## Abstract

It is commonly assumed that a reasonable estimate of the SST-forced component of the observed atmospheric circulation is given by an atmospheric GCM (AGCM) forced with the observed SST. However, there are results that find different SST-forced responses from the observed, for example for the ENSO–monsoon relationship, and suggest that these differences are due to lack of coupling to the ocean rather than atmospheric model bias unrelated to coupling. Here, the coupling issue is isolated and examined through perfect model experiments. A coupled atmosphere–ocean GCM (CGCM) simulation and an AGCM simulation forced by the SST from the CGCM are compared to examine whether the SST-forced responses are the same. This question cannot be addressed directly, since the SST‐forced response of the CGCM is a priori unknown. Therefore, two indirect tests are applied, based on the assumption that the noise decorrelation time scale is short compared to a month.

The first test is to compare the time-lagged linear regressions of the atmospheric fields onto several SST indices (defined as the area-averaged SST anomalies in the tropics or extratropics), with SST leading the atmosphere by a month. The second test is to compare the time lagged linear covariances of several atmospheric indices (including two monsoon indices and a North Atlantic Oscillation index) and SST, with the SST leading the atmosphere by a month. Both tests find that the SST-forced responses are the same in the CGCM and SST-forced AGCM. These tests can be extended to compare the SST-forced responses between different AGCMs, CGCMs, and observations.

## 1. Introduction

Atmospheric GCMs (AGCMs) forced by the observed SSTs are often used to estimate the SST-forced response of the observed atmosphere. However, some studies suggest that this approach is intrinsically flawed. Kumar et al. (2005) found that an AGCM forced by observed SST disagreed with the observed relationship between SST and Indian monsoon rainfall (IMR); however, the observed relationship was reproduced if the AGCM was coupled with a mixed-layer ocean model, except with SST specified in the tropical Pacific. Copsey et al. (2006) showed that the observed relationship between SST trend and sea level pressure (SLP) trend in the Indian Ocean was reversed in an AGCM simulation forced by the observed SST. Meng et al. (2012) found a similar result fromforcing an AGCM with observed SST. These results suggest that the forced response to observed SST in the AGCM is different from the SST-forced response in the observations.

Perfect model experiments, in which the AGCM is forced with SST from the coupled GCM (CGCM) instead of from observations, allow testing the hypothesis in a situation where it is not necessary to consider the effects of model bias and observational uncertainties. Wu and Kirtman (2004) found that the regressions of atmospheric fields onto the IMR index were opposite in sign in the CGCM and SST-forced AGCM. Wu et al. (2006) showed that the local correlations of the monthly latent heat flux and SST or SST tendency were different between the CGCM and AGCM in the subtropics and midlatitudes. These results may indicate that the SST-forced responses are different in perfect model experiments. Gallimore (1995) also reached a similar conclusion. On the other hand, Compo and Sardeshmukh (2009) used a set of linear anomaly equations to represent the coupled and uncoupled systems, and found that the ensemble mean atmospheric responses to prescribed SSTs in the uncoupled system were consistent with those in the coupled system. Meng et al. (2012) found no inconsistency in the SLP trends in a CGCM simulation made with no external forcing and an AGCM forced by the SST from the CGCM.

The SST-forced response from the CGCM or from the AGCM cannot be found directly, as weather noise (the internal variability of the atmosphere that is not SST forced) cannot be eliminated from the individual model simulations. The methodology of extracting the SST-forced response as the ensemble mean of an Atmospheric Model Intercomparison Project (AMIP)-like AGCM ensemble has been employed by many investigators (Harzallah and Sadourny 1995; Saravanan 1998; Frankignoul 1999; Schneider and Fan 2007; Hurrell et al. 2009; Fan and Schneider 2012; Schneider and Fan 2012). The interactive ensemble coupling procedure (Kirtman and Shukla 2002; Schneider and Fan 2007; Kirtman et al. 2009, 2011; Fan and Schneider 2012; Schneider and Fan 2012) is relevant to understanding the climate variability only if the SST-forced responses in the CGCM and AGCM are the same.

Frankignoul (1985) showed that lagged covariance was a useful tool to distinguish between cause and effect. If the atmosphere acts as noise forcing on the SST and the SST has no feedback interaction with atmosphere, then the lagged covariance between SST and atmospheric variables is negligible when SST leads by more than the atmospheric persistence time, but not when the atmosphere leads the SST. Using a singular value decomposition of the covariance matrix between SST and an atmospheric field from observations, Czaja and Frankignoul (1999) found that significant anomalies of the atmospheric circulation were related to the previous SST anomalies in the North Atlantic, reaching maximum when SST led the atmosphere by 3–5 months.

Using a linear model, Barsugli and Battisti (1998) showed that an essential difference between coupled and uncoupled models was that the stochastic forcing of the atmosphere forced SST in the coupled model when the atmosphere led the ocean, but there was no relationship between the stochastic forcing and the SST in the uncoupled model. This leads to differences between the coupled and uncoupled models in the lagged regressions between the atmosphere and SST when the atmosphere leads the ocean, but not when the ocean leads the atmosphere beyond the atmospheric noise decorrelation time.

Recently, Chen et al. (2013) found that the statistics of the atmospheric weather noise were indistinguishable between the CGCM and the AGCM forced by the SST from the CGCM (simply AGCM in the following). The SST-forced response was determined as the ensemble mean of an ensemble of AGCMs forced by the same SST (e.g., AMIP-type simulations; Gates et al. 1999), and this response was assumed to be the same in the CGCM and AGCM. There was no restriction for the SST-forced response to be linear. The results are consistent with the interference mechanism due to weather noise forcing of SST operating in the CGCM but not the AGCM (e.g., Hasselmann 1976; Barsugli and Battisti 1998), only if the SST-forced response is taken to be the same in both models. Colfescu et al. (2013) reexamined Copsey et al. (2006) in perfect model simulations with time-varying external forcing, and found that the inability of the AGCM, forced by observed SST and external forcing, to reproduce the observed SLP trends in the Indian Ocean could be due to model bias unrelated to coupling rather than to lack of coupling.

The motivation of the current study is to test whether the SST-forced responses are the same or different in the CGCM and AGCM. A test was described in Chen et al. (2013), based only on simultaneous regressions of an atmospheric index with other fields. In that test, the CGCM and AGCM would have the same forced response if the difference in the regressions was small and the noise was spatially uncorrelated at sufficiently large horizontal scales. However, as the internal variability in the atmosphere is well known to have very large-scale spatial structures, this is not a particularly useful test. Here we apply potentially more useful tests using time-lagged regressions and covariances, extending the arguments of Frankignoul and Hasselmann (1977), Frankignoul (1985), and Czaja and Frankignoul (1999). In these tests, the CGCM and AGCM are found to have the same SST-forced response if the difference in the test quantity is not different from zero and the noise is assumed to behave like multivariate red noise (e.g., Newman et al. 2011) with decorrelation time scale short compared to a month. Note that these tests *do not* involve explicit calculation of the SST-forced responses.

## 2. Models and experiments

The CGCM and AGCM used in the study are the same as in Chen et al. (2013). Thus, the models and experiments are only briefly described in this section.

### a. Models

The CGCM is the Community Climate System Model, version 3 (CCSM3; Collins et al. 2006a), from the National Center for Atmospheric Research. It couples the atmosphere, ocean, land, and ice models through a flux coupler without flux correction. We choose the version “T42_gx1v3” consisting of a spectral atmosphere truncated at total wavenumber 42 with 26 levels in the vertical, and 1° horizontal resolution in the ocean and ice models.

The AGCM is the stand-alone Community Atmosphere Model, version 3 (CAM3; Collins et al. 2006b), the same as the atmospheric component in CCSM3. The version used in this study has the same horizontal and vertical resolutions as the CGCM. We use the stand-alone version rather than going through the flux coupler in order to include thermodynamic sea ice. Settings have been chosen in order to match the stand-alone CAM3 with the version in CCSM3 [see details in Chen et al. (2013)].

### b. Experiments

First, the CGCM with constant external forcing was integrated for 300 years, and was considered as the control simulation. Then the AGCM, forced by the time-varying daily SSTs from the CGCM, was also integrated for 300 years. One hundred years of daily average output were produced for selected variables from the CGCM and AGCM simulations.

### c. Tests

The lagged regressions of the atmospheric fields onto the SST, with SST leading the atmosphere by much longer than the noise decorrelation time, will be the same in the CGCM and AGCM if their SST-forced responses are the same, as there are no time-varying external forcings on these time scales, such as ozone changes, aerosols, solar effects, and greenhouse gases. In the case of the SST leading the atmosphere by less than the noise decorrelation time, or SST being simultaneous with or lagging the atmosphere, the regressions can differ between CGCM and AGCM even when the SST-forced responses are the same. The statistical tests find whether the SST-forced responses are the same/different between CGCM and AGCM.

Test 1: Compare the time-lagged linear regressions of the atmospheric fields onto several SST indices, with the SST leading the atmosphere by much longer than the noise decorrelation time.

Test 2: Compare the time-lagged linear covariances of several atmospheric indices and the SST field, with the SST leading the atmosphere by much longer than the noise decorrelation time.

The tests described here reinterpret and extend those presented in Wang et al. (2005), which considered the local lag correlations between the SST and rainfall.

#### 1) Test 1

As discussed in Chen et al. (2013), we decompose an atmospheric variable (*V*) into the part due to SST forcing (i.e., the nonlinear SST-forced response) and the part due to the internal variability that is not SST-forced (i.e., weather noise), that is,

where the superscripts *F* and *N* denote the SST-forced response and weather noise, respectively.

There is also oceanic noise in the CGCM simulation (e.g., Schneider and Fan 2007; Compo and Sardeshmukh 2009), as well as coupled variability (e.g., ENSO); however, since the time-varying SST used as the boundary forcing for the AGCM is the same as the CGCM SST, this study does not explicitly consider the role of oceanic noise. Then regression of an atmospheric field (*V*) onto an SST index (SSTI) is

where Reg is regression, Cov is covariance, and Var is variance.

The main assumption in our test is that there exists a time scale (*τ*_{N}) such that when the SST index leads the atmosphere by much longer than *τ*_{N}, the SST index and the noise are uncorrelated, that is,

where *t* and *τ* denote the time and time lag, and *τ*_{N} denotes the noise decorrelation time. A model of the noise that satisfies our decorrelation assumption is multivariate red noise (e.g., Newman et al. 2011). Equation (3) is taken to be valid in both the CGCM and AGCM. Then Eq. (2) can be written as

and the CGCM–AGCM difference of lagged regressions onto the SST index is

where the superscripts *C* and *A* denote CGCM and AGCM, respectively. If the difference on the lhs of Eq. (5) is zero (nonzero) when SST leads the atmosphere by much longer than *τ*_{N}, then the test indicates that the SST-forced responses are the same (different) in the CGCM and AGCM. Note that we are not relating the structure of the lagged regression to that of the SST-forced response.

When the SST index leads the atmosphere by less than *τ*_{N}, or when the SST index is simultaneous with or lags the atmosphere, the covariance of the SST index and noise is zero in the AGCM since forcing is in only one direction, from ocean to atmosphere; however, the covariance is nonzero in the CGCM when the atmospheric noise forces the ocean. In this case, the CGCM–AGCM difference of lagged regressions onto the SST index is

Hence, comparison of the lagged regressions between the CGCM and AGCM in this case, includes a contribution from the weather noise leading the SST index in the CGCM but not the AGCM. We interpret this relationship in the following as being due to weather noise forcing of the SST in the CGCM but not the AGCM, following Hasselmann (1976), Frankignoul (1985), and Barsugli and Battisti (1998). If the SST index is *not* forced by the atmospheric noise (i.e., is due to oceanic noise or coupled variability with time scale longer than a month), then the difference of regressions from Eq. (6) will be small.

In application of the tests in later sections, we assume *τ*_{N} = 1 month while using monthly mean model output. Figure 1 shows the smooth transition from negative lag to simultaneous to positive lag behavior, and characterizes the time scales over which the autocorrelation of the atmosphere affects the differences in covariances between the coupled and uncoupled models. The autocorrelation functions for the SST and the surface net heat flux (NHF) in a box region of 0°–60°N, 80°W–0° in the North Atlantic, using the daily average output from the CGCM and AGCM, are shown in Fig. 1a. The SST has very long persistence time (over 1 yr) and the NHF is decorrelated beyond a week. Coupling does not have significant influence on the persistence time of the atmosphere.

Figure 1b shows the lagged linear regression of the NHF (positive downward) onto the SST in the same box region. In the AGCM (blue solid curve), the lagged regression between the daily NHF and the prescribed SST are negative and almost symmetric in lag, indicating that the atmospheric response to the SST forcing is always damping the SST. However, the CGCM (black solid curve) shows strong asymmetry in lag. For negative lags, the atmosphere forces the ocean, peaking at time lag of a week. For positive lags, the CGCM and AGCM differ for lags shorter than a week, corresponding to the NHF decorrelation time scale. For lags much longer than a week, the CGCM lagged regression approaches that of the AGCM, indicating the same SST-forced damping in the CGCM and AGCM. Using a 30-day running mean filter of the daily data, the lagged regression for the AGCM (blue dashed curve) is nearly the same as using the daily average, while the lagged regression for the CGCM (black dashed curve) still shows strong asymmetry in lag, but peaks for the atmosphere leading the ocean by about a month. Using the monthly average output as a filter of the daily data, the lagged regression is almost entirely consistent with the 30-day running mean at each point in the CGCM (black squares), although not agreeing exactly for the AGCM (blue squares). The results are consistent with those expected from a multivariate red noise model with decorrelation time scale of about a week for the heat flux. The use of monthly means appears to be adequate to represent the regressions for lags long compared to the noise decorrelation time.

#### 2) Test 2

Covariance instead of regression is used in this test, since the atmospheric variances differ between the CGCM and AGCM because of interference between the weather noise and the response to the weather noise-forced SST in the CGCM only (Chen et al. 2013).

Following the arguments given in the preceding subsection, when SST leads the atmosphere by much longer than *τ*_{N}, the covariance of SST and the noise component of the atmospheric index are zero. If the SST-forced responses are the same, the CGCM–AGCM difference of covariances will be indistinguishable:

where Idx is an atmospheric index.

When SST leads the atmosphere by less than *τ*_{N}, or when SST is simultaneous with or lags the atmosphere, the CGCM–AGCM difference of covariances is

Differences for these lags indicate regions where the noise part of the atmospheric index forces the SST in the CGCM.

## 3. Results

This section presents the comparison of the SST-forced responses between the CGCM and AGCM. The two tests are carried out using the monthly average output, with the annual cycle removed. We apply the tests simultaneously, and with ocean leading/lagging the atmosphere by one month. In interpreting the results, we assume the noise decorrelation time is short compared to a month.

Significance tests are a *t* test for regressions/covariances and Fisher’s *z* transform for the differences of regressions/covariances. Degrees of freedom (DOF) are estimated based on lag-1 month autocorrelation functions.

### a. Test 1: Lagged regressions of atmospheric fields onto the SST indices

To examine test 1 in both tropical and extratropical regions, we define three SST indices as time series of the area-averaged SST anomalies: the Niño-3.4 index (5°S–5°N, 170°–120°W) representing the tropics, the Atlantic multidecadal variability index (AMV; 0°–60°N, 80°W–0°) encompassing both tropics and extratropics, and the North Pacific variability index (NPV; 30°–50°N, 120°E–120°W) confined to the extratropics. The time evolution and spatial patterns of these indices are displayed in Fig. 2. There are large SST anomalies over the central and eastern equatorial Pacific corresponding to the Niño-3.4 index (Fig. 2a), which has obvious interannual variability with a dominant period of 2 yr (Fig. 2b), the period of ENSO in CCSM3 (Collins et al. 2006a). Figure 2c shows large SST anomalies over the North Atlantic corresponding to the AMV index, which has both interannual and decadal variability (Fig. 2d). Corresponding to the NPV index, there are positive SST anomalies over the midlatitude North Pacific and negative SST anomalies along the western coast of North America and the tropical eastern Pacific/North Atlantic (Fig. 2e), which also has large decadal and multidecadal variability (Fig. 2f).

The lagged linear regressions of the monthly NHF anomalies onto the Niño-3.4 index are displayed in Fig. 3. When SST leads the atmosphere by a month, the NHF damps SST in the tropical Pacific in both the CGCM (Fig. 3a) and AGCM (Fig. 3d), and the differences are not significantly different from zero (Fig. 3g), indicating from Eq. (5) that the SST-forced responses of the NHF are indistinguishable between the CGCM and AGCM. When SST is simultaneous with or lags the atmosphere, the regression patterns are similar with the SST leading case (Figs. 3b,c,e,f) and the differences are also not significantly different from zero (Figs. 3h,i), suggesting from Eq. (6) that the SST over the Niño-3.4 region is not primarily forced by the atmospheric noise heat flux. Note that we are not ruling out that the atmosphere forces the ocean, just that the noise is not playing a large role in mature ENSO events in this model. For the pure coupled variability, the SST-forced signal is the same as the signal forcing the ocean [e.g., as in the Zebiak and Cane (1987) coupled atmosphere–ocean model in which there is no atmospheric noise forcing the ocean and the evolution of ENSO is sensitive to initial conditions], and this SST-forced signal will be reproduced by the SST-forced AGCM simulations. However, there is no conflict with the noise playing a role in ENSO at time scales longer than subseasonal, particularly in the transition between events. Using a linear inverse model driven by Gaussian white noise, Penland and Sardeshmukh (1995) displayed that the stochastic forcing controlled the optimal structure of the ENSO growth. Schneider and Fan (2007) found that realistically irregular Niño-3.4 variability in a CGCM was determined by the noise forcing.

Similar features can be found for the lagged regressions of the SLP anomalies onto the Niño-3.4 index (Fig. 4). The SLP and the Niño-3.4 SST, for the SST being simultaneous with, leading, or lagging the atmosphere by a month, are positively related over the western tropical Pacific and tropical Indian Ocean and negatively related over the eastern tropical Pacific, similar to the Southern Oscillation pattern (Trenberth 1984). The CGCM–AGCM differences of regressions are not different from zero (Figs. 4g,h,i), indicating from Eqs. (5) and (6) that the SST-forced responses of SLP are indistinguishable between the CGCM and AGCM and that the Niño-3.4 SST is not forced by atmospheric noise surface fluxes related to SLP.

Figure 5 shows the lagged regressions of the monthly NHF anomalies onto the AMV index. When SST leads the atmosphere by a month, the NHF damps SST to the east of 30°W (Figs. 5a,d), and the differences are not significantly different from zero (Fig. 5g), indicating from Eq. (5) that the SST-forced responses of the NHF are indistinguishable between the CGCM and AGCM. When SST is simultaneous with or lags the atmosphere by a month, the NHF in the CGCM changes to positively correlated with SST over the North Atlantic (Figs. 5b,c), and the CGCM–AGCM differences are significant over the North Atlantic using the conservative estimate of DOF (black shading in Figs. 5h,i), arising from the noise NHF forcing SST over this region in the CGCM but not the AGCM from Eq. (6).

Figure 6 shows the lagged regressions of the monthly SLP anomalies onto the AMV index. When SST leads the atmosphere by a month, the associated SLP pattern includes low pressure over the index region, and the compensation for this low pressure occurs to the north of Canada and in the tropical Indian Ocean/western and central Pacific (Figs. 6a,d). The SST-forced responses of SLP are indistinguishable between the CGCM and AGCM (Fig. 6g). When SST is simultaneous with or lags the atmosphere by a month, the SLP dipole in the CGCM over the mid- and high-latitude North Atlantic and north of Canada changes sign, comparing Fig. 6c with Fig. 6a. The SST-lagging CGCM–AGCM differences are significant over North Atlantic using the conservative estimate of DOF (black shading in Fig. 6i) and resemble the NAO pattern, with a positive anomaly in midlatitudes and negative anomalies in high latitudes. The SST-lagging CGCM–AGCM differences in the tropical Indian Ocean/western and central Pacific are not significant. These results are consistent with the SST over the North Atlantic being forced by the SLP-related noise, but the low-latitude SLP being forced by the AMV SST.

Figures 7 and 8 show the lagged regressions of the monthly NHF and SLP anomalies onto the NPV index. Similar to the results using the AMV index, when SST leads the atmosphere by a month (Figs. 7a,d), the NHF is opposite in sign to the SST pattern in Fig. 2e, and the differences are not significantly different from zero (Fig. 7g), indicating that the SST-forced responses of the NHF are indistinguishable between the CGCM and AGCM. The SLP response also satisfies test 1 (Figs. 8a,d,g). When SST is simultaneous with or lags the atmosphere by a month, the NHF in the CGCM changes sign in mid- and high-latitude North Pacific (Figs. 7b,c), and the CGCM–AGCM differences are significant over the extratropical North Pacific using the conservative estimate of DOF (black shading in Fig. 7i), arising from the noise NHF forcing SST over this region in the CGCM but not the AGCM. The SLP for these lags differ in the North Pacific and polar region (black shading in Fig. 8i), indicating that the atmospheric noise related to SLP forces the NPV SST.

Therefore, test 1, lagged linear regressions of the atmospheric fields onto the SST indices with the SST leading the atmosphere by a month, finds that the SST-forced responses are the same in the CGCM and AGCM.

### b. Test 2: Lagged covariances of the atmospheric indices and SST

Test 2 is illustrated by taking the atmospheric indices (Idx) as the monsoon rainfall indices and NAO index. These indices are chosen to highlight both tropical and extratropical regions.

The lagged covariances of the June–August (JJA) mean IMR (area averaged over 5°–20°N, 70°–100°E) and May surface temperature anomalies are strong in both the CGCM (Fig. 9a) and AGCM (Fig. 9b) in the northern Indian Ocean, central North Pacific and the tropical Pacific. The differences are not significantly different from zero (Fig. 9c), indicating from Eq. (7) that the CGCM and AGCM have the same SST-forced IMR variability.

Figures 9d and 9e display the simultaneous covariances of the IMR and surface temperature anomalies in the CGCM and AGCM, in which enhanced IMR is associated with a La Niña–like SST pattern. As pointed out in Chen et al. (2013), this association is opposite to that found by Kumar et al. (2005) using the same AGCM forced by observed SST. The CGCM–AGCM differences over ocean are significant and primarily over the tropical and subtropical western Pacific and equatorial central Pacific (Fig. 9f). We interpret this as due to noise forcing of the SST in the CGCM from Eq. (8).

It can be seen that IMR is also related to the surface temperature over land for the simultaneous case (Figs. 9d,e). Although the atmosphere is not coupled with ocean in the AGCM, it is strongly coupled with land. Hence, the land forcing in the AGCM is different from the CGCM, but we are not able to provide an interpretation of the land differences at this time. An ensemble of AGCMs with specified land surface state variables is needed to find the forced response over land, which is beyond the scope of this work.

Figure 10 is plotted in the same way as Fig. 9, with the IMR index replaced by the northwest Pacific monsoon rainfall index (NWP, defined as JJA mean rainfall area-averaged over 10°–20°N, 110°E–180°). The test indicates that the SST-forced component of the NWP index is the same in the CGCM and AGCM (Figs. 10a–c). When SST is simultaneous with the NWP index, CGCM–AGCM differences are due to the atmospheric noise forcing the SST.

Finally we examine an NAO index, an important atmospheric index in the mid- and high latitudes, defined as the December–February (DJF) mean SLP area-averaged over a northern box (65°–72°N, 15°–45°W) minus a southern box (40°–50°N, 0°–30°W). The DJF NAO is not related to a significant SST pattern in November (Figs. 11a,b), and the CGCM–AGCM differences are not significantly different from zero except in several scattered regions (Fig. 11c), indicating that the SST-forced responses of the NAO index are indistinguishable between the CGCM and AGCM. When SST and the atmosphere are simultaneous (Figs. 11d–f), significant differences are found over the North Atlantic and the tropical and subtropical western Pacific, arising from the forcing of SST by the noise component of the NAO in the CGCM. Note that there are also large correlations between the NAO and the surface temperature over land. As explained earlier, since the atmosphere is strongly coupled with land in the AGCM, the land forcing in the AGCM is different from the CGCM.

Taking the above results together—test 2, lagged linear covariances of the atmospheric indices and SST, with the SST leading the atmosphere by a month—also finds that the SST-forced responses are the same in the CGCM and AGCM.

## 4. Conclusions

In Chen et al. (2013), we *assumed* that the SST-forced responses were the same in the CGCM and the AGCM forced with SST from the CGCM. It was found that the weather noise statistics were indistinguishable between the CGCM and AGCM. In this paper we *tested* whether the SST-forced responses were the same in the CGCM and AGCM. The origin of the SST variability was not addressed explicitly in these studies, as the SST was provided by a single CGCM simulation.

Because of the existence of weather noise, the SST-forced response from the CGCM or from the AGCM cannot be found directly. Instead, we compared the forced response indirectly by two tests, based on the assumption that the noise decorrelation time was short compared to a month. Neither test found differences of the SST-forced responses between the CGCM and AGCM. The first test was time-lagged linear regressions of the atmospheric fields (NHF and SLP) onto several SST indices including the Niño-3.4 index, the AMV index, and the NPV index. When SST led the atmosphere by a month, the CGCM–AGCM differences were not significantly different from zero, indicating that the SST-forced responses were indistinguishable between the CGCM and AGCM. When SST was simultaneous with or lagged the atmosphere, differences were attributed to the atmospheric noise forcing of SST in the CGCM but not the AGCM. In regions where CGCM–AGCM differences were not found with the atmosphere leading the ocean, the interpretation was that the noise component of the atmospheric field was small. The second test was time-lagged linear covariances of the atmospheric indices (two monsoon rainfall indices and an NAO index) and SST. We found that the SST-forced responses of these indices were indistinguishable between the CGCM and AGCM, and the differences were attributed to forcing of the SST by the noise component of the atmospheric indices.

We return now to the apparent inconsistencies between coupled and uncoupled models mentioned in the introduction. The results of Kumar et al. (2005) from forcing the AGCM with observed SST can probably be attributed to model bias unrelated to coupling, as we did not find this discrepancy using the same AGCM forced with the CGCM SST. This conclusion is consistent with the diagnosis of the Copsey et al. (2006) results by Colfescu et al. (2013). The results of Gallimore (1995), Wu and Kirtman (2004), and also Kumar et al. (2005) are equivalent to simultaneous regressions, and we have shown that the nonzero differences between CGCM and AGCM in simultaneous case are explained by the atmospheric noise forcing of SST in the CGCM. The simultaneous correlation with SST tendency in Wu et al. (2006), calculated using centered finite difference, is close to lagged correlation with SST lagging the atmosphere.

The arguments given in this paper and Chen et al. (2013) are complementary. Chen et al. (2013) was based on the assumption that the SST-forced responses were the same in CGCM and AGCM and inferred the weather noise. The study here was based on the assumption that the noise decorrelation time was short compared to a month, and found, without explicit calculation of the forced response, that the SST-forced responses were the same in CGCM and AGCM. To verify the consistency of the assumptions made in the two papers, we examined the validity of the noise decorrelation assumption of this paper using the explicitly calculated monthly noise described in Chen et al. (2013), and found for several fields that the noise was not correlated on monthly or longer time scales and that the noise spectra for the monthly means were white. Therefore, the assumptions are plausible.

The SST-forced response and weather noise statistics from different AGCMs forced by the same SST can be found and compared using the same technique we have described. Similarly, the weather noise statistics for observations can be estimated assuming that the SST-forced response in the observations is that of an AGCM forced by the observed SST (e.g., Fan and Schneider 2012). It is our conjecture that if a model is realistic, its weather noise statistics should agree with those inferred for the observations, and its SST-forced response should be indistinguishable from that found from observations. Current models can be examined by these tests. The lagged regression analysis using suitably defined indices can also be used to compare the SST-forced responses in different coupled models or between the coupled models and observations.

There are remaining issues that should be addressed. The observed Madden–Julian oscillation (MJO; Madden and Julian 1971), thought to be mostly internal atmospheric variability, has decorrelation time scales longer than a month. The amplitude of the MJO in the model used in this study, CCSM3, is much too weak, and the decorrelation time scale is then unrealistically short. For models that produce realistic simulations of the MJO, the time lag for the tests may need to be increased.

## Acknowledgments

We thank the anonymous reviewers for their constructive and insightful comments on this work. The contributions of Chen and Schneider were supported by NSF Grants ATM-0653123 and AGS-1137902. Schneider was also supported by NSF Grants ATM-0830068 and ATM-0830062, NOAA Grant NA09OAR4310058, and NASA Grant NNX09AN50G. Our work is part of the CLIVAR International Climate of the Twentieth Century Project (C20C). The NCAR CISL provided computer resources for the simulations. Data analyses and plotting were done using GrADS.

### APPENDIX

#### Mathematical Framework for the Regression Analysis

Here, we present a mathematical framework, following Compo and Sardeshmukh (2009), of a set of linear anomaly equations for the coupled and uncoupled systems. Since there is no external forcing considered, the coupled system is adapted as follows, using their notation:

The atmospheric state vector is *y* and the SST state vector is *x*. The atmospheric and oceanic dynamics and interactions are represented by the matrices *L*_{αβ}, the vectors *η*_{α} denote the atmospheric and oceanic stochastic forcing, and the matrices *B*_{α} transform the stochastic forcing into dynamic forcing.

The equation for the uncoupled atmospheric system forced by the time-varying *x* from the coupled system is

Structurally, Eqs. (A3) and (A1) are identical, but the realization of the stochastic forcing differs, although it is taken to have the same statistics. For time scales longer than decorrelation time of the atmosphere, the *d*/*dt* terms on the lhs of Eqs. (A1) and (A3) are much smaller than the other terms on the rhs and can be neglected. The following applies for the magnitude of the time lag much longer than the atmospheric decorrelation time. Then the solution to Eq. (A1) is

where and . Similarly, the solution to Eq. (A3) is

that is, as the sum of an SST-forced component (*Ax*) and a noise component. The linear transformation of the noise vector is also noise vector. The SST-forced component is identical for the coupled and uncoupled systems. Using Eq. (A4), Eq. (A2) can be written as

where and .

The formal solution to Eq. (A6) is

Then the CGCM–AGCM differences at a point of lagged covariances over time *T* of *x* and *y* (*ŷ*) with a time lag of *τ* are

Since the atmospheric noise from the uncoupled system () is uncorrelated with ocean (*x*) and the atmospheric noise from the coupled system (*η*_{y}) is uncorrelated with oceanic noise (*η*_{x}) or with oceanic initial condition (*x*_{0}), Eq. (A7) is reduced to

From the property of stochastic noise, contribution from the integral only occurs when *τ* < *τ*_{atm}, with the atmospheric decorrelation time (*τ*_{atm}) taken into account. The integral is not significantly different from zero when *τ* > *τ*_{atm}. Thus, when ocean leads the atmosphere by longer than the atmospheric decorrelation time (i.e., τ > *τ*_{atm}), the rhs of Eq. (A8) is small; that is, the difference of lagged covariances between coupled and uncoupled systems is small. However, when ocean leads the atmosphere by less than the atmospheric decorelation time (i.e., *τ* < *τ*_{atm}), which includes the ocean being simultaneous with the atmosphere (*τ* = 0) and ocean lagging the atmosphere (*τ* < 0), the difference is not negligible.

If the atmosphere is still modeled as a linear system plus fast noise, but with a nonlinear relationship of SST forcing the atmosphere (i.e., *L*_{yx} is nonlinear but *L*_{yy} is still linear), the solutions to Eqs. (A1) and (A3) can still be written as in the linear case in Eqs. (A4) and (A5). The nonlinear matrix *A* (due to *L*_{yx}) does not affect the CGCM–AGCM differences in Eq. (8), and the conclusions are the same as in preceding paragraph.

To examine the effect of state-dependent noise, such as depending on the SST (e.g., Weng and Neelin 1999; Majda et al. 2009; Sura and Sardeshmukh 2009), the operator *B*_{y} in Eqs. (A1) and (A3) is taken to depend on the *x* (i.e., SST). The solutions to Eqs. (A1) and (A3) are still written as Eqs. (A4) and (A5), but the matrix *C* in this case is *x* dependent. Then the matrix *E* is also *x* dependent. The dependence on *x* of *C* and *E* does not change the conclusions concerning the lag correlations.

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## Footnotes

Current affiliation: Key Laboratory of Meteorological Disaster of Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, China.