1. Introduction
The intertropical convergence zone (ITCZ) is a band of deep convective clouds, located at the rising branch of the tropical meridional overturning (Hadley) circulation (Fig. 1), where near-surface meridional mass fluxes vanish. Since the Hadley circulation transports energy away from the ITCZ and dominates the tropical energy transport, column-integrated energy fluxes also vanish and diverge near the ITCZ, forming an atmospheric energy flux equator (EFE) (e.g., Neelin and Held 1987; Broccoli et al. 2006; Kang et al. 2008, 2009; Donohoe et al. 2013, 2014; Adam et al. 2016).
(left) Mean precipitation (mm day−1) for (top) ANN, (middle) boreal summer (JAS), and (bottom) boreal winter (JFM). Orography is shown by gray contours at 1-km intervals. (right) The respective zonal-mean precipitation over ocean (solid) and land (dashed). Data are from the Tropical Rainfall Measuring Mission Multisatellite Precipitation Analysis (Liu et al. 2012) for 1998–2014.
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-15-0710.1
The approximate collocation of the ITCZ and EFE provides a basis for relating the mean ITCZ position to the atmospheric energy transport (AET). Usually, an ITCZ and EFE north of the equator imply southward AET across the equator, away from the ITCZ, and vice versa. Indeed, the zonal-mean position of the ITCZ was found to be closely related to cross-equatorial AET in observations (e.g., Donohoe et al. 2013, 2014; Adam et al. 2016) and in climate models of varying complexity (e.g., Chiang and Bitz 2005; Broccoli et al. 2006; Yoshimori and Broccoli 2008; Kang et al. 2008, 2009; Donohoe et al. 2013, 2014).
Relating the ITCZ and EFE to the AET provides theoretical insight into, for example, the ocean’s role in setting the mean position of the ITCZ (Frierson et al. 2013; Marshall et al. 2014), the effect of extratropical forcing on the mean ITCZ position (e.g., Chiang and Bitz 2005; Broccoli et al. 2006; Yoshimori and Broccoli 2008; Kang et al. 2008; Frierson and Hwang 2012; Bischoff and Schneider 2014), and the mean ITCZ position in past climates (Donohoe et al. 2013; Schneider et al. 2014). Such studies have quantitatively related the ITCZ position to the atmospheric energy budget in the zonal mean, but they have made only qualitative inferences about zonal variations of the ITCZ position, if at all [see Chiang and Friedman (2012) and Schneider et al. (2014) for reviews].
Here we extend previous results to zonal variations of the ITCZ position, which are a central characteristic of the tropical hydrological cycle (Fig. 1). For example, seasonal meridional migrations of marine ITCZs are small relative to their continental (monsoonal) counterparts (Fig. 1); interannual variations of the ITCZ associated with ENSO are most pronounced over the Pacific but are relatively small elsewhere (e.g., Dai and Wigley 2000; Adam et al. 2016); and the eastern Pacific ITCZ bifurcates to a double ITCZ that straddles the equator during boreal spring, but such bifurcations do not usually occur elsewhere (Fig. 1, bottom; Zhang 2001; Gu et al. 2005). We will examine the extent to which energetic arguments can be extended to account for such zonally varying shifts of the ITCZ.
Using reanalysis data, we have recently shown that the zonal-mean EFE and ITCZ position covary on seasonal and interannual time scales and that seasonal and interannual variations of the ITCZ position can be quantitatively related to variations of the AET and its derivatives (Adam et al. 2016). In the present paper, we examine the relation between the zonally varying EFE and ITCZ and extend the theoretical framework developed for zonal-mean variations (Bischoff and Schneider 2014, 2016; Schneider et al. 2014; Adam et al. 2016) to regional variations. Using reanalysis data, we focus on the ITCZ in separate zonal sectors, to account for such observed features as the eastern Pacific double-ITCZ bifurcation in boreal spring, and local ITCZ shifts associated with ENSO.
The data and methods employed are described in section 2. Some theoretical aspects of the zonally varying EFE are derived in section 3. The data analysis is presented in section 4, followed by a summary and discussion in section 5.
2. Data and methods
The analysis uses four-times-daily data from the European Centre for Medium-Range Weather Forecasts (ECMWF) interim reanalysis (ERA-Interim, hereinafter ERAI; Dee et al. 2011) for 1979–2014. As in Adam et al. (2016), column-integrated fluxes are adjusted using a barotropic mass-flux correction (Trenberth 1997; see http://www.cgd.ucar.edu/cas/catalog/newbudgets/ for detailed documentation). The fluxes are calculated from pressure level ERAI data; they agree well with fluxes calculated using native ERAI model grids (Fasullo and Trenberth 2008). We also used the oceanic Niño index (ONI) obtained from version 3b of the Extended Reconstructed Sea Surface Temperature (ERSST.v3b) dataset, provided by NOAA’s National Climatic Data Center (Smith et al. 2008). For data retrieval and analysis we used the geophysical observation analysis tool (GOAT), a free MATLAB-based tool for geophysical data management (http://www.goat-geo.org).
3. Theory
Here we briefly review and extend the first- and third-order approximations, described in Bischoff and Schneider (2014, 2016), for a zonally varying EFE. See Schneider et al. (2014) for a review of the underlying concepts.
a. First-order approximation
b. Third-order approximation
A single ITCZ (and therefore single EFE) is associated with the rising branch of a tropical meridional overturning circulation transporting energy away from the ITCZ (Fig. 2a). Thus, since the column-integrated AET is positive north of the ITCZ and negative south of the ITCZ, the meridional gradient of the AET (i.e., approximately the local NEI) near a single ITCZ is positive. Conversely, negative local NEI near the equator corresponds to convergence of energy. Since eddy energy transport is weak near the equator, transport of energy toward the equator is done by the mean circulation. Therefore, for sufficiently strong convergence near the equator, the meridional overturning circulation adjusts to two meridional overturning cells straddling the equator and transporting energy toward their shared descending branch (Fig. 2b; Bischoff and Schneider 2016). In such states, precipitation is inhibited in the equatorial descending branch; precipitation occurs at the rising branches of these cells on either side of the equator. Thus, strong convergence of AET near the equator can be associated with a double-ITCZ state.
Sketch of the meridional overturning circulation and column-integrated meridional atmospheric energy transport in (a) single-ITCZ and (b) double-ITCZ states.
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-15-0710.1
However, double-ITCZ states (i.e., precipitation peaks that straddle the equator) can exist without convergence of energy near the equator (e.g., Voigt and Shaw 2015). Moreover, in the western Pacific and over the Indian Ocean during JAS (Fig. 1), precipitation peaks exist on both sides of the equator even though there is no AET convergence in these regions (Fig. 3). The double-ITCZ state shown in Fig. 2b therefore represents an extreme case in which the bifurcation from single- to double-ITCZ states is associated with negative NEI. In less extreme cases, small positive NEI near the equator may suffice to produce local precipitation maxima straddling the equator.
ANN, JAS, and JFM averages of the meridional component of the (a)–(c) divergent MSE flux 
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-15-0710.1
4. Results
a. Zonally varying EFE and ITCZ
Figure 3 shows the annual (ANN), July–September (JAS), and January–March (JFM) averages of the meridional component of the divergent MSE flux 
As an error margin of the EFE, we use one standard deviation of the differences between annually averaged 
As seen in previous studies (e.g., Chiang and Friedman 2012), the EFE migrates seasonally toward the differentially warming hemisphere (Fig. 3). It captures some aspects of the regional seasonal migration of the ITCZ. Over the Pacific, the divergent circulation is not predominantly meridional, offsetting the location of the EFE relative to the ITCZ. During boreal winter (Figs. 3c,f), the EFE splits into a North Pacific branch, which follows the North Pacific ITCZ, and a South Pacific branch, which deviates from the precipitation maximum. Discrepancies between the zonally varying EFE and ITCZ position exist either where column-integrated energy and mass fluxes do not simultaneously vanish or where the circulation is not primarily meridional. The large discrepancies observed in Fig. 3 indicate that the EFE framework is inapplicable at small regional scales, especially in the central and western Pacific.
b. Sector-mean EFE
Since the meridional derivative of 
Longitudinal dependence of the annual-mean local net energy input 
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-15-0710.1
Figure 5 shows the daily sector-mean climatology of the EFE (black) and its first-order approximations (red), on top of precipitation, for the African (Fig. 5a), Asian (Fig. 5d), and Atlantic (Fig. 5g) sectors. The respective cross-equatorial fluxes and NEI are shown in the line plots in Fig. 5. To remove regional subseasonal variability, sector means are smoothed using a 3-month running mean. [The EFE framework is not applicable on subseasonal time scales because tropical overturning and radiative relaxation times are on the order of several weeks (Adam et al. 2016).] However, the results do not qualitatively change if a 1-month running mean is used, as in Donohoe et al. (2014) and Adam et al. (2016) for the zonal-mean EFE. Equatorial means of the cross-equatorial fluxes and all derivatives are averaged between 5°S and 5°N. The first-order approximations [Eq. (6)] are insensitive to the choice of near-equatorial averages (Adam et al. 2016). However, the third-order approximate solutions [Eq. (8)] deviate if the near-equatorial averages are extended beyond 5°.
(a),(d),(g) Daily climatology of zonal-mean precipitation (colors) in the African, Asian, and Atlantic sectors. The EFE latitude ϕEFE is shown by the thick black line. The stationary first-order approximation [Eq. (6b)] is shown in red. (b),(e),(h) Cross-equatorial AET 
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-15-0710.1
The correlation between EFE and ITCZ position [Eq. (1)] variations in these sectors is maximal (R > 0.9) for precipitation that lags EFE variations by about 1 month. However, even when accounting for the lag between EFE and ITCZ variations, the EFE deviates from the ITCZ position by up to 5°. In the African and Asian sectors, the stationary first-order approximation [Eq. (6b)] is in near-perfect agreement with the EFE, indicating that a linear approximation is sufficient in these sectors. However, over the Atlantic, the stationary approximation [Eq. (6b)] deviates by several degrees from the EFE during boreal summer and autumn. By explicitly calculating the energy storage term in Eq. (5a), we find that the difference between the approximations in Eqs. (6a) and (6b) is negligible in all three sectors, indicating that near-equatorial energy storage can be neglected. The deviation of the solution in Eq. (6b) from the EFE in the Atlantic sector therefore results from the inadequacy of the first-order approximation.
In the three sectors, seasonal EFE variations are driven by variations in both cross-equatorial AET and equatorial NEI. Seasonal variations in NEI contribute roughly 25% to the EFE variations over Africa, 20% over Asia, and 100% over the Atlantic. Moreover, variations in cross-equatorial AET and equatorial NEI differ substantially between sectors. For example, 
The relation of precipitation, AET, and the 500-hPa vertical wind in the eastern Pacific (250°–280°E) is shown in Fig. 6 for December–January, March–April, June–July, and September–October (Figs. 6a–d, respectively). The nonlinearity of AET near the equator in the eastern Pacific (Fig. 4) persists year-round. In addition, because of the antisymmetry of the AET near the equator, second-order terms in the expansion of the AET around the equator are small (less than 10%) relative to the third-order terms, justifying the approximation used to derive Eqs. (7) and (8). Consistent with Eq. (7), the bifurcation from one to three AET roots occurs for small 
Eastern Pacific (250°–280°E) divergent component of the meridional AET 
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-15-0710.1
The daily climatology of the eastern Pacific sector is shown in Fig. 7, with the third-order approximation [Eq. (8)] shown in red [Eq. (8a) for ΔI > 0 and Eq. (8b) for ΔI < 0]. The third-order approximation captures the EFE only qualitatively, with an increasing error the farther the EFE is from the equator. The bifurcation from a single- to double-ITCZ state in the eastern Pacific (Fig. 7a) lags the bifurcation in the EFE by about 1–2 months. Similarly, as in the other sector means, the correlation between the seasonal variations of the northern EFE and ITCZ position is maximal for precipitation that lags EFE variations by about 1 month, as was the case for the zonal mean (Adam et al. 2016). The (lag corrected) EFE deviates from the ITCZ position by up to 10° latitude during boreal summer and in the southern branch of the boreal spring double ITCZ. Consistent with theory, a double-ITCZ state exists for ΔI < 0 (Fig. 7d; with a lag of 1–2 months) during boreal spring because of increased ocean heat uptake (i.e., reduced NEI) and vanishing cross-equatorial AET. The local NEI (solid blue) is the sum of the vertical (I0, dotted blue, negative year-round) and zonal (
(a) Daily climatology of zonal-mean precipitation (colors) in the eastern Pacific sector (250°–280°E). The EFE ϕEFE in the sector is shown as a thick black line. Convergent 
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-15-0710.1
c. EFE response to ENSO
The zonal-mean precipitation tends to increase near the equator during El Niño episodes (e.g., Dai and Wigley 2000; Adam et al. 2016), which are characterized by sea surface temperature (SST) anomalies in the equatorial Pacific. Figure 8 shows the difference in the annual-mean precipitation between typical El Niño and La Niña conditions. Like the SST anomalies, the precipitation anomaly is most pronounced over the Pacific. In the zonal mean, precipitation maxima on either side of the equator shift equatorward (poleward) in response to warm (cold) El Niño (La Niña) episodes (Adam et al. 2016).
(left) Precipitation difference between typical annual-mean El Niño and La Niña conditions. Orography is shown by gray contours at 1-km intervals. (right) The zonal mean of the precipitation difference (thick solid) and typical El Niño (thin solid) and La Niña (dashed) conditions. Typical El Niño (La Niña) conditions are defined as ONI values above 0.2 K (below −0.2 K). Data are taken from ERSST.v3b and ERAI for 1979–2014.
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-15-0710.1
Figure 9 shows the annual-mean EFE during typical El Niño (magenta) and La Niña (green) conditions. Additionally, it shows the difference (El Niño minus La Niña) in the meridional component of the divergent AET 
(left) Difference between typical annual-mean El Niño and La Niña conditions of (a) the meridional component of the divergent MSE flux 
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-15-0710.1
5. Summary and discussion
The ITCZ lies at the ascending branch of the tropical meridional overturning circulation, where near-surface mass fluxes vanish. The ITCZ position (identified as the precipitation maximum) is also located near the atmospheric EFE, where column-integrated energy fluxes diverge and vanish. Since the EFE is characterized by meridional divergence of the column-integrated atmospheric energy transport, the zonally varying EFE is found as the latitude where the meridional component of the divergent AET both vanishes and has a positive meridional gradient.
Like the ITCZ, the zonally varying EFE migrates seasonally toward the differentially warming hemisphere. Its migrations capture some aspects of regional ITCZ migrations (Fig. 3). But at some longitudes, large discrepancies exist between the ITCZ position and EFE. Moreover, in the Pacific sector, the Walker circulation is of comparable magnitude to the meridional overturning circulation, making the correspondence between the EFE and ITCZ weak (in particular over the western Pacific, where the rising branch of the Walker circulation is located). Thus, discrepancies between the column-integrated energy and mass-flux equators, as well as the requirement that the circulation is primarily meridional, limit the applicability of the theory in some regions. However, outside the western Pacific, we find that sector-mean EFE variations capture key features of ITCZ position variations.
The EFE and roughly the ITCZ position can be approximated by meridionally expanding the divergent component of the AET around the equator (Bischoff and Schneider 2014). To a first order, the ITCZ position is proportional to the divergent cross-equatorial AET and inversely proportional to the local NEI, which is the net of vertical fluxes across the surface and top of the atmosphere, and zonal fluxes across sector boundaries. The first-order approximation captures the cross-equatorial monsoonal migrations of the ITCZ over the African and Asian sectors and the limited seasonal migrations north of the equator over the Atlantic sector (Fig. 5). In these regions, seasonal ITCZ migrations are associated with both cross-equatorial AET and local equatorial NEI variations. This is in contrast to the seasonal migrations of the zonal-mean ITCZ, which are dominated by cross-equatorial AET (Donohoe et al. 2013; Adam et al. 2016). Like the zonal-mean ITCZ (Adam et al. 2016), seasonal ITCZ migrations lag EFE variations by about 1 month, for reasons that are not clear.
Since the meridional derivative of the AET at the equator is negative in the Pacific sector (i.e., approximately local NEI; Fig. 4), third-order approximations are required there. A simplified third-order approximation [Eq. (8)] is obtained by assuming second derivatives of the AET are weak near the equator (because of the approximate antisymmetry of the AET about the equator). The third-order approximation captures the single-ITCZ phase [Eq. (8a), boreal summer to boreal autumn) and the bifurcation to a double ITCZ during boreal spring [Eq. (8b)] in the eastern Pacific (Fig. 8, left) only qualitatively; the discrepancy between the AET zeros and ITCZ branches can be as large as 10° latitude (Fig. 7a). Consistent with theory, the eastern Pacific ITCZ bifurcates from single to double ITCZ when the discriminant ΔI [Eq. (7)] is negative (Fig. 7; with a lag of 1–2 months). Likewise consistent with theory, observations indicate that a double ITCZ is more likely to occur during La Niña episodes (Zhang 2001; Gu et al. 2005). The latter observation supports the theory (Bischoff and Schneider 2016) because during cold La Niña episodes ocean heat uptake increases and NEI decreases (Fig. 9), while variations in cross-equatorial AET are negligible (Fig. 9); therefore, ΔI is generally smaller.
Since local NEI is negative in the Pacific, the predictions of linear EFE theory, which relates the ITCZ position only to cross-equatorial AET (e.g., Kang et al. 2008, 2009; Donohoe et al. 2013, 2014), are invalid there. Contrary to linear EFE theory, for negative local NEI, the ITCZ latitude and cross-equatorial AET can have the same sign (Fig. 6a). Moreover, eastern Pacific ITCZ variability (which includes bifurcations from single- to double-ITCZ states during boreal spring) is driven by a complex balance between the vertical and zonal NEI components as well as variations in cross-equatorial AET.
Correlations of interannual variations in sector-mean EFE and ITCZ position were found to be statistically insignificant in the reanalysis data. However, since seasonal variations of the sector-mean EFE and ITCZ position are in qualitative agreement (Figs. 5 and 7), the low interannual correlations are likely due to weak interannual variations outside the Pacific (Fig. 9, assuming interannual variations are primarily ENSO driven; Adam et al. 2016) and the complex nature of ITCZ variability in the Pacific. In monsoonal regions, the EFE and ITCZ position were found to be collocated in simulations driven by a wide range of forcing scenarios (Shekhar and Boos 2016). The relevance of the theory in different regions over a wide range of simulated climates remains to be tested.
ENSO-related variations in the ITCZ (Fig. 8) and EFE (Fig. 9) are most pronounced in the Pacific. As in the zonal-mean case (Adam et al. 2016), these variations are characterized by an equatorward shift of the ITCZ and EFE during warm El Niño episodes and a poleward shift during La Niña episodes. These variations are driven by increased (decreased) NEI (Fig. 9b), because of decreased (increased) ocean heat uptake during El Niño (La Niña) episodes (Schneider et al. 2014; Adam et al. 2016). Paleorecords indicate that a similar contraction or expansion of the annual-mean tropical precipitation belt in the Pacific also occurs on decadal or longer time scales (e.g., Yan et al. 2015). Similarly, records indicate large local meridional shifts of the ITCZ in past climates (Schneider et al. 2014). Better understanding of the zonally varying atmospheric energy budgets of past climates may improve our understanding of associated local and zonal-mean tropical precipitation patterns.
Acknowledgments
We thank Momme Hell for his visualization advice.
APPENDIX
Third-Order Approximation
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