We now focus on the climate drivers that contribute to the mean state biases, utilizing the MSEB model as a diagnostic tool. First, we will showcase the capability and limitations of the MSEB model, subsequently we show what elements of the MSEM model lead to biases in \(\omega\), which will then be discussed in more detail for the surface heat fluxes and the advection terms.

**a. Assessment of the MSEB model’s performance**

Figure 3 presents the tropical mean \(\omega\) as diagnosed by the MSEB model. The MSEB model captures the annual mean \(\omega\) distribution for the observed and simulated data fairly well, but it does reveal an upward motion bias in the subtropics and a downward motion anomaly in the deep tropics. A potential explanation for this globally upward motion offset issue could be the absence of the transient eddy flux of MSE in the MSEB model, as discussed in FD21.

The MSEB model also captures mean bias pattern (Fig. 3d and e) with a moderate pattern correlation of 0.6. This suggests that the model does have some skill in representing these biases, but it also has some significant limitations (see the following discussion in Fig. 4). It successfully captures several prominent CMIP bias patterns, including the double-ITCZ, the cold tongue bias over the Pacific, a southward shift of ITCZ over the Atlantic, and a partial IOD-like bias over the IO (Fig. 3d). Concerning the estimation of AMIP bias, the MSEB model captures the weakening of the double-ITCZ and the disappearance of the cold tongue bias over the Pacific, the absence of the southward displaced ITCZ bias over the Atlantic, and the weakening of the bias over the western IO (Fig. 3e). Nevertheless, there are some discrepancies between the CMIP/AMIP bias (Fig. 1d and e) and the MSEB-estimated CMIP/AMIP bias (Fig. 3d and e). For instance, the CMIP bias over the eastern IO and the AMIP bias over the Atlantic.

A Taylor diagram is utilized to quantify the MSEB model's performance (Fig. 4). The MSEB model is able to capture the mean \(\omega\) based on CMIP (AMIP) simulations with a correlation of 0.65 (0.7), as well as reanalysis data with a correlation of 0.65. The pattern standard deviation ratio is slightly underestimated for the observations, whereas the estimates from the MSEB model are overestimated for both CMIP and AMIP simulations.

The mean bias pattern in the CMIP or AMIP case is approximated by the MSEB model with spatial correlations around 0.45 and the pattern standard deviation ratio is overestimates by about 30%. This overestimation is rooted in the underestimation of \(\omega\) in the observational case and the overestimation of \(\omega\) in the CMIP/AMIP cases (Fig. 4a). In combination, the moderate correlation and amplitude overestimation suggest that the MSEB model has some capability in presenting the overall biases in the model simulations, but it also some significant limitations that will limit the outcomes of this study.

While, the overall skill of the MSEB model is limited, it does have better skill in representing the large-scale regional indices, see Fig. 5. The MSEB model effectively captures CMIP mean biases in the Pacific, Atlantic, and IO, yielding correlations of approximately 0.66, 0.73, 0.76, and 0.85, respectively. For the AMIP cases, the MSEB model captures mean biases with correlations of about 0.56, 0.73, 0.43, and 0.9, respectively.

Table 2. Comparison of the bias difference between CMIP (blue) and AMIP (red) and the bias difference between actual pattern and MSEB estimate. RMS (gray) is calculated based on the observed ω pattern of Fig. 1a and 3a. Bias is calculated based on the mean circulation bias pattern of Fig. 1b-c and Fig. 3b-c. The last three rows are the regression results of Fig. 5.

We now quantify the overall strength of the \(\omega\) mean bias in the tropical ocean regions in terms of a root-mean-square (RMS) ratio, see Fig. 6a. The RMS ratio is computed by dividing the tropical ocean’s RMS of the \(\omega\) mean bias by the RMS value of the \(\omega\) mean. This measurement considers both upward and downward motion biases. We further assess the contribution of each principal component of the MSEB model to the \(\omega\) mean bias by computing the RMS value of the MSEB model with only including the bias from the respective component in the model. This will be discussed in more detail in the following sections.

First, we can note that the biases in the CMIP and AMIP simulations are both of similar magnitude and with values of ~ 40% they present a significant change in the mean tropical circulation. Further, we note that the MSEB model overestimates the RMS, but is also similar for both CMIP and AMIP simulations.

**b. Drivers of mean circulation biases**

Figure 7 shows the CMIP and AMIP mean biases in the three major components of the MSEB (\({F}_{net}\), \(Adv\), and \({GMS}_{B}\)). CMIP models tend to simulate excessive \({F}_{net}\) on both sides of the equator in the Pacific, the south Atlantic, and the equatorial IO. Conversely, models display a tendency to simulate deficient \({F}_{net}\) over the equatorial Pacific and subtropical regions. The \({F}_{net}\) biases in the AMIP simulations are of similar magnitudes, but have substantially different patterns.

The CMIP models \(Adv\) bias pattern portrays excessive \(Adv\) over the equatorial Pacific, the northern subtropical Pacific, and the SPCZ (Fig. 7c). In all other tropical ocean regions, models predict insufficient \(Adv\). Upon comparison with the pattern of \(Adv\) bias in AMIP simulations (Fig. 7d), it is apparent that the \(Adv\) bias over the SPCZ and much of subtropical regions are similar to those of the CMIP simulations, indicating that biases these regions are likely attributable to intrinsic errors in AGCMs.

Turning to the \({GMS}_{B}\) in CMIP and AMIP simulations, we find that the bias patterns in \({GMS}_{B}\) is quite different between the two kind of simulations, with stronger magnitudes in CMIP than in AMIP simulations. This indicates the majority of these biases in \({GMS}_{B}\) are due to coupling errors in CGCMs.

We assess the contribution of the biases in each of the three major components of the MSEB model to the \(\omega\) mean bias by only including the bias from the respective component in the MSEB model to approximate the \(\omega\) mean (see method Section 2d). The results are shown in Fig. 8. First, it becomes evident that \({F}_{net}\) and \(Adv\) biases are of similar strength and both exert more impact on \(\omega\) biases compared to \({GMS}_{B}\) biases. This suggests that biases in the \({GMS}_{B}\) are of minor importance.

The \({F}_{net}\) bias are again fairly different in pattern for the CMIP versus the AMIP simulations (pattern correlation of 0.34). However, the contribution of \(Adv\) bias to \(\omega\) mean bias is similar in pattern between CMIP and AMIP simulations (pattern correlation of 0.7). Despite distinct boundary settings in both simulations, the robust correlation in the \(Adv\) bias implies a persistent intrinsic error within atmospheric models.

Figure 6a quantifies the overall strength in the \(\omega\) mean bias resulting from each principal component of the MSEB model in terms of the relative RMS values. It again highlights that the \({F}_{net}\) and \(Adv\) terms emerge as the two primary sources contributing to the \(\omega\) mean bias in CMIP and AMIP simulations, whereas the \({GMS}_{B}\) bias has a weaker impact on the \(\omega\) mean bias. It is noteworthy that the MSEB model’s sensitivities yield similar outcomes for the CMIP and AMIP simulations in almost all sub-components of the model. This implies that the \(\omega\) mean bias in CGCMs is mainly rooted in intrinsic errors within AGCMs, but the coupling to the ocean does change the regional patterns, as we will further explore the analysis in the next sections.

**c. Heat flux biases**

\({F}_{net}\) in the MSEB model comprises \({F}_{top}\) and \({F}_{sur}\), each of which can be further broken down into additional components. Initially, as illustrated in Fig. 6a, the contribution of \({F}_{net}\) bias to \(\omega\) mean bias within the tropics is further subdivided into the contributions of \({F}_{top}\) and \({F}_{sur}\) biases. This reveals that the \({F}_{sur}\) bias is the primary component driving \(\omega\) mean bias. The \({F}_{sur}\) bias can be decomposed into contributions from \({SW}_{sur}\), \({LW}_{sur}\), \(SH\), and \(LH\) (as shown in Fig. S1 and S2)., revealing that \(LH\) and \({SW}_{sur}\) biases are main the contributions to the \(\omega\) mean bias in both CMIP and AMIP simulations. This again suggests these biases are primarily intrinsic errors within AGCMs.

Figure 9a and c show the sensitivity of the CMIP ω mean bias in the MSEB model to the biases in \(LH\) and \({SW}_{sur}\). It becomes evident that the \({F}_{net}\) contribution to the \(\omega\) mean bias in CMIP simulations is related to both elements with similar strength in the deep tropics, but somewhat different regional patterns. In the AMIP simulations the \(LH\) term has a stronger impact than the\({SW}_{sur}.\)

According to Fig. 6a, both components of \({F}_{top}\), \({SW}_{top}\) and \({LW}_{top}\), contribute to \(\omega\) mean bias. Given the similarity in the contributions between CMIP and AMIP cases, it suggests that \({F}_{top}\) biases are present in both CMIP and AMIP simulations. The sensitivity of the CMIP and AMIP \(\omega\) mean bias in the MSEB model to \({SW}_{top}\) and \({LW}_{top}\) shows that both terms contribute to \(\omega\) mean bias across most tropical ocean regions, except for the northern subtropical regions (Fig. 10a and c). Furthermore, we note that \({LW}_{top}\) biases tend to predominantly enhance tropical \(\omega\) mean bias patterns, whereas \({SW}_{top}\) biases tend to weaken those patterns. The compensation between these two components underscores why \({F}_{sur}\) biases exert a more substantial influence compared to \({F}_{top}\) biases.

**d. MSE advection biases**

The contributions of \(Adv\) bias to the \(\omega\) mean bias can be further broken down into horizontal wind, temperature gradient, and moisture gradient components. To begin, we deconstruct \(Adv\) into \({Adv}_{T}\) (\(-{<\varvec{v}\bullet \nabla {C}_{p}T>}_{{P}_{T}}\)) and \({Adv}_{q}\) (\(-{<\varvec{v}\bullet \nabla {L}_{v}q>}_{{P}_{T}}\)), see Fig. 6a. In general, both \({Adv}_{q}\) and \({Adv}_{T}\) biases contribute to the mean bias in CMIP and AMIP simulations, with \({Adv}_{q}\) bias playing a somewhat stronger role than \({Adv}_{T}\) bias.

Considering that the advection terms comprise of both horizontal winds, temperature and moisture gradient terms, we further decompose the sources of contribution to \(\omega\) mean biases into terms related to biases in wind field (denoted by \(\varvec{v}\varvec{{\prime }}\)) and into terms related to biases in the gradients (denoted by \(\nabla {L}_{v}q{\prime }\) and \(\nabla {C}_{p}T{\prime }\)). This result into four different biases terms: \({Adv}_{qq}\) (\(-{<\varvec{v}\bullet \nabla {L}_{v}q{\prime }>}_{{P}_{T}}\)), \({Adv}_{qv}\) (\(-{<\varvec{v}\varvec{{\prime }}\bullet \nabla {L}_{v}q>}_{{P}_{T}}\)), \({Adv}_{TT}\) (\(-{<\varvec{v}\bullet \nabla {C}_{p}T{\prime }>}_{{P}_{T}}\)), and \({Adv}_{Tv}\) (\(-{<\varvec{v}\varvec{{\prime }}\bullet \nabla {C}_{p}T>}_{{P}_{T}}\)). For the purpose of this study, we will omit the nonlinear term due to its negligible contribution to ω mean biases. The individual bias terms for CMIP and AMIP are shown in Fig. S3 and S4.

The sensitivity experiments for the four different CMIP advection biases terms indicate that upward motion biases in the north Pacific are driven by \({Adv}_{qv}\) biases, while biases in the south and equatorial Pacific region are due to the \({Adv}_{qq}\) biases (Fig. 11a, 11c, 12a, 12c). The majority of the downward motion biases stem from the \({Adv}_{Tv}\) biases, followed by \({Adv}_{TT}\) biases. The AMIP bias terms (Fig. 11b, 11d, 12b, 12d) are fairly similar to those of the CMIP biases in most regional aspects, indicating that the bias contributions from different advection terms are likely intrinsic errors of the atmospheric model.