1. Introduction
The Hadley circulation is one of the largest and most defining features of Earth’s atmosphere, with the boundary between the two Hadley cells, the intertropical convergence zone (ITCZ), containing abundant precipitating convection. The ITCZ exhibits a pronounced seasonal cycle in latitude; when it passes over land in local summer, it is associated with changes in winds and rain typically identified as monsoons. For regions on the fringes of the ITCZ’s seasonal range, such as Africa’s Sahel, small shifts in the ITCZ’s summer latitude can bring large changes in monsoon precipitation.
The ITCZ was historically thought to be locally controlled by tropical SST and land surface properties (e.g., Xie 2004). Convective quasi-equilibrium (CQE) theories for monsoon location (e.g., Emanuel 1995; Privé and Plumb 2007) typically operate in this paradigm and predict that the time-mean ITCZ is constrained to lie just equatorward of the maximum subcloud layer moist static energy. However, the ITCZ and monsoon precipitation have been shown to respond strongly to remote, high-latitude thermal forcings (e.g., Chiang and Bitz 2005), and the vertically integrated atmospheric energy budget has been used to show how ITCZ shifts are part of the anomalous meridional energy fluxes needed to balance such remote forcings (Kang et al. 2008, hereafter K08). This has led to two distinct yet not incompatible theories for ITCZ location—CQE and the vertically integrated energy budget—becoming prominent in recent years. Yet existing forms of CQE and energy budget theories for monsoons, discussed in detail below, assume that the large-scale circulation is horizontally convergent in the lower troposphere and divergent in the upper troposphere, as in a first-baroclinic mode. This assumption does not hold in most monsoon regions because dry, shallow circulations originate over adjacent deserts and penetrate into the precipitating monsoon domain (e.g., Zhang et al. 2008; Nie et al. 2010).
This study examines how well CQE and energy budget theories diagnose ITCZ location for an idealized monsoon adjacent to a desert. We begin, in the remainder of this introduction, by discussing details of CQE and energy budget theories for monsoon location, and then review their application to the example of the West African monsoon.
a. Convective quasi-equilibrium theories for monsoons
Moist convection consumes convective available potential energy (CAPE) and restores the temperature of the convecting layer to a vertical profile near that of a moist adiabat. CQE theories for convectively coupled large-scale circulations (Arakawa and Schubert 1974; Emanuel et al. 1994; Emanuel 2007) assume that this process is fast compared to the generation of CAPE by buoyancy forcings, so that on longer time scales the temperature of the convecting layer covaries with the moist static energy h of air below the base of cumulus clouds. More precisely, changes in the saturation moist static energy of the free troposphere, 
When combined with constraints on the Hadley circulation, these ideas impose a vertical structure on monsoon circulations, yielding a two-dimensional dynamical system tightly coupled to 
b. Energy budget theories for monsoons
Clear demonstration that ITCZ location could be strongly affected by an imposed high-latitude thermal forcing (Chiang and Bitz 2005) opened a new line of inquiry into the cause of ITCZ shifts [reviewed by Chiang and Friedman (2012)]. Even if the location of the ITCZ is consistent with the location of the maximum in SST or 
In the tropical atmosphere, the total energy content, which is well approximated by h, tends to be slightly higher in the upper troposphere than in the lower troposphere so that the vertically integrated energy flux has the same direction as upper tropospheric winds (e.g., Sobel 2007). The zonal mean, vertically integrated energy flux is thus expected to be directed away from the ITCZ, which has been called the energy flux equator (EFE): the latitude at which this energy flux is zero. An anomalous energy sink in the high latitudes of one hemisphere would require anomalous convergence of the vertically integrated energy flux and, if there is no anomalous flux convergence in the tropics or subtropics, a meridional shift in the EFE and thus in the ITCZ toward the opposite hemisphere. K08 and Kang et al. (2009, hereafter K09) developed this energy budget theory for ITCZ location and showed that it quantitatively described ITCZ shifts in an idealized global model, although cloud feedbacks in their model complicated what might otherwise be a prognostic theory for ITCZ location given an imposed energy source.
Although this energy budget theory does not impose a first-baroclinic structure on winds, as is done in strict versions of CQE via the assumption of a moist adiabat, it does assume that anomalous forcings do not change the relationship between the directions of the energy and mass fluxes. More precisely, energy budget theories for ITCZ location assume that there are only small changes in the gross moist stability (GMS; Neelin and Held 1987), which is a ratio that relates a circulation’s energy transport to its mass transport. Changes in the vertical structure of winds, temperature, or moisture that alter the efficiency of the circulation at transporting energy can produce a change in energy transports without a change in the circulation, or vice versa, which is equivalent to a change in the GMS (e.g., Merlis et al. 2013a). Although different definitions of the GMS have been proposed (e.g., Sobel 2007; Raymond et al. 2009), the GMS is generally positive when the time-mean circulation diverges energy away from heavily precipitating regions. The east Pacific ITCZ is a notable example because ascent there is typically shallow, in contrast with the top-heavy ascent found in a first-baroclinic mode, and is associated with a negative GMS and a time-mean circulation that converges energy into the ITCZ (Back and Bretherton 2006; Peters et al. 2008).
c. Example of the West African monsoon
The African Sahel lies on the poleward edge of the boreal summer ITCZ and receives most of its annual precipitation during the monsoonal ITCZ migration. Variability in Sahel rainfall has been large on interannual and decadal time scales, with dry years associated with equatorward shifts of the ITCZ. Early work found that Sahel rainfall variability was strongly correlated with an interhemispheric SST difference, with drought associated with colder Northern Hemisphere and warmer Southern Hemisphere SSTs (Folland et al. 1986; Palmer 1986; Janicot et al. 1996; Giannini et al. 2003; Lu 2009). This is qualitatively consistent with energy budget theories for monsoon location (Chiang and Friedman 2012), assuming that warmer SSTs are associated with larger surface energy fluxes into the atmosphere. Sahel precipitation variability is also consistent with CQE theories, which show that the 
Yet there are dynamics in West Africa that are not captured by either the CQE or energy budget theories discussed above. Sahel precipitation is correlated with surface pressure, surface air temperature, and low-level geopotential height over the Sahara in observations and in a suite of GCMs (Haarsma et al. 2005; Biasutti et al. 2009). The circulation over the Sahara is dominated by a dry, shallow heat low circulation (e.g., Rácz and Smith 1999), with near-surface horizontal winds converging about 1000 km north of the peak monsoon precipitation, ascent reaching to at least 3–4 km altitude, and divergence in the lower middle troposphere (Zhang et al. 2008). The diverging air is very dry, and one branch of this outflow from the heat low is directed toward the precipitating ITCZ at about 700 hPa. Existing CQE and energy budget theories for monsoons do not include the effects of a proximal desert heat low and its shallow divergent circulation. The shallow circulation is clearly not captured by a first-baroclinic mode, and outflow from the heat low could cause deviations from the moist adiabatic structure on which CQE is based. The outflow could also suppress deep moist convection, since convection is sensitive to lower tropospheric moisture in ways not captured by the CQE assumption that 
d. Goals
Here we examine the degree to which existing CQE and energy budget theories can describe forced variations in an idealized model of a monsoon that includes an adjacent desert. Previous studies have used idealized models of the African monsoon to assess the response to forcings, and those by Peyrillé et al. (2007) and Peyrillé and Lafore (2007) are particularly notable. Although Peyrillé et al. (2007) did not discuss energy budget theories for monsoon location, their results are consistent with the expectation that the ITCZ will move toward an anomalous energy source and away from an anomalous sink created by SST, albedo, or aerosol forcings. In the same model, Peyrillé and Lafore (2007) found that the horizontal advection of temperature and humidity accomplished by the Sahara’s shallow heat low circulation played an important role in setting the latitude of the monsoonal ITCZ. This confirms the importance of the shallow circulation in the case of the West African monsoon, motivating examination of CQE and energy budget theories in which such shallow circulations have not been considered.
Although we designed our idealized model using the West African monsoon as a reference, we expect the results to be relevant to other monsoons that lie adjacent to deserts. Australia, southern Africa, North America, and South Asia are all regions in which a precipitating convergence zone migrates poleward to the edge of a subtropical desert during local summer, although some of these regions have less zonal symmetry than West Africa in the monsoon–desert geometry (e.g., Nie et al. 2010). Shallow heat low circulations have been observed in all of these regions, and Trenberth et al. (2000) found 20% of the annual cycle variance of the global divergent mass flux to be associated with a shallow mode. Furthermore, meridional flow often deviates from a simple first-baroclinic mode in reanalysis estimates of the zonal mean Hadley circulation (Dima and Wallace 2003) and in idealized models (Nolan et al. 2007).
The next section details our model configuration and methods. Section 3 then shows how traditional CQE constraints fail to diagnose the location of the ITCZ and provides one possible way to restore their validity. That section also shows that the energy flux equator and ITCZ are collocated across a wide range of forcings, but that linearizations of energy transports across the geographic equator provide a poor estimate of ITCZ location. We close with a discussion of caveats and implications.
2. Methods
a. Model details
The fully nonhydrostatic Weather Research and Forecasting (WRF) Model, version 3.3 (Skamarock et al. 2008), is used in a three-dimensional configuration with 15-km horizontal resolution and 41 vertical levels between 0- and 30-km altitude. The model was modified to be on an equatorial β plane and to have no seasonal cycle with insolation fixed at its 15 July distribution. The diurnal cycle was retained. A sponge layer was used above 20 km to absorb upward propagating gravity waves. All integrations were spun up for 3 months, followed by 360-day integrations (corresponding to four 3-month summer seasons). Third-order Runge–Kutta with adaptive time stepping is used with an average time step of about 40 s.
We used standard WRF parameterizations of subgrid-scale physics, including the Kain–Fritsch convective parameterization (Kain and Fritsch 1990; Kain 2004) and a six-class bulk microphysics scheme based on Thompson et al. (2008). The latitude and width of the precipitation maxima are sensitive to the choice of convection scheme, but a separate integration conducted at 4-km horizontal resolution without parameterized deep convection produced a continental precipitation maximum with a similar position and width to that obtained with the Kain–Fritsch scheme (not shown). Radiative transfer was represented by the Goddard schemes, which divide the longwave and shortwave spectra into 8 and 11 bands, respectively (Chou and Suarez 1999, 2001). Vertical diffusion of heat, moisture, and momentum were handled by the Yonsei University planetary boundary layer scheme (Hong et al. 2006).
We use a simple thermal diffusion land surface model with five layers and prescribed soil moisture, thermal inertia, albedo, and roughness (Dudhia 1996). A deep soil temperature of 287 K was imposed. Although it is somewhat unusual to use a model with prescribed soil moisture for climate studies, Xie and Saiki (1999) show that interactive soil moisture introduces unforced internal variability on seasonal and longer time scales in monsoons, which is especially problematic for simulations with perpetual summer insolation. Although the effect of soil moisture on monsoons is an interesting problem in its own right (e.g., Douville et al. 2001; Douville 2002; Koster 2004), this is peripheral to our central question of whether CQE and energy budget theories describe the time mean monsoon state. Since there is no obvious reason to expect soil moisture–precipitation interactions to alter the relationship between ITCZ location and the distributions of 
We prescribed a quasi-sinusoidal SST (Fig. 1), which was chosen to approximate average boreal summer SST. It is an adjusted version of the “Control-5N” profile from Neale and Hoskins (2000), modified to reach a maximum of 28°C from 2.5°S to 5°N. Previous studies of the relation between ITCZ location and vertically integrated atmospheric energy fluxes typically used interactive SST, most commonly with a dynamically passive slab ocean (e.g., K08). Kang and Held (2012) compared the results of simulations with fixed SST and a slab ocean in detail. They illustrated how the energy budget in simulations with a slab ocean is entirely determined by top of atmosphere radiative fluxes, whereas use of fixed SST introduces surface energy fluxes that imply subsurface sources and sinks of energy. These implied sources and sinks need not balance in the global mean and can result in differing ITCZ shifts in fixed SST simulations as compared to slab ocean simulations when the same forcing is applied. However, using fixed SST allows us to crudely represent energy tendencies associated with dynamical ocean heat flux convergence, and we show this may be particularly important in near-equatorial regions in our model and in observations. Furthermore, atmospheric energy transports must still be consistent with the Hadley circulation, and there is no obvious reason to believe that the relationship between the ITCZ and the EFE would change because of the use of prescribed SST, which is consistent with the results of Kang and Held (2012).
(a) Prescribed surface albedo and (b) prescribed SST. Control integration is shown in black, and applied forcings are shown in colors, according to legend.
Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-15-0747.1
Vegetation and land surface albedo in West Africa have a large degree of zonal symmetry, and although there is some topography, the continent is relatively flat. So we specify zonally symmetric lower boundary conditions in an equatorial β plane spanning 70°S–70°N, 10°W–10°E (the domain has 1038 meridional and 149 zonal grid points at 15-km resolution). We used periodic zonal boundary conditions and closed meridional boundary conditions. Ocean exists south of 5°N and north of 32°N. From 5° to 12°N, there is grassland with standard United States Geological Survey (USGS) surface properties specified in the WRF Model. From 12° to 32°N, USGS surface properties for desert were specified. We modified the desert albedo to be 0.32, which is roughly similar to the albedo of the Sahara. We also reduced desert soil moisture to the atypically low value of 1 kg m−3; higher values produced large precipitation over the desert because of our use of specified soil moisture and perpetual summer insolation. This choice produces reasonable sensible and latent heat fluxes over the desert when compared with reanalysis data. The entire continent was set to an elevation of 1 m above sea level.
We conduct a set of integrations (Table 1) in which we force the model with either anomalous prescribed albedo or anomalous prescribed SST. A subset of these forcings is illustrated in Fig. 1, where the SST and albedo for the control (CTL) integration are shown in black. Integrations designated as DA±α have the broadband shortwave desert land surface albedo increased or decreased by α relative to the CTL integration, with α expressed as a percentage. SST anomalies that are Gaussian in latitude with standard deviation 2° and amplitude of −2 K and +2 K are applied in equatorial cold tongue (ECT2K) and midlatitude warm pool (MWP2K) integrations, respectively. Additional integrations were conducted with the amplitude of these anomalies set to 4 K. The SH2K integration uses SST increased by 2 K south of the continent, with most of this SST perturbation lying in the Southern Hemisphere. Finally, GRASS imposes a grassland across the whole continent, from 5° to 32°N, and is equivalent to DA−20 but with increased prescribed soil moisture from 12° to 32°N. Although some of these integrations test states are very different from the current state of West Africa, they provide a test of energy budget and CQE diagnostics of ITCZ location. Some integrations excluded from Fig. 1 only apply an albedo forcing over the northern (22°–32°N) or southern (12°–22°N) half of the desert. Others apply more than one forcing: CEG2K applies a cross-equatorial gradient of 2 K by superimposing the ECT2K and MWP2K forcings, and AFRICA superimposes ECT2K and DA+10 forcings. The AFRICA integration is so named because its climate most closely resembles that observed over West Africa.
Model integrations performed and their respective forcing(s).
b. Convective quasi-equilibrium metrics
c. Energy budget metrics
3. Results
a. Basic state circulation
The time and zonal mean mass streamfunction in the control integration consists of a deep, cross-equatorial Hadley cell with its ascent branch centered near 11°N, superimposed on a shallow meridional circulation with an ascent branch centered between 15° and 20°N (Fig. 2a). The streamfunction was calculated using the method of Döös and Nilsson (2011), which includes the effects of fluctuations in the surface pressure, a small but important contribution over the desert that prevents nonzero contours from intersecting the ground. The winter Hadley cell in the control integration extends from 20°S to 11°N and is stronger than expected from reanalyses (e.g., Mitas 2005; Merlis et al. 2013b); this is unsurprising because the off-equatorial continent occupies the entire zonal extent of the model. The summer Hadley cell is visible primarily in the upper troposphere between 10° and 30°N, with the heat low’s shallow meridional circulation dominating between the surface and ~650 hPa at those latitudes. Much of the time-mean outflow from the heat low is directed toward the ITCZ, advecting hot, dry air into that precipitating region. Other deviations from a first-baroclinic mode structure are seen, with additional inflow to and outflow from the ITCZ occurring near 350 and 450 hPa, respectively, consistent with the multilevel flows found by Nolan et al. (2010) in observations and in models.
Zonal and time averages. (a) Mass streamfunction (
Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-15-0747.1
A midlevel easterly jet is centered at ~500 hPa with core velocity of 12 m s−1 (Fig. 2b). This is analogous to the African easterly jet (AEJ); although it is positioned at a slightly higher altitude than in observations (~600 hPa), it is similar in strength and location (Nicholson 2013). Separate model integrations showed that the AEJ was unrealistically strong when the domain width was decreased below 10° longitude, seemingly due to insufficient momentum transports by barotropic/baroclinic eddies. A low level off-equatorial westerly jet is centered near the ITCZ, as expected in a monsoon.
The rainfall distribution shows two peaks (Fig. 2d), one associated with the ITCZ around 11°N, and a secondary peak just south of the equator associated with an equatorial “jump” in the zonal mean mass streamfunction. Pauluis (2004) explores the equatorial jump in detail, arguing that because the Coriolis force is small near the equator, friction must be balanced by a low-level cross-equatorial pressure gradient. In the absence of such a gradient, air cannot cross the equator in the boundary layer and must ascend to cross in the free troposphere. Ascent in the equatorial jump seems to be deeper in models than in observations, as evidenced by the frequency of occurrence of double ITCZs (e.g., Privé and Plumb 2007; Zhang 2001), and our model is no exception. We are interested primarily in the continental monsoon precipitation distribution, and define 
b. Convective quasi-equilibrium evaluation
In CQE theories of monsoons (e.g., Emanuel 1995; Privé and Plumb 2007), the ITCZ lies just on the equatorial side of the maximum subcloud moist static energy, 
Values of (a) 
Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-15-0747.1
There are large differences in vertical velocity and h in the free troposphere above these two 
(a) Time-mean pressure velocity (hPa h−1) and (b) h at the three latitudes indicated, corresponding to the southward 
Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-15-0747.1
We suggest that the propensity for nearly all moist convective plumes to entrain free tropospheric air (e.g., Romps 2010) makes h above the subcloud layer relevant to CQE diagnostics of ITCZ location. Although 
Precipitating convection in our model thus seems to be fairly sensitive to free-tropospheric humidity (and consequently h), in contrast to many global models with O(100 km) resolution (e.g., Derbyshire et al. 2004). This sensitivity may arise from the moderately high resolution of our model (15-km horizontal grid spacing), permitting some mesoscale convective organization, as well as a convective parameterization that incorporates a representation of enhanced convective entrainment. The Kain–Fritsch scheme (Kain and Fritsch 1990; Kain 2004) is a mass flux parameterization in which convective updrafts are represented as an entraining/detraining plume. Lateral entrainment is represented by creation of an ensemble of mixtures of in-cloud and environmental air, which are then sorted by buoyancy, with positively buoyant parcels continuing upward in the plume and negatively buoyant parcels detraining. Older versions of the parameterization produced an increase in cloud-top height as environmental humidity decreased, counter to the sensitivity seen in cloud-resolving models; that bias was attributed to insufficient entrainment (Jonkers 2005). Modern forms of the Kain–Fritsch scheme, including the version in the WRF Model version 3.3, require the mixing that occurs before the buoyancy sorting to incorporate at least 50% environmental air (Kain 2004; de Rooy et al. 2013). Although biases in how the Kain–Fritsch scheme represents entrainment surely exist (e.g., de Rooy et al. 2013), the relatively high resolution of our model and the enforced minimum entrainment in its convection scheme likely enhance its sensitivity to free-tropospheric humidity.
We now turn our attention to how applied forcings modify the control state. As a representative sample, we consider the control and four other integrations: 2-K equatorial cold tongue (ECT2K), 2-K midlatitude warm pool (MWP2K), and two desert albedo ±0.10 forcings (DA±10). Previous idealized studies of the Sahel (Peyrillé et al. 2007) have found the precipitation maximum shifts in response to local and remote forcings, and our results support this claim (Fig. 5a), as 
(a) Precipitation with 
Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-15-0747.1
These characteristics of the 
Precipitation distributions for all model integrations shown in shading, with colors identical to Fig. 2d. Integrations were sorted according to 
Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-15-0747.1
The improved correspondence between 
The center of the desert heat low moves poleward as the ITCZ moves poleward in response to forcings that cool the winter hemisphere and warm the northern subtropics (Fig. 6). Dry, midtropospheric outflow from the desert’s shallow meridional circulation is stronger when the ITCZ is located closer to the equator, and this shallow circulation and its outflow vanish in the GRASS integration (where the entire desert is replaced with grassland). In that integration the maxima of 
c. Energy budget evaluation
Shifting focus to the vertically integrated energy budget, we see that the energy flux equator, 
(a) AET for the control integration. (b) Decomposition of the TADVH into mean flow and eddy components MADVH and EADVH respectively. (c) Decomposition of the mean flow MADVH into horizontal and vertical components HADVH and VADVH respectively. Import of energy into the column is considered positive.
Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-15-0747.1
Although we do not focus on diagnostics of the GMS, we note that its sign depends on the definition used. If the GMS is assumed to be a ratio of AET to the meridional mass flux above 500 hPa, as in Hill et al. (2015), then the GMS is positive in the domain of the Hadley cell (and is poorly defined, by construction, in the ITCZ). In contrast, if the GMS is defined as the ratio of TADVH to some weighted average of the upward motion, as in Peters et al. (2008), then the GMS is positive between the equator and the poleward edge of the continental ITCZ, but is negative in the winter branch of the Hadley cell. If the GMS is defined as the negative of the ratio of energy flux convergence to moisture flux convergence (Raymond et al. 2009), the region between the equator and the ocean–grassland coast has a negative GMS. And defining the GMS in terms of the vertical advection of h by the time-mean ω, as in Neelin and Held (1987) and Sobel (2007), provides a poor approximation to the actual flow energetics because mean vertical advection accounts for only a small part of TADVH in many parts of the domain (Fig. 7c). Mean vertical advection is smaller than mean horizontal advection (i.e., 
Decomposition of TADVH into its various components reveals other notable features. Mean flow advection produces most TADVH in the ITCZ and throughout nearly all of the winter Hadley cell (Fig. 7b), but the relative importance of mean horizontal advection and mean vertical advection changes with the region (Fig. 7c). One exception lies at the ocean–grassland boundary, where a pair of oppositely signed peaks in mean and eddy advection (Fig. 7b) are associated with a strong land/sea breeze cell—mean flow advection is negative there because the mass flux crosses strong horizontal and vertical gradients of h in the lower troposphere (Fig. 2c). Interestingly, eddy advection produces most TADVH over the desert, from 12° to 32°N, where mean flow advection is of opposite sign to eddy advection and TADVH (Fig. 7b). This likely occurs because subsidence in the summer Hadley cell converges energy into the desert region, primarily in the middle and upper troposphere, while transient eddies balance the net column energy source provided by surface enthalpy fluxes and radiation. Mass streamlines of the shallow meridional circulation over the desert are largely parallel to h contours, indicating that this circulation contributes little to the vertically integrated energy budget in that region (consistent with the opposing signs of mean horizontal and mean vertical advection there).
The northward atmospheric energy transport for the same set of integrations examined in the previous subsection shows that the EFE shifts away from imposed anomalous atmospheric energy sinks and toward imposed sources (Fig. 8). Furthermore, the ITCZ lies within two degrees latitude of the EFE (cf. Figs. 5a and 8a), so the EFE provides a relatively successful diagnostic of ITCZ position. If we take the control integration EFE latitude to be 
(a) AET for five representative model integrations according to legend. A 3× zoomed inset plot from 5° to 15°N shows the location of 
Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-15-0747.1
Previous studies have inferred the position of the EFE and ITCZ from the amount of cross-equatorial energy transport (e.g., D13), but the nonmonotonicity in AET near the equator in our model complicates this sort of estimation. The relationship between EFE latitude and AET at the geographic equator is far from linear (Fig. 9) because our forcings produce large changes in slope of the near-equatorial AET (Fig. 8a). However, AET slope near the EFE itself changes little in response to the forcings. This motivates linearizing the AET about 
Atmospheric energy transport across the physical equator (diamonds) and across the control integration energy flux equator (squares) at 10°N plotted against 
Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-15-0747.1
The AET across 
The near-equatorial atmospheric energy sink in our model has a real-world analog that makes the linearization about 
ERA-Interim 1979–2014 June–September (JJAS) 
Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-15-0747.1
Decomposition of 
Column integrated longwave heating 
Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-15-0747.1
For the 2-K equatorial cold tongue (Fig. 11c), the magnitude of the feedback is approximately 6 times that of the forcing. This is largely driven by changes in mean surface winds which induce changes in ocean evaporation (e.g., Numaguti 1995; Boos and Emanuel 2008), although there is also an increase (50 W m−2) in outgoing longwave radiation caused by a local decrease in column water vapor. The large subtropical feedback on the 2-K midlatitude warm pool forcing (Fig. 11d) is also due almost entirely to changes in ocean surface evaporation, but the fact that the mean winds change little (not shown) suggests that these changes are driven by gustiness, perhaps on synoptic scales. Wind–evaporation feedbacks may be overly strong in models with prescribed SST, but wind-induced changes in upwelling or in dynamical ocean heat transports could complicate matters further (e.g., Webster and Fasullo 2003) and would not be represented in models that used a slab ocean. This makes it difficult for our experiments to either support or provide a counterexample to the idea that high-latitude forcings are more effective than tropical forcings at causing shifts in the ITCZ (Seo et al. 2014), as ocean surface energy fluxes can change greatly between our high-latitude and low-latitude forcings. Although the particular feedbacks in our idealized model may in some cases be artifacts caused by the use of fixed SST, they illustrate how local and nonlocal interactions with column energy sources can prevent the energy budget from providing a quantitatively prognostic indicator of ITCZ shifts.
Estimates of ITCZ position based on the energy budget perform well across our entire ensemble of model integrations (Fig. 12). The energy flux equator, 
Precipitation distributions as in Fig. 6: 
Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-15-0747.1
When the different estimators of ITCZ location are compared, the one based on energy transports across the geographic equator, 
A box-and-whisker plot of the different ITCZ estimator residuals. The median of the distribution is indicated by the vertical line inside the box. The box width defines the interquartile range (IQR). Outliers (blue crosses) are more than 1.5 IQR from the box boundaries, and whiskers contain all data excluding outliers.
Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-15-0747.1
4. Conclusions
This study examined how well CQE and energy budget frameworks performed in diagnosing the latitude of peak monsoon precipitation in an idealized model. This beta-plane model represented the shallow meridional circulations associated with the deserts that lie poleward of nearly all observed monsoon regions; it was used to assess the response to an ensemble of idealized forcings.
We found that the ITCZ shifts toward an imposed atmospheric energy source and away from an energy sink, consistent with previous work (e.g., K08; Chiang and Friedman 2012; Peyrillé et al. 2007). The EFE typically lies within two degrees of the ITCZ in our model, providing a good estimate of ITCZ location across a wide range of climates. In contrast, atmospheric energy transport across the geographic equator provided a poor estimate of ITCZ location because of the nonmonotonic dependence of the energy transport on latitude. This has relevance for the West African monsoon because an atmospheric energy sink exists over the equatorial east Atlantic (Fig. 10). If energy transport across a climatological mean ITCZ latitude is instead used, the linear estimate of BS14 describes the ITCZ position accurately. These differences between the geographic equator and the seasonal mean EFE are less pronounced in the global zonal mean, where the ITCZ shifts at most 7° latitude off the equator (D13). Application of the energy budget framework to regional monsoons is complicated by the fact that the zonal component of divergent energy fluxes need not be zero when averaging over a limited band of longitudes; nevertheless, our results show that energy fluxes across the geographic equator may be of limited utility for understanding the response of regional monsoons to forcings.
We also tested CQE-based estimates of ITCZ location. A traditional CQE framework, in which the maximum 
As is common in studies based on idealized models, numerous caveats exist. Our simulations used perpetual summer insolation, prescribed SST, and prescribed soil moisture. This is a consistent set of idealizations, since it is problematic to use interactive soil moisture with perpetual summer insolation (Xie and Saiki 1999); SSTs would also rise to unrealistic values with perpetual summer insolation unless an oceanic heat sink was prescribed to represent transient heat storage. Using an annual cycle of insolation with interactive SSTs and soil moisture would have required the model be run at least 4 times as long, which was not computationally feasible given the fine horizontal resolution (15 km) of our atmospheric model. We consider this relatively fine resolution to be potentially important in representing the sensitivity of deep convection to entrainment of low-h air above the LCL because it likely allows some mesoscale convective organization, in addition to the minimum entrainment enforced in the Kain–Fritsch convection scheme. But the main point is that CQE and energy budget frameworks are expected to describe tropical circulations regardless of whether insolation, SST, and soil moisture vary. The use of fixed SST creates energy sources that would not exist in a model with a slab ocean, but the atmospheric energy budget must still close so that Hadley circulation transports are consistent with ITCZ location. Furthermore, use of prescribed SST allowed for representation of the atmospheric energy sink associated with the observed equatorial cold tongue in the east Atlantic, which would not exist in a model with an energetically closed slab ocean.
An equally important caveat is that our zonally symmetric boundary conditions eliminate the possibility that time-mean zonal energy transports influence the position of the ITCZ. Such transports may be of great importance in setting the response of regional monsoons to imposed forcings, yet these transports vanish in the global zonal mean energy budget frameworks recently used to understand why monsoon precipitation shifts in response to high-latitude forcings (e.g., Chiang and Friedman 2012). This is less problematic for CQE frameworks, which are based on the local covariance of low-level h and free-tropospheric temperature.
In summary, we demonstrated that deficiencies may exist in the application of a traditional CQE framework and a linear energy budget theory to continental monsoons. Nonetheless, with modifications that account for convective entrainment of dry air and the nonmonotonicity of meridional atmospheric energy transports, CQE and energy budget theories can accurately describe ITCZ location across a wide range of climate states. Examining whether these modifications improve diagnostics of ITCZ location in observations and in global models with realistic boundary conditions seems a worthwhile goal of future work. Unfortunately, neither CQE nor the energy budget are prognostic frameworks due to the feedbacks that alter h and the column-integrated energy source, so a mechanistic and prognostic theory for the response of a regional monsoon to an arbitrary forcing remains elusive.
Acknowledgments
Both authors were supported by National Science Foundation Grant AGS-1253222. This work was supported in part by the facilities and staff of the Yale University Faculty of Arts and Sciences High Performance Computing Center. Computing support was also provided by Yellowstone (ark:/85065/d7wd3xhc), supported by NCAR’s Computational and Information Systems Laboratory. We also thank Xavier Levine and three anonymous reviewers for very insightful feedback.
APPENDIX
Energy Budget in WRF
The moist static energy budget is calculated from the WRF Model output on a fully staggered Arakawa C grid in η coordinates, which are a small variation on terrain-following σ coordinates common in many atmospheric models. Because of the demonstrated sensitivity to regridding and interpolation (Trenberth et al. 2002), it is important to perform all calculations with native model output that has undergone as little postprocessing as possible. The model does not provide the pressure velocity ω on constant pressure levels, but instead the vertical velocity w on time varying η and Z levels, and converting to ω with any time step larger than the model computational time step introduces errors of approximately 10 W m−2.
We were careful to use a finite differencing method matching that used in our model code: one-sided differences where the staggered grid allows (i.e., 
Our direct calculation of the time-mean vertically integrated energy flux convergence −TADVH agrees well with the time-mean column atmospheric MSE source 
CTL integration. (a) Time and zonally averaged 
Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-15-0747.1
These energy budget residuals likely arise from some combination of inadequate sampling, nonconservation in WRF, and the fact that moist static energy does not account for kinetic energy. Although WRF conserves energy within its integration scheme (Skamarock et al. 2008), parameterizations of subgrid-scale physics may not conserve energy perfectly. Conservation seems to be less than perfect in the boundary layer scheme and in the deep moist convection scheme; this is perhaps not surprising given that these schemes may have been intended for use on the shorter time scales of weather prediction and not for the longer integrations conducted here. The budget residual is also sensitive to the temporal resolution at which the model state was sampled. The highest practical resolution, 10 model minutes, was chosen to minimize residuals, but this is still longer than the average model time step of 40 s. Finally, we have calculated the moist static energy budget, which is only an approximation to the total energy budget. The moist static energy budget neglects conversions between potential and kinetic energy such as frictional heating in the boundary layer and the generation of kinetic energy by buoyancy.
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