I N S T I T U T D ' A E R O I M 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 C i r c u l a i r e
B • 1180 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 4 2 - 1982
A boolean approach to climate dynamics
by
C. NICOLIS
B E L G I S C H I N S T I T U U T
B
V O O R R U I M T E - A E R O N O M I ' E 3 - Ringlaan
- 1180 BRUSSEL
F O R E W O R D
T h e p a p e r e n t i t l e d " A B o o l e a n a p p r o a c h to c l i m a t e d y n a m i c s " w i l l be p u b l i s h e d in T h e Q u a r t e r l y J o u r n a l of T h e R o y a l M e t e o r o l o g i c a l S o c i e t y , 108, 1982.
A V A N T - P R O P O S
L ' a r t i c l e i n t i t u l é : " A B o o l e a n a p p r o a c h to climate d y n a m i c s "
s e r a p u b l i é d a n s T h e Q u a r t e r l y J o u r n a l of T h e R o y a l M e t e o r o l o g i c a l S o c i e t y , 108, 1982.
V O O R W O O R D
Het a r t i k e l : " A B o o l e a n a p p r o a c h to c l i m a t e d y n a m i c s " zal v e r s c h i j n e n in het t i j d s c h r i f t T h e Q u a r t e r l y J o u r n a l of T h e R o y a l M e t e o r o l o g i c a l S o c i e t y , 108, 1982.
V O R W O R T
Die A r b e i t : " A B o o l e a n a p p r o a c h to c l i m a t e d y n a m i c s " w i r d in T h e Q u a r t e r l y J o u r n a l of t h e R o y a l M e t e o r o l o g i c a l S o c i e t y " , 108, 1982 h e r a u s g e g e b e n w e r d e n .
Samenvatting
Een van de hoofdkenmerken van ons klimatologisch systeem is het bèstaan van talrijke verschijnselen van feedback en van discontinue overgangen. Een studie door middel van discontinue veranderlijken in een Boole logica kan met succes gebruikt worden om die verschijnselen te ontleden. Een dergelijke benadering wordt eerst toegepast op een- voudige voorbeelden, en de resultaten worden vergeleken met deze bekomen na toepassing van het meer bekende continue formalisme. De kwalitatieve inzichten die uit een dergelijke analyse voortvloeien, worden naderhand toegepast op meer ingewikkelde gevallen.
Zusammenfassung
Eine der Haupteigenschaften des klimatischen Systems ist die Existenz zahlreicher Rückkopplungs- und Sprungsysteme. Es wird vorgeschlagen, für die Analyse dieses Verhaltens eine unstetige Formulierung auf Grundlage der Booleschen Logik zu verwenden. Diese Methode wird zunächst auf einfache Beispiele angewendet, und ihre logischen Voraussagen mit den Ergebnissen des bekannteren stetigen Formalismus verglichen. Die qualitativen Einsichten, die eine derartige Analyse liefern kann, werden dann an Hand komplizierterer Situationen erläutert.
1. INTRODUCTION
The complexity and diversity of the radiative and transport processes going on at a variety of space and time scales in the earth- atmosphere system is well known. For this reason many relations linking the rates of the various phenomena (like, for instance energy fluxes) to the driving forces (temperature differences, etc.) are often based on the fitting of satellite data, or on analogies drawn from simple physical models. One element that appears to recur frequently is, undoubtedly, the existence of sharp transitions : jump or threshold phenomena. The dependence of surface albedo on temperature (Schneider and Dickinson, 1974) is an important example. Beyond such thresholds the rate of the process levels off to some constant value which is altogether different from the value prevailing well below threshold. Quite often, the interval of transition between these two plateau values is fairly small with
respect to the average value of the corresponding variable. Thus, a small change around the present-day mean temperature or precipitation will certainly bring dramatic alterations, as suggested also by the available record of successive glaciation periods (Lamb, 1977).
Hitherto, the modelling of the earth-atmosphere system at the climatic scale has been based primarily on simplified energy and mass balance models, or on general circulation models. It is the purpose of the present paper to show that a natural tool for handling the numerous threshold phenomena encountered in climate modelling is the boolean approach. An additional advantage, which we shall get for free, is that such an approach leads to qualitative conclusions without anticipating the validity of specific (and often poorly justified) phenomenological relationships between rates and constraints. Moreover, it allows one to assess the effect of connectivity and topology of a network of interacting elements on the system's behavior.
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Boolean logic was widely used in the study of other complex phenomena such as neural or genetic networks (Glass and Kauffman, 1972 and references therein). Of special relevance for our purposes however, is the formulation developed since 1973 by Thomas (see for instance Thomas, 1977 and 1978). We briefly summarize the main ideas.
Let x be a state variable. Traditionally, one regards the change of x in time as a continuous' process and writes a rate equation of the form
£ = f (x, A) (1)
where f is generally nonlinear in x and depends, in addition, on a number of external parameters A typical example of Eq. (1) is the mean annual energy budget of the earth-atmosphere system. The variable x then represents the mean annual surface temperature, whereas \ may stand for such quantities as the solar constant, the albedo of land or ice, and so forth.
In most problems of interest the rate function f can be split into two parts corresponding to qualitatively different mechanisms : a highly nonlinear term X ( x , \ ) describing the specific feedbacks inherent in the system's dynamics, and a quasi-linear, non-specific decay term. The latter usually ensures the existence of stationary solutions to Eq. (1). We thus write
| | = X (x, \) - k x (2)
where k is some appropriate (positive) decay constant. For the illustra- tion of Eq. (1) given above X would stand, for instance, for the effect
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of thé surface-albedo feedback and k for the infrared cooling coefficient.
Consider now the following quite typical situation. Assume that for 0 < x < Xq / X is v e r y small but that for x > Xq it suddenly becomes appreciable and saturates shortly thereafter to some threshold value X|y|ax- To capture the essence of this response, we regard X as a discontinuous, boolean function of the discontinuous boolean variable x . In other words, we consider that in the interval O < x < Xq both x and X are zero whereas for x > Xq, x and X are equal to one ( c f . Fig. 1 ) . We will say that x is "off" and "on" respectively when x is below and beyond the threshold value x . Similarly for X . It may of course happen thai Intermediate levels between O and the maximum value have to be considered. An example is analyzed later on in the present paper.
Note t h a t , because of the existence of the second term in Eq. ( 2 ) , the temporal evolution of a function and of its corresponding variable will differ by a phase-shift, usually called delay, expressing the fact that when a function X becomes "on" the corresponding variable x will build up only after some characteristic delay t depending on the value of k in Eq. ( 2 ) . Similarly, when X is "off" x will not disappear until another characteristic delay t - has elapsed ( F i g . 2 ) .
Moré generally, let x , y , z, . . . be boolean variables taking the values O or 1 according as the quantity they represent has negligible or appreciable values. The boolean functions X , Y , Z , ' . . . will take the value 1 when a certain set of "commanding variables"is building up or remains "on" and the value O otherwise. We again emphasize that the functions X , Y , Z , . . . are taken to include only the feedbacks exerted in the system. It is assumed that there always exists a quasi-linear, non specific mechanism of the decay of x , y , z, . . . This latter mechanism is not accounted for in the logical equations describing the system, but is incorporated in the delays.
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Fig. 1 . - Boolean idealisation of a sigmoidal dependence b y a step f u n c t i o n .
TIME
2 . - Schematic representation of the temporal evolution of a boolean function X and its associated variable x .
Having adopted this representation, we immediately see that a boolean state in which all variables take the same values as the corresponding functions is a stable steady state. Indeed, when such a condition is satisfied neither the variables nor the functions will need to change further in time. A s r e g a r d s the succession of the different states in time, d u r i n g the approach to the stable steady state or to some other attractor, it will be dictated by appropriate logical equations, which link X , Y , Z, . . . to x, y , z, and are the substitutes of the differential equations of e n e r g y , mass, or momentum balance written down in the traditional, continuous formalism. In writing such logical equations it will be useful to recall the following conventions :
x not x x . y x and y
x + y x or y (inclusive)
We do not d i s c u s s the general theory any further, but rather proceed with applications to concrete situations of interest in climate modelling. We shall limit ourselves here to globally averaged, zero dimensional ( O - d ) models.
In section 2 the logical formalism is applied to simple climatic models, and the results are compared with the predictions of the continuous formalism. Section 3 is devoted to more complex situations for which no continuous description has been set up so far. A brief discussion is presented in section 4.
2. L O G I C A L A N A L Y S I S OF S I M P L E M O D E L S
A. Surface albedo feedback
Let 8 be the surface temperature. Within the framework of a O - d energy balance model only the surface albedo feedback is retained, which is positive. It expresses the fact that when 8 increases, its rate of increase T is further enhanced because of the additional absorption of solar energy arising from the retreat of ice. Hence we write the logical scheme and the corresponding logical equation
T = 8 (3)
To analyse the predictions of this equation we a r r a n g e the states of the system in the form of a matrix or truth table, as follows
8 T
0
®
1
®
For each level of the variable 8, the corresponding rate of change T is deduced from the logical equation ( 3 ) . The circled elements in the matrix correspond to stable states. We see that we may obtain up to two such stable states. Of these, the high temperature state can be regarded as representing a favorable ( p r e s e n t - d a y like) climate, whereas the low temperature one is descriptive of a cool, unfavorable
climate. T h i s agrees with the prediction of the continuous model analyzed by Crafoord and Kallen (1978) or of any other O - d model where the albedo-temperature relationship saturates at low and high temperature values.
It should be mentioned that in the continuous analysis of this type of models a third, unstable steady state is obtained which lies between the two stable ones. In the discontinuous analysis unstable states can only manifest themselves whenever they can be approached through some exceptional path, like e . g . the separatrix associated to a saddle point. In our one-variable system such a path is excluded. As a result the boolean analysis cannot account for the intermediate unstable state, which is physically Insignificant anyway.
B. Effect of ice sheets
The positive feedback, eq. ( 3 ) , may in fact be attributed to the interaction between energy and ice sheet balance equations. Let SL be the mean annual latitudinal extent of ice sheets. The simplest such interactions are that 0 and SL exert a negative feedback on each other.
T h i s is represented by :
or
( 4 a )
L = 8 ( 4 b )
2
The system has now 2 = 4 states, and the analogue of the matrix of the preceding example becomes :
- 1 0 -
6 £ T L
0 0 1 1
0 1 o
1 1 0 0
1 0 Q
We f i n d t w o s t a b l e s t e a d y s t a t e s c o r r e s p o n d i n g , r o u g h l y , t o an " i c e c o v e r e d " a n d an " i c e f r e e " e a r t h . In a c t u a l f a c t , d e p e n d i n g on t h e v a l u e of t h e t h r e s h o l d s e p a r a t i n g t h e t w o r e g i m e s , " i c e c o v e r e d " may mean a climate w i t h a l a r g e ice e x t e n t , a n d " i c e f r e e " a w a r m climate w i t h small o r no ice e x t e n t . T h i s r e s u l t is in a g r e e m e n t w i t h t h e c o n c l u s i o n o f o u r p r e v i o u s e x a m p l e , t h u s i l l u s t r a t i n g a g e n e r a l t h e o r e m b y Thomas (1978) t h a t a logical loop i n v o l v i n g an e v e n n u m b e r of n e g a t i v e elements b e h a v e s q u a l i t a t i v e l y as a p o s i t i v e o n e . Note t h a t an i n i t i a l c o n d i t i o n may lead t o e i t h e r of t h e s e t w o s t a b l e s t a t e s , d e p e n d i n g on t h e r e l a t i v e v a l u e s o f t h e d e l a y s . For i n s t a n c e s u p p o s e t h a t we s t a r t w i t h a low t e m p e r a t u r e " i c e f r e e e a r t h " ( 0 , 0 ) . A c c o r d i n g t o t h e f i r s t r o w o f t h e logical m a t r i x , b o t h 8 a n d £ level w i l l t e n d t o c h a n g e . As each o f t h e s e c h a n g e s has most p r o b a b l y d i f f e r e n t d e l a y v a l u e s a s i m u l t a n e o u s s w i t c h i n g t o ( 1 , 1 ) is i m p r o b a b l e . T h u s , t h e s y s t e m w i l l t e n d t o r e l a x t o one of t h e t w o s t a b l e s t a t e s a c c o r d i n g t o
T h e 00 -» 10 p a t h w a y w i l l be p r e f e r r e d w h e n t h e d e c a y time of 6 in Eq.
( 2 ) is f a s t e r , w h e r e a s 00 01 w i l l be p r e f e r r e d in t h e case in w h i c h t h e d e c a y time o f £ is f a s t e r .
k
00
- 1 1 -
It should of course be emphasized that the interaction between the two variables is not direct, but takes place t h r o u g h such processes as ablation, precipitation etc. We come to these points presently.
C . Free oscillations
land ice sheet balance at the planetary scale has been recently suggested and analyzed in detail by Kallen et al. (1978 and 1979). One of the most important points in their model concerns the temperature dependence of the ratio of accumulation over ablation rates. They s u g g e s t that in some temperature range there must be a positive feed- back exerted by the temperature on the land ice extent, since accumula- tion implies precipitation and snow fall which is very small for v e r y low temperatures. In this way they find that the coupled atmosphere-hydro- s p h e r e - c r y o s p h e r e system can undergo self-oscillations with periods comparable to those of the glaciation times.
According to the energy balance equation set up by the authors, there exists a large temperature range in which temperature 0, exerts a positive feedback on itself via the sea ice extent, in a way analogous to the one examined in case A. In the same range however, the land ice extent £, has the opposite effect on 6. The logical scheme for the energy balance equation alone is therefore the following
A set of two coupled differential equations for the energy and
Let us outline the logical formulation of this problem.
or
T = 6 + £ (5)
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On t h e o t h e r h a n d , s i m i l a r l y to case B above, t e m p e r a t u r e e x e r t s a n e g a t i v e feedback on land ice £ t h r o u g h a b l a t i o n ; t h i s becomes e f f e c t i v e beyond a c e r t a i n t h r e s h o l d v a l u e , say 8^. In a d d i t i o n , when g l o b a l l y a v e r a g e d t e m p e r a t u r e s a t t a i n a h i g h e r l e v e l , say Q^ (0£ > 0 ^ ) , s i g n i f i c a n t e v a p o r a t i o n can t a k e place and c o n s e q u e n t l y p r e c i p i t a t i o n and snow fall rates are h i g h . T h u s , t h e o v e r a l l e f f e c t is a p o s i t i v e feedback of 0 on £ t h r o u g h accumulation. S t i l l , an excessively large size of the ice sheet will slow down its f u r t h e r increase because of t h e parabolic p r o f i l e of the sheets (Weertman, 1964).
T h e f a c t t h a t c e r t a i n feedbacks change sign a c c o r d i n g to t h e state of the system suggests to decompose 0 into at least t h r e e levels Corresponding to the two t h r e s h o l d s 0^ and 0 T h e logical schemes and the c o r r e s p o n d i n g logical equation f o r the rate of change L of the land ice e x t e n t are g i v e n by
01 £ and 02
O -
L = §1 + 02 . £ (6a)
The s u b d i v i s i o n of 0 into levels r e q u i r e s a s l i g h t modification of Eq.
( 5 ) . It is replaced b y the set of equations
T1 = ( 01 + £) + 02 (6b)
T2 = T1 • (02 + £) (6c)
The feedbacks i n t r i n s i c to each level are 01 + £ and 0^ +£, respec- t i v e l y . Eq. ( 6 b ) expresses the f a c t t h a t 0^ is automatically " o n " if 0^ is
" o n " ; s i m i l a r l y in Eq. ( 6 c ) , T2 cannot be " o n " unless T1 has already
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been activated. In other words, the system cannot get at level without getting first through level .
We are now in position to write down the logical matrix for 3 this problem. Of the 2 = 8 states of the system only 6 are acceptable, because as we saw previously 62 cannot be "on" when is "off". We thus obtain the following representation of Eqs. (6).
el 02 SL T1 T 2 L
0 0 0 1 1 1
0 0 1 0
V
1 0 0 1 1 0
1 0 1 1 0 0
1 1 0 1 1 1
1 1 1 1 1 0
We find one stable state corresponding to a v e r y low tem- perature earth with an important land ice extent. The new point however is that there exists no boolean stable state at higher tem- peratures. Instead, the system, as we will see, can perform a cycle of the form
1 1 0 — 1 1 1
T o . show this, suppose that we start with the initial condition (1 1 0).
According to the logical matrix (row before last), SL is then bound to change. Once in state 1 1 1 however, the last row commands that SL be changed again. T h i s cycle suggests therefore the possibility of an oscillation in the vicinity of the 0? level with an important advance and
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retreat of the land ice extent. In actual fact, according to the thresh- old values given by Kallen et al, this range of 0 happens to be between the present day climate and the glacial one.
A still finer description of the system, permitting a closer agreement with the quantitative results of Kallen et al. would be to include four 0 levels, thus extending the range to even higher global temperature values. One finds an additional stable state in the high temperature range as well as the possibility of a competition between this state and the self-oscillation. These conclusions are in qualitative agreement with the results of the original paper.
3. L O G I C A L A N A L Y S I S OF A C O M P L E X NETWOKK
Having illustrated the boolean method on known simple examples, we proceed to set up plausible models for more complex situations.
Consider for instance, the still debatable coupling between global temperature 0, relative humidity h, and cloudiness n. In addition to t h e positive feedback of 0 into itself discussed already in our previous examples, we expect a negative feedback of h into itself because of the slowing down of evaporation when relative humidity increases. A positive effect of h on 0 expresses the greenhouse effect and a positive effect of h on n expresses the simple idea that the cloud build up is facilitated in a medium of high h. T h e r e remain some un- certainties in the effect of 0 on h and of n on 0 (0 is taken to act on n only indirectly through h). A c c o r d i n g to Sasamori (1975), 3h/90 > 0.
On the other hand, using a general circulation model Roads (1978) suggests that the effect should be just in the opposite direction. As regards the n - 0 interaction, it actually depends on the type of cloud and on latitude (Paltridge, 1980 and references therein). Morevover, it is often suggested (Paltridge and Piatt, 1976) that n should have a globally negative effect on 0, in order to ensure stability.
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We have examined the consequences of all these conjectures.
Hereafter we only report the results in the case where 8n/80 < 0, and 9h/30 > 0. A plausible logical scheme incorporating the above interac- tions is as follows :
or T = 9 + h i n H = e.ii
N = h (7)
The corresponding matrix now comprises 2 = 8 states and is
e h ii T H N
0 0 0 1 0 0
0 0 1 0 0 0
0 1 1 1 0 1
0 1 0 0 1
l 0 0 1 1 0
l 0 1 1 1 0
l 1 1 0 1
l 1 0 1 0 1
It gives no boolean stable state but predicts, as we see presently, the possibility of a surprising variety of oscillatory behavior.
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As an e x e r c i s e , suppose t h a t the i n i t i a l state is a climate c h a r a c t e r i z e d b y a h i g h t e m p e r a t u r e , h i g h h u m i d i t y level but no ( o r small) cloud e x t e n t ( s t a t e 110). The logical m a t r i x ( l a s t r o w ) commands t h a t both h u m i d i t y and cloudiness are b o u n d to change. As the c h a r a c t e r i s t i c times ( o r d e l a y s ) of these two processes are most p r o b a b l y d i f f e r e n t , a simultaneous change of h and n is to be r u l e d o u t . I n s t e a d , the system is g i v e n the f o l l o w i n g choice :
As in Section 2B, the p a r t i c u l a r path to be followed will depend on the r e l a t i v e values of the d e l a y s , k^ and k ^ . In o t h e r w o r d s , state 100 will be reached if the decay time of h is f a s t e r than t h a t of n . in the opposite case state 111 will be reached.
Suppose now t h a t the f i r s t t r a n s i t i o n is realized and the system is at state 100. Going back to t h e logical m a t r i x we see that h is again bound to change. C o n s e q u e n t l y , the following o s c i l l a t o r y behavior is expected
110 — 100
If on the c o n t r a r y k^ > k1 and state 111 is reached, one will have the scenario
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110 - 111 101
T h e following graph summarizes the various possibilities :
4 4 0 ^ ^
4 » A
It should be by now clear that the main difficulty in the analysis of complex networks is the choice between different pathways. As pointed out above the answer to this question requires a careful examination of the delays and will be f u r t h e r discussed elsewhere in a more technical communication (Nicolis, 1982), along with the analysis of other models.
In any case one is tempted to correlate such cycles with the presence of periodicities in the climatic record (Lamb, 1977).
4. DISCUSSION
In this paper we tried to convey our conviction that a qualitative analysis of the boolean type is likely to add new insights in the understanding of the complex problems of climate modelling. Such an analysis can in no way be a substitute to more quantitative studies.
Rather, we envisage a complementary approach based on the use of both boolean and more quantitative types of models. When a complex system is considered and some rough consensus on the types of feed- backs involved is reached, it would be appropriate to f i r s t look at it as a network of interacting binary elements, and to sort out the different potentialities t h r o u g h the analysis outlined in the present paper. If some of these behaviors agree qualitatively with observation, then one
0 0
44 4
•v 4 0
- 1 8 -
can r e a s o n a b l y e x p e c t t h a t t h e f e e d b a c k s a d o p t e d in t h e logical t r e a t m e n t a r e t h e r i g h t o n e s . A more r e f i n e d s t u d y based on c o n t i n u o u s models a n d u t i l i z i n g t h e s e f e e d b a c k s w o u l d t h e n e n a b l e one to go f u r t h e r a n d r e a c h q u a n t i t a t i v e a g r e e m e n t . If on t h e o t h e r h a n d t h e outcome o f t h e boolean a n a l y s i s c o n t r a d i c t s o b s e r v a t i o n , one w o u l d h a v e a c o m p e l l i n g e v i d e n c e t h a t t h e c o n c e n s u s r e a c h e d was not j u s t i f i e d . A new set o f h y p o t h e s e s w o u l d be n e c e s s a r y , a n d t h e i t e r a t i v e p r o c e s s w o u l d c o n t i n u e u n t i l a q u a l i t a t i v e a g r e e m e n t is r e a c h e d .
Needless t o s a y , t h e c l i m a t i c p r o b l e m s a n a l y z e d in t h e p r e s e n t p a p e r have been r e l a t i v e l y simple a n d s t r a i g h t f o r w a r d . S e v e r a l e x t e n s i o n s m o t i v a t e d b y o u r r e s u l t s can h o w e v e r be e n v i s a g e d . Of p a r t i c u l a r i n t e r e s t w o u l d be t h e p o s s i b i l i t y to h a n d l e s y s t e m s i n v o l v i n g a l a r g e n u m b e r of c o u p l e d v a r i a b l e s . S u c h s i t u a t i o n s a r e i n t r a c t a b l e b y c o n t i n u o u s m o d e l s , b u t remain w i t h i n t h e l i m i t s of f e a s i b i l i t y of t h e boolean a p p r o a c h . N e v e r t h e l e s s , as t h e n u m b e r of s t a t e s i n c r e a s e s
N
r a p i d l y w i t h t h e n u m b e r of v a r i a b l e s N ( e s s e n t i a l l y l i k e 2 ) , n u m e r i c a l t e c h n i q u e s w i l l be an i n d i s p e n s a b l e complement of t h e a p p r o a c h . A p r o m i s i n g tool a p p e a r s to be t h e use of logical m a c h i n e s , l i k e t h o s e i n v e n t e d r e c e n t l y b y F l o r i n e ( 1 9 7 3 ) , Van Ham ( 1 9 7 7 ) a n d c o w o r k e r s .
ACKNOWLEDGEMENTS
We g r e a t l y b e n e f i t e d f r o m d i s c u s s i o n s w i t h R. T h o m a s . We a r e g r a t e f u l t o M. G h i l f o r a c r i t i c a l r e a d i n g o f t h e m a n u s c r i p t . T h i s w o r k is s u p p o r t e d , in p a r t , b y t h e EEC u n d e r c o n t r a c t n ° C L I - 0 2 7 - B ( G ) .
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