1. Introduction
The evolution of zonal surface temperature gradients in the tropics is of interest in the study of future and past climates (e.g., Knutson and Manabe 1995; Fedorov et al. 2006). Recently, the changes in the zonally asymmetric component of the tropical overturning circulation, the Walker circulation, have been highlighted as a robust response to warming in climate change simulations with comprehensive GCMs (Held and Soden 2006; Vecchi and Soden 2007). Here, we examine the response of these zonally asymmetric aspects of the mean tropical climate to radiative perturbations in an idealized GCM and test theoretical arguments designed to account for their changes.
How zonal surface temperature gradients evolve as the climate warms is unclear. On one hand, increased cooling associated with the upwelling of cold water on the eastern ocean margins may lead to increased zonal temperature gradients in transient warmings (Clement et al. 1996), and this mechanism appears to be important in coupled GCM simulations (Held et al. 2010). On the other hand, paleoclimate evidence from the Pliocene suggests substantially weakened or collapsed zonal temperature gradients in a climate only somewhat warmer than Earth’s current climate (e.g., Fedorov et al. 2006). [However, see Wunsch (2009) for an alternative interpretation of the data.] The multimodel mean of the fourth Intergovernmental Panel on Climate Change (IPCC) assessment models shows little change in zonal temperature gradients in the Pacific basin over the twenty-first century (DiNezio et al. 2009). As a step toward understanding the full climate dynamics that control the zonal surface temperature gradient in the tropics, we develop a scaling relationship based on the surface energy budget and compare it to the results of the idealized GCM simulations.
The connection between changes in the hydrological cycle and large-scale atmospheric circulations in response to radiative forcing, for instance, by changed greenhouse gas concentrations, can be subtle. For example, the mass fluxes in the Hadley cell and extratropical eddy kinetic energy can vary nonmonotonically as a longwave absorber is systematically varied in an idealized GCM, with implications for water vapor fluxes and the distribution of evaporation and precipitation (O’Gorman and Schneider 2008a,b; Schneider et al. 2010; Levine and Schneider 2011).
Analysis of comprehensive GCM simulations suggests the connection between changes in the hydrological cycle and upward mass fluxes is straightforward (Held and Soden 2006; Vecchi and Soden 2007). The argument is that precipitation, which is constrained by the energy balance at the surface in the global mean, increases with warming more slowly than near-surface specific humidity, which increases similarly to the rate given by the Clausius–Clapeyron relation for small changes in relative humidity; thus, the upward vertical velocity that gives rise to condensation of water vapor weakens. This scaling argument is sometimes interpreted to account for changes in the Walker circulation (e.g., Vecchi et al. 2006), in part because changes in the zonally asymmetric component of the circulation are larger than those in the zonally symmetric component (Held and Soden 2006). A difficulty in applying the hydrological cycle argument directly to local circulations, such as the Walker circulation, is that the surface energy balance constraint on the precipitation is lost, as water vapor convergence depends locally on the atmospheric circulation. A central goal of this work is to assess the extent to which the hydrological cycle arguments apply locally to the Walker circulation, and to compare this to global-mean versions of the scaling arguments to determine the extent to which they are adequate.
Here, we use an idealized GCM to systematically examine the response of the zonally asymmetric tropical surface temperature and the Walker circulation to changes in the atmosphere longwave opacity. A scaling estimate based on the surface energy balance accounts for the changes in the zonal surface temperature gradient. We assess the hydrological cycle estimate for changes in the Walker circulation by applying it locally. The scaling estimate is accurate, though it requires knowledge of the precipitation field. In contrast, the global-mean version of the scaling estimate does not in general account for the Walker circulation changes.
2. Idealized GCM
a. Model description
We use an idealized, moist primitive-equation atmospheric GCM similar to those described in O’Gorman and Schneider (2008b) and Frierson et al. (2006). Briefly, the model includes an active hydrological cycle with a simplified Betts–Miller convection scheme (Frierson 2007). The scheme relaxes over a fixed time scale convectively unstable columns to moist pseudoadiabatic temperatures with constant (70%) relative humidity. It has a gray longwave radiation scheme, and a slab-ocean surface boundary condition with the heat capacity of 1 m of water. The top-of-atmosphere insolation is an idealized, annual-mean profile, and there is shortwave absorption in the atmosphere, both of which are as described in O’Gorman and Schneider (2008b). There is no condensed water phase in the atmosphere (i.e., there are no clouds). The climate is perturbed by varying the optical depth in the gray radiation scheme by a multiplicative factor to mimic variations in the trapping of longwave radiation due to variations in greenhouse gas concentrations. The primitive equations are integrated using the spectral transform method in the horizontal with T85 resolution (but the simulation results are similar with T42 resolution). Finite differences in σ = p/ps (with pressure p and surface pressure ps) discretize the vertical coordinate, with 30 unevenly spaced levels. The time stepping is semi-implicit, with a leapfrog time step of 300 s. The simulation statistics are averaged over the last 1100 days of the simulation following a spinup period of either 700 days for an isothermal, resting initial condition or 300 days for an initial condition from an equilibrated simulation with similar optical thickness. Aside from the horizontal resolution and time step, the only difference between the simulations presented here and those presented in O’Gorman and Schneider (2008b) is that we include a representation of ocean energy flux divergence in the slab ocean and present fewer simulated climates.
b. Prescribed ocean energy flux divergence
The surface boundary condition is a slab ocean that accounts for the radiative and turbulent surface fluxes. Additionally, it has a prescribed ocean energy flux divergence that represents the influence of the ocean circulation on the surface temperature. This is commonly referred to as a Q-flux in climate modeling.
The resulting Q-flux, shown in Fig. 1, has similar magnitude and spatial scale to reanalysis estimates (e.g., Trenberth et al. 2001, their Fig. 5).
Ocean energy flux divergence, Q-flux, for which positive values indicate cooling tendencies. (top left) Longitude–latitude Q-flux contours with contour interval of 20 W m−2, and negative values are dashed. The gray rectangles indicate the averaging regions. (right) Latitude vs zonal-mean Q-flux and (bottom) equatorial Q-flux (averaged within 8° of the equator) vs longitude.
Citation: Journal of Climate 24, 17; 10.1175/2011JCLI4042.1
c. Series of simulations
3. Zonal surface temperature gradients
a. Simulation results
Figure 2 shows the simulated zonal surface temperature differences averaged over the regions indicated in Fig. 1 for each of the equilibrated climates, which are identified by the time- and global-mean (denoted 〈·〉) surface temperature 〈Ts〉. Following O’Gorman and Schneider (2008b), the reference climate with α = 1.0 is indicated by a filled symbol. The east–west surface temperature difference ΔTs decreases nearly monotonically as the climate warms and varies by more than a factor of 3 over the simulated range of climates.
Zonal surface temperature difference (west minus east in Fig. 1) vs global-mean surface temperature of GCM simulations (circles with black dashed line), and scaling estimate (8) (gray line). The averaging conventions used to evaluate the scaling estimate are described in the text.
Citation: Journal of Climate 24, 17; 10.1175/2011JCLI4042.1
The simulated zonal temperature difference in the reference climate (1.4 K) is smaller than that in Earth’s current climate (~5 K). While we have used a Q-flux that has comparable magnitude to estimates for the current climate, the lack of clouds, and the concomitant zonal asymmetry in cloud radiative forcing leads to a smaller zonal asymmetry in the equilibrated surface temperature. One remedy would be to impose a larger asymmetry in the Q-flux since in quasi-equilibrium theory there is a near equivalence between surface fluxes and radiative forcing (Sobel et al. 2010, section 2.3). However, we chose not to amplify the Q-flux asymmetry. Therefore, it is more meaningful to consider fractional changes in the simulated zonal surface temperature contrast than the magnitude of the simulated changes. Our results hence may give changes in the surface temperature contrasts that, in reality, would likely be modified by processes we ignored, such as cloud–radiative feedbacks.
b. Scaling estimate
Figure 3 shows the surface energy budget as a function of longitude for the Earth-like reference simulation. Examining the variations in longitude of the surface energy balance reveals that the dominant balance in the zonally asymmetric component is between the prescribed Q-flux and the evaporative fluxes. This is to be expected in Earth-like and warmer climates where the Bowen ratio (ratio of sensible to latent surface fluxes) is small in the tropics. An analogous ratio can be formed for the net longwave radiation at the surface (Pierrehumbert 2010), and this is also small for sufficiently warm climates.
Surface energy fluxes of the reference simulation vs longitude averaged within 8° of equator. Here, “lw” and “sw” refer to the net longwave and shortwave radiative fluxes at the surface, respectively.
Citation: Journal of Climate 24, 17; 10.1175/2011JCLI4042.1
The decrease in the zonal temperature difference in a warming scenario can be anticipated by considering the counterexample of invariant zonal surface temperature gradients: if the east–west temperature difference remained the same as the mean climate warmed, in the absence of changes in near-surface relative humidity, the east–west difference in evaporative fluxes would rapidly increase with warming. The resulting zonal asymmetry in the evaporation would be too large to be balanced by an invariant ocean energy flux, and the other terms in the surface energy balance are too small to restore equilibrium. In a transient adjustment, the excess in evaporation would lead to a reduction in the east–west surface temperature asymmetry as, for example, the large evaporation over warm regions would cool the surface. The sensitivity of the evaporative fluxes to climate change and concomitant reduction in zonal surface temperature gradients with warming was documented in the GCM simulations of Knutson and Manabe (1995), who called it “evaporative damping.”
These approximations are adequate for the longwave radiation–induced climate changes in the idealized GCM simulations, though they may not be in general. For example, in analyses of climate change simulations, Xie et al. (2010) showed that changes in surface winds must be accounted for in the surface energy budget of some regions, and DiNezio et al. (2009) showed that cloud–radiative feedbacks affect the net surface shortwave radiation.
c. Assessment of scaling estimate
Figure 2 shows the scaling estimate (8) in gray. The derivative of the saturation specific humidity with respect to temperature in Eq. (8) is evaluated with the same simplified formulation for the saturation specific humidity as is used in the GCM, using the simulated time- and zonal-mean surface temperature averaged within 8° of latitude of the equator. Evaluated over the averaging regions in Fig. 1, Δqs varies by approximately 15% over the range of simulations, and we use the mean value, 2.4 × 10−3, in the scaling estimate. The magnitude of the variations in Δqs is consistent with the magnitude of the neglected terms in the bulk aerodynamic formula. Overall, there is close agreement (maximum deviations ≈ 15%) between the scaling estimate and the results of the GCM simulations.
We note that while the simulations and scaling estimate feature rapidly decreasing zonal temperature gradients as the mean temperature increases (7.4% K−1 at the reference climate), the decreases are not as rapid as proxies of the Pliocene suggest (~15%–30% K−1). But it is possible that the neglected radiative and ocean feedbacks increase the sensitivity to changes.
4. Walker circulation
a. Simulation results
Figure 4 shows the zonally asymmetric component of the time-mean pressure velocity, Dp/Dt = ω, in the longitude–sigma plane for three simulations.2 The zonally asymmetric pressure velocity is large over the regions of perturbed Q-flux; it is small elsewhere. The pressure depth over which it has large amplitude increases with warming because the tropopause height increases; its magnitude decreases with warming. Note that a consequence of the Walker circulation’s localization over the Q-flux perturbations is that the averaging convention we have chosen is adequate to capture nearly all of the ascending and descending mass fluxes—it is a closed circulation and alternative measures, such as surface pressure differences, yield similar results.
Contours of the zonally asymmetric component of the time-mean pressure velocity 
Citation: Journal of Climate 24, 17; 10.1175/2011JCLI4042.1
Figure 5 shows that the Walker circulation mass flux in the ascending region rapidly weakens with warming. The fractional rate of decrease is 4.4% K−1 at the reference climate relative to the global-mean surface temperature. (The fractional change is larger relative to local surface temperature changes as the warming is greater in high latitudes.) The Walker circulation varies by a factor of 6 over the simulated range of climates. In this figure, the zonally asymmetric component of the pressure velocity is evaluated on the sigma surface where it is maximum, which varies from σ = 0.60 in the coldest climate to σ = 0.32 in the warmest climate. These changes roughly follow those of the zonal-mean tropical tropopause (averaged within 8° of the equator), which rises from σ = 0.24 to σ = 0.03 determined by the World Meteorological Organization (WMO) 2 K km−1 lapse rate criterion. If the Walker circulation is evaluated using fixed sigma levels in the lower troposphere, the weakening with warming is more rapid owing to the combined changes in the vertical structure of the circulation and the changes, shown in Fig. 5, that are independent of the vertical structure.
(left) Zonally asymmetric component of the pressure velocity 
Citation: Journal of Climate 24, 17; 10.1175/2011JCLI4042.1
b. Scaling estimate
Schneider et al. (2010) argued that the free-tropospheric contribution is not generally negligible—that is, an actual difference in saturation specific humidity must be considered in Eq. (13). Physically, this can occur as the result of the amplification of warming aloft for moist adiabatic stratification. However, the amplification of temperature increases at fixed pressure levels can be offset by increases in the pressure depth of the mass fluxes, as occur with a rising tropopause. A concrete example of offsetting changes in stratification and pressure depth is given by the “Fixed Anvil Temperature” hypothesis (Hartmann and Larson 2002), which posits that the temperature that convection reaches is approximately climate invariant. If this hypothesis holds precisely, the free-tropospheric saturation specific humidity at the tropical tropopause would also be climate invariant. Schneider et al. (2010) also showed that a scaling that includes the free-tropospheric contribution to the saturation specific humidity difference accounted better for changes in the upward component of zonal-mean tropical circulation in simulations with zonally symmetric forcing and boundary conditions.
Whether the free-tropospheric water vapor contributes depends on the vertical velocity profile Ω(p) that is used to separate the integrand: 
c. Assessment of scaling estimate
The left panel of Fig. 5 shows that the zonally asymmetric component of the pressure velocity in the ascent region scales with the zonally asymmetric component of the upward pressure velocity: 
The right panel of Fig. 5 shows that the hydrological cycle estimate (13) applied locally accounts for the changes in the Walker circulation. There are two variants of the local scaling in the right panel of Fig. 5: one includes the free-tropospheric saturation specific humidity contribution to Δυqs (blue) and one that neglects it 
Figure 5 also shows the global-mean scaling estimate, −g〈P〉/〈qsb〉, in orange. This is similar to the convention used in the analysis of IPCC simulations in Held and Soden (2006) and Vecchi and Soden (2007), who further approximated the saturation specific humidity changes by linearizing its dependence on temperature about the present climate (i.e., δ〈qsb〉 ≈ 0.07 × δ〈Ts〉). The global-mean scaling captures the gross magnitude of the weakening but not the detail. It tends to underestimate the circulation changes for climates colder than the reference climate; at the reference climate, the global-mean scaling decreases by 1.3% K−1, compared with 
As the local scaling convention captures the variations in the Walker circulation with climate, we can decompose it to determine the relative contributions from the precipitation changes (Fig. 6, left) and the saturation specific humidity changes (Fig. 6, right panel). Two factors—approximately constant 
(left) Precipitation vs global-mean surface temperature for zonally asymmetric tropical-mean 
Citation: Journal of Climate 24, 17; 10.1175/2011JCLI4042.1
The zonally asymmetric component of tropical precipitation 
Changes in local precipitation cannot be understood independently of the circulation changes as local water vapor convergence depends, in part, on the circulation changes. In the simulations presented here, the changes in the zonally asymmetric component of net precipitation in the deep tropics have a large dynamic component, that is, a component associated with changed winds and fixed specific humidity. The dynamic changes reduce the zonally asymmetric precipitation due to weakening surface easterlies with warming and offset the thermodynamic component of the changes, that is, the component associated with changes in specific humidity and fixed winds, which increase the zonally asymmetric precipitation with warming. This is qualitatively consistent with the multimodel mean of the fourth IPCC assessment simulations presented in Fig. 7 of Held and Soden (2006): the equatorial Pacific is a region in which the changes in net precipitation are quite different from a thermodynamic estimate that assumes fixed circulation.
The saturation specific humidity is a rapidly increasing function of temperature, so, as expected, its inverse rapidly decreases with warming. The difference between the inverse saturation specific humidity when the free-tropospheric component of Δυqs is neglected and when it is retained is small. This is a consequence of the depth of the circulation—it reaches sufficiently low pressures and temperatures that qs is negligible in that region of the free troposphere. The appendix has a more extensive discussion of the dependence of the scaling estimate on the vertical structure of the circulation and how the results of the simulations presented here relate to previous work.
Figure 6 also sheds light on why the global-mean convention may, in some cases, appear adequate. Global-mean precipitation 〈P〉 increases more rapidly than the zonally asymmetric component of tropical precipitation 
5. Conclusions
Zonal surface temperature gradients decrease rapidly as the climate warms in the idealized GCM. This can be understood by considering a simplified surface energy budget that consists of a two-term balance between the zonally asymmetric component of the Q-flux and evaporation. It leads to a scaling for the zonal difference in tropical surface temperature that depends inversely on the rate of change of saturation specific humidity (or the Clausius–Clapeyron relation), evaluated using the mean surface temperature in the tropics if the Q-flux is fixed.
The Walker circulation rapidly weakens with warming in the idealized GCM. The changes can be accounted for using locally evaluated hydrological cycle scaling estimates. The changes in the scalings can be decomposed into (i) equatorial precipitation increases being approximately zonally uniform with warming and (ii) rapid, Clausius–Clapeyron increases in near-surface saturation specific humidity with warming. The global-mean version of the hydrological cycle scale estimate does not accurately account for the changes in the Walker circulation in all climates, though it does capture gross changes due to compensating changes in precipitation and saturation specific humidity that are larger than the locally averaged quantities.
That the hydrological cycle scaling estimate requires knowledge of local precipitation changes is a limitation. Ideally, one would want to understand circulation changes independently and use that result in the water vapor budget to constrain local precipitation changes. The moist static energy budget is one approach, though it has established difficulties (Sobel 2007). Another approach is that of the Lindzen–Nigam model (Lindzen and Nigam 1987), in which the surface temperature field is used to determine pressure gradients, which in combination with a linear momentum equation determine boundary layer convergence. We have solved the Lindzen–Nigam equations using the GCM-simulated surface temperature fields as forcing and found that the changes the Lindzen–Nigam model predicted in the boundary layer mass flux underestimated those simulated by the GCM by roughly a factor of 2 over the range of climates. Devising conceptual models of how tropical circulations change with climate that adequately account for GCM simulations remains a challenge.
A limitation of this study is the prescribed and invariant Q-flux. But understanding the atmosphere-only dynamics is a necessary step toward the more realistic case that includes an interactive ocean.
Acknowledgments
We thank Simona Bordoni, Isaac Held, Yohai Kaspi, Xavier Levine, and Paul O’Gorman for helpful discussions and technical assistance. The comments of two anonymous reviewers helped clarify the presentation of our work. This work was supported by a National Defense Science and Engineering Graduate Fellowship, a National Science Foundation Graduate Research Fellowship, and a David and Lucile Packard Fellowship. The GCM simulations were performed on Caltech’s Division of Geological and Planetary Sciences Dell cluster. The program code for the simulations, based on the Flexible Modeling System of the Geophysical Fluid Dynamics Laboratory, as well as the simulation results themselves are available from the authors upon request.
APPENDIX
Dependence of Water Vapor Scaling on Vertical Structure
To understand why the free-tropospheric contribution to the saturation specific humidity difference is negligible, we examine the vertical structure of the vertical velocity, as Δυqs is an approximation of {Ω(p)∂pqs}. Also, the vertical structure of the circulation has importance beyond the hydrological cycle scaling estimate: it determines whether moist static energy is imported or exported by overturning circulations (Sobel 2007; Back and Bretherton 2006; Peters et al. 2008). If the vertical velocity profile has larger amplitude in the free troposphere, it makes the saturation specific humidity difference less sensitive to the free-tropospheric values entering Δυqs because temperature and saturation specific humidity are small there. Indeed, the vertical velocity profile is “top-heavy” in the ascending branch of the Walker circulation (solid lines in Fig. A1); that is, it has significant amplitude in the upper troposphere. In contrast, the zonal-mean vertical velocity profile has more amplitude near the surface and less amplitude in the upper troposphere (dashed lines in Fig. A1). The vertical velocity profiles of the idealized simulations are broadly consistent with the regional variations in the vertical velocity profile in Earth’s tropics (Back and Bretherton 2006).
Vertical structure of vertical velocity Ω(σ) = ω↑/min(ω↑) for simulations with 〈Ts〉 = 289.9 K (black) and 295.3 K (gray). Solid lines are zonally asymmetric pressure velocity profiles 
Citation: Journal of Climate 24, 17; 10.1175/2011JCLI4042.1
For the zonal-mean component of the simulations presented here, the hydrological cycle scaling estimate that neglects the changes in the saturation specific humidity away from the boundary layer 
Figure A1 also shows that the pressure depth over which there is significant vertical velocity amplitude increases and the tropopause pressure decreases with warming. For this reason, we vary the pressure at which the free-tropospheric contribution to Δυqs is evaluated in Fig. 5; neglecting these changes can potentially lead to different scaling behavior.
There are two ways in which the vertical velocity profile can lead to Clausius–Clapeyron-like scaling of Δυqs. First, a larger pressure depth decreases the free troposphere’s weight (e.g., in the limit of the whole troposphere, the tropopause saturation specific humidity is orders of magnitude smaller than that of the surface). Second, holding the pressure depth fixed and shifting it upward (e.g., the difference between 800 and 300 hPa compared to the difference between 900 and 400 hPa) leads to more rapid changes in the saturation specific humidity difference as the temperatures are lower and the rate of change of the saturation specific humidity is larger. These changes will appear closer to the rate given by the Clausius–Clapeyron relation if it is evaluated at a fixed level such as the surface (related issues about the sensitivity of saturation specific humidity to where it is evaluated and the possible imprecision that results are discussed in O’Gorman and Muller 2010).
In an analysis of comprehensive GCMs, Vecchi and Soden (2007) showed that changes in the global-mean upward component of the vertical velocity scaled with the hydrological cycle scaling estimate that neglects the free-tropospheric contribution to the saturation specific humidity (i.e., 
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We are not suggesting that this scaling holds in general [see, e.g., O’Gorman and Schneider (2008b) and our Fig. 6 for a counterexample in the global mean] but only for the zonal difference in the tropics.
All figures show psDσ/Dt, which differs only slightly (~0.1% in the region of interest) from Dp/Dt as σDps/Dt is small.
