Overestimated acceleration of the advective Brewer–Dobson circulation due to stratospheric cooling

Tropospheric warming and stratospheric cooling influence the vertical structure of the atmosphere. Numerous studies have analysed the thermal expansion of the troposphere, however, stratospheric cooling reverses the sign of this shift in the middle stratosphere, causing a downward shift in the upper stratosphere and mesosphere. This is a robust feature in transient climate model simulations, but its impact is commonly unappreciated. Here, we quantify the trend difference of the residual mean vertical velocity ( w‾∗ ), a proxy for diagnosing the advective Brewer–Dobson circulation (BDC) strength, which arises from implicit neglect of the shrinking distance between stratospheric pressure levels in the CCMI‐1 (Chemistry‐Climate Model Initiative part 1) data request. There, a log‐pressure formula with constant scale height is recommended to compute w‾∗ . However, stratospheric cooling in transient climate simulations causes a reduction of the geometrical distance between pressure levels and thereby also the scale height significantly decreases over time. Using the general scale height definition for the transformation, the w‾∗ trends are therefore smaller. In both cases, the units are m·s−1, but in the latter case it is the constant measure of length geopotential metres and not log‐pressure metres. We quantify that, due to the temperature dependence of log‐pressure metres, past studies that based w‾∗ trend analyses on log‐pressure w‾∗ overestimated the advective BDC acceleration by ∼ 20%. This result is consistent among the CCMI‐1 projections over the 1960–2100 period. We highlight that other diagnostics can also be affected by the neglect of the declining stratospheric pressure level distance. A detailed description of the diagnostics is necessary for consistent assessments of trends. Data processing tools should generally not include the constant scale height assumption if the data are used for trend analyses.


INTRODUCTION
There is robust observational evidence that the troposphere is warming and the stratosphere is cooling in response to the radiative forcing of anthropogenic greenhouse gas (GHG) emissions (IPCC, 2013). Changes in temperature directly influence the vertical structure of the atmosphere. This can be quantified via the hypsometric equation (assuming dry air and hydrostatic balance): (1) Here, (p) is the geopotential and Z(p) the geopotential height of pressure levels, g 0 = 9.80665 m⋅s −2 is the global average of gravity at mean sea level, T the temperature, p the pressure and R the gas constant for dry air (R = 287 J⋅kg −1 ⋅K −1 ). Equation (1) shows that the troposphere should be thermally expanding as a consequence of the positive temperature trends. This expansion has been observed and documented by means of measurements of the tropopause upward shift (Seidel and Randel, 2006). Sausen and Santer (2003) have proposed that changes in tropopause height may be a sensitive indicator of anthropogenic climate change. Gettelman et al. (2010), Kim et al. (2013) and Vallis et al. (2015) have shown that the tropopause rise is a robust fingerprint of climate change in simulations with comprehensive climate models. Analysing reanalysis data, Manney and Hegglin (2018) found out that it is induced through both radiative and dynamical processes.
On the one hand, the tropopause rise reflects the thermal expansion of the troposphere via Equation (1). On the other hand, the tropopause pressure is also decreasing over time, which means that the tropopause rise exceeds the vertical shift of pressure levels. Still, the increasing tropopause height is considered as a marker of the upward shift of the general circulation in the troposphere (Singh and O'Gorman, 2012) as well as in the lower stratosphere (Shepherd and McLandress, 2011). In the latter region, it partly overlaps with the robustly projected Brewer-Dobson circulation (BDC) acceleration (e.g., Oberländer-Hayn et al., 2016;Abalos et al., 2017;Eichinger et al., 2019).
Stratospheric cooling is consistently simulated in current chemistry-climate models (CCMs; e.g., Fu et al., 2004;Austin et al., 2009) with regard to satellite-based and radiosonde temperature observations (e.g., Randel et al., 2009;Funatsu et al., 2016;Khaykin et al., 2017;Maycock et al., 2018). This negative temperature trend reduces the upward shift of pressure levels in the stratosphere with increasing altitude so that it even reverses sign in the middle stratosphere (Šácha et al., 2019), leading to a downward shift of the upper stratosphere and mesosphere. The upward shift in the lower stratosphere together with the downward shift in the upper stratosphere result in a contraction of the stratosphere, or in other words, a reduction of stratospheric thickness. The process is even enhanced by the tropopause and stratopause shift relative to pressure levels (Laštovička et al., 2006;Lübken et al., 2009;Berger and Lübken, 2011;Šácha et al., 2019). In the mesosphere and in the lower parts of the thermosphere, the continuing GHG-induced downward shift of the geopotential height of pressure levels influences meteor heights, satellite trajectories, orbital lifetimes, propagation of radio waves and hence the performance of space-based navigational systems (Laštovička et al., 2006;Jacobi, 2014;Stober et al., 2014;Lima et al., 2015).
In the present paper, we report and quantify the uncertainty of advective BDC trends that arise from the implicit neglect of the stratospheric pressure level contraction using log-pressure coordinates, as is requested by the Chemistry-Climate Model Initiative part 1 (CCMI-1; Eyring et al., 2013;Hegglin et al., 2015;Morgenstern et al., 2017). The tropical mean of the residual mean vertical velocity (w * ) at a given pressure level is often taken as a proxy for diagnosing the stratospheric circulation strength (e.g., Butchart et al., 2010;Abalos et al., 2015;Dietmüller et al., 2018). Due to the type of vertical coordinate of most climate models ( levels), w * is commonly converted from Pa⋅s −1 to m⋅s −1 units, which are traditionally used for further analyses and are then consistent among the models. The log-pressure formula (Andrews et al., 1987) requested by CCMI-1 (Hegglin et al., 2016) includes a constant scale height for this conversion. However, the distance between pressure levels (and the scale height) significantly decreases over time in transient climate simulations as the stratosphere cools. The temperature dependence of the scale height is made explicit when calculating the conversion using the equation of state. A key difference between the two conversion formulae is that they are related to different vertical coordinate systems -log-pressure and geopotential coordinates. It has long been known that the relationship between geometric height (or geopotential height) and log-pressure height is nonlinear (e.g., Andrews et al., 1987, table 1). However, the fact that for the log-pressure height the relationship changes as the structure of the atmosphere changes (Schmidt et al., 2006) has often been overlooked in the past. Using the CCMI-1 projection simulations, the present study demonstrates that, particularly for trend studies, the log-pressure formula requested in the CCMI-1 data request can be misleading. In Section 2 of the paper, we briefly describe the analysed CCMI-1 REF-C2 model simulations. In Section 3, we explain in detail how stratospheric cooling distorts the w * trends if the unit conversion is made with constant scale height. In Section 4, an analysis of the CCMI-1 REF-C2 simulations is made to quantify this artificial enhancement of w * , which can be misinterpreted as BDC acceleration and, if not computed consistently, can also have an effect on the streamfunction and the tropical upward mass flux. As stratospheric cooling is a robust and strong pattern in climate projection simulations, also the trends of other diagnostics which are computed by means of a constant scale height will be influenced, and this is also the case in other multi-model projects. We discuss these points and some consequences for climate model analyses in Section 5 and end with concluding remarks in Section 6.
This selection is based on the availability of the variables in the British Atmospheric Data Centre (BADC) repository we require for the analysis. In particular, we used the variables w * , temperature and geopotential height. The REF-C2 simulations cover the period 1960-2100 and follow the WMO (2011) A1 scenario for ozone-depleting substances (ODSs) and the Representative Concentration Pathway (RCP 6.0) scenario (Meinshausen et al., 2011) for other greenhouse gases, tropospheric ozone (O 3 ) precursors, as well as aerosol and aerosol precursor emissions. For anthropogenic emissions, the recommendation was to use MACCity (Granier et al., 2011(Granier et al., ) until 2000, followed by RCP 6.0 emissions. For more details on the particular models and on the model set-up, refer to Morgenstern et al. (2017) and citations therein.
Due to the reversal in signs of ODS and ozone trends, the year 2000 marks a change in stratospheric dynamics Polvani et al., 2018;Eichinger et al., 2019). However, figures 3 and 4 in Šácha et al. (2019) show that, with respect to the pressure level rise, there is no substantial trend difference between shorter periods between 1960 and 2100. The multi-model mean geopotential heights at 1 hPa, at 100 hPa and their difference show no trend reversals or changes related to the periods of ODS or ozone rise and decline (see Supplementary Information). Therefore, we consider the entire period 1960-2100 for our analysis.

PHYSICAL MECHANISM
As mentioned above, tropospheric warming together with stratospheric cooling lead to an upward shift of pressure levels in the troposphere (e.g., Vallis et al., 2015). This upward shift reduces with altitude in the lower stratosphere and reverses sign in the middle stratosphere, resulting in a downward shift of pressure levels in the upper stratosphere and mesosphere in transient climate model simulations (Šácha et al., 2019). Figure 1 shows the geopotential height trends of the EMAC-L47 CCMI-1 REF-C2 simulation from 1960 to 2100 (Jöckel et al., 2016;Morgenstern et al., 2017) to illustrate the shrinking distances between pressure levels in the stratosphere due to stratospheric cooling. A positive geopotential height trend is visible in the troposphere and lower stratosphere with a maximum in the tropical tropopause region. Above the middle stratosphere, the trends are negative and the magnitude of the trend increases with altitude. This means that the pressure levels in the stratosphere move closer together. Table 1 shows the mean distance between 100 and 1 hPa and its absolute and relative trends in percentage as a proxy for the shrinking distance between stratospheric pressure levels for the REF-C2 simulations.
The multi-model-mean (MMM) climatological distance between the 1 and 100 hPa levels is 31.0 (±0.2) km in the tropical stratosphere. The distance shrinks by about 70 m per decade, which accounts for about 0.23% per decade. This is consistently simulated in the eleven REF-C2 simulations, and the inter-model standard deviations are comparably small (4 m⋅decade −1 and 0.01%⋅decade −1 ).  Akmaev and Fomichev (1998) and Schmidt et al. (2006) have noted that in CO 2 -doubling experiments, atmospheric changes can be very different for fixed height and fixed pressure. Berger and Lübken (2011) have shown that, due to the downward shift of pressure levels in the middle atmosphere, trends at geometric heights are larger than trends at pressure levels. For the BDC trends in the midlatitude lower stratosphere, this discrepancy has been quantified in Šácha et al. (2019) and a correction term has been proposed. Hence, the choice of the vertical coordinate significantly influences the statistical analysis of trends. In the following, we will show another mechanism by which trend estimates can be influenced by vertical structure changes of the atmosphere, even when the analysis is conducted on pressure levels -by the choice of log-pressure formulae. We show the impact on w * trends, where the log-pressure formula is used for converting it from Pa⋅s −1 to m⋅s −1 .
The scale height H is defined as the height for which the atmospheric pressure decreases by a factor of e in an isothermal atmosphere. Generally, H depends on the temperature T and it can be calculated by which can be derived from the barometric formula (assuming hydrostatic balance). Here, R is the gas constant 287.05 J⋅kg −1 ⋅K −1 . For the stratosphere, H is commonly chosen to be 6950 m (e.g., the CCMI-1 data request; Hegglin et al., 2015) to make the pressure levels resemble the geometric altitude. According to Equation (2), this corresponds to a stratospheric mean temperature of around 237.5 K, which is an arbitrary constant in the log-pressure system. However, when the stratosphere cools, the geometric (or geopotential) distance between pressure levels declines, according to the hypsometric equation (Equation (1)). In Equation (2), this process is reflected in the temperature dependence of the scale height and can be seen as a direct consequence of stratospheric cooling. The constant scale height assumption is commonly used in many well-established climate model diagnostics.
In the present study, we focus on the transformed Eulerian-mean (TEM) vertical velocity w * , which can be calculated by assuming hydrostatic balance (Andrews et al., 1983). Here, a is the mean radius of the Earth, the latitude and = v ′ ′ ∕( ∕ p). Andrews et al. (1987) and Hardiman et al. (2010) give details and the derivation of w * for various approximations and coordinates. The overbar stands for the zonal mean. As most models use hybrid-pressure coordinates, the vertical velocity is usually available in Pa⋅s −1 (then being denoted as ) and is being converted to m⋅s −1 .
For CCMI-1, Hegglin et al. (2015) implicitly recommend a log-pressure formula by defining a fixed scale height in their data request. The log-pressure definition of w * is wherez is the log-pressure height (Hardiman et al., 2014, equation 23). In Equation (4), a conversion of vertical velocity and vertical derivative in pressure coordinates is assumed: This is identical to computing * using Equation (3) and making the transformation afterwards (note that the averaging is made on pressure levels) with The log-pressure formula has also been requested by the previous inter-model comparison activities CCM-Val and CCMVal2 (Chemistry-Climate Model Validation Activity; SPARC, 2010). In the CCMVal data request, equation 3.5.1 of Andrews et al. (1987) (which here is Equation (4)) is explicitly asked for. However, using the general definition of the scale height (Equation (2)), the more general transformation emerges, which includes the temperature dependence and leads to the geopotential height vertical coordinate. At a leading order, the second term on the right-hand side can be neglected (( 2 )), assuming small amplitudes of disturbances . This is indeed the case in our test calculations with EMAC-L90; in the tropical upwelling region, this eddy term and also its trend are two orders of magnitude smaller than the first term with T dependence. For the lack of three-dimensional data of in the BADC repository, the term is neglected further in the study for the transformation between log-pressure and geopotential w * ; that is, the following formula is used: Dietmüller et al. (2018) have already noted in their supplement, that the CCMI-1 w * of the EMAC-L90, EMAC-L47, SOCOLv3 and NIWA/UKCA models on the BADC (British Atmospheric Data Centre) server was uploaded using Equation (8) despite the conflict with the CCMI-1 data request, that is, not using the log-pressure formula. In fact, the Unified Model-based models NIWA/UKCA and ACCESS use hybrid-height as the native vertical coordinate and have a non-hydrostatic dynamical core, leading to the w * calculation in m⋅s −1 (Morgenstern et al., 2009;Stone et al., 2016). We assume that the models use equations 10 and 12 from Hardiman et al. (2010)  The literature has so far not appreciated that Equation (6) implicitly neglects the scale height changes due to stratospheric cooling. To quantify the error in trend calculations that arises due to increasing difference between distances in log-pressure and geometric coordinates (the relation between geometric and geopotential coordinates is constant), we here calculate two different w * trends for each CCMI-1 model. For models of the groups (MRI, GEOSCCM, CMAM, CCSR/NIES, ACCESS and WACCM) that delivered to the BADC server w * transformed according to the log-pressure formula (Equation (6), in the following called w * H ), we calculate the temperature-dependent w * (in the following called w * T ) for each latitude ( ) using Equations (6) and (8) as with H = 6950 m, dependent on time (t) and pressure level (p). For EMAC-L90, EMAC-L47 and SOCOLv3 which used Equation (8) for the transformation, we calculate the log-pressure version by to receive two different w * estimates for each simulation.
We also treat ACCESS and NIWA/UKCA as the models in the latter group, because the w * from these two models is in geometric metres with a constant link to geopotential metres and thus reflects the temperature dependence. The difference between general non-hydrostatic TEM quantities and their hydrostatic analogues has been found to be negligibly small by Hardiman et al. (2010). The difference of the two respective w * trends of each model yields an estimate of the implicitly neglected stratospheric cooling effect on the w * H trends (Equation (1)). Since we only quantify the trends in particular levels in this study, the reduced vertical resolution of the CCMI-1 data to 31 levels in the vertical does not significantly affect our results.

Trends of the residual vertical velocity w *
The strength of the advective part of the BDC is often diagnosed by means of w * (e.g., Butchart et al., 2010;Abalos et al., 2015;Dietmüller et al., 2018). Therefore, the estimate of the BDC acceleration in climate projections is biased through neglect of the decreasing scale height in the stratosphere. A number of studies have calculated w * trends from the log-pressure version or from multi-model datasets and so can be affected (e.g., Garcia and Randel, 2008;Butchart et al., 2010;Palmeiro et al., 2014). The MMM tropical w * trend difference profile is shown in  (9) and (10)). To put this into perspective, the difference between the two w * trends can be boiled down to the fact that the units of the vertical velocities are different although they are written identically: w * T is in geopotential m⋅s −1 , while w * H is in log-pressure m⋅s −1 .
Most models show a minimum of the absolute w * differences in the middle stratosphere. The model spread of the w * trend differences also shows a minimum in the middle stratosphere. However, the relative differences increase with altitude up to the upper stratosphere and also the model spread of the relative differences grows with altitude. This is mainly due to the small w * H trends in the middle stratosphere ( Figure S1). The systematic difference between the two w * trends clearly demonstrates that the oversimplified assumption of a constant scale height leads to an artificially enhanced w * trend and thus to an overestimated acceleration of the advective BDC part. As the stratosphere cools, the geopotential height difference (and the distance in geometric metres) between pressure levels decreases, but remains constant for log-pressure metres (equations 1.1.7 and 1.1.8 in Andrews et al., 1987). Therefore, log-pressure metres are decreasing because geometric and geopotential metres are constant measures of distance. In other words, the original vertical velocity (in Pa⋅s −1 ) increases in absolute values, even if the velocity in geometric (or geopotential in our case) m⋅s −1 is constant.
A widely used proxy for the strength of the advective BDC is the w * value at 70 hPa (e.g., Hardiman et al., 2014;Butchart et al., 2010;Abalos et al., 2015;Dietmüller et al., 2018). Table 2 Figure S3 shows the time series of both w * estimates. The of the trend difference is 20.7%, and so the uncertainty in percentage is only around 14% of the MMM. Therefore, at 70 hPa the relative change is more consistent among the models than the absolute change; this can also be seen in Figure 2. To assess if the uncertainty connected with the determination of the turnaround latitudes has an impact on our results, we have also performed the calculations between fixed latitudes (20 • S-20 • N), yielding similar results; the MMM relative difference is 19.9% and its standard deviation 1.8%. Overall, this means that, when estimated using the w * H trend at 70 hPa, around 20% of the advective BDC acceleration in climate projection simulations is artificial, that is, due to neglecting stratospheric cooling.
The same values as in Table 2, but for 10 hPa, are provided in Table S1. The w * relative differences at 10 hPa are generally larger than at 70 hPa, the MMM of Δw * (10 hPa) is 42.9%. However, that is mainly due to the fact that in general the w * trends are much lower there ( Figure S1). Some of the w * trends at 10 hPa are not even significant (Table S1). The model spread at 10 hPa is much larger (17.3%) than at 70 hPa (2.9%). This is mainly due to the general decrease of the w * trend with altitude, which appears in the denominator when calculating the percentage difference.

Effect on residual streamfunction and mass flux
The residual mean streamfunction and the tropical upward mass flux, which are defined using w * , are among the other quantities that are widely used to study stratospheric transport. The residual mean streamfunction is calculated as with a being the mean Earth's radius, the latitude and the density. Depending on the vertical coordinate system, can either be computed using a constant scale height in the log-pressure system: (with 0 = p 0 ∕(RT 0 ) andz being the log-pressure height), or using the ideal gas law for dry air: The scaling of w * with to calculate the streamfunction (Equation (11)) can either be done with the constant scale height assumption for both w * and , that is, using the log-pressure formula, by to yield * Hc , or using the geopotential conversion, that is, dependent on temperature for both w * and , by to yield * TT . As it could have been expected, the two formulae yield identical results, because the density trend dependence cancels the w * trend dependence on the vertical coordinate. On a more general note, the cancellation can be inferred directly from dimensional analysis, because streamfunction (and mass flux) has the unit kg⋅s −1 and is therefore insensitive to the vertical coordinate. Only quantities that include metres (in the vertical direction) in their units can be affected.
However, inconsistent usage of Equation (11) can lead to streamfunction trend (and climatology) differences. In particular, scaling w * H with T would yield * and scaling w * T with c yields * To provide an estimate for that error, Figure 3 shows the MMM * trend differences (Δ * Tc−TT and Δ * HT−TT ) which emerge from this kind of inconsistent calculation of the streamfunction for the eleven CCMI-1 REF-C2 simulations. Figure 3 shows a considerable overestimation of the streamfunction trend if w * H is used and an underestimation if the constant c is used. Again, this is a direct consequence of stratospheric cooling and could be expected from the equations above. Particularly in the lower stratosphere, the trend differences are comparatively large. Next, this will be quantified more precisely by means of calculating the tropical upward mass flux trends.
The tropical upward mass flux can be computed directly from the streamfunction by for each pressure level and latitude. Hence, the same facts which apply to the streamfunction also apply to the mass flux: consistent usage of the assumptions (F TT and F Hc ) leads to the cancellation of the effect (⇒ F TT = F Hc ), but inconsistent usage (F HT and F Tc ) leads to trend (and climatology) differences. Figure 4 shows the multi-model mean trend differences of the mass flux estimates as a function of altitude with the absolute and relative values. All three estimates of the absolute mass flux trends of each model individually are provided in Figure S5. The F HT trend is larger than the F TT (= F Hc ) trend throughout the stratosphere and the F Tc trend is smaller.
In absolute values, the trend difference decreases strongly with altitude because mass flux generally is much smaller in the upper stratosphere. The relative values range between 15 and 35% and −15 and −35%, respectively, with a maximum in the upper stratosphere. There, the range between maxima and minima is very large, because the CMAM model simulates downward mass fluxes, which is consistent with the negative vertical velocities detected in CMAM. At 70 hPa (the traditional pressure level for mass flux analysis, e.g., Butchart et al., 2010), the MMM of the mass flux trend differences are 28.5% for F HT and −21.5% for F Tc ( Figure S6 and Table S2).

DISCUSSION
While tropospheric warming leads to an expansion of the troposphere (Vallis et al., 2015;Oberländer-Hayn et al., 2016;Abalos et al., 2017), stratospheric cooling contracts the pressure levels in the stratosphere (Šácha et al., 2019). Across the period 1960-2100, the CCMI-1 models simulate that the distance between 100 and 1 hPa shrinks by about 70 m⋅decade −1 . This is around 0.2%⋅decade −1 of the climatological distance and is consistent among the eleven analysed model simulations. Berger and Lübken (2011) and Šácha et al. (2019) have shown that, due to the vertical shift of pressure levels, trends computed in geopotential (or geometric) coordinates are different from trends computed in pressure levels. In the present paper we point out another trivial consequence -that in the stratosphere, the reduction of pressure level distances causes a decrease of the scale height H. However, for various dynamical diagnostics, the common calculation (or conversion) method includes a constant H. One prominent diagnostic of this type is the residual vertical velocity w * . For the CCMI-1 data request, Hegglin et al. (2015) suggest the log-pressure formula including the constant scale height 6950,m for computation of w * .
We compare the trends of stratospheric w * in the Tropics (within the turnaround latitudes) as converted with the suggested log-pressure formula with constant H (w * H ) and with a more general conversion method which includes the temperature dependence (w * T ). Dietmüller et al. (2018) have already shown that at 70 hPa, the climatological value of w * H is about 17% larger than that of w * T . Our analysis shows that also the trend of w * H is systematically larger than that of w * T throughout the stratosphere in all eleven analysed CCMI-1 simulations. The w * differences show a minimum in the middle stratosphere; this is also where the trend values are smallest, hence the relative differences grow with altitude. At 70 hPa, the two w * trend estimates differ by 20.7% in the multi-model mean, with an inter-model standard deviation of only 2.9%, showing consistency among the models. At 10 hPa, the percentage differences are generally larger (42.9%), but also the model spread is much larger (17.3%) and the trends are partly not significant. Via the hypsometric equation, the direct cause of the w * trend differences is the local temperature trend.
The pressure level shift is directly induced by the local temperature trend and scale height changes. Note that this pressure level contraction should not be mistaken for the distance reduction between tropopause and stratopause.
The stratospheric contraction as diagnosed by the distance between the tropopause and stratopause can additionally be influenced by pressure changes at these levels, but this does not have an effect on the scale height. However, it can have an influence on the BDC in a yet unquantified manner.
A number of studies have calculated w * trends (or trends of diagnostics that are based on w * ) for climate change simulations using log-pressure formulae (e.g., Garcia and Randel, 2008;Butchart et al., 2010;Palmeiro et al., 2014). The difference in the two w * trend estimates can have consequences for research on stratospheric dynamics. Based on our results, we estimate that BDC trends that are based on w * can be overestimated by about 20%. However, in order to quantitatively understand climate change-induced trends, they have to be computed as accurately as possible, otherwise studies that for example attribute stratospheric tracer trends to changes in upward transport may be biased. The difference will presumably be multiplicative, that is, for example, a 40% contribution of upward velocities to tropical lower stratospheric ozone trends would bear an 8% error in ozone trend contribution. Not only the w * trends are affected by the constant scale height assumption, which is implicit to the log-pressure formulae. The widely used Eliassen-Palm (EP) flux diagnostic in log-pressure form (Andrews et al., 1987) usesF(z) = −(H∕p 0 )F(p) for conversion of the vertical component of the EP fluxF(z) or F(p) in the CMIP6 DynVarMIP data request (Gerber and Manzini, 2016). Therefore, trend analyses of this quantity can also be biased by the effect of stratospheric cooling (also Hardiman et al., 2010). However, the magnitude of the error in trend analyses with this or other quantities must be evaluated for every individual case. The streamfunction and the tropical upward mass flux are not affected by the error in w * , because the density trend and the scale height trend cancel each other out. In a general perspective, only diagnostics with units including metres can be affected because these can be sensitive to the variable relationship between log-pressure and geometric metres. Note, however, that horizontal distances are not affected by the constant scale height assumption. Still, care has to be taken because, if the scaling for the streamfunction and mass flux calculations is done inconsistently (i.e., if a constant density is mixed with w * T , or a temperature-dependent density with w * H ), errors in the streamfunction or mass flux trend can occur also. And it is fairly easy to make this mistake, because for the CCMI data repository, some modelling groups delivered w * computed other than with the log-pressure formula, despite the request to use the log-pressure formula. Moreover, in most papers, the exact conversion method from Pa⋅s −1 to m⋅s −1 is not provided. This can also make it impossible to conclusively estimate the errors that may have been made in past studies without direct contact with the respective researchers. In the case of the mass flux trends, our estimation of the possible error is either almost 30% or around −20%, respectively for the two error sources.
We have analysed w * data of the CCMI-1 simulations, however, the log-pressure formula for conversion can be found in most of the data requests in multi-model comparison projects. The CCMVal and CCMVal-2 (SPARC, 2010) and the CMIP6  DynVarMIP (Gerber and Manzini, 2016) data requests also include the formula. In CMIP5 (Taylor et al., 2012) w * was not explicitly provided, but there are still studies that analysed w * from CMIP5 simulations. For example, Manzini et al. (2014) used w * data computed by the modelling centres, and in that case it is not possible to track which conversion equations have been applied and so an error estimation is hardly feasible. Also for the first stratospheric multi-model assessments of chemistry climate models (Austin et al., 2003), a clear documentation of equation usage could not be found. Furthermore, the log-pressure equations with constant scale height are generally common and widely used by operators of climate data processing algorithms.
Our results strongly recommend that use of a constant scale height for computing diagnostics should be avoided in climate change simulations. In particular, we call on multi-climate-model projects to consider the effect of varying geopotential height of pressure levels in their data requests. For example, for CCMI-2 it is not too late to tell the modelling centres to avoid formulae with the constant scale height assumption. Similar effects of opposite sign can also be generally expected in the troposphere which is thermally expanding.

CONCLUSIONS
Stratospheric cooling leads to decreasing distances between pressure levels in the stratosphere. In the course of current climate change, the distance between 100 and 1 hPa is decreasing by about 70 m per decade. The stratospheric scale height decreases as a consequence of the contraction of pressure levels. However, a constant scale height is assumed in the CCMI-1 data request for computation of the residual mean vertical velocity w * in the log-pressure form (Hegglin et al., 2015), a diagnostic that is often used as a proxy for the advective Brewer-Dobson circulation (BDC) strength. Analysing eleven CCMI-1 REF-C2 simulations over the period 1960-2100, our study shows that the constant scale height assumption leads to a ∼20% overestimation of the w * trends. In this diagnostic framework, stratospheric cooling therefore leads to an overestimation of the advective BDC acceleration. Other diagnostics like the vertical EP flux are also affected by the error. For the streamfunction as well as the upward mass flux, the computation has to be done consistently with the definition to avoid inaccuracies in their trends, a mistake that can easily be committed because details of the transformation are often unknown. For the data of other multi-model projects and also for individually processed data, the same log-pressure formulae have been used (e.g., SPARC, 2010; Gerber and Manzini, 2016, for CCMVal and CMIP6, respectively). Our study calls for caution when making trend analyses of dynamic or transport variables as the relationships between coordinate systems can alter through climate change. For w * , one should keep in mind that previous studies may bear a 20% error in their trend calculations, thereby overestimating the advective BDC acceleration. This also implies that the contribution estimate of stratospheric tracer trends to the BDC acceleration may be incorrect. In the long run, we encourage those responsible to change their data processing tools and especially to reformulate data requests regarding multi-model assessments. We thank the modelling groups for making their simulations available for this analysis, the SPARC/ IGAC Chemistry-Climate Model Initiative (CCMI) project for organising and coordinating the model data analysis activity and the British Atmospheric Data Centre (BADC) for collecting and archiving the CCMI model output. The ACCESS-CCM simulations were supported by the Australian Research Council's Centre of Excellence for Climate System Science (CE110001028), the Australian Government's National Computational Merit Allocation Scheme (q90) and the Australian Antarctic science grant programme (FoRCES 4012). The CCSR/NIES simulations were performed on NEC-SX9/A(ECO) and NEC-SXACE computers at the Center for Global Environmental Research (CGER), National Institute for Environmental Studies (NIES), and supported by the Environment Research and Technology Development Fund, Ministry of Environment, Japan (2-1303 and 2-1709). The EMAC simulations were conducted at the German Climate Computing Centre DKRZ through support from the Bundesministerium für Bildung und Forschung (BMBF) within the project ESCiMo (Earth System Chemistry integrated Modelling). For the WACCM results, computing resources (ark:/85065/d7wd3xhc) were provided by the Climate Simulation Laboratory at NCAR's Computational and Information Systems Laboratory, sponsored by the National Science Foundation and other agencies. The MetUM was developed by the UK Met Office. The NIWA simulations were carried out within the program CACV by the NZ Governments Strategic Science Investment Fund (SSIF) with support from the New Zealand eScience Infrastructure (NeSI), which is funded by NeSIs collaborator institutions and through the Ministry of Business, Innovation & Employments Research Infrastructure programme. Open access funding enabled and organized by Projekt DEAL.

DATA AVAILABILITY
All CCMI-1 data used in this study can be obtained through the British Atmospheric Data Centre (BADC) archive (ftp://ftp.ceda.ac.uk; accessed 1 February 2020).