pplied Mathematical Sciences, Vol. 9, 5, no. 8, 875-884 HIKRI Ltd, www.m-hikari.com http://dx.doi.org/.988/ams.5.448 n Optimal Control Problem Formulation for the tmospheric Large-Scale Wave Dynamics Sergei Soldatenko Center for ustralian Weather and Climate Research 7 Collins Street, VIC 38, Melbourne, ustralia Rafael Yusupov St. Petersburg State Polytechnical University and SPIIRS 9 Polytechnical, St. Petersburg 955, Russia Copyright 5 Sergei Soldatenko and Rafael Yusupov. This is an open access article distributed under the Creative Commons ttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. bstract In this paper, an optimal control problem is formulated for the large-scale atmospheric dynamics, which is mainly characterized by slow-moving planetary-scale waves with zonal wavenumbers to 6, also known as Rossby waves. n equivalent barotropic model of the atmosphere is used to describe the evolution of Rossby waves. Necessary conditions for optimality are obtained. Keywords: Optimal control, Rossby waves, distributed parameter system Introduction Weather modification is the man-made activity for intentionally manipulating or altering the weather and atmospheric processes. This activity has so far been considered outside the scope of control theory [, 3, 7]. For this reason the boundaries of weather modification, its goals and methods of achieving the objectives are commonly formulated in general terms and expected results are fairly vague. However, weather modification can be designed and implemented on the basis of control theory. This paper aims to formulate the optimal control problem for large-scale atmospheric dynamics, which is mainly characterized by slow-moving planetary-scale waves, also known as Rossby waves [4]. On weather maps these waves are observed as large-scale meanders of the pressure field in the
876 Sergei Soldatenko and Rafael Yusupov middle troposphere. Since Rossby waves are quasi-vertical and quasi-barotropic, in this study an equivalent barotropic model (EBM) of the atmosphere [5] is used as a tool to formulate the optimal control problem and to derive the necessary conditions for optimality. Equivalent barotropic model of the atmosphere tmospheric models, used for numerical weather prediction (NWP), are formulated as initial value problems for systems of nonlinear partial differential equations called primitive equations (PEs), which are comprised of the horizontal momentum equations, the equation of hydrostatic equilibrium, the thermodynamic equation and the continuity equation. We consider the inviscid adiabatic PEs written in the normalized isobaric coordinates ( xy,, ) [4] u u uu fv, () t P x v v uv fu, () t P y RT, u, (3) P where u, T u T S, (4) t uv is the horizontal velocity vector; is the two-dimensional gradient operator on a surface of constant ξ; is the geopotential; f is the Coriolis parameter; T is the temperature; dp dt is pressure vertical velocity, where p is pressure; R is the gas constant for dry air; S is the static stability measure, pp is the normalized pressure, P hpa is a standard pressure. The equation of the EBM is derived from the vorticity equation. Taking xof () and subtracting yof () gives the equation for the vertical component of vorticity v x u y, which represents the most interest in meteorology. Retaining the terms of order - s - yields the vorticity equation that is valid for large-scale atmospheric dynamics [4, 5]: f f f u v t x y P, (5) where we have used the continuity equation (4).
n optimal control problem formulation 877 The following basic assumptions are made in the derivation of EBM: the wind speeds vary little with height, and the variations of wind velocity with height are similar at all vertical levels. With these assumptions, at a certain vertical level the motion of the baroclinic atmosphere corresponds to the motion of a barotropic atmosphere and the vorticity equation can be written as [5]: where f f f u v, (6) t x y P is an empirical function used to describe the vertical variation of the horizontal wind speed, and is a pressure vertical velocity at the level ( z ). The level is known as the equivalent barotropic level, and the equation (6) is called as the equivalent barotropic vorticity equation. Climatological data shows that there exist two equivalent barotropic levels in the atmosphere: one around the 5 hpa (~5.5 km) level, and the other around hpa (~6 km) level. The mid-tropospheric level (5 hpa) is of particular interest for the NWP. Since the large-scale atmospheric motions are quasi-geostrophic [4], the vorticity equation (6) can be recast in terms of the geostrophic streamfunction f : f, f, (7) t P where a, b a xb y a yb x. By the definition [4, 5]: dp p p p u v gw. dt t x y Here w dz dt is the vertical velocity at the level z, is the air density and g is the gravity acceleration. The subscript refers to the level z. On the lower boundary the kinematic boundary condition requires that w. For the geostrophic motion it follows that up. Consequently, the value of reduces to p t. Substitution of this expression into (7) gives: f p, f. (8) t P t Suppose that p t and t are correlated with each other, where is the geopotential at the level. Since f p t R f t, then where R c is a correlation coefficient, and the equation (8) becomes Rc f, f. (9) t P t The coefficient Rc f P has the dimension of an inverse square length. Thus, from (9) we can obtain the following inhomogeneous Helmholtz equation: c
878 Sergei Soldatenko and Rafael Yusupov where q t, F, f q q L q F, () and L = Rc f P. We shall consider the equation () in a closed domain D of the xy - plane with a piecewise continuous boundary D. Let [, T ] be a time interval on which the solution of equation () is defined. Let the boundary curve be represented in parametric form: x, y, here and are piecewise continuous functions of the parameter. ssuming the coordinate origin coincides with the North Pole, we shall obtain: x r cos, y r sin, () where r, is a polar coordinate of a generic point on the circle. The right-hand side of () can be calculated if the geopotential field is given at the initial time x, y, x, y. Specifying const along the boundary D, t : and introducing the new dependent variable z t t, where t is the integration time step, we can obtain the following Dirichlet problem: where z L z G in D, () z on D, G F t. The problem () is solved numerically using an appropriate z t t and, therefore, the iteration method to obtain the function streamfunction: t z t Consequently, we can obtain the forecast of geopotential field f t, (3), which characterizes the Rossby wave dynamics. The forward difference (3) is used only on the first time step and then a central difference is applied. 3 Statement of the optimal control problem When the equation (6) is equal to zero on the right-hand side, we refer to it as a barotropic vorticity equation. Thus, (6) is a forced barotropic equation in which the forcing term is determined by the vertical motion at the lower boundary. The control objective is to manipulate the phase velocity of Rossby waves. It is known that the natural forcing of Rossby waves in the atmosphere is mainly of orographic or thermal origin. These two effects are formally described by the right-hand term in the vorticity equation (6), i.e. by the vertical velocity. Therefore, the variable can be chosen as a control variable. Let us consider the linearized vorticity equation on a periodic mid-latitude beta-plane [4]:
n optimal control problem formulation 879 ' ' ' u u, (4) t x L t x where f y is the latitudinal gradient of Coriolis parameter, u is a zonal unperturbed flow, u d u dy, ' is the perturbation of streamfunction. We seek a solution of the form ikxct ' x, y, t Re e cos ly. (5) Here is an amplitude of perturbation, k Lx and l Ly are the zonal and meridional wavenumbers respectively, and c is a perturbation phase speed. fter substituting (5) into (4) we can obtain the dispersion relationship for the Rossby waves: u k l u c k l L. (6) This formula shows that the atmospheric vertical structure through the factor L contributes to the dispersion properties of Rossby waves. However, the factor L depends on the correlation coefficient R, which in turn relates the vertical velocity c with the streamfunction ψ at the mid-tropospheric vertical level. Hence, the phase speed of Rossby waves is also a function of the vertical velocity. Let us now consider the following control system with the state variable, U U x, y : z z x y and the control variable z L z U G in D, z on D. (7) The term G is calculated using the geostrophic streamfunction at the initial time t : G G, f t, where f. Let us assume that the control variable U U, where U is the set of all permissible controls. The control U, from a physical standpoint, is a measure of additional vertical velocity on the bottom of the atmosphere near the ground. It is very important to note that the set U should be defined on the basis of physical and technical feasibility, taking into account the properties of the atmosphere as a physical object. Let us introduce the performance index as z z U dd, (8) D
88 Sergei Soldatenko and Rafael Yusupov where the pair zu, satisfies the equation (7) and z is the desired spatial distribution of z. The term U ( ) is proportional to the consumed energy. The optimal control problem is defined as follows: find the control U U generating the system state z X such that the performance index (8) is minimized. Here X is a set of additional constraints on the state variable z. The control problem, from a physical viewpoint, reflects the ability to manipulate phase speeds of large-scale atmospheric waves by changing a vertical velocity on the lower boundary of the atmosphere. Let us transform the equations (7) to a normal form by introducing new dependent and parametric variables [, 6]: z z z z z, z3,,, x y x y (9) z3 z3, U z G. x y L djoin the equations (9) to the performance index (8) with spatial-varying Lagrange multipliers,, 3,, and 3: z z z z z U z z3 x y x D z z3 z 3 3 3 U z G dd. () y x L y The Hamiltonian function H associated with the optimal control problem is introduced as H z z U z 3 z3 3 U z G. L () The augmented performance index is rewritten then as: z z z3 z z z3 H 3 3 dd, x x x y y y D or in the compact form z z H dd, () x y D
n optimal control problem formulation 88 where z z, z, z 3,,, 3 and,, 3 subtract the term z x y. If we add and to the integrand of (), we shall obtain: H zdd z z dd. (3) D x y D x y By the use of Green's theorem, the second integral in (3) becomes a line integral, and then takes the following form: H zdd zd. D x y D The first variation in due to variations in the control and parametric variables is: H H z U dd zd, (4) D z x y U D where,, U U. We choose the functions and to cause the first term in parentheses in the double integral in (4) to vanish: H. (5) x y z Since the function z does not change along the boundary D ( z ), its variation z on D. For z and z 3, which are not defined on the boundary D, we require the relations to hold along D. Then, taking into account (), we can obtain the following boundary conditions for the system (9): cos sin, cos sin. (6) 3 3 The first variation in then becomes: H U dd. (7) U D For a minimum, it is necessary that.this can only be achieved if
88 Sergei Soldatenko and Rafael Yusupov H U. (8) Equations (5) and (8) with the boundary conditions (7) and (6) represent the necessary conditions for optimum, which can be written as 3 z z,, x y L x y (9) 3 3, U 3, 3, 3. x y Therefore, to find the control U that minimize the performance index (8) we must solve the system of partial differential equations (9) and (9) with unknown two-dimensional variables and with given boundary conditions: z,. (3) D D D Let us consider a particular case of the optimal control problem in which the constant is equals to zero. From (9) we can obtain that:, 3 and the necessary conditions (9) for an extremum of takes the form: z z,, x y y (3) 3, 3. x Since 3, the second and third equations of the system (3) are similar to the first two equations of the system (9) and we might make the following assumptions [, 6]: 3 z, z, 3 z, (3) where is an arbitrary constant. These assumptions are compatible with the boundary conditions (3). If we substitute (3) into the first equation of the system (3) then we get: from which z z z z, xy xy z z in the whole domain D. It follows that the performance index can achieve its absolute minimum under the constraint (7).
n optimal control problem formulation 883 To find the optimal control U that minimizes, we need to solve the following problem U z L z G, z D in the domain D. Rossby waves propagate westward relative to the mean quasi-zonal flow [4], so that it is possible for them to be stationary (with respect to the surface) in a westerly atmospheric flow. If the optimal control objective is to achieve the stationarity of Rossby waves by manipulating the lower boundary vertical velocity, i.e. z t, then the control U in the domain D can be find by t solving the equation U G. t first glance the results obtained seem almost obvious. However, in this paper we rigorously prove the existence of absolute extremum of the performance index (8) under the constraint (7). If various additional constraints are imposed on the control variable U, then the performance index (8) may not achieve its absolute extremum, and then the expression for calculating the control variable U is not so obvious. 4 Conclusion Based on the optimal control theory of distributed parameter systems, an optimal control problem for Rossby wave dynamics in the atmosphere has been theoretically formulated. The equivalent barotropic atmospheric model was used as an instrument to obtain the necessary conditions for optimality. The application of the optimal control theory to manipulate weather and climate (geoengineering) represents a new multidisciplinary research area. However, this problem is extremely complex due to the uniqueness of the control object, the atmosphere, which is very sophisticated to model and predict. This paper provides a basis for further research in the field of meteorological cybernetics that is a new research area of a self-regulating cybernetic system in which the atmosphere represents the control object and the human society plays the role of controller. There is no doubt that this problem requires also the consideration of physical, technical, ethical and legal aspects and limitations. cknowledgments. This work was supported in part by the Government of the Russian Federation under Grant 74-U. References [].-L. rmand, pplications of the theory of optimal control of distributed parameter systems to structural optimization, NS CR-44, Washington, D.C., 97, 54 p.
884 Sergei Soldatenko and Rafael Yusupov [] Critical Issues in Weather Modification Research. Committee on the Status and Future Directions in U.S Weather Modification Research and Operations, National Research Council. National Research Council. National cademic Press, Washington, D.C., 3, 44 p. [3]. R. Fleming, The pathological history of weather and climate modification: Three cycles of promise and hype, Historical Studies in the Physical and Biological Sciences, 37 (6), 3-5. http://dx.doi.org/.55/hsps.6.37..3 [4]. R. Holton, n introduction to dynamic meteorology, London, Elsevier, 4, 535 p. [5] I.. Kibel, n introduction to the hydro-dynamical methods of short period weather forecasting. Pergamon Press, New York, 963, 383 p. [6] K.. Lurie, pplied optimal control theory of distributed systems, New York, Springer, 993, 499 p. http://dx.doi.org/.7/978--4757-96- [7] D. L. Mitchell and W. Finnegan, Modification of Cirrus clouds to reduce global warming, Environmental Research Letters, 4 (9), 45, -8. http://dx.doi.org/.88/748-936/4/4/45 Received: anuary 6, 5; Published: anuary 9, 5