points in $C_{p,h}$. The correctness follows straightforwardly from Section 3
and the fact that for each $p$ we report all $m$-star patterns that $p$
belongs to (not necessarily as the central point) having the maximum left
endpoint on its corresponding temporal interval. In particular we report all
$\tau$-durable $m$-star patterns and we might also report a few $\tau$-durable
$\varepsilon$-$m$-star patterns. The running time is also asymptotically the
same with Algorithm 1.

##  Appendix E Missing Material in Section 5

###  E.1. UNION

1

2$\mathcal{C}_{p}:\\{\mathcal{C}_{p,1},\mathcal{C}_{p,2},\cdots,\mathcal{C}_{p,k}\\}\leftarrow\mathsf{durableBallQ}(p,\tau,\varepsilon/2)$,
with $\textsf{Rep}_{i}$ denoting the representative point of the cover tree
node for $\mathcal{C}_{p,i}$, and $\mathsf{IT}^{\cup}_{p,i}$ denoting the
annotated interval tree for this cover tree node;

3 foreach  _$j\in[k]$_ do

4 foreach  _$q\in\mathcal{C}_{p,j}$ in descending order of $I_{q}^{+}$_ do

5 $I^{\prime}\leftarrow I_{q}\cap I_{q}$;

6 $\bar{I}=\textsc{MaxIntersection}(I^{\prime})$;

7 $H\leftarrow\textsf{newHeap}(\\{(\bar{I},I^{\prime},|I\cap
I^{\prime}|)\\})$;

8 $t\leftarrow 0$;

9 for  _$h=1\ldots\kappa$_ do

10 $(I_{x},I_{y},|I_{x}\cap I_{y}|)=H.removeTop()$;

11 $t\leftarrow t+|I_{x}\cap I_{y}|$;

12 foreach  _$I_{z}\in I_{y}\setminus I_{x}$_ do

13 $\bar{I}=\textsc{MaxIntersection}(I_{z})$;

14 $H.\textsf{insert}(\bar{I},I_{z},|\bar{I}\cap I_{z}|)$;

15

16

17 if  _$t\geq(1-1/e)\tau$_ then

18 report $(p,q)$;

19

20 else

21 break;

22

23

24

25

26Subroutine $\textsc{MaxIntersection}(I_{in})$ begin

27 $\mu\leftarrow-1$;

28 $\hat{I}\leftarrow\emptyset$;

29 foreach  _$i\in[k]$_ do

30 if  _$\phi(\textsf{Rep}_{i},\textsf{Rep}_{j})\leq 1+\frac{\varepsilon}{2}$_
then

31
$\hat{I}_{i}\leftarrow\textsc{ComputeMaxUnionD}(\mathsf{IT}^{\cup}_{p,i},I_{in})$;

32

33 if  _$|\hat{I}_{i}\cap I_{in}| >\mu$_ then

34 $\mu=|\hat{I}_{i}\cap I_{in}|$;

35 $\hat{I}=\hat{I}_{i}$;

36

37

38 return ($\hat{I}$);

39

40

Algorithm 8
ReportUNIONPair$(\smash{\mathcal{D}^{\cup}},p,\tau,\varepsilon,\kappa)$

In Algorithm 8 we show the algorithm for finding all $(\tau,\kappa)$-UNION
durable pairs. The high level idea of the algorithm follows from Algorithm 4.

Instead of $\mathsf{IT}^{\Sigma}$, we have the primitive data structure
$\mathsf{IT}^{\cup}$. Given a query interval $I_{in}$ the goal is to find the
interval $\hat{I}\in\mathcal{I}$ such that $|\hat{I}\cap I_{in}|$ is
maximized, where $\mathcal{I}=\\{I_{p}\mid p\in P\\}$. This can be found using
a variant of the interval tree $\mathsf{IT}^{\cup}$ as follows: First, among
all intervals that intersect $I_{in}^{-}$ it finds the interval
$\hat{I}_{a}\in\mathcal{I}$ with the largest right endpoint. Second, among all
intervals that intersect $I_{in}^{+}$ it finds the interval
$\hat{I}_{b}\in\mathcal{I}$ with the smallest left endpoint. Third, among all
intervals that lie completely inside $I_{in}$ it finds the longest interval
$\hat{I}_{c}\in\mathcal{I}$. In the end, we return
$\hat{I}\in\\{\hat{I}_{a},\hat{I}_{b},\hat{I}_{c}\\}$ with the longest
intersection $|\hat{I}\cap I_{in}|$. Similarly to $\mathsf{IT}^{\Sigma}$, the
data structure $\mathsf{IT}^{\cup}$ has space $O(n\log n)$, it can be
constructed in $O(n\log^{2}n)$ time, and given a query interval $I_{in}$, it
returns $\hat{I}$ in $O(\log^{2}n)$ time.

Using $\mathsf{IT}^{\cup}$, we construct $\smash{\mathcal{D}^{\cup}}$
similarly to $\smash{\mathcal{D}^{\Sigma}}$. The only difference is that for
each node in the cover tree there exist an $\mathsf{IT}^{\cup}$ data structure
(instead of $\mathsf{IT}^{\Sigma}$). The procedure
$\textsc{ComputeMaxUnionD}(\mathsf{IT}^{\cup}_{p,i},I_{in})$ returns the
interval $\hat{I}_{i}$ that has the largest intersection with $I_{in}$ and its
corresponding point in $P$ lies in $\mathcal{C}_{p,i}$.

Next, we describe the subroutine $\textsc{MaxIntersection}(I_{in})$. It simply
considers all canonical nodes in $\mathcal{C}_{p}$ running
$\textsc{ComputeMaxUnionD}(\mathsf{IT}^{\cup}_{p,i},I_{in})$ for each
$\mathcal{C}_{p,i}\in\mathcal{C}_{p}$ such that
$\phi(\textsf{Rep}_{i},\textsf{Rep}_{j})\leq 1+\varepsilon/2$. Hence
$\textsc{MaxIntersection}(I_{in})$ visits all canonical nodes that are close
to both $p$ and $q$, and among these nodes $\mathcal{C}_{p,i}$, it returns the
interval $\hat{I}=\operatorname*{arg\,max}_{\hat{I}_{i}}|\hat{I}_{i}\cap
I_{in}|$, i.e., the interval with the largest intersection with $I_{in}$. We
also notice that ComputeMaxUnionD can be easily modified so that we always
skip $I_{p}$ and $I_{q}$ from the procedure of finding the interval in
$\mathcal{I}$ with the largest intersection with $I_{in}$. From the proofs in
the previous sections we have that $\phi(p,q)\leq 1+\varepsilon/2$ while the
witness set has distance at most $1+\varepsilon$ from both $p$ and $q$.
Overall, $\textsc{MaxIntersection}(I_{in})$ finds an interval $\hat{I}=I_{s}$
such that

$|I_{s}\cap I_{in}|\geq\max_{s^{\prime}\in
P:\phi(p,s^{\prime}),\phi(q,s^{\prime})\leq 1}|I_{s^{\prime}}\cap I_{in}|,$

and

$\phi(p,s),\phi(q,s)\leq 1+\varepsilon.$

Finally, we describe
ReportUNIONPair$(\smash{\mathcal{D}^{\cup}},p,\tau,\varepsilon,\kappa)$. After
finding $\mathcal{C}_{p}$ and for each $\mathcal{C}_{p,j}$ we visit
$q\in\mathcal{C}_{p,j}$ in descending order of $I_{q}^{+}$ as we did in
Algorithm 4. For each pair $(p,q)$ we check, we run the greedy algorithm for
the max $k$-coverage problem. First, we find the interval
$\bar{I}\in\mathcal{I}$ that has the largest intersection with
$I^{\prime}=I_{p}\cap_{q}$. We create a max heap $H$ and we insert the pair
$(\bar{I},I^{\prime})$ with value $|\bar{I}\cap I^{\prime}|$. Then the
algorithm proceeds in $\kappa$ iterations. In each iteration, it finds the
pair $(I_{x},I_{y})$ in the max heap $H$ with the maximum $|I_{x}\cap I_{y}|$.
$I_{x}$ is an interval from $\mathcal{I}$, while $I_{y}$ is an uncovered
segment of $I^{\prime}$. Hence, in each iteration, it finds the interval
$I_{x}$ that covers the largest uncovered area of $I^{\prime}$. Then we add
$|I_{x}\cap I_{y}|$ in variable $t$ that maintains the overall union the
algorithm has computed. An interval $I_{x}$ might split $I_{y}$ into two
smaller uncovered segments or into one smaller uncovered segment of
$I^{\prime}$. In each case, $I_{z}\subseteq I_{y}$ represents one uncovered
segment created after adding $I_{x}$. We run $\textsc{MaxIntersection}(I_{z})$
and we find the interval $\bar{I}\in\mathcal{I}$ that covers the largest
portion of $I_{z}$ and we insert the pair $(\bar{I},I_{z})$ in max heap $H$
with value $|\bar{I}\cap I_{z}|$. We repeat the same procedure for $\kappa$
iteration. In the end we check whether $t\geq(1-1/e)\tau$. If yes, we report
the pair $(p,q)$, otherwise we skip $\mathcal{C}_{p,j}$ and continue with the
next canonical node. Overall, this algorithm gives an implementation of the
greedy algorithm for the max $\kappa$-coverage problem in our setting, using
efficient data structure to accelerate the running time.

Putting everything together, the correctness of this algorithm follows by the
correctness of the greedy algorithm for the max $\kappa$-coverage problem, the
correctness of $\textsc{MaxIntersection}(I_{in})$, and the correctness of
Algorithm 4. For a pair $(p,q)$ if we find that $t\geq(1-1/e)\tau$, then
$(p,q)$ is definitely an $((1-1/e)\tau,\kappa)$-UNION $\varepsilon$-pair. As
we argued in Algorithm 4, assume that for a $q\in\mathcal{C}_{p,j}$ we find
that $t<(1-1/e)\tau$. Then it is safe to skip $\mathcal{C}_{p,j}$ because
there is no other $(\tau,\kappa)$-UNION durable pair to report. Notice that
the approximation factor for the greedy algorithm is $1-1/e$, so if the greedy
implementation returns $t<(1-1/e)\tau$, we are sure that the pair $(p,q)$ is
not $(\tau,\kappa)$-UNION durable. Hence, any other $w\in\mathcal{C}_{p,j}$
with $I_{w}^{+}<I_{q}^{+}$ will also not be $(\tau,\kappa)$-UNION durable. In
any case, our algorithm returns all $(\tau,\kappa)$-UNION durable pairs and
might return some additional $((1-1/e)\tau,\kappa)$-UNION durable
$\varepsilon$ pairs. Hence, it holds that
$|K_{\tau,\kappa}|\leq\textsf{OUT}\leq|K_{(1-1/e)\tau,\kappa}^{\varepsilon}|$.

Next, we analyze the running time of our algorithm. As pointed out,
$\mathsf{IT}^{\cup}$ is constructed in $O(n\log^{2}n)$ time and it finds the
interval that covers the largest uncovered area of a query interval in
$O(\log^{2}n)$ time. Hence, $\smash{\mathcal{D}^{\cup}}$ is constructed in
$O(n\log^{3}n)$ time. The subroutine $\textsc{MaxIntersection}(I_{in})$ calls
$\textsc{ComputeMaxUnionD}(\mathsf{IT}^{\cup}_{p,i},I_{in})$,
$O(\varepsilon^{-O(\rho)})$ times. For each pair $(p,q)$ we check, the
subroutine $\textsc{MaxIntersection}(I_{in})$ is called $O(\kappa)$ times,
while all update operations in the max heap $H$ takes $O(\kappa\log n)$ time.
For each point $p$ we might check at most $O(\varepsilon^{-O(\rho)})$ pairs
that are not reported, one for each canonical node in $\mathcal{C}_{p}$.
Overall, Algorithm 8 runs in
$O(n\log^{3}n+(n+\textsf{OUT})\varepsilon^{-O(\rho)}\kappa\log^{2}n)$ time.


# Mid-latitude interactions expand the Hadley circulation

###### Abstract

The Hadley circulation describes a planetary-scale tropical atmospheric flow,
which has a major influence on climate. Contemporary theoretical understanding
is based upon angular momentum conservation, the basic dynamical constraint
governing the state of the flow pattern. However, despite the degree of
success in representing the Hadley circulation, the canonical theoretical
model does not treat interactions with other regions, particularly the mid-
latitudes. Here, we extend the original model of Held and Hou (1980) to
include the influence of mid-latitude large-scale atmospheric dynamics, which
we treat using the planetary-scale heat equation with a parameterized poleward
heat flux driven by synoptic eddies. The energy flux balance within the Hadley
cell includes the poleward heat flux at the poleward edge of the cell, which
is controlled by the baroclinic instability of the sub-tropical jet. We find
that an increase (decrease) in the poleward heat flux leads to a strengthening
(weakening) of tropical convection, driving an equatorward (poleward) shift of
the edge of the Hadley cell. Thus, our theoretical solutions suggest that
global warming, which can reduce the baroclinicity of the subtropical jet, can
lead to the poleward expansion of the Hadley cell due to the change in energy
flux balance within it.

The Hadley circulation is a principal dynamical feature in the tropics, acting
as an engine that transfers heat and momentum to high latitudes. At the same
time, the extratropical regions can influence the Hadley circulation. Under
global warming, high latitudes warm faster than low latitudes, reducing the
equator to pole temperature difference and the poleward heat flux in the mid-
latitudes. The theory described here shows that the decrease in the poleward
heat flux can lead to the expansion of the Hadley cell, and a poleward shift
of subtropical dry zones.

##  1 Introduction

The imbalance of energy between the tropics and the high-latitudes drives
large-scale atmospheric and oceanic motions under the influence of Earth’s
rotation (e.g., Trenberth and Solomon, 1994). These motions can be described
by solving the primitive equations, which are approximate forms of the Navier-
Stokes equations and the conservation of mass and thermal energy in a rotating
frame of reference, assuming horizontal motions dominate vertical motions for
an atmosphere of depth much less than Earth’s radius (Phillips, 1959). Their
numerical solution was first achieved by the revolutionary development of
stored program computing (e.g., Edwards, 2011). However, even with this
computational advance, the complexity of the primitive equations still masks a
great deal of understanding. This motivates theoretical approaches that model
the essential physical processes (Houghton et al., 1997).

The zonal mean dynamics of the tropical atmosphere is described by a
thermally-driven cell extending from the equator to the sub-tropics (e.g.,
Webster, 2004). Tropical convection drives upward motions with the upper
atmosphere traveling towards high-latitudes, where radiative cooling drives
downward motions in sub-tropical regions. The air returns to the tropics
through the near surface trade winds, thereby completing what is known as the
Hadley circulation. The simplest model, due to North (1975), is a forced heat
equation with a thermal eddy diffusivity $D$ that parameterizes the activity
of synoptic eddies. The poleward heat transport by turbulent synoptic eddies
is interpreted as being “diffusive” (e.g., Okubo and Levin, 2001), under the
assumption that turbulent eddies govern the equator-to-pole transport.
However, turbulent diffusion cannot characterize the entire transport process
of the large-scale convective cell extending from the tropics to the mid-
latitudes.

The central means of side-stepping the complexity and ambiguities of turbulent
diffusivity in the Hadley circulation is to replace the equations of motion
themselves with angular momentum conservation. If $a$ and $\Omega$ are Earth’s
radius and rotation rate respectively, then a “static air parcel” (i.e., no
zonal mean zonal wind) at the surface in the tropics has angular momentum
$\Omega a^{2}$, which is conserved during its journey towards the upper edge
of the Hadley cell. This is used to calculate the zonal mean structure of the
Hadley circulation (Held and Hou, 1980). Combined with energy flux
conservation inside the cell, the theory of Held and Hou (1980) deduces the
size of the Hadley cell in terms of the position of the descending edge. This
idealized model was extended by Lindzen and Hou (1988) to represent the
seasonal evolution of the Hadley circulation.

Recently Hill et al. (2022) used a different approach to deduce the size of
the Hadley cell. Numerical studies showed that angular momentum is not
conserved in the upper branch of the Hadley cell (Walker and Schneider, 2006),
which was expected due to the dynamic role of synoptic eddies leading to the
slowdown of the zonal mean zonal wind. Hence, rather than relying on angular
momentum conservation combined with thermodynamic constraints, the descending
edge is determined as the minimum latitude at which the zonal mean wind is
baroclinically unstable (Kang and Lu, 2012), and the ascending edge is
determined by calculating the latitude at which the absolute vorticity
vanishes near the tropopause (Hill et al., 2021).

The dynamics of baroclinic waves in the mid-latitudes are known to influence
the intensity and extent of the Hadley cell (Lee and Kim, 2003; Walker and
Schneider, 2006). Indeed, the dynamic and thermodynamic interaction between
the tropics and mid-latitudes is a key factor in determining both the large
scale structure and internal dynamics of the Hadley cell. Moreover,
observations and global climate models show that the extent of the Hadley cell
has increased due to global warming (Frierson et al., 2007; Lu et al., 2007;
Johanson and Fu, 2009; Choi et al., 2014). This can lead to a dramatic change
in large-scale tropical atmospheric dynamics, including the monsoon and
impacts on the desertification in sub-tropical regions. The Hadley circulation
is not a closed subsystem of the climate, and given its importance, an
appropriate theory should contain dynamic and thermodynamic interactions
between the tropics and the mid-latitudes.

Here, we extend the closed Hadley circulation model to include the interaction
with mid-latitudes. This influence is approximated by the planetary-scale heat
equation with a parameterization of the synoptic-scale turbulent heat flux.
The mutual interaction between the Hadley cell and the mid-latitude atmosphere
is considered through two thermodynamic constraints; (a) the continuity of
potential temperature at the poleward edge of the Hadley cell, and (b) the
energy flux balance inside the cell, including the poleward heat flux
calculated at the edge of the cell. Namely, the deviation of the potential
temperature at the cell edge from radiative-convective equilibrium drives the
large-scale dynamics of turbulent synoptic eddies in the mid-latitudes, and
the poleward heat flux from synoptic eddies balances the energy flux inside
the cell. Therefore, these mid-latitude interactions are viewed as an
extension of the theory of Held and Hou (1980), and a modification of the
energy flux balance model of North (1975).

##  2 Theoretical background and development

The Hadley circulation emerges from the interaction between turbulent synoptic
eddies and tropical convection. Despite this complexity, a key dynamic
constraint, angular momentum conservation, facilitates the deduction of the
meridional structure of potential temperature. The foundational theory of Held
and Hou (1980) treats the tropical regions adjacent to the mid-latitudes,
maintaining a radiative-convective equilibrium temperature, defined as
$\Theta_{E}$. This equilibrium temperature is determined through a regional
energy flux balance, accounting for local convection with a length-scale
asymptotically smaller than the synoptic scale. Notably, the original theory
does not incorporate the potential impact of synoptic eddies, generated
primarily by baroclinic instability near the edge of the Hadley circulation.
Studies highlight the influence of synoptic eddy dynamics on the dynamical
characteristics of the Hadley circulation (Lee and Kim, 2003; Walker and
Schneider, 2006). Therefore, it is important to consider the effect of mid-
latitude synoptic eddies on the overall dynamics of the Hadley circulation.

Atmospheric dynamics on the planetary scale, characterized by a length-scale
larger than the external Rossby deformation radius (3,000 km), is governed by
planetary geostrophic motion, a heat equation and hydrostatic balance (e.g.,
Moon and Cho, 2020, and refs therein). In contrast, synoptic-scale atmospheric
motions evolve according to the potential vorticity equation (Burger, 1958;
Phillips, 1963). Given the size of the Hadley circulation, falling within the
planetary scale, the leading-order dynamics should be represented by planetary
geostrophic motion. However, this motion alone cannot comprehensively capture
the dynamics of the Hadley circulation, because it does not account for
turbulent synoptic eddies. Despite the overall planetary-scale structure, the
synoptic dynamics play a crucial role in shaping the major characteristics of
the circulation. Advancements employing a multi-scale formalism, either focus
solely on the tropics (Majda and Klein, 2003), or propose that planetary
geostrophic motion, coupled with thermal forcing, establishes a balanced field
conducive to the growth of synoptic eddies (Dolaptchiev and Klein, 2013; Moon
and Cho, 2020). This latter multi-scale formalism provides a framework for
describing wave-mean flow interactions, suggesting that the Hadley circulation
is a consequence of such interactions. In this section, we elaborate on the
original Held and Hou (1980) theory within the context of planetary-scale
dynamics, incorporating angular momentum conservation. Furthermore, we extend
the treatment to include the influence of synoptic eddies in the mid-
latitudes.

For comparison and contrast, and to make this paper reasonably self-contained,
we summarize the essential known and new features here.

###  2.1 Planetary geostrophic motion and its interaction with synoptic
eddies–Background

The theory of large-scale atmospheric dynamics is rooted in the principles of
Rossby wave dynamics (Dickinson, 1978). Disturbances acting upon a mean flow
field, characterized by a background potential vorticity gradient, give rise
to large-scale waves that exert control over synoptic mid-latitude weather
patterns. The generation, propagation, and breaking of these waves
significantly influence the overall circulation of the large-scale atmosphere.
The dynamics of Rossby waves are effectively described by the quasi-
geostrophic potential vorticity equation, which provides a fundamental
framework for investigating various large-scale phenomena in the atmosphere,
primarily due to the pivotal role played by large-scale eddies (Pedlosky,
2013).

However, the understanding of the general circulation, including the Hadley
cell, entails recognizing it as an organized structure arising from the
collective influence of turbulent eddies. To effectively represent the
comprehensive structure of the general circulation, it is essential to
establish a framework capable of capturing the cumulative impact of turbulent
synoptic eddies. This requires a mathematical treatment with spatial and
temporal scales that are asymptotically larger than the synoptic scale,
thereby parameterizing the cumulative effect of turbulent eddies through the
gradient of the state variables.

In the context of the Fickian approximation, achieving scale separation is
crucial. Thus, planetary geostrophic motion, characterized by an
asymptotically larger length scale than synoptic-scale eddies, provides a
suitable setting for applying the Fickian approximation to the turbulent heat
flux induced by eddies.

###  2.2 Quasi-geostrophic motion, Planetary geostrophic motion, and the heat
equation

Quasi-geostrophic motion and planetary geostrophic motion are characterized by
a small Rossby number, so that the Coriolis effect dominates inertial effects.
Thus, both scales exhibit the same leading-order balances in momentum
conservation; geostrophic and hydrostatic balances.

In quasi-geostrophic motion, the leading-order balance does not explicitly
involve the continuity and heat equations, which are automatically satisfied
by geostrophic balance. Therefore, the leading-order balance alone cannot
uniquely determine the leading-order variables, which are the geostrophic
winds and pressure, a unique determination of which requires higher-order
dynamics, through the conservation of potential vorticity.

On the other hand, when the horizontal length-scale is comparable to the
external Rossby deformation radius $L_{D}\equiv\sqrt{gH}/f$, where $H$ is a
characteristic vertical scale, and $f$ is the mid-latitude Coriolis parameter,
the continuity and heat equations are not satisfied automatically by
geostrophic balance. Thus, using a linear meridional variation in the Coriolis
parameter, $\beta\equiv df/dy$, in the continuity equation has a leading order
effect, which leads to the Sverdrup relationship (Sverdrup, 1947). Note that
$L_{D}\approx 3000$ km, and characterizes the length scale of a Rossby wave
for a barotropic atmosphere on Earth.

Importantly, we turn to the heat equation to show a clear separation of
scales. Namely, whereas on the quasi-geostrophic scale the advection of
potential temperature appears as part of a higher-order balance, on the
planetary-scale the advection of potential temperature appears at leading-
order. Therefore, the heat equation is a central focus in planetary
geostrophic motion.

We non-dimensionalize the primitive equations with length-scale $L$, velocity-
scale $U$, and advective time-scale $L/U$, where the dimensional independent
variables $x^{*},y^{*},z^{*}$ and $t^{*}$ have their usual meanings as zonal,
meriodional, vertical and time coordinates respectively. The dimensional
pressure is $P^{*}=P^{*}(x^{*},y^{*},z^{*},t^{*})$ and the dimensionless
pressure is $p=p(x,y,z,t)$, and they are related through
$P^{*}=P^{*}_{S}(z^{*})+\rho^{*}_{S}(z^{*})Uf_{0}Lp$, where $P^{*}_{S}(z^{*})$
and $\rho^{*}_{S}(z^{*})$ are horizontally averaged (the subscript $S$)
reference vertical profiles of pressure and density respectively, $f_{0}$ is
the Coriolis parameter at a reference latitude, and thus
$\rho^{*}_{S}(z^{*})Uf_{0}L$ is a scale for the anomalous pressure originating
from geostrophic balance. The dimensional density, $\rho^{*}$, and potential
temperature, $\Theta^{*}$, are non-dimensionalized using the geostrophic and
hydrostatic balances, giving $\rho^{*}=\rho^{*}_{S}(z^{*})[1+\epsilon F\rho]$
and $\Theta^{*}=\Theta^{*}_{S}(z^{*})[1+\epsilon F\Theta]$, where $\rho$ and
$\Theta$ are dimensionless. Here, the Rossby number is $\epsilon\equiv U/fL$,
and the Froude number is $F\equiv f^{2}L^{2}/gH=L^{2}/L_{D}^{2}$, with $g$ the
acceleration due to gravity. Based on this scaling the primitive equations
become

$\displaystyle\epsilon\frac{Du}{Dt}-(1+\beta y)v=-\frac{\partial p}{\partial
x},$ $\displaystyle\epsilon\frac{Dv}{Dt}+(1+\beta y)u=-\frac{\partial
p}{\partial y},$ $\displaystyle\frac{\partial u}{\partial x}+\frac{\partial
v}{\partial y}+\frac{1}{\rho_{S}}\frac{\partial}{\partial z}(\rho_{S}w)=0,$
$\displaystyle\rho=\frac{1}{\rho_{S}}\frac{\partial}{\partial
z}(\rho_{S}p),\qquad\text{and}$ $\displaystyle\epsilon
F\frac{D\Theta}{Dt}+\frac{1}{\Theta_{S}}\frac{d\Theta_{S}}{dz}w=Q,$ (1)

where $\vec{u}=(u,v,w)$ is the velocity field,
$\frac{D}{Dt}\equiv\frac{\partial}{\partial t}+u\frac{\partial}{\partial
x}+v\frac{\partial}{\partial y}+w\frac{\partial}{\partial z}$, and
$Q=Q(x,y,z,t)$ is the dimensionless equator to pole radiative-convective
energy flux, which drives the meridional temperature gradient, and is
$O(\epsilon^{2})$. A more detailed development can be found in Pedlosky
(2013).

As noted above, large-scale atmospheric dynamics is characterized by a small
Rossby number $\epsilon\ll 1$, which is the basis for a regular perturbation
analysis, expanding the dynamic and thermodynamic variables as
$x=x_{0}+\epsilon x_{1}+O(\epsilon^{2})$, yielding the leading-order behavior
of geostrophic and hydrostatic balance as

$\displaystyle v_{0}=\frac{\partial p_{0}}{\partial x},\quad
u_{0}=-\frac{\partial p_{0}}{\partial
y},~{}~{}\text{and}~{}~{}\rho_{0}=\frac{1}{\rho_{S}}\frac{\partial}{\partial
z}(\rho_{S}p_{0}).$ (2)

Next, we seek to extract a leading-order balance from the heat equation, which
is determined by the choice of $L$.

####  2.2.1 Length scale selection and the heat equation

Selecting $L$ as the internal Rossby deformation radius, which is
approximately $1000$ km, yields $F$ of approximately $0.1$, which is also the
same order of magnitude as $\epsilon$, and the static stability term
$\frac{1}{\Theta_{S}}\frac{d\Theta_{S}}{dz}$. Therefore, the temperature
advection term is $\epsilon F\frac{D\Theta}{Dt}=O(\epsilon^{2})$, so that
under adiabatic conditions the leading-order vertical velocity $w_{0}$ is
essentially zero, which characterizes quasi-geostrophic motion.

Quasi-geostrophic motion is based on leading order hydrostatic and geostrophic
balances. However, for a given set of boundary conditions, this cannot fully
determine all of the fields described by Eqs. (1), resulting in so-called
geostrophic degeneracy. Hence, in order to determine the leading-order
variables, one must consider dynamics at $O(\epsilon)$. This leads to quasi-
geostrophic potential vorticity conservation under adiabatic conditions, which
captures the essentials of quasi-geostrophic motion.

Alternatively, selecting $L$ to be of order the external Rossby deformation
radius, $L_{D}$, gives $F$ = $O(1)$. This leads to $\epsilon
F\frac{D\Theta}{Dt}$ and $\frac{w}{\Theta_{S}}\frac{d\Theta_{S}}{dz}$ being
the same order of magnitude. Therefore, on this length-scale, geostrophic
degeneracy vanishes, thereby eliminating the need to consider higher-order
dynamics (Pedlosky, 2013; Moon and Cho, 2020).

####  2.2.2 Planetary geostrophic motion–multi-scale analysis

Planetary geostrophic motion is comprised of geostrophic and hydrostatic
balance, the Sverdrup relation arising from continuity, and the heat equation,
and is given in non-dimensional form as follows:

$\displaystyle u_{L}=-\frac{\partial P_{L}}{\partial y},\quad
v_{L}=\frac{\partial P_{L}}{\partial
x},\quad\rho_{L}=-\frac{1}{\rho_{S}}\frac{\partial}{\partial
z}(\rho_{S}P_{L}),$ $\displaystyle\frac{1}{\rho_{S}}\frac{\partial}{\partial
z}(\rho_{S}w_{L})-\beta_{L}v_{L}=0,\qquad\text{and}\qquad$
$\displaystyle\frac{\partial\Theta_{L}}{\partial
t}+u_{L}\frac{\partial\Theta_{L}}{\partial
x}+v_{L}\frac{\partial\Theta_{L}}{\partial
y}+w_{L}\left(S(z)+\frac{\partial\Theta_{L}}{\partial z}\right)=Q_{L},$ (3)

where the subscripts $L$ denote planetary-scale. Thus, the pressure is
$P_{L}$, the potential temperature is $\Theta_{L}=\frac{\partial
P_{L}}{\partial z}$, the zonal, meridional, and vertical velocities are
$u_{L}$, $v_{L}$, and $w_{L}$ respectively, and the planetary-scale thermal
forcing is $Q_{L}$. The $\beta$-effect is characterized by $\beta_{L}$, the
hemispheric average vertical density profile is $\rho_{S}$, and the dry static
stability of the entire hemisphere is given by $S(z)$.

Central here is the heat equation for $\Theta_{L}$ under the influence of the
velocity field, and the thermal forcing $Q_{L}$. Therefore, planetary
geostrophic motion yields a balanced flow in response to a specified
planetary-scale thermal forcing (Pedlosky, 2013; Moon and Cho, 2020).

A complete description of the dynamics of the large-scale atmosphere must
treat the interaction between the planetary and synoptic scales, which
requires a multi-scale formalism, the crux of which lies in expressing
thermodynamic variables $P$, $\Theta$, and $\rho$ as a linear sum of planetary
and synoptic scales (Moon and Cho, 2020). For instance,
$P^{*}=P^{*}_{S}(z^{*})+\rho^{*}_{S}(z^{*})f_{0}L_{D}UP_{L}(X,Y,z,\tilde{t})+\rho^{*}_{S}(z^{*})f_{0}LUP(x,y,z,t)$,
where as above $L_{D}$ is the external Rossby deformation radius ($\approx
3000\,\text{km}$), and $L$ is the length for the synoptic scale ($\approx
1000\,\text{km}$). The symbols $X$, $Y$, and $\tilde{t}$ ($x$, $y$, and $t$)
denote the space and time variables for planetary-scale (synoptic-scale)
motions. The multi-scale formalism leads to

$\displaystyle u_{L}=-\frac{\partial P_{L}}{\partial Y},\quad
v_{L}=\frac{\partial P_{L}}{\partial X},\quad u_{0}=-\frac{\partial
p_{0}}{\partial y},\quad\text{and~{}~{}}v_{0}=\frac{\partial p_{0}}{\partial
x},$ $\displaystyle\frac{1}{\rho_{S}}\frac{\partial}{\partial
z}(\rho_{S}w_{L})=\beta_{L}v_{L},$
$\displaystyle\frac{\partial\Theta_{L}}{\partial\tilde{t}}+u_{L}\frac{\partial\Theta_{L}}{\partial
X}+v_{L}\frac{\partial\Theta_{L}}{\partial
Y}+w_{L}\left(\frac{\partial\Theta_{L}}{\partial z}+S(z)\right)$
$\displaystyle=-\left(\frac{\partial}{\partial
t}+u_{L}\frac{\partial}{\partial x}+v_{L}\frac{\partial}{\partial
y}\right)\Theta_{0}-\left(u_{0}\frac{\partial}{\partial
X}+v_{0}\frac{\partial}{\partial Y}\right)\Theta_{L}$
$\displaystyle-\frac{\partial}{\partial
x}(u_{0}\Theta_{0})-\frac{\partial}{\partial
y}(v_{0}\Theta_{0})-w_{1}\left(\frac{\partial\Theta_{L}}{\partial
z}+S(z)\right)+Q_{L},\quad\text{and~{}~{}}$
$\displaystyle\frac{\partial}{\partial
t}\nabla^{2}P_{0}+(u_{L}+u_{0})\frac{\partial}{\partial
x}\nabla^{2}P_{0}+(v_{L}+v_{0})\frac{\partial}{\partial y}\nabla^{2}P_{0}$
$\displaystyle+\beta\frac{\partial P_{0}}{\partial
x}=\frac{1}{\rho_{S}}\frac{\partial}{\partial z}(\rho_{S}w_{1}).$ (4)

Here, the subscripts $0$ and $1$ denote the leading order and $O(\epsilon)$
synoptic scale variables, respectively. The planetary scale continuity
equation transforms into the Sverdrup relation. Both scales obey geostrophic
and hydrostatic balances, and both scales collectively contribute to the heat
flux balance. In terms of planetary-scale motion, synoptic-scale motions act
as external forcing for the spatio-temporal evolution of planetary-scale
potential temperature. Finally, planetary-scale motion acts as a mean field
for the horizontal vorticity of synoptic-scale flows (Moon and Cho, 2020).

The general circulation in the atmosphere is treated using zonally-averaged
fields on the hemispheric spatial scale. In particular, the Hadley circulation
is a consequence of tropical convection and turbulent synoptic eddies in mid-
latitudes. Since the formulation is based on the beta plane, it is challenging
to fully account for the influence of tropical convection. The Held and Hou
(1980) theory employs angular momentum conservation to establish a connection
between tropical convection and the dynamics of mid-latitudes. Therefore, the
dynamics of the Hadley circulation can be effectively described by combining
angular momentum conservation, planetary geostrophic motion, and synoptic
eddies.

Considering the extent of the Hadley circulation, its dynamics should be
described by a planetary-scale treatment. However, a description of the
zonally-averaged circulation must account for the collective influence of the
turbulence of synoptic eddies, which must be incorporated in the framework of
planetary geostrophic motion. Therefore, we treat the effect of synoptic
eddies by taking the spatio-temporal and zonal averages of synoptic variables,
which leads to

$\displaystyle u_{L}=-\frac{\partial P_{L}}{\partial Y},$
$\displaystyle\Theta_{L}=\frac{\partial P_{L}}{\partial z},$
$\displaystyle\frac{1}{\rho_{S}}\frac{\partial}{\partial
z}(\rho_{S}\langle\overline{w}_{1}\rangle)=-\epsilon\frac{\partial^{2}}{\partial
Y^{2}}\langle\overline{u_{0}v_{0}}\rangle,\qquad\text{and}$
$\displaystyle\frac{\partial\Theta_{L}}{\partial\tilde{t}}=Q_{L}-\epsilon^{1/2}\frac{\partial}{\partial
Y}\langle\overline{v_{0}\Theta_{0}}\rangle-\langle\overline{w}_{1}\rangle\left(\frac{\partial\Theta_{L}}{\partial
z}+S(z)\right),$ (5)

where $\langle X\rangle$ and $\overline{X}$ denote zonal and time averages
respectively, and $\epsilon$ is the planetary scale Rossby number. It appears
from the spatial average of the synoptic spatial derivative $\partial/\partial
y$, which becomes $\epsilon^{1/2}\partial/\partial Y$ wherein
$L/L_{D}=\epsilon^{1/2}$. This treatment consists of geostrophic balance
containing the westerlies in the mid-latitudes, and heat flux balance, which
has three components: thermal forcing $Q_{L}$, poleward heat flux by synoptic
eddies, and adiabatic warming or cooling by vertical motion. This vertical
motion is driven by eddy momentum convergence, which is equivalent to a
horizontal vorticity flux.

Within this framework the general circulation of the large-scale atmosphere
can be understood by examining how the thermal forcing, $Q_{L}$, is
distributed by turbulent synoptic eddies. In particular, the Hadley
circulation can be described by combining the above framework with angular
momentum conservation.

####  2.2.3 Radiative convective equilibrium temperature

Within the planetary geostrophic framework, we seek to distinguish the roles
played by thermodynamics and fluid dynamics in the distribution of the thermal
forcing, specifically in the form of shortwave radiance. Within the framework
of quasi-geostrophic dynamics, the role of radiative-convective equilibrium is
indirectly characterized as thermal forcing. The potential temperature
exhibits a tendency to relax towards the radiative-convective equilibrium
temperature with a characteristic response time-scale $\tau$ (Held and Suarez,
1994). This process can be systematically derived within the context of
planetary geostrophic motion, as this motion is directly influenced by the
residual effects of local radiation and convection.

The radiative-convective temperature, $\Theta_{E}$ , is a consequence of the
local balance between radiative heat flux and convection. The actual
temperature differs from the radiative-convective temperature due to the
presence of large-scale atmospheric flow. The primary function of this large-
scale atmospheric flow is to transfer surplus energy from low-latitudes to
high-latitudes. Consequently, one anticipates that the real temperature is
lower (higher) than the radiative-convective equilibrium temperature in the
low-latitudes (high-latitudes). However, the strength of the seasonal cycle in
surface temperature indicates that the dominant factor influencing the local
temperature is the local energy flux balance, primarily determined by
shortwave radiation. As a result, the radiative-convective equilibrium
temperature is much greater than the difference between the real temperature
$\Theta_{L}$ and $\Theta_{E}$.

In the heat equation, the external thermal forcing $Q_{L}$ represents local
radiative-convective processes. It is reasonable to assume that
$Q_{L}=Q_{L}(\Theta_{L})$, so that the definition of radiative-convective
equilibrium is $Q_{L}(\Theta_{E})=0$. Therefore, $Q_{L}(\Theta_{L})\simeq
Q_{L}(\Theta_{E})+\partial
Q_{L}/\partial\Theta_{L}|_{\Theta_{L}=\Theta_{E}}(\Theta_{L}-\Theta_{E})=-(\Theta_{L}-\Theta_{E})/\tau$,
which is the sensitivity of the local radiative-convective equilibrium, with
response time-scale of $\tau$.

We define the anomalous potential temperature as
$\eta_{L}\equiv\Theta_{L}-\Theta_{E}$, and use the condition
$\Theta_{E}\gg\eta_{L}$, so that the heat equation from (5) can be written as

$\displaystyle\frac{\partial\eta_{L}}{\partial\tilde{t}}=-\frac{1}{\tau}\eta_{L}-\frac{\partial}{\partial
Y}\langle\overline{v_{0}\Theta_{0}}\rangle-\epsilon^{-1/2}\langle\overline{w}_{1}\rangle\left(\frac{\partial\Theta_{E}}{\partial
z}+S(z)\right),$ (6)

where $\Theta_{L}\simeq\Theta_{E}+\epsilon^{1/2}\eta_{L}$, so that
$\frac{\partial\Theta_{L}}{\partial\tilde{t}}\simeq\epsilon^{1/2}\frac{\partial\eta_{L}}{\partial\tilde{t}}$
and $Q_{L}(\Theta_{L})\simeq-\frac{\epsilon^{1/2}}{\tau}\eta_{L}$. Note that
all terms are $O(\epsilon)$, apart from the last term proportional to
$\langle\overline{w}_{1}\rangle$, which is $O(\epsilon^{1/2})$ because
$\langle\overline{w}_{1}\rangle\sim O(\epsilon)$, in accordance with the
relationship between $\langle\overline{w}_{1}\rangle$ and the eddy momentum
flux $\langle\overline{u_{0}v_{0}}\rangle$. Importantly, this lower order
term, associated with ageostrophic vertical motion $w_{1}$, emerges from
horizontal vorticity dynamics, which is closely linked to eddy momentum
convergence; a factor necessary for accelerating the zonal mean zonal wind in
the upper atmosphere. As will be notable below in our analysis of the zonally
symmetric heat equation (3.1.2), this lower order term is associated with
eddy-driven mid-latitude jet streams.

In the context of the perturbation expansion presented above, we discern that
the anomalous potential temperature $\eta_{L}$ arises from the intricate
dynamics of the large-scale atmosphere, which orchestrates the transfer of
excess tropical energy to higher latitudes. Whereas tropical convection is
responsible for negative $\eta_{L}$, the large-scale mid-latitude dynamics
driving poleward eddy heat flux is responsible for the emergence of positive
$\eta_{L}$. The governing equation (6) implies that a steady state results
from a delicate thermal balance involving the negative feedback from local
radiative convective processes, poleward eddy heat flux, and adiabatic warming
or cooling induced by eddy momentum flux. This framework simply expresses the
general circulation as being intricately linked to the thermodynamic processes
embedded in radiative-convective equilibrium and the large-scale atmospheric
dynamics that generate turbulent eddy heat and momentum fluxes.

##  3 Extending Held and Hou theory using planetary geostrophic motion

The leading-order dynamics of the Hadley circulation are encapsulated by
angular momentum conservation. Surface air in the tropics possesses the
maximum angular momentum, and in the nearly inviscid limit angular momentum is
nearly conserved within the Hadley cell. This conservation shapes the zonal
mean zonal wind from the tropics to the sub-tropics, and subsequently
determines the meridional structure of potential temperature through the
thermal wind balance (Held and Hou, 1980). In this context, we extend the
original theory using the formalism of planetary-scale motions.

Whereas the leading-order momentum equation in the mid-latitudes is
geostrophic balance, in the tropics that balance is disrupted because the
Coriolis effect becomes negligible. This distinction in the leading-order
dynamics underlies the challenges in expressing the dynamical interaction
between the tropics and the extra-tropics, which is the origin of using
angular momentum conservation as a bridge to dynamically connect the tropics
to the mid-latitudes.

The angular momentum of the large-scale atmosphere is given by $M=\Omega
a^{2}\cos^{2}\theta+u^{*}a\cos\theta$, where $\Omega$ and $a$ are the angular
velocity and radius of the Earth respectively, $\theta$ is the latitude, and
$u^{*}$ is the zonal velocity. In the tropics, the angular momentum of
stationary air parcels on the surface is $M=\Omega a^{2}$, so that the zonal
velocity that conserves angular momentum is

$\displaystyle u^{*}_{M}\equiv\frac{\Omega
a\sin^{2}\theta}{\cos\theta}\simeq\Omega a\theta^{2},$ (7)

where we use the small angle approximation ($\sin\theta\simeq\theta$ and
$\cos\theta\simeq 1$). Thus, in a zonally symmetric atmosphere, the zonal wind
$u^{*}_{M}$ can determine the meridional structure of potential temperature
through basic balances.

The principal action of the Hadley circulation is to redistribute the low
latitude heat surplus towards high latitudes. Consequently, it is essential to
determine the latitudinal distribution of potential temperature, primarily
governed by local radiative-convective processes. Thus, as discussed above,
the focus is on the anomalous potential temperature with respect to the
radiative-convective equilibrium potential temperature, which is
$\Theta_{E}=1-\frac{2}{3}\Delta_{H}P_{2}(\sin\theta)$. Here, $\Delta_{H}$ is
the fractional change in the equator to pole potential temperature,
$P_{2}(x)=\frac{1}{2}(3x^{2}-1)$ is the second Legendre polynomial, and
$R=\frac{gH\Delta_{H}}{\Omega^{2}a^{2}}$ is the thermal Rossby number, with
$H$ the vertical length-scale. This leads to the following scaling of the
thermodynamic variables; $P^{*}=P_{S}(z)+\rho_{s}(z)\Omega^{2}a^{2}P_{L}$,
$\rho^{*}=\rho_{s}\left(1+\frac{\Delta_{H}}{R}\rho_{L}\right)$ and
$\Theta^{*}=\Theta_{S}(z)(1+\Theta_{L})$. Finally, the zonal velocity
$u^{*}=\Omega au_{L}$ is scaled by $\Omega a$. This specific scaling is chosen
to recover the results as Held and Hou (1980).

Planetary geostrophic motion under zonally symmetric thermal forcing is given
by

$\displaystyle u_{L}\theta=-\frac{1}{2}\frac{\partial P_{L}}{\partial\theta},$
(8) $\displaystyle\Theta_{L}=\frac{\Delta_{H}}{R}\frac{\partial
P_{L}}{\partial z},\qquad\text{and}$ (9)
$\displaystyle\frac{\partial\eta_{L}}{\partial\tilde{t}}+u_{L}\frac{\partial\eta_{L}}{\partial\phi}=-\frac{1}{\tau}\eta_{L}-\frac{\partial}{\partial\theta}\overline{v_{0}\Theta_{0}},$
(10)

where $\phi$ is the longitude and, as above, the subscripts $L$ and $0$ denote
planetary-scale and leading-order synoptic-scale variables respectively (Moon
and Cho, 2020). In equation (8), we neglect the geometric term
$u_{L}^{2}\tan\theta$ for simplicity, but note that this does not influence
the overall qualitative results, and for numerical accuracy this term can be
included.

We neglect the meridional, $v_{L}$, and vertical, $w_{L}$, velocities due to
zonal symmetry, and the dynamic mean field depends solely on the zonal wind
$u_{L}$, which satisfies geostrophy (8). The key time-evolving equation is the
heat equation (10), the zonal average $\langle\cdot\rangle$ of which is

$\displaystyle\frac{\partial\eta_{L}}{\partial\tilde{t}}=-\frac{1}{\tau}\eta_{L}-\frac{\partial}{\partial\theta}\langle\overline{v_{0}\Theta_{0}}\rangle.$
(11)

We note that in traditional quasi-geostrophic wave-mean field theory variables
are written as the sum of the zonal-averages (the means) and departures from
those averages viz. $\theta_{0}=\langle\theta_{0}\rangle+\theta^{\prime}_{0}$.
In the multi-scale formalism, the planetary-scale variables are treated as the
means and the synoptic-scale variables the departures from those means, which
then have the traditional interpretation in a zonally symmetric atmosphere.

The thermal wind balance obtained by combining equations (8) and (9) is

$\displaystyle\frac{\partial u_{L}}{\partial
z}\theta=-\frac{R}{2\Delta_{H}}\frac{\partial\Theta_{L}}{\partial\theta},$
(12)

and integrating both sides in the vertical from 0 to 1 gives

$\displaystyle
u_{M}\theta=-\frac{R}{2\Delta_{H}}\frac{\partial\Theta_{L}}{\partial\theta}.$
(13)

Using equation (7), where $u^{*}_{M}=\Omega au_{M}$, we have
$u_{M}\simeq\theta^{2}$, so that integration of equation (13) gives

$\displaystyle\Theta_{L}(\theta)=\Theta_{\text{eq}}-\frac{\Delta_{H}}{2R}\theta^{4},$
(14)

where $\Theta_{\text{eq}}$ is the potential temperature at the equator.

Figure 1: (a) A schematic meriodional section of the Hadley circulation theory
of Held and Hou (1980). Within the Hadley cell the angular momentum of surface
air in the tropics is maximal and nearly conserved as it moves upward and
poleward towards the edge of the cell at $\theta=\theta_{H}$. It is assumed
that the mid-latitude atmosphere is simply defined by a prescribed radiative-
convective equilibrium temperature, $\Theta_{E}$, and the poleward heat flux
from tropics to the mid-latitudes is negligible. (b) A schematic of the simple
model described here, which connects the Hadley cell to the mid-latitudes. The
principal dynamics within the Hadley cell is same as in Held and Hou (1980),
but is connected to the mid-latitudes by approximating planetary-scale
atmospheric dynamics using a diffusive model.

The determination of the edge of the Hadley cell ($\theta=\theta_{H}$) and the
equatorial potential temperature ($\Theta_{\text{eq}}$) within the meridional
structure of the zonal mean zonal wind and potential temperature is not
straightforward. Namely, the challenge associated with localizing the
boundaries is due to the ascent of air from the tropics as it moves towards
high latitudes, but sinking in the mid-latitudes instead of completing the
journey to the polar regions.

Rather than tracking the vertical movement of air, Held and Hou (1980) impose
two key constraints on potential temperature to ascertain the cell edge and
equatorial potential temperature (Fig. 1a). Firstly, continuity of potential
temperature demands that $\Theta_{L}=\Theta_{E}$ at $\theta=\theta_{H}$.
Secondly, for a steady-state Hadley cell, energy flux balance must be
maintained. Integrating equation (11) yields

$\displaystyle\frac{\partial}{\partial\tilde{t}}\int_{0}^{\theta_{f}}\eta_{L}\,d\theta=-\frac{1}{\tau}\int_{0}^{\theta_{f}}\eta_{L}\,d\theta-\langle\overline{v_{0}\Theta_{0}}\rangle|_{\theta=\theta_{f}}.$
(15)

The cell is assumed to be in quasi-isolation, and thus
$\langle\overline{v_{0}\Theta_{0}}\rangle|_{\theta=\theta_{H}}\simeq 0$. In
the extra-tropics, where baroclinic instability is absent, there is no
significant contribution from synoptic eddies in transferring thermal energy
poleward. Thus, the energy flux balance within the cell reduces to

$\displaystyle\int_{0}^{\theta_{H}}\Theta_{L}\,d\theta=\int_{0}^{\theta_{H}}\Theta_{E}\,d\theta,$
(16)

so that with the prescribed $\Theta_{E}$ and $\Theta_{L}$ in equation (16),
the edge of the Hadley cell is given as

$\displaystyle\theta_{H}=\left(\frac{5}{3}R\right)^{1/2}=\left(\frac{5}{3}\right)^{1/2}\left(\frac{gH\Delta_{H}}{\Omega^{2}a^{2}}\right)^{1/2}.$
(17)

The continuity of potential temperature and the steady-state energy flux
balance are the two essential assumptions that form the basis of the Held and
Hou (1980) theory. The resulting values of $\theta_{H}$ and $\Theta_{0}$
define the meridional profiles of potential temperature $\Theta_{L}$, and thus
determine the poleward heat flux within the cell
$\langle\overline{v_{0}\Theta_{0}}\rangle$.

Assuming self-similarity of the synoptic scale potential temperature
$\Theta_{0}$ and zonal wind $u_{0}$, one can deduce the meridional structure
of the momentum flux $\langle\overline{u_{0}v_{0}}\rangle$ and the surface
wind, providing the easterlies in the tropics. We note again that the small-
angle approximation is used for simplicity. A comprehensive treatment involves
use of full spherical coordinates, but our aim here is to provide general
qualitative insight.

###  3.1 The interaction of the Hadley circulation with the mid-latitudes

The Held and Hou (1980) theory does not treat the influence of mid-latitude
synoptic eddies. In the vicinity of the Hadley cell edge, the subtropical jet
undergoes baroclinic instability, initiating the life cycle of synoptic
eddies. Baroclinic instability results in the amplification of synoptic waves,
causing lower-level poleward heat flux. Saturated upper-level synoptic eddies
break on the equatorward flank of the jet stream, generating eddy momentum
flux convergence and intensifying the zonal mean zonal wind. Synoptic eddy
breakage creates downward motion associated with the eddy momentum flux and
thus drives adiabatic warming.

The intricate dynamics of synoptic eddies underlie the transfer of the heat
surplus from the Hadley cell to the mid-latitudes. The two constraints of the
Held and Hou (1980) theory used to determine the potential temperature at the
equator and the edge of the Hadley cell discussed above, originate in the heat
equation. The main consequence of neglecting the energy transfer by synoptic
eddies near the Hadley cell edge is the equal area constraint of equation
(16), stating that the latitudinal integral of potential temperature equals
that of the radiative convective equilibrium. However, treating mid-latitude
synoptic eddy activity is synonymous with accounting for heat transfer from
the Hadley cell to the mid-latitudes, which drives a non-zero mid-latitude
anomalous potential temperature, and this must be determined in a manner
consistent with continuity of potential temperature at the Hadley cell edge.

####  3.1.1 The Hadley cell

Angular momentum conservation determines the dynamic and thermodynamic
structure of the Hadley cell as in Held and Hou (1980). Consequently, the
upper-level zonal mean zonal wind $u^{*}_{M}$ is given by equation (7). Hence,
as shown above, using the thermal wind balance of equation (12) the potential
temperature is given by equation (14). Therefore, the anomalous potential
temperature $\eta_{L}(\theta)$ is

$\displaystyle\eta_{L}(\theta)=\Theta_{\text{eq}}-\left(1+\frac{1}{3}\Delta_{H}\right)+\Delta_{H}\theta^{2}-\frac{\Delta_{H}}{2R}\theta^{4}$
(18)

The constant of integration is the potential temperature at the equator,
$\Theta_{\text{eq}}$, which remains undetermined at the level of the thermal
wind balance; it is determined by treating the interaction between the Hadley
cell and the mid-latitudes through the heat flux balance, as discussed next.

####  3.1.2 The Mid-latitudes

The mid-latitudes range from the northern edge of the Hadley cell,
$\theta=\theta_{H}$, to the North Pole, $\theta=\theta_{f}$. Thus, extension
of the Held and Hou (1980) theory to incorporate the influence of synoptic
eddies near the high-latitude edge of the Hadley cell requires the
determination of the meridional distribution of the anomalous potential
temperature throughout the mid-latitudes as shown in Fig. 1(b). The primary
objective is to account for the impact of the Hadley cell to mid-latitude eddy
energy flux, which generates positive mid-latitude anomalous potential
temperature. To obey the constraint of potential temperature continuity at the
Hadley cell edge, a simple mid-latitude thermodynamic model is needed to
redistribute the energy received from the Hadley cell.

We emphasize that our aim is not to construct an analytical model that
captures realistic mid-latitude circulation. Rather, here we provide a
simplified treatment of the synoptic eddy driven energy flux between the
Hadley cell and the mid-latitudes, with a particular emphasis on its impact on
the intensity and size of the Hadley cell. For simplicity, as discussed above,
we make the $\beta$-plane approximation and use a Cartesian meridional
coordinate system, with $y=0$ ($y=L_{A}$) corresponding to $\theta=\theta_{H}$
($\theta=\theta_{f}$), as shown in Fig. 1(b). We use $y$ as the independent
variable in the mid-latitudes and $\theta$ in the tropics and match these
appropriately as described below. Thus, we rewrite the zonally symmetric heat
equation (11) as

$\displaystyle\frac{\partial\eta_{L}}{\partial\tilde{t}}$
$\displaystyle=-\frac{1}{\tau}\eta_{L}-\frac{\partial}{\partial
y}\langle\overline{v_{0}\Theta_{0}}\rangle$
$\displaystyle\equiv-\frac{1}{\tau}\eta_{L}+D\frac{\partial^{2}\eta_{L}}{\partial
y^{2}},$ (19)

which as noted above neglects ageostrophic vertical motion and thus provides a
very simple model for redistributing heat flux originating from the Hadley
cell. Namely, the poleward heat flux is assumed to be Fickian and hence
$\langle\overline{v_{0}\Theta_{0}}\rangle=-D\frac{\partial\eta_{L}}{\partial
y}$ with an eddy diffusivity $D$. The Fickian approximation for the poleward
eddy heat flux is discussed extensively by Held (2000) and has been validated
against observational reanalysis data (Barry et al., 2002). Moreover, the
Fickian approximation can be modified to incorporate the eddy memory effect,
known as the minimal $\tau$ approximation, thereby capturing the quasi-
oscillatory behavior of the poleward heat flux in large-scale mid-latitude
dynamics (Moon et al., 2021). Clearly, this approximation is valid when
planetary variables evolve much more slowly than the synoptic eddy turnover
time-scale, and is thus appropriate in treating steady state poleward eddy
heat flux, for which equation (3.1.2) becomes

$\displaystyle D\frac{\partial^{2}\eta_{L}}{\partial
y^{2}}-\frac{1}{\tau}\eta_{L}=0,$ (20)

and is governed by the following boundary conditions, as shown in Fig. 1 (b).
The poleward edge of the Hadley cell is at $y=0$ ($\theta=\theta_{H}$), and
the North Pole is at $y=L_{A}$ ($\theta=\pi/2$). The boundary condition at
$y=0$ is that $\langle\overline{v_{0}\Theta_{0}}\rangle$ is continuous, and
hence

$\displaystyle\lim_{\theta\to\theta_{H}-}\frac{\partial\eta_{L}}{\partial\theta}=\lim_{y\to
0+(\theta\to\theta_{H}+)}\frac{\partial\eta_{L}}{\partial y}.$ (21)

The meridional heat flux vanishes at the North Pole, so that

$\displaystyle-D\frac{\partial\eta_{L}}{\partial y}|_{y=L_{A}}=0.$ (22)

####  3.1.3 Interaction between the tropics & mid-latitudes

The anomalous potential temperature in the tropics is given by equation (18),
and in the mid-latitudes the solution to equation (20) is
$\eta_{L}(y)=K_{1}e^{y/\sqrt{D\tau}}+K_{2}e^{-y/\sqrt{D\tau}}$, with two
undetermined constants $K_{1}$ and $K_{2}$. The boundary conditions at $y=0$
and $y=L_{A}$ give

$\displaystyle-K_{1}+K_{2}$
$\displaystyle=2\Delta_{H}\sqrt{D\tau}\left(\frac{\theta^{3}_{H}}{R}-\theta_{H}\right)\qquad\text{and}$
(23) $\displaystyle K_{2}$
$\displaystyle=K_{1}\text{exp}\left(2\frac{L_{A}}{\sqrt{D\tau}}\right),$ (24)

from which we find

$\displaystyle
K_{1}=\frac{2\Delta_{H}\sqrt{D\tau}}{e^{2\frac{L_{A}}{\sqrt{D\tau}}}-1}\left(\frac{\theta^{3}_{H}}{R}-\theta_{H}\right)\qquad\text{and}$
(25) $\displaystyle
K_{2}=2\Delta_{H}\sqrt{D\tau}\frac{e^{2\frac{L_{A}}{\sqrt{D\tau}}}}{e^{2\frac{L_{A}}{\sqrt{D\tau}}}-1}\left(\frac{\theta^{3}_{H}}{R}-\theta_{H}\right).$
(26)

To uniquely determine the meridional structure of the anomalous potential
temperatures $\eta_{L}$, we require two constraints to determine the two
unknowns $\Theta_{\text{eq}}$ and $\theta_{H}$. The first is the continuity of
potential temperature,

$\displaystyle\lim_{\theta\to\theta_{H}-}\eta_{L}=\lim_{\theta\to\theta_{H}+}\eta_{L},$
(27)

and the second is energy flux balance in the Hadley cell. Integrating (20)
with respect to $\theta$ gives

$\displaystyle
D\frac{\partial\eta_{L}}{\partial\theta}|_{\theta=\theta_{H}}-\frac{1}{\tau}\int_{0}^{\theta_{H}}\eta_{L}d\theta=0,$
(28)

which when combined with (27) gives

$\displaystyle
2\Delta_{H}\sqrt{D\tau}\text{coth}\left(\frac{L_{A}}{\sqrt{D\tau}}\right)\left(\frac{\theta^{3}_{H}}{R}-\theta_{H}\right)$
$\displaystyle=-\frac{2\Delta_{H}}{5R}\theta^{4}_{H}+\frac{2}{3}\Delta_{H}\theta^{2}_{H}-2D\tau\Delta_{H}\left(\frac{\theta^{2}_{H}}{R}-1\right),$
(29)

and

$\displaystyle\Theta_{\text{eq}}$
$\displaystyle=\frac{\Delta_{H}}{10R}\theta^{4}_{H}-\left(\frac{2D\tau\Delta_{H}}{R}+\frac{1}{3}\Delta_{H}\right)\theta^{2}_{H}$
$\displaystyle+\left(1+\frac{1}{3}\Delta_{H}\right)+2D\tau\Delta_{H}.$ (30)

Therefore, equations (3.1.3) and (3.1.3) determine the two unknown
coefficients $\theta_{H}$ and $\Theta_{\text{eq}}$.

Integrating the stationary form of the first version of heat flux balance in
(3.1.2) yields the meridional structure of the poleward heat flux
$\overline{v_{0}\Theta_{0}}$ within the Hadley cell as

$\displaystyle\overline{v_{0}\Theta_{0}}$
$\displaystyle=-\frac{1}{\tau}\int_{0}^{\theta}\eta_{L}d\theta^{\prime}$
$\displaystyle=-\frac{1}{\tau}\left[\Theta_{\text{eq}}\theta-\left(1+\frac{1}{3}\Delta_{H}\right)\theta+\frac{1}{3}\Delta_{H}\theta^{3}-\frac{\Delta_{H}}{10R}\theta^{5}\right].$
(31)

On the other hand, in the mid-latitudes ($\theta>\theta_{H}$) we have

$\displaystyle\overline{v_{0}\Theta_{0}}=-D\frac{\partial\eta_{L}}{\partial
y}=-\sqrt{\frac{D}{\tau}}\left(K_{1}e^{\frac{y}{\sqrt{D\tau}}}-K_{2}e^{-\frac{y}{\sqrt{D\tau}}}\right).$
(32)

Akin to Held and Hou (1980), to obtain an estimate of the momentum fluxes, we
assume that the profiles of $u_{0}$ and $\Theta_{0}$ are self-similar, which
leads to

$\displaystyle\overline{u_{0}v_{0}}\simeq\frac{u_{L}(z=1)-u_{L}(z=0)}{\Theta_{L}(z=1)-\Theta_{L}(z=0)}\overline{v_{0}\Theta_{0}}\simeq\frac{\theta^{2}}{\Delta_{v}}\overline{v_{0}\Theta_{0}},$
(33)

where $\Delta_{v}\equiv\Theta_{L}(z=1)-\Theta_{L}(z=0)$ represents the
vertical stability of atmosphere. From the zonal momentum equation, the
surface stress in the tropics can be deduced as

$\displaystyle\frac{\partial}{\partial\theta}\overline{u_{0}v_{0}}\simeq-
Cu_{L}(z=0),$ (34)

and similarly in the mid-latitudes,

$\displaystyle\frac{\partial}{\partial y}\overline{u_{0}v_{0}}\simeq-
Cu_{L}(z=0),$ (35)

from which the trade wind and the mid-latitude surface wind are calculated
respectively, and $C$ is a constant drag coefficient.

##  4 Poleward heat flux and the Hadley cell dynamics

Equation (3.1.3) gives the poleward edge of the Hadley cell, $\theta_{H}$, as
a function of $D\tau$ and the thermal Rossby number $R$, which is proportional
to $H$; viewed as the height of the tropical tropopause. This height is
controlled by tropical radiative-convective equilibrium in the upper
troposphere, stratospheric circulation, and the interaction with mid-latitude
eddies (e.g., Fueglistaler et al., 2009), and thus may be linked, directly or
indirectly, to recent global warming. The eddy diffusivity $D$ is related to
baroclinic instability of the subtropical jet stream, which generates poleward
heat transport through synoptic eddies (North, 1975; Taylor, 1980), and
$\tau^{-1}$ is proportional to the overall climate sensitivity (Moon et al.,
2021). Therefore, $\tau$ it is closely related to the magnitude of positive
feedbacks in climate system (Moon and Wettlaufer, 2017).

The interannual variability of the height of the tropical tropopause is $50$ m
(Randel et al., 2000), and has not shown an increase sufficient to change the
size of the Hadley cell (Seidel and Randel, 2006). Therefore, the influence of
global warming on the size of the Hadley cell may be reflected through the
eddy diffusivity $D$ and the response time scale $\tau$. It is known that mid-
latitude baroclinic eddies can control the size and intensity of the Hadley
cell (Walker and Schneider, 2006; Levine and Schneider, 2015), and these
eddies are generated by jet stream instabilities (Pierrehumbert and Swanson,
1995). Thus, there is substantial baroclinic eddy activity of the subtropical
jet near the edge of the Hadley cell.

Therefore, the eddy diffusivity (a) represents the role of synoptic eddies on
the poleward heat transfer; particularly near the poleward edge of the Hadley
cell, and (b) has a magnitude closely related to the planetary baroclinicity,
which is proportional to the meridional potential temperature gradient (Barry
et al., 2002). Although the large-scale atmospheric dynamics of the
subtropical jet is too complicated to be fully described by a constant $D$
with meridional temperature gradient, the overall impact of synoptic eddies on
the Hadley cell may be captured by an eddy diffusivity (North et al., 1981;
Barry et al., 2002).

Figure 2: The change in (a) the position of the Hadley cell $\theta_{H}$, and
(b) the anomalous potential temperature $\eta_{L}(\theta=0)$ at the equator,
as a function of the eddy diffusivity $D$.

Reanalysis data and model simulations show that the mid-latitude eddy heat
flux decreases under the influence of the Arctic Amplification (Chemke and
Polvani, 2020), implying that $D$ changes with global warming, and hence in
the context of our model can lead to a change in the Hadley circulation. We
fix $\tau$ and $R$ and solve (3.1.3) for a range of eddy diffusivities. For
planetary geostrophic motion, Moon et al. (2021) estimated the dimensionless
synoptic eddy diffusivity as $D\simeq 1/5\pi^{2}\sim 0.03$, to explain the
quasi-oscillatory behavior of the baroclinic annular mode.

Here we vary $D$ from $0.001$ to $0.03$ to observe the effect of decreasing
the poleward heat flux in the mid-latitudes under global warming. In Fig. 2
(a) we show that the poleward edge of the Hadley cell, $\theta_{H}$, moves
poleward as $D$ decreases, changing rapidly for small $D$ and asymptoting for
larger values. It is expected that the Hadley cell moves further poleward
during summer when the baroclinicity is weak (Hu et al., 2018). In Fig. 2 (b)
we see that the negative anomalous potential temperature at the equator,
$\eta_{L}(\theta=0)$, becomes more negative as $D$ increases.

Figure 3: For the range of eddy diffusivities $D$, shown in the insets, we
plot (a) the anomalous potential temperature $\eta_{L}=\Theta_{L}-\Theta_{E}$;
(b) the poleward heat flux, from equations (3.1.3) and (32).; (c) upper level,
$u_{L}(z=1)$, and (d) the lower level, $u_{L}(z=0)$, zonal mean zonal wind,
from the tropics to the pole. In the Hadley cell
($0\leq\theta\leq\theta_{H}$), $u_{L}(z=1)$= $u_{M}$, whereas outside of the
Hadley cell it is calculated from the thermal wind balance. The lower level
wind, $u_{L}(z=0)$, is determined from equations (34) and (35).

Fig. 3(a) shows the meridional structure of the anomalous temperature from the
tropics poleward; negative in the tropics, increasing to a positive maximum at
the edge of the Hadley cell, and then slowly decreasing towards the polar
regions. When $D$ is smaller (larger), $\eta_{L}$ at the edge of the cell
becomes larger (smaller) and decreases rapidly (slowly) with increasing
latitude. Fig. 3(b) shows that the poleward heat flux,
$\overline{v_{0}\Theta_{0}}$, is maximal in the middle of the Hadley cell and
decreases in the mid-latitudes. As expected, for smaller $D$ the overall
magnitude of the heat flux is smaller.

Following the thermal wind balance, the meridional structure of the potential
temperature determines the zonal mean zonal wind $u_{L}$. Fig. 3(c) shows the
upper level zonal mean zonal wind $u_{L}(z=1)$, which decreases at the edge of
the Hadley cell as $D$ increases. The surface zonal mean zonal wind,
$u_{L}(z=0)$, shown in Fig. 3(d), reflects the intensity of the Hadley cell
and the magnitude of $\eta_{L}$ at the equator, measured by the magnitude of
the tropical trade winds. As $D$ increases, the surface easterlies in tropics
strengthen. This implies that the increase in the poleward heat flux by
synoptic eddies in mid-latitudes can control the intensity of the Hadley cell.
As the extent of the Hadley cell decreases (increases) due to the increase
(decrease) of the poleward heat flux, the intensity of the cell increases
(decreases).

##  5  Discussion

We have extended the Hadley circulation model of Held and Hou (1980) by
including a simple extratropical thermodynamic treatment that reveals the
linkage between the tropics and the mid-latitudes. In particular, we find that
the poleward heat flux driven by baroclinic instability near the poleward edge
of the Hadley cell can change the dynamical features of the Hadley
circulation, including the meridional extent of the cell.

The role of the poleward heat flux at the edge of the cell can be considered
as follows. In the limit $D\ll 1$, let $\delta\equiv\sqrt{D\tau}$ and consider
the perturbative solution
$\theta_{H}\simeq\theta^{0}_{H}+\delta\theta^{1}_{H}$ of (3.1.3), which leads
to $\theta^{0}_{H}=\left(\frac{5}{3}R\right)^{1/2}$ and $\theta^{1}_{H}=-1$.
This is equivalent to
$\theta_{H}\simeq\left(\frac{5}{3}R\right)^{1/2}-\sqrt{D\tau}$. Therefore, for
$R$ fixed, the deviation is $\Delta\theta_{H}\simeq-\sqrt{D\tau}$.

The poleward heat flux at the edge of the Hadley cell
$\overline{v_{0}\Theta_{0}}|_{\theta=\theta_{H}}$ is proportional to the
degree of baroclinic instability of the subtropical jet. This can be measured
by the growth rate of baroclinic waves, which according to the Eady problem is
$\sigma_{max}=0.31\frac{f}{NH_{e}}\Delta\overline{U}$, where $N$ is the static
stability, $H_{e}$ the tropopause height at the edge of the cell, and
$\Delta\overline{U}$ is the shear of the zonal mean zonal wind. The poleward
heat flux is generated during the growth stage of synoptic waves by baroclinic
instability. Hence,
$\overline{v_{0}\Theta_{0}}|_{\theta=\theta_{H}}\propto\sigma_{max}$, so that
$D\propto\overline{v_{0}\Theta_{0}}|_{\theta=\theta_{H}}\propto 1/H_{e}$.
Therefore, the width of the Hadley cell decreases as $\sqrt{D\tau}$ increases,
and the increase of $H_{e}$ leads to the decrease in $D$, implying that the
width of the Hadley cell increases.

Lu et al. (2007) considered a scaling of $\theta_{H}$ suggested by Held
(2000), wherein one defines the poleward edge of the Hadley cell as the
latitude at which the vertical shear of the zonal wind is baroclinically
unstable, giving $\theta_{H}\propto\left(\frac{NH_{e}}{\Omega
a}\right)^{1/2}$. It is known that $H_{e}$ is highly correlated with the
expansion of the Hadley cell (Lu et al., 2007), and that baroclinic eddy
fluxes are central in determining the extratropical tropopause height and
thermal stratification (Schneider, 2004). Therefore, turbulent synoptic
motions initiated by baroclinic instability due to the vertical shear of the
zonal wind near the edge of the cell are crucial in determining the scaling of
$\theta_{H}$.

Here, we directly estimate the change of the Hadley cell size due to the
degree of baroclinicity of the subtropical jet. As the tropopause height
$H_{e}$ increases and the vertical shear of the jet decreases, the size of the
Hadley cell increases. This is consistent with the scaling used by Lu et al.
(2007), however our scaling comes from an exact extension of the original
theory that includes interaction with the mid-latitudes rather than a
phenomenological conjecture.

Global climate model simulations show that during El Niño events the Hadley
circulation strengthens and contracts toward tropics, whereas under global
warming it weakens and expands poleward (Lu et al., 2008). During El Niño, the
heat is concentrated in the tropics and baroclinicity near the edge of the
Hadley cell increases. The theory described here suggests the same
intensification and contraction. However, under global warming the temperature
in the mid-latitudes increases more than in the tropics, which leads to a
decrease in baroclinicity near the edge of the Hadley cell. Idealized dynamic
core simulations also show that the enhanced surface heat flux in the tropics
leads to the intensification of the Hadley cell (Son and Lee, 2005).
Reanalysis data and climate model simulations consistently show a weakening
and widening of the Hadley circulation due to a decrease in mid-latitude
temperature gradients (Adam et al., 2014). These findings are consistent with
the theory presented here.

Our rationale for neglecting the influence of the eddy momentum flux in the
heat equation is as follows. The steady state planetary-scale equations are

$\displaystyle-\frac{\partial}{\partial\theta}\langle\overline{v_{0}\Theta_{0}}\rangle-\frac{S}{\epsilon^{1/2}}\langle\overline{w_{1}}\rangle-\frac{1}{\tau}\eta_{L}=0.\qquad\text{and}$
(36) $\displaystyle\left(\frac{\partial}{\partial
z}-\frac{1}{H}\right)\langle\overline{w_{1}}\rangle=-\epsilon\frac{\partial^{2}}{\partial\theta^{2}}\langle\overline{u_{0}v_{0}}\rangle,$
(37)

where the eddy momentum flux is represented by the averaged vertical velocity
$\langle\overline{w_{1}}\rangle$, and Eq. (37) is derived from the temporal
and zonal mean of the quasi-geostrophic potential vorticity equation. The
small planetary-scale Rossby number $\epsilon$ arises from the spatial average
of the synoptic derivatives, and
$-\frac{1}{H}=\frac{1}{\rho_{S}}\frac{\partial\rho_{S}}{\partial z}$, with $H$
the density scale height, assumed to be constant (Moon and Cho, 2020).
Consequently, $\frac{S}{\epsilon^{1/2}}\langle\overline{w_{1}}\rangle\sim
O(\epsilon^{1/2})$, and the contribution of the vertical velocity is
asymptotically smaller than that of the eddy heat flux, which is the key point
we develop here, by revisiting the energy flux balance inside the Hadley cell.
Integrating Eqs. (36) and (37) and eliminating the averaged vertical velocity
we obtain

$\displaystyle\frac{1}{\tau}\int_{0}^{1}\int_{0}^{\theta_{H}}\eta_{L}d\theta
dz=-\int_{0}^{1}\langle\overline{v_{0}\Theta_{0}}\rangle|_{\theta_{H}}dz$
$\displaystyle-\epsilon^{1/2}SH\int_{0}^{1}\frac{\partial}{\partial\theta}\langle\overline{u_{0}v_{0}}\rangle|_{\theta_{H}}dz,$
(38)

using $w_{1}=0$ at $z=0$ and $z=1$.

The excess energy in the Hadley cell is transported to the high latitudes,
where we assume that the eddy heat and momentum fluxes become negligible, and
hence we have

$\displaystyle\int_{0}^{1}\int_{0}^{\pi/2}\eta_{L}d\theta dz=0,$ (39)

so that

$\displaystyle\int_{0}^{1}\int_{0}^{\theta_{H}}\eta_{L}d\theta
dz=-\int_{0}^{1}\int_{\theta_{H}}^{\pi/2}\eta_{L}d\theta dz.$ (40)

Because the surplus energy in the tropics is transported to the high
latitudes, the high latitude integrals of $\eta_{L}$ should be positive, and
hence

$\displaystyle\int_{0}^{1}\int_{0}^{\theta_{H}}\eta_{L}d\theta dz<0.$ (41)

Therefore, because (a) the eddy heat flux
$\langle\overline{v_{0}\Theta_{0}}\rangle$ in mid-latitudes generated by
baroclinic instability is positive, implying that synoptic waves are generated
to transfer the surplus of energy to high latitudes, and (b) the eddy momentum
flux changes from positive to negative when transecting the jet stream, so
that $\frac{\partial}{\partial\theta}\langle\overline{u_{0}v_{0}}\rangle<0$,
then

$\displaystyle-\int_{0}^{1}\langle\overline{v_{0}\Theta_{0}}\rangle|_{\theta_{H}}dz$
$\displaystyle<0\qquad\text{and}$
$\displaystyle-\epsilon^{1/2}SH\int_{0}^{1}\frac{\partial}{\partial\theta}\langle\overline{u_{0}v_{0}}\rangle|_{\theta_{H}}dz$
$\displaystyle>0.$ (42)

Taken together with the constraint (41), we have

$\displaystyle\left|\int_{0}^{1}\langle\overline{v_{0}\Theta_{0}}\rangle|_{\theta_{H}}dz\right|>\left|\epsilon^{1/2}SH\int_{0}^{1}\frac{\partial}{\partial\theta}\langle\overline{u_{0}v_{0}}\rangle|_{\theta_{H}}dz\right|,$
(43)

so that the influence of the eddy heat flux on the expansion or contraction of
the Hadley cell is modulated by the eddy momentum flux. This is consistent
with the life cycle of mid-latitude baroclinic waves. The poleward heat flux
generated by baroclinic waves weakens the jet stream, but the eddy momentum
flux induced by wave breaking near the critical latitudes strengthens the jet
stream.

The essential point is that the magnitude of the poleward heat flux at the
edge of the Hadley cell controls its size and intensity. Clearly, the
inequality (43) could be divined from the small $\epsilon$ limit of (38), but
the role of the eddy heat flux is the same. However, a precise calculation of
the size of the Hadley cell requires treating both the eddy momentum flux and
the eddy heat flux.

Figure 4: Schematic showing the impact of changing the poleward eddy heat
flux, $\Delta\text{E}_{\text{eddy}}$, at the edge of the Hadley cell,
$\theta_{H}$. Hemispheric thermal energy conservation implies that
$S_{1}=S_{2}+\Delta\text{E}_{\text{eddy}}$, and the figure shows the changes
in $S_{1}=S_{1}(\Delta\text{E}_{\text{eddy}})$ and
$S_{2}=S_{2}(\Delta\text{E}_{\text{eddy}})$, indicated by the red arrows, as
$\Delta\text{E}_{\text{eddy}}$ (a) increases or (b) decreases (see text for
more discussion). An increase (decrease) in the poleward heat flux
$\Delta\text{E}_{\text{eddy}}$ leads to an increase (decrease) in
$S_{1}-S_{2}$, thereby strengthening (weakening) tropical convection and
leading to an equatorward (poleward) shift of the edge of the Hadley cell,
$\theta_{H}$. Note, for ease of demonstration the profiles of potential
temperature are linear in latitude $\theta$, but, although they are not
monotonic, as seen in figure 3(a), the essential behavior shown above is the
same.

Our simple extension of the Held and Hou (1980) theory incorporates the heat
transfer enhanced by synoptic eddies near the edge of the Hadley cell. The
original theory relies on the equal area constraint of equation (16),
rewritten here as

$\displaystyle\int_{0}^{\theta_{H}}\eta_{L}d\theta=0.$ (44)

As noted above, this constraint assumes that the Hadley cell is in quasi-
isolation. However, if synoptic eddies transport heat out of the Hadley cell
to mid-latitudes, we can modify the constraint as

$\displaystyle-\frac{1}{\tau}\int_{0}^{\theta_{H}}\eta_{L}d\theta=\Delta\text{E}_{\text{eddy}},$
(45)

where, $\Delta\text{E}_{\text{eddy}}$ is the synoptic eddy driven heat flux
from the Hadley cell to the mid-latitudes, including contributions from
adiabatic warming associated with upper level eddy momentum flux.

In the tropics, robust tropical convection transports an excess of heat
towards higher latitudes, resulting in a negative anomaly,
$\eta_{L}=\Theta_{L}-\Theta_{E}$, as shown in Fig. 3(a). Thus, a tropical
region in a latitude range from $0$ to $\theta^{*}$ has positive
$-\frac{1}{\tau}\eta_{L}$. Let
$S_{1}=-\frac{1}{\tau}\int_{0}^{\theta^{*}}\eta_{L}d\theta$,
$-S_{2}=-\frac{1}{\tau}\int_{\theta^{*}}^{\theta_{H}}\eta_{L}d\theta$, and
$-S_{3}=-\frac{1}{\tau}\int_{\theta_{H}}^{\pi/2}\eta_{L}d\theta$, where
$S_{1}$, $S_{2}$, and $S_{3}$ are all positive. Hemispheric energy
conservation implies that $S_{1}=S_{2}+S_{3}$, which is equivalent to an equal
area constraint across the entire hemisphere:

$\displaystyle-\frac{1}{\tau}\int_{0}^{\frac{\pi}{2}}\eta_{L}d\theta=0,$ (46)

thereby establishing that $S_{3}=\Delta\text{E}_{\text{eddy}}$.

The consequences of the variation of the turbulent heat transport at the edge
of the Hadley cell are shown in Fig. 4. When $\Delta\text{E}_{\text{eddy}}$
decreases (red arrows), $S_{3}$ and thus $S_{1}-S_{2}$ decrease. This
reduction is associated with the weakening of tropical convection and the
associated anomalous potential temperature, and a poleward shift in the Hadley
cell associated with an increase in $\theta_{H}$. By parity of reasoning, the
opposite scenario is realized when $\Delta\text{E}_{\text{eddy}}$ increases.

##  6  Conclusion

The key dynamical features of the Hadley cell, including its meridional
extent, the intensity of sub-tropical jets, and the tropical trade winds, are
all influenced by unstable baroclinic waves in the mid-latitudes. The basic
dynamical constraint for the Hadley circulation is angular momentum
conservation. This constraint introduces a departure of the potential
temperature from radiative-convective equilibrium near the poleward edge of
the Hadley cell, which is used to drive large-scale atmospheric dynamics in
the mid-latitudes. In particular, this departure drives baroclinic instability
near the poleward edge of the Hadley cell, which results in poleward heat flux
by synoptic eddies and is treated with a thermal eddy diffusivity $D$.

We have extended the Hadley circulation model of Held and Hou (1980) to
include the interaction between the tropics and the mid-latitudes. Their model
scaled the meridional size of the Hadley cell using the thermal Rossby number
$R$. Numerical simulations, however, reveal that mid-latitude baroclinic
synoptic eddies influence the dynamical structure and meridional extent of the
Hadley cell. The simple theory presented here captures this behavior.

A central question concerns the impact of global warming on the response of
the Hadley circulation. Our simple theory shows the importance of large-scale
mid-latitude atmospheric dynamics. Because the high-latitudes warm faster, the
mean baroclinicity in the mid-latitudes is impacted non-uniformly. In this
context, recently observed changes in large-scale atmosphere dynamics may
reflect stronger interactions between the tropics and the mid-latitudes. This
highlights the importance of having a unified theoretical framework linking
the dynamics between these regions.

###### Acknowledgements.

We gratefully acknowledge support from the Swedish Research Council Grant No.
638-2013- 9243. No data have been used in this publication apart from the
simple solutions plotted in Figures 2 and 3.

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