## 1. Introduction

The Atlantic and Pacific storm tracks of the Northern Hemisphere climate system are often modeled separately, although there are no obvious physical mechanisms that would account for complete or nearly complete separation. Separation of the sectors in models is solidly supported by observations. The two familiar teleconnection patterns, the Pacific–North America (PNA) and the North Atlantic Oscillation (NAO), which describe separate modes of variability over the Pacific and Atlantic, respectively, emerge objectively from an empirical orthogonal function (EOF) analysis of observed data (Wallace and Gutzler 1981). In an alternative description of the Northern Hemisphere flow in terms of an index cycle, in which “blocked” flow patterns alternate irregularly with “zonal” patterns, observations indicate that there are two blocking centers in the two sectors, respectively, each with its own independent index cycle (e.g., Rex 1950; Treidl et al. 1981; Tibaldi and Molteni 1990).

Yet the major landmasses between the Atlantic and Pacific Oceans do not create barriers, arising from topographic, hydrological, or thermal properties, sufficient to divide a flow that can be approximated as a single reentrant channel flow into two. Relationships between the sectors of some sort are indeed to be expected. It might be imagined, for instance, that a blocking event in one sector would impact the circulation in a way that would affect the likelihood of blocking in the opposite sector. The nature and magnitude of such relationships are the subjects of the present work.

The relationship between the Atlantic and Pacific sectors appears typical of a situation that is common in climate modeling. Simple models are constructed for climate subsystems that are defined geographically or functionally, although it is well known that the subsystems are not autonomous components of the global climate system. Low-dimensional chaos is invoked to explain the irregular oscillations of these subsystems, as with the midlatitude index cycle (Charney and Devore 1979; Legras and Ghil 1985; de Swart 1988), or the El Niño–Southern Oscillation (ENSO) in the Tropics (e.g., Vallis 1986; Jin et al. 1994, Tziperman et al. 1995). The input from other climate subsystems is temporarily ignored, although the chaotic behavior is expected to be sensitive to such input.

One therefore seeks to understand the interactions of coupled climate subsystems in a way that accounts for their semiautonomy and also predicts measurable relationships. A relevant property of loosely coupled chaotic systems is that they tend to fall into synchronized motion along their strange attractors, irrespective of initial conditions, for a wide variety of system dynamics and coupling configurations (Fujisaka and Yamada 1983; Afraimovich et al. 1986; Pecora and Carroll 1990; Pecora et al. 1997). In the general case, synchronization is defined by a one-to-one correspondence between states that are different, but the correspondence is fixed. In a previous study (Duane et al. 1999), it was shown that a form of partial synchronization governs the relationship between low-order (10 variable) chaotic models of the Northern and Southern Hemisphere midlatitude systems, which are loosely coupled due to exchange of Rossby waves through the Tropics. The partial synchronization results in small simultaneous correlations between modeled blocking events in the two hemispheres, a behavior that agrees with observations. It was suggested that similar relationships could give rise to a large new class of teleconnections between pairs of subsystems of the global climate.

While the models of the midlatitude systems clearly exhibit a tendency toward synchronization, the claimed confirmation of this tendency in observations is questionable because 1) the rather small correlations might have some other explanation, and 2) the highly idealized 10-variable truncation may not capture the physics of the full fluid dynamical equations or their coupling. Other pairs of climate subsystems might provide better examples of synchronization. The relationship between the Atlantic and Pacific sectors of the Northern Hemisphere is naively expected to be much stronger than the relationship between the two midlatitude systems that are separated by the Tropics. Therefore, an attempt has been made to find synchronization between simple coupled models of zonal/ blocked flow vacillation in the two sectors. Unlike the previous north–south work, fully resolved models were used for the present study, although other simplifications were retained. Strong synchronization was indeed found, but the detailed correspondence between states defining the synchronization implies *anti*correlation between blocking events. Further, the magnitudes of these correlations are small, as in the north–south case.

The synchronization study provides a perspective on a recent debate about whether Atlantic sector variability and Pacific sector variability are both part of a circumpolar Arctic Oscillation (AO). Thompson and Wallace (1998) suggested that the North Atlantic Oscillation should be viewed as a component of the AO, which has a large signature in the Pacific sector as well, and resembles the dominant mode of variability in the lower stratosphere. There are indeed strong correlations between the Atlantic midlatitudes and the Arctic. However, as was pointed out by Deser (2000), correlations between the Atlantic midlatitudes and the Pacific midlatitudes are still weak. The present study suggests that the weakness of the intersectorial relationship stems simply from the physical distance between the sectors and the long advective time scales that result.

Another general question addressed by the study concerns the role of low-order chaos in the modeling of systems that arguably are not low-order chaotic. In the case of ENSO, for example, it has been argued that the observed variability is more likely the result of the forcing of a linear system by stochastic noise (e.g., Penland and Magorian 1993) than of low-order nonlinear dynamics. The point of view taken here is that the difference is not qualitative, in a situation where the “noise” arises from deterministic processes. The fully resolved models that were used here are not thought to possess attractors of very low dimension. However, we suggest that the existence of an inertial manifold relegates the highest-order modes to slave status and makes a low-order chaotic paradigm broadly relevant.

In this paper, which extends a previously published note (Duane and Tribbia 2001), we represent each sector using a channel model of zonal/blocked flow vacillation of Vautard et al. (Vautard et al. 1988; Vautard and Legras 1988) that is reviewed in the next section. The synchronized chaos paradigm and its previous application to geophysical fluid dynamics is also reviewed. Then we show in section 3 that two Vautard–Legras channels can be coupled so as to represent a single channel with two sectors, without introducing an arbitrary boundary between the sectors. In section 4, we explain how synchronization between the two channels implies a coherence between the two sectors that preserves a weak intersectorial anticorrelation effect predicted by the Vautard–Legras model, and why this effect is independent of noise and order of truncation. To compare the model results to observations, one must properly describe the nonsynchronous relationships between the large-scale components of the flow, as discussed in section 5. Although the anticorrelation predicted by the simplified model cannot therefore be observed directly, a more robust prediction of the model study is indeed confirmed in both observations and in general circulation model (GCM) data. The discussion of the general role of chaos synchronization in the climate system in the concluding section leads to a suggestion of scenarios in which the effects of synchronization may be prominent.

## 2. Background

### a. The quasigeostrophic channel model of midlatitude blocking

*q*:

*i*= 1, 2,

*ψ*is streamfunction, and the Jacobian

*J*(

*ψ,*· ) = (∂

*ψ*/∂

*x*)(∂ · /∂

*y*) − (∂

*ψ*/∂

*y*)(∂ · /∂

*x*) gives the advective contribution to the Lagrangian derivative

*D*/

*Dt.*Equation (1) states that potential vorticity is conserved on a moving parcel, except for forcing

*F*

_{i}and dissipation

*D*

_{i}. The discretized potential vorticity is

*q*

_{i}

*f*

_{0}

*βy*

^{2}

*ψ*

_{i}

*R*

^{−2}

_{i}

*ψ*

_{1}

*ψ*

_{2}

^{i}

*f*(

*x,*

*y*) is the vorticity due to the earth's rotation at each point (

*x,*

*y*),

*f*

_{0}is the average

*f*in the channel,

*β*is the constant

*df*/

*dy*, and

*R*

_{i}is the Rossby radius of deformation in each layer. Periodic boundary conditions are imposed in the longitudinal

*x*dimension. Following Vautard and Legras, we use an expression for dissipation of the form

*D*

_{p}) for modes with zonal wavenumbers 0, 1, 2, and 3. This form of the dissipation term, model parameter values, and other details are based on those of Vautard et al. (1988), except that the width of the channel is half the length and a second channel (not shown in the figures), with flow in the opposite direction, is used to join the upper and lower latitudinal boundaries, to enforce the free-slip boundary conditions used by Vautard et al. If the forcing is chosen to be a relaxation term,

*F*

_{i}

*μ*

_{0}

*q*

^{*}

_{i}

*q*

_{i}

*q*

^{*}

_{i}

*ψ** shown in Fig. 1a. For parameter values such as those given in appendix A, the model will then vacillate chaotically between two relatively stable flow regimes that naturally divide state space, illustrated for instance by the flows in Figs. 1b and 1c, which are identified with the zonal and blocked phases of the midlatitude index cycle, respectively.

The periodic boundary conditions are taken here to represent the actual topology of the midlatitude region. However, since there is only one jet and one blocking center, the model represents either an active Pacific sector adjoined to a passive Atlantic sector, or vice versa.

*q*

_{i}/∂

*t*) vanishes. One such solution is a blocked flow configuration that can be taken to define the blocking regime. (Dynamical regimes can also be defined by local maxima of the PDF over the space of flows, with similar results.) An analysis of the balance of terms for the equilibrated blocked flow solution reveals the mechanism that maintains the block. The term

*L*and

*S*denote large- and small-scale components, respectively, defined by arbitrarily partitioning a Fourier expansion.) This feedback effect is primarily responsible for maintaining the blocking dipole. In Fig. 2, it is seen that the spatial pattern of FL (after a transformation

*q*→

*ψ*) closely matches the streamfunction anomaly in the blocked state and so acts to maintain this anomaly against dissipation. (In the upper layer, the FL pattern is shifted slightly upstream relative to the anomaly pattern, but the difference is balanced by the large-scale advection terms.) Thus emerges the well-known explanation of blocking as the result of interscale interactions, with the eddies that are engendered by a blocked flow configuration in turn acting to maintain that configuration. A qualitatively similar picture of two-way interscale interactions also follows from observational studies of the maintenance of storm track anomalies (Branstator 1995).

A less familiar inference from the Vautard–Legras analysis is that the interscale interactions tend to suppress blocking in the passive sector, to a small degree, when the active sector is blocked. The FL term would inhibit a streamfunction anomaly (±) pattern similar to that shown in Fig. 2, but which is translated downstream by half a channel length, since the extrema in the fourth longitudinal quadrant of level 2 are slightly larger in magnitude than the extrema in the third quadrant. Physically, the inhibition effect occurs because potential vorticity is conserved on moving parcels except for forcing and dissipation, so that potential vorticity that is transferred between scales in one sector is not available for transfer downstream.

However, the inhibition effect does not by itself imply anticorrelation between blocking events in opposite sectors, in a symmetrical model with jets in both sectors. As shown by the counterexample in section 4a, anticorrelation in a model with both Atlantic and Pacific jets requires an additional coherence, since the small mutual inhibition effect may be obscured by nonlinear effects of combining the two forcings. It is therefore necessary to explicate the construction of a double-jet model to draw inferences from the Vautard–Legras analysis for Atlantic–Pacific relationships.

### b. Synchronized chaos in geophysical fluid dynamics

Low-order chaotic models have been proposed for various climate subsystems that exhibit irregular vacillation, such as ENSO and the midlatitude index cycle, that arguably capture key features of the subsystem dynamics. Indeed, the proper role for chaos in theoretical climate dynamics, if any, appears to be in the description of such subsystems rather than in the specification of a low-dimensional attractor for the global climate, a point emphasized by Lorenz (1991).^{1}

*synchronized chaos*(or

*chaos synchronization*). Pairs of systems of ordinary differential equations in many variables that are coupled through only a few of those variables often fall into synchronized motion along their strange attractors irrespective of differences in initial conditions. For instance, a pair of Lorenz systems linked symmetrically by exchange of their

*X*variables only:

*t*→ ∞:

*X*−

*X*

_{1}→ 0,

*Y*−

*Y*

_{1}→ 0, and

*Z*−

*Z*

_{1}→ 0. Synchronization also occurs if the

*Y*variables are coupled in a manner analogous to (6), but not if the

*Z*variables are so coupled.

*generalized synchronization,*we have two different dynamical systems

**x**∈

*R*

^{N}and

**y**∈

*R*

^{N}. If the dynamics are modified so as to couple the systems:

*R*

^{N}→

*R*

^{N}such that ‖Φ(

**x**) −

**y**‖ → 0 as

*t*→ ∞. Identical synchronization may be transformed to generalized synchronization simply by a change of variables in one system, but not the other, that is, a change in the description of one of the systems (Rulkov et al. 1995). In this situation, the correspondence function Φ is known a priori. Generalized synchronization may be difficult to detect without prior knowledge of Φ.

Synchronized chaos also occurs in pairs of systems of partial differential equations, such as a pair of Ginzberg–Landau systems which synchronize when coupled only at a discrete set of points (Kocarev et al. 1997). Additional examples and mathematical properties of synchronized chaotic systems are given in appendix B.

*S*accumulates information about the difference between the states of the two Lorenz subsystems. As Γ → ∞ in (9d), with

*·*finite,

*S*→

*c*(

*X*−

*X*

_{1}). In this limit, (9) reduces to a form similar to (6) but where the exchange of

*X*variables is incomplete. Each of the Lorenz systems is partially driven and partially autonomous, with the degree of autonomy given by the quantity 1 −

*c.*It is found that this configuration also synchronizes for a range of values of

*c,*including

*c*= 1.

In the general case of the coupled system (9) with finite Γ, the subsystems exchange information more slowly: If *X* and *X*_{1} are slowly varying, then *S* asymptotes to *c*(*X* − *X*_{1}), over a time scale 1/Γ. Thus, while *c* may be interpreted as the degree of coupling, Γ is an inverse time lag in the coupling dynamics.

Trajectories of (9) are depicted in Fig. 3 by plotting *Z*(*t*) − *Z*_{1}(*t*), for *c* = 1 and decreasing values of Γ. For large Γ, the case represented in Fig. 3a, the subsystems synchronize. As Γ is decreased, corresponding to increased time lag, increasingly frequent bursts of desynchronization are observed, as in Fig. 3b, until at Γ = 1 in Fig. 3c, no portion of the trajectory is synchronized. The case where Γ = 1 is the physically relevant situation in which the time scale of the delay due to the coupling is of the same order as the time scale of the intrinsic dynamics of the subsystems. In this case, although there are no macroscopically discernible periods of complete synchronization, the trajectories hug the *X* axis more closely than they would in the uncoupled case (Fig. 3d), a situation described as *partial synchronization.*

It was shown previously (Duane 1997) that an ordinary differential equation (ODE) system with a coupling similar to that in (9) with Γ ≈ 1 describes the interaction of two 10-component models of the Northern and Southern Hemisphere midlatitude circulation, each obtained by truncating the barotropic vorticity equation. The physical coupling is due to the exchange of Rossby waves through narrow areas of the Tropics where the prevailing winds are westerly. Less severe truncations also give a coupling of similar form (Duane et al. 1999).

The interhemispheric correlations due to partial synchronization in the north–south model are small, typically with values less than 0.2, a prediction that agrees with observations, perhaps fortuitously. In the Atlantic– Pacific case studied here, one might naively expect larger correlations, since there is no natural barrier separating the two sectors analogous to the band of tropical easterlies in the north–south case. For the same reason, a different construction is needed, since the separate sectors are not readily described as semiautonomous systems.

## 3. The two-sector midlatitude channel model as two coupled channels

A two-sector model is easily constructed by using a potential vorticity field *q** in (4) derived from a streamfunction *ψ** [via (2)] that would describe a flow with two jets upstream of corresponding blocking centers in each sector, as shown in Fig. 4, in place of the single-jet flow shown in Fig. 1a. However, the relationship of such a two-sector model to a pair of single-sector Vautard–Legras models is not immediately clear, since there is no natural way to impose boundary conditions at a midline in the two-sector model so as to partition it into two coupled sectors. We therefore describe an alternative construction of the two-sector channel model, achieved by coupling two single-sector channels of the active–passive type described earlier, with jets in opposite sectors.

**k**on each quantity indicates the wavenumber

**k**spectral component. The superscripts

*A*and

*B*in (10) and (11) designate the two separate channel models, each of which has two layers. The layer index

*i*has been suppressed in (10) and (11), and will be suppressed henceforth. (Unlike the forcing terms, the dissipation terms

*D*

^{A,B}will consistently be assumed to have the same form in the two channels.)

*k*

_{x0}= 3 and

*k*

_{y0}= 2 (in units of waves per channel length or width, respectively) to distinguish between “eddies” and the large-scale flow, following Vautard and Legras (1988).

*q̂*= (

*q*

^{A}+

*q*

^{B})/2 of the solutions of (10a) and (10b), for strong coupling

*c*= 1/2, is the solution of a model with the average forcing:

*q*

^{A*}and

*q*

^{B*}. As the coupling is increased from

*c*= 0 to

*c*= 1/2, the two separate Vautard–Legras models are merged continuously. [For intermediate values, one might imagine that each channel weakly feels the effect of a jet in the “passive” sector, but this effect is not strictly equivalent to a relaxation forcing of the form (4) with a weak jet. These intermediate values are of interest for the mathematical study of the bifurcation that leads to synchronized behavior, although the coupled system defined by such values has no direct physical interpretation.] This construction has introduced extra, nonphysical degrees of freedom, corresponding to the passive sector in each channel. The decomposition of the two-sector channel is not unique but arguably has the simplest form permitted by the nonlinearities in the Jacobian terms. It is shown in the following section that the nonphysical degrees of freedom are dynamically constrained, and essentially disappear in the fully coupled model.

## 4. Anticorrelation in blocking activity from synchronization of the flow

### a. Generalized synchronization of the coupled channels

The flow fields in the coupled channels governed by (10) are found to synchronize, regardless of differences in initial conditions, as seen in Fig. 5. The two channels cannot exhibit *identical* synchronization because the forcing terms on the right-hand sides of the two equations (10) are different. Close inspection of Fig. 5 indeed reveals persistent small differences between the flow fields in the two channels. The situation is that of generalized synchronization, as known to occur in pairs of systems of ordinary differential equations (Rulkov et al. 1995) when small differences in parameters are introduced. Synchronization is commonly robust against such parameter variations: there is still a relationship between the states of the two systems, but the correspondence deviates slightly from the identity. Likewise here, at *c* = 1/2 we have *ψ*^{A} ≈ *ψ*^{B} ≈ *ψ̂**c* = 0 to *c* = 1/2, the dynamics of each channel changes so as to incorporate an approximate, “virtual” counterpart of the dynamics of the sector that is forced in the other channel.

It is desired to characterize the relationship between the two sectors in terms of the familiar blocked and zonal flow regimes. Blocking can be defined in each sector of the two-jet channel analogously to the definition for a single-jet channel (see Fig. 4). It is found that blocking in one sector anticorrelates with blocking in the other sector, as shown in Fig. 6 for several variants of the model. Anticorrelation is seen to be robust against changes in the shape and position of the jets, unless the jets are skewed latitudinally, a case that will be discussed in section 5.

The anticorrelation in blocking activity follows from generalized synchronization of the two constituent single-jet channels. Instances of generalized synchronization for which the correspondence is very different from the identity can be obtained from identical synchronization by a change of variables in one of a pair of identically synchronized systems, that is, simply by changing the *description* of one of the systems, as in the original examples given by Rulkov et al. (1995). In the present case, the correspondence between the two channels is near to the identity in the representation suggested by Fig. 5, but corresponding states in the two channels are described very differently. While the state of channel *A* is described as blocked or zonal depending on the flow in the first half of the channel, in the area denoted by the solid box in Fig. 5, just downstream of the channel's driving jet, the state of channel *B* is similarly described depending on the flow in the second half of the channel. A change of variables in one channel corresponding to the 180° shift in the zonal direction that is appropriate to describe the rough Atlantic–Pacific symmetry turns near-identical synchronization into a correspondence far from the identity that is manifest as anticorrelation in blocking activity.

As discussed in section 2a for the case of an uncoupled channel with one jet, blocking activity is weakly anticorrelated in the two halves of either channel, although blocking occurs less frequently in the passive sector in the uncoupled case. Synchronously coupled channels therefore exhibit interchannel anticorrelation as well. Since blocking is defined nonlinearly in terms of the flow field, anticorrelation in the average field *q̂* does not follow generally from anticorrelation within the two parallel channels separately, but does follow in the synchronous case where *q̂* ≈ *q*^{A} ≈ *q*^{B}.

Synchronization of the two channels implies coherence between the chaotic blocked/zonal flow vacillations in the two sectors of a single channel. Without synchronization, anticorrelation in a two-jet model is not a necessary consequence of anticorrelation in a single-jet model. As a counterexample, illustrated in Fig. 7, we consider a model with wavenumber-independent forcing *F*_{k} = *μ*_{0}(*q*^{*}_{k}*q*_{k}). With a jet in only one sector, anticorrelation between blocking activity in the two sectors is indeed observed. For forcing coefficient values above about 0.32, consistently negative correlations are seen in Fig. 7a. (The values of *μ*_{0} used in the model considered here, with all modes forced, are smaller than those in the model with free eddies.) The inference of anticorrelation from the eddy-forcing analysis in section 2a is thus confirmed.^{2} In the double-jet model, on the other hand, the correlations plotted in Fig. 7b for corresponding parameter values are positive. Examining the corresponding two-channel model given by *F*^{(A,B)}_{k}*μ*_{0}[*q*^{(A,B)*}_{k}*q*^{(A,B)}_{k}

To further elucidate the role of synchronization, one might consider a simple mechanical analogy. Imagine a flexible rubber seesaw that is somehow forced at one end. The up–down positions of the two ends anticorrelate, though not perfectly. Now imagine that the seesaw is forced at both ends. The anticorrelation effect may or may not be preserved, depending on the composition of the seesaw and the nature of the forcing. One seeks to describe additional requirements for anticorrelation to be preserved in the doubly forced seesaw. It is suggested that a sufficient condition for the required coherence is of the following form: Tie two parallel identical seesaws together in a certain way, using the same material of which the seesaws are made. The tie should not cause additional bending in either seesaw, but should merely adjust the relative timing of the swings. Force the two parallel seesaws at opposite ends. If the end positions of each individual seesaw anticorrelate and the two linked parallel seesaws synchronize, despite potential conflicts between the two forcings, then there will also be anticorrelation between the two end positions of a single doubly forced flexible seesaw.

### b. The synchronization mechanism

*β*= 0, in which case no Rossby waves exist, blocking events in the two sectors are still found to anticorrelate, as shown in Fig. 8. Rather, the anticorrelation appears to be due simply to the exchange of the small-scale eddy components of the flow, since it is the small-scale components that are left unconstrained by the forcing (11) with the wavenumber dependence (12). Coupling of the eddies is essential to both synchronization and the anticorrelation effect, as shown by the counterexample described earlier. The role of the eddies is established by explicitly restricting the coupling to the small-scale components, replacing (10) by

*q*

_{eddy}is constructed from the spectral components

*q*for which |

_{k}*k*

_{x}| >

*k*

_{x0}or |

*k*

_{y}| >

*k*

_{y0}. The channels still synchronize, as shown in Fig. 9. If, on the other hand, we allow only the large-scale components of the flow to take part in the advective coupling:

*q*

_{large}≡

*q*−

*q*

_{eddy}, synchronization does not occur, as seen in Fig. 10.

The essential role of the eddies in synchronization would appear to be related to the upscale cascade of energy in quasi-two-dimensional fluid dynamics. The synchronization behavior described here is expected to be widespread in such systems, but would not be expected in a pair of fully three-dimensional models.

*q*

_{medium}is constructed from the spectral components

*q*

_{k}for which |

**k**| ≤

*k*

_{noise}and either |

*k*

_{x}| >

*k*

_{x0}or |

*k*

_{y}| >

*k*

_{y0}. Additional coupling in the forcing terms is introduced so as to induce synchronization with a minimum number of components coupled. Equations (11) and (12) are replaced by

The channels thus coupled through the medium-scale eddies alone still synchronize, as shown in Fig. 11, where the uncoupled noise modes are defined as those with wavenumbers higher than *k*_{noise} = 15.^{3} In this configuration, only about one-fourth of all the modes in the numerical model, which is truncated at *k*_{x} = 31, *k*_{y} = 23 (in waves per channel length) are directly coupled. To verify that the flow fields in Fig. 11 are generally synchronized, one can remove the skew in the forcing terms, so that *q*^{A*} = *q*^{B*}, and inquire as to whether the resulting flow fields for the modified model are *identically* synchronized. The resulting flow fields shown in Fig. 12 are indeed identical. For a decreased value of *k*_{noise}, however, the synchronization effect disappears, as seen in Fig. 13 for *k*_{noise} = 10. Coupling of additional modes up to about *k* = 15 is required. Thus, there are three ranges of wavenumbers of interest: low wavenumbers of modes that are ineffective in synchronizing the channels, medium wavenumbers defining the synchronizing eddies, and high wavenumbers defining modes that need not be coupled to affect synchronization. That synchronization occurs without coupling the highest spatial frequencies used in the model also establishes that the model is fully resolved for the purposes of the present study and that the inclusion of more modes would not change the results.^{4}

Formally, the two-sector model is recovered by coupling all modes in the advective terms and only medium-scale modes in the forcing terms. The synchronized solutions are then approximate solutions of a model with the average forcing (13) as before, since the terms in the forcing that couple the two channels cancel for arbitrary *c*′. However, including the extra small-scale modes in the advective coupling preserves the synchronization effect and does not tightly constrain these modes. We therefore continue to refer in this section to Eq. (16), in which only the medium-scale modes are coupled.

*ψ*

*ψ*

_{master}

*ψ*

_{slave}

*ψ*

_{master}∈

_{1}, a finite-dimensional subspace,

*ψ*

_{slave}∈

_{2}, an infinite-dimensional subspace,

_{1}⊥

_{2}, and

_{1}⊕

_{2}=

*ψ*

_{slave}

*ψ*

_{master}

*derived*from variables that are already coupled in a given representation. Coupling the original variables with sufficient strength should be as effective as coupling the derived variables, provided that the function defining the derived variables is well behaved. If the function

*c*= 1/2—the value required to generate the two-sector model via (13)—is insufficient to affect synchronization with coupling of the master modes alone.

When the forcing terms *F*^{A} and *F*^{B} are different in form, due to longitudinal skew between the two jets, the master–slave functions ^{A} and ^{B} are also different. In this case *ψ*^{A}_{slave}*ψ*^{B}_{slave}*ψ*^{A}_{master}*ψ*^{B}_{master}

Noise enters the intersectorial relationship in several ways, but does not qualitatively affect our conclusions. (Noise in this context refers only to elements that are not essential to the sectorial dynamics—as distinct from the case of linear stochastic models that are driven by noise.) Chaos synchronization is usually robust against small amounts of noise in the coupled systems and in the coupling signal, hence its usefulness in communications schemes (Pecora et al. 1997). The same dynamical mechanisms that allow two coupled systems to reach synchronization from arbitrary initial conditions in the first place typically resist noise-induced perturbations also. Where the sources of noise are physical differences in the dynamical systems associated with the two jets, the slaved components are expected to compensate these differences, as they compensate the difference in jet locations.

### c. Weakness of the intersectorial relationship

Contrary to what might be naively expected, the anticorrelation between blocking events in the two sectors is small and remains so for various changes in system configuration, as shown in Fig. 6. The relationship is in fact no larger in magnitude than the correlation or anticorrelation between blocking events in the midlatitude systems of the Northern and Southern Hemispheres reported by Duane et al. (1999), although these two hemispheres would appear to have a weaker connection than that between the two sectors of the same midlatitude channel. Evidently, the magnitude of the correlation is not affected by the mass transport between the sectors or by the fact that the two sectors are always coupled along the entire length of any boundary between them (in contrast to the Northern and Southern Hemisphere systems, which are mostly separated by a region of tropical easterlies, only exchanging Rossby waves through the narrow, intermittent “westerly ducts”).

In the north–south case, the smallness of the relationship was shown to be a result of the explicit time lag in the coupling between the two hemispheric systems, resulting from the time required for Rossby waves from one hemisphere to reach interior points of the opposite hemisphere. The coupling time lag results in a relationship of partial synchronization that is very far from complete synchronization. Although in the present case the two channels are completely synchronized, the intersectorial advective time lag within each channel system separately appears to explain the smallness of the anticorrelation values found here. That the eddy contribution to the streamfunction tendency is much weaker in the sector opposite to the jet seems to be a result simply of the physical distance between the sectors and the proportionately long advective time scales. Specifically, these time scales are long in comparison to dissipative time scales, so that eddies produced by the forcing jet are dissipated or transformed before reaching the opposite sector. An alternative picture is that of a range of velocities for the transmission of information between corresponding points in the two sectors, and hence a range of time scales that is increased proportionately for longer distances, resulting in interference which dampens the effect of eddies on blocking in the opposite sector as compared to their effect in the originating sector. The synchronization of the channels preserves the coherence of the sectors that is already present in each of the single-jet models, but does not increase it, so that the anticorrelation in the two-jet model remains small.

The relationship between the leading modes of variability in the Atlantic and Pacific sectors might be similarly explicated. While the cycles associated with the NAO and the PNA cannot be equated to the blocked/ zonal flow vacillation examined in the model study here, the essential characteristic of the synchronously coupled two-sector model is a correspondence between the detailed flow states of two coextensive models, each with one active and one passive sector. Such a correspondence can yield relationships between any regimes or indices defined from these flow states. Weak relationships between the two sectors of each model are preserved, but not enhanced, when the models are merged. The construction used in this study is especially general because it does not require the imposition of an arbitrary boundary between the sectors. That the Atlantic–Pacific relationship is quite weak in observations [despite Thompson and Wallace's (1998) claims to the contrary] would thus follow simply from the physical distance between features such as the sectorally local upper-tropospheric jets, themselves arising through processes that are external to the model.

## 5. Comparison with observations

The prediction that exchange of eddies between the Atlantic and Pacific sectors causes anticorrelation of blocking activity cannot be directly compared with observations because the treatment of the large-scale flow in the model is unrealistic. In reality, the strength and positions of the upper-tropospheric jets vary considerably. The relationships between the large-scale jet flow in the two sectors must therefore be considered, before predictions can be made about correlations in blocking activity.

*T*at all altitudes, across a given meridian

*λ*

_{0}, is

*ρ*and

*u*are density and zonal velocity, respectively, at longitude

*λ*

_{0}, latitude

*ϕ,*and height

*z*or equivalently pressure level

*p,*

*a*is the earth's radius,

*g*is the gravitational constant, and a hydrostatic approximation,

*dp*=

*gρdz,*is used to convert to pressure coordinates in the second equality. We compute the vertical density

*p*

_{0}= 300 mb, in a midlatitude band between 40° and 80°N:

*λ*

_{0}= 10°W in the Atlantic sector (solid line) and for

*λ*

_{0}= 175°E in the Pacific sector (dashed line). The fluctuations in transport are seen to be comparable in magnitude to the mean value.

Variations in jet position can also be readily incorporated in the channel model. Flows with the two jets in latitudinally skewed positions, as shown for example in Fig. 17, can be used to define the forcing streamfunction and corresponding potential vorticity field *q** in (11). Blocking correlations for a random assortment of skew values are plotted as circles in Fig. 6. In contrast to other changes in model configuration with results shown in Fig. 6, the introduction of a latitudinal skew in jet position tends to destroy the anticorrelation effect. The effect of skew can be understood if the inhibition effect is traced to the approximate conservation of potential vorticity on moving parcels, so that interscale transfer of potential vorticity is less likely downstream of a blocking region where it has already occurred. Such an effect would indeed be expected to weaken in regions of the passive sector that are not directly downstream.

The size of the intersectorial correlating effect in the real atmosphere due to coupling of fluctuations in the large-scale flow is not known. Nor is it known whether this correlating effect or the anticorrelating effect of the eddies is dominant. However, because the relationship between the fluctuations in the two hemispheres seems to be based on a principle of mass conservation, this relationship is expected to persist in the presence of latitudinal skew.^{5} Therefore, whatever the net correlation when the jets are at equal latitude, this value is expected to increase when the jets are skewed [as occurs at a low significance level in Fig. 16 for the skewed case (circles)]. The hypothesis that anticorrelation is weakened during periods of jet skew can thus be compared directly to observed data. Using a standard diagnostic for blocking described in appendix C, and obtaining the position of each of the two jets from the maximum of 300-mb zonal wind speed at nonblocked longitudes (i.e., where the jet is not divided by a blocking high pressure center), we computed the intersectorial correlation on days of given skew between jet positions, for boreal winters over a 40-yr period. The results plotted in Fig. 18 show significant increase in correlation as the skew increases from 0° to 10°, as expected.

An alternative theory of the correlation in blocking activity might be based on a tendency for the jets to covary in latitudinal position, instead of a tendency to covary in magnitude as in the model. But if the correlating effect were due to intersectorial continuity in jet position, the effect would not be likely to increase as the skew increases. Thus in such an alternative theory, as in a theory based on correlated jet magnitudes, the observed increase seems to come from a weakening of the eddy-induced anticorrelation.

To achieve greater confidence in the observed relationship between blocking correlation and jet skew, the analysis was repeated with data obtained from a general circulation model (GCM). Results are shown in Fig. 19 for 10^{6} days ≈ 2738 yr of perpetual winter, as simulated by the Community Climate Model. The correlations increase with skew value as predicted, with narrower error bars than for the observational data. While the increase occurs over a different range of skew values than for the observational data, the difference may be related to the fact that the GCM gives an unrealistically low blocking rate in the Pacific sector.

The behavior of the intersectorial correlation in blocking activity provides preliminary confirmation of the anticorrelation effect in the synchronized chaos model. Since the anticorrelation effect appears to be robust against configurational changes other than latitudinal jet skew and since the magnitude of the correlation caused by large-scale coupling is unknown, other ways to test the theory are not obvious at present. However, the preliminary test appears valid. It is expected that the variation in eddy-induced anticorrelation with jet skew and the relative independence of the large-scale correlation effect will be maintained in the presence of additional thermal and topographic detail. For instance if the form for the dissipative terms (3) is made to vary with longitude, so as to simulate the effects of continents and oceans, but the variation is the same in the two channels, then synchronization is maintained. The behaviors of the real atmosphere and the GCM depicted in Figs. 18 and 19, respectively, likely reflect the synchronized channel model dynamics.

## 6. Summary and conclusions

While the confirmation of the synchronizing effect in observations is preliminary, its ubiquity in quasi-two-dimensional fluid dynamical models has here been firmly established. In particular, it is now clear that synchronization due to exchange of medium-scale eddies occurs independently of the resolution of the model. Synchronization in truncated models such as the one discussed by Duane et al. (1999) is therefore representative of the behavior of fully resolved models.

More generally, it has been shown that a low-order chaotic paradigm is representative of the behavior of a high-dimensional system when that system has an inertial manifold. Where there is an inertial manifold that limits the effective dimensionality, albeit at a high value, the behavior of high-dimensional systems qualitatively resembles that of low-order ones. It is suggested that similar reasoning might provide an ex post facto justification for the historical use of low-order chaos to represent various phenomena in climate, even when those phenomena seem to be more accurately represented as high-dimensional systems driven by stochastic noise.

Synchronization is observed in fully resolved fluid dynamical models, as in simple systems of ordinary differential equations, for a variety of coupling configurations. Coupling two quasigeostrophic channel models via their advective terms results in a synchronizing tendency, regardless of any separation between the jets in the two channels. That synchronization occurs even when the jets are in different sectors in the two channels implies a coherence between different sectors of a single channel with two jets.

Coupling of the medium-scale eddies affects synchronization by slaving the smallest-scale eddies so as to compensate for differences in the forcing terms, as would follow from the existence of an inertial manifold for each channel. While the resulting reduction in dimensionality is expected to be important in the behavior of the climate system generally, the effects on correlations between the spatially separated subsystems that have been examined here and previously by Duane et al. (2001) are small. Despite the strong interaction between two sectors of the same channel, the time lag in the interaction diminishes the magnitude of the correlations, obscuring the novel order in the climate system implied by the synchronization tendency.

To find scenarios in which the synchronization tendency would lead to larger correlations, one might look either for faster coupling or longer time scales in the subsystem dynamics to which the coupling time is referenced. An arena in which time scales are characteristically longer than those considered here is that of oceanic processes. Especially if such processes are coupled via faster atmospheric processes, one might find strong synchronization effects. One candidate for such a configuration, that is deserving of investigation, would consist of the pair of subtropical and subpolar gyres in either ocean, coupled through an atmospheric wind stress mechanism.^{6}

Alternatively, fast coupling might be achieved by choosing a pair of systems that are contiguous or are defined on spatially overlapping domains, but that remain functionally distinct. A good candidate is the pairing of the Indian monsoon system and the ENSO system. The two systems overlap via convective and wind stress activity over the western Pacific, and are known to correlate intermittently. That is, strong monsoons tend to occur in La Niña years and weak monsoons in El Niño years (Yasunari 1990; Webster and Yang 1992), but the relationship disappears at irregular intervals (Torrence and Webster 1999). Since chaos synchronization commonly degrades through on–off intermittency (Ashwin et al. 1994; Venkataramani et al. 1996; Duane 1997), the observed intermittency may be a vestige of synchronization. The possibility that detailed synchronization underlies the correlations observed on long time scales merits investigation.

As for the Atlantic–Pacific relationship, to which this study was addressed, our findings support the conventional view, based on EOF analysis, that the modes of variability in the two sectors are largely distinct. Although the PNA and NAO teleconnection patterns cannot be described simply in terms of the blocked/zonal index cycle, the same factors that limit the size of the correlation effect in the case of blocking—stemming from the advective time scales associated with the spatial distances between the sectors—will reduce the effects of any synchronization tendency in the coupled Atlantic–Pacific dynamics. On the other hand, the chaos synchronization paradigm provides an understanding of the small relationship that does exist.

## Acknowledgments

The authors thank Grant Branstator for useful discussions. Part of this work was supported under NSF Grant 0327929.

## REFERENCES

Afraimovich, V. S., N. N. Verichev, and M. I. Rabinovich, 1986: Stochastic synchronization of oscillation in dissipative systems.

,*Radiophys. Quant. Electron***29****,**795–803.Ashwin, P., J. Buescu, and I. Stewart, 1994: Bubbling of attractors and synchronisation of chaotic oscillators.

,*Phys. Lett. A***193****,**126–139.Branstator, G., 1995: Organization of storm track anomalies by recurring low-frequency circulation anomalies.

,*J. Atmos. Sci***52****,**207–226.Brown, R., N. F. Rulkov, and N. B. Tufillaro, 1994a: The effects of additive noise and drift in the dynamics of the driving on chaotic synchronization.

,*Phys. Lett. A***196****,**201–205.Brown, R., N. F. Rulkov, and N. B. Tufillaro, 1994b: Synchronization of chaotic systems: The effects of additive noise and drift in the dynamics of the driving.

,*Phys. Rev. E***50****,**4488–4508.Charney, J. G., and J. G. DeVore, 1979: Multiple flow equilibria in the atmosphere and blocking.

,*J. Atmos. Sci***36****,**1205–1216.Deser, C., 2000: On the teleconnectivity of the “Arctic Oscillation.”.

,*Geophys. Res. Lett***27****,**779–782.de Swart, H. E., 1988: Low-order spectral models of the atmospheric circulation: A survey.

,*Acta Appl. Math***11****,**49–96.Duane, G. S., 1997: Synchronized chaos in extended systems and meteorological teleconnections.

,*Phys. Rev. E***56****,**6475–6493.Duane, G. S., 2003: Synchronized chaos in climate dynamics.

*Proc. 7th Experimental Chaos Conference,*San Diego, CA, AIP Conference Proceedings 676, 115–126.Duane, G. S., and J. J. Tribbia, 2001: Synchronized chaos in geophysical fluid dynamics.

,*Phys. Rev. Lett***86****,**4298–4301.Duane, G. S., P. J. Webster, and J. B. Weiss, 1999: Co-occurrence of Northern and Southern Hemisphere blocks as partially synchronized chaos.

,*J. Atmos. Sci***56****,**4183–4205.Fujisaka, H., and T. Yamada, 1983: Stability theory of synchronized motion in coupled-oscillator systems.

,*Prog. Theor. Phys***69****,**32–47.Huygens, C., 1673:

*Horoloquim Oscilatorium*. Apud F. Muget.Jin, F-F., D. Neelin, and M. Ghil, 1994: ENSO on the devil's staircase.

,*Science***264****,**70–72.Kalnay, E., and Coauthors, 1996: The NCEP/NCAR 40-Year Reanalysis Project.

,*Bull. Amer. Meteor. Soc***77****,**437–471.Kocarev, L., Z. Tasev, and U. Parlitz, 1997: Synchronizing spatiotemporal chaos of partial differential equations.

,*Phys. Rev. Lett***79****,**51–54.Legras, B., and M. Ghil, 1985: Persistent anomalies, blocking and variations in atmospheric predictability.

,*J. Atmos. Sci***42****,**433–471.Lorenz, E. N., 1991: Dimension of weather and climate attractors.

,*Nature***353****,**241–244.Parlitz, U., and L. Kocarev, 1997: Using surrogate data analysis for unmasking chaotic communication systems.

,*Int. J. Bifurcations Chaos***7****,**407–413.Pecora, L. M., and T. L. Carroll, 1990: Synchronization in chaotic systems.

,*Phys. Rev. Lett***64****,**821–824.Pecora, L. M., T. L. Carroll, G. A. Johnson, D. J. Mar, and J. F. Heagy, 1997: Fundamentals of synchronization in chaotic systems, concepts, and applications.

,*Chaos***7****,**520–543.Penland, C., and T. Magorian, 1993: Prediction of Niño 3 sea surface temperatures using linear inverse modeling.

,*J. Climate***6****,**1067–1076.Rex, D. F., 1950: Blocking action in the middle troposphere and its effect upon regional climate. Part I: An aerological study of blocking action.

,*Tellus***2****,**196–211.Rulkov, N. F., M. M. Sushchik, and L. S. Tsimring, 1995: Generalized synchronization of chaos in directionally coupled chaotic systems.

,*Phys. Rev. E***51****,**980–994.Strogatz, S. H., 2003:

*Sync: The Emerging Science of Spontaneous Order*. Hyperion, 338 pp.Temam, R., 1988:

*Infinite-Dimensional Dynamical Systems in Mechanics and Physics*. Springer-Verlag, 500 pp.Thompson, D. W. J., and J. M. Wallace, 1998: The Arctic Oscillation signature in the wintertime geopotential height and temperature fields.

,*Geophys. Res. Lett***25****,**1297–1300.Tibaldi, S., and F. Molteni, 1990: On the operational predictability of blocking.

,*Tellus***42A****,**343–365.Torrence, C., and P. J. Webster, 1999: Interdecadal changes in the ENSO–monsoon system.

,*J. Climate***12****,**2679–2690.Treidl, R. A., E. C. Birch, and P. Sajecki, 1981: Blocking action in the Northern Hemisphere: A climatological study.

,*Atmos.– Ocean***19****,**1–23.Tsonis, A. A., and J. B. Elsner, 1989: Chaos, strange attractors, and weather.

,*Bull. Amer. Meteor. Soc***70****,**14–23.Tziperman, E., M. A. Cane, and S. Zebiak, 1995: Irregularity and locking to the seasonal cycle in an ENSO prediction model as explained by the quasi-periodicity route to chaos.

,*J. Atmos. Sci***52****,**293–306.Vallis, G. K., 1986: El Niño: A chaotic dynamical system.

,*Science***232****,**243–245.Vautard, R., and B. Legras, 1988: On the source of midlatitude low-frequency variability. Part II: Nonlinear equilibration of weather regimes.

,*J. Atmos. Sci***45****,**2845–2867.Vautard, R., B. Legras, and M. Déqué, 1988: On the source of midlatitude low-frequency variability. Part I: A statistical approach to persistence.

,*J. Atmos. Sci***45****,**2811–2843.Venkataramani, S. C., B. R. Hunt, and E. Ott, 1996: Bubbling transition.

,*Phys. Rev. E***54****,**1346–1360.Wallace, J. M., and D. S. Gutzler, 1981: Teleconnections in the geopotential height field during the Northern Hemisphere winter.

,*Mon. Wea. Rev***109****,**784–812.Webster, P. J., and S. Yang, 1992: Monsoon and ENSO: Selectively interacting systems.

,*Quart. J. Roy. Meteor. Soc***118****,**877–926.Yasunari, T., 1990: Impact of Indian monsoon on the coupled atmosphere/ocean system in the tropical Pacific.

,*Meteor. Atmos. Phys***44****,**29–41.

## APPENDIX A

### Parameters of the Single-Sector Quasigeostrophic Channel Model

Values of the parameters that define the quasigeostrophic channel model given by Eqs. (1) and (3) in a reentrant channel of length *L* and width *W* are given next, with the various dissipation rates expressed as inverse relaxation times.

*L*length of channel 28 000 km*W*width of channel 14 000 km*f*_{o}planetary vorticity 1.0 × 10^{−4}s^{−1}*β**df*/*dy*at 45° latitude 1.6 × 10^{−11}m^{−1}s^{−1}*R*_{1}Rossby radius of deformation in upper layer 450 km*R*_{2}Rossby radius of deformation in lower layer 450 km internal friction acting on upper layer (1.1 days)*ν*^{int}_{1}^{−1} internal friction acting on lower layer (5.6 days)*ν*^{int}_{2}^{−1}*ν*^{Ek}Ekman damping, acting on lower layer (0.64 days)^{−1}*α*superviscosity coefficient in*x*dimension (relaxation time = 0.9 h for smallest-scale modes) 7.97 × 10^{13}km^{8}s^{−1}*α*′ superviscosity coefficient in*y*dimension 0.0 km^{8}s^{−1}

*D*

^{p}in (3) are

*q*

^{mn}

_{i}

*q*are given by

*d*

_{m|m=0,1,2,3}are

*F*

_{i}=

*μ*

_{0}(

*q*

^{*}

_{i}

*q*

_{i}) is given by

*q*

^{*}

_{i}

*ψ*

^{*}

_{i}

The equations are considered over the domain 0 ≤ *x* < *L* and 0 ≤ *y* < 2*W,* with periodic boundary conditions in both *x* and *y.* The region 0 ≤ *y* < *W* defines the “channel,” with a “second channel” in the region *W* ≤ *y* < 2*W,* where the streamfunction *ψ** forces a reverse flow.

The numerical integrations of the model (as for the two-sector model) were effected with a spectral transform method, with a time step of 0.38 h.

## APPENDIX B

### Phenomenology of Synchronized Chaos

*Y*

_{1},

*Z*

_{1}-subsystem to the master

*X,*

*Y,*

*Z*-subsystem. Equation (6) is a bidirectionally coupled version of (B1).

*diffusively,*as with a pair of bidirectionally coupled Rossler systems:

*α*parametrizes the coupling strength. (The coupling term used in the two-channel model with coefficient

*c*′ is of similar diffusive form.) Contrary to naive expectations, the tendency to synchronize does not increase monotonically with

*α,*but synchronization will indeed occur for sufficiently large

*α.*

Synchronization reduces the effective dimension of the phase space by half. Once synchronized, the six-dimensional system (B2) evolves in a three-dimensional hyperplane, and is further constrained, as *t* → ∞, to evolve on a strange attractor within that hyperplane. [For the coupled Lorenz systems (B1), an auxiliary variable *X*_{1} could be added, satisfying *Ẋ*_{1} = *σ*(*Y*_{1} − *X*_{1}), to define a system that is six-dimensional before synchronization.) With *generalized synchronization* of nonidentical systems, as discussed in section 2b, the hyperplane becomes a *synchronization manifold* defined by an invertible correspondence function Φ : *R*^{N} → *R*^{N}. The *N*-dimensional manifold in 2*N*-dimensional space is *p,* Φ(*p*)]|*p* ∈ *R*^{N}}. The synchronization manifold is dynamically invariant: If *x*(*t*) is a trajectory of a system such as (B1) or (B2), for *x* ∈ *R*^{2N}, and *x*(*t*_{1}) ∈ *x*(*t*_{2}) ∈ *t*_{2} > *t*_{1}. That is, a perfectly synchronized system remains synchronized.

It is commonly not the existence, but the stability of the synchronization manifold that distinguishes coupled systems exhibiting synchronization from those that do not [such as (B2) for different values of *α*]. A set of *N* Lyapunov exponents can be defined for perturbations in the *N*-dimensional space that is transverse to the synchronization manifold *h*^{⊥}_{max}*h*^{⊥}_{max}*local* stability properties, the size of the basin of attraction for the synchronized regime remains unknown. As *h*^{⊥}_{max}*blowout bifurcation.* For small positive values of *h*^{⊥}_{max}

*k*:

*X,*

*Y,*

*Z*→

*U,*

*V,*

*W*is not smooth (Pecora et al. 1997).

*D*-dimensional generalized Rossler systems (each equivalent to a Rossler system for

*D*= 3) will synchronize for any

*D,*no matter how large, when coupled via only one of the

*D*variables:

*D*greater than about 40, and a large number of positive Lyapunov exponents that increases with

*D.*(A system with more than one positive Lyapunov exponent is called

*hyperchaotic.*)

Suggestions that chaos synchronization could be useful for secure communications were based on the idea that a time series in the one variable coupling the two systems would be difficult to distingusish from noise, especially for high-dimensional systems such as (B4) (e.g., Parlitz and Kocarev 1997). A signal composed of such a time series might be meaningful only to a receiving system with parameters identical to those of the sending system, or at least with a known correspondence function between states of the the two systems in the synchronized regime. The practical utility of chaos synchronization for secure communications has still not been established.

The existence of synchronized chaos in naturally occurring systems was made more plausible by demonstrations of synchronization in spatially extended systems governed by partial differential equations (PDEs). Kocarev et al. (1997) performed numerical experiments with a large class of PDEs in one space and one time dimension (e.g., complex Ginzberg–Landau, Gray– Scott reaction–diffusion, and Kuramoto–Sivashinsky). They coupled pairs of such models at a discrete set of points in space, and in a discrete-time impulsive approximation to diffusive coupling. Their finding of synchronization with coupling thus limited presaged the present results on synchronization effected by bandpass eddies in the quasigeostrophic system.

## APPENDIX C

### Definition of Blocking and Blocked Longitudes in Observed Data

*λ,*at each instant of time

*t*:

*z*(

*ϕ,*

*λ,*

*t*) is the 5-day averaged geopotential height at time

*t,*at longitude

*λ,*and latitude

*ϕ,*

*t*if

*λ*and for at least one value of Δ in (C3).

Blocked longitudes are those within 20° of any *λ* for which (C4) is satisfied.

The FL term (contours) for (top) level 1 and (bottom) level 2 as computed by Vautard and Legras (1988), representing the contribution of the eddy components of the flow to the to the total streamfunction tendency (∂*ψ*/∂*t*), for a flow state that is typical of the blocked flow regime. The plus and minus signs represent the streamfunction anomaly in this state. The FL term tends to maintain the anomaly pattern, but would inhibit a similar pattern translated downstream by half a channel length, since the extrema in the fourth longitudinal quadrant of level 2 are slightly larger in magnitude than the extrema in the third quadrant. The units are 10 m^{2} s^{−2} with a contour interval of 10 m^{2} s^{−2} in level 1 and 5 m^{2} s^{−2} in level 2. Plus and minus signs indicate areas where the positive and negative streamfunction anomalies, respectively, exceed 1.5 × 10^{7} m^{2} s^{−1} in level 1 and 0.7 × 10^{7} m^{2} s^{−1} in level 2. [Reprinted from Vautard and Legras (1988)]

Citation: Journal of the Atmospheric Sciences 61, 17; 10.1175/1520-0469(2004)061<2149:WATASC>2.0.CO;2

The FL term (contours) for (top) level 1 and (bottom) level 2 as computed by Vautard and Legras (1988), representing the contribution of the eddy components of the flow to the to the total streamfunction tendency (∂*ψ*/∂*t*), for a flow state that is typical of the blocked flow regime. The plus and minus signs represent the streamfunction anomaly in this state. The FL term tends to maintain the anomaly pattern, but would inhibit a similar pattern translated downstream by half a channel length, since the extrema in the fourth longitudinal quadrant of level 2 are slightly larger in magnitude than the extrema in the third quadrant. The units are 10 m^{2} s^{−2} with a contour interval of 10 m^{2} s^{−2} in level 1 and 5 m^{2} s^{−2} in level 2. Plus and minus signs indicate areas where the positive and negative streamfunction anomalies, respectively, exceed 1.5 × 10^{7} m^{2} s^{−1} in level 1 and 0.7 × 10^{7} m^{2} s^{−1} in level 2. [Reprinted from Vautard and Legras (1988)]

Citation: Journal of the Atmospheric Sciences 61, 17; 10.1175/1520-0469(2004)061<2149:WATASC>2.0.CO;2

The FL term (contours) for (top) level 1 and (bottom) level 2 as computed by Vautard and Legras (1988), representing the contribution of the eddy components of the flow to the to the total streamfunction tendency (∂*ψ*/∂*t*), for a flow state that is typical of the blocked flow regime. The plus and minus signs represent the streamfunction anomaly in this state. The FL term tends to maintain the anomaly pattern, but would inhibit a similar pattern translated downstream by half a channel length, since the extrema in the fourth longitudinal quadrant of level 2 are slightly larger in magnitude than the extrema in the third quadrant. The units are 10 m^{2} s^{−2} with a contour interval of 10 m^{2} s^{−2} in level 1 and 5 m^{2} s^{−2} in level 2. Plus and minus signs indicate areas where the positive and negative streamfunction anomalies, respectively, exceed 1.5 × 10^{7} m^{2} s^{−1} in level 1 and 0.7 × 10^{7} m^{2} s^{−1} in level 2. [Reprinted from Vautard and Legras (1988)]

Citation: Journal of the Atmospheric Sciences 61, 17; 10.1175/1520-0469(2004)061<2149:WATASC>2.0.CO;2

The difference between the simultaneous states of two Lorenz systems with time-lagged coupling, as specified in Eq. (9), represented by *Z*(*t*) − *Z*_{1}(*t*) vs *t* for various values of the inverse time lag Γ illustrating (a) complete synchronization, (b) intermittent or “on–off” synchronization, (c) partial synchronization, and (d) decoupled systems. Average Euclidean distance 〈*D*〉 between the states of the two systems in *X,* *Y,* *Z* space is also shown. The trajectories are generated by adaptive Runge–Kutta numerical integrations with *σ* = 10.0, *ρ* = 28.0, and *β* = 8/3

The difference between the simultaneous states of two Lorenz systems with time-lagged coupling, as specified in Eq. (9), represented by *Z*(*t*) − *Z*_{1}(*t*) vs *t* for various values of the inverse time lag Γ illustrating (a) complete synchronization, (b) intermittent or “on–off” synchronization, (c) partial synchronization, and (d) decoupled systems. Average Euclidean distance 〈*D*〉 between the states of the two systems in *X,* *Y,* *Z* space is also shown. The trajectories are generated by adaptive Runge–Kutta numerical integrations with *σ* = 10.0, *ρ* = 28.0, and *β* = 8/3

The difference between the simultaneous states of two Lorenz systems with time-lagged coupling, as specified in Eq. (9), represented by *Z*(*t*) − *Z*_{1}(*t*) vs *t* for various values of the inverse time lag Γ illustrating (a) complete synchronization, (b) intermittent or “on–off” synchronization, (c) partial synchronization, and (d) decoupled systems. Average Euclidean distance 〈*D*〉 between the states of the two systems in *X,* *Y,* *Z* space is also shown. The trajectories are generated by adaptive Runge–Kutta numerical integrations with *σ* = 10.0, *ρ* = 28.0, and *β* = 8/3

The forcing streamfunction *ψ̂**q̂** ≡ (*q*^{A*} + *q*^{A*})/2 via (2)] for the two-sector model (13), with one jet in each sector. At any given time, the flow in each sector is labeled as blocked or zonal, depending on whether the minimum, taken over all longitudes *x* in the box in that sector, of the difference in streamfunction *ψ* latitudinally across the box, is less than or greater than 0.01 (in the units of Fig. 1), respectively

The forcing streamfunction *ψ̂**q̂** ≡ (*q*^{A*} + *q*^{A*})/2 via (2)] for the two-sector model (13), with one jet in each sector. At any given time, the flow in each sector is labeled as blocked or zonal, depending on whether the minimum, taken over all longitudes *x* in the box in that sector, of the difference in streamfunction *ψ* latitudinally across the box, is less than or greater than 0.01 (in the units of Fig. 1), respectively

The forcing streamfunction *ψ̂**q̂** ≡ (*q*^{A*} + *q*^{A*})/2 via (2)] for the two-sector model (13), with one jet in each sector. At any given time, the flow in each sector is labeled as blocked or zonal, depending on whether the minimum, taken over all longitudes *x* in the box in that sector, of the difference in streamfunction *ψ* latitudinally across the box, is less than or greater than 0.01 (in the units of Fig. 1), respectively

(a), (b) The forcing *ψ**, and (c)–(f), the evolving flow *ψ,* in the parallel channel model with only advective coupling [given by (10), (11), and (12), with *c* = 1/2, *c*′ = 0, *μ*_{0} = 8.8 × 10^{−4} s^{−1}, and other parameters as listed in appendix A], represented in the manner of Fig. 1, for the indicated numbers *n* of time steps in a numerical integration. One time step is ≈0.4 h. Near-identical synchronization occurs by (e), (f) the last time step shown. The solid-line boxes designate the regions in the two channels used to label a given flow as blocked or zonal. Blocking activity in the dashed-line box in channel *A*, which is nearly the same as in the solid-line box in channel B after synchronization, anticorrelates with blocking in the solid-line box in channel *A*. (Similarly for the dashed-line box in channel *B*)

(a), (b) The forcing *ψ**, and (c)–(f), the evolving flow *ψ,* in the parallel channel model with only advective coupling [given by (10), (11), and (12), with *c* = 1/2, *c*′ = 0, *μ*_{0} = 8.8 × 10^{−4} s^{−1}, and other parameters as listed in appendix A], represented in the manner of Fig. 1, for the indicated numbers *n* of time steps in a numerical integration. One time step is ≈0.4 h. Near-identical synchronization occurs by (e), (f) the last time step shown. The solid-line boxes designate the regions in the two channels used to label a given flow as blocked or zonal. Blocking activity in the dashed-line box in channel *A*, which is nearly the same as in the solid-line box in channel B after synchronization, anticorrelates with blocking in the solid-line box in channel *A*. (Similarly for the dashed-line box in channel *B*)

(a), (b) The forcing *ψ**, and (c)–(f), the evolving flow *ψ,* in the parallel channel model with only advective coupling [given by (10), (11), and (12), with *c* = 1/2, *c*′ = 0, *μ*_{0} = 8.8 × 10^{−4} s^{−1}, and other parameters as listed in appendix A], represented in the manner of Fig. 1, for the indicated numbers *n* of time steps in a numerical integration. One time step is ≈0.4 h. Near-identical synchronization occurs by (e), (f) the last time step shown. The solid-line boxes designate the regions in the two channels used to label a given flow as blocked or zonal. Blocking activity in the dashed-line box in channel *A*, which is nearly the same as in the solid-line box in channel B after synchronization, anticorrelates with blocking in the solid-line box in channel *A*. (Similarly for the dashed-line box in channel *B*)

Correlation between binarized (blocked vs zonal) states of the two sectors of the model illustrated in Fig. 4, with the binarization as defined in the Fig. 4 caption, in numerical integrations of (13) over 4 × 10^{5} time steps with arbitrarily chosen initial conditions. The horizontal axis is the forcing strength *μ*_{0} (in units of 7.3 × 10^{−5} s^{−1}). Correlations for the two-sector model as previously defined are denoted by × symbols, squares are for a variant with one sector two-thirds as long as the other, triangles are for one sector half as long as the other, asterisks are for shortened jets in both sectors, plus signs are for lengthened jets in both sectors, and circles are for a variant in which the latitudinal positions of the two jets are skewed arbitrarily by shifts ranging from a channel half-width to a channel width. Significant anticorrelation is found in all model variants, for *μ*_{0} not too small, except in the case of latitudinally skewed jets

Correlation between binarized (blocked vs zonal) states of the two sectors of the model illustrated in Fig. 4, with the binarization as defined in the Fig. 4 caption, in numerical integrations of (13) over 4 × 10^{5} time steps with arbitrarily chosen initial conditions. The horizontal axis is the forcing strength *μ*_{0} (in units of 7.3 × 10^{−5} s^{−1}). Correlations for the two-sector model as previously defined are denoted by × symbols, squares are for a variant with one sector two-thirds as long as the other, triangles are for one sector half as long as the other, asterisks are for shortened jets in both sectors, plus signs are for lengthened jets in both sectors, and circles are for a variant in which the latitudinal positions of the two jets are skewed arbitrarily by shifts ranging from a channel half-width to a channel width. Significant anticorrelation is found in all model variants, for *μ*_{0} not too small, except in the case of latitudinally skewed jets

Correlation between binarized (blocked vs zonal) states of the two sectors of the model illustrated in Fig. 4, with the binarization as defined in the Fig. 4 caption, in numerical integrations of (13) over 4 × 10^{5} time steps with arbitrarily chosen initial conditions. The horizontal axis is the forcing strength *μ*_{0} (in units of 7.3 × 10^{−5} s^{−1}). Correlations for the two-sector model as previously defined are denoted by × symbols, squares are for a variant with one sector two-thirds as long as the other, triangles are for one sector half as long as the other, asterisks are for shortened jets in both sectors, plus signs are for lengthened jets in both sectors, and circles are for a variant in which the latitudinal positions of the two jets are skewed arbitrarily by shifts ranging from a channel half-width to a channel width. Significant anticorrelation is found in all model variants, for *μ*_{0} not too small, except in the case of latitudinally skewed jets

Behavior of an alternative model with wavenumber-independent forcing. Correlations between binarized (blocked/zonal) states of the two sectors of the model for various initial conditions and forcing strengths, displayed as in Fig. 6, have different signs for forcing with (a) one jet and (b) with two jets. (c) The corresponding two-channel model does not synchronize by the time step shown (*n* = 100 000).

Behavior of an alternative model with wavenumber-independent forcing. Correlations between binarized (blocked/zonal) states of the two sectors of the model for various initial conditions and forcing strengths, displayed as in Fig. 6, have different signs for forcing with (a) one jet and (b) with two jets. (c) The corresponding two-channel model does not synchronize by the time step shown (*n* = 100 000).

Behavior of an alternative model with wavenumber-independent forcing. Correlations between binarized (blocked/zonal) states of the two sectors of the model for various initial conditions and forcing strengths, displayed as in Fig. 6, have different signs for forcing with (a) one jet and (b) with two jets. (c) The corresponding two-channel model does not synchronize by the time step shown (*n* = 100 000).

Correlation between binarized states of the two-sector model as in Fig. 6, but with the model parameter *β* set to 0. The binarized states of the two sectors anticorrelate in this case also

Correlation between binarized states of the two-sector model as in Fig. 6, but with the model parameter *β* set to 0. The binarized states of the two sectors anticorrelate in this case also

Correlation between binarized states of the two-sector model as in Fig. 6, but with the model parameter *β* set to 0. The binarized states of the two sectors anticorrelate in this case also

The flow for two channels coupled advectively, shown for time step *n* = 40 000, as in Fig. 5, but with the large-scale components decoupled, according to (14). The channels near-identically synchronize with only the smaller scales coupled

The flow for two channels coupled advectively, shown for time step *n* = 40 000, as in Fig. 5, but with the large-scale components decoupled, according to (14). The channels near-identically synchronize with only the smaller scales coupled

The flow for two channels coupled advectively, shown for time step *n* = 40 000, as in Fig. 5, but with the large-scale components decoupled, according to (14). The channels near-identically synchronize with only the smaller scales coupled

The flow for two channels coupled advectively, as in Fig. 9, but with the large-scale components coupled and the other components decoupled, according to (15), at time step *n* = 100 000. The channels do not synchronize

The flow for two channels coupled advectively, as in Fig. 9, but with the large-scale components coupled and the other components decoupled, according to (15), at time step *n* = 100 000. The channels do not synchronize

The flow for two channels coupled advectively, as in Fig. 9, but with the large-scale components coupled and the other components decoupled, according to (15), at time step *n* = 100 000. The channels do not synchronize

The evolving flow for two coupled channels as in Figs. 5c–f, but with coupling both in the advective terms (with *c* = 1/2) and in the forcing terms (with *c*′ = 0.025), and with only the medium scales coupled, according to (16). The cutoff wavenumber *k*_{noise} = 15 defines the smallest scales that are not coupled. Other parameters are unchanged. The channels are near-identically synchronized at each time step shown

The evolving flow for two coupled channels as in Figs. 5c–f, but with coupling both in the advective terms (with *c* = 1/2) and in the forcing terms (with *c*′ = 0.025), and with only the medium scales coupled, according to (16). The cutoff wavenumber *k*_{noise} = 15 defines the smallest scales that are not coupled. Other parameters are unchanged. The channels are near-identically synchronized at each time step shown

The evolving flow for two coupled channels as in Figs. 5c–f, but with coupling both in the advective terms (with *c* = 1/2) and in the forcing terms (with *c*′ = 0.025), and with only the medium scales coupled, according to (16). The cutoff wavenumber *k*_{noise} = 15 defines the smallest scales that are not coupled. Other parameters are unchanged. The channels are near-identically synchronized at each time step shown

Flow results for a coupled-channel model with only medium-scale components coupled, with *k*_{noise} = 15 and other parameters as in Fig. 11, but with (a), (b) identical forcing streamfunctions in place of the longitudinally skewed flows shown in Figs. 5a, b. (c), (d) By the time step shown (*n* = 2000), the channels are synchronized identically

Flow results for a coupled-channel model with only medium-scale components coupled, with *k*_{noise} = 15 and other parameters as in Fig. 11, but with (a), (b) identical forcing streamfunctions in place of the longitudinally skewed flows shown in Figs. 5a, b. (c), (d) By the time step shown (*n* = 2000), the channels are synchronized identically

Flow results for a coupled-channel model with only medium-scale components coupled, with *k*_{noise} = 15 and other parameters as in Fig. 11, but with (a), (b) identical forcing streamfunctions in place of the longitudinally skewed flows shown in Figs. 5a, b. (c), (d) By the time step shown (*n* = 2000), the channels are synchronized identically

The flow for two channels with only medium-scale coupling, identical forcing, and other parameters as in Fig. 12, but with a somewhat smaller cutoff wavenumber *k*_{noise} = 10. At time step *n* = 50 000, much later than the time considered in the previous case, the channels are not synchronized

The flow for two channels with only medium-scale coupling, identical forcing, and other parameters as in Fig. 12, but with a somewhat smaller cutoff wavenumber *k*_{noise} = 10. At time step *n* = 50 000, much later than the time considered in the previous case, the channels are not synchronized

The flow for two channels with only medium-scale coupling, identical forcing, and other parameters as in Fig. 12, but with a somewhat smaller cutoff wavenumber *k*_{noise} = 10. At time step *n* = 50 000, much later than the time considered in the previous case, the channels are not synchronized

Difference between the flow fields in Figs. 11a and 11b, depicting the two channels of the generally synchronized model with medium-scale components coupled and longitudinally skewed forcing. High spatial frequencies are evident in the regions of the jets (see Figs. 5a,b) where the difference between the forcing terms is largest

Difference between the flow fields in Figs. 11a and 11b, depicting the two channels of the generally synchronized model with medium-scale components coupled and longitudinally skewed forcing. High spatial frequencies are evident in the regions of the jets (see Figs. 5a,b) where the difference between the forcing terms is largest

Difference between the flow fields in Figs. 11a and 11b, depicting the two channels of the generally synchronized model with medium-scale components coupled and longitudinally skewed forcing. High spatial frequencies are evident in the regions of the jets (see Figs. 5a,b) where the difference between the forcing terms is largest

Vertical density (in pressure coordinates) of total zonal mass transport between 40° and 80° at 300 mb (denoted

Vertical density (in pressure coordinates) of total zonal mass transport between 40° and 80° at 300 mb (denoted

Vertical density (in pressure coordinates) of total zonal mass transport between 40° and 80° at 300 mb (denoted

Correlation between the binarized states of a two-sector model as in Fig. 6, but with the modified forcing (24) that leaves the magnitudes of the jets unconstrained. The anticorrelation found with the original forcing (11) is absent

Correlation between the binarized states of a two-sector model as in Fig. 6, but with the modified forcing (24) that leaves the magnitudes of the jets unconstrained. The anticorrelation found with the original forcing (11) is absent

Correlation between the binarized states of a two-sector model as in Fig. 6, but with the modified forcing (24) that leaves the magnitudes of the jets unconstrained. The anticorrelation found with the original forcing (11) is absent

The forcing streamfunction *ψ̂*

The forcing streamfunction *ψ̂*

The forcing streamfunction *ψ̂*

Correlation between blocking activity in the Atlantic (100°W–100°E) and Pacific (100°E–100°W) sectors, in observed meteorological data, on days of given latitudinal skew between Atlantic and Pacific jet positions. The values plotted are based on the NCEP– NCAR reanalysis data (Kalnay et al. 1996) for boreal winters 1958– 98. Daily jet position is defined as the position of maximum eastward zonal wind at the 300-mb level, at nonblocked longitudes. Blocking and blocked longitudes are defined in appendix C. Correlations are shown for the full 40-yr period (solid line) and for five 8-yr segments (five broken lines). The partitioning demonstrates that the increase in correlation as the skew increases from 0° to 10° is statistically significant. (The decrease in significance for higher skew values is due to the small number of days representing those values in the dataset)

Correlation between blocking activity in the Atlantic (100°W–100°E) and Pacific (100°E–100°W) sectors, in observed meteorological data, on days of given latitudinal skew between Atlantic and Pacific jet positions. The values plotted are based on the NCEP– NCAR reanalysis data (Kalnay et al. 1996) for boreal winters 1958– 98. Daily jet position is defined as the position of maximum eastward zonal wind at the 300-mb level, at nonblocked longitudes. Blocking and blocked longitudes are defined in appendix C. Correlations are shown for the full 40-yr period (solid line) and for five 8-yr segments (five broken lines). The partitioning demonstrates that the increase in correlation as the skew increases from 0° to 10° is statistically significant. (The decrease in significance for higher skew values is due to the small number of days representing those values in the dataset)

Correlation between blocking activity in the Atlantic (100°W–100°E) and Pacific (100°E–100°W) sectors, in observed meteorological data, on days of given latitudinal skew between Atlantic and Pacific jet positions. The values plotted are based on the NCEP– NCAR reanalysis data (Kalnay et al. 1996) for boreal winters 1958– 98. Daily jet position is defined as the position of maximum eastward zonal wind at the 300-mb level, at nonblocked longitudes. Blocking and blocked longitudes are defined in appendix C. Correlations are shown for the full 40-yr period (solid line) and for five 8-yr segments (five broken lines). The partitioning demonstrates that the increase in correlation as the skew increases from 0° to 10° is statistically significant. (The decrease in significance for higher skew values is due to the small number of days representing those values in the dataset)

Correlation between blocking activity in the Atlantic and Pacific sectors vs latitudinal jet skew, as in Fig. 18, but for synthetic data obtained from the Community Climate Model, integrated over 10^{6} days ≈ 2738 yr, in a perpetual winter simulation. The solid line shows correlations for the full run; the five broken lines are for five 2 × 10^{5} day segments. There is a statistically significant increase in correlation as the skew increases from 0° to 45°

Correlation between blocking activity in the Atlantic and Pacific sectors vs latitudinal jet skew, as in Fig. 18, but for synthetic data obtained from the Community Climate Model, integrated over 10^{6} days ≈ 2738 yr, in a perpetual winter simulation. The solid line shows correlations for the full run; the five broken lines are for five 2 × 10^{5} day segments. There is a statistically significant increase in correlation as the skew increases from 0° to 45°

Correlation between blocking activity in the Atlantic and Pacific sectors vs latitudinal jet skew, as in Fig. 18, but for synthetic data obtained from the Community Climate Model, integrated over 10^{6} days ≈ 2738 yr, in a perpetual winter simulation. The solid line shows correlations for the full run; the five broken lines are for five 2 × 10^{5} day segments. There is a statistically significant increase in correlation as the skew increases from 0° to 45°

^{*}

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

^{1}

Lorenz cites Tsonis and Elsner (1989) as the original source of this suggestion.

^{2}

Vautard et al. (1988) are not specific about the form of their forcing term, only giving the exact form of the resulting baroclinic jet, but a wavenumber-independent forcing seems more likely. Both setups give intersectorial anticorrelation.

^{3}

Eddy-induced synchronization for the simpler coupling scheme defined by *c* = 0, *c*′ > 0 was illustrated previously (Duane and Tribbia 2001).

^{4}

A related suggestion is that the synchronization of two dynamically identical channels, directionally coupled, provides an alternative conceptual framework for meteorological data assimilation in an interpretation of the two channels as “truth” and “model,” respectively. That the highest spatial frequencies need not be coupled, in this interpretation, corresponds to the fact that a discrete set of observation points is usually sufficient to update a weather prediction model (Duane 2003).

^{5}

The specific relationship between jet fluctuations in either sector and blocking in the same sector previously posited by Duane and Tribbia (2001)—that blocking is less likely when the jet is stronger— does not appear to be correct. Blocking activity in fact tends to increase slightly with total mass transport.

^{6}

The authors thank Eli Tziperman for this suggestion.