## 1. Introduction

The climate system can be divided into several component subsystems; for example, the atmosphere, the ocean, sea ice, etc. On short enough timescales, these component subsystems can often be studied in isolation. On longer timescales, the interaction between the different subsystems, say between the atmosphere and the ocean, becomes more important. On interannual timescales, for example, coupled ocean–atmosphere interaction associated with the El Niño–Southern Oscillation plays a dominant role in the climate system. The nature and importance of ocean–atmosphere interaction on decadal and longer timescales is less well understood. One reason for this is the lack of sufficiently long data records. Even if adequate data were available, observational analysis cannot always answer questions related to causality, such as Does the atmosphere force the ocean on interdecadal timescales, or vice versa? To answer such questions, one often needs to resort to models of the ocean–atmosphere system. The most comprehensive of these models, the coupled general circulation models (GCMs), are still fairly expensive to use as far as multicentury integrations are concerned, although some long integrations have been carried out (e.g., Delworth et al. 1993; Manabe and Stouffer 1996). Furthermore, because of their inherent complexity, it is often easier to use GCMs to simply simulate phenomena rather than to actually understand the mechanisms responsible for the phenomena.

In this study, we use an ocean–atmosphere model with an intermediate degree of complexity—simpler than GCMs but more complex than, say, box models—to investigate climatic variability on interdecadal timescales. [We shall use the term “interdecadal” to refer to, rather loosely, timescales of O(10–100) yr.] The model we use is the one previously described by Saravanan and McWilliams (1995), hereafter SM95. It is an idealized ocean–atmosphere model consisting of a global two-level atmospheric model coupled to a zonally averaged sector ocean model. Although SM95 used the model primarily to study the multiple equilibrium structure of the mean climate and forced climate transitions, the model also exhibits natural oscillations on interdecadal timescales. The purpose of this study is to analyze the properties of this interdecadal oscillation and to elucidate the mechanisms behind it.

The coupled model described by SM95, although highly simplified, does incorporate several processes that are believed to play an important role in midlatitude ocean–atmosphere interaction; for example, horizontal transport of heat and moisture by stationary and transient eddies in the atmosphere, and the meridional transport of heat and salt by the oceanic thermohaline circulation. We hope that by explaining the mechanisms responsible for interdecadal variability in this model, we may increase our understanding of such variability in the real climate system as well.

We shall restrict our attention to *intrinsic* variability in the extratropics. By “intrinsic variability” we mean variability that is locally attributable to either the ocean or the atmosphere; that is, we exclude the variability associated with radiative effects of volcanic emissions, solar flux variability, anthropogenic effects, etc. We hope to address the following questions regarding intrinsic extratropical variability on interdecadal timescales:

- Is the variability driven primarily by atmospheric processes, oceanic processes, or coupled ocean–atmosphere interactions?
- What determines the frequency spectrum of such variability?
- What determines the spatial structure of such variability?

Observational studies suggest that upper-oceanic temperature variability in the middle latitudes may be caused primarily by atmospheric forcing on timescales of up to a few decades (e.g., Cayan 1992; Deser and Blackmon 1993), although intrinsic oceanic variability may become more important on longer timescales (e.g., Kushnir 1994). In the region with the best data coverage, the North Atlantic, negative SST anomalies occur concurrently with positive anomalies in surface wind speed and vice versa. The indication is that stronger surface winds are associated with the cooling of the ocean through increased sensible and latent heat fluxes. The subsurface structure of observed interdecadal temperature variations in the North Pacific also seems to indicate that the variability originates at the surface and penetrates downward (Deser et al. 1996).

Ocean modelers have taken several different approaches to studying the ocean’s role in interdecadal variability. One class of ocean modeling studies involves using the observed surface heat flux and wind stress anomalies to force the ocean. For example, Miller et al. (1994) were able to reproduce some of the observed interdecadal variations in the Pacific using such an approach. However, one cannot necessarily conclude from such studies that the atmosphere forces the ocean, because the observed flux anomalies used to force the model could already contain information about oceanic feedbacks.

Another class of ocean modeling studies has tended to focus on self-sustaining thermohaline oscillations that have been seen in ocean models subject to a variety of simplified surface boundary conditions (e.g., Weaver and Sarachik 1991; Winton and Sarachik 1993; Greatbatch and Zhang 1995). Since these boundary conditions, such as the traditional “mixed” boundary conditions, incorporate rather simplified representations of atmospheric feedbacks, it is not clear to what extent the real coupled ocean–atmosphere system is capable of supporting such self-sustained oscillations (Chen and Ghil 1995). A third class of ocean modeling studies has relied on stochastic surface forcing to excite oceanic modes of variability on decadal to centennial timescales (e.g., Mikolajewicz and Maier-Reimer 1990; Mysak et al. 1993; Weisse et al. 1994; Capotondi and Holland 1997). Our study suggests that this approach may turn out to be the most relevant for understanding interdecadal variability in the coupled system, although it is not clear what the most appropriate surface boundary conditions would be for an uncoupled ocean model. There have also been some recent studies of interdecadal variability in coupled ocean atmosphere GCMs (e.g., Delworth et al. 1993; Latif and Barnett 1994; Robertson 1996), although there is no clear consensus on the primary mechanism behind such variability. Latif and Barnett (1994) have proposed a mechanism based on ocean–atmosphere interaction that relies upon a fairly large atmospheric response to SST anomalies.

Our analysis of interdecadal variability in the idealized coupled model shows that although the model itself is deterministic, it is useful to conceptualize certain aspects of atmospheric variability as being stochastic in nature. This motivates us to carry out stochastically forced integrations of the ocean model. These integrations demonstrate that the preferred spatial patterns of very low frequency in the atmosphere force an oscillatory mode in the ocean with its own preferred spatial pattern. The feedback from the ocean affects the amplitude of the variability but does not appear to control the spatial patterns associated with it. In section 2, we briefly describe the coupled model and its climatology. The data analysis techniques are described in section 3. Interdecadal variability present in the coupled-model integration is analyzed in detail in section 4. In section 5, we describe several uncoupled integrations using the atmospheric/oceanic component of the coupled model to study different aspects of ocean–atmosphere interaction. Integrations using linear and stochastically forced versions of the ocean model are described in section 6, followed by concluding remarks in section 7.

## 2. Model description

The coupled model used in this study is the same as that used by SM95, where it is described in considerable detail. We summarize below some of the important features of the coupled model and its climatology.

### a. Atmospheric model and land surface processes

The atmosphere is modeled using the moist primitive equations. The domain is global, with T21 spectral truncation in the horizontal and two pressure levels in the vertical, corresponding to 250 and 750 mb, respectively. The prognostic variables are horizontal velocity **u** = (*u,* *υ*), temperature *T,* and specific humidity *q.* Simplified parameterizations are used to represent unresolved physical processes. Shortwave and longwave radiation are represented using a gray atmosphere approximation. The seasonal cycle of insolation is included. Planetary albedo is parameterized as a linear function of surface temperature. A moist adiabatic adjustment parameterization is used for precipitation. Rayleigh friction is used in the lower model level to represent frictional drag at the surface. Surface fluxes of sensible heat and evaporation are parameterized using simple bulk aerodynamic formulations. Highly scale selective (^{8}_{H}

### b. Oceanic model

We consider a single-sector ocean basin interacting with the global atmosphere (Fig. 1). The ocean sector occupies 30% of the surface area, which is somewhat larger than that of the Atlantic basin and somewhat smaller than that of the Pacific basin. The ocean model is two-dimensional—that is, all fields are assumed to have no variations in the zonal direction, leaving only meridional and vertical variations. The governing equations of the ocean model are the two-dimensional Boussinesq equations, in the infinite Prandtl number limit, with temperature *T* and salinity *S* as the prognostic variables. The meridional transport streamfunction Ψ is determined diagnostically from the *T–S* distribution. The ocean model has a flat bottom, with 48 equally spaced levels in the vertical extending from 5000-m depth to the surface. The meridional domain consists of 64 equal-area grid intervals from pole to pole. Although highly idealized, this two-dimensional representation can capture several important features of the oceanic thermohaline circulation with some degree of realism, as shown by previous studies (e.g., Marotzke et al. 1988; Wright and Stocker 1991; SM95). Important deficiencies of the model are that there is no explicit role for wind-driven ocean dynamics, and the role of sea ice is neglected.

### c. Coupling and model climatology

The ocean model is driven by heat and freshwater fluxes from the atmosphere, zonally averaged across the ocean basin. The wind stress from the atmospheric model does not affect the ocean. The zonally uniform temperature at the top level of the ocean model, the “SST,” serves as the lower boundary condition for the atmospheric model. Surface fluxes for the atmosphere are computed over the entire globe, over both land and sea grid points. The atmospheric and the oceanic models are both integrated with a 1-h time step, and the flux exchange between them occurs synchronously. No flux adjustments were used. The climatology of the coupled model is analyzed in some detail by SM95, in the context of multiple equilibria and transitions in the climate system. In this study, we consider only one of the three coupled equilibrium states obtained by SM95, which is referred to as the pole-to-pole (PP) equilibrium. We chose this state because its oceanic circulation resembles that of the present-day thermohaline overturning in the Atlantic basin.

Some important features of the northern winter climatology of the atmospheric model (for the PP state) are shown in Fig. 2. The 250-mb streamfunction *ψ*_{250} shows a midlatitude jet that is strongest near the baroclinic region on the western boundary of the ocean basin (Fig. 2a). Consistent with that, the transient eddy kinetic energy at 250 mb shows a “storm-track” region with maximum eddy activity slightly downstream of the jet maximum (Fig. 2b).

The annual mean climatology of the ocean model is shown in Fig. 3. There is a basin-wide overturning circulation, with a narrow sinking region in the northern high latitudes, and a broad upwelling region elsewhere (Fig. 3a). The strength of the overturning is about 60Sv, which—considering that the basin is about twice as wide as that of the Atlantic—may be somewhat stronger than the actual magnitude of overturning associated with the Atlantic thermohaline circulation. The thermocline is too diffuse, and the lateral gradients in the deep ocean are rather strong (Fig. 3b). The salinity exhibits “realistic” subtropical upper-ocean maxima, but the polar minimum in the Southern Hemisphere is rather strong (Fig. 3c).

## 3. Analysis technique

Our datasets usually consist of 4000 yr of annually averaged data for several atmospheric–oceanic variables, derived from coupled and uncoupled model integrations. To focus attention on the interdecadal timescale, we carry out much of our analysis after applying a nonrecursive 201-point filter (Otnes and Enochson 1978) on the once-yearly data, to retain variability in the 10–100-yr band. We shall refer to this filtered data as *bandpass* data. We describe below two techniques that we will be using to analyze the data: one is standard principal component analysis (PCA); the other is a singular-value-decomposition technique that we have developed to analyze relationships between two variables at multiple lags.

### a. Principal component analysis

Our analysis will make considerable use of empirical orthogonal functions (EOFs) of the bandpass data. Consider a field *f*(**x**_{j}, *t*_{n}), where **x**_{j} denotes the spatial gridpoint locations with corresponding area-weights *w*^{2}_{j}*t*_{n}, *n* = 0, . . . , *N* − 1 denotes the sampling time, with *N* denoting the total number of samples. We compute the area-weighted covariance matrix, *N*^{−1} Σ_{n} *w*_{j}*f*(**x**_{j}, *t*_{n})*f*(**x**_{j′}, *t*_{n})*w*_{j′}, and its eigenvectors or EOFs. We denote the *m*th EOF of *f* by *E*_{m}[*f*; **x**_{j}], or simply *E*_{m}[*f*], and the corresponding eigenvalue or associated variance as ^{2}_{m}*E*_{m}[*f*] would refer to the set of EOFs for that particular integration.) The EOFs are normalized so that Σ_{j} *E*_{l}[*f*; **x**_{j}]*E*_{m}[*f*; **x**_{j}] = *δ*_{lm}. However, when we display the EOFs, we always display the “dimensional” EOF pattern, ε_{m}*E*_{m}[*f*; **x**_{j}]/*w*_{j}, so as to give an idea of the typical variability associated with the pattern. The temporal coefficients (or principal components) *P*_{m}[*f*; *t*] associated with each EOF are defined as *P*_{m}[*f*; *t*] = Σ_{j} *E*_{m}[*f*; **x**_{j}]*f*(**x**_{j}, *t*).

*index,*and compute lead–lag correlations with respect to it. Consider an index

*g*(

*t*

_{n}) of zero mean and unit variance. We define the covariance pattern

*C*[

*f*;

**x**

_{j},

*τ*] of a field

*f*(

**x**

_{j},

*t*

_{n}), lagging the index by time

*τ*, as follows:Since

*g*is a normalized time series, the covariance patterns are the same as regression patterns. As in the case of EOFs, we always display the unweighted spatial pattern of covariance,

*C*[

*f*;

**x**

_{j},

*τ*]/

*w*

_{j}, in the figures. Since the EOFs of

*f*form a complete basis, we may expand the covariance pattern as follows:where the covariances in EOF space,

*C*

_{m}[

*f*;

*τ*], are defined asand corresponding correlation coefficients in EOF space,

*γ*

_{m},[

*f*;

*τ*], are defined asMost of our covariance calculations were carried out after truncating the bandpass data for each field to the domain spanned by its first 18 EOFs.

### b. Multiple-lag singular-value decomposition

One of the primary goals of our analysis will be to describe correlated behavior between oceanic and atmospheric fields. Since standard principal-component analysis is designed to capture the variance associated with a single variable, it is not really convenient for studying the correlated behavior between different variables. Bretherton et al. (1992) have recently reviewed the meteorological applications of the standard linear algebraic technique of singular-value decomposition (SVD) to detect highly “correlated” spatial patterns between two different fields. These applications usually involve SVD of the covariance matrix between the time series of two different fields at a specified lag. Therefore, we shall refer to it as “single-lag SVD.”

Since we will be interested in relationships between two variables at different lags, we describe below a variant of the SVD approach that can be applied over a range of lags. The technique involves SVD of the covariance matrix between the time series of a vector field *f* and the time series of a scalar index *g* at different lags. We shall refer to it as “multiple-lag SVD” or MSVD to distinguish it from single-lag SVD. Unlike the SVD approach of Bretherton et al. (1992), which deals with the spatial dimension in both variables, MSVD deals with the spatial dimension in one variable and the lag-time dimension in the other variable. The MSVD technique combines some of the ideas behind single-lag SVD and a different technique, singular spectrum analysis (SSA), which was developed to study periodicities in a single time series [see the review by Vautard et al. (1992)].

*f*and index

*g*over a set of time lags

*τ*

_{k}= (

*k*− 1)Δ

*t, k*=

*k*

_{0}, . . . ,

*k*

_{1}, where Δ

*t*denotes the sampling interval. In the terminology of SSA,

*K*=

*k*

_{1}−

*k*

_{0}+ 1 is called the

*window length*or

*embedding dimension,*and should be chosen based on the approximate period of the oscillatory phenomenon of interest (cf. Vautard et al. 1992). Consider the truncated

*M*-dimensional EOF space of field

*f.*The lagged covariance matrix in EOF-space

*C*

_{mk}≡

*C*

_{m}[

*f*;

*τ*

_{k}], with

*M*rows and

*K*columns, as defined by (3) may be decomposed using SVD as follows (cf. Strang 1988):such that the singular values

*λ*

_{l}are in descending order (

*λ*

_{1}≥

*λ*

_{2}≥ . . . ≥

*λ*

_{M}). Here we have assumed that

*M*≤

*K,*since we can always choose to truncate the EOF expansion as severely as we wish. We shall refer to the (left, right) singular vector pair (

_{ml},

_{kl}) associated with the

*l*th singular value as the

*l*th singular mode. Each singular mode is associated with a unique fraction of the total squared covariance (cf. Bretherton et al. 1992), which may be referred to as the squared-covariance fraction (SCF) and is given byThis is analogous to the variance fraction associated with EOFs. Therefore, the first few singular modes can be thought of as being the modes that most concisely describe the spatiotemporal structure of the lag-lead covariance between

*f*and

*g.*

*l*th singular mode in gridpoint space, denoted by

_{l}[

*f, g*;

**x**

_{j}] or simply

_{l}[

*f*], may be expressed as a linear combination of the EOFs; that is,We shall refer to

_{l}[

*f*] as the

*l*th spatial singular pattern (SSP) and to the corresponding

_{kl}as the

*l*th temporal singular pattern (TSP). One may think of the SSPs as being like a “rotation” of the EOFs of

*f*that best captures the covariance with respect to index

*g.*The expansion coefficients

*σ*

_{l}associated with each SSP are computed as linear combinations of the principal components

*P*

_{m}as follows:Since the SSPs are orthogonal in the truncated EOF-space representation, each SSP is associated with a unique portion,

*ϑ*

^{2}

_{l}

*f,*given byjust as each EOF is associated with variance

^{2}

_{m}

*ϑ*

_{l}

_{l}[

*f, g*;

**x**

_{j}]/

*w*

_{j}, so as to give an idea of the typical variability associated with the pattern. One can also show from (5) that the covariance between

*σ*

_{l}and

*g*at lag

*τ*

_{k}is given simply by

*λ*

_{l}times the

*l*th TSP. Thus, the correlation between

*σ*

_{l}and

*g*at lag

*τ*

_{k}would be given by

*λ*

_{l}

_{kl}/

*ϑ*

_{l}.

## 4. Interdecadal variability in the coupled integration

A preliminary analysis of the variability associated with the PP equilibrium state of the coupled ocean–atmosphere model was presented by SM95, based on a 1000-yr coupled integration. We have now carried out a subsequent 4000-yr extension of that coupled integration, hereafter referred to as the *F*_{s}), evaporation (*E*), and precipitation (*P*). To limit data volume, only annual mean data were saved for the prognostic variables and the surface fluxes over the 4000-yr period. Additionally, seasonal mean data, including atmospheric eddy statistics and radiative fluxes, were saved over a shorter (1000-yr) period.

We now focus our attention on variability in the

### a. Coupled oceanic variability

We computed EOFs for the oceanic variables Ψ, *T, S* using 10–100-yr bandpass-filtered data. The raw (unfiltered) data were then projected on the EOFs to obtain the projection time series, from which the variance spectrum was computed. We used raw data, rather than filtered data, to ensure that any peaks in the spectrum are not simply an artifact of the time filter. The spectra were computed by removing any linear trend from the unfiltered 4000-yr time series, applying a Welch data window, and then using a fast Fourier transform. The resulting spectrum was smoothed by “binning,” using a bin size of 15 frequency values.

Although the EOFs of oceanic variables were computed over the entire model domain, only the Northern Hemisphere is shown, because that is where practically all the variance is localized. The dominant EOFs of Ψ and *T* and the corresponding variance spectra are shown in Figs. 5 and 6. Most of the bandpass variance of Ψ (∼80%) is associated with *E*_{1}[Ψ], which shows localized structure in the northern mid-/high latitudes, near the sinking region (Fig. 5a). The amplitudes are fairly small, of the order of 1 Sv (Sv ≡ 10^{6} m^{3} s^{−1}). The corresponding variance spectrum shows a prominent peak in the 30–50-yr period range (Fig. 6a). Also shown in the figure are the 5% and 95% a posteriori confidence intervals for a reference “red noise” spectrum, obtained by fitting a first-order autoregressive [AR(1)] process to the time series. We see that there is a statistically significant spectral peak in the 30–50-yr range. Figure 6 also shows a reference spectrum computed by fitting a second-order autoregressive [AR(2)] process to the time series (Wilks 1995). We see that the spectrum is rather well described by this AR(2) process, which has a “pseudoperiodicity” of about 40 yr and autocorrelation decay timescale of about 30 yr.

Next we consider the *T* field, where almost half the bandpass variance is associated with the first two EOFs (Figs. 5b,c). Here, *E*_{1}[*T*] is confined mostly to the upper ocean, in the northern mid-/high latitudes, and has a dipolelike structure with extrema at 45°N and 70°N. Here, *E*_{2}[*T*] is essentially a high latitude mode in the vicinity of the sinking region and has fairly deep structure. The variance spectrum of *E*_{1}[*T*] is dominated by a peak in the 30–50-yr range (Fig. 6b). The variance spectrum of *E*_{2}[*T*] also shows a weak peak in the 30–50-yr range, but set against a background of red noise. It is worth noting that the north–south dipole structure of SST associated with *E*_{1}[*T*] bears a qualitative resemblance to the dipolelike modes of decadal variability noted in the observed SST field in the North Atlantic (e.g., Deser and Blackmon 1993; Kushnir 1994).

In addition to the EOFs of Ψ and *T,* the EOFs of salinity *S* were also computed. Both *T* and *S* affect the momentum field through their contribution to the buoyancy, which is proportional to *α**T* − *β**S* in the ocean model (*α* is the thermal expansion coefficient and *β* is the haline contraction coefficient). Figure 6b also shows the variance spectrum of *E*_{1}[*S*], scaled by *α*^{2}/*β*^{2} to make it comparable to the temperature variance. Here, *E*_{1}[*S*] does not exhibit a spectral peak in the 30–50-yr period range, and the overall variance on interdecadal timescales is considerably weaker when compared to the first two temperature EOFs. The other EOFs of salinity also show similar features. We conclude that salinity variations are likely to play only a minor role in the dynamics of interdecadal variability in this model. Therefore, we exclude salinity from the rest of our analysis, although we shall briefly revisit this issue when discussing uncoupled ocean-only integrations (section 5c.).

We proceed now to characterize this interdecadal signal in the oceanic temperature field using simple correlation analysis. Since *E*_{1}[Ψ] is rather dominant, and has a well-defined maximum in its variance spectrum, we find it convenient to define an *index* based upon the first principal component *P*_{1}[Ψ], computed from bandpass data. We shall refer to this time series, after normalizing it to have zero mean and unit variance, as the Ψ_{c}

To understand the lead-lag relationship between *T* and Ψ, we compute correlations with respect to the Ψ_{C}*P*_{1}[Ψ] (Fig. 6) has an autocorrelation decay timescale of 30 yr. As a conservative estimate, this would suggest that the 4000-yr time series of the Ψ_{C}*t* test. We also computed correlations using a synthetic statistically independent time series created by swapping two halves of the Ψ_{C}_{C}

The strongest correlations between *T* and Ψ (Fig. 7) show that *P*_{1}[*T*] is correlated with the Ψ_{C}*P*_{2}[*T*] is correlated at zero lag, with an approximate quadrature relationship between *P*_{1}[*T*] and *P*_{2}[*T*]. In other words, *E*_{1}[*T*] and *E*_{2}[*T*] represent two phases of a propagating mode, with propagation in the same sense as that of advection due to the mean meridional overturning. Since the dominant EOFs of Ψ and *T,* namely *E*_{1}[Ψ], *E*_{1}[*T*], and *E*_{2}[*T*], are highly correlated, they may be said to constitute a well-defined “oceanic interdecadal oscillation,” which is characterized by the Ψ_{C}

The advective nature of the interdecadal oscillation is also seen in the covariance between SST and the Ψ_{C}*τ* ≈ −22 and moves slowly northward until about *τ* ≈ −10; beyond about 60°N, the anomaly intensifies, and the northward propagation speeds up; the anomaly finally “disappears” at *τ* ≈ 0. In the meantime, a weaker positive anomaly has formed around 45°N at *τ* ≈ −8 and moves northward in a similar fashion. The evolution of the SST anomalies clearly has an advective character, and is consistent with the structure of the time-averaged meridional circulation (Fig. 3a). Although the SST anomalies are fairly small (≤0.2K), they occur in a zonally uniform manner over a fairly wide ocean basin. Hence, they represent a planetary-scale SST pattern, and the associated change in the oceanic heat content is quite significant.

The streamfunction variability associated with the interdecadal oscillation seems to be localized in the northern mid-/high latitudes, between about 50°N and 75°N, that is, over a region with a length scale of about 2800 km. The upper branch of the mean meridional overturning is associated with northward velocities of ∼0.6 cm s^{−1} in this region, implying an advective timescale of about 15 yr. The fact that this is about one-half the period associated with the interdecadal oscillation suggests a plausible mechanism for the period, based upon the following two processes:

- advection of the mean temperature field by the streamfunction anomalies; and
- advection of temperature anomalies by the mean streamfunction.

Since the interdecadal oscillation is very weak, one may suppose that the nonlinear advection of temperature anomalies by the streamfunction anomalies is likely to be small. This supposition is confirmed quantitatively in section 6, where it is shown that a linearized version of the ocean model can reproduce the interdecadal oscillation quite well.

The proposed mechanism for the oscillatory behavior is as follows:

- Say a deep negative temperature anomaly exists in the polar sinking region, for example, as in the positive phase of
*E*_{2}[*T*]. This would be associated with enhanced meridional overturning, as suggested by the strong simultaneous correlation between*E*_{2}[*T*] and*E*_{1}[Ψ] at zero lag (Fig. 7). - Stronger meridional overturning would result in warmer waters from the south being advected north of 50°N, resulting in a positive
*T*anomaly. Since the mean meridional temperature gradients are strongest near the surface, the anomalies would also have maximum amplitudes near the surface, as evidenced by the structures of*E*_{1}[*T*] and*E*_{2}[*T*] (Figs. 5b,c). The positive*T*anomalies would be advected to the sinking region by the mean meridional circulation, and eventually reach 75°N after about 15 yr or so. This would result in weaker meridional overturning, and the whole process can then repeat itself starting from step 1, but with the opposite sign.

Although the above mechanism may explain the timescale of the oscillation, it does not completely explain the structure of *E*_{1}[*T*], whose features extend farther south, almost to 30°N. We believe this aspect of the oceanic interdecadal variability is directly related to the spatial structure of atmospheric forcing; this will become clearer in the context of the stochastic experiments discussed in section 6. Furthermore, the AR(2) fit (Fig. 6) is associated with strong decay—with a timescale of about 30 yr—implying that the oscillation is strongly damped.

Another feature of the above mechanism is that in step 1, although *E*_{2}[*T*] is fairly localized meridionally, the associated streamfunction anomaly *E*_{1}[Ψ] is more spread out. In our idealized ocean model, this nonlocal response is associated with the inverse Laplacian relationship between buoyancy anomalies and Ψ anomalies in the infinite Prandtl number limit. In the real ocean–atmosphere system, one may expect such a nonlocal response to arise from the pressure gradient forces induced by the buoyancy anomalies. The nonlocality of the Ψ response is also similar to that present in simple “loop oscillation” models of the thermohaline circulation (cf. Welander 1985; Winton and Sarachik 1993), where a local buoyancy anomaly produces a torque affecting the whole loop. An important difference is that in our case the timescale is not determined by the transit time around the entire loop, but only over a portion of the loop. That is, the idealized model exhibits an “incomplete loop oscillation.” Note also that *E*_{1,2}[*T*] do not show *T* anomalies closing a circuit through the deep ocean (Figs. 5b,c). The actual overturning timescale associated with the mean meridional circulation would be of the order of centuries, since the upwelling velocities in the subtropics are very weak [O(10 m yr^{−1})].

The mechanism for the interdecadal oscillation suggested above bears some similarities to the advective mechanisms for decadal variability in sector ocean GCMs described by Weaver and Sarachik (1991) and Greatbatch and Zhang (1995), in that the period is related to the time it takes to advect anomalies from the middle to the high latitudes. The oscillation seems to be predominantly thermal in nature in our case (see Fig. 6), as it was in Greatbatch and Zhang (1995; Fig. 5), whereas salinity fluctuations play a dominant role in the oscillations described by Weaver and Sarachik (1991). Although the use of a linear equation of state may have affected our results, we believe that our choice of a “cold water” value of 0.8 × 10^{−4} K^{−1} for the thermal expansion coefficient, which would correspond to linearizing about surface temperature values of ∼2°C, should help improve the validity of our conclusions, in so far as middle/high latitude processes are concerned.

### b. Coupled atmospheric variability

We begin our analysis of atmospheric variability associated with the oceanic interdecadal oscillation by focusing on the heat flux into the surface *F*_{s}. Unlike the relationship between oceanic *T* and the Ψ_{C}_{C}*F*_{s} and Ψ_{C}*τ* = −30, . . . , 10 for the MSVD analysis.

The dominant SSP of *F*_{s}, _{1}[*F*_{s}] is shown in Fig. 9. Also shown in the plot title are the squared covariance fraction or SCF for the SSP, the variance *ϑ*^{2}_{l}*σ*_{l} and the Ψ_{C}_{1}[*F*_{s}] is associated with an SCF value of 90% and thus captures most of the covariance. It is also significantly correlated with the Ψ_{C}*τ* = −8 yr, that is, simultaneous with the *E*_{1}[*T*] (Fig. 7). The relationship between _{1}[*F*_{s}] and the Ψ_{C}*τ* = −8, which is indicative of a simultaneous correlation between _{1}[*F*_{s}] and *E*_{1}[*T*] at near-zero lag.

The spatial structure of _{1}[*F*_{s}] (Fig. 9) is dominated by a tilted dipole structure localized in the mid-/high latitudes near the western boundary of the ocean, with each “pole” flattened along the SE–NW direction. The relationship between _{1}[*F*_{s}] and *E*_{1}[*T*] is such that in regions where the *F*_{s} anomaly is positive, the SST anomaly is also positive and vice versa (Figs. 5b, 9).

Next, we study the relationship between the atmospheric flow and the oceanic interdecadal oscillation. As in the case of surface heat flux, most of the correlated signal occurs in the extratropical regions of the Northern Hemisphere. Therefore, one may expect that the atmospheric dynamics associated with the interdecadal oscillation is geostrophic to a good approximation and that the atmospheric flow is well described by just two fields—the horizontal streamfunction *ψ* on the two model levels, 750 mb and 250 mb. As in the case of *F*_{s}, we use MSVD analysis and restrict our attention to the first SSP of *ψ*_{750} and *ψ*_{250} (Fig. 11) because the other SSPs are not very highly correlated with the Ψ_{C}

The first SSP of low-level streamfunction, _{1} [*ψ*_{750}], shows a predominantly monopolar structure centered around 55°N near the western boundary of the ocean basin, which will be referred to as the WO (western ocean) pattern. There is also a strong dipolar structure over land, located downstream of the monopole, which will be referred to as the EL (eastern land) pattern. The sign and location of the *ψ*_{750} anomalies is such that negative *F*_{s} anomalies tend to coincide with northwesterly wind anomalies and vice versa. As will be discussed later, the dominant contribution to the *F*_{s} anomalies comes from the northern winter season. This suggests that anomalous low-level advection of cold continental airmasses over the ocean is responsible for the strong *F*_{s} anomalies near the western boundary of the ocean basin.

Although the atmospheric model, being formulated in pressure coordinates, has no surface pressure variations, one may think of *ψ*_{750} as being analogous to the surface pressure, in that it determines the low-level flow. It is interesting to note that, with this interpretation, the WO pattern seen in _{1}[*ψ*_{750}] is reminiscent of the decadal surface pressure pattern seen in observational studies of interdecadal variability over the North Atlantic (e.g., Deser and Blackmon 1993).

The first SSP of upper-level streamfunction, _{1}[*ψ*_{250}] shows broad zonal structure with three distinct features superimposed—a dipole off the western boundary of the ocean basin, a dipole structure mostly over land downstream of the ocean basin, and a monopole north of 60°N lying in between.

The spatial structure of the dominant SSPs of *F*_{s} and *ψ*_{750} (Figs. 9, 11a) shows that much of the atmosphere–ocean heat-flux exchange occurs near the western boundary of the ocean basin, in the storm-track region. To determine how different atmospheric processes contribute to the *F*_{s} anomaly structure shown in Fig. 9, we computed the different terms in the atmospheric heat budget, such as the latent heat flux (which is negatively proportional to the evaporation), the temperature tendency due to low-level advection by the time-mean flow, the temperature tendency due to low-level transient eddy heat-flux convergence, as well as radiative fluxes at the top of the atmosphere. To determine the seasonal dependence, we computed the bandpass covariances for these terms separately for each of the four seasons, using seasonally averaged data over a 1000-yr period of the *τ* = −8, that is, when the maximum correlations between the atmospheric flow and the oceanic interdecadal oscillation tend to occur. The covariances were also spatially averaged over a 20° longitude band over the ocean adjoining the western boundary, where the maximum flux anomalies occur (Fig. 9).

Of the four seasons, we find that the northern winter makes the largest contribution to the dominant annual mean *F*_{s} anomaly structure shown in Fig. 9. The covariant part of the atmospheric heat budget associated with the interdecadal oscillation is dominated by the anomalies in advective temperature tendency due to the low-level seasonal mean flow (Fig. 12). The low-level transient eddy heat-flux convergence provides only a small contribution, which is actually of the opposite sign. The contribution to the heat budget from anomalies in top of the atmosphere radiation (not shown) is very small. In terms of how the atmospheric heat content anomalies are actually communicated to the ocean surface, latent heat fluxes seem to be responsible for about half the *F*_{s} anomalies in the midlatitudes, with the remaining part coming from surface radiative and sensible heat-flux anomalies. Note also that the local wintertime *F*_{s} anomalies are fairly large, with maximum values exceeding 15 W m^{−2}.

## 5. Uncoupled integrations

The analysis of the oceanic interdecadal oscillation in the coupled integration, and its relationship to the atmospheric variability, indicates that the interdecadal oscillation may primarily be an oceanic response to atmospheric forcing, with ocean advection playing an important role in determining the structure of the response. However, it is very difficult to draw definitive conclusions regarding these and other mechanistic issues relating to ocean–atmosphere interaction by analyzing the coupled variability alone. To reach more definitive conclusions, we need to consider the behavior of the atmosphere or the ocean in isolation, that is, without feedbacks from the other component. To this end, we carry out several uncoupled integrations using the atmospheric and oceanic components of the coupled model. The boundary conditions for these integrations are derived from the coupled integration and chosen to represent different aspects of ocean–atmosphere interaction.

### a. Atmosphere-only integrations

The ocean forces the atmosphere through SST variations. Therefore, we designed a set of three atmospheric integrations, referred to as

At the next level of approximation, in the

The third integration involves coupling the atmosphere to a 100-m-thick slab ocean or “mixed layer.” A seasonally varying horizontal heat transport term, derived from the

Each of the above three integrations was carried out for a period of 4000 yr. All three integrations have essentially the same mean climate as the

### b. Ocean-only integrations

We also carried out several ocean-only integrations, analogous to the atmosphere-only integrations, to isolate the roles of different aspects of atmospheric forcing. Since the atmosphere forces the ocean through surface fluxes of heat and freshwater, we used pure flux boundary conditions to force the ocean model (i.e., no restoring for temperature or salinity). Four different ocean-only integrations were carried out, referred to as _{T}, _{S}, and *F*_{s} and *E* − *P*) at the surface; _{T}, forced by the time-varying *F*_{s} only, with *E* − *P* held fixed at its climatological annual mean value; _{S}, forced by time-varying *E* − *P* only, with *F*_{s} held at its climatological annual mean value.

The initial condition for all three integrations was the steady state obtained by “spinning up” the ocean model using the climatological annual mean values of both *F*_{s} and *E* − *P* as the flux boundary conditions. The ocean model was integrated without the seasonal cycle, with a time step of 10 days. The annual mean surface flux values were linearly interpolated between successive years to provide the surface forcing at each time step.

The fourth ocean integration,

### c. Uncoupled oceanic variability

We begin our analysis of the uncoupled integrations by considering the ocean-only integrations, because some features of the *E*_{1}[Ψ] for the _{T}, and _{S} integrations. Note that for the *E*_{1}[Ψ] and the first two EOFs of *T* (not shown) are also very similar to that seen in the

The differing roles played by the surface thermal and freshwater forcing can be seen by comparing the variance spectra of *E*_{1}[Ψ] for the _{T}, _{S} integrations (Fig. 13a). The _{T} integration demonstrates that variability in the surface thermal forcing alone is capable of reproducing much of the variability associated with the interdecadal oscillation, but not all of it. The variability in the surface freshwater flux seems to produce an interdecadal response with much weaker variability (Fig. 13a). The spatial structure of *E*_{1}[Ψ] for both _{T} and _{S} integrations (not shown) is very similar to that in the coupled integration. This suggests that the interdecadal oscillation is a “preferred” mode of variability in the ocean that can be forced in different ways, although the surface thermal forcing appears to be primarily responsible for exciting it in the

Although the surface flux–forced *E*_{1}[Ψ] shows spectral peaks in the 30–50-yr range, just as in the *E*_{1}[Ψ] and the two dominant EOFs of *T* in the

Since an oceanic interdecadal oscillation, very similar to that seen in the coupled integration, is present in the _{C}*P*_{1}[Ψ] computed from the bandpass data, after normalization to have zero mean and unit variance, as the Ψ_{AO}

### d. Uncoupled atmospheric variability

Consider the atmospheric response to a prescribed pattern of SST anomalies associated with the oceanic interdecadal oscillation. Analyzing the variability in the atmosphere-only _{C}_{C}

The dominant SSPs of the lower-/upper-level streamfunction (Figs. 14a,b) show some similarities to the corresponding SSPs for the *ψ*_{750}, the dominant feature is the dipolar pattern east of the ocean basin, which closely resembles the EL pattern seen in the *ψ*_{250}, a more zonally extended dipolar pattern is present in the eastern part of the domain, bearing some similarity to the features seen over the same domain in the

What are missing in the SSPs of the *ψ*_{750} and the dipolar pattern in *ψ*_{250} adjacent to the western boundary of the ocean basin. Given that the mean zonal midlatitude winds are westerly, the *ψ*_{750} and *ψ*_{250} being coincident.

To find the missing upstream patterns, we turn to the *free* modes of the atmosphere, to distinguish them from the at least partially *forced* modes of the

We have already seen how the free modes of the atmosphere are capable of exciting the oceanic interdecadal oscillation in the _{AO}_{1}[*ψ*_{750}] shows a dominant monopolar pattern near the western boundary of the ocean basin (Fig. 14c), which is very similar to the WO pattern seen in the _{1}[*ψ*_{250}] shows a dipolar structure near the western boundary of the ocean basin—also seen in the

To obtain a more quantitative description of the relationship between the SSPs of the *ψ*_{750} and *ψ*_{250} in the *free* modes of the atmosphere. [It is worth recalling that EOFs are not generally the same as the dynamical normal modes of the atmosphere (North 1984).] We restrict the domain of the EOFs to the Northern Hemisphere because atmospheric variability in the Southern Hemisphere shows only weak correlations with the dominant modes of oceanic variability. Figures 15a–c show the first, second, and the fourth free modes for *ψ*_{750}. The first mode closely resembles _{1}[*ψ*_{750}] of the _{1}[*ψ*_{750}] of the *ψ*_{250} (Figs. 15d–f), the first mode has large-scale structure throughout the hemisphere, extending to the Tropics. The second mode is confined to the middle/high latitudes and is quite similar to _{1}[*ψ*_{250}] of the

_{1}[

*ψ*

_{750}] of the

*m*th free mode,

*p*

^{C}

_{m}

*E*

^{A}

_{m}

**x**

_{j′}] denote the free modes as defined above,

*E*

^{C}

_{m′}

**x**

_{j′}] denote the EOFs of

*ψ*

_{750}in the

*S*

^{C}

_{m′1}

*ψ*

_{750}with respect to the

*ψ*

_{C}

The projections of the variance associated with _{1}[*ψ*_{750}] on the free modes for the

The projections of the variance associated with _{1}[*ψ*_{250}] on the free modes also lead to similar conclusions (Figs. 16d,f). The first two free modes capture over 70% of the variance for all three integrations. Once again, the first free mode appears to play a dominant role in forcing the ocean (i.e.,

An important conclusion is that the basis set for low-frequency atmospheric variability is essentially the same for these different integrations, but the phase relations among the free modes—hence the preferred spatial patterns—are all quite different due to their different dynamical relation to the oceanic variability.

Next we consider the spectral properties of the free modes and compare them to the spectral properties of analogous modes in the coupled integration. We computed the Northern Hemispheric EOFs 1 and 2 of *ψ*_{250} for the

Finally, we consider an integral measure of the variance on interdecadal timescales, to highlight the dependence of the overall level of atmospheric variability on the strength of ocean–atmosphere coupling. The measure we use is the area-weighted global average of bandpass variance. We pick two fields that show contrasting behavior, the surface heat flux *F*_{s}, and the low-level atmospheric temperature *T*_{750}. Figure 18 shows the global average of interdecadal bandpass variance associated with these fields for the *F*_{s} for the different integrations, with

One normally thinks of the

When the atmosphere exchanges heat fluxes with an ocean of finite heat capacity (as in the *F*_{s} anomaly because of the air–sea temperature gradient. The resulting surface heating would produce a positive SST anomaly, and an associated reduction in the air–sea temperature difference. This negative feedback would, in general, have a damping effect on the amplitude of surface heat-flux variability. Thus one would expect the *F*_{s} variance, and the *F*_{s} variance. This argument also resolves a puzzling aspect of the ocean-only integrations discussed earlier, that the surface fluxes from the atmosphere-only

The variance of *T*_{750} (Fig. 18) also shows a modest dependence on the oceanic heat capacity, but more or less in the opposite sense to that of *F*_{s}. This should not be surprising, since weaker heat exchange with the ocean would allow atmospheric temperature anomalies to persist longer. This effect is also seen in other atmospheric variables, such as *ψ*_{750} and *ψ*_{250}, which tend to have slightly higher interdecadal variance in the coupled integration, as compared to the uncoupled integration (e.g., see Fig. 17). A similar effect, though much less pronounced, is seen in the 1000-yr GCM integrations described by Manabe and Stouffer (1996). They find that the standard deviation of 5-yr mean surface air temperature anomalies tends to be larger when the atmospheric GCM is coupled to a mixed-layer ocean (as compared to the coupled ocean–atmosphere GCM), and smaller when the atmospheric GCM is forced by climatological SSTs.

## 6. Linear and stochastic integrations

Since the Ψ, *T* variations associated with the oceanic interdecadal oscillation are fairly small compared to the spatial variations in the time mean state, one may hypothesize that the interdecadal oscillation is essentially a linear perturbation about the climatological mean state of the ocean. To test this hypothesis, we constructed a linearized version of the ocean model by expanding the nonlinear advective terms in the *T* and *S* equations (see SM95) into the mean and perturbation parts, and retaining the linear diffusion terms. The linearization was carried out about the climatological mean state of the ocean for the coupled integration. Using this linearized model, we carried out a 4000-yr ocean-only integration (the _{l} integration) forced by the anomalies of *F*_{s} and *E* − *P* from the *E*_{1}[Ψ] for the _{l} integration is almost the same as that of the *T* (not shown) are also virtually identical to that of the

We also carried out another integration with the linearized ocean model coupled to the atmospheric model, but with the advective tendency associated with the oceanic mean flow acting on *T* and *S* perturbations set to zero. We did this to isolate the role of mean flow advection in the ocean. We found that this integration did not exhibit an interdecadal oscillation in the oceanic variables, but the lag-covariance relationship between the atmospheric flow and oceanic thermal anomalies (not shown) was quite similar to that in the

An important feature of the oceanic interdecadal oscillation, as demonstrated by the

The scenario of a stochastic atmosphere forcing the ocean was considered by Hasselmann (1976). In its simplest context, the argument is that the ocean will act as an integrator of white noise atmospheric forcing, resulting in a red noise response. There are no preferred timescales in the system (except for the damping timescale for the ocean, which determines the transition from red noise to white noise behavior in the response). In our case, the ocean–atmosphere interaction involves preferred timescales in the ocean and preferred spatial structures in both systems. We would like to explore to what extent these can be reproduced in an ocean model forced by stochastic surface fluxes. Similar experiments have been carried out by other researchers using more complex ocean models.

Since we know from the _{T} integration that surface heat fluxes provide the dominant forcing (Fig. 13a), we force the ocean model using time-varying surface heat fluxes only, keeping the surface freshwater flux constant at its time mean value. We would like to explore the role of spatial correlations in atmospheric variability in exciting the oceanic interdecadal oscillation. So we construct two different synthetic stochastic time series of surface heat-flux anomalies, *F*^{′}_{sw}*y*_{j}, *t*_{n}) and *F*^{′}_{sww}*y*_{j}, *t*_{n}), for each of the 4000-yr *t*_{n} and at each meridional grid point *y*_{j} at the ocean surface. The difference between *F*^{′}_{sw}*F*^{′}_{sww}*F*^{′}_{sw}*F*^{′}_{sww}

*E*

_{m}[

*F*

_{s};

*y*

_{j}] are the EOFs of the zonally averaged

*F*

_{s}over the ocean basin, computed from the unfiltered 4000-yr time series for the

^{2}

_{m}

*s*

^{2}is the variance of

*F*

_{s}averaged over all grid points. Here,

*W*

_{m}(

*t*

_{n}),

*W*

_{j}(

*t*

_{n}) denote independent Gaussian white noise random variables with zero mean and unit variance. They have the property that

*W*

_{m}(

*t*

_{n}) is uncorrelated with

*W*

_{m}(

*t*

_{n−1}) and so on. Note that, by construction, the EOFs and associated variances of

*F*

^{′}

_{sw}

*F*

_{s}; that is, this synthetic times series preserves the spatial correlation and variance structure of

*F*

_{s}. The

*F*

^{′}

_{sww}

_{w}and

_{ww}, were carried out using the two synthetic time series as the surface forcing.

The _{w} integration reproduces the spectral peak associated with *E*_{1}[Ψ] in the coupled integration quite well, but the variance amplitude seems to be somewhat larger (Fig. 19). We believe that this discrepancy is related to the damping effect of the ocean on the surface heat fluxes. This effect is implicitly present in the surface fluxes used to force the _{l} integration because they are taken from the *T* of the _{w} integration (not shown) are also quite similar to those of the

The importance of the preferred spatial structures associated with the surface forcing is illustrated by the _{ww} integration, which is forced by white noise in space and time. As we note in Fig. 19, eliminating the spatial correlations in the forcing substantially reduces the variance associated with the interdecadal oscillation, although it is still excited. The spatial structure of *E*_{1}[Ψ] is very similar to that of the *E*_{1}[*T*], however, shows monopolar structure in SST (Fig. 20b), whereas dipolar structure was seen in the *E*_{2}[*T*] also shows some differences when compared to the *T* is consistent with the advective mechanism for the interdecadal oscillation suggested in section 4a. Hence, we conclude that the EOFs of the _{ww} integration shown in Fig. 20 represent an oscillatory mode of the ocean with a time scale of 30–40 yr, analogous to the free modes of the atmosphere described in section 5d

Since we find that several different kinds of surface flux forcing (i.e, the _{T}, _{S}, _{w}, _{ww} integrations) can excite the oceanic interdecadal oscillation, we ask the following question: what kind of surface heat flux forcing will most efficiently excite this oscillatory mode in the ocean? To answer this question, we computed the covariance between the first principal component *P*_{1}[Ψ] of the oceanic streamfunction and the surface heat flux forcing *F*_{s} for the _{ww} integration. Since the surface forcing is white noise in space and time, the spatial structure of the covariance field should identify that portion of the white noise, which actually excites the oceanic oscillatory mode.

Figure 21 shows the spatial structure of the covariance at lag *τ* = −8 (when maximum correlations of about 0.25 occur). Although somewhat noisy, the “most efficient” surface heat flux forcing pattern for exciting the oceanic interdecadal oscillation is characterized by large negative values in the high latitudes, starting from zero values around 50°N and decreasing poleward. Also shown in the figure is the spatial structure of EOF1 of zonally averaged *F*_{s} for the

## 7. Summary and discussion

Ocean–atmosphere interaction on decadal and longer timescales is an important component of climate variability. In this study we have carried out a detailed analysis of this interaction in a very long integration of an idealized ocean–atmosphere model. The idealized model has several deficiencies. For example, the zonally averaged nature of the ocean model ignores some important details such as the wind-driven gyres and boundary currents. The zonal averaging also tends to smear out the SST anomalies, thus weakening them. The simple land surface parameterization ignores the potentially important role of evaporative processes over land in determining the climate. Furthermore, we have not considered the possible role of sea ice in forcing interdecadal variability, which could turn out to be important (cf. Deser and Blackmon 1993). Nevertheless, the idealized model does qualitatively represent many of the important processes in midlatitude ocean–atmosphere interaction.

The idealized model exhibits spontaneous interdecadal oscillations of the thermohaline circulation, with a preferred timescale of 30–40 yr. The oscillation shows a north–south dipole structure in SST in the mid-/high latitudes, which is qualitatively similar to the observed decadal variability in the North Atlantic (e.g., Deser and Blackmon 1993; Kushnir 1994). By analyzing the atmospheric flow patterns associated with the oceanic interdecadal oscillation, and through mechanistic ocean-only integrations, we have shown that thermal forcing of the ocean by the atmosphere is primarily responsible for exciting the interdecadal oscillations in the ocean. Anomalous low-level heat advection by the atmospheric stationary eddy field is the dominant contributor to the surface heat-flux anomalies that force the ocean. Low-level atmospheric transient eddies have a damping effect on the surface heat-flux variability. Such an effect was also seen in the observational study of Trenberth and Hurrell (1994). Changes in the radiative flux at the top of the atmosphere are rather weak. In other words, interdecadal variability in the atmosphere is characterized by horizontal redistribution of heat by the quasi-stationary flow.

The oceanic component of the interdecadal variability has many characteristics of a linear subcritically damped oscillatory mode. The half-period of the oscillation appears to be related to the time period for the advection of temperature anomalies from the middle to the high latitudes by the thermohaline circulation. Such an advective mechanism could well operate in the real ocean–atmosphere system, although the relevant timescales would most likely differ from the 30–40-yr timescale found in the idealized model, because its geography and thermohaline circulation are only qualitatively realistic. Similar advective mechanisms have been proposed by Weaver and Sarachik (1991) and Greatbatch and Zhang (1995) to explain self-sustaining decadal timescale oscillations in a sector ocean GCM.

The interdecadal oscillations in our idealized model bear a superficial resemblance to those seen in the coupled GCM integration described in Delworth et al. (1993), in that the modulation of the strength of the thermohaline circulation is a prominent feature. The box model study of Griffies and Tziperman (1995) associated these interdecadal oscillations with a single damped oscillatory thermohaline eigenmode excited by stochastic surface forcing. Our conclusions are broadly similar to those of Griffies and Tziperman, in that oceanic interdecadal variability in the idealized model has a linear noise-driven character to it; however, salinity fluctuations do not seem to play an important role in our case.

Atmospheric variability on interannual and longer timescales in the idealized model is dominated by a small number of preferred spatial patterns, analogous to the teleconnection patterns described by Wallace and Gutzler (1981). The very low frequency modes of the atmosphere appear to have no preferred timescales, and are associated with an essentially white noise spectrum (e.g., Feldstein and Robinson 1994). They may be thought of as being random “climate noise” arising from high-frequency daily weather fluctuations (Madden 1976). One may therefore approximate very low frequency atmospheric variability as being a white noise stochastic process, but with spatially coherent structures. If one assumes that there exist one or more linear normal modes of the ocean with preferred spatial patterns, and possibly preferred timescales, then those oceanic modes with surface excitation patterns that most closely match the preferred atmospheric spatial patterns will be the ones that are most efficiently excited.

In the context of a simple harmonic oscillator, the term resonance refers to forcing the system at its preferred frequency. We may use the term “spatial resonance” to refer to the forcing of a system with its preferred spatial pattern. In the idealized model, partial spatial resonance between the dominant mode of surface heat-flux forcing and a single oceanic mode appears to be responsible for exciting the oceanic interdecadal oscillation. It seems likely, given their correspondence to observed variability, that the modes of ocean–atmosphere interaction seen in the idealized model will persist in more realistic models. However, the real climate system is likely to possess a greater variety of free modes in each component subsystem than are seen in the idealized model. The preferred modes of interaction between the atmosphere and the ocean would then correspond to those free modes that are closest to spatial resonance. To identify these modes of interaction, one could use stochastically forced ocean models to compute the preferred spatial patterns of surface excitation, and then try and match them with the atmospheric modes of variability.

There are two types of oceanic feedback that can affect the atmospheric variability. *Local feedback*: when the ocean responds to atmospherically generated surface heat-flux anomalies by forming SST anomalies, it would result in reduced air–sea temperature gradients and decreased surface heat-flux anomalies. *Nonlocal feedback*: SST anomalies produced by atmospheric forcing can be horizontally advected by the oceanic flow and produce an atmospheric response elsewhere. The parameter dependence of the local and nonlocal feedbacks is analyzed by Saravanan and McWilliams (1997), using a one-dimensional model of advective ocean–atmosphere interaction. To understand the local oceanic feedback, it is useful to introduce the notion of an *effective* oceanic heat capacity (e.g., see Wigley and Raper 1991), even for uncoupled integrations. In integrations of atmospheric models with prescribed SST, one effectively assumes an infinite heat capacity for the underlying ocean. An interactive ocean, on the other hand, would be associated with a finite effective heat capacity, whose actual value would depend not just upon the mixed layer depth, but also on the advective/diffusive processes that act on longer timescales. Although this distinction may not be important on short timescales (say, compared to the thermal relaxation timescale of the oceanic mixed layer), it cannot be ignored on decadal or longer timescales, when the oceanic feedback can have a damping effect on surface flux variability.

In our idealized coupled model integrations, we do find evidence for the local oceanic feedback described above, but not much evidence for the nonlocal feedback. The ocean temperature in the coupled model exhibits an oscillation with a timescale of 30–40 yr, and there are SST anomalies associated with this oscillation that have an advective character. One may therefore expect to see a spectral peak in the atmospheric variability associated with this oscillation, in response to the SST anomalies. However, we were unable to find any statistically significant spectral peaks in the atmospheric variables. This suggests that the nonlocal feedback associated with advected SST anomalies is too weak to be detected in the presence of stochastic background noise in the atmosphere. This differs from the conclusions of Latif and Barnett (1994), who have argued that there exist unstable coupled ocean–atmosphere modes of variability in the midlatitudes. However, our results are consistent with several GCM studies of atmospheric response to midlatitude SST anomalies, which find a relatively weak response (e.g., see Kushnir and Held 1996, and references therein). Since many of the studies, including ours, have used low horizontal resolution (R15 or T21) atmospheric models, it has been suggested that this weak atmospheric response may be a result of the transient eddy fluxes not being represented properly (cf. Kushnir and Held 1996). One of us (RS; manuscript in preparation) has carried out an analysis of atmospheric response to midlatitude SST anomalies using extended integrations of a T42-resolution GCM (National Center for Atmospheric Research Community Climate Model, version 3). This analysis also suggests that the atmosphere responds only weakly to midlatitude SST anomalies.

The notion of an effective oceanic heat capacity brings up some possible limitations on using AMIP-style integrations, where an atmospheric model is forced by observed SSTs, to simulate interdecadal variability. In regions where atmospheric variability is primarily ocean driven, such as in the Tropics, using realistic SST boundary conditions should certainly improve the simulation of tropical atmospheric variability and remotely forced extratropical variability. However, intrinsic extratropical variability in the surface fluxes could be overestimated by AMIP-style integrations, because the feedback from the ocean is ignored. Our experiments using the idealized model do show such an effect. Our results also suggest that coupling an atmospheric model to slab (or mixed layer) ocean models, which have relatively small effective heat capacities compared to the deep ocean, could result in much weaker surface flux variability, and lead to overestimation of atmospheric variability on decadal timescales.

Although the atmospheric variability in the coupled integration does not show significant spectral peaks, it does show robust correlations with oceanic variability. The correlations between the atmospheric flow and the oceanic interdecadal oscillation show extratropical spatial patterns that fall into two distinct types: *forcing* patterns and *response* patterns. Figure 22 schematically illustrates this aspect of midlatitude ocean–atmosphere interaction. The forcing patterns are associated with the stochastic components of atmospheric variability and tend to occur near the western boundary of the ocean basin, in the vicinity of the storm-track region. The surface heat-flux anomalies associated with the forcing patterns produce SST anomalies, which in turn excite the atmospheric response patterns. The response patterns tend to occur downstream of the SST anomalies and have roughly equivalent barotropic vertical structure away from the forcing region, as one may expect from a linear quasi-stationary wave response to thermal forcing. However, the forcing patterns appear to be less equivalent barotropic in their vertical structure. Both the forcing and the response patterns are expressible as linear combinations of a small number of atmospheric *free* modes, that is, the modes of variability of the uncoupled atmosphere.

It is interesting to compare the above scenario of midlatitude ocean–atmosphere interaction with the observational study of Wallace et al. (1990). Wallace et al. computed the correlation between the expansion coefficient of EOF1 of winter-mean SST in the North Atlantic and the 500-mb heights over a 39-yr period, and obtained a correlation pattern that closely resembled the North Atlantic Oscillation (NAO) pattern (cf. Wallace and Gutzler 1981). However, when they carried out a similar analysis using the SST tendencies, rather that the SSTs, they obtained a pattern resembling the west Atlantic (WA) teleconnection pattern. The centers of the WA pattern lie westward (i.e., upstream) and a bit southward of the NAO pattern. If we assume that the SST tendencies reflect atmospheric forcing, as Wallace et al. argue, then we can identify the *forcing* pattern of the idealized model with the observed WA pattern and the *response* pattern with the observed NAO pattern (Fig. 22). Although the idealized model tends to exaggerate the spatial separation between these two types of patterns, it does capture the qualitative differences between the two.

Wallace et al. (1990) also found a similar westward shift of the correlation patterns with respect to SST tendencies (as compared to the correlation patterns with respect to the SST itself) in the North Pacific region as well, suggesting that this distinction between upstream and downstream modes may be a generic property of midlatitude ocean–atmosphere interaction. The study of Palmer and Sun (1985) also found some differences between the model response patterns and the observed correlations between the SST and atmospheric flow patterns. The model response was substantially weaker (for the same amplitude of the SST anomaly as observed), and tended to occur 10°–15° longitude farther downstream. Any observational study of the relationship between the SST anomalies and the atmospheric flow would find it difficult to distinguish between the forcing and the response patterns. AMIP-style GCM integrations would only allow us to identify the response patterns. A complementary type of GCM integration, where fluxes from an uncoupled atmospheric integration with the same time mean climatology are used to force an ocean model, can be used to unambiguously identify the forcing patterns.

To summarize, we note that an idealized ocean–atmosphere model can exhibit spontaneous interdecadal variability in the extratropics, with qualitatively realistic features, such as the dipolar structure in the SST anomalies (e.g., Deser and Blackmon 1993), and the differences between the forcing and the response patterns in the atmospheric flow (Wallace et al. 1990). Although the model itself is deterministic, it is useful to conceptualize the atmospheric variability arising from weather excitation as having a significant stochastic component, associated primarily with variations in the quasi-stationary flow. An interesting feature of this variability is the partial spatial resonance between the spatial patterns of surface fluxes associated with very low-frequency atmospheric variability, and the spatial patterns of surface excitation associated with the oceanic modes of variability. Such interactions could be an important feature of interdecadal climate fluctuations in the real climate system. Negative thermal feedback from the ocean damps surface heat-flux amplitudes and leads to slightly increased atmospheric variability in the coupled system, when compared to uncoupled atmospheric integrations.

## Acknowledgments

We wish to acknowledge discussions with G. Branstator, A. Capotondi, C. Penland, P. Sardeshmukh, and J. Tribbia that helped us better understand the stochastic aspects of climate variability. We would also like to thank C. Deser and Y. Kushnir for useful conversations, and T. Delworth for useful comments.

## REFERENCES

Bretherton, C. S., C. Smith, and J. M. Wallace, 1992: An intercomparison of methods for finding coupled patterns in climate data.

*J. Climate,***5,**541–560.Capotondi, A., and W. R. Holland, 1997: Decadal variability in an idealized ocean model and its sensitivity to surface boundary conditions.

*J. Phys. Oceanogr.,***27**, 1072–1093.Cayan, D. R., 1992: Latent and sensible heat flux anomalies over the northern oceans: Driving the sea surface temperature.

*J. Phys. Oceanogr.,***22,**859–881.Chen, F., and M. Ghil, 1995: Interdecadal variability of the thermohaline circulation and high-latitude surface fluxes.

*J. Phys. Oceanogr.,***25,**2547–2568.Delworth, T., S. Manabe, and R. J. Stouffer, 1993: Interdecadal variations of the thermohaline circulation in a coupled ocean–atmosphere model.

*J. Climate,***6,**1993–2011.Deser, C., and M. L. Blackmon, 1993: Surface climate variations over the North Atlantic Ocean during winter: 1900–1989.

*J. Climate,***6,**1743–1753.——, M. A. Alexander, and M. S. Timlin, 1996: Upper-ocean thermal variations in the North Pacific during 1970–1991.

*J. Climate,***9,**1840–1855.Feldstein, S. B., and W. A. Robinson, 1994: Comments on ‘Spatial structure of ultra-low frequency variability in a simple atmospheric circulation model.’

*Quart. J. Roy. Meteor. Soc.,***120,**739–735.Greatbatch, R. J., and S. Zhang, 1995: Interdecadal oscillation in an idealized ocean basin forced by constant heat flux.

*J. Climate,***8,**81–91.Griffies, S. M., and E. Tziperman, 1995: A linear thermohaline oscillator driven by stochastic atmospheric forcing.

*J. Climate,***8,**2440–2453.Hasselmann, K., 1976: Stochastic climate models: Part I. Theory.

*Tellus,***28,**473–485.Kushnir, Y., 1994: Interdecadal variations in North Atlantic Sea surface temperature and associated atmospheric conditions.

*J. Climate,***7,**141–157.——, and I. M. Held, 1996: Equilibrium atmospheric response to North Atlantic SST anomalies.

*J. Climate,***9,**1208–1220.Latif, M., and T. P. Barnett, 1994: Causes of decadal climate variability over the North Pacific and North America.

*Science,***266,**634–637.Madden, R. A., 1976: Estimates of the naturally occurring variability of time-averaged sea-level pressure.

*Mon. Wea. Rev.,***104,**942–952.Manabe, S., and R. J. Stouffer, 1996: Low-frequency variability of surface air temperature in a 1000-year integration of a coupled atmosphere–ocean–land surface model.

*J. Climate,***9,**376–393.Marotzke, J., P. Welander, and J. Willebrand, 1988: Instability and multiple steady states in a meridional plane model of the thermohaline circulation.

*Tellus,***40A,**162–172.Mikolajewicz, U., and E. Maier-Reimer, 1990: Internal secular variability in an ocean general circulation model.

*Climate Dyn.,***4,**145–156.Miller, A., D. R. Cayan, T. P. Barnett, N. E. Graham, and J. M. Oberhuber, 1994: Interdecadal variability of the Pacific Ocean: Model response to observed heat flux and wind stress anomalies.

*Climate Dyn.,***9,**287–302.Mysak, L. A., T. F. Stocker, and F. Huang, 1993: Century-scale variability in a randomly forced two-dimensional thermohaline ocean model.

*Climate Dyn.,***8,**103–116.North, G. R., 1984: Empirical orthogonal functions and normal modes.

*J. Atmos. Sci.,***41,**879–887.Otnes, R. K., and L. Enochson, 1978:

*Applied Time Series Analysis.*Vol. 1. John Wiley and Sons, 449 pp.Palmer, T. N., and Z. Sun, 1985: A modeling and observational study of the relationship between sea surface temperature in the north west Atlantic and the atmospheric general circulation.

*Quart. J. Roy. Meteor. Soc.,***111,**947–975.Robertson, A. W., 1996: Interdecadal variability over the North Pacific in a multi-century climate simulation.

*Climate Dyn.,***12,**227–241.Saravanan, R., and J. C. McWilliams, 1995: Multiple equilibria, natural variability, and climate transitions in an idealized ocean–atmosphere model.

*J. Climate,***8,**2296–2323.——, and ——, 1997: Advective ocean–atmosphere interaction: An analytical stochastic model with implications for decadal variability.

*J. Climate,*in press.Strang, G., 1988:

*Linear Algebra and its Applications.*Harcourt, Brace, and Jovanovitch, 505 pp.Trenberth, K. E., and J. W. Hurrell, 1994: Decadal atmosphere–ocean variations in the Pacific.

*Climate Dyn.,***9,**303–319.Vautard, R., P. Yiou, and M. Ghil, 1992: Singular-spectrum analysis: A toolkit for short, noisy chaotic signals.

*Physica D,***58,**95–126.Wallace, J. M., and D. S. Gutzler, 1981: Teleconnections in the geopotential height field during the Northern Hemisphere winter.

*Mon. Wea. Rev.,***109,**784–812.——, C. Smith, and Q. Jiang, 1990: Spatial patterns of atmosphere–ocean interaction in the northern winter.

*J. Climate,***3,**990–998.Weaver, A. J., and E. S. Sarachik, 1991: Evidence for decadal variability in an ocean general circulation model: An advective mechanism.

*Atmos.–Ocean,***29,**197–231.Weisse, R., U. Mikolajewicz, and E. Maier-Reimer, 1994: Decadal atmosphere–ocean variations in the Pacific.

*J. Geophys. Res.,***99,**12411–12421.Welander, P., 1985: Thermohaline effects in the ocean circulation and related simple models.

*Large-Scale Transport Processes in the Oceans and the Atmosphere,*D. L. T. Anderson and J. Willebrand., Eds., NATO ASI Series, Reidel, 163–223.Wigley, T. M. L., and S.C. B. Raper, 1991: Internally generated natural variability of global-mean temperatures.

*Greenhouse-Gas-Induced Climatic Change: A Critical Appraisal of Simulations and Observations,*M. E. Schlesinger, Ed., Elsevier, 471–482.Wilks, D. S., 1995:

*Statistical Methods in the Atmospheric Sciences.*Academic Press, 467 pp.Winton, M., and E. S. Sarachik, 1993: Thermohaline oscillations induced by strong steady salinity forcing of ocean general circulation models.

*J. Phys. Oceanogr.,***23,**1389–1410.Wright, D. G., and T. F. Stocker, 1991: A zonally averaged ocean model for the thermohaline circulation: I. Model development and flow dynamics.

*J. Phys. Oceanogr.,***21,**1713–1724.

Model integrations (4000 yr each).

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*thinsp;The National Center for Atmospheric Research is sponsored by the National Science Foundation.