## 1. Introduction

### a. The problem

Coupled variability of the atmosphere–ocean system is generated when the atmosphere responds to a forcing from the ocean, or when the ocean responds to a forcing from the atmosphere. The response could feed back to the subsystem, from which the forcing originates, leading to a two-way interaction process. The response could also have no effect on the subsystem, from which the forcing originates, leading to a one-way action. Because of the lack of feedbacks, the forcing is independent of the other subsystem. The resulting variability is sometimes referred to as the forced variability. When the forcing originates from the atmosphere, which has a short memory, it can be treated as a stochastic white noise forcing for the ocean.

Both types of interactions have to be considered, when one wants to model the oceanic variability. Normally, this is done by using either a fully coupled atmosphere–ocean general circulation model (GCM), or a simple atmospheric model coupled to an oceanic GCM. While the former is expected to produce more realistic air–sea interactions, the latter is superior because of low computational costs, and is therefore often preferred.

The often used simple atmospheric models are ad hoc models based on simple assumptions. It is for instance argued that, since the atmosphere has a fast adjustment timescale, atmospheric variations can be considered as steady-state responses to changes in the ocean. These responses feed back and drive the oceanic circulation. An example of ad hoc models based on this assumption is the atmospheric component of the hybrid coupled model introduced by Latif and Villwock (1990). It describes wind stress responses to SST changes and models a two-way interaction originating from the ocean. On the other hand, it is also sometimes assumed that the atmosphere can be considered as a noise generator, thereby emphasizing the role of one-way stochastic forcing in exciting oceanic variations. Based on this assumption, additive noises are included, for example, in the experiments of Mikolajewicz and Maier-Reimer (1990).

Even though both assumptions are physically based, it is not clear, to what extent are they valid, and whether they are consistent with the air–sea interactions produced by a fully coupled atmosphere–ocean GCM. Furthermore, it is also not clear how to validate the simple atmospheric models, which are used to replace computationally expensive atmospheric GCMs. In order to answer these questions, one needs first to find a way that translates the complex nature of air–sea interactions into simple signatures. This is the primary goal of this paper.

### b. A measure of air–sea interactions

*x*and

*y*and their corresponding mean seasonal cycle by 〈

*x*〉 and 〈

*y*〉. Frankignoul and Hasselmann (1977) and Frankignoul (1985) suggested to consider the cross-covariance function

*R*

_{xy}(

*τ*) between

*x*and

*y,*defined by

*R*

_{xy}

*τ*

*x*

_{t}

*x*

*y*

_{t+r}

*y*

*τ*corresponds to the lags when

*x*leads (follows)

*y.*If air–sea interactions produce coupled variations of

*x*and

*y, R*

_{xy}(

*τ*) would be nonzero at some time lags. Moreover, since a cross-covariance function is generally not symmetric about the zero lag, it can distinquish the situation when atmosphere leads from that when ocean leads.

Since the coupling between the atmosphere and the ocean is directly controlled by SST and surface fluxes of momentum, heat, and fresh water, this paper will concentrate on the cross-covariance functions between surface fluxes and SST. Hereinafter, *y* represents SST, and *x* is one of the surface fluxes.

In Frankignoul (1985), a stochastic climate model was derived for the extratropical SST. In a recent study (Frankignoul et al. 1998), both the magnitude and the sign of heat flux feedbacks in the eastern part of the North Atlantic were estimated from observed lagged correlation functions. The study suggested that turbulent surface heat flux contributes both to generate and to dampen SST variability, once it is generated.

Different from Frankignoul (1985) and Frankignoul et al. (1998), in which emphasis was made to derive a stochastic SST model and to understand the generation of extratropical SST variations, no attempt is made here to derive any model of SST from the full system. Instead, the characterization of air–sea interactions, based on cross-covariance function suggested by Frankignoul and Hasselmann (1977) and Frankignoul (1985), is extended. The extension is done by considering cross-covariance functions derived from various conceptual coupled models, which deal not only with extratropical but also tropical interactions (section 2a), and by considering the general features automatically carried by a lagged cross-covariance function (section 2b).

### c. Data

In section 3, the results of section 2 are used to interpret the cross-covariance functions obtained from a 300-yr integration with the coupled ECHAM3/LSG GCM. Ideally, this should be done for the observational data. However, since the shortness of the available time series makes the estimation of true cross-covariance functions difficult, it is easier to first apply the result of section 2 to cross-covariance functions that are essentially free of sample noises. The problem is demonstrated in Fig. 1, which shows two estimates of the lagged cross-correlation function between SST and net heat flux at a grid point in the ECHAM3/LSG model. They are derived from the same time series, but one uses 20 yr and the other 300 yr of the records. Since the estimator of a lagged covariance function is consistent [i.e., as the available time series becomes longer, the mean squared error of the estimator goes to zero and the estimator converges to the true lagged covariance function (Jenkins and Watts 1968)], the simple structure shown at the bottom of Fig. 1 is expected to be close to the true cross-correlation function, whereas the considerably more complicated structure shown of Fig. 1 (top) is a result of sample noises. By using long model data, the interpretation of cross-correlation functions in section 3 follows directly from the comparison with the theoretically obtained cross-correlation functions given in section 2.

Without examining the realism of the modeled air–sea interactions, the results of section 3 cannot be directly applied to air–sea interactions in the real world. A detailed verification will be considered elsewhere. A comparison based on observations considered by Frankignoul indicates that the air–sea interactions in the model are consistent with those found in the observations. This comparison is briefly described in section 4.

The coupled ECHAM3/LSG model and the integration with this model are described by Voss (1996) and Voss et al. (1998). Here, only a brief description is given. The atmospheric component is the ECHAM3 model with a triangular truncation at wavenumber 21 (T21) and 19 levels in the vertical (Roeckner et al. 1992). The surface fluxes of momentum, heat, and freshwater are parameterized using bulk formulas, whereby using the definitions of drag coefficient and transfer coefficient for heat given by Louis (1979). The oceanic component is the Hamburg Large-Scale Geostrophic (LSG) model (Maier-Reimer et al. 1993). The version used has 11 vertical levels and is integrated on an E grid with an effective horizontal resolution of about 4° × 4°. The two components are coupled through the air–sea fluxes of momentum, heat, and freshwater. Flux corrections (Sausen et al. 1988) are applied to avoid climate drift in the coupled system. The corrections are the differences between the mean fluxes derived from the observations [wind stress from Hellerman and Rosenstein (1983); heat flux from the Comprehensive Ocean–Atmosphere Data Set described by Woodruff et al. (1987);and freshwater flux from Levitus (1982)] and the mean fluxes provided by the uncoupled ECHAM3 model. These differences are calculated for each calendar month and included as constant terms in the coupled integration. Using flux corrections is therefore equivalent to coupling the atmosphere and the ocean with anomalies of the fluxes computed relative to the mean states of the two uncoupled subsystems.

Some features of the variability in the integration with the ECHAM3/LSG model are given by Timmermann et al. (1998), von Storch et al. (2000a, b). El Niño–Southern Oscillation (ENSO)-type of variability is produced by the ECHAM3/LSG model. Apart from the fact that the timescale of ENSO is longer in the model than in the observations (Timmermann et al. 1998), the overall spectral shape of Niño-3 SST is comparable with the observations (von Storch et al. 2000a). This indicates that the relative strength of interannual and interseasonal variations in the model is realistically simulated. The variability in the deep ocean is, to a considerable extent, consistent with that obtained from long integrations with other atmosphere–ocean GCMs (von Storch et al. 2000b).

Figure 2 shows annual mean surface fluxes produced by the coupled model. The flux corrections (Fig. 3), which are added to the fluxes shown in Fig. 2, represent essentially the annual differences between the “observed” and the modeled fluxes. The observed net heat fluxes were parameterized by restoring SST to the observed near-surface air temperature. The diagnostic freshwater fluxes, which were determined by relaxation to climatological sea surface salinity of Levitus (1982) in an ocean-only integration, were treated as the observed freshwater fluxes. These fluxes significantly differ from those derived from sophisticated parameterizations, such as those used by Oberhuber (1988). Discussion on flux corrections used in the ECHAM3/LSG integration is given by Schiller et al. (1997). Large differences between the observed and the modeled fluxes are found for wind stress at high latitudes, for heat flux near the western boundaries of the North Pacific and the North Atlantic and along the Antarctic coast, and for freshwater flux over the North Atlantic and a large part of the Pacific. Since the air–sea interactions described by *R*_{xy}(*τ*) involve only anomaly couplings, the model deficiency in producing observed mean fluxes may not be substantial for the time-varying air–sea interactions. This issue will be addressed elsewhere.

## 2. Interpretation of a lagged cross-correlation function

### a. Conceptual coupled models

*x*and

*y.*It is assumed that both

*x*and

*y*can be decomposed into two components:

*x*

*x*

*x*

*y*

*y*

*y*

*y*

*x*

It is assumed that internal dynamics can be approximately described by a first-order autoregressive [AR(1)] process [for definition, see von Storch and Zwiers (1999)], and responses can be characterized by linear forcing–response relations. It is beyond the scope of this paper to verify these assumptions. A more rigorous deviation of a coupled system similar to one of the conceptual models is given in Frankignoul (1985). Nevertheless, it is worth mentioning that some basic features of the conceptual models do not depend on the assumptions made. The assumption of linear forcing–response relation does not necessarily imply that the governing dynamics are purely linear. What counts here is whether the coupled system reveals linear behaviors. For instance, even though the response of the atmosphere to a SST anomaly is described by a set of nonlinear equations, the solution may be approximately linearly related to an imposing SST anomaly.

#### 1) Conceptual coupled model I

*x**(

*y*) and

*y**(

*x*). Since the existence of

*x**(

*y*) and

*y**(

*x*) relies on variations in the other subsystem,

*x*′ and

*y*′ (which are the source of the variability) cannot be both zero. Consider the case, in which

*x*′ = 0. One has

*x*

*x*

*y*

*y*

*y*

*y*

*x*

*x*

_{t}

*α*

_{x}

*y*

_{t−τx}

*y*

_{t}

*λ*

_{y}

*y*

_{t−1}

*s*

_{t}

*α*

_{y}

*x*

_{t−τy}

*α*

_{x}

*y*

_{t−τx}

*x*

_{t}to a forcing from subsystem

*earlier*time

*t*−

*τ*

_{x}. The amplitude of

*α*

_{x}measures the strength of the response. Here

*τ*

_{x}is positive, and characterizes the timescale needed for

*x*to respond to

*y.*Analogously, the term

*α*

_{y}

*x*

_{t−τy}

*y*to a forcing from

*x,*where

*τ*

_{y}and |

*α*

_{y}| characterize, respectively, the response time and the strength of the response. The two response terms, which represent

*x**(

*y*) and

*y**(

*x*) in model (3), ensure that the interaction works in both directions. The internal process responsible for

*y*′ is approximated by an AR(1) process in model (4). The

*λ*

_{y}characterizes the inertia of subsystem

*s*is a white noise forcing. The coupled variability of model (4) is generated by

*y*through the responses of

*x,*since

*y*

_{t+τ}and taking expectation on the result, one finds

*R*

_{xy}

*τ*

*α*

_{x}

*γ*

_{y}

*τ*

*τ*

_{x}

*γ*

_{y}(

*τ*) is the autocovariance function of

*y.*The

*R*

_{xy}(

*τ*) in Eq. (5) is symmetric about and has its largest amplitude at

*τ*= −

*τ*

_{x}, that is, when

*y*leads

*x*by

*τ*

_{x}. The amplitude of

*R*

_{xy}(

*τ*) depends on the strength of the response of

*x,*|

*α*

_{y}|, and the variance

*γ*

_{y}(0) of

*y,*from which the coupled variability originates. The decay is controlled by the autocovariance functions of

*y,*which is a function of

*λ*

_{y},

*α*

_{y},

*τ*

_{x}, and

*τ*

_{y}. If

*τ*

_{x}+

*τ*

_{y}equals one, the equation of

*y*is an AR(1) process. The

*γ*

_{y}and therefore also

*R*

_{xy}decay exponentially. If

*τ*

_{x}+

*τ*

_{y}is larger than one, the second equation in model (4) reduces to a higher-order AR process. Here

*γ*

_{y}, and therefore also that of

*R*

_{xy}, could then be oscillatory.

*x*in model (4). One has

*y*

_{t}

*α*

_{y}

*x*

_{t−τy}

*R*

_{xy}

*τ*

*α*

_{y}

*γ*

_{x}

*τ*

*τ*

_{y}

*x*leads

*y*by

*τ*

_{y}. Equations (5) and (7) show that, if variations in one subsystem are completely described by linear steady-state responses to forcings from the other subsystem,

*R*

_{xy}(

*τ*) would be symmetric and peaks at a time lag led by the subsystem, from which the forcings originate. Such interactions may occur in the tropical Pacific. Two cross-covariance functions of this type of interactions are given in Fig. 4a.

#### 2) Conceptual coupled model II

*x**(

*y*) and

*y**(

*x*). Consider the case in which

*y**(

*x*) ≠ 0 and

*x**(

*y*) = 0. One has

*x*

*x*

*y*

*y*

*y*

*x*

*x*

_{t}

*λ*

_{x}

*x*

_{t−1}

*r*

_{t}

*y*

_{t}

*λ*

_{y}

*y*

_{t−1}

*s*

_{t}

*α*

_{y}

*x*

_{t−τy}

*x*′ and

*y*′, are approximated by AR(1) processes with parameters

*λ*

_{x}and

*λ*

_{y}, and white noise

*r*and

*s,*respectively. The

*α*

_{y}

*x*

_{t−τy}

*y*to a forcing from

*x*and represents

*y**(

*x*) in model (8). Here

*α*

_{y}and

*τ*

_{y}have the same meaning as in model (4). The absence of

*x**(

*y*) in the first equation in model (8) suggests that

*x*is independent of the state of

*x,*since

*y*does not affect

*x.*

*x*

_{t−τ}and taking expectation on the result. To be consistent with model (8), the covariance between

*x*and

*s*is set to zero. One finds then

*σ*

^{2}

_{x}

*x.*

The *R*_{xy}(*τ*) vanishes when *y* leads. If the response time *τ*_{y} of *y* is not zero, *R*_{xy}(*τ*) would also be zero for positive *τ* with *τ* < *τ*_{y}. The shape of *R*_{xy}(*τ*) for *τ* ≥ *τ*_{y} is controlled by *λ*_{y} and *λ*_{x}. The *R*_{xy}(*τ*) with two combinations of *λ*_{x} and *λ*_{y} is plotted in Fig. 4b. If *λ*_{x}/*λ*_{y} ≪ 1, that is, the memory of *R*_{xy} could peak at *τ* = *τ*_{y} and decays exponentially for *τ* > *τ*_{y} (the upper curve in Fig. 4b). If both subsystems have long memories, *R*_{xy}(*τ*) could peak at a time lag longer than *τ*_{y} and the decay would not be exponential (the lower curve in Fig. 4b). The amplitude of the cross covariances is again proportional to the variance of the variable responsible for the coupled variability, *σ*^{2}_{x}*α*_{y}|.

In the above consideration, if *R*_{xy}(*τ*) vanishes at time lags followed by the subsystem, which provides the one-way stochastic forcing. Extratropical wind stress anomalies may act as one-way stochastic forcings for SST variations.

#### 3) Conceptual coupled model III

*x**(

*y*) and

*y**(

*x*) in model (2), respectively.

With *τ*_{x} = 0, *τ*_{y} = 1, and *x* representing a stochastic forcing from the turbulent surface heat flux, the equation of *y* in model (11) is formally identical to the stochastic equation of SST considered by Frankignoul (1985) and Frankignoul et al. (1998). Different from the SST equation of Frankignoul, in which s represents a forcing from wind stress and heat flux, the variations of SST in model (11) are not entirely generated by the atmosphere and s represents oceanic internal dynamics. With different s, the total variability of SST is differently generated, but not so much the part of the SST variability coupled to *x.* For the latter, the term *α*_{y}*x*_{t−τy}

If *τ*_{x} and *τ*_{y} are both one, model (11) represents a multivariate AR(1) process. The process parameter is a matrix **A***λ*_{x} and *λ*_{y} being its diagonal and *α*_{x} and *α*_{y} its off-diagonal elements. The cross-covariance function of a multivariate AR(1) process is given in von Storch (1995). If **A***R*_{xy}(*τ*) will reveal oscillatory behaviors. If at least one of *τ*_{x} and *τ*_{y} is larger than one, model (11) would represent a higher-order AR process. The *R*_{xy}(*τ*) would have a more complicated form. It becomes difficult to obtain an analytical solution for *R*_{xy}(*τ*). Fortunately, the basic features of the coupled variability in model (11) can be studied, without considering the precise expression of *R*_{xy}(*τ*).

*y*

_{t+τ}and the second one by

*x*

_{t−τ}and taking expectation on the result, one finds

Here *R*_{xy}(*τ*) is the sum of two cross-covariance functions. One is of the type described by Eqs. (5) or (7), which characterizes the coupled variability generated by two-way interactions, and the other of the type described by Eq. (10), which characterizes the coupled variability generated by one-way stochastic forcings. The detailed shapes of *R*_{x*y} and *R*_{x′y}, or of *R*_{xy*} and *R*_{xy′}, may differ from those shown in Figs. 4a and 4b, but the basic features of these cross-covariance functions remain unchanged. The *R*_{x′y} (*R*_{xy′}) remains zero when *y* (*x*) leads. As long as the response of one subsystem to the other can be described by a linear relation, *R*_{x*y} (*R*_{xy*}) would remain symmetric about and peak at a time lag leaded by *y* (*x*). The relative importance of *R*_{x*y}(*τ*) [*R*_{xy*}(*τ*)] and *R*_{x′y}(*τ*) [*R*_{xy′}(*τ*)] determines the shape of *R*_{xy}(*τ*).

Two possible shapes of *R*_{xy}(*τ*) are shown in Fig. 4c. The first one (dashed line) corresponds to the situation, in which *R*_{x′y} and *R*_{x*y} (or *R*_{xy′} and *R*_{xy*}) have the same sign. The larger the amplitude of *R*_{x′y} (*R*_{xy′}) is, the less symmetric *R*_{xy} would be. The second one (solid line) corresponds to the situation, in which *R*_{x′y} and *R*_{x*y} (or *R*_{xy′} and *R*_{xy*}) have opposite signs, so that an antisymmetric *R*_{xy} is obtained.

### b. General features of a lagged cross-correlation function

This section discusses two properties that reflect the total effect of processes involved in the interaction between SST and atmospheric fluxes. For instance, a positive anomaly of sensible heat flux, *Q*_{s}, can lead to an increase of SST. A positive surface temperature anomaly on the other hand can lead to a stronger upward heat flux. A SST anomaly can induce changes in the atmospheric circulation and therefore changes in wind speed, which in turn alters *Q*_{s}, and so on. All these processes are, to some extent, coupled to each other. Their total effect is imprinted in *R*_{xy}(*τ*) derived from time series produced by these processes. Two features of this total effect are considered here.

The first one indicates the relative importance of *x* and *y* in generating the coupled variability. According to the definition of *R*_{xy}(*τ*), *R*_{xy}(*τ*) at positive time lags describes the cross covariances obtained when *x* leads *y.* These covariances can only be generated by *x.* Similarly, *R*_{xy}(*τ*) at negative time lags describes the cross covariances obtained when *y* leads *x,* and can only be generated by *y.* When the amplitude of *R*_{xy}(*τ*) at positive (negative) time lags is larger than that at negative time lags, *x* (*y*) plays a more dominant role in generating the coupled variability.

This feature is consistent with the results of conceptual models. For conceptual models I and II, *R*_{xy}(*τ*) has larger amplitude at time lags when the subsystem leads, from which the coupled variability originates. In conceptual model III, the above statement normally also holds, even though the coupled variability is now generated by both subsystems.

The second feature concerns the feedback involved in an interaction. If no feedback is involved, *R*_{xy} would be zero at either positive or negative time lags. On the other hand, when feedbacks are at work, *R*_{xy} is generally nonzero at both positive and negative lags. It can have the same sign at all time lags, as *R*_{xy}(*τ*) shown in Fig. 4a or the dashed curve in Fig. 4c, or opposite signs at positive and negative time lags, as the solid line in Fig. 4c.

If *R*_{xy} has the same sign at all time lags, 〈*x*_{t−τ1}*y*_{t}*y*_{t}*x*_{t+τ2}*y*_{t−τ2}*x*_{t}*x*_{t}*y*_{t+τ1}*τ*_{1} and *τ*_{2} are arbitrary positive numbers. This suggests that a positive (negative) anomaly of *x* at some earlier time tends to remain positive (negative) after having interacted with *y*_{t}. Equivalently, a positive (negative) anomaly of *y* at some earlier time tends to remain positive (negative) after having interacted with *x*_{t}. An interaction associated with such a cross-covariance function does not drastically alter an earlier anomaly, and will be referred to as an interaction with weak feedback. On the other hand, if *R*_{xy} has opposite signs at positive and negative time lags, an initial positive (negative) anomaly tends to become negative (positive) after having interacted with the other subsystem. An interaction with such a cross-covariance function strongly affects an earlier anomaly and will be referred to as an interaction with strong feedback.

It is noted, that an amplification of an earlier anomaly, corresponding to a positive feedback initiated by an instability process, cannot be described by conceptual models of section 2a, which deal only with stationary time series. However, the tendency of preserving the sign of earlier anomalies is consistent with an air–sea interaction with positive feedback suggested by instability studies. Similarly, the tendency of reversing the sign of earlier anomalies is consistent with the notation of negative feedback.

The two general features discussed here do not provide information about how the coupled variability is generated. In order to obtain such information, a concrete coupled model and its associated lagged cross-correlation function, as described in section 2a, has to be considered.

## 3. Lagged cross-covariance functions obtained from the ECHAM3/LSG integration

This section applies the result of section 2 to air–sea interactions in the coupled ECHAM3/LSG GCM. The two features discussed in section 2b are considered in sections 3a and 3b, respectively. Furthermore the modeled cross-correlation functions are compared with those of the conceptual models of section 2a so that more specific statements about the generation of the coupled variability can be made. In order to be able to compare the result of different fluxes, cross-correlation function *C*_{xy} is considered, where *y* represents monthly anomalies of SST or surface temperature of sea ice (if sea ice is formed) and *x* a flux of momentum, heat, or freshwater. Anomalies are obtained by removing the mean seasonal cycle.

Apart from interpreting the shapes of cross-correlation functions using the results of section 2, the sign of correlations, which was not considered in section 2, is also discussed. Some processes, capable of producing the observed signs of correlations, are suggested. The suggestions are, however, speculative and have to be further verified elsewhere.

### a. Relative dominance of the atmosphere and the ocean

As suggested by section 2b, the relative dominance of the atmosphere and the ocean in generating coupled variations can be assessed by considering the time lag, at which the amplitude of *C*_{xy}(*τ*) is maximal. The *C*_{xy} has larger amplitude at positive than at negative lags when the coupled variability is generated by *x,* and vice versa when it is generated by *y.* The dark areas in Fig. 5 indicates regions, where the maximum value of |*C*_{xy}(*τ*)| is found when atmosphere leads, whereas the gray areas indicates regions, where the maximal value of |*C*_{xy}(*τ*)| is found when ocean leads. White areas are regions where the maximum of |*C*_{xy}| is either small (smaller than 0.05), indicating the lack of coupled variability, or it occurs at the zero lag, indicating *x* and *y* varying in phase.

The most striking feature is the dominant dark color in Figs. 5a–c. Apart from some tropical and polar regions, both heat and momentum fluxes play the dominant role in generating the coupled variations of SST and fluxes. This is particularly true for the momentum flux in the extratropics. Maps of *C*_{xy} at various time lags (Figs. 6 and 7) suggest that *C*_{xy} is near zero over almost the entire extratropical oceans when ocean leads, but reaches about 0.3 to 0.4 when atmosphere leads.

The differences between the amplitudes of *C*_{xy} at positive and negative time lags are much smaller for heat flux (Fig. 8) than for momentum flux. When separately considering the four components of the total heat flux, that is, latent and sensible heat flux, *Q*_{l} and *Q*_{s}, and solar shortwave and thermal longwave radiation, *Q*_{sw} and *Q*_{lw}, different roles in generating the coupled variability are found. The coupled variations of SST and radiative heat flux (i.e., solar plus thermal radiation) are essentially generated by the flux (Fig. 9b). In contrast, it is difficult to tell what is more important in generating coupled variations of SST and turbulent (i.e., sensible plus latent) heat flux (Fig. 9a). The patchy structure in Fig. 9a results from the fact that the amplitudes of *C*_{xy}(*τ*) at positive lags are comparable to those at negative lags. One may interpret this as the situation in which the roles of the atmosphere and the ocean in generating the coupled variability are comparable. Different from the white areas, in which maximal amplitude of *C*_{xy}(*τ*) occurs at the zero lag, the maximal amplitude of *C*_{xy}(*τ*) is found at nonzero lags in Fig. 9a.

*σ*

_{Qt}

*σ*

_{Qr}

*σ*

_{Q}, and the cross-correlation functions with turbulent, radiative, and total heat flux by

*C*

_{Qt}

*C*

_{Qr}

*C*

_{Q}. One has

*σ*

_{Qt}

*σ*

_{Qr}

In some tropical and polar regions, the importance of the atmosphere weakens. In the Tropics, the light gray areas in Fig. 5c suggest the dominant role of the ocean in generating the coupled variability of SST and heat flux. The ocean is also important in generating the coupled variability of SST and wind stress over the tropical western Pacific and in some regions in the tropical Indian Ocean (Figs. 5a,b).

The relation between the freshwater flux and the SST (Fig. 5d) is more complicated. When separating the effect of evaporation from that of precipitation, one finds that SST plays a more dominant role in generating the coupled variability of SST and precipitation in the Tropics (Fig. 9d). An exception is found north of Australia, where the maximum of the annual mean precipitation, resulting mainly from the monsoon in the ECHAM3 model, is located. In the extratropics, there is a tendency for having maximal correlations between SST and precipitation at positive time lags, indicating the dominant role of the atmosphere in generating coupled SST–precipitation variations. The coupled variability of SST and evaporation is generated in a similar manner as the coupled variability of SST and latent heat flux. The maximum amplitude of *C*_{xy}(*τ*) when atmosphere leads is comparable to that when ocean leads.

Over most of the Arctic Ocean and in some regions along the Antarctic coast, it is impossible to determine the relative dominance of the atmosphere and the ocean. A typical cross-correlation function in these regions is shown in Fig. 10 for SST and zonal wind stress. The *C*_{xy} peaks at the zero lag and becomes essentially zero at nonzero lags. It can be shown that such a cross-correlation function is obtained, when *x* (*y*) responds to *y* (*x*) almost instantaneously and the memories of both *x* and *y* are essentially zero. In most areas, in which *C*_{xy}(*τ*) is similar to that shown in Fig. 11, *y* represents the temperature over sea ice. Such a temperature seems to be strongly affected by the atmosphere, which has a short memory. In the following section, cross-correlation function similar to Fig. 10 will not be discussed, since the lack of memory diminishes all possible feedbacks.

### b. Nature of air–sea interactions

*τ*

_{o}is a positive number.

An interaction with no feedback is characterized by a large or small value of |*r*|. If a flux generates variations in SST, which, once being generated, do not feed back to the atmosphere (Fig. 4b), *C*_{xy}(−*τ*_{o}) would be close to zero, leading to a large amplitude of *r.* On the other hand, if SST generates variations in a flux, which, once being generated, do not feed back to the ocean, *C*_{xy}(*τ*_{o}) would be close to zero, and *r* is close to zero.

An interaction with weak feedback is characterized by a positive *r* with moderate amplitude, since the corresponding cross-covariance function is significantly nonzero and has the same sign for both positive and negative time lags, as Fig. 4a and the dashed line in Fig. 4c. On the other hand, an interaction with strong feedback is characterized by a negative *r* with moderate amplitude, since *C*_{xy} is antisymmetric and significantly nonzero at both positive and negative time lags (solid line in Fig. 4c).

A consideration of various cross-correlation functions suggests that interactions with different feedbacks can be most appropriately described by *τ* with *τ*_{o} being one month. The values of *τ* are plotted in Fig. 11 for the fluxes of zonal and meridional momentum, heat, and freshwater flux. Black and white areas indicate interactions with no feedback, dark gray areas with strong feedback, and light gray with weak feedback.

In addition to Fig. 11, lagged cross-covariance functions at selected grid points are calculated and compared with those derived in section 2a. The comparison suggests to what extent the coupled variability of the ECHAM3/LSG model can be considered as being generated in the way quantified by the conceptual coupled models.

#### 1) Zonal momentum flux

At mid- to high latitudes, the large amplitude of |*r*| in Fig. 11a indicates that the zonal wind stress generates variations in SST, which, once being generated, do not feed back to the flux. Figure 12 shows a typical cross-correlation function in these regions, which resembles *R*_{xy}(*τ*) shown in Fig. 4b. It is essentially zero at all time lags when ocean leads, confirming the lack of SST feedbacks. Moreover, the memory of the atmosphere can be considered as zero (i.e., smaller than one month), so that *R*_{xy} peaks when atmosphere leads by one month (as the upper curve in Fig. 4b) and decays exponentially for larger time lags.

Figure 12 shows negative correlations at positive time lags. An inspection of Fig. 6 suggests that this is a feature typical for the extratropics. Since these are also regions with strong mean eastward stress (Fig. 2), an anomaly of eastward stress increases the total zonal stress, leading to a stronger wind speed. The latter can produce a negative SST anomaly through stronger mixing with deeper layer, or through stronger upward latent heat flux through increased evaporation.

In the Tropics, feedbacks become important (Fig. 11a). The nature of feedbacks, however, is far from uniform. Figure 13 shows cross-correlation functions at four grid points in the equatorial western Pacific, central, and eastern Pacific, respectively. In the far west of the equatorial Pacific (Fig. 13a), *C*_{xy}(*τ*) has the largest amplitude when ocean leads, indicating the dominant role of the ocean in generating the coupled variability. Correlations are positive, when ocean leads, but negative when atmosphere leads. When a positive SST anomaly generates eastward wind stress anomalies, their feedback tends to reverse the sign of the earlier SST anomaly. One deals with an interaction with strong feedback.

In the western Pacific (Fig. 13b), *C*_{xy}, which resembles the type of *R*_{xy}(*τ*) shown in Fig. 4a, describes coupled variations generated by two-way interactions. It is essentially symmetric at *τ* = 1 when atmosphere leads by one month, indicating that the coupled variability results from the oceanic response to an atmospheric forcing. Anomalous eastward (westward) wind stresses lead to positive (negative) SST anomalies, likely due to anomalous equatorial downwelling (upwelling) through Ekman transport and horizontal advection of the mean SST gradient. Comparable positive correlations at positive time lags are also found in the central Pacific (Fig. 13c). However, different from the two-way interaction indicated by Fig. 13b, SST anomalies in the equatorial central Pacific feed back little to the atmosphere, leading to smaller correlations at negative time lags in Fig. 13c. The oceanic feedback changes and results in negative correlations at negative lags in the eastern Pacific (Fig. 13d). Here, the cross-correlation function has opposite signs at positive and negative time lags, indicating an interaction with strong feedback.

Overall, the largest correlations are found in the western and central tropical Pacific when atmosphere leads. This suggests that the coupled variability of SST and zonal stress in the tropical Pacific of the ECHAM3/LSG model is, to a large extent, dominated by oceanic responses to wind stress changes. The oceanic feedback is stronger in the western equatorial Pacific (Fig. 13b) than in the central Pacific (Fig. 13c). It is noted that the tendency of preserving the sign of the earlier anomalies is consistent with an air–sea interaction with positive feedback suggested by instability studies. Such an interaction plays an important role in initiating the growth of anomalies during an El Niño event. The feedbacks, which tend to reverse the sign of an earlier anomaly and can therefore affect the transition from a warm to cold (cold to warm) event, are located, according to Figs. 13a and 13d (and also the dark gray tropical areas in Fig. 11a), near in the far west and the far east of the tropical Pacific. The smallness of the correlations indicates the possible involvement of processes operating over long distances.

#### 2) Meridional momentum flux

To a large extent, the *τ* distribution in Fig. 11b is similar to that in Fig. 11a. One-way stochastic forcings from meridional momentum flux dominate the generation of the coupled variability at mid- and high latitudes, and two-way interactions become important in the Tropics. However, the black areas are much more patchy in Fig. 11b than in Fig. 11b. Furthermore, the physical processes involved are likely to be different from those responsible for the correlations between SST and zonal wind stress.

The situation is most clearly demonstrated in Fig. 7. Although the amplitude of *C*_{xy}(*τ*) drastically changes with changing *τ,* the overall pattern remain the same. Positive correlations are found in the most areas of the Northern Hemisphere, whereas negative correlations are found in the most areas of the Southern Hemisphere. A possible explanation for this feature is that meridional wind stress affects the SST through advection processes in the atmosphere. Anomalous equatorward (poleward) wind stresses lead to advection of colder (warmer) air from higher (lower) latitudes, which produces stronger upward (downward) sensible heat flux, and from that negative (positive) SST anomalies.

After being generated, the reaction of the extratropical SST anomalies to the atmosphere differs from that of the tropical SST anomalies. While there is essentially no feedback from the extratropical ocean, significant feedbacks are found in the Tropics. This is demonstrated in Fig. 14. The upper two cross-covariance functions are obtained from the extratropical time series and the lower two from the tropical time series. In both regions, anomalies of poleward (equatorward) wind stress are associated with positive (negative) SST anomalies, and the largest correlations are found when atmosphere leads. At time lags when ocean leads, there is no correlation in the upper-two diagrams, but notable correlations in the lower-two diagrams. The shapes of Figs. 14c and 14d suggest that the interaction tends to preserve the sign of an earlier anomaly, indicating the involvement of a weak feedback. Apart from few dark gray spots resulting from small correlations, such behavior is found in almost the entire Tropics (Fig. 11b), and can be identified as a common feature of all tropical oceans.

#### 3) Heat flux

Figure 11c suggests that, apart from some polar and tropical regions, the lagged cross-correlation function between SST and net heat flux, *Q,* is essentially antisymmetric, indicating the involvement of a strong feedback. A typical cross-correlation function is given in Fig. 1. Anomalous downward (upward) heat fluxes generate positive (negative) SST anomalies, which in turn induce anomalous upward (downward) heat fluxes, favoring a removal of the original positive (negative) heat flux anomalies.

The interaction with strong feedback shown in Fig. 11c is further studied by considering which component of the total heat flux, *Q*_{s}, *Q*_{l}, *Q*_{sw}, and *Q*_{lw}, produces the largest covariance with SST. For this purpose, standard deviations of *Q*_{s}, *Q*_{l}, *Q*_{sw}, and *Q*_{lw} have to be considered. In general, *Q*_{l} and *Q*_{sw} have the largest standard deviation. Large variability of *Q*_{l} is found in the extratropics, whereas variability of similar strength of *Q*_{sw} is found over the tropical Indian Ocean and the tropical western Pacific.

Figure 15 shows cross-correlation functions between SST and the four components of the total heat flux at the grid point considered in Fig. 1. The dashed curve is the cross-correlation function between SST and *Q* shown in Fig. 1, and the other curves are correlations between SST and *Q*_{s}, *Q*_{l}, *Q*_{sw}, and *Q*_{lw}, respectively. The largest correlations are found between SST and *Q*_{l} (solid line in Fig. 15b), which strongly resembles that between SST and total heat flux. Since *Q*_{l} has the largest standard deviation in the extratropics, Fig. 15b suggests that the latent heat flux is most important in producing the antisymmetric shape of *C*_{xy}(*τ*) observed in Fig. 11c.

Taking the relative amplitudes of correlations and standard deviations into account, *Q*_{sw} is the second most important component of *Q,* which contributes to the generation of covariances between SST and total heat flux. Different from *C*_{xy} for *Q*_{l} (solid line in Fig. 15b), *C*_{xy} for *Q*_{sw} (solid line in Fig. 15c) indicates the involvement of a weak feedback. Correlations are essentially positive and have larger amplitudes when atmosphere leads. Stronger (weaker) incoming solar radiation, presumably related to less (more) clouds, lead to positive (negative) SST anomalies, which, once being generated, feeds back little to *Q*_{sw}.

The situation in the Tropics is depicted in Fig. 16, which shows cross-correlation functions between SST and *Q*_{s}, *Q*_{l}, *Q*_{sw}, and *Q*_{lw}, respectively, obtained from the time series at a grid point in the central equatorial Pacific. As in Fig. 15, the dashed curve represents *C*_{xy}(*τ*) between SST and *Q.* The *C*_{xy}(*τ*) between SST and *Q*_{sw} (solid line in Fig. 16c) is more similar to *C*_{xy}(*τ*) between SST and *Q* than the other cross-correlation functions. Based on the fact, that the standard deviation of *Q*_{sw} is larger than those of *Q*_{s}, *Q*_{l}, and *Q*_{lw} in the Tropics, Fig. 16 suggests that covariances between SST and *Q* result, to a large extent, from the interactions with *Q*_{sw}.

The dashed curve in Fig. 16 can be considered as the sum of *R*_{x′y} and *R*_{x*y}, where *R*_{x*y} has large negative covariances at all lags and peaks at *τ* = −1, *R*_{x′y} has small positive covariances at positive lags and peaks at *τ* = 1, and *R*_{x*y} dominates *R*_{x′y}. The resulting *C*_{xy} suggests the involvement of a weak feedback. Positive (negative) SST anomalies lead to more (less) clouds, and from that less (more) downward solar radiation. There is a tendency for downward solar radiation to generate positive SST anomalies. However, the feedback of *Q*_{sw} to the SST, presumably through changes in the atmospheric circulation, is strong enough to mask this process. Consistent with the suggestion that the SST–*Q*_{sw} relation results from changes in cloud cover, *C*_{xy}(*τ*) for *Q*_{sw}(*τ*) (solid line in Fig. 16c) is similar to *C*_{xy}(*τ*) for *Q*_{lw} (solid line Fig. 16d), but has the opposite sign. More clouds lead to weaker incoming solar radiation, but stronger thermal radiation by the sea surface. These relations are consistent with the thermostat effect discussed by Ramanathan and Collins (1991).

#### 4) Freshwater flux

Figure 11d shows that the interaction between SST and freshwater flux changes its nature drastically from region to region. A more clear picture is obtained in Fig. 17 where evaporation and precipitation are separately considered.

The *r* distribution for evaporation (Fig. 17a) is almost identical to that of the latent heat flux (not shown). This reflects the direct relation between evaporation and latent heat flux. As for latent heat flux, an antisymmetric *C*_{xy}(*τ*) for evaporation indicates the involvement of a strong feedback. However, since the standard deviation of precipitation is comparable to, or larger than, that of evaporation, the antisymmetric form of *C*_{xy}(*τ*) described in Fig. 17a is hardly noticeable in Fig. 11d for the total freshwater flux.

In the equatorial regions, the SST–precipitation interaction involves a strong feedback over the Indian Ocean and the western Pacific, but a weak feedback over the equatorial central Pacific and the Atlantic. The latter is further demonstrated in the top of Fig. 18. Apart from a small local minimum at *τ* = 1, the cross-correlation function resembles that shown in Fig. 4a. Variations of precipitation appear, to a large extent, as linear responses to changes in SST. The similarity of the two curves in the top of Fig. 18, which represent *C*_{xy}(*τ*) between SST and total freshwater flux and *C*_{xy}(*τ*) between SST and precipitation, respectively, indicates the minor role of evaporation for the cross-correlation function between SST and total freshwater flux in the equatorial regions.

In the subtropical regions, *r* has some large values in Fig. 11d, but moderate values in Fig. 17b. The situation is further quantified in the bottom of Fig. 18, which shows the *C*_{xy}(*τ*) for total freshwater flux (dashed line), the *C*_{xy}(*τ*) for evaporation (the antisymmetric solid curve), and the *C*_{xy}(*τ*) for precipitation (the symmetric solid curve). At time lags when ocean leads, the negative correlations between SST and evaporation (the antisymmetric solid line) cancel, to a large extent, the positive correlations between SST and precipitation (the symmetric solid line). Thus, even though feedbacks are involved in both SST–evaporation and SST–precipitation interactions, *C*_{xy}(*τ*) for the total freshwater (dashed line in bottom of Fig. 18) resembles the cross-covariance function shown in Fig. 4b.

Cancellations are also found at mid- and high latitudes, where correlations between SST and precipitation are positive when ocean leads and negative when atmosphere leads, but the opposite is true for correlations between SST and evaporation.

### c. Seasonal dependence

The results of sections 3a and 3b are derived from monthly anomalies, without taking the seasonal dependence into account. In order to quantify the possible seasonal dependence, similar calculations are carried out for seasonal-dependent data, whereby December, January, and February representing the northern winter; March, April, and May the northern spring; June, July, and August the northern summer; and September, October, and November the northern fall. The resulting cross-correlation functions are noisier than those considered before, mainly due to change in sampling interval from a month to a year within the considered time series. Nevertheless, the results indicate significant seasonal dependences of air–sea interactions in the model. Some salient features are summarized below.

Extratropical interaction between wind stress and SST, in which wind stress generates the coupled variability and a SST anomaly, once being generated, does not feedback, is obtained in all seasons. However, the strength of this interaction varies with the seasonal cycle. This is particularly true for meridional wind stress. Correlations between SST and meridional wind stress become smaller during the summer season, likely caused by a decrease in meridional temperature gradient, and from that a decrease in meridional advection of temperature through wind during this part of year.

Seasonal dependence of tropical interactions between zonal wind stress and SST is also evident. The cross-correlation function in the tropical western Pacific in Fig. 13b, which is nearly symmetric about *τ* = 1 and indicates a linear response of SST to wind forcing, is only found during the northern winter. In the northern summer and fall, the cross-correlation function is symmetric about *τ* = 0, indicating that SST and wind stress vary in phase.

Extratropical interaction between heat flux and SST is characterized, outside the summer season, by the antisymmetric cross-correlation function between latent heat flux and SST. During the summer season, the role of atmospheric fluxes in generating SST anomalies weakens. The cross-correlation function between SST and latent heat flux at the same location as in Fig. 15b shows a nearly symmetric shape centered around *τ* = −1 in June, July, and August, indicating that the coupled variability is characterized by a linear response of latent heat flux to SST forcing. Moreover, since the extratropical variance of solar radiation can be comparable, or even larger than that of latent heat flux during the summer season, the role of solar radiation in generating coupled variability enhances during this part of year.

## 4. Discussions

The results of this paper suggest severe problems of the ad hoc approaches discussed in section 1. When integrating an oceanic GCM using the wind stress model of Latif and Villwock (1990), the coupled variability originates entirely from the ocean. The cross-correlation functions derived from integrations of these models are expected to have largest values when ocean leads. This contradicts with the results of section 3. Figure 5 shows that, in the coupled model, the largest correlations between SST and wind stress are found when atmosphere leads. The other ad hoc approach, which treats atmosphere as a noise generator and includes additive noise into the ocean-only system, must also be considered with care. Cross-correlation functions resulting from this approach are expected to be near zero when ocean leads. Such cross-correlation functions are only found for extratropical wind stress in the coupled model. For heat and freshwater fluxes, feedbacks are involved. In particular, the cross-correlation functions in the Tropics of the coupled model have large amplitude when ocean leads.

Apart from the ad hoc approaches, efforts are also made to formulate more sophisticated and physically based models that represent feedbacks between SST and the fluxes. Examples of these models are the coupled model of Zebiak and Cane (1986), the simple atmospheric model of Kleeman and Power (1995), and the advective mixed layer model of Seager et al. (1995). An analysis of cross-correlation functions derived from integrations with these models can shed some light on the nature of the air–sea interactions produced by these models.

In general, the air–sea interactions identified in the coupled ECHAM3/LSG model can be used to verify simple atmospheric models, which are designed to drive an oceanic GCM in the place of an atmospheric GCM. If such a simplified coupled system produces the same interactions found in the fully coupled GCM, the cross-covariance functions in the simple coupled model should be similar to those found in the fully coupled GCM.

The results shown in section 3 are consistent with a few studies concerning cross-correlation functions derived from observations. For instance, Fig. 12 is similar to the cross-covariance function between the dominant EOF coefficients of SST and sea level pressure over the North Pacific (Fig. 20 in Frankignoul 1985). This indicates that, in both the observations and the coupled integrations, the extratropical SST variations, once being generated, do not feed back to the atmospheric circulation. Consistency is also found for the cross-covariance function between SST and turbulent heat flux. The observed cross-covariance function between SST and turbulent heat flux in the eastern part of the North Atlantic (Fig. 3 in Frankignoul et al. 1998) is very similar to curves in Figs. 15a and 15b, suggesting that the same interaction process is at work in both observations and the model simulation.

The result of section 2 can also be applied to other types of interactions. For instance, the atmospheric fluxes can also generate variations in, for example, thermohaline circulation or gyre circulation. Since these variations cannot directly feed back to the atmosphere (though they may indirectly feed back through SST), it is conceivable, that their feedbacks are weaker, at least when monthly data are considered, than those discussed in this paper. One can quantify such interactions by considering cross-covariance functions between fluxes and variables, which are characteristic for the thermohaline and gyre circulations.

The air–sea interactions considered in this paper suffer from some limitations. First, the analysis concentrates on interactions on interseasonal timescales by using monthly data. When considering interactions on much longer or shorter timescales, the time series should be prepared accordingly, that is, by using much higher temporal resolution or by smoothing the data. Secondly, remote interactions are poorly described. Consider the situation in which wind stress anomalies at a location *i* are generated by SST anomalies originating in some other places. If these SST anomalies are recognizable at the location *i,* the remote interaction would be partly described by the cross-correlation function at the location *i.* The small correlations in the far western and eastern tropical Pacific in Figs. 13a and 13b may represent these type of remote interactions. On the other hand, remote SST anomalies can also induce changes in the atmosphere and, through teleconnections, changes in the wind stress at the location *i.* In this case, wind stress variations at location *i* are generated by SST anomalies at a location *j* ≠ *i.* Such interactions cannot be captured by local cross-covariance functions and have to be described by correlations between time series at different locations.

## 5. Summary

A cross-covariance function *R*_{xy}(*τ*) between an atmospheric variable *x* and an oceanic variable *y* can, by definition, distinguish the situation when atmosphere leads from the situation when ocean leads. Because of that, a particular type of interactions is characterized by a well-defined class of *R*_{xy}(*τ*). By considering conceptual coupled models, various types of interactions and their associated cross-covariance functions are identified.

There are two types of coupled variations. One is generated by two-way interactions. The variability of one subsystem, resulting from responses to forcings from the second subsystem, feeds back to the second subsystem. Assuming linear forcing–response relation, such a two-way interaction is characterized by a symmetric *R*_{xy}(*τ*). The second type of coupled variations is generated by one-way stochastic forcings. The variability of one subsystem does not feed back to the second one. The state of the second subsystem is independent of that of the first so that the forcing can be considered as being stochastic. The resulting *R*_{xy}(*τ*) is zero at time lags followed by the subsystem, which provides the forcing.

The shape of *R*_{xy}(*τ*) indicates the feedbacks involved in an interaction. An interaction with no feedback is described by a cross-covariance function, which is zero at either positive or negative time lags, whereas an interaction with feedback is described by a cross-covariance function, which is nonzero at all time lags. Depending on the sign of *R*_{xy}(*τ*) at positive and negative time lags, the interaction described by *R*_{xy}(*τ*) can involve a weak or strong feedback. The former has the same sign at both positive and negative time lags and tends to preserve the sign of an earlier anomaly, whereas the latter has opposite signs at positive and negative time lags and tends to reverse the sign of an earlier anomaly. In all cases, the relative amplitude of *R*_{xy}(*τ*) at positive and negative time lags suggests the relative dominance of the atmosphere and the ocean in generating the coupled variability.

Using this knowledge of *R*_{xy}(*τ*), air–sea interactions in an integration with the coupled ECHAM3/LSG model are quantified by considering cross-covariance functions between SST and surface fluxes of heat, freshwater, and momentum. Two conclusions are drawn. First, the atmosphere plays the dominant role in generating the coupled variability of SST and momentum flux and of SST and heat flux over most of the ocean. For the coupled variability of SST and freshwater flux, the atmosphere and the ocean play nearly equal role in generating of the variability. The second conclusion concerns the feedbacks involved in the interactions and is summarized below.

In the extratropics, eastward (westward) wind stress generates, through stronger (weaker) mixing and stronger (weaker) evaporation, negative (positive) SST anomalies. Equatorward (poleward) wind stress generates negative (positive) SST anomalies, through advection of colder (warmer) air temperature from the higher (lower) latitudes. The SST anomalies, once being generated, do not feed back to the atmosphere, leading to near zero *C*_{xy}(*τ*) when SST leads. In the Tropics, interactions with feedbacks are found. Whereas similar feedbacks are identified for meridional wind stress over almost all tropical oceans, the feedback associated with the zonal wind stress changes its nature from location to location. In the western Pacific, SST anomalies respond to zonal wind stress forcings in a linear manner and feed back to the atmosphere. The interactions tend to preserve the sign of an earlier anomaly, and are consistent with positive feedbacks responsible for initiating the growth of anomalies during an ENSO event. In the far western and eastern Pacific, interactions with strong feedbacks are found. The interactions tend to reverse the sign of an earlier anomaly and are consistent with negative feedbacks responsible for the transition from a warm to cold (cold to warm) event.

The coupled variability of SST and heat flux is characterized by antisymmetric shapes of cross-covariance functions. Variations in the total heat flux generates SST anomalies, which feed back and tend to reverse the sign of the earlier heat flux anomalies. This strong feedback originates from the latent heat flux. The coupled variability of SST and evaporation is generated by the similar interaction process. In the tropical-central Pacific, variations of precipitation appear essentially as linear responses to changes in SST.

The above results hold only for monthly anomalies without considering the seasonal dependence. If seasonal-dependent data are considered, the results have to be modified, because of changes induced by, for example, seasonal variations in the mean state and seasonal changes in the variances of various components of heat and freshwater fluxes.

## Acknowledgments

I thank Bin Wang and Claude Frankignoul for helpful discussions and Reinhard Voss for proving the GCM data. Thanks also to the two anonymous reviewers for constructive comments.

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Annual mean of fluxes of zonal, meridional momentum, heat, and freshwater, averaged over 300 yr, produced by the ECHAM3 model. Units: Pa for wind stress, W m^{−2} for heat flux, and m s^{−1} for freshwater flux. Contour intervals are 0.05 Pa for zonal and 0.025 Pa for meridional stress, 50 W m^{−2} for heat flux, and 310^{−8} m s^{−1} for freshwater flux. Solid (dashed) lines indicate positive (negative) values.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Annual mean of fluxes of zonal, meridional momentum, heat, and freshwater, averaged over 300 yr, produced by the ECHAM3 model. Units: Pa for wind stress, W m^{−2} for heat flux, and m s^{−1} for freshwater flux. Contour intervals are 0.05 Pa for zonal and 0.025 Pa for meridional stress, 50 W m^{−2} for heat flux, and 310^{−8} m s^{−1} for freshwater flux. Solid (dashed) lines indicate positive (negative) values.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Annual mean of fluxes of zonal, meridional momentum, heat, and freshwater, averaged over 300 yr, produced by the ECHAM3 model. Units: Pa for wind stress, W m^{−2} for heat flux, and m s^{−1} for freshwater flux. Contour intervals are 0.05 Pa for zonal and 0.025 Pa for meridional stress, 50 W m^{−2} for heat flux, and 310^{−8} m s^{−1} for freshwater flux. Solid (dashed) lines indicate positive (negative) values.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Annual mean of corrections for fluxes of zonal, meridional momentum, heat, and freshwater. The corrections are constant terms (apart from the annual cycle) and represent essentially the differences between the observed and the modeled fluxes. Units and contour intervals are as in Fig. 2. Solid (dashed) lines indicate positive (negative) values.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Annual mean of corrections for fluxes of zonal, meridional momentum, heat, and freshwater. The corrections are constant terms (apart from the annual cycle) and represent essentially the differences between the observed and the modeled fluxes. Units and contour intervals are as in Fig. 2. Solid (dashed) lines indicate positive (negative) values.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Annual mean of corrections for fluxes of zonal, meridional momentum, heat, and freshwater. The corrections are constant terms (apart from the annual cycle) and represent essentially the differences between the observed and the modeled fluxes. Units and contour intervals are as in Fig. 2. Solid (dashed) lines indicate positive (negative) values.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-covariance functions obtained from conceptual models. Amplitudes are arbitrary. The time lags on the *x* axis are in months, when the time step of the models considered in section 2a is 1 month.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-covariance functions obtained from conceptual models. Amplitudes are arbitrary. The time lags on the *x* axis are in months, when the time step of the models considered in section 2a is 1 month.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-covariance functions obtained from conceptual models. Amplitudes are arbitrary. The time lags on the *x* axis are in months, when the time step of the models considered in section 2a is 1 month.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Relative dominance of SST and fluxes of (a) zonal and (b) meridional momentum flux, (c) heat, and (d) freshwater in generating coupled variability, as indicated by the time lag *τ*_{max}, at which the maximal amplitude of *R*_{xy} is found. Black (gray) areas are areas in which the maximal amplitude of *R*_{xy}(*τ*) is found when atmosphere (ocean) leads, indicating the dominance of the flux (SST). In the white areas, *τ*_{max} is zero or maximal amplitude of *R*_{xy} is smaller than 0.05.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Relative dominance of SST and fluxes of (a) zonal and (b) meridional momentum flux, (c) heat, and (d) freshwater in generating coupled variability, as indicated by the time lag *τ*_{max}, at which the maximal amplitude of *R*_{xy} is found. Black (gray) areas are areas in which the maximal amplitude of *R*_{xy}(*τ*) is found when atmosphere (ocean) leads, indicating the dominance of the flux (SST). In the white areas, *τ*_{max} is zero or maximal amplitude of *R*_{xy} is smaller than 0.05.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Relative dominance of SST and fluxes of (a) zonal and (b) meridional momentum flux, (c) heat, and (d) freshwater in generating coupled variability, as indicated by the time lag *τ*_{max}, at which the maximal amplitude of *R*_{xy} is found. Black (gray) areas are areas in which the maximal amplitude of *R*_{xy}(*τ*) is found when atmosphere (ocean) leads, indicating the dominance of the flux (SST). In the white areas, *τ*_{max} is zero or maximal amplitude of *R*_{xy} is smaller than 0.05.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Distributions of cross-correlation function *C*_{xy}(*τ*) between SST and zonal momentum flux at time lags *τ* = −1, 0, 1 month. (top) *C*_{xy}(−1), (middle) *C*_{xy}(0), and (bottom) *C*_{xy}(−1).

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Distributions of cross-correlation function *C*_{xy}(*τ*) between SST and zonal momentum flux at time lags *τ* = −1, 0, 1 month. (top) *C*_{xy}(−1), (middle) *C*_{xy}(0), and (bottom) *C*_{xy}(−1).

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Distributions of cross-correlation function *C*_{xy}(*τ*) between SST and zonal momentum flux at time lags *τ* = −1, 0, 1 month. (top) *C*_{xy}(−1), (middle) *C*_{xy}(0), and (bottom) *C*_{xy}(−1).

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Same as Fig. 6, but for meridional momentum flux.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Same as Fig. 6, but for meridional momentum flux.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Same as Fig. 6, but for meridional momentum flux.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Same as Fig. 6, but for total heat flux.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Same as Fig. 6, but for total heat flux.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Same as Fig. 6, but for total heat flux.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

As Fig. 5, but for (a) turbulent and (b) radiative heat flux, (c) evaporation, and (d) precipitation.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

As Fig. 5, but for (a) turbulent and (b) radiative heat flux, (c) evaporation, and (d) precipitation.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

As Fig. 5, but for (a) turbulent and (b) radiative heat flux, (c) evaporation, and (d) precipitation.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-correlation function between SST and zonal and momentum flux at (75°N, 56°E). Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-correlation function between SST and zonal and momentum flux at (75°N, 56°E). Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-correlation function between SST and zonal and momentum flux at (75°N, 56°E). Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Distribution of *r* = *C*_{xy}(1)/*C*_{xy}(−1) for (a) zonal and (b) meridional momentum flux, (c) heat flux, and (d) freshwater flux. Areas with large and small values of *r* (black and white areas) indicate interaction with no feedback. Areas with negative *r* with moderate amplitude (dark gray areas) indicate interaction with strong feedback. Areas with positive *r* with moderate amplitude (light gray areas) indicate interaction with weak feedback.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Distribution of *r* = *C*_{xy}(1)/*C*_{xy}(−1) for (a) zonal and (b) meridional momentum flux, (c) heat flux, and (d) freshwater flux. Areas with large and small values of *r* (black and white areas) indicate interaction with no feedback. Areas with negative *r* with moderate amplitude (dark gray areas) indicate interaction with strong feedback. Areas with positive *r* with moderate amplitude (light gray areas) indicate interaction with weak feedback.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Distribution of *r* = *C*_{xy}(1)/*C*_{xy}(−1) for (a) zonal and (b) meridional momentum flux, (c) heat flux, and (d) freshwater flux. Areas with large and small values of *r* (black and white areas) indicate interaction with no feedback. Areas with negative *r* with moderate amplitude (dark gray areas) indicate interaction with strong feedback. Areas with positive *r* with moderate amplitude (light gray areas) indicate interaction with weak feedback.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-correlation function between SST and zonal momentum flux at (47°S, 118°E) in the Southern Ocean. Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-correlation function between SST and zonal momentum flux at (47°S, 118°E) in the Southern Ocean. Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-correlation function between SST and zonal momentum flux at (47°S, 118°E) in the Southern Ocean. Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-correlation functions between SST and zonal momentum flux in the tropical Pacific at (3°N, 146°E), (3°N, 163°E), (3°N, 180°E), and (3°N, 107°W). Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-correlation functions between SST and zonal momentum flux in the tropical Pacific at (3°N, 146°E), (3°N, 163°E), (3°N, 180°E), and (3°N, 107°W). Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-correlation functions between SST and zonal momentum flux in the tropical Pacific at (3°N, 146°E), (3°N, 163°E), (3°N, 180°E), and (3°N, 107°W). Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-correlation functions between SST and meridional stress at (47°N, 166°E), (53°N, 112°W), (3°S, 124°W), and (14°N, 152°W). Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-correlation functions between SST and meridional stress at (47°N, 166°E), (53°N, 112°W), (3°S, 124°W), and (14°N, 152°W). Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-correlation functions between SST and meridional stress at (47°N, 166°E), (53°N, 112°W), (3°S, 124°W), and (14°N, 152°W). Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-correlation functions in the extratropics at (42°N, 180°E) between SST and sensible heat flux, *Q*_{s}, latent heat flux, *Q*_{l}, solar radiation, *Q*_{sw}, and long wave radiation, *Q*_{lw}. Also shown in (a)–(d) is the cross-correlation function between SST and total heat flux, *Q,* (dashed line). Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-correlation functions in the extratropics at (42°N, 180°E) between SST and sensible heat flux, *Q*_{s}, latent heat flux, *Q*_{l}, solar radiation, *Q*_{sw}, and long wave radiation, *Q*_{lw}. Also shown in (a)–(d) is the cross-correlation function between SST and total heat flux, *Q,* (dashed line). Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Lagged cross-correlation functions in the extratropics at (42°N, 180°E) between SST and sensible heat flux, *Q*_{s}, latent heat flux, *Q*_{l}, solar radiation, *Q*_{sw}, and long wave radiation, *Q*_{lw}. Also shown in (a)–(d) is the cross-correlation function between SST and total heat flux, *Q,* (dashed line). Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Same as Fig. 15, but for a grid point at (3°N, 180°E).

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Same as Fig. 15, but for a grid point at (3°N, 180°E).

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

Same as Fig. 15, but for a grid point at (3°N, 180°E).

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

As Fig. 11, but for (a) evaporation and (b) precipitation.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

As Fig. 11, but for (a) evaporation and (b) precipitation.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

As Fig. 11, but for (a) evaporation and (b) precipitation.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

(top) The cross-correlation functions between SST and precipitation and between SST and freshwater flux at (3°S, 108°E). The similarity between the two curves indicates the negligible role of evaporation at this grid point. (bottom) The cross-correlation functions at (19°N, 219°E) between SST and net freshwater flux (dashed curve in the bottom), between SST and evaporation (solid symmetric curve in the bottom), and between SST and precipitation (solid antisymmetric curve in the bottom). Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

(top) The cross-correlation functions between SST and precipitation and between SST and freshwater flux at (3°S, 108°E). The similarity between the two curves indicates the negligible role of evaporation at this grid point. (bottom) The cross-correlation functions at (19°N, 219°E) between SST and net freshwater flux (dashed curve in the bottom), between SST and evaporation (solid symmetric curve in the bottom), and between SST and precipitation (solid antisymmetric curve in the bottom). Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2

(top) The cross-correlation functions between SST and precipitation and between SST and freshwater flux at (3°S, 108°E). The similarity between the two curves indicates the negligible role of evaporation at this grid point. (bottom) The cross-correlation functions at (19°N, 219°E) between SST and net freshwater flux (dashed curve in the bottom), between SST and evaporation (solid symmetric curve in the bottom), and between SST and precipitation (solid antisymmetric curve in the bottom). Time lags are in months.

Citation: Journal of Climate 13, 19; 10.1175/1520-0442(2000)013<3361:SOASII>2.0.CO;2