2.1 A Model of Urban Warming and Heat Island
This section introduces the attributes/characteristics model based on the Gorman–Lancasterian theory where goods are regarded as a composition of characteristics.
3 All trace gases such as greenhouse gases (GHGs) can be interpreted as
gaseous attributes in our framework, since they infinitesimally compose the urban atmosphere as an urban public good. Characteristics of urban climate are in order: pollution substances, solar radiation, cloudiness, precipitation, temperature, absolute and relative humidity, and wind velocity. These attributes compose the urban planetary boundary layer. This paper focuses upon heat as an attribute, since it is the main cause of urban warming. Before rushing into our theoretical model, let me introduce some basic concepts of urban warming and heat islands.
Let us consider a city where we analyze urban warming and heat islands and the related problems occurring in urban and suburban areas. For the sake of simplicity, assume that our city is composed of two areas, i.e., the urban area and the suburban area, which are divided into many regions, \(\beta \in \mathbf{B}=\{1,\ldots,B\}\): the set of regions. Assume that any size whatever can be chosen for a region.
Let there be N residents indexed by \(i\in \mathbf{N}=\{1,\ldots,N\}\): the set of inhabitants who live in both urban and suburban areas. Each individual emits heat and trace gases when consuming goods or services indexed by j and its set is \(\mathbf{J}=\{1,\ldots,J\}\). For the sake of simplicity, it is assumed that each producer j supplies only one good j which is composed of C characteristics indexed by \(c\in \mathbf{C}=\{1,\ldots,C\}\): the set of tangible attributes. Denote as \(q_{jc}\) an amount of attribute c embodied in one unit of good j. There are producers: e.g., offices, hotels, and manufacturers.
Suppose that resident i chooses a landscape gardener named ℓ to have a part of his/her land planted with trees and flowers, and \(\boldsymbol{\varLambda }=\{1,\ldots,\varLambda \}\) is the set of gardeners. Define \(q_{\ell s}\) as a biomass of species s in one cube meter supplied by landscape gardener ℓ, and \(\mathbf{S}=\{1,\ldots,S\}\) as the set of species of flora and fauna as biological attributes. Let \(x_{ij}\) be resident i’s consumption of good j, and \(V_{i\ell }\) be his/her demand for the plants supplied by landscape gardener ℓ, thus, \(x_{i}= ( x_{i1},\ldots,x_{iJ},V_{i\ell } ) \) is his/her consumption vector. There is also a local government whose task is to reduce heat emissions by making effective use of an urban warming tax scheme as defined below. No need to mention, every inhabitant, producer, and landscape gardener resides in some house or construction in region β, so that an index β will be omitted hereafter in almost all the cases, except for describing some variables related to region β.
Different from von Thünen (
1826), or the usual urban economic theory, our city is hypothesized as follows:
H1.
The city is formed in a heterogeneous plain, where the climate differs among its regions.
H2.
The city is not necessarily circular and its center is called the Central Business District.
H3.
Its urban and suburban transportation systems are available in any direction whatsoever.
H4.
Residents commute to work for an office in the CBD or in suburbs.
H5.
Landscape gardeners plant in a part of the lands of residents and producers.
H6.
Manufacturers produce goods in the suburban area.
The urban atmosphere is regarded as a complex of gaseous attributes including GHGs, which are to be mainly generated by production and consumption activities. The amount of gases such as N2 and O2 are stationary, so I can focus upon heat and trace gases as attributes in this paper. Let \(\mathbf{G}=\{C+1,\ldots,C+G\}\) be the set of trace gases which compose the urban atmosphere.
Taking urban warming into consideration, let us extend and generalize the framework developed by Sato (
2006,
2008). Inhabitants emit heat and gases,
\(q_{ih}\geq 0\) and
\(q_{ig}\geq 0\),
\(\forall g\in \mathbf{G}\), which are resident
i’s unit emissions of heat and gas. Hence,
\(q_{ih}V_{i\ell }\) and
\(q_{ig}V_{i\ell }\),
\(\forall g\in \mathbf{G}\), are individual
i’s emitted quantities of heat and gas in his/her consumption of
\(V_{i\ell }\) units of greening service
ℓ. Resident
i’s consumption of gas and heat as attributes are, respectively, given by
$$ z_{ig}=q_{ig} \biggl( \sum_{j\in \mathbf{J}}x_{ij}+V_{i\ell } \biggr) , \quad \forall g\in \mathbf{G} $$
(1)
and
$$ z_{ih}=q_{ih} \biggl( \sum_{j\in \mathbf{J}}x_{ij}+V_{i\ell } \biggr) . $$
(2)
When producing one unit of good, each producer cannot choose but jointly emit trace gases as vexing by-products,
\(q_{jg}\geq 0\), which is producer
j’s unit emission of gas. Thus,
\(q_{jg}x_{j}\) is
j’s amount emitted of gas when it produces
\(x_{j}\) units of good
j.
\(q_{jg}x_{j}\) and
\(q_{jh}x_{j}\) producer
j’s amounts of gas and heat in his/her production of goods. One observes therefore
$$ z_{jg}=q_{jg}x_{j}. $$
(3)
Similarly, when producing one unit of good, each producer must jointly emit heat as an annoying by-product,
\(q_{jh}\geq 0\), which is producer
j’s unit emission of heat. Thus,
\(q_{jh}x_{j}\) is producer
j’s amount emitted of heat when it produces
\(x_{j}\) units of good
j. One obtains therefore
$$ z_{jh}=q_{jh}x_{j}. $$
(4)
Landscape gardeners also emit heat and gases,
\(q_{\ell h}\geq 0\) and
\(q_{\ell g}\geq 0\),
\(\forall g\in \mathbf{G}\), which are gardener
ℓ’s unit emissions of heat and any gas. Hence,
\(q_{\ell g}V_{\ell }\) and
\(q_{\ell h}V_{\ell }\) are gardener
ℓ’s emitted quantity of a gas and heat, respectively, in his/her provision of
\(V_{\ell }\) units of greening service
ℓ. We have
$$ z_{\ell g}=q_{\ell g}V_{\ell } $$
(5)
and
$$ z_{\ell h}=q_{\ell h}V_{\ell }. $$
(6)
Let
\(z_{i0}\) be resident
i’s available time that he/she possesses, by which he/she utilizes other characteristics. In other words, any characteristic cannot be utilized without using time. Note that dioxide (CO
2) can be reduced by
\(\varPi _{g}\) which is an amount of CO
2 fixed in trees, i.e., the net carbon gain.
4 Amounts of each tangible attribute embodied in the goods and intangible attributes in the atmosphere, which are consumed by metropolitan agents are given for any
\(c\in \mathbf{C}\), for any
\(g\in \mathbf{G}\), and for heat:
$$\begin{aligned} z_{ic} =&\sum_{j\in \mathbf{J}}q_{jc}x_{ij} , \end{aligned}$$
(7)
$$\begin{aligned} z_{g} =&\sum_{i\in \mathbf{N}}z_{ig}+\sum _{j\in \mathbf{J}}z_{jg}+\sum _{\ell \in \boldsymbol{\varLambda }}z_{\ell g}-\chi _{g}\varPi _{g} , \\ \chi _{g} =&1\quad \text{if }g=\mathrm{CO_{2}}\quad \text{and}\quad \chi _{g}=0\quad \text{if }g\neq \mathrm{CO_{2}} , \end{aligned}$$
(8)
and
$$ z_{h}=\sum_{i\in \mathbf{N}}z_{ih}+\sum _{j\in \mathbf{J}}z_{jh}+\sum _{\ell \in \boldsymbol{\varLambda }}z_{\ell h}. $$
(9)
In the above equations, \(z_{ic}\) means the consumption of tangible attributes which compose goods, while \(z_{g }\) and \(z_{h}\) represent the total amount of a trace gas and heat emitted by all residents, producers and gardeners. Note also that the values of \(z_{g}\), \(\forall g\in \mathbf{G}\), and \(z_{h}\) can be measured in tons or kilojoules. Heat and gases are generated both in the consumption and production of goods and services. Every inhabitant is made to consume not only his/her emissions but also the quantity emitted by the rest of the city. When he/she uses goods, he/she emits heat and gases. They are already released when the goods are made by producers and greening services are provided by landscape gardeners.
The above three equations may be interpreted as characteristics availability functions which convert commodities into attributes. In the framework of this paper, any good j can be recognized as \((x_{j},q_{j1},\ldots ,q_{jC+G})\) and any greening service ℓ can be represented as \((V_{\ell },q_{\ell C+1},\ldots ,q_{\ell C+G},q_{\ell C+G+1},\ldots ,q_{\ell C+G+S})\). The amount of any characteristic associated with each good and service can be regarded as a parameter that is objective and common to all consumers, i.e., it has the public-good property. Thus, the inhabitants as consumers must behave as “price and quality takers,” since they can only change their consumption of \(z_{ic}\), \(z_{ih}\), and \(z_{ig}\), via the choice of \(x_{ij}\) and \(V_{i\ell }\), given the price and the quality of each good and service. Producers and gardeners can choose the composition of attributes embedded in their goods and their greening services.
The following notation is used in what follows.
Al:
planetary albedo (0.3) determining how much of the incoming energy is reflected by the atmosphere
Ω:
solar constant (1372 W m−2)
ε:
emissivity (assumed to be 1)
σ:
Stefan–Boltzmann constant (\(5.67 \times 10^{-8}\mbox{ W}\,\mbox{m}^{-2}\,\mbox{K}^{-4}\)) [the outgoing flux is \(\varepsilon \sigma T^{4}\) by the Stefan–Boltzmann Law]
\(\varUpsilon _{\beta }\):
anthropogenic heat stocks at region β [J]
\(\dot{T}_{\beta }^{t}\):
temperature increase at region β at time t [°C]
\(c_{p}\):
specific heat capacity [1004.2 J/kg °C]
M:
atmospheric density [1.293 kg m−3]
Ab:
a coefficient (0.3) determining how much of the energy which is not absorbed by the surface of the earth
2.2 Global Warming Function
It is observed that warming of cities is mainly due to heat emissions and partly due to global warming, and the GHGs’ concentrations affect the latter. These effects therefore must be incorporated in the model of urban and global warming.
As was mentioned in Sect.
2.1,
\(z_{g}\),
\(\forall g\in \mathbf{G}\), is the total quantity of each GHG released all over the world. A part,
\(\alpha _{g}z_{g}\),
\(0<\alpha _{g}<1\),
\(\forall g\in \mathbf{G}\), of an aggregate emission of trace gas
g, is observed to go to the atmosphere and the rest,
\((1-\alpha _{g})z_{g}\), is perceived to be absorbed by the oceans and forests as carbon sinks, if
g is carbon dioxide. Of this amount, about 43 % of the CO
2 emissions are observed to be absorbed. The mass of the
gth GHG staying in the atmosphere is
\(\alpha _{g}z_{g}\),
\(\forall g\in \mathbf{G}\). The disintegration rate or an inverse of an atmospheric lifetime of each trace gas is denoted as
\(\mu _{g}\),
\(0<\mu _{g}<\alpha _{g}\),
\(\forall g\in \mathbf{G} \).
One problem of the GHGs is that they are not flows (emissions), but stocks (concentrations). Let
\(t\in{}[0,\infty)\) be the time argument. Denote
\(t_{0}\) as a base year. Let
\(\nu_{g}>0\),
\(\forall g\in\mathbf{G}\), be a conversion parameter from mass (GtC/year) to concentration (ppm), then the latter at time
t is represented by
$$ \zeta_{g}^{t}=\int_{t_{0}}^{t} \nu_{g}\bigl(\alpha_{g}^{\tau}-\mu_{g}^{\tau} \bigr)z_{g}^{\tau}\,d\tau, \quad \forall g\in\mathbf{G}. $$
(10)
So one observes a vector of GHGs’ concentrations as stocks:
$$ Z^{t}=\bigl(\zeta _{1}^{t},\ldots,\zeta _{G}^{t}\bigr). $$
(11)
In the sequel, let me follow Greiner (
2004a,
2004b) for global warming in an endogenous growth model, originally due to Roedel (
2001).
The most basic energy balance model is presented as follows:
(12)
This is the temperature without the greenhouse effect. Following Greiner (
2004a,
2004b), the difference between the outgoing flux and the incoming radiative flux is given by
$$ F=\frac{19.95}{109}. $$
(13)
The average surface temperature on earth with the greenhouse effect can be calculated by
(14)
Thus, the above calculations lead us to conclude that the greenhouse effect for the earth is 33 °C. Next we incorporate the effect of GHGs’ concentrations to global warming. Let
\(\zeta _{\text{CO}_{2}}\) be an actual concentration of CO
2 at time
t and
\(\zeta _{\text{CO}_{2}}^{0}\) its concentration at the Pre-industrial Revolution: e.g., the former is 377 ppm in 2004 and the latter is 280 ppm in 1750. IPCC (
1990) assumed that the radiative forcing (W m
−2) of carbon dioxide is given by
$$ \varXi ^{t}\bigl(\zeta _{\mathrm{CO_{2}}}^{t}\bigr)=5.35\ln \bigl( \zeta _{\mathrm{CO_{2}}}^{t}/\zeta _{\mathrm{CO_{2}}}^{1750} \bigr) . $$
(15)
For the sake of simplicity, we take two main GHGs: CO
2 and nitrous oxide. Let
\(\rho _{g}\) be a Global Warming Potential and
\(\mu _{g}\) is an inverse of an atmospheric lifetime compared with CO
2 (
\(\rho _{\mathrm{CO_{2}}}=1\) and
\(\mu _{\mathrm{CO_{2}}}=1\)). According to Michaelis (
1990), nitrous oxide’s contribution to global warming is
\(\rho _{g}\mu _{g}=58/12=29/6\) compared with CO
2. We remark that the global warming depends on the stocks (concentrations)
\(Z^{t}\) of GHGs:
$$ \varXi ^{t}\bigl(Z^{t}\bigr)=5.35\sum _{g\in \mathbf{G}}\rho _{g}\mu _{g} \ln \bigl( \zeta _{g}^{t}/\zeta _{g}^{1750} \bigr) . $$
(16)
It is observed that the average surface temperature on earth with global warming can be calculated by
(17)
Compare the two values of temperature, i.e., 15 °C and 17 °C. The difference is 2 °C, which is attributable to the effect of GHGs’ concentrations between the years 1750 and 2004.
The above arguments lead us to define the
Global Warming Function,
$$ W\bigl(\varXi ^{t}\bigl(Z^{t}\bigr)\bigr)= \biggl\{ \frac{\varOmega (1-\mathit{Al})\mathit{Ab}+\varXi ^{t}(Z^{t})}{4\varepsilon \sigma F} \biggr\} ^{{1}/{4}} , $$
(18)
which represents the average surface temperature on earth with the global warming due to the buildup of GHGs’ concentrations.
5
2.3 Heat Island Integral
The problem of how to represent anthropogenic heat stocks in any block was analyzed in Sato (
2006) who introduced the concept of
Heat Island Integral. Here a new version is proposed. In effect, there are differences in the temperature of building surface, back alleys, roof-tops, streets, and green tracts of land, which are directly exposed to the solar radiation. These differences of the surface temperature of the ground coverage can be measured by utilizing infrared cameras or remote sensing.
A formula is proposed for heat as a stock in this subsection. It is the microclimate surrounding a construction, which most influences any agent who resides or works in any region. However, climatical incidents depend not only upon the heat stocks in each region, but also upon those in the entire city. More precisely, it is the sum of developable areas of the ground coverage, e.g., streets, tree crowns, roof-tops, and walls of the buildings which exist in region β. Denote \(\kappa _{\beta }^{r}\) and \(\delta _{\beta }^{r}(u,v)\) as the area and the height of construction r in region β, respectively, and R as the set of constructions. Suppose β is an urban commercial region and \(\beta ^{\prime }\) is a suburban residential region.
The Riemann sum has led me to propose a concept of
Heat Island Integral (
HII) between regions
β and
\(\beta ^{\prime }\):
$$\begin{aligned} \mathit{HII} =&c_{p}M\sum_{r\in \mathbf{R}} \biggl\{ A_{\beta }\dot{T}_{\beta }\iint_{\kappa _{\beta }^{r}}\delta _{\beta }^{r}(u,v)\,du\,dv \\ &{} -A_{\beta ^{\prime }}\dot{T}_{\beta ^{\prime }}\iint_{\kappa _{\beta ^{\prime }}^{r}}\delta _{\beta ^{\prime }}^{r}(u,v)\,du\,dv \biggr\} . \end{aligned}$$
(19)
Needless to say, the existence condition of this multiple integral is that the functions are continuous and compact in the domains \(\kappa _{\beta }^{r} \) and \(\kappa _{\beta ^{\prime }}^{r}\), and it is easily seen that this condition is satisfied. Region β is called a Heat Island if \(\mathit{HII}>0\). Naturally, it is necessary to consider meteorological conditions peculiar to region β, such as the Foehn phenomenon, the convergence of sea breezes, and the transport of the warmed air from other regions, as well as the configuration of the region such as being a basin. Region β is called a Cool Island if \(\mathit{HII}\leq 0\).
As
\(\dot{T}_{\beta }=\varUpsilon _{\beta }/c_{p}A_{\beta }M\) by physics,
HII can be rewritten as
$$ \mathit{HII}=\sum_{r\in \mathbf{R}} \biggl\{ \varUpsilon _{\beta } \iint_{\kappa _{\beta }^{r}}\delta _{\beta }^{r}(u,v)\,du\,dv-\varUpsilon _{\beta ^{\prime }}\iint_{\kappa _{\beta ^{\prime }}^{r}}\delta _{\beta ^{\prime }}^{r}(u,v)\,du\,dv \biggr\} . $$
(20)
2.4 Urban Warming Function
Let \(z_{ih\beta }\) be heat emitted by resident i and \(z_{jh\beta }\) (\(z_{\ell h\beta }\), resp.) is heat emissions of producer j (landscape gardener ℓ, resp.) at region β. Any city is composed of regions with anthropogenic heat stocks, \(\varUpsilon _{\beta }=\varUpsilon _{\beta }(E_{\beta })\), where \(E_{\beta }=\sum_{i\in \mathbf{N}}z_{ih\beta }+\sum_{j\in \mathbf{J}}z_{jh\beta }+\sum_{\ell \in \boldsymbol{\varLambda }}z_{\ell h\beta }\), \(\forall \beta \in \mathbf{B}\). With \(E=(E_{1},\ldots ,E_{B})\), the total heat stocks, \(\varUpsilon (E)=\sum_{\beta \in \mathbf{B}}\varUpsilon _{\beta }(E_{\beta })\), may affect agents in any city.
As above, an increase in temperature of an airshed of region
β is given by the following equation:
\(\forall \beta \in \mathbf{B}\)
$$ \dot{T}_{\beta }^{t}=\frac{\text{anthropogenic heat stocks}}{\text{specific heat capacity}\times \text{airshed of region }\beta \times \text{atmospheric density}}. $$
(21)
In August 2004, the 23 wards of Tokyo generated an anthropogenic heat stocks of 2106.5 TJ/day. It was observed that warm air occupied up to an altitude of 400 m in the summer of Tokyo of which area is 621 km
2. We choose this as an airshed and the altitude of 400 m, then the average daily temperature increase is given by
(22)
Adding the above equation to the model to compute the average surface temperature on earth, and one observes
(23)
The difference between this temperature and the average temperature is 3.26 °C. It is considered to be attributable urban warming of the average daily anthropogenic heat stocks generated from the 23 wards of Tokyo in August 2004.
The greenhouse effect, global warming, and urban warming of any region
\(\beta \in \mathbf{B}\) is involved in the function:
$$ U_{\beta }^{t}= \biggl\{ \frac{\varOmega (1-\mathit{Al})\mathit{Ab}+\varXi ^{t}(Z^{t})}{4\varepsilon \sigma F} \biggr\} ^{{1}/{4}}+\frac{\varUpsilon _{\beta }^{t}(E_{\beta }^{t})}{c_{p}A_{\beta }M} , $$
(24)
which represents the average surface temperature of any region
β. In the bracket the denominator is given, and the second term varies according to each region. With the influx and outflux of energy to earth, the concentrations of GHGs and anthropogenic heat stocks define the
Urban Warming Function
\(U_{\beta }^{t}\). Note that
\(\varXi ^{t}\) and
\(\varUpsilon _{\beta }^{t}\) are functions of stocks of GHGs and anthropogenic heat, respectively. The following assumption is needed.
Urban warming is a typical example of a public good which is non-rival but excludable. Its impact on each resident varies from region to region, which can be treated as a regional public good. The global warming function depends upon the anthropogenic heat stocks in region β, \(\varUpsilon _{\beta }\), and the concentrations Z composed of gaseous attributes of the urban atmosphere.