a. Model development

The model we propose to account for the basic effects of coupling between the atmosphere and the ocean in the midlatitudes is displayed schematically in Fig. 2. It is a one-dimensional thermodynamic model for the upper (slab) ocean coupled to a graybody atmosphere. The single spatial dimension represents a typical point in the midlatitudes, although we also interpret the model variables in terms of individual horizontal modes of the atmosphere–ocean system. There is an assumed random forcing associated with the ubiquitous dynamical motions in the midlatitude atmosphere. The equations for this model, linearized about the climatological mean state, are as follows [see appendix A; also cf. Schopf (1985)]:

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Subscripts “a” and “o” refer to atmosphere and ocean respectively; T̃ is the anomalous temperature; γ the heat capacity; λ_s the linearized coefficient of combined latent, sensible, and longwave heat flux (values of λ_sa and λ_so differ only slightly); and λ_a, λ_o the radiative damping of each component to space. Surface heat fluxes are calculated using the surface air temperature T̃_s. The term F̃ represents the dynamical component of the forcing, which we take to be stochastic.

We assume that the surface air temperature anomaly is linearly related to the free atmosphere temperatureanomaly, T̃_s = cT̃_a. We take the Fourier transform (t → ω) of Eqs. (1) and (2) and divide through by λ_sa to yield

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We have made the following substitutions: σ = γ_aω/λ_sa, a = λ_a/λ_sa + c, d = λ_o/λ_so + 1, β = (γ_o/γ_a)(λ_sa/λ_so), and F = F/λ_sa. A tilde denotes a time-domain variable, and an unadorned variable the corresponding Fourier transform variable. Explicit reference to the independent variables t and σ will be used only for emphasis or to avoid confusion. The derivation of Eqs. (1) and (2) from a more detailed energy balance model is presented in appendix A, where reasonable values of the model parameters are also justified. These parameter values, shown in Table 1, will be referred to as the“standard parameters” and are used in the examples to follow.

In the analysis that follows we will split the dynamical forcing into an SST-forced deterministic part and a purely random part as follows: F = (b − 1)T_o + N, where b is a real constant. We have assumed that the dynamical response is proportional to the SST anomaly at the low frequencies of interest. We assume that the power spectrum of N is independent of the coupling to the ocean, hence “inherent” to the atmosphere. When substituted into Eq. (3), the total “thermal” and “dynamical” response becomes bT_o, and we will refer to b as the “atmospheric response parameter.” In actuality the temperature response of the free atmosphere to diabatic heating is accomplished largely by dynamical adjustment; therefore we will focus only on the total response in the rest of this paper.

With the above assumptions about the atmospheric response, Eqs. (3) and (4) are in the standard form of a two-variable linear system (with stochastic forcing only in the atmosphere equation)²:

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The coefficients a and d represent damping of the atmosphere and ocean respectively, and the coefficients b and c represent the coupling between atmosphere and ocean. At this point it is useful to define a coupling coefficient

αbc,

which represents the feedback due to atmosphere–ocean coupling, and a stability parameter

zadα,

which results from the competition between this feedback and damping. Note that the atmospheric damping parameter, a, contains a dependence on the parameter c.

b. Methodology

The design of the numerical experiments in B95 will be repeated using the simple model presented in this paper. This design consisted of three model runs as follows.

1. Coupled: The coupled model was run first.

2. Uncoupled: (a) The atmosphere model was run with SST fixed to be the zonal mean of the climatology of the coupled run. (b) The slab mixed layer model was integrated in diagnostic mode, forced with the time history of winds and temperatures from run 2a.

3. MOGA [Midlatitude Ocean, Global Atmosphere, after Lau and Nath (1994)]: (a) The atmosphere model was run with SST prescribed to be the time history of SSTs from the coupled run. (b) The slab mixed layer model was integrated in diagnostic mode, forced with the time history of winds and temperatures from run 3a.

Equations (6) and (7) can be used to model the coupled, uncoupled, and MOGA experiments as follows. The “coupled” model (denoted by superscript C) solves Eqs. (6) and (7) as a coupled set:

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Given N^C, one can solve for T^C_a and T^C_o. We have introduced the abbreviated notation σ_a = (iσ + a) and σ_o = (iβσ + d). For the “uncoupled” system, denoted by superscript U, we get the following set of equations:

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Equation (10) was obtained by setting T_o = 0 in Eq. (6), since SST is fixed at climatological values. Given N^U, one can solve for T^U_a. Equation (11) is the diagnostic equation for a “slave ocean” driven by the uncoupled atmosphere, which we solve to get T^U_o. Finally, to model the prescribed SST experiment (MOGA, denoted by superscript M), we set T_o = T^C_o in the atmosphere equation and use the MOGA atmosphere to force a “slave” mixed layer ocean, yielding the following equations:

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In addition, we assume that N^M and T^C_o are uncorrelated. Recall that the direct effect of the prescribed SST on the dynamical forcing, including any reorganization of the atmospheric eddy fluxes, is accounted for by the dynamical response term (b − 1)T^C_o. This latter assumption will simplify the calculation of power spectra for the case of prescribed SST and is, in fact, central to the success of this simple model.

We have assumed that the natural variability of the atmosphere is unaffected by coupling to the surface. Mathematically, N^M, N^U, and N^C should be considered as separate realizations of the same random process, with the same power spectra. The superscripts on N included in Eqs. (8)–(13) only clarify the above assumption and will be dropped for all the following calculations. The models in Eqs. (8)–(13) are formally equivalent to linear Markov processes. The uncoupled system can be cast as one-dimensional Markov processes: T^U_a as a first-order process, and T^U_o as a second-order process. The coupled system can be cast as a two-dimensional, first-order Markov process, and the MOGA system as a four-dimensional, first-order Markov process, subsuming the coupled system. However, we have chosen to keep the models in a form that highlights the physical parallelism between the systems.

The methodology of the coupled, uncoupled, and MOGA experiments as expressed in Eqs. (8)–(13) illuminates the effect of coupling the atmosphere and ocean in the midlatitudes. The results of the uncoupled experiment, along with the diagnostic SST field, serve as the “uncoupled null hypothesis” against which we test for the basic effects of thermal coupling. Our analysis of the MOGA experiment has bearing on understanding the role of midlatitude SST anomalies in seasonal climate prediction. In addition, the MOGA experiment aids in the interpretation of the role of the midlatitudes in the simulations doneas part of the Atmospheric Model Intercomparison Project (Gates 1992), in which the global historical SST field was specified as a boundary condition for realistic AGCMs. These diagnostic mixed layer model experiments provide a “common currency” for comparing the atmospheric variability in the coupled, uncoupled, and MOGA runs in terms of the effect of the atmosphere on SST variability.

c. Comments on the stochastic forcing and the coupling parameters

The noise forcing N(σ) and the coupling parameters b and c represent the distillation of complicated atmospheric dynamics and deserve further comment. Figure 3, adapted from B95, shows that the low-frequency portion of the spectrum of atmospheric temperature from a long integration of the two-level model is well approximated by Eq. (16), assuming white noise forcing and a total damping timescale of about 2 days. The salient features reproduced are the flat spectrum at low frequencies and the drop-off toward higher frequencies. The damping timescale is about half that of the “standard parameters,” with the discrepancy probably due to neglecting the dynamical damping term discussed in appendix A, although the choice of an effective heat capacity of the atmosphere is also an issue. Equation (10) is not meant to be a sophisticated model of atmospheric low-frequency variability but only an approximate representation that we use as a starting point to illustrate the effects of coupling.

We reiterate that nonlinear potential vorticity dynamics is driving the low-frequency variability of the atmosphere. The “noise” therefore results from the nonlinearities in the equations of motion, as well as from the representation of a multidimensional systemin terms of one variable. For this simple model, we focus solely on the thermal component of the potential vorticity forcing because numerical experiments lead us to believe that the greatest effect of coupling will be on temperature (and associated thermal wind) variance, which we refer to as “thermal variance.” For simplicity, in the analytic calculations presented in sections 3b–d we will assume N to be white noise of unit amplitude.

In general, the coupling coefficient α will depend on the vertical structures of the uncoupled atmospheric variability, of the stochastic forcing, and of the diabatic effects associated with coupling, and on how well these project on one another. These dependencies in turn are a function of the horizontal structure of low-frequency variability. In light of this complexity we view α as a characteristic coupling coefficient for the entire coupled system. We have chosen the vertical mean temperature as the appropriate free atmosphere temperature variable because of the dominant role played by equivalent barotropic, quasi-stationary structures in low-frequency variability. Later we shall see how deep and shallow atmospheric modes can be interpreted in terms of varying strengths of this coupling coefficient.

a. Power spectra in the coupled, uncoupled, and MOGA systems

As noted above, the stochastic model can be used to interpret coupled, uncoupled, and MOGA experiments. The stochastic model allows us to predict how coupling affects variance (power) in the three experiments. In the coupled case (superscript C) we solve the coupled set of Eqs. (8) and (9) for the power spectral densities P^C_a(σ) = |T^C_a|² and P^C_o(σ) = |T^C_o|² as follows:

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For the uncoupled (superscript U) case we solve Eq. (10) for the atmospheric power spectrum. The power spectrum for T^U_o, the slave ocean temperature, follows from the diagnostic ocean equation (11). In that case we get for the power spectra:

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For the MOGA case (superscript M), we solve Eq. (12), assuming that N and T^C_o are uncorrelated. Thus, when we calculate the power spectrum of T^M_a the cross-terms between N and T^C_o are zero. As in the uncoupled case, the MOGA atmosphere is used to force a slave mixed layer ocean. The power spectra for the MOGA case are as follows:

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Plots of these quantities are shown in Fig. 4 for the standard parameters, with N assumed to be white noise with unit amplitude. Note that the uncoupled atmosphere power is pure “red noise”—the result of a first-order Markov process with autocorrelation time of a⁻¹, or in dimensional units, τ_a = γ_a(λ_sa + λ_a)⁻¹. For the standard parameters τ_a ≈ 4 days.

We are now in a position to draw some general conclusions about variance in coupled and uncoupled runs. The MOGA power spectrum for either atmosphere or ocean temperature can be expressed in terms of the respective coupled and uncoupled power spectra as follows:

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At this stage we have not assumed anything about the shape of the spectrum of the forcing term N. We can see that at all frequencies, both the coupled and MOGA runs have more variance than the uncoupled run, at least for reasonable values of α. The comparison of the MOGA and coupled runs is more complicated. For low frequencies, βσ² < ad, the coupled variance exceeds the MOGA variance. For higher frequencies, the direct forcing by the SST anomalies exceeds the internal variance. However, due to the long timescales associated with the ocean there is little power at these higher frequencies.

It is instructive to consider the ratios of variance between different systems in the limit as σ → 0 (or equivalently, ω → 0) as follows:

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where z = ad/α as before. We have defined the ratio δ^C,U = P^C(0)/P^U(0), with the other ratios defined analogously, and have dropped the subscripts because these ratios are the same for the atmosphere and ocean variables. (This equality of variance ratios does not necessarily hold in more realistic models, as discussed in section 4a.) The ratio of power at low frequencies depends only on the stability parameter z. A plot of the above power ratios as a function of z is shown in Fig. 5. The standard parameters correspond to z = 2.42, and this value is indicated by a vertical line in the plot.

As previously mentioned, z is a stability parameter for the coupled system. For z > 1, the coupled system is stable, and the power in the system is maintained by the stochastic forcing. Large values of the stability parameter, z ≫ 1, correspond to either large damping, large negative atmospheric feedback, or inefficient coupling between the free atmosphere and the surface. In this case the MOGA variance approaches the uncoupled variance, while the coupled variance approaches the uncoupled variance, though much more slowly. In the limit as z → 1, which is off the scale of Fig. 5, the coupled and MOGA variances become equal, and both approach infinity. This limit corresponds to the unrealistic case where positive atmospheric feedback completely counteracts damping. Even in the case where we allow noatmospheric dynamical feedback (i.e., we allow only the “thermal” response of the atmosphere so that b = 1, resulting in z = 1.25 for the standard parameters), we get excessively large ratios between coupled and uncoupled runs. For z < 1, we have linearly unstable atmosphere–ocean interaction for which the stochastic model proposed here is inappropriate.

b. Integrated variance

The differing effect that coupling has on the atmosphere versus the ocean can be seen if we take the integral over all frequencies in Eqs. (14)–(19) to get the total variance. We will assume white noise forcing of unit amplitude (i.e., |N| = 1) to simplify the integration. The results for the uncoupled case are the simplest (see e.g., CRC Math Tables):

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where d̃ = d/β. For the coupled system we will show only the ratio of the total coupled power to the total uncoupled power, where we define

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(the corresponding ratios for oceanic variance and for the MOGA system are defined analogously). The integrals for the coupled and MOGA systems are more complicated than for the uncoupled system. The exactresults are shown in appendix B. If we expand the exact results in the small parameter ε = d/(aβ), keeping only the leading order in ε, we get

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where z is as before; Δ^C,U_o is plotted as a function of z in Fig. 6. For the standard parameters, Δ^C,U_a = 1.03 and Δ^C,U_o = 1.71. The parameter ε is proportional to γ_aγ⁻¹_o, the ratio of the effective heat capacity of the atmosphere to that of the ocean mixed layer. Therefore, ε is inversely proportional to the mixed layer depth h. The corresponding ratios for the MOGA case are as follows:

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Because the bulk of atmospheric variability lies in higher frequencies where coupling has little effect, the ratio of coupled to uncoupled total atmospheric variance is near unity. On the other hand, the bulk of model oceanic variance is at low frequencies where the effect of coupling is strong, so the ocean ratio is substantially greater than 1. This reflects what happens in the two-level model of B95, as the total atmospheric variance is only slightly affected by coupling, but the oceanic variance is substantially increased by coupling (see section 4c). The effect of coupling on the total oceanic power is strictly independent of ε in the model presented here, and thus is independent of the mixed layer depth.

c. Surface flux spectra

We can compute the power spectra of surface fluxes in a straightforward manner from the model equations (8)–(13), so the formulas are not shown here. For the coupled system, we define Q^C_↕ ≡ T^C_s − T^C_o = cT^C_a − T^C_o to be scaled symmetric surface flux. For convenience, we are neglecting the terms in the surface fluxes, that are not proportional to the air–sea temperature difference. If these terms were included, the fluxes that force the ocean would strictly go to zero for ω → 0, but otherwise the discrepancy is small. For the MOGA system, there are two sets of fluxes, those that force theatmosphere (with SSTs specified from the coupled model), Q^M_↑ = T^M_s − T^C_o, and those that force the diagnostic ocean, Q^M_↓ = T^M_s − T^M_o. Likewise, for the uncoupled system we have Q^U_↑ = T^U_s and Q^U_↓ = T^U_s − T^U_o. These power spectra are plotted in Fig. 7. At low frequencies the fluxes that force the ocean satisfy Q^C_↕ > Q^M_↓ > Q^U_↓, as in B95. Given the simplicity of our ocean model, it is therefore not surprising that the resulting SST variance follows the same ordering. Also, at low frequencies Q^U_↑ > Q^C_↕. This relation is a manifestation of the reduced thermal damping in the coupled system, and it agrees with the numerical model results of B95 and Bhatt et al. (1998). Finally, Q^M_↑ > Q^U_↑ at low frequencies. That is, there is a large excess of power in the low-frequency fluxes that the MOGA atmosphere sees, compared to those that the uncoupled and coupled atmospheres see. This result was somewhat unexpected in the numerical experiments of B95, particularly in light of the fact that both the atmospheric and oceanic thermal variance of the MOGA run were intermediate between the uncoupled and coupled runs. In the present model, this excess is a direct result of assuming that a large fraction of the“natural variability” is independent of the prescribed SST anomalies. The implications forone-way forced experiments using numerical models are discussed in section 4b.

d. Lag covariance

Figure 8 shows the autocorrelation functions for T_o and T_a for the three systems with the standard parameters. The autocorrelation functions are simply the normalized Fourier transforms of the power spectra shownin Fig. 4. Clearly, coupling significantly increases persistence of both atmospheric and oceanic temperature anomalies. Likewise, the lag covariance between T_o and T_a is the Fourier transform of T_aT^*_o, where the asterisk denotes the complex conjugate. The derivation of formulas for lag-covariance from Eqs. (8)–(13) is straightforward and is not shown. The lag-covariance function can then be used to derive the lag-linear regression function, which is shown in Fig. 9. Atmospheric variance, which is strongly dependent on averaging period, does not appear in the denominator of the linear regression, reducing the sensitivity of this measure to the averaging period used.

Four curves are shown in Fig. 9. The uncoupled case (curve a) shows strong asymmetry in lag. This asymmetry is characteristic of a system with large thermal inertia (the ocean), forced by a stochastic system with a fast decorrelation time (the atmosphere), as seen in Hasselmann (1976). The MOGA and coupled systems (curves b and c) show a progressively larger component that is symmetric in lag. The regression between the prescribed SST and the MOGA atmosphere (curve d) is almost entirely symmetric in lag. This symmetry is the result of the assumption that the effects of SST anomalies on the atmosphere are instantaneous. The analogous lag correlation functions (for pentad means) are similar in shape to those in Fig. 9, but with maximum correlations of around 0.4 for curves a–c and 0.1 for curve d. These curves are consistent with the lagged linear regression maps for these four cases shown in B95. For the standard parameters, the lag-correlation peaks for atmosphere leading ocean by about 10 days, a little more than twice the decorrelation time of the atmosphere.

a. General comments about coupling to slab mixed layers

To put the effects of coupling in a larger framework, consider an atmospheric GCM coupled to a slab mixed layer ocean of constant depth h. The value of h determines the nature of the lower boundary condition on the atmospheric thermodynamic equation. In the extreme case of h → 0, the lower boundary condition becomes an instantaneous surface energy balance (SEB). The other extreme, h → ∞, corresponds to fixed SST, where atmospheric temperature anomalies of all frequencies are damped equally by surface fluxes. The fixed SST case should exhibit the most damping due to surface fluxes, and the SEB case the least. For intermediate values of h the coupling acts to damp only high-frequency atmospheric temperature anomalies. For timescales longer than τ_ML, the SST can adjust to atmospheric temperature anomalies and the surface fluxes are reduced to near zero. As h → 0, the timescale of the mixed layer becomes comparable to or shorter thanthe synoptic timescale in the atmosphere. Because baroclinic conversion is the ultimate source for much of midlatitude variability, the entire spectrum of variability may be affected. Therefore, we have avoided consideration of the SEB case. Instead we chose the other extreme, h → ∞ (the “uncoupled” system, with fixed SST) as the basis for comparison with the other systems.

In the coupled and uncoupled cases, Eqs. (8)–(11), the stochastic dynamics of the atmosphere is the only source for low-frequency variability in the coupled model. In the MOGA framework however, the prescribed SST anomalies act as an additional external forcing at low frequencies [see Eqs. (12) and (13)], so we expect that the MOGA run will have more variance than the uncoupled run with its fixed, zonally symmetric SST. The question is “How much more?” It seems reasonable to assume, as we have done, that the bulk of the midlatitude nonlinear atmospheric variability is uncorrelated with the prescribed SST anomalies, at least for SST anomalies of modest amplitude. Evidence of this in a more realistic model is seen, for example, in the predictability study of Miller and Roads (1990), where inclusion of actual SST anomalies in midlatitudes did little to enhance predictability. Hence, it is not surprising that the MOGA atmosphere temperature variance is less than the coupled model variance. For example, Eq. (24) with the standard values for the parameters indicates that the low-frequency variance should be 1.93 times greater in the coupled framework than in the MOGA framework.

Barsugli (1995) argues that large responses to SST anomalies are not generic to atmosphere–ocean coupling, but rather depend on the special circumstances involving such factors as the climatological mean stationary waves, the position of the SST anomalies, or land–ocean temperature contrasts. Two commonly seen responses of realistic AGCMs to prescribed SST anomalies are a surface-trapped low over a warm SST anomaly and an equivalent barotropic high downstream from a warm SST anomaly (Palmer and Sun 1985; Peng et al. 1995). Both of these responses are consistent with the simple model presented in this paper, provided that the interaction is stable. A large barotropic response would indicate that SST anomalies couple effectively to natural barotropic variability of the atmosphere, probably through precipitation or eddy fluxes, resulting in a large effective value of b. Some models (e.g., Latif and Barnett 1994; Palmer and Sun 1985) show evidence of a large positive feedback; that is, α > ad. Such a linearly unstable system is not amenable to treatment by the present stochastic model and, like any linearly unstable system, requires a mechanism for (statistical) equilibration in order to be a complete theory. The instability in the above experiments appears to arise largely because of a strong atmospheric response to SST anomalies. One would expect to see two phenomena in the presence of such strong upward coupling, which, as far as we know, are not present in the observations: First, lag–lead correlations between SST and the free atmosphere would be largely symmetric about zero lag. Second, prescription of midlatitude SST anomalies would significantly enhance predictability of the atmosphere in hindcast studies.

b. One-way forced experiments and spurious surface fluxes

In Fig. 7 we show that atmospheric models with prescribed SST can have large spurious surface fluxes at low frequencies. The largest spurious fluxes, defined as Q^M_↑/ Q^C_↕, occur for small values of b and for values of α near the stability boundary (i.e., α ≈ ad). These spurious fluxes do not present a serious problem for making short-term forecasts, as these forecast models are initialized with the correct atmospheric state from the coupled system (i.e., from observations). However, a problem may arise for seasonal predictions where SST is specified, whether specified as strict persistence of observed SST anomalies or whether these anomalies are damped back to climatology over some empirical mixed layer timescale. Specifying midlatitude SST anomalies constrains the amplitude of atmospheric temperature anomalies through the spurious (damping) surface fluxes. It would be better to make seasonal forecasts using an ensemble of runs of AGCMs coupled to mixed layer models than to use fixed boundary conditions.

The same problem arises in climate simulation studies in which the history of global SSTs is specified, or in establishing “control run” statistics for studies of decadal variability. The true variability in the coupled system is not necessarily well approximated by forcing the atmosphere with realistic SSTs for models with small or moderate values of the atmospheric response parameter b.

Numerical experiments in which atmospheric quantities are prescribed as the forcing for an ocean modelsuffer from a different problem. The merits of specifying surface air temperatures and winds versus specifying surface heat fluxes has been debated in the literature. Both methods suffer from problems, as both surface fluxes and air temperature are strongly coupled to the ocean temperature. If the flux forcing comes from an uncoupled or MOGA experiment, then extreme SST variance at low frequencies will result because of spurious surface fluxes. The excessive low-frequency flux forcing is often compensated by applying an arbitrary feedback term proportional to the SST anomaly. Neglecting wind-forced surface flux variance, this method is linearly equivalent to specifying surface air temperatures and winds, as long as the correct value of the feedback is chosen. This is seen in the following extension of Eq. (2):

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However, GCM boundary layer formulations usually contain a strong nonlinear dependence on the static stability of the boundary layer. Computing the forcing from atmospheric surface air temperature and wind, using a sophisticated nonlinear surface flux parametrization avoids the assumption of linearity, but the complexity of some GCM boundary layer formulations may make this an undesirable choice in practice.

The power spectrum of surface fluxes used to force a slab mixed layer model must tend toward zero at low frequencies, or else the temperature variance would be unacceptably large. If a more complicated ocean model is forced with spurious low-frequency fluxes, this forcing will be compensated by other oceanic processes—entrainment, advection, convection, diffusion—at these timescales. In fact, substantial low-frequency surface flux variance in a coupled model or in observations is a signature of such processes.

A potentially serious problem arises if one uses surface fluxes diagnosed from observations to force ocean models. Observational errors, which project onto all frequencies, can lead to potentially large errors in low-frequency temperature variance (e.g., Ronca and Battisti 1997). Adding a simple linear relaxation term, as discussed above, is not entirely satisfactory, as there exists some frequency below which the temperature variance is determined primarily by the error variance in the surface fluxes. In effect, random (uncorrelated in time) observational error in surface fluxes results in a low-frequency limit below which one is unable to model temperature variance reliably.

c. Comparison with more complete atmosphere and ocean models

Barsugli (1995) performed uncoupled, MOGA, and coupled integrations in which the atmosphere was modeled by the global, two-level primitive equations with parameterized convection and radiation [similar to themodel of Held and Suarez (1978)]. SST was either prescribed or simulated using a 50-m slab ocean model. The model geometry was idealized to be a global ocean. The coupled stochastic model qualitatively reproduces the ratio of total variance between coupled, MOGA, and uncoupled runs in B95, as shown in Table 2. The spectra of ocean and atmosphere temperature fields and surface flux fields from B95 are reproduced in Fig. 10. These are qualitatively consistent with the results from the stochastic energy balance model, as seen in a comparison of Fig. 10a to Fig. 4, and Fig. 10b to Fig 7.

The results from the three experiments in B95 support the hypothesis for the twofold effects of coupling: the reduction of thermal damping at low frequencies and the relatively weak direct effect of forcing by SST anomalies. There are, however, two major discrepancies between the variance ratios predicted by Eqs. (22)–(24) from the stochastic coupled model and those calculated from the two-level GCM of B95. First, the low-frequency variance enhancement in the atmosphere is generally less than that of the ocean (see Table 2), and the enhancement decreases as one passes from the surface to 250 mb (not shown). Clearly, in the two-level model there can exist variability with a vertical structure that has no signature in surface temperature, and thus does not participate in coupling. This additional atmospheric variability dilutes the enhancement of variance one would expect from a univariate model such as we have presented. The second major discepancy is seen in the wavenumber-frequency spectra of B95—enhancement of variance is a strong function of zonal wavenumber. The explanation lies in the modal structure of variability in the two-level model. There are two horizontal modes that participate strongly in the coupling, a deep, equivalent barotropic mode with zonal wavenumber k = 4, and a shallow k = 1 mode. The former is the dominant mode of low-frequency variability in the uncoupled case and is merely enhanced by the coupling to the mixed layer. The latter appears strongly in the coupled run but only very weakly in the uncoupled run, and can be explained as a surface-trapped advective, coupled mode (Frankignoul 1985; B95). The k = 1 mode also appearsstrongly in the MOGA run spectrum, suggesting a large value of α associated with this mode. These discrepancies, and the associated dynamical reasons for them, are indicative of what to expect from the analysis of more realistic models.

Manabe and Stouffer (1996) present results from the GFDL R15 AGCM in which the AGCM was (i) integrated using the prescribed annual cycle of SST and (ii) coupled to a 50-m slab mixed layer, and (iii) coupled to an ocean GCM. They found that coupling acted to enhance the variance of SST and surface air temperature in the midlatitudes, with the greatest increase occurring at lowest frequencies (see their Figs. 1–5). In Fig. 1 we have reproduced their Fig. 4b, showing the ratio of standard deviations between their coupled GCM and fixed SST runs. The coarse contour interval in their figures makes it difficult to determine quantitatively the extent of the agreement between their GCM and the stochastic model. However, their figures indicate that the ratio of coupled-to-uncoupled variance (derived by squaring their standard deviations) in surface air temperature in midlatitudes is roughly 2.5 for 5-yr averaged fields, whereas the stochastic model yields a ratio between 2.4 (period 5 yr) and 2.9 (period infinity). Significantly, the ratios of variance between the models with the ocean GCM and the slab mixed layer are near unity over much of the open oceans (their Fig. 4a). Furthermore, the GFDL coupled AGCM/OCGM yields midlatitude spectra of surface air temperature and SST (their Fig. 12) that are remarkably similar in structure to that from thesimple stochastic model. However, the GFDL model yields slightly more variance in SST than in surface air temperature (unlike the stochastic model) for periods greater than about 20 years, presumably due to variability associated with changes in the oceanic thermohaline circulation.

Bladé (1997) used the same atmosphere GCM as in Manabe and Stouffer (1996) to examine the effect of midlatitude coupling on the variability in the midlatitude atmosphere. In this study, the control integration was compared to an integration in which the atmosphere was coupled to a 50-m slab ocean in the midlatitudes of both hemispheres; both integrations featured perpetual January solar forcing. Bladé reported that the influence of midlatitude coupling significantly enhanced the midlatitude lower tropospheric temperature and circulation variance at the lowest frequencies. For example, the ratio of coupled-to-uncoupled power in the 850-mb air temperature over the midlatitude oceans at lowest frequencies is approximately 1.6. Bladé (1997) furthermore noted that the effect of midlatitude coupling in the full GFDL GCM was quantitatively similar to that found in the idealized two-level GCM studies of B95.

Bhatt et al. (1998) examined the differences in the atmospheric circulation anomalies that result from coupling the NCAR CCM1 R15 AGCM to the variable-depth mixed layer model of the North Atlantic Ocean. Both coupled and control integrations were 35 yr long and include a seasonal cycle. Compared to the control integration, the variance of the wintertime averaged surface air temperature in the coupled integration increased by a factor of about 2.3. The net surface heat fluxes decreased by a factor of about 0.6. Coupling also enhanced the persistence of the principal mode of variability in the atmosphere. These results are in agreement with predictions from the stochastic model.

While the simple model we present was developed to explain enhancement of internally generated midlatitude low-frequency variability, it is also applicable to externally forced variability. Lau and Nath (1996) reported the results from a series of experiments with the GFDL R15 AGCM on the effect of midlatitude coupling on tropically forced midlatitude variability. In these experiments they examined the portion of the midlatitude variability in the ocean and atmosphere that is associated with ENSO via atmospheric teleconnections (“the atmospheric bridge”) by prescribing observed tropical Pacific SST anomalies under the AGCM. The effect of the coupling in the midlatitudes on the ENSO-forced midlatitude variability was determined by examining differences between two ensembles of integrations: the first ensemble held midlatitude SST fixed at climatological values (TOGA), while the second ensemble allowed the midlatitude atmosphere to interact with a 50-m slab ocean (TOGA-ML). The principal results of Lau and Nath’s study are qualitatively consistent with the stochastic model results. Specifically, the ENSO-related variance in the surface air temperature over theNorth Pacific Ocean, which is predominantly at low frequencies, increased about fourfold due to coupling.³ The variance and persistence of the 500-mb and surface atmospheric circulation anomalies associated with the ENSO were also enhanced by the midlatitude atmosphere–ocean coupling, consistent with the enhancement in the temperature variance via the thermal-wind relation. The structure of the midlatitude ENSO-related anomalies was found to be insensitive to midlatitude coupling.

In a similar investigation to Lau and Nath (1996), Alexander (1992) concluded that “air–sea interaction primarily acts to damp ocean anomalies,” whereas we come to the opposite conclusion. The resolution of this seeming contradiction is that he compared his coupled SSTs to SSTs from an ocean model forced by surface fluxes from his uncoupled atmosphere integrations (dT_o/dt = Q^U_↑ in our notation). As discussed in section 3c, this latter methodology produces spuriously large SST variability, and does not serve to illuminate the basic role of coupling. In fact, what Alexander (1992) calls“partially coupled” SST is equivalent to our “uncoupled SST,” and indeed shows about half the temperature signal compared to his coupled SST away from the continents in winter, thus confirming our viewpoint.

d. Caveat concerning the use of this model to interpret GCM results

The main virtues of the model in Eqs. (8)–(13) are its simplicity and flexibility. One is tempted to use it to quantitatively diagnose output from atmospheric GCMs coupled to slab mixed layer oceans. Caveat emptor: the model as it stands is oversimplified in several important aspects, making it unsuitable to use for quantitative diagnosis. We believe that the major oversimplifications are as follows: the assumption of a first-order Markov process for an uncoupled atmosphere model, the assumption of purely thermal coupling, and the neglect of modal structure in the atmosphere. We will discuss these below.

The uncoupled atmosphere model of Eq. (10) does not generally capture the spectrum of low-frequency variability in more realistic GCMs. However, the exact form taken by the uncoupled atmosphere equation is not critical to understanding the basic effects of atmosphere–ocean coupling. For example, the 850-mb temperature power spectrum from and uncoupled run of the GFDL R15 AGCM (Bladé 1997), taken at grid points over the North Pacific Ocean, cannot be fit by Eq. (10). However, an ad hoc fit to the GCM spectrum may be obtained using Eq. (10) with an 8-day damping timescale, assuming that there is an additional background white noise variance. The qualitative results of this paper still hold in this case, and the quantitative calculations may be carried through with only slight modifications.

While purely thermal effects seem to explain much of the phenomenology of atmosphere–ocean coupling in realistic GCM experiments, it is not the whole story, even if the ocean is a slab mixed layer. Low-frequency surface wind variability and nonlinear moisture variability may modulate surface fluxes independent of the air–sea temperature difference. If these effects play a significant role, then the surface temperature (and moisture) variability will differ substantially from SST variability at low frequencies. Wind variability also drives Ekman currents in the ocean, which we have neglected. These effects are discussed in Liu (1993) and Miller (1992) and provide a complementary approach to ours.

Finally, atmospheric variability has vertical and horizontal structure. A single variable is at best a crude representation of what is inherently a multivariate system. It is tempting to interpret the single atmospheric temperature variable in Eqs. (8)–(13) as representing a single mode of atmospheric variability. Indeed we have done so in this paper in interpreting the results of B95. However, it must be kept in mind that the total variability (say, at a grid point) may be the amalgam of modes with widely differing behavior under coupling. In addition, atmosphere–ocean coupling may allow for interaction between “uncoupled” atmospheric modes through diabatic or nonlinear heating anomalies associated with SST anomalies.