the role of neutral singular vectors in midlatitude air–sea coupling...

88 VOLUME 16J O U R N A L O F C L I M A T E

q 2003 American Meteorological Society

The Role of Neutral Singular Vectors in Midlatitude Air–Sea Coupling

JASON C. GOODMAN* AND JOHN MARSHALL

Program in Atmospheres, Oceans, and Climate, Department of Earth, Atmospheric, and Planetary Sciences,Massachusetts Institute of Technology, Cambridge, Massachusetts

(Manuscript received 21 June 2001, in final form 4 March 2002)

ABSTRACT

The role of ‘‘neutral vectors’’ in midlatitude air–sea interaction is studied in a simple coupled model. Neutralvectors—the right singular vectors of the linearized atmospheric model tendency matrix with the smallest singularvalues—are shown to act as pattern-specific amplifiers of ocean SST anomalies and dominate coupled behavior.

These ideas are developed in the framework of a previously developed analytical coupled model, whichdescribed the mutual interaction across the sea surface of atmospheric and oceanic Rossby waves. A numericalmodel with the same physics is developed that permits the consideration of nontrivial background conditions.It is shown that the atmospheric modes that are least damped, and thus the patterns most easily energized bystochastic forcing, are neutral vectors.

1. Introduction

Low-frequency variability of the atmosphere in themidlatitudes, generated primarily by internal atmospher-ic dynamics, is dominated by a small number of char-acteristic patterns, of which the North Atlantic Oscil-lation (NAO) and the Pacific–North America (PNA) pat-tern are the most prominent (see, e.g., Wallace and Gut-zler 1981; Barnston and Livezey 1987; Kushnir 1994).The underlying ocean is driven by these stochastic-in-time but coherent-in-space patterns of air–sea flux var-iability (see Cayan 1992). The ocean integrates over the(almost) white temporal ‘‘noise’’ producing a ‘‘red’’power spectrum in oceanic variables (see, e.g., Fran-kignoul and Hasselmann 1977; Deser and Blackmon1993; Sutton and Allen 1997; Czaja and Marshall 2001).If midlatitude SSTs can feed back on the modes of var-iability themselves, then coupled interactions can occur,thus reddening atmospheric spectra, too. Observationssuggest that SST does have a small but discernible im-pact on, for example, the NAO (Czaja and Frankignoul1999; Czaja and Marshall 2001), in agreement with themodeling studies of Rodwell et al. (1999), Mehta et al.(1999), and Watanabe and Kimoto (2000).

On seasonal to interannual timescales, thermodynam-

* Current affiliation: Department of Geophysical Sciences, Uni-versity of Chicago, Chicago, Illinois.

Corresponding author address: Dr. Jason C. Goodman, Departmentof Geophysical Sciences, University of Chicago, 5734 S. Ellis Ave.,Chicago, IL 60640.E-mail: [email protected]

ic aspects of the coupled system can be rather welldescribed by stochastic climate models appropriatelymodified to allow for (i) feedback between SST and airtemperature, as in Barsugli and Battisti (1998) or Breth-erton and Battisti (2000); and (ii) reduced damping time-scales of air–sea interaction resulting from the reemer-gence process1 (e.g., Watanabe and Kimoto 2000). Sev-eral authors have developed simplified coupled modelsthat, in addition to thermodynamics, also include at-mosphere and/or ocean dynamics (Frankignoul et al.1997; Jin 1997; Weng and Neelin 1998; Neelin andWeng 1999; Goodman and Marshall 1999; Cessi 2000;Gallego and Cessi 2000; Marshall et al. 2001; Ferreiraet al. 2001).

The research described here attempts to identify whatcharacterizes those dynamical modes of the atmospherethat couple most strongly with the underlying ocean.We find that ‘‘neutral vectors’’—the right singular vec-tors of the linearized atmospheric model’s tendency ma-trix with the smallest eigenvalues (see Marshall andMolteni 1993; Navarra 1993)—act as pattern-specificamplifiers of ocean SST anomalies and dominate thecoupled behavior. We believe that this is a rather generalresult that pertains not just to the simple model studieshere, but to more complex models and perhaps also tothe real coupled system.

We begin with a review of the mathematical devel-

1 Thick winter ocean mixed layer temperature anomalies are cov-ered in the summertime by a thin seasonal mixed layer. The followingfall, these anomalies are exposed as the summer mixed layer erodes.Thus, winter SST anomalies seem to disappear in summer, and ‘‘re-emerge’’ the following winter.

1 JANUARY 2003 89G O O D M A N A N D M A R S H A L L

opment of neutral vectors and the ‘‘optimal forcing pat-terns’’ that excite them (section 2). This section recaststhe work of Marshall and Molteni (1993) and Navarra(1993) in the light of coupled atmosphere–ocean inter-action. In section 3, we illustrate the connection betweenneutral vectors and coupled modes using a numericalformulation of the coupled model considered (in ana-lytical form) by Goodman and Marshall (1999, hereafterGM99). A substantial review of the dynamics of theGM99 model may be found in the appendix, along withour procedure for discretizing and solving the numericalmodel. This numerical model enables us to considercoupled dynamics in more complicated geometries andbackground states than the analytical GM99 model; wediscuss the effect of these elaborations in section 3a. Insection 3b, we analyze the behavior of the coupled mod-el using singular vectors and find that the atmosphere’sfirst neutral vector is the atmospheric component of theleast-damped coupled mode; as a result, it dominatesthe stochastically forced system. These ideas suggestways to use neutral vectors to investigate interannualatmosphere–ocean coupling in the observations and inmore complex models, as described in section 4.

2. Neutral vectors and coupled modes

We begin by presenting the concepts underlying neu-tral vectors. The mathematical formalism presented hereis a review of that found in Marshall and Molteni (1993)and Navarra (1993); our contribution here is the dem-onstration that these patterns are important in under-standing atmosphere–ocean coupled modes.

a. Neutral vectors

Marshall and Molteni (hereafter MM) were interestedin atmospheric wave patterns that tended to persist in agiven state for long periods of time. They attempted tocompute patterns of maximum persistence by beginningwith a forced three-layer quasigeostrophic potential vor-ticity (QGPV) model, which we schematize as

]q 5 M (C) 1 f,

]t

where C is a vector representing the model stream-function, q is the model potential vorticity, M is a non-linear tendency operator, and f is a potential vorticitysource term. The model can be linearized by expressingq and C as the sum of a given time-mean basic stateplus a perturbation. The first-order perturbation equationcan be written as

]q 5 MC 1 f, (1)

]t

where q and C now represent perturbations about aspecified basic state, and M is a perturbation tendencymatrix. Marshall and Molteni were interested in the free,

unforced perturbations that displayed the smallest timetendency. Free, unforced waves obey

]q 5 MC. (2)

]t

To find the modes with the smallest time tendency, MMattempted to minimize this expression:2

] ]q, q7 8]t ]t

2l 5 , (3)^C, C&

where the angle brackets ^a, b& represent the inner prod-uct of a and b. This expression minimizes the size ofthe mode’s tendency, normalized by the magnitude ofthe mode itself. Inserting (2) into (3), we seek to min-imize

†^MC, MC& ^M MC, C&2l 5 5 ,

^C, C& ^C, C&

where M† is the adjoint of M. The C that minimize lwill be the eigenvectors Cn of M†M with minimum ei-genvalue :2ln

2†^M MC , C & ^l C , C &n n n n n2l 5 5 .^C , C & ^C , C &n n n n

Marshall and Molteni call these smallest eigenvectorsof M†M the neutral vectors of the atmospheric model.They are the right singular vectors of M with the smallestsingular values. At least one of the neutral vectors close-ly resembles a leading empirical orthogonal function(EOF) of the observed wintertime streamfunction fields(i.e., the NAO). We demonstrate and discuss the con-nection between EOFs and neutral vectors in a com-panion paper (Goodman and Marshall 2002). Neutralvectors are important because they identify the mostprominent patterns of variability in the system from adynamical framework. EOFs identify the most prevalentpatterns in the data, but do not explain why those pat-terns appear. That neutral vectors have a close connec-tion to EOFs is perhaps not surprising: the neutral vec-tors are, by design, the most stable and persistent wavepatterns, so it makes sense that these patterns should beprevalent in observations. In fact, one can demonstrate(Goodman and Marshall 2002; Navarra 1993) that sub-ject to certain restrictions, a mathematical identity existsbetween neutral vectors and the EOFs of a stochasticallyforced linearized model.

What are the dynamics of a neutral singular vectors?The matrix M describes Rossby wave propagation,

2 Actually, MM wrote their equations as streamfunction tendencies,and minimized ^(]/]t)C, (]/]t)C&/^C, C&. This has the advantage ofallowing l to be interpreted as an inverse timescale, but since modeswith small streamfunction tendency must also have small PV ten-dency, the difference should be otherwise unimportant. The techniqueused here is computationally simpler and faster, and will make furtherdevelopments more lucid.


downstream advection, and dissipation terms of theQGPV equation. For MC to be small, dissipation mustbe weak (implying large-scale patterns), and there mustbe a near balance between the propagative and advectiveterms. Thus we can expect that neutral vectors will be‘‘resonant’’ structures that can readily be excited.

Singular vectors are often applied to investigate therole of nonnormal growth to the production and main-tenance of transient perturbations to the atmosphere,beginning with the study of Farrell (1982). A relatedproblem considers the problem of optimal excitationpatterns (see Farrell 1989; Molteni and Palmer 1993).The linear version of the principal interaction pattern/principal oscillation pattern (PIP/POP) techniques in-troduced by Hasselmann (1988) also use singular vec-tors to find patterns that best describe a system’s ten-dency. The technique has also been used to find optimalamplification patterns in classic, nonrotating fluid dy-namics (Andersson et al. 1999; Luchini 2000). However,in all the aforementioned applications the focus is onthe singular vectors with the largest singular values—the most rapidly evolving and changing modes. Hereour emphasis is on the physical relevance of the singularvectors with the smallest singular values.

b. Relevance of neutral vectors to coupled interaction

While MM were interested in the long-term stabilityof atmospheric wave patterns, we are interested in at-mospheric patterns that are involved in coupled air–seainteraction. In particular, we wish to consider the pos-sibility of a mutually coupled interaction, in whichocean SST anomalies force the atmosphere, which thenfeeds back on the ocean through wind stress, generatingvariability on interannual timescales. This type of in-teraction has been explored in simple models by GM99,Marshall et al. (2001), Gallego and Cessi (2000), Neelinand Weng (1999), and many others. We believe thatneutral vectors are prime candidates for involvement inthis class of interaction. Our reasoning is simple: if cou-pled modes are to occur, the atmosphere must respondstrongly to the relatively small variations in atmosphericforcing generated by sea surface temperature anomalies.As we will now demonstrate, neutral vectors are theatmospheric modes that respond most strongly to PVforcing anomalies in a linearized model.

We return to the model in (1), but now we look atthe forced, stationary response to an applied PV forc-ing f:

0 5 MC 1 f.

What pair of forcing and response will have the largestresponse per unit forcing? We want to find the C andf that will maximize

^C, C&22l 5 .

^f, f&

Since f 5 2MC, this is equivalent to minimizing

^MC, MC&2l 5 .

^C, C&

This is exactly the condition required for the neutralvectors. Thus, the neutral vectors are not only the moststationary modes in the unforced time-evolving model,they are also the forced, stationary modes that exhibitthe largest response to external forcing.

Interestingly, this means we can not only find theneutral vectors, Cn, but also the optimal forcing patternsfn that maximally excite them, by solving MCn 1 fn 5 0.The Cn are the right singular vectors of M; the fn arethe left singular vectors.

3. Neutral vectors and coupled modes in a simplecoupled model

We now consider neutral modes in the context of aparticular model of coupled interaction: that of GM99.A summary of this model maybe be found in the ap-pendix. In brief, the model describes the linear inter-action of traveling oceanic Rossby waves with forcedstationary atmospheric planetary waves. The ocean isdriven by wind stress and heat fluxes, while the at-mosphere responds to thermal forcing from SST anom-alies. The model produces a spontaneously growingcoupled wave of subdecadal period and basin-fillinghorizontal wavelength. The positive feedback that pro-duces this self-amplifying coupled mode is discussed inthe appendix and at length in GM99.

As discussed in GM99, the coupled mode’s atmo-spheric behavior is close to that of an atmospheric ‘‘freemode’’ in which there is an approximate balance be-tween westward Rossby wave propagation and eastwardPV advection by the mean atmospheric flow. This onlyoccurs at a narrow range of wavelengths, so the coupledgrowth mechanism is highly scale selective.

Note that the balance exhibited by the atmosphericpart of the coupled mode is exactly the balance satisfiedby the neutral vectors. The M operator in (1) in largepart describes the advection of PV anomalies by themean flow and the advection of mean vorticity by theanomalous flow. The C is a neutral vector when it caus-es cancellation of these terms, resulting in a small MC.Thus, only a small forcing is necessary to excite a largeatmospheric response for this mode.

Of course, a large atmospheric response is only partof the story. In order to produce a mutually coupledinteraction, SST anomalies must be actually able to pro-duce a forcing that resembles the optimal forcing pat-terns. Moreover, the atmospheric response must drivethe ocean in such a way as to enhance the original SSTanomalies. The large atmospheric response to forcingexhibited by neutral vectors is a necessary but not suf-ficient condition for a coupled mode.

We will now demonstrate that the atmospheric com-ponent of the GM99 coupled model has neutral vectorstructure. This is not very remarkable (and has already


been described) for the extremely simplified case ofuniform basic-state winds, as used in the GM99 model;however, we will also demonstrate a connection betweenneutral vectors and a coupled mode in the more realisticcase of nonuniform basic-state winds. We will also dem-onstrate that when this model is excited by stochasticforcing, the atmospheric response is dominated by thecoupled mode/neutral vector behavior. To enable thiscomparison, we have formulated the GM99 dynamicsas a numerical model, discussed in the appendix.

a. Coupled dynamics with coastlines and varicosebackground flow

Before computing neutral vectors and comparingthem with the coupled modes, it is important to notethat the more realistic geometry enabled by the numer-ical coupled model causes some differences in behaviorfrom the simple GM99 scheme. These extensions be-yond GM99 must be understood in order to proceed,and are interesting in their own right.

The new features of the numerical model are the ad-dition of coastlines to the ocean domain and the use ofa nonuniform, varicose background flow in the atmo-sphere. We will consider the effects of these changesone at a time.

1) OCEAN BASIN

GM99’s ocean and atmosphere had no boundaries.Here, we restrict the ocean domain to a basin 6000 kmwide. The atmosphere lies within a 25 000-km zonallyreentrant channel. ‘‘SST’’ anomalies over land are de-fined to be zero. Basic-state winds in the model’s twoatmospheric levels are 14 and 5 m s21. Other parametersare as specified in GM99.

We compute the eigenvalues of the model tendencyoperator for this single-basin model. Figure 1 shows theeigenvalues with the most positive real part. The growthrate of each eigenvector is given by its position on thex axis of the top panel; its frequency is given by itsposition on the y axis. Each complex-conjugate pair ofeigenvalues corresponds to a pair of eigenvectors thatare identical but for a 908 phase shift; phase propagationis always westward. Below the eigenspectrum, we dis-play the eigenvector pattern corresponding to the right-most eigenvalue pair. While the original GM99 modelcontained coupled growing modes (which would appearto the right of the origin in this figure), we see that whencoastlines are added, all modes are damped. However,the structure of the rightmost, ‘‘least-damped’’ mode isquite similar to the coupled mode predicted by GM99.It displays equivalent barotropic wavenumber-3 struc-ture in the atmosphere with matching wavelengths inthe ocean, and high pressure over warm water—pre-cisely the arrangement that grows most quickly in theanalytical model. The wavelength of the mode seen in

Fig. 1 matches the wavelength of maximum growth ofthe GM99 model.

The process that causes damping is easy to under-stand. SST anomalies in the GM99 model are dynam-ically connected to propagating oceanic baroclinic Ross-by waves, and thus propagate westward. These Rossbywaves are dissipated by interaction with the westernboundary. A warm SST patch at the western side of thebasin will excite the wavenumber-3 pattern character-istic of the atmosphere’s equilibrated mode. This patternwill provide a wind stress forcing at the eastern side ofthe basin that creates a cool ‘‘child’’ SST anomaly tothe west of the ‘‘parent.’’ For a growing mode to occur,the parent must bring the amplitude of the child up toits own amplitude before the parent is destroyed at thewestern boundary. This condition cannot be met withthe choice of so narrow a basin (we have found that thecritical basin width for these parameters is about 15 000km), and so each parent produces a child weaker thanitself, and the mode gradually dies away.

There are now no growing modes in this system; itonly supports damped modes. However, if this systemis excited with white stochastic forcing (as from at-mospheric synoptic eddies), the least-damped modeshould retain the most energy, and be the most prom-inent. Thus the feedback mechanism described in GM99remains important in understanding the behavior of thepresent system. We will test this claim in section 3a(3).

The NAO and other observed patterns of interannualvariability do not exhibit the rapid growth and pure-tone oscillations observed in GM99; instead, the NAO’stime series spectrum (Hurrell and van Loon 1997) ispredominantly reddish, with some apparent enhance-ment of power at interannual frequencies. In GM99, wenoted the unrealistically rapid growth, and speculatedthat the unmodeled damping processes would counteractit. Such damping processes are readily observed in thepresent model.

2) VARICOSE ATMOSPHERIC BACKGROUND FLOW

We now attempt to study the effect of a more realisticatmospheric background flow on the model physics andproceed with a schematic formulation for the atmo-spheric stationary wave pattern. The midlatitude at-mospheric flow at intermediate height exhibits a jet thatconstricts over the western shores of the Atlantic andPacific, and is spread out over the eastern shores of thebasins. We schematize this pattern by specifying a basic-state wind field that looks like Fig. 2. The zonally ormeridionally averaged wind speed is constant and iden-tical to the values used in the previous experiment.

For this experiment, we illustrate the behavior of thecoupled model by integrating it forward in time. Themodel is initialized with random SST and ocean stream-function anomalies of arbitrary amplitude, and steppedforward using a simple Euler forward scheme, with a30-day time step. Figure 3 shows the evolution of the


FIG. 1. GM99 model with parallel channel atmospheric background flow, ocean confined to6000-km-wide basin. (a) Eigenvalues of the coupled model’s tendency matrix. (b)–(e) Real partof eigenvector associated with the eigenvalue with largest real part.

model with an ocean basin identical to that in section3a(1), and with basic-state winds as shown in Fig. 2.The upper four panels of Fig. 3 show a snapshot ofanomalies of upper- and lower-layer atmosphericstreamfunction, SST, and ocean streamfunction. TheHovmoeller diagrams in the lower left show the evo-lution of SST anomalies and of lower-level atmosphericperturbations. Perturbations decay to zero, as before,due to the destruction of oceanic anomalies at the west-ern boundary. Oscillation and damping rates are similarthe parallel-flow model, and the atmospheric wave pat-tern again generally shows equivalent barotropic wave-number-3 behavior, although some additional elementsare present. The atmospheric wave no longer propagateswestward in phase with SST anomalies; instead, it re-mains more or less fixed at a particular longitude, and

fluctuates in sign. The varicose background flow locksthe atmospheric response to a particular longitude,which is more reminiscent of observed patterns of low-frequency variability.

The SST pattern of this system is more complex thanthe simple propagating wave pattern of previous ex-periments, but generally shows westward-propagatingpatches of warm and cool water.

3) RESPONSE TO STOCHASTIC FORCING

In section 3a(1), we noted that when confined to abasin, the coupled mode no longer grew, but remainedthe least-damped mode. We argued that the least-damped mode would be most susceptible to excitationby stochastic forcing. In this section, we test this claim.


FIG. 2. Basic-state streamfunction pattern used in the varicose back-ground flow experiment. The basic-state streamfunction is given by therelation c 5 U0y 1 0.1UO(Ly/2)F(x, y), where F(x, y) 5 sin(2py/Ly)sin2(py/Ly) sin[4p(x 2 x0)/Lx] sin2[2p(x 2 x0)/Lx] when x . x0 andx , (Lx/2 1 x0), and F(x, y) 5 0 elsewhere. Here UO is the wind velocityin either level from the uniform-flow experiment, x0 is 106 m, and Lx andLy are the zonal and meridional extents of the channel.

We consider the case of a varicose background at-mospheric flow over an ocean basin, as in section 3a(2),and carry out a long forward integration of the model,as in Fig. 3. But now, at each time step, after computingthe atmospheric response to SST (c1, c2), we randomlygenerate an additional stochastic streamfunction com-ponent (c1s, c2s), add it to the response, and use theresult to force the SST and dynamic ocean parts of thecoupled model [Eq. (A3) and Eq. (A4) in appendix,respectively].

The stochastic components of the atmospheric fieldsare chosen to very roughly mimic the structure and am-plitude of transient eddies in the atmosphere. At eachmodel time step (Dt 5 1 month) we generate a Gaussianrandom streamfunction field, spectrally truncated tozonal wavenumber 8 and meridional wavenumber 3.Thus the smallest wavelengths are around 3000 km. Wemultiply this by sin(py/Ly)0.7 to bring the eddy amplitudeto zero at the northern and southern walls, thus avoidingboundary condition problems. We multiply this field byan amplitude factor of 180 geopotential meters (gpm)in the upper layer and 120 gpm in the lower layer, tocreate an equivalent barotropic streamfunction pattern.These fields drive the ocean through the air–sea heatflux term and wind stress curl.

In Fig. 4, we show snapshots of the model state, andits evolution through time. The atmospheric componentsplotted here are only the deterministic responses to SST;the additional random component is not shown in thesefigures. As we would expect, the stochastic model’s evo-lution is much less orderly. However, the patterns ofatmospheric response show the wavenumber-3 by wave-number-1 mode previously identified as a coupled mode,though the resemblance is not always as strong as inthis snapshot. SST anomalies show westward propa-gation, and both atmospheric and oceanic variablesshow strong interannual variability. The amplitude ofSST anomalies is of the order of a degree or so, withatmospheric responses of a few tens of geopotential me-ters; these amplitudes are similar to, but somewhat larg-er than, those observed on interannual timescales in theAtlantic (see Czaja and Marshall 2001). These ampli-tudes are determined entirely by the strength of sto-

chastic forcing, but we chose a stochastic amplitudebased on observed wintertime synoptic activity, and didnot tune it to improve the resemblance of the coupledresponse to observations.

To more clearly isolate the dominant patterns of var-iability, we perform an EOF analysis on the upper-at-mospheric streamfunction field—the EOFs are shownin Fig. 5. The first EOF explains 56% of the nonsto-chastic part of the atmospheric variability; the secondEOF explains 35%, and the third EOF, 5%. The firstEOF shows a predominantly zonally symmetric pattern,with no clear pattern to its variability and only a weakcovarying oceanic pattern (not shown). This mode reactsstrongly to SST anomalies, but is unable to excite amutually coupled interaction. We will revisit this patternin section 2a. The second and third EOFs are much moreinteresting. They show wavenumber-3 structure; the pat-terns are essentially identical to the atmospheric com-ponent of the least-damped coupled mode depicted inFig. 3. [The shape of the atmospheric pattern in section3a(2) varies periodically; it tends to oscillate betweena state resembling EOF 2 and a state resembling EOF3.] A 5-yr cycle is apparent in the amplitude timeseriesof EOF 2, the timescale set by the propation speed ofSST anomalies in the ocean controlled here by Rossbywave dynamics; a similar frequency is present (thoughless obvious) in the time series of EOF 3. The amplitudeof EOF 2 is about 10% of the total atmospheric am-plitude, including both stochastic and deterministic con-tributions. As anticipated, the least-damped coupledmode previously discussed does, in fact, explain a largeamount of the stochastic model’s variability.

We now discuss how the coupled interaction can beinterpreted and understood in terms of neutral vectors.

b. Interpretation of coupled interaction in terms ofneutral vectors

To compute the neutral vectors of the atmosphericcomponent of the model, in both constant zonal flow[section 3a(1)] and varicose flow [section 3a(2)] ar-rangements, we use an Arnoldi (Lehoucq et al. 1998)technique to yield the smallest five singular vectors ofthe model atmosphere’s tendency matrix M. To performthis computation, we must specify a particular innerproduct in (3): here, we define ^a, b& as the sum overall model grid points of the products of a and b evaluatedat each grid point. Other, differently weighted innerproducts are possible, but our results are not sensitiveto this choice.

The first five neutral vectors for the constant zonalflow model (not shown) only vary meridionally, andhave no zonal structure. As such, they are unaffectedby zonal advection or Rossby wave propagation: theyare ‘‘neutral’’ in a rather trivial way. The first neutralvectors with zonal structure are the sixth and seventhones; they display wavenumber-3 structure zonally, andwavenumber-1 zonally, and are 908 out of phase with


FIG. 3. Snapshots and evolution of the numerical model, run with a basin ocean and varicose backgroundflow. The model was initialized with random values in SST and ocean streamfunction. (a), (b) The upper-and lower-atmospheric streamfunction anomalies (c1, c2), expressed as geopotential height (in m) of pressuresurfaces. (c), (d) The SST anomalies, in K, and ocean streamfunction anomalies, in m2 s21. (e), (f) Time–longitude sections of SST and c2, taken at the latitude of the small ‘‘3’’ in the upper panels. (g), (h) Atime series of SST and c2, taken at the 3.

each other. Their structure matches that of the coupledmode found in section 3a(1).

In Fig. 6, we show the three leading neutral modesfor the atmosphere with a varicose background flowdiscussed in section 3a(2). The first neutral vector is azonal mode analogous to the modes with no zonal struc-ture found above. The second and third neutral vectorsdisplay structures nearly identical to the structure of the

coupled mode in section 3a(2). Neutral vectors 1, 2, and3 are identical in pattern and order to the EOFs of thestochastically forced coupled model (see Fig. 5).

This provides a demonstration that the neutral vectorscontinue to determine the behavior of the atmosphericcomponent of the coupled system, even when the at-mosphere has a complicated background flow.

How are these neutral vectors excited in the coupled


FIG. 4. Same as Fig. 3, but with stochastic forcing added. The rms amplitude of stochastic atmosphericperturbations is 180 gpm in upper layer, 120 gpm in lower layer. Only the deterministic part of the at-mospheric perturbation is shown here. Atmospheric perturbations are expressed in gpm; units of SST areK, units of ocean streamfunction are m2 s21.

model? This model has a very simple atmospheric heat-ing scheme: there is a linear connection between SSTand baroclinic PV forcing as discussed in the appendix.Thus, when the coupled model’s SST has a strong pro-jection onto the baroclinic part of an optimal forcingpattern fn, we see a corresponding strong atmosphericresponse of the corresponding neutral vector.

Figure 7 more clearly displays the interaction betweenatmospheric neutral vectors and evolving SST anoma-lies in the GM99 model. At the top of the figure, we

show the upper-level streamfunction of neutral vectors2 and 3 for the varicose background flow model; theseare the neutral vectors that resembled the coupled mod-el’s behavior. The upper time series shows the projectionof the model’s atmospheric state onto the first five neu-tral vectors. Since the model ocean has a western bound-ary, the model response is damped. We have thus mul-tiplied the projection values by et/3 (where t is given inyr), to counteract the exponential decay and zoom in onthe longer-term variations. We clearly see the oscillation


FIG. 5. EOFs of the upper-layer atmospheric streamfunction for the stochastically forced coupled model run depictedin Fig. 4. (a)–(c) EOF patterns; (d)–(f) corresponding amplitude time series. (a), (d) Pattern and time series for firstEOF, which explains 56% of the deterministic variability. (b), (e) Second EOF, explaining 35% of the deterministicvariability. (c), (f) Third EOF, explaining 5% of the deterministic variability.

of the model state between neutral vectors 2 and 3, asdescribed in the previous section. The lower pair ofcontour plots show the baroclinic part of the optimalforcing pattern associated with these neutral vectors.The lower time series shows the projection of the ther-mal forcing anomalies generated by SST onto the firstfive optimal forcing patterns, rescaled as with the neutralvector time series. The match is identical, as a conse-quence of the linearity of the atmospheric response op-erator.

From Fig. 7, we can describe the behavior of thecoupled mode. As the model’s SST pattern evolves ac-cording to ocean dynamics, it projects alternately ontotwo different optimal forcing patterns. This projectionexcites a large atmospheric neutral vector response,which then provides a wind stress to further modify theSST.

Neutral vectors 2 and 3 are active in the coupledinteraction. Patterns that are less neutral (modes 4 andup) respond only weakly to SST forcing, undergo amuch weaker coupled interaction, and are more rapidlydamped. As mentioned earlier, a strong atmospheric re-

sponse to thermal forcing is a necessary, but not suf-ficient, condition for a mutually coupled mode: neutralvector 1 responds strongly to SST forcing, but does notfeed back onto the model ocean in a way that leads topositive feedback.

One point of concern regarding this description is therobustness of the optimal forcing patterns. In Fig. 7, theoptimal forcing patterns show complicated fine structurein some areas (particularly near x 5 5000 km, y 5 3200km, in the ‘‘pinched’’ part of the background flow; there,we find an alternating positive/negative banded patternat the grid-scale level). This may be due to the minimaleddy viscosity used in this model. If most of the SSTforcing pattern’s projection onto this pattern occurs inthis fine-structure region, we should be concerned thatthe stability of the coupled mode is sensitive to smallchanges in the model domain. However, we find (figurenot shown) that the bulk of the projection of SST ontothe forcing pattern occurs in the broad ‘‘wings’’ to thenorth and south in forcing pattern 2, and to the northeastand southeast in pattern 3.


FIG. 6. Maps of the first three neutral vectors of the atmospheric model with a varicose background flow aspictured in Fig. 2. (a)–(c) Upper-level streamfunction. (d)–(f) Lower-level streamfunction.

4. Conclusions

Neutral vectors are the singular vectors of a linearizedmodel’s tendency matrix with smallest singular value.We have demonstrated mathematically that neutral vec-tors are also the patterns that respond most stronglywhen the model is externally forced. Moreover we haveshown that neutral vector dynamics (an approximatebalance between advection of the mean PV by a flowperturbation and advection of the perturbation’s PV fieldby the mean flow) are responsible for the atmosphericpart of the coupled growing mode found in Goodmanand Marshall’s (1999) simple atmosphere–ocean cou-pled model.

We proceeded to illustrate the generality of this con-nection by building a numerical model that obeys thecoupled physics desribed in GM99, but that allows formore complex geometry and nonuniform basic-stateflows. The added complexities change the system’s be-havior (a western boundary current damps the mode;varicose atmospheric background flow locks the at-mospheric mode to particular longitudes), but the cou-pled mechanism described by GM99 remains important.The atmospheric patterns involved in the coupled in-teraction are still neutral vectors.

In our coupling mechanism, the optimal forcing pat-tern/neutral vector pair can be viewed as a mechanismthat accepts a small SST thermal forcing from the oceanand returns a large atmospheric response, which maytranslate into a large feedback onto the ocean. However,this is only half the story: the ocean must be able toaccept the forcing provided by the atmosphere and re-turn (some nontrivial projection onto) the neutral vec-

tor’s optimal forcing pattern in order for a mutuallycoupled interaction to occur. Optimal forcing pattern/neutral vector pairs may play a key role as pattern-selective amplifiers in a coupled atmosphere–ocean sys-tem.

The atmospheric model used in this study is linearand steady state. High-frequency variability and thenonlinear feedback of this variability onto the mean floware not modeled explicitly. However, synoptic variabil-ity is crucial to the coupling mechanism. It stochasticallyforces the slowly varying coupled system; the coupledmode responds most strongly to this forcing, and be-comes prominent. More complicated eddy–mean flowinteractions lie outside the scope of this simple model.We discuss the feasibility of incorporating eddy–meanflow interaction into a neutral vector calculation in acompanion paper (Goodman and Marshall 2002).

The ocean component of the numerical coupled modeldeveloped for this work could easily be extended to letus impose nonzero basic-state currents (a double-gyreflow, for example), but we have not considered thisimportant case here. An imposed ocean current wouldadvect SST and oceanic PV anomalies around the oceanbasin. Even if the ocean currents dominate the Rossbywave propagation seen here, a coupled oscillation is stillpossible. As SST anomalies are transported around agyre, they will excite neutral vector responses from theatmosphere. If the surface fluxes caused by any neutralvector act to enhance the original SST anomaly pattern,a growing coupled mode will still occur, with the mode’soscillation period now given by the gyral recirculationtime, rather than the Rossby wave period.


FIG. 7. Projections of neutral vectors and optimal forcing patterns onto a forward model run. (a), (b) Upper-layerstreamfunction pattern for neutral vectors 2 and 3; respectively, of the atmospheric model with a varicose jet backgroundflow. (c) The projection of the time evolution of the coupled model’s atmosphere onto singular vectors 1–5. (d) theprojection of SST-induced atmospheric thermal forcing onto the first five optimal forcing patterns. (e), (f) The baroclinicpart of the PV forcing that excites neutral vectors 2 and 3 (‘‘optimal forcing patterns’’; see text).

Our model is extraordinarily crude, and so we shouldnot expect the particular shapes and patterns of the cou-pled mode and the atmospheric neutral vectors to cor-respond in detail with observed patterns, although thebroad correspondence is promising. However, the phys-ical mechanism of coupling (excitation of neutral vec-tors by SST forcing) can be applied in much more re-alistic situations. Our results suggest that neutral vectorsare likely to be important for atmosphere–ocean inter-action in a very general sense.

In a companion paper (Goodman and Marshall 2002),we return to Marshall and Molteni’s (1993) three-layerQG model, to look more closely at the connection be-tween neutral vectors and EOFs, and to compute optimalforcing patterns associated with neutral vectors. If wecan identify a forcing pattern that induces a particularmode of atmospheric variability, then it can be used tosee whether the patterns of SST that covary with, forexample, the NAO pattern, are those that are capableof exciting that pattern.


FIG. A1. Vertical structure of the coupled model defining the keyvariables of the GM99 model.

The identification of neutral vectors is not limited tosimple QG models with trivial model physics. The ten-dency matrix M can be arbitrarily complex: it could evenrepresent an entire atmospheric general circulation mod-el, linearized about some suitable basic state. The M†matrix is then the adjoint of this model. It is thus pos-sible to find the neutral vectors of an entire linearizedGCM, along with the corresponding optimal forcing pat-terns. This is, as one might imagine, a computationallyintensive task. However, the implementation could bemade easier through the use of an automatic tangentlinear/adjoint compiler (Marotzke et al. 1999), whichcan automatically generate adjoint model code from theforward source.

The neutral vector concept can be generalized to al-most any model physics, and will be relevant to theinvestigation of atmosphere–ocean coupled modeswhenever the atmosphere responds to SST forcinganomalies in an essentially linear way, and when a largeatmospheric response will produce a large forcing ofthe ocean by the atmosphere.

Neutral vectors will respond strongly to any forcingsource, regardless of its origin. Thus, they may shedsome light on observations that the same observed pat-terns of variability (NAO, PNA, etc.) dominate atmo-spheric variability on both short (intramonthly) and long(interannual) timescales. It is possible that the long-termvariability is simply the low-frequency tail of a high-frequency stochastic process, but it is also possible thatthese patterns represent atmospheric neutral vectors re-sponding both to random high-frequency forcing bysynoptic eddies and to coherent low-frequency forcingdriven by SST.

Acknowledgments. This work was supported by theNOAA Office of Global Programs.

APPENDIX

The GM99 Coupled Model

a. Dynamical framework

In this section, we summarize the results of Goodmanand Marshall (1999, hereafter GM99), which providesthe foundation upon which our ideas are developed.

The GM99 model (Fig. A1) incorporates a 1½-layerquasigeostrophic ocean anomaly model in an unboundeddomain. In this model, undulations of the layer interfacelead to surface currents and propagating Rossby waves.A basic state at rest is prescribed. The QG ocean forcesa simple mixed layer model, which incorporates storage,advection, air–sea transfer, and mixed layer entrainmentprocesses. The sea surface temperature anomalies pro-duced by this mixed layer model thermally excite a two-level quasigeostrophic atmospheric anomaly model,whose basic state has constant uniform zonal winds ofdiffering magnitudes in the two layers. Basic-state

winds are chosen to resemble wintertime conditions: theannual cycle is not resolved. The atmospheric pressureanomalies drive the QG ocean through surface windstress, allowing mutual coupling of atmosphere andocean.

This physics is expressed in the following set of linearpartial differential equations that represent first-orderbalances for linear perturbations about a basic state.

• Atmosphere (upper level):

J(c , Q ) 1 J(C , q )1 1 1 1

g SSTa5 (c 2 c ) 2 . (A1)1 22 [ ]L ra a• Atmosphere (lower level):

J(c , Q ) 1 J(C , q )2 2 2 2

g SSTa5 2 (c 2 c ) 2 . (A2)1 22 [ ]L ra a• SST:

]SST 5 2g [SST 2 r (c 2 c )]o a 1 2]t

2 J(c , SST) 2 g (SST 2 r c ). (A3)o e o o

• Dynamic ocean:

] ] 3 12q 1 b c 5 a¹ c 2 c . (A4)o o 2 11 2]t ]x 2 2

In the above, C1 and C2 are the basic-state atmo-spheric streamfunctions; c1, c2, co are the streamfunc-tion anomalies in the upper- and lower-atmospheric lay-ers and in the ocean; ua 5 ra(c1 2 c2) is the atmospherictemperature; Q1 and Q2 are the basic-state atmosphericPV fields; and q1, q2, and qo are the QG potential vor-ticity anomalies in the atmosphere and ocean. andSSTSST are the unperturbed and anomalous sea surface tem-perature; the term on the right-hand side of Eq. (A4) isthe mechanical forcing of the ocean by the winds ex-trapolated down to the surface (a is a drag coefficient);terms on the right-hand sides of Eqs. (A1) and (A2)represent a baroclinic thermal forcing via relaxation ofthe atmospheric temperature anomaly ua to a value set


FIG. A2. Phase relationships between ocean and atmosphere forthe fastest growing coupled mode found by GM99. The symbols Hand L denote highs and lows of atmospheric pressure, with the am-plitude of the pressure anomaly increasing with height. The symbolsW and C denote warm and cold SST, and the undulating line indicatesthe depth of the thermocline. Note the high (low) pressure abovewarm (cold) water, and the phase match between wind stress andcurrent.

by SST; and the forcing terms for SST represent air–sea heat flux, advection of the mean SST gradient, andentrainment, in that order. The parameters ra and ro areconversion constants for translating streamfunctionsinto temperatures via the thermal wind equation, in theatmosphere and ocean, respectively. All notations anddefinitions are identical to those in GM99.

In GM99, simplifications are made such that this setof equations is linear and has constant coefficients.Thus, they can be solved by inserting plane-wave so-lutions and solving for the dispersion relation. Withina narrow band of wavelengths, a coupled growing waveis possible. As discussed in GM99, this wave has afrequency of about 5–8 yr and an e-folding growth time-scale of about 2 yr, given plausible (but rather uncertain)parameters.

Figure A2 illustrates the structure and physics of thisgrowing mode. The mixed layer entrains fluid from thestratified ocean interior into itself. When the thermoclineis bowed upward, this fluid will be anomalously cool;when the thermocline is bowed downward, it will bewarm. This leads to a correlation between a deep ther-mocline and warm SST, and vice versa. Even in theabsence of entrainment, this correlation still exists dueto geostrophic advection in the mean SST gradient—see section 3b(1) of GM99 for details.

Suppose the thermocline is bowed upward, causinga cold SST anomaly. For particular zonal and meridionalwavelengths, this will generate an equivalent-barotropiclow pressure anomaly in the atmosphere, centered abovethe cool SST. This low pressure anomaly will induce acyclonic wind stress curl to the ocean, causing upwardEkman pumping, raising the already shallow thermo-cline, and making SST even colder. This is a coupledpositive feedback. While this feedback is occurring, the

thermocline anomaly propagates westward as a Rossbywave.

The growth mechanism can only operate when theatmospheric response is characterized by equivalent bar-otropic highs (lows) over warm (cold) water. This occursonly in a rather narrow band of wavelengths for whichdownstream advection balances upstream Rossby wavepropagation; that is, for winds, potential vorticity gra-dients, and scales that are close to that of a neutral stateof the atmosphere (i.e., free stationary Rossby waves).For reasonable parameter choices, this scale-selectivegrowth mechanism selects a wavelength close to that ofthe NAO pattern.

In order to allow solution by hand, the GM99 modelis oversimplified in many ways. Among these are itsconstant zonal basic-state winds, the lack of a reentrantatmospheric geometry, the use of only two levels in theatmosphere, the lack of coastal boundaries in the ocean,and the absence of basic-state currents in the ocean. Inthe next section, we develop a model in which we canrelax some of these simplifications.

b. Numerical model formulation

We begin with the same differential equations (A1)–(A4) as in GM99. In GM99, we assumed that C1, C2,Q1, and Q2 were linear functions of the meridional co-ordinate (i.e., the basic-state flows were constant andpurely zonal) to make analytic solution tractable. Herewe allow for the possibility that the ocean is of restrictedeast–west extent, and study the effect of nonuniformatmospheric flows. The important case of nonzero basic-state ocean currents will not be considered here.

For arbitrary basic states, the system of Eqs. (A1)–(A4) must be solved numerically. We do this by formingfinite-difference forms of the equations above. To allowsolution in a rectangular oceanic domain, we must adda Stommel (1948) frictional term to Eq. (A4), permittinga western boundary current to form. We must also becareful to write the finite-difference forms of the ad-vection operators in (A1) and (A2) so that they conservePV exactly. We also added a small amount of vorticitydiffusion and dissipation of PV anomalies into the mod-el atmosphere for numerical stability.

The model is discretized using a standard finite-dif-ference scheme, with c- and q-points coincident, andusing a centered difference scheme for first derivativesand a five-point stencil for the Laplacian operator. Wespecify a periodic channel geometry in the atmospherewhose zonal and meridional extents are 25 000 and 7200km, respectively. The north and south boundary con-ditions in the atmosphere are designed to allow no PVflux through the walls. The meridional grid spacing is360 km; the zonal grid spacing is 550 km in the at-mosphere and 275 km in the ocean. Upon discretization,the SST and ocean streamfunction equations can be writ-ten in the form


C1 ] SST C25 P , (A5) [ ]]t C SSTo Co

where SST, Co, C1, and C2 are vectors containing thediscretized elements of the SST, ocean streamfunction,and atmospheric upper- and lower-layer streamfunctionanomaly fields. The matrix P is a sparse matrix repre-senting the finite-differenced forms of the differentialoperators in (A4) and (A3). We must find C1 and C2using the side constraints provided by (A1) and (A2):

C g 2SST1 aM 5 , (A6)2[ ] [ ]C L r SST2 a a

where M is a matrix incorporating discretized forms ofthe advection and dissipation operators acting on theatmospheric state vectors. It is identical to the M usedin (1) and throughout section 2.

c. Time evolution and eigenspectrum

We can step the system forward in time by inverting(A6) to find C1 and C2 at each time step from SST,then plugging those into (A5) to get the rate of changeof SST and Co, which can then be advanced to the nexttime step using a simple Euler forward scheme.

We can also solve for the eigenvectors of the system’stendency matrix; this allows us to compute frequenciesand growth/decay rates for the coupled mode, as donein GM99 for the simpler case of plane-parallel flow. Weneed not write out the tendency operator as an explicitmatrix: the Arnoldi (Lehoucq et al. 1998) algorithm cancompute the eigenvectors and eigenvalues of a sparselinear operator that is specified as an algorithm like thatdescribed in the previous paragraph. In addition, theArnoldi technique can solve for a small number of ei-genvalues of desired characteristics, rather than findingthe entire eigenspectrum. Since coupled modes that arerapidly damped will not be observable in nature, wesolve for the eigenvalues with the most positive realpart.

REFERENCES

Andersson, P., M. Berggren, and D. S. Henningson, 1999: Optimaldisturbances and bypass transition in boundary layers. Phys. Flu-ids, 11, 134–150.

Barnston, A., and R. E. Livezey, 1987: Classification, seasonality andpersistence of low-frequency circulation patterns. Mon. Wea.Rev., 115, 1083–1126.

Barsugli, J. J., and D. S. Battisti, 1998: The basic effects of atmo-sphere–ocean thermal coupling on middle-latitude variability. J.Atmos. Sci., 55, 477–493.

Bretherton, C., and D. Battisti, 2000: An interpretation of the resultsfrom atmospheric general circulation models forced by the timehistory of the observed sea surface temperature distribution.Geophys. Res. Lett., 27, 767–770.

Cayan, D., 1992: Latent and sensible heat flux anomalies over the

northern oceans: The connection to monthly atmospheric cir-culation. J. Climate, 5, 354–369.

Cessi, P., 2000: Thermal feedback on windstress as a contributingcause of climate variability. J. Climate, 13, 232–244.

Czaja, A., and C. Frankignoul, 1999: Influence of the North AtlanticSST on the atmospheric circulation. Geophys. Res. Lett., 26,2969–2972.

——, and J. Marshall, 2001: Observations of atmosphere–ocean cou-pling in the North Atlantic. Quart. J. Roy. Meteor. Soc., 127,1893–1916.

Deser, C., and M. Blackmon, 1993: Surface climate variations overthe North Atlantic ocean during winter: 1900–1989. J. Climate,6, 1743–1753.

Farrell, B. F., 1982: The initial growth of disturbances in a baroclinicflow. J. Atmos. Sci., 39, 1663–1686.

——, 1989: Optimal excitation of baroclinic waves. J. Atmos. Sci.,46, 1193–1206.

Ferreira, D., C. Frankignoul, and J. Marshall, 2001: Coupled ocean–atmosphere dynamics in a simple midlatitude climate model. J.Climate, 14, 3704–3723.

Frankignoul, C., and K. Hasselman, 1977: Stochastic climate models.Part II: Application to sea surface temperature variability andthermocline variability. Tellus, 29, 284–305.

——, P. Müller, and E. Zorita, 1997: A simple model of the decadalresponse of the ocean to stochastic wind forcing. J. Phys. Ocean-ogr., 27, 1533–1546.

Gallego, B., and P. Cessi, 2000: Exchange of heat and momentumbetween the atmosphere and ocean: A minimal model of decadaloscillations. Climate Dyn., 16, 479–489.

Goodman, J., and J. Marshall, 1999: A model of decadal midlatitudeatmosphere–ocean coupled modes. J. Climate, 12, 621–641.

——, and ——, 2002: Using neutral singular vectors to study low-frequency atmospheric variability. J. Atmos. Sci.,, 59, 3206–3222.

Hasselmann, K., 1988: PIPs and POPs: The reduction of complexdynamical systems using principal interaction and oscillationpatterns. J. Geophys. Res., 83 (D9), 11 015–11 021.

Hurrell, J. W., and H. van Loon, 1997: Decadal variations in climateassociated with the North Atlantic oscillation. Climatic Change,36, 301–326.

Jin, F.-F., 1997: A theory of interdecadal climate variability of theNorth Pacific ocean–atmosphere system. J. Climate, 10, 324–338.

Kushnir, Y., 1994: Interdecadal variations in North Atlantic sea sur-face temperature and associated atmospheric conditions. J. Cli-mate, 7, 141–157.

Lehoucq, R., D. Sorensen, and C. Yang, 1998: ARPACK Users’Guide: Solution of Large-Scale Eigenvalue Problems with Im-plicitly Restarted Arnoldi Methods. SIAM Publications, 140 pp.

Luchini, P., 2000: Reynolds-number-independent instability of theboundary layer over a flat surface: Optimal perturbations. J.Fluid Mech., 404, 289–309.

Marotzke, J., R. Giering, K. Q. Zang, D. Stammer, and T. Lee, 1999:Construction of the adjoint MIT ocean general circulation modeland application to Atlantic heat transport sensitivity. J. Geophys.Res., 104 (C12), 29 529–29 547.

Marshall, J., and F. Molteni, 1993: Toward a dynamical understandingof planetary-scale flow regimes. J. Atmos. Sci., 50, 1792–1818.

——, H. Johnson, and J. Goodman, 2001: A study of the interactionof the North Atlantic Oscillation with ocean circulation. J. Cli-mate, 14, 1399–1421.

Mehta, V. M., M. J. Suarez, J. V. Manganello, and T. L. Delworth,2000: Oceanic influence on the North Atlantic Oscillation andassociated Northern Hemisphere climate variations: 1959–1993.Geophys. Res. Lett., 27, 121–124.

Molteni, F., and T. Palmer, 1993: Predictability and finite-time insta-bility of the northern winter circulation. Quart. J. Roy. Meteor.Soc., 119, 269–298.

Navarra, A., 1993: A new set of orthonormal modes for linearizedmeteorological problems. J. Atmos. Sci., 50, 2569–2583.


Neelin, J. D., and W. Weng, 1999: Analytical prototypes for ocean–atmosphere interaction at midlatitudes. Part I: Coupled feedbacksas a sea surface temperature dependent stochastic process. J.Climate, 12, 697–721.

Rodwell, M. J., D. P. Rowell, and C. K. Folland, 1999: Oceanicforcing of the wintertime North Atlantic oscillation and Euro-pean climate. Nature, 398, 320–323.

Stommel, H., 1948: The westward intensification of wind-drivenocean currents. EOS, Trans. Amer. Geophys. Union, 99, 202–206.

Sutton, R. T., and M. R. Allen, 1997: Decadal predictability of North

Atlantic sea surface temperature and climate. Nature, 338, 563–566.

Wallace, J. M., and D. S. Gutzler, 1981: Teleconnections in the geo-potential height field during the Northern Hemisphere winter.Mon. Wea. Rev., 109, 784–812.

Watanabe, M., and M. Kimoto, 2000: Ocean atmosphere thermalcoupling in the North Atlantic: A positive feedback. Quart. J.Roy. Meteor. Soc., 126, 3343–3369.

Weng, W., and J. D. Neelin, 1998: On the role of ocean–atmosphereinteraction in midlatitude interdecadal variability. Geophys. Res.Lett., 25, 167–170.

the role of neutral singular vectors in midlatitude air–sea coupling...

Documents