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On a new fluctuation-dissipation theorem in climate dynamics

by

V

C. NICOLIS

B E L G I S C H I N S T I T U U T V O O R R U I M T E - A E R O N O M I E 3 - Ringlaan

B • 1180 BRUSSEL

FOREWORD

T h e paper entitled "On a new fluctuation-dissipation theorem in climate dynamics" will be published in Developments in Atmospheric Sciences.

A V A N T - P R O P O S

L'article intitule : "On a new fluctuation-dissipation theorem in climate dynamics" sera publie dans Developments in Atmospheric Sciences.

VOORWOORD

Het artikel : "On a new fluctuation-dissipation theorem in climate dynamics" zal verschijnen in het tijdschrift Developments in Atmospheric Sciences.

VORWORT

Die Arbeit : "On a new fluctuation-dissipation theorem in climate dynamics" wird in Developments in Atmospheric Sciences heraus- gegeben werden.

O N A NEW F L U C T U A T I O N - D I S S I P A T I O N T H E O R E M I N C L I M A T E D Y N A M I C S

b y

C . N I C O L I S

A b s t r a c t

In recent y e a r s , a f l u c t u a t i o n - d i s s i p a t i o n theorem l i n k i n g the time correlation of f l u c t u a t i o n s of climatic v a r i a b l e s to the matrix monitoring the linear r e s p o n s e to an external f o r c i n g , h a s been e x t e n s i v e l y u s e d in climate d y n a m i c s . In the p r e s e n t p a p e r an e x t e n d e d f l u c t u a t i o n - d i s s i p a t i o n theorem is s u g g e s t e d , w h i c h r e f e r s to the

p r o p e r t i e s of the random f o r c e s r a t h e r t h a n t h o s e of the climatic v a r i a b l e s . T h e implications of t h i s theorem are a n a l y z e d in the f r a m e - w o r k of an e n e r g y - b a l a n c e model. In p a r t i c u l a r , some c h a r a c t e r i s t i c p r o p e r t i e s of the noise associated with climatic v a r i a b i l i t y are identified.

Résumé

A u c o u r s de ces d e r n i è r e s années un théorème de f l u c t u a t i o n - d i s s i p a t i o n reliant la fonction de corrélation temporelle d e s f l u c t u a t i o n s climatiques aux coefficients de r é p o n s e linéaires a été i n t r o d u i t en d y n a m i q u e climatique. D a n s le p r é s e n t t r a v a i l on p r o p o s e une e x t e n s i o n de ce théorème f a i s a n t i n t e r v e n i r les p r o p r i é t é s d e s f o r c e s aléatoires au lieu de celles d e s v a r i a b l e s climatiques. Les c o n s é q u e n c e s de ce n o u v e a u théorème s o n t a n a l y s é e s et illustrées s u r u n modèle de bilan é n e r g é t i q u e .

Samenvatting

Gedurende de laatste j a r e n w o r d t een f l u c t u a t i e - d i s s i p a t i e theorema, dat de tijdscorrelatie van klimaatschommelingen in v e r b a n d b r e n g t met de coëfficiënten van lineaire respons, v e e l v u l d i g g e b r u i k t in de klimaatdynamica. In d i t w e r k w o r d t een u i t b r e i d i n g van d i t theorema v o o r g e s t e l d , die veeleer de eigenschappen van w i l l e k e u r i g e k r a c h t e n naar voor b r e n g t i . p . v . deze van de klimaatschommelingen. De gevolgen van d i t nieuw theorema worden geanalyseerd en g e ï l l u s t r e e r d op een model van energetische balans. In het bijzonder worden bepaalde

k a r a k t e r i s t i e k e eigenschappen van het g e l u i d , geassocieerd met klimaat- v a r i a b i l i t e i t , u i t g e l e g d .

Zusammenfassung

In letzter Zeit ist eines S c h w a n k u n g - D i s s i p a t i o n T h e o r e m , dass die Zeitcorrelation von Klimatschwankungen in V e r b i n d u n g setz mit die Koeffizienten von linear A n t w o r t , viel b e n u t z t in die Klimatdynamik.

In diese A r b e i t ist eine A u s b r e i t u n g von diesem Theorem d a r g e s t e l l t , die vielmehr die Eigenschäppe von Z u f a l l s k r ä f t e t u t a u f t r e t e n s t a t t diese von die K l i m a t v e r ä n d e r l i c h e r n . Die Gefolge von diesem neuem Theorem sind a n a l y s i e r t und i l l u s t r i e r t am eines Modell von E n e r g i e - B i l a n z . Besonders sind manche c h a r a k t e r i s t i s c h e Eigenschäfte von G e r ä u s c h , assoziiert mit K l i m a t v a r i a b l i t ä t , e r k l ä r t .

1. I N T R O D U C T I O N

Climate variability is an undisputable fact, and its effect both on the short term behavior and on the large scale transitions of the climatic system is widely recognized (Lorenz, 1976). On the other, hand, its quantitative characterization is an extremely complex statistical problem. Indeed, climate dynamics is an ensemble of nonlinear processes involving a large number of coupled variables, which evolve under conditions far away from thermodynamic equilibrium. Even at a labo- ratory scale, such processes are poorly understood. For instance the statistical foundations of hydrodynamic instabilities (see e . g . Swinney and Gollub, 1981) or of chemical instabilities (Nicolis and P r i g o g i n e , 1977) are still in their infancy.

A n important step toward the quantitative formulation of climate variability was accomplished by Leith (1975, 1978), whose ideas were further d i s c u s s e d by Bell (1980) and North et al. (1981) among others. We briefly summarize the main points.

Let {x } be a set of climatic variables. On the one side, their evolution is subjected to continuous random deviations from some macro- scopic average, owing to the intrinsic stochasticity of climate. A n d on the other side, these same variables are capable of responding to any change of the external environment (like e . g . the solar output, the earth's orbital characteristics, the C 02 level e t c . . . ) in the form of a deterministic signal. B u t the local dynamics of { xa} has presumably no way to "know" whether the perturbation to which it responsed was due to the system's intrinsic stochasticity or to the external f o r c i n g . Hence, the two responses must somehow be related. We call this relation the fluctuation-dissipation theorem. Leith wrote it in the form :

- 3 -

< 6 xa ( 0 ) 6 Xp(x) > = I < 6 xa( 0 ) 6 x ^ ( 0 ) > . gy p ( t ) ( 1 )

T h e l e f t hand side of Eq. ( 1 ) f e a t u r e s t h e t i m e - c o r r e l a t i o n matrix of t h e fluctuations {6><a} in t h e absence of e x t e r n a l f o r c i n g . In t h e r i g h t hand side we have t h e static q u a d r a t i c average of these same fluctuations multiplied by t h e matrix g ( x ) , which monitors t h e linear response of f x 1 to a small change of the e x t e r n a l e n v i r o n m e n t . Specifically, t h e

1 or

time integral of g „ ( t ) may be viewed as some sort of t r a n s p o r t

Of p

coefficient,

00

( 2 a ) ƒ « « P ( x ) d T

relating t h e response 6<xa> to the e x t e r n a l f o r c i n g , :

6 <x > = I D Q 6 Fr (2 b>

a p a0 p

T h e b r a c k e t s in Eqs ( 1 ) and ( 2 b ) denote t h e a v e r a g i n g over a statistical ensemble d e s c r i p t i v e of t h e system. Presumably, both the time-correlation functions and t h e static q u a d r a t i c averages of { 6 xa} are accessible from meteorological d a t a . Hence, from Eq. ( 1 ) one can evaluate the response function g ( t ) and subsequently determine t h a n k s to Eqs ( 2 ) , t h e system's sensitivity to a v a r i e t y of e x t e r n a l signals.

T h i s , according to L e i t h , is t h e main usefulness of f l u c t u a t i o n - dissipation like relations in climate dynamics.

In view of t h e enormous complexity of t h e e a r t h - a t m o s p h e r e system, it is only unavoidable t h a t many of t h e assumptions needed to a r r i v e at the basic relation, Eq. ( 1 ) , are not f u l l y j u s t i f i e d . Let us examine t h e most crucial of them in some d e t a i l .

F i r s t , in t h e original d e r i v a t i o n ( L e i t h , 1975, 1978) the Liouville equation f o r a conservative flow is t h e s t a r t i n g p o i n t . Now, t h e dynamics of t h e climatic variables is a d i s s i p a t i v e , non c o n s e r v a t i v e d y n a m i c s . As well known the passage from t h e microscopic degrees of freedom evolving according to t h e Liouville equation to t h e macro- variables obeying to such conservation laws as t h e N a v i e r - S t o k e s equations is a complex process. Additional assumptions are t h e r e f o r e needed. Specifically, it is postulated t h a t t h e p r o b a b i l i t y ensemble of t h e macro variables is g i v e n b y a Gaussian d i s t r i b u t i o n and t h a t t h e process is ergodic. T h i s is c e r t a i n l y t r u e when t h e system evolves in t h e v i c i n i t y of a single globally stable steady state. If on t h e c o n t r a r y t h e e a r t h - a t m o s p h e r e system is capable of performing t r a n s i t i o n s between d i f f e r e n t t y p e s of climatic states ( L o r e n z , 1976, N o r t h et al.

1981) this assumption should b r e a k down : T h e p r o b a b i l i t y ensemble is in this case multi-humped (Nicolis and Nicolis, 1981) and t h e process becomes "metrically almost i n t r a n s i t i v e " ( L o r e n z , 1 9 7 6 ) .

Second, relation ( 1 ) involves two unknown sets of quantities : t h e correlation functions and t h e static q u a d r a t i c averages of the fluctuations. As a r e s u l t , it cannot give to us information on the source of intrinsic randomness of t h e climatic system. M o r e o v e r , a d i f f i c u l t y arises when one is interested in long term climatic changes, like those on which one focuses in t h e framework of e n e r g y - b a l a n c e models. T h e measurability of t h e above mentioned two sets of average quantities becomes t h e n questionable, and as a result t h e usefulness of a fluctuation-dissipation relation of t h e form given in Eq. ( 1 ) is r e d u c e d . -

T h e purpose of t h e present paper is to present some results c o n t r i b u t i n g to a partial resolution of the above mentioned d i f f i c u l t i e s . T h e s t a r t i n g point is to realize t h a t climatic v a r i a b i l i t y is the r e s u l t of f l u c t u a t i o n s . T h e latter are incorporated into t h e traditional description based on t h e phenomenological balance equations t h r o u g h t h e addition of random f o r c e s , as suggested by Hasselmann ( 1 9 7 6 ) :

- 5 -

dx

= f0 ( { xp} ) - d i v Ja + Fa( r , t ) ( 3 a )

H e r e f a r e t h e s o u r c e o r s i n k t e r m s , J t h e f l u x of x , a n d F t h e a a a a r a n d o m f o r c e associated to x ^ . T h e p r o b l e m of o b t a i n i n g i n f o r m a t i o n on t h e s t a t i s t i c s of climatic v a r i a b l e s amounts t h e r e f o r e t o f i n d i n g t h e s t a t i s t i c a l p r o p e r t i e s of t h e s e r a n d o m f o r c e s . A s , b y d e f i n i t i o n , t h e l a t t e r a r e not accessible e x p e r i m e n t a l l y , one m u s t r e s o r t to p h y s i c a l l y p l a u s i b l e models a n d t e s t t h e i r v a l i d i t y a p o s t e r i o r i , on t h e basis of t h e b e h a v i o r t h e y p r e d i c t f o r t h e o r i g i n a l climatic v a r i a b l e s .

O n e , w i d e l y a d o p t e d , assumption is t h a t t h e r a n d o m f o r c e s d e f i n e a Gaussian w h i t e noise in time :

< Fa (t, t ) F p(t' f ) > = Qa p( r , r ' ) 6 ( t - f ) ( 3 b )

I n o r d e r t o c h a r a c t e r i z e t h e s t a t i s t i c s c o m p l e t e l y , it remains h o w e v e r to o b t a i n i n f o r m a t i o n on t h e c o v a r i a n c e m a t r i x { Qap ( £ ' £ ' ) } • T h i s i n t u r n

can o n l y be a c h i e v e d if t h e p h y s i c a l mechanisms at t h e o r i g i n of f l u c t u a - tions a r e i d e n t i f i e d . So f a r t h i s a s p e c t has been completely o v e r l o o k e d in t h e l i t e r a t u r e : e i t h e r t h e a u t h o r s a s s i g n an a r b i t r a r y s t r u c t u r e to { Qap } ( H a s s e l m a n n , 1976; Nicolis a n d N i c o l i s , 1981; S u t e r a , 1 9 8 1 ) o r , a t b e s t , t h e y i n c o r p o r a t e in t h e d e s c r i p t i o n some c o n s t r a i n t s a r i s i n g f r o m s y m m e t r y a r g u m e n t s ( N o r t h a n d C a h a l a n , 1 9 8 1 ) . As a r e s u l t , in systems i n v o l v i n g s e v e r a l v a r i a b l e s one is c o n f r o n t e d w i t h a l a r g e n u m b e r of u n k n o w n p a r a m e t e r s . T h i s o b s c u r e s c o n s i d e r a b l y t h e u n d e r - s t a n d i n g of t h e mechanisms w h i c h a r e at t h e o r i g i n of climatic v a r i a b i l i t y .

O u r work provides a f i r s t step toward a more fundamental characterization of the noise associated with climatic variability. T h e basic idea is the following : Fluctuations originate, typically, in the form of localized, small scale events and as a result they cannot perceive the nonequilibrium constraints conferring a nonlinear character to the overall dynamics. It is therefore h i g h l y plausible that such properties as microscopic reversibility and fluctuation-dissipation like relations still hold for the random forces descriptive of these fluctua- tions, even if they may be compromized for the state variables them- selves.

The search of such a fluctuation-dissipation theorem of the

"second kind" will be the central point of our work. A s a b y p r o d u c t , this procedure will enable us to determine the characteristics of the fluctuations of the state variables and t h u s obtain information on the type of noise characterizing climate variability.

The general formulation of these ideas, which are patterned after the Landau-Lifshitz theory of fluctuations in fluid dynamics, is presented in Section 2. In Section 3 we apply the theory to a zonally averaged energy balance climate model and derive expressions for the correlation matrix of the effective random forces. Section 4 is devoted to the detailed analysis of the implications of this result, u s i n g the two-mode truncation of the model of Section 3. In particular, we show that the globally averaged surface temperature constitutes a non- Markovian process with red noise spectrum. Some comments and s u g g e s t i o n s are presented in Section 5.

2. A F L U C T U A T I O N - D I S S I P A T I O N T H E O R E M FOR E N E R G Y - B A L A N C E M O D E L S : G E N E R A L F O R M U L A T I O N

In order to avoid unnecessary complications and present the ideas as clearly as possible we focus, in the major part of this

p a p e r , on e n e r g y balance climate models. We do not comment on the r a n g e of v a l i d i t y of the d e s c r i p t i o n a f f o r d e d b y s u c h models, r e f e r r i n g to N o r t h et al (1981) f o r a recent s u r v e y . We w a n t to emphasize however t h a t most of the b a s i c ideas e x p r e s s e d below are model- i n d e p e n d e n t .

T h e g e n e r a l form of the evolution equation in an e n e r g y balance model is :

Here e ( e n e r g y per u n i t s u r f a c e ) is the e n e r g y content of a column h a v i n g a u n i t c r o s s section a n d a h e i g h t of the o r d e r of the d e p t h of the mixed l a y e r , c is the heat c a p a c i t y of t h i s column, T the s u r f a c e t e m p e r a t u r e , f the radiation b u d g e t ( d i f f e r e n c e between incident solar f l u x a n d o u t g o i n g i n f r a r e d r a d i a t i o n ) :

Finally £ is the space coordinate, a n d J the e n e r g y f l u x .

We would now like to e x p r e s s the mechanism b y w h i c h f l u c t u a - t i o n s a p p e a r in the balance equation ( 4 ) . A s d i s c u s s e d e x t e n s i v e l y in the t h e o r y of i r r e v e r s i b l e p r o c e s s e s ( s e e e . g . P r i g o g i n e , 1961; L a n d a u a n d L i f s h i t z , 1959) f l u c t u a t i o n s o r i g i n a t e from the d y n a m i c s of the d i s s i p a t i v e d e g r e e s of freedom. In o r d e r to i d e n t i f y the latter we h a v e to t u r n to the e n t r o p y balance equation associated to E q . ( 4 ) a n d c o n s t r u c t the e x p r e s s i o n of the rate of d i s s i p a t i o n per unit time - the e n t r o p y p r o d u c t i o n . For an e n e r g y balance model of the k i n d c o n s i d e r e d

E =

' «> "

I

f ( T , = F^s - F^{i r} (5)

in the p r e s e n t Section t h i s w a s accomplished in Nicolis a n d Nicolis (1980). T h e r e s u l t is

In other w o r d s the r a d i a t i v e terms f ( T , r ) do not c o n t r i b u t e to the d i s s i p a t i v e b e h a v i o r of the s y s t e m in t h i s a p p r o x i m a t i o n . We are t h e r e f o r e forced to conclude t h a t the o n l y w a y f l u c t u a t i o n s can influence the e n e r g y balance equation is t h r o u g h the heat f l u x J. In the f r a m e w o r k of the deterministic d e s c r i p t i o n it is c u s t o m a r y to e x p r e s s J b y a F o u r i e r t y p e law, the d i f f e r e n c e b e i n g t h a t the p r o p o r t i o n a l i t y coefficient is here the e d d y d i f f u s i v i t y r a t h e r t h a n the molecular heat c o n d u c t i v i t y . In the p r e s e n c e of f l u c t u a t i o n s the 1 situation will c h a n g e , a n d we will h a v e

J t ) = - D' V T + I ijr, t ) (7)

D' is the e d d y d i f f u s i v i t y which f o r simplicity is t a k e n to be both r a n d t - i n d e p e n d e n t a n d 4 ( r , t ) is the s p o n t a n e o u s heat f l u x a r i s i n g from f l u c t u a t i o n s . S u b s t i t u t i n g into E q . ( 4 ) we obtain a s t o c h a s t i c differential equation of the form g i v e n in E q . ( 3 a ) :

c t * = f ( T , r ) + D' V²T + F Q:, t ) (8a)

o t

w h i c h d i s p l a y s the random force term

entropy p r o d u c t i o n = - 1 (6)

O = - d i v l (;r, t ) (8b)

We w a n t now to e s t a b l i s h t h e p r o p e r t i e s of t h e r a n d o m f i e l d j ^ ( r , t ) f r o m w h i c h t h e p r o p e r t i e s of t h e e f f e c t i v e r a n d o m f o r c e F will follow a u t o m a t i c a l l y . Let us f o r g e t f o r a moment t h a t we a r e d e a l i n g w i t h t h e e a r t h - a t m o s p h e r e system a n d c o n s i d e r a simple h e a t c o n d u c t i n g f l u i d o p e r a t i n g in t h e v i c i n i t y of t h e r m o d y n a m i c e q u i l i b r i u m . I n t h i s case a f l u c t u a t i o n - d i s s i p a t i o n t h e o r e m l i n k i n g ^ ( r , t ) to t h e p h e n o m e n o - logical c o e f f i c i e n t D' was d e r i v e d b y L a n d a u a n d L i f s h i t z ( 1 9 5 9 ) on i n t u i t i v e g r o u n d s a n d f u r t h e r j u s t i f i e d b y Fox a n d U h l e n b e c k ( 1 9 7 0 ) . I t has t h e f o r m :

< j£ (< f t ) > = 0 ( 9 a )

< J i t } > = 2 S D' ^ «An ^ " £ ') 6 ( t " t , }

( 9 b )

w h e r e K is t h e B o l t z m a n n ' s c o n s t a n t a n d f t h e e q u i l i b r i u m t e m - B

p e r a t u r e . I n o t h e r w o r d s d i f f e r e n t components of t h e random p a r t of t h e h e a t f l u x a r e u n c o r r e l a t e d b e t w e e n t h e m s e l v e s , t h e v a l u e s of a g i v e n component at d i f f e r e n t space r e g i o n s a r e also u n c o r r e l a t e d , a n d f i n a l l y t h e whole process c o n s t i t u t e s a Gaussian w h i t e noise in t i m e .

As p o i n t e d o u t in t h e I n t r o d u c t i o n , f l u c t u a t i o n s o r i g i n a t e s p o n t a n e o u s l y in t h e f o r m of l o c a l i z e d , s h o r t scale e v e n t s i n d e p e n d e n t l y of t h e d i s t a n c e f r o m t h e s t a t e of t h e r m o d y n a m i c e q u i l i b r i u m . We t h e r e f o r e e x p e c t t h a t t h e a b o v e p r o p e r t i e s will still hold u n d e r t h e h i g h l y n o n e q u i l i b r i u m c o n d i t i o n s c h a r a c t e r i z i n g t h e climatic system ( s e e Nicolis a n d P r i g o g i n e , 1977; K e i z e r , 1978 f o r a g e n e r a l d i s c u s s i o n of t h i s p o i n t ) . We t h e r e f o r e w r i t e

I

K

h

< j£ t } j m t f ) > ⁼ °^{2 6}£m " ^{6 ( t} " ^

In other words we require to have the same spatial and temporal properties as in equilibrium but allow for a different numerical factor,

2 - 2

ct , which replaces the factor 2 K_ D' T in Eq. ( 9 b ) and expresses the

B 2

strength of fluctuations far from equilibrium. In the sequel a will be treated as a parameter. Our analysis will focus solely on the consequences of the fact that j is delta-correlated both in space and time. Note that, because of Eq. ( 8 b ) , thie effective random force F appearing in the energy balance equation ( 8 a ) , is not delta correlated in space. T h e consequences of this rather important property are examined in Section 4.

Eqs. (10) are somewhat reminiscent of the formalism developed recently by North and coworkers (North et al, 1981; North and Cahalan, 1981). However, to our knowledge these authors have not incorporated in their description the physical constraint that, the effective random force F is the divergence of a random vector field and that the latter is a white noise both in space and time. T r u e , our expressions may be considered as a special case of a general covariance matrix {Qap(jC'JC'^ t h e r a n d o m f°rc e field ( c f . Eq. ( 3 b ) ) . However, in the theory of fluctuations it is important to become gradually as specific as possible in order to incorporate the maximum amount of information available from first principles.

Finally, let us emphasize that the procedure outlined above can be extended readily to any other kind of conservation equation

like, for instance, the momentum balance equation :

(10a) (10b)

- 1 1 -

in which p is the mass density, % is the velocity field, p the hydro- static pressure, a the dissipative part of the stress tensor, and d/dt the hydrodynamic derivative. For an incompressible fluid the analogue of Eq. (7) is :

dv. 6 v .

= " n ( a r ^{+ + s}i j

J ¹

r| being the shear viscosity and S.. the fluctuating stress tensor. Eqs.

(9) are now to be replaced by (Landau and Lifshitz, 1959) :

< S . . (t, t ) > = 0

< ^Si j U . ^S£^m U ' » f ) > = 2 k^b T n (6^{i A} 6^{j m +} 6^ira

fixing the statistics of the effective random force present in the momentum balance equation. The transition from this near-equilibrium result to the highly nonlinear regime of climate dynamics would naturally follow the same lines as above and would lead to an equation similar to (10b), in which the factor 2K^BTn is replaced by a simple parameter a² expressing the strength of fluctuations far from equilibrium. Some properties of the noise of the vector field % are

reported in Nicolis et al. (1983) along with an application to the Lorenz model.

3 . A P P L I C A T I O N T O A Z O N A L L Y - A V E R A G E D MODEL

We now simplify f u r t h e r t h e model i n t r o d u c e d in t h e p r e v i o u s Section b y t a k i n g into account o n l y t h e e n e r g y t r a n s f e r in t h e m e r i - dional d i r e c t i o n . In t h i s z o n a l l y - a v e r a g e d , one-dimensional ( 1 - d ) model

Eqs. ( 8 ) t a k e t h e form ( N o r t h , 1975)

c = f ( T , X) + D ( 1 - x2) f l - i ( 1 - x2)1 7 2 j ( x , t ) 5t 9x ox R dx

= f ( T , x ) + D ( 1 - x2) g + F ( x , t ) ( 1 1 )

T o a r r i v e at E q . ( 1 1 ) one has also to p e r f o r m a c h a n g e to s p h e r i c a l coordinates on t h e e a r t h ' s s u r f a c e , x is t h e sine of t h e l a t i t u d e , R t h e e a r t h ' s r a d i u s , and D an e f f e c t i v e heat d i f f u s i o n c o e f f i c i e n t d e f i n e d b yN

D = D ' / R2.

In o r d e r to f i n d t h e f o r m of t h e g e n e r a l i z e d f l u c t u a t i o n - dissipation t h e o r e m , Eqs. ( 1 0 ) , in t h e s e c o o r d i n a t e s , we f i r s t o b s e r v e t h a t in t h e f r a m e w o r k of o u r 1 - d model, o n l y t h e localization in t h e 8 - d i r e c t i o n needs to be e x p r e s s e d . T h e c p - d e p e n d e n c e contained in t h e delta f u n c t i o n 6 ( r - r ' ) should t h e r e f o r e merely e n t e r as p a r t of t h e Jacobian of t h e t r a n s f o r m a t i o n to s p h e r i c a l c o o r d i n a t e s , namely R Acpcos 6 . T h u s 2

• U - f J - T T R Aq> cos 8

2

T h e size of Aq> has to be chosen in such a way t h a t R Aqp r e d u c e s to t h e u n i t s u r f a c e element f o r which t h e e n e r g y balance e q u a t i o n is u s u a l l y w r i t t e n . F u r t h e r m o r e ,

- 1 3 -

6(x - x ' ) =f 6(sin 6 - sin 9 ' ) = 6(9 - 9 ' )

We obtain therefore

6(£ - = - J c o s 9 6(x - x ' ) = 6(x - x ' ) (12) R Acp cos 0

and hence (cf. Eq. (10b)) :

< j (x, t ) j(x* , f ) > = a2 6(x - x ' ) 6 ( t - t ' ) (13)

From this relation as well as Eq. (8b) we may compute the auto- correlation function of the effective random force

< F(x, t ) F(x* , f ) > =

= ( 2 , 2 ( , . x2)1'2 ( 1 - X '2)1 / 2 6 ( x - X ' ) ( 1 4 )

e

As usual (North, 1975) we expand both the temperature and the random force fields in series of Legendre polynomials :

T = I T ( t ) P ( x ) ( 1 5 a ) n n n

F = I F ( t ) P ( x ) ( 1 5 b ) n n n

I n s e r t i n g r e l a t i o n ( 1 5 b ) i n t o E q . ( 1 4 ) m u l t i p l y i n g b y Pm( x ) p f l(x' ) a n d

i n t e g r a t i n g o v e r x a n d x ' we o b t a i n :

2 2 < F F „ > = ( | )2 6 ( t - f ) .

2A + 1 2m + 1 m £ V R

ƒ d x d x ' P^m( x ) ( 1 - x ² )^{1 / 2} P^£( x ' ) ( 1 - x '² )^{1 / 2} 6 ( x - x ' )

I n t e g r a t i n g b y p a r t s a n d p e r f o r m i n g t h e d e l t a f u n c t i o n we f i n a l l y o b t a i n :

< F F „ > = q2 6 *r 6 ( t - t ' ) ( 1 6 a )

M Si TII Jim

w i t h

= . ( . + 1) ( | )2 ( 1 6 b )

S e v e r a l i m p o r t a n t conclusions can be d r a w n f r o m t h e s e e x p r e s s i o n s . F i r s t , t h e r a n d o m f o r c e s associated to d i f f e r e n t L e g e n d r e modes a r e u n c o r r e c t e d . S e c o n d , t h e r e is no random f o r c e a c t i n g d i r e c t l y on t h e e q u a t i o n f o r t h e g l o b a l l y a v e r a g e d t e m p e r a t u r e T^q, as q = 0 f r o m E q . ( 1 6 b ) . A n d t h i r d t h e i m p o r t a n c e of t h e f l u c t u a t i o n s

- 1 5 -

I

d e p e n d s o n t h e o r d e r o f t h e L e g e n d r e m o d e . F o r t h e f i r s t f e w modes q is e x p e c t e d t o be s m a l l , b e c a u s e o f t h e p r e s e n c e o f t h e i n v e r s e s q u a r e o f t h e e a r t h ' s r a d i u s i n E q . ( 1 6 b ) . F o r h i g h e r m h o w e v e r q

3

r a p i d l y i n c r e a s e s , v a r y i n g r o u g h l y as m . Now as w e l l k n o w n h i g h e r o r d e r L e g e n d r e modes r e p r e s e n t l o c a l i z e d d i s t u r b a n c e s . We r e a c h t h e r e f o r e a v e r y n a t u r a l c o n c l u s i o n , n a m e l y t h a t t h e i m p o r t a n c e o f f l u c t u a t i o n s d e p e n d s o n t h e i r s p a c e s c a l e . T h i s is r e m i n i s c e n t o f t h e p h e n o m e n o n o f n u c l e a t i o n f r e q u e n t l y e n c o u n t e r e d i n p h a s e t r a n s i t i o n s . T h e a b o v e p o i n t s w o u l d o f c o u r s e h a v e b e e n m i s s e d c o m p l e t e l y i f we h a d p r o c e e d e d f o r m a l l y , k e e p i n g a g e n e r a l , u n s p e c i f i e d s t r u c t u r e f o r t h e c o v a r i a n c e m a t r i x o f t h e r a n d o m f o r c e s .

4. T H E T W O - M O D E T R U N C A T I O N

In t h i s S e c t i o n we e x p l o r e t h e c o n s e q u e n c e s o f E q s . ( 1 6 a ) a n d ( 1 6 b ) o n t h e t w o - m o d e t r u n c a t i o n o f E q . ( 4 ) , i n o r d e r t o o b t a i n e x p l i c i t i n f o r m a t i o n o n t h e s t a t i s t i c s o f t h e t e m p e r a t u r e f i e l d . A s well k n o w n ( N o r t h e t a l . 1981) t h e f i r s t t w o L e g e n d r e modes p r o v i d e a r e a s o n a b l y g o o d d e s c r i p t i o n o f t h e l a r g e scale f e a t u r e s o f t h e m e r i d i o n a l t e m p e r a t u r e d i s t r i b u t i o n a n d h e a t f l u x .

T h e s t a r t i n g e q u a t i o n s a r e o b t a i n e d b y i n s e r t i n g ( 1 5 b ) „ i n t o ( 4 ) , m u l t i p l y i n g b y Pm(x) > i n t e g r a t i n g o v e r x , a n d r e c a l l i n g t h a t Fq = 0 . We t h u s h a v e :

dT

C d T = ^{Q H}o ^{( x}s^} " ^{( A + B} V dT

C d t = Q H2 ( XS} " ( B + 6 D ) T2 + F2( t )

( 1 7 a )

( 1 7 b )

Here Q is the solar constant divided by 4, A and B are cooling coefficients arising in the parameterization of the infrared radiation flux :

Fi r = A + BT (X, t)

and H H_ are the first two Legendre moments of the coalbedo a(x) o 2

multiplied by the mean annual distribution, S(x), of radiation reaching the top of the atmosphere (North, 1 9 7 5 ) . Both Hq and H2 depend on the position of the ice edge xg, which is related to Tq and T^ through the Budyko boundary condition (Budyko, 1 9 6 9 )

T (t) + T (t) P.(x ) = - 10°C (18) o ^ z s

Let T*. T , x* be a reference steady-state solution of Eqs.

o 2 s

(17) representative, say, of the present-day climate. We first focus on the short term variability of this climate. To this end it is sufficient to analyze the linear response by introducing the small deviations

6 = T - T* • o o o

0 = T - T* (19) 2 2 2

| = x - x* . s s

- 1 7 -

Linearizing Eqs. (17), (18) with respect to these quantities we obtain :

* = " T* P ' U [ 6o + 92 P2 2 2

= a 8 + p 0, (20) o 2

and

de

c = (Q H' a - B ) 6 + Q H' p 6 (21a) dt o o o z

de0

c ^ = Q H' a 6o - (B + 6D - Q H' P) e2 + F2( t ) (21b)

where the prime denotes differentiation of H H^, P2 with respect to their argument.

The first point we make in connection with these equations is motivated by a result of one of the authors (Nicolis, 1980) concerning the passage from 1-d to 0-d climate models. Specifically, by eliminating 62 in terms of 6o from Eq. (21b), we obtain a closed equation for 6q. Contrary to the deterministic analysis reported (Nicolis, 1980), this equation contains now an effective noise term, as depends oh the random force F2( t ) . To evaluate the characteristics of this noise we first solve Eq. (21b) with the initial condition that e2 = 0 at t = - We find :

<J

t

e2(t) = e "^{Y t} i ƒ d f [Q H^ a 0^o( f ) + F²(t')] e^{Y t}' (22)

- 0 0

w i t h

Y = i (B + 6D - QH^ p) (23)

We t h u s o b t a i n a s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n f o r 6^q :

de t

± = - (Q H ; a - B) 6⁰ + \ Q² H ; RJa p f d f e "Y ( t _ t , ) ^ ( f )

C C J

dt - oo

+ ^ Q H ; p ƒ d f e "Y ( t"t , ; ) F2( t ' ) (24)

c -oo

Let us d e n o t e b y <t> ( t ) t h e last t e r m of t h e r i g h t h a n d s i d e . T h e o c o r r e l a t i o n f u n c t i o n of t h i s e f f e c t i v e noise t e r m is :

< <t> ( t ) 4> ( t ' ) > = o o

' Q H l S \2 f f - 7 ( t - x ) - 7 ( f - T )

- f - / d x¹ ƒ d t²e ^! e < F²( t¹) F²( t²) >

c ' -oo -oo

U t i l i z i n g E q . (16a) f o r m = 2 we o b t a i n a f t e r some a l g e b r a :

< 4> ( t ) • ( to o ¹) > =

QH 0P 2 1_ ~yI t - t ' f i2 2 V

2 1 - y t - t l

= r tto 2y e " ' , (25)

-19-

T h i s relation is c h a r a c t e r i s t i c of an O r n s t e i n - U h l e n b e c k process ( S o o n g , 1 9 7 3 ) . In o t h e r w o r d s , t h e e f f e c t i v e noise acting on t h e globally averaged surface t e m p e r a t u r e 6^q is not a white noise, c h a r a c t e r i s t i c of a diffusion process, b u t a colored noise h a v i n g an exponentially decaying time correlation f u n c t i o n . In as much t h e only Markovian process with continuous realization is a diffusion process, it t h e r e f o r e follows ( A r n o l d , 1974) t h a t 0^q is a n o n - M a r k o v i a n process. In other w o r d s , if fluctuations a r e incorporated in a reduced description ( l i k e for instance in a zero-dimensional climate model) involving a limited number of variables ( l i k e for instance t h e mean a v e r a g e surface tem- p e r a t u r e ) , one should be p a r t i c u l a r l y careful in t h e modelling of t h e corresponding random f o r c e s . On t h e other h a n d , t h e pair ( 9^q, 0²) is still Markovian and t h u s amenable to a description in terms of a b l - v a r i a t e F o k k e r - P l a n c k equation.

In o r d e r to see the repercussions of this p r o p e r t y on t h e characteristics of 6 and 0_ themselves, we solve Eqs. ( 2 1 ) by the

O c.

method of Fourier t r a n s f o r m s . Defining

(26a)

(26b)

we obtain by a s t r a i g h t f o r w a r d algebra :

§2( W ) = ^ Q «2 " ® o( w ) + ^ f2( U ) ) ( 2 ? )

and

/ 00

du) ^e"^{1 U ) t} e^o(u>)

- 0 0

/

00 ' .

du> e "1 U ) t 02(u»)

- 0 0

e (w) = - Q H* p

F2( W )

o - ' - o r 2 „ , u ( . _N . 2

Q H' H' a3 + 7(QH' a-B) + u + iw (7 + B - QH' a)

o 2 0 ' H o

(28)

For simplicity we have normalized our time scale so that c = 1. From relation (28) we can evaluate th«

noting that from Eq. (16a) one has

relation (28) we can evaluate the autocorrelation function of 6o(u>),

< F2(u>) F2(u)') > = q* ^ 6(u) + W ) (29)

By switching back to the time variables one finally obtains :

f„s2 |t-t'|

2n Z y-oo (K + a) ) + u" L' 2 2 (QHoP) 2 rco e 1 '

< e (t) 0 (tO O „_ I J1) = 2 q„ / dw~ ^(V Y~2~ , J. ² 2

(30)

where we set

K = Q2 H' HI ap + Y (Q H' a - B) o / o

L = -y +' B - Q IT Of

(31)

- 2 1 -

T h e integral in Eq. ( 3 0 ) can be performed b y t h e calculus of residues.

T h e result is

Q H'3 ,

< 0 ( t ) 6 ( f ) > = — ; r r — q .

° ° J L | (L + 4K) 2

1 - u ^ t - t1) 1 - u ( t - t1)

— e + e

2w 1 2u,

(32)

Here u)^, u^ are t h e singularities of t h e i n t e g r a n d of Eq. ( 3 0 ) :

^ •= \ [ L + ( L2 + 4 K )1 / 2]

u>2 = \ [ | l | - ( L2 + 4 K )1 / 2]

2

If L + 4K > 0 , expression ( 3 2 ) describes t h e superposition of two decaying exponentials with d i f f e r e n t characteristic times. T h i s is to be contrasted with t h e O r n s t e i n - U h l e n b e c k process, whose auto- correlation function is a single decaying exponential ( c f . Eq. ( 2 5 ) ) . If the e f f e c t i v e noise acting on t h e equation f o r 6q were a Gaussian white noise, t h e autocorrelation function of 6 would be of this latter t y p e . o T h e d i f f e r e n c e with the present situation, Eq. ( 3 2 ) , is t h e r e f o r e due to the fact t h a t t h e e f f e c t i v e noise acting on 0q is not a white noise, which in t u r n implies t h a t 6 is a n o n - M a r k o v i a n process. o

Fig. 1 illustrates the d i f f e r e n c e between t h e power spectra of the two t y p e s of stochastic process. Both spectra are characteristic of red noise processes which are commonly observed in atmospheric phenomena (Gilman et a l , 1962; Hasselmann, 1 9 8 1 ) . However, in t h e

0)

Fig. 1.- Power spectrum associated with the autocorrelation function of the mean surface temperature. Eq. (32) (curve (a)) and an Orstein-Uhlenbeck

2 - 2 - 1

process (curve (b)), for q = 1_ Kcal cm yx . The parameters of the - 2 - 1 model described in Eq. (21) are as follows : Q = 252.76 Kcal cm yr , A = 159.24 Kcal c m "2 y r "1, B = 1.17 Kcal c m "2 y r "1, D = 0.44 Kcal c m "2

y r "and a(x) = 0.38 for x > x , T* = 14.9°C, T * = - 28.2°C, x* = 0.96. 1, S(x) = 1 - 0.477 P„(x), a(x) = 0.697 - 0.00779 P „ ( x ) for x < x 2 s

' S O 4 s

case analyzed in t h e p r e s e n t paper ( c u r v e ( a ) ) t h e spectrum is localized more s h a r p l y in t h e low f r e q u e n c y range as compared to t h e spectrum c h a r a c t e r i z i n g t h e O r n s t e i n - U h l e n b e c k process ( c u r v e ( b ) ) . T h i s reflects t h e additional " f i l t e r i n g " achieved b y t h e spatially in- homogeneous f l u c t u a t i o n s . T h e evaluation of t h e power spectrum of 8 using Eq. ( 2 7 ) , leads to similar conclusions. T h e situation would be altogether d i f f e r e n t f o r L + 4K < 0 , as in this case t h e correlation 2 function of 6q would e x h i b i t damped oscillations. H o w e v e r , f o r the parameter values usually adopted in t h e deterministic analysis of Eqs.

2

( 1 7 ) , L + 4K t u r n s out to be positive. T h i s possibility has t h e r e f o r e to be ruled o u t .

A v e r y i n t e r e s t i n g case aribub when K 0 . One can easily see from t h e deterministic version of Eqs. ( 2 1 ) and t h e definitions ( 3 1 ) t h a t t h e limit corresponds to t h e occurrence of a climatic t r a n s i t i o n in the form of bifurcation of new branches of solutions of t h e initial non-

* )

linear equations . From Eq. ( 3 2 ) it t h e n follows t h a t t h e second exponential acquires a long time tail and a d i v e r g e n t amplitude. In other w o r d s , a climatic change appears to be "signalled" by an enhancement of t h e amplitude and lifetime of- t h e correlation f u n c t i o n of t h e fluctuations of the climatic v a r i a b l e s . T h i s is reminiscent of t h e phenomenon of critical divergencies familiar from phase t r a n s i t i o n s . A c t u a l l y , this conclusion is characteristic of a mean-field t h e o r y of phase transitions ( M a , 1 9 7 6 ) , w h e r e b y only t h e effect of long w a v e - length spatial fluctuations is t a k e n into account.

* ) Note t h a t f o r t h e parameters values f i t t i n g t h e p r e s e n t climate t h e case K -»• 0 cannot be realized f o r t h e e n e r g y balance model corresponding to Eqs. ( 1 7 ) .

5. D I S C U S S I O N

O u r principal goal in this work was to incorporate into t h e description of fluctuations in climate dynamics c e r t a i n physical constraints suggested by t h e statistical t h e o r y of i r r e v e r s i b l e processes. T h i s was achieved b y a p p l y i n g a f l u c t u a t i o n - d i s s i p a t i o n t y p e of relation to t h e random forces appearing in t h e deterministic balance equations. C l e a r l y , t h e above idea holds f o r q u i t e general situations and can t h u s be applied to complex sets of coupled evolution equations, as long as t h e local description of the processes involved remains a legitimate approximation. In the p r e s e n t paper however we p r e f e r r e d to keep t h e formalism as simple and explicit as possible, and f o r t h i s reason we c a r r i e d out t h e main p a r t of t h e analysis f o r t h e p a r t i c u l a r - although quite r e p r e s e n t a t i v e - c l a s s of z o n a l l y - a v e r a g e d e n e r g y balance models. T h i s provided us with extensive information on t h e c h a r a c t e r i s - tics of t h e fluctuations of t h e t e m p e r a t u r e f i e l d .

We have f i r s t shown t h a t when a Legendre expansion of t h e noise and t e m p e r a t u r e field is performed t h e r e is no random force associated with t h e equation for t h e globally averaged surface tem- p e r a t u r e . As a b y p r o d u c t , in t h e framework of a two-mode t r u n c a t i o n , it follows t h a t the noise source

F ( x , t ) = F2( t ) P2( x )

is more important in high latitudes ( P² - » 1 ) t h a n in equatorial latitudes ( P2 1 / 2 ) . T h i s might provide a plausible i n t e r p r e t a t i o n for t h e observations (see f o r instance L e i t h , 1978 and q u i t e r e c e n t l y , N o r t h et a l , 1982).

- 2 5 -

nr

A second result of i n t e r e s t is t h a t t h e e f f e c t i v e noise acting on t h e equation for t h e globally a v e r a g e d surface t e m p e r a t u r e 6 is an

o

O r n s t e i n - U h l e n b e c k r a t h e r than a Gaussian white noise. As a r e s u l t , 6 o is a n o n - M a r k o v i a n process. I t also displays t h e characteristics of red noise and is picked around tu = 0 more s h a r p l y t h a n if t h e e f f e c t i v e noise were w h i t e . These f e a t u r e s should play an important role in t h e i n t e r p r e t a t i o n of power spectra associated with climatic v a r i a b i l i t y .

F i n a l l y , we have seen t h a t a t r a n s i t i o n in t h e climatic system in t h e form of a bifurcation of new branches of solutions is reflected by t h e emergence of p e r s i s t e n t , high amplitude correlations. We believe t h a t this p r o p e r t y should provide a useful index of climatic change.

T h e specific application considered in Section 4 was limited to a linear response analysis around t h e p r e s e n t climate. On t h e o t h e r hand t h e fluctuation-dissipation theorem of the second kind utilized in the present work should remain valid in t h e nonlinear range since, as we repeatedly emphasized, it reflects t h e short range c h a r a c t e r of t h e fluctuation sources. It follows t h a t t h e results of Sections 2 and 3 provide also t h e basis of an analysis in which t h e multiplicity of solutions of t h e f u l l y nonlinear model, Eqs. ( 1 7 ) , is t a k e n into account.

I T h i s would allow us to investigate for instance, the passage times between t h e d i f f e r e n t climatic states (see Nicolis and Nicolis, 1981 f o r the estimation of these times in a zero-dimensional model). M o r e o v e r , it would be h i g h l y desirable to set up a suitable description of localized, short wavelength f l u c t u a t i o n s . We will r e p o r t on these problems in f u t u r e w o r k .

ACKNOWLEDGEMENT

T h i s work is s u p p o r t e d , in p a r t , by the EEC under contract n ° C L I - 0 2 7 - B ( G ) .

R E F E R E N C E S

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F O X , R . F . a n d U H L E N B E C K , G . E . , 1970. C o n t r i b u t i o n s to n o n - e q u i l i b r i u m t h e r m o d y n a m i c s . I . T h e o r y of h y d r o d y n a m i c a l f l u c t u a - t i o n s . P h y s . F l u i d s , 13 : 1 8 9 3 - 1 9 0 2 .

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