1. Introduction
Climatologically over the annual cycle, the poleward, descending edges of the Hadley cells vary meridionally by ≲ 5° latitude about their annualmean positions, considerably less than the ∼15°S–15°N range of the shared, ascending edge (cf. Fig. 4 of Adam et al. 2016). These result in a pronounced seasonal cycle of zonalmean rainfall in the deep tropics versus more consistently dry conditions in the subtropics. Regional hydrological deviations from the zonal average are pronounced—with, for example, intense Indian summer monsoon rainfall spanning roughly the same latitudes as the Sahara and Arabian Deserts (Rodwell and Hoskins 1996)—nevertheless, we focus on the zonalmean dynamical problem, seeking a minimal explanation for the differing annual cycles of the Hadley cell descending and ascending edges (henceforth φ_{d} and φ_{a}, respectively, and formally defined below in terms of the mass overturning streamfunction).
For φ_{d}, our starting point is the theory of Kang and Lu (2012, henceforth KL12), whose own starting point is that of Held (2000, henceforth H00) for the annualmean φ_{d} that assumes the Hadley cells terminate where their zonal wind profiles become baroclinically unstable. KL12 extend the H00 model in two key ways. First, they generalize from the annual mean to the annual cycle by accounting, albeit diagnostically, for offequatorial φ_{a}. For angularmomentumconserving (AMC) zonal winds as assumed by H00, ascent off the equator results in less positive zonal winds at each latitude (Lindzen and Hou 1988, hereafter LH88) and thus to baroclinic instability onset occurring farther poleward than for equatorial ascent. All else equal, this would cause φ_{d} to be farther poleward in solsticial seasons when φ_{a} is farther offequator than in equinoctial seasons. An example of this framework’s utility is Hilgenbrink and Hartmann (2018), who interpret changes in φ_{d} throughout the annual cycle caused by ocean heat transports in terms of changes in φ_{a}.
Second, KL12 relax the H00 assumption of strictly AMC winds by assuming that the Rossby number (Ro) is uniform throughout each Hadley cell’s upper branch but not necessarily unity. Its formal definition follows below, but for now Ro is exactly unity for AMC winds and exactly zero if zonal winds themselves are zero, and KL12 derive an analytical expression for the meridional profile of zonal wind under uniform 0 < Ro ≤ 1. In simulations (Walker and Schneider 2006) and reanalysis data (Schneider 2006), Ro is regularly below unity and typically smaller in the equinoctial and summer cells than in the crossequatorial winter cell (Bordoni and Schneider 2008; Schneider and Bordoni 2008). By diagnosing a bulk value of Ro for each cell and meteorological season in addition to φ_{a}, KL12 provide closed expressions for the Northern and Southern Hemisphere descending edge latitudes in all four seasons.^{1}
For φ_{a}, in Hill et al. (2021) we presented a scaling for φ_{a} by assuming that it is set by the meridional extent of supercritical radiative forcing in the summer hemisphere (the meaning of which we expand upon below). In essence, the present study simply replaces the diagnosed φ_{a} in the KL12 model with this predictive scaling. The one departure from Hill et al. (2021)—which implicitly treated thermal inertia as negligible by relating the seasonally varying φ_{a} to the contemporaneous insolation—is accounting for thermal inertia’s damping and delaying of the φ_{a} annual cycle via the analytical model of Mitchell et al. (2014). The result is a unified theory for φ_{a} and both hemispheres’ φ_{d} with only two proportionality constants as well as Ro diagnosed (potentially with distinct Ro values required for each cell and season).
Separately, in general circulation model (GCM) simulations with differing planetary rotation rates (Ω) the solsticial, crossequatorial Hadley cell expands into both the summer and winter hemispheres as Ω decreases (e.g., Faulk et al. 2017; Singh 2019). But whereas in an Earthlike regime the summer φ_{a} and winter φ_{d} are comparably poleward, in slowly rotating cases the summer φ_{a} is farther poleward than the winter φ_{d}_{:} the crossequatorial Hadley cell becomes increasingly lopsided about the equator [e.g., Fig. 5 of Guendelman and Kaspi (2018) and Fig. 12 of Guendelman and Kaspi (2019)].^{2} Moreover, Guendelman and Kaspi (2019) empirically find distinct bestfit powerlaw exponents for the two edges, close to φ_{a} ∝ Ω^{−2/3} and φ_{d} ∝ Ω^{−1/2}. We will use our theory along with idealized GCM simulations to explain these exponents and how they relate to the lopsidedness of the crossequatorial cell.
In the following sections we

derive and describe fixedRo zonal wind, angular momentum, and depthaveraged potential temperature fields (section 2);

present our unified theory, which essentially combines the KL12 model for φ_{d} with the Hill et al. (2021) theory for φ_{a} (section 3);

and test our theory against idealized GCM simulations, first over the annual cycle in one moist model and second for the solsticial Hadley circulation across rotation rates in one dry and two moist models (section 4).
We then conclude with summary and discussion (section 5).
2. UniformRo fields
Figure 1 shows example u_{Ro}, M_{Ro}, and
3. Combined theory for Hadley cell ascending and descending edges
We now use these fixedRo fields to derive an expression for φ_{d} given values of Ro and φ_{a} that closely follows KL12. We then introduce within it our scaling for φ_{a}, yielding our unified theory for φ_{d} and φ_{a}. We then incorporate the influence of surface thermal inertia on the seasonal cycles of φ_{a} analytically and on φ_{d} more empirically.
a. Baroclinic instability onset theory for the Hadley cell edge with Ro < 1
We have performed a 2D parameter sweep over φ_{a} and (BuΔ_{v }/Ro)^{1/4}, from 0° to 90° in 1° increments for φ_{a} and from 0 to 2 in 0.01 increments for (BuΔ_{v }/Ro)^{1/4}, solving (11) numerically for each pair of parameter values. The results are shown as shaded contours in Fig. 2.^{6} Recalling that φ_{Ro,ann} ≡ (BuΔ_{v}/Ro)^{1/4}, the right vertical axis shows the equivalent values of φ_{Ro,ann} up to 90°, above which the smallangle solution obviously is nonsensical but the full solution retains its validity. The value of φ_{d} increases monotonically with φ_{a} and with (BuΔ_{v}/Ro)^{1/4}. Close to the vertical axis of Fig. 2, φ_{Ro,ann}/φ_{a} ≫ 1, and φ_{d} ≈ φ_{Ro,ann}: φ_{a} is negligibly offequator. Close to the horizontal axis, φ_{Ro,ann}/φ_{a} ≪ 1, and thus φ_{d} ≈ φ_{a}. This regime usefully describes cases where the summer cell effectively disappears, as in all of the perpetual solstice simulations we will discuss below. For the winter hemisphere, the interpretation is that baroclinic instability onset occurs just poleward of −φ_{a}, near enough that it can be approximated as −φ_{a}. For intermediate values, if, e.g., φ_{a} = φ_{Ro,ann}, then φ_{d} is displaced 27% poleward of φ_{a}. We note that φ_{d} ≥ φ_{a}; the descending edge latitude is always at or poleward of the ascending edge latitude. This is appropriate for the summer hemisphere φ_{d} but will prove imperfect for the winter φ_{d} in the idealized GCM simulations across rotation rates discussed below.
Numerical solutions of (11) for values of φ_{a} and of (BuΔ_{v}/Ro)^{1/4}, with φ_{a} sampled from 0° to 90° in 1° increments and (BuΔ_{v}/Ro)^{1/4} (which is dimensionless) from 0 to 2.0 in 0.01 increments. The right vertical axis labels are the φ_{Ro,ann} solutions from 0° to 90° corresponding to the given (BuΔ_{v}/Ro)^{1/4} values. Areas in white indicate that the simple numerical algorithm used did not converge, but clearly they correspond to φ_{d} very near the pole. Contours are from 5° to 90° in 5° increments according to the color bar. Overlaid gray contours are the corresponding smallangle solutions obtained using (12), likewise from 5° to 90° in 5° increments.
Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JASD210328.1
Numerical solutions of (11) for values of φ_{a} and of (BuΔ_{v}/Ro)^{1/4}, with φ_{a} sampled from 0° to 90° in 1° increments and (BuΔ_{v}/Ro)^{1/4} (which is dimensionless) from 0 to 2.0 in 0.01 increments. The right vertical axis labels are the φ_{Ro,ann} solutions from 0° to 90° corresponding to the given (BuΔ_{v}/Ro)^{1/4} values. Areas in white indicate that the simple numerical algorithm used did not converge, but clearly they correspond to φ_{d} very near the pole. Contours are from 5° to 90° in 5° increments according to the color bar. Overlaid gray contours are the corresponding smallangle solutions obtained using (12), likewise from 5° to 90° in 5° increments.
Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JASD210328.1
Numerical solutions of (11) for values of φ_{a} and of (BuΔ_{v}/Ro)^{1/4}, with φ_{a} sampled from 0° to 90° in 1° increments and (BuΔ_{v}/Ro)^{1/4} (which is dimensionless) from 0 to 2.0 in 0.01 increments. The right vertical axis labels are the φ_{Ro,ann} solutions from 0° to 90° corresponding to the given (BuΔ_{v}/Ro)^{1/4} values. Areas in white indicate that the simple numerical algorithm used did not converge, but clearly they correspond to φ_{d} very near the pole. Contours are from 5° to 90° in 5° increments according to the color bar. Overlaid gray contours are the corresponding smallangle solutions obtained using (12), likewise from 5° to 90° in 5° increments.
Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JASD210328.1
b. Incorporating theory for φ_{a}
Using (12) requires knowledge of φ_{a}, which KL12 diagnose. Hill et al. (2021) derive a prognostic theory for φ_{a} as determined by the meridional extent of supercritical forcing, based on the following arguments. If no largescale meridional overturning circulation existed, local radiative–convective equilibrium (RCE) must prevail at each latitude in the time mean. But given the resulting meridional temperature gradients driven by the meridional distribution of insolation, this hypothetical RCE state would generate zonal wind fields through gradient balance that are symmetrically unstable from the equator to some latitude in the summer hemisphere; the presence of this instability defines the tropical supercritical forcing extent (Plumb and Hou 1992; Emanuel 1995). A largescale meridional overturning circulation therefore must emerge spanning at least the supercritical latitudes, and in axisymmetric atmospheres that circulation must be the Hadley cells. In eddying atmospheres, in principle the circulation that emerges over the supercritical region could be predominantly macroturbulent as in the extratropics rather than Hadleylike, but in practice the opposite occurs: the ascending edge latitude is poleward of—and, crucially, proportional to—the supercritical extent.
For the solsticial, crossequatorial Hadley cells in the simulations analyzed by Hill et al. (2021) the bestfit value of c_{a} ranges from 1.3 to 2.6 across three idealized GCMs.^{7} For Ro_{th}, Hill et al. (2021) show that for solsticial seasons one can attain an accurate estimate with φ_{m} set to 90° by tuning the value of Δ_{h}. Doing so, the nonstandard sinφ_{m} term drops out and the Ro_{th} definition becomes the more conventional Ro_{th} = BuΔ_{h}. But the sinφ_{m} dependence is necessary for understanding the annual cycle as will be discussed further below (see also Fig. 15 of Guendelman and Kaspi 2020).
c. Influence of surface thermal inertia on the φ_{a} seasonal cycle
Because insolation varies seasonally, any nonzero thermal inertia of the surface mixed layer damps and delays the surface thermal response, the more so the larger the mixed layer heat capacity. This is true of the dynamically equilibrated climate but also the hypothetical latitudebylatitude RCE state that determines φ_{a}. We therefore now define an “effective” thermal forcing based on the analytical model of Mitchell et al. (2014) which leaves the functional form of (13) for φ_{a} intact but modifies the φ_{m} term within Ro_{th} to be damped and delayed in its seasonal excursions. The appendix below provides the derivation, presenting here only the end results of how φ_{m} is modified.
First consider the unmodified φ_{m} annual cycle. For Earth’s presentday insolation, during equinoctial seasons there is only one maximum in φ_{m}, but during solsticial seasons there are two maxima in the summer hemisphere, a local one near 44° and the global maximum at the summer pole (see, e.g., Fig. 1 of Hill et al. 2021). Though the polar maximum is relevant for the globalscale Hadley cells in other planetary atmospheres (Singh 2019), for Earth we can comfortably consider the midlatitude maximum at solstice to be φ_{m}. An advantage of this choice is that, combining the equinoctial and solsticial seasons, this yields a nearly sinusoidal annual cycle of φ_{m} (not shown): φ_{m} ≈ φ_{m}_{,ann} cos[ω_{orb}(t − t_{solst})], where φ_{m}_{.ann} = 44° is the annual maximum value of φ_{m}, ω_{orb} is the orbital frequency, t is the time of year, and t_{solst} is the time of year of northern summer solstice.
Next, cf. Mitchell et al. (2014) we define the ratio of the seasonal time scale to the thermal inertial time scale as α ≡ (ω_{orb}τ_{ti})^{−1}, where
For the time scale of φ_{d} relative to the seasonal cycle, we proceed much more empirically. Insofar as φ_{d} is determined by the combination of φ_{a} and Ro as we have posited, one relevant time scale is that of the upperlevel zonal wind adjustment: it takes a finite amount of time for the zonal winds in the descending branch to adjust to a change in φ_{a} in the opposite hemisphere. The other is the time scale of changes in Ro; however, we lack clear intuition for what controls Ro and thus this time scale (see the discussion below for some speculation); moreover a timeinvariant Ro seems to fit the simulation well as shown below. We find empirically that the best fit to the simulated φ_{d} occurs by lagging our predicted φ_{d} based on the contemporaneous Ro and φ_{a} by roughly one month, which seems not radically too long nor short.
4. Simulation results
We now assess these theoretical arguments against simulations in three idealized GCMs. After describing the models and simulations, we consider the annual cycles of φ_{d} and φ_{a} in an Earthlike aquaplanet control simulation, followed by their behaviors across a wide range of rotation rates in all three GCMs.
a. Description of models and simulations
Details of the model formulations and simulations are provided by Hill et al. (2021). Briefly, the dry model (Schneider 2004) approximates radiative transfer via Newtonian cooling, with the equilibrium temperature field that temperatures are relaxed toward being the forcing field from LH88 but maximizing at the North Pole (i.e., setting φ_{m} = 90°). The relaxation field is statically unstable, and a simple convective adjustment scheme relaxes over a fixed time scale the temperatures of unstable columns toward a lapse rate of γΓ_{d}, where Γ_{d} = g/c_{p} is the dry adiabatic lapse rate with c_{p} the specific heat of dry air at constant pressure, and γ = 0.7 mimics the stabilizing effects of latent heat release that would occur in a moist atmosphere (though the model is otherwise dry). Four simulations are performed, three with the Δ_{h} parameter that determines the horizontal temperature gradients of the forcing set to 1/15 and with the planetary rotation rate set to 0.25, 1, or 2 × Earth’s value, and another with Δ_{h} = 1/6 and Earth’s rotation rate. The Δ_{h} = 1/6 value is conventional (LH88), but Hill et al. (2021) show that, for φ_{m} = 90°, Δ_{h} = 1/15 is the best fit to numerically simulated latitudebylatitude RCE.
The moist simulations are those originally presented by Faulk et al. (2017) and Singh (2019). The Faulk et al. (2017) simulations use the idealized aquaplanet model of Frierson et al. (2006) featuring a slabocean lower boundary with a 10m mixed layer depth. They are forced either with an annual cycle of insolation approximating that of presentday Earth or with insolation fixed at northern summer solstice. The annual cycle simulations include planetary rotation rates ranging from 1/32 to 4 × Earth’s by factors of 2, while the three perpetual solstice simulations are at 1, 1/8, or 1/32 × Earth’s rotation rate. The Singh (2019) simulations use an idealized aquaplanet close to that of O’Gorman and Schneider (2008), itself a modified version of the Frierson et al. (2006) model. All of these simulations use a timeinvariant, solsticial insolation forcing as in the second subset of the Faulk et al. (2017) simulations, with rotation rates ranging from 1/8 to 8 × Earth’s.
The simulated values of φ_{d} are diagnosed conventionally as the latitude at which the massoverturning streamfunction at the level of the cell center reaches 10% of its maximum value, with an additional cosφ weighting factor that accounts for constricting latitude circles moving poleward (Singh 2019). The 10% threshold is needed rather than a zero crossing for cases with large Hadley cells, in which the Ferrel cells and/or summer Hadley cell can be nonexistent and the streamfunction samesigned (albeit very weak) all the way to either pole. For φ_{a}, the same 10% threshold is used in the perpetualsolstice simulations and in the annual cycle simulations for months in which the summer Hadley cell has effectively vanished. In months where both Hadley cells are well defined, φ_{a} is taken as the average of the inner edges of the two cells computed using this 10% criterion (which is approximately the latitude of the streamfunction zero crossing; not shown).
b. Annual cycles of φ_{a} and φ_{d}
Before presenting the simulation results, we delineate three regimes regarding the relative importance of Ro versus φ_{a} in determining φ_{d}. First is where Ro predominates: by (12) if φ_{a} is small relative to φ_{Ro,ann} = (BuΔ_{v}/Ro)^{1/4} throughout the annual cycle, then the annual cycle of φ_{d} is determined by the annual cycle of Ro (provided that Δ_{v} and H are constant across seasons). Second is intermediate, with both Ro and φ_{a} influential as in the compensation regarding the winter φ_{d} found by KL12 in CMIP3 simulations: φ_{a} is farthest poleward at solstice, acting to move φ_{d} poleward, but Ro is largest in the solsticial crossequatorial cell, acting to move the winter φ_{d} equatorward. Third is where φ_{a} predominates, as we now show holds for the seasonally forced simulation at Earth’s rotation rate of Faulk et al. (2017): φ_{a} variations (which are well predicted by supercriticality provided thermal inertia is accounted for) with Ro = 1 assumed throughout the annual cycle account for the annual cycle of the winter φ_{d}.
Figure 3 shows the climatological annual cycles of φ_{a} (solid red curve), of φ_{d} in both hemispheres (solid blue curves), and of the meridional overturning streamfunction at 500 hPa (color shading), as well as theoretical estimates described below for each cell edge. The simulated cells are Earthlike in their total meridional extent (φ_{d} varies over 21.3°–27.7°N and 21.7°–26.5°S) and annual cycle phasing, with φ_{a} migrating into either summer hemisphere with a ∼1.5month lag behind the insolation. However, the φ_{a} excursions are excessive, 25.7°S–23.2°N, resulting in an excessively rapid transition from equinoctial to solsticial regimes, approaching closer to the squarewave prediction of LH88 for axisymmetric atmospheres than Earth’s more sinusoidal variations (Dima and Wallace 2003).^{8} This results in the summer cell being too weak, such that φ_{d} is only well defined in the winter halfyear for either hemisphere (December–May for the Northern Hemisphere, June–November for the Southern Hemisphere). These discrepancies seem attributable to the rather shallow 10m surface mixed layer, which promotes excessive seasonality (Donohoe et al. 2014; Wei and Bordoni 2018). Nevertheless, the variations in φ_{d} and φ_{a}—in particular that the φ_{d} ranges are comparable to Earth’s and several times smaller than that of φ_{a}—lead us to consider this useful enough for Earth.
In the seasonally forced, Earthlike aquaplanet simulation, climatological annual cycle of Hadley cell streamfunction at 500 hPa in shading according to the color bar, as well as Hadley cell edges and theories for the Hadley cell edges as indicated in the legend. The effective φ_{m} is the effective insolation maximum for that month given the damping and delaying effects of thermal inertia on the seasonally varying forcing. The insolation maximum (yellow stars) is lagged by 1 month from its actual value to facilitate comparison with φ_{a}, and in the 2 months nearest solstice the maximum is near the summer pole and thus not shown.
Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JASD210328.1
In the seasonally forced, Earthlike aquaplanet simulation, climatological annual cycle of Hadley cell streamfunction at 500 hPa in shading according to the color bar, as well as Hadley cell edges and theories for the Hadley cell edges as indicated in the legend. The effective φ_{m} is the effective insolation maximum for that month given the damping and delaying effects of thermal inertia on the seasonally varying forcing. The insolation maximum (yellow stars) is lagged by 1 month from its actual value to facilitate comparison with φ_{a}, and in the 2 months nearest solstice the maximum is near the summer pole and thus not shown.
Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JASD210328.1
In the seasonally forced, Earthlike aquaplanet simulation, climatological annual cycle of Hadley cell streamfunction at 500 hPa in shading according to the color bar, as well as Hadley cell edges and theories for the Hadley cell edges as indicated in the legend. The effective φ_{m} is the effective insolation maximum for that month given the damping and delaying effects of thermal inertia on the seasonally varying forcing. The insolation maximum (yellow stars) is lagged by 1 month from its actual value to facilitate comparison with φ_{a}, and in the 2 months nearest solstice the maximum is near the summer pole and thus not shown.
Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JASD210328.1
The insolation maximum is near the summer pole during the core solsticial months and within the tropics otherwise (yellow stars, lagged by 1 month to ease comparison with φ_{a}). The effective φ_{m} accounting for thermal inertia (orange stars) is damped and lagged from the actual insolation as described above. φ_{a} stays within the tropics always, with φ_{a} ≈ φ_{m} to a reasonable degree. The supercriticality scaling (13) yields an even better prediction (dotted dark red curve), calculated as follows. Hill et al. (2021) perform a 2D parameter sweep of φ_{m} and Δ_{h} to determine best fits of the LH88 thermal forcing profile against numerical simulations of latitudebylatitude RCE under solsticial forcing; for φ_{m} ≈ 44° the best fit over the tropics occurs for Δ_{h} = 1/8 (approximately twice that of the Δ_{h} = 1/15 value for φ_{m} = 90°).^{9} We therefore take Δ_{h} = 1/8. With Δ_{h} set, the proportionality constant c_{a} is left effectively as a fitting parameter, which by eye provides the best fit for c_{a} ≈ 1.9.^{10} The prediction is clearly not without empiricism, but nevertheless, we are pleased with the overall accuracy against the simulated φ_{a}.
We then use this theoretically computed φ_{a} as just described to predict φ_{d}. Due to an inadvertent loss of zonalwind data from the Faulk et al. (2017) simulations, we are not able to directly diagnose Ro. Instead, we assume Ro = 1, which provided 0 ≤ Ro ≤ 1 yields the equatorwardmost possible φ_{d} predictions, all else equal. Even still, this yields a φ_{d} prediction poleward of the simulated φ_{d} values (not shown), which we correct for by setting c_{d} = 0.75 in (15). We then shift the results later in time by one month, resulting in a fairly accurate fit to the simulations (dotted blue curve). In the concluding section below we provide speculative arguments to justify this equatorward displacement and 1month phase lag of φ_{d} compared to φ_{a} (which in turn lags the insolation by ∼1.5 months). The φ_{d} prediction is only marginally improved if the actual simulated φ_{a} values are used rather than our predicted φ_{a} (not shown).
Because the monthly variations of φ_{d} are comparable to those of the comprehensive GCMs shown by KL12 (≲5° about their annual means), we infer that muted annual cycles of φ_{d} relative to that of φ_{a} can emerge via different mechanisms even restricting to Earthlike conditions. On the one hand are the comprehensive GCMs analyzed by KL12: the seasonal Ro values (that KL12 indirectly diagnose as a fitting parameter) span 0.45–1, and φ_{a} presumably varies closer to the realworld value and thus less than in our aquaplanet simulations. On the other hand is our aquaplanet simulation: φ_{d} variations (provided c_{d} and the 1month lag from φ_{a} are accounted for) appear determined almost entirely by the seasonality of φ_{a} with Ro treated as constant.
The theoretical predictions are also relatively insensitive to a reasonable range of parameter values. By (12), using conventional values of H = 10 km and Δ_{v} = 1/8, then for Earth Bu ≈ 0.46 and BuΔ_{v} ≈ 0.06. Then varying φ_{a} over 0°–15°, and Ro over the KL12reported Ro range of 0.45–1 yields a φ_{d} range of 28.0° to 35.9° (for φ_{a} = 0, Ro = 1 and φ_{a} = 15°, Ro = 0.45, respectively). Moreover, the φ_{d} range is fairly insensitive to Ro if φ_{a} is held fixed and likewise to φ_{a} if Ro is held fixed: if Ro is fixed at unity, the φ_{a} = 15° prediction moves equatorward only by 1.7°, and conversely if φ_{a} is at the equator the Ro = 0.45 prediction moves poleward by only 2.1°.
c. Relative behaviors of solsticial φ_{a} and φ_{d} across rotation rates
Figure 4a shows the winter φ_{d} for all the simulations as a function of
(a) Latitude of the winter hemisphere descending edge of the crossequatorial Hadley cell, φ_{d}, in idealized aquaplanet simulations of Faulk et al. (2017) and Singh (2019), and in the idealized dry simulations of Hill et al. (2021) as a function of the thermal Rossby number to the onefourth power, each signified by different symbols as indicated in the legend. The solid lines show the linear best fit for φ_{d} as a function of
Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JASD210328.1
(a) Latitude of the winter hemisphere descending edge of the crossequatorial Hadley cell, φ_{d}, in idealized aquaplanet simulations of Faulk et al. (2017) and Singh (2019), and in the idealized dry simulations of Hill et al. (2021) as a function of the thermal Rossby number to the onefourth power, each signified by different symbols as indicated in the legend. The solid lines show the linear best fit for φ_{d} as a function of
Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JASD210328.1
(a) Latitude of the winter hemisphere descending edge of the crossequatorial Hadley cell, φ_{d}, in idealized aquaplanet simulations of Faulk et al. (2017) and Singh (2019), and in the idealized dry simulations of Hill et al. (2021) as a function of the thermal Rossby number to the onefourth power, each signified by different symbols as indicated in the legend. The solid lines show the linear best fit for φ_{d} as a function of
Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JASD210328.1
Bestfit exponents of powerlaw scalings for the winter and summer edges of the crossequatorial solsticial Hadley cell in each set of simulations, as well as the bestfit slope and intercepts for each simulation set against the theoretical Ro_{th} power law. The slope for φ_{d} amounts to an approximation of c_{d} and that of φ_{a} an approximation of c_{a}; the latter is reported with the additional 2^{−1/3} factor included to facilitate direct comparison with c_{d}. Simulations are restricted to those for which Ro_{th} < 2, since the theoretical predictions of 1/3 and 1/4 for the winter and summer edges, respectively, assume small angle and thus small Ro_{th}. The dry LH88forced simulations do not include the Δ_{h} = 1/6 case. S19 stands for Singh (2019), and F17 stands for Faulk et al. (2017). The last row lists the diagnosed c_{d} and c_{a} values for the annual cycle in the Faulk et al. (2017) Earthlike simulation.
Table 1 includes bestfit powerlaw exponents for φ_{d} and for φ_{a} against Ro_{th} computed for each set of simulations by linear regression in log–log space. For all sets of simulations, the inferred exponent is larger and closer to 1/3 for φ_{a} than for φ_{d}_{,} which is closer to 1/4. The dry simulations exhibit the largest exponents for both, 0.41 and 0.3, respectively, and the Faulk et al. (2017) seasonally forced and perpetualsolstice simulations, respectively, give the smallest exponents, 0.28 and 0.21. The average of the bestfit exponents across the four simulation sets are nearly identical to the scalings, 0.26 and 0.33.
As Ro_{th} increases beyond ∼1, the simulated φ_{d} level off, never exceeding ∼70°. The full, nonsmallangle expression (11) solved numerically with φ_{a} = 0 and all parameters except Ω set to Earthlike values (dotted gray curve) qualitatively captures this. This contrasts with φ_{a}—shown in Fig. 4b as a function of
If Ro and the forcing isentropic slope Δ_{v}/Δ_{h} are both approximately unity, it can be shown from (13) and (16) that φ_{a} = φ_{d} provided that
The unfilled yellow triangle in Fig. 4 shows the dry, LH88forced simulation at Earth’s rotation rate with Δ_{h} = 1/6 rather than Δ_{h} = 1/15 as in the other three dry simulations. As is the case for φ_{a} (Hill et al. 2021), φ_{d} is somewhat separated from the power law of the Δ_{h} = 1/15 cases. Strictly speaking, in the φ_{a} ≈ 0 limit of (11), φ_{d} is independent of Δ_{h}. But, while small, φ_{a} ≠ 0 in the simulations, and since an increase in Δ_{h} moves φ_{a} poleward, it is qualitatively consistent that φ_{d} moves poleward as a result. Given that the annual cycle amounts to a variation in Δ_{h} sin φ_{m}, it is worth noting that the slope between the Δ_{h} = 1/15 and Δ_{h} = 1/6 cases at Earth’s rotation rate is shallower than that inferred across rotation rates at Δ_{h} = 1/15, which qualitatively coheres with c_{d} being smaller for the annual cycle than across rotation rates in the Faulk et al. (2017) simulations (Table 1). A caveat, however, is that the deviation of the Δ_{h} = 1/6 case is modest; whether Δ_{h} = 1/6 cases at different rotation rates or other Δ_{h} values would actually yield a different slope remains an open question.
d. Validity of the uniformRo assumption
Rossby number in the Singh (2019) simulations at the 300hPa level, computed either conventionally using (3) or the generalized form (17) that accounts for the tilting of streamlines. Overlaid are the cell edges φ_{a} and φ_{d}, with −φ_{a} also shown to ease comparison of the relative poleward extents of φ_{a} and φ_{d}. Rossby number values outside of the Hadley circulation are shown as thinner curves, since they are less relevant. They are also masked near the equator where division by the Coriolis parameter makes them less meaningful, within 12° on either side for the slowest rotation rate and by 1° less on either side for each subsequent rotation rate.
Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JASD210328.1
Rossby number in the Singh (2019) simulations at the 300hPa level, computed either conventionally using (3) or the generalized form (17) that accounts for the tilting of streamlines. Overlaid are the cell edges φ_{a} and φ_{d}, with −φ_{a} also shown to ease comparison of the relative poleward extents of φ_{a} and φ_{d}. Rossby number values outside of the Hadley circulation are shown as thinner curves, since they are less relevant. They are also masked near the equator where division by the Coriolis parameter makes them less meaningful, within 12° on either side for the slowest rotation rate and by 1° less on either side for each subsequent rotation rate.
Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JASD210328.1
Rossby number in the Singh (2019) simulations at the 300hPa level, computed either conventionally using (3) or the generalized form (17) that accounts for the tilting of streamlines. Overlaid are the cell edges φ_{a} and φ_{d}, with −φ_{a} also shown to ease comparison of the relative poleward extents of φ_{a} and φ_{d}. Rossby number values outside of the Hadley circulation are shown as thinner curves, since they are less relevant. They are also masked near the equator where division by the Coriolis parameter makes them less meaningful, within 12° on either side for the slowest rotation rate and by 1° less on either side for each subsequent rotation rate.
Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JASD210328.1
The generalized Rossby number is close to unity over a large fraction of the crossequatorial Hadley cell extent in all cases. (Both forms are masked out near the equator, as specified in the caption, where division by the Coriolis parameter makes them less physically meaningful.) The difference made by the vertical advection term is particularly large in the ascending branches. For either version, we subjectively identify two regimes over the descending branch. Slowly rotating cases have Ro_{gen} relatively uniform or even increasing slightly from the equator to the winter descending edge. More rapidly rotating cases have Ro_{gen} decreasing poleward, approaching zero in the vicinity of the winter descending edge, but there is considerable scatter in the value of Ro_{gen} at the edge. Despite this variation in the Rossby number across the simulations, it is evidently small enough that taking the bulk Ro_{gen} value as fixed in our scalings does not introduce major error.
5. Conclusions
a. Summary
We have introduced a unified theory for the latitudes of all three Hadley cell edges—the equatorward, ascending edge (φ_{a}) shared by the two Hadley cells as well as each cell’s poleward, descending edge (φ_{d})—throughout the annual cycle by combining two previous theories. First we predict φ_{a} using our recent theory based on the meridional extent of lowlatitude supercritical forcing (Hill et al. 2021). We then essentially plug this φ_{a} into the theory for φ_{d} based on baroclinic instability onset of KL12 that uses the seasonally varying φ_{a} and an assumed uniform Rossby number (Ro) within each Hadley cell’s upper branch. The new theory predicts that φ_{d} is displaced poleward when Ro decreases or as φ_{a} moves poleward, and φ_{a} varies with the thermal Rossby number (Ro_{th}) to the onethird power. But in the smallangle limit reasonable for Earth, the dependence on φ_{a} drops out and the scaling for φ_{d} predicts a onefourth power dependence on the planetary Burger number, or equivalently on Ro_{th} if only the planetary rotation rate (or any other term appearing in both Bu and Ro_{th}) are varied. The mixed layer’s thermal inertia acts to damp and delay φ_{a} relative to the insolation annual cycle, which we account for via an “effective” forcing annual cycle based on the formalism of Mitchell et al. (2014).
In an Earthlike, seasonally forced idealized aquaplanet simulation with a relatively shallow, 10m mixed layer ocean depth, φ_{a} migrates rapidly to ∼25° into either summer hemisphere, and this seasonal cycle is well captured by the supercriticalitybased scaling. The summer cell is too weak for the summer φ_{d} to be meaningful, but the winter φ_{d} varies by only ≲5° latitude about its mean position in either hemisphere. Our combined theory predicting φ_{a} and φ_{d} captures this behavior with Ro kept at unity as in the original H00 model, but requires in place of Ro variations that the φ_{d} prediction be lagged by 1 month from that of φ_{a}—which in turn is lagged by ∼1.5 months from the insolation.
In simulations across a wide range of planetary rotation rates in three idealized GCMs, both φ_{d} and φ_{a} adhere to the respective powerlaw exponents predicted by our theory in the relevant small thermal Rossby number regime. This, combined with a smaller proportionality constant for φ_{d} compared to φ_{a}_{,} helps explain why at very slow rotation rates the solsticial Hadley cell ascends essentially at the summer pole but descends considerably equatorward of the winter pole, ∼70°, rather than being roughly symmetric in extent about the equator as for more rapidly rotating cases including Earth.
b. Discussion
How might a predictive theory for Ro be constructed? Hoskins et al. (2020) offer an intriguing perspective relating to the frequency of deep convection in the ascending branch. They argue that only when convection is sufficiently deep will there be uppertropospheric meridional outflow that travels nearly inviscidly (i.e., with Ro ≈ 1) toward either pole; at times and longitudes where deep convection is absent, they argue Ro ≈ 0. Under those conditions, the timemean, zonalmean Ro field becomes a function of the spatial and temporal occurrence of deep convection in the ascending branch. This contrasts with the conventional, extratropically focused approach to Ro, wherein it is controlled by stresses from subtropical and extratropical eddies propagating into the deep tropics and breaking (Walker and Schneider 2006; Schneider 2006).
Vallis et al. (2015) speculate that Rossby waves are generated at the latitude of baroclinic instability onset, that these Rossby waves then propagate equatorward and break, and that the Hadley cell terminates at this wavebreaking latitude rather than the instability onset latitude. This equatorward displacement may relate to our need for the c_{d} < 1 parameter value to fit the φ_{d} annual cycle in the seasonally forced aquaplanet simulation. And this additional step—with some finite time scale required for the overall process of Rossby wave development, propagation, and breaking—could contribute to the lag of φ_{d} relative to φ_{a} in the Earthlike seasonal cycle simulation. At the same time, across rotation rates the best fit c_{d} parameters exceed unity in some cases (Table 1), which is harder to square with this Rossby wave–based mechanism of Vallis et al. (2015).
The physical credibility of the twolayer model’s critical shear criterion for baroclinic instability has been fairly questioned; a series of studies utilize a more comprehensive treatment of baroclinic instability to argue that φ_{d} occurs where the vertical extent of baroclinic eddies spans a sufficient fraction of the troposphere (Korty and Schneider 2008; Levine and Schneider 2011, 2015). The same studies also incorporate the influence of moisture on the effective static stability Δ_{v} (Levine and Schneider 2011, 2015).
Though we have relied on Ro being uniform over the upper branch of each Hadley cell (cf. KL12), the baroclinic instability criterion is computed latitude by latitude, and as such strictly speaking the behavior of Ro equatorward of the instability onset latitude is irrelevant. This contrasts with the equalarea model appropriate for axisymmetric atmospheres, which depends on the meridional integral of the difference between the RCE and dynamically equilibrated potential temperature fields over the expanse of the cell. In principle one could solve the equalarea model with our fixedRo temperature field (6) as a means of indirectly introducing eddy influences into it.
Under annualmean forcing in two dry and one moist idealized GCM, Mitchell and Hill (2021) find that φ_{d} scales as Ω^{−1/3} in all three models. This could be squared with our Ω^{−1/4} scaling for φ_{a} = 0 if Ro scales as Ω^{−2/3}. By eye from their Figs. 8 and 10, Ro does indeed follow an exponent close to this in two of the models—the same dry GCM we use and the moist GCM used by Faulk et al. (2017). But a simpler dry dynamical core (Held and Suarez 1994) shows no clear dependence of Ro on Ω. Mitchell and Hill (2021) also put forward an “omega governor” mechanism which operates in the case that static stability and the effective heating (diabatic plus eddy heat flux convergence) averaged over the descending branch do not change. Under those conditions, the poleward extent and mass overturning rate of the Hadley cell must vary in tandem: the cell weakens if it narrows, and it widens if it strengthens. Prior to any adjustment by φ_{d}, if φ_{a} moves poleward then the cell widens, which under the omega governor would act to strengthen the overturning. One can imagine that strengthening causing Ro to increase, insofar as parcels then traverse the upper branch more rapidly and hence are less exposed to eddy stresses. The increase in Ro would, all else equal, act to move φ_{d} equatorward, countering the direct influence of φ_{a} moving poleward. (This apparently is not important for the annual cycle simulation discussed above where Ro = 1 throughout the annual cycle perform suitably.)
The Mitchell et al. (2014) model we use for the effective forcing annual cycle in the presence of thermal inertia is based on radiative equilibrium rather than RCE. It also considers only the equilibration of the ocean surface mixed layer rather than the coupled nearsurface atmosphere–ocean. Cronin and Emanuel (2013) derive expressions for the time scale of equilibration to the RCE state in a coupled ocean–atmosphere column but do not consider the latitudinally nor seasonally varying problem. It could be useful to combine these approaches, ideally arriving at an analytical model for the effective seasonally varying forcing for moist atmospheres.
Our theory could be further tested in numerous ways: against reanalysis data for the climatological annual cycle of the Hadley cells, against reanalysis data for interannual variability and trends, against comprehensive climate model simulations of global warming (cf. KL12), and against simulations of other terrestrial planetary atmospheres. For the global warming problem, a useful starting point would be diagnosing seasonal, climatological bestfit Ro and φ_{a} values for each Hadley cell across comprehensive GCMs in preindustrial simulations in the CMIP6 archive. These could then be compared to diagnosed φ_{d} climatological values in the same simulations and forced changes in CMIP6 simulations under increased CO_{2}, although care must be taken in interpreting, e.g., changes in Δ_{h}, which is strictly a parameter of the hypothetical latitudebylatitude RCE state, not the dynamically equilibrated state that the archived simulations represent. We look forward to such tests.
Other authors also have considered a uniform Ro in the tropical upper troposphere. Becker et al. (1997) find that a uniform Ro = 0.5 approximation [their Fig. 7 and Eq. (28)] adequately captures the vorticity distribution in the descending branch of the winter Hadley cell in their simplified, dry GCM. ZuritaGotor and Held (2018) discuss the absolute vorticity distribution corresponding to uniform Ro.
This strictly applies to nonaxisymmetric atmospheres. It does not emerge clearly in simulations of axisymmetric atmospheres, which on theoretical grounds should exhibit φ_{d} poleward of −φ_{a} (Hill et al. 2019).
Davis and Birner (2022) present what amounts to (4) (i.e. u_{amc} multiplied by a lessthanunity constant) from heuristic grounds albeit without reference to Ro; it should be noted that they also challenge the physical validity of baroclinic instability onset determining Hadley cell extent.
H00 uses the symbol R to denote the planetary Burger number, which elsewhere (Held and Hou 1980; Hill et al. 2019) is used for the thermal Rossby number. To prevent confusion, we use the more explicit notation Bu for the planetary Burger number and Ro_{th} for the thermal Rossby number.
The tropopause depth H, which for the H00 theory is strictly the local tropopause height, is assumed horizontally uniform and unmodified by the largescale circulation from its forcing value corresponding to latitudebylatitude radiative–convective equilibrium; see Hill et al. (2020) for justification.
Over most of the parameter space, this quartic equation has only two solutions, which correspond to ±φ_{d}. Two additional solutions straddling very close to zero can appear for sufficiently large φ_{a} but are not physically meaningful.
Hill et al. (2021) report values for c_{a} of 1.0, 1.7, and 2.1 for the Faulk et al. (2017), Singh (2019), and Hill et al. (2021) simulations, respectively, but these implicitly incorporate the 2^{−1/3} ≈ 0.8 factor in (13). We separate it out from c_{a} for better consistency with (13).
In fact the annualmean rainfall and Hadley cells both show a double ITCZ resulting from this rapid jumping of the ascent fairly deep into either summer hemisphere (not shown).
For the annualmean rather than solsticial RCE state, Hill et al. (2020) diagnose a similar Δ_{h} ≈ 1/8 value based on numerical RCE simulations under annualmean insolation. This value is considerably larger than the solsticial one, which suggests that Δ_{h} would be even larger under equinoctial forcing. But we do not attempt to account for this seasonality in Δ_{h}.
This value of c_{a} is ∼45% larger than the bestfit value of 1.31 (Hill et al. 2021) for the solsticial φ_{a} across the Faulk et al. (2017) seasonally forced simulations with different rotation rates—a neither trivial nor orderofmagnitude difference, suggesting that the proportionality is moderately influenced by different processes in these two distinct contexts. The 1.9 value is also less than the values of 2.2 and 2.6 diagnosed across rotation rates for, respectively, the simulations of Singh (2019) and the dry simulations of Hill et al. (2021).
The extents of this linear regime (as well as the individual φ_{a} and φ_{d} values) differ appreciably between the two moist models, which are very similarly formulated, for reasons we do not understand. It spans
Acknowledgments.
We are grateful to Sean Faulk and Martin Singh for sharing the data from their simulations and for many valuable discussions. We thank Ilai Guendelman and three anonymous reviewers for helpful comments. S.A.H. acknowledges financial support from NSF Award 1624740 and from the Monsoon Mission, Earth System Science Organization, Ministry of Earth Sciences, Government of India. J.L.M. acknowledges funding from the Climate and LargeScale Dynamics program of the NSF, Award 1912673.
Data availability statement.
Data from the Singh (2019) simulations are available for download at Singh (2021). Data from the Faulk et al. (2017) simulations are available for download at Faulk (2021). Data from the Hill et al. (2021) simulations are available for download at Hill (2021).
APPENDIX
Model of Effective Insolation Annual Cycle Given Surface Thermal Inertia
Here we present the analytical model of Mitchell et al. (2014) for, given nonzero surface thermal inertia, the “effective” annual cycle of radiativeequilibrium temperatures, which are damped and lagged from the insolation (see also appendix A of Lee and Mitchell 2021). The notation and derivations are slightly modified—most notably, we use real numbers throughout rather than using complex numbers to represent annual oscillations—but the model is ultimately identical to that of Mitchell et al. (2014).
REFERENCES
Adam, O., T. Bischoff, and T. Schneider, 2016: Seasonal and interannual variations of the energy flux equator and ITCZ. Part I: Zonally averaged ITCZ position. J. Climate, 29, 3219–3230, https://doi.org/10.1175/JCLID150512.1.
Becker, E., G. Schmitz, and R. Geprags, 1997: The feedback of midlatitude waves onto the Hadley cell in a simple general circulation model. Tellus, 49A, 182–199, https://doi.org/10.3402/tellusa.v49i2.14464.
Bordoni, S., and T. Schneider, 2008: Monsoons as eddymediated regime transitions of the tropical overturning circulation. Nat. Geosci., 1, 515–519, https://doi.org/10.1038/ngeo248.
Byrne, M. P., and R. Thomas, 2019: Dynamics of ITCZ width: Ekman processes, nonEkman processes, and links to sea surface temperature. J. Atmos. Sci., 76, 2869–2884, https://doi.org/10.1175/JASD190013.1.
Caballero, R., R. T. Pierrehumbert, and J. L. Mitchell, 2008: Axisymmetric, nearly inviscid circulations in noncondensing radiativeconvective atmospheres. Quart. J. Roy. Meteor. Soc., 134, 1269–1285, https://doi.org/10.1002/qj.271.
Cronin, T. W., and K. A. Emanuel, 2013: The climate time scale in the approach to radiativeconvective equilibrium. J. Adv. Model. Earth Syst., 5, 843–849, https://doi.org/10.1002/jame.20049.
Davis, N. A., and T. Birner, 2022: Eddymediated Hadley cell expansion due to axisymmetric angular momentum adjustment to greenhouse gas forcings. J. Atmos. Sci., 79, 141–159, https://doi.org/10.1175/JASD200149.1.
Dima, I. M., and J. M. Wallace, 2003: On the seasonality of the Hadley cell. J. Atmos. Sci., 60, 1522–1527, https://doi.org/10.1175/15200469(2003)060<1522:OTSOTH>2.0.CO;2.
Donohoe, A., D. M. W. Frierson, and D. S. Battisti, 2014: The effect of ocean mixed layer depth on climate in slab ocean aquaplanet experiments. Climate Dyn., 43, 1041–1055, https://doi.org/10.1007/s0038201318434.
Emanuel, K. A., 1995: On thermally direct circulations in moist atmospheres. J. Atmos. Sci., 52, 1529–1534, https://doi.org/10.1175/15200469(1995)052<1529:OTDCIM>2.0.CO;2.
Faulk, S., 2021: Output from the Faulk, Bordoni, and Mitchell 2017 J. Atmos. Sci. idealized aquaplanet simulations, version 1. Zenodo, accessed 21 December 2021, https://doi.org/10.5281/ZENODO.5796259.
Faulk, S., J. Mitchell, and S. Bordoni, 2017: Effects of rotation rate and seasonal forcing on the ITCZ extent in planetary atmospheres. J. Atmos. Sci., 74, 665–678, https://doi.org/10.1175/JASD160014.1.
Frierson, D. M. W., I. M. Held, and P. ZuritaGotor, 2006: A grayradiation aquaplanet moist GCM. Part I: Static stability and eddy scale. J. Atmos. Sci., 63, 2548–2566, https://doi.org/10.1175/JAS3753.1.
Guendelman, I., and Y. Kaspi, 2018: An axisymmetric limit for the width of the Hadley cell on planets with large obliquity and long seasonality. Geophys. Res. Lett., 45, 13 213–13 221, https://doi.org/10.1029/2018GL080752.
Guendelman, I., and Y. Kaspi, 2019: Atmospheric dynamics on terrestrial planets: The seasonal response to changes in orbital, rotational, and radiative timescales. Astrophys. J., 881, 67, https://doi.org/10.3847/15384357/ab2a06.
Guendelman, I., and Y. Kaspi, 2020: Atmospheric dynamics on terrestrial planets with eccentric orbits. Astrophys. J., 901, 46, https://doi.org/10.3847/15384357/abaef8.
Held, I. M., 2000: The general circulation of the atmosphere: 2000 program in geophysical fluid dynamics. WHOI Tech. Rep. WHOI200103, 54 pp.
Held, I. M., and A. Y. Hou, 1980: Nonlinear axially symmetric circulations in a nearly inviscid atmosphere. J. Atmos. Sci., 37, 515–533, https://doi.org/10.1175/15200469(1980)037<0515:NASCIA>2.0.CO;2.
Held, I. M., and M. J. Suarez, 1994: A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models. Bull. Amer. Meteor. Soc., 75, 1825–1830, https://doi.org/10.1175/15200477(1994)075<1825:APFTIO>2.0.CO;2.
Hilgenbrink, C. C., and D. L. Hartmann, 2018: The response of Hadley circulation extent to an idealized representation of poleward ocean heat transport in an aquaplanet GCM. J. Climate, 31, 9753–9770, https://doi.org/10.1175/JCLID180324.1.
Hill, S. A., 2021: Output of dry idealized GCM simulations under perpetual solsticelike forcing, version 1. Zenodo, accessed 21 December 2021, https://doi.org/10.5281/ZENODO.5796233.
Hill, S. A., S. Bordoni, and J. L. Mitchell, 2019: Axisymmetric constraints on crossequatorial Hadley cell extent. J. Atmos. Sci., 76, 1547–1564, https://doi.org/10.1175/JASD180306.1.
Hill, S. A., S. Bordoni, and J. L. Mitchell, 2020: Axisymmetric Hadley cell theory with a fixed tropopause temperature rather than height. J. Atmos. Sci., 77, 1279–1294, https://doi.org/10.1175/JASD190169.1.
Hill, S. A., S. Bordoni, and J. L. Mitchell, 2021: Solsticial Hadley cell ascending edge theory from supercriticality. J. Atmos. Sci., 78, 1999–2011, https://doi.org/10.1175/JASD200341.1.
Hoskins, B. J., G.Y. Yang, and R. M. Fonseca, 2020: The detailed dynamics of the June–August Hadley cell. Quart. J. Roy. Meteor. Soc., 146, 557–575, https://doi.org/10.1002/qj.3702.
Kang, S. M., and J. Lu, 2012: Expansion of the Hadley cell under global warming: Winter versus summer. J. Climate, 25, 8387–8393, https://doi.org/10.1175/JCLID1200323.1.
Korty, R. L., and T. Schneider, 2008: Extent of Hadley circulations in dry atmospheres. Geophys. Res. Lett., 35, L23803, https://doi.org/10.1029/2008GL035847.
Lee, H.I., and J. L. Mitchell, 2021: The dynamics of quasistationary atmospheric rivers and their implications for monsoon onset. J. Atmos. Sci., 78, 2353–2365, https://doi.org/10.1175/JASD200262.1.
Levine, X. J., and T. Schneider, 2011: Response of the Hadley circulation to climate change in an aquaplanet GCM coupled to a simple representation of ocean heat transport. J. Atmos. Sci., 68, 769–783, https://doi.org/10.1175/2010JAS3553.1.
Levine, X. J., and T. Schneider, 2015: Baroclinic eddies and the extent of the Hadley circulation: An idealized GCM study. J. Atmos. Sci., 72, 2744–2761, https://doi.org/10.1175/JASD140152.1.
Lindzen, R. S., and A. V. Hou, 1988: Hadley circulations for zonally averaged heating centered off the equator. J. Atmos. Sci., 45, 2416–2427, https://doi.org/10.1175/15200469(1988)045<2416:HCFZAH>2.0.CO;2.
Mitchell, J. L., and S. A. Hill, 2021: Constraints from invariant subtropical vertical velocities on the scalings of Hadley cell strength and downdraft width with rotation rate. J. Atmos. Sci., 78, 1445–1463, https://doi.org/10.1175/JASD200191.1.
Mitchell, J. L., G. K. Vallis, and S. F. Potter, 2014: Effects of the seasonal cycle on superrotation in planetary atmospheres. Astrophys. J., 787, 23, https://doi.org/10.1088/0004637X/787/1/23.
O’Gorman, P. A., and T. Schneider, 2008: The hydrological cycle over a wide range of climates simulated with an idealized GCM. J. Climate, 21, 3815–3832, https://doi.org/10.1175/2007JCLI2065.1.
Plumb, R. A., and A. Y. Hou, 1992: The response of a zonally symmetric atmosphere to subtropical thermal forcing: Threshold behavior. J. Atmos. Sci., 49, 1790–1799, https://doi.org/10.1175/15200469(1992)049<1790:TROAZS>2.0.CO;2.
Rodwell, M. J., and B. J. Hoskins, 1996: Monsoons and the dynamics of deserts. Quart. J. Roy. Meteor. Soc., 122, 1385–1404, https://doi.org/10.1002/qj.49712253408.
Schneider, T., 2004: The tropopause and the thermal stratification in the extratropics of a dry atmosphere. J. Atmos. Sci., 61, 1317–1340, https://doi.org/10.1175/15200469(2004)061<1317:TTATTS>2.0.CO;2.
Schneider, T., 2006: The general circulation of the atmosphere. Annu. Rev. Earth Planet. Sci., 34, 655–688, https://doi.org/10.1146/annurev.earth.34.031405.125144.
Schneider, T., and S. Bordoni, 2008: Eddymediated regime transitions in the seasonal cycle of a Hadley circulation and implications for monsoon dynamics. J. Atmos. Sci., 65, 915–934, https://doi.org/10.1175/2007JAS2415.1.
Singh, M. S., 2019: Limits on the extent of the solsticial Hadley cell: The role of planetary rotation. J. Atmos. Sci., 76, 1989–2004, https://doi.org/10.1175/JASD180341.1.
Singh, M. S., 2021: Output of aquaplanet idealized GCM simulations under perpetual solsticelike forcing from Singh 2019, J. Atmos. Sci., version 1. Zenodo, accessed 21 December 2021, https://doi.org/10.5281/ZENODO.5796278.
Vallis, G. K., P. ZuritaGotor, C. Cairns, and J. Kidston, 2015: Response of the largescale structure of the atmosphere to global warming. Quart. J. Roy. Meteor. Soc., 141, 1479–1501, https://doi.org/10.1002/qj.2456.
Walker, C. C., and T. Schneider, 2006: Eddy influences on Hadley circulations: Simulations with an idealized GCM. J. Atmos. Sci., 63, 3333–3350, https://doi.org/10.1175/JAS3821.1.
WattMeyer, O., and D. M. W. Frierson, 2019: ITCZ width controls on Hadley cell extent and eddydriven jet position and their response to warming. J. Climate, 32, 1151–1166, https://doi.org/10.1175/JCLID180434.1.
Wei, H.H., and S. Bordoni, 2018: Energetic constraints on the ITCZ position in idealized simulations with a seasonal cycle. J. Adv. Model. Earth Syst., 10, 1708–1725, https://doi.org/10.1029/2018MS001313.
ZuritaGotor, P., and I. M. Held, 2018: The finiteamplitude evolution of mixed Kelvin–Rossby wave instability and equatorial superrotation in a shallowwater model and an idealized GCM. J. Atmos. Sci., 75, 2299–2316, https://doi.org/10.1175/JASD170386.1.