is at least qualitative evidence to support this finding. The decrease of wind speed over an urban area has been documented by Bornstein and Johnson (1976), who reported measurements of mean wind speed at various distances through New York City. They found that in the daytime the wind speed decreased steadily as the wind flowed from rural terrain towards the city centre, no doubt reflecting the progressive rise in urban canopy density. The ratio U 2 (z)/U1 (z) is less than unity at the top of the canopy and decreases monotonically down the canopy, so that the difference between the two equilibrium profiles grows with distance down the canopy. Hence, ground-level winds are especially dependent on canopy density so that the largest effects of any changes are likely to be felt at or near pedestrian level. Rotach (1995, 1999) noted that winds within street canyons are very dependent on the direction of the approach flow relative to the canyon axis. In terms of the canopy picture, this arises because the frontal area density is much larger for cross-canyon flow. This leads to much reduced wind speeds in that case as observed by Rotach (1995, 1999). Similar arguments show that the equilibrium windspeed increases in going from a denser to a less dense canopy (L c2 /h2 > Lc1 /h1 ,). Case II: (canopies of same density but different heights) Here Lc2 /h2 = Lc1 /h1 , i.e. both canopies are equally dense. This implies that a1 = a2 = a, so that the equilibrium windspeed ratio Equation (17) now simplifies to Uh2 U2 (z) = ea(1/h2 −1/h1 )z , U1 (z) Uh1 for z ≤ min(h1 , h2 ). 



(19)

First, consider the transition to a taller canopy, so that h 2 > h1 . This implies (1/h2 − 1/h1 ) < 0, so that the exponential factor is less than unity. However, as z tends to zero the exponential term tends to unity

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and hence at the bottom of the canopy U 2 (z)/U1 (z) tends to Uh2 /Uh1 , the ratio of the wind speeds at the top of the respective canopies. The wind speed ratio Uh2 /Uh1 may be estimated as follows. In a typical undisturbed atmospheric boundary layer, wind speed increases logarithmically with height. Cheng and Castro (2002a) showed that, over an urban canopy, the log law holds for spatially averaged winds even in the roughness sublayer with a mean displacement height d and roughness length z0 for the canopy. Assuming that the log law holds all the way down to the top of the obstacles, we may write Uh =

h−d u∗ ln κ z0 



=

u∗ 1 − d/h ln κ z0 /h 



(20)

Now suppose wind blows from a canopy of height h 1 to a taller canopy of height h2 , but both with the same density. Since the parameters d/h and z0 /h depend on the density but not the height of a canopy (Raupach et al., 1991; Macdonald et al., 1998; Grimmond and Oke, 1999; Macdonald, 2000), they are the same for both canopies. Hence, Equation (20) implies that Uh ∝ u∗ . The surface stress u∗ depends on the vertically integrated drag exerted by the canopies on the airflow. Since both canopies have the same density, the taller canopy exerts a larger net drag. This implies that u ∗ is larger for flow over the taller canopy. Hence, according to Equation (20), U h2 /Uh1 > 1. The arguments in the last paragraph then indicate that, equilibrium winds near the ground are amplified in the transition to a taller canopy of the same density. Further up the canopy the exponential term eventually dominates so that the relative wind speed is reduced at higher levels. These considerations are confirmed by numerical simulations using the urban canopy model. Figure 2a shows the variation of horizontal velocity with fetch at different heights for a transition from a canopy of height h1 = 10 m to a taller canopy of height h2 = 40 m. Both canopies have the same building density, and hence the same value of Lc /h = 3.5. The run is initialised so that the mean wind speed at z = 800 m is 12ms−1 . It can be seen that at z = h1 /2 the wind speed U2 in the taller canopy is distinctly less, whereas at z = h 1 /4 the wind speeds in the two canopies are of comparable magnitude. Figure 3 shows the vertical profile of the wind speed ratio at equilibrium U2 (z)/U1 (z) for flow to a taller canopy and compares it with flow to a denser canopy, both computed using the urban canopy model. As the figure shows, the wind speed increases relative to its upwind value below the height of the lower canopy, and U 2 (z)/U1 (z) attains the value of ≈ 1.4 at ground level. Indeed there is a characteristic minimum in the vertical profile of the wind speed ratio between the two regions.

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This contrasts with a transition involving only an increase of canopy density, where the wind speed decreases monotonically. To complete the analysis, consider now the transition from a taller to a less tall canopy. This corresponds to the wind blowing in the reverse direction in Figure 1. The resulting mean velocity profiles essentially follows the same trend with fetch as in case II, but in reverse (Figure 2b). Note that the equilibrium wind speed ratio in the two canopies remains the same when the wind direction reverses. The most notable new feature is that now there is an undershoot near the surface at the transition. This undershoot gives rise to negative velocities, i.e. flow reversal in the spatially-averaged flow close to the ground, as shown by the plot at a quarter of the height of the less tall canopy. This effect is discussed in the next section. 3.2. Wind speed overshoot and undershoot at a transition When wind blows from a less tall to a taller section of town, Figure 2a indicates a sudden overshoot occurs in the wind speed. This overshoot typically extends over a distance of about 100 m from the transition, and decays over a fetch of order x0 . The overshoot may lead to a doubling of the local wind speed at pedestrian level. Simulations reveal that it occurs even between two regions of similar density but of different height (Figure 2a). Conversely, the effect disappears if the wind blows from a canopy of the same height. These observations therefore suggest that the overshoot is associated with a change in mean building height rather than a change in building density. What is the mechanism for this effect? Figure 4 shows the streamlines for flow from a less tall to a taller canopy, corresponding to the velocity profiles of Figure 2a. This plot clearly shows a descent of air at the transition in the lower part of the canopy. Faster air is being forced down towards the ground. This happens from the ground up to about two-thirds of the height of the taller canopy. Beyond that point the air is forced upwards. This situation is reminiscent of the flow pattern around a tall bluff body placed immediately downstream of a low one (Gandemer, 1976; Isyumov and Davenport, 1976; Britter and Hunt, 1979). Observations of flow around tall buildings reveal a descent of faster air around the tall obstacle and an ascent above a stagnation point, which is about two-thirds of the way up the height of the obstacle. The mechanism of the overshoot phenomenon that accompanies an increase in canopy height is essentially the same as that which occurs around a tall building. When fast air impinges upon the upper part of a tall building, it is slowed down and thus gives rise to an increase in pressure. Because of the vertical velocity gradient in the incident

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wind profile, the pressure increase is smaller further down the building. This gives rise to a vertical pressure gradient, which thus forces air down from below the stagnation point to the bottom of the building. The same effect takes place when an incident sheared wind profile encounters a region of increased drag offered by a tall canopy. Several wind tunnel studies have been devoted to the wind speedup around tall buildings. Penwarden and Wise (1974) and Hunt (1975) show that a building 80 m tall can lead to an amplification of mean ground level winds by a factor of 2.7. Isyumov and Davenport (1976) report that wind speeds around the sides of tall buildings four or more times higher than their surroundings may be twice the ambient wind speed at the same height. They also found that amplification factors do not exceed unity for building heights less than twice that of the surroundings. Based on wind tunnel tests by Lawson and Penwarden (1975) and Leene (1992), Bottema (1999) suggests that the maximum local wind speed near a building is of the order of the undisturbed wind speed at roof height, a suggestion consistent with the idea that air is forced downwards from the top of the buildings. All these studies relate to the maximum local windspeeds, e.g. around building corners. The horizontally averaged wind speed that is calculated in the present model at a particular height would be less than those quoted above, but still higher than if the buildings were the same height as their surroundings. Taking this fact into account, the observations seem to confirm qualitatively the main findings of the present simulations. Figure 5 shows the streamlines corresponding to velocity profiles of Figure 2b, which depict the flow from a tall to a low canopy. There an opposite effect is manifest, an ascent of air at low levels, causing a sudden deceleration. This undershoot is particularly pronounced near the ground, where it leads to flow reversal (Figure 2b). Figure 5 shows a recirculation region associated with this effect.

4. Flow through an inhomogeneous urban canopy The morphology of real cities is usually characterised by well-defined patches of different land use types. City centres in particular usually consist of at least three main urban types: residential areas, commercial areas and downtown core areas. These three areas normally have significantly different mean building heights and mean building densities. Here realistic simulations based on measured building data for Los Angeles are performed, in order to study the combined effect of changes in canopy density and canopy height through a real city centre. Measured values of building area densities λ p and λf and mean building heights

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h for Los Angeles are shown in Table II, together with computed values of Lc /h and Lc . Note that there is a four-fold increase in mean building height h between residential and commercial areas, and a seven-fold increase between residential and downtown areas. The large building heights account for the large values of frontal area density λ f for the commercial and downtown sections. Note, however, that the values of plan area density λp are virtually the same for all three areas in Los Angeles. In the calculation of Lc , (Equation (7)) the same value of cd (= 2) is taken here for all three areas, despite the difference in aspect ratio of the buildings representative of each area. Since λ p is constant across these areas, h . (21) Lc ∝ λf Because of the large increase in h the adjustment length L c actually increases in going from the residential to the commercial to the downtown areas. Note however, that Lc /h decreases. Burian et al. (2000) in a morphological analysis of Los Angeles, map the distribution of λf in a 3 km by 3 km area of Los Angeles. Based on this map, a simulation is presented here for a northerly wind direction through this area. A northerly transect is taken through the middle of the selected region. Broadly three different areas with mean values of λf corresponding approximately to residential, commercial and downtown sections can be identified (see Burian et al., 2000). Referring to the discussion in section 2, the land use data of Burian et al. (2000) shows that a scale separation between the averaging lengthscale and the lengthscale of land use variations indeed exists along the chosen transect. To simplify analysis and numerical work, the selected region is represented as a succession of three uniform canopies with a step transition between them, as depicted in Figure 6. In the absence of morphological information upstream of the residential section, it is assumed here that winds are effectively adjusted to the residential canopy. This is justified since the value of L c for the residential area is about 50 m, compared to the available fetch of at least 1500 m (30 L c ). In the actual simulation, the domain is divided up as follows: the residential area extends from the beginning of the domain a distance of 800 m, followed by the commercial area which extends for 400 m, and finally the downtown area extends for the remaining 1200 m. Figure 7 shows plots of the velocity profiles with fetch at vertical levels corresponding to the three different canopy heights h 1 , h2 and h3 and half those heights h1 /2, h2 /2 and h3 /2. Note that the top two curves correspond to air flowing through the tallest canopy (downtown core) only, while the next two curves correspond to flow through the

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commercial and downtown areas. Only the bottom two curves depict winds through all three canopies (at heights z = h 1 /2 and z = h1 /4). Focussing our attention on these lowest curves, a number of features can be identified. First there is a region of initial deceleration, followed by a sharp overshoot at the transition from the residential to the commercial area. This overshoot is repeated at the next transition from the commercial to the downtown area. Within the commercial and downtown sections the windspeed decelerates rapidly after the overshoot within an adjustment distance. The final equilibrium wind level in the downtown area is comparable to that in the residential area. The wind speed in the commercial area rapidly decreases to a value lower than that in the residential area and that in the commercial area. The adjustment distance after the overshoot in the downtown area is about 450 m, which is equal to 3Lc . In the commercial area, the adjustment distance is 3Lc ≈ 400 m. Now, there is only about 350 m after the overshoot to the next canopy. Within this available fetch one would expect the windspeed to decelerate before the transition to the downtown section within about 150-200 m. This leaves only about 150200 m for the flow to adjust to the commercial area. Clearly there is insufficient fetch for the wind within the commercial section to attain equilibrium. Hence, there is a sharp horizontal velocity gradient within the commercial section. The overshoot takes place as a result of a transition between two urban sections, and depends in particular on how sharp the transition is. The adjustment distance and equilibrium wind speed however, are independent of the transition. This is demonstrated by performing two further simulations with the length of the commercial portion changed to 200 m and 800 m respectively (not shown here). In either case the adjustment distance and equilibrium wind speed in the downtown section remain as before. Figure 8 shows the vertical wind profiles at three different fetches in the simulated section of Los Angeles. The first one is within the residential section, 200 m before the transition to the commercial section. The second is in the middle of the commercial section and the third is 200 m into the downtown section. It can be seen that the wind speeds at the respective canopy heights h3 , h2 and h1 satisfy Uh3 > Uh2 > Uh1 . The wind speed ratio U3 (z)/U1 (z) decreases down to the height of the residential canopy, then increases sharply. In particular, the ratio U3 (z)/U1 (z) increases to a little above 1 at ground level. Higher momentum air at the top of the taller canopy is mixed down the canopy with the net effect that the wind speed at ground level in the downtown core area is of comparable magnitude to that in the residential area.

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Hence, even though it is ‘denser’ downtown, there is still a marked wind speed-up at ground level, due to the mean building height being much taller than that in the residential area (approximately seven times taller).

5. Summary This paper has shown how mean winds in inhomogeneous urban areas may be modelled using an urban canopy approach. Canopy parameters have been defined in terms of urban morphological parameters, and have been computed using values for Los Angeles and Salt Lake City. A parameter Lc /h, controls the spatially averaged mean wind speed within a homogeneous canopy, whilst the parameter x 0 = 3 Lc lnK gives the fetch needed for the wind to adjust to such a canopy. Computed values show that winds adjust within a distance of order of tens to hundreds of metres, depending on the density of the urban area. The effects of canopy inhomogeneity on mean winds has been investigated by performing simulations involving a change of canopy density and a change of canopy height respectively. When an increase in canopy density occurs, a decrease in the mean wind speed results. Winds decelerate and readjust to the new canopy on a lengthscale x 0 determined by the new canopy characteristics. The largest difference between mean winds in canopies of different densities occurs near ground level. Winds generally decrease upon encountering a taller canopy of the same density, but this effect may be reversed very near the ground, with possible speed-ups. This happens because higher momentum air at the top of the taller canopy is mixed down the canopy, thus increasing the wind speed at lower levels. The ratio of the equilibrium wind speed in the tall canopy to that of the low canopy has a characteristic minimum at the height of the low canopy. This ratio increases down the canopy and at ground level is of the order of the ratio of the wind speeds at the top of the respective canopies. In the vicinity of the transition there is an overshoot in the wind speed in the bottom part of the canopy. This overshoot is associated with high winds being forced downwards owing to a vertical pressure gradient induced when the sheared wind profile encounters the region of enhanced drag offered by the tall canopy. This mechanism is essentially the same as the one responsible for wind speed up observed in the vicinity of a tall building, an effect that has been amply documented in the wind engineering literature. These observations offer indirect, qualitative support for the results obtained in the present canopy simulations. When wind blows from a tall to a low

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canopy reverse flow appears near the ground just after the transition. This is associated with an undershoot in the mean wind speed. The morphology of real cities is characterised by changes in both building height and in building density. Land use and building data for Los Angeles show that a distinct scale separation exists between the lengthscale over which land use varies and the typical building separations within a given land use type. This separation of scales allows the canopy approach to be applied. Realistic simulations performed for a section of Los Angeles including three built-up types (residential, commercial and downtown core) reveal the following features. Transitions from residential to commercial areas, and from commercial to downtown areas, are associated with an increase in both canopy density and canopy height. At each transition there is a region of initial deceleration of the mean wind, followed by a sharp overshoot. The mean wind speed then decelerates rapidly within an ajustment distance x 0 . The final equilibrium wind level at pedestrian level in the downtown area is higher than in the commercial area, and comparable to that in the residential area. This seemingly surprising result arises because the mean building height in the downtown area is very large, being seven times the mean building height in the residential area.

Acknowledgements O.C. gratefully acknowledges funding from UWERN and NERC, grant number DST/26/39. We thank Ian Harman and the anonymous reviewers for helpful suggestions. This work forms part of the UWERN Urban Meteorology Programme.

References Belcher, S. E., Jerram, N. and Hunt, J. C. R.: 2003, Adjustment of the atmospheric boundary layer to a canopy of roughness elements. J. Fluid Mech. 488, 369-398. Bornstein, R. D., and Johnson, D. S.: 1976, Urban-rural wind velocity diferences. Atmos. Env., 11, 597-604. Bottema, M.: 1999, Towards rules of thumb for wind comfort and air quality. Atmos. Env., 33, 4009-17. Britter, R. E., and Hanna, S. R.: 2003, Flow and dispersion in urban areas. Annu. Rev. Fluid Mech., 35, 469-96. Britter, R. E., and Hunt, J. C. R.: 1979, Velocity measurements and order of magnitude estimates of the flow between two buildings in a simulated atmospheric boundary layer. J. Indust. Aerodyn., 4, 165-82. Brown, M.: 2000, Urban parameterisations for mesoscale meteorological models. Mesoscale Atmospheric Dispersion, Ed. Z. Boybeyi, Wessex Press, 193-255.

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Burian, S., Brown, M., Linger, S.: 2002, Morphological analyses using 3D building databases: Los Angeles, California. LA-UR-02-0781. Los Alamos National Laboratory, 74pp. Cheng, H. and Castro, I. P.: 2002a, Near wall flow over urban-like roughness. Boundary-Layer Meteorol., 104 229-259. Cheng, H. and Castro, I. P.: 2002b, Near-wall flow development after a step change in surface roughness. Boundary-Layer Meteorol., 105 411-432. Cionco, R. M.: 1965, A mathematical model for air flow in a vegetative canopy. J. Appl. Meteorol., 4 517-522. Coceal, O. and Belcher, S. E.: 2004, A canopy model of mean winds through urban areas. Quart. J. Roy. Meteorol. Soc., 130 1349-1372. Finnigan, J. J.: 1985, Turbulent transport in flexible plant canopies. The ForestAtmosphere Interaction, ed. BA Hutchinson, BB Hicks, Reidel, Dordrecht, 44380. Finnigan, J. J.: 2000, Turbulence in plant canopies. Annu. Rev. Fluid Mech., 32, 519-572. Gandemer, J.: 1976, Wind environment around buildings; aerodynamic concepts. Proc. 4th Int. Conf. on Wind Effects on Buildings and Structures, London. 1975. Cambridge University Press, 423-432. Grimmond, C. S. B. and Oke, T. R.: 1999, Aerodynamic properties of urban areas derived from analysis of surface form. J. Appl. Meteorol., 38, 1262-1292. Hunt, J. C. R.: 1975, Fundamental studies of flow near buildings. Proc. Conf. on Wind Effects on Models and Systems in Architecture and Building, Sept. 1973. Medical and Technical Publishing, Lancaster, England, 101-109. Isyumov, N. and Davenport, A. G.: 1976, The ground level wind environment in built up areas. Proc. 4th Int. Conf. on Wind Effects on Buildings and Structures, London. 1975. Cambridge University Press, 403-422. Lawson, T., V. and Penwarden, A. D.: 1976, The effect of wind on people in the vicinity of buildings. Proc. 4th Int. Conf. on Wind Effects on Buildings and Structures, London. 1975. Cambridge University Press, 605-622. Leene, J. A.: 1992, Building wake effects in complex situations. J. Indust. Aerodyn., 41, 2277-2288. Macdonald, R. W.: 2000, Modelling the mean velocity profile in the urban canopy layer. Boundary-Layer Meteorol., 97 25-45. Macdonald, R. W., Griffiths, R. F. and Hall, D. J.: 1998, An improved method for the estimation of surface roughness of obstacle arrays. Atmos. Env., 32, No. 11, 1857-1864. Macdonald, R. W., Carter, S. and Slawson, P. R.: 2000, Measurements of mean velocity and turbulence statistics in simple obstacle arrays at 1:200 scale. Thermal Fluids Report 2000-01, University of Waterloo, 130pp. Martilli, A., Clappier, A. and Rotach, M. W.: 2002, An urban surface exchange parameterisation for mesoscale models. Boundary-Layer Meteorol., 104 261-304. Raupach, M. R., and Shaw, R. H.: 1982, Averaging procedures for flow within vegetation canopies. Boundary-Layer Meteorol., 22 79-90. Raupach, M. R., Antonia, R. A. and Rajagopalan, S.: 1991, Rough-wall turbulent boundary layers. Appl. Mech. Rev., 44 1-25. Raupach, M. R., Finnigan, J. J. and Brunet, Y.: 1996, Coherent eddies and turbulence in vegetation canopies: The mixing layer analogy. Boundary-Layer Meteorol., 78 351-382. Rotach, M. W.: 1995, Profiles of turbulence statistics in and above an urban street canyon. Atmos. Env., 29, 1473-86.

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Rotach, M. W.: 1999, On the influence of the urban roughness sublayer on turbulence and dispersion. Atmos. Env., 33, 4001-8.

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FIGURE CAPTIONS Figure 1: Flow transition from one canopy to another. The two canopies are characterised by the parameters h 1 , Lc1 and h2 , Lc2 . Figure 2: Variation of mean horizontal velocity with fetch at different heights in the vicinity of a transition between two canopies with the same value of Lc /h = 3.5 but different heights h1 = 10 m and h2 = 40 m. (a) Flow to the taller canopy, (b) flow from the taller canopy. Solid line: z = h1 /4; dotted line: z = h1 /2; dashed line: z = h1 ; long-dashed line: z = 2h1 ; dot-dashed: z = 4h1 ; plus symbols: z = 8h1 . Figure 3: Vertical profile of the wind speed ratio at equilibrium U2 (z)/U1 (z) for (i) flow to a denser canopy (solid line) and (ii) flow to a taller canopy (dot-dashed line). In (i), the canopy densities λ f = λp are 0.1 and 0.4 respectively. In (ii), the canopy heights are h and 4h respectively. The value of h is 10 m. Figure 4: Streamlines for flow through a less tall to a taller canopy of the same density, showing a descent of faster air near ground level just after the transition at x = 0. Canopy parameters are: h 1 = 10 m, h2 = 40 m, λf = 0.2. Figure 5: Streamlines for flow to a less tall canopy of the same density, showing an ascent of air an a recirculating region near the ground just after the transition at x = 0. Canopy parameters are: h1 = 10 m, h2 = 40 m, λf = 0.2. Figure 6: Representation of a selected section of Los Angeles in a 2-d simulation as a series of three canopies. The relevant dimensions are as follows: h1 = 6.4m, h2 = 24.5m, h3 = 45.0m and L1 = 800m, L2 = 400m, L3 = 1200m. Figure 7: Variation of mean horizontal wind velocity with fetch at different heights through the selected region of Los Angeles. Solid line: z = h1 /4; dotted line: z = h1 /2; dashed line: z = h1 ; long-dashed line: z = 2h1 ; dot-dashed: z = 4h1 ; plus symbols: z = 8h1 . See text for description. Figure 8: (a) Vertical profiles of mean horizontal velocity at three different fetches in the selected area of Los Angeles. Residential area (solid line); commercial area (dotted line), downtown core area (dot-dashed line). (b) Vertical profile of wind speed ratio U 3 (z)/U1 (z) between the downtown and residential sections. Respective canopy heights of the three areas are indicated by the dotted horizontal lines.

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Table I. Morphological data and computed values of Lc /h and x0 for Los Angeles and Salt Lake City. Morphological data obtained from Burian et al. (2002) Los Angeles Land Use Type Transportation Other Urban/Built-up Commercial/Services Downtown Core Residential Mixed Urban/Built-up Industrial Mixed Ind. and Comm.

Salt Lake City

λp

λf

h(m)

Lc /h

x0 (m)

λp

λf

h(m)

Lc /h

x0 (m)

0.03 0.06 0.28 0.29 0.30 0.34 0.38 0.47

0.01 0.04 0.27 0.42 0.19 0.24 0.10 0.14

7.9 7.4 24.5 45.0 6.4 12.0 6.3 7.4

97 23.5 2.7 1.7 3.7 2.8 6.2 2.8

2300 520 200 230 70 100 120 60

– 0.02 0.27 0.33 0.21 0.19 0.27 –

– 0.01 0.20 0.17 0.18 0.14 0.12 –

– 13.8 17.9 23.6 9.6 11.2 10.8 –

– 98 3.7 4.0 4.4 5.8 6.1 –

4000 200 280 130 190 200

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Table II. Computed values of Lc /h and Lc for three areas of Los Angeles. Morphological data obtained from Burian et al. (2002) Parameter λp λf h(m) Lc /h Lc (m)

Residential

Commercial

Downtown Core

0.30 0.19 6.4 3.7 23.5

0.28 0.27 24.5 2.7 65.5

0.29 0.42 45.0 1.7 76

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