in Oscillatory Climate Model Correlation Functions and Variability

Geophys. Astrophys. Fluid Dynamics, 1985, Vol. 32, pp. 91-102 0309-1929/85/3202-0091 $18.50/0

0 1985 Gordon and Breach, Science Publishers, Inc. and OPA Ltd.

Printed in Great Britain

Correlation Functions and Variability in an Oscillatory

Climate Model

C. NlCOLlS

lnstitut d l e r o n o m i e Spatiale de Belgique, Avenue Circulaire 3, B- 1180 Bruxelles, Belgique

(Received July 24, 1984: in final form November 19, 1984)

The correlation functions of a stochastically forced climate model describing the coupling between the mean ocean surface temperature and the' extent of sea-ice are computed in the Gaussian approximation. A diagnostic relation between the variability of the surface temperature and that of the ice extent is obtained.

Information on the variance of the random forces acting on the system is also deduced.

1. INTRODUCTION

Climate models describing the coupling between the mean ocean surface temperature and the extent of sea-ice have received consi- derable attention recently. Such models predict periodic solutions with periods comparable to the characteristic time scales of quater- nary glaciations (Saltzman et al., 1981, 1982). It is therefore expected that they should provide very useful information on the mechanisms of major climatic changes.

In previous work (Nicolis, 1984a) we have shown, using suitable perturbation and scaling techniques, that one can determine analyti- cally the structure of the periodic solutions and study their response to stochastic perturbations. It is the purpose of the present paper to assess the connection between these theoretical results and the main features of climatic variability as obtained from the data.

91

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92 C. NICOLIS

I n Section 2 we recapitulate the main steps of the construction of the analytic form of the periodic solutions. In Section 3 we study the response of the oscillatory system to stochastic perturbations and, in particular, we solve the Fokker-Planck equation for the underlying probability distribution in the Gaussian approximation. Section 4 is devoted to the computation of the second moments of the proba- bility distribution. Specifically, a “diagnostic” relation between the variability of the ocean surface temperature and that of the extent of sea-ice is suggested. Moreover, an explicit relation between these quantities and the variance of the stochastic perturbations is ob- tained and illustrated by numerical estimates drawn from the data. A short discussion of the results is presented in the final Section 5.

2. BIFURCATION ANALYSIS

OF

THE SURFACE TE M PER ATU R E-S EA-I CE CO U PLI N G

Let 8 , y be respectively the deviations of the mean ocean surface temperature and of the sine of latitude of the extent of sea-ice from the steady state. The interactions between these two variables can be described by the following set of coupled equations, expressed in dimensionless variables (Saltzman et al., 1981, 1982; Nicolis, 19844:

where a and b are positive parameters. The term be describes the positive feedback of the temperature on itself via the surface albedo, CO, concentration etc.; the term -aq the negative feedback arising from the insulating effect of sea-ice on temperature; the term -q28 a non-linear restoring mechanism which begins to operate when the deviations q and 0 are no longer small; and finally the second equation stands for the mass balance of sea-ice.

Equations (1) admit only one physically acceptable steady-state solution, 19 = q = 0. A standard stability analysis shows that this state behaves like a stable focus for b< 1, and like an unstable one for b > 1. It is therefore expected that b = 1 is a bifurcation point of time periodic solutions of the limit cycle type (Hopf bifurcation). In order to bring out the essential features of this transition we transform the initial equations to the normal form of a dynamical system operating

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AN OSCILLATORY CLIMATE MODEL 9 3 in the vicinity of a Hopf bifurcation (Arnold, 1980). This is achieved by the following transformation to radial and angular variables, r and (b respectively:

0 = 2r(cos

4

^- oo sin

4),

y ⁼ 2r cos

4.

(2)

Substituting into Eqs. (1) one can see that, as long as b remains close to the value b= 1, all terms containing the coupling between r and

4

can be neglected. The equations then take the simple form

where

is the linearized frequency of the periodic motion, and

is the bifurcation parameter.

solutions:

The first equation of (3) admits the following steady-state i) r = 0, representing the steady-state in the original variables, 0 = y

= 0. This state is stable for

/3

< 0 and unstable for

/3

> 0.

ii) r, =(2P)l”, representing [cf. Eqs. (2)] the amplitude of the periodic solution in the original variables 0 and q. This state exists only beyond the Hopf bifurcation point ( b > 1 or /3>0) and is stable in this range. We refer to this phenomenon as supercritical Hopf bifurcation.

On the other hand, the second equation of (3) admits no stationary solutions. In fact one can solve it exactly, the result being

We see that the instantaneous value of

4

keeps the memory of the initial condition

40.

We expect therefore that the phase variable will have rather weak stability properties. Combining these results with

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94 C. NICOLIS

the Eqs. (2) we find the explicit form of the periodic solution in the original variables:

~ ( t ) = 2 ( 2 f l ) ” 2 [ c o s ( ~ o + f i t ) - o o s i n ( ~ , + f i t ] , q(t) =2(2fl)1’2 cos(& +fit).

( 6 )

Note that the validity of the Eqs. (3)+6) can be guaranteed only as long as fl/oo

<

^1.

Equations (3)<5) can be extended easily to the case of subcritical Hopf bifurcation predicted in the model developed by Ghil and Tavantzis (1983). In the sequel, however, we limit ourselves to Saltzman’s model which turns out to be more tractable analytically.

3. EFFECT OF FLUCTUATIONS AND R A N D O M PERTURBATIONS

In climate dynamics it is important to study the response of a system to random perturbations or to its own fluctuations, which are always present and modify continuously the deterministic evolution.

This is carried out by augmenting Eqs. (1) by the addition of random forces F,, F,:

In analogy with the statistical theory of irreversible processes (Landau and Lifshitz, 1959) it is assumed that F,,F, describe a multi-Gaussian white noise:

Equations (7) and (8) define a stochastic process referred to as a

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AN OSCILLATORY CLIMATE MODEL 95 diffsion process. It is known from probability theory that such phenomena are described by a Fokker-Planck equation for the joint probability distribution P(0, q), obeying the usual positivity and normalization conditions

As a matter of fact, in view of the analysis of the preceding section it will be more convenient to work in terms of the radial and angular variables r and

4,

respectively. Using Eqs. ( 2 ) one can rewrite (9) in the form

m Z R

0 0

J

dr

J

d44worP(r,

4)

= 1.

At this point it becomes, therefore, natural to introduce a new distribution

which is normalized according to

U3 2 R

1

^dr

f

d 4 P ( r ,

4)

^{= 1.}

0 0

It can be shown by a straightforward, though lengthy, calculation that in terms of this distribution the Fokker-Planck equation reads (Nicolis, 1984a)

where

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96 C. NICOLIS

Q++ ⁼ q i sin2

4

-2qRC sin

4

cos

4 +

^{q$ cos2}

4,

and q i , q& qRC are suitable linear combinations of q t , q i , qse:

We are interested in the solutions of Eq. (11) in the limit of weak noise. Mathematically, we express this through the following scaling:

Moreover, to ensure consistency with the range of validity of the Eqs. (3), we consider that the bifurcation parameter

p

is smaller than unity. We do not, however, scale this parameter by E, since we want to explore the regime in which the system remains at some distance from the bifurcation, thereby exhibiting a finite limit cycle of radius r,=(28)1/2.

Under these conditions we merely have to express that the radial variable r in Eq. (11) is equal to the deterministic value rs, plus a small deviation related to the fluctuations, which is expected to scale as a suitable power of E,

Note that no scaling can be applied to the angular variable

4.

Indeed, as pointed out in Section 2, this variable increases in the interval (0,271) and does not enjoy any stability property.

The (non-negative) exponent y must be chosen in such a way that both the drift and diffusion terms contribute to the evolution of P in Eq. (1 1). Indeed, should the diffusion term be negligible, Eq. (1 1) would reduce trivially to the deterministtic description. On the other hand, should the drift term be negligible, this equation would predict an erratic behavior similar to a random walk, in which no trace of the deterministic motion would subsist. It can be easily checked that these conditions imply

r=*.

The dominant terms in E therefore

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AN OSCILLATORY CLIMATE MODEL

become

97

We observe that, because of the +-dependence of the second derivative term, the variables p and

4

cannot be separated. Still, one can eliminate the &dependence by assuming that it appears in the form

Indeed,

and

where

Since Eq. (16a) no longer contains a derivative in @, the angular variable can from now on be treated as a parameter, ^Q0. Without loss of generality one can set 0, = 0 and

We can now seek solutions of Eq. (16a) in the form

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Substituting we find the conditions

d[ln N ( t ) ] / d t = 2p -&'Q(t), ^{do/& =} -4po

+

Q(t). ₍₁ 8) The first equation is fulfilled automatically by N =(2nna)- '1'. The second one yields

a(t) = e-4Bt

[

^a.

⁺

^dz e4p7Q(~)].

0

Now

Performing the integral over z we thus find, in the limit of long times

t - t o o ,

4. COMPUTATION OF THE CORRELATION FUNCTIONS

Equations (17) and (20) provide extensive information on the variability of the climatic system as described by our model. This information is summarized in the behavior of the variances and covariances

(e2)>,

( q 2 ) ,

(re).

From the Eqs. (2) and (16c) one first sees that

In other words, in the vicinity of the birfucation point, these three quuntilics arc not independent. since they are all expressed in terms

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AN OSCILLATORY CLIMATE MODEL 99 of ( r ’ ) . Specifically, we obtain the following “diagnostic” relations:

Now, some of these variances and correlation functions can be estimated from the data. For instance, if in the framework of quaternary glaciations the data allow one to reproduce the time series of paleo-temperatures from which (8’) can be inferred, then the Eqs. (22) will automatically provide information on the variability of the extent of sea-ice as well as on its correlation with the temperature variability. A similar estimation could be made for the variability of continental ice using the model by Ghil and his co- workers (KallCn et al., 1979; Ghil and Tavantzis, 1983).

A more quantitative study requires the computation of ( r ’ ) . From Eqs. (14) and (17) one has

where ( 6 r 2 ) is the variance of r around its deterministic value on the limit cycle. Using Eqs. (20) and (13)

From this relation the following conclusions can immediately be drawn:

i) The variances are periodic functions of time, with a period equal to half the period of the macroscopic oscillation [Eq. ( 6 ) ] . This illustrates the intuitively clear idea that fluctuations vary on a faster time scale compared to the macroscopic evolution.

ii) The time average of the variances over a period of the macroscopic oscillation reduces to

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100 C. NICOLIS

Far from bifurcation

( p

between 0.1 and 1 say) the deviation of ( r 2 ) around the limit cycle is thus small, since it is proportional to the effective strength of the noise Q’. On the other hand, close to bifurcation

( p

very small) it increases gradually and can eventually become comparable to the deterministic value, 2p.

An interesting application of Eqs. (21) and (24) is the following. By estimating (0’) from the data one finds ( r ’ ) and, through Eq. (24), the value of the effective intensity of noise which is not directly accessible by experiment. The result can then be used to make further predictions concerning the variability of other quantities of interest, or concerning the time-dependent behavior of the system. As we see from the third equation of (21) and from (24)

~~

(SO’)

=((I2) -4ap=Q2a/4P.

The deviation of (6’) from its value 4ap predicted deterministic analysis is therefore directly proportional strength of the noise, Q2. For the values of the parameters

(25) by the to the a and j3 given by Saltzman et al. (1982, 1984), a=6.4, p=O.5, the coefficient of proportionality is a/4P=3.2. The response to the noise is therefore manifested by a temperature variability amplifying the level of noise by a factor of three.

Saltzman and his co-workers (1982, 1984) have estimated the variance of the temperature to be about (S02),-10-60KZ. In our dimensionless quantities ( S O 2 ) = 6.10-

‘.

From this value and Eq.

(25) one can estimate the random force intensity to be

following estimate:

In other words we

Moreover from the Eqs. (22) one can determine the dimensionless variance (6~’). Using the appropriate conversion factors from the dimensionless to the original variables (Nicolis, 1984a) one finds for the numerical values adopted by Saltzman et al. (1982, 1984) the

(Sy2)d

-

predict that the variance of the extent of sea-ice

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AN OSCILLATORY CLIMATE MODEL 101 should be about five orders of magnitudes smaller than that of the temperature.

5. DISCUSSION

In this paper we have seen that the information contained in the stochastic analysis of a simple climate model can be connected with data concerning long-term variability. The basic tool enabling one to establish this connection is the second moments of the probability distribution. These quantities have been computed analytically in the Gaussian approximation (small fluctuations) and the effect of the various parameters has been brought out explicitly.

Throughout this work we considered the dynamics of an autonomous oscillator. It is, however, known that the climatic system is subjected to a variety of external forcings, corresponding to the variability of the earth‘s orbital characteristics. An asymptotic perturbative study of the effect of such forcings on the solutions of the deterministic equations has recently been carried out (Nicolis, 1984b,c). In some forthcoming work, we intend to extend the stochastic analysis outlined here to incorporate the effect of external forcings and to arrive, in this way, at a more complete view of climatic variability.

Acknowledgement

This work is supported, in part, by the EEC under contract No. CLI-106-B(RS).

References

Arnold, V., Chapitres Supplimentaires de la Theorie des Equations Differentielles Ghil, M. and Tavantzis, J., “Global Hopf bifurcation in a simple climate model,”

Kallen, E., Crafoord, C. and Ghil, M., “Free oscillations in a climate model with ice- Landau, L. D. and Lifshitz, E. M., Fluid Mechanics, Pergamon Press, Oxford (1959).

Nicolis, C., “Self-oscillations and predictability in climate dynamics,” Tellus 36A. 1-10 Ordinaires, Mir, Moscow.

SIAM J . Appl. Math. 43, 1019-1041 (1983).

sheet dynamics,” J . Atmos. Sci. 36, 2292-2303 (1979).

( 1984a).

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102 C. NICOLIS

Nicolis, C., “Self-oscillations and predictability in climate dynamics-periodic forcing and phase locking,” Tellus %A, 217-227 (1984b).

Nicolis, C., “Self oscillations, external forcings and climate predictability,” in:

Milankooitch and Climate: Understanding the Response to Orbital Forcing (eds. A.

Berger, J. Hays, J. Imbrie, G . Kukla and B. Saltzman), Dordrecht, Holland (1984~).

Saltzman, B., Sutera, A. and Evenson, A,, “Structural stochastic stability of a simple auto-oscillatory climate feedback system,” J . Atmos. Sci. 38, 494503 (1981).

Saltzman, B., Sutera, A. and Hansen, A. R., “A possible marine mechanism for internally generated long-period climate cycles,” J . Atmos. Sci. 39, 26342637 (1982).

Saltzman, B., Sutera, A. and Hansen, A. R., “Earth-orbital eccentricity variations and climate change,” in: Milankooitch and Climate: Understanding the Response to Orbital Forcing (eds. A. Berger, J. Hays, J. Imbrie, G. Kukla and B. Saltman), Dordrecht, Holland (1984).

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