Complexity

Volume 2017, Article ID 4615072, 12 pages

https://doi.org/10.1155/2017/4615072

## Weather and Climate Manipulation as an Optimal Control for Adaptive Dynamical Systems

St. Petersburg Institute for Informatics and Automation, The Russian Academy of Sciences, No. 39, 14th Line, St. Petersburg 199178, Russia

Correspondence should be addressed to Sergei A. Soldatenko; us.bps.saii@oknetadlos

Received 30 July 2016; Accepted 13 November 2016; Published 12 January 2017

Academic Editor: Dimitri Volchenkov

Copyright © 2017 Sergei A. Soldatenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

Weather and climate manipulation is defined as a deliberate intervention in the natural evolution of physical processes in the atmosphere and other components of the earth’s climate system (ECS) in order to achieve some desired results [1, 2]. Here, as usual, the ECS () is understood as a complex large-scale physical system that consists of five basic and interacting constituent complex subsystems, namely [3],(i)atmosphere (), the gaseous and aerosol envelope of the earth that propagates from the land, water bodies, and ice-covered surface outward to space,(ii)hydrosphere (), the oceans and other water bodies on the surface of our planet and water that is underground and in the atmosphere,(iii)cryosphere (), the sea ice, freshwater ice, snow cover, glaciers, ice caps and ice sheets, and permafrost,(iv)lithosphere (), the solid, external part of our planet,(v)biosphere (), the part of our planet where life exists; that is,

Each component of the ECS is characterized by a finite set of variables, usually called state variables, whose values at a given time determine the instant state of the ECS. All components of the ECS are extremely complex physical systems that in turn also are composed of constituent elements (subsystems). It should be noted that the current status of climate system studies is characterized by dramatic progress achieved over the last few decades in understanding basic physical processes and fundamental feedback mechanisms that control the climate formation and change (thanks to the advances in computing hardware and algorithms and due to a substantial increase in the volume of climatological data [4–8]). The detailed retrospective analysis of climate science can be found, for example, in [9]. However, our knowledge for climate remains incomplete due to the enormous complexity of the ECS. A deeper understanding of current climate, its variability, and change is among the greatest challenges facing modern science that can help not only to project the future climate but also to develop methods for controlling the ECS.

The objectives and methods of weather modification are distinctly different than those of climate manipulation. This distinction is due to the fact that the terms “weather” and “climate” have different meanings. Weather is defined as the daily conditions of the atmosphere in terms of such atmospheric variables as temperature, humidity, wind direction and velocity, surface pressure, clouds, and precipitation. In contrast, climate represents the ensemble of states traversed by the climate system over a sufficiently long enough temporal interval. Here, the ensemble means not only the set of system states, but also a probability measure defined on this set. Climate, roughly speaking, can be considered as the “average” weather, in terms of mean and variance, in a certain geographical location over many years.

Weather modification is aimed at altering weather conditions and weather phenomena locally or regionally. The successful examples of weather modification operations are cloud seeding to increase precipitations, the removal of fogs, stratiform and convective clouds, the suppression of hail, and lightning (e.g., [10–14]). The large-scale weather modification experiments such as the reduction of the intensity of hurricanes were less successful [15–17]. Thus weather modification aims to intentionally affect processes and phenomena only in the atmosphere, the most volatile component of the ECS. Climate manipulation known also as a climate engineering or geoengineering was suggested as a response measure on the global climate change that currently occurs on our planet [3]. Geoengineering aiming at stabilizing the global climate and preventing further warming is a planetary-scale intentional intervention on the ECS and represents a multidisciplinary problem of extreme complexity, which is currently explored mostly theoretically because the consequences and unintended side effects of climate manipulation operations are very uncertain. Geoengineering requires further consideration of not only scientific and technical issues, but also the legal and ethical aspects and limitations. It is expected that the interest in climate and weather manipulation will continue to grow, so that it makes it necessary to develop the appropriate theoretical framework. To date a number of methods and technologies were proposed by scientists and engineers to stabilize the ECS and to mitigate the global warming (e.g., [18–24]).

The realization of both weather modification and climate engineering operations is a goal-directed process, having specific objectives that should be somehow formulated and certainly achieved. Mathematically, weather and climate manipulation can be viewed as an optimization problem [25, 26]. Consequently, instead of examining weather and climate manipulation as amorphous actions without clearly defined objectives, boundaries, and methods, we can consider operations to modify weather and climate as an optimal control problem for the ECS and its components.

It is important to make the following comments concerning controllability of the ECS and its components and, in particular, the atmosphere. Controllability can be thought of as a property of a control system that indicates the ability to transfer a system from one state to another, using admissible operations (e.g., [27]). From the perspective of physics, the ECS is a controllable system, and its key driver is the incoming shortwave solar radiation. On the geological time scale, the earth’s climate has been highly variable throughout history as a result of both external forcing and internal perturbations [28] (e.g., variations in the chemical and aerosol composition of the atmosphere and aperiodic El Niño-Southern Oscillation). Since the incoming solar radiation can be viewed as a primary control variable, geoengineering seeks to reduce global warming by using physically feasible ability to manipulate the climate via changes in the amount of incoming solar radiation (by varying reflection or absorption) as well as natural feedback mechanisms that exist in the ECS. However, geoengineering is not the same as control, and thus the term “engineering” rather than “control” is currently used in the literature to emphasize this difference. Considering geoengineering within the formal framework of the optimal control theory, we are faced with a number of very hard mathematical problems, such as the proof of the controllability property for any particular climate model.

Similar problems arise when we consider weather modification. For example, we can create the artificial precipitation by cloud seeding; however the formal proof of the existence of the optimal solution and the proof of the controllability property for an applied mathematical model are problems of extreme difficulty. For this reason, in weather modification, the term “intentional (deliberate) action” is usually used instead of “control.” Thus, we have to make the distinction between control as an engineering problem and control as a mathematical problem.

In this paper, the weather and climate manipulation is examined as an optimal control problem for the climate system, which is considered as a complex adaptive dynamical system. The complexity of the climate system is discussed and illustrated using the simplified low-order coupled chaotic dynamical system. Finally, as an example, the necessary conditions of optimality are derived for the large-scale atmospheric dynamics. This shows that even a relatively simplified control problem for the atmospheric dynamics requires significant efforts to obtain the solution.

#### 2. Complexity in the Earth’s Climate System

Let us make some preliminary comments. The term “system” generally refers to a goal-oriented set of interconnected and interdependent elements that operate together to achieve some objectives [29]. The system is called complex if it possesses such characteristics as emergent behavior, nonlinearity and high sensitivity to initial conditions and/or to perturbations, self-organization, chaotic behavior, feedback loops, spontaneous order, robustness, hierarchical structure, and some others [30]. Complexity in systems appears in particular from spatiotemporal interactions between their components. These nonlinear interactions lead to the appearance of new dynamical properties (e.g., synchronous oscillations and other structural changes) that cannot be observed by exploring constituent elements individually. Let us go one step forward. Complex systems include a special class of systems that have the capacity to adopt to system’s environment. These systems are known as complex adaptive systems (e.g., [31, 32]). In complex adaptive system its parts are linked together in such a way that the entire system as a whole has the capacity to transform fundamentally the interrelations and interdependences between its components, the collective behavior of a system, and also the behavior of individual components due to the external forcing. Complex adaptive systems are dynamical systems since they evolve and change over time. These systems have a number of properties that include the following [32, 33]: coevolution, connectivity, suboptimality, requisite variety, iteration, edge of chaos and, certainly, emergence, and self-organization.

The ECS is a large-scale and peculiar dynamical system that possesses numerous unique physical, dynamical, and chemical properties (e.g., [7, 8, 33]). The study and a fortiori control of the ECS are a highly difficult problem. There is no doubt that the ECS can be viewed as a complex adaptive system. This is because of the following.(i)The ECS is a complex system combining the atmosphere, hydrosphere, cryosphere, land, and biota together with global biochemical cycles (first of all, cycles of CO_{2}, N_{2}O, and CH_{4}) and aerosols. The climate system’s components are heterogeneous thermodynamical systems characterized by specific variables that determine their states. Subsystems of the ECS have strong differences in their structure, dynamics, physics, and chemistry. They embrace processes with different temporal and spatial scales and link together via numerous coupling physical mechanisms, which can be weak and strong. Each constituent component of the ECS is also a complex and possibly also adaptive system. For example, the following weather systems could be viewed as subsystems of an overarching atmospheric system: tropical cyclones, hurricanes, mid-latitude cyclones and anticyclones, clouds, precipitations, and so forth.(ii)The ECS has a hierarchical structure. Each component of the ECS can be characterized by a specific response time. The belonging of a certain component to the ECS is determined by the ratio between the temporal scale of processes under consideration and the response time of the ECS components. For example, the atmosphere, which has a response time of about one month in the troposphere, can be considered as a sole component of the ECS for processes with temporal scales of days to weeks. In this case, oceans, land surface, and ice cover are considered as the boundary conditions and/or external forcing. If we study processes which have temporal scales of months to years, the atmosphere and ocean must be included in the ECS together with sea ice.(iii)The ECS has a large number of positive and negative feedback mechanisms, for example, ice-albedo feedback (positive feedback), water vapor feedback (positive feedback), cloud feedback (both positive and negative feedbacks), carbon cycle feedback (negative feedback), and feedback due to Arctic methane release (positive feedback).(iv)Physical and dynamical processes in the ECS cover a broad spectrum of temporal and spatial scales. Time scales are varied from seconds to decades, and spatial spectrum of dynamical processes covers molecular to planetary scales. Dynamical processes in the ECS and its components are nonlinear. Subsystems of the ECS interact with one another nonlinearly producing under certain conditions a chaotic behavior of subsystems and the overall climate system.(v)The ECS and its components inherently have emergent properties. Examples of atmospheric emergent phenomena include but not limited to clouds, large-scale eddies (cyclones and anticyclones), and small-scale vortices such as tornados. Examples of climate emergent phenomena are the El Niño-Southern Oscillation, which is quasiperiodic irregular variation in ocean surface temperature over the Pacific in tropics that strongly influences the global climate and ocean circulation patterns. Natural emergent phenomena appear spontaneously under certain favorable conditions.(vi)The ECS is thermodynamically open and nonisolated system because it exchanges energy with its surroundings. However, the ECS is a closed system for the exchange of matter with outer space. The energy that drives the ECS is a solar energy. The ECS is affected by changes in external driving forces, which imply natural causes such as solar activity variations and volcanic activities and human-made changes in chemical composition of the atmosphere. However, the impact of the ECS on the outer space is insignificant. Currently, changes in climate are mostly affected by variations in the atmospheric composition of particles and gases. In the Arctic the role of changes in albedo (reflection coefficient) is also tangible. In spite of the fact that the most significant heat-trapping gas is atmospheric water vapor, the most influential gas component to affect the climate change is CO_{2}, which comprises about 70 percentage of points of the global warming potential [3].(vii)The components of the ECS are also nonisolated systems. They act as cascading systems and interact with each other in various ways including the transfer of momentum, sensible and latent heat, gases, and particles. All together they compose the climate system, which is a unique large-scale natural system.(viii)Dynamical processes in the ECS periodically and irregularly oscillate due to both internal factors (natural oscillations) and external forcing (forced oscillations). Natural fluctuations are due to internal instability (e.g., convective, barotropic, and baroclinic instabilities) with respect to stochastic perturbations. Human impacts, both intentional and unintentional, are considered as external forcing.

Certainly, there are other specific properties of the ECS that should be taken into account while studying climate as a complex adaptive system.

With regard to weather and climate manipulation, it is necessary to underline that the response of nonlinear systems to external perturbations is completely different, both quantitatively and qualitatively, than the response of linear systems [34]. Recall that operations to manipulate weather and climate represent external forcing with respect to the ECS. This is very important in terms of assessing the effects of weather and climate manipulation operations.

To study and simulate the ECS we should assign some mathematical object that is an abstract representation of the real climate system. This object is usually known as an “ideal” model of the system of interest. The climate system model represents a set of interacting and interdepending subsystems (agents). The number of these subsystems is determined by the objectives of problem under consideration. For example, to study the large-scale climate variability the model can include the following major agents: tropical, mid-latitude and polar troposphere, stratosphere, land ice, oceans and sea ice, surface and boundary layers, hydrological cycle, clouds of various types, precipitations, aerosols, CO_{2} and CH_{4} cycles, solar radiation, and terrestrial emission. Other subsystems of the ECS (e.g., vegetation, land surface, and biota) can be considered as the boundary conditions and external forcing.

The major components of the ECS are physical continuum and their evolution can mathematically be described by a set of multidimensional nonlinear differential equations in partial derivatives:where is a state vector of a system, is a parameter vector, is a vector of spatial variables, is the time, is a nonlinear differential operator that describes the evolution of a system, and is initial conditions. Since (2) describes a continuous medium, the state vector is infinite-dimensional. The solution to such infinite-dimensional system cannot be found analytically, and we need to employ numerical methods. To obtain numerical solution, the original set of (2) is replaced with discrete spatiotemporal approximations using some appropriate technique (e.g., finite-difference method and Galerkin approach). It is very important to underline that a large number of physical processes and cycles (model agents), which also may be viewed as a system, cannot be explicitly represented in the climate model due to its discrete spatiotemporal structure. State-of-the-art models of the ECS are generally unable to realistically simulate processes on spatial scales of the order of twice the model grid length [35]. Such thermodynamical, physical, and chemical processes and cycles are parameterized, that is, expressed parametrically using simplified description. Some of these newly introduced parameters can be considered as controls (see below). By varying control parameters, we can formally manipulate the ECS and its components.

Models of the ECS due to their extreme complexity require extensive computational resources to study the behavior of climate system and its components by computer simulations. However, in order to mimic some features of the ECS behavior and to study its certain properties as a complex system we can use simplified low-dimensional models. For illustrative purposes, we consider a coupled nonlinear model, which is composed of fast (the “atmosphere”) and slow (“the ocean”) components and obtained by coupling of two versions of the original Lorenz system [36] with distinct temporal scales (e.g., [37–39]):Here lower case and capital letters represent, respectively, the “atmosphere” (fast model) and the “ocean” (slow model); , , and are the parameters of Lorenz system; and are parameters that describe the coupling strength between fast and slow models, is the amplitude scale factor, is an “uncentering” index, and is the time-scale separation parameter. For simplicity but without loss of generality we can assume that , , and . The combination of two Lorenz systems allows one to imitate the interaction between fast-oscillating atmosphere and slow-fluctuating ocean [40, 41].

The Lorenz system is one of the most well-known nonlinear dynamical systems, which derived from the equations of atmospheric Rayleigh-Bénard convection [42]. Three variables of the Lorenz model, , , and , are, respectively, the intensity of convective motion and horizontal and vertical temperature gradients. The model parameters have the following meanings: is the Prandtl number, is a normalized Rayleigh number, and is a nondimensional wavenumber. Properties of the Lorenz model are well known and described in the literature (e.g., [43]). We only note that the original Lorenz system can be studied using different values of parameters. However, for the standard parameter values , , and , the Lorenz model exhibits chaotic behavior and possesses a two-wing strange attractor, which has a fractal dimension of about 2.06 [36, 43]. For and , the chaotic behavior is detected if , where . Varying the parameter , we can observe structural changes in the system dynamics. In our computations the standard values of parameters were used. We also assume that .

The dynamics of coupled system (3) strongly depends on the parameter . This parameter, referred to as the coupling parameter, describes the interaction degree between the fast and slow systems [44]. In the existing climate models, atmosphere and ocean are commonly coupled by adjusted air-sea fluxes of heat, momentum, and fresh water using various parameterization techniques (e.g., [45]). In this context, the parameter can be viewed as a simple parameterization of the interaction between two models used in this study. Figures 1–4 show the temporal changes of fast ( and ) and slow ( and ) variables calculated by the numerical integration of the coupled model equations. Even a cursory examination of these figures reveals that when a coupling strength parameter is varied, the qualitative behavior of the system changes. Qualitative shifts in the dynamical properties of coupled system can be traced by the spectrum of conditional Lyapunov exponents. These exponents, on the one hand, characterize the average rate of exponential divergence of nearby orbits in the phase space, and, on the other hand, they are used to analyse the synchronization with coupled systems [44]. When the coupling strength parameter tends to zero, coupled system (3) has six distinct Lyapunov exponents: two positive, two negative, and two zero. The dependence of two largest Lyapunov exponents on the parameter is shown in Figure 5 [39]. These exponents are monotonically decreasing functions. Being initially positive, they approach the -axis at about and at about become negative. Thus, the phase-synchronous regime is observed when . When , the system approaches a limit cycle since all six Lyapunov exponents become negative.