A mid-latitude stratosphere dynamical index for attribution of stratospheric variability and improved ozone and temperature trend analysis

We find that wintertime temperature anomalies near 4 hPa and 5 0◦N/S are related, through dynamics, to anomalies in ozone and temperature, particularly in the tropical stra tosphere, but also throughout the upper stratosphere and me sosphere. These mid-latitude anomalies occur on timescales of up to a m nth, and are related to changes in wave-forcing. A change in the meridional circulation extends from the middle stratosphe re into the mesosphere and forms a temperature-change quadr upole from equator to pole. We develop a dynamical index based on de trended, deseasonalised mid-latitude temperature. When 5 employed in multiple linear regression, this index can acco unt for up to 40% of the total variability of temperature and o zone and a doubling of the total coefficient of determination in th e equatorial stratosphere above 20 hPa. Further, the uncert ainty on all multiple-linear regression coefficients can be reduced by up to 45% and 25% in temperature and ozone, respectively, a nd so this index is an important tool for quantifying current and f uture ozone recovery.

We thank both referees for very helpful guidance and suggestions in the background, justification and statistical analysis performed.This has led to significant improvements in the quality of the manuscript (which are already apparent in our current revisions and in our response below).Major changes in the revised manuscript have been highlighted in bold text.
We begin by highlighting major updates that both reviewers should be aware of: i) Figures 10, 11, 12 and 13 have been updated to reflect the use of AR1 instead of AR0. Figure 13 in particular has been completely changed, and numbers added, to clarify the error improvement and a change in the mean values.Figure 12 has been updated to reflect similar style and information as provided in Figure 11.ii) The point made in the manuscript about no aliasing between regressors being shown by the relative importance plots has been modified.Due to the use of AR1, for temperature, there is a redistribution of the relative importance from the original regressors without the new index to the new one, in addition to the increase in total variance accounted for.However, the fact it does not change the mean value of the regression coefficients in the trend still supports the claim that it does not alias the derived signal.Modifications to the text in the manuscript have been made to reflect this change, and discussion on this point has been added.iii) A significant amount of background have been added to the introduction.iv) We have renamed the MLSD index to the Upper-branch Brewer Dobson Circulation (UBDC) index, to reflect a more direct interpretation of what it represents, and for which it should be more easily understood; this changes the title of the manuscript.
Paper's main finding is coherence in the variability of stratospheric temperature and ozone in the tropics and extratropics and in the upper stratosphere and lower mesosphere.The authors attribute this coherence to dynamics, specifically to the stratospheric meridional (Brewer-Dobson) circulation, and propose that an index accounting for dynamical effects could be used in multiple regression analysis as additional regressor.They further build such an index using extratropical upper stratospheric temperatures and demonstrate that the index explains considerable fraction of variability in stratospheric ozone and temperatures.Although the authors present interesting analysis, they still have to show how their analysis is related to previous research and high light novel results.The use of regressors accounting for dynamical effects has been discussed in previous WMO Ozone Assessments, discussing their pros and cons.I believe that a more thorough discussion of issues associated with the use of dynamical proxies, as well as relation of the current analysis with previous studies is needed before possible publication in ACP.Please see my specific comments below.
Major comments 1. Various dynamical proxies have been used in past to explain stratospheric variability related to dynamics, see examples in Weiss, et al. 2001;Brunner et al. 2006;Mäder et al., 2007;Wohltmann et al., 2005;2007 and references therein.While a considerable fraction of variability in both ozone and temperatures can indeed be explained by these proxies, this benefit comes at the cost of attributing variability to processes which are themselves dependent on the variables to be explained (wave propagation depends on the mean state of the stratosphere), i.e. one mixes cause and effect.I suggest that these issues should be discussed in the manuscript.Relevant discussion regarding the use of dynamical proxies for attributing ozone variability can be found in Chapter 2 of WMO ozone Assessment 2011 (Sections 2.1.2 and 2.4).
We agree, and we appreciate the useful set of references that has led to an expansion of the discussion in the manuscript.Further, the new background material simply further highlights the need for such a dynamical proxy, especially in the equatorial region, since previous studies have focused on BDC proxies that operate on inter-annual timescales and longer, and while often being briefly mentioned, the monthly and shorter timescales are usually ignored (except, e.g.Chandra et al., 1986 as mentioned by the second reviewer).Previous studies have also focused on total ozone column, mid-to-high latitudes, and the mid-to-lower stratosphere.This leads to the clear conclusion that, while not a new concept, the development of a proxy accounting for noise-like dynamical events in the upper stratosphere and mesosphere is necessary, and that its application and focus on the equatorial region is new. References: 1. Brunner, D., J. Staehelin, J.A. Maeder, I. Wohltmann, and G.E. Bodeker, Variability and trends in total and vertically resolved stratospheric ozone based on the CATO ozone data set, Atmos. Chem. Phys.,6 (12), 4985-5008, doi: 10.5194/acp-6-4985-2006, 2006. 2. Mäder, J.A., J. Staehelin, D. Brunner, W.A. Stahel, I. Wohltmann, andT. Peter, Statistical modeling of total ozone: Selection of appropriate explanatory variables, J. Geophys.Res., 112, D11108, doi: 10.1029/2006JD007694, 2007. 3. Wohltmann, I., M. Rex, D. Brunner, and J. Mäder (2005), Integrated equivalent latitude as a proxy for dynamical changes in ozone column, Geophys.Res. Lett., 32, L09811, doi:10.1029/2005GL022497. 4. Wohltmann, I., R. Lehmann, M. Rex, D. Brunner, andJ. Mäder, A processoriented regression model for column ozone, J. Geophys.Res., 112, D12304, doi: 10.1029/2006JD007573, 2007. 5. Weiss, A. K., J. Staehelin, C. Appenzeller, and N. R. P. Harris (2001), Chemical and dynamical contributions to ozone profile trends of the Payerne (Switzerland) balloon soundings, J. Geophys.Res., 106(D19), 22685-22694, doi:10.1029/2000JD000106 6. WMO: Scientific Assessment of Ozone Depletion: 2010, Global Ozone Research and Monitoring Project, 52, 516, 2011.2. There are also problems with using temperature as a proxy representing extratropical wave dynamics.Stratospheric temperature is controlled by a number of processes, such as horizontal and vertical advection, diabatic heating, and not all variability is necessarily directly attributable to extratropical wave forcing.Constructing an index by maximizing correlation, as is done in this study, also maximizes the risk of mixing statistical noise with physical processes.That is why using proxies more directly related to wave activity could be a better choice.While I agree that wave activity proxies such as EP-flux divergence are difficult to calculate, one can try, for example, heat flux evaluated at 100hPa (e.g.Newman et al. 2001), which is quite easy to calculate.
Again, we agree with this assessment and indeed we will make it clear that we are mixing in physical processes from different specific sources (see previous response) in the introduction.Further, it would make sense that if wave-driving from the troposphere at mid-latitudes is one of the main drivers of variance in the equatorial upper stratosphere, then the use of EP-flux divergence (EPFD), or the heat flux at 100 hPa would represent a more physical proxy.We note, however, that while Newman et al. (2001) were successful in representing short-term dynamical fluctuations in the stratosphere with EPFD, they did not investigate effects above 10 hPa.During the analysis of our original manuscript, we investigated the relationship between 100 hPa heat flux and equatorial ozone and temperature in the upper stratosphere, but were unable to find any clear agreement, even if we considered a lag to account for the time waves take to propagate into the stratosphere and force a response.We revisited this following the reviewer's suggestion (above) and considered the correlation between indices of the NAO, AAO, ENSO, QBO, and 100 hPa heat flux (v'T') averaged between 60-90 S, 60-90 N, 45-75 N and 45-75 S. We further divided months out to consider Dec-Feb and Nov-Apr for the northern hemisphere, and Jun-Aug and May-Oct for the southern hemisphere, and either the original timeseries or detrended and deseasonalised.We compared all of these cases with the MLSD index, which has high agreement locally to the respective hemisphere and with the equatorial upper stratosphere and mesosphere.Considering just the R^2 values (i.e.correlation coefficient squared), we found agreement exceeding 0.15 only in three cases: for DJF 60-90 N after deseasonalising and detrending the data, and 0.18 and 0.19 for DJF 45-75 N with and without deseasonalising and detrending (the third value shown is also similar for a 1 month lag); there was nothing clear in the southern hemisphere, with values all close to zero.This indeed suggests that there is possibly some relationship between heat flux at 100 hPa, and we concede this was a simplistic set of checks.However, the three results showing some coherence with the MLSD index account for very little of the variance we see in temperature above 10 hPa.The source of the variance warrants further investigation beyond this manuscript, as our analysis clearly shows a relationship with changes in temperature in the upper stratosphere and mesosphere related to what appears to be a wave-forcing like response in the EPFD and stream functions (i.e.Figs 5 and 6).
We have expanded this section to a separate discussion of equatorial and higher latitude MLR analysis, which, e.g.includes other proxies such as the AO and NAO, but also see our response to Major Comment 1.
2. P2L118: I believe there are older references which show influence of dynamics on stratospheric ozone, e.g.Fusco and Salby 1999 and references therein.
We have included this reference in addition to others, and those already mentioned above.
Reference: Fusco, A. C. and Salby, M. L.: Interannual variations of total ozone and their relationship to variations of planetary wave activity, J. Clim., 12, 1619Clim., 12, -1629Clim., 12, , 1999. .3. P2L22-23: Please note that acceleration of BD circulation leads not only to increase of ozone in the extratropics but also to a decrease in the tropics, thus it is more correct to say that ozone is redistributed, not just increased.
We suggest this can be made clear with the addition of the following bold text: "The increase in ozone at mid-latitudes comes partly from ODSs reductions, but also because the BDC is expected to accelerate (Garcia and Randel, 2008;Butchart, 2014), which will reduce the time for ozone depletion to occur and lead to faster transport of ozone from the equatorial region to higher latitudes.This in turn leads to a reduction of ozone over the equator and a prevention of a full recovery over the tropics.Thus, the recovery of ozone at mid-to-high latitudes can be understood as being partly due to less ozone destruction by lower ODS concentrations, and partly due to a faster redistribution of ozone-rich air from the tropics." 4. P3L27-28: I think smoothing removes short-term variability, not long-term.Please rewrite.
The sentence was incorrectly formulated, and should make clear we remove the smoothed time series from the original; thus the following formulation should be clearer.
"…we remove all long-term variability by subtracting a timeseries that has been smoothed, with a 13-month running mean, and then deseasonalised, with monthly values, at each latitude and pressure." 5. P8L4-6: Please see Major Comment 2. I think some caution is needed when using stratospheric temperature as proxy for dynamics.
See response to major comments 1 and 2 and revisions of the text.
6. P9L6: 'Verses' -> 'versus' Done 7. Figure 10: The difference in Fig. 10b between regression results from GOZCARDS and SWOOSH from the one hand and SBUV from the other hand are interesting.It appears like dynamical variability in GOZCARDS and SWOOSH is represented by the other proxies because, after addition of the dynamical proxy, the explained variability changes only little in these data sets, and the total explained variability is quite similar in all four data sets.Do you think it is purely statistical effect or it may be related to the way these data sets are compiled?(Sorry I am not familiar with these data sets.) This is an entirely valid, and interesting, question.As you suggest, we believe (and have evidence to support) that the answer resides in the way the datasets are compiled.Indeed, looking at individual time series, it becomes clear that the earlier periods in GOZCARDS and SWOOSH at these altitudes contain high variance fluctuations that look more related to the datasets used themselves than real variability; GOZCARDS and SWOOSH use a similar source of data (SAGE II) for this period.Given the variance is on the order of that in the MLSD index, it is likely (and this is now a postulation) that the reduced improvement during this period is due to these high variance artefacts.We are tackling this problem and are due to submit an article relating to this soon.
8. Figure 11: I am puzzled by why the annual R2 for the w/ MLDS regression in the middle panel is larger than any seasonal one.The result from the w/o regression, where the annual R2 looks like the mean of seasonal results, looks more logical, is it not?
As you correctly identified, there was a mistake in how anomalies were dealt with in the regression routine.This has been corrected, and indeed the results now appear more logical.9. Captions to Figure 11: What is distribution peak?Is it the mode?
The peak is, more precisely, the median and we have added this to the description.

L. Hood (Referee) Received and published: 2 August 2016
Overall, this is a useful effort to improve statistical estimation of stratospheric ozone and temperature trends and interannual variability by accounting for a source of shortterm (month-tomonth) dynamical variability in tropical stratospheric data sets.The presentation is excellent and the figures are state-of-the art.However, the value of the adopted technique for trend estimation and its ability to "explain" a larger fraction of the variance in the observations is somewhat overstated, in my opinion.Some important revisions are needed prior to publication.

Main comments:
(1) A major claim of the paper is that inclusion of the mid-latitude stratosphere dynamical (MLSD) index can reduce the uncertainty "on all multiple linear regression coefficients ... up to 45% and 25% in temperature and ozone, respectively."First of all, the accuracy of these reduction estimates is questionable because, as mentioned on p. 11, line 12, "we do not consider use of any autoregressive modeling."In other words, serial correlation (autocorrelation) of the residuals of the MLR analysis is not accounted for.It is possible that serial correlation of the monthly residuals is increased when the MLSD index is used because the month-to-month variability is reduced.Have the authors tested whether this is the case?Accounting for any increased serial correlation would increase the uncertainty estimates.For example, application of a "pre-whitening" technique (e.g., Tiao et al. [1990]; Garny et al. [2007]) would ensure that the residuals are approximately white noise thereby yielding more reliable uncertainty estimates.Please re-do the analysis in this manner to provide such a test and yield more accurate (larger) uncertainty estimates.
These are indeed important points.Serial correlation is important, and you are correct that their consideration does indeed puff-up error bars.However, it does not change the main result and the usefulness of the index.To make this point clear we have produced a new plot, which we include in the paper, to emphasize that auto-regression will have an effect and should be considered.Further, we will replace The plot is shown below: AR0 (blue) and AR1 (yellow) are shown for cases with (thick lines) and without (thin) the MLSD index for SWOOSH (ozone, left) and SSU (Temp., right).We see the percentage change (in respective colours) at heights where we see the largest changes.In both cases, we still have a maximum improvement of up to 30% in the errors.Not shown here, but will be in the final manuscript, is that the index now increases R^2 from a maximum increase of 40% (fig 10), to nearly 60% in temperature, and between 30 and 55% for ozone (depending on the dataset used).In the 1998-2012 periods, ozone error improvements are essentially unaffected at a 25% reduction in uncertainty, but the earlier 1983-1997 period is affected, on average reducing uncertainties by around 10% to a maximum of around 15%; some regions show a small increase in error, but this likely reflects the fact the datasets show different variability on all timescales (see response to question from the other reviewer on this point).We will update the manuscript to reflect this.
[Additional note: we also considered AR2, but AR1 was sufficient to account for partial correlation at 1-month] Second, even without accounting for serial correlation, the difference in the ozone and temperature trend results with and without the MLSD term shown in Figure 13 is not very impressive.For the sake of clarity, consider only the yellow curves in the figure.The error bars for with (thick curves) and without (thin curves) the MLSD cases overlap.These are presumably 2σ error bars, right?If not, then the overlap is even larger.The error bars are roughly the same size at most levels.At 2.5 hPa, the ozone error bar appears to be about 25% smaller for the with MLSD case, which is consistent with the authors' statement.But it is not a very significant difference considering the sizes of the error bars and the large variation in the trend estimates from one pressure level to the next.For most of the other levels, the difference in size of the error bars is hard to discern.
We agree with these comments (the error bars are 2-sigma).In fact, we tried to make this clear with the grey shading in the old version of Fig 13a to highlight the altitudes where we see the largest improvement.In hindsight, the plot has such a large absolute range of profiles, that seeing this improvement is difficult.Figure 10 already shows similar results, that is the reduction in uncertainty as a function of altitude (right panels of each sub-plot) -the idea of Figure 13 was to show how it appeared in practice.By accounting for an attributable source of variability (or at least being able to show that it is not simply noise, but a clear dynamical factor) we make a step closer to better understanding those variables we are trying to determine (e.g.trend and solar cycle) -see point below.The new figure (above) reduces the absolute range and focuses in on one of the datasets.We consider this a more useful plot, and discuss and refer to other articles that do show the profiles.
(2) The other major claim of the paper is that use of the MLSD index in a regression analysis can "explain much larger fractions of the total variability."I am not sure that the word "explain" is appropriate.The dynamically induced variability is being accounted for in the MLR analysis but it is not really being explained.For example, the see-saw temperature and ozone variations between the tropics and extratropics are in many cases associated with minor and major polar stratospheric warmings in the winter hemispheres.The latter are modulated by a number of external forcings including the QBO and the solar cycle.A true explanation of the variability would therefore need to account for the external forcings that are controlling the rate of wave absorption events, which in turn produce the ozone and temperature fluctuations.I also disagree with the terminology "total coefficient of determination", which is used in place of explained variance (R2) in the text.The words "determination", "explained", and "attribution" are all misleading if the sources of the dynamical fluctuations are not identified.Please revise the introduction and conclusions section to make this clear.
We are happy to clear up terminology.As the other reviewer also pointed out, the use of temperature mixes potential sources of the variance that correlates with temperature, but is actually the underlying driver, and we have added additional text to the introduction to account for this.The point we are trying to make is that we can 'account' for variability that is physical, and not simply noise that, unconsidered, would lead to higher uncertainty in quantities we wish to determine.It is true, the index itself doesn't necessarily represent the underlying driver of the changes in the meridional flow, but it does act as a proxy and is related to a real variance in the system (which we relate through the EPFD to wave driving, as shown in the manuscript).We disagree about the use of the coefficient of determination, R^2, and would argue it is a useful quantity with which to test how much better our regression model, with the index, improves the amount of variability we can account for.By applying the bootstrapping (examples in Figs 11 and 12), we can also account for further statistical uncertainties to ensure that the improvement from the additional dynamical index is robust.

Minor comments:
(3) I agree with the other referee that the history of the ozone and temperature variations that are discussed in the paper and their application to trend analyses is not adequately summarized in the paper.The first report of the existence of such global stratospheric temperature oscillations with a change in phase between low and middle to high latitudes was by Fritz and Soules [1970].Some stratospheric dynamicists still refer to these oscillations as the "Fritz-Soules effect".See also, e.g., Andrews et al. [1987] for general discussions of their dynamical origin.Another observational study by Chandra [1986] could also be referenced.
We have added additional discussion and references as suggested by both referees (see response above to the first referee on this point).The reference by Chandra [1986] was particularly enlightening; our findings also confirm, and expand upon, the results from that study.
(4) In Figure 1 (and maybe other figures), the definitions of the diamonds in the upper right corner seem to be incorrect and are opposite to those given in the caption.
You are correct: the legend in the figure was wrong; this has been fixed; we also checked the other figures, which did not have this problem.

Changes of note to "A mid-latitude stratosphere dynamical index for attribution of stratospheric variability and improved ozone and temperature trend analysis" by William T. Ball et al
All relevant changes to the document have been highlighted in bold in the attached new version of the manuscript.
Additionally, the following major changes should be considered: -The index, which is the focus of the paper, has been renamed to be a more explanatory name: Upper-branch Brewer Dobson Circulation (UBDC) index.-The title has been changed to reflect a change in the index name: "An Upper-branch Brewer Dobson Circulation index for attribution of stratospheric variability and improved ozone and temperature trend analysis".-Figure 10 has been updated to reflect the change of statistical analysis to include AR1 autoregressive processes.-Figures 11 and 12 have been updated.
-Figure 13 from the initial manuscript has been replaced with a new one following, and as discussed in the response to, reviewers' comments.

Introduction
Trend analysis, typically using multiple linear regression (MLR), is a key approach to understand drivers of long-term changes in the stratosphere (e.g.WMO (1994), Soukharev and Hood (2006), Chiodo et al. (2014), Kuchar et al. (2015), Harris et al. (2015)).Ozone and temperature have received most attention, partly because they have the longest observational records.Temperature is important for understanding climate change, while quantifying changes in the ozone layer is necessary to estimate the impacts of elevated, or reduced, ultraviolet (UV) radiation reaching the surface, especially following the implementation of the Montreal Protocol to reduce halogen-containing ozone depleting substances (ODSs).
in the Brewer-Dobson circulation (BDC), whereby air rises in the tropics, advects polewards either on a lower, shallow-(below ∼50 hPa) or an upper, deep-branch, and descends at mid-latitudes (less than ∼60 • ) or over the poles, respectively (Birner and Bönisch, 2011).The BDC is mainly driven by mid-latitude upward propagating planetary and gravity waves that break and impart momentum, acting like a paddle to drive the circulation (Haynes et al., 1991;Holton et al., 1995;Butchart, 2014).Wave forcing depends on the mean-state of the flow, and vice-versa (Charney and Drazin, 1961;Holton and Mass, 1976); changes in either affect ozone transport by a change in the speed of the BDC that leads to adiabatic heating, or cooling, and directly affects chemistry through temperature-dependent reaction rates (Chen et al., 2003;García-Herrera et al., 2006;Shepherd et al., 2007;Lima et al., 2012).As such, ozone and temperature have an inverse relationship in the equatorial stratosphere above 10 hPa, which in turn has a dependence on dynamics (Fusco and Salby, 1999;Mäder et al., 2007;Stolarski et al., 2012), although this is not always the case in the lower stratosphere (Zubov et al., 2013).Ultimately, then, dynamical perturbations at mid-to-high latitudes can directly influence the variability of ozone and temperature (Sridharan et al., 2012;Nath and Sridharan, 2015).
The stratospheric ozone layer has been damaged by the use of ODSs and following a ban through the 1987 Montreal Protocol (Solomon, 1999), levels of ODSs have declined since their peak in 1998 (Egorova et al., 2013;Chipperfield et al., 2015), although the peak may be earlier or later depending on the location of interest.However, the rate of ozone recovery is latitude dependent, with southern mid-to-high latitudes expected to recover from elevated ODSs (WMO, 2011).The increase in ozone at mid-latitudes comes partly from ODSs reductions, but also because the BDC is expected to accelerate (Garcia and Randel, 2008;Butchart and Scaife, 2001;Butchart, 2014), which will reduce the time for ozone depletion to occur and lead to faster transport of ozone from the equatorial region to higher latitudes.This in turn leads to a reduction of ozone over the equator and a prevention of a full recovery over the tropics.Thus, the recovery of ozone at mid-to-high latitudes can be understood as being partly due to less ozone destruction by lower ODS concentrations, and partly due to a faster redistribution of ozone-rich air from the tropics.Additionally, the cooling stratosphere will slow ozone depletion and further support the increase in ozone at mid-latitudes (WMO, 2014).However, estimates of decadal trends in ozone since 1998 have a high level of uncertainty (Harris et al., 2015) because various long-term datasets provide different pictures (Tummon et al., 2015), and we do not understand much of the stratospheric variability on short timescales.Anomalous, monthly variability, like that at the equator as shown in Figs.7 and 8 in Shapiro et al. (2013), and which could be related to high latitude variability (e.g.Kuroda and Kodera (2001) and Hitchcock et al. (2013)) may simply be considered as noise in MLR trend estimates (and other regressors) where it is not accounted for, which increases the uncertainty.
In MLR analysis of the equatorial stratosphere, variability is usually described with at least six regressors that represent In summary, our aim here is to provide an index (section 4) to account for sporadic, noise-like stratospheric variability in monthly timeseries that represents rapid adjustments in the BDC and, therefore, better account for residual variance, improve estimates of trends and regressor variability, and reduce their uncertainties (section 5).We do this using model, reanalysis and observational data (section 2) to identify a source for the short-term variability (section 3).
2 Data and models

Chemistry climate model in specified dynamics mode
To investigate temperature and ozone variability in the stratosphere and mesosphere at all latitudes, without data gaps, we simulate historical ozone and temperature variations using the Chemistry Climate Model (CCM) SOlar Climate Ozone Links (SOCOL; version 3 (Stenke et al., 2013)) in specified dynamics mode, whereby the vorticity and divergence of the wind fields, temperature and the logarithm of surface pressure are 'nudged' using the ERA-Interim reanalysis (Dee et al., 2011) between 1983-2012 and up to 0.01 hPa; see Ball et al. (2016) for full nudging details.Note that we use the Stratospheric Processes and their Role in Climate (SPARC)/International Global Atmospheric Chemistry (IGAC) Chemistry Climate Model Intercomparison (CCMI) boundary conditions and external forcings (Revell et al., 2015), except for the solar irradiance input, for which we use the SATIRE-S model (Krivova et al., 2003;Yeo et al., 2014).In the following we focus on temperature and ozone variables; the former is nudged, while the latter is simulated by the CCM SOCOL.

Observations
We verify that the nudged-model output fields ozone (not nudged) and temperature (nudged) agree with observations.For ozone we use the Stratospheric Water and OzOne Satellite Homogenized (SWOOSH) ozone composite (Davis et al., 2016) for 215-0.2hPa (∼10-55 km) at all latitudes.For temperature, we compare the nudged-model output with independent measurements from the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument (Russell et al., 1999) on the Thermosphere-Ionosphere-Mesosphere-Energetics and Dynamics (TIMED) satellite, spanning 2002-2015 and for 100 to 0.00001 hPa (∼10-140 km) and latitudes out to 52  2014)), and JRA-55 (Ebita et al., 2011) and MERRA (Rienecker et al., 2011) reanalyses.All observations are re-gridded onto the SOCOL model pressure levels and latitudes.We consider monthly mean zonally-averaged data.

Equatorial ozone and temperature variability
We define short-term, 'anomalous', variability here to be that occuring on monthly, or shorter, timescales.To identify this rapid variability, distinct from behaviour on seasonal and longer timescales, we remove all long-term variability by subtracting a timeseries that has been smoothed, with a 13-month running mean, and then deseasonalised, with monthly values, at each latitude and pressure.We apply this pre-processing to all variables described in sections 3 and 4.An example equatorial (20 • S-20 • N) ozone and temperature anomaly timeseries from the CCM SOCOL at 2.5 hPa is shown in Fig. 1.
SWOOSH ozone from 1985 to 2012, and SABER temperature from 2002 to 2012 (Fig. 1), show similar anomalies to the model and have correlation coefficients (r c ) of 0.72 and 0.83 with the nudged model results, respectively; the model, therefore, reproduces observations well.The monthly temperature and ozone anomalies have a very strong relationship, especially between 0.1 and 6.3 hPa, with negative r c reaching -0.96 (Fig. 2) between 0.1 and 10 hPa, while being positive elsewhere.
To establish the coherency of the ozone-temperature relationship in the tropics, we identify 'extreme' anomalies (or 'events') as those at least at the 90th percentile from the mean in temperature and at less than the 10th percentile for ozone (and vice versa).We call 'low-T' events those that have low equatorial temperature at the same time as a high ozone concentration (blue lines at 2.5 hPa, Fig. 2), and 'high-T' for the opposite situation (red lines); the Low-T thresholds are -1.3K for temperature and +2.4% for ozone, while High-T thresholds are +1.1 K and -2.2% (these are also given the upper plot of Fig. 1).
We note that the ozone mixing ratio maximum in parts per million (ppm) is at ∼10 hPa.We use 2.5 hPa as a reference here, but other pressure levels at altitudes between 0.1 and 6 hPa give similar results.The majority of the events (45/60) occur in December-January-February (DJF; red/blue in Fig. 2) and June-July-August (JJA) (yellow/turquoise in Fig. 2).High-T and low-T months remain grouped above 10 hPa, but mix and lose coherence at altitudes below 10 hPa, implying that the events have a similar source at all altitudes above 10 hPa, but a different one below (i.e.r c is high at 25 and 40 hPa, but the events at 25 hPa are well-mixed).This indicates a likely transition between BDC branches and that the driver of variability is dynamical, which we confirm in the following.December-January-February (DJF) anomalies are identified by red (high-T) and blue (low-T) diamonds.

Mid-latitude temperature variability
To identify and locate the source of the driver behind ozone and temperature anomalies shown and described in the previous section, in Fig. 3 we correlate the 2.5 hPa equatorial temperature low-T and high-T events with detrended and deseasonalised temperature at all latitudes and pressure-levels, for DJF and JJA months (Figs. 3a and b,respectively).A quadrupole-like structure emerges with positive correlations centred around 2.5 hPa at the equator and in the winter-polar mesosphere (<0.8 hPa), and negative correlations in the winter stratosphere at mid-to-high-latitudes and in the equatorial mesosphere.The inverse correlation in the stratosphere for DJF extreme months peaks at ∼52 • N (r c = -0.92);while JJA events peak at ∼43 • S (r c = -0.93).
We find similar results when using other equatorial pressure levels near 2.5 hPa as a reference to calculate correlations.zero are given as yellow and blue contours, respectively).Equatorial temperature anomalies (∼2 K) are smaller than at high latitudes (∼5 K or more).The maximum temperature response at mid-to-high latitudes does not always reside at the same location as the peak correlation.Although the statistics are less robust, since the period is shorter, the quadrupole structure is also evident in SABER observations (Fig. 4).Thus, we can be confident that the nudged-model is giving a good representation of observations.those within regions defined by the red (high-T events) and blue lines (low-T events); red crosses are for high-T events in December, January, and February (DJF), yellow for high-T events in June, July and August (JJA), and green for 'other' high-T events.Dark blue, turquoise and blue represent DJF, JJA and 'other' low-T months (see also legends in 1.0 and 1.6 hPa plots).Correlation coefficients are given for all crosses together.The y-scale has been decreased by a factor of 30 and 5 at 0.01 and 0.05 hPa, respectively, as indicated in the plots.
The quadrupole structure is likely the result of (i) an acceleration of the BDC that adiabatically cools the equator during Low-T events as more air arrives at high-latitudes, and adiabatically heats there, and (ii) a deceleration of the BDC that adiabatically heats the equator during High-T events as less air arrives at high-latitudes leading to cooling there; both processes are associated with changes in wave activity.
We show that the mid-latitude temperature, as well as the equatorial temperature and ozone anomalies are related to variations in wave activity using the Transformed Eulerian Mean streamfunction (TEMS;  and 2, and used in Fig. 3, we find clear EPFD and TEMS anomalies centred near 55 • , slightly poleward of the mid-to-high latitude peak correlations (Fig. 3a-b).As anomalies, they do not represent a reversal of meridional air flow, but a slowing or acceleration.When high-T anomalies occur the EPFD is positive, which implies zonal mean westerly winds have accelerated and the BDC has slowed, which is confirmed by the TEMS, indicating increased equatorward flow.This will have the exact effect found, of adiabatically heating the equatorial region and cooling the mid-to-high latitudes relative to the mean state.
The opposite is the case for low-T anomalies.These results confirm that equatorial anomalies are dynamically driven and we suggest that it appears to be mainly related to, at the equator, a shift in the ozone maximum upwards during low- T events that then produces the anti-correlation seen in Fig. 2 above 10 hPa, and vice versa during high-T events.A further consequence of the circulation changes for ozone is that a temperature increase should lead to faster catalytic destruction, and therefore a decrease of ozone, and vice versa for temperature decreases, though these effects seem to be less important than the rapid profile adjustment itself.It seems surprising that these dynamical signals survive other processes, such as chemistry and radiative effects, on a monthly timescale and we suggest this warrants further Shading colours and black contours are the same as in Fig. 3 (dashed, negative; solid, positive; thick, zero).Signals at two standard deviations from zero are given as yellow contours in panels b and c.

Upper-branch Brewer Dobson Circulation (UBDC) index
The link between anomalous mid-latitude temperature changes and equatorial temperature and ozone provides a way to account for sporadic variability.When performing, e.g., an MLR analysis to understand variability in the stratosphere, such an index of monthly anomalies can account for a large proportion of variability previously unaccounted for, and drive down uncertainties on regressor estimates.We focus here on the equatorial region, but our results imply this index could be applied to other locations in the stratosphere and mesosphere.
Below, we describe how we construct an upper-branch Brewer Dobson circulation (UBDC) index based on detrended and deseasonalised temperature averaged over 43-49 • S and 2.5-6.3 hPa for June-October, and averaged over 52-57 • N and 4-10 hPa for November-May.Our index utilizes the output from the CCM SOCOL in specified dynamics mode, similar to ERA-Interim and observations, but such an index could be constructed in a similar way for any specific model.

Construction
To construct a useful upper-branch Brewer Dobson circulation (UBDC) index requires the identification of maximum correlation between the equator and each hemisphere separately, followed by a combination of information from these two regions.
We have previously considered just the extreme events, but we now consider all monthly anomalies between 1983 and 2012.
While wave-activity drives the temperature changes, it is not an easily observable quantity.Thus, temperature is a natural and simple quantity to build the index with.Additionally, we have found that the CCM SOCOL in free-running mode (i.e.without nudging) shows the same anomalous temperature-quadrupole structure as in Fig. 3 (not shown).Therefore, one can easily construct an index using model data to represent anomalous behaviour in the equatorial regions, and elsewhere where there is a quadrupole response.We identify the maximum inverse temperature correlations at mid-latitudes in both DJF and JJA by varying the reference equatorial pressure-level.We find that averaging over the nine grid-cells centred on the mid-latitude peak improves the relationship with the equatorial region.Therefore, we construct the index with anomalous temperatures averaged over 43-49 • S and 2.5-6.3 hPa in the southern hemisphere (SH) JJA months, and 52-57 • N and 4-10 hPa in the northern hemisphere (NH) DJF months.
For March-May and September-November months, we complete the UBDC index by combining November-April NH anomalies with May-October SH anomalies; this combination maximises the relationship with equatorial temperature.We plot the index derived from the CCM SOCOL in specifed dynamics mode using ERA-Interim in Fig. 7. Fig. 8 shows the SH and NH mid-latitude temperature anomalies versus the 4 hPa 20 • S-20 • N equatorial average (a and b, respectively; grey crosses represent November-April, and black May-October).The SH May-October temperature anomalies are inversely correlated with equatorial temperatures (r c =-0.70) while November-April are not (r c =0.05); the opposite is true for the NH (r c =0.02 and -0.78, respectively).The ozone-temperature events identified in Fig. 1 are highlighted with coloured circles, showing that the  equatorial anomalies are related to mid-latitude wave-driving.Fig. 8c shows the UBDC index plotted against all equatorial temperature anomalies at 4 hPa (r c =-0.74).The lower panels (d-f) show the equatorial ozone relationship with respect to midlatitude temperature and the UBDC index; the absolute correlation coefficient is lower (r c =0.65) than for equatorial temperature in the upper panels, but there is still a strong relationship.
In Fig. 9 we show the amount of variability the UBDC index can account for in nudged-model temperature anomalies everywhere , using the coefficient of determination r 2 c , or R 2 .It ranges from 0 to 1; a value of 1 is synonymous with the index accounting for 100% of the variability.In Fig. 9a the UBDC index can account for >50% of variability between 10 and 1 hPa, and above 0.05 hPa.The variability accounted for at mid-latitudes is less (up to ∼30%), even at the index source locations (white circles), because the UBDC index has almost zero agreement half of the time there (see Fig. 8). I Fig. 9b and c, the UBDC index accounts for much of the DJF/JJA variability: above 20 hPa it can account for over 70% of equatorial variability, more than 60% of polar mesospheric variability (80% in the SH), and much of polar stratosphere variability.and 6).

Improvement in MLR analysis using the UBDC index
The UBDC index leads to a large uncertainty reduction in MLR analysis.To show this, we consider MLR with or without the index focused on the equatorial region (20 • S-20 • N).In both cases we use the two QBO indices, SAOD, ENSO, a linear Focusing on the yellow bars in Fig. 13 representing results using AR1 processes, we show the equatorial decadal trend profiles of the datasets considered in Fig. 10 and the 2σ uncertainties derived from multiple linear regression with (thick lines) and without (thin lines) the UBDC index, between 25 and 0.2 hPa.A full discussion of the differences in the profiles is undertaken by Tummon et al. (2015) and Harris et al. (2015), so we do not repeat that here.We simply note that the mean decadal equatorial trends in temperature are almost unaffected by the UBDC index (right panel of Fig. 13).However, we see that the influence of the UBDC index on the mean profile of ozone in SWOOSH leads to a decrease in the ozone trend of ∼0.5% per decade, at the altitudes where the index also performs best at reducing uncertainties (Fig. 13a).This decrease may be a result of the largest anomalies after 1998 being positive (see upper plot in Fig. 1), which might introduce a slight upward bias in the trend analysis; once accounted for with the UBDC index, this bias is removed and the trend is reduced slightly.
Nevertheless, this result suggests that ozone trend estimates that do not take the short, anomalous variability into account will overestimate the decadal trends, though it is clear that the biggest uncertainties remain in the underlying datasets themselves (Harris et al., 2015).

Conclusions
We have shown that detrended and deseasonalised ozone and temperature anomalies in the tropics are strongly influenced by mid-latitude dynamical perturbations that influence temperature throughout the upper stratosphere and mesosphere of the perturbed hemisphere.The strongest correlations with these anomalies occur at latitudes around 50 • in the winter of both hemispheres, which are linked to changes in wave-forcing.
We develop a new upper-branch Brewer Dobson circulation (UBDC) index, which has the power to considerably improve the statistical significance of ozone and temperature trends, and account for much larger fractions of the total variability.Our results suggest that the index is able to improve the uncertainty of temperature and ozone estimates by up to 30 and 25%, respectively, between 0.3 and 40 hPa, and up to 60% of the total variance accounted for.While we focus on improvements in equatorial temperature and ozone, we suggest it could also be used in the analysis of other stratospheric variables, and also in other regions as well as in the mesosphere.The UBDC index should be employed in future investigations of stratospheric trends in the upper stratosphere and mesosphere.For modelling studies, this index can be extracted from pressure levels and latitudes similar to those put forward here, though the exact peak is likely to be model dependent; for future trends it may be necessary to determine the exact peak again since the regions of wave propagation and breaking may change.
In all cases considered here, the UBDC index both improves our ability to reduce uncertainties and better account for equatorial stratospheric ozone and temperature variability and, by extension, attain better estimates of trends in stratospheric and mesospheric mid-to-high latitude variability.
Fig 13 with this one, since the point of Fig 13 is to show clearly how the reduction (and any effect on mean value) works in practice -this also addresses the second main point below.
the solar cycle UV flux changes (e.g., with the F10.7cm radio flux), volcanic eruptions (stratospheric aerosol optical depth; SAOD), the El Nino Southern Oscillation (ENSO) surface temperature variations, two orthogonal modes of the dynamical quasi-biennial oscillation (QBO), and the equivalent effective stratospheric chlorine (EESC), which describes the long-term or, alternatively to applying both GHG and EESC proxies, a linear (or piece-wise linear) trend is considered.At higher latitudes, other proxies have beenused to represent dynamical indices, e.g. in the northern hemisphere, the North Atlantic Oscillation (NAO) and Arctic Oscillation (AO), which are related to surface pressure changes, though their relation is less anti-correlated with stratospheric variability than, e.g., the tropopause pressure (Weiss et al., 2001).Trends in dynamically related quantities, such as horizontal advection and mass divergence, contribute to long-term changes in ozone (Wohltmann et al., 2007).For short timescales, Wohltmann et al. (2007) note that tropospheric pressure is a physical quantity directly responsible for changes in lower stratospheric temperatures, but that it is nevertheless inferior to stratospheric temperature when accounting for ozone and temperature variance in column ozone and temperature; this can also depend on the location of tropospheric blocking events (WMO, 2014).However, longer timescales render these proxies unreliable due to additional radiative effects.Ziemke et al. (1997) identified that the use of high-latitude temperature at 10 hPa in winter-spring months, together with 200 hPa temperature at mid-latitudes all year round, were most effective at reducing residuals, though Ziemke et al. (1997) limited their study to total column ozone.In fact, several studies have identified proxies, such as temperature in the stratosphere, that can help improve MLR analysis (e.g., Ziemke et al. (1997); Appenzeller et al. (2000); Weiss et al. (2001); Mäder et al. (2007)).However, these have tended to be focused on total column ozone, the mid-to-lower stratosphere, or mid-latitude and polar regions, and thus most attention on dynamical variability remains associated with the lower branch of the BDC (e.g.Newman et al. (2001); Wohltmann et al. (2005); Brunner et al. (2006)).Further, studies of dynamical variability have also tended to focus on seasonal and inter-annual timescales, and thus any fluctuations on monthly or shorter timescales may be missed, underestimated, or driven by processes operating on different timescales.There are differing conceptual approaches to improve regression models (i.e., see WMO (2011)): use of a statistical approach (e.g.Mäder et al. (2007)); or to use proxies that can be (at least partly) physically understood (e.g.Wohltmann et al. (2007)).Both approaches have their limitations and the physical mechanisms may not be fully understood in either case.As Wohltmann et al. (2007) points out, the use of (or lack thereof) unphysical or too many regressors could lead to systematic errors -through the attribution of correlated variables -that go unnoticed because error statistics do not change, or indeed, decrease.A third approach simply considers dynamical variability as noise that leads to enhanced uncertainties on trend analysis.The identification of a correlation between two variables can be considered a first step in identifying the physical mechanism that underlies the causal link.A relationship between the proxies and processes that drive their variability need to be shown through additional information, it cannot be done simply through statistical means alone as it needs information from a physical understanding, either a priori, or following further investigation.Here, our aim is to find an index, or proxy, that represents rapid changes, on timescales of a month or less, in the upper branch of the BDC by investigating an identified association between temperature variation in the mid-latitude upper stratosphere and planetary wave-breaking.While temperature alone does not represent a complete picture of the physical driver of rapid processes, especially where the exact physical drivers remain unresolved; the use of standard dynamical proxies is, as we shall show, not enough to capture this variance.Chandra (1986) identified similar short-term dynamical variability that we identify in monthly data here, but he applied it to understand dynamical influences on upper stratospheric variability relevant to identifying 27-day solar irradiance variability -indeed showing that 27-day solar modulation was very difficult to identify due to large, rapid dynamical fluctuations -and did not extrapolate this information to improving MLR analysis, as we aim to do here.The drawback to using temperature is that it mixes processes that might have different influences on ozone(Wohltmann et al., 2005).However, it is a simple and direct measure of dynamical changes, at least on monthly or shorter timescales, that relate to rapid dynamical adjustments within the stratosphere.

Figure 1 .
Figure 1.Monthly anomalies of equatorial (20 • S-20 • N) ozone (upper; %) and temperature (lower; degrees Kelvin) at 2.5 hPa, following the subtraction of 13-month box-car smoothing and monthly-deseasonalising from the CCM SOCOL model in specified dynamics mode.SWOOSH ozone composite timeseries and SABER temperature measurements are shown in light-blue in the upper and lower plots, respectively.The dashed blue and red horizontal lines are the thresholds shown in Fig.2; thresholds for each coloured diamond are given on the right of the upper panel.June-July-August (JJA) anomalies exceeding the thresholds have orange (high-T) and turquoise (low-T) diamonds;

Figures
Figures 3c-f show temperature composites for each event type: (c) DJF Low-T, (d) JJA Low-T, (e) DJF High-T and (f) JJA High-T; all show the same temperature-quadrupole structure as in Fig. 3a-b (signals at two and three standard deviations from

Figure 2 .
Figure 2. Regression of equatorial (20 • N-20 • S) ozone and temperature anomalies (following 13-month smoothing and monthly deseasonalising) from the CCM SOCOL model in specified dynamics mode for pressure levels 0.01 to 40 hPa (∼80 -22 km).Grey crosses are for all other months in 1983/01-2012/10.Coloured crosses in each plot are determined at 2.5 hPa (lower-left, and plotted as diamonds in Fig. 1) by

Figs. 5 )
Figs. 6), which is a measure of the resolved wave-induced forcing of the mean flow (positive values imply an acceleration of the zonal mean flow and a deceleration of the BDC, and negative values the opposite).Using the events identified in Figs. 1

Figure 3 .
Figure 3. Correlation coefficient maps of zonal mean 20 • N-20 • S 2.5 hPa temperature anomalies from the SOCOL model with respect to latitude and altitude for all identified low-and high-T (a) DJF and (b) JJA events, as defined in Fig. 2. (c-f) Composite temperatures for (c) DJF low-T, (d) JJA low-T, (e) DJF high-T and (f) JJA high-T events.Dashed (solid) contours are negative (positive) with the bold line representing zero.Signals at the 2 and 3 standard deviations from zero are given as yellow and blue contours, respectively, in panels c-f.

Figure 4 .
Figure 4. (a) SABER temperature data correlation coefficient map of zonal mean 20 • N-20 • S 2.5 hPa anomalies with all latitudes and altitudes for all low-and high-T DJF and events.Composite temperatures for DJF (b) low-T and (c) high-T events, as defined in Fig 2.

Figure 5 .
Figure 5.The median of the Transformed Eulerian Mean streamfunction (TEMS) anomalies for (a) DJF Low-Temperature events, (b) JJA Low-T events, (c) DJF High-T events, (d) JJA High-T events for the same months as in Fig. 3c-f.Contours lines (solid, positive; dashed negative) and colours are given in the legend.Positive values indicate clockwise-acceleration along the contour lines; negative are anticlockwise.Data are from the SOCOL model in specified dynamics mode.

Figure 6 .
Figure 6.As for Fig. 5, but for EP-Flux Divergence.Positive values indicate increased wave-activity; negative, decreased activity.

Figure 7 .
Figure 7.The UBDC index from the CCM SOCOL model in specifed dynamics mode using ERA-Interim from 1983 to 2012.

Figure 8 .
Figure 8. (a-b) SOCOL equatorial temperature anomalies (4hPa, 20 • N-20 • S) plotted against (a) temperature means from 2.5-6.3 hPa and 43-49 • S and (b) 52-57 • N and 4-10 hPa.(d-e) As for upper panels, but equatorial ozone anomalies (4 hPa, 20 • N-20 • S) are instead plotted against high-latitude temperature anomalies.Grey crosses are for November-April months; black crosses for May-October; correlations for both periods are given in each panel.Red and blue circles identify the DJF High-T and Low-T events in Fig. 1 and 2, respectively; orange and light-blue circles similarly identify JJA events.(c,f) May-October 43-49 • S temperatures and November-April 52-57 • N temperatures are combined in the right panel (UBDC index) and plotted against equatorial (c) temperature and (f) ozone.

Figure 9 .
Figure 9. Coefficient of determination (R 2 ) maps of the upper-branch Brewer Dobson circulation (UBDC) index with SOCOL model temperature at all latitudes and altitudes for (a) all months, (b) DJF and (c) JJA months.White circles represent the approximate region that the UBDC index is derived from.

Figure 10 .Figure 11 .Figure 12 .Figure 13 .
Figure 10.Coefficient of determination summed over all regressors (R 2 c ) and the reduction in the Student's t-test-based error on regressor coefficients (%) for equatorial profiles (positive values) for (a) temperature from 1983-2005, and for ozone between (b) 1985 and 1997, and(c) 1998 and 2012 for various datasets (see legends).For R 2 , dotted lines represent estimates without the UBDC index, solid lines with, and the difference (without-UBDC minus with-UBDC) is given as negative and dashed lines.