A Theory for the Hadley Cell Descending and Ascending Edges throughout the Annual Cycle

Spencer A. Hill aProgram in Atmospheric and Oceanic Sciences, Princeton University, Princeton, New Jersey
bLamont-Doherty Earth Observatory, Columbia University, Palisades, New York

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Simona Bordoni cDepartment of Civil, Environmental and Mechanical Engineering (DICAM), University of Trento, Trento, Italy
dDivision of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California

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Jonathan L. Mitchell eDepartment of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California
fDepartment of Earth, Planetary, and Space Sciences, University of California, Los Angeles, Los Angeles, California

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Abstract

We present a theory for the latitudinal extents of both Hadley cells throughout the annual cycle by combining our recent scaling for the ascending edge latitude based on low-latitude supercriticality with the theory for the poleward, descending edge latitudes of Kang and Lu based on baroclinic instability and a uniform Rossby number (Ro) within each cell’s upper branch. The resulting expressions for all three Hadley cell edges are predictive except for diagnosed values of Ro and two proportionality constants. Thermal inertia—which damps and lags the ascent latitude relative to the insolation—is accounted for semianalytically through the Mitchell et al. model of an “effective” seasonal forcing cycle. Our theory, given empirically an additional ∼1-month lag for the descending edge, captures the climatological annual cycle of the ascending and descending edges in an Earthlike simulation in an idealized aquaplanet general circulation model (GCM). In simulations in this and two other idealized GCMs with varied planetary rotation rate (Ω), the winter, descending edge of the solsticial, cross-equatorial Hadley cell scales approximately as Ω−1/2 and the summer, ascending edge as Ω−2/3, both in accordance with our theory. Possible future refinements and tests of the theory are discussed.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Spencer Hill, spencerh@princeton.edu

Abstract

We present a theory for the latitudinal extents of both Hadley cells throughout the annual cycle by combining our recent scaling for the ascending edge latitude based on low-latitude supercriticality with the theory for the poleward, descending edge latitudes of Kang and Lu based on baroclinic instability and a uniform Rossby number (Ro) within each cell’s upper branch. The resulting expressions for all three Hadley cell edges are predictive except for diagnosed values of Ro and two proportionality constants. Thermal inertia—which damps and lags the ascent latitude relative to the insolation—is accounted for semianalytically through the Mitchell et al. model of an “effective” seasonal forcing cycle. Our theory, given empirically an additional ∼1-month lag for the descending edge, captures the climatological annual cycle of the ascending and descending edges in an Earthlike simulation in an idealized aquaplanet general circulation model (GCM). In simulations in this and two other idealized GCMs with varied planetary rotation rate (Ω), the winter, descending edge of the solsticial, cross-equatorial Hadley cell scales approximately as Ω−1/2 and the summer, ascending edge as Ω−2/3, both in accordance with our theory. Possible future refinements and tests of the theory are discussed.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Spencer Hill, spencerh@princeton.edu

1. Introduction

Climatologically over the annual cycle, the poleward, descending edges of the Hadley cells vary meridionally by ≲ 5° latitude about their annual-mean 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 zonal-mean 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 zonal-mean dynamical problem, seeking a minimal explanation for the differing annual cycles of the Hadley cell descending and ascending edges (henceforth φd and φa, respectively, and formally defined below in terms of the mass overturning streamfunction).

For φd, 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 annual-mean φd 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 off-equatorial φa. For angular-momentum-conserving (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 φd to be farther poleward in solsticial seasons when φa is farther off-equator than in equinoctial seasons. An example of this framework’s utility is Hilgenbrink and Hartmann (2018), who interpret changes in φd throughout the annual cycle caused by ocean heat transports in terms of changes in φa.

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 cross-equatorial 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 φa, KL12 provide closed expressions for the Northern and Southern Hemisphere descending edge latitudes in all four seasons.1

For φa, in Hill et al. (2021) we presented a scaling for φa 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 φa 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 φa to the contemporaneous insolation—is accounting for thermal inertia’s damping and delaying of the φa annual cycle via the analytical model of Mitchell et al. (2014). The result is a unified theory for φa and both hemispheres’ φd 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, cross-equatorial 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 φa and winter φd are comparably poleward, in slowly rotating cases the summer φa is farther poleward than the winter φd: the cross-equatorial 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 best-fit power-law exponents for the two edges, close to φa ∝ Ω−2/3 and φd ∝ 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 cross-equatorial cell.

In the following sections we

  • derive and describe fixed-Ro zonal wind, angular momentum, and depth-averaged potential temperature fields (section 2);

  • present our unified theory, which essentially combines the KL12 model for φd with the Hill et al. (2021) theory for φa (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. Uniform-Ro fields

In general, absolute angular momentum is
M=acosφ(Ωacosφ+u),
where a is planetary radius, φ is latitude, and u is the zonally averaged zonal wind. This can be considered the sum of planetary angular momentum, Mp(φ) ≡ Ωa2 cos2φ, and relative angular momentum ua cosφ. The corresponding AMC zonal wind field—for which angular momentum is meridionally uniform and equal to the planetary angular momentum value Mp(φa) at a specified latitude φa—is
uamc(φ)=Ωacosφ(cos2φacos2φ1).
For the Hadley cells, we equate φa in (2) with the cells’ ascending edge on the grounds that ascent out of the viscous boundary layer there transmits the local planetary angular momentum Mp(φa) to the comparatively inviscid free troposphere (Held and Hou 1980; LH88). This neglects the finite width of the ascent branch (Watt-Meyer and Frierson 2019; Byrne and Thomas 2019), but in principle one could compute an effective ascent latitude by averaging the planetary angular momentum over the full extent of the ascending branch, perhaps weighting by the vertical velocity out of the boundary layer at each latitude.
The Rossby number is defined as
Roζf,
where ζ ≡ −(a cos φ)−1φ(u cos φ) is zonal-mean relative vorticity and f ≡ 2Ω sin φ is the planetary vorticity (i.e., the Coriolis parameter). Absolute vorticity is given by η = f + ζ = f(1 − Ro). Absolute vorticity is proportional to the meridional gradient of angular momentum, and as such in an AMC state necessarily η = 0 and Ro = 1. In discussing the GCM simulations below, we will make use of a generalized version of Ro (Singh 2019), but (3) is the quantity used in the fixed-Ro fields we now define.
From (3), ζ = −Rof. If Ro < 1 but horizontally uniform, integrating meridionally yields the fixed-Ro zonal wind field,
uRo(φ)=Rouamc(φ),
which is simply the AMC zonal wind field scaled by Ro.3 The corresponding angular momentum field is
MRo(φ)=Ωa2[Rocos2φa+(1Ro)cos2φ],
which is a Ro-weighted average of the planetary angular momentum at the ascent latitude, Mp(φa), and the full meridional distribution of the planetary angular momentum, Mp(φ).
Finally, though it does not enter into our model for the Hadley cell edges below, we present for a Boussinesq atmosphere [see, e.g., Eq. (1) of Hill et al. (2019) for the full underlying system of equations] the depth-averaged potential temperature field in gradient balance with uRo. Denoted θ^Ro, it is given by
θ^aθ^Ro(φ)θ0=Ro2Bu{(2Ro)cos2φ+cos2φa×[4(1Ro)ln(cosφacosφ)+Rocos2φacos2φ2]},
where θ^a is the depth-averaged potential temperature at the latitude φa, θ0 is the Boussinesq reference potential temperature,
BugH(Ωa)2
is the planetary Burger number with gravity g and tropopause height H, and we have assumed that uRo occurs near the tropopause while u ≈ 0 at the surface.

Figure 1 shows example uRo, MRo, and θ^Ro fields with Ro = 1, 0.5, or 0.3 and φa = 0° or 20°. For zonal wind, irrespective of Ro and φa, uRo vanishes at φa and increases monotonically moving away from the equator. If φa ≠ 0, uRo is negative from −φa to φa, minimizing at the equator. Both moving φa off equator and decreasing Ro act to make the westerlies less positive. For example, at 30°S/N, uRo ≈ 133, 67, and 40 m s−1, respectively, for Ro = 1, 0.5, and 0.3 if φa = 0 or 71, 36, and 21 m s−1, respectively, if φa = 20°. For angular momentum, irrespective of Ro and φa, MRo at φa is equal to the local planetary angular momentum Mp(φa); if Ro = 1, all latitudes take this value. For Ro < 1, it maximizes at the equator and decreases monotonically moving poleward. For the gradient-balanced potential temperature, θ^Ro maximizes at φa irrespective of Ro and φa and for φa ≠ 0 has a local minimum at the equator. As Ro increases, the meridional temperature gradients increase in magnitude, with a deeper equatorial dip and a more equatorward shoulder poleward of φa where temperatures begin dropping rapidly toward the pole.

Fig. 1.
Fig. 1.

Example (a) zonal wind (uRo, in m s−1), (b) angular momentum (MRo, normalized by Ωa2 and thus unitless), and (c) depth-averaged potential temperature (θ^Ro, in K) fields under a uniform Rossby number. Red curves are for φa = 0 and blue curves φa = 20°S/N, while dotted, dashed, and solid curves are for Ro = 1.0, 0.5, and 0.3, respectively. Horizontal axis spacing is in sin φ.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0328.1

3. Combined theory for Hadley cell ascending and descending edges

We now use these fixed-Ro fields to derive an expression for φd given values of Ro and φa that closely follows KL12. We then introduce within it our scaling for φa, yielding our unified theory for φd and φa. We then incorporate the influence of surface thermal inertia on the seasonal cycles of φa analytically and on φd more empirically.

a. Baroclinic instability onset theory for the Hadley cell edge with Ro < 1

Following H00, the baroclinic instability criterion for the two-layer model is approximately
uΩa=BuΔvcosφsin2φ,
where u is the zonal wind in the upper layer, the wind in the lower layer has been assumed small enough to neglect, and Δv is a static stability parameter representing the bulk fractional increase in potential temperature from the surface to the tropopause.4,5 H00 applies this to the annual-mean Hadley cells by assuming on-equatorial ascent, that the zonal winds are AMC, and that the descending edge is identical to this baroclinic instability onset latitude. Formally, taking φa ≈ 0, using (2) for u in (8), and taking the small-angle limit yields the original H00 theory for φd, which we denote φH00:
φH00=(BuΔv)1/4.
Using uRo (4) rather than uamc (2) as the zonal-wind profile in (8), for Ro < 1 the predicted cell edge becomes
φRo,ann=(BuΔvRo)1/4=Ro1/4φH00,
with φH00 given by (9). This displaces the cell edge prediction poleward by, for example, ∼19% if Ro = 0.5 or ∼50% if Ro = 0.2. Using the original H = 10 km and Δv = 1/8 parameter values from Held and Hou (1980) yields Bu ≈ 0.46 and φH00 ≈ 28°, which becomes approximately 33.3° if Ro = 0.5 or 41.9° if Ro = 0.2. This poleward displacement as Ro decreases coheres with physical intuition: because the upper-layer zonal wind magnitude at each latitude decreases as Ro decreases, the two-layer baroclinic instability onset criterion is met farther poleward.
For φa ≠ 0, using (4) in (8) gives without approximation
sin4φdcos2φdsin2φasin2φdcos2φdBuΔvRo=0,
where φd is the descending edge. From (10) the last term could equivalently be written φRo,ann4. These arguments serve equally for the cross-equatorial, winter cell and the summer cell (provided it exists), depending on which cell the chosen Ro value is representative of. A corollary is that if the mean Ro value is the same in both cells, then the Hadley circulation extends equally far into either hemisphere irrespective of φa. Note that our assumption of uniform Ro throughout either cell differs slightly from KL12, who assume Ro = 1 from the summer hemisphere edge of the cross-equatorial cell to the equator, a uniform Ro ≤ 1 value in the winter hemisphere, and a uniform Ro ≤ 1 value throughout the summer cell.

We have performed a 2D parameter sweep over φa and (BuΔv /Ro)1/4, from 0° to 90° in 1° increments for φa and from 0 to 2 in 0.01 increments for (BuΔv /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 φRo,ann ≡ (BuΔv/Ro)1/4, the right vertical axis shows the equivalent values of φRo,ann up to 90°, above which the small-angle solution obviously is nonsensical but the full solution retains its validity. The value of φd increases monotonically with φa and with (BuΔv/Ro)1/4. Close to the vertical axis of Fig. 2, |φRo,ann/φa| ≫ 1, and φdφRo,ann: φa is negligibly off-equator. Close to the horizontal axis, |φRo,ann/φa| ≪ 1, and thus φdφa. 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 −φa, near enough that it can be approximated as −φa. For intermediate values, if, e.g., φa = φRo,ann, then φd is displaced 27% poleward of φa. We note that |φd| ≥ |φa|; the descending edge latitude is always at or poleward of the ascending edge latitude. This is appropriate for the summer hemisphere φd but will prove imperfect for the winter φd in the idealized GCM simulations across rotation rates discussed below.

Fig. 2.
Fig. 2.

Numerical solutions of (11) for values of φa and of (BuΔv/Ro)1/4, with φa sampled from 0° to 90° in 1° increments and (BuΔv/Ro)1/4 (which is dimensionless) from 0 to 2.0 in 0.01 increments. The right vertical axis labels are the φRo,ann solutions from 0° to 90° corresponding to the given (BuΔv/Ro)1/4 values. Areas in white indicate that the simple numerical algorithm used did not converge, but clearly they correspond to φd very near the pole. Contours are from 5° to 90° in 5° increments according to the color bar. Overlaid gray contours are the corresponding small-angle solutions obtained using (12), likewise from 5° to 90° in 5° increments.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0328.1

Making the small-angle approximation for both φd and φa in (11) yields a closed expression for φd:
φd2=φa2[12+14+BuΔvRoφa4]=φa2[12+14+(φRo,annφa)4].
This is shown as overlaid contours in Fig. 2. Only for (BuΔv/Ro)1/40.8 does the small-angle approximation error exceed a few degrees latitude, irrespective of φa.

b. Incorporating theory for φa

Using (12) requires knowledge of φa, which KL12 diagnose. Hill et al. (2021) derive a prognostic theory for φa as determined by the meridional extent of supercritical forcing, based on the following arguments. If no large-scale 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 large-scale 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 Hadley-like, but in practice the opposite occurs: the ascending edge latitude is poleward of—and, crucially, proportional to—the supercritical extent.

Formally, in the small-angle limit Hill et al. (2021) show that
φa=ca(Roth2)1/3,
where (Roth/2)1/3 is the theoretical solution for the supercritical extent and ca is the empirically determined proportionality constant. Roth is the thermal Rossby number,
RothBuΔhsinφm,
with Δh the parameter of the imposed forcing (i.e., the latitude-by-latitude RCE state) that—in conjunction with φm, the latitude at which the forcing maximizes—determines the bulk meridional temperature gradients of the forcing. The same Roth1/3 scaling emerges in the semiempirical, axisymmetric theory of Caballero et al. (2008), as well as (for sufficiently small Roth) in numerical solutions of the original LH88 axisymmetric, equal-area model of the solsticial Hadley cells (Guendelman and Kaspi 2018). The Hill et al. (2021) scaling (13) emerges from a less strict set of assumptions than Caballero et al. (2008) and presupposes neither axisymmetry nor anything about the resulting circulation’s zonal wind and energy transport fields, unlike the equal-area model. Moreover, the solsticial equal-area model (LH88) predicts implausibly large Hadley cells for Earth as φm moves more than a few degrees off equator (Hill et al. 2019) and is biased poleward against numerical simulations in an idealized GCM over a wide range of planetary parameter values (cf. Fig. 15a of Guendelman and Kaspi 2019).

For the solsticial, cross-equatorial Hadley cells in the simulations analyzed by Hill et al. (2021) the best-fit value of ca ranges from 1.3 to 2.6 across three idealized GCMs.7 For Roth, Hill et al. (2021) show that for solsticial seasons one can attain an accurate estimate with φm set to 90° by tuning the value of Δh. Doing so, the nonstandard sinφm term drops out and the Roth definition becomes the more conventional Roth = BuΔh. But the sinφm dependence is necessary for understanding the annual cycle as will be discussed further below (see also Fig. 15 of Guendelman and Kaspi 2020).

Given diagnosed values of ca and Ro for each Hadley cell, (13) in conjunction with (12) provide a theory for all three cell edges: (13) predicts φa, and using that in (12) for φd then yields
φd=cdca(Roth2)1/312+14+24/3ca4ΔvΔhsinφm1RoRoth1/3,
where we have also included the empirical fitting parameter cd that will prove necessary for the simulations analyzed below. The term Δvh sin φm, which is the ratio of the planetary-scale vertical to horizontal fractional potential temperature changes of the latitude-by-latitude RCE state, amounts to a seasonally varying bulk isentropic slope of the forcing. From (15), the poleward edge of either Hadley cell increases with increasing Roth, increasing RCE isentropic slope, or decreasing Ro. Large Roth corresponds to the large-φa limit above, φd ≈ φa, while small Roth corresponds to the small-φa limit of φd ≈ φRo,ann.

c. Influence of surface thermal inertia on the φa 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 latitude-by-latitude RCE state that determines φa. 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 φa intact but modifies the φm term within Roth to be damped and delayed in its seasonal excursions. The appendix below provides the derivation, presenting here only the end results of how φm is modified.

First consider the unmodified φm annual cycle. For Earth’s present-day insolation, during equinoctial seasons there is only one maximum in φm, 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 global-scale Hadley cells in other planetary atmospheres (Singh 2019), for Earth we can comfortably consider the midlatitude maximum at solstice to be φm. An advantage of this choice is that, combining the equinoctial and solsticial seasons, this yields a nearly sinusoidal annual cycle of φm (not shown): φmφm,ann cos[ωorb(ttsolst)], where φm.ann = 44° is the annual maximum value of φm, ωorb is the orbital frequency, t is the time of year, and tsolst 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 ≡ (ωorbτti)−1, where τti=Cml/(4σT¯3) is the thermal inertia time scale with Cml the surface mixed layer heat capacity, σ the Stefan–Boltzmann constant, and T¯ the annual-mean, global-mean temperature. The heat capacity of the mixed layer is given by Cml = ρwcwHml, where ρw is the density of liquid water, cw is the heat capacity of liquid water, and Hml is the depth of the mixed layer. Using values of ρw = 1000 kg m−3, cw = 4186 J kg−1 K−1, and taking Hml = 10 m as used in the simulation to be discussed, we have Cml ≈ 4.186 × 107 J m−2 K−1. Then using T¯=288K and σ = 5.67 × 10−8 W m−2 K−4, the thermal inertia time scale is τti ≈ 89 days. Using ωorb = 2π/365 days−1 we have α ≈ 0.65, an intermediate value: the thermal inertial and seasonal cycle time scales are comparable, meaning that the effective φm annual cycle will be moderately damped and phase lagged from the insolation. Cf. the appendix below, this yields a damping factor to φm of α/1+α20.55 and a lag of τlag=ωorb1arctan(α1)57days: roughly, the effective φm is damped by half and delayed by two months. These values will prove fairly accurate in predicting φa in the simulation discussed next, albeit with the lag empirically reduced by ∼20%, to 45 days, for a best fit.

For the time scale of φd relative to the seasonal cycle, we proceed much more empirically. Insofar as φd is determined by the combination of φa and Ro as we have posited, one relevant time scale is that of the upper-level zonal wind adjustment: it takes a finite amount of time for the zonal winds in the descending branch to adjust to a change in φa 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 time-invariant Ro seems to fit the simulation well as shown below. We find empirically that the best fit to the simulated φd occurs by lagging our predicted φd based on the contemporaneous Ro and φa 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 φd and φa 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 φm = 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 γΓd, where Γd = g/cp is the dry adiabatic lapse rate with cp 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 Δh 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 Δh = 1/6 and Earth’s rotation rate. The Δh = 1/6 value is conventional (LH88), but Hill et al. (2021) show that, for φm = 90°, Δh = 1/15 is the best fit to numerically simulated latitude-by-latitude 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 slab-ocean lower boundary with a 10-m mixed layer depth. They are forced either with an annual cycle of insolation approximating that of present-day 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 time-invariant, 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 φd are diagnosed conventionally as the latitude at which the mass-overturning 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 same-signed (albeit very weak) all the way to either pole. For φa, the same 10% threshold is used in the perpetual-solstice 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, φa 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 φa and φd

Before presenting the simulation results, we delineate three regimes regarding the relative importance of Ro versus φa in determining φd. First is where Ro predominates: by (12) if φa is small relative to φRo,ann = (BuΔv/Ro)1/4 throughout the annual cycle, then the annual cycle of φd is determined by the annual cycle of Ro (provided that Δv and H are constant across seasons). Second is intermediate, with both Ro and φa influential as in the compensation regarding the winter φd found by KL12 in CMIP3 simulations: φa is farthest poleward at solstice, acting to move φd poleward, but Ro is largest in the solsticial cross-equatorial cell, acting to move the winter φd equatorward. Third is where φa predominates, as we now show holds for the seasonally forced simulation at Earth’s rotation rate of Faulk et al. (2017): φa 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 φd.

Figure 3 shows the climatological annual cycles of φa (solid red curve), of φd 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 (φd varies over 21.3°–27.7°N and 21.7°–26.5°S) and annual cycle phasing, with φa migrating into either summer hemisphere with a ∼1.5-month lag behind the insolation. However, the φa excursions are excessive, 25.7°S–23.2°N, resulting in an excessively rapid transition from equinoctial to solsticial regimes, approaching closer to the square-wave 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 φd is only well defined in the winter half-year for either hemisphere (December–May for the Northern Hemisphere, June–November for the Southern Hemisphere). These discrepancies seem attributable to the rather shallow 10-m surface mixed layer, which promotes excessive seasonality (Donohoe et al. 2014; Wei and Bordoni 2018). Nevertheless, the variations in φd and φa—in particular that the φd ranges are comparable to Earth’s and several times smaller than that of φa—lead us to consider this useful enough for Earth.

Fig. 3.
Fig. 3.

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 φm 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 φa, 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/JAS-D-21-0328.1

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 φa). The effective φm accounting for thermal inertia (orange stars) is damped and lagged from the actual insolation as described above. φa stays within the tropics always, with φaφm 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 φm and Δh to determine best fits of the LH88 thermal forcing profile against numerical simulations of latitude-by-latitude RCE under solsticial forcing; for φm ≈ 44° the best fit over the tropics occurs for Δh = 1/8 (approximately twice that of the Δh = 1/15 value for φm = 90°).9 We therefore take Δh = 1/8. With Δh set, the proportionality constant ca is left effectively as a fitting parameter, which by eye provides the best fit for ca ≈ 1.9.10 The prediction is clearly not without empiricism, but nevertheless, we are pleased with the overall accuracy against the simulated φa.

We then use this theoretically computed φa as just described to predict φd. Due to an inadvertent loss of zonal-wind 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 equatorward-most possible φd predictions, all else equal. Even still, this yields a φd prediction poleward of the simulated φd values (not shown), which we correct for by setting cd = 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 1-month phase lag of φd compared to φa (which in turn lags the insolation by ∼1.5 months). The φd prediction is only marginally improved if the actual simulated φa values are used rather than our predicted φa (not shown).

Because the monthly variations of φd are comparable to those of the comprehensive GCMs shown by KL12 (≲5° about their annual means), we infer that muted annual cycles of φd relative to that of φa 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 φa presumably varies closer to the real-world value and thus less than in our aquaplanet simulations. On the other hand is our aquaplanet simulation: φd variations (provided cd and the 1-month lag from φa are accounted for) appear determined almost entirely by the seasonality of φa 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 Δv = 1/8, then for Earth Bu ≈ 0.46 and BuΔv ≈ 0.06. Then varying φa over 0°–15°, and Ro over the KL12-reported Ro range of 0.45–1 yields a φd range of 28.0° to 35.9° (for φa = 0, Ro = 1 and φa = 15°, Ro = 0.45, respectively). Moreover, the φd range is fairly insensitive to Ro if φa is held fixed and likewise to φa if Ro is held fixed: if Ro is fixed at unity, the φa = 15° prediction moves equatorward only by 1.7°, and conversely if φa is at the equator the Ro = 0.45 prediction moves poleward by only 2.1°.

c. Relative behaviors of solsticial φa and φd across rotation rates

From (13), φaRoth1/3, which Hill et al. (2021) show accurately describes the solsticial φa across planetary rotation rates in the simulations presently under consideration. For φd, by (11) for small Roth and thus small φa,
φdcd(BuΔvRo)1/4=cd(RothRoΔvΔh)1/4,
again incorporating the empirical fitting parameter cd. In that case, provided Ro does not change appreciably then φdBu1/4Roth1/4, where in this context we can substitute Roth for Bu since only Ω is varied and appears identically (as Ω−2) in the two nondimensional numbers. We now argue that the idealized GCM simulations reflect this modest 1/3 − 1/4 = 1/12 difference in power-law exponent in Roth for φa versus φd.

Figure 4a shows the winter φd for all the simulations as a function of Roth1/4. For each model, a best-fit line is included of φd with Roth1/4, restricting to simulations in the linear regime of φd versus Roth1/4.11 Overall the simulations follow this scaling well. Table 1 lists the slope and intercept from the linear best fits of φd against Roth1/4, with the slope amounting to a best fit for the empirical cd parameter in (15) (at least in the small-Roth limit). The inferred cd values range from 1.4 for the Singh (2019) simulations to 0.9 for the Faulk et al. (2017) perpetual solstice simulations. The value of 1.0 for the Faulk et al. (2017) seasonal cycle simulations is modestly higher than the value discussed above of 0.8 for the climatological annual cycle in the Faulk et al. (2017) simulation at Earth’s rotation rate—opposite to ca, which was larger for the annual cycle than across rotation rates. The intercepts, which in theory should be zero, range from −2.4° to 5.6° latitude and average across the simulation sets to a modest 0.3°.

Fig. 4.
Fig. 4.

(a) Latitude of the winter hemisphere descending edge of the cross-equatorial Hadley cell, φd, 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 one-fourth power, each signified by different symbols as indicated in the legend. The solid lines show the linear best fit for φd as a function of Roth1/4 for the given simulation set, restricting to Roth < 1, with red, blue, and yellow for the Singh (2019), Faulk et al. (2017), and the Δh = 1/15 dry simulations, respectively. The dotted gray curve is the numerical solution to (11) with φa = 0 and RoΔhv = 1. (b) The corresponding quantities for the summer, ascending edge φa, with the horizontal axis spacing being Roth1/3, and the dashed gray curve being the numerical solution to the full expression for the extent of supercritical forcing [(b) replicates Fig. 6 of Hill et al. (2021)].

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0328.1

Table 1

Best-fit exponents of power-law scalings for the winter and summer edges of the cross-equatorial solsticial Hadley cell in each set of simulations, as well as the best-fit slope and intercepts for each simulation set against the theoretical Roth power law. The slope for φd amounts to an approximation of cd and that of φa an approximation of ca; the latter is reported with the additional 2−1/3 factor included to facilitate direct comparison with cd. Simulations are restricted to those for which Roth < 2, since the theoretical predictions of 1/3 and 1/4 for the winter and summer edges, respectively, assume small angle and thus small Roth. The dry LH88-forced simulations do not include the Δh = 1/6 case. S19 stands for Singh (2019), and F17 stands for Faulk et al. (2017). The last row lists the diagnosed cd and ca values for the annual cycle in the Faulk et al. (2017) Earthlike simulation.

Table 1

Table 1 includes best-fit power-law exponents for φd and for φa against Roth 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 φa than for φd, 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 perpetual-solstice simulations, respectively, give the smallest exponents, 0.28 and 0.21. The average of the best-fit exponents across the four simulation sets are nearly identical to the scalings, 0.26 and 0.33.

As Roth increases beyond ∼1, the simulated φd level off, never exceeding ∼70°. The full, non-small-angle expression (11) solved numerically with φa = 0 and all parameters except Ω set to Earthlike values (dotted gray curve) qualitatively captures this. This contrasts with φa—shown in Fig. 4b as a function of Roth1/3 (reproducing Fig. 6 of Hill et al. 2021)—which in the slowly rotating regime is near the summer pole (Hill et al. 2021). In other words, as rotation rate decreases the cross-equatorial Hadley cell becomes increasingly lopsided, in the slowly rotating regime extending effectively to the summer pole but not the winter pole. As noted above, the solutions (11) and (12) for φd always predict |φd| ≥ |φa|, and as such to explain the |φd| < |φa| cases we must appeal to the empirical proportionality constants ca and cd introduced in (13) and (15) as follows.

If Ro and the forcing isentropic slope Δvh are both approximately unity, it can be shown from (13) and (16) that φa = φd provided that 21/3cd/ca=Roth1/12. Table 1 includes the slope and intercepts of φa against Roth1/3, with the slope amounting to a best fit for ca/21/3 in (13). From Table 1 then, the ratio 21/3 cd/ca takes values approximately over 0.6–1 in the four simulation sets. Given the 0.6–1 range for the left-hand side, this yields Roth ≈ 1, with φd equatorward of φa for larger Roth. This roughly captures the behavior of the simulations. Of course, the slowly rotating regime with |φa| > |φd| extends beyond the regime where the small-angle power-law scalings being appealed to are credible, but it nevertheless begins within the small-angle regime.

The unfilled yellow triangle in Fig. 4 shows the dry, LH88-forced simulation at Earth’s rotation rate with Δh = 1/6 rather than Δh = 1/15 as in the other three dry simulations. As is the case for φa (Hill et al. 2021), φd is somewhat separated from the power law of the Δh = 1/15 cases. Strictly speaking, in the φa ≈ 0 limit of (11), φd is independent of Δh. But, while small, φa ≠ 0 in the simulations, and since an increase in Δh moves φa poleward, it is qualitatively consistent that φd moves poleward as a result. Given that the annual cycle amounts to a variation in Δh sin φm, it is worth noting that the slope between the Δh = 1/15 and Δh = 1/6 cases at Earth’s rotation rate is shallower than that inferred across rotation rates at Δh = 1/15, which qualitatively coheres with cd 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 Δh = 1/6 case is modest; whether Δh = 1/6 cases at different rotation rates or other Δh values would actually yield a different slope remains an open question.

d. Validity of the uniform-Ro assumption

Finally, we assess the validity of treating the upper-tropospheric Ro as uniform within each Hadley cell. Figure 5 shows the meridional profile of Ro at 300 hPa in each simulation of Singh (2019), computed both conventionally as (3) and, following Singh (2019), in a generalized form that incorporates vertical advection of angular momentum:
Rogen1f[ζ+ωυup],
where υ is meridional velocity, ω is vertical velocity in pressure coordinates, p is pressure, and all quantities are zonal averages. This accounts for the considerable tilting of streamlines in the upper branch of the cross-equatorial cells [cf. Figs. 3 and 4 of Faulk et al. (2017), Fig. 7 of Singh (2019), and Fig. 5 of Hill et al. (2021)], which causes the conventional Ro to deviate from unity even if streamlines and angular momentum contours are everywhere parallel. Though it is the conventional Ro that appears in uRo and thus ultimately our expressions for φd, for the simulations we argue that (17) is more instructive: for the two-layer model of baroclinic instability utilized, the bulk upper-tropospheric behavior is more relevant than that at any chosen pressure level. And as streamlines begin tilting toward the surface in the descending branch, the conventional Ro at any given level decreases, while the bulk zonal velocities of the upper branch still roughly correspond to the planetary angular momentum values from where the streamlines exited the boundary layer in the ascending branch. In other words, along streamlines angular momentum is nearly conserved (see Fig. 7 of Singh 2019), which the meridional profile of Ro at a fixed pressure level cannot capture.
Fig. 5.
Fig. 5.

Rossby number in the Singh (2019) simulations at the 300-hPa level, computed either conventionally using (3) or the generalized form (17) that accounts for the tilting of streamlines. Overlaid are the cell edges φa and φd, with −φa also shown to ease comparison of the relative poleward extents of φa and φd. 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/JAS-D-21-0328.1

The generalized Rossby number is close to unity over a large fraction of the cross-equatorial 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 Rogen relatively uniform or even increasing slightly from the equator to the winter descending edge. More rapidly rotating cases have Rogen decreasing poleward, approaching zero in the vicinity of the winter descending edge, but there is considerable scatter in the value of Rogen at the edge. Despite this variation in the Rossby number across the simulations, it is evidently small enough that taking the bulk Rogen 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 (φa) shared by the two Hadley cells as well as each cell’s poleward, descending edge (φd)—throughout the annual cycle by combining two previous theories. First we predict φa using our recent theory based on the meridional extent of low-latitude supercritical forcing (Hill et al. 2021). We then essentially plug this φa into the theory for φd based on baroclinic instability onset of KL12 that uses the seasonally varying φa and an assumed uniform Rossby number (Ro) within each Hadley cell’s upper branch. The new theory predicts that φd is displaced poleward when Ro decreases or as φa moves poleward, and φa varies with the thermal Rossby number (Roth) to the one-third power. But in the small-angle limit reasonable for Earth, the dependence on φa drops out and the scaling for φd predicts a one-fourth power dependence on the planetary Burger number, or equivalently on Roth if only the planetary rotation rate (or any other term appearing in both Bu and Roth) are varied. The mixed layer’s thermal inertia acts to damp and delay φa 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, 10-m mixed layer ocean depth, φa migrates rapidly to ∼25° into either summer hemisphere, and this seasonal cycle is well captured by the supercriticality-based scaling. The summer cell is too weak for the summer φd to be meaningful, but the winter φd varies by only ≲5° latitude about its mean position in either hemisphere. Our combined theory predicting φa and φd captures this behavior with Ro kept at unity as in the original H00 model, but requires in place of Ro variations that the φd prediction be lagged by 1 month from that of φa—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 φd and φa adhere to the respective power-law exponents predicted by our theory in the relevant small thermal Rossby number regime. This, combined with a smaller proportionality constant for φd compared to φa, 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 upper-tropospheric 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 time-mean, zonal-mean 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 wave-breaking latitude rather than the instability onset latitude. This equatorward displacement may relate to our need for the cd < 1 parameter value to fit the φd 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 φd relative to φa in the Earthlike seasonal cycle simulation. At the same time, across rotation rates the best fit cd 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 two-layer 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 φd 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 Δv (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 equal-area 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 equal-area model with our fixed-Ro temperature field (6) as a means of indirectly introducing eddy influences into it.

Under annual-mean forcing in two dry and one moist idealized GCM, Mitchell and Hill (2021) find that φd scales as Ω−1/3 in all three models. This could be squared with our Ω−1/4 scaling for φa = 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 φd, if φa 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 φd equatorward, countering the direct influence of φa 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 near-surface 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 best-fit Ro and φa values for each Hadley cell across comprehensive GCMs in preindustrial simulations in the CMIP6 archive. These could then be compared to diagnosed φd climatological values in the same simulations and forced changes in CMIP6 simulations under increased CO2, although care must be taken in interpreting, e.g., changes in Δh, which is strictly a parameter of the hypothetical latitude-by-latitude RCE state, not the dynamically equilibrated state that the archived simulations represent. We look forward to such tests.

1

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. Zurita-Gotor and Held (2018) discuss the absolute vorticity distribution corresponding to uniform Ro.

2

This strictly applies to nonaxisymmetric atmospheres. It does not emerge clearly in simulations of axisymmetric atmospheres, which on theoretical grounds should exhibit φd poleward of −φa (Hill et al. 2019).

3

Davis and Birner (2022) present what amounts to (4) (i.e. uamc multiplied by a less-than-unity 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.

4

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 Roth for the thermal Rossby number.

5

The tropopause depth H, which for the H00 theory is strictly the local tropopause height, is assumed horizontally uniform and unmodified by the large-scale circulation from its forcing value corresponding to latitude-by-latitude radiative–convective equilibrium; see Hill et al. (2020) for justification.

6

Over most of the parameter space, this quartic equation has only two solutions, which correspond to ±φd. Two additional solutions straddling very close to zero can appear for sufficiently large φa but are not physically meaningful.

7

Hill et al. (2021) report values for ca 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 ca for better consistency with (13).

8

In fact the annual-mean 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).

9

For the annual-mean rather than solsticial RCE state, Hill et al. (2020) diagnose a similar Δh ≈ 1/8 value based on numerical RCE simulations under annual-mean insolation. This value is considerably larger than the solsticial one, which suggests that Δh would be even larger under equinoctial forcing. But we do not attempt to account for this seasonality in Δh.

10

This value of ca is ∼45% larger than the best-fit value of 1.31 (Hill et al. 2021) for the solsticial φa across the Faulk et al. (2017) seasonally forced simulations with different rotation rates—a neither trivial nor order-of-magnitude 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).

11

The extents of this linear regime (as well as the individual φa and φd values) differ appreciably between the two moist models, which are very similarly formulated, for reasons we do not understand. It spans Roth1 for the Singh (2019) simulations vs Roth1.5 for the Faulk et al. (2017) simulations.

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 Large-Scale 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 radiative-equilibrium 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).

We take insolation S to vary with sin2φ from a maximum latitude φm that varies over the annual cycle:
S(φ,t)S0=1+Δs3Δs[sin2φ2sinφm(t)sinφ],
where Δs controls the fractional horizontal gradients in insolation. At any given time, this expression amounts to the forcing from LH88 but with the sin2φm term omitted [it is also the same functional form as the seasonally varying Newtonian cooling temperature profile of Schneider and Bordoni (2008)]. Omitting that term does not alter the gradient-balanced wind and hence the φa prediction, but it makes the resulting derivations easier.
We then take φm to vary sinusoidally in time: sin φm = μm,ann sin(ωorbt), where ωorb is the orbital frequency, t is the time of year relative to boreal fall equinox, μm,ann = sinφm,ann, with φm,ann = 44° being the annual maximum value of φm. Thus,
S(φ,t)S0=1+Δs3Δs[sin2φ+2μm,annsin(ωorbt)sinφ].
As noted by Mitchell et al. (2014), this results in the annual mean at each latitude being independent of the seasonality (i.e., of φm and ωorb): denoting annual averages by overbars, we have S¯/S0=1+Δs/3Δssin2φ.
We assume an ocean surface mixed layer of uniform depth with heat capacity Cml, whose temperature evolution at each latitude is determined purely by radiative fluxes, warmed by the seasonally varying insolation, and cooled by longwave emission:
CmldTdt=SσT4,
where σ is the Stefan–Boltzmann constant and T is the mixed layer temperature. This implicitly treats the albedo as uniformly zero; a uniform, nonzero albedo could be incorporated by simply rescaling S0. In the annual average at each latitude we then have S¯=σT¯4. Defining T0(S0/σ)1/4 and taking Δs ≪ 1, this yields the familiar T¯/T0=1+Δh/3Δhsin2φ (e.g., Held and Hou 1980), having defined Δh ≡ Δs/4. [Note that S0 and T0 are global parameters but not the global-mean values; global-mean insolation is S0(1 + Δs/3) and global-mean temperature T0(1 + Δh/3).]
We now split the insolation and surface temperature into annual-mean and annually varying components at each latitude: S=S¯+S and T=T¯+T. We assume that |T/T¯|1, such that T4=(T¯+T)4T¯4+4T¯3T via Taylor expansion. Then, defining the thermal inertia time scale, (A3) becomes
dTdt+Tτti=SCml.
We then nondimensionalize (A4), with nondimensional quantities in hats, as T=T0T^, S=S0S^, and t=ωorb1t^. In that case, (A4) becomes
ωorbT0dT^dt^+T0τtiT^=S0CmlS^.
Next, we define the ratio of the seasonal time scale to the thermal inertial time scale as α ≡ (ωorbτti)−1. Dividing through by ωorbT0, using S0/T0=σT03, recalling τti=Cml/4σT¯3, and approximating T¯T0, we have
dT^dt^+αT^=α4S^.
Denoting β ≡ 2Δsμm,ann sin φ, then S^=βsint^ and thus
dT^dt^+αT^=αβ4sint^.
This ordinary differential equation has the solution of
T^(t)=14αβ1+α2[αsint^cost^]+cexp(αt^),
and we set the constant c = 0 since the solution is bounded for t^. This difference between a sine and cosine of the same wavelength amounts to a damped and phase-lagged sine as follows. Denoting the nondimensional phase lag τ^lag, by trigonometric identity sin(t^τ^lag)=cosτ^lagsint^sinτ^lagcost^. Setting this equal to γ(αsint^cost^), where γ is a constant to be determined, implies sinτ^lag=γ and cosτ^lag=αγ, and therefore that τ^lag=arctan(α1). Then, requiring sin2τ^lag+cos2τ^lag=1 yields γ=(1+α)1/2. Thus,
T^(t)=14αβ1+α2sin(t^τ^lag),τ^lag=arctan(α1).
For large α, thermal inertia is weak relative to the annual cycle, yielding τ^lag0 and α/1+α21: the temperatures are nearly undamped and in phase and with the insolation. For α ≪ 1, thermal inertia is relatively strong, yielding τ^lagπ/2 and α/1+α2α: the temperatures are strongly damped (by a factor of α) and in quadrature with the insolation.
Finally, for completeness the dimensional form of the full seasonally varying, radiative-equilibrium temperature field is
T(t)T0=1+Δh3{13sin2φ+6α1+α2μm,annsinφsin[ωorb(tτlag)]},τlag=ωorb1arctan(α1).

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, 32193230, https://doi.org/10.1175/JCLI-D-15-0512.1.

    • Search Google Scholar
    • Export Citation
  • 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, 182199, https://doi.org/10.3402/tellusa.v49i2.14464.

    • Search Google Scholar
    • Export Citation
  • Bordoni, S., and T. Schneider, 2008: Monsoons as eddy-mediated regime transitions of the tropical overturning circulation. Nat. Geosci., 1, 515519, https://doi.org/10.1038/ngeo248.

    • Search Google Scholar
    • Export Citation
  • Byrne, M. P., and R. Thomas, 2019: Dynamics of ITCZ width: Ekman processes, non-Ekman processes, and links to sea surface temperature. J. Atmos. Sci., 76, 28692884, https://doi.org/10.1175/JAS-D-19-0013.1.

    • Search Google Scholar
    • Export Citation
  • Caballero, R., R. T. Pierrehumbert, and J. L. Mitchell, 2008: Axisymmetric, nearly inviscid circulations in non-condensing radiative-convective atmospheres. Quart. J. Roy. Meteor. Soc., 134, 12691285, https://doi.org/10.1002/qj.271.

    • Search Google Scholar
    • Export Citation
  • Cronin, T. W., and K. A. Emanuel, 2013: The climate time scale in the approach to radiative-convective equilibrium. J. Adv. Model. Earth Syst., 5, 843849, https://doi.org/10.1002/jame.20049.

    • Search Google Scholar
    • Export Citation
  • Davis, N. A., and T. Birner, 2022: Eddy-mediated Hadley cell expansion due to axisymmetric angular momentum adjustment to greenhouse gas forcings. J. Atmos. Sci., 79, 141159, https://doi.org/10.1175/JAS-D-20-0149.1.

    • Search Google Scholar
    • Export Citation
  • Dima, I. M., and J. M. Wallace, 2003: On the seasonality of the Hadley cell. J. Atmos. Sci., 60, 15221527, https://doi.org/10.1175/1520-0469(2003)060<1522:OTSOTH>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • 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, 10411055, https://doi.org/10.1007/s00382-013-1843-4.

    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., 1995: On thermally direct circulations in moist atmospheres. J. Atmos. Sci., 52, 15291534, https://doi.org/10.1175/1520-0469(1995)052<1529:OTDCIM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • 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.

    • Search Google Scholar
    • Export Citation
  • 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, 665678, https://doi.org/10.1175/JAS-D-16-0014.1.

    • Search Google Scholar
    • Export Citation
  • Frierson, D. M. W., I. M. Held, and P. Zurita-Gotor, 2006: A gray-radiation aquaplanet moist GCM. Part I: Static stability and eddy scale. J. Atmos. Sci., 63, 25482566, https://doi.org/10.1175/JAS3753.1.

    • Search Google Scholar
    • Export Citation
  • 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 21313 221, https://doi.org/10.1029/2018GL080752.

    • Search Google Scholar
    • Export Citation
  • 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/10.3847/1538-4357/ab2a06.

    • Search Google Scholar
    • Export Citation
  • Guendelman, I., and Y. Kaspi, 2020: Atmospheric dynamics on terrestrial planets with eccentric orbits. Astrophys. J., 901, 46, https://doi.org/10.3847/1538-4357/abaef8.

    • Search Google Scholar
    • Export Citation
  • Held, I. M., 2000: The general circulation of the atmosphere: 2000 program in geophysical fluid dynamics. WHOI Tech. Rep. WHOI-2001-03, 54 pp.

    • Search Google Scholar
    • Export Citation
  • Held, I. M., and A. Y. Hou, 1980: Nonlinear axially symmetric circulations in a nearly inviscid atmosphere. J. Atmos. Sci., 37, 515533, https://doi.org/10.1175/1520-0469(1980)037<0515:NASCIA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • 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, 18251830, https://doi.org/10.1175/1520-0477(1994)075<1825:APFTIO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • 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, 97539770, https://doi.org/10.1175/JCLI-D-18-0324.1.

    • Search Google Scholar
    • Export Citation
  • Hill, S. A., 2021: Output of dry idealized GCM simulations under perpetual solstice-like forcing, version 1. Zenodo, accessed 21 December 2021, https://doi.org/10.5281/ZENODO.5796233.

    • Search Google Scholar
    • Export Citation
  • Hill, S. A., S. Bordoni, and J. L. Mitchell, 2019: Axisymmetric constraints on cross-equatorial Hadley cell extent. J. Atmos. Sci., 76, 15471564, https://doi.org/10.1175/JAS-D-18-0306.1.

    • Search Google Scholar
    • Export Citation
  • 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, 12791294, https://doi.org/10.1175/JAS-D-19-0169.1.

    • Search Google Scholar
    • Export Citation
  • Hill, S. A., S. Bordoni, and J. L. Mitchell, 2021: Solsticial Hadley cell ascending edge theory from supercriticality. J. Atmos. Sci., 78, 19992011, https://doi.org/10.1175/JAS-D-20-0341.1.

    • Search Google Scholar
    • Export Citation
  • 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, 557575, https://doi.org/10.1002/qj.3702.

    • Search Google Scholar
    • Export Citation
  • Kang, S. M., and J. Lu, 2012: Expansion of the Hadley cell under global warming: Winter versus summer. J. Climate, 25, 83878393, https://doi.org/10.1175/JCLI-D-12-00323.1.

    • Search Google Scholar
    • Export Citation
  • 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.

    • Search Google Scholar
    • Export Citation
  • Lee, H.-I., and J. L. Mitchell, 2021: The dynamics of quasi-stationary atmospheric rivers and their implications for monsoon onset. J. Atmos. Sci., 78, 23532365, https://doi.org/10.1175/JAS-D-20-0262.1.

    • Search Google Scholar
    • Export Citation
  • 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, 769783, https://doi.org/10.1175/2010JAS3553.1.

    • Search Google Scholar
    • Export Citation
  • Levine, X. J., and T. Schneider, 2015: Baroclinic eddies and the extent of the Hadley circulation: An idealized GCM study. J. Atmos. Sci., 72, 27442761, https://doi.org/10.1175/JAS-D-14-0152.1.

    • Search Google Scholar
    • Export Citation
  • Lindzen, R. S., and A. V. Hou, 1988: Hadley circulations for zonally averaged heating centered off the equator. J. Atmos. Sci., 45, 24162427, https://doi.org/10.1175/1520-0469(1988)045<2416:HCFZAH>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • 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, 14451463, https://doi.org/10.1175/JAS-D-20-0191.1.

    • Search Google Scholar
    • Export Citation
  • 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/10.1088/0004-637X/787/1/23.

    • Search Google Scholar
    • Export Citation
  • 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, 38153832, https://doi.org/10.1175/2007JCLI2065.1.

    • Search Google Scholar
    • Export Citation
  • 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, 17901799, https://doi.org/10.1175/1520-0469(1992)049<1790:TROAZS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Rodwell, M. J., and B. J. Hoskins, 1996: Monsoons and the dynamics of deserts. Quart. J. Roy. Meteor. Soc., 122, 13851404, https://doi.org/10.1002/qj.49712253408.

    • Search Google Scholar
    • Export Citation
  • Schneider, T., 2004: The tropopause and the thermal stratification in the extratropics of a dry atmosphere. J. Atmos. Sci., 61, 13171340, https://doi.org/10.1175/1520-0469(2004)061<1317:TTATTS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Schneider, T., 2006: The general circulation of the atmosphere. Annu. Rev. Earth Planet. Sci., 34, 655688, https://doi.org/10.1146/annurev.earth.34.031405.125144.

    • Search Google Scholar
    • Export Citation
  • Schneider, T., and S. Bordoni, 2008: Eddy-mediated regime transitions in the seasonal cycle of a Hadley circulation and implications for monsoon dynamics. J. Atmos. Sci., 65, 915934, https://doi.org/10.1175/2007JAS2415.1.

    • Search Google Scholar
    • Export Citation
  • Singh, M. S., 2019: Limits on the extent of the solsticial Hadley cell: The role of planetary rotation. J. Atmos. Sci., 76, 19892004, https://doi.org/10.1175/JAS-D-18-0341.1.

    • Search Google Scholar
    • Export Citation
  • Singh, M. S., 2021: Output of aquaplanet idealized GCM simulations under perpetual solstice-like forcing from Singh 2019, J. Atmos. Sci., version 1. Zenodo, accessed 21 December 2021, https://doi.org/10.5281/ZENODO.5796278.

    • Search Google Scholar
    • Export Citation
  • Vallis, G. K., P. Zurita-Gotor, C. Cairns, and J. Kidston, 2015: Response of the large-scale structure of the atmosphere to global warming. Quart. J. Roy. Meteor. Soc., 141, 14791501, https://doi.org/10.1002/qj.2456.

    • Search Google Scholar
    • Export Citation
  • Walker, C. C., and T. Schneider, 2006: Eddy influences on Hadley circulations: Simulations with an idealized GCM. J. Atmos. Sci., 63, 33333350, https://doi.org/10.1175/JAS3821.1.

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  • Watt-Meyer, O., and D. M. W. Frierson, 2019: ITCZ width controls on Hadley cell extent and eddy-driven jet position and their response to warming. J. Climate, 32, 11511166, https://doi.org/10.1175/JCLI-D-18-0434.1.

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  • 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, 17081725, https://doi.org/10.1029/2018MS001313.

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Save
  • 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, 32193230, https://doi.org/10.1175/JCLI-D-15-0512.1.

    • Search Google Scholar
    • Export Citation
  • 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, 182199, https://doi.org/10.3402/tellusa.v49i2.14464.

    • Search Google Scholar
    • Export Citation
  • Bordoni, S., and T. Schneider, 2008: Monsoons as eddy-mediated regime transitions of the tropical overturning circulation. Nat. Geosci., 1, 515519, https://doi.org/10.1038/ngeo248.

    • Search Google Scholar
    • Export Citation
  • Byrne, M. P., and R. Thomas, 2019: Dynamics of ITCZ width: Ekman processes, non-Ekman processes, and links to sea surface temperature. J. Atmos. Sci., 76, 28692884, https://doi.org/10.1175/JAS-D-19-0013.1.

    • Search Google Scholar
    • Export Citation
  • Caballero, R., R. T. Pierrehumbert, and J. L. Mitchell, 2008: Axisymmetric, nearly inviscid circulations in non-condensing radiative-convective atmospheres. Quart. J. Roy. Meteor. Soc., 134, 12691285, https://doi.org/10.1002/qj.271.

    • Search Google Scholar
    • Export Citation
  • Cronin, T. W., and K. A. Emanuel, 2013: The climate time scale in the approach to radiative-convective equilibrium. J. Adv. Model. Earth Syst., 5, 843849, https://doi.org/10.1002/jame.20049.

    • Search Google Scholar
    • Export Citation
  • Davis, N. A., and T. Birner, 2022: Eddy-mediated Hadley cell expansion due to axisymmetric angular momentum adjustment to greenhouse gas forcings. J. Atmos. Sci., 79, 141159, https://doi.org/10.1175/JAS-D-20-0149.1.

    • Search Google Scholar
    • Export Citation
  • Dima, I. M., and J. M. Wallace, 2003: On the seasonality of the Hadley cell. J. Atmos. Sci., 60, 15221527, https://doi.org/10.1175/1520-0469(2003)060<1522:OTSOTH>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • 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, 10411055, https://doi.org/10.1007/s00382-013-1843-4.

    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., 1995: On thermally direct circulations in moist atmospheres. J. Atmos. Sci., 52, 15291534, https://doi.org/10.1175/1520-0469(1995)052<1529:OTDCIM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • 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.

    • Search Google Scholar
    • Export Citation
  • 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, 665678, https://doi.org/10.1175/JAS-D-16-0014.1.

    • Search Google Scholar
    • Export Citation
  • Frierson, D. M. W., I. M. Held, and P. Zurita-Gotor, 2006: A gray-radiation aquaplanet moist GCM. Part I: Static stability and eddy scale. J. Atmos. Sci., 63, 25482566, https://doi.org/10.1175/JAS3753.1.

    • Search Google Scholar
    • Export Citation
  • 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 21313 221, https://doi.org/10.1029/2018GL080752.

    • Search Google Scholar
    • Export Citation
  • 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/10.3847/1538-4357/ab2a06.

    • Search Google Scholar
    • Export Citation
  • Guendelman, I., and Y. Kaspi, 2020: Atmospheric dynamics on terrestrial planets with eccentric orbits. Astrophys. J., 901, 46, https://doi.org/10.3847/1538-4357/abaef8.

    • Search Google Scholar
    • Export Citation
  • Held, I. M., 2000: The general circulation of the atmosphere: 2000 program in geophysical fluid dynamics. WHOI Tech. Rep. WHOI-2001-03, 54 pp.

    • Search Google Scholar
    • Export Citation
  • Held, I. M., and A. Y. Hou, 1980: Nonlinear axially symmetric circulations in a nearly inviscid atmosphere. J. Atmos. Sci., 37, 515533, https://doi.org/10.1175/1520-0469(1980)037<0515:NASCIA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • 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, 18251830, https://doi.org/10.1175/1520-0477(1994)075<1825:APFTIO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • 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, 97539770, https://doi.org/10.1175/JCLI-D-18-0324.1.

    • Search Google Scholar
    • Export Citation
  • Hill, S. A., 2021: Output of dry idealized GCM simulations under perpetual solstice-like forcing, version 1. Zenodo, accessed 21 December 2021, https://doi.org/10.5281/ZENODO.5796233.

    • Search Google Scholar
    • Export Citation
  • Hill, S. A., S. Bordoni, and J. L. Mitchell, 2019: Axisymmetric constraints on cross-equatorial Hadley cell extent. J. Atmos. Sci., 76, 15471564, https://doi.org/10.1175/JAS-D-18-0306.1.

    • Search Google Scholar
    • Export Citation
  • 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, 12791294, https://doi.org/10.1175/JAS-D-19-0169.1.

    • Search Google Scholar
    • Export Citation
  • Hill, S. A., S. Bordoni, and J. L. Mitchell, 2021: Solsticial Hadley cell ascending edge theory from supercriticality. J. Atmos. Sci., 78, 19992011, https://doi.org/10.1175/JAS-D-20-0341.1.

    • Search Google Scholar
    • Export Citation
  • 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, 557575, https://doi.org/10.1002/qj.3702.

    • Search Google Scholar
    • Export Citation
  • Kang, S. M., and J. Lu, 2012: Expansion of the Hadley cell under global warming: Winter versus summer. J. Climate, 25, 83878393, https://doi.org/10.1175/JCLI-D-12-00323.1.

    • Search Google Scholar
    • Export Citation
  • 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.

    • Search Google Scholar
    • Export Citation
  • Lee, H.-I., and J. L. Mitchell, 2021: The dynamics of quasi-stationary atmospheric rivers and their implications for monsoon onset. J. Atmos. Sci., 78, 23532365, https://doi.org/10.1175/JAS-D-20-0262.1.

    • Search Google Scholar
    • Export Citation
  • 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, 769783, https://doi.org/10.1175/2010JAS3553.1.

    • Search Google Scholar
    • Export Citation
  • Levine, X. J., and T. Schneider, 2015: Baroclinic eddies and the extent of the Hadley circulation: An idealized GCM study. J. Atmos. Sci., 72, 27442761, https://doi.org/10.1175/JAS-D-14-0152.1.

    • Search Google Scholar
    • Export Citation
  • Lindzen, R. S., and A. V. Hou, 1988: Hadley circulations for zonally averaged heating centered off the equator. J. Atmos. Sci., 45, 24162427, https://doi.org/10.1175/1520-0469(1988)045<2416:HCFZAH>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • 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, 14451463, https://doi.org/10.1175/JAS-D-20-0191.1.

    • Search Google Scholar
    • Export Citation
  • 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/10.1088/0004-637X/787/1/23.

    • Search Google Scholar
    • Export Citation
  • 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, 38153832, https://doi.org/10.1175/2007JCLI2065.1.

    • Search Google Scholar
    • Export Citation
  • 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, 17901799, https://doi.org/10.1175/1520-0469(1992)049<1790:TROAZS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Rodwell, M. J., and B. J. Hoskins, 1996: Monsoons and the dynamics of deserts. Quart. J. Roy. Meteor. Soc., 122, 13851404, https://doi.org/10.1002/qj.49712253408.

    • Search Google Scholar
    • Export Citation
  • Schneider, T., 2004: The tropopause and the thermal stratification in the extratropics of a dry atmosphere. J. Atmos. Sci., 61, 13171340, https://doi.org/10.1175/1520-0469(2004)061<1317:TTATTS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Schneider, T., 2006: The general circulation of the atmosphere. Annu. Rev. Earth Planet. Sci., 34, 655688, https://doi.org/10.1146/annurev.earth.34.031405.125144.

    • Search Google Scholar
    • Export Citation
  • Schneider, T., and S. Bordoni, 2008: Eddy-mediated regime transitions in the seasonal cycle of a Hadley circulation and implications for monsoon dynamics. J. Atmos. Sci., 65, 915934, https://doi.org/10.1175/2007JAS2415.1.

    • Search Google Scholar
    • Export Citation
  • Singh, M. S., 2019: Limits on the extent of the solsticial Hadley cell: The role of planetary rotation. J. Atmos. Sci., 76, 19892004, https://doi.org/10.1175/JAS-D-18-0341.1.

    • Search Google Scholar
    • Export Citation
  • Singh, M. S., 2021: Output of aquaplanet idealized GCM simulations under perpetual solstice-like forcing from Singh 2019, J. Atmos. Sci., version 1. Zenodo, accessed 21 December 2021, https://doi.org/10.5281/ZENODO.5796278.

    • Search Google Scholar
    • Export Citation
  • Vallis, G. K., P. Zurita-Gotor, C. Cairns, and J. Kidston, 2015: Response of the large-scale structure of the atmosphere to global warming. Quart. J. Roy. Meteor. Soc., 141, 14791501, https://doi.org/10.1002/qj.2456.

    • Search Google Scholar
    • Export Citation
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  • Fig. 1.

    Example (a) zonal wind (uRo, in m s−1), (b) angular momentum (MRo, normalized by Ωa2 and thus unitless), and (c) depth-averaged potential temperature (θ^Ro, in K) fields under a uniform Rossby number. Red curves are for φa = 0 and blue curves φa = 20°S/N, while dotted, dashed, and solid curves are for Ro = 1.0, 0.5, and 0.3, respectively. Horizontal axis spacing is in sin φ.

  • Fig. 2.

    Numerical solutions of (11) for values of φa and of (BuΔv/Ro)1/4, with φa sampled from 0° to 90° in 1° increments and (BuΔv/Ro)1/4 (which is dimensionless) from 0 to 2.0 in 0.01 increments. The right vertical axis labels are the φRo,ann solutions from 0° to 90° corresponding to the given (BuΔv/Ro)1/4 values. Areas in white indicate that the simple numerical algorithm used did not converge, but clearly they correspond to φd very near the pole. Contours are from 5° to 90° in 5° increments according to the color bar. Overlaid gray contours are the corresponding small-angle solutions obtained using (12), likewise from 5° to 90° in 5° increments.

  • Fig. 3.

    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 φm 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 φa, and in the 2 months nearest solstice the maximum is near the summer pole and thus not shown.

  • Fig. 4.

    (a) Latitude of the winter hemisphere descending edge of the cross-equatorial Hadley cell, φd, 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 one-fourth power, each signified by different symbols as indicated in the legend. The solid lines show the linear best fit for φd as a function of Roth1/4 for the given simulation set, restricting to Roth < 1, with red, blue, and yellow for the Singh (2019), Faulk et al. (2017), and the Δh = 1/15 dry simulations, respectively. The dotted gray curve is the numerical solution to (11) with φa = 0 and RoΔhv = 1. (b) The corresponding quantities for the summer, ascending edge φa, with the horizontal axis spacing being Roth1/3, and the dashed gray curve being the numerical solution to the full expression for the extent of supercritical forcing [(b) replicates Fig. 6 of Hill et al. (2021)].

  • Fig. 5.

    Rossby number in the Singh (2019) simulations at the 300-hPa level, computed either conventionally using (3) or the generalized form (17) that accounts for the tilting of streamlines. Overlaid are the cell edges φa and φd, with −φa also shown to ease comparison of the relative poleward extents of φa and φd. 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.

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