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 - Avenue Circulaire 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° 278 - 1984
Self-oscillations and predictability in climate dynamics by
C. NICOLIS
B E L G I S C H I N S T I T U U T V O O R R U I M T E A 3 - Ringlaan
B • 1180 BRUSSEL
E R 0 N 0 M I E
S E L F - O S C I L L A T I O N S AND P R E D I C T A B I L I T Y IN CLIMATE DYNAMICS
by
C. NlCOLIS
Abstract
A class of nonlinear climatic models giving rise to sustained oscillations is considered. The evolution equations are cast into a
"normal form" which allows one to distinguish between two different types of quantities : a radial variable, which obeys a closed equation and has strong stability properties; and a phase variable which is extremely labile. It is shown that as a result of these peculiar stability properties, the disturbances or random fluctuations acting on the system tend to deregulate the oscillatory behavior. It is concluded that such a phenomenon is a basic reason for progressive loss of predictability.
Résumé
Dans ce travail on étudie certaines propriétés des modèles climatiques simples donnant lieu à des solutions périodiques dans le temps. Dans ce but les équations d'évolution du système sont trans- formées en une "forme normale" faisant apparaître deux types de variables : l'amplitude de l'oscillation et sa phase. On montre que la première de ces variables est extrêmement stable vis à vis des fluctuations aléatoires d'origine interne ou externe tandis que la seconde est caractérisée par une grande labileté. Le lieu entre ce comportement et la prédictabilité climatique est analysé. Les résultats généraux sont illustrés sur un modèle tenant compte de l'interaction entre glace marine et température océanique.
FOREWORD
The article entitled "Self-oscillations and predictability in climate dynamics" will appear in Tellus.
AVANT-PROPOS
L'article intitulé "Self-oscillations and predictability in climate dynamics" sera publié dans Tellus.
VOORWOORD
Het artikel "Self-oscillations and predictability in climate dynamics" zal in Tellus gepubliceerd worden.
VORWORT
Der Artikel "Self-oscillations and predictability in climate dynamics" wird in Tellus herausgegeben werden.
S a m e n v a t t i n g
In dit w e r k w o r d e n bepaalde e i g e n s c h a p p e n v a n n o n - l i n e a i r e klimaatmodellen b e s t u d e e r d die aanleiding g e v e n tot periodieke o p l o s s i n g e n in de tijd. De e v o l u t i e v e r g e l i j k i n g e n v a n het systeem w o r d e n omgezet in een "normale v o r m " waardoor 2 soorten variabelen ontstaan : de radiaalvariabele en de fasevariabele. Er w o r d t aangetoond dat de eerste bijzonder stabiel is t . o . v . willekeurige interne of e x t e r n e schommelingen, terwijl de tweede zeer labiel is. Het v e r b a n d t u s s e n dit g e d r a g en de klimaatvoorspelling w o r d t b e s t u d e e r d . De algemene resultaten w o r d e n op een model g e ï l l u s t r e e r d , r e k e n i n g h o u d e n met de w i s s e l w e r k i n g t u s s e n zee-ijs en oceaantemperatuur.
Z u s a m m e n f a s s u n g
In diese A r b e i t werden bestimmte E i g e n s c h a f t e v o n n i c h t - lineare klimatische Modellen s t u d i e r t die A n l a s s g e b e n zu p e r i o d i s c h e L ö s u n g e n in der Zeit. Die E v o l u t i o n g l e i c h u n g e n vom S y s t e m werden ungestellt in einer "normalen Form" w o d u r c h zwei V e r ä n d e r l i c h e n entstehen : die Radialveränderliche u n d die P h a s e v e r ä n d e r l i c h e . Es wird g e z i g t d a s s die erste s e h r stabil ist mit R ü c k s i c h t auf willkürlichen internen oder e x t e r n e n S c h w a n k u n g e n , w ä h r e n d der zweite s e h r labil ist. Der Z u s a m m e n h a n g zwischen diesem Benehmen u n d der klimatische V o r a u s s a g e wird s t u d i e r t . Die Resultate werden in einem Modell illustriert, r e c h n e n d mit der W e c h s e l w i r k u n g z w i s c h e n Seeeis u n d O z e a n - temperatur.
1. I N T R O D U C T I O N
In r e c e n t y e a r s i t has been p o i n t e d o u t b y s e v e r a l a u t h o r s t h a t simple c l i m a t e models a r e c a p a b l e of p r o d u c i n g a u t o n o m o u s s u s -
3
t a i n e d o s c i l l a t i o n s . C o n s i d e r f i r s t t h e r a n g e o f 10 y e a r s w h i c h , as p o i n t e d o u t . b y B e r g e r ( 1 9 8 1 ) , is an u p p e r b o u n d o f t h e i n t r i n s i c t i m e scale of v a r i a t i o n o f sea ice e x t e n t . Saltzman a n d c o w o r k e r s ( 1 9 7 8 , 1980, 1981, 1982a) a n a l y z e d t h e i n t e r a c t i o n s b e t w e e n t h e l a t t e r a n d t h e ocean s u r f a c e t e m p e r a t u r e . T h e y s u g g e s t e d t h e e x i s t e n c e o f a n e g a t i v e f e e d b a c k loop a r i s i n g f r o m t h e p o s i t i v e ( " i n s u l a t i n g " ) e f f e c t o f sea ice on t e m p e r a t u r e , a n d f r o m t h e n e g a t i v e e f f e c t of t e m p e r a t u r e on sea i c e . T h e y c o n c l u d e d t h a t s u c h a s y s t e m s h o u l d g i v e r i s e t o a u t o n o m o u s s u s t a i n e d o s c i l l a t i o n s , a n d p r o p o s e d a mathematical model f o r c u c h a b e h a v i o r . For one set o f i n d i c a t i v e p a r a m e t e r v a l u e s t h e p e r i o d i c i t y 3 t u r n s i n d e e d o u t t o be of t h e o r d e r o f 10 y e a r s .
4 5
C o n s i d e r n e x t t h e r a n g e of 10 - 10 y e a r s , w h i c h b r i n g s us t o t h e time scale of q u a t e r n a r y g l a c i a t i o n s . Saltzman et al ( 1 9 8 2 b ) w o r k e d o u t an e x t e n d e d v e r s i o n o f t h e a b o v e m e n t i o n e d model w h i c h , f o r a n o t h e r set of p a r a m e t e r v a l u e s , g i v e s r i s e t o s e l f - o s c i l l a t i o n s of p e r i o d s c o m p a r a b l e t o t h e g l a c i a t i o n time s c a l e s . A l o n g p e r i o d autonomous o s c i l l a t i o n , i n v o l v i n g ice s h e e t d y n a m i c s has also been s t u d i e d b y Kallen et al (1979) a n d B h a t t a c h a r y a a n d G h i l ( 1 9 8 2 ) .
Let us now examine t h e c l i m a t i c r e c o r d t o see i f i t c a r r i e s some t r a c e of s u c h p e r i o d i c i t i e s . A s Saltzman et al ( 1 9 8 1 ) p o i n t o u t an u n d u l a t o r y v a r i a t i o n of sea ice e x t e n t p r e s u m a b l y o c c u r r e d at least once w i t h i n t h e l i t t l e ice a g e . A s i d e f r o m t h i s h o w e v e r , t h e r e seems t o e x i s t no c o m p e l l i n g e v i d e n c e of c l e a r c u t c y c l i c c l i m a t i c c h a n g e on t h e scale o f
q 4 5 10 y e a r s . T h e s i t u a t i o n is q u i t e d i f f e r e n t f o r t h e scale of 10 - 10 y e a r s . In p a r t i c u l a r , i t is well k n o w n (see e . g . B e r g e r , 1981) t h a t q u a t e r n a r y g l a c i a t i o n s p r e s e n t a c y c l i c c h a r a c t e r w i t h a d o m i n a n t p e r i o d i -
- 3 -
city of 105 years. T h i s difference is reflected in a s t r i k i n g manner on the structure of variance spectra of paleoclimatic time series : T h e values around the frequencies corresponding to the 10
3
year range are indeed by orders of magnitude smaller than the values corresponding to variability in the 10 - 10 range. 4 5
One is naturally tempted to correlate these facts with the apparent lack of systematic external forcings acting on the climatic
3
system in the range of 10 y e a r s , and with the presence of a systematic astronomical forcing arising from the earth's orbital variations in the range of 10^ years. In other words : Might it not be that the absence of a " s y n c h r o n i z i n g agent" such as an external forcing, compromizes the existence of a relatively well defined cyclic variability despite the potential existence of autonomous oscillations?
The purpose of the present paper is to s u g g e s t that the answer to the above question is in the affirmative, for a reason that is deeply rooted in the dynamics of any nonlinear nonconservative oscillator operating in the absence of a systematic forcing. In Section 2, the equations for the evolution of such an oscillator are cast into a
"normal form" which allows us to undertake a qualitative study largerly independent of the details of the particular model. In Section 3 it is observed that from this transformation a basic difference emerges between the order parameter, that is to say the variable that evolves at the slowest time scale, and the phase variable of the oscillator.
Specifically while the order parameter shows a pronounced stability, the phase variable is extremely labile. A s a result, there is an inherent tendency to desynchronization, chaotic behavior and loss of predictabil- ity. In Section 4 the phenomenological description is enlarged to take the fluctuations of the climatic variables into account. We show that de- synchronization is reflected by the existence of a stationary state for the probability distribution as a result of which, statistically speaking, oscillation disappears altogether. Final comments are presented in Section 5. T h r o u g h o u t the work the general ideas are illustrated on Saltzman's model oscillator.
2. NORMAL FORM AND ORDER P A R A M E T E R OF A NONLINEAR O S C I L L A T O R
Let { X . } , i = 1, . . . , N be a set of climatic variables, {X?1} a reference regime corresponding to a steady state. T h e evolution of the excess variables
x . = X. - X« (1) i l l
can be written in the generic form
dx . N
" d i = 1 3i j Xj + * i ( { XJ} ) ' 1 = ••• N ( 2 ) j = l
in which the coefficients of the linearized part a., depend on the reference state and gj contain the effects of nonlinearity. Suppose that system ( 2 ) predicts the existence of sustained oscillations. In a typical situation this will be reflected by the fact that among the N eigenvalues of the matrix {3.^}, 2 will be complex conjugate with positive real p a r t s , and the real parts of the remaining N - 2 will be negative. It is known from the qualitative theory of differential equations (see e . g . Arnold, 1980) that under these conditions the number of variables contributing effectively to the dynamics can be greatly reduced. Specifically, close to the bifurcation point and in the limit of long times we can cast eqs.
( 2 ) in the form
I t = b n n + bi 2 0 + hi <n, 0)
f = b2 1 n + b2 2 0 + h2 (n, e> (3)
i n w h i c h t h e e i g e n v a l u e s of t h e 2 x 2 m a t r i x { b j . } a r e t h e t w o p r i v - i l e g e d e i g e n v a l u e s o f {a^.} r e f e r r e d t o a b o v e . T h e " m a s t e r v a r i a b l e s " n a n d 0 a r e a p p r o p r i a t e l i n e a r c o m b i n a t i o n s o f t h e x1 s , w h i l e t h e r e m a i n i n g N - 2 v a r i a b l e s o f t h e o r i g i n a l p r o b l e m a r e e x p r e s s e d e n t i r e l y i n t e r m s o f n a r |d 0-
A p a r t i c u l a r e x a m p l e o f ( 2 ) - ( 3 ) , w h i c h w i l l be u s e d f r e q u e n t l y in t h e sequel f o r i l l u s t r a t i v e p u r p o s e s is t h e sea ice ocean s u r f a c e t e m p e r a t u r e model d e v e l o p e d b y Saltzman et al ( 1 9 8 1 ) :
i ? = " n + §
d e = - n + «1», e - «1», n2 e ( 4 )
in w h i c h all p a r a m e t e r s a r e p o s i t i v e , n is t h e d e v i a t i o n o f t h e sine of t h e l a t i t u d e of t h e sea ice e x t e n t f r o m t h e s t e a d y s t a t e , a n d 6 is t h e excess mean ocean s u r f a c e t e m p e r a t u r e . A c t u a l l y , i n s t e a d of ( 4 ) we w i l l p r e f e r t o w o r k w i t h a d i m e n s i o n l e s s f o r m r e s u l t i n g f r o m t h e f o l l o w i n g s c a l i n g t r a n s f o r m a t i o n :
1/2 0
3 / 2n , e = = ( 5 ,
T h e r e s u l t is :
? = - n + 0
dt 1
d0 = _ V l + _ 2 ' ( 6 )
dt 02 n <t»2
- 6 -
We see that the dynamical behavior depends on two dimensionless para- meters only, playing, a role analogous to that of Rayleigh or Prandtl numbers in fluid dynamics. In what follows we shall fix il^, tjjg, <J>and
<J>2 to the values given by Saltzman et al, and use ^ as bifurcation parameter.
We now return to the general case, eq. ( 3 ) . Let
ujj = u)* = p + iu>o (7)
be the eigenvalues of the linearized operator {b.._}. At p. = 0 the system undergoes a bifurcation beyond which a limit cycle is expected to emerge. In general the full problem of solving exactly eq. ( 3 ) in this range is intractable. Let us therefore limit ourselves to a qualitative analysis. To this end it is convenient to cast the dynamics in a form in which the linear part becomes separable, corresponding to a full diag- onalization of { b j j } . T h i s is achieved by a linear transformation T , which in the present case is a 2 x 2 matrix whose columns are the two right eigenvectors of { b . . } . Operating on both sides of eq. ( 3 ) with the
_ -i 'J
inverse matrix T and introducing the new (complex) variables
(8)
we obtain
g = ( M iw0) z
2 2 3
+ H2 ( Z , z * z , z* ; p) + H3 (z Z«, z z- , z , z* ; p)
(9)
-7-
and a similar equation for z*, obtained by taking the complex conjugate of both sides of eq. ( 9 ) . The functions H H ^ etc contain the quadratic, cubic etc nonlinearities of the problem. T h e y are obtained by
- 1 ' *
operating with T on h^ and b^ (eq. ( 3 ) ) and switching subsequently to the variables z , z* by the transformation law given in eq. ( 8 ) .
As it stands the system of eq. (9) is as complicated as the original system, eq. ( 3 ) . However, the form displayed in eq. (9i) is especially suitable for approximations. To see this we perform a final transformation, switching to "radial" and "angular" variables through
z = r e ^ (10)
Eqs. (9) give then rise to the following two equations for r and 4>
dr
Q= Pr + Re (r e , r e r e .
D u (2 i<|> 2 -±<f> 2 -2i<|K
T)
D
„ , 3 3 2i<J> 3 — 2i4> 3 -4i<k . , + Re H
3(r , r e
Y, r e r e
T) + . . . (11a)
d(b
A1
T„ , 2 i<t> 2 -icj> 2 -2i(|).
-7± = U) + - Im H dt o r 2
0(r e
T, r e , r e
r) 1 Y JJ / 3 3 2±0 3 -2i(f> 3 -4i<().
+ - Im H„ (r , r e , r e r e r 3
Y) ( l i b )
The simplification to be made is now intuitively clear. Indeed, close to the bifurcation point the right hand sides of eqs. (11) display a slow
3
motion part, corresponding to the terms pr, r , and a fast motion
corresponding to oscillating terms of the type r^ e~ '
n<'
), r^ e~ with
n = 1,2 etc. It may thus be expected that for the long time behavior
only the slow, "secular" part will be important. Actually, the full justification of this conjecture requires the use of a chapter of bifurca- tion theory known as theory of normal forms ( A r n o l d , 1980). As it turns out, discarding the angular dependence is a completely consistent procedure provided that the system operates relatively close to the bifurcation point, in the sense that
L&l • « 1 (12) U) o
Throughout our qualitative analysis we shall assume that in all cases of interest this inequality is satisfied» T h i s will allow us to procecd a considerable way with an analytic approach. Subsequently we will relax this condition and we will v e r i f y numerically that the qualitative results hold even far from bifurcation.
?
Summarizing, by retaining only the slow part of the motion we obtain :
jjf = ßr - u r3 + . . . . (13a)
^ = u> - v rdt o 2 + . . . . (13b)
where u and v are parameters arising from the real and imaginary parts of H^ and H^.
In bifurcation theory ( A r n o l d , 1980) eqs. (13) are known as the normal form of the initial dynamical system ( e q s . ( 3 ) ) , whereas the radial variable r is referred to as the order parameter.
Let u s i l l u s t r a t e t h i s r e s u l t f o r S a l t z m a n ' s m o d e l . A s t r a i g h t - f o r w a r d b u t t e d i o u s a l g e b r a y i e l d s
8 + iu) = -o 2 ^ - 1 + [( ^ - l )2 - 4 ( - ) ]
0O 0O
(14a)
a n d a t r a n s f o r m a t i o n m a t r i x T e q u a l to
1 1 T =
3 + i u + 1 13 - i u + 1
o o
(14b)
o r , more e x p l i c i t l y
r| = 2r cos <|>
0 = 2r { (P + 1) cos (j) - u) s i n (J) } (14c)
T h e e v o l u t i o n e q u a t i o n s in t h e ( r , <)>) r e p r e s e n t a t i o n a r e
d r - « r
dE " P r " " 2 :
f = u) (1 + p) r2
d t o 2ui ^
(15a)
(15b)
T h e f i r s t of t h e s e e q u a t i o n s allows u s to e v a l u a t e a n a l y t i c a l l y t h e r a d i u s of t h e limit c y c l e . S e t t i n g d r / d t = 0, w h i c h in t h e ( r , (J))
r e p r e s e n t a t i o n p l a c e s us o n t h e limit c y c l e , we o b t a i n :
- 1 0 -
rs. * (2 P) 1/2
(16)
For t h e n u m e r i c a l v a l u e s g i v e n in Saltzman et al (1981) f o r t h e b e h a v i o r 3
in t h e 10 y e a r r a n g e we o b t a i n p = 1 . 5 , rg = 1 . 7 3 . Going b a c k t o t h e o r i g i n a l v a r i a b l e s one can see t h a t t h i s implies an a m p l i t u d e of o s c i l l a t i o n of q equal t o r jm a x -v 0.035 in good a g r e e m e n t w i t h t h e n u m e r i c a l s i m u l a t i o n s of t h e o r i g i n a l p a p e r . We see t h e r e f o r e t h a t t h e normal f o r m a n a l y s i s g i v e s t h e r i g h t t r e n d even b e y o n d t h e immediate v i c i n i t y of b i f u r c a t i o n p o i n t .
T h e second r e l a t i o n ( 1 5 ) i n c l u d e s t w o t y p e s of c o n t r i b u t i o n to t h e a n g u l a r v e l o c i t y Q = . One a r i s i n g t h r o u g h t h e d e p e n d e n c e of t h e l i n e a r i z e d f r e q u e n c y u)o on t h e b i f u r c a t i o n p a r a m e t e r p (see e q . ( 1 4 a ) ) , and a n o t h e r a r i s i n g f r o m t h e e f f e c t of n o n l i n e a r i t i e s ,
2 2iu0
( 1 + P) r . W i t h i n t h e r a n g e of v a l i d i t y of t h e normal f o r m , e q s . ( 1 5 ) , o n l y t h e f i r s t n o n t r i v i a l c o r r e c t i o n t o t h e v a l u e of uj at t h e b i f u r c a t i o n
o
p o i n t P = 0, u)Q(0), s h o u l d be r e t a i n e d . We a r r i v e in t h i s way at t h e f o l l o w i n g e x p r e s s i o n f o r t h e a n g u l a r v e l o c i t y
° = S • •.(« * p ( ^ )
+lu ^coT . (»•>
On t h e limit c y c l e i t s e l f r = r , a n d t h e d o m i n a n t c o r r e c t i o n to t h e a n g u l a r v e l o c i t y is :
Q = = < V ° ) + i ^ O ) P (17b)
- 1 1 -
We see that as the system moves further from bifurcation the period is shortened with respect to the value 2n/u)o predicted by the linear theory. T h i s agrees qualitatively with the numerical simulations reported in Saltzman et al (1981). Indeed these authors report a periodicity of 1260 years far from bifurcation whereas the linearized period 2n/\aQ, corresponding to p = 0 can be shown to be equal to 1740 y e a r s . Fig. 1 describes the dependence of both amplitude and period in terms of p.
We obtain the right qualitative trend but, as expected from the perturbative nature of the normal form analysis, quantitative agreement fails beyond the range of small values of p.
Finally, one can compute analytically the phase lag a between H and 0 variables. From the first eq. (14c) one sees that a = arc tg (u)Q/(p + 1 ) ) . Near bifurcation this yields a ^ 320 years in excellent agreement with numerical results in this range.
3. S T A B I L I T Y OF T H E O R D E R P A R A M E T E R V E R S U S L A B I L I T Y OF T H E P H A S E . C L I M A T I C " I S O C H R O N S "
Inspection of the normal forms derived in the previous Section (eqs. (13) and (15)) reveals a fundamental difference between the evolution of the order parameter r and of the phase <(>'. The former obeys a closed equation that can be solved analytically. For Saltzman's model, starting with r = rQ at t = tQ, one easily finds
'2( t ) " = 2 P 1 _ 2 p ( t . t ) (18)
1 + D e ° with
2(3 - r 2 D =
- 1 2 -
F i g . 1 . - A n a l y t i c a l l y c o m p u t e d d e p e n d e n c e of t h e a m p l i t u d e of o s c i l l a t i o n of t h e sea ice e x t e n t ( p a r t ( a ) ) a n d of t h e p e r i o d of t h e limit c y c l e ( p a r t ( b ) ) of S a l t z m a n ' s model as a f u n c t i o n of t h e b i f u r c a t i o n p a r a m e t e r p.
D o t t e d lines a r e u s e d f o r t h e p a r t of t h e c u r v e s f o r w h i c h t h e v a l i d i t y of t h e p e r t u r b a t i v e c a l c u l a t i o n c a n n o t be g u a r a n t e e d . C r o s s e s c o r r e s p o n d to t h e v a l u e s o b t a i n e d f r o m n u m e r i c a l s o l u t i o n of S a l t z m a n n ' s e q u a t i o n s f a r f r o m b i f u r c a t i o n .
As long as p > 0 and rQ t 0, this predicts an exponential relaxation to the limit cycle. In other words the variable r enjoys asymptotic stability, in the sense that any perturbation that may act accidentally on r will be damped by the system.
The situation is very different for <J>. Being an angular variable, the latter increases continuously in the interval (0, 2n) from some initial value <|>o. If <j>o is perturbed this monotonic change will start all over again from the new value, and there will not be any tendency to reestablish the initial phase <t>0.
In order to realize better the consequences of this property, let us perform the following thought experiment. Suppose that the system runs on its limit cycle, r = r . At some moment, corresponding to a value (|>o of the phase, we displace the system to a new state (cf.
Fig. 2). Owing to the stability of the order parameter, the perturbed state will spiral to the limit cycle. Clearly however, when the limit cycle will be reached again, the phase will generally be different from the one that would characterize an unperturbed system following its limit cycle during the same time interval. Inasmuch as the state at which the system can be thrown by a perturbation is unpredictable, it therefore follows that the reset phase of the oscillator will also be unpredictable (see Fig. 3). In other words, a non linear oscillator is bound to behave sooner or later in an erratic way under the action of the perturbations.
This provides already a qualitative answer to the question raised in the I ntroduction.
We can go further and define the locus of perturbed states which, when the limit cycle will be reached again will be characterized by a given value of phase, We call this locus a climatic isochron, and the common phase <l> latent phase. These notions were first introduced by Winfree (1980) in connection with biological oscillations.
For Saltzman's model an analytic expression for <t> can be derived, by
(a) (b)
Fig. 2 . - Schematic representations of the evolution following the action of a perturbation leading from state Aq on the limit cycle to state A . Parts ( a ) and ( b ) describe the situation, respectively, in the space of the variables of the normal form and in the space of the variables, r|r 9-
Fig. 3.- Resetting of a perturbed oscillation on its limit cycle. Depending on the type of the perturbation the reset phase may differ widely from that of the unperturbed situation.
i n t e g r a t i n g e q . ( 1 5 b ) in time a n d u s i n g t h e f a c t t h a t r s a t i s f i e s e q . ( 1 5 a ) . T h e i n t e r e s t e d r e a d e r is r e f e r r e d to W i n f r e e (1980) f o r t h e d e t a i l s of t h i s d e r i v a t i o n . F o r o u r model t h e r e s u l t is :
* = < b - d > + — (1 + B ) l n — (19)
° % rs
w h e r e <t>Q is t h e i n i t i a l p h a s e on t h e limit c y c l e ( i . e . <|> = <t>0, $ = 0, r = r ) a n d ( r , <J>> a r e t h e v a l u e s of t h e r a d i a l a n d p h a s e v a r i a b l e s d e s c r i p t i v e of t h e p e r t u r b e d s t a t e . T h e r - d e p e n d e n t p a r t of t h e r . h . s . e x p r e s s e s t h e d e v i a t i o n f r o m t h e i s o c h r o n o u s m o t i o n , a n d is d u e to t h e f a c t t h a t The e q u a t i o n f o r t h e p h a s e of t h e o s c i l l a t o r ( s e e ( 1 5 b ) ) c o n t a i n s a c o n t r i b u t i o n f r o m t h e r a d i a l p a r t of t h e m o t i o n . F i g . 4 g i v e s p l o t s in t h e ( r , (|>) p l a n e c o r r e s p o n d i n g to v a r i o u s v a l u e s of <J>. A s in t h e p r e v i o u s S e c t i o n it is u n d e r s t o o d t h a t o n l y d o m i n a n t t e r m s in 0 a r e to be r e t a i n e d in o r d e r to e n s u r e c o n s i s t e n c y w i t h t h e r a n g e of v a l i d i t y of t h e normal f o r m .
A s an e x a m p l e s u p p o s e t h a t an i n i t i a l s t a t e on t h e limit c y c l e c o r r e s p o n d i n g to p h a s e <|> = - 1.114 is p e r t u r b e d . We w a n t to c h a r a c t e r i z e t h o s e p e r t u r b e d s t a t e s t h a t will r e s e t o n t h e limit c y c l e in p h a s e o p p o s i t i o n , <t> = 7t w i t h r e s p e c t to t h e u n p e r t u r b e d s i t u a t i o n . C h o o s i n g , f o r i n s t a n c e , r = 0 . 5 , we f i n d f r o m e q . ( 1 9 ) t h a t <|> = - 0 . 0 5 n e a r b i f u r c a t i o n (p = 0 . 0 0 5 ) . F r o m e q s . ( 1 4 c ) a n d ( 5 ) we c a n t h e n e x p r e s s t h e p e r t u r b e d s t a t e in t e r m s of t h e o r i g i n a l v a r i a b l e s
rj -v 0.01 , 0 ^ 0 . 1 4 > (20)
A n u m e r i c a l e x p e r i m e n t , c o n f i r m s e n t i r e l y t h i s a n a l y t i c p r e d i c t i o n : s t a r t i n g w i t h t h e i n i t i a l c o n d i t i o n ( 2 0 ) , we i n d e e d f i n d t h a t
-17-
Fig. 4.- Plot of climatic isochrons for Saltzman's model as given by eq. (19) for
1 an initial phase <j>Q = 0 and for a value of bifurcation parameter p = 0.005. Motion along these isochrons leads to a resetting on the limit cycle in phase opposition, 4> = n , with the unperturbed signal. The isochrons <t> = - n , - 3n etc. describe evolutions in which the system spirals once, twice etc. before resetting on limit cycle.
t h e system reaches t h e limit cycle w i t h a phase s h i f t p r a c t i c a l l y equal to n. It is f u r t h e r m o r e f o u n d t h a t t h e a n a l y t i c a l r e s u l t g i v e s t h e r i g h t t r e n d f a r from t h e b i f u r c a t i o n p o i n t , f o r t h e parameter values g i v e n b y Saltzman et al (1981) in t h e case of v a r i a b i l i t y in t h e 103 year r a n g e .
From a p r a c t i c a l p o i n t of v i e w , we believe t h a t t h é notion of climatic isochron will p r o v e to be i m p o r t a n t . Such plots as in Fig. 4 g i v e us i n f o r m a t i o n on the most " d a n g e r o u s " p e r t u r b a t i o n s t h a t are l i k e l y to d e r e g u l a t e t h e oscillator completely, b y i n d u c i n g a r e s e t t i n g at a value n e a r l y in phase o p p o s i t i o n w i t h respect to t h e u n p e r t u r b e d s i t u a t i o n . T h e y also show t h a t t h e r e e x i s t " b e n i g n " p e r t u r b a t i o n s f o r which no sensible v a r i a t i o n of phase is o b s e r v e d . I n t e r e s t i n g l y , b o t h small and large amplitude p e r t u r b a t i o n s may e q u a l l y well belong t o e i t h e r of t h e above two classes.
4. STOCHASTIC A N A L Y S I S
In t h i s Section we p u r s u e t h e consequences of phase l a b i l i t y . T h e main p o i n t t h a t we want to make is t h a t in any complex p h y s i c a l system t h e r e is a u n i v e r s a l mechanism of p e r t u r b a t i o n s g e n e r a t e d spontaneously b y t h e d y n a m i c s , namely the f l u c t u a t i o n s . Because of t h e complexity of t h e processes at t h e i r o r i g i n , f l u c t u a t i o n s are basically random e v e n t s . T h i s impli es t h a t t h e state v a r i a b l e s (r), 8) o r ( r , <J>) become themselves random processes. O u r p u r p o s e here is to examine w h e t h e r t h i s may a f f e c t the r o b u s t n e s s of t h e o s c i l l a t o r y regime on t h e limit c y c l e . I n t u i t i v e l y , we e x p e c t from t h e p r e v i o u s Section t h a t t h e radial v a r i a b l e r will remain r o b u s t , b u t t h e phase v a r i a b l e $ will become completely d e r e g u l a t e d . It is shown in t h i s Section t h a t t h i s is indeed t h e case. Moreover, because of t h e d e r e g u l a t i o n of <(> t h e o s c i l l a t o r y behavior will disappear a l t o g e t h e r a f t e r a s u f f i c i e n t l y long lapse of time.
-19-
Let u s incorporate the effect of fluctuations b y a d d i n g random f o r c e s F , FQ to the deterministic rate of equations ( H a s s e l m a n n , 1976).
A s usual we assume the latter to define a m u l t i - G a u s s i a n white noise :
< Fn( t ) Fn( t ' - ) > = qjj 6 ( t - t ' )
< F0( t ) F g ( t ' ) > = qg S ( t - f )
< Fn( t ) F 0 ( f ) > = qn e S ( t - f )
T h i s allows us to write a F o k k e r - P l a n c k equation for the p r o b a b i l i t y d i s t r i b u t i o n P(r|, 6, t ) of the climatic v a r i a b l e s . In general this equation is intractable. H o w e v e r , the situation is g r e a t l y simplified if one limits the a n a l y s i s to the r a n g e in which the normal form ( e q s ( 1 3 ) or ( 1 5 ) ) is valid.
T o see this we f i r s t e x p r e s s the F o k k e r - P l a n c k equation in the polar coordinates ( r , $) defined in eq. ( 1 0 ) . For S a l t z m a n ' s model we obtain, after a long calculation (see also B a r a s et al, 1982) :
9 P ( r , <t>> t ) =
3t
a_
3r a_
3ct>
+ -
w +. ~ — o 2u),
3r <rr + 2 9r 30 Pr -
\
r3 + ^ Q3 (1 + P) r2 - ^ Qr<J>
2 Q 2 Q a a w
2
(21)
where
- 2 0 -
Q<|><|> = qR s i n 2 * " 2 qRC s i n ^ c o s ^ + % C O s 2 *
2 2 2 2
Q ^ = - qR s i n 4> cos <|> + qR C (cos <f> - s i n (|)) + qc s i n <|> cos (J) 2 2 2 2
Qr r = q R cos <() + 2qR(, s i n <|> cos (() + qc s i n <|> (22)
? 2 2 2 and q ^ , q^ and qR C are suitable linear combinations of q^, qQ, q^g.
T o go f u r t h e r it is necessary to introduce a perturbation parameter in the problem. It is reasonable to choose it to be related to the weakness of the noise terms. Mathematically, we e x p r e s s this t h r o u g h the following scaling :
Qaa = e Q„
Q = e Q (23) r r r r
We next scale both the bifurcation parameter p and the deviation of the radius r from its value on the limit c y c l e , r = rg, by suitable- powers of e. We do this in order to be in accordance with the conditions of validity of the normal form (eqs (12) and ( 1 5 ) ) . However, we should
keep in mind that according to the numerical results reported in the preceeding Section, the qualitative predictions should still describe the general trend beyond the v i c i n i t y of the bifurcation point. Note that no scaling can be applied to the angular variable <)), as the latter increases in the interval ( 0 , 2n) and does not enjoy any stability p r o p e r t y . Summarizing we write, using also eq. (16) and following Baras et al
(1982) : '
- 2 1 -
f = > e2 c
r = r • peb = (2 h V 2 eC + peb s
4» = <t> (24)
T h e ( n o n n e g a t i v e ) exponents b and c are chosen in such a way t h a t b o t h t h e d r i f t and d i f f u s i o n terms c o n t r i b u t e to t h e e v o l u t i o n of P in e q . ( 2 1 ) . Indeed, should t h e d i f f u s i o n terms be n e g l i g i b l e t h e p r o b a b i l -
i t y P would be a delta f u n c t i o n a r o u n d t h e d e t e r m i n i s t i c motion, and as a r e s u l t t h e e f f e c t of f l u c t u a t i o n s would be wiped o u t . If on t h e o t h e r hand t h e d r i f t terms were n e g l i g i b l e P would e x h i b i t a p u r e l y random motion similar to t h a t of a B r o w n i a n p a r t i c l e in a f l u i d , and would t e n d to zero e v e r y w h e r e as t « . T h e best way to estimate t h e magnitude of these two terms is to i n t r o d u c e t h e conditional p r o b a b i l i t y P(<|>/p,t) t h r o u g h
I n t e g r a t i n g eq. (21) o v e r (f) and t a k i n g t h e scaling ( 2 3 ) and (24) into account one can see t h a t 3 P ( p , t ) / 8 t is of o r d e r e. P ( p , t ) is t h e r e f o r e a slow v a r i a b l e : T h i s is the p r o b a l i s t i c analogue of t h e f a c t stressed in Section 2, t h a t t h e o r d e r parameter v a r i e s at t h e slowest of all time
P ( p , <t>, t ) = P (<j>/p,t) P(p, t ) (25a)
where
2n
0
-22-
scales present in the problem. Using this property and dividing through eq. (21) with P ( p , t ) we obtain a closed equation for P(<j>/p,t) which to dominant order in e reads :
= - - „ P « / P . t ) (26)
T h i s equation admits a properly normalized stationary solution,
P(<t>/p,t) = ^ (27)
Introducing this into eq. (25a) and integrating eq. (21) over (|> we can obtain a closed equation for the slow variable P ( p , t ) . On this equation one can see that the drift and diffusion terms are of the same order of magnitude if and only if
b = c = | (28)
The equation for P ( p , t ) reads then :
3P(p,t)
=~ \n
- e ^ ! [ .
2 f c. 3 ( I ) .
p 2- I p 3
+
2
;I P U ^ ^ P (29)
2 ( ( 2 $ )
1 / 2+ p) J
-23-
where
J = i •R %-> (30)
T h i s equation admits a s t e a d y - s t a t e s o l u t i o n . S w i t c h i n g back to t h e o r i g i n a l v a r i a b l e s and parameters r , p and Q t h r o u g h t h e i n v e r s e scaling (eqs (23) and ( 2 4 ) ) we can see t h a t t h i s s t e a d y - s t a t e is of t h e f o r m
T h e terms in p a r e n t h e s i s in t h e exponential f e a t u r e the i n t e g r a l of t h e r i g h t hand side of t h e d e t e r m i n i s t i c e v o l u t i o n equation ( 1 5 a ) . T h i s q u a n t i t y is known as k i n e t i c p o t e n t i a l , and its importance in climate has been stressed b y Ghil (1976), N o r t h et al (1980) and Nicolis and Nicolis (1981). In t h e p r e s e n t case, d e n o t i n g t h e potential b y U ( r , 0 ) , we can w r i t e
P ( r ) ^ r exp 2 r & d - ^ (31).
Q K 2 8
dr 9U(r,P) d t ap
w i t h
Ps( r ) * exp [ - | U ( r , P ) ] (32)
- 2 4 -
Fig. 5 represents function (31) or (32) in terms of the variables (n,6)- We obtain a crater-like distribution. The projection of the lip of the crater is identical to the deterministic limit cycle.
Let us summarize the situation. We have shown that starting from the initial variables r|f 6
o n e c a nswitch to two combinations which
have a completely different status. The angular variable (|> which, according to the transformation laws given in Section 2, is related to q, 0 through
d. = arc tg [ -J- (p + 1 - £ ) ] (33)
wo 1
has a completely "flat" probability distribution (cf. eq. ( 2 7 ) ) . It may therefore be qualified as "chaotic", in the sense that the dispersion around its average will be of the same order as the average value it- self. T h i s is to be correlated with the remarks made in Section 3 about the labile behavior of the phase under the action of perturbations.
The situation is different as far as the statistics of the radial variable r is concerned. This variable, related to
0through
r 2
= I { n
2+ ±2- [ (P + 1) n - 6 ]
2} (34) w o
has a stationary probability distribution (eqs (31) and (32)) such that the dispersion around the most probable value r = r
gis small. Nev- ertheless, the mere fact that the probability distribution is stationary rather than time-periodic implies that a remnant of the chaotic behavior of c|> subsists in the statistics of r. In a sense we arrive at a picture whereby if an average over a large number of samples is taken, the
-25-
Fig. 5.- Schematic representation of the steady state probability distribution eq. (31). Note that the projection of the lip of the crater on the phase Diane of the variables reproduces the deterministic limit cycle.
- 2 6 -
periodicity predicted b y the deterministic a n a l y s i s will be wiped out as a r e s u l t of d e s t r u c t i v e p h a s e interference. We believe that this p r o p e r t y may be at the o r i g i n of the lack of p r o n o u n c e d systematic periodicities in the climatic r e c o r d for time scales for which no systematic external f o r c i n g is a c t i n g , like for instance in the r a n g e of 10 y e a r s . 3
It s h o u l d be pointed out that the above result is to be u n d e r - stood in an asymptotic s e n s e . In a g i v e n s y s t e m which r e p r e s e n t s a particular realization of the stochastic p r o c e s s d e s c r i b e d b y the F o k k e r - Planck equation a trace of the periodic b e h a v i o r may s u b s i s t for a substantial amount of time and show up as a peak in a power s p e c t r u m computed o v e r that interval. E v e n t u a l l y h o w e v e r , the oscillatory b e h a v i o r is b o u n d to d i s a p p e a r . How soon this will h a p p e n d e p e n d s on the s t r e n g t h , Q of the random force. T h e computer simulations on the effect of noise reported in the paper b y Saltzman et al (1981) corroborate this view.
5. D I S C U S S I O N
In this paper we e x p l o r e d some qualitative p r o p e r t i e s of autonomous oscillations in climate d y n a m i c s . O u r a n a l y s i s led us to raise some general q u e s t i o n s c o n c e r n i n g predictability. Specifically we have s h o w n that, as a result of the poor stability p r o p e r t i e s of the phase variable, the systematic oscillation t e n d s to d i s a p p e a r in the climatic record and to be g r a d u a l l y replaced b y a random looking motion.
C o r r e s p o n d i n g l y , in a time series c o r r e s p o n d i n g to a long sampling interval detailed predictability would be compromized.
It is important to realize that the o r i g i n of this mechanism of loss of predictability is completely different from the one associated with the appearance of aperiodic or chaotic solutions ( s t r a n g e a t t r a c t o r s ) of
the deterministic balance equations. More work is necessary to differentiate between the consequences of "probabilistic" and
"deterministic" chaos.
T h r o u g h o u t this work we have argued in terms of an oscillatory behavior of mean surface properties, encompassing the entire earth-atmosphere-cryosphere system. It is clear however that if spatial inhomogeneities are allowed, such a global oscillator can only result from the dynamics of localized oscillators, each of them representing the climate of a certain region of the globe. In this picture, the lability of the phase of each local oscillator will be reflected by the d e s y n c h r o n i z a - tion of this oscillator with respect to its neighbors. T h u s , if an average over a large space region ib laken, Ihe oscillatory behavior will be wiped out. The behavior of coupled Saltzman oscillators is analyzed in a forthcoming paper by the author (Nicolis 1983a).
The analysis carried out in this work is valid both for intermediate and for longer period oscillations, like those developed in connection with quaternary glaciations. In this latter case however an additional argument comes into play : Because of the existence of a systematic external forcing of comparable period, related to the earth's orbital variations, the oscillator may be entrained and t h u s be less sensitive to external disturbances or fluctuations. A s a result, a relatively well defined periodicity may exist in the climatic record. We arrive in this way at a synthesis between internally generated and external mechanisms of climatic change. In Nicolis (1982, 1983b) the behavior of forced oscillators is analyzed, and the mechanisms of phase stabilization by resonant or harmonic coupling to the forcing are explored.
A C K N O W L E D G E M E N T S
We acknowledge fruitful discussions with D r . M. Malek Mansour. T h i s work is supported, in part, by the EEC under contract n° C L I - 0 2 7 - B ( G ) .
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R E F E R E N C E S
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tions - A d d i t i v e fluctuations. T e l l u s 33, 225-237.
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North, G . R . , Cahalan, F . R . and Coakley, J . A . , 1981. E n e r g y balance climate models, Rev. Geophys. Space P h y s . , 19, 91-121.
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