# Mathematical Modelling of Climate Change and Variability in the Context of Outdoor Ergonomics

^{*}

## Abstract

**:**

## 1. Introduction

_{2}) in the atmosphere. This greenhouse gas (GHG), which is a by-product of fossil fuel combustion, is the main contributor to global warming.

## 2. Notes on Outdoor Environmental Ergonomics through the Prism of Climate Change

## 3. Climate Models as Dynamical Systems and Their Sensitivity Analysis

#### 3.1. Climate as a Complex Dynamical System

- –
- As mentioned earlier, it is a five-element large-scale physical system with several global hydrological and biochemical cycles. Its elements, being heterogeneous thermo-dynamical systems, have significant differences in their structure, dynamics, physics and chemistry. Dynamical and physical processes that occur in the ECS subsystems differ in their scales, both spatial and temporal. Elements of the ECS link together through numerous physical coupling mechanisms, both weak and strong, including feedback mechanisms. In turn, each ECS subsystem can be viewed as complex, consisting of subsystems, which themselves are composed of low-order subsystems. The atmosphere, for instance, can be divided into several vertical layers depending on its thermal stratification: the troposphere, stratosphere, mesosphere and thermosphere. The troposphere, in turn, can also be subdivided into the surface layer, boundary layer and free atmosphere based on the effect of the surface friction.
- –
- Physical and dynamical processes in ECS span a wide range of time and space scales. Temporal scales range from seconds to decades, and spatial scales range from molecular to planetary scales. The dynamics of the ECS is nonlinear. ECS elements interact with each other nonlinearly, creating, under certain conditions, chaotic behaviour of subsystems and the climate system as a whole.
- –
- ECS has a large number of feedback mechanisms, both positive and negative, that strongly affect climate formation.
- –
- ECS components are non-insulated systems that act as cascade systems and interact with each other in various ways, including through the transfer of momentum, sensible and latent heat, gases and particulate matter. Collectively, ECS elements constitute a climate system, which is a unique large-scale natural object.
- –
- Each ECS component has a specific response time, which must be considered when building ECS models. For example, the atmosphere can be considered the only component of the ECS model for dynamical processes with timescales from days to weeks since the tropospheric response time is about one month, while oceans, land surface and ice cover can be used to specify boundary conditions and/or external forcing. To explore the dynamical processes with timescales ranging from months to years, the ECS model must include the atmosphere and ocean, along with sea ice. Thus, ECS models are built from a hierarchy of models that ultimately form a complex integrated model.
- –
- The ECS is an oscillating system that is characterized by fluctuations that are under the influence of internal factors (natural oscillations), as well as due to external perturbations (forced oscillations). Naturally occurring fluctuations result from internal instabilities (e.g., hydrodynamic instabilities, such as barotropic and baroclinic instabilities) and heat transport within the climate system that is caused by the interaction of its components [7]. Intentional and unintentional human impacts belong to the class of external forcings.
- –
- Since the ECS exchanges energy with the environment, in this sense, it is a thermodynamically open and non-isolated system. However, the ECS is a closed system with regard to the exchange of matter with the environment. The main energy source that drives the ECS is solar energy. Climate is affected by variations in external driving forces that imply natural causes, such as fluctuations in solar and volcanic activities, as well as changes in the gaseous and chemical composition of the atmosphere due to anthropogenic factors. The impact of the ECS on outer space is nonessential. At present, climate change is most influenced by variations in the composition of atmospheric particles and gases. Carbon dioxide (CO
_{2}), the concentration of which in the atmosphere has been continuously increasing since the Industrial Revolution, has the greatest impact on current climate change. - –
- The ECS and its subsystems possess emergent properties, examples of which are atmospheric emerging phenomena, such as clouds, large-scale baroclinic and barotropic eddies (cyclones, hurricanes) and small-scale eddies (tornadoes). An example of an emerging climate event is the El Niño–Southern Oscillation, which is a sporadic quasi-periodic variation in ocean surface temperatures across the tropical Pacific Ocean that affects the global atmospheric circulation and ocean circulation patterns. Natural emergent phenomena occur suddenly under some favorable conditions.

- –
- These systems belong to the class of dissipative dynamical systems that possess (strange) attractors.
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- Trajectories of climate dynamical systems are generally unstable in the Lyapunov sense.
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- Some unstable trajectory, enclosed in an attractor, generates deterministic dynamical chaos.

#### 3.2. Climate System Sensitivity to External Forcing

#### 3.3. Sensitivity Analysis of a Chaotic Dynamical System

_{z}.

#### 3.4. Climate System Response to Small External Forcing Based on the Fluctuation Dissipation Relation

_{2}concentration, a simple relationship between the CO

_{2}concentration and the corresponding radiative forcing can be used [48]: $\Delta {f}_{C{O}_{2}}\left(t\right)=5.35\times \mathrm{ln}\left(c\left(t\right)/c(0)\right)$, where $c\left(t\right)$ is the CO

_{2}concentration (in parts per million volume) at time t and $c\left(0\right)$ is the reference CO

_{2}concentration. Using the formula $\Delta {\mathsf{\Theta}}_{s}=S\Delta {f}_{C{O}_{2}}$, where the sensitivity coefficient has a value of $0.8\text{}{\mathrm{K}\text{}\mathrm{W}}^{-1}{\text{}\mathrm{m}}^{2}$ [48], for a doubling of pre-industrial CO

_{2}level, we obtain $\Delta {\mathsf{\Theta}}_{s}\approx 3\text{}\xb0\mathrm{C}$. Radiative forcing of $3.7\text{}{\mathrm{W}\text{}\mathrm{m}}^{-2}$ for a doubling of CO

_{2}was used in this calculation.

## 4. Mathematical Models of Climate System

#### 4.1. General Notes on Climate Modelling

- –
- The mathematical formulation of the problem, i.e., the translation of the real-world problem into the form of mathematical equations to be solved.
- –
- Consideration of the existence and uniqueness of a solution to the climate model equations.
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- The problem of the existence of attractor and estimating its dimension.
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- Study of the attractor’s fractal structure and invariant measure of the attracting set.
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- Finite-difference approximations and their convergence, stability and consistency.
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- Climate model sensitivity theory (theorems on linear approximation for various moments, linear response theory to small perturbations, algorithms for constructing the response operator).
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- Response operator approximation methods.
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- Numerical experiments and their analysis.

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- Atmospheric radiative transfer processes (transfer of shortwave (solar) and longwave (terrestrial) radiation through the atmosphere).
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- Hydrological cycle, including cloud formation and precipitation.
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- Convection.
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- Turbulence in the boundary and surface layers.
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- Small-scale orography.
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- The gravity wave drag.
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- Small-scale diffusion and dissipation.
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- Land surface processes.
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- Mixed ocean layer processes.

#### 4.2. Governing Equations

- (a)
- Two momentum equations:

- (b)
- The thermodynamic equation:

- (c)
- The equation for specific humidity that describes the hydrological cycle:

- (d)
- The continuity equation:

- (e)
- Hydrostatic equation:

- (f)
- The equation of state:

- (a)
- Two momentum equations:

- (b)
- The thermodynamic equation:

- (c)
- The equation for the mass continuity of salinity:

- (d)
- The continuity equation:

- (e)
- Hydrostatic equation:

- (f)
- The equation of state:

#### 4.3. Using Low-Order Simple Models to Study Climate Variability

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Sensitivity functions of the variables z and Z to the parameter r, which were calculated for strong coupling between fast and slow systems ($c=0.8$).

**Figure 2.**Original (red) and pseudo (blue) trajectories for the fast z and slow Z variables, calculated for strong coupling between fast and slow systems ($c=0.8$).

**Figure 3.**Absolute sensitivity functions ${\psi}_{\lambda}$ (

**a**) and ${\psi}_{C}$ (

**b**), which were calculated for different values of the feedback coefficient f.

**Figure 4.**Power spectra of the global mean surface temperature fluctuations that were obtained from the two-box random energy balance model for different values of the feedback coefficient f. The dashed line shows the characteristic $1/{\nu}^{2}$ slope.

**Figure 5.**Relative sensitivity functions ${\psi}_{\lambda}^{R}$ and ${\psi}_{C}^{R}$, which were calculated for different values of the feedback coefficient $f$.

**Table 1.**Sensitivity functions for fast and slow variables for strong ($c=0.8$) and weak ($c=0.15)$ coupling between systems.

c | $\partial \mathit{x}/\partial \mathit{r}$ | $\partial \mathit{y}/\partial \mathit{r}$ | $\partial \mathit{z}/\partial \mathit{r}$ | $\partial \mathit{X}/\partial \mathit{r}$ | $\partial \mathit{Y}/\partial \mathit{r}$ | $\partial \mathit{Z}/\partial \mathit{r}$ |
---|---|---|---|---|---|---|

0.8 | 0.07 | 0.07 | 1.02 | 0.03 | 0.08 | 0.69 |

0.15 | −0.01 | −0.01 | 1.01 | −0.09 | −0.09 | 0.91 |

**Table 2.**Absolute and relative uncertainties in power spectrum of the GMST fluctuations caused by one-sigma uncertainties in the feedback parameter $\lambda $ and the climate system inertia parameter $C$.

Uncertainty | Period of Oscillations (yr) | |||
---|---|---|---|---|

2 | 10 | 30 | 100 | |

$\delta {\left({S}_{TT}\right)}_{\lambda}\text{}\left({\mathrm{K}}^{2}\text{}\mathrm{yr}\right)$ | ±8.66 × 10^{−8} | ±4.03 × 10^{−5} | ±7.14 × 10^{−4} | ±1.793 × 10^{−3} |

$\delta {\left({S}_{TT}\right)}_{C}\text{}\left({\mathrm{K}}^{2}\text{}\mathrm{yr}\right)$ | ±1.20 × 10^{−5} | ±2.23 × 10^{−4} | ±4.46 × 10^{−4} | ±1.17 × 10^{−4} |

${\left[\delta \left({S}_{TT}\right)/{S}_{TT}\right]}_{\lambda}\text{}(\%$) | ±0.2 | ±4.7 | ±19.6 | ±30.9 |

${\left[\delta \left({S}_{TT}\right)/{S}_{TT}\right]}_{C}\text{}(\%$) | ±29.8 | ±25.7 | ±12.2 | ±2.0 |

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**MDPI and ACS Style**

Soldatenko, S.; Bogomolov, A.; Ronzhin, A.
Mathematical Modelling of Climate Change and Variability in the Context of Outdoor Ergonomics. *Mathematics* **2021**, *9*, 2920.
https://doi.org/10.3390/math9222920

**AMA Style**

Soldatenko S, Bogomolov A, Ronzhin A.
Mathematical Modelling of Climate Change and Variability in the Context of Outdoor Ergonomics. *Mathematics*. 2021; 9(22):2920.
https://doi.org/10.3390/math9222920

**Chicago/Turabian Style**

Soldatenko, Sergei, Alexey Bogomolov, and Andrey Ronzhin.
2021. "Mathematical Modelling of Climate Change and Variability in the Context of Outdoor Ergonomics" *Mathematics* 9, no. 22: 2920.
https://doi.org/10.3390/math9222920