1. Introduction
The intertropical convergence zone (ITCZ) is a band of deep convective clouds in the tropics that marks the region of maximum precipitation on Earth. It is located around 6°N latitude in the annual and zonal mean but migrates seasonally between a more northern position in boreal summer and a more southern position in austral summer (Waliser and Gautier 1993; Adler et al. 2003; Berry and Reeder 2014). Additionally, the annual-mean position of the ITCZ has shifted on geological time scales (e.g., Haug et al. 2001; Arbuszewski et al. 2013), mimicking its seasonal migration insofar as it tends to shift toward a differentially warming and away from a differentially cooling hemisphere (e.g., Koutavas and Lynch-Stieglitz 2004; Chiang and Friedman 2012). Modeling studies with general circulation models (GCMs) have demonstrated that remote extratropical factors, such as the presence or absence of polar ice cover or high-latitude temperature variations caused by the Atlantic meridional overturning circulation, can lead to shifts of the ITCZ, generally away from a cooling and toward a warming hemisphere (e.g., Vellinga and Wood 2002; Chiang et al. 2003; Chiang and Bitz 2005; Broccoli et al. 2006; Yoshimori and Broccoli 2008).
Recent studies have focused on the role of the cross-equatorial energy transport and the atmospheric energy budget in controlling the ITCZ position. They have demonstrated that the ITCZ tends to shift southward as the northward atmospheric energy transport across the equator strengthens, for example, in response to a northern high-latitude cooling (e.g., Kang et al. 2008, 2009; Frierson and Hwang 2012; Donohoe et al. 2013, 2014). As the atmospheric energy transport generally strengthens in the direction of a cooling hemisphere, to partially compensate the cooling, the recent studies focusing on the atmospheric energy transport are broadly consistent with studies that emphasize surface temperature changes (e.g., Chiang et al. 2003; Chiang and Bitz 2005; Cvijanovic and Chiang 2013).
However, the ITCZ also shifts southward during El Niño (e.g., Dai and Wigley 2000; Berry and Reeder 2014), an observation not easily related to changes in cross-equatorial energy transport. A quantitative understanding of the factors controlling the ITCZ position, including the sensitivity of the ITCZ to changes in cross-equatorial energy transport, has remained elusive. That other factors than those hitherto considered must influence the ITCZ position is evident already for dimensional reasons: for example, it is empirically clear that the ITCZ position, which has units of length (e.g., displacement from the equator), is sensitive to the cross-equatorial atmospheric energy transport, which has units of power when integrated over latitude circles. To relate these two physical quantities, another quantity with units of power per unit length must enter any relation between ITCZ position and cross-equatorial energy transport.
Here we derive a quantitative relation between the cross-equatorial energy transport and the position of the ITCZ from the atmospheric energy balance. We show that the net energy input to the equatorial atmosphere is the natural quantity of units of power per unit of length to relate the ITCZ position to the cross-equatorial energy transport. We use an idealized GCM to demonstrate that the theoretically derived relation accurately captures the factors controlling the ITCZ position under both global and primarily tropical warming. The global and primarily tropical warming represent in an idealized manner aspects of what occurs under global warming associated with increased greenhouse gas concentrations and under more local tropical warming associated with changes in the tropical energy budget, such as occur during El Niño. We find that the ITCZ generally shifts away from the equator under global warming and toward the equator under primarily tropical warming.
However, other factors not taken into account in our simulations, such as differential changes in aerosol loading between the hemispheres (e.g., Rotstayn and Lohmann 2002; Hwang et al. 2013), likely would modulate the results we obtain in an idealized setting.
2. Theory
Atmospheric energy balance and energy transport
Equation (3) provides a first-order quantitative basis for understanding the anticorrelation between cross-equatorial energy transport and ITCZ position seen in GCM simulations (e.g., Kang et al. 2008, 2009; Frierson and Hwang 2012; Donohoe et al. 2013). Additionally, it explains how and by how much the ITCZ can shift in response to equatorial changes that may not have a signature in cross-equatorial energy transport: If the net energy input to the equatorial atmosphere in the denominator of Eq. (3) changes, the sensitivity of the ITCZ position to cross-equatorial energy transport changes, and that alone can shift the ITCZ (see Fig. 1 for an illustration). For example, the increased net energy input to the equatorial atmosphere during El Niño by itself (apart from any change in cross-equatorial energy transport) implies an equatorward shift of the ITCZ according to Eq. (3), as is in fact observed (e.g., Dai and Wigley 2000; Berry and Reeder 2014). Changes in the net energy input to the equatorial atmosphere may also explain why the ITCZ shifts as tropical cloud parameterizations are varied in GCMs (Kang et al. 2008, 2009).
Qualitative behavior of the ITCZ position (large dots) as the northward cross-equatorial atmospheric energy flux 
Citation: Journal of Climate 27, 13; 10.1175/JCLI-D-13-00650.1
The linear approximation implicit in Eq. (3) gives the exact position of the zero of the atmospheric energy flux if the energy flux varies linearly with latitude in the vicinity of the equator (as illustrated in Fig. 1). Where this is inaccurate—that is, where the energy flux varies more strongly with latitude near the equator, as it does regionally—higher-order terms in the expansion, involving higher derivatives of the atmospheric energy flux near the equator, may be needed to improve the approximation. However, in the zonal and annual mean, Eq. (3) approximates the zero of the atmospheric energy flux to within 1°–2° accuracy, according to the data provided by Fasullo and Trenberth (2008).
Before relating interhemispheric asymmetries in energy fluxes to those in surface temperatures through a series of closure approximations, we illustrate and test the energetic constraints on the ITCZ position with GCM simulations.
3. GCM simulations
We use an idealized moist GCM to be able to test the validity of the theoretical Eqs. (3) and (4) over a wider range of climates than those of Earth’s recent past and proximate future. This allows us to delineate the range of validity of the expressions more clearly than would be possible by analyzing observations or simulating climate changes with a comprehensive GCM. With the idealized GCM, we can investigate separately the effects of tropical energy input changes and cross-equatorial energy flux changes, effects that usually occur together during ENSO or global warming. Our GCM is that of O’Gorman and Schneider (2008), which is similar to that used in Frierson et al. (2006) and Frierson (2007). The GCM uses a two-stream radiation scheme without clouds or aerosols, and the lower boundary consists of a slab ocean. We explore two different energy budget perturbations, one global and one tropical: (i) we vary the longwave optical depth of the atmosphere globally, leading to global temperature changes, with global-mean surface temperatures spanning the large range from 275 to 315 K, and (ii) we vary an imposed ocean energy flux convergence in a thin band around the equator, leading to primarily tropical temperature changes, with tropical surface temperatures spanning 303–308 K. (We have verified that this zonally symmetric surface heating has the same effect in the zonal mean as a zonally more localized surface heating with the same zonal mean, which would be more representative of El Niño.) For the atmospheric circulation to be hemispherically asymmetric, we also impose a hemispherically antisymmetric ocean energy uptake/release 
a. Global warming
The contours in Fig. 2a show typical mass flux streamfunctions for a cold and a warm case in the global warming scenario. In the warmer climate, the region of maximum low-level upward mass flux and the zero contour of the mass flux streamfunction are located farther away from the equator. This accompanies an increase in tropopause height and Hadley circulation width (cf. Schneider et al. 2010; O’Gorman et al. 2011; Levine and Schneider 2011). At the same time, the precipitation maximum (Fig. 2b) lies farther away from the equator in the warmer climate, as does the collocated maximum of the low-level upward mass flux. The precipitation maximum is also strengthened in the warmer climate, primarily because of the increased near-surface specific humidity in the tropics.
(a) Mass flux streamfunction and (b) precipitation for global-mean surface temperatures Tgl = 282 K and Tgl = 298 K in the global warming scenario. The contour interval in (a) is 1.6 × 1010 kg s−1. Red/solid contours show northward mass transport, and blue/dashed contours show southward mass transport. The maximum and minimum values of the streamfunctions are 7.8 × 1010 kg s−1 and −14.5 × 1010 kg s−1 at Tgl = 282 K and 7.1 × 1010 kg s−1 and −15.8 × 1010 kg s−1 at Tgl = 298 K. The black dots in (b) mark the precipitation maximum. The ITCZ generally is farther poleward in warmer climates, and the maximum precipitation is strengthened. In addition, the Hadley cells expand and the tropopause height increases. The zero of the mass flux streamfunction moves poleward with global-mean surface temperature but remains poleward of the precipitation maximum.
Citation: Journal of Climate 27, 13; 10.1175/JCLI-D-13-00650.1
Generally, as the longwave optical depth and global-mean surface temperature increase, the ITCZ and its tropical precipitation maximum shift monotonically away from the equator (Fig. 3a). Our estimate in Eq. (3) of the ITCZ position captures this shift to within ≲2° (Fig. 3a, black dots). If the cross-equatorial energy flux 
ITCZ position in GCM simulations under (a) global and (b) tropical warming. Colors show precipitation normalized by its global maximum, with contours from 0.9 to 1.0. Black crosses indicate where the moist static energy flux is zero. Black dots show the ITCZ latitude δ calculated from Eq. (3). Magenta dots show the approximate ITCZ latitude δ calculated from Eq. (8). The horizontal axes are (a) global-mean surface temperature and (b) equatorial surface temperature. The ITCZ shifts are qualitatively different in the two series of simulations: Under global warming, the ITCZ shifts are primarily caused by hemispherically asymmetric changes in extratropical latent energy fluxes and associated changes in cross-equatorial energy flux (cf. blue line in Fig. 1). Under tropical warming, the ITCZ shifts are primarily caused by changes to the net energy input to the equatorial atmosphere (cf. red line in Fig. 1).
Citation: Journal of Climate 27, 13; 10.1175/JCLI-D-13-00650.1
In the global warming simulations, changes in δ are primarily associated with hemispherically asymmetric changes in extratropical eddy energy fluxes (Fig. 5a). Changes in the net energy input to the equatorial atmosphere by themselves would imply an equatorward shift of the ITCZ as the climate warms (Fig. 6a). Their effect (2° equatorward ITCZ shift over the range of simulations) is overcompensated by the effect of the energy flux changes (7° poleward shift). In our simulations, the net energy input to the equatorial atmosphere increases with increasing global-mean surface temperature primarily because the equatorial top-of-atmosphere outgoing longwave radiation decreases as the temperature increases. This is because the atmosphere exports more energy from the tropics to the extratropics in warmer climates, thus increasing the radiative imbalance at the top of the atmosphere at the equator. If changes in δ are regressed onto changes in the cross-equatorial atmospheric energy flux 
Decomposing the cross-equatorial energy flux into components as in Eq. (4) shows that the principal reason for the monotonic poleward shift of the ITCZ under global warming lies in a strengthening of extratropical latent energy fluxes and their hemispherically asymmetric component, which overcompensate nonmonotonic changes in extratropical dry static energy fluxes (Fig. 5a). This is in agreement with earlier studies that identified the latent energy flux as the main driver of ITCZ shifts in CO2-doubling simulations in idealized setups (e.g., Hwang and Frierson 2010; Frierson and Hwang 2012). It is also consistent with the notion that latent energy fluxes dominate the poleward energy flux in warm climates (Pierrehumbert 2002; Caballero and Langen 2005; O’Gorman and Schneider 2008; Caballero and Hanley 2012), with preexisting asymmetries in latent energy fluxes amplifying as the climate warms (Held and Soden 2006).
The response of the ITCZ location to increases of the longwave optical depth likely depends on the strength of the imposed hemispherically antisymmetric ocean energy uptake/release 
b. Tropical warming
The contours in Fig. 4a show typical mass flux streamfunctions for a cold and a warm case in the tropical warming scenario. The region of maximum low-level upward mass flux and the zero contour of the mass flux streamfunction are located closer to the equator in climates with higher equatorial surface temperatures. At the same time, the Hadley circulation is narrower and stronger, resembling qualitatively (albeit not quantitatively) El Niño conditions on Earth (Seager et al. 2003). The precipitation maximum (Fig. 4b) is located closer to the equator. The strengthened maximum precipitation in the case with a warmer equatorial surface arises because the mass flux is strengthened and the near-surface specific humidity in the tropics is increased.
(a) Mass flux streamfunction and (b) precipitation for equatorial surface temperatures Teq = 303 K and Teq = 307 K in the tropical warming scenario. Colors and contours are as in Fig. 2. The maximum and minimum values of the streamfunctions are 6.9 × 1010 kg s−1 and −11.5 × 1010 kg s−1 at Teq = 303 K and 8.8 × 1010 kg s−1 and −17.8 × 1010 kg s−1 at Teq = 307 K. The black dots in (b) mark the precipitation maximum. The ITCZ is generally located closer to the equator in simulations with higher equatorial surface temperatures, and the maximum precipitation is strengthened. In addition, the Hadley cells contract. The zero of the mass flux streamfunction moves equatorward as equatorial surface temperatures increase, but it remains poleward of the precipitation maximum.
Citation: Journal of Climate 27, 13; 10.1175/JCLI-D-13-00650.1
Generally, as the imposed ocean energy flux convergence at the equator and with it tropical surface temperatures increase, the ITCZ and its tropical precipitation maximum shift toward the equator (Fig. 3b). Our estimate in Eq. (3) of the ITCZ position again captures this shift accurately, to within ≲2° (Fig. 3b, black dots). As for the global warming simulations, if the cross-equatorial energy flux 
Interhemispheric asymmetries in energy fluxes in GCM simulations under (a) global warming and (b) tropical warming. The interhemispheric asymmetries are measured by the arithmetic mean of the fluxes at the subtropical latitude where the mean moist static energy flux 
Citation: Journal of Climate 27, 13; 10.1175/JCLI-D-13-00650.1
Increasing the width of the tropical forcing does not change the response qualitatively as long at the impact on moist static energy fluxes associated with extratropical eddies remains small. This is consistent with the linear approximation implicit in Eq. (3) because the denominator only depends on the amplitude of the net energy input at the equator.
c. Implications
The simulations illustrate how cross-equatorial energy transport and the net energy input to the equatorial atmosphere act together to determine the ITCZ position. They demonstrate that changes in the ITCZ position do not need to be correlated with changes in cross-equatorial energy transport but can be associated with changes in equatorial net energy input alone.
In reality, most changes in ITCZ position likely are a superposition of changes in tropical energy input and cross-equatorial energy transport. Such changes can be caused by a variety of processes, from changes in cloud albedo and aerosol loading to changes in ocean upwelling. For example, an increased equatorial shortwave albedo (everything else fixed) reduces the net incoming shortwave radiation 
4. Closure approximations
While the expressions for the ITCZ position in terms of energy fluxes are relatively accurate, it is desirable to relate the ITCZ position to quantities that are more easily measured or inferred, in particular for climates of the past. To do so, we relate the relevant energy fluxes through a series of closure approximations to near-surface temperatures.
a. Atmospheric eddy energy fluxes
The eddy energy flux 
In our simulations, where there are only transient eddies, Eq. (5) captures the variations of the interhemispheric asymmetry in the dry static energy fluxes [first term in Eq. (5)] to within 15% in climates with global-mean surface temperatures ≲305 K (Fig. 5). We relate the latent energy flux, directed poleward and upward approximately along isentropes from the subtropics to the extratropics (e.g., Galewsky et al. 2005), to the dry static energy flux 
For all simulations presented, this closure for eddy moist static energy fluxes is essentially indistinguishable from diffusing moist static energy directly: that is, approximating 
b. Asymmetric energy input to tropical belt
The overall error in the cross-equatorial energy transport introduced by the various approximations does not exceed 30% over the range of simulations presented here. This is a relatively good approximation given that the interhemispheric asymmetry is a small difference between the large eddy fluxes in each hemisphere (see appendix B) and given that global-mean surface temperatures vary by 40 K over the range of simulations.
c. Equatorial energy input to the atmosphere
Net energy input to the atmosphere at the equator in GCM simulations under (a) global and (b) tropical warming. Dots and solid lines show the GCM results; dashed lines show approximations in which the outgoing longwave radiation is approximated [Eq. (7)]. The right axis shows the implied ITCZ position when the numerator in Eq. (3) is taken from a reference climate with a global-mean surface temperature of 288 K and an equatorial surface temperature of 304 K. The reciprocal of the net energy input at the equator is a measure of the sensitivity of the ITCZ position to interhemispheric asymmetries in energy fluxes. Under global warming, the changes in the net energy input by themselves would imply an equatorward shift of the ITCZ in colder climates and little change in warmer climates. In fact, however, the ITCZ shifts poleward (Fig. 3a), implying changed interhemispheric asymmetries in energy fluxes (Fig. 5a) dominate the ITCZ shift. Under tropical warming, the changes in the net energy input imply an equatorward shift of the ITCZ, which in fact occurs (Fig. 3b).
Citation: Journal of Climate 27, 13; 10.1175/JCLI-D-13-00650.1
d. ITCZ position
In our simulations, Eq. (8) captures much of the overall shift of the ITCZ. Under global warming, it captures the poleward shift well for global-mean surface temperatures between 280 and 300 K (Fig. 3a); however, the poleward shift for warmer climates is not estimated accurately because interhemispheric asymmetries in dry static and latent energy fluxes are not estimated accurately (Fig. 5a). Under tropical warming, it captures the equatorward shift of the ITCZ over the entire range of simulations (Fig. 3b).
5. Conclusions and discussion
We have shown that the ITCZ position in the zonal mean is approximately determined by two factors: the cross-equatorial energy transport in the atmosphere and the net energy input to the equatorial atmosphere [Eq. (3)]. Atmospheric energy fluxes are generally directed away from the ITCZ in the zonal mean, implying that the ITCZ is located close to a zero of the atmospheric meridional energy flux, as has been previously demonstrated (e.g., Kang et al. 2008, 2009). For an ITCZ not too far from the equator, so that the atmospheric energy flux can be approximated as varying linearly with latitude between the equator and the ITCZ, it then follows that a northward energy flux across the equator implies an ITCZ in the Southern Hemisphere and that a southward energy flux across the equator implies an ITCZ in the Northern Hemisphere. How far from the equator the ITCZ is located is controlled by the divergence of the atmospheric energy flux at the equator or by the slope of the energy flux as a function of latitude (Fig. 1). Because this is equal to the net energy input to the equatorial atmosphere (neglecting energy storage in the atmosphere), it is the net energy input to the equatorial atmosphere that controls the sensitivity of the ITCZ position to cross-equatorial energy transport. The hemispheric energy balance, in turn, shows how the cross-equatorial energy transport is related to interhemispheric contrasts in extratropical eddy energy fluxes and in the tropical energy input to the atmosphere [Eq. (4)].
Relating extratropical eddy energy fluxes and the tropical net energy input to the atmosphere through a series of closure approximations to surface temperatures, among other factors, we have shown how ITCZ variations associated with energetic changes can be related to temperature changes, both in the tropics and in the interhemispheric temperature contrast [Eqs. (8) and (9)]. This reconciles perspectives on the ITCZ position that have focused on energy fluxes with those that have focused on surface temperatures and interhemispheric temperature contrasts. It shows that these perspectives are not mutually contradictory but are in fact compatible with each other and complementary.
Our theory and simulations allow us to offer unified interpretations of seemingly disparate previous results. Paleoclimatological evidence and more recent observations and simulations show that the ITCZ has shifted in the past and suggest that the ITCZ can shift with variations in atmospheric greenhouse gas concentrations, aerosol loadings, and El Niño (Dai and Wigley 2000; Sachs et al. 2009; Hwang and Frierson 2010; Hwang et al. 2013; Berry and Reeder 2014). The anticorrelation between ITCZ position and cross-equatorial energy transport has been noted in many previous studies. On the basis of that anticorrelation, however, it has been suggested that for some of the observed and inferred ITCZ shifts—for example, during the Little Ice Age—large changes in cross-equatorial energy transport would be necessary (Donohoe et al. 2013). Our results instead suggest that smaller changes in cross-equatorial energy transport may account for the observed or inferred shifts when changes in the net energy input to the equatorial atmosphere are also considered. In other words, the correlation coefficient between the ITCZ position and the cross-equatorial atmospheric energy transport (or the interhemispheric temperature contrast) depends on climate. For example, in our global warming simulations, the ITCZ position depends approximately linearly on the cross-equatorial atmospheric energy transport, moving 3°–4° poleward for every 1-PW reduction in the energy transport, quantitatively consistent with previous studies (e.g., Donohoe et al. 2013). However, this overemphasizes the importance of the cross-equatorial energy flux and conceals the role of the equatorial net energy input to the atmosphere, which varies at the same time. If the net energy input to the atmosphere is kept fixed, the ITCZ position moves 4°–6° poleward for every 1-PW reduction in the energy transport, quantitatively consistent with our first-order estimate Eq. (3).
Previous modeling studies with aquaplanet GCMs with slab oceans have shown that the ITCZ is typically located in the warmer hemisphere because the atmosphere transports energy across the equator into the colder hemisphere (Broccoli et al. 2006; Yoshimori and Broccoli 2008; Kang et al. 2008, 2009; Frierson and Hwang 2012). In this picture, high-latitude changes (e.g., in ice cover) are communicated to the tropics by large-scale extratropical eddies and ocean circulations. Differential warming in one hemisphere, even far from the equator, can then drive ITCZ shifts toward that hemisphere (Chiang et al. 2003; Donohoe et al. 2013; Cvijanovic and Chiang 2013). Our approximate quantitative expression [Eq. (8)] for the ITCZ position shows how interhemispheric temperature differences relate to the ITCZ position: namely, primarily through interhemispheric differences in extratropical meridional temperature contrasts.
Because the ITCZ is not only controlled by interhemispheric asymmetries in energy fluxes that lead to cross-equatorial energy transport but also by the net energy input to the equatorial atmosphere, the ITCZ can shift even without cross-equatorial energy transport changes: for example, when there is reduced equatorial ocean energy uptake, such as during El Niño. This shows that also tropical processes alone (e.g., changes in cloud albedo or energy uptake by the equatorial oceans) can lead to ITCZ shifts. Thus, to understand future ITCZ shifts, it is important to understand not only changes in cross-equatorial energy transport, which can be remotely triggered, but also tropical changes in equatorial upwelling, El Niño, and the equatorial cloud and aerosol distributions, as they all enter the equatorial energy balance.
We have approached the question of what sets the position of the ITCZ from an energetic perspective, leading to results that depend on, among other factors, near-surface temperatures. This leaves open the question of what determines the temperatures. Answering that question requires consideration of the angular momentum balance of the atmosphere and how it controls the circulations that accomplish the atmospheric energy transport (e.g., Walker and Schneider 2006; Schneider 2006; Schneider et al. 2010; Levine and Schneider 2011). Additionally, the ITCZ defined by the precipitation maximum is not always collocated with the zero of the atmospheric energy flux, because the maximum upward mass flux and mean precipitation maximum are determined by the meridional derivative of the mass flux streamfunction and not necessarily by where the streamfunction (or the energy flux) vanish (Donohoe et al. 2013, 2014). A completely closed theory of the tropical precipitation maximum therefore would require understanding both the angular momentum and energy balances of the tropical troposphere. Additionally, for a theory that can also be applied to seasonal variations, it will be important to have a theory of seasonal ocean energy storage and release. Particularly for monsoon regions, it is questionable whether energy fluxes alone will give a sufficiently accurate picture of ITCZ variations (Chiang and Friedman 2012). Even when they do, it may become necessary to go beyond the linear approximations of energy fluxes on which we focused in this paper.
Acknowledgments
We thank Gerald Haug for stimulating discussions during the course of this research and Simona Bordoni and Robert Wills for helpful comments on drafts of the paper. Comments by two anonymous reviewers greatly helped to improve the quality of this paper. This work was supported by NSF Grants AGS-1019211, AGS-1049201, and AGS-1003614. Idealized GCM simulations were performed on Caltech’s Division of Geological and Planetary Science CITerra computing cluster.
APPENDIX A
Description of GCM and Simulations
The GCM has a simplified representation of the hydrological cycle and radiative transfer (Frierson et al. 2006; O’Gorman and Schneider 2008). It only takes into account the liquid–vapor phase transition, it has no ice phase, and the latent heat of evaporation is fixed at Lυ = 2.5 × 106 J kg−1. It uses a two-stream gray radiation scheme with prescribed and time-invariant optical opacity profiles.
The different components of the ocean energy flux divergence used in the idealized GCM experiments. The solid black line shows the symmetric ocean energy flux divergence with amplitude 50 W m−2 given by the term in Eq. (A4a). It is used in the tropical warming experiments. The dashed–dotted blue line shows the antisymmetric ocean energy flux divergence with amplitude 100 W m−2 that is used to drive the ITCZ off the equator in all experiments. Its functional form is given by the term in Eq. (A4b). The dashed red line shows the tropical ocean energy flux divergence with amplitude −100 W m−2 that is used in the tropical warming simulations. Its functional form is given by the term in Eq. (A4c).
Citation: Journal of Climate 27, 13; 10.1175/JCLI-D-13-00650.1
Moist convection is parameterized by a simplified quasi-equilibrium scheme that relaxes temperatures to a moist adiabat and specific humidity to a profile with a fixed reference relative humidity of 70% (O’Gorman and Schneider 2008; Frierson 2007). Large-scale condensation is parameterized so that relative humidity on the grid scale does not exceed 100%. The excess water is removed as precipitation, without reevaporation of condensate.
The GCM’s dynamical core integrates the primitive equations spectrally at a horizontal spectral resolution of T85 with 30 unevenly spaced vertical σ levels. The spinup time for the simulations is 2 yr from an isothermal resting state. Time averages are taken over 4 yr after spinup.
a. Global warming simulations
In the global warming simulations, we vary the longwave optical depth τ = ητref(ϕ) by varying the rescaling factor η between 0.4 and 6.0 (18 simulations), as in O’Gorman and Schneider (2008). This results in climates with global-mean surface temperatures between 275 and 315 K. The amplitudes of the symmetric and tropical ocean energy flux divergences are taken to be zero, 
b. Tropical warming simulations
In the tropical warming simulations, the optical depth is fixed at η = 1.0, resulting in climates with an Earth-like global-mean surface temperature of 288 K. The amplitude of the symmetric ocean energy flux divergence is fixed at 
APPENDIX B
Closure Approximations for Eddy Energy Fluxes
a. Closure 1: Diffusing dry static energy and latent energy separately
We made the following assumptions in this closure:
The vertically averaged eddy flux of dry static energy can be represented by near-surface fluxes, and the potential energy flux component
The near-surface eddy fluxes can be closed diffusively invoking an average eddy velocity.
The relevant length scale for the diffusive closure is given by the energy-containing length scale of the turbulent velocity field, and mean temperatures vary over the same length scale, assumed to extend from the subtropics to near the poles. (This justifies the use of pole to subtropics temperature differences provided changes in eddy length scales with climate are ignored.)
In addition to the assumptions made for the dry fluxes, we made the following assumptions for this closure:
The near-surface turbulent fluxes can be closed via eddy velocities and gradients along isentropes.
Gradients of relative humidity along isentropes are unimportant for gradients of specific humidity along isentropes.
b. Closure 2: Moist static energy diffusion
For the inferences in section 4, both closure schemes are equally useful. The choice of closure does not affect our conclusions.
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The moist static energy h = s + l is defined as the sum of dry static energy s = cpT + gz and latent energy l = Lυq, where Lυ is the latent heat of vaporization, q is the specific humidity, and other symbols have their usual meanings.
A slightly more accurate relation is obtained if the net energy input in the denominator is not evaluated at the equator but is averaged between the equator and the ITCZ; however, given the uncertainties in the inferred net energy input, this makes little difference in Earth’s atmosphere.
