The current climate change, unlike previous ones, is caused by human activity and is characterized by an unprecedented rate of increase in the near-surface temperature and an increase in the frequency and intensity of hazardous weather and climate events. To survive, society must be prepared to implement adaptation strategies and measures to mitigate the negative effects of climate change. This requires, first of all, knowledge of how the climate will change in the future. To date, mathematical modelling remains the only method and effective tool that is used to predict the climate system’s evolution under the influence of natural and anthropogenic perturbations. It is important that mathematics and its methods and approaches have played a vital role in climate research for several decades. In this study, we examined some mathematical methods and approaches, primarily, mathematical modelling and sensitivity analysis, for studying the Earth’s climate system, taking into account the dependence of human health on environmental conditions. The essential features of stochastic climate models and their application for the exploration of climate variability are examined in detail. As an illustrative example, we looked at the application of a low-order energy balance model to study climate variability. The effects of variations in feedbacks and the climate system’s inertia on the power spectrum of global mean surface temperature fluctuations that characterized the distribution of temperature variance over frequencies were estimated using a sensitivity analysis approach. Our confidence in the obtained results was based on the satisfactory agreement between the theoretical power spectrum that was derived from the energy balance model and the power spectrum that was obtained from observations and coupled climate models, including historical runs of the CMIP5 models.

The weather and climate manipulation is examined as an optimal control problem for the earth climate system, which is considered as a complex adaptive dynamical system. Weather and climate manipulations are actually amorphous operations. Since their objectives are usually formulated vaguely, the expected results are fairly unpredictable and uncertain. However, weather and climate modification is a purposeful process and, therefore, we can formulate operations to manipulate weather and climate as the optimization problem within the framework of the optimal control theory. The complexity of the earth’s climate system is discussed and illustrated using the simplified low-order coupled chaotic dynamical system. The necessary conditions of optimality are derived for the large-scale atmospheric dynamics. This confirms that even a relatively simplified control problem for the atmospheric dynamics requires significant efforts to obtain the solution.

A low-order stochastically forced two-layer global energy balance model (EBM) admitting an analytical solution is developed for studying natural inter-annual, decadal and multi-decadal climate variability, and ultimately to better understand forced climate change. The EBM comprises upper and lower oceanic layers with a diffusive coupling, a radiative damping term including feedbacks and stochastic atmospheric forcing. The EBM is used to analyse the influence of radiative forcing, feedbacks and climate system inertia on the global mean surface temperature variance (climate variability) and to understand why Coupled Model Intercomparison Project, Phase 5 (CMIP5) models exhibit such a wide range in the level of variability in globally averaged surface air temperature. We examine the influence of the model parameters on the climate variability on different timescales. To this end, we derive the Fokker–Planck equation for the EBM and then obtain the analytical expression that quantifies the sensitivity coefficients for all model parameters. For all timescales, the most influential factors are as follows: (1) the magnitude of the stochastic forcing, (2) the feedback mechanisms, (3) the upper layer depth, (4) the diffusion parameter and (5) the lower ocean depth. Results from the EBM imply that the range of stochastic forcing in the CMIP5 climate models is around twice as important as the strength of radiative feedback or upper layer depth in causing the model-to-model spread in the magnitude of globally averaged climate model variability.

Intentionally altering natural atmospheric processes using various techniques and technologies for changing weather patterns is one of the appropriate human responses to climate change and can be considered a rather drastic adaptation measure. A fundamental understanding of the human ability to modify weather conditions requires collaborative research in various scientific fields, including, but not limited to, atmospheric sciences and different branches of mathematics. This article being theoretical and methodological in nature, generalizes and, to some extent, summarizes our previous and current research in the field of climate and weather modification and control. By analyzing the deliberate change in weather and climate from an optimal control and dynamical systems perspective, we get the ability to consider the modification of natural atmospheric processes as a dynamic optimization problem with an emphasis on the optimal control problem. Within this conceptual and unified theoretical framework for developing and synthesizing an optimal control for natural weather phenomena, the atmospheric process in question represents a closed-loop dynamical system described by an appropriate mathematical model or, in other words, by a set of differential equations. In this context, the human control actions can be described by variations of the model parameters selected on the basis of sensitivity analysis as control variables. Application of the proposed approach to the problem of weather and climate modification is illustrated using a low-order conceptual model of the Earth’s climate system. For the sake of convenient interpretation, we provide some weather and climate basics, as well as we give a brief glance at control theory and sensitivity analysis of dynamical systems.

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