# The role of tropical and extra-tropical waves in the Hadley circulation

ABS Thakur Corresponding author: ABS Thakur<EMAIL_ADDRESS>Centre
for Atmospheric and Oceanic Sciences, Indian Institute of Science, Bangalore
560012, India  Divecha Centre for Climate Change, Indian Institute of Science,
Bangalore 560012, India  Jai Sukhatme  Centre for Atmospheric and Oceanic
Sciences, Indian Institute of Science, Bangalore 560012, India  Divecha Centre
for Climate Change, Indian Institute of Science, Bangalore 560012, India  Nili
Harnik  Department of Geosciences, Tel Aviv University, Tel Aviv 69978, Israel

###### Abstract

The tropical overturning circulation is examined in a moist aquaplanet general
circulation model forced using a non-interactive sea surface temperature (SST)
distribution that varies between a present-day Earth-like profile and one that
is globally uniform. A traditional Hadley Cell (HC)-like flow is observed in
all experiments along with the poleward transport of heat and angular
momentum. In simulations with non-zero SST gradients, latent heat released
from organized convection near the equator sets up a deep tropical cell;
midlatitude baroclinic Rossby waves flux heat and angular momentum poleward,
reinforcing the thermally direct circulation. As the imposed SST gradient is
weakened, the HC transitions from a thermally and eddy-driven regime to one
that’s completely eddy-driven. When the SST is globally uniform, equatorial
waves concentrate precipitation in the tropics and facilitate the lower-level
convergence necessary for the ascending branch of the HC. Conventional
midlatitude Rossby waves become very weak, but upper-level baroclinicity
generates waves that cause equatorward transport of heat and poleward
transport of momentum. Moreover, these upper-level waves induce a circulation
that opposes the time-mean HC, thus highlighting the role of tropical waves in
driving a traditional overturning flow for uniform SSTs. In all cases,
anomalies associated with the tropical waves closely resemble those that sum
to give the upper-level zonal mean divergent outflow. Through their ability to
modulate tropical rainfall and the related latent heating, equatorial waves
cause considerable hemispheric asymmetry in the HC and impart synoptic and
intraseasonal variability to the tropical overturning circulation.

Keywords : Hadley Cell, Kuo-Eliassen, tropical waves, Rossby waves, eddy-mean-
flow interaction, equatorial convection, moist aquaplanet

##  1 Introduction

The Hadley Cell (HC) is the most prominent feature of the zonally averaged
tropical circulation. It consists of a pair of thermally direct cells with
rising motion over the deep tropics and subsidence over the subtropical
latitudes and transports energy and angular momentum from the equatorial
region to the subtropics (Peixoto and Oort,, 1992). The presence of the HC
explains the occurrence of easterlies in the tropics and partly explains the
subtropical jet in the upper troposphere. In contrast, large-scale baroclinic
eddies are at the core of the extratropical circulation theories and flux heat
and momentum from the subtropics towards the poles. These eddies arise from
the release of zonal mean available potential energy stored in the pole-
equator temperature gradient of the Earth, created by the externally imposed
differential heating from the sun (Pierrehumbert and Swanson,, 1995).
Convergence of angular momentum by these eddies drives the Ferrel cell and
explains the midlatitude westerly jets and the surface westerlies (Schneider,,
2006).

Moreover, baroclinic eddies have an influence on the strength and extent of
the Hadley circulation (Schneider,, 2006; Held,, 2019). The importance of
these midlatitude baroclinic eddies in shaping the Hadley circulation is best
appreciated by suppressing them artificially (Schneider and Lindzen,, 1977;
Schneider,, 1977; Held and Hou,, 1980). Such axisymmetric simulations yield
meridional overturning circulations similar to the annual mean HC with a
subtropical jet and surface trade winds. These studies show that the HC does
not extend to the poles, even in the absence of eddies. However, the
subtropical jet is too strong in these angular momentum-conserving flows.
Further, Davis and Birner, (2019) observed that the axisymmetric Hadley cell
remains confined to the upper troposphere and extends down to the lower
boundary only in the presence of significant viscous damping or eddy stresses.
Thus, eddies modify the wind and temperature fields to give a more realistic
tropical circulation.

Another way to assess the impact of eddies is to compare axisymmetric and
eddy-permitting simulations, identical to each other in every respect except
in their ability to resolve eddies (for example, Williams, 1988a, ; Williams,
1988b, ). Using an idealized dry GCM, Becker et al., (1997) showed that the
feedback of eddies onto the Hadley circulation is dependent on the amount of
equatorial heating. Kim and Lee, (2001) showed that while only 38% of the HC
is directly attributable to eddy fluxes, an equal amount is due to changes in
diabatic heating and surface friction induced by the eddies. Walker and
Schneider, (2006) used an idealized dry GCM to show that the Hadley
circulation strength is largely related to the eddy momentum flux divergence
at the latitude of the streamfunction maxima, over a wide range of flow
parameters. Moreover, transitions of the HC between equinoctial and solstitial
regimes in response to seasonally varying thermal forcing are sharper in eddy-
permitting simulations in comparison to axisymmetric simulations due to eddy-
mean-flow interactions (Bordoni and Schneider,, 2010). In both moist and dry
simulations, the meridional extent of the HC is highly correlated with the
latitude at which baroclinic eddies become deep enough to reach the upper
troposphere (Korty and Schneider,, 2008; Levine and Schneider,, 2015). Moist
simulations forced by fixed SST boundary conditions yield comparable strengths
for the axisymmetric and eddy-permitting circulation (Satoh et al.,, 1995;
Singh and Kuang,, 2016); but with interactive SSTs, the eddy-permitting HC is
2-3 times stronger (Singh et al.,, 2017).

From the discussion above, the influence of midlatitude baroclinic eddies on
the HC is evident. However, the influence of tropical transient modes on the
HC is not as well understood. In the zonal mean, regions of convergence and
divergence associated with these waves tend to cancel each other, but the
precipitation survives since it is a positive-only quantity (Horinouchi,,
2012). Convectively coupled equatorial waves tend to modify the structure of
the precipitation field in the tropics (Kiladis et al.,, 2009). As latent
heating related to precipitation maintains the ascending branch of the
overturning circulation, equatorial waves can influence the HC. In the summer
and winter solstitial seasons, the overturning circulations that average to
give the HC are confined to specific regions where both synoptic and
intraseasonal activity are important (Hoskins et al.,, 2020; Hoskins and
Yang,, 2021, 2023). In these regions, the maxima in Outgoing Longwave
Radiation (OLR) variability and meridional wind have characteristic structures
associated with westward-moving mixed Rossby-Gravity waves (Hoskins et al.,,
2020). Additionally, Hoskins and Yang, (2023) demonstrated that the upper-
level circulation features associated with longitudinally localized deep
convection can be described as the superposition of $n$=0 Mixed Rossby-
Gravityand $n$=1 equatorial Rossby waves. Further, the local Hadley and Walker
circulations are strengthened or weakened, based on the location of Madden-
Julian Oscillation (MJO)-related active or suppressed convection (Schwendike
et al.,, 2021). In fact, equatorial waves contribute directly to the Atlantic
HC strength through their meridional wind component and indirectly by
triggering moist convection through low-level convergence (Tomassini and
Yang,, 2022).

In idealized aquaplanet conditions, the role of equatorial waves is best
illustrated in simulations with globally uniform sea-surface temperatures
(SSTs) and solar insolation in the absence of any externally imposed
meridional gradients except for the meridionally changing effect of the
Earth’s rotation (Sumi,, 1992). Well-defined structures like the intertropical
convergence zone (ITCZ) and Hadley circulation emerge in such simulations
(Sumi,, 1992; Kirtman and Schneider,, 2000; Chao and Chen,, 2004; Barsugli et
al.,, 2005; Horinouchi,, 2012; Suhas et al.,, 2021). Here, Sumi, (1992)
observed that the organization of convection was sensitive to the rotation
rate of the Earth and reported eastward propagating convective activity in the
tropics after the setting-up of zonal mean states. Varying a tunable parameter
that affects the tropical transient variability, Horinouchi, (2012) obtained
nearly symmetric HCs with strengths that are comparable to the observed
equinoctial climatological values; moreover, the strength of these HCs were
highly correlated with the eddy kinetic energy in the tropical lower
troposphere. By modulating the latent heat of vaporization of water ($L_{v}$),
Suhas et al., (2021) mimicked atmospheres with varying degrees of moist
coupling. They observed that the strength of their simulated HC decreases with
weaker moist coupling; and there was a shift in the dominant intraseasonal
peak, from 90 to 30 days, linked with eastward propagating modes associated
with convergence and moist ascent over the equator.

Thus, it is apparent that eddies of both tropical and extratropical origin can
influence the tropical overturning circulation. So, a theory for the dynamics
of the HC should incorporate both tropical and extratropical eddies. In this
study, our primary motivation is to showcase the role of tropical waves in
contrast to extratropical waves in shaping the Hadley circulation. We use an
aquaplanet general circulation model forced by a range of zonally symmetric
lower boundary forcing conditions along with globally uniform solar flux and
fixed orbital parameters. Specifically, we vary SSTs gradually from present-
day Earth-like conditions to one with a globally uniform latitudinal profile.
Varying the SSTs in this manner gives us a passive control on the generation
of midlatitude baroclinic eddies, as the growth rate of these instabilities is
directly proportional to the meridional temperature gradient (Eady,, 1949).
Therefore, we expect to transition from an Earth-like atmosphere with dominant
midlatitude eddies to one where they are very weak or non-existent.

Apart from a fundamental insight into the role of eddies in the HC, the
varying SST simulations are relevant in the global warming scenario as a
robust response of the climate system to greenhouse gas forcing is a decrease
in pole-equator temperature difference (Vallis et al.,, 2015). Although we do
not change the Carbon Dioxide concentration in our simulations, by weakening
the meridional gradient of SST we are forcing the indirect atmospheric
response to greenhouse gas warming (Deser and Phillips,, 2009; Polvani et
al.,, 2011; Shaw and Voigt,, 2015). Also, as noted by Sumi, (1992), an
atmosphere with globally uniform SST is possible when the planet’s interior
heating by long-wave radiation is dominant in comparison to external heating
by short-wave radiation. Such an outcome is possible in a runaway greenhouse
scenario (for example, Venus; Ingersoll,, 1969; Goldblatt and Watson,, 2012).
Moreover, relatively flatter SSTs were known to have prevailed in
paleoclimates, thus there is relevance in understanding how the atmospheric
circulation would have differed in such a setting (Korty and Emanuel,, 2007;
Fedorov et al.,, 2010, 2019).

The rest of the paper is structured as follows: Section 2 presents details of
the modeling framework, the experimental setup, along with the diagnostic
tools used in this study. Section 3 presents the zonally averaged picture. The
role of the midlatitude and tropical eddies in shaping the zonal mean
circulation is discussed in Section 4. Section 5 contains a discussion of our
findings along with conclusions and future scope.

##  2 Methods and Analysis

###  2.1 Simulations

We carry out aquaplanet simulations using the Community Atmosphere Model
(CAM6) of Community Earth System Model (CESM v2.1.3; Danabasoglu et al.,,
2020). The simulations are performed using the Eulerian dynamical core which
is a global spectral model, at T42 horizontal resolution and 32 vertical
levels. The model uses the Zhang-MacFarlane convection scheme for deep
convection and the Cloud Layers Unified By Binormals (CLUBB) scheme for both
shallow convection and microphysics. All simulations are performed with
uniform solar insolation of 342 W-m-2, perpetual equinox orbital condition,
and no diurnal cycle. As we are primarily focused on the physical processes
that govern the tropical overturning circulation, the chemistry model is
turned off to reduce computational cost. Following Medeiros et al., (2016), we
make the aerosols radiatively inactive. The model is forced by zonally
symmetric sea surface temperature (SST) profiles shown in Figure 1. The SST
profiles range from approximating the present-day climate (warm equator and
cold poles, $\alpha$=1.0) to being uniform with latitude ($\alpha$=0.0). The
SST profile for the $\alpha$=1.0 run is the Qobs profile of Neale and Hoskins,
(2000). The analytical expressions for the SST profiles are as follows.

$T_{s}=\begin{cases}T_{0}\left(1-\alpha\sin^{2}(\frac{3\phi}{2})\right),&|\phi|<\frac{\pi}{3}\\\
T_{0}(1-\alpha),&\text{otherwise}\end{cases}\\\ $ (1)

Figure 1: Zonal mean profiles of the sea surface temperatures (SST) for
$\alpha\in[0,1]$ used to force the aquaplanet simulations.

As discussed in the Introduction, by holding the tropical SST fixed and
gradually warming the poles, we are emulating the weakening of the meridional
surface temperature gradients and polar amplification which are robust trends
in global warming simulations (Vallis et al.,, 2015). Indeed, a similar
reduction in SST gradients can be achieved by holding the polar SST fixed and
cooling the tropics, slowly transitioning from the present-day Earth-like
climate to a Snowball Earth (Pierrehumbert,, 2005; Voigt and Marotzke,, 2010).

Note that we do not relax to a radiative equilibrium profile. The cells and
winds develop spontaneously as a response to boundary forcing. The impact of
eddies on the Hadley circulation is analyzed using eddy-free axisymmetric
simulations. In this set, all wave components ($k>0$) are set to zero (Satoh,,
1994; Becker et al.,, 1997; Bordoni and Schneider,, 2010). The utility of this
set will be explained in Section 2.3, and the corresponding results will be
presented in Section 4. All simulations are run for 12 years, and the results
presented in this study are based on the last 11 years.

###  2.2 Zonal mean and eddy diagnostics

Here we provide a brief description of the diagnostic utilities employed in
this study to assess the zonal mean circulation. In all the expressions that
follow, square brackets ($[.]$) indicate a zonal mean while the overbar
$\overline{(.)}$ indicates a temporal mean. Primed quantities ()’ refer to
eddies.

####  2.2.1 Mass-weighted streamfunction

The Eulerian mean mass-weighted streamfunction is defined as Peixoto and Oort,
(1992),

$\Psi=\frac{2\pi a\cos\phi}{g}\int_{0}^{p_{s}}[\overline{v}]dp,$ (2)

where $a$ is the radius of the Earth, $g$ is the acceleration due to gravity
and $p_{s}$ is the surface pressure. The above definition satisfies the zonal
mean continuity equation such that
$([\overline{v}],[\overline{\omega}])=(\frac{\partial\Psi}{\partial
p},-\frac{\partial\Psi}{a\partial\phi})$.

####  2.2.2 Decomposing the eddy momentum flux into rotational and divergent
components

Treating the horizontal wind field on each pressure level as a two-dimensional
vector field, the rotational and divergent components can be obtained using
the Helmholtz decomposition. Following Zurita-Gotor, (2019), the rotational
(denoted by subscript $r$) and divergent (subscript $d$) components of the
eddy momentum flux are,

$\displaystyle u^{\prime}v^{\prime}|_{r}=u_{r}^{\prime}v_{r}^{\prime},$ (3)
$\displaystyle
u^{\prime}v^{\prime}|_{d}=u_{r}^{\prime}v_{d}^{\prime}+u_{d}^{\prime}v_{r}^{\prime}.$

The contribution from the purely divergent component
$u_{d}^{\prime}v_{d}^{\prime}$ is much smaller compared to the other
components (Zurita-Gotor,, 2019).

####  2.2.3 Rossby number

We make use of the vorticity-based bulk Rossby number formulation, as defined
in Singh, (2022). $Ro$ is computed at the latitude of the streamfunction
maxima as

$Ro=-\frac{1}{f}\frac{\int_{p_{t}}^{p_{m}}{[\overline{v}][\overline{\zeta}]\frac{dp}{g}}}{\int_{p_{t}}^{p_{m}}{[\overline{v}]\frac{dp}{g}}},$
(4)

where $p_{m}$ is the level of the streamfunction maxima and $p_{t}$ is the
tropopause level, $f$ is the Coriolis parameter and $v$ is the meridional
wind.

####  2.2.4 Vertically integrated zonal mean transport of energy and momentum

The poleward transport of moist static energy (MSE) is,

$2\pi a\cos{\phi}\int_{0}^{p_{s}}{\overline{vMSE}}\frac{dp}{g},$

where MSE consists of the dry static energy (DSE) and latent energy. The
poleward transport of momentum is defined as in Satoh et al., (1995),

$2\pi a^{2}\cos^{2}{\phi}\int_{0}^{p_{s}}{\overline{uv}}\frac{dp}{g}.$

####  2.2.5 Eliassen-Palm flux

The generation, propagation and dissipation characteristics of Rossby wave
activity are explored through the Eliassen-Palm (EP) flux. As in Edmon et al.,
(1980), the zonal mean EP flux ($\boldsymbol{F}$) is,

$\boldsymbol{F}\equiv\\{F_{\phi},F_{p}\\}=a\cos{\phi}\biggl{\\{}-[\overline{u^{\prime}v^{\prime}}],f\frac{[\overline{v^{\prime}\theta^{\prime}}]}{\partial_{p}\overline{[\theta}]}\biggr{\\}}.$
(5)

Here, primed quantities denote deviations from a zonal mean. When WKBJ theory
is valid, $\boldsymbol{F}$ ($\approx\boldsymbol{c_{g}}\mathcal{A}$) is the
advective flux of wave activity ($\mathcal{A}$) by the group velocity
($\boldsymbol{c_{g}}$). For frictionless and adiabatic flow, the wave activity
follows the conservation relation (Edmon et al.,, 1980),

$\frac{\partial\mathcal{A}}{\partial t}+\boldsymbol{\nabla\cdot F}=0,$

where,

$\boldsymbol{\nabla\\!\cdot
F}=\frac{1}{a\cos{\phi}}\frac{\partial}{\partial\phi}(F_{\phi}\cos{\phi})+\frac{\partial}{\partial
p}(F_{p})=(a\cos{\phi})\overline{v^{\prime}q^{\prime}}.$ (6)

####  2.2.6 Residual Circulation

As defined in Edmon et al., (1980), the residual velocity is,

$[\overline{v^{*}}]=[\overline{v}]-\partial_{p}\biggl{(}\frac{[\overline{v^{\prime}\theta^{\prime}}]}{\partial_{p}\overline{[\theta}]}\biggr{)}.$

Therefore, the residual mean meridional circulation streamfunction is given
by,

$\Psi^{*}=\frac{2\pi a\cos\phi}{g}\int_{0}^{p_{s}}[\overline{v^{*}}]dp.$ (7)

###  2.3 Estimating the eddy influence on the overturning circulation

To study the impact of eddies on tropical overturning circulation, we utilize
the Kuo-Eliassen equation (Kuo,, 1956; Chemke and Polvani,, 2019; Kang et
al.,, 2019; Zaplotnik et al.,, 2022). This is a second-order linear elliptic
partial differential equation that is generally derived using the quasi-
geostrophic approximation (Peixoto and Oort,, 1992). Specifically, the quasi-
geostrophic approximation is applied to the zonally averaged thermodynamic and
momentum equations written in flux form. Using the zonally averaged continuity
equation, a streamfunction ($\psi$) is introduced as
$([v],[\omega])=\biggl{(}\frac{g}{2\pi
a\cos{\phi}}\frac{\partial\psi}{\partial p},-\frac{g}{2\pi
a\cos{\phi}}\frac{\partial\psi}{a\partial\phi}\biggr{)}$ to represent the
zonal mean non-divergent flow in the pressure-latitude plane. The zonal mean
thermal wind relation ($f\frac{\partial[u]}{\partial
p}=\frac{R}{p}\frac{\partial[T]}{\partial y}$) is used to combine the momentum
and thermodynamic equations into the Kuo-Eliassen equation, which reads,

$\displaystyle f^{2}\frac{g}{2\pi a\cos{\phi}}\frac{\partial^{2}\psi}{\partial
p^{2}}+\Gamma\frac{g}{2\pi
a}\frac{\partial}{a\partial\phi}\biggl{(}\frac{1}{a\cos{\phi}}\frac{\partial\psi}{\partial\phi}\biggr{)}=\underbrace{\frac{R}{p}\frac{\partial[Q]}{a\partial\phi}}_{S_{Q}}-\underbrace{f\frac{\partial[D]}{\partial
p}}_{S_{F}}$ (8)
$\displaystyle-\underbrace{\frac{R}{p}\frac{\partial}{a\partial\phi}\frac{1}{a\cos{\phi}}\frac{\partial}{\partial\phi}[v^{\prime}T^{\prime}]\cos{\phi}}_{S_{ehf}}-\underbrace{f\frac{1}{a\cos^{2}\phi}\frac{\partial^{2}}{\partial
p\partial\phi}[u^{\prime}v^{\prime}]\cos^{2}\phi}_{S_{emf}}.$

In the above equation, $\Gamma$ ($=-[T]\frac{\partial[\theta]}{\partial p}$)
is the static stability parameter, $Q$ is the diabatic heating rate estimated
from the thermodynamic budget, $D$ is the drag estimated from the momentum
budget, $f$ is the Coriolis parameter, $R$ is the universal gas constant and
$a$ is the radius of the Earth. Subscripts ehf and emf stand for eddy heat
flux and eddy momentum flux, respectively.

It is convenient to write Equation 8 in symbolic form as

$L\psi=S,$

where $L$ is the linear operator on the LHS, and $S$
($=S_{Q}+S_{ehf}+S_{F}+S_{emf}$) is the sum of the source terms on the RHS,
and $\psi$ is the streamfunction. As the Kuo-Eliassen equation is linear, it
allows for the diagnosis of the effect of various heat and momentum forcing
protocols on the meridional overturning circulation. We solve the above
equation for $\psi$ numerically by first converting Equation 8 into a finite-
difference equation on a pressure-latitude grid that is solved iteratively
using Gauss-Seidel’s method for faster convergence.

On long timescales, the source terms in Equation 8 tend to affect each other,
wherein the eddy fluxes will modify the diabatic heating and drag terms
(Chang,, 1996). We utilize the methodology outlined by Kim and Lee, (2001) to
estimate the true impact of the eddies on the Hadley circulation. This
approach decomposes each term in the Kuo-Eliassen equation into an
axisymmetric component (subscript a) and the corresponding deviation due to
the presence of eddies (subscript e). Thus, $L=L_{a}+L_{e}$, $S=S_{a}+S_{e}$
and $\psi=\psi_{a}+\psi_{e}$. Noting that $L_{a}\psi_{a}=S_{a}$, we get purely
eddy-driven streamfunction response as (Kim and Lee,, 2001),

$\psi_{e}=L^{-1}(S_{e}-L_{e}\psi_{a}).$ (9)

This approach requires explicit knowledge of both axisymmetric and eddy-
permitting simulations. It allows for the partitioning of the eddy influence
on the overturning circulation into direct and indirect impacts. The direct
impact of the eddies is via their heat and momentum transport. The indirect
effect is through a modification of the diabatic heating, zonal mean
temperature, static stability, and zonal mean zonal winds; this results in a
change both in the diabatic forcing ($S_{F}$) and drag ($S_{D}$), and in the
operator $L$, which depends on $\Gamma$. The effect of these changes on the
Hadley circulation is the indirect impact of the eddies. Simply using the
eddy-permitting simulation to diagnose the impact of eddies will result in the
incorrect attribution of the indirect response of the eddies to the physical
processes like drag or diabatic heating (Kim and Lee,, 2001).

We have also performed this decomposition using the extended form of the Kuo-
Eliassen equation (Zaplotnik et al.,, 2022), which does not employ the quasi-
geostrophic approximation in the momentum and thermodynamic equations before
connecting them using thermal wind. The results are largely similar to the
conventional form of the Kuo-Eliassen equation used in this study (Equation
8).

##  3 The Zonal Mean View of the Circulation

Figure 2: Eulerian mean mass streamfunction (dark contours) estimated using
Equation 2, zonal mean horizontal eddy momentum flux divergence (colours) and
angular momentum (light contours) for $\alpha\in[0,1]$. The eddy momentum flux
divergence is presented in units of m s-1day-1. Streamfunction contours are
logarithmic, with the magnitude of the lowest contour being 109 kg-s-1. The
magnitude doubles for successive contours. Contours of angular momentum are
spaced at $0.1\Omega a^{2}$ with the $0.95\Omega a^{2}$ contour plotted
additionally. The red curve in each panel shows the tropopause height
(Reichler et al.,, 2003).

###  3.1 Tropical overturning flow

The Eulerian mean mass-weighted streamfunction is shown in Figure 2 with black
contours. The HC for the $\alpha$=1.0 simulation is similar to that obtained
by other moist aquaplanet simulations (Satoh et al.,, 1995; Williamson et
al.,, 2013; Singh and Kuang,, 2016; Medeiros et al.,, 2016), and the present-
day Earth during the equinoctial seasons (Dima et al.,, 2005). For all
$\alpha$, the temporally and zonally averaged tropical overturning flow
consists of rising motion over the equator and descent in the subtropics. In
all simulations, the overturning mean flow is accompanied by a poleward flux
of DSE and an equatorward flux of latent energy; the tropical MSE transport is
determined by the compensation between these components (Figure S1). As the
equator-to-pole SST gradient decreases (i.e., smaller $\alpha$), we observe
that the HC tends to be more localized towards the equator. In all cases, we
see a well-formed Ferrel Cell poleward of the tropical HC. The intensity of
the tropical circulation decreases with $\alpha$, and the maximum value of the
streamfunction reduces by an order of magnitude when $\alpha$ goes to zero.

In the cases with a non-zero SST gradient, deep convection preferentially
organizes over the near-equatorial regions that have warmer SSTs; the
circulation is then sustained by the equatorward flux of moisture by the mean
flow (Figure S1). We also obtain an HC with globally uniform SSTs. Its
strength is comparable to that observed in previous studies (Kirtman and
Schneider,, 2000; Barsugli et al.,, 2005; Horinouchi,, 2012; Shi and
Bretherton,, 2014; Suhas et al.,, 2021); though it is relatively weaker in the
upper troposphere. A wavenumber-frequency diagram of tropical variability
shows the presence of an eastward-moving Kelvin wave, a westward Mixed Rossby-
Gravity wave along with a prominent low-frequency MJO-like mode (Figure 3a,b;
see also Pritchard and Yang,, 2016; Suhas et al.,, 2021)111We refer to this
large-scale, eastward propagating system as an MJO-like mode as it has certain
features in common with the Madden-Julian Oscillation, but there are
differences as have been noted in other aquaplanet experiments (Das et al.,,
2016; Shi et al.,, 2018).. These waves tend to concentrate the precipitation
in the tropics, whereafter convection and circulation interactively adjust to
set up a conventional HC (Horinouchi,, 2012).

Figure 3: For the globally uniform SST case. (a) Wavenumber-frequency power
spectra of the vertical pressure velocity ($\omega$) at 850 mbar, calculated
following Wheeler and Kiladis, (1999). (b) Composite map of the Madden-Julian
Oscillation (MJO, Zhang,, 2005). Colours represent $\omega$ at 500 mbar, while
the contours and quivers represent the streamfunction and horizontal wind
velocity averaged over the 150-300 mbar layer. Composites are obtained by
regression onto the first two principal components of the 850 minus 150 mbar
$\Delta\chi$ field, as outlined by Adames and Wallace, (2014). Continental
boundaries are drawn in panel (b) for reference. (c) Vorticity-based bulk
Rossby number (Equation 4) comparison for all values of $\alpha$.

Neglecting the vertical mean advection and eddy terms (Walker and Schneider,,
2006; Geen et al.,, 2020), the temporally and zonally averaged zonal momentum
balance in the upper troposphere can be written as

$f(1-Ro)[\overline{v}]=(\cos^{2}\phi)^{-1}\partial_{y}[\overline{u^{\prime}v^{\prime}}\cos^{2}{\phi}].$

Here, $Ro$ is the vorticity-based bulk Rossby number (Equation 4), and other
symbols have their usual meaning. The above equation implies a balance between
zonal mean absolute vorticity flux by the mean flow (LHS) and the horizontal
eddy momentum flux divergence (RHS). As $Ro\to 0$, the circulation responds to
the eddy momentum flux divergence in the upper troposphere. As $Ro\to 1$, the
circulation gets uncoupled from the eddy fluxes and responds directly to the
thermal driving. In our experiments, $Ro$ decreases monotonically from about
$0.689$ for $\alpha$=1.0 to $0.027$ for $\alpha$=0.0 (Figure 3c). This implies
that with decreasing $\alpha$, the HC transitions from a regime that is both
thermally and eddy-driven to one that is strongly influenced by eddies. The
regime change with $\alpha$ can be identified from the angular momentum
contours as well. For the larger $\alpha$ runs, contours of constant angular
momentum appear to be advected by the upper tropospheric flow. As seen in
Figure 2, streamlines cross angular momentum contours towards the edge of the
HC due to angular momentum transport out of the HC by eddies. As the HC moves
closer to the low Rossby number regime with weakening SST gradients, angular
momentum contours become more vertical, similar to those in the extratropics.

Figure 4: Anomalous temperature (colours), zonal mean zonal wind (contours)
and Eliassen-Palm flux (Equation 5; arrows) for $\alpha\in[0,1]$. Temperature
anomalies are computed by removing the global mean from every pressure level.
Zonal wind contours are logarithmic in nature. The smallest magnitude
represented is $1$ ms-1, and the magnitude doubles for successive contours.
The number of data units per unit length of the arrows is shown on the top-
right of each panel. The red curve in each panel shows the tropopause height
(Reichler et al.,, 2003). Prior to the presentation, the EP flux vectors were
scaled as $\\{\frac{k\cos{\phi}}{\pi}F_{\phi},\frac{k}{10^{5}}F_{p}\\}$ where
$k=\cos{\phi}\sqrt{\frac{1000}{p}}$

.

###  3.2 Eddy momentum and EP flux

The colours in Figure 2 show the divergence of the horizontal eddy momentum
flux (EMF). For pronounced SST gradients, eddy momentum tends to maximize at
the tropopause because typical baroclinic eddies reach that level and break
near the subtropical edge of the HC (Ait-Chaalal and Schneider,, 2015). The EP
fluxes for the $\alpha$=1.0 run (arrows in the top-left panel of Figure 4)
resemble the present-day atmosphere; wave activity moves upwards in the
midlatitudes, then turns equatorward near the tropopause which acts as a
turning surface for baroclinic Rossby waves (Ait-Chaalal and Schneider,,
2015). A rotational-divergent partition of the EMF (Figure S2) reveals that
this is due to the rotational component, typically associated with midlatitude
eddies. The EMF convergence in the deep tropics is due to the divergent
component, linked with the MJO-like mode (see Figures 3a,b, S2 and S7; Zurita-
Gotor,, 2019).

As $\alpha$ decreases, the change in the EMF structure is quite striking.
Distinct from the solitary maxima in the simulations with non-zero SST
gradient, the EMF has a double maxima structure in the flat SST run, one each
in the upper and lower troposphere. The lower tropospheric maximum is weaker
and appears to be due to waves generated near the surface that remain confined
to the lower troposphere below the 800 mbar level. The upper tropospheric
maximum is associated with the waves that are generated near the subtropical
tropopause that propagate downwards and equatorwards; these waves cause
equatorward eddy heat transport but poleward momentum transport. The Eady
growth rate suggests that an upper-level source of baroclinicity is present in
all simulations (see Figure S3; Yuval and Kaspi,, 2016). Waves generated from
this upper-level instability become evident when the traditional lower-level
baroclinic waves become very weak (note that the scale of the EP flux vectors
for the $\alpha$=0.0 run is smallest in Figure 4). A sequence of these waves
on the 200 mbar surface is shown in Figure 5; quite clearly, they ride on the
sharp background meridional potential vorticity (PV) gradient and as they
evolve, distort and overturn, they act to mix PV across latitudes. A Hovmöller
diagram of the meridional wind, averaged over 30-40∘N, suggests that these
waves have a westward phase speed of $\sim$$2.1$ ms-1 and eastward group speed
of $\sim$$4.8$ ms-1 (not shown; Chang,, 1993).

###  3.3 Zonal mean zonal wind

The zonal mean zonal wind for the $\alpha$=1.0 simulation (contours in the top
left panel of Figure 4) is characterized by a strong subtropical jet,
midlatitude westerlies, and low-level tropical easterlies. These features are
largely similar to the present-day Earth’s atmosphere. However, the upper
tropical superrotation222Superrotation refers to zonal mean zonal winds that
have an angular momentum greater than the angular momentum of the surface
(Held,, 1999). is distinct from that of the Earth’s troposphere. It occurs due
to the action of divergent eddy momentum fluxes in the deep tropics
($u^{\prime}v^{\prime}|_{d}$; see Figure S2 Zurita-Gotor,, 2019). In the
Earth’s upper tropics, deceleration by the seasonally reversing HC exceeds the
acceleration by tropical eddies and leads to the prevalent zonal mean
easterlies (Lee,, 1999; Dima et al.,, 2005). As there is no seasonal cycle,
our perpetual equinox simulations superrotate because the mean flow
deceleration is too weak to offset the equatorward convergence of momentum by
these tropical waves. This leads to the presence of an isolated angular
momentum maximum over the deep tropics in all our simulations (Figure 2; Liu
and Schneider,, 2011).

Figure 5: Instantaneous snapshots of Ertel’s Potential Vorticity on the 200
mbar surface (colours, in potential vorticity units or PVU) for the
$\alpha$=0.0 simulation. Few PVU contours have been marked out separately, for
clarity. Continental boundaries are drawn in each panel for reference.

As the imposed SST gradient weakens, lower tropical easterlies and sub-
tropical westerlies decrease in strength in concert with the weakening HC
intensity. This indicates that there is a decrease in the uptake of angular
momentum by the atmosphere from the surface. The thermally-driven subtropical
jet and the eddy-driven midlatitude jet begin to separate near $\alpha$=0.4.
The subtropical westerly jet disappears altogether for $\alpha$=0.0 run, for
which there are distinct columns of zonal mean easterlies in the tropics and
westerlies in the midlatitudes, both of which extend from the surface to the
tropopause. The deceleration over the equator, and above the surface in the
tropics, is achieved by the divergence of rotational eddy momentum flux (see
Figures 2 and S2). In contrast to the other runs, the vertically integrated
transport of momentum by the eddies is poleward even in the tropics (Figure
S4), confirming the eddy-driven nature of the tropical easterlies in the flat
SST run. In accord with Horinouchi, (2012), the EP flux structure for the
globally uniform SST case suggests that convergence of momentum by both upper
and lower tropospheric eddies is important for the midlatitude westerly flow.
Although the imposed SST is globally uniform and the near-surface temperature
gradients are very weak, the small vertical shear in the zonal wind profile
near the extratropical surface (Figure 4) is sufficient for the near-surface
baroclinicity to be important (see the Eady growth rate in Figure S3).

###  3.4 Thermal structure

Colours in Figure 4 show the latitudinally anomalous zonal mean temperature.
The tropics remain relatively warmer than the other regions for
$\alpha$$\geq$0.2, mimicking the imposed SST distribution because the fixed-
SST boundary condition strongly constrains the lower tropospheric thermal
structure (Singh and Kuang,, 2016). However, the subtropics become anomalously
warm in the flat SST simulation. The extratropical mean flow and transient
waves flux DSE equatorward; this leads to a net accumulation of heat near the
HC edge (Figure S1). As the zonal mean DSE transport is dominated by the
upper-level circulation (Suhas et al.,, 2021), the warm region is confined to
the middle and upper troposphere around the HC edge (see Figure 2).

The solid red lines in Figures 2 and 4 show the tropopause height estimated
using the WMO definition (Reichler et al.,, 2003). In the present-day Earth
and in the $\alpha=1.0$ case, while the tropical tropopause height is set by
moist convection, the extratropical tropopause is set by the depth of
baroclinic eddies (Schneider,, 2004). The tropical tropopause height remains
constant as $\alpha$ decreases, but the extratropical tropopause rises; the
tropopause is almost flat for $\alpha$=0.0. This can also be seen in the
vertical profiles of temperature (Figure S5). The near coincidence of the
tropical-extratropical thermal stratifications suggests that convection
becomes increasingly important in extratropical dynamics for the runs with
flatter SSTs (Ait-Chaalal and Schneider,, 2015; Suhas et al.,, 2021). Further,
the lower tropospheric slope of the $\theta_{e}$ profiles shows a tendency
towards increasing convective instability with decreasing SST gradients
(Figure S5).

Our simulations also exhibit a meridionally inverted temperature gradient in
the upper troposphere lower stratosphere (UTLS) region, where the tropics are
cooler than the poles (Figure 4). Such a temperature distribution arises due
to the vertical thermal stratification of the atmosphere. Typically,
temperature decreases with height in the troposphere and increases with height
in the stratosphere, with a temperature inversion at the tropopause. In all
our simulations, the tropical tropopause is slightly higher than in the
extratropics. Thus, at $\sim$170 or 180 mbar, where the tropics are still
cooling with height, the extratropics have started warming (Figure S5). Hence,
the tropics are cooler than the poles in the UTLS region. This ubiquitous UTLS
temperature gradient is consistent with the upper-level source of
baroclinicity discussed in Section 3.2.

##  4 Role of the eddies in driving the overturning circulation

As described in Section 2.3, we bring out the role of the eddies by
decomposing the eddy-permitting HC into an axisymmetric response and the
changes induced by the eddies. The axisymmetric response is captured from
another set of simulations where the model setup is the same as the eddy-
permitting one, except that the model is truncated at the zonal wavenumber
$k$=0.

Figure 6: Eddy-permitting (left), axis-symmetric (middle) and eddy-induced
(right) streamfunctions estimated from the Kuo-Eliassen equation (Section 2.3)
for different SST profiles. The eddy-induced circulation has been estimated
following the methodology of Kim and Lee, (2001). Contours in light shade are
drawn additionally to draw attention to the weaker features of the overturning
circulation. The contours are logarithmic in nature, with the lowest magnitude
being 109 kg-s-1, and the magnitude doubles for successive contours. The red
curve in each panel of the first two columns indicates the tropopause height
(Reichler et al.,, 2003) for the respective run. The third column shows the
tropopause height for the eddy-permitting run.

The eddy-permitting, axisymmetric, and eddy-induced cells for varying SST
gradients are shown in the first, second, and third columns of Figure 6. In
the tropics, for $\alpha$=1.0, the axisymmetric circulation is stronger in the
upper troposphere, while the eddy-permitting one is broader and more intense
in the lower troposphere. Thus, the eddies strengthen the axisymmetric
circulation in the lower troposphere but weaken it in the upper troposphere
(top row of Figure 6). As $\alpha$ decreases, the axisymmetric circulation
becomes dominant in the tropics. Further, the eddy-induced circulation has
oppositely directed shallow upper tropospheric cells; with decreasing
$\alpha$, these cells extend lower down to the surface, resulting in the eddy-
driven circulation that opposes the axisymmetric circulation.

To understand the eddy-induced circulation in greater detail, we decompose it
into the circulations forced by the eddy heat flux, eddy momentum flux, drag,
and diabatic heating terms (see Equation 9 and Section 2.3). These are shown
in Figure 7, and the diabatic heating rate estimated as a residual from the
thermodynamic budget is shown in Figure 8. Note that the summation of all the
columns of Figure 7 gives the right-hand column of Figure 6. Clearly, the
direct influence of eddy heat and momentum fluxes is secondary; the nature of
the entire eddy-driven cell in the tropics follows the eddy-induced diabatic
heating term. The diabatic heating field (Figure 8) for the $\alpha$=1.0
axisymmetric run has a large isolated peak in the upper troposphere, while
that for the eddy-permitting run is somewhat moderate in intensity in the
upper troposphere but extends down to the lower troposphere. Here, eddy heat
fluxes modify the diabatic heating field and weaken the meridional temperature
gradient, which results in a weaker eddy-permitting HC in the upper
troposphere. Further, eddies distribute the diabatic heating over a deeper
layer, thereby strengthening the HC in the lower troposphere (Figure 7). Such
behavior is consistent with Kim and Lee, (2001), who showed that the presence
of eddies strengthens the HC below, and weakens it above the level of maximum
convective heating. This is also consistent with Becker et al., (1997), who
showed that the axisymmetric HC is stronger than the eddy-permitting one in
the presence of prescribed convective heating in the tropics and weaker in the
absence of it. Further, midlatitude eddies flux momentum out of the tropics,
decelerating the flow and making it more easterly; this increases the surface
drag. As the approximate zonal momentum balance near the surface is between
the Coriolis force and the surface drag, a direct cell is induced to offset
the effect of enhanced drag (Figure 7).

Figure 7: Decomposition of the impact of eddies on the overturning
streamfunction for different SST profiles. Contours in light shade are drawn
additionally to draw attention to the weaker features of the overturning
circulation. The contours are logarithmic in nature, with the lowest magnitude
being 109 kg-s-1. The magnitude doubles for successive contours. The red curve
in each panel indicates the tropopause height (Reichler et al.,, 2003) for the
eddy-permitting run.  Figure 8: Zonal mean profiles of the diabatic heating
rate in K-day-1 for (left) axisymmetric and (right) eddy-permitting runs for
the (top)$\alpha$=1.0 and (bottom)$\alpha$=0.0 runs. The diabatic heating
field is estimated as a residual from the thermodynamic energy budget.

With weaker imposed SST gradients, the isolated upper tropospheric maxima of
diabatic heating in the axisymmetric runs weakens and spreads over the depth
of the troposphere (Figure 8). Thus, the ”reverse” cells confined to the upper
troposphere for $\alpha$=1.0 systematically descend and displace the direct
lower tropospheric circulation (Figures 6 and 7). The thermally indirect
tropical circulation induced by the eddies opposes the direct axisymmetric
component; the resultant eddy-permitting circulation is thermally direct and
is weaker than the individual axisymmetric and eddy-induced components. We
also note the eddy-driven thermally indirect Ferrel cell that is centered near
$30^{\circ}$ and a direct cell further poleward. Indeed, in this Eulerian mean
representation, for all $\alpha$, the overturning flow poleward of the
subtropics is almost entirely accounted for by the eddy component.

To summarize so far, our axis-symmetric comparisons and the Kuo-Eliassen
decomposition clearly show that eddies play an important role in shaping the
strength and structure of the HC. However, eddies can be of tropical or
extratropical origin and may exert their influences on the HC differently.
Next, we will explicitly consider the impact of midlatitude eddies on the HC
followed by the impact of tropical eddies.

###  4.1 Impact of midlatitude eddies

Figure 9: Temporally varying Kuo-Eliassen streamfunction regressed against
$u^{\prime}v^{\prime}_{max}$ time series.

As eddy momentum fluxes tend to maximize at the edge of the HC where Rossby
waves break (Vallis et al.,, 2015), we use the maximum value of the EMF to
construct an index of extratropical eddy variability (Zurita-Gotor and
Álvarez-Zapatero,, 2018). We form a standardized time-series of
$\frac{u^{\prime}v^{\prime}_{NH,max}-u^{\prime}v^{\prime}_{SH,min}}{2}$, and
regress the temporally varying streamfunction estimated from the Kuo-Eliassen
decomposition (Equation 8) against it. This isolates the co-variability
between the HC and extratropical baroclinic Rossby waves. The extrema of the
EMF are computed over all latitudes in each hemisphere, over all levels, and
at every instance of the model output. For reference, the time-averaged
position of the NH EMF extrema has been plotted in the rightmost column of
Figure S4. Using the extrema from both NH and SH yields a hemispherically
symmetric picture of the mid-latitude eddy influence. Note that, as the index
of midlatitude variability is uncorrelated with the axisymmetric component,
regressing Equation 9 against the index yields the same results as that for
Equation 8.

Figure 9 shows the results of the regression. As eddy momentum fluxes tend to
be largest in the upper troposphere (Ait-Chaalal and Schneider,, 2015), the
near-surface correlation between the regression index and the streamfunction
is poor. For conventional SSTs, the midlatitude-eddy-induced circulation (left
column of Figure 9) acts to strengthen the prevalent time-mean HC (left column
of Figure 6). This effect gradually weakens with weaker SST gradients, and the
midlatitude-eddy-induced circulation abruptly reverses when the imposed SST is
globally uniform. These results also suggest that, for non-zero SST gradients,
the midlatitude-eddy-induced circulation opposes the full-eddy-induced
circulation (right column of Figure 6). Further, the midlatitude-eddy-induced
circulation is always at least an order of magnitude smaller than both the
time-mean HC and the full-eddy-induced circulation; the weak feedback may be
due to the absence of stationary eddies in our simulations which can have a
large feedback onto the Hadley circulation (Becker and Schmitz,, 2001).

Further examination of the midlatitude-eddy-driven response for the larger
$\alpha$ runs, suggests that the regression captures the co-variability
between the Hadley and Ferrel cells as well (Zurita-Gotor and Álvarez-
Zapatero,, 2018). As noted above, for conventional SSTs, the midlatitude
eddies induce a circulation in the same sense as the prevalent HC (Figure 9;
first row). The reinforcing nature of this response extends through to all
components of Equation 8. Midlatitude baroclinic eddies diverge momentum away
from the subtropics where they break; this eddy momentum forcing can be
balanced by the Coriolis force on the mean meridional circulation, thereby
inducing a direct circulation in the tropics (see also Held and Phillips,,
1990). As discussed in Section 4, these eddies modify the heat budget;
dynamical cooling in the subtropics due to the poleward fluxing of heat by the
eddies is balanced by the adiabatic heating in the sinking branch of the mean
flow response, again inducing a direct circulation.

The strength of the induced circulation weakens with smaller SST gradients.
The extratropical response decreases while the tropical response becomes
largely dominated by the diabatic heating component. For $\alpha$=0.2, a drag-
dominated indirect lower tropospheric response emerges that has a magnitude
comparable to that of the upper tropospheric direct circulation. In the runs
with weaker SST gradients, eddies that originate in the midlatitudes are able
to propagate into the deep tropics (see EP fluxes in Figure 4). They generate
a negative momentum forcing right above the equator that is brought down to
the surface by this reversed circulation. This negative eddy momentum forcing
then aids the uptake of angular momentum from the surface. When the SST
becomes globally uniform, the lower tropospheric indirect component
strengthens, but the upper tropospheric diabatic component abruptly
transitions into a strong oppositely signed response. This response in the
globally uniform SST case is coherent with the EP flux vectors that descend
downwards from the tropopause, indicating equatorward heat transport by the
eddies (Figure 4). The mean flow response offsets the heating perturbation
with ascending motion that causes cooling.

Though the regression is aimed at the midlatitude waves, the results in this
section indirectly also highlight the importance of the tropical waves for the
mean tropical circulation. This is most readily seen in the globally uniform
SST case. Since the midlatitude-eddy-induced HC tends to oppose the time-mean
HC, the influence of the tropical waves must dominate. Along with the noted
differences between the full eddy-induced circulation and the midlatitude-
eddy-induced circulation in simulations with non-zero SST gradients, this
makes a compelling case for the tropical wave influence on the Hadley
circulation. We elucidate the role of the tropical waves in Hadley circulation
in the next section.

###  4.2 Impact of tropical eddies

Figure 10: First two empirical orthogonal functions (EOFs) of the time-varying
zonal mean overturning streamfunction ($\Psi$, Eq. 2) for eddy-permitting
runs, and variance preserving spectra of the corresponding principal
components (PCs) for (top) $\alpha=1.0$ and (bottom) $\alpha=0.0$. The
percentage variance explained by the respective EOFs is mentioned in the
corresponding panel. Prior to the EOF decomposition, the midlatitude eddy
signal was regressed out using the time series from Section 4.1, and the
streamfunction values were weighted by the square root of the cosine of
latitude along the horizontal and by the incremental change in pressure at
each pressure level along the vertical (Dima and Wallace,, 2003). The PC
spectra were computed using Welch’s method (Welch,, 1967).  Figure 11:
Composite maps of the upper-level divergent meridional wind for (top)
$\alpha$=1.0 and (bottom) $\alpha$=0.0 simulations. Composites are made over a
period where PC1 (left) and PC2 (right) attain a large value. In each panel,
colours show the unfiltered anomaly while the contours show the tropical wave-
filtered anomalies; solid contours indicate positive values while dashed
contours indicate negative values, and the zeroth contour is dotted. In each
panel, the contours are such that they contain eleven values, are centered at
zero, and span half of the range indicated by the respective colour bar. The
wave-filtering is performed to isolate the Kelvin wave, mixed Rossby Gravity
wave, and the MJO-like mode, consistent with the wavenumber-frequency spectra
(Figure S11) and the PC spectra (Figure 10). The filtering is performed prior
to the compositing and over a longer period that includes the compositing
period, consistent with Wheeler and Kiladis, (1999). Continental boundaries
are drawn in each panel for reference.

As discussed in the Introduction, convectively coupled equatorial waves can
modulate the HC through their influence on tropical precipitation and the
associated latent heat release. Given the influence of midlatitude baroclinic
waves noted above, the HC must exhibit variability on time scales that are
typically associated with equatorial waves along with the synoptic scale
variability associated with midlatitude waves. Regression onto an index of
tropical variability (wave-filtered precipitation or outgoing longwave
radiation as in, for example, Wheeler et al.,, 2000) did not yield
interpretable results. Hence, the tropical wave influence on the HC is
assessed via an Empirical Orthogonal Function (EOF) decomposition of the
temporally varying zonal mean overturning streamfunction (Dima and Wallace,,
2003). To facilitate a direct attribution to equatorial waves, the midlatitude
eddy signal has been regressed out prior to the EOF decomposition.

Figure 10 shows the first two EOFs and the temporal spectra of the associated
Principal Components (PCs) for the Earth-like and the globally uniform SST
simulations. The most striking feature of the EOFs is that the dominant mode
of variability is a hemispherically asymmetric cross-equatorial cell. This
mode is somewhat similar to the cross-equatorial variability of the real-world
HC associated with the seasonal cycle (Dima and Wallace,, 2003). In both
simulations, this mode is associated with prominent peaks in both the synoptic
and intraseasonal timescales. Specifically, EOF1 for the $\alpha$=1.0 run
shows a prominent 30-day peak along with synoptic-scale peaks between 7 and 9
days. The PC for $\alpha$=0.0 has a large peak at the 8-day period along with
a secondary 80-day peak. The hemispherically asymmetric mode explains about
20% of the variance for the Earth-like $\alpha$=1.0 run but accounts for more
than 50% of the variance for the globally uniform SST case. Thus, as posited,
the imprint of tropical waves is seen in the timescales of variability of the
HC. This also suggests that, even in the Earth-like simulation with
equatorially symmetric SSTs and perpetual equinox conditions, the tropical
circulation is quite transient, and the equatorial symmetry associated with
the Eulerian mean HC (Figure 2) is likely a result of temporal averaging.

In both simulations, the second EOF is equatorially symmetric and explains a
similar amount of variance ($\sim$15%) in both cases. Much like EOF1, the PCs
of the second mode also exhibit power in the synoptic and intraseasonal bands,
but they are less pronounced. In general, the peaks in the PC spectra can be
due to multiple waves present in the equatorial waveguide (see Wheeler-Kiladis
spectra in Figure S7). In particular, for $\alpha=0$, the synoptic-scale peaks
may be due to the Mixed Rossby Gravity wave or the Kelvin wave, whereas the
intraseasonal peaks are due to the MJO-like mode that is ubiquitous in all our
simulations (Figure S7).

Apart from their signature in the EOFs, the influence of tropical waves in
driving the variability in the tropical overturning circulation is clarified
using composite maps of unfiltered and tropical-wave-filtered divergent
meridional wind (vd) anomalies shown in Figure 11; the unfiltered anomalies
are shown in colours while the filtered anomalies are shown in contours. We
consider the divergent component of the meridional wind because the rotational
component would sum to zero when integrated over a latitude circle (Zhang and
Wang,, 2013; Sun et al.,, 2019); also, the zonal mean overturning circulation
is an ensemble of such localized divergent circulations. The composite is
constructed during a period when the respective PCs attained a high value.
Figure 11 shows that the unfiltered and tropical wave-filtered vd anomalies
have a close resemblance. When the composite is constructed using PC2 for the
$\alpha$=1.0 run, the anomalies are characterized by a quadrupole-like
structure straddling the equator and are possibly associated with MJO-like
convection (Monteiro et al.,, 2014). This is consistent with the observations
of Schwendike et al., (2021); they showed that the phase-dependent
fluctuations of MJO-related convection can modulate the local Hadley and
Walker circulations. For $\alpha$=0.0, the tilted planetary-scale outflows
dominate the tropics in the composites made using both PC1 and PC2. Further,
for all the SST cases, the pattern correlation between the full anomalies and
the tropical-wave-filtered anomalies lies between 0.7 and 0.8.

As the divergent outflows result from convection, vd anomalies are closely
linked with precipitation anomalies. Tropical waves can modulate the
precipitation field in the tropics (see the latitudinal variability of the
wave-filtered precipitation field in Figure S6) and, thus, influence the
ascending branch of the HC through low-level convergence. The tropical wave
influence on the HC, through off-equatorial ascent, is most clearly seen in
the hemispherically asymmetric modes. Their influence on the symmetric modes
(EOF2 of $\alpha$=1.0 & 0.0) can be visualized clearly when considering these
equatorially symmetric EOFs with a flipped sign, as the sign of the EOF is
arbitrary. Further, the equatorward flux of divergent eddy momentum by the
MJO-like mode (see the discussion in Sections 3.2 and 3.3) also triggers a
compensatory mean flow response in the same sense as the conventional HC.

###  4.3 The Transformed Eulerian Mean Circulation

To tie things together, we now employ the Transformed Eulerian Mean formalism
(TEM; Andrews and Mcintyre,, 1976) to develop a complementary picture of the
zonal mean atmospheric circulation that accounts for the motion induced by
eddies. The time mean residual circulation (Figure 12) is defined as the
conventional mean meridional circulation corrected by the quasi-linear Stokes
drift due to atmospheric waves (Becker,, 2012); it closely approximates the
diabatic circulation, especially in the upper troposphere and lower
stratosphere (Held and Schneider,, 1999; Juckes,, 2001; Birner,, 2010). For
the usual pole-to-equator SST profile, the residual circulation shows a direct
hemisphere wide cell as is true of the troposphere (Holton and Hakim,, 2012).
In the midlatitude lower troposphere, the action of eddies can be seen as the
residual circulation abruptly transitions from vertical descent
($\omega^{*}>0$) to quasi-horizontal polewards flow. Here, the residual
circulation and eddy fluxes are strongly coupled as the Coriolis torque on the
residual circulation balances the EP flux convergence (Birner,, 2010). As
discussed in Held and Schneider, (1999), the TEM exhibits an infinite return
flow near the surface because the eddy heat flux vanishes only in a very thin
layer that’s unresolved in global climate models. Another reason for such
behaviour could be that the quasi-linear Stokes’ drift contains the static
stability in the denominator, which becomes very small in the boundary layer
(Held and Schneider,, 1999).

Figure 12: EP flux vectors (quivers), its divergence (contours) and the
residual circulation (colours) for the moist eddy-permitting runs. The EP flux
divergence is normalised to constrain the range between $-1$ and $1$.

As $\alpha$ decreases, the eddy heat fluxes become small, and the residual
flow looks similar to the conventional Eulerian mean. In the case where
$\alpha$=0.0, an equatorward $v^{*}$ is induced by the isolated centers of EP
flux divergence in the subtropical upper troposphere. This is reminiscent of
the ”downward control” phenomena in the stratosphere (Vallis,, 2017), but in a
reversed sense. In the real atmosphere, upward propagating wave activity
deposits zonal momentum, creating a region of EP flux convergence where Rossby
waves break. This induces an overturning circulation that connects the wave-
breaking region with the frictional bottom boundary (Vallis,, 2017). However,
for our $\alpha$=0.0 case, the waves are generated locally due to the upper-
level instability that leads to a region of EP flux divergence that induces an
equatorward mean flow and consequent thermally indirect cells in the
subtropics.

Comparing the Eulerian mean and residual circulations for the $\alpha$=0.0
case (Figures 2 and 12 respectively), we see that the reversed midlatitude
circulation is not an artifact of averaging; this is quite different from the
upper-tropospheric branch of the $\alpha$=1.0 Eulerian mean Ferrel Cell that
is absent in the residual circulation. For the $\alpha$=0.0 run, the indirect
midlatitude cells are present in the isoentropic mass streamfunction too (not
shown; Held and Schneider,, 1999), indicating that both adiabatic heating
through latent heat release and mechanical eddy driving are important for
their formation and maintenance. In the tropics, the diabatic circulation is
limited only to the mid and lower troposphere. The upper-tropospheric tropical
cells are artifacts of the Eulerian mean circulation and are probably eddy-
driven. This ties in with the Rossby number-based observation that the HC in
the globally uniform SST simulation is eddy-driven (see Section 3.1).

##  5 Summary and Discussions

We have examined the tropical circulation using an aquaplanet GCM forced by
fixed zonally symmetric SSTs, spatially uniform solar flux, and fixed orbital
parameters. Our primary motivation is to isolate the influence of midlatitude
and tropical waves in shaping the Hadley circulation. To this end, we vary the
SST forcing from a present-day Earth-like warm-equator cold-poles
configuration to one which is globally uniform, passively controlling the
generation of midlatitude waves.

A conventional HC-like flow is observed in all our experiments, with large-
scale ascent over the equator and descent over the subtropics. The
circulations transport dry static energy poleward and latent energy
equatorward; the mean overturning circulation largely dictates the transport
in the equatorial region. The strength and meridional extent of the HC
decrease with weakening SST gradients. When forced by a non-zero SST gradient,
organized convection preferentially occurs over latitudes with warmer SSTs and
sets up a deep tropical cell. With globally uniform SSTs, the necessary
equatorial convergence is provided by tropical waves. A vorticity-based bulk
Rossby number suggests that the HC transitions from a thermally and eddy-
driven regime to one that is completely eddy-driven. In all simulations,
Ferrel cells develop in the midlatitudes that have a sense opposite to that of
the tropical Hadley circulation.

All our simulations are super-rotating due to the lack of a seasonal cycle and
have colder tropics with warmer poles near the tropopause. In simulations with
non-zero SST gradients, the tropospheric zonal wind and temperature
distributions are similar to that of the Earth. The zonal wind shows
subtropical westerlies, prominent midlatitude eddy-driven jets, and tropical
easterlies; the temperature distribution mimics the imposed SST, and eddies
control the extratropical tropopause height (Held and Schneider,, 1999). With
weakening SST gradients, the subtropical westerlies and the midlatitude eddy-
driven jets begin to separate, and the extratropical tropopause rises. When
the SSTs are globally uniform, easterlies form through the depth of the
tropical troposphere with barotropic westerly jets in the subtropics. In this
setup, the subtropics are anomalously warmer than the tropics and poles, and
convection determines the extratropical tropopause height.

Our simulations support a full spectrum of tropical waves in the equatorial
waveguide. These waves modulate convection and rainfall variability in the
tropics. The extratropical waves undergo a significant change with SST
gradients. For present-day Earth-like SST profiles, as is expected, Rossby
waves generated near the extratropical surface propagate upwards toward the
tropopause, which acts like a turning surface and guides them meridionally
(Ait-Chaalal and Schneider,, 2015). They transport heat and angular momentum
away from the HC edge, controlling the strength and extent of the HC (Walker
and Schneider,, 2006; Levine and Schneider,, 2015). As the SST gradients
weaken, the strength of these waves decreases. When the SST is globally
uniform, traditional Rossby waves stay confined to the lower troposphere, and
we find waves generated near the tropopause due to upper-level baroclinic
instability. Like traditional Rossby waves, they transport momentum polewards,
but their heat flux is equatorward, consistent with the warmer subtropics in
this regime. This structure of the midlatitude waves in the flat SST run
causes a double maximum in the eddy momentum flux distribution, one each in
the upper and lower troposphere.

The Kuo-Eliassen equation is used to diagnose the impact of atmospheric waves
on the HC. In particular, we follow Kim and Lee, (2001) and utilize an
additional set of simulations where eddies are suppressed artificially. This
allows us to consider the tropical overturning circulation as axisymmetric and
eddy-modified components (Kim and Lee,, 2001). The presence of eddies in the
system changes the thermodynamic and momentum balances and modifies the
surface friction and diabatic heating fields. The direct effect of eddy heat
and momentum fluxes is found to be secondary. In the present-day Earth-like
case, eddies strengthen the axisymmetric component of the HC in the lower
troposphere but weaken it in the upper troposphere. With weaker SST gradients,
the weakening influence of eddies extends throughout the depth of the
troposphere.

We then consider the influence of midlatitude baroclinic eddies and tropical
eddies on the HC separately. The heat and momentum transport by midlatitude
baroclinic eddies triggers compensatory transports by the mean flow. In cases
where the imposed pole-to-equator SST gradient is large, the induced flow
reinforces the HC. When the SSTs are globally uniform, the eddies induce a
thermally indirect circulation that is consistent with the equatorward heat
transport by waves generated from the upper-level instability. In contrast,
equatorial waves modify the precipitation field in the tropics and modulate
the HC ascent branch by their influence on latent heating and low-level
convergence. Via an Empirical Orthogonal Function decomposition of the
temporally varying mass-weighted streamfunction, we find that the leading
modes of variability exhibit considerable hemispheric asymmetry. In fact, in
the regime of globally uniform SSTs, a dominant cross-equatorial cell accounts
for more than 50% of the total variance. This suggests that the tropical
circulation is quite transient, and the equatorial symmetry associated with
the Eulerian mean HC (Figure 2) is likely a result of temporal averaging. The
hemispherically asymmetric modes are associated with the westward-moving Mixed
Rossby-gravity wave, the eastward-moving Kelvin, and an MJO-like mode.
Composite maps suggest that the upper-level anomalous flow is closely
associated with tropical waves, suggesting that tropical waves play an
important role in the transient variability of the HC. Further, the
equatorward eddy momentum flux transport by the MJO-like mode can also force a
mean flow response in the same sense as the prevalent HC.

Finally, we construct the residual circulation to develop a wave-corrected
view of the tropical overturning circulation. For the Earth-like case, the
residual circulation is a thermally direct hemisphere-wide cell; it closely
resembles the Hadley circulation in the tropics but has a ”U-shape” in the
midlatitudes, indicating that the residual circulation and the eddy fluxes are
strongly coupled. In the flat SST case, the tropical residual circulation is
shallow, extending only up to the middle troposphere, reaffirming the eddy-
driven nature of the HC. The divergence of EP flux associated with upper-level
instability induces subtropical cells in a manner opposite to the ”downward
control” mechanism in the stratosphere. The flat SST simulation serves as an
intermediate case to eddy momentum flux exploration in normal and reversed
equator-to-pole temperature gradient experiments that are relevant in the
present-day Earth and high obliquity exoplanets, respectively (Ait-Chaalal and
Schneider,, 2015; Kang et al.,, 2019). Indeed, a natural extension of the
present work would be the inverse SST gradient simulations which are relevant
in understanding the atmospheric dynamics of high obliquity exoplanets (Lobo
and Bordoni,, 2020).

A possible way to quantitatively estimate the influence of tropical waves on
the strength of the HC would be to simulate an atmosphere without them.
However, this appears difficult to achieve because these waves are fundamental
modes of the tropical atmosphere; they exist due to the Earth’s rotation near
the equator (Matsuno,, 1966). Horinouchi, (2012) conducted an experiment where
the eddy horizontal winds over the deep tropics were preferentially damped; he
observed that the simulated HC was weaker (see their Section 4b). However,
this approach would also dampen other waves that can propagate into the deep
tropics, thereby negating their interaction with the mean flow. As the zonal
mean HC is an ensemble of longitudinally localized divergent circulations
(Zhang and Wang,, 2013; Sun et al.,, 2019), damping the eddy horizontal winds
would dampen these circulations as well. Further, tropical waves are very
sensitive as each wave behaves differently to changes in parameters that
control the behavior of convection in the model (Frierson,, 2007; Peatman et
al.,, 2018; Suhas and Sukhatme,, 2020; Suhas et al.,, 2021). On reducing the
trigger for moist convection, Horinouchi, (2012) observed a weakening of the
HC along with the disappearance of most of the tropical wave spectra except
for some long-period variability. On varying the latent heat of water vapour,
thereby changing the degree of moist coupling, Suhas et al., (2021) made a
similar observation regarding tropical wave variability. Further, it has been
observed that large-scale modes of low-frequency tropical variability are
intimately tied to the background moisture gradient (Sukhatme,, 2013, 2014).
Changing such parameters impacts the circulation non-trivially, thereby making
the attribution to equatorial waves more challenging.

Historically, evaluation of the effect of eddies on the HC has been performed
with dry models that use Newtonian relaxation towards a radiative equilibrium
temperature profile (Held and Hou,, 1980; Walker and Schneider,, 2006; Bordoni
and Schneider,, 2010). Such an arrangement allows the occurrence of small-
scale dry convection at every latitude (Satoh,, 1994) and cannot explain the
occurrence of the subtropical dry zones (Rodwell and Hoskins,, 1996, 2001).
However, the real atmosphere is moist, and the release of latent heat pushes
the temperature stratification towards moist adiabatic, which is stable to dry
convection (Satoh,, 1994). In a moist atmosphere where convection releases
latent heat, convection and circulation interactively adjust each other
(Horinouchi,, 2012); a dry atmosphere with Newtonian relaxation precludes such
an interactive adjustment. In addition, midlatitude storms exhibit greater
strength and larger asymmetry in vertical velocity in the presence of moisture
(Booth et al.,, 2015; O’Gorman,, 2011), and their feedback onto the mean flow
should be closer to reality in our moist simulations.

The occurrence of zonal mean Hadley circulations in aquaplanets with spatially
uniform SST and insolation (both eddy-permitting and axisymmetric) has been
known for long (Sumi,, 1992; Kirtman and Schneider,, 2000; Barsugli et al.,,
2005; Horinouchi,, 2012; Pritchard and Yang,, 2016; Suhas et al.,, 2021).
However, we present some discussion here for completeness. Typically,
convection self-organizes into large clusters in idealized models with
homogeneous boundary conditions (Muller et al.,, 2022), in non-rotating models
with a rectangular domain (Bretherton et al.,, 2005), and even on a non-
rotating sphere (Arnold and Randall,, 2015). With rotation, the atmosphere
supports equatorial waves that can concentrate the precipitation in the
tropics (Inertia Gravity waves in the axisymmetric setting; see Chao and
Chen,, 2004). Convection and circulation interactively adjust each other to
form a prominent meridional circulation even in the absence of any imposed
thermal gradients (Horinouchi,, 2012). Thereafter, equatorward moisture flux
by the mean flow makes the equatorial region a preferred location for
sustained deep convection.

Noticeable differences exist in the simulated circulations between our
$\alpha$$=$$0.0$ run and the $L_{v}$$=$$1.0$ run of Suhas et al., (2021).
Their simulations were performed using the Finite Volume dynamical core of CAM
5.3 of CESM 1.2.2 on an f09 grid ($\sim 1^{\circ}$). They obtained a steady
and deep HC that extended upto 200 mbar (see their Figure 1b), shallow
midlatitude jets extending from the surface only upto 500 mbar, and warmer
poles than the equatorial region (their Figure 2b); their vorticity-based
local Rossby numbers suggest that the HC is partly eddy- and partly thermally-
driven. In contrast, our flat SST HC is eddy-driven and appears transient in
the upper troposphere; the midlatitude westerly jets are deep but narrow, and
the subtropics are anomalously warmer than the other regions. Apart from the
known differences between the two dynamical cores (Williamson,, 2008) and
possible divergences occurring between the two due to version differences, the
Finite Volume core has an unclosed angular momentum budget (Toniazzo et al.,,
2020). The spurious sinks of angular momentum tend to decelerate the flow and
force secondary circulations with the same sign as the HC, whereby the
simulated HC in CAM-FV has a larger mass flux (Toniazzo et al.,, 2020). These
spurious sinks of momentum are possibly responsible for the differences in the
zonal wind and temperature distribution (through the thermal wind relation)
between our $\alpha$=0.0 run and the $L_{v}$$=$$1.0$ run of Suhas et al.,
(2021). Another striking difference is in the simulated transient variability
beyond the subtropics; Tropical Cyclone (TC)-like vortices dominate the
extratropics in their run, whereas our run shows only some such vortices (not
shown). This is likely due to inadequate horizontal resolution in our runs,
similar to previous studies that also used a coarse horizontal resolution
(Sumi,, 1992; Barsugli et al.,, 2005; Kirtman and Schneider,, 2000;
Horinouchi,, 2012); compared to previous studies, Shi and Bretherton, (2014)
obtained the TC-like vortices in their simulations using a finer horizontal
grid resolution. Consistent with our results, Suhas et al., (2021) also
observed transient Rossby wave activity in the tropics and subtropics
connected with the upper-level instability (personal communication).

The changing nature of the driving of the HC with the weakening SST gradients
is of potential interest in greenhouse gas warming scenarios where the
decrease in meridional surface temperature gradient is a robust trend (Adam et
al.,, 2014; Vallis et al.,, 2015). In addition, the tropical upper troposphere
becomes anomalously warm due to a decreasing saturated lapse rate (Vallis et
al.,, 2015). This causes the baroclinicity to weaken in the lower troposphere
but strengthen in the upper troposphere (Wu et al.,, 2011; Yuval and Kaspi,,
2016). As the growth rate of baroclinic instability depends on the imposed
meridional temperature gradient (Eady,, 1949), the corresponding baroclinicity
and strength of the baroclinic waves in our simulations decrease in concert
with weakening gradients of SST (Kodama and Iwasaki,, 2009). The upper-level
baroclinicity is tied to the near tropopause reverse temperature gradient
that’s ubiquitous in all our simulations. It is possible that these upper-
level waves are present in all our simulations but are overshadowed by the
stronger waves generated in the lower troposphere for present-day Earth-like
SST profiles. In fact, Horinouchi, (2012) obtained a similar upper-level
instability but attributed it to the shear instability of the angular momentum
conserving zonal wind. In the present-day Earth’s atmosphere, prominent
regions of EP flux divergence have been observed on the poleward flanks of the
jet core in both hemispheres (Birner et al.,, 2013), and a tropopause-level
planetary wave source can trigger waves that force Sudden Stratospheric
Warmings (Boljka and Birner,, 2020).

As discussed in the Introduction, an understanding of the zonal mean
overturning circulation has developed in two well-separated regimes. While the
eddy-mean flow interaction related to baroclinic midlatitude waves has been
studied with Earth-like warm equator-cold poles thermal forcing (Schneider,,
2006; Walker and Schneider,, 2006; Korty and Schneider,, 2008; Levine and
Schneider,, 2015), the role of tropical waves has been indirectly elucidated
with spatially uniform thermal forcing (Sumi,, 1992; Kirtman and Schneider,,
2000; Horinouchi,, 2012; Barsugli et al.,, 2005; Suhas et al.,, 2021). Our
fixed SST simulations, with the intermediate thermal gradients, connect the
two regimes and cover the gap that has existed so far in the literature.
Indeed, the atmosphere-ocean is a coupled system, and the atmospheric
circulation should be able to modify the SST distribution (Xie and Philander,,
1994). In this regard, fixed SST aquaplanet simulations are an idealization
but form an important first step in developing a more complete understanding
of the drivers of the tropical overturning circulation (Held,, 2005; Jeevanjee
et al.,, 2017; Maher et al.,, 2019).

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