JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, C07025, doi:10.1029/2005JC002897, 2005
Passive mechanism of decadal variation of
thermohaline circulation
Liang Sun
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui, China
Mu Mu
LASG, Institute of Atmospheric Physics, Chinese Academy of Science, Beijing, China
De-Jun Sun and Xie-Yuan Yin
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui, China
Received 24 January 2005; revised 8 March 2005; accepted 31 March 2005; published 29 July 2005.
[1] The decadal variation of thermohaline circulation (THC) is investigated in a simple
coupled ocean-atmospheric two-box model with the new approach of conditional
nonlinear optimal perturbation (CNOP). First, the nonlinear optimal initial perturbations
with different constrains are found by this approach. There are two different types of
perturbations in the nonlinear regime. One is the freshwater flux perturbation, which is the
CNOP of THC and has stronger amplification. The other is salinity flux perturbation,
whose amplification is weaker. Freshwater (salt) perturbations weaken (enhance) the mean
circulation and hence weaken (enhance) the stability of THC. Second, the passive
variabilities of THC are investigated by superposing the initial perturbations to the
thermohaline circulation. The passive variabilities found in this model are due to
nonnormal and nonlinear growth of initial perturbations. These variabilities, measured as
recovering time of perturbations, can cause decadal variability of THC. The results of
this paper suggest that CNOP approach is applicable to the investigation of the
dependence of the THC sensitivity on the background climate state and is of potential
application to the interpretation of past climate change linked to THC variations in
nonlinear regime.
Citation: Sun, L., M. Mu, D.-J. Sun, and X.-Y. Yin (2005), Passive mechanism of decadal variation of thermohaline circulation,
J. Geophys. Res., 110, C07025, doi:10.1029/2005JC002897.
1. Introduction
[2] The climate variation in the world has captured the
public’s eyes in recent years, in which the thermohaline
circulation (THC) plays an important role. As a major heat
budget of the North Atlantic region, there is the warm
anomaly over the northern North Atlantic (about 10C) due
to THC [Rahmstorf, 2002]. The variation of THC has
received a great deal of attention, especially on the decadal
and interdecadal variabilities of THC. Though the short
length of the time series of sea surface temperature (SST)
and sea level pressure (SLP), observations indeed link
interdecadal sea surface temperature variability to the overturning meridional circulation variability [Kushnir, 1994].
From long instrumental records of central England temperatures, interdecadal (15- and 25-year) oscillation is related
to THC [Plaut et al., 1995]. Besides, there are numerous
instrumental studies and observations of THC, addressing
the decadal variability in North Atlantic region (see Dijkstra
[2000] or Marshall et al. [2001] for a comprehensive
Copyright 2005 by the American Geophysical Union.
0148-0227/05/2005JC002897$09.00
review). One of the central questions is whether this
variability can be understood from instabilities of the mean
flow.
[3] To investigate the problem theoretically, numerical
models from simple two-box models to complex global
ocean models are used to simulate the phenomena. The
physical mechanisms of THC variability have been
explained in several different ways. Bigg et al. [2003]
pointed out that it is useful to define two types of internally
generated variabilities. Passive variability involves the
modulation by slower components of the climate system
(such as the oceans) of (typically random) variability in the
faster components (such as the atmosphere). Active variability is caused by variations and instabilities of the
dynamics of the climate system, and by coupled interaction
between components of the climate system (such as the
atmosphere and ocean).
[4] To understand the active variability of THC, Chen
and Ghil [1995] and Delworth and Mann [2000] suggested the THC variability is self-sustained of the climate
system. Chen and Ghil [1995] showed that the interdecadal oscillation likely arises when a critical parameter value
is crossed through a Hopf bifurcation. After that, some
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perturbations are addressed in section 4. Finally, the results
are discussed in section 5.
2. Model and Method
Figure 1. The coupled atmosphere-ocean boxes mode
[after Lohmann et al., 1996].
recent work [de Verdière and Huck, 1999; Huck et al.,
1999, 2001; Te Raa and Dijkstra, 2002] follow the same
way, and find an interdecadal oscillatory timescale instability raising from their complex ocean models. In addition, this oscillatory mode in [Te Raa and Dijkstra, 2002]
ocean model has a timescale of 65-year in well agree with
70-year SST variability in North Atlantic [Delworth and
Greatbach, 2000].
[5] Other variabilities can be seen as the stochastic
forcing [Griffies and Tziperman, 1995; Timmermann and
Lohmann, 2000; Vélez-Belchı́ et al., 2001], or the recovering
process of THC after being disturbed from equilibrium,
which is the passive variability of THC. This transient
amplification might be the decadal variation of THC
[Lohmann et al., 1996; Lohmann and Schneider, 1999].
Tziperman and Ioannou [2002] studied the nonnormal
growth of perturbations on the thermally driven state and
identified two physical mechanisms associated with the
transient amplification of these perturbations. One such
mechanism, with a transient amplification timescale of a
couple of years, involves an interaction between the THC
induced by rapidly decaying sea surface temperature
anomalies and the THC induced by the slower-decaying
salinity mode. The second mechanism of transient amplification involves an interaction between different slowly
decaying salinity modes and has a typical growth timescale of decades. For the wide range of transient amplification timescale, this can be used to interpret the different
decadal timescale variations.
[6] One goal of this paper is to investigate the mechanism of passive decadal variability of THC. The decadal
variation of THC, which is caused by temperature and
salinity perturbations, is studied. For this purpose, a new
nonlinear approach is applied in this paper to investigate
the problem within a simple model context. This approach
is called conditional nonlinear optimal perturbation
(CNOP) method [Mu et al., 2002, 2003]. Here CNOP is
applied to a coupled ocean-atmosphere two-box model,
which was developed by Lohmann et al. [1996]. The
model used and the approach of CNOP are outlined in
section 2. After that, the nonlinear optimal growth perturbations of THC are obtained with this method in section 3.
Next, the passive variabilities of THC due to those
[7] The model used in this paper was developed by
Lohmann et al. [1996], which is after Stommel’s two-box
model [Stommel, 1961]. As shown in Figure 1, the model
has three hierarchies: atmosphere, upper ocean and deep
ocean hierarchy. The atmosphere has a low latitude box and
a high latitude box, which provides freshwater and heat flux
forcing between the ocean and atmosphere interface. Both
the SST and the salinity of the North Atlantic Ocean
decrease at the higher latitude. Thus the buoyancy forcing
drives the thermohaline circulation. The strength of THC is
assumed to be proportional to the meridional density
difference, i.e., = c(aT bS), where T and S are
temperature difference and salinity difference between high
and low latitude ocean, c, a and b are density, thermal and
haline expansion coefficients specified in Appendix A. The
steady state of THC is estimated as (T, S) = (13.5 K,
1.5 psu), where the mean overturning flow rate of THC is
= 14 Sv. The time evolution of temperature and salinity
are governed by two terms. One is the local thermal
(salinity) forcing due to air-sea interaction, the other is the
horizontal advection through the THC. The evolution of the
perturbation superposed on the mean flow is governed by
the nonlinear differential equations
dx
¼ Ax hb; xix
dt
ð1Þ
where x = (T0, S0) represents the perturbations on the basic
state of temperature and salinity, h, i is inner product of
two vectors, the parameter matrix A and vector b can be
found in Appendix A. The linear term in equation (1)
represents the local air-sea forcing, where the atmospherically meridional transports are parameterized as
diffusion. As the perturbation changes the buoyancy
between south and north ocean boxes, the dimensionless
flow rate y0 = hb, xi = c(aT0 bS0) is proportional to
the change of overturning flow due to perturbations
[Stommel, 1961; Lohmann et al., 1996]. So the dimensionless flow rate y0 represents the change of horizonal
advection rate, i.e the horizonal advection term depends on
the initial perturbation nonlinearly. If there is more
freshwater entering the North Atlantic Ocean, the perturbation has S0 < 0 and y0 < 0, which is called freshwater
perturbation. If there is less freshwater entering the North
Atlantic Ocean, the perturbation has S0 > 0 and y0 > 0,
which is called salinity perturbation. Since Lohmann’s box
model has considered the ocean-atmosphere action in
current climate, it is suitable to use this model to
investigate the variation of THC.
[8] In this paper, the nonlinear approach CNOP is
employed to explore the mechanism of THC variation.
Different with those linear approaches, such as empirical
orthogonal function (EOF) analysis and linear singular
vector [Lorenz, 1965], CNOP can provide the optimal
growing perturbation under given condition. This method
has been successfully used to the study of predictability
problem of El Nino [Mu et al., 2002, 2003; Duan et al.,
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Figure 2. The evolution J of linear (dashed) and nonlinear
(solid) perturbations versus azimuth angle q. The linear
evolution is symmetric to q = p, while the nonlinear
evolution is asymmetric.
2004]. In general, the model governing the motions of the
atmosphere or ocean is assumed as follows:
dx
¼ F ðx; t Þ
dt
ð2Þ
where x = (x1(t), x2(t), . . ., xn(t)), F is a nonlinear operator.
Suppose there is an initial perturbation x0 at time t = 0, its
evolution reaches xt at time t. Then the magnitude of
perturbations
at time t is J(x0) = kxtk, where the norm
pffiffiffiffiffiffiffiffiffiffiffi
kxk =
hx; xi. The perturbation x0d is called the
conditional nonlinear optimal perturbation with constraint
condition kx0k d, if and only if:
J ðx0d Þ ¼ max J ðx0 Þ
kx0 k
d
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perturbation x0 is written as x0 = (T 00 , S 00 ) = (d cos q, d sin
q), where d is the magnitude of initial perturbation and q the
angle of the initial perturbation vector with the T axis. Then
d cos q is temperature perturbation component, and d sin q is
salinity perturbation component. In the following discussion, we have time t = 10, which represents 10 years of
evolution for perturbations.
[11] Given the constrain d = 0.5, the magnitude J of both
linear and nonlinear perturbations is shown in Figure 2. It is
clear that the opposite perturbation vectors (their azimuth
angle q having a difference of p) have the same growth in
linear case (dashed line in Figure 2. While in the nonlinear
case (solid curve in Figure 2), they have different growth.
For the initial salinity flux perturbations (where q < p), the
growth of nonlinear perturbation is smaller than the linear
evolution. For the initial freshwater flux perturbations
(where q > p), the evolution of nonlinear perturbation is
larger than the linear evolution. The perturbation (where q =
4.79), which has largest growth (J = 1.096), is called
constrained nonlinear optimal perturbation (CNOP) in this
case (d = 0.5). The perturbation (where q = 1.63), which has
local maximal growth (J = 0.387), is salinity flux perturbation. In this case, the CNOP correctly provides the optimal
initial condition for transient growth of THC. In addition, it
is clear from the results that the thermohaline circulation is
more sensitive to fresh water flux perturbation, which
agree well with former researches [Chen and Ghil, 1995;
Lohmann, 2003; Mu et al., 2004]. Generally speaking, the
linear evolution is symmetric in Figure 2, while the
nonlinear evolution is asymmetric.
[12] In addition, the evolution of the CNOP J is calculated under different constrains. The curve of evolution J
versus initial magnitude d is shown in Figure 3. It is clear
that J increases nonlinearly from J = 0.126 to J = 2.42 as
initial magnitude increases linearly form d = 0.1 to d = 0.7.
The larger the initial perturbation is, the sharper the slope of
the curve is. For comparison, the growth of the fastest linear
singular vector is also shown in the plot (dash line). It is
ð3Þ
Since the initial problem equation (2) cannot be solved
analytically, we obtain the solution numerically; see Mu et
al. [2004] for details. The CNOP is the initial perturbation
whose nonlinear evolution reaches the maximal value at
certain time t with the constraint conditions. The CNOP
can be regarded as the most nonlinearly unstable initial
perturbation superposed on the basic state. With the same
constraint conditions, the larger the nonlinear evolution of
the CNOP, the more unstable the basic state is.
[9] One may note that the norm used here is quite
different from Tziperman and Ioannou [2002]. They
employed the overturning stream function as measurement,
which would lead a constant shift in the salinities of all
boxes. We simply use L2 norm of vector as the measurement
of magnitude of perturbations, which can avoid such
constant shift.
3. CNOP of THC
[10] In this section, the property of initial perturbations
on the thermohaline circulation is investigated. The initial
Figure 3. The optimal growth of linear (dashed) and
nonlinear (solid) versus magnitude of initial perturbations.
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Figure 4. The mechanism of nonlinear feedback for the
thermohaline circulation.
clear that the growth of linear singular vector increases
linearly from J = 0.115 to J = 0.808, as the magnitude of
initial perturbation increases from d = 0.1 to d = 0.7. The
difference of growth between linear and nonlinear optimal
perturbations is relatively small when d is small. This means
the linear approximation might be valid for small magnitude
perturbations.
[13] To understand why the thermohaline circulation is
more sensitive to fresh water flux perturbation, we derive
the evolution equation of y0 = hb, xi from equation (1)
d 0
y ¼ hb; Axi y02
dt
ð4Þ
Integrating the above equation, we find
y0 ðt Þ ¼ y0 ð0Þ þ
Z
Z
t
t
y02 dt
LðxÞdt 0
ð5Þ
0
where L(x) = hb, Axi is the linear part of equation (4). It is
well known that the linear term in equation (5) determine
the linear stability of the steady state [Stommel, 1961].
[14] It can be seen from equation (5) that the nonlinear
term is always negative for any type of perturbation. As a
result, the effect of nonlinear term is to enhance the initial
perturbation if it is freshwater flux type, i.e., y0(0) < 0;
whereas to damp the initial perturbation if it is salinity flux
type, i.e., y0(0) > 0. Therefore with the consideration of
nonlinear effect, the freshwater flux type perturbation actually grows faster than that in the linear case, and the salinity
flux type perturbation grows slower correspondingly. That
is why the thermohaline circulation is more sensitive to
freshwater flux perturbation. Only by nonlinear analysis can
we explain this phenomena.
[15] The result discussed above can be generalized to
other dynamical systems like equation (1). Noting that the
result depends on the negatively defined nonlinear term, for
a dynamical system which nonlinear term is negatively
defined, its nonlinear feedback is similar and the discussion
above is still valid. However the nonlinear behavior would
be more complex than equation (1).
[16] The physical mechanism behind the loss of stability
of the steady state is often discussed in terms of the saltadvection feedback [Marotzke, 1996]. [Marotzke, 1996] and
[Bigg et al., 2003] pointed out that the linear term has a
negative feedback, or more precisely the mean circulation
tends to eliminate the anomalous temperature and salinity
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gradients. Here we find that the nonlinear term may have
either positive or negative feedback, which depends on the
type of initial perturbation, i.e., the sign of y0(0), as shown
in Figure 4. For freshwater flux perturbation (y0 < 0), the
nonlinear term enhances the initial perturbation. This is a
positive feedback. The stronger the freshwater perturbation
is, the stronger the nonlinear feedback destabilizes the
steady state. For salinity flux perturbation (y0 > 0), the
nonlinear term damps the initial perturbation. This is a
negative feedback. The stronger the salinity perturbation
is, the stronger the nonlinear feedback stabilizes the steady
state. Nonlinear mechanism hence makes the steady state
more stable to salinity perturbations.
[17] In model experiments, salinity perturbations were
usually applied to present-day climate states of the THC
to study its sensitivity, here we give an physical interpretation how the meridional transportation in North Atlantic
Ocean can be amplified (dumped) by the nonlinearly
positive (negative) feedback. The discussion might be
applied to understand the past climate change, whereas
the variation of thermohaline circulation is linked to the
large freshwater or salinity perturbation and nonlinear
feedback may play important role.
[18] We have identified the optimal initial conditions (the
CNOPs) for the growth of initial perturbations of THC.
These initial perturbations, being of freshwater flux type,
are very important for the decadal variations of THC. In the
nonlinear cases, the perturbations will cause larger variations than in the linear cases. Since the linear method cannot
distinguish between the evolution of freshwater type and
salinity type perturbations, it cannot identify the real optimal mode between two equally amplified linear optimal
perturbations, the choice of linear optimal modes in analysis
may not be reliable. Moreover it is not sure whether the
linear approximation is still valid. However the nonlinear
approaches can provide more reasonable results and are thus
more reliable.
4. Decadal Variation of THC
[19] In this section, the passive variabilities of THC is
investigated by superposing initial perturbations to the
thermohaline circulation. As shown above, CNOPs can
identify the most unstable freshwater perturbations which
induce the most strong passive variabilities of the thermohaline circulation. Accordingly those optimal perturbations
may cause longer timescale variations than others. Here we
use CNOPs to investigate the decadal variation of THC.
First we choose a positive number e to measure the recovery
of a perturbation. For an initial perturbation of THC, it
usually takes a certain time to decay to a smaller magnitude
less than or equal to d. Obviously this time period depends
on both the magnitude of initial perturbations and the
pattern. Naturally, we use CNOPs to define recovering time
td, which is the time period the CNOP with initial constraint
condition d takes to recover to . In this section, the
recovering index is fixed as = 0.01. If the magnitude of
perturbations decays to be smaller than , the thermohaline
circulation can be regarded as recovering to steady state. As
mentioned above, to calculate the CNOP of THC, we
choose time t = 10, which represents 10 years of evolution
for perturbations.
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Figure 5. The recovering time td versus magnitude d of
initial perturbation for = 0.01. Dashed curve, linear results;
solid curve, nonlinear results.
[20] As shown in Figure 5, the recovering time td versus
initial magnitude of perturbation d are calculated with linear
and nonlinear perturbations. When the initial magnitude of
perturbation is relatively small, the difference of results is
relatively small. For example, d = 0.075, the linear and
nonlinear results are 25.7 years and 26.4 years, respectively.
When d = 0.15, the linear and nonlinear results are 30.4 years
and 31.7 years, respectively. As d increases, the difference
between the linear and nonlinear results are remarkable.
For example, d = 0.3, the linear and nonlinear results are
34.6 years and 37 years, respectively. From those results,
we can draw a conclusion that if the magnitude of
perturbation is less than 10% of the magnitude of the
steady state, the linear dynamics is a good approximation.
In this case, Tziperman and Ioannou’s assumption is valid
that the linearized dynamics might be able to describe it
relatively well. However, the nonlinear effect must be
considered for large initial perturbations.
[21] Now we consider the mechanism of this transient
growth. Figure 6 shows the property of the variation of
THC by integrating equation 1 for the evolutions of the
CNOPs. The plots show the evolution of perturbation
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temperature T0, salinity S0 and the magnitude J, with given
various initial magnitude. As shown in Figure 6a, the
temperature of perturbation has a very sharp decrease,
although d only varies from 0.02 to 0.05. The time interval
for this fast decrease is about 2 years. After that it undergoes a relatively long increase range, which is about
decadal timescale for the initial perturbation with respect
to magnitude of d. This typical process also occurs in
salinity perturbations and the magnitude of CNOP as shown
in Figures 6b and 6c. These agree well with the former
results that larger perturbation magnitude induces longer
recovering time. [Lohmann and Schneider, 1999] were the
first who found out that the transient growth is related to the
nonorthogonality of the eigenmodes of THC. Also physical
interpretations were given that the transient amplification
mechanism is due to the interaction of two (or more)
nonorthogonal eigenmodes with different decay times.
When the initial magnitude is larger enough, the evolution
of CNOP will be different from that with the smaller initial
perturbations. This is shown in Figure 7a. For initial
magnitude d = 0.8, different from others, the temperature
decreases sharply in about first 3 years. After that, it
undergoes a relatively slow decrease down to 4C, and
finally has a more quick descend so that it will not recover
to the steady state. This can also be seen from the evolution
of J in Figure 7c, where J increases continually. There is a
critical value of d, say dc, such that the perturbation might
be asymptotically nonlinear unstable, when d > dc. This
critical value is also calculated under present parameters,
which is about 0.791. Then it might induce a jump from
present thermal-driven sate to a salinity-driven state, if an
initial perturbation has a larger magnitude than dc.
[22] The physical mechanism of these decadal variabilities can be understood from Figure 4. As pointed out
above, the competition of the local forcing and transport
control the thermohaline circulation. A freshwater flux
perturbation, which enter into the North Atlantic Ocean,
weakens the thermohaline circulation. Then the salt and
thermal mass flow, which enters into North Atlantic Ocean
from low latitude, is less than before. Thus it takes a longer
time to convey enough salt and mass flow northward, and
consequently the horizontal advection is weaker, so the
freshwater perturbation decays more slowly. A salinity flux
Figure 6. The evolution of (a) temperature T0, (b) salinity S0, and (c) magnitude J of CNOPs. The solid,
dashed, and dash-dot curves represent d = 0.02, 0.04, and 0.05, respectively.
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Figure 7. The evolution of (a) temperature T0, (b) salinity S0, and (c) magnitude J of CNOPs. The solid,
dashed, and dash-dot curves represent d = 0.6, 0.7, and 0.8, respectively.
type perturbation enhances the thermohaline circulation.
Then the salt and thermal mass flow, which enters into
North Atlantic Ocean from low latitude, is more than
before. Thus it takes a shorter time to convey enough salt
and mass flow northward, so the salinity perturbation
decays more quickly for the stronger horizontal advection.
Generally, The stronger the perturbation of the same flux
type is, the longer the recover time is.
[23] Furthermore, the recovering time td of variable
azimuth angle q are also calculated, for different fixed
initial magnitudes. As shown in Figure 8a, given fixed d =
0.02, the freshwater type of perturbation has a maximal
recovering time of 16.5 years, though the salinity ones has
16.2 years. Most of perturbations have their recovering time
of approximate 14 years, the rest have shorter recovering
time varying from 1 year to 10 years. In this case, both types
of perturbations are similar because the linear dynamics is
dominant. When the perturbations increase, their recovering
times increase monotonously. As the initial perturbations
are 0.04 and 0.05, the maximal recovering time are 22 years
and 23.8 years, respectively. Most of perturbations have
approximately 20 years recovering time correspondingly.
The plot in the Figure 8b is similar to Figure 8a. As d = 0.1,
most of perturbations have approximately 25 –30 years
recovering time. However, there are still some perturbations
have short recovering time as long as several years when the
azimuth angle q are approximately 0 or p. The difference
between the freshwater and salinity flux perturbations
becomes remarkable in this figure. For instance, when d =
0.5, the maximal recovering time of salinity flux perturbation and freshwater flux perturbation are 37.2 years and
48.6 years, respectively. The perturbations related to
CNOP are always the slowest recovering perturbations
no matter what the initial magnitude is. The salinity flux
perturbations have different trends comparing to freshwater
ones as the increasing of initial magnitude, which is shown
in Figure 8c. It is clear that when the initial magnitude d >
0.6, most salinity flux perturbations have a saturated
recovering time of about 40 years. However, the freshwater
perturbations have a quickly increasing recovering time. For
instance, when d = 0.6, the most slowly recovering time is
about 53 years, and they are 67 years and 96 years when d =
0.7 and d = 0.79, respectively. These timescales are from
multidecadal to century scales. However, they cannot be
longer in this model. When the initial perturbations are
larger than a critical dc = 0.791, the freshwater perturbation
Figure 8. The plot of recovering time td versus initial azimuth angle q. The initial magnitude of
perturbations vary (a) from d = 0.02 to d = 0.05, (b) from d = 0.1 to d = 0.5, and (c) from d = 0.6 to
d = 0.79.
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Figure 9. The evolution of magnitude J of perturbations:
(a) salinity flux type with q = p/2 and (b) freshwater flux
type with q = 3p/2. The solid, dashed, dash-dot, and long
dashed curves represent d = 0.6, 0.7, 0.8, and 0.9,
respectively.
related to CNOP will develop quickly, and the thermohaline
circulation cannot recover back to the original state.
[24] The saturation of td in Figure 8c for the salinity flux
perturbation is quite interesting. To understand this
phenomena, the evolution of salinity type perturbations is
investigated by chosen different magnitude perturbations
with fixed q = p/2. It is clear from Figure 9a that the larger
the initial salinity flux perturbation is, the faster the
perturbation damps. When t = 10, the three different
perturbations are similarly small, even their magnitude are
relatively larger at the beginning. So that for the salinity flux
perturbation d = 0.8 > dc, it damps to 0.058 at t = 10, less
than 10% of initial magnitude. To compare this to the
freshwater flux perturbations, the evolution of CNOPs are
replotted in Figure 9b. The freshwater flux perturbations
grows faster as d increases. For the freshwater flux
perturbation d = 0.8 > dc, it increases to 3.9 at t = 10,
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near to 5 times of initial magnitude. These differences
don’t exist in linear case, which can be seen clearly from
Figures 2 and 3. In these cases, THC is unstable for large
freshwater perturbations, but may be still stable for even
very larger salinity perturbations. So the magnitude of
initial perturbation is not the only pivotal factor, the type
of initial perturbation is equally important. To our
knowledge this phenomenon has not been reported by
other authors and this result is new compare to previous
investigations. Now an interesting question is what is the
mechanism of this saturation? The answer is that the
nonlinear feedback distinguishes the freshwater perturbations from the salinity perturbations. The saturation of
recovering time is totally a nonlinear phenomenon, and
can be explained by nonlinear feedback of THC as
follows.
[25] Generally speaking, the larger the initial perturbation is, the longer the recovering time is. However, for
salinity flux type perturbations, there is negative feedback
due to nonlinearity (see Figure 4). The larger the initial
perturbation is, the stronger the negative feedback effects
is. So this negative feedback process will shorten the
recovering time and finally result in saturation of recovering time. If the nonlinear effect had been ignored, the
phenomenon of saturation of recovering time would have
disappeared.
[26] From the above results, the recovering time in the
simulation are most of decadal timescale, provided the
perturbation magnitude is less than dc = 0.791. It is about
20-year variation of THC under some relatively small
perturbations, which is d = 0.04 as estimated by [Lohmann
et al., 1996]. This timescale somewhat agrees with the SST
variability in central England [Plaut et al., 1995] and Great
Salinity Anomaly (GSA) in the Northern North Atlantic
[Dickson et al., 1988]. Bentsen et al. [2004] also indicated
that the simulated 20-year period variability with the fully
coupled Bergen Climate Model (BCM) is robust and real.
The variation of North Atlantic overturning could be as
larger as 5% [Goosse et al., 1999] to 20% [Wadley and
Bigg, 2002], so the magnitude of perturbation d in this study
is reasonable.
5. Summary and Discussion
[27] The mechanism of THC variation is still an open
problem for scientists. The transient amplification of initial
perturbations is important for thermohaline variability.
Within a simple two-box model, we have addressed the
conditional nonlinear optimal initial perturbations(CNOPs)
which lead to transient growth of THC. Those perturbations,
being of freshwater flux type, can cause decadal variations
of THC. Our new approach of CNOP employed in this
paper has two distinguish characters.
[28] First, CNOP can distinguish the freshwater flux type
perturbations from salinity ones (Figure 2). The linear
evolution is symmetric to q = p, while the nonlinear
evolution is asymmetric. The break of symmetric here is
due to nonlinearity of the circulation. The nonlinear feedback mechanism of THC was also explored. The nonlinear
feedback, depending on the initial type of perturbations, can
be positive or negative (Figure 4). That is also the reason
why the thermohaline circulation is more sensitive to fresh
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water flux perturbation. Through the nonlinear feedback of
THC, we can explain some results in global circulation
models. [Chen and Ghil, 1995; de Verdière and Huck,
1999; Te Raa and Dijkstra, 2002] clearly showed that the
increasing of horizonal mixing coefficient of heat can
stabilize the circulation. We show here that increasing of
horizontal advection (y0 > 0) can also stabilize the
thermohaline circulation according to Figure 4. This
means the stronger horizonal transport can damp the
amplification of the perturbation in the thermohaline
circulation. This nonlinear feedback mechanism and the
effects of nonlinearity on the variation of THC suggest
that there is possibility to link the climate change to THC
variations by investigations of large freshwater perturbations in the nonlinear regime.
[29] Second, the results of this paper also provide an
examination on whether the linear assumption is valid
[Lohmann et al., 1996; Huck et al., 2001; Tziperman and
Ioannou, 2002] by using a nonlinear model. [Tziperman
and Ioannou, 2002] assumed that current THC variability is
most likely of a fairly small amplitude (5 – 10% of the mean
value) and that linear dynamics approximation was valid.
Since the steady state of the thermohaline circulation
= (13.5 K,
S)
corresponding to present climate has (T;
1.5 psu), the perturbations of d < 0.075 have 5% maximal
variance of the steady state. For d < 0.15, they are 10% of
the steady state. It can be seen from Figures 3 and 5 that
the difference between linear and nonlinear cases is not
significant in this regime. So their approximation is valid
for these cases.
[30] We also examined the mechanism of thermohaline
variability due to transient growth of THC anomalies in a
simple meridional two-box model. The mechanism of the
decadal variations is due to the interaction of two nonorthogonal eigenmodes, which had been found by Lohmann
and Schneider [1999] and Tziperman and Ioannou [2002].
The difference between ours and theirs is that their transient
amplification decays in several years. It seems that the
mechanism in the model of Lohmann et al. [1996] is more
sensitive than that in the model of Tziperman and Ioannou
[2002]. Lohmann and Schneider [1999] have investigated
the decadal variability of THC with this model. Their
primary studies showed that some freshwater flux perturbation would lead to a decadal variation of the thermohaline
circulation, and the stochastic optimals have been analyzed
in the linear framework. We showed here that the optimal
perturbations must be freshwater flux type, and they could
induce decadal variations of THC, though their timescale
are of O(20) years or longer.
[31] One significant phenomenon found in this investigation is the saturation of recovering time td, which is
essentially nonlinear, and can be understood as saltadvection feedback [Marotzke, 1996; Mu et al., 2004].This
saturation demonstrates that the thermohaline circulation is
strongly stable under those salinity flux perturbations
(where q < p) comparing to the fresh water perturbations.
[32] The thermohaline circulation also plays a pivotal role
for understanding past climate variability. The glacial states
differ from present-day climate states and are weaker in the
strength of the thermohaline circulation [Lohmann et al.,
1996; Ganopolski and Rahmstorf, 2001; Prange et al.,
2002; Romanova et al., 2004]. In addition, it is more
C07025
sensitive for the past climate states than present ones, so
the nonlinear feedbacks would be an important factor.
Romanova et al. [2004] emphasized the importance of the
background hydrological cycle, which may provide a possible link between the low latitudes and the ocean circulation on paleoclimatic timescales. In order to understand
these linkages and feedbacks, more intense investigations
are required.
Appendix A
[33] The matrix A and vector b in equation (1) are defined
as
0
caT 0 q 0
B
V
A¼@
caS 0
p
V
1
0 ca 1
cbT 0
C
B V C
V
A; b ¼ @ cb A ðA1Þ
cbS 0 þ 0
V
V
where c = 17 106 m3s1, a = 0.15 K1, b = 0.8 psu1, p =
4.04 102, q = 3.67 103. In this paper, the
corresponding parameters in the equilibrium are (T0, S0) =
(13.5 K, 1.5 psu), 0 = 14 106 m3s1, the deep sea
box volume is V = 1.514 1014m3, and the parameters in
cb
dimensionless flow rate ca
V = 0.053125, V = 0.28333.
[34] Acknowledgments. We thank the reviewers for constructive
comments. This work was supported by the National Natural Scientific
Foundation of China (40233029 and 40221503) and KZCX2-208 of the
Chinese Academy of Sciences.
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M. Mu, LASG, Institute of Atmospheric Physics, Chinese Academy of
Science, Beijing 100029, China. ([email protected])
D.-J. Sun, L. Sun, and X.-Y. Yin, Department of Modern Mechanics,
University of Science and Technology of China, Hefei, Anhui 230027,
China. ([email protected]; [email protected]; [email protected])
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