1. Introduction
Why does the shared, ascending edge of Earth’s Hadley cells sit around 15° latitude in the summer hemisphere, instead of say 1.5° or at the summer pole? Results from idealized general circulation model (GCM) simulations suggest that neither limit is as outlandish as may initially seem. For the former, an O(1) increase in the surface–atmosphere system’s thermal inertia time scale leaves the ascending branch insufficient time to migrate more than a few degrees off the equator before the insolation maximum moves back toward the opposite hemisphere (e.g., Donohoe et al. 2014). For the latter, the insolation distribution that ultimately drives the general circulation maximizes at the summer pole, and an O(1) decrease in the planetary rotation rate yields nearly pole-to-pole solsticial Hadley circulations (e.g., Williams and Holloway 1982).
Although increasing the system’s thermal inertia (or hastening the annual cycle) pulls the solsticial ascending branch equatorward, decreasing it (or slowing the annual cycle) does not push the branch much poleward—even in the limit of time-invariant solsticial forcing (e.g., Faulk et al. 2017, hereafter F17; Zhou and Xie 2018; Singh 2019, hereafter S19). This suggests the presence of a dynamical constraint emanating from the time-mean forcing at solstice.
Several theories exist of direct or indirect relevance to this fundamental property of the general circulation, but each is limited in one or more substantive ways. The energetic framework for the position of the intertropical convergence zone (ITCZ; e.g., Kang et al. 2008; Schneider et al. 2014) is diagnostic1 and not always accurate, even qualitatively (e.g., Hill 2019). The solsticial equal-area model of Lindzen and Hou (1988, hereafter LH88) is predictive but inaccurate over much of the relevant parameter space, even restricting to axisymmetric atmospheres for which it is strictly applicable (cf. Hill et al. 2019)—though we will make ample use of the analytical forcing profile introduced by LH88. A recent theory for the ascending edge based on slantwise convective neutrality (S19) is quantitatively accurate across the idealized GCM simulations against which it has been tested, but it is diagnostic. Here, we will pursue an alternative, predictive theory based on the extent of supercritical radiative forcing.
A supercritical latitude is one at which, supposing no large-scale overturning circulation existed, the resulting state of latitude-by-latitude radiative–convective equilibrium (RCE) would possess impermissible distributions of angular momentum and absolute vorticity (Plumb and Hou 1992; Emanuel 1995)—that is, distributions that violate Hide’s theorem (Hide 1969).2 A large-scale overturning circulation must therefore span at minimum all supercritical latitudes (Held and Hou 1980). Recent studies using idealized dry, axisymmetric (Hill et al. 2019) and moist, eddying (F17; S19) GCMs explore the qualitative utility of the supercritical forcing extent as a predictor of the solsticial Hadley cell extent as planetary rotation rate is varied. But they fall short of deriving a closed, analytical expression for the solsticial supercritical forcing extent.
An attractive feature of the supercritical forcing extent is that its interpretation as setting the minimum extent of a large-scale circulation holds equally for axisymmetric and zonally varying atmospheres: by definition, RCE implies the absence of any large-scale circulation, and therefore over those latitudes where RCE cannot be sustained some circulation has to emerge. At the same time, it does not specify the nature of the large-scale circulation that emerges, in particular whether even Hadley like at all or instead strongly macroturbulent as in the extratropics. Using the supercritical forcing extent as a theory specifically for the Hadley cell ascending edge, therefore, entails some additional empirical justification. A more beneficial corollary of this dynamical agnosticism, though, is that the supercritical forcing extent’s validity does not depend on the resulting Hadley cells being in one of the two limiting regimes of the zonal momentum budget—angular momentum conserving or eddy dominated. Such limit-based approaches will always be incomplete for the simple reason that Earth’s solsticial Hadley cells do not consistently adhere to one or the other limit (e.g., Schneider 2006; Bordoni and Schneider 2008).
For annual-mean forcing, an analytical expression for the extent of supercritical forcing has been known for decades thanks to Held and Hou (1980), who assume an RCE depth-averaged temperature profile varying simply as sin2φ, where φ is latitude.3 For solsticial forcing, then, a natural starting place is the analytical RCE profile from LH88 that moves the global maximum of the RCE temperature field off the equator but retains the simple sin2φ meridional dependence as in Held and Hou (1980). In fact, a cruder sinφ approximation will prove adequate.
This paper addresses these issues by showing that
conceptually, supercritical forcing extent can constitute a meaningful theory for the solsticial Hadley circulation ascending latitude in zonally varying atmospheres, provided certain empirical claims are established (section 2);
the LH88 forcing usefully approximates latitude-by-latitude RCE under solsticial forcing with respect to fields relevant to the Hadley cells (section 3);
a simple, approximate analytical solution exists for the supercritical forcing extent at solstice based on the LH88 forcing (section 4); and
the cross-equatorial Hadley cell extent obeys this simple scaling in previously reported moist idealized GCM simulations as well as newly performed dry idealized GCM simulations (section 5).
2. Supercritical forcing: Basis and interpretation in eddying atmospheres
a. Solsticial insolation
Figure 1 shows the diurnally averaged insolation distribution on the day of boreal summer solstice for Earth’s present-day orbit (all results are equally applicable to austral summer). Insolation is zero in the polar night region spanning the winter high latitudes. Moving northward, it increases, reaching ~386 W m−2 at the equator, but with steadily decreasing slope up to a local maximum of ~485 W m−2 near 43°N. From there it decreases modestly to a local minimum of ~478 W m−2 near 62°N and finally increases monotonically from there to its global maximum of ~525 W m−2 at the north pole. Figure 1 also shows insolation for longer averaging periods of 30 and 90 days centered on northern summer solstice. Differences across the three averaging periods are modest.
Insolation (W m−2) for averaging windows centered on northern summer solstice of 1, 30, and 90 days. It is computed using the “daily_insolation” function of the climlab package (Rose 2018), and is based on the methods of Berger and Loutre (1991).
Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0341.1
b. Conceptual basis of supercritical forcing extent
If RCE prevailed at each latitude, then large-scale meridional and vertical velocities would vanish. The large-scale zonal velocity field would be in gradient balance (i.e., thermal wind balance but also including the nonlinear metric term) with the temperature field that is determined by the interactions between radiative and convective processes at each latitude. But this exhibits physically untenable features, most obviously at the equator where the Coriolis parameter vanishes: no gradient-balanced solution is attainable with a nonzero cross-equatorial insolation gradient (which occurs at all times other than equinox; LH88).
Away from the equator in the summer hemisphere where RCE temperature increases moving poleward, gradient balance yields upper-tropospheric easterlies (assuming zonal wind is small at the surface due to drag) that draw angular momentum below its local planetary value. If sufficiently strong, these easterlies can cause the RCE angular momentum field, denoted Mrce, to be increasing poleward, thereby changing the sign of the Mrce meridional gradient and thus of the RCE absolute vorticity, denoted ηrce (Plumb and Hou 1992). Symbolically, this implies fηrce < 0 (Emanuel 1995), where f ≡ 2Ωsinφ is the planetary vorticity (i.e., the Coriolis parameter) with planetary rotation rate Ω. That cannot be a time-mean solution for multiple reasons [see Adam and Paldor (2009) and Hill et al. (2019) for details]: it implies local extrema in Mrce, which cannot be sustained in the presence of nonzero viscosity; it is the sufficient condition for symmetric instability; and, near the tropopause where vertical velocity vanishes, a change in sign would require the absolute vorticity to pass through a fixed point (i.e., where ∂tηrce = 0) that occurs at ηrce = 0 in the vorticity equation. A large-scale circulation must emerge spanning at minimum all such latitudes, which are referred to as supercritical. Equivalently, where ηrce = 0 in the summer hemisphere constitutes the minimal extent of the large-scale circulation in that hemisphere.4
c. Supercritical forcing in eddying atmospheres
Supercritical forcing extent has not figured centrally in theories for Earth’s solsticial Hadley cell ascending edge for reasons that seem plausible in passing but that falter under scrutiny.
First is the notion that supercriticality is meaningful in axisymmetric atmospheres only and is in principle inapplicable to macroturbulent atmospheres. One can see how this would emerge. Supercriticality (though not referred to as such) was popularized by Held and Hou (1980) as an intermediate step in developing their highly influential axisymmetric, angular-momentum-conserving model for the annual-mean Hadley cells. For solstice, the fηrce < 0 facet was presented by Plumb and Hou (1992) also in a purely axisymmetric context [though soon extended to moist, zonally varying contexts by Emanuel (1995)]. Moreover, the marginally critical state of ηrce = 0 corresponds to uniform Mrce, which, with its homogeneous angular momentum distribution, might sound like a description of the axisymmetric (and nearly inviscid) angular-momentum-conserving model.
But the angular momentum that is spatially homogeneous in the angular-momentum-conserving model is that of the dynamically equilibrated state, M, and crucially Mrce ≠ M. By definition, the latitude-by-latitude RCE state is one in which there is no large-scale circulation, zonally symmetric or otherwise. Irrespective of whether the Hadley cells in the dynamically equilibrated state end up perfectly homogenizing angular momentum, or are totally controlled by eddies, or (most likely) something in between, latitude-by-latitude RCE cannot be sustained over any latitude that is supercritically forced. Therefore, at least in the narrow sense regarding the minimal extent of a large-scale circulation of some kind, supercritical forcing extent is meaningful in all rotating atmospheres.
This leads to a second concern: whether in practice the supercritical forcing extent usefully predicts, much more specifically, the location of the Hadley cell ascending edge. In simulations for which supercritical extent has been explicitly computed, the ascending edge latitude sits poleward of the ηrce = 0 latitude (F17; Hill et al. 2019; S19).5 As such, to be a useful predictor, the supercritical forcing extent must scale proportionally with the actual ascending cell edge latitude. As section 5 will demonstrate—albeit empirically—this does in fact hold in a diverse range of idealized GCM simulations.
3. Latitude-by-latitude RCE under solsticial forcing
a. Numerical simulations
We use the climlab single-column model (Rose 2018) to simulate solsticial latitude-by-latitude RCE. Each single-column simulation is forced with insolation corresponding to present-day, boreal summer solstice at a specified latitude, with the chosen latitudes in 1° increments spanning from equator to the pole in the summer hemisphere and from the equator to 55° in the winter hemisphere. Apart from using solsticial rather than annual-mean insolation, the setup is identical to that of Hill et al. (2020), to which readers are referred for more details.
Temperature as a function of latitude and pressure from the solsticial RCE simulation, as indicated in the color bar. The gray line at 200 hPa indicates the level at which the temperature is used to compute the gradient-balanced wind.
Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0341.1
The solid curves in Fig. 3 show the simulated meridional profiles of temperature averaged from the surface to 200 hPa and of the inferred 200-hPa zonal wind, absolute angular momentum, and absolute vorticity. The depth-averaged temperature field (shown as a deviation from its 45°S–45°N mean) retains the extrema locations of the insolation and varies meridionally by roughly 25 K from the equator to the summer pole and 75 K from the equator to the region of polar night. The inferred gradient wind is westerly throughout the winter hemisphere and asymptotes toward infinity approaching the equator; it is undefined in a narrow range of the summer hemisphere near the equator, poleward of which very strong easterlies gradually weaken, turning to weak westerlies around 40°N. This zonal wind field causes the angular momentum field to deviate sharply from its planetary value (overlain in Fig. 3c). Angular momentum is undefined from the equator to ~5°N and increases to a local maximum near ~15°N, poleward of which it tends toward the planetary value as u weakens and the distance from the rotation axis diminishes. The absolute vorticity field changes sign at the angular momentum maximum ~15°N, and this constitutes the poleward extent of supercritical forcing in the summer hemisphere.
In solid red, results from numerical simulations of latitude-by-latitude radiative–convective equilibrium, compared to approximations thereto as dashed curves. Dashed yellow corresponds to the analytical forcing profile given by Eq. (4), and dashed purple corresponds to the further-simplified forcing that is linear rather than quadratic in sinφm − sinφ. (a) Vertically averaged temperature or potential temperature, shown as deviation from 45°S to 45°N mean; (b) gradient-balanced zonal wind at the tropopause; (c) absolute angular momentum at the tropopause; and (d) absolute vorticity at the tropopause.
Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0341.1
b. Analytical approximation
With g, Ω, and a set to appropriate Earth values, there are still four free parameters between (4) and (5), namely, H, θ0, φm, and Δh—enough to potentially overtune to the numerically simulated RCE fields of interest. Appendix A details our procedure for choosing these values; in short, we choose conventional values of H = 10 km and θ0 = 300 K and then perform a two-dimensional parameter sweep over Δh and φm values to find best fits to the simulated RCE temperature field over 45°S–45°N (rather than directly for the ηrce = 0 point of ultimate interest). Fortunately, provided
The resulting
4. Analytical expression for solsticial supercritical forcing extent
Figure 4a shows the supercritical forcing extent, i.e., where (9) vanishes, solved numerically, if φm = 90° as Roth is varied over 0 < Roth < 1.5, and Fig. 4b shows the same but with BuΔh = 0.1 as φm is varied from equator to pole. Figure 4 also shows numerical solutions for the small-angle approximation (11) and the analytical expression (12). For the given φm = 90° (Fig. 4a), the true zero crossing and the approximation thereto move poleward monotonically with Roth. The approximation (12) captures the exact expression reasonably well even for Roth ~ 1, though it is consistently equatorward of the exact value by a modest degree. Similarly, for a reasonably Earthlike BuΔh ~ 0.1, the zero crossing moves poleward most rapidly as φm moves off the equator by a few degrees and increases more gradually poleward thereof (Fig. 4b). In the small-angle approximation, for example, the maximum value of 23.6° occurs for φm = 90°, but it is displaced only 2° equatorward thereof for φm moved all the way to 55°N. The approximate solution again is accurate though biased slightly equatorward for large φm.
(a),(b) Supercritical forcing extent under the forcing given by (4) as a function of different parameters, with the full numerical solution, the small-angle numerical solution, and the analytical solution given by (12) as indicated in the legend in (b). In (a) φm = 90° and solutions are shown as a function of Roth. In (b) BuΔh = 0.1 and solutions are shown as a function of φm.
Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0341.1
5. Ascending edge latitude in idealized GCM simulations
Among other things, (14) implies φa ∝ Ω−2/3. Here we present evidence that this accurately characterizes the moist idealized GCM simulations originally presented by F17 and S19 as well as newly performed simulations in an idealized dry GCM. Each of the three idealized GCMs—the idealized aquaplanet of Frierson et al. (2006) for F17, the version thereof as modified by O’Gorman and Schneider (2008) for S19, and the dry idealized GCM of Schneider (2004)—are widely used and documented in the literature. As such, we leave details of model formulation in appendix C and describe here only the salient properties specific to the simulation sets used here.
a. Description of simulations
For the F17 simulations, insolation follows the present-day Earth annual cycle, diurnally averaged, using a 360-day calendar. Across the simulations, planetary rotation rate is varied by factors of 2 from 4 × to 1/32 × ΩE, where ΩE is Earth’s value, as well as one with 1/6 × ΩE, with all other planetary parameters taking their standard Earth values. The simulations are run at T42 horizontal spectral resolution, with 25 levels unevenly spaced in the σ vertical coordinate, and for ten 360-day years. Results are averaged over the 30 days centered on northern solstice across the last 8 years.7 Three additional simulations, at 1, 1/8, and 1/32 × ΩE, are forced with time-invariant solsticial rather than seasonally varying insolation, and we present averages over the final 8 years of these 10-yr integrations.
The simulations of S19 were run at T42 spectral horizontal resolution with 30 unevenly spaced σ levels. Rather than seasonally varying insolation, these simulations are forced at all times by the diurnally averaged insolation occurring at present-day northern solstice. Planetary rotation rate is varied across the simulations, one each for 8, 4, 3, 2, 3/2, 1, 3/4, 2/3, 1/2, 1/4, and 1/8 × ΩE. The simulations span 6 × 360 = 2160 days, and results are averaged over the final 720 days.
In the idealized dry GCM, radiative transfer is approximated by Newtonian cooling toward a prescribed equilibrium temperature profile, which thereby defines the hypothetical latitude-by-latitude RCE temperature field. As such, we set its meridional structure to be (4), with θ0 = 300 K, φm = 90° and Δh = 1/15, the same values as used in section 3. Simulations are performed with planetary rotation rates of 2, 1, and 1/4 × ΩE with Δh = 1/15. One additional sensitivity test is performed at Earth’s rotation rate with Δh = 1/6 as in LH88 (though the largest φm used by that study was 8°). All simulations ran for 1440 days, with averages taken over the final 360 days. We refer to these as the LH88-forced simulations.
For all simulations, we compute the Hadley cell ascending edge latitude using the definition of S19, as described in appendix C. We diagnose Roth for each simulation using the appropriate value of Ω, standard Earth values of a and g, and the sinφm = 1 and Δh = 1/15 best-fit values inferred from the latitude-by-latitude RCE simulations. For the approximate RCE tropospheric depth H, we infer it to be ~10 km for the F17 and S19 simulations based on the explicit latitude-by-latitude RCE simulation performed by S19.8 For the LH88-forced simulations, we infer H directly from the imposed equilibrium temperature field, yielding ~7 km (not shown).
b. Simulation results
Figure 5 shows the mass overturning streamfunctions from the four LH88-forced simulations, each normalized by the solsticial Hadley cell’s overall maximum overturning rate occurring at the cell center. This facilitates comparison of the cell spatial structures across simulations in the face of large variations in strength, over an order of magnitude between the 2 × ΩE case and the 1 × ΩE, Δh = 1/6 case. In the three Δh = 1/15 cases, the weakness of the cross equatorial forcing gradient results in an equatorial jump of near-surface streamlines out of the boundary layer (cf. Pauluis 2004).
Mass overturning streamfunction normalized by its maximum value in each of the LH88-forced simulations. Each panel corresponds to the simulation as labeled in the panel’s top-left corner, where ΩE is Earth’s rotation rate. The blue dot indicates the solsticial cell maximum in the free troposphere, and the adjacent number indicates the mass overturning strength (109 kg s−1) at that point. The vertical solid orange line in each panel is the simulated φa computed using (C1). The vertical dashed red line is the approximation thereto from a linear best fit in
Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0341.1
Comparing to the streamfunctions of F17 (their Figs. 3 and 4) and S19 (his Fig. 3), at Earth’s rotation rate there are differences in detail, but to first order the simulated cells are similar. At 1/4 times Earth’s rotation rate, there is more heterogeneity across the simulation sets, with the F17 cell extending the least far poleward and the S19 cell extending the farthest poleward. Across all the simulations for each model the cross-equatorial Hadley cell grows as the planetary rotation rate decreases (as was shown by F17 and S19 and as expected for the LH88-forced simulations).
Figure 6 shows the ascending edge latitude in each simulation as a function of
Cross-equatorial Hadley cell edge in the summer hemisphere in idealized aquaplanet simulations of F17 and S19 and in the idealized dry simulations of the present study as a function of the thermal Rossby number to the one-third power, each signified by different symbols as indicated in the legend. The solid lines show the linear best fit to φa as a function of
Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0341.1
The slopes of the linear best fits in
Unfilled squares in Fig. 6 show φa in the three F17 perpetual solstice cases. The ascending edge at 1/32 × ΩE is nearly identical for either insolation treatment, but in the 1 × and 1/8 × ΩE time-invariant forced cases φa is a few degrees poleward from that of corresponding seasonally varying case. This difference is not large, and the perpetual solstice F17 φa values still sit equatorward of the corresponding S19 ones. We lack an explanation for this difference between the perpetual solstice simulations of F17 and S19, which is somewhat surprising given seemingly modest differences in model formulation. The slope and y intercept of the linear fit for the F17 1× and 1/8 × ΩE time-invariant forced cases (1.1 radians per
We have also computed best fit power-law exponents by a standard least squares fit to each simulation set in (logRoth, logφa) space, again restricting to Roth < 1. For the F17, S19, and Δh = 1/15 LH88-forced sets, the best fit Ω exponents are 0.28, 0.34, and 0.41, respectively—all reasonably close to the 1/3 power predicted by (14), and nearly identical to it, at 0.34, in their average. The exponent inferred for the 1 × and 1/8 × ΩE F17 perpetual-solstice simulations is 0.30, slightly closer to the 1/3 value than the 0.28 value from the F17 seasonal cycle simulations. Again given the uncertainties, this small difference may or may not be physically meaningful.
Finally, the overlain dotted curve in Fig. 6 shows the numerical solution to the full expression for φc, (10), which does not assume small Roth or small φ. This lower bound qualitatively captures the leveling off of φa in the F17 and S19 simulations with large Roth where the cells become nearly pole to pole.
c. LH88-forced case with Δh = 1/6
The unfilled triangle in Fig. 6 corresponds to the LH88-forced simulation at Earth’s rotation rate in which Δh = 1/6 rather than 1/15 as in the others (but still with φm = 90°). The 2.5-times increase in Δh increases Roth accordingly, and the ascending edge latitude does move poleward, but not enough to fall along the same scaling as the Δh = 1/15 cases. This suggests that modifying Δh at a fixed rotation rate excites one or more mechanisms that influence φa that the supercritical forcing extent does not account for. This could constitute an important limitation to our theory’s applicability to e.g., changes under global warming. Adjudicating this would require additional simulations and analyses beyond the scope of the present study, but we do speculate on one potential candidate, namely, influences of Δh on zonally asymmetric eddy processes.
In the Δh = 1/6 case, the northern subtropics to extratropics exhibit a very long-lasting wave-3 pattern that propagates westward but persists for hundreds of days (not shown). The wave is very regular. It spans meridionally over ~20°–60°N, and its three centers are located between 30° and 40°N. By contrast, in the Δh = 1/15 case, the summer hemisphere zonally asymmetric circulation outside of the tropics is much more Earthlike, with most commonly a wave-4 structure, but with individual lows and highs growing, decaying, and moving relative to each other, while on average being advected by the mean easterlies (not shown). Such qualitatively distinct extratropical circulations in the summer hemisphere could very well impart very different influences on the Hadley circulation.
6. Relationship to slantwise convective neutrality constraint
Separately, F17 show that the latitude of the ITCZ, defined as the latitude of maximum zonal-mean precipitation, in their seasonal-cycle simulations scales as Ω−0.63, very close to the Ω−2/3 scaling predicted by (12) (cf. their Fig. 6). This ITCZ latitude is equatorward of the cell edge (cf. their Fig. 5) and could in principle exhibit a unique scaling with Roth to that of the cell edge. Instead, evidently these hydrological and dynamical markers of the ascending branch position vary in proportion to one another and, in turn, the supercritical forcing extent.
7. Summary
We have presented a new theory for the latitude of the ascending edge of Earth’s Hadley circulation during solsticial seasons and tested the theory’s predictions against simulations in idealized GCMs. The theory posits that the ascending edge latitude is determined by the meridional extent of supercritical forcing. A supercritically forced latitude is one at which, supposing no large-scale overturning circulation existed, the resulting state of latitude-by-latitude RCE would generate time-mean distributions of angular momentum and/or absolute vorticity that are impermissible. It directly follows that a large-scale circulation must exist that spans at the very least all latitudes that are supercritically forced. The resulting overturning circulation, however, can and typically does span poleward of this lower bound, leading to our empirical ansatz that the ascending edge latitude is proportional to the supercritical forcing extent. Despite this empiricism, we argue that the resulting theory—which is predictive and largely accurate with respect to the simulations we test it against—offers advantages over other existing theories relevant to the problem.
We use a single-column model to simulate RCE at individual latitudes under Earth’s present-day solsticial insolation, and by concatenating the simulations together we infer gradient-balanced zonal wind, angular momentum, and absolute vorticity distributions. We then use a simple analytical expression (4), originally from LH88, to approximate the simulated RCE depth-averaged temperature field as quadratic in sinφ with its maximum located in the summer hemisphere. The resulting expression for the absolute vorticity zero crossing, i.e., the supercritical forcing extent, can be solved analytically in the Earth-relevant limit. The solution states that the ascending edge latitude is proportional to
We examine the ascending edge latitude in simulations performed in two variants of an idealized, moist GCM and an idealized dry GCM, across each of which planetary rotation rate is varied. Under solsticial conditions, in each model the cross-equatorial Hadley cell expands meridionally as the rotation rate decreases, and for diagnosed Roth values up to order unity, this expansion follows the
Future work could further test the predictions of our scaling (14). In particular, simulations varying parameters other than the planetary rotation rate that appear in the thermal Rossby number would be valuable. It is conceivable that changes in planetary radius, the gravitational constant, or tropopause height could induce processes not incorporated into the scaling that cause the solsticial Hadley cell ascending edge to deviate from the
We do not rest satisfied with a theory whose accuracy is qualitative, whose justification is semiempirical, and whose strict interpretation is as a lower bound rather than a precise prediction. We do consider it a useful step forward.
Acknowledgments
We are very grateful to Sean Faulk and Martin Singh for sharing the data from their simulations and for many valuable discussions. 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. We thank Martin Singh and two anonymous reviewers for helpful comments.
APPENDIX A
Choice of Free Parameters in the LH88 Forcing Approximation
For a wide range of φm values spanning from the subtropics to the summer pole, reasonably accurate approximations to the numerical RCE simulations (at least with respect to the fields of relevance to supercritical forcing) can be found by tuning the value of Δh. We perform a two-dimensional parameter sweep of (4), for 1° ≤ φm ≤ 90° in 0.1° increments and 0.01 ≤ Δh ≤ 0.3 in 0.01 increments. For each profile, we compute
Figure A1 summarizes the results of these calculations, showing as a function of φm the minimum root-mean-square error, the corresponding Δh value, the corresponding value of the product Δh sinφm, and the corresponding supercritical extent. The error in the analytical meridional temperature gradient field relative to the simulated one over 45°S–45°N is minimized for φm = 36° with Δh ≈ 0.145 ≈ 1/7. Moving equatorward thereof, the best-fit Δh increases, and the error metric increases considerably. Moving poleward thereof, the best-fit Δh decreases, and the error metric levels off at only slightly higher values.
Results from two-dimensional parameter sweep of (4), in φm and Δh, with respect to the accuracy of the fit to the meridional temperature derivative field over 45°S–45°N from the numerical simulations of solsticial RCE. (a) The minimum root-mean-square error (RMSE) obtained as a function of φm. (b) The Δh value corresponding to that minimum RMSE value. (c) The product Δh sinφm using those values. (d) The latitude where ηrce = 0 using those values.
Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0341.1
This decrease in the best-fit Δh value as φm is increased leads to the product Δh sinφm remaining remarkably constant across the profiles with φm ≥ 36°. This is important, because sinφm only appears multiplied by Δh in the analytical expressions shown below for the supercritical forcing extent (though Δh separately appears on its own). In other words, the LH88 approximations to the true RCE state, which might otherwise seem degenerate in φm and Δh, effectively collapse into a single solution in Δh sinφm space, at least with respect to the supercritical forcing extent.
APPENDIX B
Absolute Vorticity Zero Crossing for
APPENDIX C
Formulation of Idealized GCMs Used
The simulations of F17 were performed in the Frierson et al. (2006) idealized aquaplanet GCM. This model’s spectral dynamical core solves the primitive equations on the sphere with no topography and a water-covered surface. The sigma vertical coordinate is defined according to the local surface pressure, σ = p/ps. Simplified gray radiative transfer is used with a prescribed, time-invariant, meridionally uniform longwave optical depth field, no shortwave absorption in the atmosphere, and a prescribed, uniform surface albedo. Surface turbulent fluxes of latent heat and sensible heat are calculated via standard bulk aerodynamic formulas. The surface approximates the thermodynamic effects of the ocean’s upper, well-mixed layer. Its temperature tendency is determined by the net downward radiative plus turbulent flux into the surface along with the prescribed heat capacity, which corresponds to a water depth of 10 m. There is no prescribed ocean heat flux divergence (i.e., “Q flux”).
Moist convection is parameterized using the convective adjustment scheme of Frierson (2007), based on so-called Betts–Miller schemes (Betts 1986; Betts and Miller 1986), that relaxes the humidity and temperature profiles of convectively unstable columns toward a moist adiabat with a prescribed 70% relative humidity over a fixed 2-h time scale. Neither water vapor nor cloud radiative feedbacks operate, the former because the prescribed longwave optical depth field does not depend on water vapor. The latter is because there are no clouds—liquid water generated either through the convective parameterization or by gridscale saturation is immediately precipitated out to the surface.
We refer readers to F17 and Frierson et al. (2006) for further details on the model formulation. We refer readers to S19, Frierson et al. (2006), and O’Gorman and Schneider (2008) for further details on the model formulation.
We perform additional simulations in the dry idealized GCM of Schneider (2004). This model uses the same spectral dynamical core as the moist simulations just described, with horizontal resolution T85 and 20 unevenly spaced sigma levels. The vertical dependence of its Newtonian relaxation temperature field approximates the radiative equilibrium temperature profile of a semigray atmosphere in the troposphere, and it more crudely represents the stratosphere as an isothermal layer of 200 K extending to the model top. The Newtonian relaxation time scale is 50 days in the free atmosphere, 7 days at the surface, and varies linearly in σ within the planetary boundary layer with prescribed top at σ = 0.85.
Within the troposphere, the equilibrium temperature profile is statically unstable over much of the troposphere, and at each time step any statically unstable column triggers a convective adjustment procedure. The convective adjustment relaxes statically unstable columns over a uniform 4-day time scale toward a prescribed lapse rate of Γ = γΓd, where Γ is the lapse rate, Γd = g/cp is the dry adiabatic lapse rate, and γ = 0.7. The γ term acts to mimic the stabilizing effects of latent heat release by moist convection while retaining the simplicity of an otherwise dry fluid. The two dissipative processes are a conventional ∇8 hyperdiffusion and a quadratic drag on the zonal and meridional winds within the boundary layer. Additional details of the model formulation are described by Schneider (2004), and note that various additional modifications made by Hill et al. (2019)—in particular making the model axisymmetric—are not employed in the present study.
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By diagnostic, we mean that the theory requires knowledge of one or more fields from the dynamically equilibrated state that is nominally being predicted. By predictive, we mean that the theory requires knowledge only of fields related to the forcing, thereby yielding a true prediction of the dynamically equilibrated state. Naturally, all else equal, a predictive theory is preferable.
Particularly in extratropical contexts, the term “supercriticality” is sometimes used in reference to the isentropic slope. In this manuscript, however, supercriticality always refers to Hide’s theorem.
Of course, in the annual mean the ascending edge will reliably sit near the equator (potentially as a double ITCZ straddling the equator), and the utility of the supercritical forcing extent is as a lower bound for the location of the poleward, descending Hadley cell edges.
A latitude is also supercritical if Mrce > Ωa2 or Mrce < 0 (Held and Hou 1980). But in the summer hemisphere, at least for Earth, the ηrce = 0 point sits poleward of these conditions, save perhaps for just after spring equinox when the Mrce = Ωa2 point can be farther (cf. Figs. 3 and 4 of Hill et al. 2019). Henceforth we take the summer hemisphere supercritical forcing extent as identical to where ηrce = 0.
Earth’s extratropics, which are nominally subcritical by this definition throughout the annual cycle, obviously are not in a state of latitude-by-latitude RCE. There, the hypothetical RCE state is unstable in other ways, of most relevance baroclinically. Such baroclinic instability—and with it an extratropical dynamical regime—could in principle extend into the supercritically forced region, pushing the solsticial Hadley cell ascending latitude equatorward thereof (much as it limits the Hadley descending, poleward edges; cf. Held 2000; Korty and Schneider 2008; Kang and Lu 2012).
A third mathematically possible case, 0 < φc ≪ Roth ≪ sinφm, yields a physically nonsensical result.
This deviates from the procedure of F17, who vary their 40-day solsticial averaging window across simulations based on the seasonal timing of the ITCZ poleward migration into the summer hemisphere. Results are qualitatively insensitive to this difference (not shown).
Specifically, from Fig. 5 of S19, over the summer hemisphere latitudes relevant to supercriticality the troposphere-average temperature (
In equating the tropopause depth in the φS19 expression—which corresponds to the dynamically equilibrated state—with that in the φc expression—which corresponds to the latitude-by-latitude RCE state—we are implicitly assuming that the emergence of the circulation does not substantially change this depth.
