Initial Conditions and Resilience in the Atmospheric Boundary Layer of an Urban Basin

Abstract

The possibilities of micrometeorological resilience in urban contexts immersed in a basin geographical configuration are investigated. For this purpose, time series data with measurements of meteorological variables (temperature, magnitude of wind speed and relative humidity) and atmospheric pollutants (PM_2.5, PM₁₀, CO) are analyzed through chaos theory, calculating the coefficient of Lyapunov (λ), the correlation dimension (Dc), the Hurst coefficient (H), the correlation entropy (S_K), the fractal dimension (D) and the Lempel–Ziv complexity (LZ). Indicators are built for each measurement period (2010–2013 and 2017–2020), for each locality studied and located at different heights. These indicators, which correspond to the quotient between the entropy resulting from the meteorological variables and that of the pollutants, show sensitivity to height. Another important indicator, for identical measurement conditions, arises from the calculation of the fractal dimensions of the meteorological variables and that of the pollutants, which allows for comparative studies between the two periods. These indicators are conclusive in pointing out that, in a large city with basin geographical characteristics, subjected to an intensive urbanization process, there is no micrometeorological resilience and a great variation occurs in the initial conditions.

1. Introduction

1.1. Resilience

Human interaction with nature has always had consequences [1]. The difference with today’s times is that generally, in the past, nature could reduce or absorb these effects in relatively short times. Various factors have strained this capacity to the point that the disturbances, which now occur continuously and in short periods of time, are not assimilated in the short term.

Resilience is the term used in the ecology of communities and ecosystems to indicate their ability to absorb disturbances, keeping their structural, dynamic and functional characteristics practically intact; being able to return to the situation prior to the disturbance after the cessation of it [2]. Thus, resilience is a measure of the magnitude of disturbances that a system can absorb to move from one equilibrium to another.

The idea of resilience can arise from a reductionist approach; the system is limited to the sum of the interactions between its parts. However, when a complex system manifests emergence, that is, a general behavior that cannot be described by the sum of the interactions of its components, a systems-based approach can be more fruitful. It is observed in nature, for example, that the tendency for the mixed phase of arctic clouds to persist despite their inherent microphysical instability suggests emergent qualities. It also evokes self-organizing properties, defined as “internal, local interactions of processes that give rise to global order”. The nature of these interactions results in a robust system that is resistant to disturbance.

Self-organization prevails in a variety of natural and man-made factors of systems. Examples include oscillating chemical reactions, behavior among bird populations, predator–prey interactions in the fields of ecology and cloud physics [3], light amplification by stimulated emission of radiation, and computer network theory. In the subtropics and mid-latitudes, cloud fields associated with mesoscale convection often occur as closed or open cellular patterns, exhibiting system-wide ordering that selects the preferred state based on, among other things: environmental conditions or due to external forces. Small perturbations often strengthen the robustness of the state by allowing further exploration of phase space in the proximity of the attractor, while large perturbations can cause the system to transition from one preferred state to another [4].

1.2. Cities and Risk

Risk, threat, vulnerability, and disaster are terms from the resilience literature that are related to one another. For example, risk can be defined in terms of hazards and vulnerability. The probability of a dangerous event occurring only becomes a risk if it can negatively affect people, communities or systems, that is, if these actors are susceptible to the impacts of threats (vulnerability). Disaster risk, for its part, refers to the possible negative effects of a hazard, which are determined in relation to risk exposure and the ability to act on it [5].

Urban resilience can be understood as the permanent capacity of cities to absorb, adapt, transform and prepare for shocks and stands out in the economic, social, institutional and environmental dimensions, with the objective of maintaining the functions of a city and improving the response to future crises [5].

From another perspective, it is argued that economic activities are sustainable only if the ecosystems that support life, and on which they are dependent, have an adequate level of resilience. Approaching the problem from a multivariate perspective is an arduous task. At present we have been learning that the environmental dimension is decisive for the viability of a city [6]. Within this dimension, this study will limit the problem to urban centers: the meteorology of the limit layer of the atmosphere and the disturbance due to air pollution by particulate matter (PM_2.5 and PM₁₀) and carbon monoxide (CO). Many questions arise from the previous considerations: is it possible to build indicators of disturbance of the atmosphere? What kind of processes can be induced? Can a system return to its initial condition if the disturbance is increasing over time and incorporates new elements such as buildings, change in surface roughness, decay of humidity, etc.?

Through the chaos theory, applied to time series data of urban micrometeorology [7,8,9,10,11,12,13] and of atmospheric pollutants, is it possible to obtain information on the resilient behavior of the atmospheric boundary layer, comparing different periods?

1.3. Entropy

As noted, all systems present in nature are subjected to irreversible processes [14,15,16]. This occurs when a system and its surroundings cannot return to their initial state [17]. It is observed that the phenomenon of irreversibility according to Prigogine [17] has a constructive character, highlighting the “creative role of time”. Therefore, it supposes, at least at the macroscopic level, a kind of anti-entropy: the non-equilibrium universe is a connected universe. This reveals to us that in the phenomena of nature there is connectivity and irreversibility. A magnitude that accounts for systems that have these characteristics is entropy. This is a fundamental variable that has different definitions and is measured in different ways, depending on the different fields in which it is applied [16,18,19].

From a historical perspective, in classical physics the entropy of a physical system is proportional to the amount of energy not available to do physical work. In quantum mechanics, von Neumann extended the notion of entropy to quantum systems by means of the density matrix [20]. In probability theory, the entropy of a random variable measures the uncertainty regarding the value that should be assumed by the variable [21]. In information theory, the compression entropy of a message (a computer file) quantifies the amount of information that is carried by the message in terms of the best lossless compression ratio [22]. In dynamic systems theory, entropy quantifies the exponential complexity of systems or the average flow of information per unit of time [23]. In sociology, entropy is the natural decay of the structures (such as laws, organization, and conventions) of a social system. Additionally, in the usual sense, entropy means disorder or chaos. By saying that entropy is a measure of disorder, it is implied that the highest entropy is the greatest disorder, which appears when it is considered as a measure of the unavailability of a system’s energy to do work. It is a parameter that represents the state of disorder of an atomic, ionic, or molecular system. It is also a measure of disorder in the universe or simply a measure of disorder in any system [24,25].

1.4. Kolmogorov Entropy and Its Relation to Information Loss

Following what has been indicated by Farmer, one of the basic differences between chaotic and predictable behavior is that chaotic trajectories continuously generate new information while predictable trajectories do not [26,27]. Metric entropy makes this notion more rigorous. In addition to providing a good definition of chaos, metric entropy provides a quantitative way to describe how chaotic a dynamical system is. In the Kolmogorov entropy [28,29], S_K is the average loss of information [22] when “l” (side of the cell in units of information) and τ (time) become infinitesimal [30]:

$${\mathrm{S}}_{\mathrm{K}}=-\underset{\mathsf{\tau }\to 0}{\lim } \underset{\mathrm{l}\to 0}{\lim }\underset{\mathrm{n}\to \infty }{\lim }{\left(\mathrm{n}\mathsf{\tau }\right)}^{-1}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{P}}_{\mathrm{i}}{\mathrm{logP}}_{\mathrm{i}}$$

(1)

S_K has units of information bits per second and bits per iteration in the case of a discrete system [31,32]. The limit process of Equation (1) develops according to the order: (i) n → ∞, (ii) l → 0, canceling the dependency of the selected partition (n is the number of cells or partitions) and (iii) τ → 0, for continuous systems. The Kolmogorov entropy difference, ΔS_K = S_Kn+1 − S_Kn between two neighboring cells, gives the required complementary information about the cell (i_n+1) in which the system will be in the future. The difference gives the loss of information, in time, of the system [30].

In summary, for the calculation of the Kolmogorov entropy, it is first verified that the entropy is between zero and infinity (0 < S_K < ∞), which allows verifying the presence of chaotic behavior. If the Kolmogorov entropy is equal to 0, no information is lost and the system is regular and predictable. If S_K$\to$∞, the system is completely random, and it is impossible to make any predictions. Second, the amount of information required to predict the future behavior of two interacting systems is determined, in this case, the atmosphere and the hourly concentration of pollutants (PM₁₀, PM_2.5 and CO). In this way, the rate at which the system loses (or outdates) information over time can be estimated. Finally, the horizon of maximum predictability of the system can be established. This horizon is a limit frontier from which it is not possible to make predictions or formulate new scenarios [29]. The loss of information can be determined according to the equation:

$$〈∆\mathrm{I}〉=〈{\mathrm{I}}_{\mathrm{NEW}}-{\mathrm{I}}_{\mathrm{OLD}}〉=\frac{-\mathsf{\lambda }\left({\mathrm{i}}_0\left(\mathrm{t}\right)\right)}{\log 2}$$

(2)

λ is the Lyapunov exponent, <ΔI> in bits/h, is the loss information. Two types of <ΔI> can be calculated: one for the contribution of pollutants, and another for the sum of the loss of information of each meteorological variable (MV): temperature (T), magnitude of wind speed (WS) and relative humidity (HR) [30].

2. Materials and Methods

2.1. Materials

In Chile, there is an online public network for monitoring meteorological and pollutant variables [33] that has extensive periods of data records that form time series. The field recording of data serves as a starting line for this research. To analyze the data, we followed the line of chaos theory applied to time series [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41]. Many authors have approached the treatment of time series in this way [23,42,43,44,45,46,47,48,49,50,51]. From the extensive data records, measurements of meteorological variables (magnitude of wind speed, relative humidity, temperature) and air pollutants (PM_2.5 particulate matter, PM₁₀ and carbon monoxide CO) constituted in time series for different periods were selected in the first approximation [51,52,53,54,55]. The analysis of these series, through a software that explores their chaoticity, allows to determine if their nature is chaotic from the ranges of five parameters characteristic of chaos theory: Lyapunov’s coefficient (λ > 0), Hurst’s Coefficient (0.5 < H < 1), correlation dimension (Dc < 5), correlation entropy (S_K > 0), and complexity Lempel–Ziv (LZ > 0). Finally, the comparison between the entropies resulting from the meteorological variables and that of the pollutants gives an indicator of variation by morphological geography.

2.2. Methods

The flowchart, Figure 1, indicates the stages of the procedure to be followed once the time series of the pollutants and the meteorological variables have been obtained [30].

2.3. Study Area

The city of Santiago is located at 33.5° S and 70.8° W. It contains a population of 7,508,334 inhabitants, which represents 40% of the total population of the country, on a surface of approximately 641 km². It is located in the middle of the country, at a height of about 520 m.a.s.l. The altitude above sea level increases from west to east. It is surrounded by two mountain chains: The Andes and the Coastal Mountain range. Its climate is Mediterranean. The driest and warmest months are from December to February, reaching maximum temperatures of about 35 °C in the shade (air temperature in the sun), Figure 2. Given its topography and the dominant meteorological conditions, there is in general a strong horizontal and vertical dispersion of pollutants generated by an important number of sources in the city (heating, vehicles, industries, etc.), especially during autumn (20 March–21 June) and winter (21 June–23 September) [51,56,57,58,59,60]. The emissions have a tendency to increase given the increasing population density, which implies an increase in fixed and mobile sources. In addition, the number of vehicles has increased rapidly in recent years.

3. Results

Two tables (

Table 1. Presents the entropies of the meteorological variables (MVs) for the two periods (2010–2013 and 2017–2020) according to the same heights.

Station	m.a.s.l. (m)	Series 1	Series 2
		2010–2013	2017–2020
		SK,MV (1/h)	SK,MV (1/h)
EML	784	1.334	1.284
EMM	709	1.452	1.205
EMO	469	1.197	0.993
EMS	698	1.289	1.250
EMV	485	1.202	0.994
EMN	570	1.262	1.145

and

Table 2. It presents the entropies of the pollutants (Ps) in the two periods of the study (2010–2013 and 2017–2020) carried out in the same locations according to their own heights.

Station	m.a.s.l. (m)	Series 1	Series 2
		2010–2013	2017–2020
		SK,P (1/h)	SK,P (1/h)
EML	784	1.542	1.577
EMM	709	1.550	1.406
EMO	469	1.210	1.630
EMS	698	1.377	1.702
EMV	485	1.431	1.220
EMN	570	1.531	1.479

) were built, one for the entropies of the meteorological variables and another for the entropies of the pollutants. Data were extracted from references [49] (page 175) and [51](page 11). The heights, with respect to sea level, are the same for the two periods (2013–2013 and 2017–2020) of the study and correspond to that of the monitoring stations.

3.1. Meteorological Variables

Figure 3 presents the distribution of entropies according to heights and periods of measurement of the meteorological variables.

3.2. Pollutants

Figure 4 presents the distribution of entropies according to heights and periods of the pollutants.

3.3. Relationship between S_K,MV and S_K,P

It can be establish a relationship between C_K= S_K,MV/S_K,P =${\left(\sum {\mathrm{S}}_{\mathrm{K}.\mathrm{MV},\mathrm{i}}/\sum {\mathrm{S}}_{\mathrm{K}.\mathrm{P},\mathrm{i}}\right)}_{\mathrm{COMMUNES}}$ and the height of the different monitoring stations of meteorological and pollutant variables and, in addition, of the measurement periods, what appears in

Table 3. Percentage variations of the C_K.

h (m.a.s.l.)	2010–2013	2017–2020	% Variation
	CK1 (Series 1)	CK2 (Series 2)
784 (EML)	0.865	0.814	6
709 (EMM)	0.937	0.857	7
485 (EMV)	0.834	0.815	3
570 (EMN)	0.824	0.774	7
698 (EMS)	0.936	0.734	22
469 (EMO)	0.989	0.609	38

and is plotted in Figure 5.

Figure 5 represents the relationship between C_K and the height:

4. Discussion

A scalar time series is one-dimensional. There are important aspects of the scale of measurement and they are related to the expression of the size and to the topological dimension: the value of the magnitude depends on the value of the scale, but the magnitude of the topological dimension is independent of the scale (for example, length, area and volume will appear more fractional due to scale changes).

As fractals are made up of smaller and smaller elements of themselves, a set of recursive or self-similar points at any scale is arrived at. This set determines a geometry that is contained in a certain dimension. Mandelbrot, points out that a fractal is, by definition, a set whose Hausdorff–Besicovitch dimension is strictly greater than its topological dimension. Thus, the key mathematical property of an essentially fractal object is that its fractal metric dimension is a rational number greater than its topological dimension. The dimension of the time series graph (the trace) is the fractal dimension of the graph generated by the time series. If the series is a scalar observable, then this dimension will be between 1.0 for a very smooth curve and 2.0 for a very erratic one [23]. As the fractal dimension increases, the geometry of the fractal object contained in the time series is more complex, the described process is more complex or more chaotic. Considering that D = 2 − H, where D is the fractal dimension and H is the Hurst exponent; due to the values obtained, the fractal dimension of the time series for the period 2010–2013 are closer to the line and those for the period 2017–2020 are further from the line (

Table A1. All the values of the Hurst exponent (H), the fractal dimension (D) for all the variables of interest by studied community and the two periods (2010–2013 and 2017–2020).

		PM10	PM2.5	CO	T	HR	WV
EML	2010–2013
	H	0.967	0.973	0.959	0.989	0.991	0.976
	D	1.033	1.027	1.041	1.011	1.009	1.024
	2017–2020
	H	0.922	0.963	0.933	0.915	0.942	0.975
	D	1.078	1.037	1.067	1.085	1.058	1.025
EMM	2010–2013
	H	0.972	0.977	0.981	0.991	0.990	0.980
	D	1.028	1.023	1.019	1.009	1.010	1.02
	2017–2020
	H	0.906	0.983	0.933	0.917	0.941	0.976
	D	1.094	1.017	1.067	1.083	1.059	1.024
EMN	2010–2013
	H	0.972	0.974	0.953	0.989	0.991	0.968
	D	1.028	1.026	1.047	1.011	1.009	1.032
	2017–2020
	H	0.929	0.960	0.933	0.916	0.942	0.973
	D	1.071	1.04	1.067	1.084	1.058	1.027
EMO	2010–2013
	H	0.965	0.955	0.937	0.992	0.989	0.968
	D	1.035	1.045	1.063	1.008	1.011	1.032
	2017–2020
	H	0.936	0.925	0.933	0.919	0.942	0.974
	D	1.064	1.075	1.067	1.081	1.058	1.026
EMS	2010–2013
	H	0.969	0.973	0.953	0.990	0.992	0.957
	D	1.031	1.027	1.047	1.010	1.008	1.043
	2017–2020
	H	0.921	0.975	0.933	0.915	0.942	0.976
	D	1.079	1.025	1.067	1.085	1.058	1.024
EMV	2010–2013
	H	0.967	0.970	0.952	0.989	0.989	0.956
	D	1.033	1.03	1.048	1.011	1.011	1.044
	2017–2020
	H	0.931	0.966	0.933	0.919	0.942	0.975
	D	1.069	1.034	1.067	1.081	1.058	1.025

, Appendix).

Each of the time series studied had a large and positive Lyapunov exponent, which means that each is chaotic and will usually have a strange attractor with non-integer dimension, as shown by the calculations. In the case of the time-delay embedding space, it is a reconstructed state space chosen with the minimum dimension for which important dynamical and topological properties are preserved.

If the Hurst coefficient is between 0.5 < H < 1, the time series are persistent, that is, characterized by long-term memory effects. Theoretically, what happens at a given moment affects the future. There is no characteristic time scale, which is the key property of fractal time series. In terms of chaotic dynamics there is sensitivity about the initial conditions. The persistent series are the most common found in nature.

The calculations presented in Table A1 [30,51] (in the Appendix) show that the Hurst exponent and the Lempel–Ziv complexity of the wind speed magnitude is, in general, higher in the period 2017–2020 compared to the period 2010–2013. The wind shows a more turbulent behavior for the period 2017–2020.

According to the Kolmogorov cascade effect, the fully developed turbulent medium is characterized by two quantities, the average rate of energy dissipation, ε_d, and the kinematic viscosity ν. The dimensionality of ε_d is energy/time/mass, or ${\mathrm{L}}^2{\mathrm{T}}^{-3}$, and the dimensionality of ν is ${\mathrm{L}}^2{\mathrm{T}}^{-1}$. We can combine these two quantities to find the length scale:

$${\mathrm{l}}_{\mathrm{k}}={\left(\frac{{\mathsf{\nu }}^3}{{\mathsf{\epsilon }}_{\mathrm{d}}}\right)}^{1/4}$$

(3)

known as the Kolmogorov or dissipational length scale. The length scale is the size of the smallest eddies in the fluid. Eddies smaller than the Kolmogorov scale rapidly dissipate their kinetic energy by viscous heat and disappear [28]. The eddies of size l_K rotate with a speed:

$${\mathrm{u}}_{\mathrm{K}}={\left({\mathsf{\nu }\mathsf{\epsilon }}_{\mathrm{d}}\right)}^{1/4}$$

(4)

and dissipates its energy in a time, roughly, equal to

$${\tau }_K={\left(\frac{\nu }{{\epsilon }_d}\right)}^{1/2}$$

(5)

In Earth’s atmosphere, ${\epsilon }_d=10\frac{c{m}^2}{s^3}$ and ν$~ 0.1\frac{c{m}^2}{s}.$ Thus, the smallest eddy in the Earth’s atmosphere is $l_K ~ 0.1 cm$, which is significantly larger than the mean free path λ $~ {10}^{-4}\mathrm{cm}$. Smaller eddies rotate around the surroundings with speed $u_K~1\frac{cm}{s}$, dying according to a decay time of ${\tau }_K~0.1 s$.

The Kolmogorov entropy of the time series of this study has units ${\mathrm{S}}_{\mathrm{K}}\left[\frac{\mathrm{bits}}{\mathrm{h}}\right]$. When transforming their units by means of Landauer’s principle,${ \mathrm{S}}_{\mathrm{K}}\left[\frac{\mathrm{J}}{\mathrm{K} \mathrm{h}}\right]$ remains, which is equivalent to energy/(temperature time), dimensionally:

$$\left[\frac{\mathrm{M}}{\mathrm{T}}\right]\left[\frac{{\mathrm{L}}^2}{{\mathrm{t}}^3}\right]~\left[\frac{\mathrm{M}}{\mathrm{T}}\right]*\mathrm{average} \mathrm{power} \mathrm{dissipation} \mathrm{rate}$$

In general, the rate of change of ${\mathrm{C}}_{\mathrm{K}}{=\mathrm{S}}_{\mathrm{K},\mathrm{MV}}{/\mathrm{S}}_{\mathrm{K},\mathrm{P}}$ with height is:

$$\frac{\mathrm{d}}{\mathrm{dh}}{\mathrm{C}}_{\mathrm{K}}=\frac{\mathrm{d}}{\mathrm{dh}}\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}=\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}–{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}{}^2}=\frac{\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}–\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}$$

(6)

Series 1: 450 m < h < 550 m, very close to the ground, for the period 2010–2013:

$$\frac{\mathrm{d}}{\mathrm{dh}}\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}=\frac{\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}–\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}<0$$

(7)

Figure 5 exhibits, in a first approximation, a strong variation of C_K in certain ranges of height, which could be compatible with the generation, by increases in building height in the period 2017–2020 (Series 2 furthest from equilibrium ($\frac{S_{K,MV}}{S_{K,P}}~1$) between the entropies of the selected pollutants and the meteorological variables), of a more turbulent wind regime in the basin.

Since S_K,P is always greater than zero, inequality (7) is satisfied if:

$$\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}<\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}}$$

(8)

with:

$$\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}>0$$

(9)

$$\frac{1}{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}<\frac{1}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}}$$

(10)

$$\frac{{\mathrm{d} \ln \mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}<\frac{{\mathrm{d} \mathrm{lnS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}}$$

(11)

Very close to the ground, the variation with the height of C_K is negative and of very large magnitude, it decreases towards a relative minimum. Urban weather is more stable or “strong”, which makes it less vulnerable to the effect of air pollutants. As an initial condition, this “strength” has a detrimental effect on air pollutants. As height increases, this variation smooths out. The variation with the height of the ln of the entropy of the meteorological variables is less than the variation with the height of the ln of the entropy of the pollutants, that is, the meteorology of the period 2010–2013 is dynamically more persistent and balances the polluting system generated through the term σ (with units of length⁻¹):

$$\frac{{\mathrm{d} \ln \mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}+\mathsf{\sigma }=\frac{{\mathrm{d} \mathrm{lnS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}}$$

(12)

It is possible to reduce the system, in the form:

$$\mathrm{d}\ln {\mathrm{S}}_{\mathrm{K},\mathrm{MV}}+\mathsf{\sigma }\mathrm{dh}={\mathrm{d} \mathrm{lnS}}_{\mathrm{K},\mathrm{P}}\to \mathrm{d}\ln {\mathrm{S}}_{\mathrm{K},\mathrm{MV}}-{\mathrm{d} \mathrm{lnS}}_{\mathrm{K},\mathrm{P}}=-\mathsf{\sigma }\mathrm{dh}$$

(13)

$$\mathrm{d} \ln \left(\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}\right)=-\mathsf{\sigma }\mathrm{dh}\to \frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}={\mathrm{e}}^{-\mathsf{\sigma }\mathrm{h}}\to {\mathrm{S}}_{\mathrm{K},\mathrm{MV}}={\mathrm{S}}_{\mathrm{K},\mathrm{P}}{\mathrm{e}}^{-\mathsf{\sigma }\mathrm{h}}$$

(14)

Verification, differentiating with respect to h:

$$\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}=\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}}{\mathrm{e}}^{-\mathsf{\sigma }\mathrm{h}}-{\mathrm{S}}_{\mathrm{K},\mathrm{P}}{\mathsf{\sigma }\mathrm{e}}^{-\mathsf{\sigma }\mathrm{h}}$$

$$\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}=\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}}\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}-{\mathrm{S}}_{\mathrm{K},\mathrm{P}}\mathsf{\sigma }\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}$$

$$\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}=\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}}\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}-{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}\mathsf{\sigma }$$

$$\frac{1}{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}=\frac{1}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}}-\mathsf{\sigma }\to \frac{{\mathrm{d} \ln \mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}+\mathsf{\sigma }=\frac{{\mathrm{d} \mathrm{lnS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}}$$

Series 2: 450 m < h < 550 m, very close to the ground, for the period 2017–2020:

$$\frac{\mathrm{d}}{\mathrm{dh}}\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}=\frac{\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}-\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}>0$$

(15)

from where:

$$\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}>\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}} \mathrm{with} \frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}>0$$

(16)

$$\frac{1}{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}>\frac{1}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}\frac{{\mathrm{dS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}}\to \frac{{\mathrm{d} \ln \mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{\mathrm{dh}}>\frac{{\mathrm{d} \mathrm{lnS}}_{\mathrm{K},\mathrm{P}}}{\mathrm{dh}}$$

(17)

In this case, it follows:

$$\frac{d \ln S_{K,MV}}{dh}=\delta +\frac{d \ln S_{K,P}}{dh}$$

(18)

Very close to the ground, the variation with height of C_K is positive and of very large magnitude and grows towards a relative maximum. Urban weather is more variable, turbulent or less “strong”, making it more vulnerable to the effect of more stable air pollutants. As height increases, this variation smooths out. The variation with the height of the ln of the entropy of the meteorological variables is greater than the variation with the height of the ln of the entropy of the pollutants, that is, the pollutants are dynamically more persistent, balancing the system with the term, according to height, δ (with units of length⁻¹).

Some turbulence breaks out around 700 m, more moderate than between 450 and 500 m, and at higher altitudes the processes represented by both series tend to stabilize at different C_K’s depending on the study periods. It is possible to model, in a very approximate way, according to Equations (19) and (20) the processes described by C_K in each period:

2010–2013 (Series 1):

$${\mathrm{C}}_{\mathrm{K}}\left(\mathrm{h}\right)=\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}=0.8+3.5{*\mathrm{e}}^{-6\left(\mathrm{h}-0.2\right)}*\sin \left(3.5*\left(2*\mathrm{h}-0.2\right)/6\right)$$

(19)

2017–2020 (Series 2):

$${\mathrm{C}}_{\mathrm{K}}\left(\mathrm{h}\right)=\frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}=0.75-1.5{*\mathrm{e}}^{-6\left(\mathrm{h}-0.2\right)}*\sin \left(3.5*\left(2.2*\mathrm{h}-0.2\right)\right)$$

(20)

obtaining Figure 6.

Figure 7 is the graphical representation of Equations (19) and (20). It allows to visualize the effect on C_K of the heights close to the Earth’s surface where the density of the atmosphere is higher with higher kinematic viscosity. The opposite happens when the height is greater, although the volume is also greater, the kinematic density is lower, so the energy exchanges are smoother and tend to balance compared to the atmosphere closer to the ground.

Figure 7 conforms to a phenomenology of the type described by the Kolmogorov cascade, as represented in Figure 8.

In addition, the analysis of the ratio between the entropies of the urban meteorology variables and those of the pollutants shows sensitivity according to the measurement periods. If the system is considered in the period 2010–2013, as shown in the representation in Figure 5 of the ratio with the height, it is the initial condition of the total period of the study. The variation of this initial condition, due to systematic anthropic disturbance over time, as shown in Figure 5 of the 2017–2020 period, produces a different behavior of the system. Variations of the initial condition, induced by the increasing contamination of the atmosphere, determine different evolutions of the system with a “relatively asymptotic” behavior of C_K as the height increases. It stabilizes at higher altitudes by increasing the volume of the atmosphere and the proximity to the upper atmospheric layer (which has a different thermodynamics).

Although urban meteorology has its natural modes of relationship (with solar radiation, volcanoes, geography, seas, etc.) and entropy variations, with respect to the historical average, the interactions were naturally processed in ancient and resilient periods. Human activity is a disruption and contributes to the imbalance of the system [30,57,58,59,60,61,62,63,64,65,66].

5. Conclusions

The data from the two time periods, 2010–2013 and 2017–2020, from urban meteorology measurements (three time series) and pollutants (three time series) correspond to a continuous and hourly instrumental record. The analysis applied to the set of variables that the system represents proves that it is a complex system, of non-linear dynamics, very sensitive to variations in the initial conditions. Small variations in these initial conditions can imply large differences in future behavior, making long-term prediction impossible. The variation with the height of C_K indicates that it varies strongly and positively (dC_K/dh) at low altitudes in the period 2017–2020 (intensification of high-rise buildings), unlike the period 2010–2013 (dC_K/dh < 0). The process declines asymptotically with height, with a gap between periods, but the 2017–2020 period is the one with the greatest drop. According to

Table A2. Comparative table of the amount of wind speed according to persistence, H, and LZ complexity, where > means an increase, and = means the same value.

Stations	EML	EMM	EMV	EMN	EMS	EMO
Periods	H; LZ	H; LZ	H; LZ	H; LZ	H; LZ	H; LZ
2010–2013	0.976; 0.320	0.980; 0.558	0.956; 0.325	0.968; 0.286	0.957; 0.293	0.968; 0.538
2017–2020	0.975 (=); 0.551 (>)	0.976 (=); 0.557 (=)	0.975 (>); 0.544 (>)	0.973 (>);0.539 (>)	0.976 (>); 0.556 (>)	0.974 (>); 0.537 (=)

of the Appendix, in the period 2017–2020 there is a greater complexity and persistence of the wind (greater turbulence). The entropy, the Hurst exponent, the fractal dimension, and the loss of information indicates the impossibility of returning the studied system (corresponding to a measurement context and a geography) to its initial condition because there are elements of perturbation, continuous action, and that are the result of human activity. From the perspective of the ground measurements and the general mathematical model of analysis, a return to the initial condition is not concluded, which is one of the basic characteristics of a resilient natural system, in this case referring to the period of this study of the atmosphere. The idea of a resilient city, as an end in itself, makes no sense as it is inserted in its interaction and connectivity with nature.