**Research article**
21 Dec 2021

**Research article** | 21 Dec 2021

# A simple model of ozone–temperature coupling in the tropical lower stratosphere

William J. Randel Fei Wu Alison Ming and Peter Hitchcock

^{1,2},

^{1},

^{3},

^{4}

**William J. Randel et al.**William J. Randel Fei Wu Alison Ming and Peter Hitchcock

^{1,2},

^{1},

^{3},

^{4}

^{1}National Center for Atmospheric Research, Atmospheric Chemistry Observations and Modeling Laboratory, Boulder, CO, USA^{2}COSMIC Program, University Corporation for Atmospheric Research, Boulder, CO, USA^{3}Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK^{4}Department of Earth and Atmospheric Sciences, Cornell University, Ithaca, NY, USA

^{1}National Center for Atmospheric Research, Atmospheric Chemistry Observations and Modeling Laboratory, Boulder, CO, USA^{2}COSMIC Program, University Corporation for Atmospheric Research, Boulder, CO, USA^{3}Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK^{4}Department of Earth and Atmospheric Sciences, Cornell University, Ithaca, NY, USA

**Correspondence**: William J. Randel (randel@ucar.edu)

**Correspondence**: William J. Randel (randel@ucar.edu)

Received: 17 Jun 2021 – Discussion started: 08 Jul 2021 – Revised: 25 Sep 2021 – Accepted: 21 Oct 2021 – Published: 21 Dec 2021

Observations show strong correlations between large-scale ozone
and temperature variations in the tropical lower stratosphere across a wide
range of timescales. We quantify this behavior using monthly records of
ozone and temperature data from Southern Hemisphere Additional Ozonesonde (SHADOZ) tropical balloon measurements
(1998–2016), along with global satellite data from Aura microwave limb sounder and GPS radio
occultation over 2004–2018. The observational data demonstrate strong
in-phase ozone–temperature coherence spanning sub-seasonal, annual and
interannual timescales, and the slope of the temperature–ozone relationship
(*T* $/$ O_{3}) varies as a function of timescale and altitude. We compare the
observations to idealized calculations based on the coupled zonal mean
thermodynamic and ozone continuity equations, including ozone radiative
feedbacks on temperature, where both temperature and ozone respond in a
coupled manner to variations in the tropical upwelling Brewer–Dobson
circulation. These calculations can approximately explain the observed
(*T* $/$ O_{3}) amplitude and phase relationships, including sensitivity to timescale and altitude, and highlight distinct balances for “fast” variations
(periods < 150 d, controlled by transport across background
vertical gradients) and “slow” coupling (seasonal and interannual
variations, controlled by radiative balances).

Large-scale ozone and temperature variations in the tropical lower stratosphere exhibit strong correlations across a range of timescales. This behavior is well known for the annual cycle in the lower stratosphere (Chae and Sherwood, 2007; Randel et al., 2007) and for interannual variations linked to the quasi-biennial oscillation (QBO) (e.g., Hasebe et al., 1994; Baldwin et al., 2001; Witte et al., 2008; Hauchecorne et al., 2010) and El Niño–Southern Oscillation (ENSO; Randel et al., 2009). Abalos et al. (2012, 2013) and Gilford et al. (2016) also note strong temperature–ozone correlations in this region across a range of timescales. Calculations have shown that the radiative effects of ozone feed back onto and enhance temperature variations, and this topic has been well studied in relation to the annual cycle in the tropical lower stratosphere (Chae and Sherwood, 2007; Fueglistaler et al., 2011; Ming et al., 2017; Gilford and Solomon, 2017) and also by Forster et al. (2007) and Polvani and Solomon (2012) for decadal-scale trends. Yook et al. (2020) showed that ozone feedback is an important contribution to tropical stratospheric thermal variability in global models. Birner and Charlesworth (2017) and Dacie et al. (2019) have demonstrated strong sensitivity of tropical stratospheric temperatures to ozone using idealized one-dimensional model calculations, following the earlier results of Thuburn and Craig (2002). Charlesworth et al. (2019) extended that work to study transient ozone–temperature feedbacks, highlighting larger effects for low-frequency variations (periods longer than about half a year).

The dominant mechanism for strong ozone–temperature correlations in the
tropical lower stratosphere is relatively simple: namely, variations in
upwelling (i.e., fluctuations in the tropical Brewer–Dobson circulation)
acting on the strong background vertical gradients of both ozone and
potential temperature, leading to correlated variability. This behavior was
quantified from observations and model simulations in Abalos et al. (2012, 2013), highlighting the control of upwelling for forcing transient
variations in temperature, ozone and other trace species with strong
vertical gradients, such as carbon monoxide (CO). The radiative feedback of
ozone to temperature imparts further complexity to this simple system, which is the focus of this work. Here we update the observational evidence of
ozone–temperature coupling based on long records of tropical balloon
measurements from SHADOZ (Thompson et al., 2003), focusing on annual and
interannual variability. We also analyze over a decade of continuous
satellite measurements to quantify ozone–temperature coherence and phase in
the tropical stratosphere over a continuous range of timescales. We compare
the observational results with calculations based on the coupled zonal mean
thermodynamic and ozone continuity equations, simplified to approximate the
balances in the tropical lower stratosphere, and including ozone feedback on
temperature. Our goal is to explain the salient features of
temperature–ozone (*T*–O_{3}) coupling from observations in a relatively simple
framework, including the frequency and altitude dependences of the (*T* $/$ O_{3})
amplitude and phase relationships. These results are a complement to the
recent analyses of Birner and Charlesworth (2017) and Charlesworth et al. (2019), based on a very different model.

## 2.1 SHADOZ ozone and temperature

The Southern Hemisphere Additional Ozonesonde (SHADOZ) network consists of
∼ 12 stations covering a range of longitudes over the latitude
band ∼ 10^{∘} N–20^{∘} S, with measurements beginning
in 1998 (Thompson et al., 2003). Recent reprocessing of the data is discussed
in Witte et al. (2017) and Thompson et al. (2017). The SHADOZ balloons measure
ozone and pressure–temperature–wind profiles with an effective vertical
resolution of ∼ 50–100 m. The data used here are sampled with
0.5 km vertical spacing, and we focus on altitudes 15–30 km. We analyze data
from SHADOZ stations with long and continuous records, updated from Randel
and Thompson (2011). There are typically 2–4 observations per month at each
of the SHADOZ stations, which we combine into simple monthly averages. The
stratospheric segment of the ozone profile exhibits a high degree of
longitudinal symmetry (Thompson et al., 2003; Randel et al., 2007; Randel and
Thompson, 2011), and we combine monthly average results from all stations to
provide approximate zonal average monthly means of ozone and temperature,
with data covering 1998–2016.

## 2.2 Aura microwave limb sounder ozone and GPS temperature

Satellite ozone measurements from the Aura microwave limb sounder (MLS) are
analyzed for the period September 2004–May 2018. We use retrieval version
4.2 (Livesey et al., 2018). Data are available for standard pressure levels
(12 per decade) covering from 316 hPa to above 1 hPa; the vertical resolution of
the grid is ∼ 1.3 km, but the resolution of the MLS
measurements is closer to ∼ 3 km (i.e., the data are
oversampled). Data quality for MLS v4.2 ozone is discussed in Livesey et al. (2018). Our analyses focus on the latitude band 10^{∘} N–S, and we
calculate zonal mean values for 5 d (pentad) averages. Some isolated data
gaps are filled by linear interpolation in time. This provides a long and
continuous time series of MLS ozone covering 998 pentads (4990 d).

Temperature data are obtained from GPS radio occultation, which provides
high-quality and high vertical resolution (∼ 1 km)
measurements over 10–30 km and near-global sampling (Anthes et al., 2008).
We combine measurements from several different GPS satellites for the period
overlapping the MLS ozone data (September 2004–May 2018) and construct
pentad time series from data over 10^{∘} N–S to match the MLS ozone time
series discussed above. We focus on altitude levels close to the MLS ozone
grid. The time series analyzed here are an update of the data analyzed in
Randel and Wu (2015), and further details are discussed there.

## 2.3 Spectrum analysis

We include spectrum and cross-spectrum analysis of the satellite-derived
ozone and temperature time series to quantify frequency-dependent
relationships. Spectra are calculated by direct Fourier transform of the
998 pentad time series for both ozone and temperature, resolving periods of
4990 to 10 d with a frequency resolution of $\mathrm{\Delta}\mathit{\omega}=(\mathrm{2}\mathit{\pi}/\mathrm{4990}$ d). Calculations are based on standard formulas in Jenkins
and Watts (1968). Power spectra are smoothed with a Gaussian-shaped
smoothing window with a half-width of 2Δ*ω*. Temperature–ozone
amplitude ratios, coherence squared spectra and phase spectra are calculated using a
wider bandwidth (10Δ*ω*) to enhance statistical stability.
This results in approximately 10 independent Fourier harmonics for each
spectral estimate, and the resulting 95 % significance level for the coherence squared (coh^{2})
statistic is 0.45. The high- and low-frequency ends of the spectra are
smoothed using one-sided Gaussian smoothing so that significance levels are
somewhat higher.

## 3.1 Coupled thermodynamic and ozone continuity equations

We explore the coupling of ozone and temperature based on the zonal mean thermodynamic and ozone continuity equations, simplified to approximate behavior in the tropical lower stratosphere, namely neglecting mean meridional advection and eddy forcing terms. The zonal mean thermodynamic equation in transformed Eulerian-mean coordinates using a log-pressure vertical coordinate (Andrews et al., 1987) is as follows:

Here *T* is zonally averaged temperature, (*v*^{*}, *w*^{*}) are components of the
residual meridional circulation, *S* is a stability parameter and *Q* is the
zonal mean diabatic heating rate. We note that all variables in the
equations are zonal mean quantities, but no overbars are used in the
notation. In the tropical lower stratosphere the *v*^{*} and eddy forcing terms
are relatively small (Abalos et al., 2013), and thus the approximate
thermodynamic balance is as follows:

In this work we specify the zonal mean diabatic forcing Q with two
components $Q={Q}_{\text{relaxation}}+{Q}_{\text{ozone}}$, representing radiative
relaxation and ozone forcing of temperature, respectively. We assume
radiative relaxation is proportional to temperature, ${Q}_{\text{relaxation}}=-\mathit{\alpha}(T-{T}_{\text{eq}})$, with *T*_{eq} an equilibrium temperature and
*α* an inverse radiative damping timescale (Andrews et al., 1987;
Hitchcock et al., 2010). *α* is obtained from the results of Hitchcock
et al. (2010) as discussed below. In addition, correlated variations in ozone
produce a positive radiative feedback on temperature (e.g., Fueglistaler et al., 2011; Gilford et al., 2017; Ming et al., 2017), and while this is in
general a non-local effect (in altitude), for simplicity we model the
temperature tendency as proportional to the local ozone anomaly: ${Q}_{\text{ozone}}=\mathit{\beta}(X-{X}_{\text{eq}})$. Here *X* is zonal mean
ozone mixing ratio, *X*_{eq} is a background equilibrium ozone value
and *β* is a constant derived from radiative transfer calculations
(described below). Based on these simplified assumptions, the zonal mean
thermodynamic equation becomes

Assuming harmonic time expansions of the form $T\left(t\right)=\sum {T}_{\mathit{\sigma}}{\mathrm{exp}}^{i\mathit{\sigma}t}$, with *σ* the angular frequency (2*π* per period)
and likewise for *w*^{*}(*t*) and *X*(*t*), and assuming *T*_{eq}, *X*_{eq} and *S* are constant in time, Eq. (3) can be rewritten as an equation
for each harmonic component:

A similar analysis is applied to the zonal mean ozone continuity equation (Andrews et al., 1987, Eq. 9.4.13):

Here *P*−*L* represents chemical ozone production minus loss terms. In contrast
to the thermodynamic balance discussed above, the eddy terms for ozone
transport in the tropical lower stratosphere are not negligible, and there
is a maximum during boreal summer near the tropopause related to transport
from the subtropical monsoon circulations (Konopka et al., 2009, 2010; Abalos
et al., 2013). This contribution is relatively large below ∼ 80 hPa (18 km). However, for simplicity in our idealized calculations the eddy
terms are neglected here, along with the *v*^{*} term. This yields

where we have defined ${X}_{z}=(\partial X/\partial z)$. In the tropical lower stratosphere ozone production minus loss is
positive and is relatively constant in time, with a small semiannual
variation in production following solar inclination (Abalos et al., 2013). We
parameterize ozone loss as $L=-\mathit{\delta}(X-{X}_{\text{eq}})$,
where *δ* is the inverse lifetime of ozone, obtained from model
calculations as described below. Assuming a constant production rate (*P*) and
a constant background ozone gradient *X*_{z}, the harmonic expansion
of Eq. (6) is then given by the following simple balance:

We show idealized model calculations below including realistic ozone damping estimates (along with results for no damping), which demonstrate that ozone damping has a relatively small influence for the majority of results.

The balances in the simplified equations (Eqs. 4 and 7) are driven by
temperature and ozone responses to imposed vertical velocity variations (*w*^{∗})
in the tropical lower stratosphere, as is observed and derived from model
simulations (Abalos et al., 2012, 2013). Temperature is furthermore
influenced by radiative damping (*α* term) and the radiative response
to ozone changes (*β* term), while ozone balance includes damping
(*δ* term). Equations (4) and (7) can be combined to eliminate the
${w}_{\mathit{\sigma}}^{\ast}$ dependence to obtain a single equation relating temperature
and ozone harmonic components, in particular the temperature $/$ ozone ratio as
a function of frequency:

with ${\mathit{\beta}}^{\prime}=({X}_{z}/S)\mathit{\beta}$. This can be rewritten as follows:

with $A=(S/{X}_{z})({\mathit{\sigma}}^{\mathrm{2}}+\mathit{\alpha}({\mathit{\beta}}^{\prime}+\mathit{\delta}\left)\right)/({\mathit{\sigma}}^{\mathrm{2}}+{\mathit{\alpha}}^{\mathrm{2}})$ and $B=(S/{X}_{z})\mathit{\sigma}(\mathit{\alpha}-({\mathit{\beta}}^{\prime}+\mathit{\delta}\left)\right)/({\mathit{\sigma}}^{\mathrm{2}}+{\mathit{\alpha}}^{\mathrm{2}})$.

Here ($S/{X}_{z})$ is a key parameter related to the ratio of
background stability (potential temperature gradient) to ozone vertical
gradient, which is derived directly from the time average temperature and
ozone profile data; the vertical profile of ($S/{X}_{z})$ is shown in
Fig. 1c. We note some small (∼ 10 %) seasonal and
interannual variations to the individual *S* and *X*_{z} terms in the
tropical lower stratosphere, but these follow each other and the ratio
$(S/{X}_{z})$ is more nearly constant. Equation (8) can be rewritten as
expressions for (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ amplitude and phase:

Our analyses focus on evaluating the quantity (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ as a metric for temperature sensitivity to ozone as a
function of frequency (and altitude), and below we test results from this
idealized model with (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ amplitude and
phase derived from observations. We note that the phase is defined such that
positive values denote temperature variations leading ozone in time. The
observational data are based on measurements from SHADOZ and MLS and/or GPS in the
deep tropics over ∼ 10^{∘} N–S, and hence they represent this
tropical average. We note that Stolarski et al. (2014) and Tweedy et al. (2017) highlight distinct ozone behavior in southern tropics vs. northern
tropics in the region up to ∼ 18 km due to influence of the
boreal summer monsoons. This could potentially impact our comparisons close
to the tropopause but should have less influence above ∼ 18 km.

The (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ ratio in Eq. (8) is a generally
complex function of *σ*, *α*, *β*^{′}, *δ* and
($S/{X}_{z})$, but it is useful to consider the high- and
low-frequency limits (compared to the inverse timescales *α* and
(${\mathit{\beta}}^{\prime}+\mathit{\delta}\left)\right)$. For high frequencies (*σ*≫*α*, (${\mathit{\beta}}^{\prime}+\mathit{\delta}\left)\right)$, the (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ ratio simplifies to ∼ ($S/{X}_{z})$;
i.e., the temperature and ozone anomalies are in phase, with a ratio simply
related to the background gradients in potential temperature and ozone. For
the low-frequency limit (*σ*≪*α*, (${\mathit{\beta}}^{\prime}+\mathit{\delta}\left)\right)$, (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})\sim (S/{X}_{z}\left)\right(({\mathit{\beta}}^{\prime}+\mathit{\delta})/\mathit{\alpha})$. For *δ*
smaller than *β*^{′} (as suggested by Fig. 1a), the ratio simplifies to
($\mathit{\beta}/\mathit{\alpha})$. This expresses an in-phase balance of ozone and
temperature associated with the *α* and *β* radiative terms in
the thermodynamic equation (Eq. 3); i.e., heating from ozone anomalies
balances radiative cooling. The simple model predicts that the lower-frequency limit will occur for frequencies lower than (${\mathit{\beta}}^{\prime}+\mathit{\delta})$, corresponding to periods longer than about 150 d at 24 km (using the
values in Fig. 1a).

The effect of ozone feedback on temperature is given by the *β* term in
Eq. (3), quantified by the *β*' term in the coupled equations (Eq. 8).
Below we directly evaluate this influence by comparing calculations with
${\mathit{\beta}}^{\prime}=\mathrm{0}$, which explicitly quantifies the ozone feedback on temperature
in our simplified framework. The simple model suggests this influence will
be seen at low frequencies; in the absence of ozone feedback the (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ low-frequency limit reduces to ($S/{X}_{z}\left)\right(\mathit{\delta}/\mathit{\alpha})$.

## 3.2 Estimating *α*, *β* and *δ* from model calculations

Our calculations use a vertical profile of *α* in the tropical
stratosphere derived by Hitchcock et al. (2010), as shown in Fig. 1a. These
results are based on regressions derived from radiative heating rates and
temperatures output from a chemistry–climate model. We note that there are
several uncertainties inherent in these calculations, including factors such
as tropospheric clouds influencing lower stratospheric heating rates and
dependence on the vertical scale of temperature perturbations (Hartmann et al., 2001; Hitchcock et al., 2010). The overall structure and magnitude of
*α* used here is consistent with other published estimates, e.g.,
Newman and Rosenfield (1997) and Randel and Wu (2015).

We estimate vertical profiles of the parameter *β* from radiative
transfer calculations using a modified version of the Morcrette (1991)
radiation scheme (Zhong and Haigh, 1995). The calculations use realistic
background temperature, ozone and water vapor profiles, and carbon dioxide
is assumed to be well mixed with a volume mixing ratio of 360 ppmv.
Shortwave heating rates are calculated as diurnal averages, including
realistic surface albedo, and all calculations assume clear-sky conditions.
*β* is derived by applying a 0.1 ppmv perturbation to the ozone field
at each vertical level and calculating the ratio of the instantaneous
heating rate change at that level to the amplitude of the ozone
perturbation. The resulting profile of *β* is shown in Fig. 1b, with
typical values of 0.3–1.0 (K/d/ppmv), decreasing in altitude away from the
tropopause. The vertical structure of ${\mathit{\beta}}^{\prime}=({X}_{z}/S$)
*β* is included in Fig. 1a, showing a magnitude somewhat smaller than
*α* throughout the profile. This in turn implies a positive
(${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}}{)}_{\text{phase}}$ from Eqs. (8) to (9b) (including a
realistic small *δ*); i.e., temperature leads ozone in the coupled
response based on these parameters, although as shown below the phase
difference turns out to be small. Vertical profile of the quantity ($\mathit{\beta}/\mathit{\alpha})$ (zero frequency limit for (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$, for small *δ*) is included in Fig. 1c, showing a decrease from the
tropopause to the middle stratosphere with values substantially smaller than
($S/{X}_{z})$.

We derived an estimate of the damping rate *δ*(*z*) for ozone from
simulations of the Whole Atmosphere Community Climate Model (WACCM; Marsh et al., 2013), which includes a sophisticated stratospheric ozone chemical
scheme. These calculations use daily zonal average output of ozone amount
(*X*) and photochemical ozone loss rate (*L*) as a function of latitude
and altitude, and we take an annual average of their ratio: $\mathit{\delta}\left(z\right)=(L/X)$, averaging results over 10^{∘} N–S. The resulting vertical
profile of *δ* is shown in Fig. 1a, showing very small damping (long
ozone lifetimes) in the lower stratosphere, increasing to slightly larger
values in the middle stratosphere (damping timescale of ∼ 30 d at 30 km). Calculations below show idealized model results including
these realistic values of *δ*, and for comparison we also include
results for *δ*=0. Including realistic values of ozone damping has
almost no influence on calculations in the lower stratosphere because of the
very small damping. Damping can have a small but noticeable effect at higher
altitudes for lower-frequency variations (increasing the ${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}}$ ratios) but still only accounts for a
∼ 10 % effect.

## 4.1 Annual and QBO variability in SHADOZ ozone and temperature

The approximately monthly sampling of SHADOZ data allows for characterization of the annual cycle and interannual variations of tropical stratospheric ozone and temperature. There is a relatively large annual cycle in ozone and temperature in the tropical lower stratosphere over ∼ 16–22 km, with relative maxima during boreal summer and temperature slightly leading ozone in time. Figure 2 shows this behavior for the 19 km level, near the peak of the annual cycle. This correlated ozone–temperature behavior is mainly a response to the annual cycle in tropical upwelling (Randel et al., 2007); horizontal transport from the boreal summer monsoons also contributes to the seasonal maximum in ozone close to the tropopause (Konopka et al., 2009, 2010; Stolarski et al., 2014; Tweedy et al., 2017), but mean upwelling is the dominant mechanism above 18 km (Abalos et al., 2013). Above 23 km, the annual cycle becomes small and the dominant seasonal variation becomes semiannual in both ozone and temperature. We note that the seasonal variations in Fig. 2 are very similar based on the MLS ozone and GPS temperature data (not shown).

Interannual anomalies in ozone and temperature from SHADOZ data over 1998–2016 are shown in Fig. 3, derived by simply subtracting the mean annual cycle. In Fig. 3 ozone anomalies are shown in terms of ozone density (DU/km) instead of mixing ratio, in order to emphasize variations throughout the lower stratosphere. As is well known, there are strong downward propagating anomalies in ozone and temperature linked to the QBO; the ozone and temperature anomalies are approximately in phase over ∼ 17–27 km, and the variations in ozone are small above 27 km due to a transition from dynamical control in the lower stratosphere to photochemical control above ∼ 27 km (e.g., Chipperfield and Gray, 1992; Park et al., 2017). Episodic ENSO events also result in correlated ozone–temperature variations in the tropical lower stratosphere for levels from the tropopause to ∼ 22 km (Randel et al., 2009; Calvo et al., 2010). The constructive interference of QBO and ENSO effects can result in large anomalies near and above the tropopause (e.g., Diallo et al., 2018), as seen for the SHADOZ data in 1999–2000 and 2015–2016.

Figure 4 shows the ozone–temperature correlation derived from the deseasonalized SHADOZ data (from Fig. 3) as a function of altitude and time lag. Strong positive correlations (> 0.8) are found over 17–27 km, as expected from Fig. 3. The strongest correlations occur near zero time lag, but the lag correlations are skewed towards positive lags, which is a signature of temperature leading ozone anomalies by a small amount, similar to the annual cycle in Fig. 2.

A scatterplot of the SHADOZ temperature vs. ozone deseasonalized anomalies
at 24 km over 1998–2016 is shown in Fig. 5, highlighting the strong observed
correlation. The slope of the (*T* $/$ O_{3}) variations is near 6.1 (K/ppmv). This
slope changes as a function of altitude (as shown below), and this is one of
the quantities that we aim to understand from a simple perspective.

## 4.2 Satellite observations

We use MLS and GPS satellite data to quantify ozone–temperature correlations
over a continuous range of time scales from days to over a decade. Time
series of zonal mean GPS temperatures and MLS ozone over the Equator
(10^{∘} N–S) at 24 km (31 hPa for MLS) are shown in Fig. 6 for pentad
averages covering September 2004–March 2018. Visual inspection of Fig. 6
shows the clear signature of the QBO (as in Fig. 3) and strong correlations
of ozone and temperature across all scales of variability, including both
long- and short-term fluctuations.

Power spectra for temperature and ozone at 24 km are shown in Fig. 7a. In
these and the following spectral plots, the ordinate shows the wave period
(from 10 to 4990 d) using a logarithmic frequency axis in order to more
clearly separate low- and high-frequency behavior. The spectra for both
quantities show the most power at low frequencies, with peaks linked to the QBO,
annual and semiannual cycles. At altitudes below 24 km the annual cycle is
more pronounced, while above 24 km the semiannual cycle is larger. Power
decreases systematically at periods shorter than semiannual for both ozone
and temperature in Fig. 7a. Temperature-ozone coherence squared (coh^{2})
at 24 km is shown in Fig. 7b, highlighting significant values over nearly
the entire range of periods longer than ∼ 20 d. There is a
relative minimum in coh^{2} near the semiannual cycle, and this could
possibly be related to the semiannual variation in low-latitude ozone
photochemical production noted above, which adds additional ozone
variability that is less coherent with temperature. The reason for the lack
of coherence at the shortest resolved timescales (< 20 d) is
unknown but could be related to the very low power in both data sets (Fig. 7a)
and poorer temporal resolution of these time scales based on pentad data.
There is a relatively small phase difference between ozone and temperature
over all frequencies, as shown below. Similar behavior is found for
temperature-ozone coh^{2} and phase for all altitudes over 17–27 km. Above
29 km there is a strong coh^{2} maximum for the semiannual oscillation
(∼ 180 d period), where ozone and temperature are
approximately out of phase (not shown). This behavior is due to the
transition to photochemical control of ozone and the impact of temperature
on the odd-oxygen (O_{x}) loss rate (Brasseur and Solomon, 2005).

We next compare frequency-dependent (${X}_{\mathit{\sigma}}/{T}_{\mathit{\sigma}})$
amplitude and phase between observations and results from the idealized
model calculations (Eq. 9a–b). Figure 8a compares observed and modeled 24 km (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ amplitude as a function of
frequency. Observations are based on the GPS temperature and MLS ozone results,
where (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ is calculated from the
respective harmonic coefficients and the ratio is smoothed in frequency.
Model results are shown with and without including ozone damping effects,
which has relatively small influence, along with calculations neglecting
ozone feedback effects on temperature (*β*=0). Additionally, Fig. 8a
includes the (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ ratio estimated from
deseasonalized SHADOZ anomalies (from Fig. 5) which are mainly associated
with the QBO (∼ 28-month period). The observed (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ ratio shows a systematic change over the frequency
range, with approximately a factor of 2 decrease in the ratio for low
frequencies (periods > 150 d) compared to high frequencies.
The idealized model results show a similar (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ frequency dependence, albeit with substantial disagreement on the
detailed shape of the transition region between semiannual and interannual
(QBO) periods, with a slower transition in the model. This disagreement is
not understood but might be related to the semiannual ozone photochemical
production term over the Equator discussed above, which is not included in
the model calculations. The overall systematic change with frequency in Fig. 8a corresponds to the change from ozone–temperature coupling via transport
(high frequency) to radiative balance (low frequency). Including the ozone
damping (*δ*) slightly improves the agreement at low frequencies.

Differences between the full model and *β*=0 results in Fig. 8a
quantify the ozone feedback on temperature in the coupled system. Ozone
radiative feedback is mainly important for low-frequency (“slow”)
variability and becomes increasingly important for the longest timescales.
For example, for the QBO time period (28 months) the ozone feedback
increases the (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ ratio at 24 km by
approximately 40 %, with even larger effects at lower frequencies. We note that
this increasing importance of ozone feedbacks for low frequencies is
consistent with the results of Charlesworth et al. (2019).

The observed and modeled (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ phase
relationship as a function of frequency is shown in Fig. 8b, showing
approximately in-phase behavior across all frequencies in both cases. The
model (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ phase is slightly positive (as
expected from Sect. 3) and in approximate agreement with observed
results. Similar behavior to Fig. 8 is found in the satellite data for all
altitudes over 19–27 km. Neglecting ozone feedbacks (*β*=0) gives
worse agreement with observations.

Vertical profiles of observed and modeled (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ amplitude are shown in Fig. 9 for three different frequency bands, corresponding to “fast” frequencies (30–60 d period, Fig. 9a), the annual cycle (Fig. 9b) and the QBO (Fig. 9c). In addition to observations from the MLS and GPS data, we include the corresponding ratios calculated from SHADOZ ozone and temperature data for the annual cycle (e.g., Fig. 2a), calculated as the ratio of the respective ozone and temperature maximum-minimum values over the annual cycle, for altitudes 19–23 km where the annual cycle is distinct in the data. Figure 9c includes SHADOZ results for deseasonalized anomalies, which are mainly linked to the QBO and derived from regression as in Fig. 5. These SHADOZ results in Fig. 9b–c agree well with the corresponding estimates from MLS and GPS satellite data. The fast frequencies (Fig. 9a) are governed by vertical transport with a (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ vertical profile close to ($S/{X}_{z})$ (Fig. 1c), and the model shows a good fit to the observed vertical structure, at least up to ∼ 27 km. The annual cycle (Fig. 9b) is close to the cross-over between high- and low-frequency behavior, and the model again shows approximate agreement to observations over altitudes where the annual cycle is large (∼ 19–23 km). This agreement helps confirm the interpretation that the annual cycles in tropical stratospheric temperature and ozone (e.g., Fig. 2) can be interpreted as coupled responses to the annual cycle in tropical upwelling, with ozone feeding back on temperature. For the lower-frequency QBO variations (Fig. 9c) the idealized model shows good agreement with the (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ amplitude from both the satellite data and SHADOZ throughout the profile. Our conclusions from these comparisons is that the idealized model can quantitatively explain the observed (${T}_{\mathit{\sigma}}/{X}_{\mathit{\sigma}})$ amplitude and phase relationships in the tropical lower stratosphere, including their dependence on frequency and altitude.

Observations show strong correlations between ozone and temperature in the
tropical lower stratosphere, and calculations show that the ozone radiative
feedbacks significantly enhance temperatures, e.g., by ∼ 30 %
for the annual cycle (e.g., Ming et al., 2017). This ozone feedback
significantly enhances thermal variability in global model simulations (Yook
et al., 2020). The goals of this work include providing an update of
observational evidence for *T*–O_{3} coupling and simplified understanding based
on idealized zonal mean theory. The excellent long-term tropical ozonesonde
measurements from SHADOZ demonstrate approximate in-phase *T*–O_{3} correlations
for the annual cycle (Fig. 2) and for interannual anomalies (Figs. 3–5),
which are dominated by the QBO. Long-term continuous satellite measurements
from zonal average MLS and GPS data agree well with these results for annual
and interannual variations, and furthermore demonstrate strong *T*–O_{3}
coherence for faster sub-seasonal variability (Figs. 6–7b). This coherent
behavior is observed throughout the lower to middle stratosphere,
∼ 17–27 km, with *T*–O_{3} anomalies approximately in phase over
all altitudes. A key result is that the observed (*T* $/$ O_{3}) ratio changes as a
function of frequency, with approximately half the ratio for low frequencies
(annual cycle and longer) compared to faster variability (Fig. 8a). The
(*T* $/$ O_{3}) ratio also depends on altitude, with much larger ratios for levels in
the lower stratosphere (Fig. 9). These are the key observational
characteristics of *T*–O_{3} coupling that we seek to explain.

We compare observations to results from idealized zonal mean theory,
assuming vertical advection from the upward Brewer–Dobson circulation
controls thermal balances and ozone transport, i.e., neglecting mean
meridional advection and eddy transport terms. This is a reasonable
approximation in the tropical lower stratosphere above the tropical tropopause layer (TTL) (Abalos et al., 2013), although eddy transport from monsoon circulations makes important
contributions to ozone tendencies during boreal summer at and below
∼ 18 km (Konopka et al., 2009, 2010; Stolarski et al., 2014).
Thermodynamic balance includes linear radiative damping (*α*) and
ozone feedback (*β*) terms, and the coupled equations (including linear
ozone damping *δ*) can be solved analytically to calculate the (*T* $/$ O_{3})
ratio as a function of frequency and altitude, dependent on model parameters
*α*, *β* and *δ* and the ratio of background gradients
expressed as ($S/{X}_{z})$. In general, ozone damping is a minor
influence over most of the domain because of the long ozone lifetimes. The
model balances highlight two timescales for *T*–O_{3} coupling, including fast
variability, where the (*T* $/$ O_{3}) ratio is determined by the background vertical
gradients ($S/{X}_{z})$, and slow timescales determined by radiative
balance ($\mathit{\beta}/\mathit{\alpha}$ in the zero-frequency limit). The idealized
model shows a functional frequency dependence for the (*T* $/$ O_{3}) ratio similar
to observations (Fig. 8a), although there is disagreement in the transition
region where the observations show a more rapid (*T* $/$ O_{3}) transition with a
slight minimum near the semiannual period. This detail is not well
understood but could be influenced by a semiannual ozone photochemical
production term in the equatorial region related to solar inclination
(Abalos et al., 2013) that is not included in our model, along with neglected
eddy transport effects, especially near the tropopause. This semiannual
ozone production may also explain the relative minimum in *T*–O_{3} coherence
squared near this frequency found in Fig. 7b. The vertical profiles of
(*T* $/$ O_{3}) ratio agree well with the observations for both fast (Fig. 9a) and
slow (Fig. 9b–c) timescales, enhancing confidence in a simple
understanding.

Ozone feedback on temperature is easy to quantify in our model by comparing
results neglecting the feedback (*β*=0). Results show important ozone
feedbacks for low-frequency variations (i.e., the slow regime), and the
feedback becomes increasingly important at lower frequencies (e.g., Fig. 8a).
We note that the frequency-dependent *T*–O_{3} ozone feedback shown here is
consistent with the results of Charlesworth et al. (2019), which show larger
ozone radiative impacts on tropopause temperatures for low frequencies.

One further aspect of coupled *T*–O_{3} behavior can be deduced from Eq. (3),
noting that the *α* and *β* terms are closely coupled by observed
*T*–O_{3} correlations in the lower stratosphere. Using the empirical
approximation $\mathrm{\Delta}X\sim \mathrm{\Delta}T({X}_{z}/S)$, where
$\mathrm{\Delta}X=\mathit{\beta}(X-{X}_{\text{eq}})$ and likewise
for Δ*T*, the combined terms ($-\mathit{\alpha}\mathrm{\Delta}T+\mathit{\beta}\mathrm{\Delta}X)$ in Eq. (3) can be rewritten as ($-\mathit{\alpha}\mathrm{\Delta}T+{\mathit{\beta}}^{\prime}\mathrm{\Delta}T)$. These terms can be combined into −*α*_{eff}Δ*T*, with ${\mathit{\alpha}}_{\text{eff}}=(\mathit{\alpha}-{\mathit{\beta}}^{\prime})$
representing an “effective” thermal damping timescale combining both
radiative relaxation and ozone feedback effects. Using realistic *α*
and *β*^{′} values (Fig. 1a), *α*_{eff} is significantly smaller
than *α* alone; i.e., the ozone feedback increases the radiative
damping timescale compared to radiative relaxation alone. This result is
consistent with the effective timescales inferred by Fueglistaler et al. (2014) and with the coupled chemistry–climate model calculations in Yook et al. (2020), their Fig. 6. Alternatively, since the ozone radiative feedback
arises primarily from transport effects, its effect can also be viewed as an
enhancement to the dynamical heating.

It is worthwhile to appreciate the limitations associated with the idealized
model calculations, especially uncertainties related to the parameters
*α* and *β*, which control the low-frequency model behavior.
While Hitchcock et al. (2010) show that linear regression on temperature
captures ∼ 80 % of the variance in modeled radiative heating
rates in the tropical lower stratosphere, the broad spectrum of vertical
scales in this region can introduce additional uncertainties in estimating
*α*. Our calculations of ozone heating via the *β* term in Eq. (3)
neglects the effects of non-local ozone changes, which will also depend in
detail on the vertical scale of perturbations. In spite of these caveats,
the overall agreement between model and observations demonstrates that the
idealized zonal mean theory (quantifying coupled *T*–O_{3} response to variations
in the Brewer–Dobson circulation) is a valid perspective to understand the
strong *T*–O_{3} coupling in the real atmosphere.

SHADOZ data were obtained from the SHADOZ website https://tropo.gsfc.nasa.gov/shadoz/ (last access: 10 December 2021). MLS ozone data were obtained from https://mls.jpl.nasa.gov/index-eos-mls.php (last access: 10 December 2021). GPS temperatures were obtained from the COSMIC Data Analysis and Archive Center (CDACC) website https://cdaac-www.cosmic.ucar.edu/ (last access: 10 December 2021).

The study was conceived by WJR, and data analysis was performed by FW. AM and PH provided input for the idealized model calculations and contributed to interpretation of results. The paper was written by WJR, with editing from AM and PH.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The National Center for Atmospheric Research is sponsored by the U.S. National Science Foundation. This work has been partially supported by the COSMIC NSF-NASA Cooperative Agreement under grant no. 1522830, and by the NASA Aura Science Team under grant no. 80NSSC20K0928. AM would like to acknowledge support from the Leverhulme Trust as an early career fellow. We thank Marta Abalos, Rolando Garcia, Paul Konopka, and Lan Luan for discussions and comments that significantly improved the manuscript and two anonymous reviewers for constructive reviews.

This research has been supported by the NASA Earth Sciences Division (grant no. 80NSSC20K0928).

This paper was edited by Rolf Müller and reviewed by two anonymous referees.

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- Abstract
- Introduction
- Data and analyses
- Simplified zonal mean theory
- Ozone and temperature observations
- Comparisons with idealized model calculations
- Summary and discussion
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References

- Abstract
- Introduction
- Data and analyses
- Simplified zonal mean theory
- Ozone and temperature observations
- Comparisons with idealized model calculations
- Summary and discussion
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References