 Research Letter
 Open Access
 Published:
The partitioning of poleward energy transport response between the atmosphere and Ekman flux to prescribed surface forcing in a simplified GCM
Geoscience Lettersvolume 5, Article number: 22 (2018)
Abstract
Recent studies have indicated that ocean circulation damps the atmospheric energy transport response to hemispherically differential energy perturbations, thereby muting the shifts of the InterTropical Convergence Zone (ITCZ). Here, we focus on the potential role of Ekman heat transport in modulating this atmospheric response. An idealized representation of Ekmandriven heat transport (F_{E}) is included in an aquaplanet slab ocean coupled to a gray radiation atmospheric model. We first alter the strength of F_{E} in the control climate by tuning the gross stability of the Ekman layer S_{E}. For a wide range of F_{E}, the total poleward transport of energy remains nearly unchanged, but the ocean transports an increasing share for larger S_{E}. The control climate is then perturbed by adding surface cooling in the Southern Hemisphere and warming in the Northern Hemisphere. The Ekman coupling damps the atmospheric energy transport response, as in previous coupled model experiments with full ocean dynamics. The ratio of the changes in Ekman to atmospheric energy transport is determined by the ratio of the gross stability in the Ekman layer to the atmosphere in the control climate, and is insensitive to the amplitude and location of forcing. We find that an unrealistically large S_{E} is needed to reproduce the ratio of the changes in crossequatorial oceanic to atmospheric energy transport in fully coupled models. The limited damping effect of Ekman transport highlights the need to examine the roles of deep circulation and subtropical gyres, as well as ocean heat uptake processes.
Background
The InterTropical 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 crossequatorial 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 crossequatorial 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 Δ_{s} is set to 1.4 as in Frierson et al. (2006), whereas KSX18 uses Δ_{s} of 0.8 to mimic the mean SST profile in GFDL AM2 (Anderson et al. 2004) coupled to an aquaplanet slab ocean under the annualmean 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 spinup period of 10 years.
The lower boundary is a 50m 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 Ekmandriven heat fluxes, with the meridional Ekman mass transport (M_{E}) computed from the surface zonal wind stress τ_{x}:
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 M_{E} 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).
The mass transport by a deep return flow is assumed to be equal and opposite to that in the surface mixed layer M_{E}. Then, the heating induced by the total Ekman transport (denoted H_{E}) can be obtained by the convergence of the heat transports by the surface Ekman flow at the surface slab temperature T_{s} and by the deep return flow at a lower temperature T_{d}
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., H_{E} < 0), while warming is induced by a convergence in the subtropics to midlatitudes (i.e., H_{E} > 0), inferring that the model reproduces the effect of the winddriven subtropical cells.
The magnitude of the heating induced by the Ekman transport H_{E} is proportional to T_{s} − T_{d}, which is equivalent to the gross stability of the ocean Ekman layer. The return flow temperature (T_{d}) is diagnosed from the surface temperature (T_{s}), following the 1.5layer scheme in Codron (2012), as
where T_{f} is the freezing temperature of seawater (= 271.3 K). This formulation ensures that T_{d} remains below the surface temperature T_{s} and above the freezing temperature T_{f}. The difference between T_{s} and T_{d} becomes small at high latitudes (because T_{s} is close to T_{f}); 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 T_{s} and T_{d}. We alter the α parameter from 0.65 to 0.94, which produces T_{s} − T_{d} on the equator to be 10.45 K and 1.90 K, respectively. For simplicity, the notation T_{s} − T_{d} 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 T_{s} − T_{d} 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 F_{S} 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 crossequatorial implied ocean heat transport F_{S0} 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 crossequatorial atmospheric energy transport response of similar magnitude between the two models (Kang et al. 2009; Seo et al. 2014). For a reference case with F_{S0} = 4.6 PW, we examine the sensitivity to the gross stability of the ocean Ekman layer T_{s} − T_{d} 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 φ_{0} ranges from 8° to 72°, and the maximum amplitude of the surface heating anomalies S is adjusted to ensure that the crossequatorial transport F_{S0} is fixed at 1.5 PW (Additional file 1: Figure S1d). The experiments with varying F_{S0} and φ_{0} are run with α = 0.70 and 0.90, which, respectively, corresponds to T_{s} − T_{d} = 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
where R is the net downward topofatmosphere (TOA) radiative flux, H_{E} is the Ekmaninduced heating computed by Eq. (2), S is the prescribed surface heating anomaly, and F_{A} is the atmospheric energy transport, which is the vertically integrated meridional moist static energy transport. The Ekmaninduced heating H_{E} can be expressed as a convergence of the meridional heat transport by the Ekman flow (denoted F_{E}), 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 H_{E} is W m^{−2}, and the unit for F_{E} is W. The magnitude of the Ekmaninduced heating is proportional to the gross stability of the Ekman layer (i.e., T_{s} − T_{d}), which is controlled by altering α in Eq. (3). There is no Ekmandriven heating (i.e., H_{E} = 0) when α = 1. In our range of T_{s} − T_{d}, 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 F_{E} between 0.76 and 2.40 PW, respectively (dashed lines in Fig. 1c). The cases with T_{s} − T_{d} 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).
The difference in the atmospheric energy balance between the perturbed and control experiments can be written as (after expressing H_{E} and S as the convergence of the corresponding meridional transports)
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), F_{S} = 0 at all latitudes. The equation states that a prescribed forcing S is compensated by a threeway 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 F_{S} 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 timemean states of CNTs with varying α. A decrease in α (or an increase in T_{s} − T_{d}) results in a greater energy transport by the Ekman flux (i.e., larger F_{E}), which then leads to a reduction in atmospheric energy transport (i.e., smaller F_{A}), as shown in Fig. 1c. The increase in F_{E} is compensated by the decrease in F_{A}, so that the total transport remains nearly constant with T_{s} − T_{d} (Fig. 2a). A smaller F_{A} is accomplished by a weakening of the Hadley circulation mass transport M_{A} (solid lines in Fig. 1a). As a result, the tropical easterlies become weaker, which then induces a weaker Ekman mass transport M_{E} (dashed lines in Fig. 1a). This reduction in both M_{A} and M_{E} with T_{s} − T_{d} can be clearly seen in Fig. 2b. The weaker M_{A} with larger T_{s} − T_{d} also causes less moisture convergence in the equatorial region, which leads to a flatter ITCZ (Fig. 1d). For a sufficiently large T_{s} − T_{d}, 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 M_{A} 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, M_{A} ≈ M_{E}. 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(T_{s} − T_{d}), which is altered via the α parameter. We can also deduce the gross stability using
where S_{A} and S_{E} represent the gross stability of the atmosphere and ocean Ekman layer, respectively. Figure 2c compares S_{A} and S_{E} at 10° latitude in CNT as a function of T_{s} − T_{d} (or α). As we discuss later, there is ambiguity in the treatment of M_{E} near the equator; thus, 10° latitude is used to represent the tropics. Note that we prefer to discuss in terms of T_{s} − T_{d} rather than α, because it is more physically based and α is directly related to T_{s} − T_{d} via Eq. (3). With increasing T_{s} − T_{d}, S_{E} increases linearly, whereas S_{A} remains constant. For T_{s} − T_{d} > 4.30 K, the gross stability of the Ekman layer S_{E} exceeds that of the atmosphere S_{A}, which results in more of the energy transported by the Ekman layer than transported by the atmosphere; that is, F_{E} > F_{A} (Fig. 2a). An estimate of the gross stability of the atmosphere assuming M_{A} = M_{E} (dashed red line in Fig. 2c) closely follows the actual S_{A} (solid red line) due to the similarity between M_{A} and M_{E} (Fig. 2b).
Response to surface heating anomalies
Figure 3a compares the energy transport anomalies by the atmosphere δF_{A} and Ekman layer δF_{E} in response to the prescribed extratropical surface heating anomalies with F_{S0} = 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 T_{s} − T_{d}. The prescribed northward heat transport F_{S} is compensated by the anomalous southward energy transport by both the atmosphere and Ekman flow (i.e., δF_{A} < 0 and δF_{E} < 0). The response is linear to the forcing amplitude F_{S0} (not shown). The Ekman transport becomes more effective at compensating the prescribed forcing with increasing T_{s} − T_{d}, whereas less energy is compensated by the atmospheric transport.
We first discuss how the Ekman layer responds to compensate for the prescribed forcing. The prescribed northward heat transport F_{S} strengthens the southern Hadley circulation and weakens the northern Hadley circulation (so that more energy is transported southward to compensate for F_{S}). 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 δM_{E} ≈ 0, because M_{E} in both the perturbed and control experiments approaches zero near the equator following Eq. (1). The latitudinal pattern of δM_{E} closely determines that of δF_{E} (dashed in Fig. 4c). A convergence of δF_{E} warms the region between 40°S and 5°S (δH_{E} > 0) and a divergence of δF_{E} cools the region between 5°N and 40°N (δH_{E} < 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, δF_{E} 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 T_{s} − T_{d} value in the extratropics.
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, δM_{E} is constrained to zero at the equator following Eq. (1), which leads to δF_{E} ≈ 0 regardless of T_{s} − T_{d} (Fig. 4c). However, there is ambiguity in the treatment of equatorial flow, which affects the strength of equatorial positive feedback. In cases where M_{E} is assumed to be proportional to meridional winds near the equator, as in Codron (2012), δM_{E} should become positive at the equator, which would result in an even stronger equatorial positive feedback than that observed in our case. Instead, the nearequatorial flow can be treated to follow the local Sverdrup balance such that M_{E} = − ∂_{y}τ_{x}/β, where β = df/dφ, in which case δM_{E} 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 offequatorial region at 10°S/N.
Figure 3b shows the ratio of Ekman to atmospheric energy transport response at 10° as a function of T_{s} − T_{d}. This ratio increases with T_{s} − T_{d}, and the anomalous Ekman energy transport begins to dominate over the anomalous atmospheric energy transport when T_{s} − T_{d} = 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 δF_{A} ≈ S_{A}δM_{A} and δF_{E} ≈ S_{E}δM_{E} where S_{A} and S_{E} indicate the respective CNT estimates for the different values of T_{s} − T_{d}. Further assuming δM_{A} ≈ δM_{E} and using M_{A} ≈ M_{E} in CNT (as confirmed in Fig. 2b) yields
Equation (5) tells that the ratio δF_{E}/δF_{A} can be predicted from the CNT experiment. That is, the increase in δF_{E}/δF_{A} with T_{s} − T_{d} 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 F_{S0} (red bars) or the forced latitude band φ_{0} (blue bars) is varied, which are run with two different values of α, corresponding to T_{s} − T_{d} = 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 δF_{E}/δF_{A} is nearly insensitive to F_{S0} and φ_{0}, confirming that the ratio depends solely on T_{s} − T_{d} or the gross stability of the Ekman layer in CNT.
A larger δF_{E}/δF_{A}, and thus a smaller δF_{A}, is expected to lead to a smaller shift in the tropical precipitation. The zonalmean profile of the tropical precipitation response (Additional file 1: Figure S2) closely follows that of the Ekman heating response δH_{E} (Fig. 4d). The precipitation response between 10°S/N and 30°S/N becomes smaller for a larger T_{s} − T_{d} due to stronger subtropical negative feedback. Hence, the centroid of zonalmean precipitation between 20°S and 20°N (or 30°S–30°N) decreases with T_{s} − T_{d} (Additional file 1: Figure S3), as would be expected. However, the precipitation response equatorward of 10° becomes larger with T_{s} − T_{d} (Additional file 1: Figure S2) due to stronger equatorial positive feedback; thus, the centroid of zonalmean precipitation between 10°S and 10°N increases with T_{s} − T_{d} (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 Ekmandriven 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 (T_{s} − T_{d}) is altered. For a larger T_{s} − T_{d}, there is a larger damping in the atmospheric energy transport response, or equivalently δF_{E}/δF_{A} increases. The ratio of Ekman to atmospheric energy transport response δF_{E}/δF_{A} 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 δF_{E}/δF_{A} 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 F_{S0} 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 T_{s} − T_{d} 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 crossequatorial 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 crossequatorial 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 crossequatorial 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 T_{s} − T_{d} 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 T_{s} − T_{d}, the ratio of Ekman to atmospheric energy transport response δF_{E}/δF_{A} 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 δF_{E}/δF_{A} based on Eq. (5). The Ekman energy transport F_{E} 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 T_{s} − T_{d} = 9 K, and the atmospheric energy transport F_{A} is taken from Donohoe et al. (2014) where the same reanalysis is used (Additional file 1: Figure S4). The reanalysis exhibits the ratio F_{E}/F_{A} 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 crossequatorial 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.
Abbreviations
 ITCZ:

InterTropical Convergence Zone
 SST:

sea surface temperature
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Authors’ contributions
SMK and FC designed the project. FC provided the 1.5layer ocean model. YS performed the model experiments. SMK and YS analyzed the data. SMK led the writing of the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The authors thank ShangPing Xie and Ken Takahashi for the helpful discussions during the early stage of work.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The dataset supporting the conclusions of this article is available in the PANGEA repository in https://doi.pangaea.de/10.1594/PANGAEA.886149.
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Funding
S.M.K. and Y.S. were supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2016R1A1A3A04005520). F.C. was supported by the French National Research Agency (ANR) project MORDICUS (ANR13SENV0002).
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Keywords
 Ekman coupling
 Energy transport partitioning
 Gross stability