Influence of Volumetric Geometry on Meteorological Time Series Measurements: Fractality and Thermal Flows

Abstract

This work analyzes the behavior of the boundary layer subjected to stresses by obstacles using hourly measurements, in the form of time series, of meteorological variables (temperature (T), relative humidity (RH), and magnitude of the wind speed (WS)) in a given period. The study region is Santiago, the capital of Chile. The measurement location is in a rugged basin geography with a nearly pristine atmospheric environment. The time series are analyzed through chaos theory, demonstrating that they are chaotic through the calculation of the parameters Lyapunov exponent (λ > 0), correlation dimension (D_C < 5), Kolmogorov entropy (S_K > 0), Hurst exponent (0.5 < H < 1), and Lempel–Ziv complexity (LZ > 0). These series are simultaneous measurements of the variables of interest, before and after, of three different volumetric geometries arranged as obstacles: a parallelepiped, a cylinder, and a miniature mountain. The three geometries are subject to the influence of the wind and present the same cross-sectional area facing the measuring instruments oriented in the same way. The entropies calculated for each variable in each geometry are compared. It is demonstrated, in a first approximation, that volumetric geometry impacts the magnitude of the entropic fluxes associated with the measured variables, which can affect micrometeorology and, by extension, the climate in general. Furthermore, the study examines which geometry favors greater information loss or greater fractality in the measured variables.

1. Introduction

The interaction between the atmosphere and various geographic surface features, such as mountains, basins, and coasts, which are not of regular geometry, significantly influences climate patterns. These interactions create localized variations in temperature, relative humidity, precipitation, and wind, shaping the diverse climates observed around the world [1,2]. Nature, in general, does not build in regular shapes, such as straight parallelepipeds, pyramids, cylinders, spheres, etc. (Appendix A). Hills and mountains, for example, are of irregular geometry, fractals, and the atmosphere, over millions of years, has adapted its connectivity to this irregular geometry and has evolved its behavior based on this objective reality. In a certain way, there is a language that communicates and gives the moments in which the atmosphere makes certain processes possible, such as rain, etc. We know of the influence of the tilt of the Earth’s axis on the seasons of the year and the effect of the relative distance between the Earth and the Sun on the climate. But, to a first approximation, under very localized initial conditions in the boundary layer of the variables of temperature, relative humidity, and wind speed, in conjunction with the geographic morphology, the set of variables achieves the complex connectivity required to make rain possible from the higher-energy clouds that are well above the boundary layer. But this precursor, or “call,” to precipitation has a very specific character. Communication can be hampered and blocked by pollution, urban densification, changes in soil roughness, housing development in mountainous and coastal areas, the albedo of construction materials, etc. [3,4,5]. High-energy clouds carrying the heaviest rainfall will then randomly unload in areas where they find the appropriate signals. This can be deep in the ocean, in distant geographic locations, etc., traveling great distances, leaving vast geographies with water deficits.

Observation tells us that in general, many of the other living beings on the planet do not build their homes in regular geometry: termites, bees, spiders, moles, birds, beavers, wasps, etc. They keep open, not blocked by their constructions, the geography–atmosphere dialogue, something that is mutually beneficial as a not yet understood evolutionary symbiosis of species–geography–atmosphere.

Humans are the only ones who build their homes using regular geometric structures: buildings, sheds, silos, houses, temples, etc. It is no longer a mystery that cities contribute to climate change with thermal islands, the artificial change in soil roughness comparable to the thickness of the boundary layer, high-albedo construction materials, urban planning that reduces relative humidity, makes winds more turbulent, and increases temperatures, etc. [6,7]. In a crude form, the purpose is, through these measurable regular geometries, to optimize space and economic profit. But we cannot ignore the aspects of harmony and order linked to practical themes, which found some of their most notable precursors in ancient Greece, such as the Pythagorean school and the golden rule. This rule significantly influenced the construction of temples, homes, art, etc. for centuries [8]. Fiction, from its perspective, also provides us with striking angles [9]. These approaches, of many, to the essential topic of human housing constructions transcend all history. Finally, over time, changes in lifestyles, diet, and work have led to a modern approach to housing, driven by products like pollution and garbage. The issue has become more complex with the addition of subjective valuations that have increased costs. Our interaction with geography, and the atmosphere in particular, became dramatic. Is it possible that fractal cities integrated with fractal geographic morphology are a solution? Would this allow us to reconstruct the fragmented dialogue between humanity, geography, and the atmosphere? Technology has the potential to make such an idea possible. What does seem clear is that the artificial, regular, “geomorphological” geometry created by humans is not contributing to improving the connectivity of nature’s characteristic processes, operating almost like a Tower of Babel by obstructing past connectivity. Currently, it is known that the real estate sector, including construction and building operations, is one of the most important sources of pollution worldwide, consuming 36% of global energy and producing 39% of CO₂ emissions [10,11].

1.1. Turbulence Due to Obstacles

Figure 1 presents, approximately, a simplified version of the wind flow before and after an obstacle.

Figure 2 shows the interaction of the wind flow with a straight parallelepiped obstacle before and after and the possible location of instruments measuring temperature (T), relative humidity (RH), magnitude of wind speed (WS), and pressure (P) in the turbulent zone.

It is possible to study the interactions with other volumetric geometries by changing the parallelepiped for a cylinder and for an irregular volumetric figure whose effective section, A, is contained in the straight parallelepiped and in the cylinder, as shown in Figure 3.

1.2. Thermal and Mechanical Turbulence

Figure 4 is an approximate representation of the behavior of thermal and mechanical turbulence.

Thermal or convective turbulence, also known as free convection, consists of ascending currents of warm air and descending currents of cold air due to the effect of buoyancy.

Updrafts form near the ground due to the heating of the air near the surface. This heating causes the air to become warmer than the surrounding air, and it begins to rise because it is less dense. This process occurs at several nearby points, forming several curtain-like updrafts. The intersection of these currents generates a single upward movement that we can identify as a plume with a diameter of about 100 m.

At higher levels within the boundary layer, these plumes tend to group together to form larger updrafts, on the order of a kilometer in diameter, known as thermals. Some of these updrafts, provided they have sufficient moisture content to reach the condensation stage, become visible through the formation of cumulus clouds.

Mechanical turbulence, shown in Figure 5, occurs when obstacles, such as buildings, uneven terrain, or trees, interfere with the normal flow of wind.

1.3. Turbulence Dissipation

Like all motion, turbulence is associated with a certain kinetic energy, which varies depending on the size of the eddy. This turbulent kinetic energy is not conserved and is continuously dissipated internally due to the viscosity of the fluid. Dissipation causes the disappearance of turbulence and occurs in smaller eddies.

Therefore, recalling the mechanism of inertial turbulence generation, which involves the loss of energy from larger eddies to form smaller ones, and taking into account that energy is dissipated in the latter, for turbulence to exist, larger-scale mechanical and thermal generation processes must continuously exist. Otherwise, once turbulence dissipates in small eddies, it would cease to exist.

2. Materials and Methods

2.1. Study Area and Equipment

The study area, where the geometric volumes were located, corresponds to a high mountain area called San Alfonso, Cajón del Maipo, Chile, shown in Figure 6. It is located at an altitude of 1115 m above sea level. The Cajón del Maipo is an Andean canyon in the upper Maipo River basin with glacial erosion, with an average elevation greater than 900 m above sea level. The area is characterized by hills, cliffs, and massifs. The climate is temperate with rain in winter and dry summers, with an average annual temperature of 20 °C. The approximate coordinates of the Cajón del Maipo are 33°38′00″ S, 70°21′00″ W. It is located at an approximate distance of 48 km from the Metropolitan Region of Santiago de Chile. The mountainous area is very sparsely populated. Sparsely populated refers to a geographic area with a low population density and limited activity, that is, with few inhabitants compared to its land area. This occurs for various reasons, such as adverse environmental conditions, access difficulties, undeveloped land belonging to private individuals, absence of buildings (especially high-rise buildings), very little industrial or social activity, etc. The population density is less than 0.26 inhabitants per hectare (10,000 m²). Similarly, the level of air pollution from the concentration of particulate matter (PM₁₀, PM_2.5) and gases (CO, CO₂, O₃, etc.) is well below the minimum risk level.

The weather stations used in this study were new (unused), as shown in Figure 7, so their factory certification is valid, as they had not been subjected to the rigors of the environment. These measuring devices are German-made and bear the PCE Instruments designation. This means they comply with U.S. and European regulations (U.S. EPA and WMO equivalent). They are built for outdoor use and have very low power consumption. The device requires two AA 1.5 V LR6 alkaline batteries; the battery life is a minimum of 24 months for the temperature and humidity sensor. It is equipped with solar panels, making the device relatively self-sufficient. Its operation and data-logging systems were connected to software installed on a PC that allowed for control by second, minute, hour, day, week, and month. The measurement record consists of hourly time series. Although some of them presented missing data, the percentage of missing data does not exceed 2% of the total data in a typical time series. Moving linear regression was used to complete the missing data. The evolution of the time series over the entire measurement period did not show any outliers. The device was used under ambient conditions (temperature, humidity, etc.) within the limit range indicated in the specifications: temperature range, −40 to 50 °C; relative humidity range, 1 to 99%; wind speed range, 0 to 50 m/s.

The equipment used in the measurements, shown in Figure 7, corresponds to meteorological stations with a power source provided by solar panels and a fastening system that allows them to be fixed to a mast.

2.2. Mathematical Methods

2.2.1. Entropies Associated with Heat Fluxes and Turbulence

The entropy balance equation [12] is derived from the Gibbs relation [13], assuming local equilibrium, such that the entropy per unit volume, s, is written in the form of

$${\displaystyle \frac{\partial \mathrm{s}}{\partial \mathrm{t}}}=\nabla ·{\overrightarrow{\mathrm{J}}}_{\mathrm{S}}+\mathrm{\sigma },$$

(1)

where $\nabla ·{\overrightarrow{\mathrm{J}}}_{\mathrm{S}}$ is the divergence of the entropy flux per unit volume through the surface whose boundary encloses the system. Equation (1) denotes the entropy production, and ${\overrightarrow{\mathrm{J}}}_{\mathrm{S}}$ is the entropy flow, defined as [13]

$${\overrightarrow{\mathrm{J}}}_{\mathrm{s}}=\mathrm{s}\overrightarrow{\mathrm{v}}+{\displaystyle \frac{1}{\mathrm{T}}}{\overrightarrow{\mathrm{J}}}_{\mathrm{q}}-\sum _{\mathrm{k}}{\displaystyle \frac{{\mathrm{\mu }}_{\mathrm{k}}}{\mathrm{T}}}{\overrightarrow{\mathrm{J}}}_{\mathrm{K}}$$

(2)

where s = S/V, as noted, is the entropy per unit volume for a total density ρ, $\overrightarrow{\mathrm{v}}$ is the vector representation of velocity, ${\overrightarrow{\mathrm{J}}}_{\mathrm{q}}$ heat flux [14], T is the temperature on an absolute scale, ${\mathrm{\mu }}_{\mathrm{K}}$ is the chemical potential for the k-th component species (measured in Gibb/mol), and ${\overrightarrow{\mathrm{J}}}_{\mathrm{K}}$ is the diffusive mass flux of the k-th component species. If the product is considered as

$$\nabla ·{\displaystyle \frac{1}{\mathrm{T}}}{\overrightarrow{\mathrm{J}}}_{\mathrm{q}}=-{\displaystyle \frac{1}{T^2}}\nabla T·{\overrightarrow{\mathrm{J}}}_{\mathrm{q}}+{\displaystyle \frac{1}{T}}\nabla ·{\overrightarrow{\mathrm{J}}}_{\mathrm{q}}$$

(3)

for steady-state heat,

$$\nabla ·{\displaystyle \frac{1}{\mathrm{T}}}{\overrightarrow{\mathrm{J}}}_{\mathrm{q}}=-{\displaystyle \frac{1}{T^2}}\nabla T·{\overrightarrow{\mathrm{J}}}_{\mathrm{q}}$$

(4)

According to Fourier’s Law, ${\overrightarrow{\mathrm{J}}}_{\mathrm{q}}=-k\nabla T$, with k thermal conductivity of the substance, we obtain

$$\nabla ·{\displaystyle \frac{1}{\mathrm{T}}}{\overrightarrow{\mathrm{J}}}_{\mathrm{q}}={\displaystyle \frac{k}{T^2}}{\nabla T}^2={\displaystyle \frac{\partial s_{ heat}}{\partial t}}$$

(5)

This is a term that describes the heat flow from a region with a high temperature to a region with a lower temperature in a substance. This quantity is the entropic heat flow per volume and is expressed in SI units of J/Ksm³.

2.2.2. Entropy Fluxes Due to Turbulence

Turbulent flow with time-averaged values is considered a laminar flow with the presence of additional forces originating from turbulence, as shown in Figure 8.

On the surface element dAx, the stresses are [15,16]

$${\mathrm{\tau }}_{\mathrm{x}\mathrm{x}}={\displaystyle \frac{\mathrm{d}{\mathrm{F}}_{\mathrm{x}}}{\mathrm{d}{\mathrm{A}}_{\mathrm{x}}}}{\mathrm{\tau }}_{\mathrm{x}\mathrm{y}}={\displaystyle \frac{\mathrm{d}{\mathrm{F}}_{\mathrm{y}}}{\mathrm{d}{\mathrm{A}}_{\mathrm{x}}}}{\mathrm{\tau }}_{\mathrm{x}\mathrm{z}}={\displaystyle \frac{\mathrm{d}{\mathrm{F}}_{\mathrm{z}}}{\mathrm{d}{\mathrm{A}}_{\mathrm{x}}}}$$

(6)

On the surface element dAy, the stresses are

$${\mathrm{\tau }}_{\mathrm{y}\mathrm{x}}={\displaystyle \frac{\mathrm{d}{\mathrm{F}}_{\mathrm{x}}}{\mathrm{d}{\mathrm{A}}_{\mathrm{y}}}}{\mathrm{\tau }}_{\mathrm{y}\mathrm{y}}={\displaystyle \frac{\mathrm{d}{\mathrm{F}}_{\mathrm{y}}}{\mathrm{d}{\mathrm{A}}_{\mathrm{y}}}}{\mathrm{\tau }}_{\mathrm{y}\mathrm{z}}={\displaystyle \frac{\mathrm{d}{\mathrm{F}}_{\mathrm{z}}}{\mathrm{d}{\mathrm{A}}_{\mathrm{y}}}}$$

(7)

On the surface element dAz, the stresses are

$${\mathrm{\tau }}_{\mathrm{z}\mathrm{x}}={\displaystyle \frac{\mathrm{d}{\mathrm{F}}_{\mathrm{x}}}{\mathrm{d}{\mathrm{A}}_{\mathrm{z}}}}{\mathrm{\tau }}_{\mathrm{z}\mathrm{y}}={\displaystyle \frac{\mathrm{d}{\mathrm{F}}_{\mathrm{y}}}{\mathrm{d}{\mathrm{A}}_{\mathrm{z}}}}{\mathrm{\tau }}_{\mathrm{z}\mathrm{z}}={\displaystyle \frac{\mathrm{d}{\mathrm{F}}_{\mathrm{z}}}{\mathrm{d}{\mathrm{A}}_{\mathrm{z}}}}$$

(8)

It can be expressed as a nine-element matrix that accounts for the distribution of stresses and strains in the continuous medium:

$$\left[\mathrm{\tau }\right]=\left(\begin{array}{ccc}{\mathrm{\sigma }}_{\mathrm{x}\mathrm{x}}& {\mathrm{\sigma }}_{\mathrm{x}\mathrm{y}}& {\mathrm{\sigma }}_{\mathrm{x}\mathrm{z}}\\ {\mathrm{\sigma }}_{\mathrm{y}\mathrm{x}}& {\mathrm{\sigma }}_{\mathrm{y}\mathrm{y}}& {\mathrm{\sigma }}_{\mathrm{y}\mathrm{z}}\\ {\mathrm{\sigma }}_{\mathrm{z}\mathrm{x}}& {\mathrm{\sigma }}_{\mathrm{z}\mathrm{y}}& {\mathrm{\sigma }}_{\mathrm{z}\mathrm{z}}\end{array}\right)=\left(\begin{array}{ccc}{\mathrm{\sigma }}_{\mathrm{x}\mathrm{x}}& {\mathrm{\tau }}_{\mathrm{x}\mathrm{y}}& {\mathrm{\tau }}_{\mathrm{x}\mathrm{z}}\\ {\mathrm{\tau }}_{\mathrm{y}\mathrm{x}}& {\mathrm{\sigma }}_{\mathrm{y}\mathrm{y}}& {\mathrm{\tau }}_{\mathrm{y}\mathrm{z}}\\ {\mathrm{\tau }}_{\mathrm{z}\mathrm{x}}& {\mathrm{\tau }}_{\mathrm{z}\mathrm{y}}& {\mathrm{\sigma }}_{\mathrm{z}\mathrm{z}}\end{array}\right)$$

(9)

The surface stress tensor can be broken down into two others, a compressive normal stress tensor and a viscous stress tensor. This is possible because in a fluid at rest, there can be no shear stresses (if there were, it would cease to be at rest, by definition of a fluid), so, in this particular case of fluids at rest, the stress tensor is reduced to

$$\left(\begin{array}{c}{\mathrm{f}}_{\mathrm{s}\mathrm{x}}\\ {\mathrm{f}}_{\mathrm{s}\mathrm{y}}\\ {\mathrm{f}}_{\mathrm{s}\mathrm{z}}\end{array}\right)=\left(\begin{array}{ccc}-\mathrm{p}& 0& 0\\ 0& -\mathrm{p}& 0\\ 0& 0& -\mathrm{p}\end{array}\right)\left(\begin{array}{c}{\mathrm{n}}_{\mathrm{x}}\\ {\mathrm{n}}_{\mathrm{y}}\\ {\mathrm{n}}_{\mathrm{z}}\end{array}\right)=\left[\mathbf{p}\right]\overrightarrow{\mathrm{n}}$$

(10)

where −p represents the normal compressive stress supported by the fluid volume. It can therefore be written in general terms that the surface stress tensor is the sum of a normal stress tensor plus a viscous stress tensor,

$$\stackrel{̿}{\mathrm{\tau }}\overrightarrow{\mathrm{n}}={\overrightarrow{\mathrm{f}}}_{\mathrm{s}}=-\left[\mathbf{p}\right]\overrightarrow{\mathrm{n}}+\stackrel{̿}{\mathrm{\tau }\mathrm{\prime }}\overrightarrow{\mathrm{n}}$$

(11)

being $\stackrel{̿}{\tau \prime }$, the viscous stress tensor. Therefore, this tensor can be expressed as follows:

$$\stackrel{̿}{\mathrm{\tau }}=\left(\begin{array}{ccc}-\mathrm{p}+{\mathrm{\tau }}_{\mathrm{x}\mathrm{x}}^{\mathrm{\prime }}& {\mathrm{\tau }}_{\mathrm{y}\mathrm{x}}& {\mathrm{\tau }}_{\mathrm{z}\mathrm{x}}\\ {\mathrm{\tau }}_{\mathrm{x}\mathrm{y}}& -\mathrm{p}+{\mathrm{\tau }}_{\mathrm{y}\mathrm{y}}^{\mathrm{\prime }}& {\mathrm{\tau }}_{\mathrm{z}\mathrm{y}}\\ {\mathrm{\tau }}_{\mathrm{x}\mathrm{z}}& {\mathrm{\tau }}_{\mathrm{y}\mathrm{z}}& -\mathrm{p}+{\mathrm{\tau }}_{\mathrm{z}\mathrm{z}}^{\mathrm{\prime }}\end{array}\right)$$

(12)

$$\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l} \mathrm{e}\mathrm{f}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{s}\left\{\begin{array}{c}{\mathrm{\tau }}_{\mathrm{x}\mathrm{x}}=2\mathrm{\mu }{\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{x}}}+{\displaystyle \frac{2}{3}}\mathrm{\mu }(\nabla \mathrm{\ast } \overrightarrow{\mathrm{v}})\\ {\mathrm{\tau }}_{\mathrm{y}\mathrm{y}}=2\mathrm{\mu }{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{y}}}+{\displaystyle \frac{2}{3}}\mathrm{\mu }(\nabla \mathrm{\ast } \overrightarrow{\mathrm{v}})\\ {\mathrm{\tau }}_{\mathrm{z}\mathrm{z}}=2\mathrm{\mu }{\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{z}}}+{\displaystyle \frac{2}{3}}\mathrm{\mu }(\nabla \mathrm{\ast } \overrightarrow{\mathrm{v}})\end{array}\right.$$

(13)

where $\overrightarrow{\mathrm{v}}=\mathrm{u}\widehat{\mathrm{i}}+\mathrm{v}\widehat{\mathrm{j}}+\mathrm{w}\widehat{\mathrm{k}}$ and

$$\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{r} \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{s}\left\{\begin{array}{c}{\mathrm{\tau }}_{\mathrm{x}\mathrm{y}}={\mathrm{\tau }}_{\mathrm{y}\mathrm{x}}=\mathrm{\mu }\left({\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{y}}}+{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{x}}}\right)\\ {\mathrm{\tau }}_{\mathrm{y}\mathrm{z}}={\mathrm{\tau }}_{\mathrm{z}\mathrm{y}}=\mathrm{\mu }\left({\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{z}}}+{\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{y}}}\right)\\ {\mathrm{\tau }}_{\mathrm{z}\mathrm{x}}={\mathrm{\tau }}_{\mathrm{x}\mathrm{z}}=\mathrm{\mu }\left({\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{x}}}+{\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{z}}}\right)\end{array}\right.$$

(14)

The tensor elements are, for laminar flow, incompressible and, neglecting the second viscosity coefficient in normal stresses,

$$\begin{array}{ccc}{\mathrm{\tau }}_{\mathrm{x}\mathrm{x}}^{\mathrm{\prime }}=2\mathrm{\mu }{\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{x}}}& {\mathrm{\tau }}_{\mathrm{y}\mathrm{y}}^{\mathrm{\prime }}=2\mathrm{\mu }{\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{y}}}& {\mathrm{\tau }}_{\mathrm{z}\mathrm{z}}^{\mathrm{\prime }}=2\mathrm{\mu }{\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{z}}}\end{array}$$

(15)

$$\begin{array}{ccc}{\mathrm{\tau }}_{\mathrm{x}\mathrm{y}}=\mathrm{\mu }\left({\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{y}}}+{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{x}}}\right)& {\mathrm{\tau }}_{\mathrm{y}\mathrm{x}}=\mathrm{\mu }\left({\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{y}}}+{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{x}}}\right)& {\mathrm{\tau }}_{\mathrm{z}\mathrm{x}}=\mathrm{\mu }\left({\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{x}}}+{\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{z}}}\right)\end{array}$$

(16)

$$\begin{array}{ccc}{\mathrm{\tau }}_{\mathrm{x}\mathrm{z}}=\mathrm{\mu }\left({\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{x}}}+{\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{z}}}\right)& {\mathrm{\tau }}_{\mathrm{y}\mathrm{z}}=\mathrm{\mu }\left({\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{y}}}+{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{z}}}\right)& {\mathrm{\tau }}_{\mathrm{z}\mathrm{y}}=\mathrm{\mu }\left({\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{y}}}+{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{z}}}\right)\end{array}$$

(17)

Heat dissipation can be obtained due to viscosity [17,18],

$$\mathrm{\varnothing }=\mathrm{\mu }\left\{2\left({\left({\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{x}}}\right)}^2+{\left({\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{y}}}\right)}^2+{\left({\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{z}}}\right)}^2\right)+{\left({\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{y}}}+{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{x}}}\right)}^2+{\left({\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{z}}}+{\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{x}}}\right)}^2+{\left({\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{z}}}+{\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{y}}}\right)}^2\right\}$$

(18)

The quantity is expressed in units of power over volume, in the International System W/m³, and also cal/sec m³. Because s = S/V, S is the entropy associated with heat dissipation by viscosity and V is the volume:

$$∆\mathrm{s}={\displaystyle \frac{\mathrm{\delta }\mathrm{Q}}{\mathrm{T}}}\to {\displaystyle \frac{∆\mathrm{s}}{\mathrm{d}\mathrm{t}}}={\displaystyle \frac{1}{\mathrm{T}}}{\displaystyle \frac{\mathrm{\delta }\mathrm{Q}}{\mathrm{d}\mathrm{t}}}\to {\displaystyle \frac{\partial {\mathrm{s}}_{\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}}}{\partial \mathrm{t}}}={\displaystyle \frac{\frac{\mathrm{\delta }\mathrm{Q}}{\mathrm{d}\mathrm{t}}}{\mathrm{T}}}\approx {\displaystyle \frac{\mathrm{\varnothing }}{\mathrm{T}}}$$

(19)

The entropy flow due to heat dissipation by viscosity is written as

$${\displaystyle \frac{\partial s_{heat dissipation}}{\partial t}}={\displaystyle \frac{\varnothing }{T}}={\displaystyle \frac{\mu }{T}}\left\{2\left({\left({\displaystyle \frac{\partial u}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial v}{\partial y}}\right)}^2+{\left({\displaystyle \frac{\partial w}{\partial z}}\right)}^2\right)+{\left({\displaystyle \frac{\partial u}{\partial y}}+{\displaystyle \frac{\partial v}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial u}{\partial z}}+{\displaystyle \frac{\partial w}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial u}{\partial z}}+{\displaystyle \frac{\partial w}{\partial y}}\right)}^2\right\}$$

(20)

with units, in the International System, of J/K s m³. The entropic flux associated with these two processes is

$${\displaystyle \frac{\partial \mathrm{s}}{\partial \mathrm{t}}}={\displaystyle \frac{\partial {\mathrm{s}}_{\mathrm{H}\mathrm{e}\mathrm{a}\mathrm{t}}}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial {\mathrm{s}}_{\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}}}{\partial \mathrm{t}}}$$

(21)

In full,

$${\displaystyle \frac{\partial \mathrm{s}}{\partial \mathrm{t}}}={\displaystyle \frac{\mathrm{k}}{{\mathrm{T}}^2}}{(\nabla \mathrm{T})}^2+{\displaystyle \frac{\mathrm{\mu }}{\mathrm{T}}}\left\{2\left({\left({\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{x}}}\right)}^2+{\left({\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{y}}}\right)}^2+{\left({\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{z}}}\right)}^2\right)+{\left({\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{y}}}+{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{x}}}\right)}^2+{\left({\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{z}}}+{\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{x}}}\right)}^2+{\left({\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{z}}}+{\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{y}}}\right)}^2\right\}$$

(22)

The dynamics of the atmosphere are complex and connected. The connection between the air movements of its neighboring layers is made by means of vertical velocities w, which is decisive in any explanatory model. In large-scale movements, the vertical velocity approaches a few cm/s, and the continuity equation indicates that the horizontal divergence does not exceed 10⁻⁴ s⁻¹, a value compatible with that obtained from the vorticity equation. On the small scale (horizontal scale close to 10 km), the vertical components are comparable to the horizontal magnitude of the wind speed (5.3–7.4 m/s, moderate on the Beaufort scale) [19,20].

2.2.3. Kolmogorov–Sinai Entropy

Following Farmer’s arguments, one of the basic discrepancies to consider when comparing chaotic and predictable behavior is that chaotic paths continuously form new information, while predictable ones do not [21,22]. The entropy metric, or Kolmogorov–Sinai entropy (S_K = dS/dt), provides an upper bound on the measure of information gain. The entropy metric was initially defined by Shannon [23] in an independent account of dynamical systems. The concept is used in dynamical systems, where Kolmogorov [24] and Sinai [25] clarified that it is a topological invariant.

By applying Pesin’s identity [26], the addition of Lyapunov exponents greater than zero is found to be the Kolmogorov–Sinai (or invariant) entropy, the K-S (or invariant) entropy, or the metric (or invariant) entropy. As noted, Kolmogorov [27] applied the concept of entropy to dynamical systems, and Sinai [25] provided proof. More specifically, via Ruelle’s inequality [28], the K-S entropy is a lower measure of the addition of the Lyapunov exponents, but they equalize if the natural measurement is continuous along the unsteady directions, which is common in chaotic flows [29,30].

The correlation entropy, K₂ [30,31], can be written as

$${\mathrm{K}}_2=\underset{\mathrm{m}\to \mathrm{\infty }}{\lim }\underset{\mathrm{r}\to 0}{\lim }\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }\mathrm{l}\mathrm{o}\mathrm{g}{\displaystyle \frac{\mathrm{C}\left(\mathrm{m},\mathrm{r}\right)}{\mathrm{C}\left(\mathrm{m}+1,\mathrm{r}\right)}}$$

(23)

The correlation entropy, K₂, is a lower bound on the Kolmogorov entropy, S_K, i.e., K₂~S_K [31]. This entropy, in the same way as that of Kolmogorov, determines whether a temporary series of experimental data has regular, chaotic, or random behavior. In Equation (23), C (m, r) is the correlation sum of the reconstructed trajectory of a time series for a given embedding dimension m and is used to estimate the correlation dimension [30]. It is defined as

$$\mathrm{C}\left(\mathrm{r}\right)={\displaystyle \frac{2}{\mathrm{n}\left(\mathrm{n}-1\right)}}\sum _{\mathrm{j}=1}^{\mathrm{n}}\sum _{\mathrm{i}=\mathrm{j}+1}^{\mathrm{n}}\mathrm{\Theta }\left(\mathrm{r}-{\mathrm{r}}_{\mathrm{i}\mathrm{j}}\right)=\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }{\displaystyle \frac{2}{\mathrm{n}(\mathrm{n}-1)}}\sum _{\mathrm{i}\ne \mathrm{j}}^{\mathrm{n}}\mathrm{\Theta }\left(\mathrm{r}-\sqrt{\sum _{k=0}^{m-1}{({\mathrm{X}}_{\mathrm{i}-\mathrm{k}}-{\mathrm{X}}_{\mathrm{j}-\mathrm{k}})}^2}\right)$$

(24)

C(r) can be interpreted as the number of points inside of all circles of radius r normalized to 1, when r is large enough to include all points without double counting; n is the number of data; $\mathrm{\Theta }$ is the Heaviside function (or step function); ${\mathrm{r}}_{\mathrm{i}\mathrm{j}}$ is the spatial distancing between two points with subscripts i and j, usually in an m-dimensional time-delay embedding by the Euclidean norm ${\mathrm{r}}_{\mathrm{i}\mathrm{j}}= \sqrt{\sum _{\mathrm{k}=0}^{\mathrm{m}-1}{({\mathrm{X}}_{\mathrm{i}-\mathrm{k}}-{\mathrm{X}}_{\mathrm{j}-\mathrm{k}})}^2}$; and r is a real number whose choice requires attention; if r tends to 0, then C(r) may lose meaning, and if r tends to large values, the information provided by C(r) is not useful. Equation (24) can also be written as

$$\mathrm{C}\left(\mathrm{r}\right)=\underset{n\to 0}{\lim }{\displaystyle \frac{1}{n^2}}\left[\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r} \mathrm{o}\mathrm{f} \mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{s}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{x}}_{\mathrm{j}}\right)\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h} \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t} \sqrt{\sum _{\mathrm{k}=0}^{\mathrm{m}-1}{({\mathrm{X}}_{\mathrm{i}-\mathrm{k}}-{\mathrm{X}}_{\mathrm{j}-\mathrm{k}})}^2}<r\right]$$

(25)

C(r) is calculated by varying r from 0 to the largest possible value of ${\mathrm{r}}_{\mathrm{i}\mathrm{j}}$. In an equivalent way, you can think of C(r) as the probability that two different points randomly chosen are closer than the distance r. For properly small values of r and n that are very large, C(r) takes the functional form of a power of powers:

$$\mathrm{C}\left(\mathrm{r}\right)~{\mathrm{r}}^{{\mathrm{D}}_{\mathrm{C}}},$$

(26)

D_C is the correlation dimension (or correlation exponent). In the logarithmic approach for Equation (26), if log (C(r)) is graphed against log (r), the slope of this line will be the correlation dimension. In chaotic time series studies, it is accepted that a good indication of the chaotic nature of the series is that the correlation dimension, DC, saturates at values less than 5 [32]. Meanwhile, if it saturates at values greater than 5, it implies essentially random data.

The ideas presented above are related to the typical entropy of thermodynamics (and, by extension, heat) [30], which indicates the disorder of a system, as it allows for measuring the separation of adjacent trajectories towards other regions of the spatial state. The difference with thermodynamic entropy is that the K-S entropy is made up of time units with exponent −1 (or inverse iterations) and indicates the average rate at which predictions are missed. The reciprocal gives an estimate of the expected time for a relatively reliable prediction [30]. A typically random system has infinite entropy, and a periodic system has zero entropy.

Shannon entropy is an information creation proportion as a chaotic system changes. The strong dependence on initial conditions means that two points extremely close in a state space move further apart over time. Over time, knowledge about the initial condition expands as initially insignificant digits increase their influence on its specification. A chaotic system is a source of new information [29]. Information is a measure of the surprising nature of an event or the distinguishability of a configuration, such that entropy, always positive, accounts for information that is not yet available and can have high values in a chaotic system. This process can be visualized as a loss of information; predictions of an initial state become less accurate over time.

If the measurements of temperature, relative humidity, and wind speed magnitude form time series, these can be analyzed with chaos theory [30,31,32] and used to determine, first, if the parameters of interest are in the appropriate ranges: Lyapunov exponent (λ > 0), correlation dimension (D_C < 5), Kolmogorov entropy (S_K > 0), Hurst exponent (0.5 < H < 1), and Lempel–Ziv complexity (LZ > 0). Various investigations have shown the value of the chaotic analysis of time series [32,33,34,35,36,37,38,39], and, for the case treated here, all of the time series analyzed showed that their parameters of interest are in the domains of validity. The flowchart shows the procedure applied (Figure 9).

3. Results

Mandelbrot [40,41] pointed out that fractals are, in many ways, more natural, and therefore better intuitively understood by humans, than artificially smoothed objects based on Euclidean geometry. Might we add that the atmosphere interacts differently with fractal objects? What is the basis of this difference? And what magnitude would allow us to quantitatively assess this atmosphere–fractal objects relationship?

We know that one of the major problems facing humanity is atmospheric warming, which causes climate change due to human activity. That is, we generate more heat than the atmosphere can naturally process [42,43,44,45]. Direct observation of practically all human cities shows structures (used as family homes, industrial warehouses, buildings, etc.) with geometric shapes of parallelepipeds (in all of their variations: straight, irregular, cubic, etc.) and cylinders. Cones and pyramids are very rare cases. Therefore, it seems reasonable to focus on these types of geometric volumes and their effect on the micrometeorology of the boundary layer.

Is it possible to develop a test to verify, comparatively, which human geometric construction generates the greatest heat flow into the atmosphere? As a first step towards this investigation, three geometric volumes can be used: a right parallelepiped, a right cylinder, and a miniature artificial mountain constructed to resemble real mountains known to admit fractal representation. The three geometric volumes were made of the same material, but the cylinder was red to see if this aspect makes any difference. They were exposed in a natural environment without any restrictions, measuring meteorological variables of temperature, relative humidity, and wind speed magnitude with two meteorological stations arranged at the same height and separation distance, in front and behind, with respect to the volumetric object. These hourly measurements, with a total of approximately 1650 h for each geometry, formed time series that can be analyzed using chaos theory, being of interest to obtain the value of the maximum Kolmogorov entropy [30].

Parallelepiped

The parallelepiped is placed on flat, dry, non-rough terrain with constant sunlight and radiation, with sparse shrub and tree vegetation, as shown in Figure 10. The measurement period is approximately 2.3 months, during the summer season.

Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16 show the time series of temperature, wind speed magnitude, and relative humidity for the front and back of the obstacle.

Table 1. Contains the Lyapunov exponents (λ), the correlation dimension (D_C), the Kolmogorov entropy (S_K), the Hurst exponent, the Lempel–Ziv complexity, and the summation of Kolmogorov entropies ($\sum {\mathrm{S}}_{\mathrm{K}}$) of time series of temperature (T), relative humidity (RH), and wind speed (WS) measured on the front and back faces of a rectangular parallelepiped.

Station 3 (South)	T (°C)	RH%	WS (m/s)	∑SK
λ (bits/h)	0.624 ± 0.070	0.689 ± 0.153	0.998 ± 0.082
DC	3.209 ± 0.832	1.625 ± 0.167	3.556 ± 1.419
SK (bits/h)	0.295	0.232	0.319	0.846
H	0.8897442	0.9150111	0.9707184
LZ	0.29121	0.01763	0.44480
Station 4 (North)	T (°C)	RH%	WS (m/s)
λ (bits/h)	0.646 ± 0.077	0.596 ± 0.141	1.145 ± 0.082
DC	2.529±0.419	1.792 ± 0.194	3.065 ± 1.308
SK (bits/h)	0.284	0.435	0.416	1.135
H	0.8784227	0.9149694	0.9619154
LZ	0.33342	0.02468	0.48136

is a summary of the chaotic parameter calculations, performed using CDA software (Professional Version 2.2) [33], on time series of temperature, relative humidity, and wind speed magnitude, measured in the vicinity of a geometric obstacle in the shape of a parallelepiped.

$$∆{\mathrm{S}}_{\mathrm{K},\mathrm{P}\mathrm{A}\mathrm{R}\mathrm{A}\mathrm{L}\mathrm{L}\mathrm{E}\mathrm{L}\mathrm{E}\mathrm{P}\mathrm{I}\mathrm{P}\mathrm{E}\mathrm{D}}=1.135-0.845=\mathrm{O}.29$$

${\mathrm{\Delta }\mathrm{S}}_{\mathrm{K},\mathrm{P}\mathrm{A}\mathrm{R}\mathrm{A}\mathrm{L}\mathrm{L}\mathrm{E}\mathrm{L}\mathrm{E}\mathrm{P}\mathrm{I}\mathrm{P}\mathrm{E}\mathrm{D}}$ is the difference in entropic flux calculated from the measurements made on the opposite faces of the geometric volume.

In general, the distribution of data for meteorological variables in the other two geometric volumes studied (cylinder and fractal mountain) acquires a configuration like that presented in Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, so they will be omitted.

Cylinder

The cylinder is located on flat, dry, even terrain with constant sunlight and sparse shrub and tree vegetation, as shown in Figure 17. The measurement period is approximately 2.3 months, during the summer season.

Table 2. Contains the Lyapunov exponents (λ), the correlation dimension (D_C), the Kolmogorov entropy (S_K), the Hurst exponent, the Lempel–Ziv complexity, and the summation of Kolmogorov entropies ($\sum {\mathrm{S}}_{\mathrm{K}}$) for the time series of temperature (T), relative humidity (RH), and wind speed (WS) measured on the front and back faces of a cylinder.

Station 3 (South)	T (°C)	RH%	WS (m/s)	∑SK
λ (bits/h)	0.378 ± 0.069	0.164 ± 0.038	0.197 ± 0.024
DC	1.293 ± 0.208	1.042 ± 0.023	1.136 ± 0.229
SK (bits/h)	0.410	0.167	0.136	0.713
H	0.9917651	0.9951348	0.9964377
LZ	0.14447	0.07588	0.08366
Station 4 (North)	T (°C)	RH%	WS (m/s)
λ (bits/h)	0.342 ± 0.034	0.199 ± 0.024	0.123 ± 0.018
DC	1.141 ± 0.306	1.011 ± 0.130	1.154 ± 0.212
SK (bits/h)	0.076	0.127	0.486	0.689
H	0.992838	0.9936692	0.9946308
LZ	0.04665	0.03964	0.04125

is a summary of the chaotic parameter calculations, performed using CDA software, for the time series of temperature, relative humidity, and wind speed, measurements taken near a geometric obstacle in the form of a cylinder.

$$∆{\mathrm{S}}_{\mathrm{K},\mathrm{C}\mathrm{Y}\mathrm{L}\mathrm{I}\mathrm{N}\mathrm{D}\mathrm{E}\mathrm{R}}=0.713-0.689=0.024$$

${\mathrm{\Delta }\mathrm{S}}_{\mathrm{K},\mathrm{C}\mathrm{Y}\mathrm{L}\mathrm{I}\mathrm{N}\mathrm{D}\mathrm{E}\mathrm{R}}$ is the difference in entropic flux calculated from the measurements made on the opposite faces of the geometric volume.

Fractal Mountain

The fractal mountain is located on flat, dry, and even terrain, with constant sunlight and sparse shrub and tree vegetation, as shown in Figure 18. The measurement period is approximately 2.3 months, during the summer season.

Table 3. Contains the Lyapunov exponents (λ), the correlation dimension (D_C), the Kolmogorov entropy (S_K), the Hurst exponent, the Lempel–Ziv complexity, and the summation of Kolmogorov entropies ($\sum {\mathrm{S}}_{\mathrm{K}}$) of the time series of temperature (T), relative humidity (RH), and wind speed magnitude (WS) measured on the front and back faces of a fractal mountain.

Station 3 (South)	T (°C)	RH%	WS (m/s)	∑SK
λ (bits/h)	0.635 ± 0.083	0.003 ± 0.001	0.929 ± 0.093
DC	1.309 ± 0.747	1.945 ± 0.184	1.469 ± 0.922
SK (bits/h)	0.275	0.373	0.314	0.964
H	0.8676284	0.9123346	0.9669798
LZ	0.28706	0.02186	0.40289
Station 4 (North)	T (°C)	RH%	WS (m/s)
λ (bits/h)	0.594 ± 0.080	0.003 ± 0.001	1.074 ± 0.092
DC	1.646 ± 0.730	1.746 ± 0.255	2.709 ± 1.087
SK (bits/h)	0.240	0.325	0.417	0.982
H	0.8677429	0.9123546	0.9561365
LZ	0.27775	0.01312	0.53779

is a summary of the chaotic parameter calculations, performed using CDA software, in the time series of temperature, relative humidity, and wind speed magnitude, measurements taken near a geometric obstacle in the form of a fractal mountain.

$$∆{\mathrm{S}}_{\mathrm{K},\mathrm{M}\mathrm{O}\mathrm{U}\mathrm{N}\mathrm{T}\mathrm{A}\mathrm{I}\mathrm{N}}=0.982-0.964=0.018$$

${\mathrm{\Delta }\mathrm{S}}_{\mathrm{K},\mathrm{M}\mathrm{O}\mathrm{U}\mathrm{N}\mathrm{T}\mathrm{A}\mathrm{I}\mathrm{N}}$ is the difference in entropic flux calculated from the measurements made on the opposite faces of the geometric volume.

Figure 19 is a comparison chart of the maximum entropic fluxes along two opposite faces of an object with a volumetric geometry: S _{s 1, p} = entropic flux along face 1 of the parallelepiped; S _{o s 1, p} = entropic flux along the face opposite side 1 of the parallelepiped; S _{s 1, c} = entropic flux along face 1 of the cylinder; S _{o s 1, c} = entropic flux along the face opposite side 1 of the cylinder; S _{s 1, fm} = entropic flux along face 1 of the fractal mountain; S _{o s 1, fm} = entropic flux along the face opposite side 1 of the fractal mountain. The decreasing values of ΔS indicate the differences in entropic fluxes between the faces of each volumetric geometry.

A basic characteristic to consider on the surface is the albedo, which is a measure of a surface’s ability to reflect solar radiation. It is expressed as a percentage, where a high value (100%, maximum value) indicates that the surface reflects a lot of light. Conversely, a low value (for example, 10%) indicates that the surface absorbs a lot of light. The albedo also depends on the nature of the material that makes up the surface [46]. In the case of the present investigation and according to the volumetric geometry used, the following was found.

Parallelepiped: It was constructed with veneered wood, whose albedo fluctuates between 0.15 and 0.25, a low value indicating that the surface reflects little sunlight, having a low effect on the environment.

Metal cylinder: The albedo of metals varies between 0.5 (matte metal, with little ability to reflect sunlight) and 0.98 for polished metal (with a great ability to reflect sunlight back into the environment).

For the fractal mountain constructed with white fabric, the fabric’s albedo is generally high at 0.95, reflecting a lot of sunlight back into the environment.

In this study, the surfaces with the highest albedo, the cylinder and the fractal mountain, show the lowest entropic flux into the environment. Therefore, in the context of this presentation, albedo does not have a significant impact on the results.

Table 4. Average (< >) values of temperature (T), relative humidity (RH), and wind speed magnitude (WS) measured at the front and back of the volumetric geometry.

	Front			Back
Volumetric Geometry	<T>	<RH>	<WS>	<T>	<RH>	<WS>
Parallelepiped	11.71	60.04	0.17	11.31	63.44	0.23
Cylinder	18.60	51.91	0.48	18.60	51.92	0.48
Fractal mountain	12.79	57.98	0.24	12.92	58.21	0.33

presents the average temperature, relative humidity, and wind speed magnitude on the front and back of each volumetric geometry.

When comparing the ΔS values according to the volumetric geometries used,

$${\displaystyle \frac{{\mathrm{\Delta }\mathrm{S}}_{\mathrm{K},\mathrm{P}\mathrm{A}\mathrm{R}\mathrm{A}\mathrm{L}\mathrm{L}\mathrm{E}\mathrm{L}\mathrm{E}\mathrm{P}\mathrm{I}\mathrm{P}\mathrm{E}\mathrm{D}}}{{\mathrm{\Delta }\mathrm{S}}_{\mathrm{K},\mathrm{F}\mathrm{R}\mathrm{A}\mathrm{C}\mathrm{T}\mathrm{A}\mathrm{L}\mathrm{M}\mathrm{O}\mathrm{U}\mathrm{N}\mathrm{T}\mathrm{A}\mathrm{I}\mathrm{N}}}}=16.11$$

$${\displaystyle \frac{{\mathrm{\Delta }\mathrm{S}}_{\mathrm{K},\mathrm{C}\mathrm{Y}\mathrm{L}\mathrm{I}\mathrm{N}\mathrm{D}\mathrm{E}\mathrm{R}}}{{\mathrm{\Delta }\mathrm{S}}_{\mathrm{K},\mathrm{F}\mathrm{R}\mathrm{A}\mathrm{C}\mathrm{T}\mathrm{A}\mathrm{L}\mathrm{M}\mathrm{O}\mathrm{U}\mathrm{N}\mathrm{T}\mathrm{A}\mathrm{I}\mathrm{N}}}}=1.33$$

By constructing

Table 5. Contains the volumetric geometry (VG) and the entropic flux variation (ΔS).

Volumetric Geometry (VG)	ΔS
Parallelepiped	0.29
Cylinder	0.024
Fractal Mountain	0.018

, which groups the volumetric geometry (VG) data studied and the entropic flux data for each case, Figure 20 can be constructed.

Figure 20 shows the asymptotic decay according to the volumetric geometry of the entropic (heat) flow.

It is possible to estimate the amount of information that allows for the future behavior of an interactive system to be, approximately, predictable, and, similarly, the speed with which the system loses information over time. The loss of information (<ΔI> in (bits/h)) can be determined according to the equation

$$<∆\mathrm{I}>=<{\mathrm{I}}_{\mathrm{N}\mathrm{E}\mathrm{W}}-{\mathrm{I}}_{\mathrm{O}\mathrm{L}\mathrm{D}}>=-\mathrm{\lambda }{\left(log2\right)}^{-1}$$

(27)

λ is the maximum Lyapunov exponent. <ΔI> was calculated as the sum of the partial contribution of each MV (meteorological variables: T, WS, RH). The fractal dimension was also determined:

D = fractal dimension = 2–H

(28)

where H is the Hurst exponent.

Table 6. Information loss and fractal dimension according to volumetric geometry and measuring station.

Parallelepiped
Station 3	T (°C)	RH%	WS (m/s)	${\displaystyle \overline{\sum }}$
<ΔI> (bits/h)	−2.073	−2.288	−3.315	−2.560
D	1.110	1.085	1.029	1.075
Station 4	T (°C)	RH%	WS (m/s)
<ΔI> (bits/h)	−2.146	−1.979	−3.804	−2.643
D	1.122	1.085	1.038	1.082
Cylinder
Station 3	T (°C)	RH%	WS (m/s)
<ΔI> (bits/h)	−1.256	−0.545	−0.654	−0.818
D	1.008	1.005	1.004	1.006
Station 4	T (°C)	RH%	WS (m/s)
<ΔI> (bits/h)	−1.136	−0.661	−0.409	−0.735
D	1.007	1.006	1.005	1.006
Fractal Mountain
Station 3	T (°C)	RH%	WS (m/s)
<ΔI> (bits/h)	−2.109	−0.010	−3.086	−1.735
D	1.132	1.088	1.033	1.084
Station 4	T (°C)	RH%	WS (m/s)
<ΔI> (bits/h)	−1.973	−0.010	−3.568	−1.850
D	1.132	1.088	1.044	1.090

contains a summary of the information loss and fractal dimension calculations for the different volumetric geometries studied.

The average fractal dimension of the meteorology is greater in the fractal mountain, and the cylinder loses information the fastest, while the parallelepiped causes a slower loss of meteorological information, which makes it closer to a deterministic system.

4. Discussion

The results obtained are promising for elucidating the effect of regular geometries on urban micrometeorology. The measurements were taken in an area with a relatively “pristine” or unpolluted environment. This condition may allow for the environmental response to intervention or disturbance in the form of a volumetric geometry, even if this geometry is small in size, to be complex or chaotic. Fractals can be viewed as indicators of form. They are based on a flow of repeating geometric patterns at different scales. The systems involved in the study, geographic configuration and meteorology, are fractals. The measured time series (of temperature, relative humidity, and wind speed magnitude) are used to calculate the maximum entropic fluxes originating in front of and behind the obstacles (three cases: two with regular geometry and one with fractal geometry). Calculating the difference in entropic fluxes for each case allows for comparing their reactions by obstacle type: non-fractal and fractal geometries. The maximum inflections of the entropic flow surfaces [47,48] are revealed in the form of thermal fluxes, similar to the energy dissipated in a mechanical system interacting with another. One possible interpretation, from a metric perspective, is that it may represent a measure of the departure of the fractal manifold metric from the Euclidean metric. In mathematics, a manifold is the standard geometric object that generalizes the intuitive notion of a curve (one-manifold) and a surface (two-manifold) to any dimension and over diverse fields (not only the real fields, but also complex and matrix fields). Entropy is a concept associated with disorder and information. In the context of Ricci flow [49], entropy is used as a function that evolves under the flow, allowing one to study the evolution of the geometry of a spacetime and to understand its long-term behavior. Perelman [50] introduced an entropy formula that evolves under Ricci flow, which is crucial to understanding how the flow handles singularities or maximal inflections.

The entropy time Equation (22), or entropic flux, describes the behavior of scalar quantities, such as temperature; it states that high-temperature concentrations will disperse until a uniform temperature is reached throughout the object. However, the achievement of uniformity at the surface will depend on the geometry of the object. Similarly, Ricci flow describes the behavior of a tensor quantity, the Ricci curvature tensor. Hamilton’s idea [51] was that under Ricci flow, high-curvature concentrations would disperse until a uniform curvature is reached over the entire three-dimensional manifold.

The process studied can be assumed with regular volumetric geometries that are deformed, evolving towards fractal geometry; if the initial condition of the entropic surface of urban meteorology originated by the Euclidean volumetric geometry of the parallelepiped contains high inflections, the evolution towards the fractal geometry of the regular geometry has an effect similar to the Ricci flow on the entropic surface, as it disperses the inflections until reaching the minimum inflection in the fractal mountain (fractal geometry), as shown in Figure 21.

Nature, from the perspective developed in this work, is fractal, including urban meteorology time series (temperature, relative humidity, wind speed), trees, coastal edges, mountains, geographic basins, living creatures, pollutant time series, natural processes, etc. Even the ancestral dwellings of primitive peoples tended towards fractal structures, as shown in Figure 22.

The presence of obstacles in the form of a parallelepiped, a cylinder, and a fractal mountain affects the micrometeorology [19,20] measured before and after a natural air flow interacted with them, producing turbulence. This turbulence can be related to the small-scale manifestation of the Kolmogorov cascade [52,53,54,55]. Close to the ground, the finer eddies disperse their energy as heat due to viscosity. Thus, in the turbulent zone, two quantities are manifested: the average energy dissipation rate, ε_D, and the kinematic viscosity, ν. The relationship between these two quantities gives the length scale

$${\left({\mathrm{l}}_{\mathrm{k}}\right)}^4={\mathrm{\nu }}^3{{\mathrm{\epsilon }}_{\mathrm{D}}}^{-1}$$

The Kolmogorov entropy calculated from the time series in this study has units ${\mathrm{S}}_{\mathrm{K}}\left[{\displaystyle \frac{\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}}{\mathrm{h}}}\right]$. When transforming its units [56], it remains ${\mathrm{S}}_{\mathrm{K}}\left[{\displaystyle \frac{\mathrm{J}}{\mathrm{K}\mathrm{h}}}\right]$, which is equivalent to Energy/(Temperature time), dimensionally:

$$\left[{\displaystyle \frac{\mathrm{M}}{\mathrm{T}}}\right]\left[{\displaystyle \frac{{\mathrm{L}}^2}{{\mathrm{t}}^3}}\right]~\left[{\displaystyle \frac{1}{\mathrm{T}}}\right]\mathrm{\ast }\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e} \mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$

This study of the influence of geometric volumes on the boundary layer shows that some of them favor heat flow towards the boundary layer. By estimating the average energy dissipation rate and the small-scale kinematic viscosity, a regime of variations in heat and entropy fluxes can be established [55,57,58]. Other types of fluids have shown similar behavior [58,59,60].

According to the results obtained, a first approximation shows that the geometric shape that dissipates the most heat into the atmosphere is the parallelepiped, a geometric shape that corresponds to more than 90% of housing, industrial plants, recreational areas, etc. that make up modern cities around the world. Three-dimensional constructions and entropy are linked through various aspects of physics, including thermodynamics, material science, and quantum mechanics. Entropy and entropy flow, a measure of disorder or randomness in a system, can be visualized and analyzed in the context of three-dimensional structures.

The fractal mountain in the experiment, whose irregular surface faces function as a radiator, allows for heat exchange between two media, one of which is ambient air. The radiator makes it possible to mitigate and dissipate heat from a system, thus preventing overheating. This system commonly operates by convection but also by radiation. The radiator can also be related to a heat exchanger that transfers or, in certain cases, receives heat to or from the ambient air.

In general, a metric space is a set associated with a distance function; that is, this function is defined on the set, fulfilling properties attributed to distance, such that, for any two points in the set, they are at a certain distance assigned by the function. The best-known example of a metric space is three-dimensional Euclidean space with its usual notion of distance. Fractal geometry is a source of some strange metric spaces, because a fractal, as noted, is a geometric object whose basic structure, fragmented or apparently irregular, is repeated at different scales.

A mountain can be considered a fractal. Fractals are geometric structures whose shape repeats at different scales, and mountains, as shown in Figure 23, like other natural structures, such as clouds, coastlines, and rivers, exhibit this property of self-similarity.

Ricci flow [51] is a process that deforms Riemannian metrics, similarly to heat diffusion, smoothing out the irregularities of the metric. On surfaces [61], Ricci flow tends to convert the metric into one of constant Gaussian curvature [62], Theorem 1.6, which is related to the uniformization theorem [63]. Entropic surfaces, in this context, refer to surfaces that exhibit varying maximum entropy properties or disorder in their curvature distribution affected by their interaction with regular and fractal geometry.

According to [47,48], a surface of maximum entropy variables according to measurements in a geomorphology exhibits a non-uniform curvature distribution, with areas of high and low curvature, representing states of greater disorder or entropy. These types of surfaces were determined for the periods 2010/2013, 2017/2020, and 2029/2022 in the geomorphology of the Santiago de Chile basin [47,48]. These surfaces, in a thermodynamic state space, represent the entropy quantities of the polluting systems and the urban meteorology systems. In the three periods, they allow for visualizing, in a first approximation, the evolution of the entropy surface of the polluting system and its influence on the entropy surface of urban meteorology under different conditions (droughts, heat waves, heat islands, urban densification, deforestation, basin geomorphology, etc.) for an extensive period of measurement. Entropic surfaces change as systems evolve (changes of state); their roughness reveals information about the system’s tendency to increase or decrease entropy, as well as the conditions under which entropy reaches a maximum or minimum, facilitating the understanding of physical and chemical processes.

Localized time series measurements (whether of pollutants and urban meteorology) have maximum entropy and an associated metric [36]. The entropy metric is a measure of the complexity or chaos of a dynamical system described by the time series, where a higher value indicates greater complexity (Equation (1), [36]). Essentially, the entropy metric is a measure of the growth rate of information in a dynamical system [36]. An entropic surface metric, in the sense given by Ruppeiner [64,65], is a way of measuring the distance between two points in the state space of a thermodynamic system based on the entropy geometry. In other words, it describes how the entropy of a system changes as we move from one state to another.

5. Conclusions

As a first approach, three unprotected artificial volumetric geometric structures were considered, located in a pristine high-mountain area and subject to the influence of climatic conditions typical of the area (summer season). Measurements were taken in front of and behind each geometric structure (over a period of approximately two months) in the form of time series of temperature, relative humidity, and wind speed. The chaotic parameters (λ, Dc, S_K, H, LZ, <ΔI>, D) were calculated for each time series, demonstrating that all of the series were chaotic; the correlation dimension, D_C, saturates at values less than five, rejecting the possibility of random data (D_C > 5), and from their Kolmogorov entropies (S_K), it was determined which geometric structure dissipates, more or less, heat and entropic flows towards the atmosphere. The measurements showed that the irregular geometric volume, resembling a scaled-down mountain, has a greater capacity to attenuate thermal and entropic fluxes into the atmosphere than the smoothed regular geometric volumes analyzed. This demonstrates, extending this result to the macro level, the important role that geographic fractality plays in the planet’s climate; geomorphology, which is fractal, is essential in climate regulation. The experiment developed also shows that the scaled fractal mountain favors an increase in the fractal dimension of the measured micrometeorological time series (Table 6; all series are fractal). In addition to the growing concern over global warming, one must also add the thermal effect that causes the loss of the original geographic fractality. Extensive and increasing global changes in the Earth’s surface roughness, superimposing standard regular geometries as the primary construction design unit, can have a deteriorating effect on the boundary layer and its micrometeorology. Based on the recorded data and their analysis, the geometric volumetry of buildings, basically artificial, in cities favors a specific form of pollution on the atmospheric boundary layer that can be generically designated as geometric pollution and manifests itself as thermal flow.