1. Introduction

The climate system exhibits variability on many timescales. Two simple models of the spectrum of climatic variability that are often used are white noise and red noise. For example, atmospheric spectra are often modeled as being white noise for timescales longer than a month or so (after filtering out the seasonal cycle). Hasselmann (1976) used such a model to argue that an important feature of the climate system is the red noise oceanic response to white noise atmospheric forcing. Neither white- nor red-noise spectra have any preferred timescales. However, real climatic spectra do show preferred timescales (or spectral peaks). Some of these preferred timescales are associated with periodicities in the external forcing, for example, the diurnal and annual cycles. Some others may be traced back to an intrinsic timescale associated with a single component of the climate system, such as the atmosphere or the ocean, with feedbacks from other components playing only a secondary role. An example is the Madden–Julian oscillation in the tropical atmosphere. The remaining preferred timescales of the climate system must arise through interactions between two or more components of the climate system, as in for example, the El Niño–Southern Oscillation (ENSO) phenomenon. Such timescales are associated with “coupled” modes of climatic variability, whereas the preferred timescales of a single component subsystem may be said to arise from “uncoupled” modes.

In recent years, there has been increasing attention devoted to climatic variability on decadal to centennial timescales. Understanding the mechanisms behind this variability is crucial to explaining regional climate shifts and also for distinguishing between natural and anthropogenic climate change. An important question regarding decadal variability is whether there are any preferred timescales. Some observational studies suggest that there are indeed such timescales (e.g., Deser and Blackmon 1993; Levitus et al. 1994). A recent study by Sutton and Allen (1997) suggests that there may be a decadal timescale associated with the propagation of SST anomalies along the Gulf Stream. However, observational datasets are often too short in duration to allow definitive conclusions to be drawn.

Meanwhile, a variety of mechanisms have been put forward for selecting preferred timescales of decadal variability. Latif and Barnett (1994) have proposed a coupled mechanism that depends upon a rather strong atmospheric response to sea surface temperature (SST) anomalies. Many of the other proposed mechanisms are not so strongly coupled in nature and typically depend upon the existence of one or more oscillatory ocean modes. In some cases, the oscillatory ocean mode is unstable, and expresses itself spontaneously even with steady surface forcing (Weaver et al. 1991; Weaver and Sarachik 1991; Winton and Sarachik 1993). In other cases, all the oscillatory modes are damped, and stochastic forcing at the surface is required to excite them (Mikolajewicz and Maier-Reimer 1990; Mysak et al. 1993; Griffies and Tziperman 1995; Capotondi and Holland 1997; Saravanan and McWilliams 1997). The coupled general circulation model (GCM) study of Delworth et al. (1993) suggests that on multidecadal timescales, irregular oscillations of the oceanic thermohaline circulation may force atmospheric variability. A common feature of these oscillatory modes, whether damped or not, is that they appear to involve basin-scale interactions. A prototypical example of such a mechanism is the thermohaline “loop” oscillator described by Welander (1985). It is also possible that preferred timescales may be generated by local mechanisms, that is, not involving basin-scale interactions. For example, integration of white-noise freshwater fluxes over a localized region could generate decadal timescale variability (Weisse et al. 1994).

A related issue regarding variability on decadal timescales is the role of coupled versus uncoupled modes. In the Tropics, coupled modes seem to play a dominant role, at least on interannual timescales, but the situation in the midlatitudes is less clear. Only part of the midlatitude atmospheric variability appears to be related to coupled variability in the Tropics. For example, Hoerling et al. (1995) compared observations and ensemble GCM integrations and estimated that a significant fraction of the interannual low-frequency variability in the midlatitudes is due to “internal” dynamics, that is, unrelated to tropical SST variability. There is no clear consensus on the role of coupled ocean–atmosphere modes in the midlatitudes. Early studies of local ocean–atmosphere interaction (on monthly timescales) by Davis (1976) and Frankignoul and Hasselmann (1977) suggested a red-noise type of correlation between atmospheric forcing and SST patterns, with the ocean responding passively to atmospheric forcing. A recent study by Battisti et al. (1995), who used a mixed-layer model of the North Atlantic to simulate SST variability on interannual timescales, also concluded that atmospheric forcing of the ocean played the dominant role. The analyses of North Atlantic interannual variability coupled GCMs by Delworth (1996) and Saravanan (1998) also reached a similar conclusion. The idealized coupled model study of Saravanan and McWilliams (1997) concluded that spatial correlations in the atmospheric forcing played an important role. GCM studies of atmospheric response to realistic values of midlatitude SST anomalies find that the response is often quite weak, and sometimes inconsistent (e.g., Lau and Nath 1994). This is in contrast to the arguments of Latif and Barnett (1994) for the existence of coupled modes of variability. Yet another viewpoint is taken by some ocean modeling studies (e.g., Weaver et al. 1991; Weaver and Sarachik 1991), which tend to highlight the possible role of oceans in interdecadal variability.

In this study, we introduce a simple analytical model of advective ocean–atmosphere interaction, in a linear stochastic framework, to address some of the issues raised above. More complex numerical mixed-layer representations of the ocean, incorporating both stochastic forcing and advection, have been used to fit statistical models to observed data. Lemke et al. (1980) have fitted a stochastic model incorporating horizontal transport to observed polar sea-ice variability. Frankignoul and Reynolds (1983) describe the use of a local stochastic model, including the effects of advection by the observed mean current, to predict the statistical characteristics of observed SST anomalies in the North Pacific on timescales of several months. They find that mean advection has only a small effect in general, although in regions of large currents, the advection effects were important at lower frequencies. Herterich and Hasselmann (1987) have fitted a more general nonlocal stochastic model, incorporating advection and diffusion, to observed SST anomalies over the same region. In a different context, Penland and Matrosova (1994) have applied stochastic inverse modeling techniques to ENSO. Our goal, in contrast, is to use a simple model to explore the parameter space of coupled ocean–atmosphere interaction in the spirit of Hasselmann (1976) and identify the various regimes of advective ocean–atmosphere interaction.

One of the important results of this study is that advective ocean–atmosphere interaction can serve as a mechanism for selecting preferred timescales of variability. The mechanism acts over the region of atmospheric flux exchange with the ocean, and it does not require the presence of oscillatory modes either in the atmosphere or in the ocean. Our study also highlights the differences between local and nonlocal oceanic feedback to atmospheric forcing and its dependence on the coupling parameters. Section 2 briefly discusses aspects of the atmospheric and oceanic general circulation thatmotivate the one-dimensional model. Section 3 describes the model equations, properties of the white- noise forcing, and the analytical solution to the equations. The frequency spectrum of oceanic variability is discussed in section 4. The atmospheric variability spectrum and the role of oceanic feedback are analyzed in section 5, followed by concluding remarks in section 6.

2. Phenomenological motivation for the model

Consider midlatitude ocean–atmosphere interaction on decadal timescales. The associated spatial scales of interest are typically quite large, of the order of thousands of kilometers (Deser and Blackmon 1993; Kushnir 1994; Levitus et al. 1994). This is slightly smaller than gyre or basin scales in the ocean and somewhere in between planetary and synoptic scales in the atmosphere. Atmospheric variability on timescales of the order of a month or longer in the extratropics is characterized by the presence of a small number of large-scale “standing wave” modes with well- defined spatial structures, such as the well-known teleconnection patterns, the Pacific–North American (PNA) pattern and the North Atlantic Oscillation pattern (cf. Wallace and Gutzler 1981). In other words, low-frequency variability in the atmosphere is characterized by spatial coherence. In contrast, atmospheric variability on synoptic timescales (except for blocking events) tends to be spatially incoherent, as far as the slow-moving ocean is concerned; for example, there is no preferred spatial location for the “highs” and “lows” of an extratropical cyclone, such as there is for the PNA pattern. There are various theories that attempt to explain the origin of these patterns, based on barotropic energy dispersion, nearly stationary states of the atmosphere, instability of the climatological mean state, coherent feedback from the transient eddies, and so on (e.g., see Branstator 1992 and references therein).

The temporal evolution of these modes of low-frequency atmospheric variability is less well understood. Extended integrations of atmospheric models suggest thesemodes preserve their spatial structure over a broad band of frequencies (Nitsche and Wallace 1995). The variance spectrum of these modes appears to be essentially “white” for periods longer than one or two years (Feldstein and Robinson 1994; Nitsche and Wallace 1995; Saravanan and McWilliams 1995, 1997). In other words, uncoupled atmospheric variability on very long timescales is characterized by the lack of temporal coherence.

To illustrate the spatially coherent/temporally incoherent nature of intrinsic midlatitude atmospheric variability, we present some results from multidecadal integrations using a state-of-the-art atmospheric GCM. The model we use is the Community Climate Model, Version 3 (CCM3), developed at the National Center for Atmospheric Research (NCAR), which is described in Kiehl et al. (1996). We analyze three 45-yr integrations that were carried out using CCM3 at the standard T42 horizontal resolution, using observed SST values as the surface boundary condition in the tropical belt between 30°S and 30°N. The observed SST values used were derived from monthly mean values for the period January 1950–December 1994. Outside the tropical belt, the climatological annual cycle of SST was used as the surface boundary condition. (The observed and climatological SST values were blended over a 10°-wide transition region.) Therefore, all atmospheric variability in these integrations is either intrinsically generated or forced by tropical SST variability and is not related to any midlatitude SST variability.

The three 45-yr integrations of CCM3 were started from slightly different initial conditions, forming three independent realizations of the atmosphere. The annually averaged net surface heat flux into the North Atlantic region was computed for each of the 45 yr of the three integrations. Figure 1 shows the first three empirical orthogonal functions (EOFs) of the surface heat flux computed from the combined 135-yr dataset. The spatial structure of the EOFs as shown includes the variance information, so that the value at each grid point represents the typical root- mean-square (rms) heat flux deviation associated with theEOF. These EOFs represent the intrinsic modes of extratropical surface forcing associated with the atmospheric variability, that is, in the absence of any oceanic feedbacks. Note that the three EOFs account for a significant fraction (57%) of the total variance and have fairly large-scale spatial structure, with maximum amplitudes biased toward the western part of the North Atlantic basin. Note also that the first two EOFs have dipolar structures, with the axis of the dipole oriented approximately parallel to the western boundary of the ocean basin. Similar north–south-oriented dipole patterns of heat flux variability have been noted in the hindcast study of North Atlantic SST variability by Battisti et al. (1995), in coupled GCM studies (Delworth 1996; Saravanan 1998), and also in the idealized coupled model study of Saravanan and McWilliams (1997). The time series of the principal component (or expansion coefficient) associated with EOF1 of surface heat flux is shown in Fig. 2 for one of the integrations. Although 45 yr of data are insufficient to definitively establish the spectral characteristics of the principal components, there isnot much temporal coherence from year to year. (The other principal component time series show similar features.)

Next we consider the spatial structure of annually averaged SST variability in the North Atlantic region. We computed the EOFs of observed SST between the years 1903–94 using the GISST2 dataset (Rayner et al. 1995). The SST data were regridded on the CCM3 model grid, annually averaged, and the linear (least squares) trend was removed before the EOF computation. (There are trends in the dataset that may be attributable either to changes in the observing systems or to long-term climate change.) Figure 3 shows the first three EOFs of the annual mean SST. The EOFs show nearly basin-scale spatial structures, with the first having monopolar structure and the second and the third having dipolar structures. It is also interesting to note that these SST patterns are qualitatively similar to the spatial patterns seen in the surface heat flux EOFs (Fig. 1), although there are some phase differences in the spatial structures, and the patterns are ordered differently. The time series of the principal components associated with the EOFs of SST (not shown) exhibit interannual to interdecadal timescales. The spatial and temporal structure of observed interannual and longer timescale variability in the North Atlantic SST has been analyzed in detail in recent studies by Deser and Blackmon (1993), Kushnir (1994), and Battisti et al. (1995), who find similar patterns of variability. In particular, Deser and Blackmon (1993) suggest that there is a dipolar mode of SST variability associated with a timescale of about 10 yr. Several coupled modeling studies also find a dominant dipolar pattern of SST variability (Delworth 1996; Saravanan and McWilliams 1997; Saravanan 1998).

Stochastic models of SST variability (e.g., Hasselmann 1976; Frankignoul and Hasselmann 1977) have tended to emphasize the interaction between the atmosphere and the oceanic mixed layer on timescales of several months. Although such models may be appropriate for studying seasonal to interannual variability, oceanic variability on decadal and longer timescales is likely to involve parts of the ocean that lie below the mixed layer. For example,Deser et al. (1996) find that the interdecadal variability signal in the central North Pacific originates at the surface and descends into the main thermocline to a depth of about 400 m. The advective role of the time–mean flow within the thermocline seems to be important in determining the evolution of the temperature anomalies. Levitus et al. (1994) found evidence for decadal patterns of variability at a depth of 120 m in the North Atlantic. Saravanan and McWilliams (1997) also found that the signal of interdecadal temperature variability in an idealized coupled model extended several hundred meters below the ocean surface. Stochastic models of interdecadal variability will have to take this relatively deep vertical structure and the role of time–mean advection into account.

To estimate the magnitude of the depth-averaged advective flow in the North Atlantic ocean, we turn to ocean GCMs, because it is difficult to find observational datasets of current measurements at various depths with enough spatial and temporal coverage to compute basin-wide climatologies. We analyzed the annual mean North Atlantic circulation in two different GCMs: (i) a “coarse” resolution (3° latitude–longitude grid) NCAR ocean model whose circulation is documented by Large et al. (1997), and (ii) a “fine” resolution (¹/₃°) model developed under the Community Modeling Effort (CME), whose circulation is documented in Bryan et al. (1995). Since the western boundary of the North Atlantic basin does not lie exactly along the north–south direction, we decomposed the horizontal velocity u into components (u*, υ*), where u* denotes the alongshore velocity (approximately parallel to the western boundary of the ocean basin) and υ* denotes the cross-shore velocity, that is, in the perpendicular direction. One may think of u* as roughly corresponding to the velocity along the path of the Gulf Stream. Figure 4 shows u*, vertically averaged over the top 500 m, for the annual mean circulation in the NCAR ocean model. (The great circle that defines the direction of u* is also shown in the figure.) The values of u* range from several centimeters per second near the Florida coast to a few centimeters per second after the boundary current separates.

Motivated by the spatial structure of the observed SST anomalies on interdecadal timescales (Fig. 3), we averaged u* over a box encompassing the broad maxima shown in Fig. 4 and obtained a mean value of 2 cm s⁻¹. The length of the box (along the great circle) is 5000 km, giving a timescale of about 8 yr to transit along the length of the box. This advective timescale is at least an order of magnitude larger than what one would estimate from, say, the surface velocities associated with the narrow Gulf Stream jet. Sutton and Allen (1997), in their observational study of the propagation of SST anomalies along the path of the Gulf Stream, find a similar decadal advective timescale. We also estimated u* using data from the fine-resolution CME model of the North Atlantic. Although the circulation in that model (not shown) has finer-scale features, the mean value of u* over the aforementioned rectangular region is again about 2 cm s⁻¹.

3. Model formulation and solution

a. Model equations

We now construct a model of ocean–atmosphere interaction that explores the consequences of some of the important features of midlatitude climate variability on decadal timescales as noted above: the spatial coherence and temporal incoherence of atmospheric variability, the dipolar meridional structures in the surface heat fluxes, the relatively deep vertical structure of oceanic temperature anomalies, and the advective role of the ocean circulation.

Our model can be considered as being a heuristic representation of a basin-wide meridional dipole atmospheric pattern, such as the North Atlantic oscillation, interacting with the zonally averaged overturning circulation in the ocean (Fig. 5). Note that the thick line denoting the ocean circulation in Fig. 5 is not drawn as a closed loop, because we choose to restrict our attention to scenarios where the oceanic temperature anomalies are significantly attenuated over the time it takes for a complete circuit of the meridional overturning cell. For example, in three-dimensional circulations, surface water masses may be mixed or convected below the surface somewhere along the gyre and the effective circuit may be much longer than what the two-dimensional gyre streamlines would suggest. Furthermore, the overturning timescales associated with thermohaline circulations could be quite long, on the order of centuries. This motivates us to use inflow boundary conditions for theocean model, so that unperturbed waters flow, into the localized region containing the atmospheric pattern, interact with it, and then flow out. In scenarios where there is memory associated with the gyre or overturning circulation, a reentrant boundary condition, incorporating delays and attenuation, would be more appropriate.

We consider a one-dimensional “slab” atmosphere along the meridional (y) direction, extending from y = 0 to L (Fig. 6). We take the atmosphere to be characterized by a single variable, its temperature T_a. One may think of it as representing the zonally and vertically averaged midlatitude atmosphere over an ocean basin, such as the region of the atmosphere over the extratropical North Atlantic basin (e.g., along the great circle shown in Fig. 4). Neglecting the zonal and vertical structure of atmospheric variability is a severe approximation and is made primarily for reasons of simplicity. The mean atmospheric flow in the midlatitudes is strongly zonal. According to the theory of quasi-stationary atmospheric waves (e.g., Held 1983; Frankignoul 1985), the relationship between SST anomalies and the associated atmospheric response is not necessarily local in the zonal direction. In fact, Frankignoul (1985) shows that the nonlocal nature of atmosphere–ocean interaction can sometimes give rise to eastward propagating SST anomalies, even in the absence of ocean advection. However, we have chosen to construct our one- dimensional model along the meridional direction, where the mean atmospheric flow is quite weak and the atmospheric response would tend to be more in phase with the SST anomaly.

Assume that the slab atmosphere exchanges heat with a one-dimensional slab ocean beneath it, having thickness H and temperature T_o. We prefer to use the term“slab ocean,” rather than call it a mixed layer, because we expect that on decadal timescales, layers of the real ocean beneath the mixed layer will also play a role in determining the variability (e.g., see Deser et al. 1996). Therefore, the slab ocean may be considered a vertically averaged representation of a portion of the upper ocean. An important process that we ignore is the role of vertical advection in the ocean. In particular, the slab ocean approximation would certainly break down in regions of deep convection. We assume further that the heat exchange between the atmosphere and the ocean is proportional to the local temperature difference between the two. This may not be too bad an approximation at large spatial scales in the atmosphere (see Frankignoul 1985). Note that this relationship does not necessarily imply that the maximum heating is in phase with the SST anomaly, because the atmospheric temperatures also need to be taken into account.

If we assume that the slab ocean is motionless, then there would be no nonlocal spatial interactions and we would essentially end up with a red-noise stochastic climate model of the kind described by Hasselmann (1976), with the spatial coherence of atmospheric variability simply being mirrored by the red-noise oceanic response. Horizontal advection in the ocean model permits nonlocal interactions and can lead to interesting deviations from a simple red-noise response. We consider the simplest form of oceanic advection—uniform advection in one dimension—and assume the slab ocean moves with a constant horizontal velocity, V. Henceforth, we shall use T_a, T_o to denote anomaly temperatures (i.e., deviations from a mean oceanic value), because advection of the mean ocean temperature by the mean circulation V does not affect the variability. At y = 0, we use the following inflow boundary condition for the ocean temperature anomaly T_o:

T_ot(1)

Following the approach taken by Hasselmann (1976), we may write the stochastic equations for the atmospheric temperature anomaly T_a(y, t) and the oceanic temperature anomaly T_o(y, t) as follows:

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where t denotes time, F is the heat flux into the ocean, and κ is an exchange coefficient relating the heat flux to the local air–sea temperature difference; C_a is the heat capacity of the atmosphere; C_o = ρ_wC_wH is the heat capacity of the slab ocean, with ρ_w and C_w denoting the density and specific heat of water; α is an intrinsic damping coefficient for the atmosphere representing all dissipative processes except heat exchange with the ocean, which is represented by F; and ε_a and ε_o denote stochastic forcing of the atmosphere and the ocean, respectively. A similar one-dimensional set of equations, with the addition of a diffusive term but without the stochastic or advective terms, has been used by Marotzke and Pierce (1997) to model the interaction between a diffusive atmosphere and a slab ocean.

In addition to the intrinsic atmospheric damping coefficient α, it is convenient to define the auxiliary damping coefficients

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where μ represents the damping of atmospheric temperature anomalies by heat exchange with the ocean and λ represents the damping of oceanic temperature anomalies by heat exchange with the atmosphere. This allows us to rewrite (2)–(4) as

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Although we shall be dealing with the nondimensional solutions to (6)–(7) and exploring regions of parameter space, it is useful to identify reference values for the model parameters so as to orient ourselves in parameter space. Estimates of the exchange coefficient κ are quite uncertain and scale dependent. We choose a value of 40 W m⁻² K⁻¹, based on linearizations of bulk formulas for air–sea flux exchange (e.g., Haney 1971; Frankignoul 1985). This implies that μ⁻¹ ≈ 3 days. The value of λ depends crucially on the choice of slab ocean thickness H. For typical mixed-layer thickness of about 50 m, we have λ⁻¹ ≈ 2 months. This would be appropriate for studying atmosphere–ocean interaction on seasonal to interannual timescales. On decadal timescales, the thermal variability in the ocean tends to have deeper vertical structure (e.g., Deser et al.1996), and a larger value for the slab ocean depth would be more appropriate. For example, if H = 500 m (say), then λ⁻¹ ≈ 20 months. The advection velocity V would be a vertically averaged representation of oceanic flow over the thickness of the slab. For example, if we assume zonally averaged meridional transport of 20 Sverdrups (Sv) above 500 m depth in a basin that is 5000 km wide, then the associated zonal mean meridional velocity would be about 1 cm s⁻¹, which we choose to be a reference value for V. This may be appropriate for zonally averaged thermohaline flows but may be an underestimate when the wind-driven flow is also included (e.g., Fig. 4 suggests a somewhat higher value). For the domain length scale, we choose L = 5000 km, as suggested by the great circle path shown in Fig. 4.

We choose C_a = 10⁷ J m⁻² K⁻¹, which is the mass of the atmosphere times its specific heat per unit area. The intrinsic atmospheric damping timescale α represents a variety of dissipative processes such as horizontal eddy and mean flow heat transport and radiative heat loss to space. Estimates of its value range from a few days to several tens of days and are also scale dependent. Models of quasi-stationary waves in the atmosphere typically use damping timescales of 5–10 days to mimic the diffusive effects of transient eddies (Held 1983; Borges and Sardeshmukh 1995). For example, Lorenz (1973) estimates a decorrelation timescale of about 12 days for atmospheric states, and Branstator et al. (1993) estimate a timescale of about 6 days. This timescale differs considerably from those that are derived from eddy diffusion coefficients used in atmospheric energy balance models coupled to ocean models (Rahmstorf and Willebrand 1995; Marotzke and Pierce 1997). For spatial scales of O(5000 km) that we are interested in, arguments based on eddy diffusion coefficients for zonally averaged meridional heat fluxes would suggest a timescale of O(100 days) (e.g., Marotzke and Pierce 1997). One plausible explanation for this discrepancy is that it arises because the meridional eddy heat transport arguments ignore the predominantly zonal advective atmospheric heat transport—especially near the western boundaries of ocean basins—and may thus be underestimating the atmospheric dissipation. Zonally averaged fluctuations in the atmosphere tend to persist considerably longer than zonally asymmetric fluctuations. Therefore, the energy balance models may not be appropriate for modeling zonally asymmetric basin-scale ocean–atmosphere interaction on decadal timescales, such as in the vicinity of the North Atlantic basin, although they could be quite useful on longer timescales, such as the paleoclimatic timescales. Since we are primarily interested in modeling the very low- frequency atmospheric fluctuations associated with the quasi-stationary waves, we shall use the reference value α⁻¹ = 9 days, with the caveat that it is a rather uncertain value.

We will be interested in timescales of the order of years or longer. Since the atmospheric damping timescales are much shorter than that, we choose to use the following approximation to (6), for timescales ≫max(α⁻¹, μ⁻¹):

αμT_aμT_oε_a(8)

where the atmosphere is assumed to be in instantaneous equilibrium with the ocean temperature and the stochastic forcing. We then combine (7) and (8) to obtain a single governing equation for the ocean temperature:

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where λ_eff is the effective damping coefficient for the ocean when coupled to an interactive atmosphere. For the reference parameter values discussed above, we have μ/α ≈ 2, giving λ⁻¹_eff ≈ 3λ⁻¹. This means that coupling to an interactive atmosphere significantly reduces the SST anomaly damping timescale (as compared to an atmosphere with fixed properties). A somewhat similar effect was noted by Marotzke and Pierce (1997) in a coupled linear diffusive model. In their case, coupling the atmosphere to the ocean reduced the effective horizontal diffusivity by a factor, C_a/C_o, as compared to the horizontal diffusivity of the uncoupled atmosphere. This feedback is also seen in atmospheric GCM experiments with fixed anomalies, where the flux exchange coefficient κ may be effectively reduced from 40 W m⁻² K⁻¹ to about 20 W m⁻² K⁻¹ (e.g., Frankignoul 1985). For a shallow slab with H = 50 m, we have λ⁻¹_eff ≈ 6 months, and for a deep slab with H = 500 m, we have λ⁻¹_eff ≈ 5 yr.

Although stochastic forcing associated with mesoscale ocean eddies is likely to be important for mesoscale ocean variability (e.g., Voorhis et al. 1976), it is not clear whether it can play an important role on the basin scales that are of interest to us. Furthermore, the spatial and temporal characteristics of very low frequency variability in the ocean are less well known than of the atmosphere. In this study, for the sake of simplicity, we choose to set

ε_oy, t(10)

Since ε_a and ε_o appear additively on the rhs of (9), we should be able to easily generalize the results obtained using stochastic atmospheric forcing to the case with stochastic oceanic forcing as well.

For timescales ≪λ⁻¹_eff, it is instructive to consider the behavior of (9) in the limit of very small and very large values of V. Since we expect the atmospheric forcing to be spatially coherent and temporally incoherent, we take ε_a(y, t) = W(t)f(y). In the slow advection limit (V → 0), we may approximate (9) as

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The oceanic response is the time integral of the atmospheric forcing; for example, white-noise forcing would give a red-noise response (Hasselmann 1976). The spatial structure of the response is the same as that of the forcing. In the fast advection limit (V → ∞), we may approximate (9) as

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The oceanic response is weak (∝V⁻¹) and is the spatial integral of the atmospheric forcing. The temporal structure of the response is the same as that of the forcing. Note that there is a certain symmetry between the slow and fast advection limits, if we interchange the space and time coordinates.

b. White-noise forcing

The choice of the white-noise forcing term ε_a(y, t) in (2) is motivated by the lack of temporal coherence seen in uncoupled atmospheric variability on very long timescales (e.g., Feldstein and Robinson 1994; Nitsche and Wallace 1995; Saravanan and McWilliams 1997). As for the spatial structure of ε_a, we consider atmospheric forcing with spatial structure f(y) localized in the meridional domain y = [0, L]. We can expand f(y) using a Fourier series as follows:

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We note that atmospheric variability tends to be dominated by a small number of spatial patterns, as discussed earlier (e.g., see Fig. 1). Therefore, we choose to consider two prototypical spatial structures for the forcing, cos(2πy/L) and sin(2πy/L) (Fig. 6). These would correspond to the first two terms with nontrivial spatial structure in the Fourier series expansion (13). Note that we can do this without loss of generality, because we are dealing with a linear system of equations. The atmospheric stochastic forcing is then expressed as

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where A_f and B_f are independent Gaussian white-noise random variables, with zero mean, that represent the temporally incoherent nature of atmospheric forcing.

Note that because boundary condition (1) is not reentrant, there is no preferred timescale (or length scale) in the ocean model by itself, although there is a preferred velocity scale. Thus, mechanisms such as the loop oscillator described by Welander (1985) do not operate in this model. The atmospheric model too has no preferred timescale, although there is a preferred spatial scale of forcing. To anticipate one of the main conclusions of this study, we note that the oceanic velocity scale combined with the atmospheric spatial scale can potentially give rise to a preferred timescale in the coupled system.

To describe the spectral properties of the white-noise forcing, we define the Fourier transform f̂(ν) of a function f(t) as follows:

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with ν denoting the frequency.

For any two stationary real time series f(t) and g(t), we may define the ensemble-average covariance cov(f, g; τ), with g lagging f by time τ, as follows:

f, gτftgtτ(16)

where 〈 〉 denotes ensemble averaging over different realizations of the random variables. Using the Fourier transform, (16) may be expressed as

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with (*) denoting complex conjugation. We assume that the white-noise random variables A_f and B_f satisfy the following:

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where â and b̂ are constants representing the strengths of the cosine/sine components of white-noise forcing and δ denotes the Dirac delta function.

In principle, one could use (17) and (18) to compute covariances analytically, usually through contour integration in the complex plane. In practice, however, (17) turns out to be too cumbersome to use in all but the simplest of covariance computations, because the expressions for the resulting analytical contour integrals are rather complicated. We find it much easier to evaluate covariances using a discretized Fourier transform version of (17). The integrand 〈f̂*(ν′)ĝ(ν)〉 typically falls off quite rapidly as |ν| → ∞, so that the discretized version of (17) converges fairly quickly. Details of this discretization are given in appendix A.

c. Scaling and solution

The simplicity of the coupled model described above allows analytical solution of the governing equations. To find the general solution to (9), it is convenient to apply the Fourier transform, which gives the following equation for T̂_o(y, ν):

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The form of (19) motivates us to nondimensionalize all lengths by L, and all timescales by the advective timescale T_adv ≡ L/V. Temperature is assumed to be scaled by a temperature scale T_scale, which does not need to be explicitly specified because our equations are linear in T. The surface heat flux F will be scaled by κT_scale. It is convenient to scale the stochastic forcing ε_a by αT_scale, although this means that the nondimensionalized stochastic forcing amplitude cannot be considered to be constant if α is allowed to vary. Hereafter, all variables may be assumed to be nondimensional, unless otherwise stated.

After substituting the Fourier transform of (14) in (19) and nondimensionalizing, we obtain

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The nature of the solutions to (20) is essentially controlled by a single nondimensional parameter Γ, defined by

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For the choice of parameter values L = 5000 km, V = 1 cm s⁻¹, the characteristic advective timescale T_adv ≈ 16 yr. For the North Atlantic, the relevant horizontal velocity scale may be at least twice as large (Fig. 4), implying a timescale of about 8 yr. This suggests that advection will play an important role primarily on decadal or longer timescales. If we take the atmospheric properties to be fixed, Γ can be thought of as representing the ratio of the strength of horizontal advection to the thermal inertia of the ocean (i.e., the slab thickness). Therefore, we may refer to Γ as the “advection–damping ratio.” We anticipate the following two regimes of behavior for the solutions to (20): (i) A slow–shallow regime corresponding to Γ ≪ 1, where the depth of penetration of thermal anomalies is small and thermal damping effects dominate over advection; and (ii) a fast–deep regime corresponding to Γ ≫ 1, where thermal anomalies penetrate quite deeply and the thermal damping effects are weaker than advection. For λ_eff = 2 yr and V = 1 cm s⁻¹, the values of Γ can range from 0.25 to 2.5 for values of H ranging from 100 m to 1000 m, thus straddling both regimes. We note from (9) that λ_eff is quite sensitive to the choice of the atmospheric damping timescale α⁻¹; hence the range of interest for Γ is even larger. Since we have chosen a fairly short timescale for our reference value of α⁻¹, a longer timescale would imply a larger value of Γ.

The form of the boundary condition (1) motivates theuse of the Laplace transform in the y direction, defined by

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Transforming (20), we obtain an equation for T̃_o(s, ν):

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Here we have extended the model domain to 0 ≤ y < ∞ to simplify the computations. Since there is no upstream influence in this model, this does not make any difference to the solution in the domain 0 ≤ y ≤ 1. The inverse Laplace transform of (23) can be computed quite easily by contour integration in the complex s plane (Dennery and Krzywicki 1967) around the poles s = i2π, −i2π, −2πξ, where

View Expanded

The solution is given by

View Expanded

The first two terms in the square brackets of (25) do not decay in the y direction and represent the local response to the forcing. The third term in the square brackets, which is proportional to exp(−2πy/Γ), may be thought of as being the “spatial transient response” to a forcing turned on at y = 0.

We note that there is a certain degree of symmetry between the response to cosine forcing (Â_f) and the response to sine forcing (B̂_f) in (25). For simplicity we shall henceforth restrict ourselves to the case with zero cosine forcing. Furthermore, since we are dealing with a linear system, we may choose the sine forcing to have unit amplitude, that is, we choose the following parameters for the nondimensional version of (18):

â²b̂² = 1.(26)

In other words, we shall restrict our attention to the oceanic response to a “dipole” atmospheric forcing of the form sin(2πy) from now on. Appendix B discusses sensitivity of the oceanic variability to this choice by considering monopole and tripole structures for the forcing as well.

4. Oceanic variability

b. Frequency spectrum

Consider the oceanic response over all frequencies. We define σ²_o, the total variance of T_o integrated over all frequencies, as follows:

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The spatial structure of σ²_o and its Γ dependence are shown in Fig. 9. Note that the amplitude of the oceanic variance decreases very rapidly with increasing Γ. For Γ ≪ 1, the structure of σ²_o essentially reflects dipolar atmospheric forcing, with a minimum in variance at y ≈ 0.5. For Γ ≫ 1, σ²_o has an almost monotonic spatial structure, with a narrow region of rapid initial increase in variance downstream of y = 0, followed by a rather slow increase farther downstream.

The spatial structure of σ²_o for Γ ≥ O(1) as seen in Fig. 9 does not suggest any natural way to decompose the oceanic response into different modes. Unlike the sin(2πy) atmospheric forcing function, the oceanic response does not satisfy any simple boundary condition at y = 1. However, we recall that our choice of the atmospheric forcing function was motivated by a Fourier series expansion in the interval y = [0, 1] (13). This motivates us to try and decompose the frequency-dependent oceanic response T̂_o(y) using a Fourier series expansion in the interval y = [0, 1]. Such an expansion should allow us to study how the oceanic feedback affects the atmospheric variability, because we will be using the same basis functions for both the atmospheric and the oceanic variables. Note that since the oceanic response is not periodic, the Fourier expansion will have to deal with a jump discontinuity at y = 0. We shall evaluate a posteriori the convergence properties of such an expansion, as measured by whether a small number of modes capture a large fraction of the variance.

The Fourier expansion of T̂_o(y, ν), as given by (25)–(26) may be written as

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where n denotes the “meridional wavenumber.” The Fourier components Â⁽ⁿ⁾_o and B̂⁽ⁿ⁾_o may be computed from (25) as

View Expanded

where δ_nn′ denotes the Kronecker delta function.

Recall that the atmospheric white-noise forcing (14) is confined to meridional wavenumber n = 1. However, the oceanic response can be spread over a range of meridional wavenumbers, because advection can create spatial structures that differ from the sin(2πy) forcing structure. Figure 10 shows the variance associated with the different Fourier components Â⁽ⁿ⁾_o and B̂⁽ⁿ⁾_o, integrated over all frequencies ν using (17) and expressed as a fraction of the total T̂_o variance, ∫ dy σ²_o(y). We see that in the slow–shallow regime, almost all the variance is concentrated in B̂⁽¹⁾_o, that is, the response has the same structure as the forcing. In the fast–deep regime, the fractional variance associated with B̂⁽¹⁾_o diminishes to about 40% or less, and the fractional variance associated with Â⁽¹⁾_o increases significantly, appearing to saturate around 30%. There is also some“leakage” of variance into the higher meridional wavenumbers, but it is quite small for Γ ≤ 16. Nevertheless, Â⁽¹⁾_o and B̂⁽¹⁾_o together capture a large fraction of the totalvariance of T̂_o, for a range of Γ values. Therefore, we focus on the two dominant modes of oceanic response to atmospheric forcing: the direct mode of response B̂⁽¹⁾_o and the spatially orthogonal mode of response Â⁽¹⁾_o. Hereafter, we shall denote them simply as B̂_o and Â_o, respectively. That is,

Â_oνÂ⁽¹⁾_oνB̂_oνB̂⁽¹⁾_oν(30)

There are two ways to explain the presence of an orthogonal mode of response. One being that in the limit of very fast advection, as discussed in section 3a, the oceanic response is the spatial integral of the atmospheric forcing, B_f sin2πy. Another interpretation is that the oceanic response is a propagating mode, which would project onto two spatially orthogonal standing wave modes. The presence of orthogonal modes of oceanic response raises some interesting possibilities, because these oceanic modes can, in turn, excite orthogonal atmospheric modes. This means that oceanic advection can mediate interactions between spatially orthogonal atmospheric modes. It also means that the coupled ocean–atmosphere system is likely to include modes of the uncoupled atmosphere plus the corresponding orthogonal modes. Of course, the strength of the orthogonal atmospheric response will depend on the strength of the ocean–atmosphere coupling.

We now consider the frequency-variance spectrum 〈|B̂_o(ν)|²〉 and 〈|Â_o(ν)|²〉 of the direct and orthogonal modes of response. In the slow–shallow regime, the direct mode shows a red-noise type of spectrum that flattens out as ν → 0 (Fig. 11a). The overall variance of the direct mode decreases monotonically with increasing Γ. In the fast–deep regime, the overall direct response is weaker by an order of magnitude or more, but there is a broad peak in the vicinity of ν = 1. Inother words, the fast–deep regime shows a preferred frequency corresponding to the timescale L/V. One may think of this preferred frequency as arising from a kind of “advective resonance,” because if one were to apply periodic boundary conditions at y = 0, 1, one would get truly resonant behavior for ν = 1. The orthogonal mode (Fig. 11b) shows a similar tendency for frequency selection in the fast–deep regime, but the spectral peak near ν = 1 is less pronounced. Interestingly, the maximum orthogonal response occurs not for Γ → 0 or for Γ → ∞, but for Γ = 1, that is, the orthogonal mode seems to be most efficiently excited for Γ = O(1). The phase relationship between the atmospheric forcing and the direct/orthogonal modes of oceanic response is discussed in appendix C.

One often tends to associate spectral peaks in the variance of a dynamical system with the existence of oscillatory normal modes. We have shown above that stochastically forced advective ocean–atmosphere interaction can result in spectral peaks in the oceanic variance, even when there are no identifiable oscillatory normal modes of the uncoupled or coupled systems. However, the spectral peak is discernible only for sufficiently large value of the advection–damping ratio, that is, Γ ≳ 1. Depending upon the actual advection/damping ratio in different regions of the ocean, this frequency selection mechanism may or may not turn out to be important. As suggested by Fig. 4, the advective timescale for the North Atlantic upper ocean is of the order of a decade. By the arguments of section 3a, Γ is likely to be of O(1) or even larger. This means that the frequency selection mechanism discussed above could play a role in decadal variability in the North Atlantic.

5. Atmospheric variability

a. Frequency spectrum

We now focus on how the feedback associated with the oceanic response affects the atmospheric variability itself. Using the nondimensional Fourier-transformed versions of (4) and (8), scaled as described in section3c, we may write the total atmospheric response T̂_a(y, ν) and the surface flux F̂(y, ν) as

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(Recall that the scaling for the stochastic forcing includes the factor α.) The first numerator term in the rhs of (31) represents the direct atmospheric response to white-noise forcing in the absence of the ocean, whereas the second numerator term and the denominator are due to feedback from the ocean. Since T̂_o and ε̂_a are correlated, as discussed in the previous section, the two terms can “interfere” constructively or destructively with each other, depending upon their relative complex phase. The relative contribution of the oceanic feedback to overall atmospheric variability is controlled by the ratio μ/α, which we may refer to as a “coupling parameter.” It is the ratio of the intrinsic atmospheric damping timescale to the timescale for heat exchange with the ocean. The case μ/α ≪ 1 would correspond to weak coupling, and the case μ/α ≫ 1 to strong coupling between the atmosphere and the ocean. (Recall that our choice of reference values for μ and α, as discussed in section 3a, imply a reference value of μ/α ≈ 3.)

We may decompose T̂_a(y, ν) using a Fourier series expansion in the interval 0 < y < 1, as in (28), to obtain the contributions from different meridional wavenumbers n, Â⁽ⁿ⁾_a(ν) and B̂⁽ⁿ⁾_a(ν). Restricting ourselves to the dipole-forcing case â² = 0, b̂² = 1, as in the previous section, we shall once again make the identification B̂_a(ν) ≡ B̂⁽¹⁾_a(ν), which would correspond to the direct mode of atmospheric variability, with sin(2πy) structure, and Â_a(ν) ≡ Â⁽¹⁾_a(ν), corresponding to the orthogonal mode of atmospheric variability, with cos(2πy) structure. Using (31) and the linearity of the Fourier series expansions, B̂_a, Â_a may be expressed as

View Expanded

where we have assumed for the sake of simplicity that Â_f ≡ 0, that is, there is no atmospheric forcing with cos(2πy) structure. (If such forcing did exist, and were coherent with B̂_f, it could lead to more complex interactions than are considered here.) From (34) we see that the orthogonal mode of atmospheric variability is simply a scaled version of the orthogonal mode of oceanic response shown in Fig. 11b. The direct mode of atmospheric variability is more interesting, because it interacts with the oceanic direct mode.

Since the direct mode of oceanic response shows a preferred frequency in the fast–deep regime, we may ask whether this is reflected in the atmospheric variability. Figure 12 shows the frequency spectrum of the direct mode of atmospheric response, 〈|B̂_a(ν)|²〉 for μ/α = ¹/₃, 3. In the slow–shallow regime, we see weak monotonic decay in power with increasing ν, indicating red-noise type of behavior. In the fast–deep regime, there is a significant peak near ν = 1 and white-noise structure for ν → ∞. The amplitude of the peak decreases with increasing Γ, with the peak being most prominent for Γ = O(1). The value of μ/α does not seem to affect the qualitative structure of the spectra in the slow–shallow regime. However, in the fast–deep regime, the overall variance level decreases with increasing μ/α, and the spectral peak becomes more prominent. One may think of the frequency spectrum for the fast–deep regime as consisting of a white-noise direct atmospheric response to stochastic atmospheric forcing, with a peak arising from the oceanic feedback superimposed on it. The strength of the oceanic feedback, like the oceanic variability, decreases with increasing Γ. However, its relative strength, compared to the direct atmospheric response to stochastic forcing, increases with increasing μ/α.

Next we consider the direct mode of the surface heat flux B̂_F, which may be computed using (32) and Fourier decomposition as

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Figure 13 shows the frequency spectrum of B̂_F for μ/α = 3. It is interesting to note that, in a sense, the B̂_F spectrum is the “mirror image” of the corresponding B̂_a spectrum (Fig. 12b), with elevated levels of atmospheric temperature variability corresponding to decreased surface flux amplitudes and vice versa. In particular, the B̂_F spectrum is rather like “blue noise” for Γ ≫ 1 (the slow–shallow regime). This simply reflects the fact that the air–sea temperature gradient (T_a − T_o) approaches zero for timescales longer than the damping timescale (λ⁻¹_eff) associated with the slab ocean. Of course, the deep ocean has much longer timescales associated with it, which may mask the blue-noise spectrum associated with the slab ocean response. For Γ ≫ 1, the B̂_F spectrum is more like white noise, but with a trough at ν ≈ 1, corresponding to the peak in B̂_a spectrum. This should not be too surprising since a decrease in the surface heat flux would imply decreased thermal damping of atmospheric temperature anomalies, allowing them to be more persistent.

To summarize, the frequency selection mechanism that characterizes the oceanic response in the fast–deep regime expresses itself in the atmospheric variability, through oceanic feedback associated with the surface fluxes. However, the spectral peak near ν = 1 tends to be less prominent in the atmospheric frequency spectrum, because it is superimposed upon a strong white- noise background. This means that it will be more difficult to detect a statistically significant spectral peak in the atmospheric variables than in the oceanic variables.

b. Types of oceanic feedback

There are two different types of oceanic feedback that can affect the atmospheric variability in our simplified model of ocean–atmosphere interaction.

1) Local feedback

This is a thermodynamic feedback that would occur even in the absence of ocean advection. If there is a positive atmospheric temperature anomaly over a region of uniform SST, the associated air–sea temperature difference would result in positive heat flux into the ocean and lead to the formation of a positive SST anomaly.This would result in a decreased air–sea temperature difference, and hence decreased surface heat fluxes, and increased persistence of the atmospheric temperature anomaly. This feedback can explain features such as the increase in overall atmospheric temperature variance associated with decreased surface heat flux amplitudes (Figs. 12b, 13). A recent study by Barsugli and Battisti (1998) presents a detailed discussion of this feedback. However, this local feedback cannot by itself explain the emergence of a spectral peak in the atmospheric variability for Γ ≫ 1 (Fig. 12).

2) Nonlocal feedback

This is a dynamic feedback where ocean advection plays an important role. The “constructive interference” between the spatial patterns of T_a and T_o that gives rise to the spectral peak in T_a variance (for the direct mode) is an example of such a feedback. The timescale at which this occurs is determined by the length scale associated with the atmospheric forcing and the velocity scale associated with the ocean advection. Another example of this type of feedback would be the atmospheric response to orthogonal modes of ocean variability, which would have a different spatial structure from the original atmospheric forcing.

The local feedback described above can easily be explored in the case with no advection (V ≡ 0). Since it is somewhat difficult to understand the parameter dependence of the overall variability amplitudes using the nondimensional equations, because parameters such as α appear in several places, we temporarily revert to the dimensional form of the equations. We consider two limits:(i) infinite heat capacity ocean (C_o → ∞) and (ii) zero heat capacity ocean (C_o → 0). Case (i) can be considered as representing an atmospheric model forced by prescribed SST at the surface, because the ocean acts like an infinite heat reservoir. Case (ii) can be considered as representing an atmospheric model coupled to a very shallow mixed layer. For case (i), we may approximate (8) as

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because T_o → 0 as C₀ → ∞ for finite F. For case (ii), we may approximate (8) as

View Expanded

because F → 0 as C_o → 0. Thus, the atmospheric variability would be proportional to (α + μ)⁻² for an infinite heat capacity ocean and proportional to α⁻² for a zero heat capacity ocean. In other words, one would expect an atmospheric model coupled to a mixed layer to show more variability (but weaker surface heat fluxes) than an atmospheric model forced by prescribed SST. Such an effect has been noted on decadal timescales in an idealized ocean–atmosphere model (Saravanan andMcWilliams 1997). Marotzke and Pierce (1997) have also highlighted the role of finite oceanic heat capacity in determining the damping timescales of thermal anomalies.

c. Forcing the atmosphere with “observed” SST: AMIP-style integrations

Consider now a conceptual experiment where we carry out a long integration with our coupled ocean–atmosphere model. We use the “SST” from such an integration as the lower boundary condition for a subsequent integration of a stand-alone atmospheric model, with different initial conditions. Integrations of this kind have been carried out by the Atmospheric Model Intercomparison Project (AMIP) using realistic atmospheric general circulation models and observed SST. Therefore, we shall refer to the stand-alone atmospheric integration described above as an AMIP-style integration. We then ask the following questions: (i) How well does the AMIP-style integration capture the amplitude of variability in the coupled integration? (ii) How well does the AMIP-style integration capture the correlations between the atmospheric flow and the SST?

In our simple atmospheric model, the total atmospheric response T̂_a(y, ν) for the AMIP-style integration can be expressed quite simply by rewriting (31) as

View Expanded

where (T̂_o)_coupled represents the time series of SST from the coupled integration and is a surrogate for the observed SST used in an AMIP integration.

An important question is whether the statistical properties of ε̂_a, the stochastic atmospheric forcing, depend significantly on the SST variability. Studies of the atmospheric response to prescribed midlatitude SST anomalies indicate that the response is rather weak (e.g., Lau and Nath 1994). There are also statistically significant relationships between midlatitude atmospheric variability and tropical SST anomalies; however, ensemble GCM integrations indicate that much of the interannual atmospheric variability in the midlatitudes may nevertheless be “intrinsic,” that is, independent of any SST variability (Hoerling et al. 1995). This suggests that to a first approximation, we may assume that the statistical properties of ε̂_a depend only on the climatology of SST, which is the same for both the coupled and AMIP-style integrations. In other words, we assume that ε̂_a is statistically independent of (T̂_o)_coupled, that is, 〈ε̂^*_a(ν′)(T̂_o(ν))_coupled〉 ≡ 0, but otherwise has the same statistical properties as in the coupled integration.

As in the previous section, we restrict ourselves to the dipole-forcing case â² = 0, b̂² = 1. Using the analog of (33), we may write the frequency-variance spectrum of the direct mode of atmospheric response as

View Expanded

The first term in the numerator of the above expression represents the unit variance associated with the white- noise forcing and the second term represents the scaled variance associated with the SST from the coupled integration. The terms appear additively because the cross term involved in computing the modulus of the rhs of (36) vanishes due to the assumed statistical independence of T̂_o and ε̂_a. In the limit μ/α → 0, the intrinsic atmospheric variability dominates. In the limit μ/α → ∞, atmospheric variability forced by the SST dominates. In both limits—trivially for the first limit, and by design for the second—the AMIP-style integration should provide a good estimate of the variability in the coupled integration. However, this may not necessarily be true for μ/α = O(1). Figure 14 shows the frequency-variance spectrum of the direct mode of atmospheric response, for the coupled and AMIP-style integrations. For ν = O(1), we see that the AMIP-style integration underestimates the variability for Γ = ¹/₄. For Γ = 4, the AMIP- style integration estimates the high-frequency (ν ≫ 1) variability fairly well but significantly attenuates the spectral peak near ν = 1 seen in the coupled integration. For ν → ∞, the AMIP-style integration should provide a good estimate of the variability in the coupled integration, because the SST tends to be in quadrature with the atmospheric forcing (Fig. C1).

Consider a stochastically excited positive T_a anomaly in the coupled integration. Through surface flux exchange, this would tend to produce a positive T_o anomaly. In doing so, the air–sea temperature gradient (T_a −T_o) and the associated surface heat flux would be reduced, thus allowing the T_a anomaly to persist longer. However, specifying the same T_o anomaly in the AMIP- style integration does not permit this interaction to occur, because the SST is not allowed to vary in response to the atmospheric forcing. This means that the statistics of atmospheric variability in the AMIP-style integration will not necessarily be identical to that of the coupled integration. This point is also made in the study by Barsugli and Battisti (1998).

The normalized covariance cov(B̂_a, B̂_o; τ) between the atmospheric and oceanic direct modes is shown in Fig. 15 for the coupled and AMIP-style integrations. The maximum absolute value of the covariance in the coupled integration is used as the common normalization factor. For the AMIP-style integration, the covariance actually has the same structure as the autocovariance of the coupled SST variability, cov(B̂_o, B̂_o; τ), because the SST is uncorrelated with the atmospheric white-noise forcing. For the coupled integration, the covariance is a weighted sum of cov(B̂_f, B̂_o; τ) and cov(B̂_o, B̂_o; τ). We see that the AMIP-style integration tends to underestimate the covariance at lag zero and almost misses out the negative minimum in the covariance near τ ≈ 0.4 seen in the coupled integration.

6. Summary and discussion

On interannual and longer timescales, intrinsic (i.e., uncoupled) extratropical atmospheric variability appears to be fairly well described by a white-noise frequency spectrum (Feldstein and Robinson 1994; Nitsche and Wallace 1995; Saravanan and McWilliams 1997). One may think of very low frequency atmospheric variability as random “climate noise” arising from high-frequency daily weather fluctuations. We can therefore approximate it as having stochastic time dependence, even though the equations of motion may themselves be deterministic. Although it may have no preferred timescales, very low frequency atmospheric variability is dominated by a small number of preferred spatial structures (cf. Wallace and Gutzler 1981). In this study, we have constructed a one-dimensional stochastic ocean–atmosphere model that explores the consequences of the aforementioned features of atmospheric variability. The model also incorporates the effect of horizontal advection by the time–mean ocean circulation, which could play a significant role in climate variability on decadal and longer timescales.

The simplicity of the one-dimensional model implies that some important atmospheric and oceanic processes are neglected. In particular, nonlocal atmospheric processes, such as zonal advection, are not explicitly considered. However, those nonlocal atmospheric processes that interact to produce the spatially coherent patterns of low-frequency variability are implicitly included through the stochastic forcing parameterization. This means that even fluctuations in the boundary layer heat and moisture transports that are ultimately responsible for the surface heat flux anomalies are implicitly included, to the extent that they are driven by the large-scale atmospheric low-frequency variability.

The equations governing the one-dimensional stochastic model are simple enough that they may be solved analytically using Laplace and Fourier transforms. It turns out that the nature of the analytical solutions is essentially controlled by a single nondimensional parameter, the advection–damping ratio Γ, which measures the strength of horizontal ocean advection in relation to the thermal damping effects. Depending upon the value of Γ, the solutions to the one-dimensional model fall into two regimes: (i) the slow–shallow regime (Γ ≪ 1) where advection effects are weak and the oceanic variability can be described as being red-noise (Hasselmann 1976), and (ii) the fast–deep regime (Γ ≫ 1), where horizontal advection dominates over thermal damping. We note that the thermal damping timescale of the ocean would increase with increasing depth, because of increased heat capacity. Therefore, shallow structures of temperature variability in the ocean would correspond to the first regime, and deeper structures would correspond to the second regime. Deser et al. (1996) suggest that oceanic temperature variability on decadal timescales could extend over several hundred meters of depth in the ocean, which would imply weaker thermal damping.

The qualititative difference in the time evolution of oceanic temperature anomalies for the two regimes isshown in Fig. 16, for two values of Γ. Here we have numerically solved the equations of the analytical model to produce a single realization of the stochastic time evolution of oceanic temperature for each value of Γ. (The analytical results represent the average over an “infinite” ensemble of such numerical solutions.) Note that for the slow–shallow regime (Fig. 16a), the spatial structure of oceanic temperature variability simply reflects the dipolar standing wave atmospheric forcing. In the fast–deep regime (Fig. 16b), one canclearly see advected features in the oceanic temperature variability, corresponding to a nondimensional timescale of O(1). However, given the stochastic nature of the system, there is no precise time period associated with the variability.

An interesting feature of the fast–deep regime is that the spectrum of oceanic variability can no longer be simply described as being red noise; there is actually a spectral peak in the variability corresponding to the advective timescale T_adv = L/V, where V is the horizontal advective velocity scale in the ocean and L is a length scale associated with atmospheric variability. The spectral peak tends to be more prominent in the oceanic variability than in the atmospheric variability; the strong background white-noise variability makes it more difficult to detect the peak in the atmosphere. It is worth noting that in our simple model, neither the atmosphere nor the ocean, when uncoupled, supports oscillatory modes of variability. The preferred timescale of variability arises only when the atmosphere and the ocean are coupled.

A heuristic explanation of the rigorous analytical results for the origin of the preferred timescale is shown in Fig. 17. Consider a dipolar standing wave pattern of atmospheric variability with white-noise temporal structure. This may be viewed in spectral space as a random superposition of oscillations with all possible periods. Let us focus on the component with period L/V. At time t = 0, the atmospheric forcing would excite an SST anomaly shown by the solid sinusoidal curve. At t = L/2V, in the fast–deep regime, the SST anomaly would be displaced by a distance L/2, as shown by the dashed curve. At the same time, the spectral component of atmospheric forcing with period L/V would also have changed sign, leading to positive reinforcement of the SST anomaly. Of course, white-noise atmospheric forcing would also contain spectral components at all other possible periods, but these would not interact coherently with the ocean. In the slow–shallow regime with strong damping effects, the SST anomaly would be significantly attenuated at t = L/2V, as shown by the dotted curve. Therefore, the positive reinforcement effect would be less important.

There are several different ways in which preferred timescales can arise in extratropical oceanic variability. As discussed in the introduction, it can occur through unstable modes of extratropical air–sea interaction, or through oscillatory modes of the oceanic circulation. We argue that advective ocean–atmosphere interaction can also serve as a mechanism for generating preferred timescales in the ocean. This mechanism, unlike many other mechanisms, does not rely upon having a strong atmospheric response to midlatitude SST anomalies. It is not based upon a closed atmosphere–ocean “feedback loop,” but simply on spatially coherent atmospheric forcing of an advective ocean. Of course, this mechanism generates preferred timescales only if the advection–damping ratio Γ ≳ 1. For example, in the idealized coupled model study of Saravanan and McWilliams (1997), the advective velocity scale is about 0.6 cm s⁻¹ and the depth scale for the thermal anomalies is about 500 m, which implies that Γ ≈ 0.75. This means that the effects of damping are marginally stronger than that of advection, and nonlocal oceanic feedback is likely to be somewhat weak, making it difficult to detect any preferred timescales associated with this feedback. However, for the upper oceanic circulation for the extratropical North Atlantic (say over the top 500 m), using an advective velocity scale of 2 cm s⁻¹ (e.g., see Fig. 4), suggests that Γ ≳ 2.5, depending upon the intrinsic atmospheric damping scales. The associated advective timescale is of O(10 yr). This suggests that advective ocean–atmosphere interaction could potentially play an important role in decadal climate variability in the North Atlantic region. It is interesting to note that the time evolution of oceanic temperatures in the fast–deep regime of our stochastic model (Fig. 16b) bears a qualitative resemblance to the observed time evolution of SST anomalies along the path of the Gulf Stream, as described in the study by Sutton and Allan (1997) Perhaps this model could explain the advective propagation of SST anomalies and the preferred decadal timescales noted in their study.

The one-dimensional coupled model simulates notonly the characteristics of the oceanic response to atmospheric forcing, but also the effects of the feedback from the ocean on atmospheric variability. The strength of this feedback is controlled by the coupling parameter μ/α, which is the ratio of the intrinsic atmospheric damping timescale α⁻¹ to the timescale for heat exchange with the ocean μ⁻¹. There are two types of oceanic feedback that affect the atmospheric variability. Local feedback: when the ocean responds to atmospherically generated surface heat flux anomalies by forming SST anomalies, it results in reduced air–sea temperature gradients and decreased surface heat flux; this means that atmospheric thermal anomalies with the same structure as the SST anomalies tend to be more persistent. Nonlocal feedback: SST anomalies produced by atmospheric forcing can be horizontally advected by the oceanic flow and produce an atmospheric response elsewhere. The nature of this nonlocal feedback would depend upon the strength and spatial structure of the ocean circulation.

The strength of the local feedback described above would depend upon the effective heat capacity of the ocean. The larger the oceanic heat capacity, the smaller the SST response to surface heat fluxes and hence a weaker feedback. In the limit of infinite heat capacity, which corresponds to prescribed SSTs, the feedback would vanish. On timescales of the order of several months, the effective oceanic heat capacity can be taken to be the same as that of the mixed layer. However, on longer timescales, oceanic layers below the mixed layer and advective–diffusive processes that damp thermal anomalies would also play a role in determining the effective oceanic heat capacity.

The role of the effective oceanic heat capacity in determining the surface heat flux variability brings up some interesting issues with regard to the so-called AMIP type of integrations, where observed SSTs are used as the surface boundary condition for extended integrations of an atmospheric GCM. The hope is that using these “perfect” surface boundary conditions should enable the atmospheric model to faithfully simulate the variability of the coupled ocean–atmosphere system. However, the atmosphere is forced by surface fluxes of heat and moisture, and not directly by the SST. Specifying the SST is just a convenient way of getting these fluxes right. This is guaranteed to work when the evolution of SST is deterministic, as in the seasonal cycle, for example. However, part of the observed SST variability may itself be a response to nondeterministic (stochastic) atmospheric forcing. Specifying this response component of SST variability as the boundary condition for an atmospheric model integration will not reproduce the original atmospheric forcing conditions.

In our simple coupled model, we are able to carry out AMIP-type integrations analytically, by using the statistical distribution of SSTs derived from the analytical coupled integrations to force the uncoupled version of thesimple atmospheric model. The coupling parameter μ/α plays an important role in determining how well the AMIP-type integrations can capture the true coupled variability. In the limit μ/α → ∞, the AMIP-type integrations do reproduce true coupled variability, because the ocean–atmosphere coupling is strong, as it is likely to be in the Tropics. AMIP-type integrations also produce the right estimate for the atmospheric variability in the limit μ/α → 0, because the ocean–atmosphere coupling becomes so weak that the SST variability has no effect on the atmosphere. However, for intermediate values for μ/α [i.e., μ/α = O(1)], which would be more characteristic of the extratropics, AMIP-type integrations would tend to underestimate the variability associated with the coupled system (see also Saravanan and McWilliams 1997).

To summarize, we have shown that a one-dimensional analytical stochastic model can lead to some important insights into the nature of ocean–atmosphere interaction, despite its extreme simplicity. The model envisages a scenario where spatially coherent and temporally incoherent atmospheric low-frequency variability plays a dominant role in forcing the ocean on decadal timescales. It shows how atmospheric spatial structures and oceanic advection can combine to produce preferred timescales of variability in certain parameter regimes. Such interactions could be responsible for the advective propagation of SST anomalies and the decadal timescales noted in observational studies (e.g., Sutton and Allan 1997). The analytical stochastic model also brings out the distinction between two types of oceanic feedback: the local thermodynamic feedback, and the nonlocal dynamic feedback.