Geophys. Astrophys. Fluid Dynamics, 1985, Vol. 34, pp. 65-82 0309-1929/85/3402-0065 Sl8.S0/0
0 Gordon and Breach, Science Publishers, Inc. and OPA Ltd.
Printed in Great Britain
Correlation Functions and Variability in a Periodically
Forced Oscillatory Climate
C . NlCOLlS
lnstitut d'Adronomio Spatiale de Belgique, 3 Avenue Circulaire, 1 180 Brussels, Belgium
(Received March 28,1985: infinal form June 10,1985)
The effect of fluctuations in a periodically forced climate model involving the coupling between sea ice and mean ocean temperature is considered. The transitions between phase-locked (periodic) and quasi-periodic solutions arc studied by a singular perturbation method of solution of the stochastic differential equations, as well as by numerical simulations. The variances and covariances of the principal climatic variables are computed.
1. INTRODUCTION
One of the most promising dynamical scenarios of long term climatic change is provided by the models describing the coupling between mean ocean temperature and the extent of sea ice (Saltzman et al., 1981, 1982) or of continental ice (KallCn et al., 1979; Ghil and Tavantzis, 1983). Indeed, such models predict periodic solutions whose periods are comparable to the characteristic time scales of quaternary glaciations, the well-known climatic episodes which have occurred intermittently during the last million years.
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66
c.
NIcoLrsIn a recent series of papers (Nicolis, 1984a,b) we determined analytically, using suitable perturbation and scaling techniques, the structure of the periodic solutions of Saltzman’s model, as well as their response to stochastic or to periodic perturbations. Sub- sequently (Nicolis, 1985) we computed the variances and correlation functions of the principal climatic variables. We obtained a diagnostic relation linking the variability of mean ocean temperature and ice extent, as well as information enabling one to infer from the data, the variance of the random forces acting on the system. Our principal objective in the present paper is to incorporate in the analysis both the external periodic forcing, accounting for the change in the solar output induced by the earth’s orbital variability, and the stochastic perturbations accounting for the fluctuations generated by the system’s internal dynamics.
In Section 2 we recall the main features of the periodically forced Saltzman model and recapitulate the method of construction of the analytic form of the solutions, which involve both phase-locked and quasi-periodic regimes. In Section 3 we study the response to stochastic perturbations. We derive a Fokker-Planck equation for the underlying probability distribution, and in Section 4 we solve it in the Gaussian approximation around the phase-locked state. In Section 5 we report the results of numerical simulations concerning the transitions between the phase-locked and the quasi-periodic states induced by the fluctuations. We end this paper with some comments on the implications of the results in understanding the unpredictability inherent in climate dynamics.
2. THE M E A N OCEAN TEMPERATURE-SEA ICE SYSTEM IN THE PRESENCE
OF
A PERIODIC FORCINGLet 6, q be the deviations respectively of the mean ocean temper- ature and of the sine of the latitude of the sea-ice extent from a reference state corresponding to the present climate. The interactions between these two variables have been studied in a series of papers by Saltzman and co-workers (1981, 1982). These authors have also considered the coupling between these dynamics and external forc- ings, especially periodic forcings of astronomic origin, and shown that to a first approximation it could be treated as an additive
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PERIODICALLY FORCED CLIMATE MODEL 67 coupling (Saltzman et al., 1984). The resulting dynamical system can be described by the following set of coupled equations, expressed in dimensionless variables (Nicolis, 1984b):
where a and b are positive parameters, w the frequency of the external forcing and q,, qe the system-forcing coupling coefficients. A standard stability analysis of the unforced system shows that the unique physically acceptable steady state solution O=q = O behaves like a stable focus for h < 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 (supercritical Hopf bifurcation?). In previous work we showed that the essential features of this transition and, especially, its interaction with an external forcing can be brought out most conveniently by transforming the initial dynamics to a normalform (Arnold, 1980; Guckenheimer and Holmes, 1983).
For the model described by (1) this is achieved by the following change of variables:
q = 2x, 8 = 2(x - o o y ) , (2) where oo = (a- 1)’’’ is the linearized frequency of the self-oscillatory motion. Assuming that the system remains near the bifurcation point b = 1, and setting
one obtains the following set of equations for the normal form
?In the models developed by Ghil and co-workers (1979, 1983) a small-amplitude limit cycle bifurcates subcritically and subsequently gives rise to a finite amplitude oscillation (hard excitation). In the sequel we limit ourselves to Saltzman’s model which is easier to handle analytically. It would, of course, be interesting to extend our analysis to Ghil type models.
variables x and y:
dx/dt=/3x-w0y-$(x2 + y 2 ) ( x + 3 w , ' y ) + q x ~ ~ ~ o t , dy/dt = o , x + / 3 ~ - $ ( ~ ~ + y ~ ) ( y - 3 3 w , ~ x ) + q , ~ 0 ~ o t .
(4)
Suppose that the system is operating in the vicinity of a resonance with the forcing's periodicity. For small values of
p,
qx and qy these equations can then be solved by singular perturbation methods. As it turns out, the dominant contribution to their solution can be written in the following form (Nicolis, 1984b):x = E'/~[A(T) cos at
+
B(z) sin ot], y=~'/~[A(z) sinat-B(z)cosot].( 5 )
Here E is a smallness parameter related to the amplitude of the forcing, the distance from the bifurcation point, and the distance from the resonance (Rosenblat and Cohen, 1981):
In addition to a time dependence following the external forcing, x and y are thus allowed to vary on a different, slow time scale z
related to the system's intrinsic dynamics. The coefficients A, B expressing this additional time dependence have been shown to obey to the following set of equations (Nicolis, 1984b):
dA/dz = PA - cijB -+(A2
+
B2)(A - 30, B)+&
d B / d z = f l B + 6 A - K A 2
+
B2)(B+3&'A)-$ijy.(7)
Figure 1 depicts the steady-state solutions of Eqs. (7). We see that there is one branch of stable solutions in the form of an isola [Figure 1 branch (a)] and two branches of unstable solutions [Figure 1, (b) and (c)]. As it turns out the lowest branch (c) is surrounded by a stable limit cycle solution.
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PERIODICALLY FORCED CLIMATE MODEL 69
FIGURE 1 Schematic representation of stationary solution branches of Eqs. (7) as a function of the distance from resonance 6. Full and dotted lines represent stable and unstable states respectively.
Translated in terms of the original variables q and 8, Eqs. (2) and ( 5 ) , these results mean that the system can respond to the periodic forcing in the following ways:
i) By synchronization and phase locking, reflected by a periodic solution in which the phase shift between system and forcing remains constant for all times. This regime can be qualified as
“predictable” in the sense that small perturbations of the phase will tend to die out.
ii) By a periodic variation of the phase shift between system and forcing, reflected by a quasi-periodic solution. In this regime the perturbation of the phase will not die out, and as a result the system is expected to behave in an unpredictable fashion.
In the region of Figure 1 between the two limit points, Q and Q , of the isola branch both of these regimes coexist as stable solutions.
The question that we raise now is whether, under the effect of fluctuations, the system can perform transitions between the two
types of states with an appreciable rate. If so, the predictability of the regime of branch (a) will be heavily compromised. In order to answer this question we turn to a stochastic analysis incorporating the effect of stochastic perturbations into the periodically forced system of Eqs. (1). This analysis should lead us td a theory capable of describing fluctuations around both the periodic and the quasi- periodic solutions of the underlying model.
3. EFFECT OF STOCHASTIC PERTURBATIONS O N THE PERIODICALLY FORCED M E A N OCEAN TEMPERATURE-SEA ICE SYSTEM
As usual, fluctuations are modelled by random force terms added to the phenomenological equations of evolution. Thus, instead of Eqs.
(l), one writes
dq/dt = - q
+
8+
4, cos at+
F,( t),dO/dt = - aq
+
be- q20+
40 cos a t+
F,(t). (8)The stochastic forcings F,, Fe are required to have a zero mean value, but are otherwise left unconstrained at this stage.
Switching to the normal form variables x and y [Eqs. (2)] we can easily transform Eqs. (8) to [compare also with (4)]:
where we have set
This set of stochastic differential equations for x and y can be solved by the singular perturbation method introduced in the preceding section, provided that F, and F , are also assumed to be scaled by the smallness parameter E defined in (6a):
F , =
EF,,
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PERIODICALLY FORCED CLIMATE MODEL 71 Introducing the forcing time scale, T=wt and a slow one z, related to the distance from bifurcation and resonance [cf. Eqs. ( 6 ) ] , we can thus write Eqs. (9) in the form
0 (3xlaT
+
E2i3 ax/& = PE2’3X+
(Q&2/3 - w)y-%x2 + y 2 ) ( y - 3001x)
+
&gy cos T +EFy(?;
z).The next step is to expand the unknown quantities x, y in terms of
E. In view of the scaling adopted in Eqs. (6) and (11) the following development is chosen:
Substituting in (12) we obtain, to order c1I3, a homogeneous system of equations:
The solution of this problem is the harmonic oscillator:
xi = A (z) cos T
+
B(z) sin T,y, = A(z) sin T - B(z) cos T (15) where A , B remain undetermined at this stage but are expected to depend, in general, on the slow time scale z which does not appear in Eq. (14).
The next order, c2I3 leads to an equation identical to (14), and, therefore, adds nothing new. To order E, on the other hand, we
obtain:
w ( d x 3 / d T + y 3 ) = - ax,/az +PX,
+
O?Y,-gx: +
y:)(yl -3w,’x1)+
&cos T +Fy(T
7).This is an inhomogeneous system of equations for x 3 , y 3 . It admits a solution only if a solvability condition, expressing the absence of terms growing unboundedly in time, is satisfied (Sattinger, 1973).
Such terms may arise by the following mechanism. To obtain (x3, y 3 ) we have to “divide” the right hand side of (16) by the differential- matrix operator
But this is precisely the operator that appears in (14). As we have seen previously, this equation possesses a nontrivial null space, that is nontrivial eigenvectors [given by Eqs.
(191
corresponding to a zero eigenvalue. On dividing by such an operator one may therefore introduce singularities, if the right-hand side of (16) contains contri- butions lying in its null space. The solvability condition allows one to rule out this possibility by requiring that the right-hand side of (16), viewed as a vector, be orthogonal to the eigenvectors of the operator (17). The latter are [see Eqs. (15)]:(cos
I:
sin T ) and (sin T, - cos T ) . (18) The scalar product to be used in the definition of orthogonality is the conventional scalar product of vector analysis, supplemented by an averaging over a period of the forcing. After a lengthy calculation one finds a set of equations similar to Eqs. (7), but augmented with respect to the latter by terms involving the averages ofF,,F,
over the external forcing. Specifically, we obtain the following set ofDownloaded by [The University Of Melbourne Libraries] at 20:26 19 June 2016
PERIODICALLY FORCED CLIMATE MODEL 73 stochastic differential equations for A and B:
where
2 n 0
~ A ( z ) = & I - '
f
dT(FXcos T+Fysin T )where the bars over F,, F, are defined in (10) and (1 1).
We now turn to the evaluation of the effective random forces $ A ( ~ ) .
First, in view of our previous requirement that the averages of F, and F, are zero, we immediately deduce that
As regards the properties of the time correlation functions we first obtain, from (20a),
2%
0
= & - 2
1
dTdT'{$(F,(II:z)F,(T',z'))cos Tcos T'We want this quantity to be non-zero, since fluctuations should affect the internal dynamics, which is essentially described by the evolution of the amplitudes A and B [cf. Eqs. (19)]. This means that the random forces F,, F, of the original equations should depend on the forcing time scale in such a way that the integration over a period of the forcing on the right-hand side of (22) produces a non- vanishing result. Since the evolution induced by the external forcing alone is non-dissipative,
F,,F,
should not behave as a random process with respect to the time scale ‘I: Rather, they should be subjected to an entrainment at the frequency of the forcing, while behaving in a more complex way in the slow time scale z in which the internal dynamics is taking place. We express this idea by the following relation:We assumed, without loss of generality, that initially F,,F, were in phase with the forcing. As regards Y,,Yo, we will assume that they define a Gaussian white noise:
Using Eqs. (23) and (24) we may now evaluate explicitly the right- hand side of (22). We obtain, after an elementary calculation,
and, similarly,
Equations (19), together with the properties of the effective ran- dom forces expressed by Eqs. (25a)-(25c), define a class of stochastic
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PERIODICALLY FORCED CLIMATE MODEL 75 phenomena known as dijliusion processes, whose study amounts to the solution of a Fokker-PIanck equation for the underlying proba- bility distribution (Arnold, 1974). The specific form of this equation for our system is:
4. SOLUTION IN THE GAUSSIAN APPROXIMATION We will not attempt to solve this equation by keeping the full non- linearities in the coefficients of the first derivative terms. Rather, we will analyze the behavior of the fluctuations in the vicinity of the steady state solution (Ao, Bo) of the equations for A, B in the absence of fluctuations (see discussion at the end of Section 2). Setting
A = A o + u , B = B o + u , (27) and keeping only terms linear in the deviations u, u, we obtain from
On comparison with Eqs. (7) one may verify that the aij are nothing but the coefficients entering the linear stability analysis of the solution (Ao,Bo). As shown in previous work (Nicolis, 1984b) the points in parameter space at which stability changes take place are given by a,, +a,,=O (Hopf bifurcation), or a11a22-alzazl=0 (limit point bifurcation).
From Eqs. (28) and (29) one can immediately compute the variances and the covariance of the variables u and u. It suffices to multiply both sides successively by uz,uz and uu and integrate over the range of variation of both u and u, by requiring P and its derivatives to vanish sufficiently rapidly at the boundaries of this domain. One thus obtains:
The steady-state solution of these equations has the following explicit form:
where we have set
Now, as we pointed out in the comment following Eqs. (29), D tends to zero when the reference state (&Do) loses its stability. For our model this happens (cf. Figure 1) when branch (a) terminates at the limit points of the isola branch. We conclude that the disappearance
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PERIODICALLY FORCED CLIMATE MODEL 77 of the periodic, phase-locked state should be accompanied by a violent enhancement of the fluctuations. This extends our previous result established in the absence of externaI periodic forcing (Nicolis, 1985). A similar situation arises in equilibrium phase transitions, as well as in non-equilibrium transitions such as hydrodynamic and chemical instabilities (see e.g. Nicolis and Prigogine, 1977).
From the explicit solution, (31a)-(31d), and from Eqs. (27), ( 5 ) and (2) linking the variables u,u to the original variables q , 0 one can deduce the explicit form of the variance of surface temperature, (0’) and sea-ice extent, (1’). One finds, after a straightforward calculation,
( q 2 ) = 4 [ ~ , , + ( u 2 ) cos’ w t + ( v 2 ) sin2 ot+ ( u u ) sin2wt],
(e2)=4[K,+(u2)(cosWt--WOsin~t)2+ (v2)(sinot+oocosot)2
+
2(uu)(cos ot-ao sin wt)(sin wt+
oo cos ot)],+
(u2)(sin ot+
oo cos wt) sin wt+
(uu)[(cos ot - oo sin ot) sin ot+(sin cot
+
wo cos at) cos ot]), (32)where K,,, KO, K,,, are related to the characteristics of the periodic (phase-locked) solution of q, 8 in the absence of fluctuations:
K,, = (Ao cos ot
+
Bo sin at)’,x,,, = A$(cos ot - wo sin at) cos ot
+
B$(sin wt+
oo cos wt) sin ot+
AoBo[(cos cot - wo sin at) sin ot+(sin ot+
wo cos at) cos ot].We see that in the phase-locked state the variances of the original variables are periodic functions of time, with a period equal to half the period of the forcing. This illustrates the intuitively clear idea that fluctuations vary on a faster time scale than do the macroscopic variables.
5. TRANSITIONS BETWEEN PHASE-LOCKED AND QUASI-PERIODIC SOLUTIONS
The analysis following the establishment of the Fokker-Planck equation [Eq. (26)] was limited to the vicinity of the phase-locked state. We now adopt a more global point of view and inquire into the relative stability of the phase-locked and quasi-periodic states against stochastic perturbations. Indeed, even though each of these states may be stable according to the deterministic description, fluctuations will induce a tunnelling, forcing the system to switch back and forth between different regimes.
In order to visualize the dynamics in a transparent way we introduce a Poincart surface of section (Guckenheimer and Holmes, 1983). Remember that a periodically foiced system of two variables q and 8 evolves, effectively, in a three-dimensional space since one can always express the forcing through q cos X, dX/dt = o, thereby introducing its phase X as a third variable. One can map the continuous dynamical system thus defined into a discrete one by following the points at which the trajectories cross (with a slope of prescribed sign) the plane cos X = C, corresponding to a given value of the forcing. One obtains in this way a Poincark map, that is to say, a recurrence relation linking the values of q,8 at the (n+l)st intersection to their values at the nth intersection and to the intensity of the noise.
In the absence of fluctuations, the PoincarC map will possess an invariant set which will be the intersection of the attractor of the full continuous system and the PoincarC surface of section. Thus, a periodic regime (phase-locked state) will give rise to a fixed point and a quasi-periodic one to a closed curve. We want now to see how this situation is modified by the stochastic perturbations induced by the fluctuations.
Figure 2 describes the results of a numerical solution of the
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PERIODICALLY FORCED CLIMATE MODEL 79
rl
0 -
-1
- 2
I I I I I
. ...
1 -
- .
. :. , ,. .,,.,. L.. .. ,. . .
i . :.
-
.-
1 I I I I
stochastic differential equations (8), with an initial condition close to the phase-locked state [curve (a) of Figure
11
and for parameter values far from the limit points of the isola branch. Instead of a single fixed point we now observe on the Poincark map a cloud of points corresponding to small excursions performed by the system as a result of the stochastic forcing. For the period of the simulation, no jump to the quasi-periodic solution (which coexists with the phase-locked one for this parameter range) was observed.In Figure 3 we report the results of the stochastic simulation in a parameter range closer to the limit point of the isola branch. We again start near the phase-locked solution, but for the noise strength and the same time interval as before, the system now spends a non- negligible amount of time on a part of state space belonging to the quasi-periodic solution. Finally, in Figure 4 the parameter values are such that the only stable regime is the quasi-periodic solution. We see that, because of the fluctuations, the system performs excursions around the deterministic attractor (closed curve), which practically fill an annular region around the attractor.
PERIODICALLY FORCED CLIMATE MODEL 79
FIGURE 2 Poincark map of the periodically forced Saltzman oscillator in the presence of stochastic noise and for a period of simulation of t=4500. Parameter values used b = 2, n = 6.4, 6.1 = 2.82, qq = 0.07, qe = 1.7 and u,, = bg = 0.5. Initial con- ditions are chosen close to the phase locked-state [branch (a) of Figure 11.
stochastic differential equations (8), with an initial condition close to the phase-locked state [curve (a) of Figure
11
and for parameter values far from the limit points of the isola branch. Instead of a single fixed point we now observe on the Poincark map a cloud of points corresponding to small excursions performed by the system as a result of the stochastic forcing. For the period of the simulation, no jump to the quasi-periodic solution (which coexists with the phase-locked one for this parameter range) was observed.In Figure 3 we report the results of the stochastic simulation in a parameter range closer to the limit point of the isola branch. We again start near the phase-locked solution, but for the noise strength and the same time interval as before, the system now spends a non- negligible amount of time on a part of state space belonging to the quasi-periodic solution. Finally, in Figure 4 the parameter values are such that the only stable regime is the quasi-periodic solution. We see that, because of the fluctuations, the system performs excursions around the deterministic attractor (closed curve), which practically fill an annular region around the attractor.
2 -
T
- 2
I I I I I I I
- -
0 -
-
- -
-
. .-
.
I I 1 I I I I-5 0 s 0
FIGURE 3 PoincarQ map of the periodically forced Saltzman oscillator in the presence of stochastic noise. Period of simulation and parameters as in Figure 2, except 0=2.42. Initial conditions close to the limit point Q.
FIGURE 3 PoincarQ map of the periodically forced Saltzman oscillator in the presence of stochastic noise. Period of simulation and parameters as in Figure 2, except 0=2.42. Initial conditions close to the limit point Q.
FIGURE 4 PoincarC map of the periodically forced Saltzman oscillator in the presence of stochastic noise. Period of simulation and parameters as in Figure 2, except 0=2.27 (region left of the limit point Q is where the quasi-periodic solution is the only stable regime).
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PERIODICALLY FORCED CLIMATE MODEL 81 6. CONCLUDING REMARKS
In this paper we have developed a theory of fluctuations of periodically forced climate models governed by Eqs. (1). In addition to the stochastic behavior of the periodic solutions, our formalism has enabled us to handle fluctuations around quasi-periodic solutions.
To our knowledge this calculation is one of the first attempts in this direction reported in the literature. Our results provide some new elements for understanding the origin of unpredictability in climate dynamics. While a phase-locked regime is a perfectly coherent, predictable state from the deterministic point of view, we have seen that in the presence of stochastic perturbations it becomes partially unpredictable since fluctuations can induce transitions toward the quasi-periodic state. This mechanism could provide an explanation of some of the complex features of the paleoclimatic variance spectra inferred from geological data. For instance, the new time scale associated to thc characteristic passage time of these stochastic transitions may turn out to be related to those observed in the low frequency part of the paleoclimatic record, usually attributed to systematic effects arising from orbital variations.
Acknowledgement
This work is supported, in part, by the EEC under contract no. CLI-106-B(RS).
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