The effect of nonlinearity in CO2 heating rates on the attribution of stratospheric ozone and temperature changes

. An analysis of the attribution of past and future changes in stratospheric ozone and temperature to anthropogenic forcings is presented. The analysis is an extension of the study of Shepherd and Jonsson (2008) who analyzed chemistry-climate simulations from the Canadian Middle Atmosphere Model (CMAM) and attributed both past and future changes to changes in the external forcings, i.e. the abun-dances of ozone-depleting substances (ODS) and well-mixed greenhouse gases. The current study is based on a new CMAM dataset and includes two important changes. First, we account for the nonlinear radiative response to changes in CO 2 . It is shown that over centennial time scales the radiative response in the upper stratosphere to CO 2 changes is signif-icantly nonlinear and that failure to account for this effect leads to a signiﬁcant error in the attribution. To our knowledge this nonlinearity has not been considered before in attribution analysis, including multiple linear regression studies. For the regression analysis presented here the nonlinearity was taken into account by using CO 2 heating rate, rather than CO 2 abundance, as the explanatory variable. This approach yields considerable corrections to the results of the previous study and can be recommended to other researchers. Second, an error in the way the


Introduction
There has been considerable interest in global-mean stratospheric cooling as a fingerprint of ozone depletion (see e.g. Chap. 5 of WMO, 2007), since in the stratosphere, global-mean temperature is radiatively controlled. The traditional approach (e.g. Ramaswamy et al., 2001;Shine et al., 2003) has been to attribute the stratospheric cooling observed in the recent past to a combination of CO 2 increases and ozone depletion, the latter being attributed to the increase in the stratospheric abundance of ozone-depleting substances (ODS) up to about 1998. Shepherd and Jonsson (2008; hereafter SJ08) pointed out that this approach to attribution is not really correct, because ozone is not a forcing agent but rather is an internal property of the atmosphere, which (like water vapour) itself responds to anthropogenic forcings. Moreover ozone and temperature interact, especially in the upper stratosphere, so part of the past ozone changes in this region have been attributable to the increase in CO 2 . In particular, CO 2 -induced cooling (which increases ozone) has masked some of the ODS-induced ozone depletion. SJ08 performed a theoretical analysis which untangled the ozone-temperature feedback and showed that the attribution of temperature changes to CO 2 and ODS changes could be quite different from that due to CO 2 and ozone changes. They illustrated their point by analyzing chemistry-climate simulations from the Canadian Middle Atmosphere Model (CMAM), which exhibited distinct linear trends in both temperature and ozone over 1975-1995 and 2010-2040, these two time periods representing the periods of approximately linear ODS buildup and ODS decline, respectively. The orthogonality of the CO 2 and ODS changes between these two time periods -with CO 2 increasing steadily, but ODS rising Published by Copernicus Publications on behalf of the European Geosciences Union. and then falling -allowed a simple statistical attribution of the ozone and temperature changes. SJ08 found that while about 10% of the ODS-induced upper stratospheric ozone depletion over the period of rapid ODS increases  was masked by CO 2 -induced cooling, the traditional approach towards temperature attribution is largely valid over this period. In the future, however, full account needs to be taken of the CO 2 -induced effects on ozone, as (according to SJ08) they will drive about 40% of the upper stratospheric ozone recovery over 2010-2040. Unfortunately, the CMAM CCMVal-1 simulations used by SJ08 underestimated the impact of the CO 2 increase in the recent past, due to a problem related to the middle atmosphere radiation scheme. (This should be borne in mind when analyzing CMAM CCMVal-1 simulations.) This problem has been fixed in the recently completed CMAM CCMVal-2 simulations, and we therefore analyze those simulations here. A detailed description of the changes made to the radiation scheme is given in the Appendix as a documentation for other model groups using the same scheme. In addition, the analysis is expanded to account for the nonlinear relation between CO 2 abundance and heating rates, which results in a nonlinear response of temperature to CO 2 changes over centennial time scales. To our knowledge this effect has not previously been taken into account in statistical analysis of middle atmosphere trends (such as analysis using Multiple Linear Regression). The consequences of this nonlinearity for the attribution of upper stratospheric ozone and temperature changes are analysed.

Methods
The CMAM is a fully interactive three-dimensional chemistry-climate model (de Grandpré et al., 2000;Scinocca et al., 2008). Its performance in the SPARC CCMVal-1 model intercomparison was assessed by Eyring et al. (2006) and Waugh and Eyring (2008), where CMAM was found to be one of the better-performing models. The CCMVal-2 simulations analyzed here were forced with the SPARC CCMVal "REF2" scenario described in Eyring et al. (2005). The analysis is based on the average of an ensemble of three simulations.
There are three main differences between the CMAM versions used for the CCMVal-1 and CCMVal-2 simulations. The first and most important difference in the present context was the correction of the middle atmosphere radiation scheme. The details of this correction are described in the Appendix. In the CMAM simulations considered in SJ08 (CCMVal-1) the impact of the CO 2 change over 1975-1995 on temperature was underestimated by about a factor of 2-2.5 in the upper stratosphere, which means that the cooling over this period was unrealistically small. With the radiation scheme corrected, the CMAM temperature trends in the near past are now in line with other model results and observations (Shine et al., 2008).
The second difference was the use of an interactive ocean, rather than specifying sea-surface temperatures (SSTs) from a different coupled atmosphere-ocean model. However, using different SSTs would not be expected to lead to changes in global-mean stratospheric temperature and ozone, except perhaps in the very lowest part of the stratosphere (Fomichev et al., 2007).
The third difference was minor changes in the chemistry and transport schemes to address some deficiencies identified in the earlier version of CMAM. For the most part these would not be expected to lead to discernible effects in globalmean temperature or ozone, especially in the upper stratosphere. However, a larger set of representative ODS was used, which allowed a more realistic representation of their fractional release of chlorine, and this somewhat changed the time evolution of total inorganic chlorine.
Just as with the simulations analyzed by SJ08, globalmean ozone and temperature were found to evolve linearly in time at all stratospheric altitudes over the time periods 1975-1995 and 2010-2040. Thus, the simple statistical analysis of SJ08 is applicable. However, a deficiency in SJ08 is the assumption that temperature (and ozone) changes depend linearly on CO 2 changes. This assumption is only valid for small CO 2 changes. In fact, over centennial time scales the CO 2 heating rate sensitivity to CO 2 varies significantly, as is shown in Fig. 1. For example, over the extent of the CCM-Val simulations, 1960-2100, the CO 2 mixing ratio increases from about 310 ppmv to about 700 ppmv and the CO 2 heating rate sensitivity decreases by about a factor of 1.7 around the stratopause (4-0.5 hPa). In the current attribution analysis we account for this effect by using the CO 2 heating rate as the explanatory variable (as opposed to the CO 2 abundance itself).
Note that while the CO 2 mixing ratio increases by roughly 50% over the analysis period 1975-2040 (from about 320 ppmv to about 480 ppmv), the CO 2 heating rates change by only about 10-15% (see Fig. A1 in the Appendix). So in the context of a linear attribution analysis the heating rate changes can be considered to be a small perturbation, while this is not the case for the CO 2 abundance. Moreover, the absolute CO 2 heating rate changes over 1975-2040 are generally less than 1 K/day (Fig. A1 in the Appendix), for which it is predicted that the temperature response in the middle atmosphere should be linear (Fomichev et al., 2004).
Our attribution uses the expressions (15)-(18) of SJ08. If CO 2 is used as the explanatory variable, then r (the ratio of the CO 2 increase over 2010-2040 to that over 1975-1995) is given by 3.0. If the CO 2 heating rates are used instead as explanatory variables, then r is given instead by 2.5. (Note that while the heating rates themselves vary significantly with height, r varies much less and is within 2.5±0.1 over 15-0.5 hPa.) The ratio of the ODS decrease over 2010-2040 to the increase over 1975-1995, denoted by s, is 0.44, which is about 20% smaller than the value from the CCMVal-1 simulations used in SJ08.
Atmos. Chem. Phys., 9,[8447][8448][8449][8450][8451][8452]2009 www.atmos-chem-phys.net/9/8447/2009/ 3 Results Figure 2 shows the global mean ozone trend over both time periods as well as the attribution of the trend to changes in ODS and CO 2 . The maximum ozone trend in the past (panel a) occurs at 2 hPa and is about 5%/decade, which is a little less than the 6%/decade found in SJ08. The difference results from the increased (i.e. corrected) past CO 2 -induced cooling in these simulations. Both values are reasonably consistent with observations, given the uncertainties (see discussion in SJ08). In terms of attribution, it is found here that the increase in ozone from the CO 2 -induced cooling masked about 20% of the ODS-induced ozone depletion in the upper stratosphere in the past. In the future (panel b), the ODS decline and continued CO 2 -induced cooling act in the same sense, to increase ozone, and contribute equally over nearly the entire stratosphere to ozone recovery -or stated differently, the ODS-induced ozone recovery is doubled by the effects of CO 2 -induced cooling. Thus, in the future, ODS changes will no longer dominate the evolution of ozone in the upper stratosphere. In fact, the ozone recovery trend in the future is predicted to be fully two-thirds the magnitude of the ozone depletion trend in the past (3.5%/decade versus 5%/decade), even though the ODS-induced recovery trend is less than one-third the magnitude of the ODS-induced depletion trend. Figure 3 is the same as Fig. 2 but for temperature. The maximum cooling in the past (panel a) occurs at 1 hPa and is about 1.7 K/decade, which is considerably larger than the 1.3 K/decade found in SJ08; this results from the increased (i.e. corrected) past CO 2 -induced cooling in these simulations, and brings CMAM more in line with observations (Shine et al., 2008), although it should be kept in mind that the observed trends around the stratopause are associated with large uncertainties (Randel et al., 2009). In terms of attribution, it is found here that ODS-induced ozone depletion and CO 2 -induced cooling have contributed approximately equally to upper stratospheric cooling, though with the ODS effect being a strong function of altitude; e.g. the ODS effect is 30% more than that of CO 2 at 1 hPa, and only 50% of it at 10 hPa. Note that if CO 2 abundance (as opposed to CO 2 heating rate) is used as the explanatory variable, this result changes drastically (Fig. 4): in that case the contribution from ODS is much larger, such that the ODS-induced past cooling is estimated to equal that from CO 2 throughout the middle stratosphere up to 5 hPa, and to be about 80% more at 1 hPa. Therefore, multiple linear regression using CO 2 abundance as an explanatory variable over time scales encompassing both ozone depletion and ozone recovery must be used with care in the upper stratosphere. In the future (Fig. 3, panel b), CO 2 -induced cooling dominates the temperature trend, with the ODS-induced ozone recovery taking about a 20% bite out of the cooling trend in the upper stratosphere.  Fig. 1. Sensitivity of the CO 2 heating rate to CO 2 mixing (derivative of the CO 2 heating rate with respect to CO 2 mixin tio) as a function of CO 2 mixing ratio and pressure in the m atmosphere. The heating rates were calculated according to th rameterization of Fomichev et al. (1998), with the suggested u outlined in the Appendix. The arrows at the top of the plot ind the years by which a given CO 2 amount is achieved in the CC "REF2" scenario. Fig. 1. Sensitivity of the CO 2 heating rate to CO 2 mixing ratio (derivative of the CO 2 heating rate with respect to CO 2 mixing ratio) as a function of CO 2 mixing ratio and pressure in the middle atmosphere. The heating rates were calculated according to the parameterization of Fomichev et al. (1998), with the suggested update outlined in the Appendix. The arrows at the top of the plot indicate the years by which a given CO 2 amount is achieved in the CCMVal "REF2" scenario.

Discussion
Our analysis has focused on the changes in stratospheric global mean ozone and temperature due to CO 2 and ODS changes, as these have been shown to be the primary forcing agents for past changes and are expected to be the dominant agents in ozone recovery (Chaps. 5 and 6 of WMO, 2007). The expected effects of CH 4 , N 2 O and water vapour are discussed in SJ08. It is also worth mentioning that while the main effect of ODS changes is a chemical modulation of ozone mixing ratios, ODS also have a direct radiative effect. However, in the global-mean upper stratosphere, this contribution is negligible (Forster and Joshi, 2005). Our results confirm the general point made by SJ08 that the ozone-temperature feedback needs to be taken into account when attributing changes in stratospheric ozone and temperature to their true forcing agents, which are ODS and well-mixed greenhouse gases. However, the quantitative details of the current results are different from those in SJ08, e.g. the masking of upper stratospheric ODS-induced ozone depletion in the past by CO 2 increases is now estimated to be 20% (instead of 10%) and the contribution of CO 2 increases to the future ozone recovery is now about 50% (instead of 40%). The current analysis was performed using www.atmos-chem-phys.   and future . The simulated ozone mixing ratio trend (%/decade) for the selected periods is shown in black while its contributions from CO 2 and ODS are shown in green and blue, respectively. The grey shaded areas indicate the 99% confidence intervals for the linear trend fits for the respective periods.   and future . The simulated ozone mixing ratio trend (%/decade) for the selected periods is shown in black while its contributions from CO 2 and ODS are shown in green and blue, respectively. The grey shaded areas indicate the 99% confidence intervals for the linear trend fits for the respective periods. The green and blue shaded regions indicate the 99% confidence intervals for the CO 2 and ODS attribution estimates (for details see SJ08). The approximate altitudes given on the right-hand side vertical axes are log-pressure heights assuming a constant vertical scale height of 7 km. CO 2 heating rates rather than CO 2 abundance as the explanatory variable, as it was found that the nonlinearity of the relation between the two was sufficiently large over timescales encompassing ozone depletion and recovery that using CO 2 abundance led to erroneous results. This fact should be taken into account in multiple linear regression studies, which traditionally use CO 2 abundance as the explanatory variable.
There are significant compensatory effects between the two major changes in the analysis presented here compared to that presented in SJ08, namely the correction of the CO 2 heating rate in the radiation scheme and the inclusion in the analysis of the nonlinear response of CO 2 heating rates to CO 2 changes. The differences between the attribution results presented here and those in SJ08 thus do not reflect the importance of the nonlinear dependence of the radiative response to CO 2 changes; the latter can be clearly seen by comparing Figs. 3a and 4 in this paper.
Atmos. Chem. Phys., 9, 8447-8452, 2009 www.atmos-chem-phys.net/9/8447/2009/  Fig. 4. Same as Fig. 3a but using CO 2 mixing ratio (as opposed to CO 2 heating rate) as the explanatory variable for the attribution analysis. Fig. 4. Same as Fig. 3a but using CO 2 mixing ratio (as opposed to CO 2 heating rate) as the explanatory variable for the attribution analysis.

Appendix A
To calculate CO 2 heating rates in the middle atmosphere, the CMAM uses the matrix parameterization of Fomichev et al. (1998). This parameterization provides matrix coefficients for a set of reference CO 2 amounts (150, 360, 540 and 720 ppmv); to calculate the heating rates for an arbitrary CO 2 concentration, an interpolation of matrix coefficients between the reference points is applied. However, the matrix interpolation method suggested in Fomichev et al. (1998) provides inaccurate sensitivity of the heating rates to CO 2 changes near the reference point of 360 ppmv. The matrix interpolation method suggested in Fomichev et al. (1998) utilizes the fact that the CO 2 dependence is present in matrix coefficients A ij in both explicit and implicit forms and can be expressed as c v ×F (c v ), where c v is the CO 2 volume mixing ratio and F is some combination of second order exponential integrals. Based on this fact, linear interpolation of the values log(A ij /c v ) is recommended. While this method provides heating rates for arbitrary CO 2 concentrations within the parameterization accuracy of 0.3 K/day, it produces a slight discontinuity in the heating rate dependence on CO 2 near the reference points, which leads to an inaccurate sensitivity of the heating rates to CO 2 changes, in particular near the reference point of 360 ppmv, as can be seen in Fig. A1 (dashed curves).
The original interpolation procedure has been replaced with a standard second order spline interpolation of A ij . This correction provides a smooth dependence of the heating rate on CO 2 concentration throughout the entire range of CO 2 changes (Fig. A1, solid curves). The corrected interpolation procedure has been used for the CMAM CCMVal Heating rate for CO 2 as a function of CO 2 mixing and pressure in the middle atmosphere. The vertical dashed-d lines indicate the reference CO 2 amounts in the parameteriz of Fomichev et al. (1998). The dashed curves show values ach with the original method for interpolation of matrix coefficient details see text), which were used for the CMAM CCMVal-1 lations. The solid curves show the corrected heating rates ach with spline interpolation of matrix coefficients, which were for the CMAM CCMVal-2 simulations. The arrows at the top o plot indicate the years by which a given CO 2 amount is achiev the CCMVal "REF2" scenario. Heating rate for CO 2 as a function of CO 2 mixing ratio and pressure in the middle atmosphere. The vertical dashed-dotted lines indicate the reference CO 2 amounts in the parameterization of Fomichev et al. (1998). The dashed curves show values achieved with the original method for interpolation of matrix coefficients (for details see text), which were used for the CMAM CCMVal-1 simulations. The solid curves show the corrected heating rates achieved with spline interpolation of matrix coefficients, which were used for the CMAM CCMVal-2 simulations. The arrows at the top of the plot indicate the years by which a given CO 2 amount is achieved in the CCMVal "REF2" scenario.
simulations, which bring the CMAM temperature trend for the near past in line with other model results and observations (Shine et al., 2008).