Fluctuation-Dissipation Theorem and Intrinsic Stochasticity of Climate.

1L NUOVO CIMENTO VoL. 8 C, N. 3 Maggio-Giugno 1985

Fluctuation-Dissipation Theorem and Intrinsic Stochasticity of Climate.

C. ]-'~ICOLIS

Institut d'Adronomie Spatiale de Belgiq~te 3 Avenue Circcdaire, 1180 Bruxelles, Belgium J . P . B o o ~ a n d G. ~TICOLIS

_~acuItd des Sciences de l' Universitd Libre de Bruxelles Campus Plaine (C.P. 231), 1050 Bruxelles, Belgium (ricevuto il 12 Febbraio 1985)

S n m m a r y . - - IrL climate dynamics the effect of internally generated

fluctuations is usually described by augmenting the balance equations through the addition of random ]orees. In this paper the properties of these forces arc investigated. A ]luctuation-dissipation theorem relating their covariance matrix to the phenomenological coefficients such as eddy dif- fusivity is proposed. The theorem is subsequently used to identify the statistical properties of the climatic variables themselves, and thus to characterize climatic variability from the standpoint of the statistical theory of irreversible proeesscs. Applications to a simple thermal con- vection problem and to a zonally averaged energy-balance model arc presented; the possibility of experimental verification is discussed.

PACS. 92.60. - Meteorology.

l . - I n t r o d u c t i o n .

S t o c h a s t i c analysis, i n c o r p o r a t i n g statistical fluctuations in t h e description of climatic p h e n o m e n a , is i m p o r t a n t for u n d e r s t a n d i n g climate variability.

E v e r since the classical s t u d y of B r o w n i a n motion b y EINSTEIN (1905) a n d L)~XaEVI~- (1908), it us c u s t o m a r y to c a r r y out such a n analysis b y a d d i n g

2 2 3

224 C. N I C O L I S , J'. P . B O O N and G . N I C O I , I S

random ]orces to t h e p h e n o m e n o l o g i c a l b a l a n c e cquations. One t h u s obtains a set of expressions of t h e f o r m

(1) ~tx~ --~ ]~({x~}) - - d i v J~ + F~(r, t ) .

H e r e {x~} r e p r e s e n t c l i m a t i c v a r i a b l e s (~) like for i n s t a n c e t e m p e r a t m ' c a n d c o n v e c t i o n v e l o c i t y fields, 1~ arc t h e solcree or sink t e r m s , J~ t h e flux of x~

a n d ~ a t h e r a n d o m force a s s o c i a t e d to x~.

T h e p r o b l e m of c l i m a t i c v a r i a b i l i t y c a n be f o r m u l a t e d as finding t h e second- 9 rod h i g h e r - o r d e r s t a t i s t i c s of t h e c l i m a t i c v a r i a b l e s . According to eq. (1), this should b e possible if one could d e t e r m i n e 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 r a n d o m forces. H o w e v e r , this is n o t a n e a s y t a s k in general. True, if climate d y n a m i c s were a n e a r - e q u i l i b r i u m p h e n o m e n o n , it would reduce to a s t r a i g h t - f o r w a r d p r o b l e m since, t h e n , t h e r a t e e q u a t i o n s c a n be lincarized a n d an i m p o r t a n t result of s t a t i s t i c a l m e c h a n i c s , t h e fluctuation-dissipation theorem, p r o v i d e s us w i t h explicit expressions for t h e c o v a r i a n c c m a t r i x a n d all f u r t h e r p r o p e r t i e s of 2'~.

L e t us outline t h e m a i n steps of t h e p r o c e d u r e l e a d i n g to t h e fluctuation- d i s s i p a t i o n t h e o r e m in n e a r - e q u i l i b r i u m s y s t e m s .

i) F i r s t , one writes t h e e n t r o p y b a l a n c e a s s o c i a t e d with eqs. (1). :From i r r e v e r s i b l e t h e r m o d y n a m i c s , it is k n o w n t h a t one h a s t h e g e n e r a l f o r m (-~)

dS/d$ ~ e n t r o p y flux + e n t r o p y p r o d u c t i o n ,

where t h e e n t r o p y p r o d u c t i o n , P, d e t e r m i n e s t h e r a t e of dissipation p e r unit t i m e a n d a s s u m e s t h e following r e m a r k a b l e f o r m :

(2) 1 ' - ~ ~ J~ X .

H e r e J~ a r e t h e fluxes a s s o c i a t e d w i t h t h e v a r i o u s i r r e v e r s i b l e processes t a k i n g place in t h e s y s t e m , such as h e a t a n d m a s s t r a n s f e r , viscous dissipation, ...;

a n d X~ r e p r e s e n t t h e t h e r m o d y n a m i c forces, or c o n s t r a i n t s , giving rise to these fluxes: t e m p e r a t u r e or c o n c e n t r a t i o n g r a d i e n t s , stress tensor, ....

ii) W h e n the d i s s i p a t i v e processes p r e s e n t in t h e s y s t e m are identified, t h r o u g h eq. (2), one is able to i n c o r p o r a t e t h e effect of t h e fluctuations. Spe- cifically, one first observes t h a t c o n s e r v a t i o n of x~ in t h e absence of sources

(1) K. HASSELMANX: Tellus, 28, 473 (1976).

(2) I. Pmc-oc-I~-)~: l~troduction to Thermodynamics o] Irreversible Processes (John Wiley, New York, ~ . Y., 1967).

F L ] r C T ] ~ A T I O N - I ) I S S I P A T I O N T H E O R : E M A N D I N T R I R ' S I C S T O C I I A S T I C I T ~ ~ O F C L I M A T ] ' ~ 225 or sinks requires t h a t (see eq. (1))

(3)

So j~ r e p r e s e n t t h e fluctuating fluxes, w h i c h a r e to be a d d e d to the a b o v e - m e n t i o n e d d e t e r m i n i s t i c ones. l~otice t h a t eq. (3) i m p o s e s on the r a n d o m forces 2'~ t h e severe r e s t r i c t i o n t h a t o n l y i r r e v e r s i b l e processes d i s s i p a t i n g e n e r g y c a n give rise to a r a n d o m force. F o r i n s t a n c e , if t h e flux J~ c o r r e s p o n d s to e n e r g y or m a s s t r a n s f e r v i a l a m i n a r flow, t h e n J~ = 0 a n d F~ = 0. ]~hrrther e x a m p l e s of t h e i m p o r t a n t role of d i s s i p a t i o n will be seen in t h e s u b s e q u e n t sections.

iii) Using cqs. (3), one c a n n o w w r i t e eqs. (1) in t h e f o r m

(4)

where j : d a r e t h e n o n d i s s i p a t i v e fluxes. One t h e n requires t h a t in t h e l i m i t of infinitely long t i m e , t h e p r o b a b i l i t y d i s t r i b u t i o n of t h e v a r i a b l e s {x~} reduces to t h e e q u i l i b r i u m d i s t r i b u t i o n , as r e q u i r e d b y s t a t i s t i c a l m e c h a n i c s (3,~).

T h e m a j o r conclusion e m e r g i n g f r o m such a n a n a l y s i s is t h a t , n e a r equi- l i b r i u m , t h e f l u c t u a t i n g fluxes define a G a u s s i a n w h i t e noise in t i m e , a n d a r e fully u n c o r r e l a t e d in space

(5) { <j~,(r, t)> = o,

<jc,(r,

t'))

This reflects t h e f a c t t h a t f l u c t u a t i o n s originate as localized small-scale e v e n t s . Oll t h e o t h e r h a n d , as a rule t h e v a r i a b l e s {x~} t h e m s e l v e s display correlations b o t h in space a n d t i m e as c a n be seen f r o m t h e solution of t h e s t o c h a s t i c dif- f e r e n t i a l e q u a t i o n s (1) or (4).

T h e s t u d y of c l i m a t e d y n a m i c s i n c o r p o r a t e s a n e n s e m b l e of highly non- l i n e a r processes involving a large n u m b e r of coupled v a r i a b l e s which evolve u n d e r conditions f a r f r o m t h e r m o d y n a m i c equilibrium. E v e n at l a b o r a t o r y scale, t h e r e processes are p o o r l y u n d e r s t o o d . Therefore, one would e x p e c t a t first sight t h a t t h e t h e o r y of f l u c t u a t i o n s o u t l i n e d a b o v e should b r e a k down.

This is c e r t a i n l y t r u e as f a r as 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 climatic v a r i a b l e s (x~} are concerned. I n d e e d , one e x p e c t s t h a t u n d e r f a r f r o m equilibrium con- ditions (x~} s h o u l d e x h i b i t correlations with m u c h longer range, in space a n d t i m e t h a n a t equilibrium. On t h e o t h e r h a n d t h e s t o c h a s t i c forcing, to which t h e fluctuations of (x~} c o n s t i t u t e t h e response, is still likely to originate in t h e (3) L. O.'~SAGER: Phys. Rev., 37, 405 (1931); 38, 2265 (1931).

(4) L . D . LAN])A1Y ~md E. ]~. LIFStIITZ: Fluid Mechanics (Pergamon Press, Oxford, 1959), Chapt. 17.

226 C. N I C O L I S , J . P. BOON and G. N I C O L I S

f o r m of localized, small-scale events which could n o t (, sense >> t h e overall con- s t r a i n t s d r i v i n g t h e d y n a m i c s a w a y f r o m equilibrium. }~or i n s t a n c e , t h e space- t i m e scales of e m e r g e n c e of a surface t e m p e r a t u r e a n o m a l y are s h o r t c o m p a r e d to t h e s o m e t i m e s global scale of t h e c l i m a t i c r e s p o n s e i n d u c e d b y t h e t e m - p e r a t u r e a n o m a l y . I t is, therefore, p l a u s i b l e t h a t m o s t of t h e p r o p e r t i e s of t h e r a n d o m fluxes m e n t i o n e d a b o v e - - p a r t i c u l a r l y t h e l a c k of correlations in t i m e

a n d space, see eq. ( 5 ) - - w o u l d still hold.

T h e s e a r c h of such a generalized fluctuation-dissipation theorem is t h e m a i n p u r p o s e of t h e p r e s e n t work. T h i s p r o g r a m s h o u l d n o t b e c o n f u s e d with a r e c e n t l y suggested 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 which involves d i r e c t l y t h e p r o p e r t i e s of t h e c l i m a t i c v a r i a b l e s r a t h e r t h a n those of t h e r a n d o m forces (5.~).

We consider in t h e sequel t w o t y p e s of p r o b l e m s of c l i m a t i c i n t e r e s t , for which we m a k e use of t h e s t a t i s t i c a l t h e o r y of i r r e v e r s i b l e processes to s t u d y t h e c o n s t r a i n t s i m p o s e d on t h e fluctuations. In sect. 2 we a n a l y s e t h e r m a l convection, a n i m p o r t a n t p a r t of t h e circulation of a t m o s p h e r e s a n d oceans, on t h e basis of t h e n o n l i n e a r B o u s s i n e s q equations (~) s u p p l e m e n t e d w i t h ap- p r o p r i a t e r a n d o m forces. We i n t r o d u c e a d d i t i o n a l simplifications, first sug- gested b y SALTZMAN (s) a n d LOlCE~Z (~), which a m o m l t to k e e p i n g o n l y three F o u r i e r m o d e s of t h e original p r o b l e m . T h e inclusion of fluctuations leads to a n a u g m e n t e d set of e q u a t i o n s r e m i n i s c e n t of s y s t e m s s t u d i e d r e c e n t l y b y o t h e r a u t h o r s (~o,~1). These a u t h o r s a d o p t e d t h e ad hoc a s s u m p t i o n t h a t t h e noise s t r e n g t h s are i d e n t i c a l in all t h r e e equations. H e r e we use a m o r e basic a n d m o r e g e n e r a l e p p r o a c h to derive explicit e x p r e s s i o n s for 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 r a n d o m forces u s i n g a generalized fluctuation-dissipation t h e o r e m . Consequences of t h e r e s u l t s oll t h e s t a t i s t i c s of t h e v e l o c i t y a n d t e m p e r a t u r e fields are a n a l y s e d in sect. 3; these r e s u l t s e x h i b i t e x p e r i m e n t a l l y verifiable features. I n sect. 4 we c a r r y out a s i m i l a r a n a l y s i s on a different p r o b l e m , n a m e l y a n e n e r g y b a l a n c e m o d e l i n c o r p o r a t i n g t h e surface albedo feedback. G e n e r a l conclusions are d r a w n in sect. 5.

2. - I n t e r n a l f l u c t u a t i o n s i n t h e L o r e n z - S a l t z m a n m o d e l for R a y l e i g h - B e n a r d c o n v e c t i o n .

A full-scale a n a l y s i s of c l i m a t i c v a r i a b i l i t y involves, p c r force, a v e r y large m u n b c r of v a r i a b l e s coupled t h r o u g h b a l a n c e e q u a t i o n s of t h e t y p e (1) or (4).

(5) C.E. L~zIz1~: J. Atmos. Sei., 32, 2022 (1975).

(6) C . E . L~ITIt: Nature (Loud.on,) 276, 352 (1978).

(~) S. C~A~DRAS~KHA~: Hydrodynamic a~d Hydromag~etic Stability (Oxford University Press, London, 1961).

(8) B. SALTZ~AN: J. Atmos. Sci., 19, 329 (1962).

(9) E . N . LoR~=~z: J. Atmos. Sci., 20, 130 (1963).

(10) A. SUT~A: J. Atmos. Sci., 37, 245 (1980).

(11) A. ZIPPELIVS and ~ . Li)CK~: J. State Phys., 24, 345 (1981).

F L I I C T U A T I O N - D I S S I P A T I ( ) _ N " T H E O R E M A N D I N T R I N S I C S T O C H A S T I C I T Y O F C L I M A T : E 227 l~he s t u d y of s u c h a l a r g e set of e q u a t i o n s in t h e p r e s e n c e of s t o c h a s t i c p e r - t u r b a t i o n s c a n only be p e r f o r m e d n u m e r i c a l l y : this is a n e n o r m o u s t a s k , a n d t h e c h a n c e s t h a t it could l e a d to a c l e a r - c u t identificution of t h e m a j o r effects of t h e f l u c t u a t i o n s are v e r y slim. F o r t h i s r e a s o n we focus in this section on a specific p h e n o m e n o n t a k i n g p a r t in c l i m a t e d y n a m i c s , n a m e l y t h e r m a l con- vection. I n a d d i t i o n to its p r e s e n c e iI~ t h e c i r c u l a t i o n p a t t e r n s of t h e a t m o s - p h e r e a n d of t h e oceans, c o n v e c t i o n is also i m p o r t a n t b e c a u s e it c o n s t i t u t e s t h e p r o t o t y p e of s y s t e m s c a p a b l e of h a v i n g u n p r e d i c t a b l e b e h a v i o u r . I t cap- tures, therefore, a l b e i t on a m o r e m o d e s t scale, one of t h e e s s e n t i a l c h a r a c - teristics of a t m o s p h e r i c d y n a m i c s a n d c l i m a t e . I n f a c t , t h e first m o d e l showing how u n p r e d i c t a b i l i t y c ' m arise f r o m d e t e r m i n i s t i c d y n a m i c s was c o n s t r u c t e d b y LOliENZ (~) in a n a t t e m p t to s i m p l i f y t h e a n a l y s i s of t h e r m a l convection.

Consider a h o r i z o n t a l fluid l a y e r h e a t e d f r o m below a n d s u b j e c t to t h e g r a v i t a t i o n a l field. T h e m a s s d e n s i t y ~o is a s s u m e d to b e c o n s t a n t e x c e p t in t h e t e r m of the m o m e n t u m b a l a n c e e q u a t i o n e x p r e s s i n g t h e coupling b e t w e e n the v e r t i c a l v e l o c i t y field ~nd t h e d e n s i t y . We also neglect e n e r g y dissipation arising f r o m d i s s i p a t i v e stresses on t h e s y s t e m . These are t h e basic a s s u m p - tions c o n t a i n e d in t h e B o u s s i n e s q e q u a t i o n s for c o n v e c t i o n (v), i . e .

(6a) (6b) with

( 6 c )

(6d)

OoC=(O,T + ( v . V ) T ) : AV2T + FT, oo(~v + (v'V)v) : - - V p - - ~ogi + v V ~ v -~- F,

V - v = 0 ,

__ ~ , ( ~ ) = Oo(1 - ~ ( T - - T o ) ) .

H e r e T, v, o a n d p are t h e t e m p e r a t u r e , velocity, m a s s de~lsity a n d pres- slu-e fields, r e s p e c t i v e l y ; oo a n d To d e n o t e reference v a l u e s a t t h e lower b o u n d a r y of t h e l a y e r , C~ is t h e specific h e a t a t c o n s t a n t p r e s s u r e a n d ~ t h e coefficient of t h e r m a l e x p a n s i o n , ;. a n d ~ are t h e h e a t c o n d u c t i v i t y a n d s h e a r v i s c o s i t y coefficients, r e s p e c t i v e l y ; g is t h e a c c e l e r a t i o n of g r a v i t y , I. t h e u p w a r d - d i r e c t e d u n i t v e c t o r a n d FT a n d F~ t h e r a n d o m forces describing t h e i n t e r n a l l y gen- e r a t e d sources of f l u c t u a t i o n s (see eq. (11)).

W e first discuss t h e s t r u e t m ' e of t h e r a n d o m forces. 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 , see eq. (3), c o n s e r v a t i o n of e n e r g y a n d m o m e n t u m requires 2'z a n d F~ to be t h e divergence of a v e c t o r a n d of a t e n s o r field, r e s p e c t i v e l y , j a n d s :

{

(7) F ~ ( r , t) = - - V . s ( r , t) .

Moreover, n e a r t h e r m o d y n a m i c e q u i l i b r i u m , t h e p r o p e r t i e s of j a n d s are es-

~ C. N I C O L I S , J . P . B00.% ~ ,%rid G. N I C O L I S

sentially g i v e n b y eqs. (5). Statistic,tl m e c h a n i c s allows one to i d e n t i f y t h e s t r u c t u r e of t h e ma.trix in eq. (5), a n d one h a s (4)

(8) a n d

(9)

<1(,-, t)> = o ,

<j~(r,

t)jt(r',

< s ( r , t)> = o ,

@ / ~ ( r , t ) . % . , ( r ' , t ' ) > o - '

H e r e T is t h e L c m p e r a t u r e of the r e f e r e n c e s t a t e a r o u n d which fluetlmtions arc g e n e r a t e d , k B is B o l t z m a n n ' s c o n s t a n t , 5 is t h e K r o n e c k e r s y m b o l , a n d t h e b r a c k e t s d e n o t e an e q u i l i b r i u m e n s e m b l e a v e r a g e . The p r e s e n c e of delta-func- tions in b o t h space a n d t i m e reflects t h e f a c t t h a t fluctuations are generated b y small scale, localized e v c n t s w i t h o u t m e m o r y . Note, h o w e v e r , t h a t t h e r a n d o m forces, ~'T a n d F. t h e m s e l v e s are not d e l t a - c o r r e l a t e d in space, while t h e y e x h i b i t G a u s s i a n white noise in time. As will be seen below, despite t h e locality of t h e r a n d o m fluxes, t h e s t a t e v a r i a b l e s t h e m s e l v e s ( I ' a n d v) will e x h i b i t h i g h l y nonlocal correlations in b o t h space a n d time.

I n t h e original h y d r o d y n a m i c f l u c t u a t i o n t h e o r y of L a n d a u a n d Lifshitz, t h e reference s t a t e is t h e r m o d y n a m i c equilibrium. I i e r e we are i n t e r e s t e d in b e h a v i o u r f a r f r o m e q u i l i b r i u m a n d , p a r t i c u l a r l y , in t h e f a c t t h a t eqs. (6) l)resent a sequcnce of b i f u r c a t i o n s f r o m r e g u l a r to periodic convection a n d u l t i m a t e l y to c h a o t i c c o n v e c t i o n (for a s u r ~ e y see, e . g , ref. U')). As e x p l a i n e d in t h e i n t r o d u c t i o n , e v e n u n d e r these c i r c u m s t a n c e s , we still e x p e c t fluctuation sources to arise f r o m localized, small-scale events. So (see also ref. (~3)), we a s s u m e t h a t eqs. (8) a n d (9) c a n be e x t e n d e d to t h e n o n l i n e a r d o m a i n of ir- reversible processes, with a n o n e q u i l i b r i u m reference s t a t e , t a k e n h e r e as t h e l a m i n a r c o n v e c t i o n s t a t e (i.e. t h e s t a t e j u s t b e y o n d t h e first b i f u r c a t i o n point).

I n t r o d u c i n g dimensionless q u a n t i t i e s into eqs. (6a), (6b) a n d (7), one o b t a i n s (10a) ~,.0" = R w * - - (v*. V*) 0* + V*~0 * - - V*. j * ,

(10b) ~ , * v * - - P ( V * p * --' V*-Ov * : - O * I ~ ) - - ( v * . V * ) v * - - V * . s * .

H e r e 0* is the dimensionless m e a s u r e of t h e t e m p e r a t u r e d e v i a t i o n f r o m the (1..) H . L . S w I N ~ Y and J. P. G-OLLUB (Editors) : Hydrodynamic Instabilities and Tran- sition to Turbulence (Springcr-Vcrlag, Berlin, 1981).

(a3) C. NICOLIS: On a new ]luctq~ation-dissipation theorem in climate dynamics, in New Perspectives in Climate Modelling, edited by A. B1.;RGER and C. NICOLIS (Rcidel, Dor- drecht, 1983).

F L U C T U A T I O N - D I S S I P A T I O N Ttl].;OlgiEM A N D I N ' r l / I N S I C STOCtIASTICITY O F CLIMAT:F, 229 linear profile, To--fiz. The asterisks indicate t h a t both d e p e n d e n t variables (O*,w*, v*) a n d i u d e p e n d c n t variables (r*, t*) arc dimensionless; w* is the vertical c o m p o n e n t of v*; R a n d 1' are the Raylci~'h 'rod P r a n d t l n u m b e r s , respectively,

(11) R - : ~flgd'(vx) -~ , P : : ~'~-~,

where u is t h e k i n e m a t i c viscosity (v----rtoo~), u t.he t h e r m a l diffusivity (u -- ),(ooC~)--~), d the height of the l a y e r a n d fl tile average t e m p e r a t m ' e gn'adient across the fluid layer, j* a n d ~* are the sealed noise fields given b y

{ J* = J~ v>:: Co C~,) -~ ,

8~'2 0 o ~ 2 - - i

~* ~ (~ ) 9

E q u a t i o n s (10a), (10b) are dilticult to h a n d l e because of t h c presence of the highly nonlinear t e r m s (v*'V*)O* and (v*.V*)v*, r e s p e c t i v e l y ; so we, proeeed b y p e r f o r m i n g t h e following operations:

i) We neglect the n o n l i n e a r c o n v e c t i o n t e r m s ((v*.V*)v*) in eq. (10b), which a m o u n t s to a s s u m i n g t h a t the P r a n d t l n u m b e r P is large (note t h a t c e r t a i n bifurcation schemes m a y be excluded f r o m t h e anMysis b y this assump- tion).

it) We a p p l y twice the curl o p e r a t o r ou eq. (10) a n d p r o j e c t the r e s u l t on t h e z-axis. The pressure t e e m t h u s vanishes ~nd one is left with an e q u a t i o n which involves only the v e r t i c a l c o m p o n e n t the velocity w* a n d t h e excess t e m p e r a t u r e .

iii) We focus ou a p a r t i c u l a r t y p e of c o n v e c t i v e state, c h a r a c t e r i z e d b y a two-dimensional roll p a t t e r n . As a r e s u l t all variables now depend only on x* a n a z*.

iv) We a c c o u n t for the large extension of the s y s t e m along the horizontal direction by imposing periodic b o u n d a r y conditions, a, n d we consider t h a t the l a y e r is stress-free a t tile two vertiea, l b o u n d a r i e s , so t h a t (7)

( 1 3 )

0 . ( o ) - - 0 . ( 1 ) = o , w * ( 0 ) - - w * ( 1 ) = o ,

~'-w,'@z*'- = o @ z* = 0, z * = l

Consequently 0* a n d w* can be F o u r i e r e x p a n d e d as

( ] 4 )

== O,n,(t ) exp [imk*x*] sin (nnz*) , 0"(:~:*, z*, t*) ~. * *

m n

w*(x*, z*, t*) --- ~ w*,(t*) exl) [iml~*x*j sin ( n m * ) .

mn

230 c . N I C O L I S , J . P . BOO.N ~ l ) . d . G . N I C O L I S

k* represents the c h a r a c t e r i s t i c wave n u m b e r of t h e convection cells (k* will be considered as a knowil q u a n t i t y , d e t e r m i n e d b y t h e geometrical conditions).

K e e p i n g only the first n o n t r i v i a l modes in t h e analysis which a m o u n t here to w~, 0o* a n d 0~*, one obtains a f t e r some r a t h e r h e a v y algebra (see the ap- pendix)

2~--- a X + a Y

+r

( ] 5 ) F = r x - - Y - - x g . i-

r

Z - - - b Z - ~ X Y + r

H e r e X , u Z c o r r e s p o n d to w*~, 0~* a n d 0o* , re:~pectively, b y the scaling indicated in eq. (A.3); a is the P r a n d t l n u m b e r , r the r a t i o of t h e Rayleigh n u m b e r to its critical v a l u e a n d Cx, Cr, Cz are effective r a n d o m forces definint~ a white noise, with v a r i a n c e given in eqs. (A.6)-(A.8)

2 (~/~r I ,

(16) <r162 -~ q, ,~ 5(t - - t ) i, j =- X , I ~, Z .

E q u a t i o n s (15) are the L o r e n z - S a l t z m a n equations including now internal- f l u c t u a t i o n t e r m s . I t is known t h a t , a l t h o u g h t h e L o r e n z - S a l t z m a n model does not describe q u a n t i t a t i v e l y t h e b e h a v i o r of e x p e r i m e n t a l convection at v e r y large R, it contains the essential q u a l i t a t i v e features. Most i m p o r t a n t f r o m our s t a n d p o i n t is t h a t one is in a position to fully express t h e constraints imposed on t h e effective r a n d o m forces by t h e s t a t i s t i c a l t h e o r y of irreversible processes. The p r o c e d u r e leading to eqs. (16) is outlined in the appendix.

i t should be stressed t h a t t h e s t r e n g t h s q~, q~, q~ are unequal. This will not affect q u a l i t a t i v e l y the line'~r-response properties described in the n e x t section, b u t is likely to introduce new features in t h e nonlinear domain. Indeed, in t e r m s of s t o c h a s t i c t h e o r y t h e s y s t e m does not satisfy the potential conditions, because of the difference in t h e s t r e n g t h of the r a n d o m forces. As a result, t h e role of fluctuations will not necessarily be r e s t r i c t e d to inducing aperiodic flip-flop between various s i m u l t a n e o u s l y stable a t t r a c t o r s (stable stationary, periodic or chaotic states) p r e d i c t e d b y the d e t e r m i n i s t i c equations. I n fact, fluctuations m a y now m o d i f y t h e conditions of a p p e a r a n c e of these a t t r a c t o r s , b y shifting the t h r e s h o l d values of t h e various p a r a m e t e r s like t h e Rayleigh n u m b e r , as well as t h e i r location a n d s t r u c t u r e in p h a s e space.

3 . - L i n e a r r e s p o n s e t h e o r y .

L e t (X,, Y,, Z ) be a s t e a d y - s t a t e solution of eqs. (15) in the a b s e n c e of fluctuations. W h e n r < 1, t h e r e is only one such solution t h a t is a s y m p t o t i c a l l y

FL1JCTL'ATION-DISSIPATION TII~0R)~M AND I N T R I N S I C STOCHASTICITY OF CLIMAT:E ~.31

s t a b l e (9,14)

(17) X = Y : - Z - - - - 0 .

A c c o r d i n g t o t h e definitions of W*l, 01" a n d 0o* , t h e s t e a d y - s t a t e s o l u t i o n g i v e n b y (17) c h a r a c t e r i z e s a p u r e l y c o n d u c t i v e s t a t e (no c o n v e c t i o n ) . F o r r > 1, (17) r e m a i n s a m a t h e m a t i c a l s o l u t i o n , b u t t h e c o r r e s p o n d i n g s t a t e h~s n o w lost its s t a b i l i t y . T w o n e w s t a b l e b r a n c h e s of s o l u t i o n s e m e r g e in t h i s r a n g e w h i c h b i f u r c a t e f r o m (17) a t t h e c r i t i c a l v a l u e r ---- ] a.nd a r c g i v e n b y (18) X ~ Y : i ( b ~ ) t ~ Z - ~ , 8 ~ - - r - - ] .

T h e s t e a d y - s t a t e as defined b y s o l u t i o n s (18) is t h e c o n v e c t i o n s t a t e w h i c h in t u r n loses its s t a b i l i t y a t a s e c o n d b i f u r c a t i o n p o i n t r ---- rr (~)

(19) r:, == o'(~ l_ b + 3)((~ - - b - - 1) -1

w i t h t h e c o n d i t i o n t h a t ~ be sufficiently large, a > b - - ] . A t t h i s p o i n t , t h e b i f u r c a t i o n leads to u n d a m p e d o s c i l l a t o r y solutions w i t h f r e q u e n c y

(20) ~ ---- ~: (b(a + rT)) 89 .

T h e s e s o l u t i o n s b i f u r c a t e s u b c r i t i c a l l y for r < rT a n d arc, t h e r e f o r e , u n s t a b l e (15,16).

F i g u r e 1 s u m m a r i z e s t h e a b o v e - d e s c r i b e d b i f u r c a t i o n p h e n o m e n a . T h e s e c o n d b i f u r c a t i o n p o i n t m a r k s t h e t r a n s i t i o n t o w a r d s t h e o n s e t of c o m p l e x d y n a m i c s

s

i I % i

I rT

Fig. l. - Bifurcation scheme of solutions; r denotes the ratio of the Rayleigh number to its critical value at the first bifurcation point.

(14) C. SPAP~OW: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors (Springer-Vcrlag, New York, ~'. Y., 1983).

(15) p . C . MAa-CTIN : in .Proceedings o] the International Con]erence on Statistical Mechanics (North-IIolland, Amsterdam, 1975).

(16) j . H~SDX~- and H. HcCRAc~_'~': The Hop] Bi]ureation and its Applications (Springcr-Verlag, Berlin, 1976).

2 3 2 C. N I C O L I 8 , J . P. BOON a n d o . N1COLIS

in t h e f o r m of chaotic b e h a v i o u r ; the c o r r e s p o n d i n g set of solutions (for r > rz) t h a t appears at finite distance f r o m the s t e a d y state defined by (18) are a s y m p - totically stable. Note also t h a t oscillatory bchaviom' appears already below rT;

indeed when r reaches a v a l u e rv, with 1 < r, < re, d a m p e d oscillations arise whose d a m p i n g rate decreases to zero, as r tends to re, where their f r e q u e n c y reaches the value g2~.. Note however, t h a t no b i f u r c a t i o n occurs at r = rv.

We now investigate t h e b e h a v i o u r of the s y s t e m in the v i c i n i t y of the two b i f u r c a t i o n points. More specifically we consider the statistical properties of t h e t e m p e r a t n r e a n d v e l o c i t y fields a r o m l d r = 1 a n d r = rz. We first linearize eqs. (15) a r o u n d the s t e a d y - s t a t e solution ( X , I z , Z ) . After time- F o u r i e r t r a n s f o r m a t i o n of t h e quantities X, :Y, Z a n d e x , e r , ez, i.e. using

-t-co

x(t) =faro

- - o n

a n d similarly for t h e o t h e r quantities, we o b t a i n from eqs. (15) a n d (16)

(21)

( ~ - - i ~ o ) 2 - - ~ - ~ ,

( z - - r)J? -r (1 - io)) ~., + x Z = r , - - Y X - - X ~ 4 ( b - - i o , ) Z = r

E q u a t i o n s (21) c o n s t i t u t e the set t h a t we now s t u d y for v a r i a b i l i t y a r o u n d t h e trivial state (17) a n d the n o n t r i v i a l state (18) successively.

F o r the sake of conciseness, we r e s t r i c t t h e discussion, w i t h o u t loss of gen- erality, to the b e h a v i o u r of the t h e r m a l m o d e ~o~.

r < 1: Variability around the conduction state, X, = l z = Z = 0. F r o m cqs. (20), one finds t h a t t h e a m p l i t u d e is given b y

(22)

where Po is t h e c h a r a c t e r i s t i c p o l y n o m i a l

(23) Po(z) = z 2 + ((~ + 1)z - - qe, z = - - io~,

with e = r - - I < 0. The explicit expressions for the two modes, i.e. for z+

a n d z_, are easily c o m p u t e d f r o m (23). More interesting is t h e b e h a v i o u r in t h e v i c i n i t y of t h e bifurcation point. Close to r = 1, for small values of e, one obtains

(2~) z _ ~ _ _ - - r -~, z+__~_(q+l) + a e ( a + l ) - z .

We observe t h a t the mode c o r r e s p o n d i n g to z_ exhibits a slower a n d slower

F L L ' C T U A T I O N - ] ) I S S I P A T I O N TIIEOIr A N D I . N T R I N S I C S T O C I I A S T I C I T Y OF C L I M A T E 2 3 3

d e c a y r a t e as r a p p z o a c h e s u n i t y (z_ ~ 0 as e -+ 0). This is the a n a l o g u e of critical slowb~g dowtt, in e q u i l i b r i u m p h a s e t r a n s i t i o n s . On t h e o t h e r h a n d , t h e m o d e corrcspondirtg to z+ keeps a finite d e c a y r a t e , which increases to t h e v a l u e : - I a t r - - 1. I n t h e l i t e r a t u r e of fluid d y n a m i c s z_ a n d z_ a r e r e f e r r e d to as t h e t h e r m a l a n d t h e v o r t i e i t y mod% since t h e y a r c d o m i n a t e d , r e s p e c t i v e l y , b y h e a t c o n d u c t i o n a n d viscous effects W).

An i n t e r e s t i n g q u a n t i t y f r o m t h e e x p e r i m e n t a l view p o i n t as a useful i n d e x of v a r i a b i l i t y is the, p o w e r s p e c t r u m (see, e.g., ref. (,s)). _Noting t h a t

(25) <~+.oCx=,> -~ 1 q~6((o.'~. (,/)

7~

a n d s i m i l a r l y for Cr~ a n d ~.~, we o b t a i n f r o m the, e n s e m b l e a v e r a g e d c o m p l e x c o n j u g a t e s p r o d u c t of (22)

w i t h

it r*q2x q- (w* -- a~)q~

(27) ?<~ = - ^-

( , , ~ - - z ~ ) ( ~ , , ~ - - z ~ . ) '

w h e r e z~_ = 89 (a T 1)4-~ [(a + 1) 2 4- 4as]i a r e t h e r o o t s of t h e p o l y n o m i a l Po(z), eq. (23). T h c s p e c t r u m 7(~ is ,~ s i n g l e - p e a k e d c u r v e e c n t r e d a r o u n d o> = 0.

T h e p e a k i n t e n s i t y r e a d s

(28) y(~ ~- 0) - - - 1 (r~q2x q- a~q~)(a2e~)_ , ,

which d e p e n d s oll t h e P r a n d t l n u m b e r (a), t h e noise s t r e n g t h f a c t o r s (q~, q~) a n d t h e v a l u e of t h e R ~ y l e i g h n u m b e r ('Ii == ( R e - - R ) / R ~ ) . On t h e ~ppro~ch to t h e first b i f u r c a t i o n p o i n t (R = Re) f r o m below (r < 1), t h e p e a k i n t e n s i t y diverges as e -2. H o w e v e r , t h e p e a k i n t e n s i t y is not ~n e~sily accessible q u a n t i t y f r o m t h e e x p e r i m e n t a l p o i n t of view. I n this r e s p e c t a. p r e f e r a b l e q u a n t i t y is t h e integra.ted i n t e n s i t y

-I-z~

y<o, ---jd(o ^r(o,(,o),

(29)

--r

(1~) j . p . BOO.~-: Itydrody~amic instability: structure and chaos, in Scattering Techniques Ap21ied to Supramolec~dar and No~equilibrium Systems ( P l e n u m Pn'~ss, N e w Y o r k , N. Y., 1981).

(is) j . p . B o o n a n d S. Y I e : Molecular Hydrodynamics, (.~r N e w York, . N . Y . , 1980).

16 - 1l Nuovo Oimenlo C.

234 c..-~coz,~s, g. P. BOON and G. NICOLAS w h i c h is e a s i l y c o m p u t e d h e r e f r o m (27). O n e finds

(30) ~(o)__ q~ § r r q ~ §

e a ( a §

So t h e i n t e f f r a t e d s p e c t r u m h a s a r e g u l a r p a r t ( t h e e q u i l i b r i u m v a l u e of 7 (o) at r ~ 0) a n d a c r i t i c a l p a r t t h a t d i v e r g e s like e ~ as t h e first b i f u r c a t i o n p o i n t is a p p r o a c h e d . T h e c r i t i c a l b e h a v i o r of t h e i n t e g r a t e d s p e c t r u m is a conse- q u e n c e of t h e c r i t i c a l s l o w i n g - d o w n of t h e t h e r m a l d i f f u s i v i t y m o d e z_, see (24).

r > 1: Variability around the convective state, X ~ ]z--_-L.v/be, Z, ~ - e . i n s e r t i n g t h e s t e a d y - s t a t e e x p r e s s i o n s (38) i n t o t h e set of eqs. (21) a n d solv- i n g t h e l a t t e r for t h e a m p l i t u d e of t h e t h e r m a l m o d e , one o b t a i n s

(31)

w i t h t h e c h a r a c t e r i s t i c p o l y n o m i a l

(32) _P +(z) : z3 ~,- ((~ ~ b ~- l ) z 2 -f- b(a ~- r ) z -4- 2b~ , z = - - i(o,

w h e r e ~ ~- r - - 1 > 0. T h e r e a r e n o w t h r e e m o d e s t h a t e m e r g e as t h e r o o t s of t h e c h a r a c t e r i s t i c e q u a t i o n /)+(z) ~- 0. W h e n r does n o t e x c e e d t h e v a l u e r , , t h e t h r e e r o o t s a r e real. A l t h o u g h t h e cubic c a n be s o l v e d f o r m a l l y t h e explicit e x p r e s s i o n s of t h e r o o t s a r e n o t v e r y e n l i g h t e n i n g as t o t h e i r p h y s i c a l c o n t e n t . I n t h i s r e s p e c t a c o m p u t a t i o n of t h e r o o t s to first o r d e r in e is v a l u a b l e in o r d e r t o i n v e s t i g a t e t h e m o d e b e h a v i o u r in t h e v i c i n i t y of r ---- 1 w h e n t h e b i f u r c a t i o n p o i n t is a p p r o a c h e d f r o m a b o v e . O n e finds

2ae b((r - - :l ) e (2(~ - - b) e

('~3) Z o ~ g § z ~ ( a § § ( ~ § z 2 ~ - b ( r § '

w h i c h shows t h a t t h e t h e r m a l m o d e (zo) e x h i b i t s c r i t i c a l s l o w i n g - d o w n (zo -+ 0 w h e n e -~ 0) a c c o r d i n g t o t h e s a m e p o w e r l a w as for t h e a p p r o a c h to r ~ 1 f r o m below, eq. (24), b u t w i t h a n a m p l i t u d e f a c t o r twice as l a r g e , due to t h e e m e r g e n c e of a t h i r d m o d e w h e n r > 1.

T h e p o w e r s p e c t r u m for t h e t h e r m a l m o d e is c o m p u l e d a l o n g t h e s a m e lines as in t h e p r e v i o u s case (r < 1) to y i e l d

(34)

w h e r e

C(~ -_-- - < F ~ : ~ , > = r ( + ) ( ~ ) 5 ( ~ + ~') ,

(35) ~(+)(w) : 1 (b~(2 - - r) ~ § w 2) q~ § ( ~ + c~)(b ~ § w~)q~ § be(a 2 § w~)q~

(co~ -t- z~)(~o~ § z'~)(o)~ + z~)

F L U C T U A T I O N - D I S S I P A T I O N TIIEOI~:F~I A.NI) I N T R I N S I C S T O C I I A S T I C I T Y OF CLIMAT:E 2 3 5

with

(36)

zo+zl+z2:cr-[-b-~-l,

ZoZ~ + z~z~ .~- Z~Zo - - b(a + r ) , Zo zl z., = 2(~be .

I n the r a n g e 1 < r < r~, t h e s p e c t r u m y(+)(o~) is a single-peaked f u n c t i o n of o) c e n t r e d a r o u n d ~o = 0, t h u s e x h i b i t i n g t h e s a m e q u a l i t a t i v e s t r u c t u r e as below r = 1. T h e p e a k i n t e n s i t y is g i v e n b y

(37) y ( e ) ( o ) = O ) = l ~ ( r - - 2 ) 2 q ~ c x 2 q ~ . q w

~t~

4be]"

On the a p p r o a c h to t h e b i f u r c a l i o n p o i n t r - - - - 1 f r o m a b o v e , the s t r o n g e s t divergence causes t h e p e a k i n t e n s i t y to blow u p like e --~, in t h e s a m e w a y as w h e n t h e b i f u r c a t i o n p o i n t is a p p r o a c h e d f r o m below (r < 1 ) . . N - o t c also t h e difference in t h e a m p l i t u d e f a c t o r ( c o m p a r e (37) w i t h (28)).

I I e r e a g a i n a m o s t i n t e r e s t i n g q u a n t i t y , f r o m t h e t h e o r e t i c a l view p o i n t as well as f r o m t h e e x p e r i m e n t a l view p o i n t , is t h e i n t e g r a t e d i n t e n s i t y , which is o b t a i n e d b y f r e q u e n c y i n t e g r a t i o n of (35) to yield

(38)

(a b -~- 1)b(a ~- r) 1 ~ 2(r ~-

w i t h ~ ~-- r z - - r , where rT, t h e v a l u e of r a t t h e second b i f u r c a t i o n p o i n t , is g i v e n b y (19).

The i n t e g r a t e d i n t e n s i t y y(+) e x h i b i t s i n t e r e s t i n g f e a t u r e s : it contains r e g u l a r a n d critical p a r t s . T h e r e is a critical p a r t r e-x t h a t diverges when r a p p r o a c h e s t h e first b i f u r c a t i o n p o i n t a n d a n o t h e r critical p a r t ac 5 -1 which blows u p on t h e a p p r o a c h to t h e second b i f u r c a t i o n p o i n t ( r - + r~.). T h e first divergence arises f r o m t h e b e h a v i o u r of t h e t h e r m a l m o d e which b e c o m e s critical w h e n r -+ 1 (see zo (32)) ; the c r i t i c a l b e h a v i o u r in t h e v i c i n i t y of t h e second b i f u r c a t i o n p o i n t h a s its origin in t h e v a n i s h i n g of t h e d a m p i n g of t h e c o n j u g a t e m o d e s t h a t e m e r g e f r o m r ---- r~. At this p o i n t oscillatory b c h a v i o u r sets in as t w o roots of t h e c h a r a c t e r i s t i c e q u a t i o n k)+(z) ~ 0 b e c o m e c o m p l e x conjugates. Corre- s p o n d i n g l y t h e p o w e r s p e c t r u m y~+)(o~) c o n t a i n s one cerLtral liim a ~ d t w o side p e a k s w h e n r exceeds t h e value r~. W h e n r increases, t h e c e n t r a l p e a k b r o a d e n s a n d its line w i d t h reaches t h e v a l u e Aa~ ---- a - I b + 1 a t r ---- rr; s i m u l t a n e o u s l y t h e satellite p e a k s n a r r o w in w i d t h a n d are s y m m e t r i c a l l y s h i f t e d a w a y f r o m t h e c e n t r a l p e a k . A t r -~ r r , t h e c e n t r a l line a p p e a r s ~s a b r o a d b a c k g r o u n d

2 3 6 c. NICOLIS, J. P. B O O N a R d G. NICOLI~

w i t h l o w p e a k i n t e n s i t y

1 [[rT--2~q2x q~ q: .]

(39a) ? $ ' ( w ---- 0) - - T ~ l \ r r _ _ ] ] ~-~ 4- ( r ~ _ l ) ~ ~ b ( r ~ - - : l ) J

a n d t h e s a t e l l i t e p e a k s b e c o m e ~ - f u l l c t i o ~ s l o c a t e d a t ~ o - ~ • e q . (20), as i n d i c a t e d b y t h e 5 -~ b e h a v i o u r of t h e s p e c t r u m i n t e n s i t y

(39b)

y(+l(w = +f2T) - A / B ,

A = [b(r - - 2) 2 ~- a -!- r~] q~ + (c; + b + r~)[a" § b(a ~- rr)] q~ +

~- s[(; 2 !- b((; -i- rT)] q~ , B - ~b[4#" -i- b(~ + rr)] (~2.

T h e e h a l l g e s i n t h e p o w e r s p e c t r u m b e t w e e n t h e f i r s t a n d t i l e s e c o n d b i - f u r c a t i o n a r e i l l u s t r a t e d i n fig. 2. M o s t i n t e r e s t i n g is t h e a p p e a r a n c e of s i d e p e a k s w h e u r b e c o m e s l a r g e r t h a n r~. T h e s e s h i f t e d s p e c t r a l l i n e s a r e t h e m a n -

S

2

b)

c)

- 2 0 -10

c~)

e)

0 10 ~) 20 --20 -10 0 I 0 w 20

F i g . 2. - P o w e r s p e c t r u m of t h e r m a l m o d e b e t w e e n first a n d s e c o n d b i f u r c a t i o n p o i n t s , r = 1 a n d r = rz, r e s p e c t i v e l y . T h e case i l l u s t r a t e d h e r e is f o r w a t e r at 20 ~ t h e v a l u e s of t h e p a r a m e t e r s a r e : b = 8/3, a = 27/4, qx = 0.12, qr = 0.75,

qz

a) r = 1.02, b) r = 1.1, c) r = 1.5, d) r = 2, e) r = 15, ]) r = 27.

F L U C T ~ _ ~ & T I O N - ] ) I S S I I ' A T I O N T I t E O R ~ M A N D I N T R I N ' S I C S T O C I I A S T I C I T Y O F C L I M A T E 237 i f e s t a t i o n of t h e d a m p e d oscillations which arise f r o m m o d e coupling a n d m a r k t i m onset of a r e g i m e with p r o p a g a t i n g w a v e s in t h e s y s t e m , h~ote t h a t p r o p - a g a t i n g m o d e s also exist in t h e region r < 0, w h e n It! exceeds a c e r t a i n v a l u e It*I, a n d h a v e been o b s e r v e d e x p e r i m e n t a l l y (x~). So it would be i n t e r e s t i n g t o p r o b e e x p e r i m e n t a l l y t h e region r > r . w h e r e p r o p a g a t i n g m o d e s with a dif- f e r e n t n a t u r e are p r e s e n t in t h e c o n v e c t i v e regime. N o t e t h a t t h e accessible r a n g e of R a y l e i g h n u m b e r where e x p e r i m e n t s could be p e r f o r m e d to d e t e c t t h e s e p r o p a g a t i n g m o d e s does n o t e x t e n d all t h e w a y to rT b e c a u s e of t h e in- v e r t e d b i f u r c a t i o n (see fig. 1) as a consequence of which t h e n a r r o w i n g of t h e s'~tcllites will n o t be o b s e r v a b l e in t h e v i c i n i t y of t h e second b i f u r c a t i o n . E x - p e r i m e n t a l m e a s u r e m e n t s o f t h e d e t a i l e d s t r u c t u r e of t h e p o w e r s p e c t r u m (or e v e n of t h e i n t e g r a t e d i n t e n s i t y ) would also p r o v i d e a v a l u a b l e t e s t of v a l i d i t y for t h e generalized 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 p r e s e n t e d in sect. 2.

N o t e t h a t t h e p e w e r s p e c t r u m of t h e v e l o c i t y fluctuations, Cxx , c a n be c o m p u t e d along t h e s a m e lines as d e s c r i b e d above. I n t e r e s t i n g i n f o r m a t i o n c a n be o b t a i n e d f r o m t h e p e a k i n t e n s i t y of t h e p o w e r s p e c t r u m Cxz (~o ~ 0).

I n d e e d t h e diffusion coefficient D of s u s p e n d e d p a r t i c l e s can be e x p r e s s e d b y m e a n s of F a x 6 n ' s t h e o r e m (see rcf. (,.o,2~)) in t e r m s of t h e fluid v e l o c i t y field fluctuations, t h r o u g h t h e i r p o w e r s p e c t r u m a t zero f r e q u e n c y . F l u i d - d y n a m i c s m e a s u r e m e n t s a r e c o n v e n i e n t l y p e r f o r m e d b y laser light s p e c t r o s c o p y to p r o b e fluid m o t i o n v i a t h e light s c a t t e r e d b y s u s p e n d e d (dust or seeded) p a r - ticles. T h u s e n h a n c e m e n t of h y d r o d y n a m i c fluctuations n e a r b i f u r c a t i o n points c a n be p r o b e d b y m e a s u r i n g t h e diffusion coefficient of such suspensions, e.g., it was shown t h a t D diverges like e - , w h e n r --> 1 for r < 1 (# -~ ~, ref. (22);

/~ ~ 2, ref. (23)). This r e s u l t c a n n o w be e x t e n d e d to i n v e s t i g a t e t h e v i c i n i t y of t h e first t r a n s i t i o n when t h e b i f u r c a t i o n p o i n t is a p p r o a c h e d f r o m a b o v e . T h e p r e s e n t t h e o r y p r e d i c t s t h a t t h e diffusion coefficient b e h a v e s critically w i t h t h e s a m e p o w e r law (e-~') for r > a as well as for r < l , b u t shows no divergence on t h e a p p r o a c h to the second b i f u r c a t i o n p o i n t as Czx (co = O) is well b e h a v e d a t r ~ r r . (An e x p r e s s i o n s i m i l a r to (39a) is o b t a i n e d for the p o w e r s p e c t r u m (at ~o ~ 0) of t h e v e l o c i t y field fluctuations).

4. - I n t e r n a l f l u c t u a t i o n s i n a z o n a l l y - a v e r a g e d e n e r g y - b a l a n c e m o d e l .

In this section we briefly a n a l y s e t h e effect of i n t e r n a l f l u c t u a t i o n s for a simple e n e r g y - b a l a n c e m o d e l i n c o r p o r a t i n g t h e s u r f a c e - a l b e d o f e e d b a c k a n d (t~) J . P . Boox, C. A.LLAIN and P. LALLEMAND: Phys. Rev. Lett., 43, 199 (1979).

(2o) R. Z w ~ z t G - J. Res. Nat. Bur. Stand., Sect. B, 68, 143 (1964).

(~1) p. Mnzul~ and D. BEDXAI~X: Physica, 76, 235 (1974).

(z2) H. •. K. LEKKERKERKEIr and J . P . Boo.~ : Phys. Rev. Lett., 36, 724 (1976).

(23) j . B . LAS~OVXA and J . P . Boo.~: Phys. Rev. A, 14, 1583 (1976).

238 C. N I C O L I S , J . P. BOON" ~ n ( ~ G. N [ C O L I S

the e n e r g y t r a n s f e r in t h e m e r i d i o n a l direction (24,25). The only variable r e t a i n e d in t h e description is the surface t e m p e r a t u r e T. Choosing a spherical co-ordinate s y s t e m on t h e E a r t h ' s surface a n d d e n o t i n g by s, a a n d Z t h e sine of the l a t i t u d e , the E a r t h ' s radius, a n d t h e effective h e a t diffusion coefficient, respectively, we write lhe e n e r g y balance equation in t h e form

(40)

l [ e r c ~ is t h e he~t c a p a c i t y of a c o l u m n with a unit cross-section a n d a height of t h e order of the d e p t h of the m i x e d l a y e r ; IT is the r a d i a t i o n b u d g e t (dif- ference between incident solar flux a n d outgoing infra-red r~diation); ~nd ~ . t h e effective r a n d o m force. According to eq. (3)

(41a) . ~ ( s , t ) - - - - V . J ( s , t )

or, ir~ t h e chosen spherical co-ordinate s y s t e m ,

(11b) .~T(s, t) -- - - a-~ ~,(1 --s~}~J(s, t ) .

The r a n d o m ttux correlation f u n c t i o n is o b t a i n e d from eq. (4) b y switching to spherical co-ordinates to yield

(,J2) <J(s, t ) J ( s ' , t')> - - qZb(~. - - s') b(t - - t') ,

where q2 is a q u a n t i t y similar to t h e s t r e n g t h factors in eqs. (15) a n d (16). I t follows t h a t

(43) < y~(.~, t)r t')> = q~a(t - - t ' ) ,

~ r ==- q 2 a - 2 ~,(1 - - s2) 1 ~,, (1 - - s'2)t b ( s - - s ' ) ,

i.e. t h e r a n d o m forces t h e m s e l v e s ( ~ r ) are not delta-correlated in space. ~N~otice t h a t , despite the highly singular c h a r a c t e r of the r a n d o m force, t h e t h e r m a l response will b e h a v e in ~ perfectly regular fashion.

We proceed b y e x p a n d i n g b o t h the t e m p e r a t u r e a n d r a n d o m force fields i~1 L e g e n d r e p o l y n o m i a l series

(44) T ---- ]~ r.(t)P.(,~,), ~ = Z ~ . ( t ) 2 n ( s ) .

ft

I n s e r t i n g (44) in eq. (43), m u l t i p l y i n g b y P m ( s ) - P z ( s ' ) a n d i n t e g r a t i n g over s

(24) G.R. NOrTh: J . A t m o s . S c i . , 32, 2033 (1975).

(35) G.R. NORTH, F . R . Cr and J.A. COAKLEr: Rev. Geophys. S p a c e P h y s . , 1 9 , 91 ( 1 9 8 1 ) .

F L K C T U A T I O N - D I S S I I ) A T I O N T I I E O R E M A N D I N T R I N S I C S T O C I I A S T I C I T Y OF C L I M A T ~ 239 and s', we obtain, assuming s y m m e t r i c hemispheres,

(45a) with (45b)

<.~-.(t) . ~ ( t ' ) > = q~Smt6(t - - t')

q~ = 8 9 1)m(m q-1)q~/a ~.

I n t e r e s t i n g conclusions can be d r a w n f r o m these results. First, the r a n d o m forces associated with different L e g e n d r e modes are u n c o r r e l a t e d . Second, t h e r e is no r a n d o m force a c t i n g d i r e c t l y on the e q u a t i o n for t h e globally a v e r a g e d t e m p e r a t u r e To, because it follows f r o m eq. (45b) t h a t qo = 0. Still, To keeps a s t o c h a s t i c c h a r a c t e r t h r o u g h its coupling with t h e modes T~, etc., which are affected d i r e c t l y b y a r a n d o m force. T h i r d , t h e i m p o r t a n c e of the fluctuations increases r a p i d l y w i t h t h e o r d e r of t h e L e g e n d r e mode. I n a s m u c h as higher- order L e g e n d r c modes r e p r e s e n t localized d i s t u r b a n c e s , we, t h e r e f o r e , see t h a t t h e s t r e n g t h of fluctuations depends on t h e i r space scale. This is r e m i n i s c e n t of the p h e n o m e n a of nucleation f r e q u e n t l y e n c o u n t e r e d in p h a s e t r a n s i t i o n s (26).

To o b t a i n the statistical p r o p e r t i e s of the t e m p e r a t u r e field f r o m those of t h e r a n d o m force, one would h a v e to solve t h e stochastic differential equa- tions (41a) with (41b). This is a diffmult t a s k because of t h e nonlinearities i n v o l v e d in t h e r a d i a t i v e t e r m fz. One can, however, o b t a i n explicit results b y t r u n c a t i n g the infinite set of coupled equations for T ( t ) keeping only the first two modes T O a n d T2, a n d b y linearizing a r o u n d a reference s t a t e cor- r e s p o n d i n g to t h e p r e s e n t - d a y climate. Such a n analysis has been r e p o r t e d elsewhere (xs). We s i m p l y m e n t i o n t h a t t h e power spectra o b t a i n e d f r o m t h e t i m e - F o u r i e r t r a n s f o r m s of To a n d T~ p r e s e n t a s t r u c t u r e similar to t h a t given b y cq. (26) a n d i l h l s t r a t e d in fig. 2. Such forms arc c h a r a c t e r i s t i c of red noise spectra which are c o m m o n l y observed in a t m o s p h e r i c p h e n o m e n a (27.2s), w h e r e b y m o s t of t h e power is c o n c e n t r a t e d in t h e low-frequency range.

T h e p r e s e n c e of t h e f u n c t i o n 6 ( t - - t ' ) in eq. (43) is p r i m a r i l y responsible for this quite general p r o p e r t y of climatic spectra. _Nevertheless, t h e spatial s t r u c t u r e of t h e noise plays a n i m p o r t a n t role in t h e following respect. I n t h e absence of spatially inhomogeneous fiuctuations the t e m p o r a l F o u r i e r t r a n s f o r m of t h e correlation f u n c t i o n of the m e a n surface temperatLtrc would r e d u c e to a L o r e n t z i a n , exhibiting t h e c h a r a c t e r i s t i c (z 2 + w~) -~ dependcnce. T h e presence of spatial modes modifies this s t r u c t u r e a n d gives rise to correlation functions such as given in eqs. (27) a n d (35), which h a v e an a l t o g e t h e r different f r e q u e n c y dependence.

(2s) G. NtCOLIS and I. PRIGOGIXE: Sell-organization in Non-eq~tilibvium Systems (John Wiley, New York, N.Y., 1977).

(27) D . L . GILAMN, F. J. FUGI, IST~Ir and J. M. MITCHELL: J. Atmos. Sci., 20, 182 (1963).

(~s) K. HASSEL:~XXN: in Glimatic Variations a~d Variability: .Facts and Theories, edited by A. Brzl~GV~l~ (Reidel, Dordrecht, 1981).

240 c. NIcor.~s, z. P. BOON and o. xmoms

5. - Concluding comments.

B y a p p l y i n g a generalized fluctuation-dissipation t h e o r e m to the r a n d o m forces a p p e a r i n g in t h e balance equations of t h e climatic s y s t e m , we o b t a i n extensive information on t h e c h a r a c t e r i s t i c s of the fluctuations of the climatic variables. The f o r m u l a t i o n outlined in t h e p r e s e n t p a p e r is quite general, a n d two specific applications h a v e been considered, where we a n a l y s e the v i c i n i t y of ~ given clima.tic state, i t would be i n t e r e s t i n g to e x t e n d this analysis so as to t a k e into a c c o u n t the m u l t i p l i c i t y of solutions of t h e fully nonlinear problems. The most i m p o r t a n t p o i n t to investigate in this f r a m e w o r k would be the passage times b e t w e e n different climatic states. ~t would also be desirable to develop a suit~ble description of localized, short wave-length fluctuations.

Finally, we n o t e t h a t t h e role of s t o c h a s t i c p e r t u r b a t i o n s in more c o m p l e x models like spectral models involving several modes could now be considered in the light of t h e the q u a l i t a t i v e t r e n d s suggested b y t h e analysis given in t h e p r e s e n t paper.

The work of C. 17icolis is s u p p o r t e d b y the E E C u n d e r c o n t r a c t n u m b e r CLI-027-8(G). One of us ( J P B ) ~cknowledges financial s u p p o r t b y the (( F o n d s National de la Recherche Scieutifique ,~ (Fh-RS, Belgium).

~ . P P:E N D I X

The Lorenz equations in the presence o f fluctuations.

We first e x p a n d t h e s t o c h a s t i c fields j*, ~.s* (see eqs. (10a), (J 0b)) in F o u r i e r series, analogously to ~qs. (i4). l-[owe, ver, since their effect oll t h e balance equations a p p e a r s t h r o u g h t h e diverge~ace o p e r a t o r (for j*) a n d t h e V • • o p e r a t o r (for s*), care should be t a k e n in t h e series expansion to satisfy t h e p a r i t y a n d o t h e r s y m m e t r y r e q u i r e m e n t s . F o r instance,

(A.la) j* --- ~ / ~ , exp [imk*x*] sin (n~z*) ,

m n

b u t since j* a p p e a r s in t h e e n e r g y e q u a t i o n ~s 8j*/~z*

(A.lb) j* = ~ / ~ , exp [imk*x*] cos (n~z*) .

F I A J C T U A T I O N - ] ) I S S I P A T I O N T I I J , I O R E M A X D I N T R I N S I C S T O C I I A S T I C , I T Y O F C L I M A T } ] 241 Similarly, one h a s

(A.2a) s ~ : ~ / ~ % c x p [imk*x*] cos (n~z*) , (A.2b) s~. = ~ ].,. e x p '* ~ [imk*x*] sin (n:~z*) ,

on,?

( A . f c ) .%~ = ~ ] ~ e x p [imk*x*] cos '* *~ (n~z*) .

m n

I n s e r t i o n of t h e ~ b o v c expa.nsions a n d of eqs.(14) r e d u c e s t h e n o n l i n e a r p a r t i a l differential e q u a t i o n s (10a) a n d (10b) to ~rn infinite set of coupled non- linear s t o c h a s t i c o r d i n a r y differential e q u a t i o n s for t h e m o d e a m p l i t u d e s 0~, a n d w*,. Following SALTZ.~AN (3) a n d LOaENZ ('), we t l ' m t c a t e t h e set a n d r e s t r i c t to t h e first t h r e e n o n t r i v i a l m o d e s , which a m o u n t s to k e e p i n g w~*~, 0~:3 a n d 0~*~. I n t r o d u c i n g t h e redfaced q~antities~

(A.3)

one o b t a i n s (A.4a) (A.4b) (AAc)

X - k*(~'*~ ~- ~)-'Wl*~, 2 v - - - - ' 2 i ~ k * ~ ( k * ~ + 7~2)-a0. , Z = '2~k*2(k .2 + ~ 2 ) - 3 0 " 2 , r = k * 2 ( k .2 -t- z z ) - 3 R , b ---- 47~2(k .2 + ~ ) - 1 ,

X = - r ~ X ~ - ~ Y t - C x ,

~ - = r X - - Y - - X Z - ' . - C y , 2 = - - bZ + X Y - ' - r w h e r c Cx, r a n d Cz a r e given below, see (A.5).

These a r e t h e L o r e n z - S a l t z m a n e q u a t i o n s inchlding now i n t t v n a l flltetuation t e r m s . I t is k n o w n t h a t t h e L o r e n z - S a l t z m a n e q u a t i o n s do n o t describe cor- r e c t l y t h e b e h a v i o u r of e x p e r i m e n t a l c o n v e c t i o n a t v e r y large r (r <rT); how- ever, t h e y c o n t a i n t h e essential f e a t u r e s for t h e p u r p o s e of t h e p r e s e n t analysis.

Most i m p o r t a n t is t h a t one is in a p o s i t i o n to fully express t h e c o n s t r a i n t s i m p o s e d on t h e effective r a n d o m forces b y ltsil}g st~tistieal t h e o r y of irrever- sible processes. I n d e e d , t h e f a c t t h a t t h e r a n d o m force is t h e divergence of a r a n d o m v e c t o r or t e n s o r field is a l r e a d y b u i l t into eqs. (A.4). I n addition using eqs. (8), (9), (A.]) a ~ d (A.2), one c~n d e t e r m i n e t h e s t a t i s t i c a l p r o p - erties of t h e t h r e e effective r a n d o m forces in eqs. (AA)

iA.5)

r = 2 V ~ ( k *~ + ~2)-3 [:~k,2(l~i _ / ~ ) _ i k * / ~ ] , Cy = 2 V ~ k * 2 ( l ~ *~ - t n~)-' ( i k * l ~ l - ~1~) , r = 4~1~*~(1~'*~ § : ~ ) - " l & 9

242 c . K I C O L I S , J . 1% BOOK a n d G, NICOL]8

B e c a u s e t h e c o m p u t a t i o u is r a t h e r l o n g a n d t e d i o u s , i t is n o t d i s p l a y e d h e r e ; w e m e r e l y o u t l i n e t h e g e n e r a l p r o c e d u r e . B y e x p a n d i n g e q s . (8) a n d (9) in F o u r i e r s e r i e s o n e o b t a i n s t h e t w o - t i m e c o r r e l a t i o n f u n c t i o n s of t h e F o u r i e r c o m p o n e n t s of t h e r a n d o m f o r c e s ~ / ~ , , 1 ~ . F o r 1 ~ o n e finds t h a t a l l c o r r e - l a t i o n s of t h e f o r m ( ] , , ~ ( t ) ] , , . ( t ) ? ; ~ J ' " i 4- j v a n i s h ; on t h e o t h e r h a n d f o r ]~, o n e h a s a n o r t v a ~ d s h i n g c o r r e l a t i o n ( f ~ ( t ) ]{{(t')). I t f o l l o w s t h a t t h e e f f e c t i v e r a n d o m f o r c e s Cx, C r , Cz a r c s u p e r p o s i t i o n s of t h r e e r a n d o m p r o c e s s e s 1~, ] ~ , ] ~ , t w o of w h i c h a r e c o r r e l a t e d .

So w e f i n a l l y o b t a i n t h e f o l l o w i n g n o n v a n i s h i n g c o n t r i b u t i o n s :

(A.6)

w i t h

(A.7)

w i t h

~ n d

(A.S)

w i t h

( r Cx(t')> = q~6(t - - t ' )

4 z : k *~ § 2 ( k *~ + n ~) ( r 1 6 2 ~-- q~b(t - - t')

q~ =: l 6~).l~'nT2(a~t2Cv) -2 (o~gd 4) ~k*S(k .2 q- x~) -7

( r 1 6 2 } - - q=,(~(t - - $')

q~ - ' 2 ~ ( k .2 - t ~z2) -~ q 2 .

T h e s e r e s u l t s s h o w t h a t , in g e n e r a l , t h e s t r e n g t h s of t h e r a n d o m f o r c e s a r e u n e q u a l . A n e v a l u a t i o n f r o m n u m e r i c a l v a l u e s d e s c r i b i n g l a b o r a t o r y con- d i t i o n s ( T : 2 9 3 K , d~-- 1 c m , ~ =: t g . c m -~, : r 10 - d K -~, v ~ 1 0 - ~ s t o k e s ,

~r ~ 1 . 5 . 1 0 -~ c m 2 s -1) y i e l d s qx/qz - 0.12 a n ( i q r / q z --=- 0.75.

9 R [ A S S U N T O (')

Nella dinamica del elima si deserivo solitamente l'effetto delle fluttuazioni generate i a t e r n a m e n t e a u m e n t a n d o le equazioni del bilancio con l ' a g g i u n t a di forze easuali.

In qucsto lavoro si studiano lc p r o p r i e t ~ di qucste forze. Si propone uu t c o r e m a di flut- tuazione-dissipazionc clue corrcla la m a t r i c e di covarianza ai eoefficicnti fenomenologiei come la diffusivit~ t u r b o h m t a . I1 t e o r c m a ~ usato suecessivamente pcr identificare le p r o p r i e t h statistiehc delle v a r i a b i l i climatiche stesse e per c a r a t t e r i z z a r e la variabilit~

climatie, a dal punto di vi.~ta della t e o r i a s t a t i s t i c a dei processi irreversibili. Si presen- l a n e applicazioni a un p r o b l e m a semplice di convezione t e r m i c a e a d un modello di bilancio d ' e n e r g i a m e d i a t e a zone,; si diseute la possibiliti~ di verifiea sperimentale.

(*) Traduzione a eura della Redazio~,e.

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