## 1. Introduction

Our working hypothesis in this paper is that in CMIP preindustrial (PI) simulations and, more generally, in the absence of variable external forcing, the Atlantic multidecadal oscillation (AMO) sea surface temperature (SST) is a response to white noise forcing reddened by the heat capacity of the ocean mixed layer. The noise source may be the atmosphere or the ocean. The North Atlantic Oscillation (NAO) is a well-known source of atmospheric noise, but it is not the only atmospheric noise the ocean feels. Ocean noise includes the variations in the mixed layer, which are largely induced from wind and buoyancy forcing from above, along with high-frequency fluctuations in heat transport convergences into the mixed layer that may be forced by the wind stress as well as by internal ocean fluctuations, such as eddies. It will turn out that most of the white noise forcing appears to come from the atmosphere. The important finding is that the forcing may all be noise. No systematic long-period influence from the ocean circulation is needed.

Our hypothesis is motivated by findings in Clement et al. (2015, hereafter C15). C15 showed that CMIP3 atmospheric GCMs coupled to slab oceans produce the same spatial patterns of the AMO SST as fully coupled models and that both do a reasonable job of simulating the observed pattern. The slab ocean configuration does not have an active ocean circulation; the only “ocean circulation” is the unvarying climatological *q* flux that maintains the SST climatology. Beyond that, the slab models have a specified mixed layer depth and do not include the physics to vary the mixed layer depth or allow mixing from below.

The temporal behavior of the AMO in models also motivates our noise-driven hypothesis. The spectra in Fig. 1 look like red noise and do not show any pronounced multidecadal peaks. The spectra in C15’s Figs. 2 and 3 include the multimodel mean (MMM) and individual CMIP3 models, showing that the featureless spectra in our Fig. 1 are not an idiosyncrasy of the one model shown here, CESM1(CAM5). Additional examples of red noise drawn from CMIP5 PI runs appear in Fig. 2 of Ba et al. (2014) and Fig. 2 of Peings et al. (2016).^{1} In addition, the spectra in C15 and in Fig. 1 show no structural difference between the AMO in the coupled models and that in the slab ocean model. Since the atmospheric forcing in these PI runs is more or less white in the atmosphere, and since adding an active ocean makes no essential difference, it suggests that whatever forcing comes from the ocean is also more or less white noise. On the basis of these results, C15 concluded that the ocean circulation is not an essential driver of the AMO.

The experimental protocol that leads to this conclusion is in the mold of the classic model-based attribution study: in one set of numerical experiments we include factor *X*, and in the other set we exclude it. Features that are the same in the two sets are not attributable to *X*. This sort of counterfactual test of causality goes back at least to David Hume in the 18th century: *Y* is caused by *X* if and only if *X* was not to occur, then *Y* would not occur (Hannart et al. 2016). In the present case *Y* = “AMO-like variability” and *X* = “ocean circulation together with most of mixed layer physics.” In the slab ocean model experiments, *X* does not occur, but *Y* does occur, so we conclude that ocean circulation (*X*) is not the cause of AMO-like variability (*Y*). Perhaps the real world is different, but this is what the models tell us in the absence of time varying external forcing.

Nonetheless, in reality and in the fully coupled models there surely is an ocean circulation and mixed layer physics that influence the SSTs in the North Atlantic. Thus, it is a given that the heat budgets determining SST cannot be exactly the same in the slab models as they are in the coupled models, but this need not mean that the AMO mechanism differs in an essential way from the suggestion in C15 that the AMO is driven by atmospheric white noise. In their model studies of decadal variability in the Atlantic, Fan and Schneider (2012) and Schneider and Fan (2012) conclude that the primary forcing is atmospheric noise, though they leave open a possible role for the ocean circulation. Others go further in their advocacy for the idea that the low-frequency ocean circulation, most prominently the Atlantic meridional overturning circulation (AMOC), is an essential player in multidecadal Atlantic SST variability, an idea with august beginnings in Bjerknes (1964) and Kushnir (1994), continuing to Zhang et al. (2016) and O’Reilly et al. (2016).^{2} This does not go so far as to eliminate a role for the atmosphere, but it does anoint low-frequency heat convergence organized by the ocean circulation as the key driver of the AMO. Advocates often point to a negative relationship between surface heat flux and SST at long periods. They interpret this as showing the atmosphere responding to SST changes that must be driven by ocean heat flux convergence that is low frequency, which implies that this convergence is organized by low-frequency variations in ocean circulation. Recent examples include Gulev et al. (2013), Brown et al. (2016), Zhang et al. (2016), O’Reilly et al. (2016), and Drews and Greatbatch (2016). All these analyses are based on low-pass filtered data. We will show below that these low-passed results admit other interpretations, ones that do not require any nonrandom or dominant influence from the ocean circulation.

We are less certain that the observed AMO is red noise. There may be a real multidecadal peak, as was argued by Schlesinger and Ramankutty (1994), for example, but the record is too short to be certain. Even if real, a peak need not be a sign of internal variability since the instrumental period of observations has been marked by external forcing due to volcanic eruptions, solar variability, anthropogenic aerosol, and greenhouse gases. Models run with historical forcing have more power at low frequencies than preindustrial runs. This may be seen in C15 (cf. their Fig. 2c with their Figs. 2a,b) for multimodel means and Fig. 2 of Peings et al. 2016 for individual models. Our hypothesis, pursued further in Murphy et al. (2017) and Bellomo et al. (2017), is that the observations are best explained as largely a response to external forcing from aerosol and greenhouse gases. We will return to this issue in the discussion section, but throughout the rest of the paper we will concern ourselves with variability generated within the models’ climate system in PI and slab ocean model (SOM) runs.

We begin in section 2 with the simple point model for sea surface temperature *T* that contains only a damping term and white noise forcing from both atmosphere and ocean. Such a model and its application to SST have a more or less unbroken 40-yr lineage going back to Hasselmann (1976) and Frankignoul and Hasselmann (1977). Among the long line of papers that follow, we single out Frankignoul et al. (1998) for its derivation of the simple model in Eq. (1) from a complete heat equation and for its derivation of some of the covariances in Eqs. (A2) and (A3). In this simple model, the damping term arises from the tendency of surface fluxes, especially the turbulent latent and sensible heat fluxes, to adjust to remove departures from the equilibrium value of *T* at zero total heat flux. It is generally accepted that the atmospheric forcing is white in time or nearly so (Wunsch1999; Stephenson et al. 2000; and see Fig. 2). The oceanic heat flux also appears to be nearly white in time, though this is less widely appreciated. Figure 2 shows the spectrum of oceanic heat flux derived from a PI run of the fully coupled model CESM1(CAM5) along with spectra of the surface flux *T*, and the temperature tendency ^{3} As in many other calculations of oceanic heat flux (e.g., Zhang et al. 2016), it is actually the residual *h* is mixed layer depth taken as 50 m). Figure 2 shows it to be approximately white. (The figure also shows that at low frequencies the ocean and atmospheric heat fluxes have the same power. At low frequencies, the temperature tendency is very small. Consequently, these fluxes must sum to near zero and so are equal and opposite. The implications of this will be considered below.). Recent observational results show that the ocean has power at all observed frequencies (Lozier 2012), though these series are not long enough to say whether or not the spectrum is truly white out to multidecadal periods.

(a) Spectra from a PI run of the CESM1(CAM5) coupled mode for quantities averaged over the

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

(a) Spectra from a PI run of the CESM1(CAM5) coupled mode for quantities averaged over the

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

(a) Spectra from a PI run of the CESM1(CAM5) coupled mode for quantities averaged over the

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

We put the simple model forward as a way of understanding the complex model simulations. In section 2, after defining the model, we spend some time on the properties of low-pass filters, especially when applied to white or red noise. Analytic results for the unfiltered and low-pass (LP) filtered correlations that appear in the figures are derived in the appendix. These results are summarized and discussed at the end of section 2. In section 3, we first see if this simple model can account for properties of the coupled and slab models, particularly correlations involving SST and heat fluxes. We then consider results in the literature that are held up as evidence that the ocean driving is important for the AMO in order to see if our analysis of the simple red noise *T* model can account for them. We close this section with a consideration of low-frequency forcing that is distinct from white noise. A discussion section that includes some consideration of the twentieth-century observational record follows in section 4.

We use monthly CMIP5 data (found at http://cmip-pcmdi.llnl.gov/cmip5/data_portal.html). The CESM1(CAM5) monthly averages are available online (http://www.cesm.ucar.edu/projects/community-projects/LENS/data-sets.html). The models used and the length of each simulation are reported in Table 1. The surface flux

Coupled models included here in the CMIP5 MMM. The second column gives the length of the available PI simulation for each model. The next column is the number of (nonindependent) 140-yr-long samples used in correlation statistics here and in Fig. 8. (For MIROC5, we delete the first 85 yr when the model is still spinning up.) All statistics are for the *T*′ is the variance of *ρ* are the low-pass correlations between temperature and the surface heat flux

## 2. Noise-forced model

### a. The model

*T*is temperature,

*a*and

*b*. Denoting expected value of the covariance of

*δ*is the Dirac delta function and

The assumption in Eq. (3) that ^{4} For the present, our goal is to make a very simple model and note where it fails convincingly enough to demand the addition of other processes. That it lacks the verisimilitude one hopes for in a GCM is obvious. The question is whether such simple and transparent physics is adequate to account for much of the behavior of complex models.

*a*or

*b*), and, regardless of the division, the expected value of the total noise forcing is fixed:

We will want to compare various correlations involving temperature and heat fluxes in the noise forced model [Eq. (1)] with results from GCMs. Rather than just find these by numerical simulations of the model equation, we derive analytic results for expected values for correlations of unfiltered variables in section a of the appendix. In keeping with the fact that the forcing introduces no special time scales, all of these correlations are substantially different from zero only for leads and lags within a few *e*-folding time scales of

### b. Low-pass linear filtering

*L*applied to a time series

*L*to denote both the operator

*n*-yr running mean

*n*years)

^{−1}. In our context, “low pass” means that the cutoff period is long compared to the damping time. We formalize this condition as

*L*. Then

*f*and

*g*, we define notation for the low-pass covariance by

*R*with their unfiltered covariance [Eq. (7)]. The inverse transform of

*R*is the autocovariance function for the filter

*R*is real and an even function

*τ*appears only in the combination

*n*-month running mean filter

*s*as the argument instead of

*τ*is that

*s*makes the size of terms more transparent. The transform function corresponding to

*b*merely small (i.e.

*b*very close to zero (

### c. Summary of analytic results for the NFM

The model presented in this section is quite simple: a one-dimensional temperature [Eq. (1)] forced by white noise from the atmosphere *T* has a red spectrum

*T*). The former just returns the filter autocovariance since the Fourier transform of white noise = 1 [viz., Eq. (8)]. If red noise is involved, the fact that the filter is low pass means that

*f*with amplitude

*A*,

*R*is essentially white noise, and the filtering returns the autocovariance of the filter or its derivatives. In particular, the lagged correlation of

*T*with itself

Figure 3 displays the autocovariance *R* is an even function, ^{5} Since we are dealing with low-pass filters, we expect that *R*, which is an even function, to be positive and relatively large, and *τ* where maxima, minima, and zeros occur depends on

(a) The autocorrelation function *R* is calculated by applying the same filter software used throughout to a white noise sequence. The agreement among the curves is predicted by Eqs. (11a), (11b), and (A11f) if the temperature in the coupled models is red noise.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

(a) The autocorrelation function *R* is calculated by applying the same filter software used throughout to a white noise sequence. The agreement among the curves is predicted by Eqs. (11a), (11b), and (A11f) if the temperature in the coupled models is red noise.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

(a) The autocorrelation function *R* is calculated by applying the same filter software used throughout to a white noise sequence. The agreement among the curves is predicted by Eqs. (11a), (11b), and (A11f) if the temperature in the coupled models is red noise.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

The effect of changing the filter cutoff is illustrated in Fig. 4. The top panel shows the correlation of *T* when *T* is generated from Eq. (1)]. The different curves are for filter cutoff periods of 5, 10, 20, and 30 yr. The lead or lag where the extrema occur move linearly with the cutoff period, as expected for red or white noise. The middle panel of Fig. 4 shows the correlation for a low-frequency signal *T*, which gives a peak when the tendency leads by a quarter period (15 yr in this example) is a robust feature and does not change when the filter length changes. In contrast, if the peak does move when the filter length changes, as in the top panel of Fig. 4, then this is indicative of white or red noise, and the peak is an artifact of the filter. Such movement may be used as a diagnostic to distinguish noise from a true low-frequency signal.

(top) The correlation *T* for the

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

(top) The correlation *T* for the

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

(top) The correlation *T* for the

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

## 3. Comparison of the white noise–forced model with GCM and observational results

### a. Is the white noise–forced model relevant for GCMs?

Our first task is to establish that the white noise–forced model (NFM hereafter) presented in the previous section is relevant for interpreting the GCM results. We focus on lead–lag correlations involving temperature and heat fluxes, which appear often in the literature. The right side of Fig. 5 shows the CAM5-SOM lead–lag correlations for the unfiltered results in blue and the low-pass results in red. For comparison, the left side of Fig. 5 shows the correlations from the NFM when ^{−2} K^{−1} for a 50-m deep mixed layer, which is close to but smaller than the values estimated by Park et al. (2005) and Frankignoul and Kestenare (2002) from observed data using the methods devised by Frankignoul et al. (1998).

Lead–lag correlations for (top)–(bottom) ^{−2} K^{−1} = (0.5 yr)^{−1} for a 50-m mixed layer. (right) The CAM5-SOM for the

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

Lead–lag correlations for (top)–(bottom) ^{−2} K^{−1} = (0.5 yr)^{−1} for a 50-m mixed layer. (right) The CAM5-SOM for the

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

Lead–lag correlations for (top)–(bottom) ^{−2} K^{−1} = (0.5 yr)^{−1} for a 50-m mixed layer. (right) The CAM5-SOM for the

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

The similarities between the corresponding curves in the SOM and the NFM with

The right side of Fig. 6 is as the right side of Fig. 5, but for the coupled model CESM1(CAM5) and the CMIP5 MMM for the models in Table 1. We estimate *α* to be 20 W m^{−2} K^{−1}, which is comparable to observed values (Frankignoul et al. 1998; Park et al. 2005) and slightly higher than for the slab ocean. For a 50-m mixed layer this corresponds to a damping time of 0.325 years or slightly less than 4 months. The NFM case in Fig. 6 we show as comparable has *a*. As with the SOM results in the top two panels of Fig. 5 and the NFM results in Fig. 5, we find that the fully coupled model results for temperature agree quite well with the white noise–forced model. Thus, as with the SOM, the temporal structure of temperature *T* in CESM and in the MMM are consistent with a damped response to white noise forcing. Both the unfiltered and LP filtered *T* (top panel of Fig. 6) have the same structure as the comparable SOM (Fig. 5) or the NFM (in either Fig. 5 or Fig. 6). As suggested by the analysis in section 2, given that the unfiltered *T* from the coupled model looks like red noise with a damping time short compared to the multidecadal time scales of interest for the AMO, the LP filtered correlation *R*, while the LP *R*. The MMM looks to be the same. The red noise structure of the unfiltered data has been bleached by the filtering.

As in Fig. 5, but for (left) the simple model with parameters ^{−2} K^{−1} = (4 months)^{−1} for a 50-m mixed layer; and (right) the CESM1(CAM5) fully coupled model and the CMIP5 MMM.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

As in Fig. 5, but for (left) the simple model with parameters ^{−2} K^{−1} = (4 months)^{−1} for a 50-m mixed layer; and (right) the CESM1(CAM5) fully coupled model and the CMIP5 MMM.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

As in Fig. 5, but for (left) the simple model with parameters ^{−2} K^{−1} = (4 months)^{−1} for a 50-m mixed layer; and (right) the CESM1(CAM5) fully coupled model and the CMIP5 MMM.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

Zhang et al. (2016) interpret the LP lagged correlations between *T* and ^{6} while “with 20- or 30-year LF, the multi-model mean correlation between

With the correlation *τ*. The NFM *a* so the symmetric term

Thus, there is nothing so far to rule out the hypothesis that the coupled model SST response is primarily a response to white noise forcing from the atmosphere with additional random forcing from the ocean. The ocean need not do anything systematic and need not be the dominant forcing. However, there is a considerable literature offering arguments that the AMO is forced by ocean circulation. We next review some of that literature with the aid of the simple noise-forced model.

### b. SST changes and surface heat flux

Zhang et al. (2016) studied the relation between *T*, the AMO index. They then looked at these two regressed variables at a 4-yr lead, picked because it is where *T*, they find that in the multimodel mean of a set of CMIP3 coupled models the LP tendency *T* by 4 yr in the SOM case and the coupled case (Figs. 5 and 6, second panel from the top; lag = −4 yr) it is positive, for both the GCM and the NFM. The bottom panels show that at the same lead

As noted by Clement et al. (2016), it is evident from Fig. 1 of Zhang et al. (2016) that in both the SOM and coupled simulations the temperature tendency

In the special case that the ocean forcing

Equation (15) means that, as long as there is *any* ocean forcing (*T* and *T*. There is nothing physical about this: as discussed in section 2c, it is a mathematical consequence of the fact that the autocorrelation of *T*—or *any* variable—is an even function of lag with a maximum at zero lag. So *T*. This was shown in Fig. 1 of Clement et al. (2016) and is evident in Fig. 6 here, for both the coupled model and the NFM. The arbitrary choice of lead or lag dictates the interpretation.

Neither interpretation is justified. The low-frequency relations say there is an approximate balance between heating from the atmosphere and heating from the ocean, a near equilibrium. One cannot infer causality from the sign of the correlations; they tell what the balance is at equilibrium but not what was responsible for creating the equilibrium state. Causality is no more revealed by this balance than geostrophic balance tells whether the state was achieved by the pressure adjusting to the winds or the winds adjusting to the pressure. The NFM case in Fig. 6 is an illustration. The atmospheric forcing is 6 times the ocean forcing, and yet the LP correlation between temperature and surface heating is negative at and near zero lag. The correlation remains negative even when the forcing is overwhelmingly from the surface (e.g., 95% atmosphere and only 5% ocean).

The atmosphere-only (SOM) case is singular in that the surface heat flux *T* leads, and positive when *T* lags, unlike the situation when the ocean flux is nonzero. Without question, the oceans do something in both the real coupled system and the coupled models. The issue is whether or not that something is essential for the AMO. A diagnosis of the quasi-equilibrium state does not speak to this issue.

In an important and influential paper, Gulev et al. (2013) examined the relation between SST and surface heat flux in observational data. Their intent was to examine the Bjerknes (1964) hypothesis that at short time scales the atmosphere drives the ocean, whereas at multidecadal time scales the ocean is the driver. The “surface heat flux” data they use are an estimate of the turbulent fluxes only; they omit the radiative components of atmospheric fluxes, such as variations in cloud cover (Bellomo et al. 2016). After the data were low passed by applying an 11-yr running mean, they found that the correlation at zero lag between *T* and ^{7} They interpret this to mean that the ocean is forcing the atmosphere at the low frequencies passed by the filter. In contrast, when the data are high-pass filtered, they find that the correlation is positive, which is interpreted to mean that the atmosphere is forcing the ocean. They conclude that this is evidence in support of the Bjerknes hypothesis.

We have already seen that the sign of the correlation of *T* and *T* and

The unfiltered [blue curve; Eq. (A7c)] and low-pass filtered [magenta curve; Eq.(11c)] correlation

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

The unfiltered [blue curve; Eq. (A7c)] and low-pass filtered [magenta curve; Eq.(11c)] correlation

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

The unfiltered [blue curve; Eq. (A7c)] and low-pass filtered [magenta curve; Eq.(11c)] correlation

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

We may make further use of the low-pass relation [Eq. (11c)] shown in Fig. 7 to estimate the relative magnitudes of the ocean and atmospheric forcing. O’Reilly et al. (2016) found that the low-pass correlation

Figure 8a shows the range of values of the correlation ^{8}

(a) Low-pass correlation of surface heat flux

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(a) Low-pass correlation of surface heat flux

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(a) Low-pass correlation of surface heat flux

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Figure 8a shows distributions of values in box-and-whisker format. The distributions were created from all subsamples 140 years long, the approximate length of the observational record and one allowing at least a few tens of samples for all models. The spreads are quite large, but even accounting for sampling issues the correlations are almost always negative, consistent with Eq. (15). Again, while this further demonstrates consistency with the white noise forcing hypothesis, it does not rule out other possible explanations. Moreover, even if the NFM assumptions (including

The high-pass data are computed as the difference of the unfiltered and the low-pass data. High-pass covariances are approximately equal to the unfiltered covariance for the NFM since the LP covariance is typically lower order *T*. This choice is consistent with the understanding that at short time scales the heating is driving the temperature and also with the explicit time differencing used in models. It is the choice that agrees with Gulev et al. (2013) in finding a positive correlation, but here the correlation is positive only if

Our need to choose one of the several disparate values of the unfiltered correlation at and near zero lag is a sign that the high-pass correlation is problematic. In a valuable study, O’ Reilly et al. (2016) extended Gulev et al. (2013) by calculating the same quantities in CMIP models. They found the same LP correlation between *T* and turbulent surface fluxes in observationally based estimates and in models but remark on the inability of the multimodel mean to reproduce the positive relation that they and Gulev et al. (2013) find in the fluxes estimated from observations. Cayan (1992) studied the relation between turbulent fluxes and temperature tendency and found that, for unfiltered monthly data in midlatitudes, the relation typically shows the atmosphere to be heating the ocean. Perhaps this is closer to what Bjerknes (1964) had in mind than the relation of fluxes to temperature. Figure 6 shows that we find the same positive relation between *T* and surface heat fluxes in reanalyses and CMIP3 models. (Their

^{9}Equation (15) tells us immediately that adding

*F*as ocean forcing strengthens the conclusion that at zero lag

*T*is a maximum at a quarter-wavelength lead (15 yr). In contrast to the noise-only case (dotted line), the locations of the extrema do not change if the cutoff of the LP filter changes. The third panel shows that the correlation of

*T*at zero lag is negative, as expected from Eq. (15) and the appendix.

Correlations in the simple NFM with a periodic forcing added. Equation (16) with

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Correlations in the simple NFM with a periodic forcing added. Equation (16) with

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Correlations in the simple NFM with a periodic forcing added. Equation (16) with

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0810.1

The more interesting case where the periodic forcing is added to the atmosphere—to the surface heat flux—is also illustrated in Fig. 9. The top two panels are the same as for the ocean forcing; the temperature does not care where the heat comes from and carries no telltale saying “atmosphere” or “ocean.” In this simple model, the difference between the two cases lies only in our bookkeeping, in whether the periodic forcing is counted as part of the surface heat flux or the ocean heat flux. The bottom panel of Fig. 9 shows that the LP correlation between *T* and

However, the correlation is more negative when the periodic signal is in the ocean rather than the atmosphere. The context here is a very long time series generated by a model with a simple structure. In reality and in complex models, the low-frequency signal (if any) is not simply periodic, the other forcings are not as simple as those in our NFM, and the time series are short. There is only a distant prospect of using this difference in correlation strength to detect the source of the signal in a more complex setting with many processes and a time series short compared to the signal periods.

O’Reilly et al. (2016) introduced a simple model much like ours, but the only ocean forcing they consider is a periodic low-frequency signal; that is, their model is our Eq. (16) with *any* ocean forcing, including white noise, is sufficient to make the LP correlation of surface flux and temperature negative when the flux leads. The conclusion that “these relationships … rely crucially on low-frequency forcing of SST” is too specific (O’Reilly et al. 2016, p. 2810). All that is needed from the ocean is a modicum of white noise.

## 4. Summary and discussion

We have used the simple damped noise-forced model (NFM) of Eq. (1) to help us interpret some results found in the literature that are often cited as evidence that the ocean is driving the AMO. In addition to a damping term that is linearly proportional to temperature, this model’s temperature tendency is driven by white noise forcings in the atmosphere and ocean that are independent of one another. The analytic solutions we derived for both unfiltered and low-pass filtered correlations (e.g., of temperature and temperature tendency and of heat fluxes and temperature) provide some generic results that hold for all low-pass filters. We compare the correlations in this model to those for preindustrial (i.e., constant external forcing) runs of a coupled GCM [CESM1(CAM5)], those for the CMIP5 MMM, and for an atmospheric model (CAM5) coupled to a slab ocean model (SOM).

We find that the unfiltered correlations fall off rapidly since the *e*-folding time is just the damping time, which is less than a year. This is physical, but after applying an LP filter, the structure depends on the filter, not the physics. The low-pass correlations for this model depend on the autocorrelation function of the LP filter, and their lead–lag structures depend on the cutoff period used in the filter. For example, if the peak value of the LP correlation between *T* is found at 4 yr for a 10-yr filter cutoff, then it will be at 8 yr for a 20-yr filter and 12 yr for a 30-yr filter (see Fig. 4). While the location of the peak does vary somewhat with the type of filter (e.g., Butterworth, Hamming, and running mean), the linear variation of the peak and other properties with the period of the filter is a general result for all low-pass filters. (Judging by the many papers that do not specify what type of filter is used, our literature seems to have taken to heart the notion that the type of filter does not matter.)

We find that the autocorrelation in the coupled and SOM models of temperature for the *T* in the CESM1 and the SOM are also indistinguishable from the NFM (Figs. 5, 6). We also note that Zhang et al. (2016) report that, for this correlation in the multimodel mean of 10 CMIP3 models, the peak correlation moves as in the NFM, a behavior that distinguishes the response to white noise forcing from the response to a true low-frequency forcing. A caveat is that not all CMIP models have unfiltered correlations in such good agreement with the NFM. Some do not exhibit the simple exponential decay in the autocorrelation of

We have seen that, when the underlying record is white or red noise, the LP filter will create long-period lead–lag structures constructed from the autocovariance of the filter and its derivatives. It would be good practice to check and see if there is a distinct low-frequency signal before applying a low-pass filter. The obvious check is to see if the spectrum is statistically indistinguishable from red or white noise. Another is to see if there is the tendency for the peak lag correlation (e.g., between *T*) or the lag at which a correlation crosses zero to move when the filter cutoff period is changed. It would be generally useful to try different filter time scales to see which features are robust. Another caution is that low-pass filters greatly reduce the effective sample size, so the resulting correlation estimates may be quite uncertain, as shown in Fig. 8.

A negative LP correlation between sea surface temperature *T* and surface heat flux into the ocean *T*, it must be the ocean that is driving the AMO variations. This interpretation has been applied to observations (Gulev et al. 2013; O’Reilly et al. 2016) and models (O’Reilly et al. 2016; Zhang et al. 2016; Drews and Greatbatch 2016). Here we show that this negative correlation is a necessary consequence of the surface heat balance at long time scales being in near equilibrium: that is, a state where the temperature tendency is small compared to the components of the heat fluxes that individually would induce large temperature changes. It is dictated by the negative temperature feedback associated with turbulent surface fluxes acting rapidly to balance the other heat fluxes. It tells us only that the net ocean plus atmosphere heat flux is near zero, but it is uninformative as to how the temperature got to be what it is. We show examples in Figs. 6 and 9 where the temperature is strongly driven by the atmosphere and yet the correlation between *T* and *T* and

It does imply that the ocean does something to SST. If the ocean did absolutely nothing then *T* or ^{−2} K^{−1} it follows from Eq. (12) that the ocean heating variance would have to be

Of course, there is no doubt that there is an ocean in both reality and coupled models. Here, we are saying that 1) the sign of this correlation cannot tell us whether the ocean or the atmosphere is driving the AMO; and 2) the lead–lag correlations of

Our thinking is along the lines of the paradigm in Barsugli and Battisti (1998): the AMO pattern is a response to the atmospheric forcing that least damps that forcing. The surface heat exchange between ocean and atmosphere adjusts to what the ocean imposes in order to create or maintain that pattern. In some regions, the ocean heat convergence is helpful, in others harmful, but the surface heat exchange adjusts to do what is needed. We do not prove this hypothesis in this paper but mention it here as a suggestion of how the atmospheric driving of the AMO can carry the day in the context of an active ocean circulation.

While there is a traditional idea that the ocean is most active at long periods, recent observations show clearly that the ocean has power at all frequencies. Figure 2 shows that the ocean forcing in the preindustrial CGCM is indistinguishable from white noise. We do not have a long enough observational dataset to know if this is true of the real ocean, but recent observations from the Atlantic, such as those from the Rapid Climate Change programme (RAPID) support the idea that the ocean circulation has considerable power at all observed frequencies (e.g., Lozier 2012; McCarthy et al. 2015; Zhao and Johns 2014). At present, we have no direct observational evidence to say whether or not the heat flux convergence in the North Atlantic Ocean has more power at multidecadal time scales than expected from white noise. As noted above, however, the instrumental record of SST does exhibit greater power in the unfiltered AMO (or

Understanding the AMO will be greatly helped by analysis of mechanisms in observations and coupled models, as recommended by Zhang et al. (2016). One of the few examples in the literature is Fig. 7 of Buckley and Marshall (2016), a heat budget analysis based on an analysis of the ECCO v4 ocean state estimate that is limited by being only a few decades long. It shows that over most of the North Atlantic the variance of ocean heat convergences is small compared to the local atmospheric heating, but the opposite is true along the western boundary, where the Gulf Stream turns offshore, and in parts of the subpolar gyre around Greenland and Iceland in particular. Overall, this is consistent with the idea that the SST is largely driven by the atmosphere, but it challenges us to accommodate the regions where ocean heat convergence is dominant.

The unanswered question revised by our work here is “just what is the role of the ocean in the AMO?” Unquestionably, the ocean is active. It would help to clarify what is meant by the claim that “the ocean drives the AMO.” Does it mean that ocean physics amplifies atmospheric forcing, as suggested, for example, by Delworth et al. (2016)? Would a time varying atmospheric forcing produce a stronger AMO than the same run with a slab ocean? Clement et al. (2016) indicate otherwise. Does it mean that the ocean does not amplify atmospheric noise such as the NAO but does amplify low-frequency external forcings, for example, from aerosols and greenhouse gases? Does “the ocean drives the AMO” mean that variability entirely internal to the ocean does it? At the extreme, does an ocean model generate an AMO if coupled to a climatological atmosphere, a setup that would exclude atmospheric variability? To our knowledge, no such experiment has been done with the recent generations of models. If so, we would have to conclude that the ocean alone is sufficient for the AMO, just as we concluded in C15 that the atmosphere alone is sufficient. There may be more than one way to generate the AMO. We are unaware of any recent advocacy for the proposal that the ocean alone is sufficient, though much of the thinking about Atlantic variability stems from the seminal box model papers of Stommel (1961) and Rooth (1982) and the early coupled models in which regular multidecadal oscillations were attributed to the ocean circulation (e.g., Delworth et al. 1993).

It is well to remember that the models are all imperfect, particularly in the North Atlantic, where the mean state in models has a cold bias (Wang et al. 2014). In a higher-resolution coupled model, Siqueria and Kirtman (2016) showed that interactions with the mean state produced decadal time scale variability in the North Atlantic that is absent in a version of that model with a lower resolution, one comparable to the CMIP models presented in all prior work cited here. Recently, Drews and Greatbatch (2016) showed that these surface flux diagnostics were different in a model with a corrected mean state, suggesting that improvements to the model may change the influence of the ocean. However, in that study the only evidence for the change in behavior was in the correlation of surface fluxes with temperature. As we have shown here, the correct interpretation of that diagnostic is that the ocean is doing something, but not that it is important to the simulation of the climate variability as represented by the surface temperature. A simpler and more straightforward test that an ocean circulation is essential for the model’s surface temperature response is to obtain a different SST when the active ocean is removed and replaced by a slab ocean.

## Acknowledgments

We are grateful for fruitful conversations with Yochanan Kushnir, Mingfang Ting, Claude Frankignoul, Rong Zhang, Carl Wunsch, and Michael Tippett. We thank Laure Zanna and three anonymous reviewers for helpful reviews. MAC is supported by the Office of Naval Research under the Research Grant MURI (N00014-12-1-0911) and by the National Science Foundation Grant OCE-1657209, ACC and LNM are supported by National Science Foundation Grant NSF-1304540, and KB is supported by a cooperative agreement between NASA and Columbia University (NNX15AJ05A).

## APPENDIX

### Mathematical Derivations

#### a. Unfiltered covariances and correlations

^{10}In the formulas below sgn,

*H*, and

*δ*are the standard sign, Heaviside, and impulse (Dirac delta) functions, respectively. Only a few require the inversion of the Fourier integrals:

*ε*and solving Eq. (5) accordingly (as in Miller and Cane 1989). However, this greatly complicates the calculation, and we are addicted to the efficacy of the differential calculus. Instead, it is sufficient to average the solutions over a (small) time interval

*ε*and absent a

*δ*function this averaging makes little difference:

*δ*function (which is nonzero only at

*δ*function dominates, and, with

*ε*, but it is effectively chosen by the model we are trying to emulate. If we have model output for each time step

#### b. Covariances after low-pass linear filtering

*R*is an even function so

#### c. If and covary

*ζ*’s are white in time-independent random numbers with zero mean and unit variance:

*a*and

*b*, the unfiltered covariances look exactly the same as Eqs. (A2) and (A3), but in Eq. (A3e) the last term is now

*b*is replaced by

*a*and

*b*are better measures of the atmosphere and ocean contributions. As with

*b*if

We have limited discussion to the case where

#### d. Low-frequency periodic forcing

When the periodic forcing is in the ocean, then at

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^{1}

In Peings et al. (2016), 2 of the 23 PI CMIP5 models do have significant multidecadal peaks. These models may have important ocean circulation features, though in the context of examining so many models and so many spectral bands without an a priori hypothesis, the significance of these peaks may be questioned.

^{2}

For a fuller set of references, see the review by Buckley and Marshall (2016).

^{3}

O’Reilly et al. (2016) used a region with the same limits, except that theirs goes to 60°N; we stop at 55°N to reduce the confounding influence of sea ice in winter. Gulev et al. (2013) used the average over the region east–west across the Atlantic from 35° to 50°N.

^{5}

We do not digress to discuss what happens if *R* or

^{6}

The type of filter used is not stated, but the analysis in section 2c is robust to filter type as long as it is low pass.

^{7}

Gulev et al’s (2013) sign convention is the opposite of ours in that a positive heat flux is out of the ocean. In what follows, we describe their results using our sign convention: heat flux is positive into the ocean.

^{8}

In other words, a look at the documentation for these differences did not bring enlightenment to us.

^{9}

This is not quite true for a running mean filter, which alters the signal even at frequencies deep into the passband.

^{10}

Frankignoul et al. (1998) give a different derivation of some of these results.