## 1. INTRODUCTION AND MOTIVATION

[2] Until a quarter of a century ago [*Broecker and Van Donk*, 1970; *Hays et al.*, 1976] the climates of the past had been described mostly in qualitative terms. Since then many techniques have become available to construct climatic records from geological, biological, and physical data [*Bradley*, 1999]. These proxy records show that climate variations on different timescales have been very common in the past. The enormous amount of instrumental data that has been collected over the last century and one half contributes, in turn, to a more and more complete picture of the climate system's variability.

[3] The purpose of the present review paper is to describe the role of the ocean circulation in this variability and to emphasize that dynamical systems theory can contribute substantially to understanding this role. The intended audience and the way prospective readers can best benefit from this review are highlighted in Appendix A. To facilitate diverse routes through the paper, we have included a glossary of the principal symbols in Table 1 and a list of acronyms in Table 2.

Symbol | Definition | Section |
---|---|---|

A_{H} and A_{V} | lateral and vertical friction coefficients | Appendix C |

A = H/L | aspect ratio | Appendix D |

D | equilibrium layer depth | Appendix C |

E = A_{H}/(2Ωr_{0}^{2}) | Ekman number | 2.5 |

f, f_{0} | Coriolis parameter (at latitude θ_{0}) | 2.3 |

f and p | vector field and parameter vector | 1.4 |

F_{0} | freshwater forcing coefficient | Appendix D |

F_{S} and T_{S} | pattern of freshwater and temperature forcing | Appendix D |

g, g′ = gΔρ/ρ_{0} | gravitational acceleration, reduced gravity | Appendix C |

h | upper layer thickness | 2.3 |

H | depth of the ocean basin | 2.3 |

K_{H} and K_{V} | lateral and vertical diffusion coefficients | Appendix D |

L | basin length | 2.3 |

Pr = A_{H}/K_{H} | Prandtl number | Appendix D |

Q | dimensionless wind-driven transport | 2.5 |

r_{0} | radius of the Earth | 2.3 |

R | bottom friction | Appendix C |

Ra = (gα_{T}ΔTL^{3})/(A_{H}K_{H}) | Rayleigh number | Appendix D |

Re = (δ_{I}/δ_{M})^{1/3} | Reynolds number | Appendix C |

R_{HV}^{M} = A_{V}/A_{H} | ratio of friction parameters | Appendix D |

R_{HV}^{T} = K_{V}/K_{H} | ratio of diffusivities | Appendix D |

S and T | salinity and temperature | 3.2 |

^{n} | n-dimensional space of real numbers | 1.4 |

(u, v, w) | velocity vector | Appendix C |

(U, V) = (uh, vh) | horizontal transport vector | Appendix C |

x | state vector | 1.4 |

α_{T} and α_{S} | thermal and saline expansion coefficients | Appendix D |

β | meridional variation of Coriolis parameter | Appendix C |

γ = F_{0}H/K_{V} | dimensionless measure of freshwater flux strength | Appendix D |

Γ | trajectory in phase space | 1.4 |

δ | regularization parameter | 3.3 |

δ_{I} and δ_{M} | western boundary layer thicknesses | Appendix C |

ΔT and ΔS | characteristic temperature and salinity differences | Appendix D |

ε | Rossby number | Appendix C |

η_{1}, η_{2}, and η_{3} | dimensionless parameters in box model | 3.2 |

Φ(t) | time periodic perturbation structure | 2.3 |

Ψ | meridional overturning | 3.2 |

λ | single parameter | 1.4 |

= α_{T}ΔT/(α_{S}ΔS) | buoyancy ratio | Appendix D |

μ_{1} and μ_{2} | dimensionless parameters | 3.3 |

ω and S_{ω} | frequency of oscillation and spectral power | 1.1 |

Ω | angular frequency of the Earth | Appendix C |

ρ_{0} and Δρ | background density and density difference | Appendix C |

σ = σ_{r} + iσ_{i} | complex growth factor, eigenvalue | 1.4 |

τ^{x} and τ^{y} | zonal and meridional wind stress | Appendix C |

Symbol | Definition | Section |
---|---|---|

0-D | zero-dimensional | 1.3 |

1-D | one-dimensional | 1.3 |

2-D | two-dimensional | 1.3 |

3-D | three-dimensional | 1.3 |

AABW | Antarctic Bottom Water | 3.1 |

ACC | Antarctic Circumpolar Current | 3.4 |

COADS | Comprehensive Ocean-Atmosphere Data Set | 2.6 |

EBM | energy balance model | 3.4 |

EOF | empirical orthogonal function | 3.1 |

ENSO | El Niño/Southern Oscillation | 1.1 |

GCM | general circulation model | 1.3 |

GFDL | Geophysical Fluid Dynamics Laboratory | 3.4 |

LSG | large-scale geostrophic model | 3.4 |

MOM | modular ocean model | 3.4 |

M-SSA | multichannel singular-spectrum analysis | 2.6 |

NADW | North Atlantic Deep Water | 1.2 |

NPP | northern sinking pole-to-pole flow | 3.2 |

ODE | ordinary differential equation | 1.4 |

PDE | partial differential equation | 1.4 |

PGM | planetary geostrophic model | 3.4 |

POCM | Parallel Ocean Climate Model | 2.6 |

POP | Parallel Ocean Program | 3.4 |

QG | quasi-geostrophic | 2.3 |

SA | salinity-driven flow | 3.2 |

SPP | southern sinking pole-to-pole flow | 3.2 |

SSA | singular-spectrum analysis | 2.3 |

SST | sea surface temperature | 1.1 |

TH | thermally driven flow | 3.2 |

THC | thermohaline circulation | 3.1 |

WOCE | World Ocean Circulation Experiment | 3.1 |

### 1.1. Climate Variability on Multiple Timescales

[4] An “artist's rendering” of climate variability on all timescales is provided in Figure 1a. The first version of Figure 1a was produced by *Mitchell* [1976], and many versions thereof have circulated since. Figure 1a is meant to summarize our knowledge of the spectral power *S*_{ω}, i.e., the amount of variability in a given frequency band, between ω and ω + Δω; here the frequency ω is the inverse of the period of oscillation and Δω indicates a small increment. This power spectrum is not computed directly by spectral analysis from a time series of a given climatic quantity, such as (local or global) temperature; indeed, there is no single time series that is 10^{7} years long and has a sampling interval of hours, as Figure 1a would suggest. Figure 1a includes, instead, information obtained by analyzing the spectral content of many different time series, for example, the spectrum (Figure 1b) of the 335-year long record of Central England Temperatures. This time series is the longest instrumentally measured record of any climatic variable. Given the lack of earlier instrumental records, one can easily imagine (but cannot easily confirm) that the higher-frequency spectral features might have changed, in amplitude, frequency, or both, over the course of climatic history.

[5] With all due caution in its interpretation, Figure 1a reflects three types of variability: (1) sharp lines that correspond to periodically forced variations at 1 day and 1 year; (2) broader peaks that arise from internal modes of variability; and (3) a continuous portion of the spectrum that reflects stochastically forced variations, as well as deterministic chaos [*Ghil and Robertson*, 2000; *Ghil*, 2002].

[6] Between the two sharp lines at 1 day and 1 year lies the synoptic variability of midlatitude weather systems, concentrated at 3–7 days, as well as intraseasonal variability, i.e., variability that occurs on the timescale of 1–3 months. The latter is also called low-frequency atmospheric variability, a name that refers to the fact that this variability has lower frequency, or longer periods, than the life cycle of weather systems. Intraseasonal variability comprises phenomena such as the Madden-Julian oscillation of winds and cloudiness in the tropics or the alternation between episodes of zonal and blocked flow in midlatitudes [*Ghil and Childress*, 1987; *Ghil et al.*, 1991; *Haines*, 1994; *Molteni*, 2002].

[7] Immediately to the left of the seasonal cycle in Figure 1a lies interannual, i.e., year to year, variability. An important component of this variability is the El Niño phenomenon in the tropical Pacific: Once about every 4 years, the sea surface temperatures (SSTs) in the eastern tropical Pacific increase by a few degrees over a period of about 1 year. This SST variation is associated with changes in the trade winds over the tropical Pacific and in sea level pressures [*Bjerknes*, 1969; *Philander*, 1990]; an east-west seesaw in the latter is called the Southern Oscillation. The combined El Niño/Southern Oscillation (ENSO) phenomenon arises through large-scale interaction between the equatorial Pacific and the atmosphere above. Equatorial wave dynamics in the ocean plays a key role in setting ENSO's timescale [*Cane and Zebiak*, 1985; *Neelin et al.*, 1994, 1998; *Dijkstra and Burgers*, 2002].

[8] The greatest excitement among scientists as well as the public is currently being generated by interdecadal variability, i.e., climate variability on the timescale of a few decades, the timescale of an individual human's life cycle [*Martinson et al.*, 1995]. Figure 1b represents an up-to-date “blowup” of the interannual-to-interdecadal portion of Figure 1a. The broad peaks are due to the climate system's internal processes: Each spectral component can be associated, at least tentatively, with a mode of interannual or interdecadal variability [*Plaut et al.*, 1995]. Thus the rightmost peak, with a period of 5.2 years, can be attributed to the remote effect of ENSO's low-frequency mode, while the 7.7-year peak captures a North Atlantic mode of variability that arises from the Gulf Stream's interannual cycle of meandering and intensification. The two interdecadal peaks, near 14 and 25 years, are also present in global records, instrumental as well as paleoclimatic [*Kushnir*, 1994; *Mann et al.*, 1998; *Moron et al.*, 1998; *Delworth and Mann*, 2000; *Ghil et al.*, 2002b].

[9] Finally, the leftmost part of Figure 1a represents paleoclimatic variability. The information summarized here comes exclusively from proxy indicators of climate [*Imbrie and Imbrie*, 1986]. These include coral records [*Boiseau et al.*, 1999] and tree rings for the historic past [*Mann et al.*, 1998], as well as marine sediment [*Duplessy and Shackleton*, 1985] and ice core [*Jouzel et al.*, 1991] records for the last 2 million years of Earth history, the Quaternary. Glaciation cycles, an alternation of warmer and colder climatic episodes, dominated the Quaternary era. The cyclicity is manifest in the broad peaks present in Figure 1a between roughly 1 kyr and 1 Myr. The two peaks at about 20 kyr and 40 kyr reflect variations in Earth's orbit, while the dominant peak at 100 kyr remains to be convincingly explained [*Imbrie and Imbrie*, 1986; *Ghil and Childress*, 1987; *Gildor and Tziperman*, 2001]. The glaciation cycles provide a fertile testing ground for theories of climate variability for two reasons: (1) They represent a wide range of climatic conditions, and (2) they are much better documented than earlier parts of climatic history.

[10] Within these glaciation cycles, there are higher-frequency oscillations prominent in the North Atlantic paleoclimatic records. These are the Heinrich events [*Heinrich*, 1988] with a near periodicity of 6–7 kyr and the Dansgaard-Oeschger cycles that provide the peak at around 1–2.5 kyr in Figure 1a. Rapid changes in temperature, of up to one half of the amplitude of a typical glacial-interglacial temperature difference, occurred during Heinrich events, and somewhat smaller ones occurred over a Dansgaard-Oeschger cycle. Progressive cooling through several of the latter cycles followed by an abrupt warming defines a Bond cycle [*Bond et al.*, 1995]. In North Atlantic sediment cores the coldest part of each Bond cycle is marked by a so-called Heinrich layer that is rich in ice-rafted debris. None of these higher-frequency oscillations can be directly connected to orbital or other periodic forcings.

[11] In summary, climate variations range from the large-amplitude climate excursions of the past millennia to smaller-amplitude fluctuations on shorter timescales. Several spectral peaks of variability can be clearly related to forcing mechanisms; others cannot. In fact, even if the external forcing were constant in time, that is, if no systematic changes in insolation or atmospheric composition, such as trace gas or aerosol concentration, would occur, the climate system would still display variability on many timescales. This statement is clearly true for the 3–7 days synoptic variability of midlatitude weather, which arises through baroclinic instability of the zonal winds, and the ENSO variability in the equatorial Pacific, as discussed above. Processes internal to the climate system can thus give rise to spectral peaks that are not related directly to the temporal variability of the forcing. It is the interaction of this highly complex intrinsic variability with the relatively small time-dependent variations in the forcing that is recorded in the proxy records and instrumental data.

### 1.2. Role of the Ocean Circulation

[12] We focus in this review on the ocean circulation as a source of internal climate variability. The ocean moderates climate through its large thermal inertia, i.e., its capacity to store and release heat and its poleward heat transport through ocean currents. The exact importance of the latter relative to atmospheric heat transport, though, is still a matter of active debate [*Seager et al.*, 2001]. The large-scale ocean circulation is driven both by momentum fluxes as well as by fluxes of heat and freshwater at the ocean-atmosphere interface. The near-surface circulation is dominated by horizontal currents that are mainly driven by the wind stress forcing, while the much slower motions of the deep ocean are mainly induced by buoyancy differences.

[13] The circulation due to either forcing mechanism is often described and analyzed separately for the sake of simplicity. In fact, the wind-driven and thermohaline circulation together form a complex three-dimensional (3-D) flow of different currents and water masses through the global ocean. The simplest picture of the global ocean circulation has been termed the “ocean conveyor” [*Gordon*, 1986; *Broecker*, 1991]; it corresponds to a two-layer view where the vertical structure of the flow field is separated into a shallow flow, above the permanent thermocline at roughly 1000 m, and a deep flow between this thermocline and the bottom (i.e., between a depth of roughly 1000 m and 4000 m); see Figure 2. The unit of volume flux in the ocean is 1 Sv = 10^{6} m^{3} s^{−1}, and it equals approximately the total flux of the world's major rivers. *MacDonald and Wunsch* [1996] and *Ganachaud and Wunsch* [2000] have provided an updated version of this schematic representation of the ocean circulation.

[14] In the North Atlantic, for instance, the major current is the Gulf Stream, an eastward jet that arises through the merging of the two western boundary currents, the northward flowing Florida Current and the southward flowing Labrador Current. In the North Atlantic's subpolar seas, about 14 Sv of the upper ocean water carried northward by the North Atlantic Drift, the northeastward extension of the Gulf Stream, is converted to deepwater by cooling and salinification. This North Atlantic Deep Water (NADW) flows southward, crosses the equator, and joins the flows in the Southern Ocean. The outflow from the North Atlantic is compensated by water coming through the Drake Passage (about 10 Sv) and water coming from the Indian Ocean through the Agulhas Current system (about 4 Sv). Part of the latter “Agulhas leakage” may originate from Pacific water that flows through the Indonesian Archipelago. We refer to earlier reviews [*Gordon*, 1986; *Schmitz*, 1995; *World Ocean Circulation Experiment* (*WOCE*), 2001] for more complete information on the circulation in each major ocean basin as well as from one basin to another.

[15] Changes in the ocean circulation can influence climate substantially through their impact on both the meridional and zonal heat transport. This can affect mean global temperature and precipitation, as well as their distribution in space and time. Subtle changes in the North Atlantic surface circulation and their interactions with the overlying atmosphere are thought to be involved in climate variability on interannual and interdecadal timescales, as observed in the instrumental record of the last century [*Martinson et al.*, 1995; *Ghil*, 2001]. Changes in the circulation may also occur on a global scale, involving a transition to different large-scale patterns. Such changes may have been involved in the large-amplitude climate variations of the past [*Broecker et al.*, 1985].

### 1.3. Modeling Hierarchy

[16] The climate system is highly complex. Its main subsystems have very different characteristic times, and the specific phenomena involved in each one of the climate problems alluded to in sections 1.1 and 1.2 are quite diverse. It is inconceivable therefore that a single model could successfully incorporate all the subsystems, capture all the phenomena, and solve all the problems. Hence the concept of a hierarchy of climate models, from the simple to the complex, was developed about a quarter of a century ago [*Schneider and Dickinson*, 1974; *Ghil and Robertson*, 2000].

[17] The simplest, spatially zero-dimensional (0-D) ocean models are so-called box models, used to study the stability [*Stommel*, 1961] and paleoevolution [*Karaca et al.*, 1999] of the oceans' thermohaline circulation or biogeochemical cycles [*Sarmiento and Toggweiler*, 1984; *Keir*, 1988; *Paillard et al.*, 1993]. There are one-dimensional (1-D) models that consider the vertical structure of the upper ocean, whether the oceanic mixed layer only [*Kraus and Turner*, 1967; *Karaca and Müller*, 1989] or the entire thermocline structure.

[18] Two-dimensional (2-D) models of the oceans fall into the two broad categories of “horizontal” and “vertical.” Models which resolve two horizontal coordinates emphasize the study of the oceans' wind-driven circulation [*Cessi and Ierley*, 1995; *Jiang et al.*, 1995], while those that consider a meridional section concentrate on the overturning, thermohaline circulation [*Cessi and Young*, 1992; *Quon and Ghil*, 1992, 1995; *Thual and McWilliams*, 1992].

[19] As explained in section 1.2, the oceans' circulation is essentially 3-D, and therefore general circulation models (GCMs) of the ocean are indispensable in understanding oceanic variability [*McWilliams*, 1996]. The Bryan-Cox model [*Bryan et al.*, 1974; *Cox*, 1987] has played a seminal role for the development and applications of such models; this role resembles the one played by the University of California, Los Angeles, model [*Arakawa and Lamb*, 1977] in atmospheric modeling. A number of simplified versions of the Bryan-Cox ocean GCM have been used in exploratory studies of multiple mean flows [*Bryan*, 1986; *Marotzke et al.*, 1988] and oscillatory behavior [*Weaver et al.*, 1991, 1993; *Chen and Ghil*, 1995, 1996] of the oceans.

[20] In confronting modeling results with observations one has to realize that it is the largest scales that are best and most reliably captured. This is certainly true in the atmosphere, where global observing systems have existed for half a century [*Daley*, 1991], and even more so in the oceans, where global coverage has been provided more recently by satellites and long hydrographic sections [*Ghil and Malanotte-Rizzoli*, 1991; *Wunsch*, 1996; *WOCE*, 2001]. The variability of the large scales arises from two sources: (1) the competition among the finite-amplitude instabilities and (2) the net effects of the smaller scales of motion. In this review, we concentrate mostly on the former. Dynamical systems theory provides a perfect toolkit for this type of study.

### 1.4. A Unifying Framework

[21] It is now widely understood that the climate system contains numerous nonlinear processes and feedbacks and that its behavior is rather irregular but not totally random. Dynamical systems theory studies the common features of such nonlinear systems. The theory is most fully developed for systems with a finite number of degrees of freedom [*Smale*, 1995], and the early and best known applications to atmospheric and ocean dynamics involved, not surprisingly, a small number of degrees of freedom [*Stommel*, 1961; *Lorenz*, 1963a, 1963b; *Veronis*, 1963, 1966]. This fact has led to a widespread perception that dynamical systems theory only applies to “low-order systems,” and hence its concepts are not sufficiently well known or appreciated within the community of oceanographers, meteorologists, and other geoscientists.

[22] The dynamical systems main results that are most important for the study of climate variability have been summarized by *Ghil et al.* [1991]; they involve essentially bifurcation theory [*Guckenheimer and Holmes*, 1990] and the ergodic theory of dynamical systems [*Eckmann and Ruelle*, 1985]. Bifurcation theory studies changes in the qualitative behavior of a dynamical system as one or more of its parameters changes. The results of this theory permit one to follow systematically climatic behavior from the simplest kind of model solutions to the most complex, from single to multiple equilibria, and on to periodic, chaotic, and fully turbulent solutions. Ergodic theory connects the dynamics of a system with its statistics.

[23] Here we sketch the basic concepts of bifurcation theory for a general system of ordinary differential equations (ODEs) that can be written as

[24] Here **x** is the state vector in the state space ^{n}, where *n* indicates the number of degrees of freedom. The right-hand side **f** contains the model dynamics; it depends on **x** in a nonlinear fashion, on time *t*, and on the vector **p** of *p* parameters, where typically *p* ≪ *n*. The ODE system (1) defines a dynamical system in continuous time, provided solutions exist and are unique for all times, −∞ < *t* < ∞. The system is called autonomous if **f** does not depend explicitly on time. A trajectory of the dynamical system, starting at the initial state **x**(*t*_{0}) = **x**_{0}, is a curve Γ = {**x**(*t*): −∞ < *t* < ∞} in the phase space that satisfies (1).

[25] A solution **x**(*t*) = of an autonomous ODE system is called a fixed point if

and hence a trajectory for which **x**(*t*) = at any time *t* will remain there forever. Linear stability analysis of a particular fixed point (, ) considers infinitesimally small perturbations **y**, i.e.,

linearization of (1) around then gives

where **J** is the Jacobian matrix

[26] The linear, autonomous ODE system (4) has solutions of the form **y**(*t*) = *e*^{σt}. Substituting such a solution into (4) leads to an eigenvalue problem for the complex growth factor σ = σ_{r} + *i*σ_{i}, i.e.

Those fixed points for which eigenvalues with σ_{r} > 0 exist are unstable, since the perturbations are exponentially growing. Fixed points for which all eigenvalues have σ_{r} < 0 are linearly stable.

[27] Discretization of the systems of partial differential equations (PDEs) that govern oceanic and other geophysical flows [*Gill*, 1982; *Pedlosky*, 1987, 1996] leads to a system of ODEs (1), with large *n*. In many cases the linearization (3)–(5) yields solutions that are the classical linear waves of geophysical fluid dynamics. These include neutrally stable waves, like Rossby or Kelvin waves, or unstable ones, like those associated with the barotropic or baroclinic instability of ocean currents.

[28] If the number of solutions or their stability properties change as a parameter is varied, a qualitative change occurs in the behavior of the dynamical system: The system is then said to undergo a bifurcation. The points at which bifurcations occur are called bifurcation points or critical points. A bifurcation diagram for a particular system (1) is a graph in which the variation of its solutions is displayed in the phase-parameter space. Information on the most elementary bifurcations is presented in Appendix B.

[29] Bifurcation theory goes beyond classical, linear analysis in studying the nonlinear saturation of and interactions between linear instabilities. When the interaction between several instabilities leads to irregular, apparently random behavior, ergodic theory sheds light on the statistics of this behavior.

[30] In the early 1960s it was possible to compute the first one or two bifurcations analytically and one or two more by the modest computational means of that time, all this for systems with about 10 degrees of freedom or less [*Lorenz*, 1963a, 1963b; *Veronis*, 1963, 1966]. In the mid-1970s it became possible to do so for spatially 1-D energy balance models, either discretized by spectral truncation [*Held and Suarez*, 1974; *North*, 1975; *North et al.*, 1983] or even by a full treatment of the governing PDEs [*Ghil*, 1976]. The numerical techniques to do so, so-called continuation techniques, are described elsewhere [*Doedel and Tuckermann*, 2000; *Dijkstra*, 2000]. *Legras and Ghil* [1985] were the first to apply a continuation method to a problem in geophysical fluid dynamics. Their atmospheric model had 25 spherical harmonics. In the last few years, bifurcation sequences have been computed for 2-D [*Cessi and Young*, 1992; *Quon and Ghil*, 1992, 1995; *Speich et al.*, 1995; *Dijkstra and Molemaker*, 1997] and 3-D [*Chen and Ghil*, 1996; *Ghil and Robertson*, 2000; *Dijkstra et al.*, 2001] climate models with thousands or even tens of thousands of degrees of freedom.

[31] Simplified GCMs, atmospheric, oceanic, and coupled, have thus become amenable to a systematic study of their large-scale variability. For systems that have an even larger number of degrees of freedom, such as full-scale ocean and coupled ocean-atmosphere GCMs, with *n* larger than 10^{6}, bifurcation points can be inferred by performing time-dependent forward integrations for several parameter values and monitoring changes in qualitative behavior. This type of “poor-man's continuation” has been applied successfully for models of the wind-driven [*Jiang et al.*, 1995; *Chang et al.*, 2001] and thermohaline [*Quon aand Ghil*, 1992, 1995; *Chen and Ghil*, 1996] circulation.