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

Abstract. Observations show strong correlations between large-scale ozone and temperature variations in the tropical lower stratosphere across a wide range of time scales. We quantify this behavior using monthly records of ozone and temperature data from SHADOZ tropical balloon measurements (1998–2016), along with global satellite data from Aura MLS and GPS radio occultation over 2004–2018. The observational data demonstrate strong in-phase ozone-temperature coherence spanning sub-seasonal, annual and interannual time scales, and the slope of the ozone-temperature relationship (O3/T) varies as a function of time scale 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 (O3/T) amplitude and phase relationships, including sensitivity to time scale and altitude, and highlight distinct balances for ‘fast’ variations (periods < 150 days, controlled by transport across background vertical gradients) and ‘slow’ coupling (seasonal and interannual variations, controlled by radiative balances).



Introduction
Large-scale ozone and temperature variations in the tropical lower stratosphere exhibit strong correlations across a range of time scales. 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;25 Baldwin et al, 2001;Witte et al, 2008;Hauchecorne et al, 2010) and El Nino Southern Oscillation (ENSO; Randel et al, 2009). Abalos et al (2012; and Gilford et al (2016) also note strong ozone-temperature correlations in this region across a range of time scales. Calculations have shown that the radiative effects of ozone feed back onto and enhance temperature variations, and this topic has been well-studied as related to the annual cycle in the tropical lower stratosphere https://doi.org/10.5194/acp-2021-514 Preprint. Discussion started: 8 July 2021 c Author(s) 2021. CC BY 4.0 License. (Chae and Sherwood, 2007;Fueglistaler et al, 2011;Ming et al, 2017;Gilford and Solomon, 2017), and also by Forster et al 30 (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 onedimensional 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 35 (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;, highlighting the control of upwelling for 40 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, and that 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 45 tropical stratosphere over a continuous range of time scales. 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 ozone-temperature (O3-T) coupling from observations in a relatively simple framework, including the frequency and altitude dependences of the (O3/T) amplitude and phase relationships. These results are a complement to the recent analyses of 50 Birner and Charlesworth (2017) and Charlesworth et al (2019), based on a very different model.

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 o N -20 o S, with measurements beginning in 1998 (Thompson et al, 2003). Recent 55 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 effective vertical resolution of ~50-100m. 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 60 profile exhibits a high degree of longitudinal symmetry (Thompson et al, 2003;Randel et al, 2007;Randel and Thompson, https://doi.org/10.5194/acp-2021-514 Preprint. Discussion started: 8 July 2021 c Author(s) 2021. CC BY 4.0 License. 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.

Aura MLS ozone and GPS temperature
Satellite ozone measurements from the Aura Microwave Limb Sounder (MLS) are analyzed for the period 65 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 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 o N-S, and we calculate zonal mean values for 5-day (pentad) averages. Some isolated data gaps are filled by linear interpolation in time. This provides a long, continuous time series of 70 MLS ozone covering 998 pentads (4990 days).
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 o N-S to match the MLS ozone time series discussed above. We focus on altitude levels 75 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.

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 80 for both ozone and temperature, resolving periods of 4990 to 10 days, with a frequency resolution of  = (2/4990 days).
Calculations are based on standard formulas in Jenkins and Watts (1968). Power spectra are smoothed with a Gaussianshaped smoothing window with half-width 2. Ozone-temperature amplitude ratios, coherence squared and phase spectra are calculated using a wider bandwidth (10) to enhance statistical stability. This results in approximately 10 independent Fourier harmonics for each spectral estimate, and the resulting 95% significance level for the coh2 statistic is 0.45. The high-85 and low-frequency ends of the spectra are smoothed using one-sided Gaussian smoothing so that significance levels are somewhat higher. https://doi.org/10.5194/acp-2021-514 Preprint. Discussion started: 8 July 2021 c Author(s) 2021. CC BY 4.0 License.
3 Simplified zonal mean theory

Coupled thermodynamic and ozone continuity equations
We explore the coupling of ozone and temperature based on the zonal mean thermodynamic and ozone continuity 90 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: ∂T/∂t = -v*(∂T/∂y) -w*S + eddy terms + Q (1) Here T is zonally averaged temperature, (v*, w*) are components of the residual meridional circulation, S is a stability 95 parameter, and Q is the zonal mean diabatic heating rate. In the tropical lower stratosphere the v* and eddy forcing terms are relatively small (Abalos et al, 2013), so that the approximate thermodynamic balance is: In this work we specify the zonal mean diabatic forcing Q with two components Q = Qrelaxation + Qozone, representing radiative relaxation and ozone forcing of temperature, respectively. We assume radiative relaxation is proportional to temperature, 100 Qrelaxation = - (T-Teq), with Teq an equilibrium temperature and  an inverse radiative damping time scale (Andrews et al, 1987;Hitchcock et al, 2010).  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: Qozone =  ( -eq). Here  is zonal mean ozone mixing 105 ratio, eq is a background equilibrium ozone value and  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(t) = ∑ Texp it , with  the angular frequency (2/period), and likewise for w*(t) and (t), and assuming Teq, eq and S are constant in time, Eq. 3 can be rewritten as an equation for each harmonic 110 component: A similar analysis is applied to the zonal mean ozone continuity equation (Andrews et al, 1987, Eq. 9.4.13): ∂∂t = -v*(∂/∂y) -w*(∂/∂z) + eddy terms + P -L Here P-L represents chemical ozone production minus loss terms. In contrast to the thermodynamic balance discussed above, 115 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(Konopka et al, , 2010Abalos et al, 2013). However, for simplicity in our idealized calculations the eddy terms are neglected here, along with the v* term. This yields: https://doi.org/10.5194/acp-2021-514 Preprint. Discussion started: 8 July 2021 c Author(s) 2021. CC BY 4.0 License. 120 where we have defined z = (∂/∂z). In the tropical lower stratosphere ozone production minus loss is positive and is relatively constant in time, with a small semi-annual variation in production following solar inclination (Abalos et al, 2013).
We parameterize ozone loss as L = - (eq), where  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 z, the harmonic expansion of Eq. 6 is then given by the simple balance: 125 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 (Eq. 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 130 (Abalos et al, 2012;. Temperature is furthermore influenced by radiative damping ( term) and the radiative response to ozone changes ( term), while ozone balance includes damping ( term). Equations 4 and 7 can be combined to eliminate the w* dependence to obtain a single equation relating ozone and temperature harmonic components, in particular the ozone/temperature ratio as a function of frequency: with ' = (z/S) . This can be rewritten as: Here (z/S) is a key parameter related to the ratio of ozone vertical gradient to background stability (potential temperature 140 gradient), which is derived directly from the time average temperature and ozone profile data; the vertical profile of (z/S) is shown in Fig. 1c. We note some small (~10%) seasonal and interannual variations to the individual z and S terms in the tropical lower stratosphere, but these follow each other and the ratio (z/S) is more nearly constant. Equation 8 can be rewritten as expressions for (/T) amplitude and phase: Our analyses focus on evaluating the quantity (T) as a metric for ozone-temperature coupling, and below we test results from this idealized model with (/T) amplitude and phase derived from observations. The observational data are based on measurements from SHADOZ and MLS/GPS in the deep tropics over ~10 o N-S, and hence 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) ratio in Eq. 8 is a generally complex function of 'and (z/S), but it is useful to consider the high-and low-frequency limits (compared to the inverse time scales ' and ). For high frequencies ( >> ',), the (/T) ratio simplifies to ~ (z/S), i.e. the ozone and temperature anomalies are in phase, with a ratio simply related to the 155 background gradients in ozone and potential temperature. For the low frequency limit ( ~ 0), (/T) ~ (z/S) ('+)which simplifies to() for small . This expresses an in-phase balance of ozone and temperature associated with the  and  radiative terms in the thermodynamic equation (Eq. 3), i.e. heating from ozone anomalies balances radiative cooling.

Estimating ,  and  from model calculations 160
Our calculations use a vertical profile of  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  used here is consistent 165 with other published estimates, e.g. Newman and Rosenfield (1997) and Randel and Wu (2015). We estimate vertical profiles of the parameter  from radiative transfer calculations using a modified version of the 170 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.
clear-sky conditions. 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 175 resulting profile of  is shown in Fig. 1b, with typical values of 0.3-1.0 (K/day/ppmv), decreasing in altitude away from the tropopause. The vertical structure of ' = (z/S)  is included in Fig. 1a, showing a magnitude somewhat smaller than  throughout the profile. This in turn implies a negative (T)phase from Eqns. 8-9b (including a realistic small ), i.e. ozone leads temperature 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 ( (zero frequency limit for (T), for small ) is included in Fig. 1c,  180 showing increase from the tropopause to the middle stratosphere with values substantially larger than (z/S).
We derived an estimate of the damping rate (z) for ozone from simulations of the Whole Atmosphere Community The approximately monthly sampling of SHADOZ data allows 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 195 in the tropical lower stratosphere over ~16-22 km, with relative maxima during boreal summer and ozone having a slight phase delay compared to temperature. Figure 2a shows this behavior for the 18 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(Konopka et al, , 2010Stolarski et al, 2014;Tweedy et al, 2017), but mean upwelling is the dominant 200 mechanism at and above 19 km (Abalos et al, 2013). Figure 2b shows the corresponding seasonal cycles at 24 km, highlighting mirror-image variations in ozone and temperature with a stronger semi-annual variation than that at lower altitudes (Fig. 2a) Interannual anomalies in ozone and temperature from SHADOZ data over 1998-2016 are shown in Fig. 3, derived 210 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 215 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

230
Positive lag denotes ozone leading temperature.
A scatter plot of the SHADOZ ozone-temperature deseasonalized anomalies at 24 km over 1998-2016 is shown in Fig. 5, highlighting the strong observed correlation. The slope of the (O3/T) variations is near 0.14 (ppmv/K). 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 235 perspective. Figure 5. Scatter plot of deseasonalized ozone and temperature anomalies from SHADOZ data at 24 km (data as in Fig. 3).

Satellite observations 240
We use MLS and GPS satellite data to quantify ozone-temperature correlations over a continuous range of time inspection of Fig. 6 shows a clear signature of the QBO (as in Fig. 3), and strong correlations of ozone and temperature across all scales of variability. 245 Power spectra for ozone and temperature at 24 km are shown in Fig. 7a. The spectra for both quantities show most power at low frequencies, with peaks linked to the QBO, annual and semi-annual cycles. At altitudes below 24 km the 250 annual cycle is more pronounced, while above 24 km the semi-annual cycle is larger (e.g. Fig. 2). Power decreases systematically at periods shorter than semi-annual for both ozone and temperature in Fig. 7a. Ozone-temperature 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 days. There is a relative minimum in coh 2 near the semi-annual cycle, and this could possibly be related to the semi-annual variation in low-latitude ozone photochemical production noted above, which adds additional ozone variability 255 that is less coherent with temperature. The reason for the lack of coherence at the shortest resolved time scales (<20 days) is unknown, but could be related to very low power in both data sets (Fig. 7a) and poorer temporal resolution 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 ozone-temperature coh 2 and phase for all altitudes over 17-27 km. Above 29 km there is a strong coh 2 maximum for the semi-annual oscillation (~180 days period), where ozone and temperature are approximately 260 out of phase (not shown). This behavior is due to the transition to photochemical control of ozone and the impact of temperature on the Ox loss rate (Brasseur and Solomon, 2005).  observed (T) ratio shows a systematic change over the frequency range, with approximately a factor of two increase in the ratio for low frequencies (periods > 150 days) compared to high frequencies. There is a relative maximum in the (T) 275 ratio near the semi-annual period; the cause of this feature is not understood but might be related to the semi-annual ozone photochemical production term over the equator discussed above, which is not included in the model calculations. The idealized model results show a similar (T) frequency dependence, although with substantial disagreement on the shape of the transition region between semi-annual and interannual (QBO) periods, with a much smoother transition in the model.
The systematic change with frequency corresponds to the change from ozone-temperature coupling via transport (high 280 frequency) to radiative balance (low frequency). Including the ozone damping () improves the agreement at low frequencies. The observed and modeled (T) 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) phase is slightly negative (as expected from Section 3), while the observed values are near zero or slightly positive. Similar behavior to Fig. 8

290
Vertical profiles of observed and modeled (T) amplitude are shown in Fig. 9 for three different frequency bands, corresponding to 'fast' frequencies (30-60 days period, Fig. 9a), annual cycle (Fig. 9b) and QBO (Fig. 9c). In addition to observations from the MLS/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 295 maximum-minimum values over the annual cycle, for altitudes 17-23 km where the annual cycle is distinct in the data.  . 1c), and the model shows a good fit to the observed vertical structure, at least up to ~27 km. The annual cycle (Fig. 9b) 300 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 (~17-23 km). This agreement helps confirm the interpretation that the annual cycles in tropical stratospheric ozone and temperature (e.g. Fig. 2) can be interpreted as coupled responses to the annual cycle in tropical upwelling, with ozone feeding back on temperature. There is poorer agreement in Fig. 9b for the annual cycle above 23 km, but over these altitudes the actual variability has mainly a semi-annual component. For the lower 305 frequency QBO variations (Fig. 9c) the idealized model shows good agreement with the (T) amplitude from both the satellite data and SHADOZ with a peak near 27 km, and note that including ozone damping () slightly improves the agreement above ~25 km. Our conclusions from the comparisons is that the idealized model can quantitatively explain the

Summary and discussion
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 320 al, 2020). The goals of this work include providing an update of observational evidence for O3-T coupling and simplified understanding based on idealized zonal mean theory. The excellent long-term tropical ozonesonde measurements from SHADOZ demonstrate approximate in-phase O3-T correlations for the annual cycle (Fig. 2) and for interannual anomalies , 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 O3-T 325 coherence for faster sub-seasonal variability . This coherent behavior is observed throughout the lower to middle stratosphere, ~17-27 km, with O3-T anomalies approximately in phase over all altitudes. A key result is that the observed (O3/T) ratio changes as a function of frequency, with approximately twice the ratio for low frequencies (annual cycle and longer) compared to faster variability (Fig. 8a). The (O3/T) ratio also depends on altitude, with larger ratios in the middle stratosphere (Fig. 9). These are the key observational characteristics of O3-T coupling that we seek to explain. 330 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 (Abalos et al, 2013), although eddy transport from monsoon circulations makes important contributions to ozone tendencies during boreal summer at and https://doi.org/10.5194/acp-2021-514 Preprint. Discussion started: 8 July 2021 c Author(s) 2021. CC BY 4.0 License. below ~18 km (Konopka et al, 2009(Konopka et al, , 2010Stolarski et al, 2014). Thermodynamic balance includes linear radiative damping 335 () and ozone feedback () terms, and the coupled equations (including linear ozone damping ) can be solved analytically to calculate the (O3/T) ratio as a function of frequency and altitude, dependent on model parameters and and the ratio of background gradients expressed as (z/S). In general, ozone damping is a minor influence over most of the domain because of the long ozone lifetimes. The model balances highlight two time scales for O3-T coupling, including 'fast' variability where the (O3/T) ratio is determined by the background vertical gradients (z/S) and 'slow' time scales 340 determined by radiative balance (). The idealized model shows a functional frequency dependence for the (O3/T) ratio similar to observations (Fig. 8a), although there is disagreement in the transition region where the observations have a more rapid (O3/T) transition with a relative maximum 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. 345 This semi-annual ozone production may also explain the relative minimum in O3-T coherence squared near this frequency found in Fig. 7b. The vertical profiles of (O3/T) ratio agree well with the observations for both fast (Fig. 9a) and slow (Figs. 9b-c) time scales, enhancing confidence in a simple understanding. We note that the frequency-dependent O3-T behavior shown here is consistent with the results of Charlesworth et al (2019), which indicate larger ozone radiative impacts on tropopause temperatures for low frequencies. 350 It is worthwhile to appreciate the limitations associated with the idealized model calculations, especially uncertainties related to the parameters  and 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  Our calculations of ozone heating via the  term in Eq. 3 neglects the effects of non-local ozone changes, 355 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 O3-T response to variations in the Brewer-Dobson circulation) is a valid perspective to understand the strong O3-T coupling in the real atmosphere.