## 1. Introduction

The seasonal cycle of Earth’s zonal-mean tropical circulation is characterized by a transition from an equinoctial regime, comprising a pair of Hadley cells of roughly equal strength, to a solsticial regime dominated by a single cross-equatorial cell with a rising branch in the summer hemisphere (Dima and Wallace 2003). This transition is associated with a poleward shift of the intertropical convergence zone (ITCZ) and the onset of monsoons over tropical continents (Bordoni and Schneider 2008). The latitudinal extent of the cross-equatorial solsticial Hadley cell (SHC) is therefore a key determinant of the distribution of precipitation in many tropical and subtropical regions.

The ultimate driver of the seasonal rearrangement of the tropical circulation is the variation in solar insolation associated with Earth’s orbit around the sun. But while the solsticial peak in daily mean top-of-atmosphere solar insolation occurs at the summer pole, the rising branch of Earth’s Hadley cell remains within the tropics and subtropics throughout the year. One obvious reason that the Hadley cell’s rising branch does not simply “follow the sun” is that the atmosphere and surface have nonnegligible thermal inertia. Indeed, observations and general circulation model (GCM) simulations indicate that tropical rain belts shift farther into the summer hemisphere over surfaces with lower thermal heat capacity (e.g., Wang and Ding 2008; Bordoni and Schneider 2008; Donohoe et al. 2014). However, Faulk et al. (2017) has recently shown that, in idealized simulations with a moist GCM, the rising branch of the SHC remains at subtropical latitudes even when allowed to equilibrate under perpetual-solstice forcing. Under such forcing, the effects of thermal inertia on the mean circulation are absent, highlighting the influence of other factors, such as the planetary rotation rate, on the position and extent of the SHC.

Previous studies applying perpetual-solstice forcing within a dry framework have shown that the SHC widens as the planetary rotation rate is decreased (Caballero 2008; Hill et al. 2019), but a quantitative theory for its extent remains elusive. In this work, we focus on this perpetual-solstice case within a moist framework in order to isolate the role played by planetary rotation in determining the SHC extent and the resultant distribution of precipitation. Understanding this limiting case is a prerequisite for developing a theory for the full seasonal cycle of the tropical circulation.

A useful starting point for theoretical discussions of the Hadley cell is the axisymmetric nearly inviscid models pioneered by Schneider (1977) and Held and Hou (1980). While such models neglect the effect of eddy momentum fluxes on the mean circulation (Walker and Schneider 2006; Caballero 2007, 2008; Singh and Kuang 2016; Singh et al. 2017), these effects are less important for the SHC than its equinoctial counterpart (Bordoni and Schneider 2008). In the nearly inviscid limit, the combination of angular momentum conservation and thermal wind balance within the free troposphere places a strong constraint on the thermodynamic structure of the atmosphere. If it is further assumed that the Hadley cells are energetically closed, a prediction for the width of the cells and, in the case of an off-equatorial forcing maximum, the position of the rising branch may be derived (Lindzen and Hou 1988). However, Caballero et al. (2008) found that, for the case in which the thermal forcing maximizes at the pole, nearly inviscid theory substantially overestimates the width of the SHC compared to axisymmetric simulations with a GCM. The authors instead derived a semiempirical scaling for the latitudinal extent of the SHC’s descending branch, but no theoretical constraint on the position of the rising branch was obtained.

Axisymmetric theory may also be used to investigate the onset conditions for large-scale thermally direct circulations. For example, Plumb and Hou (1992) derived a critical threshold for the strength of an isolated, off-equatorial thermal forcing maximum beyond which a hypothetical radiative–convective equilibrium (RCE) state becomes unattainable. Under the approximation of convective quasi equilibrium (Emanuel et al. 1994), this critical threshold may be expressed in terms of the boundary layer entropy distribution of the RCE state (Emanuel 1995). In principle, such a criticality condition could provide a constraint on the extent of the SHC, since an overturning circulation must extend at least over the region for which the RCE state is unattainable. One of the aims of this work is to test the applicability of this criticality condition to the solsticial circulation (see also Faulk et al. 2017; Hill et al. 2019).

A number of authors have also used diagnostic approaches in order to relate the position of the Hadley cell’s rising branch to atmospheric energy transport characteristics (e.g., Kang et al. 2008, 2009; Donohoe et al. 2013; Bischoff and Schneider 2014; Wei and Bordoni 2018) or local thermodynamic properties of the atmosphere (e.g., Lindzen and Nigam 1987; Neelin and Held 1987; Back and Bretherton 2009a,b; Nie et al. 2010). For instance, Privé and Plumb (2007a,b) found that the dividing streamline between summer and winter Hadley circulations was roughly collocated with the maximum in low-level moist static energy in simulations of an idealized monsoon circulation, with the maximum in convergence occurring somewhat equatorward of this location. More generally, the convective quasi-equilibrium view of the tropical circulation (Emanuel et al. 1994) argues that tropical precipitation belts should lie close to local maxima of boundary layer moist static energy or the related quantity of moist entropy (Neelin and Held 1987; Nie et al. 2010). But as pointed out by Faulk et al. (2017), the maximum in boundary layer entropy becomes increasingly separated from the ITCZ and the Hadley cell edge as these quantities move poleward. Indeed, the authors find that, in perpetual-solstice simulations with Earthlike parameters, the maximum in boundary layer entropy occurs at the pole, but the rising branch of the SHC remains at subtropical latitudes.

A limitation of the convective quasi-equilibrium viewpoint is that, under conditions of strong vertical wind shear, it predicts a state of moist symmetric instability, in which potential energy may be released by the motion of saturated parcels along slantwise paths oriented along angular momentum surfaces (Emanuel 1983a,b). Such slantwise convection has been recognized as being important for the structure of both tropical (Emanuel 1986) and extratropical (Emanuel 1988) cyclones, but its importance in determining the character of large-scale overturning circulations is largely unknown.

Here, we build on the study of Faulk et al. (2017), and we seek to understand the factors limiting the extent of the SHC under conditions where the thermal maximum is located at the summer pole. We conduct simulations with an idealized moist GCM forced by perpetual-solstice conditions over a range of planetary rotation rates. The simulated SHC extent decreases with increasing rotation rate, despite the fact that the highest boundary layer entropy values remain at the summer pole. These results are interpreted by constructing a predictive theory for the summer-hemisphere SHC extent based on the criticality constraint of Emanuel (1995) and a diagnostic theory based on slantwise convective neutrality within the Hadley cell. The diagnostic theory relates the summer-hemisphere SHC edge latitude to the boundary layer entropy distribution, generalizing previous constraints on tropical precipitation based on convective quasi equilibrium.

We first present the model configuration (section 2) and the basic characteristics of the simulated SHC (section 3). We then describe the predictive (section 4) and diagnostic (section 5) theories of the summer-hemisphere SHC extent and compare them to the idealized simulations. Finally, we present a summary and discussion (section 6).

## 2. Simulation design

We conduct simulations under perpetual-solstice conditions using an idealized aquaplanet GCM similar to that of Frierson et al. (2006, 2007). The model is based on the Geophysical Fluid Dynamics Laboratory Flexible Modeling System, and it includes a two-stream semigray radiation scheme and a representation of moisture with a single vapor–liquid phase transition. Additionally, the model employs the simplified quasi-equilibrium convection scheme described in Frierson (2007), a saturation adjustment scheme to prevent gridscale supersaturation, and a *k*-profile boundary layer parameterization similar to that of Troen and Mahrt (1986). The surface is assumed to be a slab ocean with a fixed depth of 2 m, and surface fluxes are computed based on bulk aerodynamic formulas, with transfer coefficients calculated based on Monin–Obukhov similarity theory.

*S*

_{TOA}is given as a function of latitude

*ϕ*by (Hartmann 1994, p. 30),

*δ*is equal to Earth’s axial tilt of 23.4°, we set

*S*

_{0}= 1367 W m

^{−2}, and

*h*

_{0}is defined by

*τ*is specified as a function of the model’s vertical sigma coordinate so that

*p*normalized by the surface pressure

We conduct a series of 11 perpetual-solstice simulations in which the planetary rotation rate is varied from

## 3. Simulated precipitation and circulation

Figure 2 shows snapshots of near-surface temperature, column water vapor, and precipitation from simulations under three different rotation rates. As expected from the imposed insolation profile, near-surface temperatures are generally highest at the north (summer) pole in all simulations, with relatively weak gradients in the Northern Hemisphere. As a result of the strong relationship between temperature and saturation vapor pressure, column water vapor values also peak at the North Pole and decrease toward the south. The latitude of the highest precipitation rates, however, decreases with increasing rotation rate from the pole in the

Snapshots of (top) temperature at the lowest model level, (middle) column water vapor, and (bottom) precipitation rate for simulations with rotation rates Ω equal to (left)

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Snapshots of (top) temperature at the lowest model level, (middle) column water vapor, and (bottom) precipitation rate for simulations with rotation rates Ω equal to (left)

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Snapshots of (top) temperature at the lowest model level, (middle) column water vapor, and (bottom) precipitation rate for simulations with rotation rates Ω equal to (left)

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

The picture above is confirmed in the time and zonal mean; the latitude of the maximum in the zonal- and time-mean precipitation, which we define as ^{−1}, exist in the tropical upper troposphere for all values of Ω simulated (Figs. 3a–c). For the slowly rotating case

(a)–(c) Streamfunction [contours; contour interval (CI) = 10^{11} kg s^{−1}] and zonal- and time-mean zonal wind (colors), (d)–(f) zonal- and time-mean precipitation rate, and (g)–(i) zonal- and time-mean boundary layer entropy

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

(a)–(c) Streamfunction [contours; contour interval (CI) = 10^{11} kg s^{−1}] and zonal- and time-mean zonal wind (colors), (d)–(f) zonal- and time-mean precipitation rate, and (g)–(i) zonal- and time-mean boundary layer entropy

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

(a)–(c) Streamfunction [contours; contour interval (CI) = 10^{11} kg s^{−1}] and zonal- and time-mean zonal wind (colors), (d)–(f) zonal- and time-mean precipitation rate, and (g)–(i) zonal- and time-mean boundary layer entropy

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Zonal- and time-mean precipitation rate as a function of sine latitude for simulations with different rotation rates as labeled (blue lines; each curve offset by 12 mm day^{−1}). Regions in which the ratio of precipitation to evaporation

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Zonal- and time-mean precipitation rate as a function of sine latitude for simulations with different rotation rates as labeled (blue lines; each curve offset by 12 mm day^{−1}). Regions in which the ratio of precipitation to evaporation

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Zonal- and time-mean precipitation rate as a function of sine latitude for simulations with different rotation rates as labeled (blue lines; each curve offset by 12 mm day^{−1}). Regions in which the ratio of precipitation to evaporation

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Given the sensitivity of the position of the precipitation maximum to small changes in the precipitation distribution, we focus here on understanding the effect of planetary rotation on the latitude of the SHC edge

The contraction of the SHC and equatorward shift of the precipitation distribution with increasing rotation rate occur in our simulations despite the fact that the zonal- and time-mean boundary layer entropy, ^{1} This is in contrast to many previous studies that have argued that tropical precipitation maxima should lie adjacent to maxima in boundary layer entropy, or the closely related quantity moist static energy (e.g., Neelin and Held 1987; Privé and Plumb 2007a,b; Nie et al. 2010). Such arguments generally rely on the assumption of convective quasi equilibrium and either the application of the weak temperature gradient (WTG) approximation (Sobel et al. 2001) to the region of strong convection in the ITCZ (Nie et al. 2010) or the assumption that the rising branch of the SHC is in a region of weak wind shear (Privé and Plumb 2007a). In our simulations, the rising branch of the SHC exists in a region of strong easterly shear and, by thermal wind balance, a region with substantial meridional gradients of temperature in the free troposphere, and thus the argument of Privé and Plumb (2007a) is inapplicable. Understanding the contraction of the SHC and the increasing separation of the SHC edge from the maximum in boundary layer entropy with increasing planetary rotation rate is the aim of the next two sections.

## 4. Predictive estimate of the summer-hemisphere SHC extent

Equation (3) defines a criticality condition on the boundary layer entropy distribution; when

To construct a simple predictive estimate for the summer-hemisphere extent of the SHC, we assume that, in the summer hemisphere, the Hadley cell extends only over latitudes for which ^{2} The temperature distribution in this RCE simulation is therefore a result of the interaction of the radiation, convection, boundary layer, and surface-flux parameterizations within each column of the GCM, but it does not depend appreciably on the planetary rotation rate.^{3}

^{4}Applying the approximation of constant

RCE solution for solsticial insolation profile. (a) Temperature at the lowest model level. (b) Saturation entropy (blue; contour interval 50 J kg^{−1} K^{−1}) and temperature anomaly relative to temperature at the lowest model level (gray). Black line in (b) shows the tropopause level, defined as the height at which the lapse rate decreases to half of its moist-adiabatic value.

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

RCE solution for solsticial insolation profile. (a) Temperature at the lowest model level. (b) Saturation entropy (blue; contour interval 50 J kg^{−1} K^{−1}) and temperature anomaly relative to temperature at the lowest model level (gray). Black line in (b) shows the tropopause level, defined as the height at which the lapse rate decreases to half of its moist-adiabatic value.

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

RCE solution for solsticial insolation profile. (a) Temperature at the lowest model level. (b) Saturation entropy (blue; contour interval 50 J kg^{−1} K^{−1}) and temperature anomaly relative to temperature at the lowest model level (gray). Black line in (b) shows the tropopause level, defined as the height at which the lapse rate decreases to half of its moist-adiabatic value.

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Figure 6 shows the two sides of (4) for three different rotation rates. Since the RCE entropy distribution is independent of rotation rate, the curvature

Supercriticality of the RCE distribution of boundary layer entropy

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Supercriticality of the RCE distribution of boundary layer entropy

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Supercriticality of the RCE distribution of boundary layer entropy

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

The dotted gray line in Fig. 4 shows the supercriticality-based prediction of the summer-hemisphere SHC edge, which we denote

A potential reason that ^{5} These results are consistent with those of Faulk et al. (2017), who found that a similar supercriticality condition underestimated the latitude of the solsticial ITCZ position in idealized aquaplanet GCM simulations. Further discussion of the limitations of estimates of Hadley cell extent based on violation of Hide’s constraint in the RCE state may also be found in Hill et al. (2019).

## 5. Diagnostic estimate of summer-hemisphere SHC extent

To obtain a more accurate estimate of the SHC extent

Streamlines (gray lines) as a function of sine latitude in simulations with planetary rotation rates of (a),(b) ^{11} kg s^{−1}), (c),(d) ^{11} kg s^{−1}), and (e),(f) ^{11} kg s^{−1}). Regions outside the outermost streamline of the SHC, where the SHC edge is defined by (2), are shaded gray. (left) Angular momentum contours (red lines) and regions where ^{−1} K^{−1}) and regions where PV* < 0 (yellow shading).

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Streamlines (gray lines) as a function of sine latitude in simulations with planetary rotation rates of (a),(b) ^{11} kg s^{−1}), (c),(d) ^{11} kg s^{−1}), and (e),(f) ^{11} kg s^{−1}). Regions outside the outermost streamline of the SHC, where the SHC edge is defined by (2), are shaded gray. (left) Angular momentum contours (red lines) and regions where ^{−1} K^{−1}) and regions where PV* < 0 (yellow shading).

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Streamlines (gray lines) as a function of sine latitude in simulations with planetary rotation rates of (a),(b) ^{11} kg s^{−1}), (c),(d) ^{11} kg s^{−1}), and (e),(f) ^{11} kg s^{−1}). Regions outside the outermost streamline of the SHC, where the SHC edge is defined by (2), are shaded gray. (left) Angular momentum contours (red lines) and regions where ^{−1} K^{−1}) and regions where PV* < 0 (yellow shading).

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

The right panels of Fig. 7 show that the isolines of saturation entropy are also roughly parallel to streamlines and angular momentum contours, particularly in the tropical upper troposphere, indicating a state close to neutral to slantwise moist convection. In fact, in broad regions of the Hadley cell’s rising branch, the simulations exhibit slight slantwise *instability*. Specifically, these regions are characterized by instability to saturated motion along angular momentum surfaces, as indicated by negative values of the saturation potential vorticity (Figs. 7b,d,f), defined

Having verified that the assumptions of slantwise neutrality and angular momentum conservation are reasonable in our simulations, we now use these assumptions to relate the Hadley cell extent to the distribution of boundary layer entropy

*α*is the specific volume and we have used hydrostatic balance

*α*may be expressed as a thermodynamic function of pressure

*p*and saturation entropy

*s** such that

*p*, we have

*b*refers to variables evaluated within the boundary layer, and the derivatives on the right-hand side are evaluated at

*M*surface as a function of temperature, which we treat here as the vertical coordinate. The author was made aware of the preceding derivation through an unpublished manuscript (K. A. Emanuel 2011, personal communication), but a similar equation is derived for angular momentum surfaces within an axisymmetric tropical cyclone in Emanuel (1986), and the present model should be seen as the application of the same reasoning to an axisymmetric Hadley cell.

For the current solsticial case, the boundary layer entropy

Distribution of angular momentum contours for the slantwise neutral solution for planetary rotation rates of (a)

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Distribution of angular momentum contours for the slantwise neutral solution for planetary rotation rates of (a)

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Distribution of angular momentum contours for the slantwise neutral solution for planetary rotation rates of (a)

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

On the winter side of the equator, the negative slope of angular momentum surfaces implies descending motion for angular momentum–conserving flow, and the assumption of slantwise neutrality is no longer appropriate. We therefore focus only on the summer hemisphere in the present manuscript, although we note that recent work has suggested that the SHC must extend at least as far into the winter hemisphere as into the summer hemisphere (Hill et al. 2019), and our summer-hemisphere solution may therefore be useful as a lower bound on the winter-hemisphere SHC extent.

Note also that nothing in (10) prevents angular momentum surfaces that emanate from the boundary layer at different latitudes from intersecting, and this occurs in the case

Given the value of ^{6} (Fig. 9). The reason for this insensitivity of

Temperature difference

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Temperature difference

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Temperature difference

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

In principle, there may be more than one latitude where (11) is satisfied, but we find that

Theoretical predictions of the Hadley cell extent based on supercriticality of the RCE state (circles) and slantwise neutral solution based on RCE state (dots) and equilibrated simulations (crosses) plotted against simulated Hadley cell extent for rotation rates between

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Theoretical predictions of the Hadley cell extent based on supercriticality of the RCE state (circles) and slantwise neutral solution based on RCE state (dots) and equilibrated simulations (crosses) plotted against simulated Hadley cell extent for rotation rates between

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

Theoretical predictions of the Hadley cell extent based on supercriticality of the RCE state (circles) and slantwise neutral solution based on RCE state (dots) and equilibrated simulations (crosses) plotted against simulated Hadley cell extent for rotation rates between

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0341.1

The slantwise neutral theory presented in this section provides a diagnostic for the Hadley cell extent under perpetual-solstice conditions. To formulate a fully closed theory, a prediction for the boundary layer entropy distribution is required. The distribution of ^{7}

In the simulations, the energy transport by the SHC is from the Northern to the Southern Hemisphere, and we would expect it to have the effect of reducing

Based on the above reasoning, we expect the summer-hemisphere SHC extent to be greater than that predicted based on the RCE entropy distribution (except when the RCE distribution predicts a global SHC), and this is indeed what is seen in the simulations (Fig. 10). Furthermore, this reasoning suggests that the extent to which the SHC edge exceeds the RCE prediction depends on the strength of the energy transport associated with the cell itself. According to this view, the SHC extent is determined in part by the strength of the SHC, and these two aspects must be solved for together in any complete theory of the Hadley circulation.

## 6. Conclusions

Using a set of idealized GCM simulations run under perpetual-solstice conditions, we have investigated the limits placed by planetary rotation on the penetration of the solsticial Hadley cell into the summer hemisphere. Consistent with previous work (Caballero et al. 2008; Faulk et al. 2017), the SHC expands as the planetary rotation rate is reduced, and at low enough values of the planetary rotation rate, the circulation is global. At Earthlike planetary rotation, however, the Hadley cell extends only roughly 30° into the summer hemisphere, suggesting that rotation may play an important role in modulating the poleward extent of Earth’s monsoon circulations.

We developed two theoretical frameworks through which to interpret our numerical results. The first theory provides a prediction of the SHC extent by assuming that the Hadley cell exists only in regions of the atmosphere for which an unattainable distribution of angular momentum would be required to maintain RCE (Plumb and Hou 1992; Emanuel 1995). This theory qualitatively reproduces the behavior of the simulations, but it generally underestimates the simulated SHC extent. We therefore argue that the RCE state and its angular momentum distribution may provide a useful lower bound on estimates of the extent of the Hadley cell (see also Hill et al. 2019).

A more quantitatively accurate estimate of the summer hemisphere SHC extent was obtained by assuming that, in the region of the SHC rising branch, the atmosphere adjusts toward a state neutral to slantwise convection. Under the additional assumption that angular momentum is conserved along streamlines, this allows for a determination of the Hadley cell extent and the shape of angular momentum contours in the summer hemisphere. This diagnostic estimate of the SHC extent quantitatively reproduces the behavior of the simulations, and it connects the summer-hemisphere SHC extent to the distribution of boundary layer entropy. Previous work based on the assumption of convective quasi equilibrium has also highlighted the importance of the boundary layer entropy distribution in determining the position of tropical rain belts (Neelin and Held 1987; Emanuel 1995; Nie et al. 2010; Hurley and Boos 2013); the diagnostic theory presented here may be considered a generalization of this convective quasi-equilibrium view to allow for slantwise convection adjustment.

While the slantwise neutral theory accurately reproduces the behavior of the summer-hemisphere SHC extent, we have not addressed the dependence of precipitation belts on rotation rate in detail. In our simulations, the latitude of maximum precipitation

The extent of the SHC in our perpetual-solstice simulations should be seen as an upper limit on the poleward seasonal migration of the Hadley cell in which the atmospheric heat capacity plays no role. In Earth’s atmosphere, transient effects are important in limiting the poleward migration of the circulation in summer, particularly in oceanic regions, making direct application of our results to observations difficult. Nonetheless, previous studies have noted that, in the summer season, Earth’s cross-equatorial Hadley circulation more closely approaches the angular momentum–conserving limit (Bordoni and Schneider 2008), and that, in the region of the Asian monsoon, the upper-tropospheric potential vorticity becomes small (e.g., Plumb 2007). These features point to the potential relevance of the slantwise neutral solution to the Asian summer monsoon. Determining the extent to which a similar constraint may relate the poleward extent of the Asian summer monsoon to the low-level entropy distribution is an interesting avenue for future work.

## Acknowledgments

I thank K. A. Emanuel, who brought the slantwise neutral solution to my attention in a graduate class on tropical meteorology, and S. A. Hill and two anonymous reviewers for their constructive comments. I acknowledge support from the ARC Centre of Excellence for Climate Extremes, support from an ARC Discovery Early Career Research Award (DE190100866), and computational support from the National Computational Infrastructure, all funded by the Australian Government.

## APPENDIX

### Derivation of Maxwell’s Relation

Here, we derive the Maxwell-type relation (7). This differs from the usual Maxwell relationship for entropy and specific volume (see Emanuel 1994, chapter 4) because it is written in terms of saturation entropy *s**, which is only a function of temperature and pressure, rather than in terms of entropy, which is a function of temperature, pressure, and total water content.

*k*and entropy

*s*, defined at saturation,

*e*is the vapor pressure,

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^{1}

Here

^{2}

To ensure an equilibrium is reached, the solsticial insolation profile is altered from that used in the full dynamical simulations such that the top-of-atmosphere insolation does not drop below 20 W m^{−2} in the RCE simulation. This alteration only affects the solution in the high southern latitudes.

^{3}

We follow O’Gorman and Schneider (2008) and use the zonal- and time-mean wind field from the equilibrated simulation with

^{4}

Using a threshold lapse rate of 2 K km^{−1} (as commonly used to define the tropopause) gives a slightly larger value for

^{5}

The exception is the case

^{6}

If