AERO NOMICA ACTA

Texte intégral

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I N S T I T U T D ' A E . R O N O M I E S P A T I A L E D E B E L G I Q U E 3 - A v e n u e Circulaire

B - 1 1 8 0 B R U X E L L E S

A E R O N O M I C A A C T A

A - N ° 2 1 8 - 1 9 8 0

O n t h e e n t r o p y balance of the e a r t h - a t m o s p h e r e s y s t e m

by

G. I\l I C O L I S and C. M I C O L I S

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 B R U S S E L

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FOREWORD

The paper entitled : "On the entropy balance of the earth- atmosphere system" will be published in The Quarterly Journal of the

Royal Meteorological Society, 106, 1980.

AVANT-PROPOS

L'article intitulé : "On the entropy balance of the earth- atmosphere system" sera publié dans I he Quarterly Journal of the

K o - y a i

Meteorological Society, 106, 1980.

VOORWOORD

Het artikel getiteld : "On the entropy balance of the earth- atmosphere system" zal verschijnen in het tijdschrift The Quarterly Journal of the Royal Meteorological Society, 106, 1980.

VORWORT

Die Arbeit : "On the entropy balance of the earth-atmosphere

system" wird in The Quarterly Journal of the Royal Meteorological

Society, 10(3/ 1980 herausgegeben werden.

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ON THE ENTROPY B A L A N C E OF THE EARTH-ATMOSPHERE S Y S T E M

by

G. NI COL IS and C. NICOLIS

Abstract

The entropy balance associated with a Budyko-Sellers climatic model is developed. It is shown that different regimes, associated with decreasing, as well as increasing values of entropy production (which measures the rate of dissipation in the system) in the course of time are possible. An explicit criterion of climatic stability is also derived, which is expressed in terms of thermodynamic quantities related to excess entropy production. The results are illustrated on simple cases involving diffusive energy transport. A comparison with Paltridge's minimum entropy exchange principle is also attempted.

Résumé

Dans ce travail on étudie le bilan entropique associé à un

modèle climatique du type Budyko-Sellers. On montre, qu'il est possible

d'avoir différents régimes climatiques associés aussi bien à une

augmentation qu'à une diminution de la production d'entropie au cours

du temps. Le problème de la stabilité des états stationnaires du système

est dès lors posé. Ensuite on déduit la forme explicite du critère de

stabilité climatique en terme de variables thermodynamiques associées à

la production d'entropie d'excès, et on illustre le résultat général sur

des cas simples. On entreprend enfin une comparaison avec l'hypothèse

de minimum d'échanges entropiques avancée récemment par Paltridge.

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Samenvatting

De entropiebalans geassocieerd met een Budyko-Sellers klimaatmodel wordt opgemaakt. Er wordt aangetoond dat verschillende toestande, overeenstemmend zowel met dalende als met stijgende waarden van de entropieproduktie (die een maat is voor dissipatiesnelheid in het systeem), mogelijk zijn in de tijd : Een expliciet kriterium voor de stabiliteit van het klimaat wordt ook afgeleid, en wordt uitgedrukt in functie van thermodynamische grootheden die in verband staan met een overvloed in de entropie-produktie. De resultaten worden geïllustreerd in een eenvoudig geval van diffusie-energie-overdracht. Een v e r - gelijking met het princiepé van Paltridge voor minimale entropie- uitwisseling wordt ook gemaakt.

Zusammenfassung

Die Entropie-entgleichung in das Budyko-Sellers klimatisches Modell ist entwickelt worden. Es wurde gezeigt dass verschiedene

Regime für steigende sowie für absteigende Entropie Produktionen (die ein Mass der Dissipation im einen System ist) mit der Zeit möglich sind.

Ein bestimmtes Criterium für klimatische Stabilität is auch entworpen worden. Thermoaynamische Eigenschaften die von dem Ubermass der Entropieproduktioi abhängen, bestimmen dieses Criterium. Die Resultaten sind für einfache Fälle für diffusives Energietransport illustriert worden. Ein Vergleich mit das Paltridge-minimum-Entropie entgleichungsprinzip is auch versucht worden.

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

T h e c o m p l e x i t y of t h e dynamical processes d e t e r m i n i n g long t e r m climatic t r e n d s is well k n o w n . N e v e r t h e l e s s , t h e need of an a p p r o a c h i n v o l v i n g o n l y a few global v a r i a b l e s is nowadays w i d e l y r e c o g n i z e d . S u f f i c e it to quote t h e e n e r g y - b a l a n c e models of the B u d y k o - S e l l e r s t y p e ( B u d y k o , 1969; S e l l e r s , 1969), w h i c h have been developed f u r t h e r b y such i n v e s t i g a t o r s as N o r t h (1975a, b ) , Ghil (1976) o r Coakley ( 1 9 7 9 ) , and w h i c h led to a q u a l i t a t i v e u n d e r s t a n d i n g of a g r e a t many f e a t u r e s of climate and its e v o l u t i o n .

Reduction of state space b y s u i t a b l y a v e r a g i n g the initial dynamical v a r i a b l e s is a w e l l - k n o w n p r o c e d u r e in many areas of p h y s i c s . The most c h a r a c t e r i s t i c example is c e r t a i n l y S t a t i s t i c a l Mechanics, w h e r e d i f f e r e n t averages have led, s u c c e s s i v e l y , f r o m the L i o u v i l l e e q u a t i o n to B o l t z m a n n - l i k e e q u a t i o n s , to M a r k o v c h a i n s , o r to macroscopic balance equations like those of h y d r o d y n a m i c s and chemical k i n e t i c s .

A second line of a p p r o a c h to t h e s t u d y of complex systems,

w h i c h is also s u g g e s t e d b y t h e s t a t i s t i c a l mechanics " p r o t o t y p e " , is the

d e v e l o p m e n t of a t h e r m o d y n a m i c d e s c r i p t i o n . T h e p r i m a r y o b j e c t i v e is to

cast some basic f e a t u r e s of t h e system in t h e p r o p e r t i e s of state

f u n c t i o n a l s - l i k e e n t r o p y ( o r more g e n e r a l l y , a L y a p o u n o v f u n c t i o n a l ; see

P r i g o g i n e et al, 1977) or e n t r o p y p r o d u c t i o n - w h i c h are l a r g e l y

i n d e p e n d e n t of t h e details of t h e i n d i v i d u a l degrees of f r e e d o m . T y p i c a l

examples of such p r o p e r t i e s are t h e Clausius i n e q u a l i t y , t h e t h e o r e m of

minimum e n t r o p y p r o d u c t i o n in t h e linear r a n g e of i r r e v e r s i b l e processes

( P r i g o g i n e , 1947), or t h e s t a b i l i t y c r i t e r i o n of steady states f a r f r o m

e q u i l i b r i u m ( G l a n s d o r f f and P r i g o g i n e , 1971). S u r p r i s i n g l y , t h i s second

line of approach is much less common in climate modelling. It is o n l y

v e r y r e c e n t l y t h a t P a l t r i d g e (1975, 1978), G o l i t s y n and Mokhov (1978)

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and N o r t h et al (1979) examined t h e possible e x i s t e n c e of a v a r i a t i o n a l p r i n c i p l e g o v e r n i n g climate. P a l t r i d g e ' s a p p r o a c h is specially s i g n i f i c a n t f o r o u r d i s c u s s i o n : B y assuming c e r t a i n r e l a t i o n s h i p s between atmo- s p h e r i c and oceanic d i s s i p a t i o n r a t e s , he showed t h a t a maximization of t h e s t e a d y - s t a t e o v e r a l l d i s s i p a t i o n rate of t h e e a r t h - a t m o s p h e r e system y i e l d s u n i q u e l y d e f i n e d spatial d i s t r i b u t i o n s of s u r f a c e t e m p e r a t u r e , cloud c o v e r and meridional e n e r g y f l u x e s , w h i c h closely resemble t h e o b s e r v e d zonally a v e r a g e d mean-annual v a l u e s .

T h e p u r p o s e of t h e p r e s e n t paper is to develop t h e e n t r o p y - balance equation associated to an e n e r g y - b a l a n c e equation of the B u d y k o - S e l l e r s t y p e . From t h i s equation we i d e n t i f y , in Section 2, t h e a p p r o p r i a t e e x p r e s s i o n s f o r t h e e n t r o p y f l u x and e n t r o p y p r o d u c t i o n w h i c h are v a l i d f o r s t a t i o n a r y as well as f o r t i m e - d e p e n d e n t states. In Section 3, we evaluate t h e t i m e - d e r i v a t i v e of t h e e n t r o p y p r o d u c t i o n and show t h a t , in g e n e r a l , it has no d e f i n i t e s i g n . As a r e s u l t the steady state solution of t h e system does not c o r r e s p o n d to a minimum of e n t r o p y p r o d u c t i o n , even if linear relations between e n e r g y f l u x and t e m p e r a t u r e g r a d i e n t are c o n s i d e r e d . T h i s p r o v i d e s an e x t e n s i o n of P r i g o g i n e ' s minimum e n t r o p y p r o d u c t i o n t h e o r e m . It also shows t h a t e n t r o p y p r o d u c t i o n can no longer s e r v e as a L y a p o u n o v f u n c t i o n a l , whose v a r i a t i o n a l p r o p e r t i e s g u a r a n t e e t h e s t a b i l i t y of t h e r e f e r e n c e s t a t e . T h i s raises t h e r e f o r e t h e q u e s t i o n of s t a b i l i t y of t h e climatic system. In t h e remaining of section 3 we show how t h i s q u e s t i o n can be t a c k l e d b y t h e methods of t h e r m o d y n a m i c s of i r r e v e r s i b l e processes.

Section 4 is d e v o t e d to t h e e x p l i c i t evaluation of t h e time course of e n t r o p y p r o d u c t i o n f o r a simple climatic model i n v o l v i n g , s u c c e s s i v e l y , an i c e - f r e e e a r t h (Section 4 A ) and a climate close to t h e p r e s e n t - d a y one (Section 4 B ) . In both cases we show t h a t e n t r o p y p r o d u c t i o n may decrease or increase in time, d e p e n d i n g on t h e initial s t a t e . T h i s c o r r o b o r a t e s t h e general r e s u l t s of t h e analysis of Section

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3, a c c o r d i n g to w h i c h the s t e a d y state climate does not appear to s a t i s f y an o b v i o u s v a r i a t i o n a l p r i n c i p l e , at least at the level of a B u d y k o - S e l l e r s t y p e of model. N e v e r t h e l e s s , some general t r e n d s appear to r e c u r c o n t i n u o u s l y . For i n s t a n c e , e n t r o p y p r o d u c t i o n t e n d s to increase w h e n e v e r t h e e q u a t o r - p o l e t e m p e r a t u r e d i f f e r e n c e becomes more p r o n o u n c e d .

Section 5 is d e v o t e d to t h e solution of t h e e n e r g y balance equation u s i n g P a l t r i d g e ' s maximum e n t r o p y p r o d u c t i o n c o n j e c t u r e . T h i s y i e l d s a meridional e n e r g y f l u x w h i c h is in f a i r agreement w i t h p r e s e n t - d a y d a t a , b u t a somewhat less s a t i s f a c t o r y t e m p e r a t u r e d i s t r i b u t i o n .

In t h e final Section 6 we d r a w t h e main conclusions of the a n a l y s i s . We p o i n t o u t t h e i n t r i n s i c v a r i a b i l i t y of t h e climatic s y s t e m , as i l l u s t r a t e d f r o m t h e d i f f e r e n t b e h a v i o r o b t a i n e d f o r t h e e n t r o p y p r o d u c t i o n b y d i f f e r e n t assumptions on t h e e n e r g y f l u x , ( s u c h as a d i f f u s i v e e n e r g y t r a n s p o r t o r a maximum e n t r o p y p r o d u c t i o n ) . It appears t h e r e f o r e t h a t t h e basic problem one is faced w i t h is to delimit t h e p r i n c i p a l f a c t o r s r e s p o n s i b l e f o r t h e selection of a p a r t i c u l a r steady state climatic regime.

2. THE MODEL. E N T R O P Y - B A L A N C E EQUATION

A one-dimensional model i n v o l v i n g meridional e n e r g y t r a n s p o r t will be adopted ( N o r t h 1975a, b ; C o a k l e y , 1979) as d e s c r i b e d in Fig. 1.

As f r e q u e n t l y done in such models, t h e a b s o r b e d p a r t of solar i n f l u x , F and t h e i n f r a r e d cooling r a t e , F

I D

are d e s c r i b e d in

S I K

terms of an e f f e c t i v e s u r f a c e t e m p e r a t u r e T , w h i c h d e p e n d s on l a t i t u d e .

T h u s , t h e f i n e s t r u c t u r e of t h e atmosphere along t h e v e r t i c a l d i r e c t i o n

is i g n o r e d .

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A basic question arising in the analysis of evolution of a physical system is to find the appropriate constitutive relationes) between the fluxes (in the present case, the meridional energy flux) and the state variables (in the present case, the surface temperature T and its gradient). The complexity of the earth-atmosphere system precludes any derivation of such laws starting from first principles. It is therefore tempting to turn to thermodynamics of irreversible processes, which provides a natural classification of physical systems according to the type of constitutive relation prevailing. As it turns out, one must first identify the proper quantities which have to be related by the constitutive relations (also known as phenomenological laws). This is done by constructing the entropy production, which plays a central role in the theory of irreversible processes. To this end, we first write the energy-balance equation for our system. It will be convenient to switch to spherical coordinates and to incorporate the square of the inverse of radius of the earth into the heat flux and the various proportionality coefficients. The only component of V surviving in a one dimensional latitudinal model is then

V

x

M l - x V

/ 2

| j (1)

where x is the sine of latitude. The balance equation takes thus the form

3e _ 3T _

T

3t

= c

3t

= F

s "

F

I R

1 V

^

or, setting

F

s

- F

i r

= f(T, x)

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C g = f(T, X) - (1 - X

2

)

1

/

2

J

x

(2)

where c is the heat capacity (or thermal inertia) coefficient and e the energy.

In order to deduce the entropy-balance equation we adopt Gibb's entropy postulate (Glansdorff and Prigogine, 1971). Namely, we assume that if the total entropy is written in the form (in the symmetric hemisphere case considered hereafter) :

0

then the reduced entropy s depends on the same variables as in thermo- dynamic equilibrium. For the system under consideration this means

s = s(e)

ds = I de = Y dT (4)

This assumption is eminently plausible. The most important climatic phenomena are those due to the transport by the oceans and the lower atmosphere. Both systems are well within the collisional regime of kinetic theory, and hence their state is expected to be close to local equilibrium.

We now combine eqs (3) - (4) with eq. ( 2 ) . We obtain :

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• 1

0 0 1

ƒ M ^ - u » - "

2

'

1

'

2

' , ]

+ ƒ „ „ T ^ _ „2,1/2 3T -1 dx J (1 - x ) x

v

' 3x 0

Performing the x-integration in the first term and using the boundary condition

J

x

= 0 at x = 0, x = ± 1 (5)

as well as eq. (1), we arrive at the expression

r

1

. .

1 dS _ , fÇT.X) I /4 T V7 T~

l

2 dt * I

d x

— T ~

+

I

d x J

x "

V

x

T

0 0

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Hence, we identify - the entropy flux

f

1 deS dt

0

= 2 f dx W f l (7)

- and the entropy production

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d .S r

1

= 2 /

dx J V T

x (8)

Note t h a t t h i s s e p a r a t i o n implies t h a t t h e f u n c t i o n f ( T , x ) , t h a t is t h e a b s o r b e d a n d o u t g o i n g r a d i a t i o n s F a n d F

I D

is e n t i r e l y a s s o c i a t e d to

S I K

n o n - d i s s i p a t i v e p r o c e s s e s . In t h i s v i e w t h e r e f o r e , t h e main r o l e of t h e r a d i a t i v e f l u x is t o c r e a t e a l a t e r a l t e m p e r a t u r e g r a d i e n t ( t h a t i s , a n o n - e q u i l i b r i u m s t a t e ) , whose m a i n t a i n a n c e is a s s o c i a t e d w i t h t h e

djS

e n t r o p y p r o d u c t i o n , e q . ( 8 ) .

We a r e now i n p o s i t i o n t o i d e n t i f y t h e v a r i a b l e s t o be r e l a t e d - 1

b y t h e c o n s t i t u t i v e e q u a t i o n s , namely J ^ a n d V^ T . Let us d i s c u s s a few r e p r e s e n t a t i v e s i t u a t i o n s ( s e e also G l a n s d o r f f and P r i g o g i n e , 1971) :

( a ) We f i r s t assume t h a t t h e s y s t e m o p e r a t e s in t h e l i n e a r r a n g e of i r r e v e r s i b l e p r o c e s s e s . T h i s w i l l be r e f l e c t e d b y t h e l i n e a r r e l a t i o n

J = L V T

- 1

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x x ,

w h e r e t h e p h e n o m e n o l o g i c a l c o e f f i c i e n t L ( L è 0) is c o n s t a n t . In t h i s . 1

r e l a t i o n , V^ T is t o be v i e w e d as a g e n e r a l i z e d t h e r m o d y n a m i c f o r c e c o n j u g a t e t o t h e e n e r g y f l u x J .

( b ) T h e p h e n o m e n o l o g i c a l c o e f f i c i e n t , L is n o t c o n s t a n t . R a t h e r , w h e n e q . ( 9 a ) is w r i t t e n in t h e F i c k i a n o r F o u r i e r f o r m :

J = - x x X V T — V T . 2 x (9b)

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t h e t r a n s p o r t c o e f f i c i e n t k = L / T is c o n s t a n t . 2

In b o t h cases ( a ) and ( b ) we have a phenomenological law reminiscent of a d i f f u s i v e mechanism of e n e r g y t r a n s p o r t . N a t u r a l l y , t h i s does not mean t h a t molecular d i f f u s i o n and heat c o n d u c t i o n are the dominant t r a n s p o r t mechanism. R a t h e r , these laws must be viewed as a phenomenological way of e x p r e s s i n g t u r b u l e n t t r a n s f e r of latent heat and sensible heat in a medium of v a r i a b l e t e m p e r a t u r e . Several a u t h o r s have discussed t h e p r o p e r t i e s of the phenomenological t r a n s f e r c o e f f i c i ent ( S t o n e , 1973; Newell, 1974; L i n , 1977), and in p a r t i c u l a r its possible dependence on both local t e m p e r a t u r e and t e m p e r a t u r e g r a d i e n t s . T h i s leads us to discuss a t h i r d t y p e of s i t u a t i o n :

( c ) T h e system operates in the nonlinear range of i r r e v e r s i b l e processes, in t h e sense t h a t t h e f l u x - f o r c e r e l a t i o n s h i p is n o n l i n e a r . One way to e x p r e s s t h i s is to t a k e the c o e f f i c i e n t s L o r A in eq. (9a) or ( 9 b ) to depend on both T and V T :

J = L ( T , I V T | ) V T "

1

X 1 X I X

(9c) or

J = - T, |V T l ) V T

X ' I X I X

C o n t r a r y to t h e case of c e r t a i n physico-chemical systems ( G l a n s d o r f f

and P r i g o g i n e , 1971) it does not seem possible to s p e c i f y t h e p a r t i c u l a r

form of n o n l i n e a r i t y i n v o l v e d in eq. ( 9 c ) . Hence, one can e n v i s a g e a

large number of d i f f e r e n t s i t u a t i o n s c o r r e s p o n d i n g to d i f f e r e n t choices

of c o n s t i t u t i v e r e l a t i o n s . All these choices may well be compatible w i t h

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p r e s e n t - d a y c l i m a t i c d a t a , if t h e c o e f f i c i e n t s i n v o l v e d in L o r \ are a d e q u a t e l y f i t t e d . A l r e a d y in t h e case of e q . ( 9 b ) , N o r t h (1975) was able to r e p r o d u c e a r e a s o n a b l e p r e s e n t - d a y 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 and f l u x b y f i t t i n g a s i n g l e p a r a m e t e r A. I t is t h e r e f o r e i m p o r t a n t t o be able t o remove somehow t h i s h i g h d e g e n e r a c y in t h e choice o f J . One way t o a c h i e v e t h i s is P a l t r i d g e ' s maximum e n t r o p y p r o d u c t i o n A n s a t z . We w i l l h a v e a d e t a i l e d look at t h i s p o s s i b i l i t y in S e c t i o n 5. In t h e f o l l o w i n g S e c t i o n we a d o p t a d i f f e r e n t a p p r o a c h . We i n t e n d t o see how f a r one can go in t h e a n a l y s i s o f t h e c l i m a t i c s y s t e m as r e p r e s e n t e d b y t h e e n e r g y balance e q u a t i o n , on t h e basis of t h e p r o p e r t i e s o f t h e r m o d y n a m i c s t a t e f u n c t i o n s l i k e e n t r o p y a n d e n t r o p y p r o d u c t i o n .

3. V A R I A T I O N A L PROPERTIES OF ENTROPY P R O D U C T I O N . L Y A P O U N O V F U N C T I O N A L S

Orve of t h e most i m p o r t a n t r e s u l t s of t h e t h e r m o d y n a m i c 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 is P r i g o g i n e ' s t h e o r e m of minimum e n t r o p y p r o d u c t i o n ( P r i g o g i n e , 1947). It a s s e r t s t h a t in p u r e l y d i s s i p a t i v e s y s t e m s in w h i c h t h e f l u x e s a n d f o r c e s a r e r e l a t e d b y l i n e a r laws of t h e f o r m ( 9 a ) , e n t r o p y p r o d u c t i o n at t h e s t e a d y s t a t e s e t t l e s to a minimum v a l u e c o m p a t i b l e w i t h t h e c o n s t r a i n t s a c t i n g on t h e s y s t e m . I t f o l l o w s t h a t t h e s e s t e a d y s t a t e s a r e s t a b l e t o w a r d all p o s s i b l e d i s t u r b - a n c e s , p r o v i d e d t h a t 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 t s e l f is s t a b l e . In o t h e r w o r d s , e n t r o p y p r o d u c t i o n acts l i k e a L y a p o u n o v f u n c t i o n a l (see e . g . C e s a r i , 1962) e n s u r i n g g l o b a l s t a b i l i t y .

Let us now see w h e t h e r t h i s r e s u l t can be e x t e n d e d , to o u r c l i m a t i c m o d e l , e q . ( 2 ) . T o t h i s e n d we e x a m i n e t h e b e h a v i o r of e n t r o p y p r o d u c t i o n as a f u n c t i o n of t i m e . To r e m a i n as l o n g as p o s s i b l e w i t h i n t h e h y p o t h e s e s of P r i g o g i n e ' s t h e o r e m we f i r s t c o n s i d e r t h e l i n e a r f l u x - f o r c e r e l a t i o n ( 9 a ) , w h e r e t h e p h e n o m e n o l o g i c a l c o e f f i c i e n t L is c o n s t a n t .

- 1 2 -

(15)

T h e balance e q u a t i o n ( 2 ) t a k e s t h e f o r m

3T ^ . 3 , , 2

n r

3 T "

1

c

a t

= f ( T

'

x )

" Bi

( 1

x } L

ST

(10)

a n d P> e q . ( 8 ) becomes :

P = 2 ƒ dx (1 - x

2

) L ) è O (11)

0

T a k i n g t h e time d e r i v a t i v e a n d r e m e m b e r i n g t h a t L is c o n s t a n t , we o b t a i n :

f = - A L / dx (1 - x 2^ 3 T '

2

)

1

3_ 1_ 3T 3x 3x 2 3 t

T

S u b s t i t u t i n g 3 T / 3 t f r o m e q . ( 1 0 ) , p e r f o r m i n g a p a r t i a l i n t e g r a t i o n and t a k i n g i n t o a c c o u n t t h e b o u n d a r y c o n d i t i o n s ( 6 ) we o b t a i n :

2 / 1 r - 1 "

2

dP 4L ƒ , 1 1 8 , .

2 N

3T

d ï

=

- — J

d x

1

1

S i

( 1

"

x )

" a ï 3x 0 T

4 L

f

1

^

1

* f T ï I

9

t i 3 T '

1

1 - J

dx

~2

f ( T

'

x ) 1

S Ï

( 1

X }

" 3 7 " J

'0

T

d . P d P - i - +

d t d t

(12)

- 1 3 -

(16)

T h e f i r s t t e r m of t h i s r e l a t i o n , d j P / d t d e s c r i b e s t h e t i m e - v a r i a t i o n of P a r i s i n g s o l e l y f r o m i n t e r n a l d i s s i p a t i v e p r o c e s s e s . If o n l y t h i s t e r m w e r e p r e s e n t r e l a t i o n ( 1 2 ) w o u l d be e q u i v a l e n t t o t h e t h e o r e m o f minimum e n t r o p y p r o d u c t i o n , d P / d t ^ 0 . In e q . ( 1 2 ) h o w e v e r we h a v e a second t e r m d

g

P / d t w h i c h d e p e n d s on t h e r a d i a t i v e f l u x f ( T , x ) a n d w h i c h has no d e f i n i t e s i g n . Its p r e s e n c e o f f e r s new p o s s i b i l i t i e s l i k e , f o r i n s t a n c e , t h e i n v e r s i o n o f t h e s i g n of d P / d t u n d e r c e r t a i n c o n d i t i o n s .

T h e reason f o r t h i s lack of u n i v e r s a l i t y , as o p p o s e d t o t h e u n i v e r s a l i t y o f P r i g o g i n e ' s t h e o r e m , is in t h e b o u n d a r y c o n d i t i o n s . In P r i g o g i n e ' s t h e o r e m , t h e l a t t e r ( f i x e d o r z e r o - f l u x c o n d i t i o n s ) r u l e o u t all s p a t i a l c o n f i g u r a t i o n s t h a t c o u l d lead t o a v a l u e of P smaller t h a n t h e s t e a d y - s t a t e o n e . In t h e p r e s e n t case h o w e v e r t h e l a t e r a l ( z e r o f l u x ) b o u n d a r y c o n d i t i o n s , e q . ( 5 ) , a r e n o t s u f f i c i e n t l y s t r i n g e n t t o e l i m i n a t e s u c h p o s s i b i l i t i e s . As a m a t t e r of f a c t , t h e o n l y e x c h a n g e s b e t w e e n s y s t e m a n d s u r r o u n d i n g s a r e a l o n g t h e v e r t i c a l d i r e c t i o n w h i c h has been l u m p e d , o w i n g t o t h e o n e - d i m e n s i o n a l c h a r a c t e r of t h e model d e s c r i b e d b y F i g . 1 and e q . ( 2 ) . As a r e s u l t , t h e r a d i a t i v e f l u x f ( T , x ) w h i c h is at t h e o r i g i n o f t h e t e r m d

g

P / d t in e q . ( 1 2 ) , acts l i k e a c o n s t r a i n t o f a new t y p e as i t is i n c o r p o r a t e d i n t o t h e s t r u c t u r e of t h e e n e r g y b a l a n c e e q u a t i o n . I n t e r e s t i n g l y , t h i s c o n s t r a i n t does n o t act d i r e c t l y as t h e d r i v i n g f o r c e f o r a d i s s i p a t i v e f l u x . R a t h e r (see also comments f o l l o w i n g e q . ( 8 ) ) , i t is a s s o c i a t e d t o a p r o c e s s o f s t o r a g e of e n e r g y . In t h i s r e s p e c t , t h e b e h a v i o r of d P / d t as d e d u c e d f r o m e q . ( 1 2 ) is somewhat r e m i n i s c e n t of t h a t o f e l e c t r i c a l . c i r c u i t s c o m p r i s i n g ' r e s i s t o r s a n d i n d u c t a n c e s . As p o i n t e d o u t b y L a n d a u e r ( 1 9 7 5 ) , in s u c h

s y s t e m s i n v o l v i n g i n e r t i a l elements in a d d i t i o n to d i s s i p a t i v e o n e s , t h e e n t r o p y p r o d u c t i o n may i n d e e d i n c r e a s e in t i m e , e v e n if t h e c i r c u i t c h a r a c t e r i s t i c s a r e c o m p l e t e l y l i n e a r .

So f a r we d i s c u s s e d s t r i c t l y l i n e a r p h e n o m e n o l o g i c a l l a w s , e q . ( 9 a ) . T h e r e s u l t s can h o w e v e r be e x t e n d e d s t r a i g h t f o r w a r d l y t o case o ( 9 b ) of a F o u r i e r t y p e of l a w . T h e balance e q u a t i o n ( 2 ) a n d t h e

e n t r o p y p r o d u c t i o n P, e q . ( 8 ) t a k e t h e f o r m :

-14-

(17)

s,auißoß|Jd oa ßujAaqo suua^sÄs |BOjwaip-co|sÄL|d lensn ßujZuaaoBJBip Z 'Bid J.0 (e) aAjno u| se 'aAjjBfiau Ä|UBSsaoau }ou si uoijonpojd Adoj^ua j.0 uojiBjJBA awn am 'sassaoojd aßBjo^s Äßjaua j.o asnsoaq

'jeqa uMogs aAeg aM 'uojiernis am azueiuiuns A^auq sn aa~|

•p uoj^oas U| pa^jodaj suo!3.B|nD|eo ipndxa am uj paijjjaA aq HjM sjMX • uojanquauoo a|qjßi|ßau e aAjß oq. paq.oadxa s; q.! 'uoseaj sim j o j -siujaq. OMi j a m ° am ublh (xe/j_e) uj japjo jagßjij e ^o sj jaAaMOij ipiqM 'wjaq. ase| am saop os 'saninqissod Mau awos sjaj.j.0 (x'j_)j.

ujjaa xn|^-3Ai3.B|pej am jo aouasajd am 'uojaoasqns snoiAajd am uj sv 'aiiui^ap aAjießau aq ujeße pinoM ip/dP 'swjaq omi isb| am inomiM

(SD

Ï S ( X

xe - I ) xe « l Ü )

x

e

mie

K

z U

x

- O ^

x

e X

XD

f

e T ^ m

px p 1

j YV

^ ( X - X) H xe

K

z

}

e

, iL ç

7 *>, l

1

Z ,YV dp

: si

;insaj |BU[^ aq_L 'a«Joj.aq sb saui| awBS am smo||o^ ip/dp uo|iBq.ndiuoD aqjL •puBjßaqui am ui ßu|q.L|ßjaM am j.o aouasajd am aq.c>N

(VI) 0 5 xe

xe

X

0

Y \ i

z x

- t)

X

P ƒ Z = d

(ex) 3E6 ( X - I) y

+

(X'X)J = O

xe ^

}

e

v J

xe

(18)

„ - - l o "

/ /

/ /

/

\ \

\ \

\

t

2 . - Possible time evolutions of entropy production compat- ible with eq. (12) and (15). C u r v e (a) : same behavior as in Prigogine's minimum entropy production theorem, (b) : Entropy production is increasing, but reference state remains stable, (c), ( d ) : Reference state is unstable, and system evolves to new steady states.

- 1 6 -

(19)

theorem. Different possibilities can be envisaged, like for instance curve ( b ) of Fig. 2. We discuss their climatic significance in Section 4.

For the purpose of our present qualitative discussion however, both situations (a) and ( b ) are indicative of the stability of the stationary climatic regime. Of more interest are therefore situations corresponding to curves ( c ) or ( d ) , which are perfectly compatible with eq. (12) or (15) and which indicate, nevertheless, that the system may evolve away from a certain reference state and tend to a new climatic configuration.

We would now like to obtain a criterion which would show when such instabilities are possible. In irreversible thermodynamics, it turns out that one cannot derive such a criterion using the variational properties of entropy production. Following Glansdorff and Prigogine (1971) we introduce a new functional related to the excess entropy around the reference state. Let us first outline the formulation in the general case where no particular constitutive relation is postulated.

A

Let T be a reference temperature, for instance that corresponding to the present-day climate. We consider a slight perturba- tion, ÔT, form this state, and set

T = T + ÔT, I ^ I « 1 (16)

T

Using eq. (4) one can easily construct the excess entropy function

\ Ô

2

s = - ÔT"

1

Ôe = - \ ~ (ÔT)

2

(17)

T

Note that

-17-

(20)

62S

= J

dx 62s S 0 ( 1 8 )

- 1

2

Because of this, we regard 6 S as a Lyapounov functional (Cesari, 1962) and we evaluate its time derivative along the motion described by

A

the balance equation ( 2 ) . Keeping in mind that T is time-dependent, we obtain :

k i ( 6

2

S ) E - 2 c ƒ ' d x l j ST § S ( . 9 a ) 0 T

with ( c f . eq. ( 2 ) )

36T _ / 3f ) a f. 2 x 1 / 2 - , r

1

Qi •>

C a t "

' I 3T fe

6 T

-

Bi ( 1 " x } 6 JX ( 1 9 b )

More explicitly :

a t i c A ) - - ^ ƒ ' ^ h ( H ) f <

s

«

2

.0

1

1

+ 2 | dx 6 J . 6 V T- 1 ( 2 0 )

) x x

0

The f i r s t term of this expression reflects the effect of radiative f l u x . The second term has the same structure as the entropy production, eq.

( 9 ) , except that we now deal with the excess flux 6 J . and the excess - 1

force 6 V

x

T . We shall refer to this combination as the excess entropy production (Glansdorff and Prigogine, 1971).

- 1 8 -

(21)

By L y a p o u n o v ' s s t a b i l i t y theorem ( C e s a r i , 1962), we conclude

j «1

t h a t T will be a s y m p t o t i c a l l y stable as long as ^ ^ ^

s

) /

h a s a s i

9

n

opposite to or

f

d x

[ : n ( M ) f

( 6 T ) 2 + 6 J

x

6 v

x

T _ 1

]

0

(21)

along all solutions of eq. ( 1 9 b ) .

To g i v e a more e x p l i c i t form to t h i s e x p r e s s i o n , we must dw J is related to

1

x

linear relation ( S t o n e , 1973) :

s p e c i f y how J is related to V T Choosing as an example, t h e non • 1 X X

J = - [ k ( x ) + k. ( JP , x ) ] V T (22)

x

L

o 1 3x '

J

x

we o b t a i n t h e e x p l i c i t form of t h e s t a b i l i t y c o n d i t i o n :

[ ^ (i 1 ) 2 (M)x t«' 2

o

T

\

1

2

+ (1 - x

2

) — I ' (V T) ( f f

1

) = 0 (23)

T

w h e r e denotes t h e d e r i v a t i v e of k^ w i t h respect to its a r g u m e n t 3 T / 8 x .

In t h i s i n e q u a l i t y , all terms b u t the f i r s t one have no d e f i n i t e s i g n . Hence, u n d e r c e r t a i n c o n d i t i o n s t h e i r sign can become n e g a t i v e

-19-

(22)

and their absolute value can exceed that of of the first* term. In this case one would have d / d t (6 S) < 0, and since (6 S) remains always 2 2 negative, by Lyapounov's theorem T would be unstable. We may refer to this situation as a climatic catastrophe. We see that it is reflected by a clearcut change in the thermodynamic properties of the system. In a sense, climatic change becomes a problem of thermodynamic stability.

Note that the terms threatening stability in eq. ( 2 3 ) are related either to the storage term f , or to the nonlinearity in the

J x

"

T

relation- ship. This is in agreement with the fact that the source of nonlinearity making bifurcations possible in the energy balance equation ( c f . eq. ( 2 ) or ( 1 9 b ) ) is, precisely, in these two terms.

Finally, it is easy to v e r i f y that the left hand side of relation ( 2 1 ) or ( 2 3 ) is closely related to the second variation of the functional recently proposed by North et al (1979) in their variational formulation of Budyko-Sellers climate models.

4. ILLUSTRATIONS

In this section we illustrate the structure of the general expressions derived so far on simple examples.

a. An ice-free earth

We f i r s t consider eq. ( 2 ) in the case of an ice-free earth. It is believed ( B u d y k o , 1977) that this was indeed the case in the mesozoic and early cenozoic eras up to the beginning of the quaternary glaciations.

We adopt relation ( 9 b ) for the energy f l u x , and the following expressions for the radiative flux terms f ( T , x ) :

J = x \ V T

- 2 0 -

(23)

f ( T , x ) = Q(1 - a) S(x) - (A + BT) (24)

w h e r e the albedo a is t a k e n to be c o n s t a n t (a r a t h e r legitimate a p p r o x i m a t i o n f o r an i c e - f r e e e a r t h ) . Q is the solar c o n s t a n t , A and B are the i n f r a r e d - c o o l i n g c o e f f i c i e n t s , and the insolation S ( x ) is a p p r o x i m a t e d b y ( C o a c k l e y , 1979) :

S(x) = 1 + S

2

P

2

(S

2

< 0) (25)

P^ being the second Legendre polynomial. Eq. ( 2 ) takes t h u s the e x p l i c i t form

c = Q(1 - «) (1 + S

2

P

2

) - (A + BT)

• A j L (1 - x

2

) g (26)

T h e solution is easily f o u n d to be

T ( x , t ) = T ( t ) + T ( t ) P2( x ) ( 2 7 a )

where t h e p l a n e t a r y t e m p e r a t u r e TQ obeys to

dT

c ^ = Q(1 - a) - (A + B

Tq)

(27b)

- 2 1 -

(24)

and the amplitude T^ to

d T ~

c ^ = Q(1 - a) S2 - (B + 6k) T2 (27C)

Both Tq / T2 are to be e x p r e s s e d in degrees c e n t i g r a d e . T h e entropy p r o d u c t i o n , eq. (14), becomes :

2 f1 2 1 fd?2 2

P - 2 \ T2 J dx (1 - x ) [(273 + T0) + T2 P2(x)]2 1 y

One can easily see (Nicolis, 1979) u s i n g the a p p r o p r i a t e numerical values f o r Q, a, S^, A , B , A that 273 + TQ >> T T h u s , the above e x p r e s s i o n can be approximated b y

P " t (273 I To( t ) J 2 ( 2 8 )

We proceed to the evaluation of d P / d t . T o simplify the p i c t u r e as much as possible we consider only those evolutions that keep the planetary temperature TQ i n v a r i a n t . T h i s is legitimate, since the equations for TQ and T^ are u n c o u p l e d . T h e time dependence of T^ is easily f o u n d to be

V 0 = T2oo + e"M t ( T20 - V ( 2 9 )

with

- 2 2 -

(25)

Q(1 - ce) S

T

2OO

= — < o

B + 6À

T£Q = initial condition

|j = i (B + 6\) > 0

It follows that

" "

M

5(273 )2

T

2

( T

20 "

X

} ( 3 0 )

0

Fig. 3 depicts the evolution of T^ and P. We see that P can decrease or increase in time, according as the initial value T

2

Q is smaller or larger than the steady-state level T

2oo

. As T

2qo

is negative, in actual fact this implies that for fixed values of the coefficients, P decreases if the initial thermal gradient, measured by J 1~

2

j, is large and it increases if the initial \T^ | is small. T h i s is quite reasonable, since the steeper the gradient, the larger the rate of dissipation will have to be.

We see in an explicit way the possibility of having maximum entropy production at the steady state for all families of initial conditions (or equivalently, for all "virtual displacements") with

T

2 0

> T

2 » '

T h e s a m e

system however can give rise to a decrease in P, for different types of initial conditions. Note that all these new possibilities do not compromise the stability of the steady-state regime, T „ . As a matter of fact, (P - P , J turns out to be a Lyapounov ^00

00

function ensuring stability both in the case of increasing and of decreasing P' (see also Fig. 2, curves ( a ) and ( b ) ) .

-23-

(26)

Fig. 3.- Time evolution of the amplitude T^ of the second Legendre mode (lower part) and of entropy production (upper part) associated with model eq. (26).

-24-

(27)

In the context of climatic history of the last 250 myr or so, one might question the assumption adopted implicitly so far that the coefficients A, B , A. do not evolve in time. Recently one of the authors (Nicolis, 1979) developed plausible scenarios of evolution of these coefficients and analyzed the consequences of such variations on the values of TQ and T ^ , using the constraint (suggested by paleoclimatic data), that the equatorial temperature, T remained practically invariant ( T = 2 5 ° C ) . We have verified that this simultaneous

v

eq

evolution of both T and the parameters does not affect the qualitative trends shown in Fig. 3. Namely, a more pronounced pole-equator thermal gradient leads to a relaxation accompanied by a decreasing entropy production, whereas the opposite is true if the pole-equator initial gradient is smaller that the steady state value.

b. The influence of ice caps

We now extend the model of the previous subsection to account for the existence of ice caps characterizing the present climate.

Eq. (26) keeps then the same general form, except that the albedo a is substituted by an expression taking into account the existence of ice edge. Specifically (North, 1975a), denoting by x

g

the position of the latter and assuming symmetric hemispheres :

1 - a = a (x, x ) b.. , x > x 0 s

a

Q

+ a

2

P

2

( x ) , x <x

s

(31)

where bp is the absorption coefficient over ice or snow 50% covered with clouds, and ag, a^

a r e

the absorption coefficients over ice-free areas obtained after analyzing the albedo distribution by Legendre series. As usual, it is assumed that at x the temperature is of - 10°C.

-25-

(28)

A n appropriate solution of eq. (26) and (31) can be found by expanding T in series of Legendre polynomials. Truncation to the second mode gives (North, 1975a) :

dT

c

dtT

=

«

H

0

( x

s

}

" (A + B V

dT

c

d T

=

Q

H

2 -

( B + x

2

T

q

+ T

2

P

2 ( Xs)

= - 10 (32)

where TQ and T^ are again expressed in degrees centigrade, and

H ( x ) = (2n + 1) f dx S(x) a(x, x ) P (x) (33) m s J s m 0

m = 0,2

Note that, contrary to the preceding subsection, these equations are coupled through the variable x .

The entropy production, eq. (14), takes the same form as eq. (28) :

P

<*>

3

¥ ( 2 7 3 V v O j '

( 3 4 )

- 2 6 -

(29)

provided we again adopt the assumption 273 + TQ » i T g h A s it t u r n s out, the solution of eq. (32) completely justifies this assumption.

In order to analyze the time dependence of P(t) we solved numerically the initial value problem for eqs. (32) using the Hamin method. We first explored ( F i g . 4) the vicinity of the steady-state solution of these equations corresponding to the present-day climate.

For the numerical values given in the caption of Fig. 4, this state corresponds to T

Q

= 14.9°C, T

2

= - 28.2°C, x

g

= 0.96 and is asymptotically stable. Fig. 4 depicts the evolution of P ( t ) induced by a perturbed pole-equator gradient, keeping the planetary temperature invariant. We see that if T

2

Q = - 30°C P decreases in time, whereas for T£Q = - 26°C P increases until the present-day climate is recovered. We arrive therefore at the same qualitative behavior as in the preceding susbsection. We thus feel that there is no support for the claim (Golitsyn and Mokhov, 1978) that the stability properties of the climate should be linked to the extremal properties of entropy production. In

Fig. 5 we report the evolution of entropy production using the same parameter values as in Fig. 4, but starting from initial conditions simulating the last major glaciation (18.000 years B . C . ) . We know that in this case the ice caps went as far down as 57° in the Northern Hemisphere, and that the planetary temperature was less by about 5°C.

The ice boundary condition (third relation ( 3 2 ) ) allows then us to compute T2• T a k i n g x

s

= sin 57° = 0.84, T

Q

= 10°C, we find T

2

= - 35.8°C. A s seen from Fig. 5, the entropy production decreases then monotonously until the present-day climate is reached.

5. C O M P A R I S O N WITH P A L T R I D G E ' S I D E A S

The main focus of this paper was on the time-dependent

properties of entropy production in the vicinity of a steady-state

climatic regime. A s is usually done in the analysis of irreversible

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Time evolution of e n t r o p y production associated with model eq. ( 3 2 ) , u s i n g two different initial conditions for T„ and T = 14.9° C . In both

d o

cases the p r e s e n t climatic regime is recovered asymptotically. Numerical values used : 0 = \ (1360) W m " Q = ^ (1360) W m "

22

, b

Q

= 0.38, a

Q

= 0.697, a

0

c = 3.138 x 10

8

J m "

2

, S

2

= - 0.477, A = 214.2 W m "

2

A = 0.591 W m "

2

.

- 0.0779,

B = 1 .575 Wm

- 2

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0 50 100 t (years)

F i g . 5 . - Time e v o l u t i o n of e n t r o p y p r o d u c t i o n associated w i t h model e q . ( 3 2 ) u s i n g an i n i t i a l c o n d i t i o n ( x

g

= 0.84, T

q

= 10° C, ~i~

2

= - 35.8° C)

s i m u l a t i n g t h e last major g l a c i a t i o n . N u m e r i c a l v a l u e s used a r e as in

F i g . 2.

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p h e n o m e n a , b o t h t h e e q u a t i o n s o f e v o l u t i o n of t h e s t a t e - v a r i a b l e s a n d t h e e n t r o p y f u n c t i o n w e r e e v a l u a t e d b y i n t r o d u c i n g s u i t a b l e c o n s t i t u t i v e r e l a t i o n s l i n k i n g f l u x e s a n d f o r c e s . T h a n k s t o t h e s e r e l a t i o n s , t h e e n e r g y - b a l a n c e e q u a t i o n became c l o s e d , a n d allowed f o r an e x p l i c i t e v a l u a t i o n o f t h e t e m p e r a t u r e p r o f i l e a c r o s s t h e s y s t e m .

A n a l t o g e t h e r d i f f e r e n t a p p r o a c h was a d o p t e d in t h e w o r k b y P a l t r i d g e ( 1 9 7 5 , 1 9 7 8 ) . His main idea is t o use an u n c o n s t r a i n e d e n e r g y balance e q u a t i o n , w h e r e b y t h e e n e r g y f l u x is n o t l i n k e d t o t h e t e m - p e r a t u r e g r a d i e n t . A t t h e s t e a d y s t a t e a n d in t h e f r a m e w o r k of t h e o n e - d i m e n s i o n a l model a d o p t e d in t h e p r e s e n t w o r k , t h i s y i e l d s :

V + J

x

=

k

( 1

"

x 2 ) V 2 J

x

= Q ( 1

"

a ( x ) ) S ( x )

"

( A + B T )

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w h e r e - V

+

is t h e a d j o i n t g r a d i e n t o p e r a t o r . As n o t e d in sec. 2, t h e i n v e r s e o f t h e r a d i u s o f t h e e a r t h has been a b s o r b e d i n t o J . From t h i s

x r e l a t i o n one may e x p r e s s T as a f u n c t i o n o f J ^ :

Q ( 1 - a ( x ) ) S ( x ) - A - J

T =

:

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B

In t h i s w a y t h e e n t r o p y p r o d u c t i o n , e q . ( 8 ) , can be w r i t t e n e n t i r e l y in t e r m s o f t h e f l u x J :

x

r

1

j

P = - 2B J dx (37)

0 273B + Q ( 1 - a ( x ) ) S ( x ) - A - J

x

- 3 0 -

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Following P a l t r i d g e , we may now seek f o r t h e f u n c t i o n d^ e x t r e m i z i n g P. We obtain in t h i s way t h e f o l l o w i n g v a r i a t i o n a l e q u a t i o n , 6 P / 6 J

x

= 0

(1 - x2)1 / 2 J ° ( x ) = ƒ g ( x ) dx - ± ƒ g1 / 2( x ) dx ( 3 8 )

0 0

w h e r e

g ( x ) = Q(1 - a ( x ) ) S(x) - A + 273B (39a)

and t h e c o n s t a n t K is a d j u s t e d to g i v e zero f l u x at t h e poles :

f

1 1 / 2 f \ A

I g ( x ) dx

K = — j ( 3 9 b ) I g ( x ) dx

0

Fig. 6 depicts t h e e n e r g y f l u x o b t a i n e d b y a p p l y i n g t h i s p r o c e d u r e and b y u s i n g t h e parameter values adopted e a r l i e r in t h e p r e s e n t w o r k . T h e r e s u l t s are reasonable, b o t h as f a r as t h e position of t h e maximum and t h e behavior near t h e poles is c o n c e r n e d . On t h e o t h e r h a n d , one can show t h a t t h e c o r r e s p o n d i n g t e m p e r a t u r e p r o f i l e g i v e s e x c e s s i v e l y h i g h values at t h e e q u a t o r and low values at t h e poles, as a l r e a d y pointed o u t b y G o l i t s y n and Mokhov (1978).

I n d e p e n d e n t l y of these technical aspects h o w e v e r , the main p o i n t to be r e t a i n e d is t h a t e n t r o p y e x t r e m i z a t i o n dispenses us from u s i n g a c o n s t i t u t i v e relation e x p r e s s i n g J in terms of 3 T / 8 x and from f i t t i n g such c o e f f i c i e n t s as X in o r d e r to obtain t h e steady state format of p r e s e n t climate. T h u s , among all possible steady states t h a t may be realized b y t h e e a r t h - a t m o s p h e r e system u n d e r a g i v e n e n e r g y i n p u t ,

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LATITUDE (degrees)

F i g . 6. L a t i t u d e d e p e n d e n c e o f t h e e n e r g y f l u x d i v i d e d b y t h e e a r t h ' s r a d i u s ,

o b t a i n e d b y t h e e n t r o p y p r o d u c t i o n e x t r e m i z a t i o n ( e q . ( 3 8 ) ) . T h e

p a r a m e t e r v a l u e s u s e d a r e t h e same as in F i g . 4.

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t h e r e is one (,cf. e q . ( 3 8 ) a n d F i g . 6) w h i c h extremize-s t h e e n t r o p y p r o d u c t i o n . O t h e r s t e a d y s t a t e s , s u c h as t h o s e e v a l u a t e d in s e c t i o n 4, a r e p o s s i b l e . T h e y h a v e , h o w e v e r , a smaller d i s s i p a t i o n r a t e t h a n t h e s t a t e J ^ e q . ( 3 8 ) . T h e s i t u a t i o n is d e s c r i b e d i n F i g . 7.

6. C O N C L U D I N G REMARKS

O u r p r i n c i p a l goal in t h i s p a p e r was t o c a s t some basic f e a t u r e s r e l a t e d t o climate a n d i t s e v o l u t i o n i n t o t h e p r o p e r t i e s of e n t r o p y a n d e n t r o p y p r o d u c t i o n . We h a v e f o u n d t h a t t h e b e h a v i o r of t h e s e q u a n t i t i e s is f a r f r o m b e i n g simple a n d u n i v e r s a l , j u s t l i k e climate i t s e l f is f a r f r o m s h o w i n g simple a n d u n i v e r s a l t r e n d s . ' ' R a t h e r , i t a p p e a r s t h a t t h e d i r e c t i o n o f c h a n g e of e n t r o p y p r o d u c t i o n is c o n d i t i o n e d b y t h e i n i t i a l s t r e n g t h o f t h e e q u a t o r - p o l e t e m p e r a t u r e g r a d i e n t as c o m p a r e d t o t h a t of t h e f i n a l s t e a d y s t a t e . N o w , a more p r o n o u n c e d t h e r m a l g r a d i e n t is c h a r a c t e r i s t i c o f g l a c i a l p e r i o d s ( N e w e l l , 1 9 7 4 ) . We may t h e r e f o r e summarize t h e r e s u l t s o f S e c t i o n 4 b y s t a t i n g t h a t t h e e v o l u t i o n t o a g l a c i a t i o n is accompanied b y an e n h a n c e d r a t e of d i s s i p a t i o n , as m e a s u r e d b y t h e e n t r o p y p r o d u c t i o n . A n a d d i t i o n a l i l l u s t r a t i o n o f t h i s c o n c l u s i o n is p r o v i d e d b y a d i r e c t c o m p a r i s o n of e q . ( 2 8 ) a n d ( 3 4 ) . U s i n g p a l e o c l i m a t i c d a t a f r o m t h e mesozoic e r a ( N i c o l i s ( 1 9 7 9 ) ) we d e d u c e t h a t f o r an e q u a t o r i a l t e m p e r a t u r e of 25° a n d a p o l a r one o f 1 5 ° ,

p

1 2

^Past ^

p a s t , 5 (273 + 22)

w h e r e a s f o r t h e p r e s e n t i n t e r g l a c i a l c l i m a t e :

- 3 3 -

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Fig. 7 . - E n t r o p y p r o d u c t i o n s u r f a c e , I as a f u n c t i o n of t h e e n e r g y f l u x , J^ and of an a v e r a g e t e m p e r a t u r e g r a d i e n t , ] A T | . : line of u n c o n s t r a i n e d s t e a d y s t a t e s . A n example of such states is ( b ) , t h e state of maximum d i s s i p a t i o n , ( a ) , ( c ) : steady states o b t a i n e d a f t e r u s i n g a c o n s t i t u t i v e r e l a t i o n . In p a r t i c u l a r state ( a ) is t a k e n to be t h e p r e s e n t - d a y climate as g i v e n b y N o r t h ' s model discussed, in section 4. Possible time- d e p e n d e n t b e h a v i o r s of e n t r o p y p r o d u c t i o n are d e s c r i b e d in t h e v i c i n i t y of p o i n t s ( a

1

) , ( c

1

) of t h e s u r f a c e I . In p a r t i c u l a r , as shown in sec.

4 B , (a

1

) is a saddle p o i n t : f o r h i g h i n i t i a l 1 AT | P t e n d s to decrease, whereas t h e opposite is t r u e f o r small initial | A T | ' s . T h i s b e h a v i o r is not t o be c o n f u s e d w i t h t h e f a c t t h a t , among all possible s t e a d y s t a t e s , ( b ) is t h e one w i t h maximum d i s s i p a t i o n . In o t h e r w o r d s , P a l t r i d g e ' s v a r i a t i o n a l p r i n c i p l e p e r t a i n s to s t e a d y - s t a t e b e h a v i o r and not to t h e e v o l u t i o n in t h e v i c i n i t y of a s t e a d y s t a t e .

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12 X. ( - 2 8 . 2 ) 2

P Present

present

(273 + 1 4 . 9 ) 2 5

We see that the change in T

2

induces about a 16fold increase of P.

Certainly, A. cannot have varied in the opposite direction by a comparable amount. T h u s , the present interglacial climate appears to be more dissipative than the climate associated with an ice-free earth.

illustrations developed in section 4B are limited by several simplifica- tions. Perhaps the most serious one-which is in fact a limitation of all diffusive models used so far in the literature - is the assumption that x

g

, adjusts instantaneously to the value of T ( x ) . This introduces an unreaslistically fast time scale into the problem. Obviously, a natural boundary condition on the ice edge must be introduced in order to allow for the ice melting or advance in a self-consistent way (see also Pollard, 1978; Nicolis, 1980). A second limitation is the two-mode truncation adopted. This does not enable us to analyze the behavior of dissipation under the effect of localized disturbances from some reference state. Such local disturbances are certainly more realistic.

The time scale of evolution is also likely to be lengthened under these conditions.

properties of steady states illustrates the considerable degeneracy associated with the modeling of the meridional f l u x . A basic problem which remains open at this time is therefore to come up with criteria determing the selection mechanisms of a particular steady state climatic regime. Paltridge (1979) suggests that the role of fluctuations is likely to be instrumental. He believes that fluctuations are capable of introducing a d r i f t in state space, driving eventually the system to the state of maximum dissipation. A general answer to this major question is Although the results of section 2 and 3 are quite general, the

The discussion of Fig. 7 in connection with the thermodynamic

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however s t i l l l a c k i n g . It may be expected t h a t t h e systematic use of thermodynamics could p r o v e useful in t a c k l i n g t h i s problem.

ACKNOWLEDGMENTS

We are g r a t e f u l to G.W. P a l t r i d g e f o r e n l i g h t e n i n g discussions and to M. Ghil f o r p o i n t i n g o u t to us t h e w o r k b y G a l i t s y n and M o k h o v .

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NICOLIS, C. (1980), "One dimensional energy balance viewed as a free b o u n d a r y - v a l u e problem", J. Atm. S c i . , submitted.

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Figure

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