2 LOW-FREQUENCY VARIABILITY OF THE
3 LARGE-SCALE OCEAN CIRCULATION:
4 A DYNAMICAL SYSTEMS APPROACH
5
Received 8 November 2002; revised 25 January 2005; accepted 18 July 2005; published XX Month 2005.
6
7
[1] Oceanic variability on interannual, interdecadal, and
8
longer timescales plays a key role in climate variability and
9
climate change. Paleoclimatic records suggest major
10 changes in the location and rate of deepwater formation in
11 the Atlantic and Southern oceans on timescales from
12 millennia to millions of years. Instrumental records of
13 increasing duration and spatial c overage docum ent
14 substantial variability in the path and intensity of ocean
15 surface currents on timescales of months to decades. We
16 review recent theoretical and numerical results that help
17 explain the physical processes governing the large-scale
18 ocean circulation and its intrinsic variability. To do so, we
19 apply systematically the methods of dynamical systems
20 theory. The dynamical systems approach is proving
21 successful for more and more detailed and realistic
22 models, up to and including oceanic and coupled ocean-
23 atmosphere general circulation models. In this approach one
24 follows the road from simple, highly symmetric model
25 solutions, through a ‘‘bifurcation tree,’’ toward the
26 observed, complex behavior of the s ys tem under
27 investigation. The observed variability can be shown to
28 have its roots in simple transitions from a circulation with
29 high symmetry in space and regularity in time to
30 circulations with successively lower symmetry in space
31 and less regularity in time. This road of successive
32 bifurcations leads through multiple equilibria to oscillatory
33 and eventually chaotic solutions. Key features of this
34approach are illustrated in detail for simplified models of
35two basic problems of the ocean circulation. First, a
36barotropic model is used to capture major features of the
37wind-driven ocean circulation and of the changes in its
38behavior as wind stress incr eases. Second, a zonally
39averaged model is used to show how the thermohaline
40ocean circulation changes as buoyancy fluxes at the surface
41increase. For the wind-driven circulation, multiple
42separation patterns of a ‘‘Gulf-Stream like’’ eastward jet
43are obtained. These multiple equilibria are followed by
44subannual and interannual oscillations of the jet and of the
45entire basin’s circulation. The multiple equilibria of the
46thermohaline circulation include deepwater formation near
47the equator, near either pole or both, as well as intermediate
48possibilities that bear some degree of resemblance to the
49currently observed Atlantic overturning pattern. Some of
50these multiple equilibria are subject, in turn, to oscillatory
51instabilities with timescales of decades, centuries, and
52millennia. Interdecadal and centennial oscillations are the
53ones of greatest interest in the current debate on global
54warming and on the relative roles of natural and
55anthropogenic variability in it. They involve the physics
56of the truly three-dimensional coupling between the wind-
57driven and thermohaline circulation. To arrive at this three-
58dimensional picture, the bifurcation tree is sketched out for
59increasingly complex models for both the wind-driven and
60the thermohaline circulation.
63 Citation: Dijkstra, H. A., and M. Ghil (2005), Low-frequency variability of the large-scale ocean circulation: A dynamical systems
64 approach, Rev. Geophys., 43, XXXXXX, doi:10.1029/2002RG000122.
1. INTRODUCTION AND MOTIVATION
67[2] Until a quarter of a century ago [Broecker and Van
68Donk, 1970; Hays et al., 1976] the climates of the past
69had been described mostly in qualitative terms. Since then
70many techniques have become available to construct
71climatic records from geological, biological, and physical
72data [Bradley, 1999]. The se proxy records show that
73climate variations on different timescales have been very
74common in the past. The enormous amount of instrumen-
1
Also at Institute for Marine and Atmospheric Research Utrecht,
Department of Physics and Astronomy, Utrecht University, Utrecht,
Netherlands.
2
Also at De´partment Terre-Atmosphe´re-Oce´an and Laboratoire de
Me´te´orologie Deynamique de CNRS/IPSL, Ecole Normale Supe´rieure,
Paris, France.
Henk A. Dijkstra
1
Department of Atmospheric Science
Colorado State University
Fort Collins, Colorado, USA
Michael Ghil
2
Department of Atmospheric and Oceanic
Sciences and Institute of Geophysics
and Planetary Physics
University of California
Los Angeles, California, USA
Copyright 2005 by the American Geophysical Union.
8755-1209/05/2002RG000122$15.00
Reviews of Geophysics, 43, XXXXXX / 2005
1of38
Paper number 2002RG000122
XXXXXX
75 tal data that has been collected over the last century and
76 one half contributes, in turn, to a more and more
77 complete picture of the climate system’s variability.
78 [3] The purpose of the present review paper is to describe
79 the role of the ocean circulation in this variability and to
80 emphasize that dynamical systems theory can contribute
81 substantially to understanding this role. The intended audi-
82 ence and the way prospective readers can best benefit from
83 this review are highlighted in Box 1/Appendix A1. To
84 facilitate diverse routes through the paper, we have included
85 a glossary of the principal symbols in Table 1 and a list
86 acronyms in Table 2.
87 1.1. Climate Variability on Multiple Timescales
88 [4] An ‘‘artist’s rendering’’ of climate variability on all
89 timescales is provided in Figure 1a. The first version of
90 Figure 1a was produced by Mitchell [1976], and many
91 versions thereof have circulated since. Figure 1a is meant
92 to summarize our knowledge of the spectral power S =
93S
w
, i.e., the amount of variability in a given frequency
94band, between w and w + Dw; here the frequency w is the
95inverse of the period of oscillation and Dw indicates a
96small increment. This power spectrum is not computed
97directly by spectral analysis from a time series of a given
98climatic quantity, such as (local or global) temperature;
99indeed, there is no single time series that is 10
7
years
100long and has a sampling interval of hours, as Figure 1a
101would suggest. Figure 1a includes, instead, information
102obtained by ana lyzing th e spectral content of many
103different time series, for example, the spectrum
104(Figure 1b) of the 335-year long record of Centr al
105England Temperatures. This time series is the longest
106instrumentally measured record of any climatic variable.
107Given the lack of earlier instrumental records, one can
108easily imagine (but cannot easily confirm) that the higher-
109frequency spectral features might have changed, in am-
110plitude, frequency, or both, over the course of climatic
111history.
t1.1 TABLE 1. Glossary of Principal Symbols
Symbol Definition Sectiont1.2
A
H
and A
V
lateral and vertical friction coefficients Appendix A3t1.3
A = H/L aspect ratio Appendix A4t1.4
D equilibrium layer depth Appendix A3t1.5
E = A
H
/(2Wr
0
2
) Ekman number 2.5t1.6
f, f
0
Coriolis parameter (at latitude q
0
)2.3t1.7
f and p vector field and parameter vector 1.4t1.8
F
0
freshwater forcing coeffficient Appendix A4t1.9
F
S
and T
S
pattern of freshwater and temperature forcing Appendix A4t1.10
g, g
0
= gDr/r
0
gravitational acceleration, reduced gravity Appendix A3t1.11
h upper layer thickness 2.3t1.12
H depth of the ocean basin 2.3t1.13
K
H
and K
V
lateral and vertical diffusion coefficients Appendix A4t1.14
L basin length 2.3t1.15
Pr = A
H
/K
H
Prandtl number Appendix A4t1.16
Q dimensionless wind-driven transport 2.5t1.17
r
0
radius of the Earth 2.3t1.18
R bottom friction Appendix A3t1.19
Ra =(ga
T
DTL
3
)/(A
H
K
H
) Rayleigh number Appendix A4t1.20
Re =(d
I
/d
M
)
1/3
Reynolds number Appendix A3t1.21
R
HV
M
= A
V
/A
H
ratio of friction parameters Appendix A4t1.22
R
HV
T
= K
V
/K
H
ratio of diffusivities Appendix A4t1.23
S and T salinity and temperature 3.2t1.24
R
n
n-dimensional space of real numbers 1.4t1.25
(u, v, w) velocity vector Appendix A3t1.26
(U, V)=(uh, vh) horizontal transport vector Appendix A3t1.27
x state vector 1.4t1.28
a
T
and a
S
thermal and saline expansion coefficients Appendix A4t1.29
b meridional variation of Coriolis parameter Appendix A3t1.30
g = F
0
H/K
V
dimensionless measure of freshwater flux strength Appendix A4t1.31
G trajectory in phase space 1.4t1.32
d regularization parameter 3.3t1.33
d
I
and d
M
western boundary layer thicknesses Appendix A3t1.34
DT and DS characteristic temperature and salinity differences Appendix A4t1.35
Rossby number Appendix A3t1.36
h
1
, h
2
, and h
3
dimensionless parameters in box model 3.2t1.37
F(t) time periodic perturbation structure 2.3t1.38
meridional overturning 3.2t1.39
l single parameter 1.4t1.40
l = a
T
DT/(a
S
DS)
buoyancy ratio Appendix A4t1.41
m
1
and m
2
dimensionless parameters 3.3t1.42
w and S(w) frequency of oscillation and spectral power 1.1t1.43
W angular frequency of the Earth Appendix A3t1.44
r
0
, Dr background density and density difference Appendix A3t1.45
s = s
r
+ is
i
complex growth factor, eigenvalue 1.4t1.46
t
x
and t
y
zonal and meridional wind stress Appendix A3t1.47
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112 [5] With all due caution in its interpretation, Figure 1a
113 reflects three types o f variability: (1) sharp lines that
114 correspond to periodically forced variations at 1 day and
115 1 year; (2) broader peaks that arise from internal modes of
116 variability; and (3) a continuous portion of the spectrum that
117 reflects stochastically forced variations, as well as deter-
118 ministic chaos [Ghil and Robertson, 2000; Ghil, 2002].
119 [6] Between the two sharp lines at 1 day and 1 year lies
120 the synoptic variability of midlatit ude weather system s,
121 concentrated at 3 –7 days, as well as intraseasonal
122 variability, i.e., variability that occurs on the timescale of
123 1–3 months. The latter is also called low-frequency atmo-
124 spheric variability, a name that refers to the fact that this
125 variability has lower frequency, or longer periods, than the
126 life cycle of weather systems. Intraseasonal variability
127 comprises phenomena such as the Madden-Julian oscill a-
128 tion of winds and cloudiness in the tropics or the alternation
129 between episodes of zonal and blocked flow in midlatitudes
130 [Ghil and Childress, 1987; Ghil et al., 1991; Haines, 1994;
131 Molteni, 2002].
132 [7] Immediately to the left of the seasonal cycle in
133 Figure 1a lies interannual, i.e., year to year, variability. An
134 important component of this variability is the El Nin˜o
135 phenomenon in the tropical Pacific: Once about every
136 4 years, the sea surface temperatures (SSTs) in the eastern
137 tropical Pacific increase by a few degrees over a period of
138 about 1 year. This SST variation is associated with changes
139 in the trade winds over the tropical Pacific and in sea level
140 pressures [Bjerknes, 1969; Philander, 1990 ]; an east-west
141 seesaw in the latter is called the Southern Oscillation. The
142combined El Nin˜o/Southern Oscillation (ENSO) phenome-
143non arises through large-scale interaction betwee n the
144equatorial Pacific and the atmosphere above. Equatorial
145wave dynamics in the ocean plays a key role in setting
146ENSO’s timescale [Cane and Zebiak, 1985; Neelin et al.,
1471994, 1998; Dijkstra and Burgers, 2002].
t2.1 TABLE 2. Glossary of Acronyms
Symbol Definition Sectiont2.2
0-D zero-dimensional 1.3t2.3
1-D one-dimensional 1.3t2.4
2-D two-dimensional 1.3t2.5
3-D three-dimensional 1.3t2.6
AABW Antarctic Bottom Water 3.1t2.7
ACC Antarctic Circumpolar Current 3.4t2.8
COADS Comprehensive Ocean-Atmosphere Data Set 2.6t2.9
EBM energy balance model 3.4t2.10
EOF empirical orthogonal function 3.1t2.11
ENSO El Nin˜o/Southern Oscillation 1.1t2.12
GCM general circulation model 1.3t2.13
GFDL Geophysical Fluid Dynamics Laboratory 3.4t2.14
LSG large-scale geostrophic model 3.4t2.15
MOM modular ocean model 3.4t2.16
M-SSA multichannel singular-spectrum analysis 2.6t2.17
NADW North Atlantic Deep Water 1.2t2.18
NPP northern sinking pole-to-pole flow 3.2t2.19
ODE ordinary differential equation 1.4t2.20
PDE partial differential equation 1.4t2.21
PGM planetary geostrophic model 3.4t2.22
POCM Parallel Ocean Climate Model 2.6t2.23
POP Parallel Ocean Program 3.4t2.24
QG quasi-geostrophic 2.3t2.25
SA salinity-driven flow 3.2t2.26
SPP southern sinking pole-to-pole flow 3.2t2.27
SSA singular-spectrum analysis 2.3t2.28
SST sea surface temperature 1.1t2.29
TH thermally driven flow 3.2t2.30
THC thermohaline circulation 3.1t2.31
WOCE World Ocean Circulation Experiment 3.1t2.32
Figure 1. (a) An ‘‘artist’s rendering’’ of the composite
power spectrum of climate variability showing the amount
of variance in each frequency range [from Ghil, 2002].
(b) Spectrum of the central England Temperature time series
from 1650 to the present. Each peak in the spectrum is
tentatively attributed to a physical mechanism; see Plaut et
al. [1995] for details. Reprinted with permission from Plaut
et al. [1995], # 1995 American Associat ion for the
Advancement of Science, http:www.sciencemag.org.
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148 [8] The greatest excitement among scientists as well as
149 the public is currently being generated by interdecadal
150 variability, i.e., climate variability on the timescale of a
151 few decades, the timescale of an individual human’s life
152 cycle [Martinson et al., 1995]. Figure 1b represents an up-
153 to-date ‘‘blowup’’ of the interannual-to-interdecadal portion
154 of Figure 1a. The broad peaks are due to the climate
155 system’s internal processes: Each spectral component can
156 be associated, at least tentatively, with a mode of interan-
157 nual or interdecadal variability [Plaut et al., 1995]. Thus the
158 rightmost peak, with a period of 5.2 years, can be attributed
159 to the remote effect of ENSO’s low-frequency mode, while
160 the 7.7-year peak captures a North Atlantic mode of
161 variability that arises from the Gulf Stream’s interannual
162 cycle of meandering and intensification. The two interde-
163 cadal peaks, near 14 and 25 years, are also present in global
164 records, instrum ental as well as paleoclimatic [Kushnir,
165 1994; Mann et al., 1998; Moron et al., 1998; Delworth
166 and Mann, 2000; Ghil et al., 2002b].
167 [9] Finally, the leftmost part of Figu re 1a represents
168 paleoclimatic variability. The information summarized here
169 comes exclusively from proxy indicators of climate [Imbri e
170 and Imbrie, 1986]. These include coral records [Boiseau et
171 al., 1999] and tree rings for the historic past [Mann et
172 al.,1998],aswellasmarinesediment[Duplessy and
173 Shackleton, 1985] and ice core [Jouzel et al., 1991] records
174 for the last 2 million years of Earth history, the Quaternary.
175 Glaciation cycles, an alternation of warmer and colder
176 climatic episodes, dominated the Quaternary era. The
177 cyclicity is manifest in the broad peaks present in Figure 1a
178 between roughly 1 kyr and 1 Myr. The two peaks at about
179 20 kyr and 40 kyr reflect variations in Earth’s orbit, while the
180 dominant peak at 100 kyr remains to be convincingly
181 explained [Imbrie and Imbrie, 1986; Ghil and Childress,
182 1987; Gildor and Tziperman, 2001]. The glaciation cycles
183 provide a fertile testing ground for t heories of climate
184 variability for two reasons: (1) They represent a wide range
185 of climatic conditions, and (2) they are much better docu-
186 mented than earlier parts of clima tic history.
187 [10] Within these glaciation cycles, there are higher-
188 frequency oscillations prominent in the North Atlantic
189 paleoclimatic records. These are the Heinrich events
190 [Heinrich, 1988] with a near periodicity of 6–7 kyr and
191 the Dansgaard-Oeschger cycles that provide the peak at
192 around 1– 2.5 kyr in Figure 1a. Rapid changes in
193 temperature, of up to one half of the amplitude of a typical
194 glacial-interglacial temperature difference, occurred during
195 Heinrich events, and somewhat smaller ones occurred over a
196 Dansgaard-Oeschger cycle. Progressive c ooling through
197 several of the latter cycles followed by an abrupt warming
198 defines a Bond cycle [Bond et al., 1995]. In North Atlantic
199 sediment cores the coldest part of each Bond cycle is marked
200 by a so-called Heinrich layer that is rich in ice-rafted debris.
201 None of these higher-frequency oscillations can be directly
202 connected to orbital or other periodic forcings.
203 [11] In summary, climate variations range from the large-
204 amplitude climate excursions of the past millennia to
205 smaller-amplitude fluctuations on shorter timescales. Sev-
206eral spectral peaks of variability can be clearly related to
207forcing mechanisms; others cannot. In fact, even if the
208external forcing were constant in time, that is, if no
209systematic changes in insolation or atmospheric composi-
210tion, such as trace gas or aerosol concentration, would
211occur, the climate system would still display variability
212on many timescales. This statement is clearly true for the
2133– 7 days synoptic variability of midlatitude weather, which
214arises through baroclinic instability of the zonal winds, and
215the ENSO variabili ty in the equatorial Pacific, as discussed
216above. Processes internal to the climate system can thus
217give rise to spectral peaks that are not related directly to the
218temporal variability of the forcing. It is the interaction of
219this highly complex intrinsic variability with the relatively
220small time-dependent variations in the forcing that is
221recorded in the proxy records and instrumental data.
2221.2. Role of the Ocean Circulation
223[12] We focus in this review on the ocean circulation as a
224source of internal climate variability. The ocean moderates
225climate through its large thermal inertia, i.e., its capacity to
226store and release heat and its poleward heat transport
227through ocean currents. The exact importance of the latter
228relative to atmospheric heat transport, though, is still a
229matter of active debate [Seager et al., 2001 ]. The large-
230scale ocean circulation is driven both by momentum fluxes
231as well as by fluxes of heat and freshwater at the ocean-
232atmosphere interface. The near-surface circulation is dom-
233inated by horizontal currents that are mainly driven by the
234wind stress forcing, while the much slower motions of the
235deep ocean are mainly induced by buoyancy differences.
236[13] The circulation due to either forcing mechanism is
237often described and analyzed separately for the sake of
238simplicity. In fact, the wind-driven and thermohaline circu-
239lation together form a complex three-dimensional (3-D)
240flow of different currents and water masses through the
241global ocean. The simplest pi cture of the global ocean
242circulation has been termed the ‘‘ocean conveyor’’ [Gordon,
2431986; Broecker, 1991]; it corresponds to a two-layer view
244where the vertical structure of the flow field is separated
245into a shallow flow, above the permanent thermocline at
246roughly 1000 m, and a deep flow between this thermocline
247and the bottom (i.e., between a depth of roughly 1000 m and
2484000 m); see Figure 2. The unit of volume flux in the ocean
249is 1 Sv = 10
6
m
3
s
1
, and it equals approximately the total
250flux of the world’s major rivers. MacDonald and Wunsch
251[1996] and Ganachaud and Wunsch [2000] have provided
252an updated version of this schematic representation of the
253ocean circulation.
254[14] In the North Atlantic, for instance, the major current
255is the Gulf Stream, an eastward jet that arises through the
256merging of the two western boundary currents, the north-
257ward flowing Florida Current and the southward flowing
258Labrador Current. In the North Atlantic’s subpolar seas,
259about 14 Sv of the upper ocean water carried northward by
260the North Atlantic Drift, the northeastward extension of the
261Gulf Stream, is converted to deepwater by cooling and
262salinification. This North Atlantic Deep Water (NADW)
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263 flows southward, crosses the equator, and joins the flows
264 in the Southern Ocean. The outflow from the North
265 Atlantic is compensated by water coming through the
266 Drake Passage (about 10 Sv) and water coming from the
267 Indian Ocean through the Agulhas Current system (about
268 4 Sv). Part of the latter ‘‘Agulhas leakage’’ may originate
269 from Pacific water that flows through the Indonesian
270 Archipelago. We refer to earlier reviews [Gordon, 1986;
271 Schmitz, 1995; World Ocean Circulation Experiment
272 (WOCE), 2001] for more complete information on the
273 circulation in each major ocean basin as well as from one
274 basin to another.
275 [15] Changes in the ocean circulation can influence
276 climate substantially through their impact on both the
277 meridional and zonal heat transport. This can affect mean
278 global temperature and precipitation, as well as their distri-
279 bution in space and time. Subtle changes in the North
280 Atlantic surface circulation and their interactions with the
281 overlying atmosphere are thought to be involved in climate
282 variability on interannual and interdecadal timescales, as
283 observed in the instrumental record of the last century
284 [Martinson et al., 1995; Ghil, 2001]. Chan ges in the
285 circulation may also occur on a global scale, involving a
286 transition to different large-scale patterns. Such changes
287 may have been involved in the large-amplitude climate
288 variations of the past [Broecker et al., 1985].
289 1.3. Modeling Hierarchy
290 [16] The climate system is highly complex. Its main
291 subsystems have very different characteristic times, and
292 the specific phenomena involved in each one of the climate
293 problems alluded to in sections 1.1 and 1.2 are quite diverse.
294 It is inconceivable therefore that a single model could
295 successfully incorporate all the subsystems, capture all the
296 phenomena, and solve all the problems. Hence the concept
297 of a hierarchy of climate models, from the simple to the
298 complex, was developed about a quarter of a century ago
299[Schneider and Dickinson, 1974; Ghil and Robertson,
3002000].
301[17] The simplest, spatially zero-dimensional (0-D) ocean
302models are so-called box models, used to study the stability
303[Stommel, 1961] and paleoevolution [Karaca et al., 1999]
304of the oceans’ thermohaline circulation or biogeochemical
305cycles [Sarmiento and Toggwe iler, 1984; Keir, 1988;
306Paillard et al., 1993]. There are one-dimensional (1-D)
307models that consider the vertical structure of the upper
308ocean, whether the oceanic mixed layer only [Kraus and
309Turner, 1967; Karaca and Mu¨ller, 1989] or the entire
310thermocline structure.
311[18] Two-dimensional (2-D) models of the oceans fall
312into the two broad categories of ‘‘horizontal’’ and ‘‘verti-
313cal.’’ Models which resolve two horizontal coordinates
314emphasize the study of the oceans’ wind-driven circulation
315[Cessi and Ierley, 1995; Jiang et al., 1995], while those that
316consider a meridional section concentrate on the overturn-
317ing, thermoha line circulation [Cessi and Young, 1992; Quon
318and Ghil, 1992, 1995; Thual and McWilliams, 1992].
319[19] As explained in section 1.2, the oceans’ circulation is
320essentially 3-D, and therefore general circulation models
321(GCMs) of the ocean are indispensabl e in understanding
322oceanic variability [McWilliams,1996].TheBryan-Cox
323model [Bryan et al., 1974; Cox, 1987] has played a seminal
324role for the development and applications of such models;
325this role resembles the one played by the University of
326California, Los Angeles, model [Arakawa and Lamb,
3271977] in atmospheric modeling. A number of simplified
328versions of the Bryan-Cox ocean GCM have been used in
329exploratory studies of multiple mean flows [Bryan, 1986;
330Marotzke et al., 1988] and oscillatory behavior [Weaver et
331al., 1991, 1993; Chen and Ghil, 1995, 1996] of the
332oceans.
333[20] In confronting modeling results with observations
334one has to realize that it is the largest scales that are best and
335most reliably captured. This is certainly true in the atmo-
Figure 2. Sketch of the global ocean circulation as a ‘‘conveyor belt.’’ Dark shaded paths indicate flow
in the surface ocean; light shaded paths indicate flow in the deep ocean. Numbers indicate volume
transport in sverdrup (1 Sv = 10
6
m
3
s
1
) (based on Schmitz [1995] but reprinted from Bradley [1999],
with permission from Elsevier).
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![Figure 6. (a) Bifurcation diagram for a barotropic shallow water model of the North Atlantic with the Ekman number as control parameter. The nondimensional Ekman number, E = AH/(2Wr0 2), where W and r0 are the angular frequency and radius of the Earth, respectively, is plotted on the abscissa, while the maximum northward volume transport is plotted on the ordinate. Solid (dotted) branches indicate stable (unstable) steady states, as usual, whereas the Hopf bifurcation points are indicated by triangles. (b and c) Contour plots of the layer thickness anomaly for two coexisting unstable solutions at E = 1.6 10 7. The contour levels are scaled with respect to the maximum value of the field; only a small part of the domain, (85 W–45 W, 24 N–51 N), is shown: ‘‘deflected’’ (Figure 5b) and ‘‘separated’’ Gulf Stream (Figure 5c). (d and e) Spatial pattern of the linear oscillatory mode at H1 in Figure 6a. Layer thickness anomalies for the real (Figure 6d) and imaginary part (Figure 6e) are shown. From Schmeits and Dijkstra [2000].](/figures/figure-6-a-bifurcation-diagram-for-a-barotropic-shallow-30np6rs9.webp)
![Figure 13. Steady states obtained in an ocean GCM limited to an idealized 60 wide sectorial configuration, subject to mixed boundary conditions. Zonal averages of the velocity field are plotted as a meridional plane stream function field. (a) Reference state obtained with equatorially symmetric, restoring boundary conditions. (b) Circulation obtained after adding a negative salinity perturbation of 1 psu south of 45 S. (c) Circulation after adding a positive salinity perturbation of 2 psu south of 45 S. (d) Circulation after adding a positive salinity perturbation of 2 psu north of 45 N. From Bryan [1986], used with permission from Nature (http://www.nature.com).](/figures/figure-13-steady-states-obtained-in-an-ocean-gcm-limited-to-9u8wf4vd.webp)
![Figure 7. (a) Time evolution of the mean monthly meridional displacement of the position of the sea surface temperature (SST) isotherm T = 15 C away from the latitude 41 N at 50 W, January 1960 to December 1997. The SST field has been spatially interpolated by cubic splines in the interval 30 N–60 N in order to compute the deviation from its mean latitude position; the data are derived from the Comprehensive Ocean-Atmosphere Data Set. Both the raw data (thin line) and data adaptively lowpass-filtered signal (bold line) are shown; the latter is based on singular-spectrum analysis (SSA) with a 16-year window, which retains the eight modes with lowest frequencies, after filtering out the annual and subannual signals. (b) Maximum entropy spectrum of the low-passfiltered time series shown in Figure 7a, given in log linear coordinates. The order of the maximum entropy method (MEM) is 40; only the powers of 10 are indicated on the ordinate. (c) Results from a 100-year simulation with a 2.5- layer shallow water model within a basin that approximates the North Atlantic in size and shape, using an idealized wind stress. Maximum entropy spectrum of the subpolargyre kinetic energy is shown. The SSA window length is 20 years, the number of modes retained is 12, and the MEM order is 40. Coordinates and labeling are the same as in Figure 7b. From Simonnet et al. [2005].](/figures/figure-7-a-time-evolution-of-the-mean-monthly-meridional-3mhjwloc.webp)