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  • View in gallery

    Schematic of the sector ocean basin on the sphere, surrounded by land. The ocean sector occupies 30% of the globe.

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    Atmospheric climatology of the coupled model in the northern winter: (a) 250-mb streamfunction (ψ250), 2 (×107) m2 s−1 contours; (b) 250-mb transient eddy kinetic energy, 30 m2 s−2 contours. [Vertical dashed lines denote the coastal boundaries.]

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    Oceanic climatology of the coupled model: (a) streamfunction (Ψ), 10 Sυ contours; (b) temperature, 2.5°C contours; (c) salinity, with 35 ppt subtracted out, 0.2 ppt contours. [Negative contours are dashed.]

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    Annual-mean T at 50°N, 750-m depth for the coupled C integration: (a) time evolution during a 1000-yr period; (b) power spectrum over a 4000-yr period. [The power spectrum was smoothed by binning, with a bin size of 15. The dashed line is a reference red-noise spectrum computed from the lag-one autocorrelation, normalized to have the same total variance. The dotted lines represent the 95% and 5% a posteriori confidence intervals.]

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    The dominant oceanic EOFs computed from the 10–100-yr bandpass data for the C integration: (a) Ψ, EOF1; (b) T, EOF1; (c) T, EOF2. [Here, the Ψ contour interval is 0.3 Sv and the T contour interval is 0.03 K. The fractional variance associated with each EOF is shown in the title.]

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    Frequency-variance spectrum of the (unfiltered) expansion coefficients associated with oceanic EOFs for the C integration: (a) EOF1 of Ψ (solid), 5% and 95% a posteriori confidence intervals for red-noise spectrum (dotted), reference AR(2) spectrum (dashed); (b) EOF1 of T (solid), EOF2 of T (dashed), and EOF1 of S (dotted, scaled by α2/β2).

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    Correlation coefficient r for P1[T] (solid) and P2[T] (dashed), at different lags (in years) with respect to the ΨC index.

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    Covariance of SST with respect to the ΨC index for the C integration, at different lags and latitudes. (The 0.05 K contours; negative contours are dashed.)

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    SSP1 of heat flux into surface {S1[Fs]} with respect to the ΨC index, for the C integration; 2 W m−2 contours. [Negative contours are dashed; vertical long dashed lines denote the coastal boundaries; the plot title contains the SSP number, and the squared covariance fraction; the subtitle contains the variance fraction, maximum correlation between SSP expansion coefficient and ΨC index, and the lag (in years) at which this correlation occurs.]

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    Lag-correlation between the ΨC index and the expansion coefficients of SSP1 of Fs (solid), ψ750 (dashed), and ψ250 (dotted) for the C integration.

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    SSPs of atmospheric streamfunction with respect to the ΨC index for the C integration: (a) S1[ψ750], 10 (×104) m2 s−1 contours; (b) S1[ψ250], 20 (×104) m2 s−1 contours. (Negative contours are dashed.)

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    Covariance of different terms in the heat budget for the northern winter with respect to the ΨC index (at lag τ = −8, averaged over four longitudinal atmospheric grid points over the ocean lying adjacent to the western boundary of the ocean basin): heat flux into surface Fs (solid), latent heat flux −E (dot–dashed), temperature tendency due to low-level mean flow advection −(u·HT)750mb (dashed), and temperature tendency due to low-level transient eddy heat flux convergence −H·(uT)750mb (dotted). (All flux terms were normalized so as to be expressible in W m−2.)

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    Frequency-variance spectrum of unfiltered expansion coefficients for EOF1 of oceanic streamfunction for different integrations: (a) C (solid), CO (dot–dashed), COT (dashed), and COS (dotted); (b) C (solid) and AO (dashed).

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    SSPs of atmospheric streamfunction: (a) S1[ψ750] for CA integration, w.r.t. ΨC index; (b) S1[ψ250] for CA integration, w.r.t. ΨC index; (c) S1[ψ750] for A integration, w.r.t. ΨAO index; (d) S1[ψ250] for A integration, w.r.t. ΨAO index. (Contours are as in Fig. 11.)

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    Atmospheric EOFs (“free modes”) of ψ750 (a), (b), (c) and ψ250 (d), (e), (f) computed from the bandpass data for the A integration (Northern Hemisphere only). [10 (×104) m2 s−1 contours for (a), (b) 5 (×104) m2 s−1 contours for (c) 20 (×104) m2 s−1 contours for (d), (e), (f).]

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    Variance associated with SSP1 of ψ750 (a), (b), (c) and ψ250 (d), (e), (f), projected on the “free modes” of the atmosphere (Fig. 15): (a), (d) C integration, ΨC index; (b), (e) CA integration, ΨC index; (c), (f) A integration, ΨAO index. [Solid bars show the projected fractional variance (p2m), with gray denoting pm < 0 and black denoting pm > 0; the hollow bars show the corresponding cumulative fractional variance (Σml=1 p2l).]

  • View in gallery

    Frequency-variance spectrum of the (unfiltered) expansion coefficients associated with EOFs 1, 2 of ψ250 in the Northern Hemisphere: (a) C integration; (b) A integration. [The solid curve represents the spectrum of EOF1; the dashed curve represents the spectrum of an AR(1) process with the same lag-one autocorrelation and total variance; dot–dashed curves represent the associated 5% and 95% a posteriori confidence intervals. The dotted curve represents the spectrum of EOF2.]

  • View in gallery

    Area-weighted global-average bandpass variance histograms for Fs (black) and T750 (gray), for the CA, A, C, and M integrations. [For each variable, the variance values for the different integrations have been normalized by the maximum variance value among all integrations.]

  • View in gallery

    Frequency-variance spectrum of unfiltered expansion coefficients for EOF1 of oceanic streamfunction for different integrations: C (solid), COl (dot–dashed), COw (dashed), and COww (dotted).

  • View in gallery

    The dominant oceanic EOFs computed from the 10–100-yr bandpass data for the COww “white noise” integration: (a) Ψ, EOF1; (b) T, EOF1; (c) T, EOF2. (The Ψ contour interval is 0.1 Sv and the T contour interval is 0.02 K.)

  • View in gallery

    Covariance between the (zonally averaged) surface heat-flux forcing Fs and the normalized principal component 1 of bandpass oceanic streamfunction, for the COww “white noise” integration, at lag τ = −8 (solid curve). (The dashed curve represents the spatial structure of EOF1 of the zonally averaged Fs for atmosphere-only A integration.)

  • View in gallery

    Schematic of the forcing/response patterns in atmospheric 250-mb streamfunction (ψ250) associated with midlatitude ocean–atmosphere interaction: upstream forcing patterns (solid ellipses), surface heat flux forcing (filled semiellipses), downstream response patterns (dotted ellipses).

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Stochasticity and Spatial Resonance in Interdecadal Climate Fluctuations

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  • 1 National Center for Atmospheric Research,* Boulder, Colorado
  • | 2 Department of Atmospheric Sciences, University of California at Los Angeles, Los Angeles, California
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Abstract

Ocean–atmosphere interaction plays a key role in climate fluctuations on interdecadal timescales. In this study, different aspects of this interaction are investigated using an idealized ocean–atmosphere model, and a hierarchy of uncoupled and stochastic models derived from it. The atmospheric component is an eddy-resolving two-level global primitive equation model with simplified physical parameterizations. The oceanic component is a zonally averaged sector model of the thermohaline circulation. The coupled model exhibits spontaneous oscillations of the thermohaline circulation on interdecadal timescales. The interdecadal oscillation has qualitatively realistic features, such as dipolar sea surface temperature anomalies in the extratropics. Atmospheric forcing of the ocean plays a dominant role in exciting this oscillation. Although the coupled model is in itself deterministic, it is convenient to conceptualize the atmospheric forcing arising from weather excitation as having stochastic time dependence. Spatial correlations inherent in the atmospheric low-frequency variability play a crucial role in determining the oceanic interdecadal variability, through a form of spatial resonance. Local feedback from the ocean affects the amplitude of the interdecadal variability. The spatial patterns of correlations between the atmospheric flow and the oceanic variability fall into two categories: (i) upstream forcing patterns, and (ii) downstream response patterns. Both categories of patterns are expressible as linear combinations of the dominant modes of variability associated with the uncoupled atmosphere.

Corresponding author address: R. Saravanan, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307.

Email: svn@ncar.ucar.edu

Abstract

Ocean–atmosphere interaction plays a key role in climate fluctuations on interdecadal timescales. In this study, different aspects of this interaction are investigated using an idealized ocean–atmosphere model, and a hierarchy of uncoupled and stochastic models derived from it. The atmospheric component is an eddy-resolving two-level global primitive equation model with simplified physical parameterizations. The oceanic component is a zonally averaged sector model of the thermohaline circulation. The coupled model exhibits spontaneous oscillations of the thermohaline circulation on interdecadal timescales. The interdecadal oscillation has qualitatively realistic features, such as dipolar sea surface temperature anomalies in the extratropics. Atmospheric forcing of the ocean plays a dominant role in exciting this oscillation. Although the coupled model is in itself deterministic, it is convenient to conceptualize the atmospheric forcing arising from weather excitation as having stochastic time dependence. Spatial correlations inherent in the atmospheric low-frequency variability play a crucial role in determining the oceanic interdecadal variability, through a form of spatial resonance. Local feedback from the ocean affects the amplitude of the interdecadal variability. The spatial patterns of correlations between the atmospheric flow and the oceanic variability fall into two categories: (i) upstream forcing patterns, and (ii) downstream response patterns. Both categories of patterns are expressible as linear combinations of the dominant modes of variability associated with the uncoupled atmosphere.

Corresponding author address: R. Saravanan, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307.

Email: svn@ncar.ucar.edu

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:

  1. Is the variability driven primarily by atmospheric processes, oceanic processes, or coupled ocean–atmosphere interactions?
  2. What determines the frequency spectrum of such variability?
  3. 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 (8H) damping is used to represent the enstrophy cascade. A slab of specified heat capacity is used to represent land. It is assumed that the land is completely dry and that any precipitation over land runs off into the ocean instantaneously. The integrated runoff from land is distributed uniformly over the ocean to close the water budget.

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 ψ250 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(xj, tn), where xj denotes the spatial gridpoint locations with corresponding area-weights w2j, and tn, 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 Σn wjf(xj, tn)f(xj, tn)wj, and its eigenvectors or EOFs. We denote the mth EOF of f by Em[f; xj], or simply Em[f], and the corresponding eigenvalue or associated variance as ε2m. (For each model integration, Em[f] would refer to the set of EOFs for that particular integration.) The EOFs are normalized so that Σj El[f; xj]Em[f; xj] = δlm. However, when we display the EOFs, we always display the “dimensional” EOF pattern, εmEm[f; xj]/wj, so as to give an idea of the typical variability associated with the pattern. The temporal coefficients (or principal components) Pm[f; t] associated with each EOF are defined as Pm[f; t] = Σj Em[f; xj]f(xj, t).

To analyze correlated behavior between different variables, we typically define a normalized time series index, and compute lead–lag correlations with respect to it. Consider an index g(tn) of zero mean and unit variance. We define the covariance pattern C[f; xj, τ] of a field f(xj, tn), lagging the index by time τ, as follows:
i1520-0442-10-9-2299-e1
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; xj, τ]/wj, in the figures. Since the EOFs of f form a complete basis, we may expand the covariance pattern as follows:
i1520-0442-10-9-2299-e2
where the covariances in EOF space, Cm[f; τ], are defined as
i1520-0442-10-9-2299-e3
and corresponding correlation coefficients in EOF space, γm,[f; τ], are defined as
i1520-0442-10-9-2299-e4
Most 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)].

Let us assume that we are interested in the correlated behavior between a field f and index g over a set of time lags τk = (k − 1)Δt, k = k0, . . . , k1, where Δt denotes the sampling interval. In the terminology of SSA, K = k1k0 + 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 CmkCm[f; τk], with M rows and K columns, as defined by (3) may be decomposed using SVD as follows (cf. Strang 1988):
i1520-0442-10-9-2299-e5
such that the singular values λl are in descending order (λ1λ2 ≥ . . . ≥ λM). Here we have assumed that MK, 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 (Sml, Tkl) associated with the lth singular value as the lth 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 by
i1520-0442-10-9-2299-e6
This 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.
The spatial pattern associated with the lth singular mode in gridpoint space, denoted by Sl[f, g; xj] or simply Sl[f], may be expressed as a linear combination of the EOFs; that is,
i1520-0442-10-9-2299-e7
We shall refer to Sl [f] as the lth spatial singular pattern (SSP) and to the corresponding Tkl as the lth 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 σl associated with each SSP are computed as linear combinations of the principal components Pm as follows:
i1520-0442-10-9-2299-e8
Since the SSPs are orthogonal in the truncated EOF-space representation, each SSP is associated with a unique portion, ϑ2l, of the variance in f, given by
i1520-0442-10-9-2299-e9
just as each EOF is associated with variance ε2m. As we do for the EOFs, we always display the “dimensional” SSP, ϑlSl[f, g; xj]/wj, so as to give an idea of the typical variability associated with the pattern. One can also show from (5) that the covariance between σl and g at lag τk is given simply by λl times the lth TSP. Thus, the correlation between σl and g at lag τk would be given by λlTkl/ϑl.

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 C integration, which will serve as the control for our analysis. Our primary dataset consists of the following fields: all atmospheric and oceanic prognostic variables, net heat flux into the surface (Fs), 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 C integration. Figure 4a shows the evolution of annually averaged temperature at a particular ocean grid point, located at 50°N at a depth of 750 m. The time series shows variability over a range of timescales, and in particular, on interdecadal timescales. The frequency-power spectrum associated with this time series (Fig. 4b) shows overall red noise structure, but also a significant spectral peak with a timescale of 30–40 yr, and a weaker peak around timescales of 20 yr or so. Since there is no externally imposed periodic forcing in the model, apart from the annual cycle, these preferred interdecadal timescales must arise intrinsically in the coupled system. We proceed to analyze this interdecadal variability in some detail, starting with the oceanic fields, where the preferred timescales are most clearly seen. (There are also interesting modes of oceanic variability on centennial and longer timescales, which we defer to a future analysis.)

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 E1[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 ≡ 106 m3 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, E1[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, E2[T] is essentially a high latitude mode in the vicinity of the sinking region and has fairly deep structure. The variance spectrum of E1[T] is dominated by a peak in the 30–50-yr range (Fig. 6b). The variance spectrum of E2[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 E1[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 αTβS 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 E1[S], scaled by α2/β2 to make it comparable to the temperature variance. Here, E1[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 E1[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 P1[Ψ], computed from bandpass data. We shall refer to this time series, after normalizing it to have zero mean and unit variance, as the Ψc index, where the subscript denotes the C integration. We shall use this as the standard index for most of our covariance calculations.

To understand the lead-lag relationship between T and Ψ, we compute correlations with respect to the ΨC index. To evaluate the statistical significance of the correlation coefficients, we need to estimate the number of degrees of freedom. The AR(2) process fitted to the spectrum of unfiltered P1[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 ΨC index has at least about 65 degrees of freedom associated with it, and that correlations of about ±0.25 are significant at the 95% level based on a t test. We also computed correlations using a synthetic statistically independent time series created by swapping two halves of the ΨC time series. This approach also indicated a significance threshold of ±0.2 for correlations. Therefore, we shall consider correlation coefficients about 0.25 or larger between a bandpass time series and the ΨC index to be statistically significant.

The strongest correlations between T and Ψ (Fig. 7) show that P1[T] is correlated with the ΨC at a lead of about 7 yr and P2[T] is correlated at zero lag, with an approximate quadrature relationship between P1[T] and P2[T]. In other words, E1[T] and E2[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 E1[Ψ], E1[T], and E2[T], are highly correlated, they may be said to constitute a well-defined “oceanic interdecadal oscillation,” which is characterized by the ΨC index.

The advective nature of the interdecadal oscillation is also seen in the covariance between SST and the ΨC index at different lags (Fig. 8). The evolution of the SST anomalies may be described as follows: a weak negative SST anomaly forms near 45°N at lag 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:

  1. advection of the mean temperature field by the streamfunction anomalies; and
  2. 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:

  1. Say a deep negative temperature anomaly exists in the polar sinking region, for example, as in the positive phase of E2[T]. This would be associated with enhanced meridional overturning, as suggested by the strong simultaneous correlation between E2[T] and E1[Ψ] at zero lag (Fig. 7).
  2. 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 E1[T] and E2[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 E1[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 E2[T] is fairly localized meridionally, the associated streamfunction anomaly E1[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 E1,2[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 Fs. Unlike the relationship between oceanic T and the ΨC index discussed above, which is quite well described by the first two EOFs, it turns out that one needs more than the first two EOFs to describe the relationship between surface atmospheric fields and the ΨC index. This motivates us to use the MSVD technique (described in section 3b) to identify the spatial singular patterns, or SSPs, that maximize the squared lag-covariance between Fs and ΨC. Since we expect periodicities of 30–40 yr based on the variance spectrum of the oceanic fields (Fig. 6), we choose the lag range τ = −30, . . . , 10 for the MSVD analysis.

The dominant SSP of Fs, S1[Fs] is shown in Fig. 9. Also shown in the plot title are the squared covariance fraction or SCF for the SSP, the variance ϑ2l (expressed as a fraction of the total variance), the strongest correlation between the SSP expansion coefficient σl and the ΨC index, and the lag at which this correlation occurs. Note that S1[Fs] is associated with an SCF value of 90% and thus captures most of the covariance. It is also significantly correlated with the ΨC index, with the maximum correlation of 0.52 occurring at lag τ = −8 yr, that is, simultaneous with the E1[T] (Fig. 7). The relationship between S1[Fs] and the ΨC index is seen more clearly in the corresponding temporal singular pattern, or TSP, which shows the lag-correlation associated with the SSP (Fig. 10). The dominant feature is the strong peak in covariance at τ = −8, which is indicative of a simultaneous correlation between S1[Fs] and E1[T] at near-zero lag.

The spatial structure of S1[Fs] (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 S1[Fs] and E1[T] is such that in regions where the Fs 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 Fs, we use MSVD analysis and restrict our attention to the first SSP of ψ750 and ψ250 (Fig. 11) because the other SSPs are not very highly correlated with the ΨC index.

The first SSP of low-level streamfunction, S1 [ψ750], 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 ψ750 anomalies is such that negative Fs anomalies tend to coincide with northwesterly wind anomalies and vice versa. As will be discussed later, the dominant contribution to the Fs 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 Fs 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 ψ750 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 S1[ψ750] 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, S1[ψ250] 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 Fs and ψ750 (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 Fs 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 C integration. We then considered the covariances at 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 Fs 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 Fs anomalies in the midlatitudes, with the remaining part coming from surface radiative and sensible heat-flux anomalies. Note also that the local wintertime Fs 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 A, CA, and M (Table 1), where we use different approximations to the SST variability in the coupled integration to force the atmosphere. The simplest is to compute the climatological annual cycle of SST in the coupled integration and use it to force the atmosphere in an annually repeating fashion. This approach is used in the A integration, where the atmosphere is forced by the annual and semiannual harmonics of SST computed from the climatology of the C integration.

At the next level of approximation, in the CA integration, we use the actual time series of SST over 4000 yr of the C integration to force the atmospheric model, starting from an arbitrary initial condition. This integration can be thought of as being like the so-called AMIP (Atmospheric Model Intercomparison Project) integrations, except that the SSTs used to force the atmosphere are taken from a coupled control integration, rather than from observations.

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 C integration, was added to the mixed-layer equations. This ensured that if the surface heat flux climatology were the same as that of the C integration, the SST climatology of the slab ocean would also be the same. We shall refer to this as the M integration.

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 C integration, by design. The associated variability, however, is not the same, as discussed in section 5d.

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 CO, COT, COS, and AO (Table 1). As is shown later (section 6), the oceanic interdecadal variability, being of small amplitude, is essentially linear in nature. Therefore, we ignore the seasonal cycle and use annual averages of surface fluxes from each of the 4000 yr of the C integration as the forcing for the first three integrations: CO, forced by the time-varying heat and freshwater fluxes (Fs and EP) at the surface; COT, forced by the time-varying Fs only, with EP held fixed at its climatological annual mean value; COS, forced by time-varying EP only, with Fs 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 Fs and EP 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, AO, was also started from the same initial condition, and forced by time-varying heat and freshwater fluxes at the surface, but these fluxes were derived from the atmosphere-only A integration as follows: the anomalous surface fluxes each of the 4000 yr of the A integration (with respect to its climatological mean value) were added to the climatological mean surface fluxes from the C integration. This procedure ensures that all the ocean-only integrations are subject to same time mean forcing at the surface.

c. Uncoupled oceanic variability

We begin our analysis of the uncoupled integrations by considering the ocean-only integrations, because some features of the AO integration will turn out to be useful in understanding the variability in the atmosphere-only integrations. Figure 13a shows the variance spectrum of E1[Ψ] for the CO, COT, and COS integrations. Note that for the CO case, the variance spectrum is virtually identical to that of the C integration. The spatial structure of E1[Ψ] and the first two EOFs of T (not shown) are also very similar to that seen in the C integration (Fig. 5). This demonstrates that given the surface fluxes, an ocean-only integration can essentially reproduce the coupled interdecadal variability.

The differing roles played by the surface thermal and freshwater forcing can be seen by comparing the variance spectra of E1[Ψ] for the COT, COS integrations (Fig. 13a). The COT 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 E1[Ψ] for both COT and COS 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 C integrations.

Although the surface flux–forced CO integration reproduces the coupled interdecadal variability rather well, it does not necessarily mean that the feedback from the ocean to the atmosphere is unimportant, since the surface fluxes may already contain information about this feedback. To determine the importance of this feedback, we consider the AO ocean-only integration, which is forced by surface flux anomalies from the atmosphere-only A integration. In this case, the variance spectrum of E1[Ψ] shows spectral peaks in the 30–50-yr range, just as in the C integration (Fig. 13b). However, variance amplitudes are considerably larger. The spatial structures of E1[Ψ] and the two dominant EOFs of T in the AO integration (not shown) are similar to that in the C integration. This leads us to conclude that the space–time structure of the surface fluxes in an uncoupled atmospheric integration has all the features necessary to excite the oceanic interdecadal oscillation. However, the overall variance associated with the fluxes is too high, indicating that the ocean provides significant negative feedback on the surface heat fluxes in the C integration (see the discussion in section 5d.).

Since an oceanic interdecadal oscillation, very similar to that seen in the coupled integration, is present in the AO integration, we may define an index to characterize it (analogous to the ΨC index). We define the first principal component P1[Ψ] computed from the bandpass data, after normalization to have zero mean and unit variance, as the ΨAO index. We shall use this index to identify atmospheric spatial patterns in the A integration that are responsible for exciting the interdecadal oscillation seen in the AO integration.

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 CA integration should allow us to identify this response. Since the SST evolution in the CA integration is identical to that of the C integration, we simply use the ΨC index to represent the oceanic interdecadal oscillation. The MSVD technique is then used to identify the atmospheric patterns that are most highly correlated with the ΨC index.

The dominant SSPs of the lower-/upper-level streamfunction (Figs. 14a,b) show some similarities to the corresponding SSPs for the C integration (Fig. 11), although the correlations are somewhat weaker. For ψ750, the dominant feature is the dipolar pattern east of the ocean basin, which closely resembles the EL pattern seen in the C integration. For ψ250, 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 C integration.

What are missing in the SSPs of the CA integration (Figs. 14a,b), in comparison to the C integration, are the features near the western boundary of the ocean basin: the WO pattern in ψ750 and the dipolar pattern in ψ250 adjacent to the western boundary of the ocean basin. Given that the mean zonal midlatitude winds are westerly, the CA integration seems to capture the downstream response to the SST anomalies in the ocean basin, but not the more upstream patterns, which presumably force the SST anomalies themselves. There is also a barotropic character to these downstream patterns, with the dominant dipolar features in ψ750 and ψ250 being coincident.

To find the missing upstream patterns, we turn to the A and AO integrations. We may consider these two integrations, taken together, as being an almost-coupled integration, where the surface flux anomalies from the atmosphere-only A integration are used to drive the ocean-only AO integration. However, the feedback from the ocean to the atmosphere is suppressed. In this sense, A/AO integration is complementary to the AMIP-style CA integration. All modes of variability in the A integration are internally generated, without any reference to oceanic variability or feedbacks. Therefore, we shall refer to these modes of variability as the free modes of the atmosphere, to distinguish them from the at least partially forced modes of the CA integration, such as the SSPs described above.

We have already seen how the free modes of the atmosphere are capable of exciting the oceanic interdecadal oscillation in the AO integration. To identify the particular free mode structures that are responsible for exciting the oceanic variability, we carry out MSVD analysis of atmospheric variables in the A integration with respect to the ΨAO index (which characterizes the interdecadal oscillation in the AO integration). Here S1[ψ750] 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 C integration (Fig. 11a). Here S1[ψ250] shows a dipolar structure near the western boundary of the ocean basin—also seen in the C integration (Figs. 14d, 11b)—against a background of weaker hemispheric-scale structure. Both of these are the upstream features that were not present in the respective SSPs of the CA integrations. In other words, the SSPs of the C integration turn out to be a superposition of the SSPs of the CA integrations, and the SSPs of the A integration.

To obtain a more quantitative description of the relationship between the SSPs of the C/CA/A integrations and the free modes of the atmosphere, we adopt the following linear algebraic approach: We choose to define the bandpass EOFs of ψ750 and ψ250 in the A integration to be 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 ψ750. The first mode closely resembles S1[ψ750] of the A/AO integration (Fig. 14c), and the fourth mode bears some resemblance to S1[ψ750] of the CA integration (Fig. 14a). Of the free modes of ψ250 (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 S1[ψ250] of the CA integration (Fig. 14b).

Since the free modes represent an orthonormal basis for all Northern Hemispheric spatial patterns, we can compute how the variance associated with the leading SSPs of the C, CA, and A/AO integrations project onto the different free modes. For example, consider S1[ψ750] of the C integration. Its projection on the mth free mode, pCm, may be computed using (5) as
i1520-0442-10-9-2299-e10
where EAm[xj] denote the free modes as defined above, ECm[xj] denote the EOFs of ψ750 in the C integration, and SCm′1 denotes the first SSP of ψ750 with respect to the ψC index.

The projections of the variance associated with S1[ψ750] on the free modes for the C, CA, and A/AO integrations are shown in Figs. 16a–c. The first five modes capture over 80% of the variance for all three integrations. This means that both the forcing of the ocean and the atmospheric response to oceanic feedbacks are mediated by the dominant free modes associated with the uncoupled atmosphere. A noteworthy feature is that the first free mode, which resembles the WO pattern seen in the C integration (Figs. 15a, 11a), dominates SSP1 of the C and A/AO integrations but makes only a small contribution to SSP1 of the CA integration. The fourth mode (Fig. 15c) dominates SSP1 of the CA integration, and is completely absent from SSP1 of the A/AO integration. This indicates that the first free mode plays a dominant role exciting the oceanic interdecadal oscillation, and the fourth mode plays a dominant role in the the atmospheric response to SST anomalies associated with the oscillation.

The projections of the variance associated with S1[ψ250] 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., C and A/AO), but much less of a role in the response to interdecadal SST variability. The second free mode seems to be predominantly a response to the SST variability, and is completely absent from SSP1 of the A/AO integration.

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 ψ250 for the C integration (not shown). Their spatial structure is virtually identical to those of the corresponding EOFs of the A integration (Figs. 15d,e), with pattern correlations exceeding 0.95. The frequency spectra associated with these EOFs have essentially white noise structure for timescales of the order of a decade or longer (Fig. 17), with the coupled integration exhibiting slightly higher levels of variance at the lowest frequencies. This means that the modes of variability in the coupled integration have essentially the same spatial structure as the free modes, and very similar spectral characteristics.

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 Fs, and the low-level atmospheric temperature T750. Figure 18 shows the global average of interdecadal bandpass variance associated with these fields for the CA, A, C, and M integrations. There is a clear progression in the variance associated with Fs for the different integrations, with CA > A > C > M. One can try to explain this by introducing the notion of an “effective” oceanic heat capacity.

One normally thinks of the CA/A type of integrations as being uncoupled, that is, with prescribed SSTs and no interaction with the ocean. However, prescribing the SSTs is equivalent to assuming that the atmosphere exchanges heat fluxes with an ocean of infinite heat capacity. At the other extreme, in the mixed-layer integration M, the atmosphere is coupled to an ocean with fairly small heat capacity (i.e., a 100-m-thick slab). The coupled integration can be thought of as being an intermediate case, associated with a larger, but still finite, heat capacity.

When the atmosphere exchanges heat fluxes with an ocean of finite heat capacity (as in the C/M integrations), a positive low-level atmospheric temperature anomaly would tend to produce a positive Fs 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 M integration, which has the smallest oceanic heat capacity, to show the least Fs variance, and the CA/A integrations to show the maximum Fs variance. This argument also resolves a puzzling aspect of the ocean-only integrations discussed earlier, that the surface fluxes from the atmosphere-only A integration excited a much stronger interdecadal oscillation in the AO ocean-only integration, when compared to the coupled integration.

The variance of T750 (Fig. 18) also shows a modest dependence on the oceanic heat capacity, but more or less in the opposite sense to that of Fs. 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 ψ750 and ψ250, 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 COl integration) forced by the anomalies of Fs and EP from the C integration. As we see in Fig. 19, the variance spectrum associated with E1[Ψ] for the COl integration is almost the same as that of the C integration. The spatial structures of EOF1 of Ψ and the first two EOFs of T (not shown) are also virtually identical to that of the C integration, confirming that the oceanic variability in the coupled integration is essentially governed by linear dynamics.

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 C integration (Fig. 11), with both the upstream forcing patterns and the downstream response patterns being present. This suggests that although the mean meridional advection in the ocean plays an important role in the interdecadal oscillation, it is not important for generating the downstream response in the atmospheric flow associated with the SST anomalies.

An important feature of the oceanic interdecadal oscillation, as demonstrated by the AO integration, is that it can be excited even by the modes of variability present in the uncoupled A integration. We have examined the variance spectra of several atmospheric variables in the A integration, and all of them are essentially “white” on interdecadal timescales (e.g., Fig. 17b). This leads us to hypothesize that even temporally incoherent (or random) atmospheric forcing can excite the oceanic interdecadal oscillation.

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 COT 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, Fsw(yj, tn) and Fsww(yj, tn), for each of the 4000-yr tn and at each meridional grid point yj at the ocean surface. The difference between Fsw and Fsww being that the former retains spatial correlations in the surface forcing and the latter does not, although both of them lack any temporal correlations. In other words, Fsw can be thought of as being “white noise in time,” and Fsww as being “white noise in both space and time.”

The stochastic flux time series are constructed as follows:
i1520-0442-10-9-2299-eq1
where Em[Fs; yj] are the EOFs of the zonally averaged Fs over the ocean basin, computed from the unfiltered 4000-yr time series for the C integration, ε2m are the variances associated with each EOF, and s2 is the variance of Fs averaged over all grid points. Here, Wm(tn), Wj(tn) denote independent Gaussian white noise random variables with zero mean and unit variance. They have the property that Wm(tn) is uncorrelated with Wm(tn−1) and so on. Note that, by construction, the EOFs and associated variances of Fsw will be the same as that of Fs; that is, this synthetic times series preserves the spatial correlation and variance structure of Fs. The Fsww time series preserves the spatially integrated variance amplitude, but does not preserve spatial correlations. The surface heat-flux anomalies were added to the time mean heat flux to produce the total surface heat-flux forcing of the ocean, as described in section 5b. Two 4000-yr ocean-only integrations, COw and COww, were carried out using the two synthetic time series as the surface forcing.

The COw integration reproduces the spectral peak associated with E1[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 COl integration because they are taken from the C integration, but is not present in surface fluxes for the stochastic integrations because all temporal correlations are eliminated. The spatial structures of EOF1 of Ψ and EOFs 1, 2 of T of the COw integration (not shown) are also quite similar to those of the C integration. This confirms that, to a good approximation, the atmospheric forcing of the ocean can be considered as being white noise in time.

The importance of the preferred spatial structures associated with the surface forcing is illustrated by the COww 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 E1[Ψ] is very similar to that of the C integration (Fig. 20a), although the amplitude is significantly weaker. Here, E1[T], however, shows monopolar structure in SST (Fig. 20b), whereas dipolar structure was seen in the C integrations, with a positive anomaly south of 50°N (Fig. 5b). This suggests that the dipolar spatial structure associated with the atmospheric heat flux is responsible for forcing the dipolar SST anomaly in the coupled integration (Figs. 9, 21). Here, E2[T] also shows some differences when compared to the C integration (Fig. 20c). However, the structure of EOFs 1 and 2 of T is consistent with the advective mechanism for the interdecadal oscillation suggested in section 4a. Hence, we conclude that the EOFs of the COww 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 CO, COT, COS, AO, COw, COww 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 P1[Ψ] of the oceanic streamfunction and the surface heat flux forcing Fs for the COww 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 Fs for the A integration, which one may think of as representing the dominant forcing associated with the free modes of the uncoupled atmosphere. Note that there is some correspondence between the two structures, especially in the higher latitudes, although the midlatitude structure is different. In other words, the atmospheric free mode forcing projects substantially onto the “most efficient” spatial pattern for exciting the oscillatory mode in the ocean. This suggests that the oceanic interdecadal oscillation is characterized by a partial “spatial resonance” effect, as discussed in the following section.

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.

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Fig. 1.
Fig. 1.

Schematic of the sector ocean basin on the sphere, surrounded by land. The ocean sector occupies 30% of the globe.

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 2.
Fig. 2.

Atmospheric climatology of the coupled model in the northern winter: (a) 250-mb streamfunction (ψ250), 2 (×107) m2 s−1 contours; (b) 250-mb transient eddy kinetic energy, 30 m2 s−2 contours. [Vertical dashed lines denote the coastal boundaries.]

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 3.
Fig. 3.

Oceanic climatology of the coupled model: (a) streamfunction (Ψ), 10 Sυ contours; (b) temperature, 2.5°C contours; (c) salinity, with 35 ppt subtracted out, 0.2 ppt contours. [Negative contours are dashed.]

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 4.
Fig. 4.

Annual-mean T at 50°N, 750-m depth for the coupled C integration: (a) time evolution during a 1000-yr period; (b) power spectrum over a 4000-yr period. [The power spectrum was smoothed by binning, with a bin size of 15. The dashed line is a reference red-noise spectrum computed from the lag-one autocorrelation, normalized to have the same total variance. The dotted lines represent the 95% and 5% a posteriori confidence intervals.]

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 5.
Fig. 5.

The dominant oceanic EOFs computed from the 10–100-yr bandpass data for the C integration: (a) Ψ, EOF1; (b) T, EOF1; (c) T, EOF2. [Here, the Ψ contour interval is 0.3 Sv and the T contour interval is 0.03 K. The fractional variance associated with each EOF is shown in the title.]

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 6.
Fig. 6.

Frequency-variance spectrum of the (unfiltered) expansion coefficients associated with oceanic EOFs for the C integration: (a) EOF1 of Ψ (solid), 5% and 95% a posteriori confidence intervals for red-noise spectrum (dotted), reference AR(2) spectrum (dashed); (b) EOF1 of T (solid), EOF2 of T (dashed), and EOF1 of S (dotted, scaled by α2/β2).

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 7.
Fig. 7.

Correlation coefficient r for P1[T] (solid) and P2[T] (dashed), at different lags (in years) with respect to the ΨC index.

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 8.
Fig. 8.

Covariance of SST with respect to the ΨC index for the C integration, at different lags and latitudes. (The 0.05 K contours; negative contours are dashed.)

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 9.
Fig. 9.

SSP1 of heat flux into surface {S1[Fs]} with respect to the ΨC index, for the C integration; 2 W m−2 contours. [Negative contours are dashed; vertical long dashed lines denote the coastal boundaries; the plot title contains the SSP number, and the squared covariance fraction; the subtitle contains the variance fraction, maximum correlation between SSP expansion coefficient and ΨC index, and the lag (in years) at which this correlation occurs.]

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 10.
Fig. 10.

Lag-correlation between the ΨC index and the expansion coefficients of SSP1 of Fs (solid), ψ750 (dashed), and ψ250 (dotted) for the C integration.

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 11.
Fig. 11.

SSPs of atmospheric streamfunction with respect to the ΨC index for the C integration: (a) S1[ψ750], 10 (×104) m2 s−1 contours; (b) S1[ψ250], 20 (×104) m2 s−1 contours. (Negative contours are dashed.)

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 12.
Fig. 12.

Covariance of different terms in the heat budget for the northern winter with respect to the ΨC index (at lag τ = −8, averaged over four longitudinal atmospheric grid points over the ocean lying adjacent to the western boundary of the ocean basin): heat flux into surface Fs (solid), latent heat flux −E (dot–dashed), temperature tendency due to low-level mean flow advection −(u·HT)750mb (dashed), and temperature tendency due to low-level transient eddy heat flux convergence −H·(uT)750mb (dotted). (All flux terms were normalized so as to be expressible in W m−2.)

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 13.
Fig. 13.

Frequency-variance spectrum of unfiltered expansion coefficients for EOF1 of oceanic streamfunction for different integrations: (a) C (solid), CO (dot–dashed), COT (dashed), and COS (dotted); (b) C (solid) and AO (dashed).

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 14.
Fig. 14.

SSPs of atmospheric streamfunction: (a) S1[ψ750] for CA integration, w.r.t. ΨC index; (b) S1[ψ250] for CA integration, w.r.t. ΨC index; (c) S1[ψ750] for A integration, w.r.t. ΨAO index; (d) S1[ψ250] for A integration, w.r.t. ΨAO index. (Contours are as in Fig. 11.)

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 15.
Fig. 15.

Atmospheric EOFs (“free modes”) of ψ750 (a), (b), (c) and ψ250 (d), (e), (f) computed from the bandpass data for the A integration (Northern Hemisphere only). [10 (×104) m2 s−1 contours for (a), (b) 5 (×104) m2 s−1 contours for (c) 20 (×104) m2 s−1 contours for (d), (e), (f).]

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 16.
Fig. 16.

Variance associated with SSP1 of ψ750 (a), (b), (c) and ψ250 (d), (e), (f), projected on the “free modes” of the atmosphere (Fig. 15): (a), (d) C integration, ΨC index; (b), (e) CA integration, ΨC index; (c), (f) A integration, ΨAO index. [Solid bars show the projected fractional variance (p2m), with gray denoting pm < 0 and black denoting pm > 0; the hollow bars show the corresponding cumulative fractional variance (Σml=1 p2l).]

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 17.
Fig. 17.

Frequency-variance spectrum of the (unfiltered) expansion coefficients associated with EOFs 1, 2 of ψ250 in the Northern Hemisphere: (a) C integration; (b) A integration. [The solid curve represents the spectrum of EOF1; the dashed curve represents the spectrum of an AR(1) process with the same lag-one autocorrelation and total variance; dot–dashed curves represent the associated 5% and 95% a posteriori confidence intervals. The dotted curve represents the spectrum of EOF2.]

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 18.
Fig. 18.

Area-weighted global-average bandpass variance histograms for Fs (black) and T750 (gray), for the CA, A, C, and M integrations. [For each variable, the variance values for the different integrations have been normalized by the maximum variance value among all integrations.]

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 19.
Fig. 19.

Frequency-variance spectrum of unfiltered expansion coefficients for EOF1 of oceanic streamfunction for different integrations: C (solid), COl (dot–dashed), COw (dashed), and COww (dotted).

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 20.
Fig. 20.

The dominant oceanic EOFs computed from the 10–100-yr bandpass data for the COww “white noise” integration: (a) Ψ, EOF1; (b) T, EOF1; (c) T, EOF2. (The Ψ contour interval is 0.1 Sv and the T contour interval is 0.02 K.)

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 21.
Fig. 21.

Covariance between the (zonally averaged) surface heat-flux forcing Fs and the normalized principal component 1 of bandpass oceanic streamfunction, for the COww “white noise” integration, at lag τ = −8 (solid curve). (The dashed curve represents the spatial structure of EOF1 of the zonally averaged Fs for atmosphere-only A integration.)

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Fig. 22.
Fig. 22.

Schematic of the forcing/response patterns in atmospheric 250-mb streamfunction (ψ250) associated with midlatitude ocean–atmosphere interaction: upstream forcing patterns (solid ellipses), surface heat flux forcing (filled semiellipses), downstream response patterns (dotted ellipses).

Citation: Journal of Climate 10, 9; 10.1175/1520-0442(1997)010<2299:SASRII>2.0.CO;2

Table 1.

Model integrations (4000 yr each).

Table 1.

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

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