Background

The Inter-Tropical Convergence Zone (ITCZ) is a meridionally narrow band of surface wind convergence and heavy precipitation located near the equator. The ITCZ shifts meridionally toward a warmed hemisphere, even for energy perturbations far from the tropics: in response to hemispherically differential energy perturbations, the Hadley circulation adjusts to transport energy in the direction of its upper branch toward the cooled hemisphere, while moisture is transported in the opposite direction by its lower branch. Thus, the ITCZ shift is proportional to changes in the cross-equatorial atmospheric energy transport (Broccoli et al. 2006; Kang et al. 2008, 2009; Seo et al. 2014; Schneider et al. 2014).

The energy transport load in the coupled system is shared by the atmosphere and the ocean; thus, fully coupled model experiments indicate that the ITCZ is relatively insensitive to extratropical energy perturbations (Deser et al. 2015; Tomas et al. 2016; Kay et al. 2016; Hawcroft et al. 2017; Xiang et al. 2018). Mechoso et al. (2016) suggest that anomalous cooling over the Southern Ocean can substantially shift the ITCZ northward in fully coupled models, depending on the capacity of a model to simulate the sensitivity of stratocumulus clouds to underlying sea surface temperatures (SSTs). Nevertheless, the ITCZ in fully coupled models is less sensitive to extratropical perturbations than that in slab ocean models with no ocean dynamics.

Several studies have attributed the damped ITCZ shifts in fully coupled models to the mechanical coupling via surface wind stress between the atmospheric Hadley circulation and the subtropical overturning cells in the ocean (Green and Marshall 2017; Schneider 2017; Kang et al. 2018). The resulting positive coupling of atmospheric and oceanic meridional energy transport ensures that the atmosphere and ocean split the cross-equatorial energy transport load, leading to a damped ITCZ response. Kang et al. (2018 hereafter KSX18) propose that a step toward resolving the issue is to develop a hierarchy of ocean models. As the simplest case, they examine the effect of meridional heat advection by surface Ekman flow in isolation. In this study, we go one step further by including the effect of the deep return flow, by implementing a representation of Ekman transport that was introduced in Codron (2012) in an aquaplanet slab ocean model. We use a gray radiation atmospheric model because our goal is to elucidate the purely dynamical response of a coupled system in which complications of water vapor or cloud radiative feedbacks are absent. The results from a more comprehensive atmospheric model will be discussed in a subsequent study. We demonstrate that the ratio of changes in ocean Ekman to atmospheric energy transport depends on the gross stability of the ocean in the control climate, which is a tunable parameter in our model.

Model description and experimental design

The atmospheric model is a simplified moist general circulation model with gray radiation in which radiative fluxes are a function of only temperature; thus, water vapor and other constituents do not affect radiative transfer (Frierson et al. 2006, 2007). The model uses a spectral dynamical core at T42 horizontal resolution and 25 vertical levels. Solar radiation is an analytical function of only latitude. There is no seasonal or diurnal cycle in the model. The atmospheric model setup is similar to that used in KSX18, except the insolation profile: the parameter that controls the meridional gradient of insolation Δs is set to 1.4 as in Frierson et al. (2006), whereas KSX18 uses Δs of 0.8 to mimic the mean SST profile in GFDL AM2 (Anderson et al. 2004) coupled to an aquaplanet slab ocean under the annual-mean insolation. The current setup produces a mean SST profile close to the AM2 under perpetual equinox conditions. All experiments are integrated for 20 years, with a spin-up period of 10 years.

The lower boundary is a 50-m aquaplanet slab ocean with no continents or lateral boundaries; hence, the SSTs evolve in response to the net surface energy fluxes and the implemented ocean heat transport. The control experiment (denoted CNT) is run with no prescribed heating or cooling in the slab ocean. In KSX18, we consider only the effect of meridional heat advection by surface Ekman flow. Here, the effect of deep return flow is additionally accounted for by the scheme developed by Codron (2012). The ocean heat transport is represented by the Ekman-driven heat fluxes, with the meridional Ekman mass transport (ME) computed from the surface zonal wind stress τx:

$$M_{\text{E}} = \frac{{ - f\tau_{\text{x}} }}{{\varepsilon^{2} + f^{2} }}.$$
(1)

f is the Coriolis parameter and the factor ɛ is added because the Ekman balance breaks down near the equator, and it has a value of \(1.0 \times 10^{ - 5} {\text{s}}^{ - 1}\), which corresponds to the Coriolis parameter f at ~ 4° latitude. The surface Ekman transport ME is directed poleward in the region of tropical easterlies, while it is directed equatorward in the region of midlatitude westerlies (dashed lines in Fig. 1a).

Fig. 1
figure 1

The time- and zonal-mean state of CNT as a function of different Ts − Td, obtained by varying α. a The mass transport by the atmospheric cells MA (solid) and by the Ekman flow ME (dashed) in Sv, b the heating induced by Ekman transport HE in W m−2, c the meridional energy transport by the atmosphere FA (solid) and by the Ekman flow FE (dashed) in PW, and d the precipitation in mm day−1

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The mass transport by a deep return flow is assumed to be equal and opposite to that in the surface mixed layer ME. Then, the heating induced by the total Ekman transport (denoted HE) can be obtained by the convergence of the heat transports by the surface Ekman flow at the surface slab temperature Ts and by the deep return flow at a lower temperature Td

$$H_{\text{E}} = - \frac{\partial }{a\cos \varphi \partial \varphi }\left\{ {CM_{\text{E}} \left( {T_{\text{s}} - T_{\text{d}} } \right)\cos \varphi } \right\},$$
(2)

where a is the Earth’s radius, φ is latitude in radians, and the specific heat capacity of water C has a value of 4180 J kg−1 K−1. For reference, this term is the last in Eq. (6) of Codron (2012). As shown in Fig. 1b, cooling is induced by a divergence of the surface Ekman flow in the deep tropics (i.e., HE < 0), while warming is induced by a convergence in the subtropics to midlatitudes (i.e., HE > 0), inferring that the model reproduces the effect of the wind-driven subtropical cells.

The magnitude of the heating induced by the Ekman transport HE is proportional to Ts − Td, which is equivalent to the gross stability of the ocean Ekman layer. The return flow temperature (Td) is diagnosed from the surface temperature (Ts), following the 1.5-layer scheme in Codron (2012), as

$$T_{\text{d}} = \alpha T_{\text{s}} + \left( {1 - \alpha } \right)T_{\text{f}} ,$$
(3)

where Tf is the freezing temperature of seawater (= 271.3 K). This formulation ensures that Td remains below the surface temperature Ts and above the freezing temperature Tf. The difference between Ts and Td becomes small at high latitudes (because Ts is close to Tf); thus, only a small amount of cooling is produced by the Ekman transport poleward of 40°S/N (Fig. 1b). The value of α determines the magnitude of the difference between Ts and Td. We alter the α parameter from 0.65 to 0.94, which produces Ts − Td on the equator to be 10.45 K and 1.90 K, respectively. For simplicity, the notation Ts − Td will be used to refer to its equatorial value. Note that the model in KSX18 which only includes the meridional heat advection by surface Ekman flow can be regarded as the same model as in this study except Ts − Td being latitudinally constant.

In a series of sensitivity experiments, the CNT climate is perturbed by various forms of surface heating anomalies (denoted S), as illustrated in Additional file 1: Figure S1. We primarily discuss the experiments where surface heating (S > 0) is prescribed poleward of 40°N and compensating surface cooling (S < 0) is prescribed poleward of 40°S (Additional file 1: Figure S1a). A positive S can be described as the convergence of an implied ocean heat transport and vice versa, such that \(S = - \frac{1}{{2\pi a^{2} \cos \varphi }}\frac{{\partial F_{\text{S}} }}{\partial \varphi },\) where FS is the implied ocean heat transport. Removing heat from the southern extratropics and adding it to the northern extratropics is equivalent to adding northward heat transport in the ocean across the equator (Additional file 1: Figure S1b). The amplitude of the prescribed surface heating is varied such that the cross-equatorial implied ocean heat transport FS0 ranges from 1.5 to 6.2 PW. Note that the high latitude forcing amplitude in a gray model must be four times as large as in a comprehensive model to produce a cross-equatorial atmospheric energy transport response of similar magnitude between the two models (Kang et al. 2009; Seo et al. 2014). For a reference case with FS0 = 4.6 PW, we examine the sensitivity to the gross stability of the ocean Ekman layer Ts − Td by varying α. We also examine the sensitivity to the latitudinal position of the forcing by prescribing surface heating anomalies at five different latitudinal bands with a latitudinal width of 16° (Additional file 1: Figure S1c). The center of the forced latitude band φ0 ranges from 8° to 72°, and the maximum amplitude of the surface heating anomalies S is adjusted to ensure that the cross-equatorial transport FS0 is fixed at 1.5 PW (Additional file 1: Figure S1d). The experiments with varying FS0 and φ0 are run with α = 0.70 and 0.90, which, respectively, corresponds to Ts − Td = 9.04 K and 3.11 K. The response to a prescribed surface heating anomaly is obtained by taking the difference between the climatology of the perturbed experiment and that of the control experiment with the same α.

Energy budget analysis

The atmospheric energy balance in a steady state in our model setup can be written as

$$R + H_{\text{E}} + S = \nabla \cdot F_{\text{A}} ,$$

where R is the net downward top-of-atmosphere (TOA) radiative flux, HE is the Ekman-induced heating computed by Eq. (2), S is the prescribed surface heating anomaly, and FA is the atmospheric energy transport, which is the vertically integrated meridional moist static energy transport. The Ekman-induced heating HE can be expressed as a convergence of the meridional heat transport by the Ekman flow (denoted FE), so that \(F_{\text{E}} \left( \varphi \right) = - \mathop \smallint \nolimits_{{- \frac{\pi }{2}}}^{\varphi } \left( {2\pi a^{2} \cos \varphi H_{\text{E}} } \right){\text{d}}\varphi\). The unit for HE is W m−2, and the unit for FE is W. The magnitude of the Ekman-induced heating is proportional to the gross stability of the Ekman layer (i.e., Ts − Td), which is controlled by altering α in Eq. (3). There is no Ekman-driven heating (i.e., HE = 0) when α = 1. In our range of Ts − Td, the cooling induced by the Ekman transport divergence \(H_{\text{E}}\) at the equator lies between 33 and 70 W m−2 (Fig. 1b), corresponding to the maximum meridional Ekman heat transport FE between 0.76 and 2.40 PW, respectively (dashed lines in Fig. 1c). The cases with Ts − Td between 3.11 and 6.10 K can be regarded as realistic states (refer to Figs. 1c and 2a), considering that the observed estimate of ocean heat uptake is 48 W m−2 near the equator (Schneider 2017), while the observed estimate of the total meridional ocean heat transport reaches 1.7 ± 0.3 PW in the Northern Hemisphere and 1.2 ± 0.5 PW in the Southern Hemisphere (Trenberth and Fasullo 2008).

Fig. 2
figure 2

a The meridional energy transport by the atmosphere FA (red), the Ekman flow FE (blue), and the sum (black) in PW, b the meridional mass transport by the atmosphere MA (red) and the Ekman flow ME (blue) in Sv, and c the gross stability of the atmosphere SA = FA/MA (red), its approximate estimate FA/ME (red dashed), and that of the Ekman flow SE = FE/ME (blue) in 104 J kg−1, as a function of Ts − Td in CNT. All estimates are computed as the average of the absolute values at 10°S and 10°N

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The difference in the atmospheric energy balance between the perturbed and control experiments can be written as (after expressing HE and S as the convergence of the corresponding meridional transports)

$$\delta R - \nabla \cdot \delta F_{\text{E}} - \nabla \cdot \delta F_{\text{A}} = \nabla \cdot F_{\text{S}} ,$$

where δ denotes the response to a prescribed surface heating anomaly S and \(\nabla \cdot\) indicates \(\frac{1}{{2\pi a^{2} \cos \varphi }}\frac{\partial }{\partial \varphi }\). In the control experiment (CNT), FS = 0 at all latitudes. The equation states that a prescribed forcing S is compensated by a three-way balance among the radiative fluxes and the meridional heat transport by the atmosphere and the Ekman flow. In the model without Ekman transport, the atmosphere is the only medium that transports energy; in which case, \(\delta F_{\text{A}}\) compensates the equatorial FS by ~ 29%  when the forcing S is prescribed in the extratropics while the rest is compensated locally by radiative fluxes. Ekman coupling is expected to reduce the burden on atmospheric energy transport. Then, the question arises as to what determines the partitioning of the energy transport response between the atmosphere and ocean Ekman flow.

Energy transport partitioning between the atmosphere and Ekman flow

Control climate

Figure 1 compares the time-mean states of CNTs with varying α. A decrease in α (or an increase in Ts − Td) results in a greater energy transport by the Ekman flux (i.e., larger FE), which then leads to a reduction in atmospheric energy transport (i.e., smaller FA), as shown in Fig. 1c. The increase in FE is compensated by the decrease in FA, so that the total transport remains nearly constant with Ts − Td (Fig. 2a). A smaller FA is accomplished by a weakening of the Hadley circulation mass transport MA (solid lines in Fig. 1a). As a result, the tropical easterlies become weaker, which then induces a weaker Ekman mass transport ME (dashed lines in Fig. 1a). This reduction in both MA and ME with Ts − Td can be clearly seen in Fig. 2b. The weaker MA with larger Ts − Td also causes less moisture convergence in the equatorial region, which leads to a flatter ITCZ (Fig. 1d). For a sufficiently large Ts − Td, the equatorial cooling induced by the Ekman flux becomes so strong (Fig. 1b) that the mean meridional atmospheric circulation descends in the equatorial region (Fig. 1a) to form a strong double ITCZ (Fig. 1d).

The mean overturning atmospheric mass transport MA is computed as the peak of the mean meridional streamfunction in the midtroposphere at each latitude. Figures 1a and 2b suggest that the peak atmospheric mass transport is generally in the Ekman balance with the surface winds; that is, MA ≈ ME. The mass and energy transports are related by the gross stability of a given fluid (Held 2001), which is measured as the energy contrast between the upper and lower branches. The gross stability of the ocean Ekman layer is C(Ts − Td), which is altered via the α parameter. We can also deduce the gross stability using

$$S_{\text{A}} = \frac{{F_{\text{A}} }}{{M_{\text{A}} }}\,{\text{and}}\,S_{\text{E}} = \frac{{F_{\text{E}} }}{{M_{\text{E}} }},$$
(4)

where SA and SE represent the gross stability of the atmosphere and ocean Ekman layer, respectively. Figure 2c compares SA and SE at 10° latitude in CNT as a function of Ts − Td (or α). As we discuss later, there is ambiguity in the treatment of ME near the equator; thus, 10° latitude is used to represent the tropics. Note that we prefer to discuss in terms of Ts − Td rather than α, because it is more physically based and α is directly related to Ts − Td via Eq. (3). With increasing Ts − Td, SE increases linearly, whereas SA remains constant. For Ts − Td > 4.30 K, the gross stability of the Ekman layer SE exceeds that of the atmosphere SA, which results in more of the energy transported by the Ekman layer than transported by the atmosphere; that is, FE > FA (Fig. 2a). An estimate of the gross stability of the atmosphere assuming MA = ME (dashed red line in Fig. 2c) closely follows the actual SA (solid red line) due to the similarity between MA and ME (Fig. 2b).

Response to surface heating anomalies

Figure 3a compares the energy transport anomalies by the atmosphere δFA and Ekman layer δFE in response to the prescribed extratropical surface heating anomalies with FS0 = 4.6 PW (Additional file 1: Figure S1a). Shown in Fig. 3 are the average values between 10°S and 10°N as a function of Ts − Td. The prescribed northward heat transport FS is compensated by the anomalous southward energy transport by both the atmosphere and Ekman flow (i.e., δFA < 0 and δFE < 0). The response is linear to the forcing amplitude FS0 (not shown). The Ekman transport becomes more effective at compensating the prescribed forcing with increasing Ts − Td, whereas less energy is compensated by the atmospheric transport.

Fig. 3
figure 3

a The meridional energy transport anomaly by the atmosphere δFA (red) and by the Ekman flow δFE (blue), and the sum (black) in PW, and b the ratio of the changes in Ekman to atmospheric energy transport δFE/δFA (black solid), the ratio of Ekman to atmospheric gross stability in CNT SE/SA (black dashed), and the ratio of Ekman to atmospheric meridional energy transport in CNT FE/FA (black dash-dot), as a function of Ts − Td. Red corresponds to the experiments with varying forcing amplitudes FS0 and blue to those with varying forced latitude bands φ0. All estimates are the average anomalies at 10°S and 10°N

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We first discuss how the Ekman layer responds to compensate for the prescribed forcing. The prescribed northward heat transport FS strengthens the southern Hadley circulation and weakens the northern Hadley circulation (so that more energy is transported southward to compensate for FS). Then, the easterlies in the southern tropics strengthen, while the easterlies in the northern tropics weaken (Fig. 4a). These changes induce a southward Ekman mass transport response throughout the tropics (Fig. 4b). The anomalous Ekman mass transport exhibits a dip at the equator, with δME ≈ 0, because ME in both the perturbed and control experiments approaches zero near the equator following Eq. (1). The latitudinal pattern of δME closely determines that of δFE (dashed in Fig. 4c). A convergence of δFE warms the region between 40°S and 5°S (δHE > 0) and a divergence of δFE cools the region between 5°N and 40°N (δHE < 0), as shown in Fig. 4d. This tropics–midlatitude component of the Ekman heat transport response partially compensates for the prescribed forcing that cools the Southern Hemisphere and warms the Northern Hemisphere, hence acting as a negative feedback. In the equatorial region, δFE diverges south of the equator and converges north of the equator (Fig. 4c), which induces cooling and warming, respectively (Fig. 4d). This equatorial component of the Ekman heat transport response amplifies the prescribed forcing, hence acting as a local positive feedback. Poleward of 40°S/N, the anomalous Ekman mass transport which arises because of the extratropical jet shift is inefficient at producing anomalous heating because of the low Ts − Td value in the extratropics.

Fig. 4
figure 4

Zonal-mean changes in a the zonal wind at 10 m in m s−1, b the meridional Ekman mass transport δME in Sv, c the meridional energy transport by the atmosphere δFA (solid) and by the Ekman flow δFE (dashed) in PW, and d the Ekman-induced heating δHE in W m−2 (solid) in the experiments with extratropical perturbations with FS0 = 4.6 PW

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In the deep tropics, the Ekman transport has limited ability to compensate for the forcing (Fig. 4c). This is because of the equatorial positive feedback, which is expected to some degree in association with strengthened equatorial upwelling in the cooled hemisphere and weakened equatorial upwelling in the warmed hemisphere. Green and Marshall (2017) indeed note that the upwelling branch of the subtropical cells remains at the equator in response to hemispherically differential energy perturbations, while there is some strengthening in the cooled hemisphere and some weakening in the warmed hemisphere (see their Fig. 8). However, the equatorial positive feedback in Green and Marshall (2017) is weak; so, their ocean energy transport response does not exhibit as sharp of an equatorial dip. In our model setup, δME is constrained to zero at the equator following Eq. (1), which leads to δFE ≈ 0 regardless of Ts − Td (Fig. 4c). However, there is ambiguity in the treatment of equatorial flow, which affects the strength of equatorial positive feedback. In cases where ME is assumed to be proportional to meridional winds near the equator, as in Codron (2012), δME should become positive at the equator, which would result in an even stronger equatorial positive feedback than that observed in our case. Instead, the near-equatorial flow can be treated to follow the local Sverdrup balance such that ME = − ∂yτx/β, where β = df/dφ, in which case δME would be negative at the equator and would lead to a weaker equatorial positive feedback than that observed in our case (or even a negative one as in the rest of the tropics). Because of this ambiguity, we discuss the effect of Ekman modulation in the off-equatorial region at 10°S/N.

Figure 3b shows the ratio of Ekman to atmospheric energy transport response at 10° as a function of Ts − Td. This ratio increases with Ts − Td, and the anomalous Ekman energy transport begins to dominate over the anomalous atmospheric energy transport when Ts − Td = 5.2 K. The fractional change in gross stability between the CNT and perturbed experiments is much smaller than that in mass transport for both the atmosphere and Ekman layer, so that δFA ≈ SAδMA and δFE ≈ SEδME where SA and SE indicate the respective CNT estimates for the different values of Ts − Td. Further assuming δMA ≈ δME and using MA ≈ ME in CNT (as confirmed in Fig. 2b) yields

$$\frac{{\delta F_{\text{E}} }}{{\delta F_{\text{A}} }} \approx \frac{{S_{\text{E}} }}{{S_{\text{A}} }} \approx \frac{{F_{\text{E}} }}{{F_{\text{A}} }}.$$
(5)

Equation (5) tells that the ratio δFE/δFA can be predicted from the CNT experiment. That is, the increase in δFE/δFA with Ts − Td is due to the increase in gross stability of the Ekman layer in CNT. Considering the simplicity of the theory, this prediction works well despite an overall slight overestimation (Fig. 3b). Figure 3b also shows the actual ratios in the experiments where either the forcing amplitude FS0 (red bars) or the forced latitude band φ0 (blue bars) is varied, which are run with two different values of α, corresponding to Ts − Td = 3.11 K and 9.04 K. The circle indicates the mean, and the bar indicates one standard deviation. One can find that the ratio δFE/δFA is nearly insensitive to FS0 and φ0, confirming that the ratio depends solely on Ts − Td or the gross stability of the Ekman layer in CNT.

A larger δFE/δFA, and thus a smaller δFA, is expected to lead to a smaller shift in the tropical precipitation. The zonal-mean profile of the tropical precipitation response (Additional file 1: Figure S2) closely follows that of the Ekman heating response δHE (Fig. 4d). The precipitation response between 10°S/N and 30°S/N becomes smaller for a larger Ts − Td due to stronger subtropical negative feedback. Hence, the centroid of zonal-mean precipitation between 20°S and 20°N (or 30°S–30°N) decreases with Ts − Td (Additional file 1: Figure S3), as would be expected. However, the precipitation response equatorward of 10° becomes larger with Ts − Td (Additional file 1: Figure S2) due to stronger equatorial positive feedback; thus, the centroid of zonal-mean precipitation between 10°S and 10°N increases with Ts − Td (Additional file 1: Figure S3). It is worth noting that the equatorial positive feedback is responsible for the large sensitivity of the precipitation centroid metric to the latitudinal range where it is computed.

Summary and discussion

Previous studies that neglected ocean dynamics suggested a strong influence of hemispherically differential thermal forcing at high latitudes on the tropical precipitation distribution. However, fully coupled model experiments indicate that this tropical precipitation response is muted as a larger fraction of the forcing is compensated by ocean energy transport rather than atmospheric energy transport in the tropics. To better understand the mechanism behind the extratropics–tropics teleconnection, we examine what controls the partitioning of energy transport between the atmosphere and ocean. In particular, we consider the effect of Ekman transport in isolation by including an idealized representation of Ekman-driven heat transport in an aquaplanet slab ocean coupled to a gray radiation atmospheric model.

Coupling of the Hadley circulation and the Ekman transport by the surface wind stress reduces the need for atmospheric energy transport. We show that the reduction depends on the gross stability of the Ekman layer, by utilizing a set of experiments where the difference between the surface temperature and the return flow temperature (Ts − Td) is altered. For a larger Ts − Td, there is a larger damping in the atmospheric energy transport response, or equivalently δFE/δFA increases. The ratio of Ekman to atmospheric energy transport response δFE/δFA is well predicted as the ratio between the gross stability of the Ekman layer and that of the atmosphere in the control experiment. Hence, the ratio is insensitive to the forcing profile, such as the forcing amplitude or the forced latitude band. However, the ratio δFE/δFA greatly depends on the forcing amplitude in KSX18 where the effect of meridional Ekman heat advection by the surface flow is considered in isolation. The ratio in KSX18 varies between 0.43 and 2.33 in the equatorial region as the forcing amplitude FS0 varies from 5.8 to 1.1 PW, respectively. A reduction in the ratio with the forcing amplitude arises because of a stronger positive extratropical feedback associated with a larger jet shift. However, the extratropical positive feedback vanishes in our model because Ts − Td becomes negligible near 50°S/N, which is consistent with the low stratification of the ocean at these latitudes.

Fully coupled model experiments show a wide range in the ratio of the changes in cross-equatorial oceanic to atmospheric energy transport for extratropical energy perturbations (Deser et al. 2015; Tomas et al. 2016; Kay et al. 2016; Hawcroft et al. 2017; Mechoso et al. 2016; Xiang et al. 2018). The lower end of the ratio is approximately 1.5, while some models even indicate that the cross-equatorial energy transport response is entirely accomplished by the ocean. Although the wide spread should partly stem from differences in the details of the experimental setup, this study suggests that the model spread in the ocean stratification in the control climate may also contribute by modulating the efficiency of the Ekman compensation. However, it is difficult to directly compare the gray radiation model and more comprehensive models because cloud radiative effects may change the picture.

A cleaner comparison can be made with Green and Marshall (2017), which employs a gray radiation atmospheric model coupled to a full ocean model. The ratio of cross-equatorial energy transport by the ocean to that by the atmosphere is 1.98 in their model. In our experiments, we consider the control climate with Ts − Td between 3.11 and 6.10 K is similar to the current Earth’s climate, as discussed in “Energy budget analysis” section. Within that range of Ts − Td, the ratio of Ekman to atmospheric energy transport response δFE/δFA lies between 0.57 and 1.55, which is smaller than that in Green and Marshall (2017). One can also compute an observed estimate of δFE/δFA based on Eq. (5). The Ekman energy transport FE is computed by taking the zonal wind stress from the NCEP/NCAR reanalysis for 1979–2009 (Kalnay et al. 1996) with an assumption of 50 m mixed layer depth and α = 0.7 that corresponds to Ts − Td = 9 K, and the atmospheric energy transport FA is taken from Donohoe et al. (2014) where the same reanalysis is used (Additional file 1: Figure S4). The reanalysis exhibits the ratio FE/FA of 0.95 at 10°S and 0.77 at 10°N, which is within the realistic range in our simple model. Our results indicate that Ekman transport alone cannot produce cross-equatorial energy transport as efficient as full ocean circulation. Factors such as deep ocean circulation and subtropical gyres other than Ekman transport should play a role in boosting oceanic compensation.