Characterising the pollutant ventilation characteristics of street canyons using the tracer age and age spectrum

Atmospheric Environment 122 (2015) 611e621

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Characterising the pollutant ventilation characteristics of street canyons using the tracer age and age spectrum K.W. Lo a, K. Ngan a, b, * a b

School of Energy and Environment, City University of Hong Kong, Kowloon, Hong Kong Guy Carpenter Asia-Pacific Climate Impact Centre, City University of Hong Kong, Kowloon, Hong Kong

h i g h l i g h t s
The ventilation of street canyons is analysed using the response to a point source.
Ventilation timescales are defined from the distribution of transit times.
Time averages can yield a misleading picture of the ventilation.
An uneven, non-uniform building array improves ventilation.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 4 March 2015 Received in revised form 7 October 2015 Accepted 8 October 2015 Available online 22 October 2015

The age of air, which measures the time elapsed between the emission of a chemical constituent and its arrival at a receptor location, has many applications in urban air quality. Typically it has been estimated for special cases, e.g. the local mean age of air for a spatially homogeneous source. An alternative approach uses the response to a point source to determine the distribution of transit times or tracer ages connecting the source and receptor. The distribution (age spectrum) and first moment (mean tracer age) have proven to be useful diagnostics in stratospheric modelling because they can be related to observations and do not require a priori assumptions. The tracer age and age spectrum are applied to the pollutant ventilation of street canyons in this work. Using large-eddy simulations of flow over a single isolated canyon and an uneven, non-uniform canyon array, it is shown that the structure of the tracer age is dominated by the central canyon “vortex”; small variations in the building height have a significant influence on the structure of the tracer age and the pollutant ventilation. The age spectrum is broad, with a long exponential tail whose slope depends on the canyon geometry. The mean tracer age, which roughly characterises the ventilation strength, is much greater than the local mean age of air. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Age of air Green's function Large-eddy simulation Pollutant dispersion

1. Introduction Ventilation refers to the process by which polluted air is replaced by fresh air. It is an important topic in indoor air quality that has attracted much attention (Etheridge and Sandberg, 1996). Ventilation is also relevant to outdoor air quality as street-level air quality depends on the rate at which pollutants leave the urban canopy layer. It is widely recognized that urban air quality may suffer on account of poor ventilation; the ‘street-canyon effect’, in which the (streamwise) aspect ratio is large and the flow aloft

* Corresponding author. School of Energy and Environment, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong. E-mail address: [email protected] (K. Ngan). http://dx.doi.org/10.1016/j.atmosenv.2015.10.023 1352-2310/© 2015 Elsevier Ltd. All rights reserved.

decouples from the flow within the canyon (Britter and Hanna, 2003; Carruthers et al., 2012; Blocken et al., 2013), is the bestknown example of this. Concrete information on ventilation timescales is valuable for end-users and researchers. Various ventilation diagnostics have been developed and applied to numerical simulations of urban domains. Examples include the air and pollutant exchange rate (ACH and BCH respectively; Liu et al., 2005), normalised flow rate (e.g. Hang et al., 2009), and exchange velocity (Bentham and Britter, 2003). A detailed summary of outdoor ventilation diagnostics may be found in Ramponi et al. (2015). Most outdoor ventilation diagnostics have focused on the roof level. Thus all the diagnostics listed above are based on roof-level fluxes. The ACH and PCH are defined from upward fluxes of mass and pollutant. The normalised flow rate is based on the mean mass

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flux. The exchange velocity is defined using roof-level fluxes and characterises the velocity difference between the canopy and the overlying atmosphere. These diagnostics reflect a traditional view of urban flow and dispersion. In accord with much of the research in the area, which highlights spatial and temporal averages (Vardoulakis et al., 2003; Li et al., 2006; Blocken et al., 2013), the ACH and PCH tacitly assume that the flow is spatially homogeneous and that the exchange velocity neglects the time-dependent ventilation process. These diagnostics, while providing a useful picture of the ventilation, have several flaws. First, ventilation is not necessarily uniform in space even though the ventilation of the canopy as a whole ultimately depends on exchange processes at the roof level (when pollutants do not escape from the sides). Second, time-averaged diagnostics do not quantify the timescales over which ventilation occurs. The latter concern is particularly important if urban computational fluid dynamics models are to be coupled to time-dependent mesoscale winds (Schlunzen et al., 2011). These concerns have been addressed in a series of papers that apply concepts from indoor air quality. In building ventilation € berg, 1983; Etheridge and theory (Sandberg, 1981; Sandberg and Sjo Sandberg, 1996), the age of a pollutant blob measures the time elapsed since it entered the domain. The age concept is attractive because it is directly relevant to the ventilation (escape or dilution) of pollutants from the urban canopy; however, the calculation of ! the mean age is not necessarily trivial. For a pollutant field cð x Þ obeying the advectionediffusion equation

vc ! ! þ u $ V c ¼ kV2 c þ S; vt

(1)

the age of the pollutant at a given point in space will depend on the !! incompressible velocity field u ð x Þ, the diffusion due to the molecular (turbulent) diffusivity k, and the (spatially varying) source ! flux Sð x Þ. In order to simplify the analysis, it is sometimes assumed that the pollutant source is spatially homogeneous (Hang et al., 2009, 2013; Hang and Li, 2011). According to the ‘homogeneous emission method’, the local mean age is then given by

tl ¼

c S

(2)

where cðx; y; zÞ is the time-averaged concentration and S is the source or pollutant emission rate. Strictly speaking tl can be interpreted as a mean age only if the pollutant field is controlled by a spatially uniform source. If S is not uniform then the mean age also depends on the Lagrangian evolution of pollutants, i.e., on timescales for transport and mixing.1 Differences between a ‘turnover time’, defined by the ratio of the mean mass or concentration to the source flux, and the mean age have been described in the building ventilation literature (Etheridge and Sandberg, 1996). Similar diagnostics have been considered by other investigators, e.g. the average residence time (Kato, 1988; Bady et al., 2008). The general applicability of tl to urban pollution problems is unclear. In many problems of interest, the pollutants are released at street level rather than uniformly throughout the domain. For this reason studies applying the homogeneous emission method have focused on the so-called ‘inhalation effect’ (Lin et al., 2014), i.e. the

1 To see this, consider two systems with identical tl and c, one with a uniform source and the other with the source localised in a corner. In the latter case age depends on the size of the domain but in the former it is independent of space. Thus, in general, tl cannot be interpreted as a mean age when the source is not uniform.

complementary problem of fresh, unpolluted air penetrating the urban canopy. It has been shown that tl is a useful diagnostic for the ‘breathability’ of regular building arrays (Buccolieri et al., 2010). An alternative approach has been developed in atmospheric science, where there has been great interest in characterising the transport and mixing of chemical constituents (e.g. Shepherd, 2002). For chemically inert species, such as SF6, transport from the troposphere to the stratosphere is analogous to ventilation of the urban canopy. In stratospheric modelling, the age of stratospheric constituents has been calculated by exploiting the linearity of the advectionediffusion equation (Hall and Plumb, 1994; Waugh and Hall, 2002). The ‘tracer age’ refers to the age of air for prescribed tracer fluxes (see Sec. 2). As any solution can be expressed in terms of the associated Green's function (see Appendix 1 for details), the age of a pollutant follows from the Green's function or ‘age spectrum’. Although they may seem abstract and of limited practical relevance, the mean tracer age and age spectrum have been influential in atmospheric science. They are now standard tools for evaluating numerical models (Waugh and Hall, 2002). The great merit of this approach is that it is based entirely on the equations of motion; in the context of urban pollutant dispersion, this means that ventilation can be analysed without any a priori assumptions. A related benefit is that the mean tracer age and age spectrum can be directly compared to observations (Hall et al., 1999). Despite the successful applications to stratospheric modelling, the Green's function approach has received little if any attention in the urban pollution literature. However, it is closely related to building ventilation theory. The mean tracer age obeys an equation that it is formally identical to one for the mean age (Etheridge and Sandberg, 1996; Holzer and Hall, 2000). This equation has not been explicitly solved in applications of building ventilation theory to urban pollutant dispersion; instead a simplified equation yielding Eq. (2) has been analysed instead. In this paper we will investigate the applicability of the tracer age and age spectrum to numerical simulations of urban pollutant dispersion. The main objectives of this study are to (i) introduce new ventilation diagnostics based on the Green's function approach (Secs. 2 and 3) and (ii) use them to characterise the influence of the street-canyon geometry. The mean tracer age, standard deviation and age spectrum will be calculated from large-eddy simulations (Sec. 4). The diagnostics will be applied to an even (i.e. parallel) street canyon (Sec. 5) as well as the less familiar uneven, non-uniform street canyon array (Sec. 6), a configuration with streamwise and spanwise inhomogeneities (Gu et al., 2011). The advantages of the new diagnostics will be assessed in Sec. 7. 2. Tracer age and age spectrum In atmospheric science the age of air has been defined using Eq. (1), which governs the evolution of a passive tracer. More precisely, the age of air is calculated from the associated Green's function; since the Green's function can be used to reconstruct any solution, this means that the approach is essentially exact (though, as explained below, simplifying assumptions are made in practice). The main elements of the Green's function approach are summarized below. The presentation follows Holzer and Hall (2000) and Waugh and Hall (2002). Further details (and brief derivations) may be found in Appendix 1. A tracer parcel is composed of irreducible elements that, owing to transport and irreversible mixing, experience different Lagrangian histories (Hall and Plumb, 1994; Waugh and Hall, 2002). These Lagrangian histories are encapsulated in the Green's function, which is nothing more than the solution to the advectionediffusion equation for a delta-function source. Physically the

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Green's function describes the relationship between a source at a !0 ! point x and a receptor at another point x . For realistic problems the Green's function is obtained numerically, i.e., by solving

0 1 Z∞ 1 ! ! 4 Ka ð x Þ ¼ @ 4 ðt  ta Þ Zð x ; tjSÞdtA  3: sa

613

(7)

0

vc ! ! ! ! þ u $ V c ¼ kV2 c þ dð x  x 0 Þdðt  t0 Þ: vt

(3)

As is well known, the Green's function depends on the boundary conditions (e.g. Duffy, 2015). The original papers in the stratospheric literature considered prescribed concentrations at the ground (Hall and Plumb, 1994; Holzer, 1999; Waugh and Hall, 2002). Subsequently the analysis was generalised to prescribed tracer fluxes (Holzer and Hall, 2000). As pollutant concentrations are not prescribed on a boundary, the latter formulation is more appropriate for urban pollution problems. In order to avoid confusion, the term ‘tracer age’ is used in this case. Henceforth we focus on prescribed fluxes and the tracer age. In the absence of pollutant sources or diffusion, a tracer parcel maintains its integrity and there are a discrete number of transit ! ! times t  t0 connecting x 0 to x . More generally, there is a continuous distribution of transit times or tracer ages for a given pollutant source. The distribution of tracer ages or age spectrum is given by

Z ! Zð x ; tjSÞ ¼

! ! ! ! d x 0 hGið x ; tj x 0 ÞhSið x 0 Þ

D

! cð x ; tÞ

4. Methodology

;

(4)

! ! where hGið x ; tj x 0 Þ and 〈S〉 denote averages of the Green's function and source flux over the pollutant-release times. Z is normalised to R satisfy Zdt ¼ 1. The density r ¼ 1. It should be noted that the tracer age and age spectrum are inherently Lagrangian (cf. Eqs. (3) and (12)). A fully Lagrangian definition is required in order to measure the time elapsed since a pollutant blob was released.

3. Ventilation diagnostics The age spectrum Z summarises the ventilation characteristics of the domain. There is relatively good ventilation for a distribution skewed towards small values of t and relatively poor ventilation for one skewed towards large values of t. Advection dominates when there is a sharply peaked distribution, while diffusion or reentrainment play an important role when there is a broad distribution of Z or one with secondary peaks. By analogy with previous work on atmospheric modelling, ventilation timescales may be defined from the age spectrum. The ! mean tracer age at x is given by the first moment of Z:

! ta ð x Þ ¼

Z∞

! tZð x ; tjSÞdt:

(5)

0

ta represents the average time required for the tracer or pollutant ! parcel to travel from the source at x0 to a location x . Other diagnostics may be defined from higher-order moments, e.g. the standard deviation 0 ∞ 11 2 Z ! ! 2 sa ð x Þ ¼ @ ðt  ta Þ Zð x ; tjSÞdtA ;

These diagnostics characterise different aspects of ventilation. ta describes the ventilation strength while sa relates to the ventilation variability. Low values of ta imply good ventilation, e.g., pollutant rapidly flushed out of the system. High values of ta may be associated with poor ventilation, but this is not necessarily the case as there will be lag effects for points that are located far from the source. The importance of lag effects can be assessed with sa. Similar diagnostics, but based on the concentration rather than the tracer age, have been used in studies of scalar mixing (Ngan and Shepherd, 1997) and indoor ventilation (Kato, 1988). ta and sa are nonlocal ventilation timescales because the Green's function is determined by the Lagrangian evolution from the source to the receptor. The ventilation of a source region can be defined from a local ventilation timescale, such as (2), but this is strictly valid only under the assumption of spatial homogeneity. In general the ventilation of a source will be spatially dependent as the separation between source and receptor need not be infinitesimal.

(6)

4.1. Numerical model The numerical simulations were performed using the large€ter, 2001; eddy simulation (LES) model, PALM (Raasch and Schro Maronga et al., 2015). PALM has been used to investigate various aspects of street-canyon flow, e.g. predictability (Lo and Ngan, 2015) and KelvineHelmholtz instability (Letzel et al., 2008). PALM solves the non-hydrostatic Boussinesq equations for the implicitly filtered variables. The governing LES equations may be found in € ter (2001) and Maronga et al. (2015). The effects of Raasch and Schro sub-grid scale motion were parameterised using the Deardorff model (Deardorff, 1980). The computational domains are illustrated in Fig. 1. The even, uniform canyon (e.g. Fig. 1a; Liu and Barth, 2002) serves as the baseline for comparison. The uneven, non-uniform canyon array2 (Fig. 1b; Gu et al., 2011) has more complicated ventilation characteristics. Here each canyon is composed of a short building H1 and a tall building H2, the ordering of the buildings varying from row to row. The two domain types are not strictly analogous. Unblocked channels exist on either side of the spanwise edges of the buildings in Fig. 1b. Thus pollutant can escape without crossing the roof level. This is not true for the uniform canyon of Fig. 1a or the non-uniform canyons studied in Gu et al. (2011). The key parameters for the single canyon are summarized in Table 1. The streamwise, spanwise and vertical dimensions of the canyon are W, L, H. The aspect ratio is unity with H ¼ W ¼ 48 m. The length of the canyon is L ≡ 2W ¼ 96 m. The domain lies in {(x,y,z): (0W  x  8W, W  y  W, 0H  z  3H)}. The centre of the canyon in the xy plane is located at (x, y) ¼ (4W, 0). The parameters for the non-uniform canyon are summarized in Table 2. The domain lies in {(x, y, z): (0W  x  12W, 1.5W  y  1.5W, 0H  z  3H)}. There are five canyons, centred at x ¼ 2, 4, 6, 8 and 10W, which we shall occasionally refer to by number. Since the buildings extend from W  y  W only, there are unblocked channels for jyj > W.

0

and the kurtosis

2

Henceforth ‘non-uniform canyon’ for brevity.

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Standard choices were made for most of the turbulence and pollutant boundary conditions, e.g. periodic boundary conditions in the spanwise direction and Neumann boundary conditions at the upper boundary. Turbulence recycling (Lund et al., 1998; Kataoka and Mizuno, 2002; Gryschka et al., 2008) was applied in order to generate a turbulent inflow for non-periodic streamwise boundary conditions; thus, a Dirichlet condition at the inflow and a radiation condition at the outflow were chosen. (The streamwise boundary conditions for the pollutant were identical except that c ¼ 0 at the inflow.) Only the treatment of the bottom boundary and solid walls differed from standard practice: while Neumann boundary conditions were adopted for the pollutant, no-slip (supplemented with a profile parameterised by the roughness length, z0 ¼ 0.1 m) was €ter, 2001; Maronga et al., chosen for the velocity (Raasch and Schro 2015). 4.2. Procedure

Fig. 1. Computational domains for (a) single canyon with unit aspect ratio; (b) uneven, non-uniform canyon array with unblocked channels. In the latter case, which is based on Case 3 of Gu et al. (2011), the heights of the tall and short buildings, H1 and H2, differ by a parameter d. x, y and z label the streamwise, spanwise and vertical directions.

Table 1 Summary of key parameters for the single street canyon. D is the grid spacing, Lx, Ly, Lz are the domain dimensions, and W, L, H are the canyon dimensions.

D

Lx

Ly

Lz

W

L

H

1m

384 m

96 m

144 m

48 m

96 m

48 m

Table 2 As in Table 1, but for the uneven, non-uniform street canyons. Dx, Dy, Dz are the grid spacings in the streamwise, spanwise and vertical directions.

Dx

Dy

Dz

Lx

Ly

Lz

H

L

W

1m

1.5 m

1m

12W

3W

3W

48 m

96 m

48 m

The heights of the short and tall buildings are controlled by a parameter d:

H1 ≡H  d;

H2 ≡H þ d:

(8)

Five cases were considered (i) d ¼ 0; (ii) d ¼ H/12; (iii) d ¼ H/6; (iv) d ¼ H/4; (v) d ¼ H/2. The parameter choices followed previous numerical studies (Letzel et al., 2008; Gu et al., 2011) and standard guidelines for urban computational fluid dynamics (Franke et al., 2007). For the uniform canyon, the grid spacing was approximately 5 times smaller than the recommended value while the distance between the walls and the lateral boundaries exceeded the recommendation of twice the building width. As in Letzel et al. (2008), an isothermal atmosphere was forced by a streamwise pressure gradient, dP/dx ¼ 0.006 Pa m1. For the single canyon, the streamwise velocity U0 ~ 2.4 ms1 at z ¼ 2H. The roughness Reynolds number Re ≡ UtH/nT ~ 600, where the values of the friction velocity Ut and the horizontally-averaged turbulent viscosity nT correspond to a height just above the street canyon. The Schmidt number Sc ¼ 1, i.e., k ≡ nT.

The Green's function was calculated numerically from Eq. (3). The tracer flux was applied for one timestep after the flow had reached statistical equilibrium. Area sources lying along the bottom boundary were used; for the non-uniform street canyon, the source was restricted to the first canyon, D ¼ {(x, y, z):(1.5W  x  2.5W, 1.5W  y  1.5W, Z ¼ 0H)}. The simulations were carried out until the pollutant field outside the canyon was nearly steady, i.e. t ~ 5000 s. Coarse-grained Green's functions for the extended source regions (Holzer, 1999) were calculated instead of the ‘elemental’ Green's functions for a single point. For the single street canyon, ! ! ! ! hGið x ; tj x 0 Þ was obtained by averaging Gð x ; t0 þ tj x 0 ; t0 Þ over 5 different release times t0 (see Appendix 2); the realisations were spaced at an interval of Dt ¼ 200 s. Replacing the time average with ! ! ! ! a single realisation, i.e. hGið x ; tj x 0 Þ≡Gð x ; t0 þ tj x 0 ; t0 Þ, did not change the results qualitatively. The mean tracer age and age spectrum were calculated from (4) and (5) using

! ! S ¼ S0 Qðt  t0 Þdð x  x 0 Þ;

(9)

where S0 is the magnitude of the source flux, Q is the Heavside step function and x0/W 2 (1.5, 2.5). S0 is arbitrary. Below we show spatial averages of ta and sa. Spanwise averages correspond to (x, z) for which the flow is not blocked by a building at any value of y:

  1 f ðx; zÞ ¼ L

ZL=2 dyf

  ! x :

(10)

L=2

Canyon averages include integration over x and z:

b f ¼

1 WLH1

W=2 Z

ZL=2 ZH1

! dx0 dydz f ð x Þ

(11)

W=2 L=2 0

0

where x ¼ 0 at the midpoint of a canyon. Using H2 or H instead of H1 to define the vertical extent of the canyon had a relatively small effect on the results (not shown). 4.3. Validation The LES simulations with the even street canyon were validated against experimental data. Fig. 2 shows the time-averaged concentration field for a line source at the bottom of the canyon. There

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615

Fig. 2. Comparison of pollutant fields: (a) numerical simulation; (b) measurements from Pavageau and Schatzmann (1999).

is good agreement with the measurements of Pavageau and Schatzmann (1999). First- and second-order turbulence statistics (not shown) are in similarly good agreement with the wind-tunnel measurements of Brown et al. (2000). The validation results were largely insensitive to turbulence recycling.

5. Uniform street canyon As the structure of the velocity field for a uniform canyon has been extensively described in the literature (e.g. Liu and Barth, 2002; Cui et al., 2004; Letzel et al., 2008; Lo and Ngan, 2015), we do not describe it here. Fig. 3 plots tracer-age statistics in the xz plane. The spanwiseaveraged mean tracer age, 〈ta〉, is shown in panel (a). It is left unnormalised so as to facilitate comparison with the results for the non-uniform canyons, but values can be compared to the canyon circulation timescale, T ~ 4H/Urms ~ 680 s, where Urms is defined by the root-mean-square average of the wind speed inside the canyon. 〈ta〉 is small near the bottom boundary and considerably larger above the roof level. Generally 〈ta〉 increases with height, reflecting the fact that more time is generally required for pollutant to travel farther from the source.

Fig. 3. Spanwise-averaged tracer ages for a single canyon. (a) mean 〈ta〉; (b) standard deviation 〈sa〉. The range of values is narrower for 〈sa〉. The y-axis has been clipped for clarity. Grey areas correspond to negligibly small concentrations.

Nevertheless the mean tracer age does not increase uniformly with height: there are large values in the centre (〈ta〉 ~ 1600 s) and an asymmetry between the downwind (〈ta〉 ~ 1200 s) and upwind (〈ta〉 ~ 1000 s) walls. The 〈ta〉 field is clearly related to the canyon flow. Trapping of pollutants by the canyon vortex increases the tracer age. Clockwise circulation within the canyon causes pollutants on the upwind wall to be older than those on the downwind wall. As the streamwise flow aloft preferentially advects pollutant downstream, the mean tracer age is essentially undefined upstream of the sharp front above the canyon where concentrations are small. The ventilation strength can be assessed by examining ta on either side of the roof level. For z ( H, 〈ta〉 ~ 1100 s; for z a H values are comparable and there is no evidence of a sharp transition around the roof level. These values are very close to the canyon average, b t a  1100 s ~2T. The standard deviation of the tracer age (Fig. 3b) represents another aspect of variability. The overall spatial structure resembles that of 〈ta〉 inside the canyon. The main difference between 〈ta〉 and 〈sa〉 is that the latter does not increase with height or downstream distance to the same extent. Physically the ventilation varies on a relatively short timescale. We associate the ventilation timescale of the canyon with 〈ta〉. Ventilation of the canyon occurs on a longer timescale than that implied by advection. If pollutants exited the canyon directly after being emitted from the ground-level source, the mean age should be around T/2 ~ 350 s. Although the ventilation is not optimal, for 〈ta〉 > T/2, pollutant trapping by the canyon vortex does not have a disastrous effect. One explanation for why 〈ta〉 > T/2 is that tracer parcels are re-entrained or make multiple revolutions around the canyon before escaping permanently (cf. Xia and Leung, 2001). The structure of the 〈ta〉 field differs significantly from the pollutant field generated by a constant tracer flux along the bottom boundary (Liu and Barth, 2002; Cui et al., 2004). Whereas the largest values of c lie near the bottom upwind corner, the largest values of 〈ta〉 lie inside the vortex. Concentrations remain high even where the ventilation is not necessarily poor, such as happens along the upwind wall, because older pollutants are continually replaced by younger pollutants. This difference between c and 〈ta〉 has implications for the applicability of the local mean age of air, tl. Applying eq. (2) yields mean ages that are about an order of magnitude smaller than 〈ta〉 (Fig. 4). Moreover the spatial structure is rather different. Since the homogeneous emission method underlying eq. (2) requires a spatially uniform source, it is not surprising that tl is much smaller. The local mean age of air assumes that the

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tracer age. There is little structure in the spanwise direction (not shown). 6. Non-uniform street canyons 6.1. Mean flow

Fig. 4. Spanwise-averaged local mean age of air 〈tl〉 corresponding to Fig. 3a.

ventilation is completely controlled by the source flux. In general the mean age also depends on transport and mixing, which do not occur instantaneously; thus 〈ta〉 > 〈tl〉, the spanwise-averaged local mean age. The homogeneous emission method provides useful information on breathability when a uniform volume source can be adopted or assumed (Buccolieri et al., 2010), but it is less suited to problems in which the source is spatially localised. tl is effectively a lower bound on the ventilation timescale. Fig. 5 shows the tracer-age distribution at the bottom, centre and top of the canyon. The spectra, which are strongly skewed towards large t, demonstrate that ventilation does not occur on a single timescale: in each case Z(t) is fairly broad, indicating that there are many routes by which pollutant reaches the height in question, as with pure diffusion in an unbounded domain. In the long-time limit the concentration at a given point decays exponentially in time. The e-folding timescale is te ~ 1100 s, which is very close to the canyon average. At the bottom of the canyon the kurtosis Ka ~ 17. The location of the initial peak represents the most obvious difference among the spectra. It can be explained in terms of the transit or advection times. At z ¼ H/24, Z peaks almost immediately due to the proximity to the source; at z ¼ H/2, the first peak occurs around tp ~ 100 s, which is roughly the time required for a parcel to be advected from the street level to the midplane; at z ¼ H, the peak shifts to tp ~ 220 s, which roughly corresponds to the time taken for a parcel to be advected from the street level to the roof top. In all cases tp < 〈ta〉. This suggests that ventilation of the canyon is not controlled by ballistic transport, but rather by the combined effects of advection and diffusion. We have also examined ta(x, y), i.e., xy sections of the mean

Fig. 5. Tracer-age distribution, Z(t), at the bottom (z ¼ H/24), centre (z ¼ H/2) and top (z ¼ H) of the single canyon. The x-axis has been clipped for clarity.

In the xz plane the structure of the time-mean velocity field is analogous to that for a uniform canyon. Summarising the main results (not shown): distinct vortices form inside each canyon for a small height difference, d, between the short and tall buildings; as d increases the circulations between the canyons begin to merge. Thus spatial heterogeneity is maximised for an intermediate value of d. In the xy plane the picture is more complicated. Fig. 6 plots the vector velocity field for each d at different z. We summarise the most important important points. For d ¼ 0 (panels aec) the velocity field is reminiscent of that for a single canyon. For d > 0 (panels deo), the flow is less unidirectional, i.e., the spanwise velocity component increases in strength: there are inflows and outflows across y ¼ W and y ¼ W, their strength depending on z and d. This complicated spanwise structure is absent from the single canyon. In general, the horizontal velocity field shows a nonmonotonic dependence on d. For example, channeling in the streamwise direction at z ¼ H/2 is weakest for d ¼ H/2. Values of the canyon circulation time T and streamwise velocity U0 are summarized in Table 3. T is shorter for d > 0. 6.2. Tracer age and age spectrum Physically one expects improved ventilation for the nonuniform canyon. As the canyon vortex may be partially disrupted by inhomogeneities in the building geometry, increased turbulence may result. This is consistent with Gu et al. (2011), who examined time-averaged pollutant fields. The actual time-dependent ventilation, however, could be more complicated. As discussed previously, c can give a misleading impression of the ventilation. Furthermore generalisations are particularly dangerous in the case of the non-uniform canyon because the dependence on d is not obvious. Fig. 6 indicates that the structure of the velocity field varies non-monotonically with d. Fig. 7 shows the spanwise-averaged mean tracer age 〈ta〉 for different d. For d ¼ 0 (Fig. 7a), the behaviour is qualitatively similar to that of the single uniform canyon, i.e. 〈ta〉 increases with height and downstream distance. Within the first canyon, however, the spatial structure is more homogeneous and values are much smaller; the increased ventilation strength arises from the streamwise channels at jyj > W. For the non-uniform canyons, i.e. d > 0 (Fig. 7bee), 〈ta〉 decreases noticeably inside the first canyon. This confirms that non-uniformity improves ventilation. Somewhat surprisingly, however, the differences among the d > 0 cases are relatively small. Just as with the single canyon, the local mean age shows little resemblance to 〈ta〉. 〈tl〉 is about an order of magnitude smaller inside the first canyon (Fig. 8). Large values are confined to the vicinity of the source at z ¼ 0. 〈tl〉 is much smaller inside the other canyons, implying that the ventilation should improve downstream. This is not supported by the mean tracer age (Fig. 7). The ventilation variability shows considerable sensitivity to d (Fig. 9). Although the spatial structure of 〈sa〉 roughly resembles that of 〈ta〉, differences among the d are clearly evident for all canyons. This shows that some aspects of ventilation are strongly affected by the unevenness or non-uniformity of the domain. Horizontal cross-sections, ta(x, y), reveal a more complicated picture than the spanwise averages analysed above. Near the street

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Fig. 6. Velocity vector plots in the xy plane for different uneven, non-uniform street canyon arrays. The first, second and third columns correspond to z ¼ H/24, H/2 and H respectively. Each row corresponds to a different d: (a), (b), (c) d ¼ 0H; (d), (e), (f) d ¼ H/12; (g), (h), (i) d ¼ H/6; (j), (k), (l) d ¼ H/4; (m), (n), (o) d¼H/2. For clarity only part of the x-domain is shown; there is qualitatively similar behaviour for canyons 3e5.

Table 3 Summary of canyon circulation timescale T and characteristic velocity U0 for the non-uniform canyons. T ~ 2(H1/Urms þ W/Urms) and U0 is the streamwise velocity at 2H. The root-mean-square canyon velocity inside the canyon, Urms, represents an average over the canyons.

d

T [s]

U0 [ms1]

0H H/12 H/6 H/4 H/2

700 520 530 460 540

2.4 2.3 1.7 1.8 2.0

level (i.e. z ¼ H/24), there are significant spanwise variations (Fig. 10). For d ¼ 0 (panel a), large values appear around y ¼ 0: they effectively represent barriers to ventilation. For d > 0 (panels bee), these barriers disappear. Nevertheless residual structures remain along the upwind wall and near the corners; the size of these structures varies non-monotonically with d, in agreement with the time-averaged velocity fields (Fig. 6). Away from the street level, ta(x, y) is larger and more spanwise structure is retained. These variations are not seen in c or tl, which have little spanwise structure (not shown). The preceding qualitative trends may be quantified using canyon averages (Fig. 11). With respect to the mean tracer age (panel a), b t a tends to be higher for d ¼ 0 but departures from naive

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Fig. 7. Spanwise-averaged mean tracer age, 〈ta〉, in the xz plane for the non-uniform street canyons. (a) d ¼ 0H; (b) d ¼ H/12; (c) d ¼ H/6; (d) d ¼ H/4; (e) d ¼ H/2.

Fig. 9. As in Fig. 7, but for the spanwise-averaged standard deviation of the tracer age, 〈sa〉. (a) d ¼ 0H; (b) d ¼ H/12; (c) d ¼ H/6; (d) d ¼ H/4; (e) d ¼ H/2. The overall pattern is similar to that for 〈ta〉, but there is greater sensitivity to d.

Fig. 8. Spanwise-averaged local mean age of air, 〈tl〉, corresponding to Fig. 7. (a) d ¼ 0H; (b) d ¼ H/12; (c) d¼ H/6; (d) d ¼ H/4; (e) d ¼ H/2.

physical intuition can also be discerned. Relatively minor modifications to building design and layout yield significantly improved air quality: b t a improves by around 50% from d ¼ 0 to d ¼ H/12 (i.e. a displacement of 4 m) inside the first two canyons. But nonuniformity does not guarantee improved ventilation. By the fifth canyon b t a is actually lowest for d ¼ 0: non-uniformity impedes ventilation. With respect to the standard deviation (panel b), similar behaviour is found in the first few canyons: b s a is largest for d ¼ 0 and smallest for d ¼ H/12. However, as seen in Fig. 9, there is greater sensitivity to d. Away from the first canyon, b t a depends on the time required for pollutant to travel downstream, which is ~50e100 s per canyon (cf. Table 3). Since the increase in 〈ta〉 exceeds this, it cannot be attributed solely to lag effects; the ventilation strength is poorer inside the downstream canyons. The ventilation variability b sa (Fig. 11b), which is not susceptible to lag effects by construction, also increases downstream. Fig. 12 shows age spectra at the street level. Inside the first canyon (panel a), Z peaks almost immediately before decaying

Fig. 10. Mean tracer age, ta(x,y), at z ¼ H/24 (street level). (a) d ¼ 0H; (b) d ¼ H/12; (c) d ¼ H/6; (d) d ¼ H/4; (e) d ¼ H/2. There is less spanwise structure for d > 0.

approximately exponentially. The improved ventilation for d > 0 arises from a decreased probability of long-timescale excursions. The decay timescales are shorter compared to the single canyon: te ¼ 540, 210, 310, 330 and 280 s for d ¼ 0 to d ¼ H/2 (in sequential order). As with the single canyon, there is fair agreement between te and bt a (cf. Fig. 11), suggesting that the ventilation is associated mostly with the combined effects of advection and diffusion. By contrast with the single canyon, secondary peaks are less prominent, even when the age spectra are evaluated at z ¼ H/2 and z ¼ H (not shown). Although te is smaller for non-uniform canyons, the kurtosis Ka ~ 30e50 inside the first canyon is about 2e3 times greater because exponential decay is established more quickly. For the downstream canyons (Fig. 12bee), the spectra broaden and shift. This is indicative of poorer ventilation, though Ka ~ 5 in

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Fig. 11. Canyon-averaged (a) mean tracer age, b t a ; (b) standard deviation, b s a . The overall trends are broadly similar for b t a and b sa .

Fig. 12. Tracer-age distributions at z ¼ H/24 for the different d. (a) canyon 1; (b) canyon 2; (c) canyon 3; (d) canyon 4; (e) canyon 5. The spectra narrow for d > 0.

the downstream canyons. For the final canyon te ranges from 770 to 470 s, which is no longer in good agreement with b t a . Given the importance of lag effects, this is not surprising. There is better agreement between b s a and te. 7. Summary and discussion New ventilation diagnostics based on the tracer-age concept (Hall and Plumb, 1994; Holzer and Hall, 2000) have been applied to the ventilation of urban street canyons. From large-eddy simulations of uniform and non-uniform street canyons, ventilation timescales have been calculated from the mean, ta, and standard deviation, sa, of the age spectrum (or tracer-age distribution). It has been shown that ventilation occurs on a timescale that is long compared to the advective timescale for both the uniform canyon and the first few non-uniform canyons; thus ventilation is not driven purely by advection. In these cases the age spectrum has a long exponential tail with a characteristic timescale that is wellapproximated by the mean tracer age. The quantification of ventilation timescales represents the most useful result from this work. On a practical level, two-dimensional cross-sections and spatial averages of the mean tracer age indicate that ventilation generally improves due to spatial inhomogeneities and the presence of unblocked streamwise channels, in agreement

with physical intuition and previous time-averaged results (Gu et al., 2011). The dependence on d, which characterises the inhomogeneity, is non-monotonic: the mean tracer age inside the first canyon attains its minimum for d ¼ H/12, which is a relatively small departure from the uniform canyon. On a more general level, the mean tracer age, standard deviation and age spectrum provide new information about urban pollutant ventilation. Existing estimates are based on time-averaged statistics or roof-level fluxes. The timescales defined by ta and sa are much longer than estimates that require a spatially homogeneous source, an assumption that is not appropriate in all cases. Hence ta, sa and the age spectrum may serve as reference points for future studies of street-canyon ventilation. They apply to complicated geometries for which the roof-level fluxes are not representative, and they may be especially useful for pollutant fields that cannot be well-represented by a time average. The approach described in this paper was originally developed for the analysis of stratospheric transport and mixing (Hall and Plumb, 1994; Waugh and Hall, 2002). Many studies have been published in the atmospheric science literature on the age of air and its variants; some of this work (e.g. interpretation of observations and evaluation of models) should be relevant to urban air quality. The mean tracer age and age spectrum would be especially valuable if the age of pollutants could be routinely measured, such as the case for stratospheric constituents (Waugh and Hall, 2002). The interplay between modelling and observations has been extremely fruitful in the stratospheric context. Although there are subtleties in the application of them (e.g. the assumption of isotropy may not be appropriate), in principle the results of this paper could have been obtained with a Lagrangian stochastic model. Such results have not been reported in the literature, but they are presently under investigation and will be described elsewhere. An advantage of the present Eulerian approach is that it is easily applied to conventional urban CFD simulations. Acknowledgements We thank the anonymous referees for helpful comments and suggestions. This work was supported financially by City University of Hong Kong through a Strategic Research Grant (Project 7004165) and Start-Up Grant (Project 7200403). Appendix 1. Derivation of the tracer age spectrum As the advectionediffusion equation, Eq. (1), is linear, it admits a ! ! Green's function, Gð x ; tj x 0 ; t0 Þ. The Green's function is the solution

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to the advectionediffusion equation with a delta-function source, Eq. (3). Using the Green's function, the concentration at a receptor ! location and time ð x ; tÞ can be written in terms of an integral ! ! connecting cð x ; tÞ to an initial location and time ð x 0 ; t0 Þ. By definition t  t0. We assume t0  0. ! Assuming cð x ; 0Þ ¼ 0, the solution takes the form (Holzer and Hall, 2000)

! cð x ; tÞ ¼

Zt

! dt0 rð x ; tÞ

Z

! ! ! ! d x 0 Gð x ; tj x 0 ; t0 ÞSð x 0 ; t0 Þ:

(12)

D

0

D denotes the region over which the pollutant source S is applied. The Green's function yields a probability distribution of ! ! transit times or tracer ages from ð x 0 ; t0 Þ to ð x ; tÞ, viz.

Z ! Zð x ; tjS; t  tÞ ¼

! ! ! ! ! d x 0 rð x ; tÞGð x ; tj x 0 ; t  tÞSð x 0 ; t  tÞ

D

! cð x ; tÞ

: (13)

The time lag t ≡ tt0. Z is the tracer-age distribution or age spectrum; it is a tracer-dependent property because it depends on the source. Eq. (13) is the most general form of the tracer-age distribution. For a steady or statistically stationary source, a simpler expression may be obtained. Taking r ¼ 1, averaging over pollutant-release times and assuming that S is steady (see Appendix 2) yields Eq. (4). Appendix 2. Statistical stationarity ! ! The Green's function Gð x ; tj x 0 ; t  tÞ depends on an initial time t0 and a final time t. In principle, therefore, the age spectrum Z must be calculated using a separate Green's function for each value of t0 (or equivalently tt). Obviously this is very expensive. Although the derivation in Appendix 1 does not require statistical stationarity, the calculations simplify in this case. With this assumption averages over the pollutant-release times are independent of t0:

! ! ! ! hGið x ; tj x 0 Þ≡hGð x ; t0 þ tj x 0 ; t0 Þi

(14a)

! ! ! ! hSið x ; tj x 0 Þ≡hSð x ; t0 þ tj x 0 ; t0 Þi;

(14b)

the angle brackets denote an average over t0. Hence the Green's function and source can be rewritten as

! ! ! ! ! ! Gð x ; tj x 0 ; t  tÞ ¼ hGið x ; tj x 0 Þ þ G0 ð x ; tj x 0 ; t  tÞ

(15a)

! ! ! ! ! ! Sð x ; tj x 0 ; t  tÞ ¼ hSið x ; tj x 0 Þ þ S0 ð x ; tj x 0 ; t  tÞ:

(15b)

hG0 S0 i

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