Nonnormal Thermohaline Circulation Dynamics in a Coupled … · 2008-04-05 · Nonnormal Thermohaline Circulation Dynamics in a Coupled Ocean–Atmosphere GCM ELI TZIPERMAN AND LAURE

Nonnormal Thermohaline Circulation Dynamics in a Coupled Ocean–Atmosphere GCM

ELI TZIPERMAN AND LAURE ZANNA

Department of Earth and Planetary Sciences, and School of Engineering and Applied Sciences, Harvard University,Cambridge, Massachusetts

CECILE PENLAND

NOAA/ESRL/Physical Sciences Division, Boulder, Colorado

(Manuscript received 23 January 2007, in final form 19 July 2007)

ABSTRACT

Using the GFDL coupled atmosphere–ocean general circulation model CM2.1, the transient amplifica-tion of thermohaline circulation (THC) anomalies due to its nonnormal dynamics is studied. A reducedspace based on empirical orthogonal functions (EOFs) of temperature and salinity anomaly fields in theNorth Atlantic is constructed. Under the assumption that the dynamics of this reduced space is linear, thepropagator of the system is then evaluated and the transient growth of THC anomalies analyzed. Althoughthe linear dynamics are stable, such that any initial perturbation eventually decays, nonnormal effects arefound to result in a significant transient growth of temperature, salinity, and THC anomalies. The growthtime scale for these anomalies is between 5 and 10 yr, providing an estimate of the predictability time of theNorth Atlantic THC in this model. There are indications that these results are merely a lower bound on thenonnormality of THC dynamics in the present coupled GCM. This seems to suggest that such nonnormaleffects should be seriously considered if the predictability of the THC is to be quantitatively evaluated frommodels or observations. The methodology presented here may be used to produce initial perturbations tothe ocean state that may result in a stricter estimate of ocean and THC predictability than the commonprocedure of initializing with an identical ocean state and a perturbed atmosphere.

1. Introduction

The present-day North Atlantic Ocean exhibits mul-tidecadal variability in sea surface temperature (SST),possibly partially caused by the large-scale thermoha-line circulation (THC) variability (e.g., Bjerknes 1964;Kushnir 1994). Numerous studies have considered thestability and variability of the THC on these time scalesusing all ranges of model complexity from simple boxmodels to general circulation models (GCMs; e.g., Del-worth et al. 1993; Griffies and Tziperman 1995). Studiesof the North Atlantic predictability find possible decad-al predictability skill of the THC and North AtlanticSST (Collins et al. 2006; Griffies and Bryan 1997).

In previous studies of weather or ENSO predictabil-ity, “nonnormal dynamics” and “transient amplifica-tion” were found to be crucial (e.g., Farrell 1988, 1989;

Farrell and Ioannou 1996; Penland and Sardeshmukh1995; Kleeman and Moore 1997). These terms refer tothe property of damped (stable) linear systems, whichdisplay no variability unless forced externally, to pro-duce strong amplification of appropriate initial pertur-bations before eventually decaying back to the back-ground state. Such behavior occurs in linear systems inwhich the operator does not commute with its Hermi-tian transposed. In other terms, the system dx/dt � Axis nonnormal if AAT � ATA, which is the case in prac-tically all systems based on fluid dynamics equations.The nonnormal dynamics of the THC was investigatedin only a few recent papers, all using fairly simple mod-els (Lohmann and Schneider 1999; Tziperman and Io-annou 2002; Zanna and Tziperman 2005).

In this work, we study the possible transient amplifi-cation of the THC in the Geophysical Fluid DynamicsLaboratory (GFDL) coupled atmosphere–ocean GCMCM2.1 (Delworth 2006; Delworth et al. 2006; Gnan-adesikan et al. 2006; Stouffer et al. 2006; Wittenberg etal. 2006; Griffies et al. 2005). The study of nonnormal

Corresponding author address: Eli Tziperman, Harvard Univer-sity, 24 Oxford St., Cambridge, MA 02138.E-mail: [email protected]

588 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 38

DOI: 10.1175/2007JPO3769.1

© 2008 American Meteorological Society

JPO3769

dynamics and transient amplification in a coupledGCM introduces nontrivial difficulties. As a first ap-proximation, we construct an EOF-based reducedspace based on temperature and salinity fields from a2000-yr control model run. We then fit a linear model tothe reduced space dynamics following the “linear in-verse modeling” approach used in the context of ENSO(e.g., Penland and Sardeshmukh 1995; Penland 1996;Penland et al. 2000; Penland and Matrosova 1998, 2001;Blumenthal 1991; Moore and Kleeman 2001). In sec-tion 2 we introduce the methodology for constructingthe reduced space and the linear fit of the model fol-lowed by an analysis of the quality of the fit. We discussthe results of the nonnormal effects in section 3 andfinally conclude in section 4.

2. Methodology

To analyze the nonnormal amplification of the THCin the GFDL CM2.1, we first need to obtain a linear-ized version of the model dynamics. Owing to the dif-ficulty of deriving the explicit linearized equations ofthe full coupled GCM, we follow the “linear inversemodeling” approach developed in the context of cli-mate research by Penland and coauthors (e.g., Penlandand Sardeshmukh 1995; Penland 1996). A reducedspace representation of the dynamics is first created byexpanding some of the model variables into empiricalorthogonal functions and by using the principal com-ponent amplitudes as the state vector (Blumenthal1991). Although principal oscillation pattern analysis(Hasselmann 1988; Von Storch et al. 1988) is the firststep of linear inverse modeling, we are not as interestedin defining the dominant eigenvectors of the linearfeedback matrix as much as we are in finding that ma-trix itself.

The use of three-dimensional (3D) multivariateEOFs for describing the dynamics and for state spacereduction is supported by the recent work of Hawkinsand Sutton (2007). Using the Hadley Centre model,they showed that multivariate EOFs provide an effec-tive description for the THC variability and could as-sociate the mechanism for the multidecadal variabilityto an internal ocean mode, dominated by changes inconvection in the Nordic seas, and affected by salinitytransport from the Arctic and North Atlantic. An al-ternative to the 3D EOFs could be the use of dynamictopography, which provides an efficient way of repre-senting the gyre dynamics, as used in the THC predict-ability study of Griffies and Bryan (1997). From previ-ous studies on nonnormal growth of the THC, we ex-pect the compensation effects of the temperature andsalinity to play a significant role in the THC amplifica-

tion. Therefore, the use of dynamic topography, whichis based on density alone, is not expected to be optimalfor the purposes of this study. However, we still usedynamic topography as a useful diagnostic of the non-normal dynamics.

Given the reduced space representation, we fit a lin-ear matrix to the evolution of the reduced state vectorand analyze its nonnormal properties.

a. Reduced space for the GFDL model

We analyze the nonnormal dynamics of the “1860-control” integration of the coupled GFDL model ver-sion CM2.1. Three-dimensional temperature, salinity,and velocity fields are available for this 2000-yr-longcontrol run. The atmospheric model resolution is 2°latitude by 2.5° longitude with 24 vertical levels. Theocean model has a resolution of 1° by 1° poleward of30° latitude and progressively increased resolution inlatitude approaching the equator with 1⁄3° resolution atthe equator. The total number of vertical levels is 50,with 22 levels evenly spaced within the first 220 m fromthe surface. The reader is invited to consult the follow-ing papers and references within for a detailed descrip-tion of the model: Delworth (2006), Delworth et al.(2006), Gnanadesikan et al. (2006), Stouffer et al.(2006), Wittenberg et al. (2006), Griffies et al. (2005).Our main interest in this work is the variability of theNorth Atlantic THC. Its strength is measured by anindex defined as the maximum overturning streamfunc-tion occurring anywhere between 20° and 80°N. Thered line in Fig. 1 shows a time series of the THC indexdeviation from its mean [the mean is 23.9 Sv (Sv � 106

m3 s�1) and the standard deviation is 1.5 Sv]. Thepower spectrum of the THC index demonstrates itsvariability with two spectral peaks at about 20 and 300yr and a possibly a weaker peak at about 60 yr (Fig. 2,Delworth and Zhang 2007). We are interested in find-ing if nonnormal THC ocean dynamics plays a role inthis variability.

Our interest being the THC variability on an inter-annual time scale and longer, we consider annually av-eraged temperature and salinity fields in the North At-lantic region (10°–80°N, 20°E–100°W). The reducedspace representation of the GCM dynamics includesonly the 3D temperature and salinity fields. To a first-order approximation, we expect the velocity field rel-evant to the large-scale THC dynamics to be mostlygeostrophic (and determined by the temperature andsalinity fields) and, therefore, we do not explicitly in-clude the velocities in the state vector.

To reduce the dataset sizes and enable the EOF cal-culation, we subsampled the temperature and salinity

MARCH 2008 T Z I P E R M A N E T A L . 589

fields to a resolution of 3° horizontally (every thirdmodel grid point) and 13 levels vertically from 5 to 4950m (every third vertical level). The temperature and sa-linity anomalies were obtained by removing their timemean at each grid point. Let the dimensional tempera-ture (similarly for salinity) at a location i and time t beT*it , and a nondimensional temperature be Tit � T*it /Wi.The weighting factor Wi may be some relative grid vol-ume �Vi /V0 � (dxi dyi dzi)/V0, the standard deviation(SD) of the temperature and salinity at a given gridpoint i, �i, or any combination of the two, for example,

Wi � �i ��Vi �V0�. �1�

Note that weighting by the SD tends to make all depthlevels more equal, eliminating the dominance of themore energetic surface. Weighting by the grid elementvolume as in the last equation tends to emphasize thedeeper ocean, where grid boxes are larger. Severalanalyses using different such combinations are dis-cussed below. Moore and Kleeman (2001) have shownthat, in order to capture most of the relevant dynamicsfor the analysis of optimal perturbations, the EOFsshould be evaluated from the correlation matrix (or,equivalently, the covariance matrix of the temperatureand salinity normalized by their SD) rather than fromthe dimensional covariance matrix. In addition, beforecalculating the EOFs, the temperature and salinityfields were smoothed by three passes of a local runninghorizontal average operator (no temporal smoothingwas used beyond the annual averaging). The smoothingresults in more easily interpretable 3D structures of theEOFs although there is the danger of misrepresentingsmall-scale critical dynamics (e.g., the Labrador Seaconvection in the current context). It appears that non-smoothed fields result in equally nonnormal operatorsand comparable transient amplification. Therefore, letthe subsampled normalized smoothed temperature and

salinity (T, S) at all grid points of the model at a giventime tn be represented by a single vector Y(tn). Themultivariate 3D EOFs of the normalized temperatureand salinity are then calculated as the eigenvectors ofthe covariance matrix C � YYT, where the angledbrackets denote time average.

The 3D EOFs generally represent large-scale spatialstructures (e.g., Figs. 3, 4), which depend, of course, onthe weighting used. Using normalization by the gridvolume and by the standard deviation at each grid pointtends to emphasize the deep variability, as seen in Fig.3 for the first EOF. The THC depends on both the deeptemperature and salinity as well as on their near-surfacesignal. The correlation coefficients between the differ-ent principal components and the THC index time se-ries (Fig. 5) show that the fifth EOF, for example, playsan important role in setting the amplitude of the THC,and its structure (shown in Fig. 4) is indeed more sur-

FIG. 2. Power spectrum of the THC index time series anomalyobtained from the GFDL coupled GCM control run. The grayshaded region indicates the 95% confidence interval.

FIG. 1. Time series of the GCM THC index anomaly in red [where the THC index is definedas the value of the maximum overturning streamfunction (Sv) occurring anywhere between20° and 80°N, and its anomaly is the deviation from its time mean], and the reconstructed THCindex, in green, calculated using the regression coefficients from the temperature and salinityPCs.


Fig 1 live 4/C

face trapped. EOF 9 (not shown) is also strongly cor-related with the THC, yet is not as surface trapped asEOF 5. Changing the weighting factor Wi used to nor-malize the temperature and salinity results in differentEOFs representing this surface-trapped signal and,therefore, in different principal components (PCs) be-ing correlated with the THC index. However, as we willsee below, some interesting conclusions regarding thenonnormal dynamics of the THC may be drawn thatare not sensitive to the weighting used.

The EOFs are found using the smoothed data fields;however, the principal components are found by pro-jecting the nonsmoothed fields on the EOFs. Effec-tively, this is a partial smoothing of the data that dis-cards the noisy part of the data that does not projectonto the first few EOFs kept for the analysis. By cal-culating the PC time series for the first N EOFs, weobtained a reduced space representation of the variabil-ity, and a new N-dimensional state vector P(tn) � Pn.An examination of the PC time series shows a nonneg-

ligible trend in some of the PCs during the first 500 yr.Therefore, we removed the first 500 yr of the data andonly use the final 1500 yr in the following analysis. Wenote that using EOFs as the basis for a reduced spacereconstruction is not necessarily the most efficient ap-proach for the purpose of extracting the nonnormaldynamics because EOFs, in general, are optimized forrepresenting the maximum variance rather than the op-timal initial perturbations that may lead to large vari-ability (Farrell and Ioannou 2001). We shall briefly re-turn to this issue later.

b. Linear inverse model and propagator

Assuming that the dynamics of reduced space aremostly linear and driven by white noise, we can esti-mate the propagator B (�0) of the evolution of the statevector from time n to time n � �0 ,

Pn��0� B��0�Pn, �2�

FIG. 3. EOF 1: vertical cross section of the zonally averaged nondimensionalized (a) temperature and (b) salinityas function of latitude and depth; horizontal section at a depth of 45 m of (c) temperature and (d) salinity as afunction of longitude and latitude; (e) and (f) as in (c) and (d) but at a depth of 1364 m.


Fig 3 live 4/C

by minimizing the variance of the residuals given byr� 0

� Pn�� 0� B(�0)Pn. The propagator for an arbitrary

lead time � is then calculated as B(�) � B(�0)�/� 0. Theassumption that the evolution of the principal compo-nent vector, Pn, may be described by linear dynamics isanything but obvious. The GCM equations are nonlin-ear, and it is more than possible that nonlinear effectsplay a role as well. We nevertheless proceed, keeping inmind this caveat.

The propagator B(�) depends, in principle, on the lagused during the fitting process, �0. We therefore usedifferent values of �0 in order to evaluate the propaga-tor in addition to trying different weighting factors Wi

and different choices for the number of EOFs N used toconstruct the reduced phase space. All of the experi-ments are summarized in Table 1. Unless noted other-wise, the results plotted are from run 1 for which weused 25 EOFs, a lag time of 1 yr, and a weighting byboth grid volume and SD.

Once the matrix B(�) is available, it is straightfor-ward to study its nonnormal dynamics, including opti-mal initial conditions and transient amplification. We

start by defining a norm by which the magnitude of thestate vector is evaluated. Let X be an N N positivedefinite symmetric matrix to serve as the “norm ker-nel.” The norm of the state vector is then given by

�P� �X2 � P�

TXP� � �B��P0�TXB��P0. �3�

Looking for the unit norm optimal initial conditions,P0, which maximize the norm of the state at time �,

FIG. 5. The correlation coefficients between the THC indexanomaly and the PC amplitudes.

FIG. 4. As in Fig. 3, but for EOF 5.


Fig 4 live 4/C

||P� ||2X, leads to the following generalized eigenvalueproblem for the optimal initial conditions (Farrell 1988;Tziperman and Ioannou 2002):

B��TXB��P0 � �XP0. �4�

We will consider two different norms: First, the energynorm, where X is the identity matrix, and then a normkernel that measures the amplitude of the THC. Toconstruct the THC norm kernel we first find the regres-sion coefficients RTHC between the PCs and the THCindex by minimizing the sum (over time) of the squareof the residuals

r�tn� � THC�tn� � RTHCT PC�tn�. �5�

The correlation between the THC index and the THCreconstructed using the regression from the PCs is ap-proximately 0.88, indicating that the PCs are an appro-priate state vector for studying the THC dynamics, andthe two are shown in Fig. 1. The good correlation be-tween the THC linearly reconstructed from the PCsand the original THC index indicates that the THC islinearly related to the principal components and henceto the temperature and salinity fields. While this is en-couraging as far as our linear framework is concerned,it still does not assure us that the PC dynamics arelinear as well and may be described by a linear modelsuch as (2).

Given the regression coefficients RTHC, we define theTHC norm kernel matrix as

XTHC � RTHCRTHCT . �6�

The correlation coefficient for each PC is shown in Fig.5. When using this norm kernel in (4), we attempt tofind the initial conditions in the reduced EOF spacethat maximize THC2(t � �) subject to the condition thatTHC2(t � 0) � 1.

Before proceeding with the analysis of the fit, weshould mention that this norm kernel is singular, beingdefined using a single vector, and it therefore requiresregularization and careful interpretation (Tziperman

and Ioannou 2002; Zanna and Tziperman 2005). Theregularization is done by adding a regular norm kernelmultiplied by a small number. That is, XTHC is replacedby XTHC � � I, for example, if the identity matrix isused. The largest eigenvalue of the nonregularizedproblem is infinite, and the corresponding initial THCis zero. As � approaches zero, the eigenvalues of theregularized problem approach infinity, but the eigen-vectors (optimal initial conditions) do not change aslong as � is not too small to cause numerical problems.However, the eigenvectors are sensitive to the form ofthe added norm kernel. If we add a matrix other thanthe identity matrix, the eigenvectors will be different.This is not surprising, given that optimal initial condi-tions are, in general, sensitive to the norm kernel used.The regularizing norm kernel constrains, in a sense, thenull space of the THC singular norm kernel, and thusthe resulting eigenvectors depend on its choice. Over-all, it seems that the resulting eigenvectors have a largescale, seemingly physical structure, and they result ingrowth of the THC as expected. We feel that the ap-proach used here of using the regular energy norm ker-nel on the one hand, and the singular but regularizedTHC norm kernel on the other, provides a useful in-formation on the nonnormal THC dynamics. An alter-native approach to dealing with this singularity, basedon using a different norm (L� rather than L2) is ex-plored in Zanna and Tziperman (2008).

c. Quality of fit

Before studying the nonnormal dynamics of the lin-ear propagator, we need to verify that our fit does notviolate any of our assumptions and that the results arenot overly sensitive to the different parameters used. Ingeneral, we found the linear fit (2) to the GFDL CM2.1to be relatively well behaved and to result in a physi-cally consistent propagator for � � 2 yr (for N � 20, 25,or 30 EOFs).

More specifically, a physical linear propagator of astable linear system needs to have eigenvalues with

TABLE 1. A summary of the different calculations used to linearly approximate the dynamics of the GFDL CM2.1 large-scale NorthAtlantic temperature and salinity variability (y: yes, n: no). The last line shows the percentage of T, S variance explained by the numberof EOFs retained in each experiment.

Parameters

Case

1 2 3 4 5 6 7 8 9 10 11 12 13

No. of EOFs 25 25 25 20 30 25 25 25 20 30 25 25 30Lead time �0 1 2 3 1 1 1 2 3 1 1 1 1 2Weight by SD y y y y y n n n n n y n yWeight by dv y y y y y y y y y y n n yVar T, S explained 99 99 99 96 99 97 97 97 93 99 95 97 99


magnitude smaller than one and with a positive realpart. Using N � 25, the smallest real part of the eigen-values of B (�0) for �0 � 1 yr is 0.77. The lag time �0 �2 yr still provides physical results, with smaller eigen-values reflecting the further decay of the modes in theadditional year. With �0 � 3 yr, two of the eigenvaluesturn out to have very small negative real parts, which isunacceptable and indicates that the fit has failed be-cause the short decay time of some of the modes cannotbe resolved using a 3-yr lag time.

The quality of the fit can be further examined bycomparing the predictability using the fitted model fordifferent values of the lag time � to the expected pre-dictability of a perfect linear model driven by whitenoise, and to the predictability of an autoregressivemodel O(1) (AR1). The latter is obtained by fitting adiagonal B to (2). In addition, the dynamics should beindependent of the value of �0 used to obtain the propa-gator (“tau test,” Penland 1996).

Consider now the prediction error covariance matrixfor a perfect model. Let Pn be the actual state derivedfrom the GCM EOFs. The residuals of the predictionare given by

r� � Pn�� B��Pn, �7�

and the prediction error covariance can be shown (Pen-land 1989) to be

C � rnrnT � �Pn�� BPn��Pn��

T � PnTBT�

� CP � BPnPn��T � Pn�� Pn

TBT � BCPBT

� CP � BPnPnTBT � BPnPn

TBT � BCPBT

� CP � BCPBT, �8�

where CP is the covariance matrix of the actual statevector from the GCM and is given by

CP � PnPnT � Pn�� Pn��

T . �9�

This “perfect model” prediction error covariance Cgiven in (8) is compared to the actual prediction errorcovariance obtained by predicting the PC state vectorusing the fitted linear propagator calculated from (2) aswell as to the prediction error covariance obtained fromthe AR1 model. The measure to be compared may bethe trace of the error covariance matrix (Fig. 6b), or theindividual diagonal elements (Figs. 6c–i). The predic-tion skill of our fitted linear model is generally betterthan that of the AR1 model (which is a stricter test thanthe commonly used comparison to persistence), al-though not as good as a perfect linear model is expectedto be. We conclude that the fitted linear model is auseful approximate representation of the large-scale T,

S dynamics in the GCM. This does not rule out a rolefor nonlinear processes, but indicates that the linearframework may, at least, be expected to be useful.

A similar derivation applies to the covariance of theTHC prediction residuals. Define the residuals as

r�THC � THCn�� RTHC

T BPn, �10�

their covariance, assuming a perfect linear model aswell as that the THC is perfectly predicted from the PCamplitudes, THCn � RT

THCPn, is then

CTHC � �r�THC�2 � THCn

2 � RTHCT BCPBTRTHC.

�11�

Figure 6a shows the prediction error covariance forthe THC based on the fitted linear model, for an AR1model, and for a perfect linear model (11). The fittedlinear model does better than the AR1 model for thefirst 10 to perhaps even 20 yr, beyond which there is notmuch of a skill. If we compare this result to Fig. 6b,which shows the trace of the prediction error covari-ance for the PCs, we see that the fitted linear model forthe PC components does better than the AR1 modelfor a longer time. We note that the assumption of alinear Markov model may not be a completely accuraterepresentation of the coupled model North Atlantic dy-namics and can lead to some of the prediction errorsobserved. Some PCs (e.g., PC1, Fig. 6c and upper panelof Fig. 7) are better predicted than others (e.g., PC5,Fig. 6g and third panel of Fig. 7). Note that the THC isstrongly correlated with PC5, which is not well pre-dicted beyond 10 to 20 yr, like the THC itself. This maybe a good place to note that PC1 and PC2 seem signifi-cantly correlated at a nonzero lag, which may be a re-flection of the oscillation mechanism and could be fur-ther explored, as done, for example, in Fig. 5 of Haw-kins and Sutton (2007).

d. Sensitivity

It is convenient to examine the sensitivity of our re-sults to the various assumptions made by examining, foreach of the experiments summarized in Table 1, themaximum amplification, �, of ||P(�)||2X as function of �calculated using (4). These maximum amplificationcurves are shown in Figs. 8 and 9 for the energy normand the THC norm respectively.

Figure 9 shows that the maximum THC amplificationoccurs between 5 and 7 yr. This result seems robust toan increase in the number of EOFs N (e.g., cf. runs 4, 1,and 5 or run 1 with run 13 in Fig. 9). Similar results areobtained if we fix both the number of EOFs and theweighting and vary the lag time �0 (e.g., cf. runs 3, 2, and


1 in Fig. 9). Some dependence of the time of maximumamplification on the weighting Wi is also seen on thisplot (see in particular run 12, which uses no weightingby SD or grid volume).

The maximum amplification curves using the energynorm, representing the predictability of the PC compo-nents rather than of the THC, are shown in Fig. 8.These curves show again a rapid increase in maximumamplification during the first 10 yr or so. Beyond thispoint, there is a wide spread of the curves, dependingmostly on the weighting strategy used.

The smallest amplification, which occurs about 7 yrafter the initial conditions, occurs when no weighting isused or when the weighting is based only on the gridvolume or only on the SD of the temperature and sa-linity variability, but not on both. Using a weightingfactor that depends on both grid element volume andSD results in the largest transient amplification. Theimportance of weighting by the standard deviation wasalso pointed out in the context of ENSO Moore andKleeman (2001). In addition, the results are sensitive to

the number of EOFs used for the reduced state recon-struction. Using 30 EOFs results in the largest amplifi-cation, while 20 and 25 EOFs results in significantly lessamplification. Unfortunately, we cannot use a largernumber of EOFs due to the short time series available.With 1500 yearly snapshots, we can produce that manyequations for the coefficients of the linear model. Using30 EOFs results in a linear propagator matrix B whosedimensions are 30 30, implying 900 elements thatneed to be solved using these 1500 equations. The re-sulting conditioning number for the coefficients of B isreasonable (less than 100), but a further increase in thenumber of EOFs will quickly result in more unknownsthan equations and therefore in an ill-posed problemfor the coefficients of the propagator. It may be pos-sible to use the monthly model output (after removingmonthly climatology) rather than the yearly averagedfields. But, while this will increase the number of equa-tions, it may not add independent information for con-straining the propagator on decadal time scales, inwhich we are interested. In addition, using monthly

FIG. 6. Quality-of-fit test for run 1 showing the error covariance matrices as a function of forecast lead time fora perfect linear model [Eq. (8), dotted], for an AR1 model (dashed) and for the fitted linear model (solid). (a) TheTHC error covariance; (b) the trace of the error covariance matrix; (c)–(i) first few diagonal elements of the errorcovariance matrix, corresponding to PCs 1 to 7.


Fig 6 live 4/C

data may require making the propagator monthly aswell to represent the additional seasonally dependentanomaly dynamics, and for these reasons we have cho-sen not to explore this direction.

There is also some sensitivity to the lag time �0 usedin (2) to calculate the fitted propagator B (�) (e.g., cf.curves corresponding to runs 1–3 in Fig. 8). For largeenough lead time � in Fig. 8 the maximum amplificationdecays to zero, as expected. The eventual decay timescale of the temperature and salinity anomalies andthen of the THC is influenced by the decay time scale ofthe slow decaying modes found to be on the order of1200 yr for the slowest mode). When using a lag time of�0 � 1 or 2 yr to evaluate the propagator [Eq. (2)], itseems likely that such a long time scale for the slowdecaying eigenmodes would not be well resolved, andwe therefore consider this part of the fit less reliable. As

a result of these sensitivities, in particular to the num-ber of EOFs used, the amplification factors found usingthe energy norm should be viewed as a lower bound onthe expected nonnormal amplification rather than aprecise estimate. Also, as mentioned above and dis-cussed in Farrell and Ioannou (2001), using the EOFsfor the reduced space reconstruction is not optimal andmay again result in an underestimate of the amplifica-tion, although some of this problem may have beencorrected by the use of the normalized temperature andsalinity as discussed above, following Moore and Klee-man (2001).

Some of the variations between the different sensi-tivity runs may be rationalized as follows: The largestgrowth in the energy norm is obtained for the largestnumber of retained EOFs and when the EOFs are nor-malized by the standard deviation. This underscores the

FIG. 7. The 1500-yr time series of different nondimensional principal components from the GCM, togetherwith short segments trying to predict each of them using our linearized model: (top) PC 1, (second) PC 2,and (third) PC 5; the segments are plotted every 10 yr and extend for 150 yr each. (bottom) A 500-yr portionof the THC time series (Sv) with segments plotted every 10 yr and extend for 10 yr each.


Fig 7 live 4/C

importance of the stochastic forcing by low-variance,unresolved nonlinearities that are weighted up by thenormalization to the evolution of the THC. Also, themost variation among the runs with �0 � 1, 2, and 3 yris found when the EOFs are normalized by the standarddeviation. This is evidence that these unresolved non-linearities are the reason for failing the � test. In con-trast, the maximum amplification curves for runs 6–10(dashed curves in Fig. 8), for which no normalization bythe SD is done and which therefore does not includethe low-variance unresolved nonlinearities, pass the �test (i.e., similarity of results for different values of �0)reasonably well in the energy norm. Thus, it seems thatthe larger resolved scales behave more linearly than dothe small unresolved scales. When the PCs are normal-ized, the small scales dominate the growth, and thisgrowth decreases with �0 (cf. runs 5 and 13 in Figs. 8 and9). The fewer EOFs are retained, the more signal istreated as unpredictable stochasticity. The linear modelparameterizes nonlinearities as a linear part plus a sto-chastic part. When the fast nonlinear signals are in-cluded in the mix of EOFs and �0 is small enough toresolve their dynamics, they show up as added growth,essentially by putting a best-fit linear model to the non-linear part.

One important sensitivity, which is not exploredhere, has to do with the model domain, which wassomewhat arbitrarily chosen as small as possible here,to minimize the problem size and enable the calculationof the EOFs. The results may be affected by the choiceof the domain. For example, if the domain was to in-clude the Arctic Ocean, the EOFs and the nonnormalgrowth effects may have been different, given that sev-eral GCMs have found significant variability in the Arc-

tic possibly affecting the THC (e.g., Jungclaus et al.2005).

We conclude that the fitted linear dynamics and de-duced transient amplification depend on some of thechoices we make during the fit. The interpretation be-low would need to carefully consider these sensitivitiesto some of the fit parameters before proceeding with aphysical interpretation and conclusions. We will seethat some of the sensitivities indicate possible limits toour analysis due to the available model output andother factors. However, other sensitivities, especially tothe weighting used, reflect actual physical effects thatshould be expected, rather than a problem with our fit.

3. Results

a. Maximum amplification curves

As seen above, the maximum amplification using theenergy norm occurs after 7 to 50 yr, depending on theassumptions used during the construction of the linearfit (Fig. 8). The nonnormal amplification of initial tem-perature and salinity anomalies under the energy normis quite impressive. Figure 8 shows an amplification ofinitial anomalies in the PC space by a factor of 5–10within 8–40 yr. All curves show a rapid amplificationduring the first 10 yr followed by a wide spread of be-havior of the amplification curves beyond this timescale. The amplification curves from all experimentsusing the THC norm kernel (Fig. 9) have a more uni-form overall appearance. All such amplification curvespeak after about 5–8 yr and then decay fairly rapidly.

The rapid amplification during the first decade seenin both the THC norm and the energy norm indicatesthat temperature, salinity, and THC anomalies are rap-idly amplified during this initial period. From a predict-

FIG. 8. Maximum amplification �, as calculated from Eq. (4), asfunction of � for all runs listed in Table 1 using the energy norm.

FIG. 9. As in Fig. 8, but for all runs using the THC norm kernel.


Fig 8 9 live 4/C

ability point of view, this means that errors in the initialconditions, which partially project on the most rapidlygrowing modes, are expected to rapidly grow as wellduring these first 10 yr, implying that much of the pre-dictability skill is expected to be lost over this timeinterval. Even though the amplification factors for theTHC and energy norm kernel (Fig. 8) are more or lesssensitive to the different parameters used, it is impor-tant to note that all curves show a rapid growth phaseduring the first 10 yr. It seems, therefore, that the rapidgrowth of anomalies and expected loss of predictabilityof the THC, temperature, and salinity on a time scale ofabout 8 yr is a robust result of this study.

The actual amplification values for the THC normkernel are large (amplification factor of a few hun-dreds). In the simpler model studies of nonnormal THCamplification (Tziperman and Ioannou 2002; Zannaand Tziperman 2005), the amplification of THC wasfound to be infinite as the initial THC anomaly inducedby the optimal initial conditions for the temperatureand salinity typically vanished. This is the result of acancellation of the initial temperature and salinityanomalies and occurs mathematically due to the singu-larity of the THC norm kernel (Tziperman and Ioan-nou 2002). The large THC amplification seen in thisstudy is due to the same effect.

b. Predictability

The fitted linear model is able to successfully predictthe THC up to about 7 yr. Predictability of both tem-perature and salinity and of the THC seems limited bythe initial 5–7-yr-long period of fast amplification seenin Figs. 8 and 9. These figures show that an optimalinitial perturbation will be amplified by a significantfactor within a period of 5–7 yr. Thus, errors in theinitial state that project on the optimal initial perturba-tions may result in a deviation of the predicted fieldsfrom their true evolution within about 5–7 yr. Whilethere is some predictability on a short time scale of afew years, the linear model is unable to predict thelonger-term oscillations corresponding to the spectralpeak of the THC index at 200–300 yr. Interestingly, thelinear model does well predicting the first and even thesecond EOFs (Figs. 6c,d and 7a,b), which have longertime scales. But this does not translate into a corre-sponding skill in predicting the slow component of theTHC oscillations (as shown by the comparison to theAR1 model in Fig. 6a). The reason for this short pre-dictability time may very well be the nonnormal ampli-fication of initial anomalies to the temperature, salinity,and hence THC. We conclude that the predictabilityof THC anomalies seems to be limited to the order of5–7 yr.

c. Optimal initial conditions and evolution tomaximally amplified state in reduced space

The optimal initial conditions and their evolution inPC space for the energy norm and the THC norm areplotted in Figs. 10a and10c. The corresponding THCanomaly is shown in Figs. 10b and 10d. Under the en-ergy norm, PC 2 is significantly amplified, as are someother PC amplitudes. The THC norm, however, leadsto the growth of PCs 2, 5, 8, and 9, some of which arestrongly correlated with the THC index as seen in Fig.5. As expected, the initial THC anomaly vanished un-der the THC norm kernel (Fig. 10d), but not under theenergy norm (Fig. 10b).

Under the THC norm kernel, the optimal initial con-ditions lead to a decaying oscillation of the THC (Fig.

FIG. 10. Evolution in PC space of the fitted linear model fromoptimal initial conditions and the corresponding THC anomalyfor (a), (b) the energy norm and (c), (d) the THC norm. Nondi-mensional units.


Fig 10 live 4/C

10d). The maximum growth seems to occur at the firstmaximum of the oscillation. One wonders if the non-normal analysis simply calculates initial conditions thatexcite this oscillation and that nonnormal effects do notactually play a role here. However, examining the pro-jection of the optimal initial conditions on the eigen-modes of the propagator B we find that many morethan just a single oscillatory mode are involved in theoptimal initial conditions, indicating that nonnormal ef-fects do play a role.

The optimal initial conditions under the energy andTHC norms project differently on the PC state vectorand therefore involve different spatial structures of thetemperature and salinity initial conditions as dictatedby the corresponding EOFs. The evolution from theoptimal initial conditions seen in the Fig. 10 are verydifferent again, reflecting a possibly different amplifi-cation dynamics. That is, different physical processesare responsible for the growth in these different cases.This is consistent with the studies using simpler models,where the physical processes may be more clearly elu-cidated (e.g., Zanna and Tziperman 2005). In these sim-pler studies one can attribute the growth to the advec-tion of temperature or salinity anomalies by the meancirculation or the advection of the mean temperature orsalinity by the anomalous circulation, etc. Our currentapproach, constrained by the GCM complexity to usinga reduced state vector, does not allow us to point tosuch specific physical processes.

d. Optimal initial conditions and maximallyamplified states in physical space

The amplification of the optimal initial conditionswhen using the energy norm involves some significantredistribution of temperature and salinity anomalies inspace, as can be seen by examining the evolution fromthe optimal initial conditions to the state at time ofmaximum amplification (Fig. 11). The plotted densityanomalies �� are calculated from the temperature andsalinity anomalies T�, S� corresponding to the optimalinitial conditions or amplified state using the long-termaverage temperature and salinity in the GCM solution,T0, S0, using the nonlinear equation of state such that�� (T0 � T�, S0 � S�) � �(T0, S0). The initial stateis very much surface trapped, while the evolved state,already at year 6 or 8, involves temperature and salinityanomalies that are spread over a wider depth range.This provides some physical insight regarding themechanism of amplification and decay of the optimalperturbations. Note that when weighting by neithergrid volume nor SD (runs 6–12, Fig. 8), the perturba-tions decay fast, after about 20 yr. With weighting byboth grid volume and SD [Eq. (1), runs 1–5 and 13, Fig.

8], the amplification takes a longer time and the decayis much slower. The reason for this is rather simple.When the weighting includes these two factors, it tendsto emphasize the deep ocean. This means that the op-timization is required to produce initial conditions thatlead to a growth of temperature and salinity anomaliesin the deep ocean. Such anomalies take a bit longer todevelop, hence the slower growth time to maximumamplification. The deep anomalies take, of course,much longer time to dissipate than surface anomalies,hence the much longer decay time seen for the runswith weighting by both SD and grid volume. The math-ematical machinery of the transient amplificationclearly results in physically sensible solutions that areconsistent with our intuition of the ocean dynamics,providing another indication that the results are likelyto be robust in spite of our many simplifying assump-tions.

It is interesting in this context that the overall struc-ture of the amplification of the THC anomalies as afunction of time is not as sensitive to the weighting andtends to decay after 10–20 yr for all weighting optionsused. This suggests, perhaps, that the THC mostly de-pends on the near-surface (upper 1 km) temperatureand salinity anomalies and is therefore not affected bythe long-lived deep temperature and salinity anomaliesdiscussed above.

In the horizontal plane, the optimal initial conditionsshow a warm and salty anomaly in the Labrador Seathat results in a dense anomaly there. This anomalypossibly results from some role played by the LabradorSea deep convection and its known effect on NorthAtlantic Deep Water and the amplitude of the THC. Inthe amplified state, after 27 yr (Fig. 11) the anomaly inthe Labrador sea vanishes and appears at 45°N in thewest Atlantic. It is also interesting to see the strongdensity anomaly that develops north of 60°N at alldepth levels, driven mostly by the salinity anomaly.Overall the structure of the optimal initial conditions isvery different from that of the maximum amplificationstate, indicating some complex dynamics that does notinvolve growth of anomalies in place but rather someadvection and nonlocalized growth.

Although the maximum amplification of the energynorm occurs after 27 yr, the temperature and salinityevolutions show a clear development of strong pertur-bations already by year 6, consistent again with thepredictability time scale for the THC. It seems that thetime scale for nonnormal growth in this model is quiterobustly 6–8 yr regardless of the norm kernel used.

The temperature anomaly for the optimal initial con-ditions and the maximally amplified state after 6 yrunder the THC norm (Fig. 12) are both states charac-


FIG. 11. Evolution from optimal initial conditions using the energy norm kernel leading to a maximum amplification after 27 yr. Eachrow corresponds to a different time in years as noted in the panel titles. From left to right in each row: temperature at 45-m depth;salinity at 45 m; vertical cross sections of the zonally averaged temperature, salinity, and density; and surface dynamic topographyrelative to 1364 m. Nondimensional units; domain boundaries as in Fig. 3.


Fig 11 live 4/C

FIG. 12. Evolution from optimal initial conditions, as in Fig. 11, except using the THC norm kernel leading to a maximum THCamplification at 6 yr.


Fig 12 live 4/C

terized by a dipole structure between 40° and 60°N,although the sign of the anomalies in the dipole re-verses during the time evolution. The salinity shows anear-surface signal in the optimal initial conditions thatdevelops into a significant anomaly in the amplifiedstate, although it seems that the density anomaly in theamplified state, ultimately responsible for determiningthe velocity field and therefore the THC amplitude, isdominated by the temperature anomaly rather than thesalinity.

The physics of THC oscillations in some GCMs hasbeen shown to involve gyre changes as well (e.g., Del-worth et al. 1993), although these gyre changes may notbe essential to the very existence of these oscillations(Griffies and Tziperman 1995). Understanding theTHC in the currently analyzed model to the degreeanalyzed in the above papers or by Hawkins and Sutton(2007) is not within the scope of this paper, but we dowant to gain some insight into the growth mechanism.Following Griffies and Bryan (1997) we use dynamictopography in order to identify some aspects of thedynamics (rightmost panels in Figs. 11 and 12). Theoptimal initial conditions under both norms show aweak dynamic topography signal off Newfoundland.This later develops into a rather different yet strongsignal for the two different norms (cf. year 10 in Figs. 11and 12). The gyre signal decays before the maximumamplification time of 27 yr in the energy norm, butseems to be getting stronger after the maximum ampli-fication time of 6 yr in the THC norm. These observa-tions raise the possibility that the gyre is involved in thegrowth as well as decay of the perturbations. The dif-ferent evolution in the horizontal plane under the twonorms again suggests different physical processes in-volved. Further understanding of the role of the hori-zontal gyres would need to await a GCM-based analysisrather than the reduced space description used here.

e. Discussion

One of the more interesting insights here has to dowith the difference from previous studies using simplermodels. As explained above, a singular THC norm ker-nel led to optimal initial conditions that correspond toa vanishing THC index, as is the case here to a goodapproximation. In those simpler, zonally averagedmodels (Tziperman and Ioannou 2002; Zanna andTziperman 2005), the THC was related to the meridi-onal density gradients in some very simple way, follow-ing the usual Stommel (1961) formulation. This meansthat the density gradients also vanish or satisfy somesimple condition that leads to a vanishing initial THC.However, in the coupled GCM analyzed here, the THCis clearly not that simply related to the density field, as

the optimal initial state for the density, shown in Figs.11 and 12, is characterized by nontrivial density gradi-ents. That the initial THC anomaly corresponding tothe optimal initial conditions for the temperature andsalinity is still very small, as implied from the largeamplification factors, is seen in Fig. 9.

As in simpler models, our analysis is able to findinitial conditions for the temperature and salinityanomaly for which the THC nearly vanishes, but whichlead to a growth at later times due to the differentevolutions of the temperature and salinity anomalies. Itis encouraging to see the nonnormal amplification op-erating both in the GCM and in the simpler models.However, the above description of the optimal initialconditions and maximum amplified state is unsatisfac-tory, as we cannot explain which physical process act tocreate the amplification as was possible using the sim-pler models. One may hope to be able to repeat theanalysis here using the actual linearized GCM equa-tions, which will enable a more explicit and satisfactoryphysical interpretation.

One thing that we can conclude from our analysisdoes shed some new light on the dynamics of THCvariability. The small amplitude of present-day THCvariability in GCMs seems to suggest that the dynamicsof this variability are mostly linear. Griffies and Tziper-man (1995) and others suggested that the THC vari-ability may be described as the excitation of a dampedoscillatory mode of the THC by stochastic atmosphericforcing. Our reduced space model does, indeed, haveseveral damped and several damped oscillatory modes.The typical decay and oscillation time scales vary fromdecades to hundreds of years. However, we find herethat the dynamics of transient amplification of the THCinvolve the interaction of more than a single mode(such interaction is necessary for transient amplifica-tion to occur). Thus the picture of how THC variabilityis excited by the atmosphere may be more complexthan previously thought.

4. Conclusions

We investigated the amplification of temperature, sa-linity, and THC anomalies, in the GFDL coupled at-mosphere–ocean general circulation model CM2.1, dueto nonnormal dynamics. The analysis was done by firstconstructing a reduced space based on EOF decompo-sition of temperature and salinity anomalies in theNorth Atlantic. Instead of describing the evolution ofthe three-dimensional temperature and salinity fields,the problem was reduced to the evolution of the am-plitudes of some 25 principal components (EOF ampli-tudes) and then to the fitting of a linear model (propa-


gator matrix) to the evolution of these principal com-ponents.

By analyzing the resulting linear model, we foundthat the nonnormal dynamics can lead to a significantgrowth of temperature, salinity, and THC anomalieswith a typical growth time of 5–10 yr. This result leadsto a possible order of magnitude estimate of predict-ability time for the THC in the model. Our analysis waslimited by the length of the control integration used tocalculate the fitted linearized model (2000 yr). It seemsthat a longer integration should allow the use of moreEOF amplitudes in the reduced state reconstruction,and our results indicate that this will likely result ineven stronger transient amplification. We thereforeview our results as a lower bound on the nonnormalityof THC dynamics in this GCM. This seems to suggestthat such nonnormal effects should be seriously consid-ered if the predictability of the THC is to be quantita-tively evaluated from models or observations. Weshould mention that the results found in this workmight be different if we were to consider the entireAtlantic basin instead of only the North Atlantic, assuggested by some numerical GCM experiments.

The suggested importance of nonnormal THC dy-namics to THC predictability means that one may needto use ensemble runs for THC prediction, following theusual practice in numerical weather prediction. Thesepredictions would need to be initialized by the singularvectors of the linearized dynamics in order to obtain areliable estimate of the prediction uncertainty. Owingto the nonnormality of the linearized dynamics, thestructure of the initial conditions will influence the am-plification of the THC after a given amount of time andone cannot arbitrarily start from some state of theocean–atmosphere and expect to estimate reliable pre-dictability limits.

Griffies and Tziperman (1995) and others suggestedthat the present-day variability of the THC may beviewed as the excitation of a damped oscillatory modeof the ocean by atmospheric stochastic forcing. The re-sults of this work seem to put this proposal in a differ-ent perspective. While stochastic atmospheric excita-tion is still likely to be important, it seems that, due tothe nonnormality of the linearized dynamics, the inter-action of several nonorthogonal ocean modes—ratherthan a single damped oscillatory mode—may be playinga very important role in amplifying the response to theatmospheric stochastic forcing. While it is unfortunatethat this picture is not as simple as that of a single modeexcited by the atmosphere, this does open up the pos-sibility to explore a richer dynamics using both modelsand observations.

The need for a long control integration for evaluating

the linear fit and nonnormal effects makes our resultstentative. It will be useful to repeat this analysis byusing a direct representation of the linearized dynamicsof the GCM, which implies using both the model and itsadjoint. This will also enable the elucidation of the pre-cise physical mechanisms of the transient growth, some-thing that was not possible in the approach taken here.Such a fuller GCM-based analysis is a technically com-plex task, but one that is not completely out of reach inthe next few years. It is also important to keep in mindthat nonlinear processes that we have parameterized asa stochastic process may be playing an important role inthe THC oscillations in the GCM, and further analysisis needed to quantify these effects.

We showed that the optimal initial conditions madephysical sense (e.g., in terms of their dependence on theweighting by grid element volume and standard devia-tion of the variability used for nondimensionalizing thetemperature and salinity). This suggests that, even be-fore using a full GCM and its adjoint to repeat theanalysis done here, an intermediate future step couldperhaps be to use the results of the present analysis toinitialize a GCM. The GCM could then be used to in-vestigate the physical growth mechanisms and the pre-dictability implications of the nonnormal dynamicsfound here. It is common practice to evaluate THCpredictability by running ensemble prediction experi-ments started by perturbing only the atmospheric state,using identical ocean states (e.g., Griffies and Bryan1997; Collins et al. 2006). This has the advantage ofevaluating, in a sense, the “best case scenario” predict-ability. Instead, perturbing the initial ocean state usingthe methodology presented here may provide a stricter,hence perhaps more realistic, estimate of the predict-ability.

Acknowledgments. It is a pleasure to thank BrianFarrell, Lenny Smith, Peter Huybers, and AndrewMoore for very useful discussions and advice, and toSteve Griffies and Ed Hawkins for their helpful re-views. We are grateful to Keith Dixon and Tom Del-worth for providing the GFDL model output used forthis work. This work was supported by the U.S. Na-tional Science Foundation Climate Dynamics ProgramGrant ATM-0351123 and by the McDonnell Founda-tion. ET thanks the Weizmann Institute, where some ofthis work was completed during a sabbatical, for itshospitality.

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