# Upwards Tropospheric Influence on Tropical Stratospheric Upwelling
Under peer review

###### Abstract

The steady response of the stratosphere to a sea surface temperature (SST)
forcing is considered in two separate theoretical models. It is first shown
that anomalies in SST impose a geopotential anomaly at the tropopause.
Solutions to the linearized quasi-geostrophic potential vorticity equations
are then used to show that the vertical length scale of a tropopause
geopotential anomaly is initially shallow, but significantly increased by
diabatic heating from radiative relaxation. This process is a quasi-balanced
response of the stratosphere to tropospheric forcing. A previously developed,
coupled troposphere-stratosphere model is then introduced and modified.
Solutions under steady, zonally-symmetric SST forcing in the linear
$\beta$-plane model show that the upwards stratospheric penetration of the
corresponding tropopause geopotential anomaly is controlled by two non-
dimensional parameters, (1) a dynamical aspect ratio, and (2) a ratio between
tropospheric and stratospheric drag. The meridional scale of the SST anomaly,
radiative relaxation rate, and wave-drag all significantly modulate these non-
dimensional parameters. Under Earth-like estimates of the non-dimensional
parameters, the theoretical model predicts stratospheric temperature anomalies
2-3 larger in magnitude than that in the boundary layer, approximately in line
with observational data. Using reanalysis data, the spatial variability of
temperature anomalies in the troposphere is shown to have remarkable coherence
with that of the lower-stratosphere, which further supports the existence of a
quasi-balanced response of the stratosphere to SST forcing. These findings
suggest that besides mechanical and thermal forcing, there is a third way the
stratosphere can be forced – through the tropopause.

Upwards motion in the tropical stratosphere, the layer of atmosphere above
where most weather occurs, is thought to be controlled by weather disturbances
that propagate upwards and dissipate in the stratosphere. The strength of this
upwards motion is important since it sets the global distribution of ozone. We
formulate and use simple mathematical models to show the vertical motion in
the stratosphere can also depend on the warming in the troposphere, the layer
of atmosphere where humans live. We use the theory as an explanation for our
observations of inverse correlations between the ocean temperature and the
stratosphere temperature. These findings imply that stratospheric cooling may
be tightly coupled to ocean warming.

##  1 Introduction

The Brewer-Dobson circulation (BDC) is a global-scale overturning circulation
in the stratosphere, characterized by air that ascends into and within the
tropical stratosphere, spreading poleward and eventually downwards in the
extratropical winter-hemisphere. This stratospheric circulation can
significantly impact tropospheric climate, most notably through its modulation
of the distribution of stratospheric ozone, which absorbs harmful ultraviolet
radiation from the sun (Dobson, 1956). The widely accepted mechanism that
explains the existence of the BDC is the principle of “downward control”
(Haynes and McIntyre, 1987; Haynes et al., 1991). This principle states that
for steady circulations, the upward mass flux across a specified vertical
level is solely a function of the zonal momentum sources and sinks above that
level; thus, processes in the middle and upper stratosphere can exert a
“downward” influence on flow in the lower stratosphere and troposphere. In the
stratosphere and mesosphere, it is primarily the dissipation of upward
propagating Rossby and gravity waves that contributes zonal momentum (Seviour
et al., 2012). The theoretical findings of Haynes et al. (1991) have been well
supported by numerical modeling evidence and withstood the test of time
(Butchart, 2014, and references therein). Thus, in the “downward control”
paradigm, wave dissipation drives the circulation.

The BDC is typically separated into two branches: a slow and deep equator-to-
pole overturning branch, and a faster shallow branch in the lower stratosphere
extending to about 50∘ latitude (Plumb, 2002; Birner and Bönisch, 2011). The
deep branch is thought to be driven by planetary scale waves breaking in the
middle and upper portions of the stratosphere, a process also known as the
extratropical pump (Holton et al., 1995). The shallow branch is thought to be
driven by sub-tropical wave-dissipation in the lower stratosphere (Plumb and
Eluszkiewicz, 1999; Plumb, 2002). In this study, we focus primarily on the
shallow branch circulation, and its connections to the tropical troposphere.

In our opinion, there are a few characteristics of the shallow branch
circulation that remain unresolved. First, calculations of residual vertical
velocities at 70-hPa indicate off-equator maxima in shallow branch upwelling
in the summer-time hemisphere (Randel et al., 2008; Seviour et al., 2012).
Even though wave-drag can force circulations non-linearly and non-locally,
wave-drag is at its annual maximum in the winter hemisphere, which is thus at
odds with the observation of tropical upwelling maximizing in the summer-time
hemisphere (Holton et al., 1995; Plumb and Eluszkiewicz, 1999). In fact, all
of the experiments performed in Plumb and Eluszkiewicz (1999, hereafter, PE99)
showed that as long as wave-drag maximizes in the winter hemisphere, upwelling
maximizes in the winter hemisphere. Only when thermal forcing was included,
did PE99 observe upwelling maximizes in the summer hemisphere. PE99 also found
that the existence of a thermally-forced circulation in the stratosphere and
the breakdown of downward control theory go together. This led PE99 to
question the generality of downward control in the deep tropics, and whether
or not thermally forced tropospheric circulations, such as the Hadley cell,
could penetrate upwards into the stratosphere. Since the Hadley circulation is
closely tied to the meridional gradient of sea surface temperature (SST)
(Emanuel, 1995), the connection between tropospheric warming and lower
stratospheric upwelling is one that perhaps deserves attention.

If tropospheric warming and stratospheric upwelling are connected, then of
particular importance is the tropical tropopause layer (TTL), which serves as
a boundary between the troposphere and stratosphere (Fueglistaler et al.,
2009). Much research has focused on the temperature in the TTL region, since
it has been linked with the concentration of water vapor in the stratosphere
(Jensen and Pfister, 2004; Fueglistaler et al., 2005; Randel et al., 2006;
Randel and Park, 2019). In the tropical stratosphere, upwelling strength is
strongly correlated with temperature, since a cold anomaly that slowly varies
in time must be maintained by adiabatic cooling against the effect of
radiative heating. Indeed, observational data suggests a strong link between
the two (Randel et al., 2006; Kerr-Munslow and Norton, 2006). Via downward-
control arguments, wave-dissipation has been historically linked with
tropopause temperature. For instance, an annual cycle in sub-tropical wave-
dissipation of equatorward propagating extra-tropical waves has been suggested
as responsible for the annual cycle in tropical tropopause temperature (which
is much larger in amplitude than that of the tropical troposphere) (Yulaeva et
al., 1994; Holton et al., 1995; Randel et al., 2002; Taguchi, 2009; Garny et
al., 2011; Kim et al., 2016). Other studies have also attempted to understand
how waves originating in the tropics (which can be excited by deep convection)
can explain various aspects of the tropopause region, including the annual
cycle in temperature (Boehm and Lee, 2003; Norton, 2006; Randel et al., 2008;
Ryu and Lee, 2010; Ortland and Alexander, 2014; Jucker and Gerber, 2017). In
this view, the strength of upwelling in the lower stratosphere is the primary
control on temperature near the tropopause.

Changes to the tropopause temperature could theoretically induce changes in
shallow branch upwelling, though a corresponding, self-consistent change in
the momentum budget must also occur to balance the changes in the meridional
circulation (Ming et al., 2016a). In the tropics, many observational studies
have found that, on a variety of space and time scales, strong cold anomalies
often occur above regions of deep convection – in essence, tropopause cooling
is associated with tropospheric heating on the local and regional scale
(Johnson and Kriete, 1982; Gettelman et al., 2002; Holloway and Neelin, 2007;
Kim and Son, 2012; Virts and Wallace, 2014; Kim et al., 2018). Some studies
have argued that convection has a limited influence on the tropopause
temperature since convection rarely penetrates the tropopause (Folkins et al.,
1999; Gettelman et al., 2002). However, other studies have suggested that
convection has a strong control on tropopause temperature, despite the rarity
of tropopause-penetrating convection (Dessler, 2002; Kuang and Bretherton,
2004). Still, there is an oft-observed link between tropopause cooling and
deep convection. In our view, there exists two theories that specifically
explain this association. Holloway and Neelin (2007) use a simple 2-D,
linearized Boussinesq model to show that a convective “cold-top” forms via
“hydrostatic adjustment” to convective heating. There is no dependence of the
temperature anomaly on the horizontal scale in this theory. In contrast, it
has also been argued that deep convection can excite a large-scale Kelvin wave
response, which also has a vertically tilted signature of tropopause cooling
(Kiladis et al., 2001; Randel et al., 2003). Most of these observational
studies, however, focus on time scales much faster than that of the Brewer-
Dobson circulation. But, there is also remarkable spatial correlation between
tropospheric warming and stratospheric cooling trends on global warming time
scales [see Fig. 1 of (Fu et al., 2006)].

If one persists with the assumption that the same mechanism responsible for
local and regional scale anti-correlations between tropospheric warming and
tropopause cooling can manifest itself at the global scale (which is not a
given), then it is perhaps unsurprising that there also exists a tight
coupling between tropospheric warming and the BDC shallow branch mass flux, at
least when using SST to characterize the tropical troposphere. In general
circulation models (GCMs) and re-analyses, there are strong correlations
between tropical-mean SST and the BDC shallow branch mass flux, across a wide
variety of time scales (Lin et al., 2015; Orbe et al., 2020; Abalos et al.,
2021). Fluctuations in tropical stratospheric upwelling have also been tied to
ENSO (El Niño Southern Oscillation), one of the dominant sources of
interannual tropical SST variability (Randel et al., 2009). In fact,
interannual variations in tropical mean SST explain 40-50% of the interannual
variability of the 70-hPa vertical mass flux (Lin et al., 2015; Abalos et al.,
2021). In addition, nearly 70% of the CMIP6 model spread in the long-term
trend of shallow branch mass flux is explained by the spread in tropical
warming (Abalos et al., 2021).

The tight coupling between tropical SST and BDC shallow branch upwelling on
interannual to climate change time scales has been explained through changes
to the wave-drag, in light of the downward-control paradigm: surface warming
leads to upper tropospheric warming and modification of the sub-tropical jets,
which can alter the upwards propagation and dissipation of mid-latitude waves
in the sub-tropics (Garcia and Randel, 2008; Calvo et al., 2010; Shepherd and
McLandress, 2011; Lin et al., 2015). While these theories (that are based on
changes to zonal-mean wave-drag) can explain how SST and shallow branch mass
flux are correlated, they were not constructed to also explain the oft-
observed local-scale anti-correlation between SST and tropopause temperature.

In this study, we will put forth an alternative explanation for the anti-
correlation between tropospheric and lower stratospheric temperature, and also
attempt to understand the degree to which zonally-symmetric tropospheric
heating can directly influence upwelling in the lower stratosphere. To start,
consider the simplified atmospheric state shown in Figure 1, which has a
troposphere in radiative convective equilibrium, with an overlying
stratosphere at rest. Suppose we impose a steady patch of positive SST anomaly
in the ocean. The increased surface enthalpy flux warms the troposphere,
following a moist adiabat. The surface pressure falls, and the geopotential at
the tropopause rises. Since there cannot be a pressure discontinuity across
the tropopause, the pressure must also rise in the lower stratosphere. How far
up does it extend, and what is the steady response in the stratosphere?

Figure 1: Schematic of a troposphere in radiative-convective-equilibrium, with
an overlying stratosphere that is at rest. The troposphere is forced with a
steady warm SST anomaly in the ocean. The troposphere warms (indicated by
color shading) following a moist adiabat, the surface pressure falls, and the
geopotential rises at the tropopause. How does the stratosphere respond to the
an imposed tropopause geopotential anomaly?

Section 2 tries to answer this conceptual question by introducing the concept
of SST forcing at the tropopause and building a zonally asymmetric framework
to understand the processes that control the upwards extent of tropopause
anomalies. It is shown that there is a quasi-balanced response of the
stratosphere to tropospheric forcing. Section 3 uses a steady, coupled
troposphere-stratosphere system to show how zonally symmetric SST anomalies
can influence tropical upwelling. Section 4 uses reanalysis data to argue for
the real-world presence of the processes described in the proposed theory.
Section 5 concludes the study with a summary and discussion.

##  2 Stratospheric Response to a Tropopause Anomaly

In this section, we introduce a simple conceptual model that will (1)
illuminate how SST forcing can induce a tropopause geopotential anomaly, and
(2) understand what parameters modulate the upwards extent of the tropopause
anomaly into the stratosphere.

To understand how the stratosphere could be forced by the troposphere, we
begin with tropospheric dynamics. In radiative-convective equilibrium, a valid
approximation is that of strict convective quasi-equilibrium, where the
saturation moist entropy, $s^{*}$, is constant with height (Emanuel, 1987;
Emanuel et al., 1994). Emanuel (1987) showed that linearized geopotential
perturbations are directly connected to linearized $s^{*}$ perturbations (note
here, for simplicity, we have ignored the small effect of water vapor on
density):

$\frac{\partial\phi^{\prime}}{\partial p}=-\bigg{(}\frac{\partial T}{\partial
p}\bigg{)}_{s^{*}}s^{*\prime}$ (1)

where prime superscripts indicate perturbation quantities. Since $s^{*}$ is
constant with height, Eq. 1 can be directly integrated in pressure to yield
(as also shown in Lin and Emanuel (2022)):

$\phi^{\prime}(p)=\phi^{\prime}_{b}+s^{*\prime}(\overline{T}_{b}-\overline{T}(p))$
(2)

where $\phi^{\prime}_{b}$ is the perturbation boundary layer geopotential,
$\overline{T}$ is the basic state temperature, and $\overline{T}_{b}$ is the
basic state boundary layer temperature. We non-dimensionalize according to:

$\phi\rightarrow H^{2}N^{2}\phi\qquad
s^{*}\rightarrow\frac{H^{2}N^{2}}{\overline{T}_{b}-[\overline{T}]}s^{*}$ (3)

where $H$ is the scale height , $N^{2}$ is the buoyancy frequency, and
$[\overline{T}]$ is the basic state vertically-averaged temperature. Dropping
primes for perturbation quantities and non-dimensionalizing yields:

$\phi(p)=\phi_{b}+\big{(}1-V_{1}(p)\big{)}s^{*}$ (4)

where $V_{1}$ is the non-dimensional first baroclinic mode (Lin and Emanuel,
2022):

$V_{1}(p)=\frac{\overline{T}(p)-[\overline{T}]}{T_{b}-[\overline{T}]}$ (5)

Eq. 5 shows that the first baroclinic mode is positive near the surface,
transitions to zero in the mid-troposphere, and is negative at the tropopause.
Evaluating Eq. 4 at the tropopause yields:

$\phi(\hat{p}_{t})=\phi_{0}-V_{1}(\hat{p}_{t})s^{*}$ (6)

where $\hat{p}_{t}$ is the non-dimensional tropopause pressure, and
$\phi_{0}=\phi_{b}+s^{*}$ is the barotropic geopotential. Note, the barotropic
geopotential is constant with height. The total geopotential is the linear sum
of the contributions of the tropospheric barotropic and baroclinic
geopotential.

Since the tropopause is colder than the mean troposphere temperature,
$V_{1}(\hat{p}_{t})$ is negative, such that for positive SST anomalies
($s^{*\prime}>0$), the tropopause geopotential anomaly will also be positive,
provided the barotropic geopotential is not less than
$V_{1}(\hat{p}_{t})s^{*}$. In the real atmosphere, baroclinic perturbations
are typically around an order of magnitude larger than barotropic ones (Lin
and Emanuel, 2022), such that for the sake of simplicity, we proceed with the
approximation that $\phi_{0}$ is small in relation to the baroclinic term. We
will relax this assumption in the next section. Then, in this simple
conceptual framework, we have a warm patch of ocean that imposes a steady
positive geopotential anomaly at the tropopause.

Next, we will consider what happens to the stratosphere subject to a steady
tropopause forcing (i.e. a steady lower boundary condition). The response of
the stratosphere to external forcing has been well-studied using theoretical
models [see Garcia (1987); Haynes et al. (1991); Plumb and Eluszkiewicz
(1999), among many others]. However, the external forcing is typically
presented in terms of being mechanical (wave-driven) or thermal in origin. We
instead impose a tropopause forcing via the SST anomaly, and use the well-
known quasi-geostrophic potential vorticity equations (QGPV), linearized about
a resting basic state on an f-plane:

$\displaystyle q^{\prime}(x,y,z)$
$\displaystyle=\frac{1}{f_{0}}\nabla_{H}^{2}\phi^{\prime}+\frac{f_{0}}{N^{2}}\frac{\partial^{2}\phi^{\prime}}{\partial
z^{2}}-\frac{f_{0}}{HN^{2}}\frac{\partial\phi^{\prime}}{\partial z}$ (7)

where $q$ is the potential vorticity (PV), $f_{0}$ is the Coriolis parameter,
$N$ is the buoyancy frequency, $\phi$ is the geopotential. Here, we are
considering perturbations large enough in scale for the quasi-geostrophic
approximation to apply. Dropping primes for perturbation quantities, assuming
wave-like solutions in the zonal and meridional [$\exp(ikx+ily)$], and non-
dimensionalizing by:

$\displaystyle\begin{split}&x\rightarrow Lx\qquad y\rightarrow Ly\qquad
z\rightarrow Hz\\\ &\phi\rightarrow H^{2}N^{2}\phi\qquad q\rightarrow
f_{0}q\qquad t\rightarrow t/f_{0}\end{split}$ (8)

where $L=NH/f$ is the Rossby radius of deformation, we obtain

$\Big{(}\frac{\partial^{2}}{\partial z^{2}}-\frac{\partial}{\partial
z}-(k^{2}+l^{2})\Big{)}\phi=q(z)$ (9)

These equations can be found in most standard textbooks, e.g. section 5.4 of
Vallis (2017). Here, we emphasize the boundary conditions:

$\displaystyle\phi(z=0)$ $\displaystyle=\phi_{T}$ (10)
$\displaystyle\frac{\partial\phi}{\partial z}(z=\infty)$ $\displaystyle=0$
(11)

where the bottom boundary condition enforces continuity of pressure across the
tropopause, given the aforementioned tropopause geopotential anomaly that is
imposed by an SST anomaly. The upper boundary condition requires the
temperature anomaly (or vertical velocity anomaly) be zero. Though $\phi_{T}$
is imposed by the troposphere, via Eq. 6, in reality, barotropic motions are
coupled to the stratosphere. Thus, we can only assume the geopotential as a
steady lower boundary condition, and solve for the stratosphere in isolation,
since we ignored the barotropic geopotential. As shall be illuminated in the
next section, the barotropic mode should really be coupled to the
stratospheric circulation.

We proceed by considering the stratospheric response to a geopotential anomaly
at the tropopause, with zero perturbation PV throughout the rest of the
stratosphere. Since imposing a geopotential anomaly at the tropopause has no
direct effect on stratospheric PV, it can be considered as the fast
stratospheric response to a tropopause geopotential anomaly. In this textbook
case, the solution is straightforward:

$\phi(z)=\exp\big{(}m_{-}z\big{)}$ (12)

where

$m_{-}=\frac{1-\sqrt{1+4(k^{2}+l^{2})}}{2}$ (13)

which shows that the geopotential anomaly decays in the vertical with a scale
inversely proportional to the horizontal scale of the anomaly. On re-
dimensionalization, the Rossby penetration depth,

$R_{d}=\frac{f_{0}L}{N}$ (14)

where $L$ is the horizontal scale, is the operative vertical scale of the
geopotential. Tropopause anomalies with large horizontal scales will extend
deeper into the stratosphere than smaller ones.

The temperature anomaly, scaling with $\frac{\partial\phi}{\partial z}$, will
also decay exponentially with height according to $R_{d}$. But how large can
the temperature anomalies get? Thermal wind balance dictates that

$g\frac{\partial\ln T}{\partial y}=-f\frac{\partial u}{\partial z}$ (15)

If we take $\partial z$ to scale as the Rossby penetration depth, then we
obtain:

$\ln T\approx\frac{Nu}{g}$ (16)

Note that $f$ drops out, which indicates that the temperature in the
stratosphere does not directly depend on $f$. It rather depends on the
magnitude of the tropopause anomaly, as well as the stratospheric
stratification. For the case of zero perturbation PV in the stratosphere, the
temperature anomaly is just the geopotential anomaly multiplied by $m_{-}$,
which is inversely proportional to the horizontal scale of the tropopause PV
anomaly. Therefore, the magnitude of the tropopause temperature perturbations
can be large for small horizontal scale anomalies, though these will be
confined to a rather shallow vertical layer near the equator (and may also not
obey the quasi-geostrophic approximation).

Next, it is instructive to consider how the stratosphere responds to the
temperature anomalies. As alluded to earlier, temperature anomalies disturb
the radiative equilibrium of the stratosphere. This must be associated with
radiative heating anomalies. In this case, PV is no longer conserved. The
response of the stratosphere can be modeled as:

$\frac{\partial q}{\partial
t}=\frac{f_{0}}{N^{2}}\frac{\partial\dot{Q}}{\partial z}$ (17)

where $\dot{Q}$ is the heating rate (thermal forcing), and is parameterized to
be a simple Newtonian radiative relaxation:

$\dot{Q}=-\alpha_{\text{r}}\frac{\partial\phi}{\partial z}$ (18)

$\alpha_{\text{r}}>0$ is the inverse time scale of the Newtonian radiative
relaxation. Hitchcock et al. (2010) found that linear radiative relaxation can
explain around 80% of the variance in longwave heating rates in a climate
model, though this is less accurate in the lower stratosphere, and dependent
on the relaxation rate having a height-dependence. Non-dimensionalizing using
Eq. 8, we obtain:

$\frac{\partial q}{\partial t}=-\gamma\frac{\partial^{2}\phi}{\partial z^{2}}$
(19)

where $\gamma=\alpha_{\text{rad}}/f_{0}$.

The effect of radiative damping on stratospheric circulations has been
thoroughly explored in a number of early theoretical studies (Garcia, 1987;
Haynes et al., 1991; Haynes and Ward, 1993). In particular, the seminal work
of Haynes et al. (1991) showed that in zonally symmetric, radiatively damped,
time-dependent systems whereby a steady mechanical forcing is instantaneously
applied, there is an adjustment to a barotropic state (in $u$) above the level
of forcing. Our set up is similar to the model outlined in section 3 of Haynes
et al. (1991), except here the steady forcing is restricted to the tropopause
geopotential – the forcing is neither wave-driven nor thermal in origin.

Figure 2: The geopotential associated with (red) a boundary PV anomaly of
$q=-1$ ($\phi_{b}$), (blue) a constant PV anomaly of $q=-1$ in the interior
($\phi_{q}$), and (yellow) the sum of the two ($\phi=\phi_{q}+\phi_{b})$. The
corresponding total PV is shown in purple. Here we assume $k_{m}=2$, and
$z_{\text{top}}=1+2\pi$.

To solve for the geopotential, the Green’s function (see the Appendix) is
convoluted with the source term under the lower boundary condition:

$q_{T}=-k_{m}\phi_{T}$ (20)

where $k_{m}=k^{2}+l^{2}$ is the total wavenumber. This can be calculated
numerically (see the Appendix for more details). Figure 2 shows the
stratospheric geopotential solutions that describe the initial and final
states after imposing a tropopause geopotential anomaly. The initial
geopotential distribution from the steady geopotential anomaly is shown as
$\phi_{b}$, and is just the zero interior perturbation PV solution mentioned
earlier in the text, where the response decays exponentially with height. The
geopotential distribution associated with the generation of anomalous PV
through diabatic heating by radiative relaxation is shown in $\phi_{q}$, while
the total geopotential is shown as $\phi=\phi_{q}+\phi_{b}$. The total
geopotential is constant with height (barotropic) above the level of forcing,
as found by Haynes et al. (1991).

A simple physical picture is painted with this conceptual model that can
provide an rather straightforward answer to the schematic shown in Figure 1.
If the troposphere is forced with a steady positive SST anomaly, a positive
geopotential anomaly forms at the tropopause. A positive tropopause
geopotential anomaly is initially accompanied with a cold anomaly in the
stratosphere, which is associated with radiative heating and rising motion. If
this process is allowed to proceed towards a steady state back to radiative
equilibrium, the geopotential and PV must eventually become constant with
height (i.e. barotropic), as implied by Eq. 18. In this way, the troposphere
can force the stratosphere, at least on the steady time scales considered
here. This also shows that the geopotential does not have to go to zero at the
upper boundary. The only requirement is that the energy density goes to zero.
Thus, the assumption of the geopotential going to zero at the upper boundary
in Holloway and Neelin (2007) seems arbitrary.

How long does it take to reach the barotropic state? Haynes et al. (1991)
showed that in the zonally symmetric case, the adjustment towards a barotropic
state above the level of forcing occurs with an upward propagation speed of
$w_{\alpha}=\alpha_{\text{rad}}R_{d}^{2}/H_{s}$. In the tropics, $w_{\alpha}$
is small, owing to the smallness of both $\alpha_{\text{rad}}$ and $R_{d}$.
For an anomaly of horizontal scale around 5000 km at a latitude of
$10^{\circ}$, and a radiative relaxation time scale of
$\alpha_{\text{rad}}=20$ days-1, $w_{\alpha}\approx O(10^{-1})$ mm s-1. This
corresponds to an upward propagation of only a few km per year. It is also
possible to numerically calculate the amount of time it takes for the system
to reach its final barotropic state, by time-stepping Eq. 19 forwards in time
while holding the lower-boundary PV fixed. For a stratosphere with a depth of
around 32-km ($z_{\text{top}}=4$ for a scale height of $H_{s}=8$ km), assuming
$\gamma=0.02$ and a Coriolis parameter akin to that at 10∘ latitude, it takes
around 3 years for the system to become barotropic.

This long relaxation time makes it unlikely that the barotropic state is ever
reached in the real stratosphere, since unsteady processes can disrupt the
simple state assumed in this model. For instance, it is unlikely that a
tropopause geopotential anomaly would remain steady on the order of years.
Furthermore, since the $\beta$-effect is not included in this simple
framework, we also ignore the possibility of the excitation of large-scale
waves (and their corresponding effects) as a part of the response to the
tropopause geopotential anomaly.

Figure 3: (Left) The diabatic heating profile ($Q/\alpha_{r}$) with height in
the stratosphere after 30 days of integration, subject to a steady tropopause
boundary forcing with a horizontal scale of around $28000$-km, and a (blue)
5-day, (red) 20-day, (yellow) 40-day. The vertical derivative of the
geopotential for the zero-PV stratospheric response to a tropopause forcing
(infinite radiative relaxation time scale) is shown in black. (Middle) and
(Bottom) are the same as top but for a horizontal scale of around $9500$-km,
and $4500$-km, respectively. We assume a scale height of $8$ km, and a
tropopause height of 16 km to convert to dimensional height.

In light of this, the intermediate states between the fast stratospheric
response [$\phi_{b}$ in Figure 2] in which the anomaly decays exponentially
with height, and the barotropic steady-state response in which the boundary
anomaly is communicated throughout the depth of the stratosphere [$\phi$ in
Figure 2], could be important. For practical purposes, the geopotential
anomaly is not as important as the associated radiative heating, which is
potentially important for tracer transport into the stratosphere. Figure 3
shows the non-dimensional diabatic heating profiles with height after 30 days
of integration, for a stratosphere subject to an imposed tropopause
geopotential anomnaly that is associated with a unitary non-dimensional
anticyclonic PV, under varying magnitudes of stratospheric radiative
relaxation rates. The diabatic heating profiles are normalized by the
radiative relaxation rate. For comparison purposes, we show the temperature
anomaly associated with the (time-independent) zero perturbation PV
geopotential solution (i.e. an infinite radiative-relaxation time scale), even
though there is no associated diabatic heating, by definition. Figure 3 shows
that after 30-days, there is non-trivial lifting (in height) of the diabatic
heating anomaly over time. The stronger the strength of radiative relaxation,
the faster the diabatic heating anomaly is communicated into the stratosphere.

These calculations show that tropospheric heating imposes a positive
tropopause geopotential anomaly, which elicits a quasi-balanced response in
the stratosphere. The fast stratospheric response is simply an anomaly that
decays in the vertical according to the Rossby penetration depth. On slower
time scales, radiative relaxation induces an upward migration of the anomaly.
The radiative relaxation rate, the horizontal scale of the anomaly, and the
Coriolis parameter all determine the upward migration rate, as shown in Haynes
et al. (1991). Thus, the ensuing, time-dependent temperature response in the
stratosphere is also tied to these parameters. In the next section, we will
elaborate on the ideas put forth in this conceptual model in a zonally-
symmetric framework, and analyze, in detail, the sensitivity of the
stratospheric response to tropospheric forcing, with regards to these
parameters.

##  3 Troposphere-Stratosphere Response to SST

In the previous section, we used a simple QGPV framework to understand how a
SST anomaly can impose a tropopause geopotential anomaly and therefore elicit
a quasi-balanced response in the stratosphere. However, we used the tropopause
as a lower boundary condition for the stratosphere when in reality, the
tropopause and stratosphere are coupled. In this section, we develop a simple,
zonally-symmetric, coupled troposphere-stratosphere model, and explore how
radiation and wave-drag can modulate the response of the stratosphere to SST
forcing.

###  3.1 Model Formulation

Lin and Emanuel (2022) formulated a linear, coupled troposphere-stratosphere
model, but in the context of unsteady equatorial waves. In that linear system,
a convecting, quasi-equilibrium troposphere was coupled to a dry and passive
stratosphere. We use the same non-dimensional system derived in Lin and
Emanuel (2022), except we only consider steady, zonally symmetric
circulations. The tropospheric system is governed by:

$\displaystyle yv_{0}-F(u_{0}+u_{1})$ $\displaystyle=0$ (21)
$\displaystyle-\frac{\partial\phi_{0}}{\partial y}-yu_{0}$ $\displaystyle=0$
(22) $\displaystyle yv_{1}-F(u_{0}+u_{1})-D_{t}u_{1}$ $\displaystyle=0$ (23)
$\displaystyle yu_{1}$ $\displaystyle=\frac{\text{d}s^{*}}{\text{d}y}$ (24)
$\displaystyle\frac{\partial v_{0}}{\partial y}+\frac{\partial v_{1}}{\partial
y}+\frac{\partial\omega}{\partial y}$ $\displaystyle=0$ (25)

where $u_{0}$ and $v_{0}$ are the barotropic zonal and meridional winds
(constant with height), $u_{1}$ and $v_{1}$ are the baroclinic zonal and
meridional winds, $\phi_{0}$ is the barotropic geopotential, $s^{*}$ is the
saturation moist entropy (that is assumed to be vertically constant, as in a
quasi-equilibrium troposphere), $D_{t}$ is a non-dimensional Rayleigh damping
coefficient, and

$F=\frac{aC_{d}|\overline{\textbf{V}}|}{\beta L_{y}^{2}h_{b}}$ (26)

is a non-dimensional surface friction coefficient (derived in Lin and Emanuel
(2022)), where $C_{d}$ is the drag coefficient, $h_{b}$ is the boundary layer
depth, $L_{y}$ is the meridional length scale, $\beta$ is the meridional
gradient of the Coriolis force, $a$ is the radius of the Earth, and
$\overline{\textbf{V}}$ is the basic state surface wind speed magnitude. The
vertical structure of the baroclinic variables are determined by $V_{1}$ (Eq.
5). Note that while there are equations for the tropospheric thermodynamics in
Lin and Emanuel (2022), they are omitted here. Since $s^{*}$ is taken to be
specified, representative of a SST forcing, there are 6 unknown variables,
($u_{0}$, $u_{1}$, $v_{0}$, $v_{1}$, $\omega$, $\phi_{0}$) and 5 equations.
The system will be completed with a formulation of boundary conditions that
will couple the troposphere system to a stratosphere (and provide the last
equation).

In the ensuing text, terms with an overlying hat are dimensional.
$\hat{D}_{t}$, the (dimensional) inverse time scale of the Rayleigh damping
coefficient is:

$\hat{D}_{t}\rightarrow\frac{\beta L_{y}^{2}}{a}D_{t}$ (27)

In Eq. 23, $D_{t}u_{1}$ acts as a relaxational wave drag on the zonal flow. It
does not act on the coupling between the troposphere and stratosphere, and is
only used to diagnose $v_{1}$ (which by definition, has a value of zero at the
tropopause). Thus, $D_{t}$ modulates the baroclinic vertical velocity profile
in the zonally symmetric meridional overturning circulation.

As formulated, the tropospheric system represents an atmosphere in which
temperature anomalies in the vertical are restricted to follow the moist
adiabat. The associated baroclinic mode, which is forced through surface
enthalpy fluxes ($s^{*}$), can then excite the barotropic mode through surface
friction (Lin and Emanuel, 2022). The barotropic mode then excites the
stratosphere. However, the stratospheric circulation becomes uncoupled with
the tropospheric circulation when $F=0$ – in this case, the tropospheric
solution simply obeys Eqs. 23-25, and the barotropic mode (as well as the
stratospheric state to tropospheric forcing) becomes ill-defined. This may
imply that friction has an outsized influence on stratospheric circulations.
This may not be true in reality, since the barotropic mode can also be coupled
to the baroclinic mode through non-linearity and vertical wind shear. Both of
these processes are not represented in this work.

The stratosphere is formulated in log-pressure coordinates and assumed to be
in hydrostatic balance [see Chapter 3 of Andrews et al. (1987)]. The steady,
linear, zonally symmetric, non-dimensional equations of the stratosphere are
also derived from the system used in Lin and Emanuel (2022), and summarized
below:

$\displaystyle yv_{s}-D_{s}u_{s}$ $\displaystyle=0$ (28)
$\displaystyle-\frac{\partial\phi_{s}}{\partial y}-yu_{s}$ $\displaystyle=0$
(29) $\displaystyle\frac{\partial v_{s}}{\partial
y}+\frac{1}{\rho_{s}}\frac{\partial(\rho_{s}w_{s})}{\partial z^{*}}$
$\displaystyle=0$ (30) $\displaystyle
w_{s}S=-\alpha_{\text{rad}}\frac{\partial\phi_{s}}{\partial z}$ (31)
$\displaystyle\rho_{s}=\exp\Big{(}\frac{H}{H_{s,s}}(1-z^{*})\Big{)}$ (32)

where subscripts denote quantities in the stratosphere, $w_{s}$ is the log-
pressure vertical velocity, $S$ is a non-dimensional stratospheric
stratification, $\rho_{s}$ is the basic state density, $H$ is the dimensional
tropopause height, $H_{s,s}$ is the dimensional scale height in the
stratosphere, the log-pressure vertical coordinate
${z^{*}\equiv-H\ln(p/p_{t})+1}$ is defined such that $z^{*}=1$ is the bottom
boundary, or the tropopause, and $\alpha_{\text{rad}}$ is the non-dimensional
radiative damping time scale in the stratosphere:

$\hat{\alpha}_{\text{rad}}\rightarrow\frac{\beta
L_{y}^{2}}{a}\alpha_{\text{rad}}$ (33)

Relaxational wave drag, $D_{s}u_{s}$, is included only in the zonal momentum
equations, as similarly used by Plumb and Eluszkiewicz (1999). It is not
necessary that $D_{s}=D_{t}$, though discontinuities in the meridional
velocity at the tropopause will occur if $D_{s}\neq D_{t}$. Furthermore, while
this form of wave drag is simplistic, it is a rather poor representation of
the response of the circulation to external forces (Ming et al., 2016b).

Finally, $S$ plays an important role in the behavior of this model, and is:

$S=\frac{N^{2}H^{2}}{\beta^{2}L_{y}^{4}}$ (34)

where $N$ is the buoyancy frequency. Note, there is no explicitly imposed
thermal or mechanical forcing in the stratosphere. Thus, we consider a
stratosphere entirely forced from the troposphere.

###  3.2 Stratospheric response to tropopause forcing

In the case of an isolated stratosphere subject to a tropopause forcing, the
stratospheric equations can be reduced to a single differential equation for
the geopotential:

$\frac{\partial^{2}\phi}{\partial
z^{2}}-\frac{H}{H_{s,s}}\frac{\partial\phi}{\partial
z}+\frac{\xi}{y^{2}}\bigg{[}\frac{\partial^{2}\phi}{\partial
y^{2}}-\frac{2}{y}\frac{\partial\phi}{\partial y}\bigg{]}=0$ (35)

where

$\xi=\frac{D_{s}S}{\alpha_{\text{rad}}}=\frac{\hat{D}_{s}}{\hat{\alpha}_{\text{rad}}}\frac{N^{2}H^{2}}{\beta^{2}L_{y}^{4}}$
(36)

is a non-dimensional term that depends on the ratio between the time scale of
wave-drag to that of radiation. This quantity is equivalent to a “dynamical
aspect ratio” that describes the ratio of the vertical to horizontal scale of
the circulation response to an imposed forcing (Garcia, 1987; Plumb and
Eluszkiewicz, 1999; Haynes, 2005; Ming et al., 2016b). As detailed in Ming et
al. (2016b), who incorporated an additional external heating in the
stratosphere, when the aspect ratio is large ($\xi>>1$), the external heating
is narrow and primarily balanced by upwelling, and when the aspect ratio is
small ($\xi<<1$), the external heating is broad and primarily balanced by
Newtonian cooling. In this study, the interpretation of $\xi$ does not have
exactly the same meaning, since we do not impose a temperature-independent
external heating to the system (which in the real world would arise from
absorption of radiation by ozone) – our simple system is instead forced via
the tropopause geopotential, and upwelling always balances Newtonian cooling.
Here, $\xi$ better describes the geopotential response with height. As we
shall see later, when the radiative time scale is much faster than the wave-
drag time scale ($\xi<<1$), the meridional derivative terms are small and the
system will become nearly barotropic in the vertical. On the other hand, when
the wave-drag time scale is much faster than the radiative time scale
($\xi>>1$), the stratospheric signature of the tropopause anomaly is muted.
Note the presence of $L_{y}$, which indicates the importance of the horizontal
scale of the anomaly.

Figure 4: (Top-row) The zonally symmetric geopotential response to an imposed
tropopause geopotential anomaly, as shown in Eq. 38, for varying values of
$\xi$. (Bottom-row) Same as the top-row except for the zonally symmetric
vertical velocity response. The red-line is the zero vertical velocity
isoline. Tropopause height is $16$-km, and stratospheric scale height is
$8$-km.

Eq. 35 can be solved numerically, discretizing the grid in the meridional and
vertical directions. The stratospheric geopotential is also subject to a zero
temperature anomaly at the top of the domain, or equivalently, zero derivative
of the geopotential. The geopotential anomaly is enforced to be zero on the
northern and southern borders. For illustrative purposes, we first solve the
equations under a fixed lower boundary condition:

$\phi(z^{*}=1)=\phi_{T}$ (37)

where

$\phi_{T}=\int_{y}y\exp\big{(}-4(y-2)^{2}\big{)}-y\exp\big{(}-4(y+2)^{2}\big{)}$
(38)

This represents a flat positive geopotential anomaly in the tropics (tropical
heating) that decays to zero in the subtropics. As will become clear later
when the solutions are coupled to the troposphere, this geopotential structure
is associated with sub-tropical jets at $y=\pm 2$.

Figure 4 shows the stratospheric response to a tropopause geopotential
anomaly, under varying values of $\xi$. Here, the numerical calculations
confirm the mathematical analysis. Indeed, for $\xi=0.1$ (i.e. when wave-drag
is very weak), radiation acts to create a nearly barotropic stratosphere, in
which motion is confined to constant angular momentum surfaces. The vertical
structure of the vertical velocity in this case is qualitatively similar to
the thermally forced vertical mode calculated in PE99 [see their Fig. 11].
When the time scale of wave-drag is faster than radiation ($\xi=100$), the
vertical penetration of the tropopause geopotential anomaly is significantly
muted. In fact, the vertical velocity anomalies only extend on the order of a
few km into the stratosphere. In this sense, the relaxational wave-drag acts
to both mute the vertical scale of the tropopause geopotential anomaly, and
sustain a meridional overturning circulation.

As elaborated on earlier, there is much existing theoretical work that shows
the response of the stratosphere to an external forcing is dependent on the
strength of wave-drag, the strength of radiative relaxation, and the aspect
ratio of the tropopause anomaly (Garcia, 1987; Haynes et al., 1991; Ming et
al., 2016b). This work is mathematically similar to and agrees with the
aforementioned studies. Unlike the others, this work emphasizes the role of
tropopause forcing on the stratosphere, and introduces the idea that there is
a quasi-balanced response in the stratosphere to tropopause forcing, via
tropospheric heating.

###  3.3 Tropospheric forcing of stratospheric upwelling

Next, we couple the stratospheric equations to the zonally symmetric
tropospheric equations, to show how tropospheric thermal forcing can influence
stratospheric upwelling. In order to couple the troposphere and stratosphere,
we use classical matching conditions: (1) continuity of pressure
(geopotential) and (2) vertical velocity at the tropopause:

$\displaystyle\phi_{s}(z^{*}=1)$ $\displaystyle=\phi_{T}$ (39) $\displaystyle
B\omega(\hat{p}_{T})$ $\displaystyle=-w_{s}(z^{*}=1)$ (40)

where $B-\frac{H_{s,t}}{H}\frac{p_{s}-p_{t}}{p_{t}}$ is a scaling coefficient
between pressure velocity and vertical velocity (Lin and Emanuel, 2022). Here,
$p_{s}$ is the surface pressure, $p_{t}$ is the tropopause pressure, and
$H_{s,t}$ is the scale height of the troposphere. Solving for $v_{0}$ using
Eqs. 25, 39, 40, and assuming zero flow at the boundaries, yields:

$v_{0}=\frac{\alpha_{\text{rad}}}{SB}\int_{y}\frac{\partial\phi_{s}}{\partial
z}\bigg{|}_{z^{*}=1}\>dy$ (41)

Here we see that under a rigid lid condition, where $S\rightarrow\infty$,
$v_{0}=0$. In addition, $B$ is proportional to the troposphere scale height,
which itself is inversely proportional to the dry stratification of the
troposphere. Hence, $SB$ can also be thought of as a scaled ratio of the
troposphere buoyancy frequency to the stratosphere buoyancy frequency. The
strength of radiative relaxation also appears in the numerator. This is
because the magnitude of the tropospheric barotropic mode is determined, in
part, by stratospheric dynamics.

Eqs. 21 and 24 are used to solve for $u_{0}$ in terms of the stratosphere and
the external forcing:

$u_{0}=y\frac{1}{\xi\gamma}\int_{y}\frac{\partial\phi_{s}}{\partial
z^{*}}\bigg{|}_{z^{*}=1}\>dy-\frac{1}{y}\frac{\text{d}s^{*}}{\text{d}y}$ (42)

where

$\gamma=\frac{FB}{D_{S}}$ (43)

is an additional non-dimensional parameter that qualitatively represents the
ratio between stratospheric and tropospheric drag (there is tropospheric wave
drag, but it does not act on the barotropic mode, only on the baroclinic
mode). $\gamma$ is not entirely independent from $\xi$, since $D_{s}$ appears
in both. Again, under the rigid lid condition, $\xi\rightarrow\infty$, such
that the barotropic zonal wind becomes only a function of the tropospheric
forcing. Note again that when $F=0$, the barotropic mode becomes ill-defined,
since it is no longer coupled to the baroclinic mode.

In order for the continuity of pressure to be satisfied, the geopotential at
the lower boundary of the stratosphere must satisfy Eqs. 6 and 39. Combining
Eqs. 6, 22, 39, and 42 yields:

$\frac{\partial\phi_{s}}{\partial
y}\bigg{|}_{z^{*}=1}\>-y^{2}\frac{1}{\xi\gamma}\int_{y}\Big{(}\frac{\partial\phi_{s}}{\partial
z}\Big{)}\bigg{|}_{z^{*}=1}\>dy=\\\
(1-V_{1}(\hat{p}_{t}))\frac{\text{d}s^{*}}{\text{d}y}$ (44)

which is an equation for the boundary geopotential entirely in terms of the
external forcing, $s^{*}$. The Rayleigh damping coefficient for stratospheric
wave-drag does not appear in the boundary condition, since

$\frac{1}{\xi\gamma}=\frac{\alpha_{\text{rad}}}{D_{s}S}\frac{D_{s}}{FB}=\frac{\alpha_{\text{rad}}}{SFB}$
(45)

When $\xi\gamma$ is large, the boundary condition simply reduces to Eq. 6,
with $\phi_{b}=0$. When $\xi\gamma$ is small, $s^{*}$ becomes a multiple of a
double integral in $y$ of the vertical derivative of the stratospheric
geopotential at the tropopause.

Incorporating Eq. 44 as the lower boundary condition is numerically tricky
given the meridional integral, since it precludes the inversion of a sparse
matrix. The integral can be removed by dividing by $y^{2}$ and differentiating
with respect to $y$, which yields:

$\frac{-2}{y^{3}}\frac{\partial\phi_{s}}{\partial
y}+\frac{1}{y^{2}}\frac{\partial^{2}\phi_{s}}{\partial
y^{2}}-\frac{1}{\xi\gamma}\frac{\partial\phi_{s}}{\partial z}=\\\
(1-V_{1}(\hat{p}_{t}))\Big{(}\frac{1}{y^{2}}\frac{\text{d}^{2}s^{*}}{\text{d}y^{2}}-\frac{2}{y^{3}}\frac{\text{d}s^{*}}{\text{d}y}\Big{)}$
(46)

where the entire equation is evaluated at $z^{*}=1$. This boundary condition
leads to a sparse matrix that can be easily incorporated into a numerical
solver.

Before continuing with the numerical solutions, we formulate the SST forcing
in the troposphere. We observe from Eq. 24 that:

$s^{*}=\int yu_{1}\>dy$ (47)

such that we can specify the baroclinic wind response to obtain a suitable
$s^{*}$ anomaly. Here, we specify:

$u_{1}(y)=-\exp\big{(}-4(y-2)^{2}\big{)}-\exp\big{(}-4(y+2)^{2}\big{)}$ (48)

which is akin to subtropical jets symmetric about the equator. Note, the
meridional baroclinic wind is:

$v_{1}=\frac{F}{y}u_{0}+\frac{D_{t}+F}{y^{2}}\frac{\text{d}s^{*}}{\text{d}y}$
(49)

Numerical evaluation of $v_{1}$ requires that the meridional derivative of
$s^{*}$ go to zero faster than $y^{2}$ in the limit of $y\rightarrow 0$,
otherwise $v_{1}$ will become unstable for small values of $y$ on the
numerical grid. However, the stratospheric solution does not depend on
$v_{1}$, so this constraint merely ensures a smoothly varying tropospheric
circulation. Thus, $u_{1}(y)$ is chosen to satisfy this constraint. We proceed
by numerically solving the stratospheric system (Eq. 35) with the modified
boundary condition shown in Eq. 46, as well as the $s^{*}$ forcing shown in
Eq. 47. See the appendix for more details on the numerical solver.

To set the non-dimensional parameters of the model, we use Earth-like
parameters of $N^{2}~{}=~{}6~{}\times~{}10^{-4}$ s-2, $H=16$ km,
$H_{s,t}~{}=~{}H_{s,s}~{}=~{}8$ km, $\beta~{}=~{}2.3~{}\times~{}10^{-11}$
$s^{-1}~{}m^{-1}$, $L_{y}~{}=~{}1200$ km (such that $y=1$ represents
approximately ten degrees of latitude), $C_{d}~{}=~{}10^{-3}$,
$|\textbf{V}|~{}=~{}3$ m s-1. Furthermore, we choose
$T_{b}~{}=~{}303~{}\>~{}\text{K}$, a surface pressure of 1000-hPa, and a
tropopause pressure of 100-hPa. The vertical temperature profile in the
troposphere follows a pseudoadiabatic lapse rate (neglecting changes to heat
capacity, see Eq. 4.7.5 of Emanuel (1994)), such that
$[\overline{T}]~{}\approx 264.5~{}\>~{}\text{K}$ and
$\overline{T}(p_{t})~{}\approx~{}176.1~{}\>~{}\text{K}$. With these values,
$V_{1}(p_{t})~{}\approx~{}-2.3$.

Figure 5: (Left): The zonally symmetric response to a SST ($s^{*}$) forcing
shown in Eq. (47). Zonal winds are shown in colors (red for westerlies),
contours show vertical motion ($w$), where 5 contours indicates a difference
of one order of magnitude. Arrows show the meridional motion. The tropopause
is shown by the thin gray line. “Earth-like” parameters of $\xi=150$,
$\gamma=30$ are used. (Right): Same as left, but zoomed in on the
stratosphere.

Since $\alpha_{\text{rad}}$ and $\hat{D}_{t}$ play critical roles in the
stratospheric response to an imposed tropopause geopotential anomaly, we will
explore the the non-dimensional space of $\xi$ and $\gamma$. Still, it is
helpful to to note the estimates of the general order of magnitudes of these
quantities in the real stratosphere. Hitchcock et al. (2010) estimated the
radiative relaxation time scale to be approximately 25 days in the lower
tropical stratosphere. The magnitude of the Eliassen Palm (EP) flux divergence
is around O(1) m s-1 $\text{day}^{-1}$ in the subtropics, but decays rapidly
as one moves equatorward into the deep tropics (Randel et al., 2008). For a
perturbation zonal wind speed of O(10) m/s, this corresponds to a Rayleigh
damping rate of around 10 days-1 and slower.

For now, we restrict the analysis to “Earth-like” parameters, with
$\hat{\alpha}_{\text{rad}}=25$ days-1, and
$\hat{D}_{s}~{}=~{}\hat{D}_{t}~{}=~{}25$ days-1. This choice leads to
$\xi\approx 150$ and $\gamma\approx 30$. Thus, $\xi\gamma$ is large, and the
tropopause geopotential can be approximated as simply a multiple of $s^{*}$.
Figure 5 shows the zonally symmetric, linear response to the prescribed,
equatorially symmetric SST forcing. We observe a meridionally shallow,
thermally direct overturning circulation in the troposphere, associated with
sub-tropical jets at $|y|=2$ that decay exponentially with height into the
stratosphere. The tropopause geopotential is elevated in the tropical region
($|y|<2$) (not shown). Associated with this elevated tropopause geopotential
is a weak, meridionally shallow, thermally indirect overturning circulation in
the stratosphere, with upwelling around an order of magnitude smaller than
peak upwelling in the troposphere. Note that the tropospheric thermally direct
overturning circulation in this model is not meant to realistically mimic the
Hadley circulation, since linear models do not capture the dynamics of the
Hadley circulation (Held and Hou, 1980). Rather, its purpose in this model is
to understand how tropopause geopotential anomalies associated with
tropospheric circulations influence the stratospheric circulation.

What is the sensitivity of the stratospheric circulation to
$\hat{\alpha}_{\text{rad}}$? Figure 6a,b shows the vertical profile of
anomalous geopotential and vertical velocity, for varying values of
$\hat{\alpha}_{\text{rad}}$. In all the solutions presented here, the
tropospheric wave drag is fixed. We first observe that for all the solutions,
the geopotential anomaly maximizes at the tropopause, and there is a
significant barotropic geopotential component associated with all of the
solutions. These positive geopotential anomalies decay as one moves upwards
into the stratosphere, but the rate at which they decay is determined by the
aforementioned parameters. When $\hat{\alpha}_{\text{rad}}=1$ day-1, we
observe a slow decay of the tropopause geopotential as one moves upwards into
the stratosphere, and large upwelling values in the lower stratosphere. In
contrast, when radiation is very slow ($\hat{\alpha}_{\text{rad}}=100$ day-1),
there is almost no penetration of the tropospheric vertical velocity into the
stratosphere. This is associated with a tropospheric vertical velocity profile
that is nearly entirely composed of the first baroclinic mode. As expected,
radiative damping plays a large role in the communication of the tropopause
forcing into the stratosphere.

Figure 6: (a) Vertical profiles of non-dimensional geopotential and (b)
vertical velocity, at y = 1.5, for varying values of radiative relaxation, at
a fixed Rayleigh damping (wave drag) of $~{}25$ days-1. Dashed lines show the
geopotential and vertical velocity associated with a pure baroclinic mode
(normalized so that the peak vertical velocity is $0.02$). (c), (d) are the
same as (a), (b), respectively, except for varying values of stratospheric
Rayleigh damping, at a fixed radiative relaxation rate of $25$ days-1.
Tropopause is defined at 16-km, and tropospheric Rayleigh damping is fixed at
$25~{}$ days-1.

The stratospheric response to a steady tropopause geopotential anomaly also
shows a strong dependence to $\hat{D}_{s}$. This is not surprising, given the
criticality of wave-drag in the zonally-symmetric solutions. Figure 6c,d shows
the solutions with varying $\hat{D}_{s}$ and a fixed radiative damping time
scale. The behavior of the coupled solutions are qualitatively similar to that
inferred from the isolated stratosphere solutions, in that faster wave-drag
time scales increase the decay of the tropopause geopotential into the
stratosphere. In addition, faster wave-drag time scales are associated with
increased upwelling in the lower stratosphere, though the differences across
the parameters shown are smaller in magnitude than that when varying the
radiative damping time scale. This result could be a result of the simple
relaxational form of wave-drag used in this study, which does not capture
detailed aspects of wave-forcing (Ming et al., 2016b). Regardless, the
numerical solutions confirm the mathematical analysis, in that both radiative
damping and wave-drag can modulate the stratospheric response to tropospheric
forcing. Note, in a similar linear system, PE99 found solutions to a
stratosphere perturbed through tropospheric thermal forcing that showed
stratospheric upwelling nearly comparable in magnitude to that of the
troposphere, which was deemed as unrealistic. In PE99, the radiative
relaxation time scale was 10 days-1 and the relaxational wave-drag time scale
was 500 days-1, which corresponds to small $\xi$, and large penetration of the
tropospheric circulation into the stratosphere.

Figure 7: (Left): Temperature anomaly right above the tropopause, per degree
of warming in the boundary layer, as a function of the radiative relaxation
and Rayleigh damping (wave-drag) time scales. Rayleigh damping time scale is
fixed in the troposphere and varied in the stratosphere. (Right): Temperature
anomaly right above the tropopause, per degree of warming in the boundary
layer, as a function of the meridional length scale, $L_{y}$ (km), for fixed
Rayleigh damping and radiative relaxation. Ordinate axis is logarithmic.

The vertical shape of the geopotential profiles above the tropopause also
allows for an estimate of the magnitude of the tropopause temperature cold
anomaly as a function of tropospheric heating. Figure 7, left, shows the
temperature anomaly right above the tropopause, per degree of warming in the
boundary layer, as a function of the radiative damping and Rayleigh damping
time scales. In general, the longer the radiative damping time scales, the
larger the temperature anomaly (as pointed out by Randel et al. (2002)). In
addition, there is also a strong dependence of the tropopause temperature
anomaly on the Rayleigh damping time scale: the faster the damping, the larger
the magnitude of the temperature anomaly. It is clear that both the magnitudes
of the Rayleigh damping (wave-drag) and radiative damping play significant
roles in modulating the temperature anomaly above the tropopause.

Interestingly, for “Earth-like” estimates of the time scale of Rayleigh
damping and radiative relaxation (O(10) days-1), the temperature anomalies
just above the tropopause are around 2-3 times the magnitude of the boundary
layer anomalies, slightly larger than what is observed in convecting regions
of the tropical atmosphere (see Fig. 5a in Holloway and Neelin (2007)). This
theory thus provides a scaling argument for the degree of tropopause cooling
that is expected per degree of boundary layer warming. Note that the
derivative of the geopotential is discontinuous across the tropopause in this
model, since we assume a instantaneous transition between quasi-equilibrium
thermodynamics in the troposphere, and dry, passive dynamics in the
stratosphere.

These theoretical results provide a potential explanation for the observed
correlation between tropical-averaged SST anomalies and tropical stratospheric
upwelling (Lin et al., 2015), as well as the anti-correlation between SST and
tropopause temperature (Holloway and Neelin, 2007; Fu et al., 2006). First, an
SST anomaly is communicated throughout the depth of the troposphere through
moist convection. Indeed, observations have found strong positive correlations
between the tropopause geopotential anomaly and the boundary layer temperature
anomaly (Holloway and Neelin, 2007). The tropopause geopotential anomaly is
initially associated with cold temperature anomalies just above the
tropopause. The strength of radiative relaxation then determines the time
scale at which the geopotential anomaly rises in the stratosphere through
diabatic heating. In the zonally-symmetric case, the presence of wave-drag,
through conservation of angular momentum, disrupts this process and induces a
meridional overturning circulation that mediates the vertical scale at which
the geopotential anomaly can rise in the stratosphere.

Our work shows that, at least in the zonally symmetric case, the ratio between
the strength of radiative relaxation and that of Rayleigh damping are
significant factors in determining the response of the stratosphere to an SST
anomaly. However, there are a number of other quantities unveiled through the
non-dimensionalization that are also important. Surface friction, for
instance, factors into $\gamma$. In general, increasing the magnitude of $F$
does little to change the behavior of the stratospheric response to
tropospheric forcing when $\xi$ is large, since $F$ only enters in $\gamma$
and $\xi\gamma$ is what matters for the tropopause boundary condition. The
tropospheric & stratospheric stratification, as well as the shape and length
scale of the SST (or tropopause) perturbation ($L_{y}$), also factor into the
non-dimensional parameters that control the vertical decay scale of tropopause
geopotential anomalies. The horizontal scale of the SST anomaly can also be
quite important, due to the dependence of $S$ on $L_{y}^{-4}$. Figure 7, right
shows the dependence of the temperature anomaly above the tropopause on
$L_{y}$. There is an approximately logarithmic scaling of the temperature
anomaly with the meridional length scale of the tropopause anomaly, at least
across the range of $L_{y}$ in the experiments. Correspondingly, the
geopotential response in the stratosphere is muted for small $L_{y}$ (not
shown). Thus, large horizontal scale tropospheric heating anomalies have a
larger penetrative depth into the stratosphere, but are also associated with
smaller (in magnitude) temperature anomalies at the tropopause.

Figure 8: (Left) Linear coefficient of geopotential at varying levels,
regressed onto regionally-averaged SST anomaly. Above 500-hPa, significant
correlations at the 1% level (two-sided) are denoted by upside-down triangles.
(Middle) Same as the left panel but for temperature. (Right) Vertical
dependence of the correlation coefficients for (blue) geopotential and (red)
temperature. The regions are (solid) the entire tropics [$20^{\circ}$S -
$20^{\circ}$N], (dashed) the Indo-Pacific region
[$40^{\circ}$E-$120^{\circ}$E], (dot-dashed) the East Pacific region
[$180^{\circ}$E-$260^{\circ}$E], and (dotted) the Atlantic region
[$80^{\circ}$E-$0^{\circ}$]. Vertical level is scaled as the logarithm of
pressure.

##  4 Tropopause forcing in reanalysis data

In this section, we evaluate the proposed theory using the ERA5 re-analysis
(Hersbach et al., 2019b, a). We use monthly fields of SST, geopotential, and
temperature, over the years 1979-2022. The Quasi-Biennial Oscillation (QBO) is
regressed out of the geopotential and temperature fields, by using the 50-hPa
zonal wind averaged over the tropics. In particular, we will analyze
correlations between metrics of tropospheric warming and stratospheric
cooling, on the global scale and the local scale.

Figure 9: Annual-average temperature anomaly at (top) 70-hPa, (middle)
100-hPa, (bottom) 500-hPa. Note the strong anti-correlation in troposphere and
lower stratospheric temperature. Anomalies are calculated by subtracting the
climatological monthly zonal mean, and averaging across the entire year.

To begin, we regress the anomalous tropical-averaged geopotential, at
different vertical levels, onto the tropical-averaged SST anomaly. Anomalies
are generated by subtracting the linear trend in each field, as well the
seasonal cycle. Figure 8, solid lines, shows the coefficients of the linear
regressions of geopotential and temperature onto SST. We first observe an
approximate moist-adiabatic structure in the tropical tropospheric
geopotential, consistent with quasi-equilibrium and the findings of previous
studies (Holloway and Neelin, 2007). We also see a large, significant
correlation ($r\approx 0.75$) between tropical-averaged SST and the
corresponding 100-hPa geopotential. The magnitude of the geopotential anomaly
maximizes at 100-hPa, which is interpreted as an approximate tropopause level,
since below this level there is warming, and above this level there is cooling
(this is not exact, since the cold-point tropopause could occur above this
level). Note the similarity to the geopotential profile shown in Figure 6,
which also maximizes around the climatological tropopause. This is indicative
of a tropopause geopotential anomaly that is induced by an SST anomaly. The
coefficient magnitudes and correlations decay with increasing height in the
stratosphere, but are still statistically significant and non-negligible even
at 20-hPa. Note, for a pure baroclinic mode anomaly, the surface geopotential
would be anti-correlated with the upper troposphere anomaly (and the SST).
Thus, when the surface geopotential is positively correlated with the upper
tropospheric anomaly, there is significant barotropic component to the
geopotential profile. We indeed observe that the tropical-averaged surface
geopotential is positively correlated with both SST and the upper tropospheric
geopotential, highlighting the role of the barotropic mode and the
troposphere’s communication with the stratosphere.

The temperature structure of the tropical troposphere is also approximately
moist-adiabatic, as also shown in Holloway and Neelin (2007). Figure 8 also
shows that the tropics-averaged 70-hPa temperature is modestly but
significantly anti-correlated ($r\approx-0.34)$ with surface temperature. We
also observe temperature anomalies at 70-hPa (lower stratosphere) to be
approximately two times larger in magnitude than that of the surface, which is
in agreement with the estimates shown in Figure 7. This is not exactly
equivalent with the quantity derived in the left portion of Figure 7, since
the poor vertical resolution of the ERA5 reanalysis prohibits an exact
determination of the tropopause height.

The same relationships are also observed on regional scales (the Indo-Pacific,
East Pacific, and the Atlantic), as shown in Figure 8. The geopotential
anomalies maximize at 100-hPa in the Indo-Pacific, at 125-hPa in the Atlantic,
and at 150-hPa in the East Pacific. Thus, the level at which the geopotential
anomaly maximizes is influenced by the mean SST of the region (the East
Pacific has the coldest climatological SSTs, while the Indo-Pacific has the
warmest). In addition, the cold anomaly associated with SST warming maximizes
above the level of maximum geopotential. The regional scale geopotential
anomalies persist upwards to around 50-hPa, though the correlations drop
significantly in magnitude, and the statistical significance ceases around
50-hPa. This means that regional and local scale variations in the lower
stratospheric geopotential (50-hPa and 70-hPa) are strongly influenced by the
tropopause geopotential in the same region.

Of course, this analysis is not definitive proof that there is a quasi-
balanced response of the stratosphere to tropopause forcing. After all, if
stratospheric temperature is modulated by tropical heating through changes to
wave-drag (Garcia and Randel, 2008; Calvo et al., 2010; Lin et al., 2015),
then one would also expect the geopotential to decay with height in the
stratosphere, as is shown in Figure 8. Perhaps what would serve as stronger
evidence for the processes described in this study is if the spatial signature
of tropospheric warming is retained in that of stratospheric cooling.

In the tropics, the surface temperature need not always be connected to
tropospheric warming, especially if the boundary layer moist static energy is
lower than the saturation moist static energy of the free troposphere. This is
possible since temperature gradients in the tropical atmosphere are weak,
owing to the smallness of the Coriolis force, such that convecting regions
more effectively modulate the free tropospheric moist static energy (Sobel and
Bretherton, 2000). Furthermore, the aforementioned analyses lose information
on spatial correlation, since the anomalies are averaged over regions. In
order to emphasize spatial variability, we compute monthly anomalies by
subtracting the climatological monthly zonal mean from the climatological
monthly mean, and then average these across all 12 months (the analysis can
also be performed on each month, as will be discussed later).

Figure 10: Grid-point 2D density histograms between the 500-hPa anomalous
temperature and the (left) 100-hPa, (middle) 70-hPa and (right) 50-hPa
anomalous temperature. Color scale is logarithmic. Linear regressions are
plotted as the dashed blue lines, with correlation coefficients shown on the
lower left of each panel.

Figure 9 shows a map of the 500-hPa temperature anomaly, a proxy for
tropospheric heating, and the anomalous temperature at 100- and 70-hPa in the
lower stratosphere. It is evident that 500-hPa temperature is an excellent
predictor of both the 100-hPa and 70-hPa temperature anomaly. Spatial
variability in the tropospheric temperature anomaly is remarkably retained in
the spatial variability of the stratospheric temperature. Furthermore, the
lower-stratospheric temperature anomalies can be rather large (upwards to
around 4 degrees in magnitude at 100-hPa and 70-hPa), though the total area
encompassed by these large anomalies is small. There is also some qualitative
evidence from the maps in Figure 9 that suggests that the magnitude of the
lower stratospheric temperature anomalies is dependent on the horizontal scale
of the tropospheric anomaly. For instance, from 60∘W to 20∘E in the Northern
Hemisphere, there is a large-scale tropospheric cold anomaly of peak magnitude
around 2 degrees. The associated temperature anomaly at 100-hPa is around 4
degrees. There is also a large-scale tropospheric warm anomaly of peak
magnitude around 2 degrees in the Asian region (60∘E to 180∘E), with 100-hPa
temperature anomalies of around -4 degrees. In contrast, smaller scale
tropospheric anomalies [(150∘W to 90∘W, 10∘S to 30∘S), (45∘W to 15∘W, 10∘S to
25∘S)] with comparatively weaker peak temperature anomalies are associated
with 100-hPa temperature anomalies that are of similar magnitude to the
100-hPa temperature anomalies of the stronger, large scale anomalies. This is
in agreement with the proposed theory. In addition, at 70-hPa, the most
prominent temperature anomalies are those associated with the large-scale
tropospheric anomalies (i.e. over the Northeast African and Asian regions).
This is also in agreement with the theory, in that the vertical depth of the
tropopause anomalies increases with the horizontal scale of the tropospheric
anomaly. Of course, the analysis here is mostly qualitative, and could be
marred by the poor vertical resolution of the reanalysis. More substantial
analysis is required to further quantify the scale dependence of the lower
stratospheric temperature anomalies, and will be pursued in future work.

The remarkable correlation between tropospheric heating and stratospheric
cooling can be further quantified by regressions of 500-hPa temperature
against lower-stratospheric temperature, among all grid points shown in Figure
9. Figure 10 shows 2-D density histograms between the 500-hPa temperature and
the 100-, 70-, and 50-hPa temperature, as well as the linear regressions. Per
degree of warming at 500-hPa, the cooling response is around 1.5 degrees at
100-hPa $(r=-0.78)$, 1 degree at 70-hPa $(r=-0.7)$, and 0.25 degrees at 50-hPa
$(r=-0.62)$. The correlations are all significant, and generally decrease in
strength as one moves up further in the stratosphere. While the monthly
anomalies shown in Figure 9 are averaged across the whole year, there is
significant seasonal variability in the pattern of 500-hPa tropospheric
temperature (not shown). The analysis can repeated by separating into each
month, and the results and interpretation remained unchanged: 500-hPa
temperature is strongly anti-correlated with lower stratospheric temperature.
It is important to note that these correlations do not suggest that there are
correlations on significantly smaller horizontal scales; as suggested by
Figure 9, the correlations merely reflect the large-scale structure of the
temperature anomalies. Regardless, these simple analyses provide strong
evidence that there is a quasi-balanced response of the stratosphere to
tropospheric thermal forcing.

##  5 Summary and discussion

In this work, we present theoretical evidence for how tropopause geopotential
anomalies, generated through tropospheric thermal (SST) forcing, can modulate
upwelling in the stratosphere. Using a conceptual model based on the
linearized QGPV equations, we show that tropospheric thermal forcing can
induce a tropopause geopotential anomaly, which subsequently elicits a quasi-
balanced response in the stratosphere. The tropopause anomalies initially have
vertically shallow structures scaled by the Rossby penetration depth (i.e. the
fast adjustment of the stratosphere). Afterwards, radiative relaxation in the
stratosphere acts to increase the vertical penetration of these anomalies. In
the steady-state limit, where radiative equilibrium is again satisfied, the
stratospheric PV becomes barotropic, though it takes on the order of years to
be achieved. The solutions are akin to those of Haynes et al. (1991), who
found that the stratosphere becomes barotropic above the level of forcing (in
this case, the tropopause).

We then formulate a zonally symmetric troposphere-stratosphere linear
$\beta$-plane model, which couples a convecting troposphere to a dry and
passive stratosphere. We show that the stratospheric response to tropospheric
forcing is controlled by two non-dimensional parameters: (1) $\xi$, a
dynamical aspect ratio (Ming et al., 2016b), and (2) $\gamma$, a ratio between
the stratospheric drag and tropospheric friction. In the limit that radiation
is much stronger than wave drag, the stratospheric response to a tropopause
forcing asymptotically becomes barotropic, while in the opposite limit, the
vertical length scale of the tropopause forcing becomes extremely small. We
find that the stratospheric response to zonally-symmetric tropospheric forcing
is largely dependent on the radiative relaxation rate, the Rayleigh damping
time scale of wave-drag, and the horizontal scale. Our analyses show that the
tropopause temperature anomaly is also modulated by all of these quantities.

We also use reanalysis data to show that tropical and regionally averaged
lower-stratospheric temperatures are modestly and negatively correlated with
SSTs in the same areas. In general, the temperature anomalies per degree of
warming in the boundary layer are approximately equivalent to the
corresponding theoretical predictions, at least when using “Earth-like”
estimates of the time scale of wave-drag and radiative relaxation.
Furthermore, we show that the spatial variability in lower-stratospheric
temperature anomalies is strongly correlated with the spatial variability in
500-hPa tropospheric temperatures. Significant correlations are seen upwards
to 50-hPa, which suggests that there is a quasi-balanced response of the
stratospheric to tropospheric forcing. This provides a scale-dependent theory
for the oft-observed anti-correlation between tropospheric warming and
stratospheric cooling (Johnson and Kriete, 1982; Gettelman et al., 2002; Fu et
al., 2006; Holloway and Neelin, 2007; Kim and Son, 2012; Virts and Wallace,
2014; Kim et al., 2018).

The widely accepted theory of tropical stratospheric upwelling is that it is
mechanically driven by sub-tropical wave-drag (Haynes and McIntyre, 1987;
Plumb and Eluszkiewicz, 1999). There is ample evidence from numerical modeling
suggesting that wave-dissipation is a dominant mechanism that modulates mean
and interannual upwelling in both the lower stratosphere and TTL (Boehm and
Lee, 2003; Norton, 2006; Calvo et al., 2010; Ryu and Lee, 2010; Gerber, 2012;
Ortland and Alexander, 2014; Kim et al., 2016; Jucker and Gerber, 2017, among
many others). Of course, it is theoretically impossible to have flow across
angular momentum contours without some momentum source. We emphasize that in
no way does this work attempt to disprove the role sub-tropical wave drag has
in modulating tropical stratospheric upwelling. In this model, even though
wave-drag acts as a Rayleigh damping, as in the linear system described in
PE99, it is an important modulator of the upwelling response.

As shown in this study, the vertical penetration of the geopotential anomaly
(and the rate at which the stratospheric circulation crosses angular momentum
surfaces) is strongly a function of the wave drag. If the wave-drag is a
function of the zonal mean state, which could vary in time in part due to
wave-forcing (Cohen et al., 2013; Ming et al., 2016b), then the vertical
penetration of the tropopause anomaly (and thus, its subsequent effect on
upwelling) would also vary in time. In this view, stratospheric wave-drag is,
as countless studies have shown, a significant modulator of tropical
upwelling. However, wave drag alone may not suffice to explain certain
features of the behavior of the lower stratosphere, the foremost of which is
the inverse correlation between SST and lower stratospheric temperature
anomalies, in both the zonal and meridional directions.

Our work, like PE99, investigates how tropospheric thermal forcing can
modulate stratospheric upwelling. In addition to mechanical and thermal
forcing, this suggests a third way in which the stratosphere can be forced –
through the tropopause via tropospheric thermal forcing. In fact, the
theoretical analysis shown in PE99 finds that in the tropics, “the existence
of a thermally driven circulation and the breakdown of downward control go
together” (if one accepts that what they define as viscosity is representative
of large-scale drag). However, their calculation of the linear response to
tropospheric thermal forcing exhibited large and unrealistic vertical
penetration of the tropospheric circulation into the stratosphere. This work
shows that this is likely a result of their assumptions of the strength of
radiative relaxation ($\alpha_{\text{rad}}=10$ days-1) and viscosity
($\hat{D}=500$ days-1). With $S=O(10^{2})$, this is equivalent to $\xi\approx
3$. In this regime, our theory predicts extensive penetration of the
tropospheric circulation into the stratosphere, as in Figure 4 and 6.

In general, it is difficult to infer causality from diagnostic relations. For
example, in Transformed-Eulerian Mean equations (derived, for instance, in
Andrews et al. (1987)), it is not clear how much of the wave-drag is an
external forcing, as opposed to a response to a circulation that has a
different forcing. Of course, variations in wave-drag that are independent of
those of the circulation support the idea that waves can force the
circulation. This aspect of the stratosphere has been well studied. But what
if wave-drag acted purely as a response to the circulation? (Note that these
ideas are at opposite ends of the spectrum with regards to the extent waves
drive the circulation)? Then, at least in our framework, the causality becomes
very clear – SST forces the stratosphere by imposing a tropopause geopotential
anomaly. Of course, one could take the wave-drag term ($-D_{s}u_{s}$) and use
it to diagnose the associated upwelling response. However, that does not imply
that waves are the forcing mechanism of the circulation.

There are a few pieces of observational evidence that could be interpreted to
be in favor of the proposed theory. As stated earlier, the spatial variability
of lower-stratospheric temperature is strongly correlated with that of the
troposphere. In addition, there is a strong observed anti-correlation of
temperature trends in the troposphere and the lower stratosphere (Fu et al.,
2006). These long-term trends are highly correlated on a grid-point by grid-
point basis, suggesting that the zonal and meridional structure of
tropospheric warming may be important to that of stratospheric cooling. In
contrast, wave-drag, in its classical arguments, can only explain departures
of temperature from the zonal-mean (Andrews et al., 1987). This is by no means
a small feat, since the annual cycle in tropical-averaged temperature near the
tropopause is around 8K, around a factor of two larger than the peak
temperature anomalies shown in Figure 9 (Chae and Sherwood, 2007).

However, the quasi-balanced response of the stratosphere to tropopause forcing
could serve as a potential explanation for a few outstanding issues. For
instance, it can explain why there is peak tropical upwelling on the summer-
side equator Rosenlof (1995). It could also help to explain the observed
connection between boundary layer temperature anomalies and lower
stratospheric temperature anomalies, as well as the high correlations between
tropical SST and the upwelling strength of the shallow BDC branch, which is
observed on all time scales (Lin et al., 2015; Abalos et al., 2021). Numerical
modeling suggests that strengthening of the sub-tropical jets changes the
upward propagation of waves (Garcia and Randel, 2008; Calvo et al., 2010;
Shepherd and McLandress, 2011), ultimately strengthening the wave-driven
stratospheric upwelling, although the exact specifics seem to vary from model
to model (Calvo et al., 2010; Simpson et al., 2011). In the zonally symmetric
coupled troposphere-stratosphere theory analyzed in this work, an equatorial
SST anomaly is not only associated with strengthening of the sub-tropical jets
(which no doubt could change the sub-tropical distribution of wave-drag in the
real-world), but also a strengthening of the tropopause geopotential. As such,
the theory proposed in this work does not have to be mutually exclusive with
those based on wave-drag.

Besides the inclusion of a relaxational wave-drag (shown to be a poor
assumption), our work stays silent on how the momentum budget must change in
order to balance changes in the meridional circulation (Ming et al., 2016b).
However, there would undoubtedly be a large scale wave response to steady
tropospheric heating (Gill, 1980). Thus, disentangling the effects of heating
from the ensuing wave-response is quite complicated, as the two occur in
concert. While other studies have analyzed the wave-response to tropospheric
heating (Ortland and Alexander, 2014; Jucker and Gerber, 2017) (as well as its
subsequent effects on the stratospheric circulation), we have instead focused
on the steady response to tropospheric heating. In general, however, when
tropical tropospheric heating is used to generate a wave response, it is
difficult to separate the tropopause forcing mechanism described in this study
from wave driving. For instance, Jucker and Gerber (2017) used idealized GCM
simulations to show that the inclusion of a tropical warm pool significantly
changed the annual-mean temperature of the tropical tropopause (and more
importantly, more so than mid-latitude land-sea contrast and orographic
forcing). However, the imposition of a warm pool will both intensify the
tropopause anti-cyclone over the region, and trigger a large-scale wave
response. According to the analysis shown in this study, the increased
tropopause geopotential will act to cool the tropopause and induce more
upwelling (as would increased wave-drag from the large-scale wave response).
Separately, Ortland and Alexander (2014) forced equatorial waves by
prescribing time-varying latent heating anomalies in a primitive equation
model, and found that stationary waves and weakly westward propagating waves
are most responsible for driving residual-mean upwelling in the TTL. Again,
tropospheric heating will induce a tropopause geopotential anomaly, such that
the steady tropospheric forcing is not separated from the wave response.
Regardless, both of the modeling results in Ortland and Alexander (2014) and
Jucker and Gerber (2017) show that at least in numerical models, the seasonal
cycle in upwelling in the tropical tropopause layer cannot be explained by
tropospheric thermal forcing.

It is only fair for these conclusions to be discussed alongside the
assumptions posited in this model. In this model, we assume that there is an
instantaneous transition between tropospheric, quasi-equilibrium dynamics, and
passive, dry stratospheric dynamics. In reality, the presence of the TTL could
dampen the upwards influence of tropospheric forcing. The assumption of a
moist adiabatic lapse rate all the way to the tropopause is one that is has
mixed observational evidence, which suggests that the free tropospheric
temperature anomalies, per degree of warming in the boundary layer,
approximately follow a moist adiabat up to around 200-hPa, after which
temperature anomalies transition to being out of phase with lower tropospheric
temperature anomalies [see Figure 8 and Holloway and Neelin (2007)] (though
some of this may be owing to time averaging with a vertically moving
tropopause). While the proposed theory can predict the magnitude of the
tropopause temperature anomalies with respect to boundary layer warming, it
does not include a transition layer. The presence of a transition layer could,
in theory, dampen the vertical penetration of thermal forcing in the
troposphere. This will be the subject of future research.

Finally, we also assume a fixed tropopause height that interfaces the two
regimes, as in PE99. This makes the analysis mathematically tractable. Indeed,
one would expect tropospheric temperature to affect tropopause height (Held,
1982; Lin et al., 2017). The relaxation of both of these assumptions will be
the subject of future research, but requires a theory for how moist convection
interacts with the transition layer. More research is necessary to understand
the role of convection in modulating the behavior of the transition layer.

The analysis carried out in section 4 uses the ERA5 reanalysis dataset, which
has very coarse vertical resolution near the tropopause. Since tropopause
anomalies can decay in the vertical very quickly, especially for anomalies
with small horizontal scale, further insight into the processes outlined in
this study would be impaired by the poor vertical resolution of the
reanalysis. This could be mitigated by the use of GPS radio-occultation (RO)
measurements, provided by the COSMIC mission (Anthes et al., 2008). The high
vertical resolution of GPS-RO measurements could be leveraged in future work.
Furthermore, while we focused on large-scale tropospheric anomalies in this
work, there are also numerous mesoscale convective systems, usually with
anticyclones at their tops, that might also be able to contribute to tracer
transport into the stratosphere. Higher resolution observational data, such as
that provided by GPS RO measurements, could also be useful to evaluate this
possibility.

###### Acknowledgements.

The author thanks Adam Sobel and Peter Hitchcock for comments and suggestions
on earlier versions of this work. The authors also thank two anonymous
reviewers and Peter Haynes for their helpful suggestions, which greatly
improved the manuscript. In particular, the authors are grateful for Peter
Haynes’s suggestions on the formulation of the coupled boundary condition. J.
Lin gratefully acknowledges the support of the National Science Foundation
through the NSF-AGS Postdoctoral Fellowship, under award number AGS-
PRF-2201441. The monthly-mean ERA5 data for sea-surface temperature is
available at https://cds.climate.copernicus.eu/cdsapp#!/dataset/reanalysis-
era5-single-levels-monthly-means via DOI: 10.24381/cds.f17050d7 Hersbach et
al. (2019b). The monthly-averaged ERA5 data for temperature and geopotential
are available at
https://cds.climate.copernicus.eu/cdsapp#!/dataset/reanalysis-era5-pressure-
levels-monthly-means via DOI: 10.24381/cds.6860a573 Hersbach et al. (2019a).
All code to generate the data from the theoretical models are available at
https://github.com/linjonathan/steady_coupled_trop_strat. Details on Solutions

###  .1 Solutions to Conceptual Model in Section 2

The general solution to the homogeneous version of Eq. 9 ($q(z)=0$) is:

$\displaystyle G(z)=A\exp(m_{+}z)+B\exp(m_{-}z)$ (50)

where $m_{\pm}=\frac{1\pm\sqrt{1+4(k^{2}+l^{2})}}{2}$. Note, since
$k^{2}+l^{2}>0$, $m_{+}>0$ and $m_{-}<0$ for all $k>0$ and $l>0$. We next
define the Green’s function, which satisfies

$LG(z,\lambda)=\delta(z-\lambda)$ (51)

and is

$G(z,\lambda)=\begin{cases}A\exp(m_{+}z)+B\exp(m_{-}z),&\text{for
}0<z<\lambda\\\ C\exp(m_{+}z)+D\exp(m_{-}z),&\text{for
}\lambda<z<z_{\text{top}}\\\ \end{cases}$ (52)

where $z_{\text{top}}$ is assumed to be the top of the domain. The lower
boundary condition requires that:

$A+B=\phi_{T}$ (53)

and the upper boundary condition requires that:

$Cm_{+}\exp(m_{+}z_{\text{top}})+Dm_{-}\exp(m_{-}z_{\text{top}})=0$ (54)

Note that we choose to explicitly include $z_{\text{top}}$ in Eq. 54, since
numerically evaluating the Green’s functions requires $z_{\text{top}}<\infty$.
Continuity of $G$ across $\lambda$ requires:

$\displaystyle
A\exp(m_{+}\lambda)+B\exp(m_{-}\lambda)=C\exp(m_{+}\lambda)+D\exp(m_{-}\lambda)$
(55) $\displaystyle\lim_{\epsilon\rightarrow 0}\frac{\partial G}{\partial
z}\bigg{|}_{z=\lambda-\epsilon}^{z=\lambda+\epsilon}-\lim_{\epsilon\rightarrow
0}G\Big{|}_{z=\lambda-\epsilon}^{z=\lambda+\epsilon}=1$ (56)

Eqs. 53-56 are solved to obtain:

$\displaystyle
A=\frac{\phi_{T}-\frac{1}{m_{d}}(\exp(-m_{-}\lambda)-\frac{m_{+}}{m_{-}}\exp(-m_{+}\lambda+m_{d}z_{\text{top}}))}{1-\frac{m_{+}}{m_{-}}\exp(m_{d}z_{\text{top}})}$
(57)

where

$m_{d}=m_{+}-m_{-}=\sqrt{1+4(k^{2}+l^{2})}>0$ (58)

$B$, $C$, and $D$ are then obtained using Eqs. 53, 54, and 55.

The Green’s function can be convoluted with the source term ($q$) to obtain
the geopotential:

$\phi(z)=\int_{0}^{\infty}G(z,\lambda)q(\lambda)d\lambda$ (59)

###  .2 Numerical Solver for Coupled System

Here, we elaborate on the numerical solver of the coupled troposphere-
stratosphere system (Eq. 35, 46), given forcing in $s^{*}$. We approximate the
meridional and vertical derivatives with second-order and sixth-order central
finite differences, respectively. Since our specified $s^{*}$ forcing is
equatorially symmetric, we only have to discretize $y$ from equator to pole,
and impose a Neumann boundary condition at the equator. However, $y$ appears
in the denominator in both Eq. 35 and 46). We circumvent this issue by
numerically evaluating the equator at $\epsilon=10^{-5}$ (three orders of
magnitude smaller than the meridional grid spacing). $y$ is evenly discretized
from $y_{\text{max}}$ to $\epsilon$, where $y_{\text{max}}=-10$. $z$ is evenly
discretized from the tropopause ($z^{*}=1$) to the domain top,
$z^{*}_{\text{top}}=7$. The boundary conditions are:

$\displaystyle\phi(y=y_{\text{max}},z^{*})$ $\displaystyle=0$ (60)
$\displaystyle\frac{\partial\phi}{\partial y}(y=\epsilon,z^{*})$
$\displaystyle=0$ (61) $\displaystyle\frac{\partial\phi}{\partial
z}(y,z^{*}=z^{*}_{\text{top}})$ $\displaystyle=0$ (62)

as well as the aforementioned Eq. 46 on the boundary $z^{*}=1$. The solutions
are ensured to solve the original linear system of equations, as well as the
boundary conditions, within numerical error. Finally, we use the findiff
Python package to solve the system numerically (Baer, 2018).

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