Trend analysis, typically using multiple linear regression (MLR), is a key approach to understand drivers of long-term changes in the stratosphere e.g.. 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).

Ozone and temperature variations in the stratosphere are directly modulated by changes in solar flux, particularly in the UV see e.g.and references therein. Ozone concentration also responds to changes 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 . 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 . Wave forcing depends on the mean state of the flow, and vice versa ; 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 . As such, ozone and temperature have an inverse relationship in the equatorial stratosphere above 10 hPa, which in turn has a dependence on dynamics , although this is not always the case in the lower stratosphere . Ultimately, then, dynamical perturbations at mid-to-high latitudes can directly influence the variability of ozone and temperature .

The stratospheric ozone layer has been damaged by the use of ODSs, and, following a ban through the 1987 Montreal Protocol , levels of ODSs have declined since their peak in 1998 , 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 . The increase in ozone at mid-latitudes comes partly from ODS reductions but is also because the BDC is expected to accelerate , 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 . However, estimates of decadal trends in ozone since 1998 have a high level of uncertainty because various long-term datasets provide different pictures , 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 , and which could be related to high-latitude variability e.g., 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 the solar cycle UV flux changes (e.g. with the F10.7 cm radio flux); volcanic eruptions (stratospheric aerosol optical depth: SAOD); the El Niño 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 influence of ODSs on ozone concentration and temperature. A greenhouse gas (GHG) proxy is sometimes also considered, or, alternatively to applying both GHG and EESC proxies, a linear (or piece-wise linear) trend is considered.

At higher latitudes, other proxies have been used to represent dynamical indices, e.g. in the Northern Hemisphere (NH), 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 . Trends in dynamically related quantities, such as horizontal advection and mass divergence, contribute to long-term changes in ozone . For short timescales, 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 . However, longer timescales render these proxies unreliable due to additional radiative effects. 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, was most effective at reducing residuals, though 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.. 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.. 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: use of a statistical approach e.g. or use of proxies that can be (at least partly) physically understood e.g.. Both approaches have their limitations, and the physical mechanisms may not be fully understood in either case. As 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 needs 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 BDC changes, we show that its variance is highly associated with wave driving and that it can act as a proxy for such 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. 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 . 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.

In summary, our aim here is to provide an index (Sect. ) to account for sporadic, noise-like stratospheric variability in monthly time series 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 (Sect. ). We do this using model, reanalysis, and observational data (Sect. ) to identify a source for the short-term variability (Sect. ).

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 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 utilises 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.

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 trend (mentioned in Sect. ), and the F30 radio flux as a proxy for solar variability, as this is superior to the F10.7 cm radio flux . We consider the use of “AR2” auto-regressive modelling through the procedure of in all cases; see for a discussion of AR. The use of second-order auto-regression was determined after assessing the regression analysis using a Durbin–Watson test, which showed that AR1 was necessary but not sufficient to account for auto-correlation in the residuals and that AR2 was sufficient. In Fig. a, we show the combined ability of the regression model to account for variance, i.e. the total R2 of all regressors, for 1983–2005 in SSU temperature observations (red), and JRA-55 (blue) and MERRA (yellow) reanalyses. R2 without the UBDC index (dotted lines; Fig. a) shows that only up to ∼ 45 % (R2=0.45) of the stratospheric variability above 10 hPa can be accounted for. However, with the UBDC index (solid lines) R2 is > 0.8 (MERRA ∼ 0.7), and, in all cases, the use of the UBDC index peaks at ∼ 5 hPa, with R2 increasing by 0.45–0.60, or an improvement of up to 60 % (see negative values in the left panel of Fig. a, i.e. Rw/oUBDC2-Rw/UBDC2). In the right panel of Fig. a, we show the relative change in regressor uncertainty (σw/o UBDC2-σw/ UBDC2)/σw/o UBDC2×100, where σ is based on Student's t test. The uncertainty estimates on the regressors decrease by up to ∼ 35 (SSU), ∼ 35 (JRA-55), and ∼ 10 % (MERRA). In addition, the index increases R2 above 0.4 hPa in the mesosphere.

To check whether the UBDC index increases the amount of variance of the total accounted for, in Fig.  we calculate the relative importance of each regressor without (blue) and with (red) the UBDC index by decomposing R2 seefor a comprehensive review of this technique, which depends on the order in which regressors are considered, unless the regressors are orthogonal, which is usually not the case for the 3 decades we consider here see e.g.. We use the robust Lindeman–Gold–Merenda (LGM) measure , which determines relative importance by averaging over all possible, n!, ordering of regressors (720 for six regressors, 5040 for seven). In Fig. a we show the relative importance of each regressor, as well as the total, in representing the variance in SSU temperature at 4.6 hPa; curves represent the complete distributions resulting from 10 000 bootstrappings of averages over orderings. At 4.6 hPa the UBDC index accounts for ∼ 61 % of temperature variance when considered in addition to the others, partly at the expense of decreasing the relative importance of the other regressors. Together, the UBDC leads to a ∼ 44 % increase in the total variance accounted for, from 38 to 82 % (peak values: solid white lines); we see similar results at the other two pressure levels (Fig. c). Figure b shows that seasonal MLR analysis is enhanced: March–April–May (MAM), JJA, and DJF peaks increase by more than double the 68 % confidence intervals, i.e. by an additional ∼ 22, ∼ 40, and ∼ 33 %, respectively; September–October–November (SON) months do not improve much (∼ 10 %).

Similar results are found for ozone. Figure b and c show the ozone composites (see Sect. ) split into 1985–1997 and 1998–2012 time periods, reflecting those often used to investigate ozone trends e.g.. There are significant differences in R2 between the ozone composites from the MLR analysis, which reflects the fact that different equatorial decadal trends are found between the ozone datasets and in solar signal profiles extracted with MLR, which may be related to the way datasets in the composites have been merged together. While smaller for ozone than temperature, an improvement is found in representing variance (R2 ∼ 0.55), and errors reduce by up to ∼ 20 % (smaller for the pre-1998 period). While the UBDC index leads to an increase in the variance accounted for in equatorial temperature variability above 40 hPa (Fig. a), it only increases above 10 hPa for ozone; this tallies with the strong relationship between temperature and ozone shown in Fig. , which also breaks down at 10 hPa. Furthermore, Fig.  shows the relative importance of each regressor using violin plots , as well as the total, for all four ozone composites for the 1998–2012 period at 1.6 hPa. The format is the same as for Fig. , though we only show relative importance for each regressor, as well as the total, for the case that includes the UBDC index. We see that at this pressure level most of the variance is given by QBO2 and UBDC indices. The UBDC accounts for between 30 and 55 % of the variance in equatorial ozone, depending on altitude and whether the composite is Stratospheric Aerosol and Gas Experiment (SAGE-II)-based (GOZCARDS and SWOOSH) or SBUV-based; the results here suggest the data the composites are based upon affect the relative contributions to the total variance (see further discussion below).

We note that the UBDC index influences the relative importance of most of the other regressors by less than the 68 % confidence interval (dashed white lines in Fig. ), and only in the case of the trend does the relative importance get decreased by a larger margin, albeit within 95 %. Figure b shows that the mean values of the SSU MLR results are almost completely unaffected (red dots are with the UBDC index; circles are without), and the MERRA and JRA-55 are only marginally affected, and therefore the UBDC index does not alias with the estimated trend in temperature. The relative importance of regressors for ozone is affected only slightly in the same way as temperature (not shown), so the larger effect on the SSU relative importance may be data dependent.

In Fig.  we show the equatorial decadal trend profiles of the datasets considered in Fig.  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 ozone profiles is undertaken by and , so we do not repeat that here. We simply note that the mean decadal equatorial trends in temperature are affected only slightly by the UBDC index (left panel of Fig. ). However, we see that the influence of the UBDC index on the mean profile of ozone leads to a decrease in the ozone trend of ∼ 0.5–1 % per decade in all the ozone composites, at the altitudes where the index also performs best at reducing uncertainties (Fig. a). This decrease may be a result of the largest anomalies after 1998 being positive (see upper plot in Fig. ), 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 .

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 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 equatorial temperature and ozone trend estimates by up to 35 and 20 %, respectively, between 0.5 and 50 hPa – and higher in the mesosphere, although there is a strong altitude dependence – and up to 60 % of the total variance can be accounted for. We also find that this result is data dependent, with the reanalysis products seeing less improvement than the observations. 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 improves our ability both to reduce uncertainties and to 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.

We provide all MLR results on the Mendeley Data portal . SABER/TIMED temperature data can be found at http://saber.gats-inc.com/. GOZCARDS ozone data can be found at https://gozcards.jpl.nasa.gov/. SWOOSH ozone data can be found at http://www.esrl.noaa.gov/csd/groups/csd8/swoosh/. SBUV ozone data can be found at http://acd-ext.gsfc.nasa.gov/Data_services/merged/. ERA-Interim reanalysis data can be found at http://www.ecmwf.int/en/research/climate-reanalysis/era-interim. MERRA reanalysis data can be found at http://disc.sci.gsfc.nasa.gov/mdisc/data-holdings/merra/merra_products_nonjs.shtml. JRA-55 reanalysis can be found at http://jra.kishou.go.jp/JRA-55/index_en.html. SSU data can be found at http://www.star.nesdis.noaa.gov/smcd/emb/mscat/products.php/. The SOCOL model code is available on request.