# Simple photochemical modelling of $\ce{NO_{X}}$ pollution in a street canyon

L. Soulhac  S. Fellini  C.V. Nguyen  P. Salizzoni  Univ. Lyon, INSA Lyon,
CNRS, Ecole Centrale de Lyon, Univ. Claude Bernard Lyon 1, LMFA, UMR5509,
69621, Villeurbanne France  Univ. Lyon, Ecole Centrale de Lyon, CNRS, Univ.
Claude Bernard Lyon 1, INSA Lyon, LMFA, UMR5509, 69130, Ecully, France

## Abstract

To predict pollutant concentration in urban areas, it is crucial to take into
account the chemical transformations of reactive pollutants in operational
dispersion models. In this work, we derive and discuss different NO-NO_2-O_3
chemical street canyon models with increasing complexity and we analytically
evaluate their applicability in different urban contexts. We then evaluate the
performance of the models in predicting NO_2 concentration at different
locations within an urban district by comparing their predictions with
measurements acquired in a field campaign. The results are in line with
analytical speculations and give indications as to which model to use
according to the conditions of the urban street canyon. In courtyards with
limited ventilation and without direct emissions, the performance of the
photostationary model is satisfactory. On the other hand, the application of a
non-photostationary model significantly improves the predictions in urban
canyons with direct vehicular emissions. The applicability of the proposed
models in operational tools at the city scale is finally discussed.

## Keywords

Photochemical smog, Urban air quality, Non-photostationary chemical model, Air
pollution measurement campaign

## Introduction

The time scales related to pollutant transfer over large urban agglomerations
range from a few minutes to several hours. During this period, a large number
of physico-chemical processes take place and determine the concentration of
pollutants in the urban atmosphere (Sillman, 1999). When the focus is on
dispersion at the local district scale, the rate of turbulent transport is
considerably high compared to the rate of chemical transformation and most of
the atmospheric compounds can be treated as inert tracers. There are however
chemical reactions which are sufficiently fast to significantly affect the
concentration of pollutants during their residence time in the streets. This
is notably the case for $\ce{NO_{X}}$, i.e. the nitrogen oxides that are most
relevant for air pollution.

The emissions of $\ce{NO_{X}}$ result from combustion processes, especially
from motor vehicle engines or from power stations and industries. They are
therefore a tracer of anthropogenic activity in urban areas and their trends
are used to assess the effectiveness of regulations on air pollution, or to
evaluate the effects of sudden changes in emissions, such as during COVID-19
restrictions (e.g., Toscano and Murena, 2020; Lovarelli et al., 2020; Misra et
al., 2021). It is generally assumed that the partition of $\ce{NO_{X}}$ at the
point of emission is approximately between 10% to 15% for $\ce{NO_{2}}$
(nitrogen dioxide) and 85% to 90% for $\ce{NO}$ (nitrogen monoxide)
(Ntziachristos et al., 2000). Acute exposure to $\ce{NO_{X}}$ causes
respiratory disease and compromises lung functioning when inhaled at high
concentrations. Children are the most vulnerable, with a demonstrated
increased incidence of childhood asthma due to $\ce{NO_{2}}$ emissions from
vehicular traffic (Khreis et al., 2017; Anenberg et al., 2022). Despite being
the major contributor to $\ce{NO_{X}}$, $\ce{NO}$ is less toxic than
$\ce{NO_{2}}$. However, as most radicals, it is extremely unstable and forms
$\ce{NO_{2}}$ through photochemical oxidation. Nitrogen dioxide is then
converted back to $\ce{NO}$ as a result of photolysis which also leads to the
regeneration of ozone ($\ce{O_{3}}$). When the photostationary state is
reached, these reactions result in a cycle with zero net chemistry and the
chemical compounds reach the equilibrium composition, which can be easily
derived in terms of kinetic reaction parameters by the Leighton relation
(Leighton, 1961). Deviations from this state occur when (i) the residence time
of polluants in the reference volume (i.e. the street) is shorter than the
time needed for reaching the photostationary equilibrium, (ii) turbulent
motions mix the reactants so slowly that they remain segregated rather than
reacting (Li et al., 2021), (iii) the transformation of nitrogen monoxide into
$\ce{NO_{2}}$ is altered by the role of complex reactions with radicals
resulting from the oxidation of Volatile Organic Compounds ($\ce{VOC}$s) and
$\ce{CO}$ (Jenkin and Clemitshaw, 2000). The concentrations of $\ce{NO}$ and
$\ce{NO_{2}}$ are also affected by reactions involving the hydroxyl radical
and leading to the production of nitric acid.

The coupling of turbulent and chemical dynamics to assess photochemical
pollution in urban areas has been explored extensively in the past two decades
by means of Computational Fluid Dynamics (CFD) simulations. Baker et al.
(2004) extended a Large Eddy Simulation (LES) for turbulent flow in a street
canyon with a simple $\ce{NO_{X}}$-$\ce{O_{3}}$ chemical model. The same
reaction scheme was adopted by Baik et al. (2007), who instead used Reynolds-
averaged Navier–Stokes (RANS) simulations. By introducing a photostationary
state defect index, both studies highlighted the regions of a street canyon
most prone to chemical instability. The chemistry of $\ce{VOC}$ has been
included in RANS simulations by Kwak and Baik (2012) and Kim et al. (2012),
while Bright et al. (2013) combined LES simulations with a detailed chemical
reaction mechanism (Reduced Chemical Scheme) comprising 51 chemical species
and 136 reactions. Similarly, Garmory et al. (2009) used the Stochastic Fields
(FS) method to simulate turbulent reacting flows with a chemistry model
comprising 28 species. These studies showed that the effect of turbulent
fluctuations (i.e. segregation) on the chemistry is significant for species
with the highest transformation rates. They also showed that increasing
chemical complexity (i.e. simulating $\ce{VOC}$ chemistry) could contribute to
additional but modest $\ce{NO_{2}}$ and $\ce{O_{3}}$ formation in the canyon.

CFD simulations, coupled with detailed chemical models, provide an accurate
prediction but are computationally expensive and require a large amount of
detailed input data. To simulate air quality in large urban domains,
consisting of hundreds to thousands of streets, a more efficient way is
adopting simplified modelling approaches (Vardoulakis et al., 2007). These are
usually Gaussian-Lagrangian models integrated with box models to simulate the
concentration in the street canyons. In these operational tools,
photostationarity is a convenient assumption as it allows the modelling of
$\ce{O3}$ and $\ce{NO_{X}}$ as inert tracers and to subsequently apply
photochemical equilibrium in the streets. This is the case of the Canyon Plume
Box Model (CPBM) (Yamartino and Wiegand, 1986), and the street network model
Sirane (Soulhac et al., 2017).

Another widespread approach is the adoption of empirical models to estimate
$\ce{NO}$-$\ce{NO_{2}}$ conversion (Ravina et al., 2022). These are based on a
photostationary assumption but are optimised to fit observed concentrations.
Hirtl and Baumann-Stanzer (2007) investigated the performances of the two
empirical conversion schemes after Romberg et al. (1996) and after Derwent and
Middleton (1996), when implemented in the Gaussian model Atmospheric
Dispersion Modelling System (ADMS) and in the LAgrangian Simulation of Aerosol
Transport (LASAT) model. These dispersion models turned out to be quite
successful in predicting average concentrations measured in street canyons.

A step forward in modeling the interaction between the time scales of chemical
reactions and those of transport is represented by the model ADMS-Urban
(McHugh et al., 1997; Carruthers et al., 2000). In ADMS-Urban, $\ce{NO_{X}}$
chemistry can be modelled by the Generic Reaction Set (GRS) (Azzi et al.,
1992; Venkatram et al., 1994) photochemical scheme which includes seven
chemical reactions. The GRS chemical model is applied to the emitted
pollutants after transport and dispersion. The chemistry calculation for the
receptor is split in two steps: the first considers the contribution from far
sources (source-receptor travel time greater than 150 s), while the second one
includes the contribution from the nearest sources (source-receptor travel
time less than 150 s) (CERC, 2022). In this way, the model takes into account
the travel time of the pollution plume and it assumes a time -or distance-
dependence on the generation of $\ce{NO_{2}}$.

Finally, the Operational Street Pollution Model (OSPM) (Palmgren et al., 1996;
Berkowicz et al., 1997) is a street canyon model which includes NO-NO_2-O_3
chemistry by means of a non-photostationary model that takes into account the
interaction between the chemical reaction rates and the residence time of the
pollutants in the street.

The overview above suggests that operational modeling of reactive pollutant
concentration at the urban scale requires an adequate description of (i) the
chemistry, (ii) the turbulent transport, (iii) the interaction between these
two processes, all while minimizing the computational cost and required input
data in order to be applied to hundreds to thousands of streets. To date,
empirical relationships and photostationary models are the most commonly used
for operational purposes while non-photostationary schemes are rarely
implemented. This is especially true for street network models, where, to our
knowledge, the non-photostationary scheme has not yet been implemented.
Furthermore, the existing literature lacks a coherent formulation of the
different photochemical models, with a clear statement of the underlying
assumptions and a concurrent validation with real data. To fill these gaps, in
this work, we derive, compare and validate three models for $\ce{NO_{X}}$
photochemical pollution that can be efficiently implemented in street network
models at the city scale. In the analytical derivation, we focus on the time
scales of pollutant transformation and transport in order to highlight the
range of application of the different models. To verify the reliability of the
different schemes we compare the model outputs to field data. The main
objective is to evaluate whether the application of a non-photostationary
model can bring substantial advantages in the prediction of pollutant
concentration in the streets, with respect to photostationary models.

The formulation of a photochemical model, adopting box-model approach, is
presented in 1. A general presentation of the measurement campaign is given in
Section 2. Results are discussed in Section 3, while the conclusions are drawn
in Section 4.

##  1 $\ce{NO}-\ce{NO_{2}}-\ce{O_{3}}$ chemical street model

To maximize computational efficiency, minimize input data while providing a
satisfactory description of pollution in the urban area, city-scale
operational models, such as street network models (Soulhac et al., 2011),
generally provide a single concentration value for each street. This can be
notably achieved adopting a bow model a the street scale, which provides
spatially averaged pollutant concentration by computing a pollutant budget
over the volume of the street. In order to simulate photochemical pollution,
the pollutant budget has to take into account the terms of chemical production
and of chemical destruction (Soulhac et al., 2011) as well as those related to
the turbulent fluxes at the street edges and at the top of the street.

To write the budget of photochemical pollutants in the street, we start by
considering the simplified chemical scheme involving NO, NO_2 and O_3:

$\ce{NO_{2}}\stackrel{{\scriptstyle
k_{1}}}{{\longrightarrow}}\ce{NO}+\ce{O^{\bullet}}$ (1)

$\ce{O_{2}}+\ce{O^{\bullet}}\stackrel{{\scriptstyle
k_{2}}}{{\longrightarrow}}\ce{O_{3}}$ (2)

$\ce{NO}+\ce{O_{3}}\stackrel{{\scriptstyle
k_{3}}}{{\longrightarrow}}\ce{NO_{2}}+\ce{O_{2}}$ (3)

It is known (Seinfeld, 1986) that the second equation is much faster than the
first and the third ones, so that the constants $k_{1}$ and $k_{3}$ are the
limiting parameters of these chemical reactions. The constant rate $k_{1}$
($\ce{NO_{2}}$ photolysis rate) depends on the intensity of solar radiation,
whilst $k_{3}$ depends on air temperature. These dependences can be modeled by
the following relations (Kasten and Czeplak, 1980; Seinfeld, 1986):

$\left\\{\begin{array}[]{l}k_{1}=\frac{1}{60}(0.5699-[9.056\cdot
10^{-3}(90-\zeta)]^{2.546})\left(1-0.75\left[\frac{Cld}{8}\right]^{3.4}\right)\,(\mbox{s}^{-1})\\\
k_{3}=1.325\cdot
10^{6}\exp\left(-\frac{1430}{T}\right)\,(\mbox{m}^{3}\mbox{mol}^{-1}\mbox{s}^{-1})\end{array}\right.$
(4)

where $\zeta$ is the solar elevation in degrees, $T$ is the air temperature in
Kelvin and $Cld$ is the cloud coverage in Oktas. These meteorological
parameters vary over time. In operational dispersion models, the time-
dependence of the meteorological parameter is usually modelled assuming a
quasi-steady approach, i.e. assuming steady condition of time step of 1 hours.
Cloud coverage and temperature are measured during the day at meteorological
stations, while the solar elevation is a function of the day of the year, the
local hour and the site latitude (e.g., Soulhac et al., 2011). Note that
$k_{1}$ is set equal to 0 at night, when the solar elevation angle is
negative. More sophisticated models for $k_{1}$ and $k_{3}$ are available in
the literature, but they are generally not adapted for operational purposes
(Seinfeld, 1986).

Referring to Eqs. 1-3, the production and destruction terms for each chemical
species are related to the molar concentration by the following expressions:

$\begin{array}[]{ll}P_{NO}=k_{1}[\ce{NO_{2}}]&D_{NO}=k_{3}[\ce{NO}][\ce{O_{3}}]\\\
P_{NO_{2}}=k_{3}[\ce{NO}][\ce{O_{3}}]&D_{NO_{2}}=k_{1}[\ce{NO_{2}}]\\\
P_{O_{3}}=k_{1}[\ce{NO_{2}}]&D_{O_{3}}=k_{3}[\ce{NO}][\ce{O_{3}}]\\\
&\end{array}$ (5)

where $[\ce{\cdot}]$ represents the molar concentration (mol/$\mbox{m}^{3}$)
of the compound.

We include the production and destruction terms in the street box model
formulated in Soulhac et al. (2011). Neglecting wet and dry deposition
phenomena, the budget of time-averaged concentration of NO_2, NO and O_3 for a
single street-canyon of length $L$, width $W$ and height $H$, can be written:

$Q_{\ce{NO_{2}}}-u_{d}\mathcal{S}_{h}\left([\ce{NO_{2}}]-[\ce{NO_{2}}]\\!^{r}\right)-U\mathcal{S}_{v}\left([\ce{NO_{2}}]-[\ce{NO_{2}}]\\!^{c}\right)+k_{3}[\ce{NO}][\ce{O_{3}}]\mathcal{V}-k_{1}[\ce{NO_{2}}]\mathcal{V}=0$
(6)
$Q_{\ce{NO}}-u_{d}\mathcal{S}_{h}\left([\ce{NO}]-[\ce{NO}]\\!^{r}\right)-U\mathcal{S}_{v}\left([\ce{NO}]-[\ce{NO}]\\!^{c}\right)+k_{1}[\ce{NO_{2}}]\mathcal{V}-k_{3}[\ce{NO}][\ce{O_{3}}]\mathcal{V}=0$
(7)
$-u_{d}\mathcal{S}_{h}\left([\ce{O_{3}}]-[\ce{O_{3}}]\\!^{r}\right)-U\mathcal{S}_{v}\left([\ce{O_{3}}]-[\ce{O_{3}}]\\!^{c}\right)++k_{1}[\ce{NO_{2}}]\mathcal{V}-k_{3}[\ce{NO}][\ce{O_{3}}]\mathcal{V}=0$
(8)

where $\mathcal{V}=LWH$ is the volume of the street, $\mathcal{S}_{h}=LW$ is
its horizontal area and $\mathcal{S}_{v}=WH$ is its vertical cross section.
The velocities $U$ and $u_{d}$ are the mean velocity along the street and the
exchange rate at roof level (e.g., Soulhac et al., 2008; Salizzoni et al.,
2009; Fellini et al., 2020) and they drive the longitudinal and vertical
pollutant fluxes entering and leaving the street volume. For each of the three
chemical compounds NO_2, NO and O_3, $Q$ is the molar emission in the street,
$[\cdot]\\!^{r}$ is the concentration in the atmosphere above roofs, and
$[\cdot]\\!^{c}$ is the concentration in the flow advected within the canopy
at the upwind intersection of the street. We point out that the source of
ozone is not included in the budget (i.e. $Q_{\ce{O_{3}}}=0$) since direct
ozone emissions in the streets are rare. Eqs. 6-8 can be reformulated by
highlighting the time scales associated with the terms of transport and
chemical reaction. For example, for $\ce{NO_{2}}$ we can write:

$\dfrac{Q_{\ce{NO_{2}}}}{\mathcal{V}}-\dfrac{[\ce{NO_{2}}]-[\ce{NO_{2}}]\\!^{r}}{\tau_{v}}-\dfrac{[\ce{NO_{2}}]-[\ce{NO_{2}}]\\!^{c}}{\tau_{h}}+\dfrac{[\ce{NO}]}{\tau_{3}}-\dfrac{[\ce{NO_{2}}]}{\tau_{1}}=0$
(9)

with

$\left\\{\begin{array}[]{l}\tau_{v}=\dfrac{H}{u_{d}}\\\
\tau_{h}=\dfrac{L}{U}\\\ \tau_{1}=\dfrac{1}{k_{1}}\\\
\tau_{3}=\dfrac{1}{k_{3}[\ce{O_{3}}]}\end{array}\right.$ (10)

To determine the order of magnitude of the different terms in Eq. 9 we can
roughly estimate the time scales involved, based on the data collected and
simulated for the city of Lyon (France) (Soulhac and Salizzoni, 2010; Soulhac
et al., 2012). The depth $H$ and length $L$ of street canyons vary in the
ranges 15-30 m and 20-150 m, respectively. The wind speed within the streets
$U$ and the typical turbulent exchange velocity $u_{d}$ can reasonably be
assumed in the ranges 0.1-5 m/s and 0.01-0.22 m/s, respectively (Salizzoni et
al., 2009; Soulhac et al., 2011) when the free stream wind above the city is
between 1.5 m/s and 8 m/s (Météo-France data for the period 1981–2006) From
these data, we obtain that $\tau_{v}$ ranges in 68-3000 s, and $\tau_{h}$ in
4-1500 s. Typical values of $k_{1}$ and $k_{3}$ can be estimated by means of
Eq. 4 by varying the cloud coverage $Cld$ between 0 and 8, the temperature $T$
in $5^{o}$C-$30^{o}$C and $\zeta$ in $10^{o}$-$90^{o}$ (in this analysis we
consider only daytime). The concentration of ozone can be taken in the range
25-75 ppb (data measured at Saint-Exupery station for the year 2008). These
data provide $\tau_{1}$ in the range 105-1850 s and $\tau_{3}$ in the range
54-247 s. Moreover, we introduce an average time scale $\tau_{s}$ related to
the pollutant wash-out from the street:

$\tau_{s}=\left(\dfrac{1}{\tau_{h}}+\dfrac{1}{\tau_{v}}\right)^{-1}$ (11)

and we find that $\tau_{s}$ varies approximately in the range 4-1000 s. This
analysis shows that there is an overlap between the timescales associated to
chemical reactions and the characteristic residence times of pollutants within
the street. Consequently a modeling approach combining chemistry and
advection-diffusion processes must be adopted, as neither of the two processes
can be neglected.

Finally, we define the average background concentration $[\cdot]\\!^{b}$ as:

$[\cdot]\\!^{b}=\dfrac{\dfrac{[\cdot]\\!^{c}}{\tau_{h}}+\dfrac{[\cdot]\\!^{r}}{\tau_{v}}}{\dfrac{1}{\tau_{h}}+\dfrac{1}{\tau_{v}}}$
(12)

.

In this way, Eq. 9 can be simplified using only the background concentration
$[\ce{NO_{2}}]\\!^{b}$:

$\dfrac{Q_{\ce{NO_{2}}}}{\mathcal{V}}-\dfrac{[\ce{NO_{2}}]-[\ce{NO_{2}}]\\!^{b}}{\tau_{s}}+k_{3}[\ce{NO}][\ce{O_{3}}]-k_{1}[\ce{NO_{2}}]=0$
(13)

The same formulation is valid for NO and O_3 so that a system of 3 equations
(Eq. 13 and the two analogous balances for NO and O_3) describes in a compact
way the dynamics of the three chemical compounds. In what follows, we will
examine the solution of this system of equations adopting different scenarios
related to the relative importance of the different time scales involved.

###  1.1 Passive scenario

As a first step, we consider the case of a passive pollutant, whose
concentration is generally referred to as $[\cdot]\\!^{*}$. For NO_2 Eq. 13
simplifies as:

$\dfrac{Q_{\ce{NO_{2}}}}{\mathcal{V}}-\dfrac{[\ce{NO_{2}}]\\!^{*}-[\ce{NO_{2}}]\\!^{b}}{\tau_{s}}=0.$
(14)

This scenario corresponds to the case of reaction times that are extremely
long (i.e. $\tau_{1}\rightarrow\infty$, $\tau_{3}\rightarrow\infty$) so that
the terms of chemical production and destruction are negligible for the budget
in the street. The solution is given by:

$[\ce{NO_{2}}]\\!^{*}=[\ce{NO_{2}}]\\!^{b}+\dfrac{\tau_{s}Q_{\ce{NO_{2}}}}{\mathcal{V}}$
(15)

By analogy, the relative solution for the ‘passive’ NO concentration reads:

$[\ce{NO}]\\!^{*}=[\ce{NO}]\\!^{b}+\dfrac{\tau_{s}Q_{\ce{NO}}}{\mathcal{V}}$
(16)

and for O_3 concentration, assuming no emission of ozone:

$[\ce{O_{3}}]\\!^{*}=[\ce{O_{3}}]\\!^{b}.$ (17)

The concentration $[\cdot]\\!^{*}$ takes into account all contributions to
pollution deriving from advective transport only, i.e. the direct emission
into the street and the transport of pollutants to the street both from
adjacent street and from the atmosphere above the roofs. Thus, Eqs. 15-17 can
be seen as the general solutions for a dispersion model able to provide
passively advected concentrations in the streets.

###  1.2 Photostationary chemical model

Let us now consider that the reactive pollutants in the control volume (i.e.
within the street canyon) have the necessary time to reach the photochemical
equilibrium. This corresponds to assume that the characteristic time scales of
the chemical reactions $\tau_{1}$ and $\tau_{3}$ are small compared to the
residence time of pollutants within the street (i.e. $\tau_{1}$ and
$\tau_{3}\longrightarrow 0$). Under these assumptions, the advective and
source terms in Eq. 13 become negligible compared to the production and
destruction terms and the balance equation is simplified as follows:

$k_{3}[\ce{NO}]\\!^{\infty}[\ce{O_{3}}]\\!^{\infty}-k_{1}[\ce{NO_{2}}]\\!^{\infty}=0$
(18)

where the photostationary concentrations have been referred to as
$[\cdot]\\!^{\infty}$. Eq. 18 is known as the Leighton relationship (Leighton,
1961), whose formulation could also be obtained from the budget of NO_2 or O_3
(see Eqs. 7 and 8).

The conservation of N and O species lead to the following relations, which are
valid for passive, photostationary or non-photostationary concentrations:

$[\ce{NO}]+[\ce{NO_{2}}]=[\ce{NO}]\\!^{*}+[\ce{NO_{2}}]\\!^{*}=[\ce{NO}]\\!^{\infty}+[\ce{NO_{2}}]\\!^{\infty}=\phi_{N},$
(19)

$[\ce{O_{3}}]+[\ce{NO_{2}}]=[\ce{O_{3}}]\\!^{*}+[\ce{NO_{2}}]\\!^{*}=[\ce{O_{3}}]\\!^{\infty}+[\ce{NO_{2}}]\\!^{\infty}=\phi_{O},$
(20)

where $\phi_{N}$ and $\phi_{O}$ are constants defining the proportion of the
different species, whatever the chemical history of the pollutants reaching
the street canyon. We note that $\phi_{N}$ and $\phi_{O}$ can be easily
computed from the results of the passive model providing the concentrations
$[\cdot]^{*}$ (Section 1.1) which take into account all the pollutant
contributions reaching the canyon (i.e. both direct emissions and transported
pollutants).

Combining Eqs. 18 to 20 (see e.g., Soulhac et al. (2011)) provides the
solution:

$[\ce{NO_{2}}]\\!^{\infty}=\dfrac{b-\sqrt{b^{2}-4c}}{2}$ (21)

with:

$\left\\{\begin{array}[]{l}b=\dfrac{k_{1}}{k_{3}}+[\ce{O_{3}}]\\!^{*}+[\ce{NO}]\\!^{*}+2[\ce{NO_{2}}]\\!^{*}=\dfrac{k_{1}}{k_{3}}+\phi_{N}+\phi_{O}\\\\[11.38092pt]
c=\left([\ce{O_{3}}]\\!^{*}+[\ce{NO_{2}}]\\!^{*}\right)\left([\ce{NO}]\\!^{*}+[\ce{NO_{2}}]\\!^{*}\right)=\phi_{O}.\phi_{N}\end{array}\right.$
(22)

Eq. 22 illustrates that NO_2 concentration depends only on $k_{1}/k_{3}$,
$\phi_{N}$ and $\phi_{O}$. This highlights that the chemical history of the
background concentration, that is included in passive concentrations
$[\cdot]\\!^{*}$ by equations 15 to 17, has no influence on the
photostationary solution because this solution corresponds to an infinite
reaction time, which offsets the initial repartition between NO, NO_2 and O_3.

Once $[\ce{NO_{2}}]\\!^{\infty}$ is known, Eqs. 19 and 20 provide
$[\ce{NO}]\\!^{\infty}$ and $[\ce{O_{3}}]\\!^{\infty}$.

###  1.3 Non-photostationary chemical model

To find the general solution for the full chemical street model (Eq.13), we
take the difference between Eqs. 13 and 14:

$-\dfrac{[\ce{NO_{2}}]-[\ce{NO_{2}}]\\!^{*}}{\tau_{s}}+k_{3}[\ce{NO}][\ce{O_{3}}]-k_{1}[\ce{NO_{2}}]=0$
(23)

By introducing Eqs. 19 and 20 in Eq. 23:

$-\dfrac{[\ce{NO_{2}}]-[\ce{NO_{2}}]\\!^{*}}{\tau_{s}}+k_{3}\left([\ce{NO}]\\!^{*}+[\ce{NO_{2}}]\\!^{*}-[\ce{NO_{2}}]\right)\left([\ce{O_{3}}]\\!^{*}+[\ce{NO_{2}}]\\!^{*}-[\ce{NO_{2}}]\right)-k_{1}[\ce{NO_{2}}]=0$
(24)

and rearranging, the equation for $[\ce{NO_{2}}]$ is finally:

$[\ce{NO_{2}}]^{2}-b^{\prime}[\ce{NO_{2}}]+c^{\prime}=0$ (25)

with

$\left\\{\begin{array}[]{l}b^{\prime}=b+\dfrac{1}{k_{3}\tau_{s}}\\\\[11.38092pt]
c^{\prime}=c+\dfrac{[\ce{NO_{2}}]\\!^{*}}{k_{3}\tau_{s}}\end{array}\right.$
(26)

We obtain the non-photostationary solution for $[\ce{NO_{2}}]$ is then:

$[\ce{NO_{2}}]=\dfrac{b^{\prime}-\sqrt{{b^{\prime}}^{2}-4c^{\prime}}}{2}$ (27)

This expression is very similar to the photo-chemical model implemented in
OSPM (Palmgren et al., 1996; Berkowicz et al., 1997) but generalized for a
street canyon with a longitudinal advection velocity $U$ and a vertical
turbulent exchange rate $u_{d}$, therefore suitable for implementation in
street network models. Also in this case, once $[\ce{NO_{2}}]$ is known, Eqs.
19 and 20 provide $[\ce{NO}]$ and $[\ce{O_{3}}]$.

The solution in Eqs. 26-27 can be discussed according to the asymptotic values
for the street residence time scale $\tau_{s}$, as shown in Fig. 1. If
$\tau_{s}\rightarrow 0$ (i.e. $\tau_{s}\ll\tau_{1}$ and $\tau_{3}$ ),
according to Eq. 24, we have that
$[\ce{NO_{2}}]\rightarrow[\ce{NO_{2}}]\\!^{*}$. It means that the pollutants
have no time to react and the concentration of the chemical species is
provided by the passive solution (Eqs. 15-17). On the other hand, if
$\tau_{s}$ tends to infinity (i.e. $\tau_{s}\gg\tau_{1}$ and $\tau_{3}$ ),
then $b=b^{\prime}$ and $c=c^{\prime}$ in Eq. 26 and the concentration
$[\ce{NO_{2}}]$ tends to $[\ce{NO_{2}}]\\!^{\infty}$. It means that the
pollutants have an infinite time to react and the final concentration of the
chemical species is provided by the photostationary solution (Eq. 21).

For intermediate values of $\tau_{s}$, the solution is in-between, with only a
partial conversion from NO to NO_2, compared to the photostationary limit. We
can remark that, unlike the photostationary case, Eq. 26 includes
$[\ce{NO_{2}}]\\!^{*}$ independently of the constants $\phi_{N}$ and
$\phi_{O}$. This adds a dependence of the solution on the chemical history of
the background concentration.

Figure 1: Trend of the non-photostationary solution ($[\ce{NO_{2}}]$) towards
the solution for the passive model ($[\ce{NO_{2}}]^{*}$) and towards the
photostationary solution ($[\ce{NO_{2}}]^{\infty}$) as a function of the
characteristic time of advective transport ($\tau_{s}$), and of the two
characteristic times of reaction. Panel (a) shows different curves as a
function of $\tau_{1}$, indicated in the legend and by the dashed vertical
lines. Similarly, panel (b) shows the curves as a function of $\tau_{3}$
indicated in the legend and by the dashed vertical lines.

To provide further interpretations, Eq. 24 can be rewritten in the form:

$\underbrace{[\ce{NO_{2}}]\\!^{*}}_{\alpha}\chi^{2}-\underbrace{\left(\dfrac{k_{1}}{k_{3}}+[\ce{O_{3}}]\\!^{*}+[\ce{NO}]\\!^{*}+\dfrac{1}{k_{3}\tau_{s}}\right)}_{\beta}\chi+\underbrace{\left(\dfrac{[\ce{NO}]\\!^{*}[\ce{O_{3}}]\\!^{*}}{[\ce{NO_{2}}]\\!^{*}}-\dfrac{k_{1}}{k_{3}}\right)}_{\gamma}=0$
(28)

with

$\chi=\dfrac{[\ce{NO_{2}}]-[\ce{NO_{2}}]\\!^{*}}{[\ce{NO_{2}}]\\!^{*}}=\dfrac{\beta-\sqrt{\beta^{2}-4\alpha\gamma}}{2\alpha}.$
(29)

where $\chi$ represents the rate of increase of NO_2 concentration with
respect to its passive value $[\ce{NO_{2}}]\\!^{*}$. The term $\gamma$ can be
seen as a quantification of the non-photostationarity of the passive
concentrations of NO, NO_2 and O_3. Consequently, if the passive
concentrations are already close to the photostationary equilibrium
($\gamma\simeq 0$), then the solution tends to $\chi\simeq 0$, which means
that the final concentration is close to the passive one.

###  1.4 Validation strategy

As a further step, we test the analytical solutions presented in the previous
sections against field data. In doing so, we will consider three different
models: the first one (Model 1) is the photostationary model (Eq. 21) where
the transformation rates $k_{1}$ and $k_{3}$ are assumed to be constant with
time, whatever the temperature and radiative conditions. This solution can be
adopted when the meteorological information (temperature, intensity of solar
radiation) is missing and to minimize the computational cost. The sensitivity
of the model to the value adopted for the $k_{1}/k_{3}$ ratio is discussed in
the following section. The second one (Model 2) is a photostationary model
with transformation rates that vary during the day according to the
meteorological conditions. To this aim, Eq. 4 is applied to estimate $k_{1}$
and $k_{3}$ with the parameters $T$, $Cld$ and $\zeta$ varying over the day.
The third one (Model 3) is the model derived for the non-photostationary
conditions in Eq. 27, with parameters $k_{1}$ and $k_{3}$ again varying during
the day.

The models presented in sections 1.2 and 1.3 are designed to apply the
chemical scheme (Eqs. 1-3) as a post-calculation after the application of a
transport and dispersion model able to provide the advected concentrations in
the street, i.e. $[\ce{NO}]\\!^{*}$, $[\ce{NO}_{2}]\\!^{*}$, and
$[\ce{O}_{3}]\\!^{*}$. In order to validate only the chemical models, avoiding
the influence of errors due to the dispersion simulation, we have considered a
virtual perfect dispersion model by using the measured concentrations in the
streets as input data for the chemical models. According to Eqs. 21 and 27,
the results of the photochemical models depend on $k_{1}/k_{3}$, $\phi_{N}$,
$\phi_{O}$ and $[\ce{NO_{2}}]^{*}$. As mentioned above, the ratio
$k_{1}/k_{3}$ can be taken as a constant or estimated from meteorological data
and as a function of time. Equations 19 and 20 show that the conserved
quantities $\phi_{N}$ and $\phi_{O}$ can be computed directly from the
measured concentrations at the monitoring stations, which correspond to
$[\ce{NO}]$, $[\ce{NO}_{2}]$, and $[\ce{O}_{3}]$. Once $\phi_{O}$ is known,
$[\ce{NO_{2}}]^{*}$ results from Eq. 20 by subtracting the measured background
concentration of ozone (see Eq. 17). The other parameter of the non-
photostationary model is the time scale $\tau_{s}$. In the validation, this
parameter was adjusted to optimize the correlation coefficient between the
model and measured concentrations. The resulting values will be discussed in
the following section.

##  2 On-site measurements

The field data were measured during the LYON6 campaign which took place
between the 9th and the 24th July 2001 in the 6th arrondissement in Lyon
(France) and was handled by COPARLY (Comité de Coordination pour le contrôle
de la Pollution Atmosphérique), the local authority for traffic and air
pollution management, in collaboration with the Fluid Mechanics and Acoustics
Laboratory (LMFA) in École Centrale de Lyon. The campaign consisted of local
measurements of vehicular traffic, air pollution and weather conditions.

The meteorological data were collected by two stations within the urban area
(Fig. 2) and by a third one located 7 km from the studied district, and
positioned away from any building that could directly influence the
measurements. To have a representative dataset over the study district, the
measurements from the three different stations were combined together. The
reference temperature was measured by the sensors located within the urban
area. To avoid local effects, cloud cover, precipitation, wind speed and wind
direction were provided by the station outside the urban area. However, a
correction to the wind intensity to take into account the difference in
surface roughness was applied as detailed in Soulhac et al. (2012). The
temporal evolution of the resulting meteorological dataset is represented in
Fig. 3.

Hourly concentration of nitrogen monoxide, nitrogen dioxide and ozone were
measured by three monitoring stations, referred to as ‘Station 1’, ‘Station 2’
and ‘Station 3’. Station 1 was located in a busy street canyon. Station 2 and
Station 3 were located inside school courtyards, far away from polluting
source. In this regard, we point out that the models derived in Section 1 are
valid for both street canyons and urban courtyards as the fundamental
assumption underlying the box model (Eq. 6-8) is the decoupling between the
dynamics in the street and the dynamics above the roofs (Salizzoni et al.,
2011). The urban courtyard differs from the street canyon by the absence of
direct emissions and street intersections at the ends. For courtyards,
therefore, the wash-out time scale $\tau_{s}$ (Eq. 11) corresponds to the rate
of vertical exchange at roof level ($\tau_{h}$) and $Q_{NO}$ and $Q_{NO_{2}}$
are equal to zero.

In Station 2 and 3, the analyzers were placed at 2 m from the ground and in
the middle of the courtyard. The concentration was measured over approximately
6 days (see Fig. 3). In Station 1, the analyzer was also placed 2 m off the
ground but a few centimeters from a building wall, and the concentration was
measured over 15 days. In real street canyons there are several factors (e.g.,
traffic, vegetation, building geometries) that increase the mixing of
pollutants therefore inducing a concetration field that is more homogeneous
than that observed in controlled case studies in wind tunnels and numerical
simulations, where a highly spatially inhomogeneous concentration field is
predicted (Buccolieri et al., 2009; Marucci and Carpentieri, 2019; Fellini et
al., 2020). For this reason, and considering that the focus of the proposed
models is on hourly averaged pollution, the concentration in the canyon can be
assumed sufficiently homogeneous and therefore less sensitive to the
positioning of the sensor within the street. This is in line with previous
validation studies of street network models (Soulhac et al., 2012, 2017) in
which an influence of the sensor position on the agreement between the model
and the measurements was not observed.

To estimate the concentration levels of background pollutants, other three
monitoring stations located outside the district were used. Refer to Soulhac
et al. (2012) for a complete description of the measurement campaign and
simulation set-up.

Figure 2: Location of the three meteorological stations, of the suburban
stations for measuring pollution background and of the three pollution
monitoring stations inside the study district. Figure 3: Temporal evolution of
the meteorological parameters for the two-weeks campaign obtained by
integrating the data from the different meteorological stations.

##  3 Results

The meteorological data collected during the field campaign directly provide
the temporal evolution of the ratio $k_{1}/k_{3}$, estimated using Eq. 4. As
shown in Fig. 4-a, this ratio is far from being constant with time, and varies
from a maximum of about 0.9 $\mu mol/m^{3}$ (20 ppb) and a minimum close to 0
during the night, with an average value of 0.3 $\mu mol/m^{3}$ (6.8 ppb). This
is due to the variation over time of temperature, cloud coverage and solar
elevation (see Eq. 4). As stated in Eq. 18, this ratio equals the ratio
$[\ce{NO}][\ce{O3}]/[\ce{NO2}]$ when the pollutants are in photostationary
equilibrium. By using the measurements from the three pollution monitoring
stations, we test this condition in Fig. 4-b and c. Results show that, for
Station 2 and Station 3, the ratio $[\ce{NO}][\ce{O3}]/[\ce{NO2}]$ agrees well
with the trend of the ratio $k_{1}/k_{3}$. This is in line with the analysis
performed in Fig. 1: in sites sheltered from direct vehicular emissions and
with long residence times, the photostationary equilibrium (Eq. 18) is a
reliable assumption. Conversely, for the busy street canyon (Station 1), the
ratio $[\ce{NO}][\ce{O3}]/[\ce{NO2}]$ is generally higher than $k_{1}/k_{3}$
(Fig. 4-c). This is due to the fact that, at the emission, $[\ce{NO_{X}}]$ are
mainly constituted by $\ce{NO}$, that is progressively transformed in
$\ce{NO_{2}}$ until photostationary equilibrium is reached. For this reason,
close to the source the ratio $[\ce{NO}][\ce{O3}]/[\ce{NO2}]$ is expected to
be higher than that corresponding to the equilibrium.

Figure 4: Evolution of the ratio $k_{1}/k_{3}$ as predicted by Eq. 4 (a), and
comparison with the ratio $[\ce{NO}][\ce{O3}]/[\ce{NO2}]$ for the site within
the street canyon -Station 1- (b) and for the two sites in the courtyards
-Station 2 and Station 3- (c).

This first analysis shows that the photostationary model has some limitations
when applied to busy street canyons with direct vehicular emissions.

To clarify this point, we assess the performance of the three photochemical
models (derived in Section 1 and retrieved in Section 1.4) in predicting the
concentration of $[\ce{NO_{2}}]$ in the three measurement stations. For the
sake of comparison, we include in our analysis the empirical model developed
by Dixon et al. (2001). Starting from the work of Derwent and Middleton
(1996), Dixon et al. (2001) developed a new NO_X-NO_2 relationship based on a
larger dataset collected across multiple sites:

$\frac{\ce{[NO_{2}]}}{[\ce{NO_{X}}]}=A+B\log([\ce{NO_{X}}])+C\log([\ce{NO_{X}}])^{2}+D\log([\ce{NO_{X}}])^{3}+E\log([\ce{NO_{X}}])^{4}.$
(30)

where the polynomial constants take the following values at urban sites:
$A=-3.08308$, $B=+7.472477$, $C=-5.11636$, $D=+1.381938$, $E=-0.12919$. For
$\ce{[NO_{2}]}<15$ ppb, one should use $\ce{[NO_{2}]}/[\ce{NO_{X}}]=60$ ppb.
The model is reffered as DDM model in the following.

The four models are applied in a quasi-steady approximation, therefore
describing temporal evolution of all variables (meteorological, emissions,
background concentration) as the succession of stationary states lasting 1
hour.

The simplest photostationary model (Model 1) assumes that the ratio
$k_{1}/k_{3}$ is constant over time and for the different urban locations.
According to Seinfeld (1986), we assume as a typical value for this ratio 10
ppb. This value is in line with the time average of the trend estimated by
means of Eq. 4 and reported in Fig. 4.a. Moreover, we have tested the
sensitivity of the model results to this constant by calculating the relative
variation of the mean $\ce_{NO_{2}}$ concentration over the simulated period
for different $k_{1}/k_{3}$ ratios. The results are reported in Table 1. The
concentration is very sensitive to variations in the ratio $k_{1}/k_{3}$ and,
as stated by Eq. 18, it increases as the ratio decreases. Moreover, the
sensitivity of the model is higher for Station 2 and Station 3 with respect to
Station 1.

| $k_{1}/k_{3}$ | 2 | 5 | 10 | 15 | 20 | 25
---|---|---|---|---|---|---|---
Relative variation of $\ce{NO}_{2}$ concentration | Station 1 | 14% | 8% |  Ref (0 %) | -7% | -14% | -20%
Station 2 | 18% | 11% | -10% | -19% | -28%
Station 3 | 20 % | 12 % | -11 % | -21% | -31%
Table 1: Sensitivity of the output from Model 1 as a function of the ratio
$k_{1}/k_{3}$.

The results provided by the the three photo-chemical models are shown in Fig.
5, where measured and simulated $\ce{NO2}$ concentrations are plotted for the
three monitoring stations. Moreover, following Chang and Hanna (2004), we
assessed the performance of the models by means of multiple statistical
indices:

  * 1.

the Relative Error:
RE$=\overline{\left(\frac{2|C_{m}-C_{p}|}{C_{m}+C_{p}}\right)}$;

  * 2.

the Fractional Bias:
FB$=2(\overline{C_{m}}-\overline{C_{p}})/(\overline{C_{m}}+\overline{C_{p}})$;

  * 3.

the Normal Mean Square Error:
NMSE$=\overline{(C_{m}-C_{p})^{2}}/\overline{C_{m}}\,\overline{C_{p}}$;

  * 4.

the Mean Geometric bias:
MG$=\exp[\overline{\ln(C_{m})}-\overline{\ln(C_{p})}]$;

  * 5.

the Geometrical mean squared Variance:
VG$=\exp[\overline{\ln(C_{m})-\ln(C_{p})}^{2}]$;

  * 6.

the correlation coefficient:
R=$\frac{\overline{(C_{m}-\overline{C_{m}})(C_{p}-\overline{C_{p}})}}{\sigma_{Cm}\sigma_{Cp}}$;

  * 7.

the ‘fraction in a factor of 2’: fraction of the data for which $0.5\leq
C_{p}/C_{m}\leq 2$,

where $C_{m}$ and $C_{p}$ are the measured and predicted concentrations, and
$\sigma_{Cm}$ and $\sigma_{Cp}$ their standard deviations. A perfect model
would have MG, VG, R and FAC2=1, and FB, NMSE=0. Following Chang and Hanna
(2004), the performances of a dispersion model can be defined as ‘good’ when
the following criteria are satisfied: $|\mbox{FB}|\leq 0.3$,
$\sqrt{\mbox{NMSE}}\leq 2$, $0.7\leq$ MG $\leq 1.3$, VG $\leq 1.6$, FAC2 $\geq
0.5$. In Hanna and Chang (2012), the same authors suggest a relaxation of
these thresholds for application in urban areas. While all the statistical
indices are used to assess the performance of the four models, we recall that
only the correlation coefficient $R$ is used as a criterion to uniquely
determine the value $\tau_{s}$ that maximizes the correlation between the
results of the non-photostationary model (Model 3) and the experimental data
(see Section 1.4).

Panel a in Fig. 5 compares the measured and predicted concentrations for
Station 1, which corresponds to the busy urban canyon with vehicular
emissions. The photostationary model with constant $k_{1}/k_{3}$ (Model 1)
predicts with a good approximation the measured data but tends to overestimate
$\ce{NO2}$ for mean to high concentration values. This is confirmed by the
negative fractional bias in Table 2. This overestimation is also observed for
low concentrations when a variable $k_{1}/k_{3}$ ratio is implemented in the
photostationary model (Model 2). The slight loss of performance of Model 2
compared to Model 1 is highlighted by the statistical metrics in Table 2, with
the increase in the absolute value of the fractional bias and the decrease in
MG from 0.91 to 0.84. On the other hand, a noticeable improvement in the
prediction is observed by applying the non-photostationary model (Model 3).
The value of $\tau_{s}$ that maximizes the correlation coefficient $R$ is
found to be equal to 89 s. This value is comparable with the time scale of the
chemical reactions and thus confirms the need to adopt a non-photostationary
solution (see Fig. 1). The scatter plot in Fig. 5 shows that the dispersion of
the points around the bisector decreases with respect the photostationary
models, as well as the relative error (RE) in Table 2. Finally, the approach
proposed by Dixon et al. (2001) (DDM model) fairly predicts low to medium
concentrations but tends to cut the highest concentration values. The same
trend was observed by Vardoulakis et al. (2007) by applying the model by
Derwent and Middleton (1996), whose prediction is almost comparable to the DDM
model for concentration up to 500 ppb. Vardoulakis et al. (2007) suggested
that this underestimation of the NO_2 concentration was because the empirical
relationship was derived using measures that do not always reflect the typical
NO_2/NO_X vehicle emission ratio of the case study.

Panels b and c in Fig. 5 compares the measured and predicted concentrations
for Station 2 and 3, which correspond to the stations located within
courtyards. Model 1 provides slightly scattered results and noticeably
underestimates NO_2 concentration in Station 3 (FB=0.18 in Table 2). The
adoption of a variable $k_{1}/k_{3}$ ratio (Model 2) improves the performance
of the photostationary model. This is highlighted by the reduction in the
relative error (RE), and by the trend towards 1 of the R metric. On the other
hand, the adoption of the non-photostationary model (Model 3) does not bring
further improvements. For both monitoring stations, the value of $\tau_{s}$
that maximizes the correlation coefficient $R$ tends to infinity. As depicted
in Fig. 1, this means that the results provided by the non-photostationary
model ($[\ce{NO}_{2}]$) correspond to those provided by the photostationary
one ($[\ce{NO_{2}}]\\!^{\infty}$). This suggests that the pollutant
concentrations in these sites already reached the photochemical equilibrium,
as foreseen in Fig. 4). Finally, the approach proposed by Dixon et al. (2001)
performs worse than the three physically-based models also for Station 2 and
3. Differently from Station 1, here the predictions are significantly
underestimated (FB=0.21 and 0.28) also for low to medium concentration values.

Despite differences in performance, we finally notice that the statistical
metrics in Table 2 are within the validity ranges suggested by Chang and Hanna
(2004) for all models.

In addition to NO_2 concentrations, the models derived in Section 1 (and
presented in Section 1.4) provide NO and O_3 concentrations. Fig. 6 shows
that, in a general way, the concentrations of NO are well simulated by the
proposed models. As observed for NO_2 concentrations, the non-photostationary
model (Model 3) outperforms Model 2 for Station 1, while for Stations 2 and 3
adopting the photostationary model with variable $k_{1}/k_{3}$ is sufficient
to maximize correlation. The DDM model shows good agreement for Station 1 for
high concentration values, while for medium-low values (between 50 and 100
ppb) the error is significant. In accordance with Eq. 19, this behaviour
reflects the results found for NO_2 concentrations. We also observe that for
stations 2 and 3, the DDM model fails to reproduce low NO concentrations.
Regarding the prediction of ozone, the same considerations made for NO and
NO_2 are valid: Model 3 brings significant improvements in results for the
busy street canyon (Station 1), while the photostationary assumption (Model 2)
holds when predicting concentrations in stations far from direct emissions
(Station 2 and 3).

| RE | FB | NMSE | MG | VG | R | FAC2
---|---|---|---|---|---|---|---
| Station 1 |  [HTML]EBF1DE0.18 |  [HTML]EBF1DE-0.12 |  [HTML]EBF1DE0.05 |  [HTML]EBF1DE0.91 |  [HTML]EBF1DE1.01 |  [HTML]EBF1DE0.96 |  [HTML]EBF1DE1.00
| Station 2 |  [HTML]DCE6F10.13 |  [HTML]DCE6F10.04 |  [HTML]DCE6F10.03 |  [HTML]DCE6F11.06 |  [HTML]DCE6F11.00 |  [HTML]DCE6F10.94 |  [HTML]DCE6F11.00
M1 | Station 3 |  [HTML]F2DCDB0.17 |  [HTML]F2DCDB0.18 |  [HTML]F2DCDB0.07 |  [HTML]F2DCDB1.18 |  [HTML]F2DCDB1.03 |  [HTML]F2DCDB0.96 |  [HTML]F2DCDB1.00
| Station 1 |  [HTML]EBF1DE0.18 |  [HTML]EBF1DE-0.15 |  [HTML]EBF1DE0.04 |  [HTML]EBF1DE0.84 |  [HTML]EBF1DE1.03 |  [HTML]EBF1DE0.97 |  [HTML]EBF1DE1.00
| Station 2 |  [HTML]DCE6F10.07 |  [HTML]DCE6F1-0.05 |  [HTML]DCE6F10.02 |  [HTML]DCE6F10.95 |  [HTML]DCE6F11.00 |  [HTML]DCE6F10.97 |  [HTML]DCE6F11.00
M2 | Station 3 |  [HTML]F2DCDB0.10 |  [HTML]F2DCDB0.10 |  [HTML]F2DCDB0.02 |  [HTML]F2DCDB1.09 |  [HTML]F2DCDB1.01 |  [HTML]F2DCDB0.99 |  [HTML]F2DCDB1.00
| Station 1 |  [HTML]EBF1DE0.09 |  [HTML]EBF1DE0.00 |  [HTML]EBF1DE0.01 |  [HTML]EBF1DE1.01 |  [HTML]EBF1DE1.00 |  [HTML]EBF1DE0.99 |  [HTML]EBF1DE0.99
| Station 2 |  [HTML]DCE6F10.07 |  [HTML]DCE6F1-0.05 |  [HTML]DCE6F10.02 |  [HTML]DCE6F10.95 |  [HTML]DCE6F11.00 |  [HTML]DCE6F10.97 |  [HTML]DCE6F11.00
M3 | Station 3 |  [HTML]F2DCDB0.09 |  [HTML]F2DCDB0.10 |  [HTML]F2DCDB0.02 |  [HTML]F2DCDB1.09 |  [HTML]F2DCDB1.01 |  [HTML]F2DCDB0.99 |  [HTML]F2DCDB1.00
| Station 1 |  [HTML]EBF1DE0.21 |  [HTML]EBF1DE0.05 |  [HTML]EBF1DE0.09 |  [HTML]EBF1DE1.04 |  [HTML]EBF1DE1.00 |  [HTML]EBF1DE0.86 |  [HTML]EBF1DE1.00
| Station 2 |  [HTML]DCE6F10.27 |  [HTML]DCE6F10.21 |  [HTML]DCE6F10.08 |  [HTML]DCE6F11.29 |  [HTML]DCE6F11.07 |  [HTML]DCE6F10.93 |  [HTML]DCE6F11.00
DDM | Station 3 |  [HTML]F2DCDB0.32 |  [HTML]F2DCDB0.28 |  [HTML]F2DCDB0.12 |  [HTML]F2DCDB1.38 |  [HTML]F2DCDB1.11 |  [HTML]F2DCDB0.96 |  [HTML]F2DCDB1.00
Table 2: Performance statistics for the four investigated models (M1, M2, M3,
DDM) in predicting the measured NO_2 concentration in the three measurement
stations. Station 1 is to the busy street canyon, while Station 2 and 3 are
courtyards. Figure 5: Comparison between the measured $\ce{NO_{2}}$
concentrations in Station 1 (a), Station 2 (b) and Station 3 (c) against the
concentrations computed with the three different photochemical models (M1, M2
and M3) and with the Derwent-Middleton model. Each point corresponds to one
hour average concentration. Station 1 is to the busy street canyon, while
Station 2 and 3 are courtyards. Figure 6: Comparison between the measured
$\ce{NO}$ concentrations in Station 1 (a), Station 2 (b) and Station 3 (c)
against the concentrations computed with the three different photochemical
models (M1, M2 and M3) and with the Derwent-Middleton model. Each point
corresponds to one hour average concentration. Station 1 is to the busy street
canyon, while Station 2 and 3 are courtyards. Figure 7: Comparison between the
measured $\ce{O_{3}}$ concentrations in Station 1 (a), Station 2 (b) and
Station 3 (c) against the concentrations computed with the three different
photochemical models (M1, M2 and M3). Each point corresponds to one hour
average concentration. Station 1 is to the busy street canyon, while Station 2
and 3 are courtyards.

##  4 Conclusions

In this work we have derived different box models to simulate the
concentration of $\ce{NO}$, $\ce{NO_{2}}$ and $\ce{O_{3}}$ in a street canyon.
Starting from a mass balance in the street, we have first defined a model for
a passive tracer, then for chemical species at photostationary equilibrium and
finally for the non-photostationary state. Prediction from the simulations
were compared with concentration measurements acquired during a field
campaign. Results showed that the photostationary models adequately reproduce
the pollutant concentration in canyons far from direct vehicular emissions.
However, the implementation of a parameterization for the reaction rates
according to the metereological conditions is crucial. In busy streets, the
photostationary equilibrium is not yet fully achieved and the non-
photostationary model performs better. Finally, empirical models such as
Dixon-Derwent-Middleton relationship fail to reproduce concentration peaks in
busy canyons and underestimate NO_2 concentrations at photochemical
equilibrium. These results show that the photostationary model with
meteorology-based parameters is satisfactory in reproducing the concentrations
in different urban scenarios. However, the non-photostationary model brings
significant improvements in busy street canyons.

Differently from previous studies, the chemical models presented here include
a description of the longitudinal and vertical ventilation processes and are
therefore suitable for application to a network of streets with pollutant
fluxes at street intersections. This paves the way for their implementation in
operational street network models such as Sirane.

Furthermore, the adoption of a coherent formulation and the analysis of the
balance equations in terms of characteristic transport and reaction times
clarify the processes involved, the physico-chemical assumptions, and the
limits of their validity. This information is critical to understanding,
developing, and improving the parametric models used in existing air quality
simulation software. In this regard, a desirable development is the treatment
of non-photostationarity outside the urban canopy, i.e. over rooftops or on
high-emission roads in open terrain.

Finally, we notice that the diffusion of low-cost sensors provides nowadays
large databases of pollutant concentration in cities. The inclusion of more
accurate transport and reaction models in operational tools for urban air
pollution is in line with this growing availability of data that can be used
for validation and data assimilation. In this sense, this work highlights the
feasibility of implementing non-photostationary models in simulation tools at
the city scale and paves the way for further application and validation.

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[e:mainplus,color=Gray,->] (d-3-6) to (d-5-6);
[e:mainplus,color=Gray,->] (d-5-6) to (d-7-6);
[e:mainplus,color=Gray,->] (d-7-6) to (d-9-6);
[e:mainplus,color=Gray,->] (d-9-6) to (d-11-6);
[e:mainplus,color=Gray,->] (d-11-6) to (d-13-6);
[e:mainplus,color=Gray,->] (d-13-6) to (d-15-6);
[e:mainplus,color=Gray,->] (d-15-6) to (d-17-6);
[e:mainplus,color=Gray,->] (d-17-6) to (d-19-6);
[e:mainplus,color=Gray,->] (d-19-6) to (d-21-6);
[e:mainplus,color=Gray,->] (d-21-6) to (d-23-6);
[e:mainplus,color=Gray] (d-23-6) to (d-24-6);

[e:mainplus,color=Gray,->] (d-1-7) to (d-3-7);
[e:mainplus,color=Gray,->] (d-3-7) to (d-5-7);
[e:mainplus,color=Gray,->] (d-5-7) to (d-7-7);
[e:mainplus,color=Gray,->] (d-7-7) to (d-9-7);
[e:mainplus,color=Gray,->] (d-9-7) to (d-11-7);
[e:mainplus,color=Gray,->] (d-11-7) to (d-13-7);
[e:mainplus,color=Gray,->] (d-13-7) to (d-15-7);
[e:mainplus,color=Gray,->] (d-15-7) to (d-17-7);
[e:mainplus,color=Gray,->] (d-17-7) to (d-19-7);
[e:mainplus,color=Gray,->] (d-19-7) to (d-21-7);
[e:mainplus,color=Gray,->] (d-21-7) to (d-23-7);
[e:mainplus,color=Gray] (d-23-7) to (d-24-7);

[e:mainplus,color=Gray,->] (d-1-8) to (d-3-8);
[e:mainplus,color=Gray,->] (d-3-8) to (d-5-8);
[e:mainplus,color=Gray,->] (d-5-8) to (d-7-8);
[e:mainplus,color=Gray,->] (d-7-8) to (d-9-8);
[e:mainplus,color=Gray,->] (d-9-8) to (d-11-8);
[e:mainplus,color=Gray,->] (d-11-8) to (d-13-8);
[e:mainplus,color=Gray,->] (d-13-8) to (d-15-8);
[e:mainplus,color=Gray,->] (d-15-8) to (d-17-8);
[e:mainplus,color=Gray,->] (d-17-8) to (d-19-8);
[e:mainplus,color=Gray,->] (d-19-8) to (d-21-8);
[e:mainplus,color=Gray,->] (d-21-8) to (d-23-8);
[e:mainplus,color=Gray] (d-23-8) to (d-24-8);

[e:mainplus,color=Gray,->] (d-1-9) to (d-3-9);
[e:mainplus,color=Gray,->] (d-3-9) to (d-5-9);
[e:mainplus,color=Gray,->] (d-5-9) to (d-7-9);
[e:mainplus,color=Gray,->] (d-7-9) to (d-9-9);
[e:mainplus,color=Gray,->] (d-9-9) to (d-11-9);
[e:mainplus,color=Gray,->] (d-11-9) to (d-13-9);
[e:mainplus,color=Gray,->] (d-13-9) to (d-15-9);
[e:mainplus,color=Gray,->] (d-15-9) to (d-17-9);
[e:mainplus,color=Gray,->] (d-17-9) to (d-19-9);
[e:mainplus,color=Gray,->] (d-19-9) to (d-21-9);
[e:mainplus,color=Gray,->] (d-21-9) to (d-23-9);
[e:mainplus,color=Gray] (d-23-9) to (d-24-9);

[e:mainplus,color=Gray,->] (d-1-10) to (d-3-10);
[e:mainplus,color=Gray,->] (d-3-10) to (d-5-10);
[e:mainplus,color=Gray,->] (d-5-10) to (d-7-10);
[e:mainplus,color=Gray,->] (d-7-10) to (d-9-10);
[e:mainplus,color=Gray,->] (d-9-10) to (d-11-10);
[e:mainplus,color=Gray,->] (d-11-10) to (d-13-10);
[e:mainplus,color=Gray,->] (d-13-10) to (d-15-10);
[e:mainplus,color=Gray,->] (d-15-10) to (d-17-10);
[e:mainplus,color=Gray,->] (d-17-10) to (d-19-10);
[e:mainplus,color=Gray,->] (d-19-10) to (d-21-10);
[e:mainplus,color=Gray,->] (d-21-10) to (d-23-10);
[e:mainplus,color=Gray] (d-23-10) to (d-24-10);

[e:mainplus,color=Gray,->] (d-1-11) to (d-3-11);
[e:mainplus,color=Gray,->] (d-3-11) to (d-5-11);
[e:mainplus,color=Gray,->] (d-5-11) to (d-7-11);
[e:mainplus,color=Gray,->] (d-7-11) to (d-9-11);
[e:mainplus,color=Gray,->] (d-9-11) to (d-11-11);
[e:mainplus,color=Gray,->] (d-11-11) to (d-13-11);
[e:mainplus,color=Gray,->] (d-13-11) to (d-15-11);
[e:mainplus,color=Gray,->] (d-15-11) to (d-17-11);
[e:mainplus,color=Gray,->] (d-17-11) to (d-19-11);
[e:mainplus,color=Gray,->] (d-19-11) to (d-21-11);
[e:mainplus,color=Gray,->] (d-21-11) to (d-23-11);
[e:mainplus,color=Gray] (d-23-11) to (d-24-11);

[e:mainplus,color=Gray,->] (d-1-12) to (d-3-12);
[e:mainplus,color=Gray,->] (d-3-12) to (d-5-12);
[e:mainplus,color=Gray,->] (d-5-12) to (d-7-12);
[e:mainplus,color=Gray,->] (d-7-12) to (d-9-12);
[e:mainplus,color=Gray,->] (d-9-12) to (d-11-12);
[e:mainplus,color=Gray,->] (d-11-12) to (d-13-12);
[e:mainplus,color=Gray,->] (d-13-12) to (d-15-12);
[e:mainplus,color=Gray,->] (d-15-12) to (d-17-12);
[e:mainplus,color=Gray,->] (d-17-12) to (d-19-12);
[e:mainplus,color=Gray,->] (d-19-12) to (d-21-12);
[e:mainplus,color=Gray,->] (d-21-12) to (d-23-12);
[e:mainplus,color=Gray] (d-23-12) to (d-24-12);

[e:mainplus,color=BritishRacingGreen] (u-1-1) to (u-2-1);
[e:mainplus,color=BritishRacingGreen,<-] (u-2-1) to (u-4-1);
[e:mainplus,color=BritishRacingGreen,<-] (u-4-1) to (u-6-1);
[e:mainplus,color=BritishRacingGreen,<-] (u-6-1) to (u-8-1);
[e:mainplus,color=BritishRacingGreen,<-] (u-8-1) to (u-10-1);
[e:mainplus,color=BritishRacingGreen,<-] (u-10-1) to (u-12-1);
[e:mainplus,color=BritishRacingGreen,<-] (u-12-1) to (u-14-1);
[e:mainplus,color=BritishRacingGreen,<-] (u-14-1) to (u-16-1);
[e:mainplus,color=BritishRacingGreen,<-] (u-16-1) to (u-18-1);
[e:mainplus,color=BritishRacingGreen,<-] (u-18-1) to (u-20-1);
[e:mainplus,color=BritishRacingGreen,<-] (u-20-1) to (u-22-1);
[e:mainplus,color=BritishRacingGreen,<-] (u-22-1) to (u-24-1);

[e:mainplus,color=BritishRacingGreen] (u-1-2) to (u-2-2);
[e:mainplus,color=BritishRacingGreen,<-] (u-2-2) to (u-4-2);
[e:mainplus,color=BritishRacingGreen,<-] (u-4-2) to (u-6-2);
[e:mainplus,color=BritishRacingGreen,<-] (u-6-2) to (u-8-2);
[e:mainplus,color=BritishRacingGreen,<-] (u-8-2) to (u-10-2);
[e:mainplus,color=BritishRacingGreen,<-] (u-10-2) to (u-12-2);
[e:mainplus,color=BritishRacingGreen,<-] (u-12-2) to (u-14-2);
[e:mainplus,color=BritishRacingGreen,<-] (u-14-2) to (u-16-2);
[e:mainplus,color=BritishRacingGreen,<-] (u-16-2) to (u-18-2);
[e:mainplus,color=BritishRacingGreen,<-] (u-18-2) to (u-20-2);
[e:mainplus,color=BritishRacingGreen,<-] (u-20-2) to (u-22-2);
[e:mainplus,color=BritishRacingGreen,<-] (u-22-2) to (u-24-2);

[e:mainplus,color=BritishRacingGreen] (u-1-3) to (u-2-3);
[e:mainplus,color=BritishRacingGreen,<-] (u-2-3) to (u-4-3);
[e:mainplus,color=BritishRacingGreen,<-] (u-4-3) to (u-6-3);
[e:mainplus,color=BritishRacingGreen,<-] (u-6-3) to (u-8-3);
[e:mainplus,color=BritishRacingGreen,<-] (u-8-3) to (u-10-3);
[e:mainplus,color=BritishRacingGreen,<-] (u-10-3) to (u-12-3);
[e:mainplus,color=BritishRacingGreen,<-] (u-12-3) to (u-14-3);
[e:mainplus,color=BritishRacingGreen,<-] (u-14-3) to (u-16-3);
[e:mainplus,color=BritishRacingGreen,<-] (u-16-3) to (u-18-3);
[e:mainplus,color=BritishRacingGreen,<-] (u-18-3) to (u-20-3);
[e:mainplus,color=BritishRacingGreen,<-] (u-20-3) to (u-22-3);
[e:mainplus,color=BritishRacingGreen,<-] (u-22-3) to (u-24-3);

[e:mainplus,color=BritishRacingGreen] (u-1-4) to (u-2-4);
[e:mainplus,color=BritishRacingGreen,<-] (u-2-4) to (u-4-4);
[e:mainplus,color=BritishRacingGreen,<-] (u-4-4) to (u-6-4);
[e:mainplus,color=BritishRacingGreen,<-] (u-6-4) to (u-8-4);
[e:mainplus,color=BritishRacingGreen,<-] (u-8-4) to (u-10-4);
[e:mainplus,color=BritishRacingGreen,<-] (u-10-4) to (u-12-4);
[e:mainplus,color=BritishRacingGreen,<-] (u-12-4) to (u-14-4);
[e:mainplus,color=BritishRacingGreen,<-] (u-14-4) to (u-16-4);
[e:mainplus,color=BritishRacingGreen,<-] (u-16-4) to (u-18-4);
[e:mainplus,color=BritishRacingGreen,<-] (u-18-4) to (u-20-4);
[e:mainplus,color=BritishRacingGreen,<-] (u-20-4) to (u-22-4);
[e:mainplus,color=BritishRacingGreen,<-] (u-22-4) to (u-24-4);

[e:mainplus,color=BritishRacingGreen] (u-1-5) to (u-2-5);
[e:mainplus,color=BritishRacingGreen,<-] (u-2-5) to (u-4-5);
[e:mainplus,color=BritishRacingGreen,<-] (u-4-5) to (u-6-5);
[e:mainplus,color=BritishRacingGreen,<-] (u-6-5) to (u-8-5);
[e:mainplus,color=BritishRacingGreen,<-] (u-8-5) to (u-10-5);
[e:mainplus,color=BritishRacingGreen,<-] (u-10-5) to (u-12-5);
[e:mainplus,color=BritishRacingGreen,<-] (u-12-5) to (u-14-5);
[e:mainplus,color=BritishRacingGreen,<-] (u-14-5) to (u-16-5);
[e:mainplus,color=BritishRacingGreen,<-] (u-16-5) to (u-18-5);
[e:mainplus,color=BritishRacingGreen,<-] (u-18-5) to (u-20-5);
[e:mainplus,color=BritishRacingGreen,<-] (u-20-5) to (u-22-5);
[e:mainplus,color=BritishRacingGreen,<-] (u-22-5) to (u-24-5);

[e:mainplus,color=BritishRacingGreen] (u-1-6) to (u-2-6);
[e:mainplus,color=BritishRacingGreen,<-] (u-2-6) to (u-4-6);
[e:mainplus,color=BritishRacingGreen,<-] (u-4-6) to (u-6-6);
[e:mainplus,color=BritishRacingGreen,<-] (u-6-6) to (u-8-6);
[e:mainplus,color=BritishRacingGreen,<-] (u-8-6) to (u-10-6);
[e:mainplus,color=BritishRacingGreen,<-] (u-10-6) to (u-12-6);
[e:mainplus,color=BritishRacingGreen,<-] (u-12-6) to (u-14-6);
[e:mainplus,color=BritishRacingGreen,<-] (u-14-6) to (u-16-6);
[e:mainplus,color=BritishRacingGreen,<-] (u-16-6) to (u-18-6);
[e:mainplus,color=BritishRacingGreen,<-] (u-18-6) to (u-20-6);
[e:mainplus,color=BritishRacingGreen,<-] (u-20-6) to (u-22-6);
[e:mainplus,color=BritishRacingGreen,<-] (u-22-6) to (u-24-6);

[e:mainplus,color=BritishRacingGreen] (u-1-7) to (u-2-7);
[e:mainplus,color=BritishRacingGreen,<-] (u-2-7) to (u-4-7);
[e:mainplus,color=BritishRacingGreen,<-] (u-4-7) to (u-6-7);
[e:mainplus,color=BritishRacingGreen,<-] (u-6-7) to (u-8-7);
[e:mainplus,color=BritishRacingGreen,<-] (u-8-7) to (u-10-7);
[e:mainplus,color=BritishRacingGreen,<-] (u-10-7) to (u-12-7);
[e:mainplus,color=BritishRacingGreen,<-] (u-12-7) to (u-14-7);
[e:mainplus,color=BritishRacingGreen,<-] (u-14-7) to (u-16-7);
[e:mainplus,color=BritishRacingGreen,<-] (u-16-7) to (u-18-7);
[e:mainplus,color=BritishRacingGreen,<-] (u-18-7) to (u-20-7);
[e:mainplus,color=BritishRacingGreen,<-] (u-20-7) to (u-22-7);
[e:mainplus,color=BritishRacingGreen,<-] (u-22-7) to (u-24-7);

[e:mainplus,color=BritishRacingGreen] (u-1-8) to (u-2-8);
[e:mainplus,color=BritishRacingGreen,<-] (u-2-8) to (u-4-8);
[e:mainplus,color=BritishRacingGreen,<-] (u-4-8) to (u-6-8);
[e:mainplus,color=BritishRacingGreen,<-] (u-6-8) to (u-8-8);
[e:mainplus,color=BritishRacingGreen,<-] (u-8-8) to (u-10-8);
[e:mainplus,color=BritishRacingGreen,<-] (u-10-8) to (u-12-8);
[e:mainplus,color=BritishRacingGreen,<-] (u-12-8) to (u-14-8);
[e:mainplus,color=BritishRacingGreen,<-] (u-14-8) to (u-16-8);
[e:mainplus,color=BritishRacingGreen,<-] (u-16-8) to (u-18-8);
[e:mainplus,color=BritishRacingGreen,<-] (u-18-8) to (u-20-8);
[e:mainplus,color=BritishRacingGreen,<-] (u-20-8) to (u-22-8);
[e:mainplus,color=BritishRacingGreen,<-] (u-22-8) to (u-24-8);

[e:mainthin,color=Gray,->] (b-2-1) to (b-2-2);
[e:mainthin,color=Gray,->] (b-2-2) to (b-2-3);
[e:mainthin,color=Gray,->] (b-2-3) to (b-2-4);
[e:mainthin,color=Gray,->] (b-2-4) to (b-2-5);
[e:mainthin,color=Gray,->] (b-2-5) to (b-2-6);
[e:mainthin,color=Gray,->] (b-2-6) to (b-2-7);
[e:mainthin,color=Gray,->] (b-2-7) to (b-2-8);
[e:mainthin,color=Gray,->] (b-2-8) to (b-2-9);
[e:mainthin,color=Gray,->] (b-2-9) to (b-2-10);

[e:mainthin,color=Gray,->] (b-4-1) to (b-4-2);
[e:mainthin,color=Gray,->] (b-4-2) to (b-4-3);
[e:mainthin,color=Gray,->] (b-4-3) to (b-4-4);
[e:mainthin,color=Gray,->] (b-4-4) to (b-4-5);
[e:mainthin,color=Gray,->] (b-4-5) to (b-4-6);
[e:mainthin,color=Gray,->] (b-4-6) to (b-4-7);
[e:mainthin,color=Gray,->] (b-4-7) to (b-4-8);
[e:mainthin,color=Gray,->] (b-4-8) to (b-4-9);
[e:mainthin,color=Gray,->] (b-4-9) to (b-4-10);

[e:mainthin,color=Gray,->] (b-6-1) to (b-6-2);
[e:mainthin,color=Gray,->] (b-6-2) to (b-6-3);
[e:mainthin,color=Gray,->] (b-6-3) to (b-6-4);
[e:mainthin,color=Gray,->] (b-6-4) to (b-6-5);
[e:mainthin,color=Gray,->] (b-6-5) to (b-6-6);
[e:mainthin,color=Gray,->] (b-6-6) to (b-6-7);
[e:mainthin,color=Gray,->] (b-6-7) to (b-6-8);
[e:mainthin,color=Gray,->] (b-6-8) to (b-6-9);
[e:mainthin,color=Gray,->] (b-6-9) to (b-6-10);

[e:mainthin,color=Gray,->] (b-8-1) to (b-8-2);
[e:mainthin,color=Gray,->] (b-8-2) to (b-8-3);
[e:mainthin,color=Gray,->] (b-8-3) to (b-8-4);
[e:mainthin,color=Gray,->] (b-8-4) to (b-8-5);
[e:mainthin,color=Gray,->] (b-8-5) to (b-8-6);
[e:mainthin,color=Gray,->] (b-8-6) to (b-8-7);
[e:mainthin,color=Gray,->] (b-8-7) to (b-8-8);
[e:mainthin,color=Gray,->] (b-8-8) to (b-8-9);
[e:mainthin,color=Gray,->] (b-8-9) to (b-8-10);

[e:mainthin,color=Gray,->] (b-10-1) to (b-10-2);
[e:mainthin,color=Gray,->] (b-10-2) to (b-10-3);
[e:mainthin,color=Gray,->] (b-10-3) to (b-10-4);
[e:mainthin,color=Gray,->] (b-10-4) to (b-10-5);
[e:mainthin,color=Gray,->] (b-10-5) to (b-10-6);
[e:mainthin,color=Gray,->] (b-10-6) to (b-10-7);
[e:mainthin,color=Gray,->] (b-10-7) to (b-10-8);
[e:mainthin,color=Gray,->] (b-10-8) to (b-10-9);
[e:mainthin,color=Gray,->] (b-10-9) to (b-10-10);

[e:mainthin,color=Gray,->] (b-12-1) to (b-12-2);
[e:mainthin,color=Gray,->] (b-12-2) to (b-12-3);
[e:mainthin,color=Gray,->] (b-12-3) to (b-12-4);
[e:mainthin,color=Gray,->] (b-12-4) to (b-12-5);
[e:mainthin,color=Gray,->] (b-12-5) to (b-12-6);
[e:mainthin,color=Gray,->] (b-12-6) to (b-12-7);
[e:mainthin,color=Gray,->] (b-12-7) to (b-12-8);
[e:mainthin,color=Gray,->] (b-12-8) to (b-12-9);
[e:mainthin,color=Gray,->] (b-12-9) to (b-12-10);

[e:mainthin,color=Gray,->] (b-14-1) to (b-14-2);
[e:mainthin,color=Gray,->] (b-14-2) to (b-14-3);
[e:mainthin,color=Gray,->] (b-14-3) to (b-14-4);
[e:mainthin,color=Gray,->] (b-14-4) to (b-14-5);
[e:mainthin,color=Gray,->] (b-14-5) to (b-14-6);
[e:mainthin,color=Gray,->] (b-14-6) to (b-14-7);
[e:mainthin,color=Gray,->] (b-14-7) to (b-14-8);
[e:mainthin,color=Gray,->] (b-14-8) to (b-14-9);
[e:mainthin,color=Gray,->] (b-14-9) to (b-14-10);

[e:mainthin,color=Gray,->] (b-16-1) to (b-16-2);
[e:mainthin,color=Gray,->] (b-16-2) to (b-16-3);
[e:mainthin,color=Gray,->] (b-16-3) to (b-16-4);
[e:mainthin,color=Gray,->] (b-16-4) to (b-16-5);
[e:mainthin,color=Gray,->] (b-16-5) to (b-16-6);
[e:mainthin,color=Gray,->] (b-16-6) to (b-16-7);
[e:mainthin,color=Gray,->] (b-16-7) to (b-16-8);
[e:mainthin,color=Gray,->] (b-16-8) to (b-16-9);
[e:mainthin,color=Gray,->] (b-16-9) to (b-16-10);

[e:mainthin,color=Gray,->] (b-18-1) to (b-18-2);
[e:mainthin,color=Gray,->] (b-18-2) to (b-18-3);
[e:mainthin,color=Gray,->] (b-18-3) to (b-18-4);
[e:mainthin,color=Gray,->] (b-18-4) to (b-18-5);
[e:mainthin,color=Gray,->] (b-18-5) to (b-18-6);
[e:mainthin,color=Gray,->] (b-18-6) to (b-18-7);
[e:mainthin,color=Gray,->] (b-18-7) to (b-18-8);
[e:mainthin,color=Gray,->] (b-18-8) to (b-18-9);
[e:mainthin,color=Gray,->] (b-18-9) to (b-18-10);

[e:mainthin,color=Gray,->] (b-20-1) to (b-20-2);
[e:mainthin,color=Gray,->] (b-20-2) to (b-20-3);
[e:mainthin,color=Gray,->] (b-20-3) to (b-20-4);
[e:mainthin,color=Gray,->] (b-20-4) to (b-20-5);
[e:mainthin,color=Gray,->] (b-20-5) to (b-20-6);
[e:mainthin,color=Gray,->] (b-20-6) to (b-20-7);
[e:mainthin,color=Gray,->] (b-20-7) to (b-20-8);
[e:mainthin,color=Gray,->] (b-20-8) to (b-20-9);
[e:mainthin,color=Gray,->] (b-20-9) to (b-20-10);

[e:mainthin,color=Gray,->] (b-22-1) to (b-22-2);
[e:mainthin,color=Gray,->] (b-22-2) to (b-22-3);
[e:mainthin,color=Gray,->] (b-22-3) to (b-22-4);
[e:mainthin,color=Gray,->] (b-22-4) to (b-22-5);
[e:mainthin,color=Gray,->] (b-22-5) to (b-22-6);
[e:mainthin,color=Gray,->] (b-22-6) to (b-22-7);
[e:mainthin,color=Gray,->] (b-22-7) to (b-22-8);
[e:mainthin,color=Gray,->] (b-22-8) to (b-22-9);
[e:mainthin,color=Gray,->] (b-22-9) to (b-22-10);

[e:mainthin,color=Gray,<-] (b-3-1) to (b-3-2);
[e:mainthin,color=Gray,<-] (b-3-2) to (b-3-3);
[e:mainthin,color=Gray,<-] (b-3-3) to (b-3-4);
[e:mainthin,color=Gray,<-] (b-3-4) to (b-3-5);
[e:mainthin,color=Gray,<-] (b-3-5) to (b-3-6);
[e:mainthin,color=Gray,<-] (b-3-6) to (b-3-7);
[e:mainthin,color=Gray,<-] (b-3-7) to (b-3-8);
[e:mainthin,color=Gray,<-] (b-3-8) to (b-3-9);
[e:mainthin,color=Gray,<-] (b-3-9) to (b-3-10);

[e:mainthin,color=Gray,<-] (b-5-1) to (b-5-2);
[e:mainthin,color=Gray,<-] (b-5-2) to (b-5-3);
[e:mainthin,color=Gray,<-] (b-5-3) to (b-5-4);
[e:mainthin,color=Gray,<-] (b-5-4) to (b-5-5);
[e:mainthin,color=Gray,<-] (b-5-5) to (b-5-6);
[e:mainthin,color=Gray,<-] (b-5-6) to (b-5-7);
[e:mainthin,color=Gray,<-] (b-5-7) to (b-5-8);
[e:mainthin,color=Gray,<-] (b-5-8) to (b-5-9);
[e:mainthin,color=Gray,<-] (b-5-9) to (b-5-10);

[e:mainthin,color=Gray,<-] (b-7-1) to (b-7-2);
[e:mainthin,color=Gray,<-] (b-7-2) to (b-7-3);
[e:mainthin,color=Gray,<-] (b-7-3) to (b-7-4);
[e:mainthin,color=Gray,<-] (b-7-4) to (b-7-5);
[e:mainthin,color=Gray,<-] (b-7-5) to (b-7-6);
[e:mainthin,color=Gray,<-] (b-7-6) to (b-7-7);
[e:mainthin,color=Gray,<-] (b-7-7) to (b-7-8);
[e:mainthin,color=Gray,<-] (b-7-8) to (b-7-9);
[e:mainthin,color=Gray,<-] (b-7-9) to (b-7-10);

[e:mainthin,color=Gray,<-] (b-9-1) to (b-9-2);
[e:mainthin,color=Gray,<-] (b-9-2) to (b-9-3);
[e:mainthin,color=Gray,<-] (b-9-3) to (b-9-4);
[e:mainthin,color=Gray,<-] (b-9-4) to (b-9-5);
[e:mainthin,color=Gray,<-] (b-9-5) to (b-9-6);
[e:mainthin,color=Gray,<-] (b-9-6) to (b-9-7);
[e:mainthin,color=Gray,<-] (b-9-7) to (b-9-8);
[e:mainthin,color=Gray,<-] (b-9-8) to (b-9-9);
[e:mainthin,color=Gray,<-] (b-9-9) to (b-9-10);

[e:mainthin,color=Gray,<-] (b-11-1) to (b-11-2);
[e:mainthin,color=Gray,<-] (b-11-2) to (b-11-3);
[e:mainthin,color=Gray,<-] (b-11-3) to (b-11-4);
[e:mainthin,color=Gray,<-] (b-11-4) to (b-11-5);
[e:mainthin,color=Gray,<-] (b-11-5) to (b-11-6);
[e:mainthin,color=Gray,<-] (b-11-6) to (b-11-7);
[e:mainthin,color=Gray,<-] (b-11-7) to (b-11-8);
[e:mainthin,color=Gray,<-] (b-11-8) to (b-11-9);
[e:mainthin,color=Gray,<-] (b-11-9) to (b-11-10);

[e:mainthin,color=Gray,<-] (b-13-1) to (b-13-2);
[e:mainthin,color=Gray,<-] (b-13-2) to (b-13-3);
[e:mainthin,color=Gray,<-] (b-13-3) to (b-13-4);
[e:mainthin,color=Gray,<-] (b-13-4) to (b-13-5);
[e:mainthin,color=Gray,<-] (b-13-5) to (b-13-6);
[e:mainthin,color=Gray,<-] (b-13-6) to (b-13-7);
[e:mainthin,color=Gray,<-] (b-13-7) to (b-13-8);
[e:mainthin,color=Gray,<-] (b-13-8) to (b-13-9);
[e:mainthin,color=Gray,<-] (b-13-9) to (b-13-10);

[e:mainthin,color=Gray,<-] (b-15-1) to (b-15-2);
[e:mainthin,color=Gray,<-] (b-15-2) to (b-15-3);
[e:mainthin,color=Gray,<-] (b-15-3) to (b-15-4);
[e:mainthin,color=Gray,<-] (b-15-4) to (b-15-5);
[e:mainthin,color=Gray,<-] (b-15-5) to (b-15-6);
[e:mainthin,color=Gray,<-] (b-15-6) to (b-15-7);
[e:mainthin,color=Gray,<-] (b-15-7) to (b-15-8);
[e:mainthin,color=Gray,<-] (b-15-8) to (b-15-9);
[e:mainthin,color=Gray,<-] (b-15-9) to (b-15-10);

[e:mainthin,color=Gray,<-] (b-17-1) to (b-17-2);
[e:mainthin,color=Gray,<-] (b-17-2) to (b-17-3);
[e:mainthin,color=Gray,<-] (b-17-3) to (b-17-4);
[e:mainthin,color=Gray,<-] (b-17-4) to (b-17-5);
[e:mainthin,color=Gray,<-] (b-17-5) to (b-17-6);
[e:mainthin,color=Gray,<-] (b-17-6) to (b-17-7);
[e:mainthin,color=Gray,<-] (b-17-7) to (b-17-8);
[e:mainthin,color=Gray,<-] (b-17-8) to (b-17-9);
[e:mainthin,color=Gray,<-] (b-17-9) to (b-17-10);

[e:mainthin,color=Gray,<-] (b-19-1) to (b-19-2);
[e:mainthin,color=Gray,<-] (b-19-2) to (b-19-3);
[e:mainthin,color=Gray,<-] (b-19-3) to (b-19-4);
[e:mainthin,color=Gray,<-] (b-19-4) to (b-19-5);
[e:mainthin,color=Gray,<-] (b-19-5) to (b-19-6);
[e:mainthin,color=Gray,<-] (b-19-6) to (b-19-7);
[e:mainthin,color=Gray,<-] (b-19-7) to (b-19-8);
[e:mainthin,color=Gray,<-] (b-19-8) to (b-19-9);
[e:mainthin,color=Gray,<-] (b-19-9) to (b-19-10);

[e:mainthin,color=Gray,<-] (b-21-1) to (b-21-2);
[e:mainthin,color=Gray,<-] (b-21-2) to (b-21-3);
[e:mainthin,color=Gray,<-] (b-21-3) to (b-21-4);
[e:mainthin,color=Gray,<-] (b-21-4) to (b-21-5);
[e:mainthin,color=Gray,<-] (b-21-5) to (b-21-6);
[e:mainthin,color=Gray,<-] (b-21-6) to (b-21-7);
[e:mainthin,color=Gray,<-] (b-21-7) to (b-21-8);
[e:mainthin,color=Gray,<-] (b-21-8) to (b-21-9);
[e:mainthin,color=Gray,<-] (b-21-9) to (b-21-10);

[e:mainthin,color=Gray,<-] (b-23-1) to (b-23-2);
[e:mainthin,color=Gray,<-] (b-23-2) to (b-23-3);
[e:mainthin,color=Gray,<-] (b-23-3) to (b-23-4);
[e:mainthin,color=Gray,<-] (b-23-4) to (b-23-5);
[e:mainthin,color=Gray,<-] (b-23-5) to (b-23-6);
[e:mainthin,color=Gray,<-] (b-23-6) to (b-23-7);
[e:mainthin,color=Gray,<-] (b-23-7) to (b-23-8);
[e:mainthin,color=Gray,<-] (b-23-8) to (b-23-9);
[e:mainthin,color=Gray,<-] (b-23-9) to (b-23-10);

A sketch of the digraph $U_{II}$ with its crosses, the top strip of bounded height we aim to keep unused, and the section of the left slice which is used for the roots of $\Bidirected{K_t}$ (highlighted in red).

As the final step of the proof we construct a $\Bidirected{K_t}$-butterfly minor grasped by $U_{II}$ in $D_{II}$.
Since $\Split{D_{II}}=B$, <ref> can then be applied to find a $K_{t,t}$-matching minor grasped by $W$ in $B$, which will conclude the proof of our claim.

Let us position the roots of the vertices for our $\Bidirected{K_t}$ on the path $P^1_{\frac{3}{2}d_1}$.
For each pair $\Brace{i,j}\in[1,t]^2$, we will construct a path $J_{i,j}$ starting at the head of $f_i^j$ and eventually ending at the tail of $e_j^i$.
Since the roots are ordered from left to right, we start out with a family of disjoint paths in $U_{II}$ which are ordered from left to right as

\begin{align*}
\end{align*}

while moving downwards in the current embedding of $U_{II}$.
We grow the paths $J_{i,j}$ until each of them ends on some $P^1_{\Fkt{h_1}{i,j}}$ where $\Fkt{h_1}{i,j}>\Fkt{h_1}{i',j'}$, if $\Brace{i,j}$ is lexicographically smaller than $\Brace{i',j'}$.
Then we grow the paths along their respective $P^1_{\Fkt{h_1}{i,j}}$, until each of them ends on a switched vertical cycle $Q_{\Fkt{h_2}{i,j}}$ of $U'_{t+1}$ where $\Fkt{h_2}{i,j}>\Fkt{h_2}{i',j'}$, whenever $\Brace{i,j}$ is lexicographically smaller then $\Fkt{h_2}{i,j}$.
Afterwards we move each path along its respective horizontal cycle upwards, until each $J_{i,j}$ reaches some $P^1_{\Fkt{h_3}{i,j}}$, where $\Fkt{h_3}{i',j'}<\Fkt{h_3}{i,j}$, if $J_{i,j}$ is further to the right than $J_{i',j'}$ while moving upwards, and each $P^1_{\Fkt{h_3}{i,j}}$ is above the top path of $T'_1$, while avoiding the top strip of height $4\Fkt{f_P}{t}$.
This is possible by the choice of our parametrisation.
Next move each $J_{i,j}$ towards the right along its $P^1_{\Fkt{h_3}{i,j}}$ until the following requirements are met:

* the path $J_{1,1}$ ends on a horizontal cycle of $N_{L,1}$, which lies left of $T_1'$,

* the two paths $J_{1,2}$ and $J_{1,3}$ meet exactly the two horizontal cycles of $N_1'$ which contain the tail of $L_1$ and $R_1$ respectively, and

* the paths $J_{1,4},\dots,J_{t,t}$ end on horizontal cycles of $N_{R,1}$ such that $J_{i,j}$ ends on a cycle left of the cycle on which $J_{i',j'}$ ends, if $J_{i,j}$ passes $T_1'$ below $J_{i',j'}$.

Now, by growing the paths downwards, we may route $J_{1,2}$ and $J_{1,3}$ through the paths $L_1$ and $R_1$, thereby swapping their relative positions within the whole path family.
We may then either grow the paths further downwards and then go right again, or use the switched horizontal cycles of $G_1$, to grow them back upwards within $U_{II}$.
Either way, we are able to route the now neighbouring paths $J_{1,2}$ and $J_{1,4}$ (or $J_{2,1}$ in case $t=3$) through $T_2'$.
Since we have $t^4$ such crosses in total, we may repeat this process until, after passing through $T_{t^4}'$, the paths are in the following order when moving downwards in our current embedding of $U_{II}$:

\begin{align*}
\end{align*}

We can now use $G_{t^4}$ to route all of these paths upwards and finally use the top strip of height $4\Fkt{f_P}{t}$, which we kept untouched so far, to route the paths back to the roots of our vertices.
With the changed order we are no able to grow each path $J_{i,j}$ such that it indeed ends on the tail of $e_{j,i}$, while making sure that our paths stay pairwise vertex disjoint.
Hence we have found a $\Bidirected{K_t}$-butterfly minor as desired and the proof of our claim is complete.$\blacksquare$

From <ref> it follows that, in case all $t^4$ slices $\Split{N_i}$ are not $\Perimeter{\Split{N_i}}$-flat in $B$ with respect to $A$, then there exists a $K_{t,t}$-matching minor grasped by $W$ in $B$.
In this case we are done and thus we may assume that there exists some $i\in[1,t^4]$ such that $\Split{N_i}$ is indeed $\Perimeter{\Split{N_i}}$-flat in $B$ with respect to $A$.
Since $N_i$ is of width $r$, $\Split{N_i}$ contains a conformal matching $r$-wall $\widetilde{W}$ such that $\Perimeter{\widetilde{W}}=\Perimeter{\Split{N_i}}$ and thus our proof is complete.
At last let us combine all assumptions on the $d_j$ to obtain the following bound on $\Fkt{\WallBound}{t,r}$:

\begin{align*}
\Fkt{\WallBound}{t,r}\leq\Brace{2^{140}t^{76}}^{2^{10}t^8+2^{12}}\Brace{r+2^{32+t^{30}}}.
\end{align*}

Please note that many of the exponential parts of the bounds in $\ApexBound$ and $\WallBound$ are due to specific difficulties encountered in the digraphic setting.
Indeed, even the exponential lower bound on $f_W$ necessary for <ref> can probably be made polynomial by simply diving deeper into the increased freedom provided by the possibility of switching perfect matchings along directed cycles.
At this point it is not clear whether the exponential part of Phase II in the proof of <ref> can also be made polynomial.
To do this, it is probably necessary to develop a technique for removing long jumps that does not rely on tilings.

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