MATHEMATICAL MODELS OF LIFE SUPPORT SYSTEMS Vol. I - Mathematical Models for Prediction of Climate - Dymnikov V.P.

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1 MATHEMATICAL MODELS FOR PREDICTION OF CLIMATE Institute of Numerical Matematics, Russian Academy of Sciences, Moscow, Russia. Keywords: Modeling, climate system, climate, dynamic system, attractor, dimension, stability, energy, stocasticity. Contents 1. Introduction 2. Matematics for climate modeling 2.1. Ideal" Model of Climate 2.2 Finite dimensional Approximation of Differential Climate Models 3. Climatic models 3.1 Zero dimensional Models 3.2 One dimensional Models 3.3 Two-dimensional Models 3.4. Tree dimensional Models Two-Layer Quasi-geostropic Model General Circulation Model wit Prescribed Distribution of Sea Surface Temperature and Pack Ice Global Climate Models based on te Coupled Models of te Atmospere and Ocean Circulation 3.5. Regional Climate Models 3.6 Dynamic-Stocastic Climate Models 4. Predictability of climate canges 5. Conclusions Glossary Bibliograpy Biograpical Sketc Summary Numerical simulation of global climate models is te major metod for examining te canges of contemporary climate under te impact of antropogenic influences. General circulation models of te atmospere and ocean are te basis of global models of climate. Te development of tese models in te last decades was spurred by vigorous development of numerical matematics and computing facilities. Tere is a specific feature of te modeling of climate and especially its future canges: it is impossible to carry out direct pysical experiments wit te climatic system to determine its sensitivity to small external impacts. It is very important, in tis connection, to examine te sensitivity of te climatic system on te basis of te teory of attractors of dissipative semidynamic systems. At present, serious results were obtained in tis direction, owever many problems remain unsolved. Te solution of tese problems would allow one to approac te grand problem: te control of climate.

2 1. Introduction Te prediction of climate canges induced by antropogenic processes is one of te most important callenges for science in te 21st century. Antropogenic impacts on te climatic system include te combustion of fossil fuels leading to te increase in te CO2 concentration in te atmospere, canges in te concentration of small gaseous species tat control te ozone concentration in te atmospere, deforestation and desertification resulting in canges in albedo, and many oter impacts. Unlike many oter problems of pysics, tese ave a distinguising feature: tey do not allow a direct pysical experiment. Moreover, adequate laboratory experiments also seem to be questionable because of specific features of te climate system. In fact, from te standpoint of large scale atmosperic processes, te atmospere is a tin layer wit 3 4 te ratio between vertical and orizontal scales ε = H/L ; at te same time, vertical distribution of parameters in tis layer is very important. Terefore, te metod of numerical simulation of global climate models is te main researc tool in tis case. We need several definitions to describe tese models. Te climatic system is understood as including te atmospere, ocean, cryospere, land, and biota. Te state of te climatic system is caracterized by a set of distributed parameters: temperature, pressure, umidity, velocity of wind in te atmospere and of currents in te ocean, concentration of gaseous components, etc. Te climate is an ensemble of states passed by climatic system for a sufficiently long period. Generally speaking, te coice of suc a period is a problem: in wat follows, we always specify te time interval wen we make strict statements. From a pysical standpoint, to examine te climate of te real climatic system, we ave only a portion of te trajectory several decades long, during wic adequate field measurements were carried out. Naturally, we can restore some caracteristics of te climatic system for longer time intervals; owever, tis concerns only individual caracteristics rater tan a sufficiently complete set caracterizing te state of climatic system. Te construction of modern climate models is based on te following principles. Te model is constructed so as to sequentially account for all te processes participating in te formation of climate, even if te contribution of some processes to te total energy of te overall process is relatively small. Suc an approac is conditioned, in particular, by te fact tat te teory of climate s sensitivity to small external impacts is not yet completely developed. It is assumed tat te Navier Stokes equations for a compressible fluid describe te dynamics of te atmospere and ocean (in describing te ocean s dynamics, as a rule, incompressibility is assumed).

3 In modern models, by virtue of te computational facilities, te Reynolds equations are employed rater tan te Navier Stokes equations. Te Reynolds equations are te Navier stokes equations averaged over certain temporal and spatial scales observing certain commutation rules for te averaging operators. It is assumed tat a closure procedure is principally possible at a certain level of accuracy: te expression of subscale processes (te scales smaller tan te average scale) troug te caracteristics of large scale processes. It is assumed tat te equations of classic equilibrium termodynamics are locally valid. As a rule, modern climate models use te ydrostatic approximation to describe large-scale atmosperic and oceanic motions: te vertical pressure gradient is compensated by te gravity force. Te use of suc an approximation involves a number of furter simplifications: it is required tat te energy conservation law must be satisfied in te absence of external sources of energy and dissipation. In particular, te eart s radius is taken to be constant and te components of Coriolis force wit vertical velocity component are disregarded. Te ydrostatic approximation reduces te system of tree dimensional Navier-Stokes equations to te system of 2.5 dimension, wic is significant in formulating te teorems concerning unique solvability of tese equations on an arbitrarily finite time interval. Te climate model formulated on te basis of tese principles (more precisely, its finite dimensional version) makes it possible to carry out numerical experiments reproducing te contemporary climate and investigate te sensitivity of te model climate to small canges in te parameters caracterizing te external impacts. However, a question arises: wat sould te climate model reproduce and wit wat accuracy to ensure tat its sensitivity to small canges in external impacts be close to te sensitivity of real climate system? A partial answer to tis question can be obtained in te framework of te teory of dissipative semidynamic systems. It is important to empasize tat eac modern model of a concrete pysical penomenon is a reflection of contemporary compreension of te related process. Global climate models wic describe a great number of diverse pysical processes and teir interactions are not an exception in tis sense. At present, climate models experience a period of vigorous development spurred by an explosive development of computer tecnologies. Terefore, we do not consider in detail te parameterization of subscale processes in modern climate models, since tese parameterizations are being developed continuously. It is sufficient to mention te parameterization of te formation and transport of clouds and teir interaction wit radiation. We also avoid presenting te stimulation results for concrete caracteristics of contemporary climate, since tese results are being continuously improved. Consequently, te focus ere is on te establised matematical results of te investigation of climate models and to te scientific directions tat are of great importance in te istory of climate studies. Historically, general circulation models of te atmospere and ocean were a basis for te development of climate models. Te first numerical model of te general circulation of te atmospere was constructed

4 by N.A. Pyllips. He also as proposed te σ -coordinate system wic is very popular nowadays. Systematic investigation of te role of various pysical factors in forming te circulation of te atmospere was started by J. Smagorinsky. Te first coupled circulation models of te atmospere and ocean were constructed by S. Manabe and K. Bryan. Numerical metods for solving te ydrodynamic equations of te atmospere and ocean were developed due to significant contributions of te international scientific community. A. Arakawa proposed te computational metod preserving two quadratic invariants for a two-dimensional incompressible fluid. S.Orszag proposed te metod of spectral mes transformation, wic made te spectral metods widely used in te general circulation models of te atmospere. G. Marcuk s Siberian scool developed a wole class of implicit metods for solving te ydrodynamic equations governing te atmospere and ocean and, in particular, te splitting metod. Qualitative examination of te termoydrodynamic equations of te atmospere and ocean was started by te works of te scientists of Russian and Frenc scools. 2. Matematics for Climate Modeling 2.1. Ideal" Model of Climate Ideal model of climatic system tat generates te observed trajectory of climatic system possesses te following properties: Te ideal model is described by te system of partial differential equations and belongs to te class of dissipative semidynamic systems. Te ideal model as a global attractor. Te trajectory generated by te ideal model lies on its attractor. Trajectories on te model s attractor are unstable in te sense of Lyapunov. Te dynamics on te attractor is caotic and ergodic, i.e., te trajectories originated from almost all points of te attractors are everywere dense on te attractor. Te last assumption is very important, since it opens a possibility to compute te attractor s caracteristics possessing only one trajectory (realization). Assumptions 2 and 3 allow te reformulation of te concept of climate troug te caracteristics of an attractor of ideal model. In particular, if te sense of a sufficiently long time interval is understood as an infinite interval, ten te climate is understood as an attractor (set) wit an ergodic invariant probabilistic measure generated by caotic dynamics of te ideal model specified on tis attractor. Te closeness of certain climate model to te ideal model sould be regarded as te closeness of te above caracteristics of te attractors of climate models. In reality, te goodness of te solution is measured in terms of closeness only at certain instants of time or even of teir projections on certain subspaces of te original pase space. Most of climate models used in climate investigations can be reduced to te canonical form.

5 ϕ + K( ϕϕ ) = Sϕ + f, t ϕ = ϕ t=0 0 (1) wit te elp of generally nonlinear transformations. Here, ϕ is te vector function caracterizing te state of te system,ϕ Φ; Φ is te pase space of te system wic is regarded as a Hilbert space wit te inner product(, ) ; ƒ is te external impact wic can depend on te solution; S is positive definite operator describing te dissipation in te system: ( S, ) ( ) ϕϕ μ ϕϕ,,μ> 0. Κ( ϕ ) is te skew-symmetric operator linearly depending on te solution: ( Κ( ϕ) ϕ ϕ ), = 0. It is clear tat, wen fs, 0, tere is te quadratic conservation law in te system: ϕ ( ϕ, ϕ ) = 0. (2) t System (1) belongs to te class of dissipative systems, since it as te absorbing set: f ϕ. (3) μ Qualitative examination of eac concrete climate model means te establisment of te following statements: Global teorem of solvability for (1). Te existence of global attractor. Te estimation of te attractor s dimension. Te examination of te attractor s structure and its stability. In wat follows, we discuss tese statements wen formulating eac new climatic model. 2.2 Finite dimensional Approximation of Differential Climate Models Equations of climate models are strongly nonlinear, terefore, te only way to examine te beavior of teir solutions, attractor structures, and response to small perturbations of external impacts is to construct teir finite-dimensional approximations.

6 Strictly speaking, eac concrete finite-dimensional approximation of differential climate model sould be regarded as an individual climatic model, since it is necessary to solve te parameterization problem for te subscale processes. Te coice of spatial resolution determines spatial scale; te latter implies te solution of an inverse closure problem, wic is also an individual problem. Form tis standpoint, it is convenient to consider te approximation problem sequentially: first, te system of ordinary differential equations is first approximated wit respect to spatial variables and ten in time. Let a spatial finite-dimensional version of system (1) ave te form ( ) dϕ + K ϕ ϕ = f Sϕ dt ϕ = ϕ ϕ Φ t= 0 0, Te solvability problem for system (4) is not complicated any more. We require tat finite dimensional approximations of te operators K,S also possess te properties of skew-symmetry and positive definiteness, respectively: ( ) μ ( ) K ϕ, ϕ Φ = 0, S ϕ, ϕ ϕ, ϕ (5) It follows from (5) tat, wen exists in (4): ( ) d, = 0 dt ϕ ϕ Φ (Here, (, ) (4) f,s 0, an analogue of te energy conservation Φ is te inner product in Euclidean space). Moreover, (5) implies te existence of an absorbing set for (4) and te existence of a global attractor in tis case for te finite-dimensional system (4) becomes an almost trivial fact. Strictly speaking, wen constructing finite-dimensional approximations of system (4), one must prove te teorems about te closeness of attractors of systems (4) and (5). Suc closeness must be proved in te Hausdorff metrics: H ( ) ( ) = ( ) ( ) dist A, A max dist A, A, dist A, A, were ( ) = y x ρ ( x,y ) dist A, B sup inf, ( xy, ) B A ρ is te metric of te space R N, A is te finite-dimensional attractor of system(4), and A is te attractor of system(5).

7 Here, a new definition of te approximation of finite-difference scemes arises: te approximation of asymptotic sets. At present, unfortunately, te teorems of closeness of te attractors of differential problems and teir finite-dimensional operators ave not been proved for all well-known climate models. Te establisment of closeness between invariant measures on attractors seems to be a muc more complicated problem. Since suc teorems are also absent, te construction of finite-dimensional approximations of differential climatic models is a nontrivial task. Te approximation of energy intensive large-scale processes and te conception of a group of invariants of climate model in te absence of dissipation and forcing is te basis of suc constructions. Of course, te total energy of system is te basic invariant; it can be reduced to quadratic form by nonlinear transformations. Tere exists a number of invariants, wose conservation seems to be quite necessary in climatic (finite-dimensional) models. Tese are te mass conservation law and te conservation law for te angular momentum wit respect to te eart s axis of rotation tat controls te distribution of winds near te eart s surface. Te atmospere and ocean can be regarded as quasi-two dimensional; consequently, asymptotic conservation laws are very important, being te conservation laws in te two-dimensional approximation. Two quadratic invariants, te energy and te enstropy, determine, in a two dimensional fluid, te energy distribution wit respect to spatial scales. Tis is an important climate caracteristic associated wit te cascade of energy from small scales to larger ones. In fact, it determines te required accuracy of te parameterization of subscale processes. 3. Climatic Models In te introduction, te climate was defined as an ensemble of states passed by te climatic system over a sufficiently long time interval. In te ligt of tis definition, te climate model is understood as a model tat reproduces tis ensemble wit acceptable accuracy. Consequently, te model tat reproduces only certain caracteristics of te ensemble cannot be regarded, strictly speaking, as a climatic model. However, following te establised tradition in te teory of climate, we regard all te models wic reproduce some caracteristics of climate as climate models. Here, te climate models are classified wit respect to te dimension of te space in wic tey are constructed. 3.1 Zero dimensional Models Te effective temperature can be defined by writing down te radiation balance equation for te wole system. Tis is possible since te system receives an overwelming portion of its energy from te sun. Assuming tat te climactic system radiates as a gray body, te radiation balance equation is written as 4 ( ) γσ T = 1 α I, (6) e e

8 were Ie = 1/4,I is te integral flux of Solar radiation, αe is te effective albedo of climate system, and γ is te grayness coefficient. 3.2 One dimensional Models Te class of one-dimensional models includes radiation models, radiation-convection models, and diffusion convection models. Te first two-models are obtained by averaging te eat transport equation over orizontal coordinates: dt dt ( z ) ( z ) = ε (7) Here, ε ( z ) denotes radiative eat fluxes distributed along te vertical coordinate. In te case of radiation-convection models, it also accounts for te vertical mixing by te mecanism of effective convection. Te models obtained by averaging te eat flux equation over te vertical coordinate and longitude belong to anoter type of models. After certain closure procedure of te eddy eat fluxes, te eat transport equation takes te form: T 1 T = K( ϕ) + f ( T, ϕ), t cosϕ ϕ ϕ were f is te term describing te eat fluxes. Te representation of te effective diffusion coefficient is a key factor in tis model. In principle, it must depend on T ϕ, taking tus into account te development of baroclinic vortices in te regions wit ig temperature gradient TO ACCESS ALL THE 23 PAGES OF THIS CHAPTER, Visit: ttp:// (8) Bibliograpy Arakawa A. (1966). Computational design for long-term numerical integration of te equations of fluid motions: two dimensional incompressible flow. Part I. J. Comp. Pys., 1 pp [Tis researc, regarded as classic nowadays, described te construction principles of finite-difference scemes for te equations tat govern te dynamics of a two-dimensional incompressible fluid; te scemes possessed two quadratic invariants], Filatov A.N. (1997). Matematics of climate modeling. Birkäuser, Boston, 264 p. [In tis book, for te first time, results of te investigation were systematically presented concerning models

9 of te circulation of te atmospere and ocean and based on te metods of a qualitative teory of differential equations] Journal of climate. (1998). Climate Systems Model Special Issue. v. 11, N 6. [Tis special issue contains results of te modeling of contemporary climate by a modern NCAR climate model] Kraicnan R.H. (1959) Classical fluctuation relaxation teorem. Pys. Rev.,, 109,pp Lions J.L., Manley O.P., Temam R.,Wang S. (1997). Pysical interpretation of te Attractor Dimension for te primitive equations of Atmopseric Circulation. J.Atm. Sci., v.54, N9, pp Manabe S., Bryan K. (1969). Climate calculations wit a combine ocean-atmosperic model. J. Atm. Sci., 26, pp [Tis was te first article tat presented results of modeling te climate wit te elp of a coupled model of te general circulation of te atmospere and ocean] Marcuk G. I.,, and Zalesny V.B. (1987). Matematical models in Geopysical Fluid Dynamics and Metods of teir Realization Leningrad, Gidrometeoizdat, 296 pp. (In Russian) [Tis book describes efficient numerical metods for solving te dynamic equations for te atmospere and ocean] Orszag S. (1970). Transform metod for calculation of vector coupled sums. Application to te spectral form of te vorticity equation. J. Atm. Sci., 27, pp [In tis article, te metod of spectral grid transformation was presented tat made spectral metods most widely used wen solving te equations of te atmospere s fluid dynamics] Pilips N.A. (1956). Te general circulation of te atmospere: a numerical experiment. Quart. J. Roy. Met.Soc. v.82, N 352, pp [In tis work, te problem of general circulation of te atmospere was formulated for te first time on te basis of differential equations of fluid dynamics] Smagorinsky J. (1963). General circulation experiment wit te primitive equations. I. Te basic experiment. Montly Wea. Rev., v. 93, N 3, pp [In tis study, te fundamentals of numeric modeling of te general circulation of te atmospere were formulated for te first time on te basis of primitive dynamic equations for te atmospere] Biograpical Sketc Dymnikov Valentin Pavlovic is a, member of te Russian Academy of Sciences and Director of Institute of Numerical Matematics, Moscow, Russia. He specializes in te filed of geopysical fluid dynamics, modeling of te general circulation of te atmospere and climate, and numerical matematics. He is one of te pioneers of te teory of caotic attractors of climatic models. He publised more tan 160 scientific works including 11 monograps and text books. He olds te cair of "Matematical Modeling of Pysical Processes" at Moscow Pysical and Tecnological Institute. Academician Dymnikov was awarded by te Fridman Prize of te Russian Academy of Sciences.