Journal of Control Science and Engineering

Volume 2016, Article ID 1547462, 9 pages

http://dx.doi.org/10.1155/2016/1547462

## The Determination of Feasible Control Variables for Geoengineering and Weather Modification Based on the Theory of Sensitivity in Dynamical Systems

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

Received 25 November 2015; Revised 2 May 2016; Accepted 23 May 2016

Academic Editor: Ai-Guo Wu

Copyright © 2016 Sergei A. Soldatenko and Rafael M. Yusupov. 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

Geophysical cybernetics allows for exploring weather and climate modification (geoengineering) as an optimal control problem in which the Earth’s climate system is considered as a control system and the role of controller is given to human operators. In mathematical models used in climate studies control actions that manipulate the weather and climate can be expressed via variations in model parameters that act as controls. In this paper, we propose the “instability-sensitivity” approach that allows for determining feasible control variables in geoengineering. The method is based on the sensitivity analysis of mathematical models that describe various types of natural instability phenomena. The applicability of this technique is illustrated by a model of atmospheric baroclinic instability since this physical mechanism plays a significant role in the general circulation of the atmosphere and, consequently, in climate formation. The growth rate of baroclinic unstable waves is taken as an indicator of control manipulations. The information obtained via calculated sensitivity coefficients is very beneficial for assessing the physical feasibility of methods of control of the large-scale atmospheric dynamics and for designing optimal control systems for climatic processes. It also provides insight into potential future changes in baroclinic waves, as a result of a changing climate.

#### 1. Introduction

Weather modification is the operation of deliberately altering the atmosphere that leads to changes in the natural evolution of physical and dynamical atmospheric processes. Predominantly, weather modification is successfully accomplished via cloud seeding in order to affect precipitation for the purpose of the local water supply (e.g., [1–4]). Many countries currently practice cloud seeding operationally. The next successful example of weather modification is fog dispersal to improve visibility at airports by heating or seeding [5]. Some attempts taken in different countries to reduce damage from hazardous weather events such as hurricanes, strong tornado and thunderstorm winds, hail, lighting, and floods, unfortunately, have not been so successful. The detailed retrospective review and the current status of weather modification with the findings and relevant concepts were considered, for example, in [6–8]. Over the last decades, the weather modification has transitioned from local scale operations to a global weather modification known also as geoengineering. Geoengineering, or climate engineering, is a deliberate and purposeful large-scale modification of the Earth’s climate system (ECS) and, first of all, the atmosphere, which is the most unstable and fast-changing element of the ECS [9]. Geoengineering was offered by scientific community as a response to global warming, which is happening. Since mankind is causing global warming by anthropogenic CO_{2} emissions [10], the most obvious idea to reduce the consequences of global climate change is a sequestration of anthropogenic greenhouse gas (GHG) emissions. However, this is unlikely achievable in foreseeable future due to the continuing growth of the world economy and population. Scientists and engineers proposed several solutions to stabilize the global climate (e.g., [11, 12]). These solutions can be divided into two main categories: carbon dioxide removal technologies (CDR) and solar radiation management (SRM). CDR technologies include, for example, various engineered solutions that remove GHGs directly from the atmosphere, using big machines or chemical absorbents. SRM techniques aim mainly to reflect some percentage of the solar radiation back into the outer space. This can be achieved by changing the planetary albedo (increasing the reflectiveness of the Earth’s surface or clouds), dispersing small particles into the stratosphere, or deploying mirrors in the upper atmosphere to reflect sunlight. However, all of these technologies introduce uncertainties and unexpected consequences that must be explored.

Let both weather modification and climate engineering hereafter be referred to as the geoengineering. Realization of geoengineering projects is a purposeful process, that is, the process imminently connected with a specific objective that can be formulated in various ways. In this context geoengineering is per se the process of controlling the ECS. However, geoengineering is still considered outside of the scope of control theory as an* intentional action to influence natural processes.* Meanwhile in the late 1970s a uniform methodology for control geophysical processes, including processes occurring in the ECS, was formulated on the basis of the ideas from cybernetics by one of the authors of this paper [13]. In this monograph, the concept of geophysical cybernetics was introduced as the new research area within the control theory. Geophysical cybernetics explores a self-regulating feedback cybernetic system, in which the ECS is considered as the control object and the role of the controller is given to human operators. From the standpoint of geophysical cybernetics, climate and weather manipulation represents an optimal control problem, which aims to synthesize the control law that ensures the achievement of the desired results that can be expressed in terms of extremal problem [13, 14]. In our previous publications [14–16] an optimal control problem for the ECS has been conceptually formulated in both probabilistic and deterministic manner. Let us emphasize that the ECS is a unique natural physical system with a large number of specific attributes [17], which makes the control problem for this system extremely complex. The development of physically feasible methods to control the ECS requires the determination of feasible control mechanisms and variables. The theory of sensitivity in dynamical systems [18, 19] serves as a theoretical instrument for solving this problem.

This paper presents a short description of the ECS as a control system, emphasizing its unique physical properties relevant to the control problems of large dynamical systems. It is also noted that the amount of energy released during processes that drive the ECS is orders of magnitudes greater than human capabilities, which makes the control of the ECS via direct interventions very problematic. However, some physical and dynamical processes occurring in the ECS are inherently unstable, which apparently allows one to implement geoengineering projects using significantly less energy resources in comparison with the energy of natural processes. Thus, the study of various types of instabilities in ECS is very important for developing an appropriate control strategy. The instability of natural physical processes is mathematically studied as a problem of finding the necessary conditions for the growth of the infinitesimal perturbations. Since these conditions are expressed in terms of some model parameters, by using sensitivity analysis, we can, first, determine parameters that can be considered as controls and, second, make a conclusion on the hypothetical possibility of control of physical process under consideration. To illustrate this “instability-sensitivity” approach we, as an example, consider the atmospheric baroclinic instability as the controlled object since this type of hydrodynamic instability plays a substantial role in the formation of large-scale atmospheric eddies in the extratropical atmosphere and, therefore, in the general circulations of the atmosphere and climate [20]. To explore the response of baroclinic instability to geoengineering interventions a multilayer geostrophic model of the atmosphere is applied. The growth rate of baroclinic unstable waves is taken as an indicator of control manipulations. The information obtained via calculated sensitivity coefficients is very beneficial for assessing the physical feasibility of methods of control of the large-scale atmospheric dynamics and for designing optimal control systems for climatic processes.

#### 2. Climate System as a Unique Control System

The ECS is a unique and peculiar natural physical system that is extremely difficult to control since it possesses a number of specific properties, including but not limited to the following [14, 17, 21, 22].

(i) The ECS is a complex, interactive system with a wide variety of positive and negative feedback mechanisms. The ECS consists of the atmosphere, ocean, sea-ice, land surface, and other bodies of water and includes global carbon cycle, chemistry, and aerosols. These natural subsystems have substantial differences in their physical and chemical properties, structure, and behavior; they can be strongly or weakly coupled and linked together by means of coupling physical mechanisms.

(ii) Dynamical and physical processes in the ECS occur over a broad spectrum of scales in both space and time. Time scales are varied from seconds (turbulent fluctuations) to dozens of years (climate change and variability). Since the ECS is a global system its spatial spectrum of motions covers molecular to planetary scales. It is important that dynamical processes in the atmosphere and ocean are nonlinear and chaotic.

(iii) Processes in the ECS oscillate due to both internal factors (natural oscillations) and external forcing (forced oscillations). Natural oscillations are due to the internal instability of ECS with respect to stochastic infinitesimal disturbances. Human impact on the ECS, both intentional and unintentional, is considered as external forcing.

The ECS has certainly a number of other specific attributes that make it a unique and complex physical system. One of the most effective instruments in studying the ECS is mathematical/numerical modeling. Climate mathematical models used in a variety of applications are commonly deterministic and derived from a set of multidimensional nonlinear differential equations in partial derivatives, which are the equations of fluid dynamics and thermodynamics. Models also take into consideration the specific properties of ECS as well as its cycles such as water, nitrogen, oxygen, and carbon cycles.

An optimal control problem for the ECS remains poorly studied due to its relative novelty and enormous complexity [15, 16]. To develop a general framework for optimal control of the ECS, the following should be taken into consideration.

(i) The ECS is a spatially distributed system, therefore the control actions for manipulating this system should also be distributed in space. However, implementation of such controls is weakly developed.

(ii) Processes in the ECS possess enormous energy potential. It is hardly possible to provide control actions whose energy is comparable to the energy of natural physical processes. Therefore, the identification of sensitive points, in which the ECS is in an unstable state, is a critical issue.

(iii) Large-scale and huge energy of climate-driven processes impose very strict requirements for the accuracy and reliability of control systems, since even minor errors in control actions can be disastrous.

(iv) Processes in the ECS are interconnected; therefore changes in the dynamics of some processes can result in uncontrollable consequences.

(v) Control actions to perform geoengineering and weather modification operations must be physically feasible and executable.

Consequently, discussing geoengineering within the scope of optimal control theory, we are faced with a number of problems including the problem of choosing variables that can be considered as controls. Mathematical climate models incorporate a certain number of physical processes responsible for the transformation of energy. These processes represent natural control mechanisms of the ECS that can be considered as potential artificial control mechanisms. Unfortunately, these physical mechanisms cannot be explicitly identified and represented in climate models due to models’ discrete spatial-temporal structure and can only be described parametrically under some simplified assumptions. Some of the newly introduced parameters together with parameters that describe the external forcing can be considered as* control variables (parameters).* Then geoengineering actions can mathematically be represented via variations in parameters that act as controls. By using sensitivity analysis, we can explore the influence of variations in control parameters on the behavior of the ECS and, therefore, evaluate the hypothetical possibility of various physical mechanisms to control the ECS. However, we need to keep in mind that control methods should be, on the one hand, physically feasible and, on the other hand, technically executable. Many physical processes occurring in the ECS are inherently unstable [20, 23]. This gives the possibility to control the ECS via natural instability mechanisms. Apparently, the atmosphere, which is the most rapidly moving and changing components of the ECS, represents the most suitable system to be controlled.

#### 3. The Model of Baroclinic Instability in the Atmosphere

Let us consider the set of the so-called primitive equations in isobaric coordinate system commonly used in modelling large-scale atmospheric flows [23]:Here and are the horizontal velocity components in the directions and , respectively (the horizontal coordinates,* x* and* y*, are directed eastward and northward); is pressure vertical velocity, where is pressure; is the geopotential;* T* is the temperature;* R* is the gas constant for dry air;* f* is the Coriolis parameter; is the gravity acceleration; is the dry adiabatic lapse rate; is the reference state lapse rate. We will employ the -plane approximation so that the Coriolis parameter is represented as , where is a standard value of the Coriolis parameter at the mid-latitude and is the latitudinal gradient of .

Let us make some comments regarding the set of equations (1)–(5). These are nonlinear differential equations that are used to describe adiabatic large-scale atmospheric dynamics. Equations (1) and (2) are the momentum equations, which mathematically express Newton’s second law of motion. Equation (3) is a hydrostatic equation. The continuity equation (4) expresses the conservation of mass and (5) is a thermal energy equation representing the first law of thermodynamics.

Baroclinic instability is commonly explored by linearizing the model equations around some unperturbed reference (basic) flow and then solution of the problem can be found using initial-value or eigenvalue approaches. We suppose that the atmospheric reference flow, defined by , , , , , is geostrophic (i.e., the Coriolis force and pressure gradient forces are in balance) and satisfies the following relations:where . The reference state (6) is a solution of (1)–(5) that describes the zonal flowwhich matches the specified distribution of the zonally averaged temperature and represents thermal wind balance. Let us underline that the geostrophic approximation used in this study is valid to high accuracy for the large-scale atmospheric flows. Then, with geostrophic assumption, (1)–(5) reduced to the vorticity equation and the thermodynamic equation [23]:Here is a geostrophic stream function and is the static stability parameter, where is the gravity acceleration. To conserve the mass, the following boundary conditions are used for the pressure velocity [23]: where is a standard pressure at the Earth’s surface.

In this paper, we will consider an analytical solution of the baroclinic instability problem using an eigenvalue approach for a two-layer model. The model vertical structure is shown in Figure 1. For this model the boundary condition (10) gives . Applying quasi-geostrophic vorticity equation (8) to the 750 and 250-hPa surfaces and approximating the derivative by finite differences one can obtain the resulting vorticity equations at levels 1 and 3:where is the horizontal velocity and hPa. The thermodynamic energy equation (10) is applied at level 2:Thus, we have a system of three equations (11)-(13) in the three variables , , and . To study the instability of the basic zonal flow with respect to infinitesimal perturbations, these equations are linearized around the basic state (6). Let us assume thatSubstituting (14) into (11)–(13), definingand eliminating the variable yield the following perturbation equations:where . We will seek normal mode solutions of the following form:where and are the amplitude of perturbations,* k* is a wavenumber, and is a complex phase velocity. By substituting (17) into (16), after some algebraic manipulation we can obtain the following equation for the phase speed of baroclinic waves:where Perturbations will grow exponentially if has an imaginary part . This will occur if .