The Sun is a prime driver of various processes in the climate system. From observations of the Sun's variability on decadal or centennial timescales, it is possible to identify temporal patterns and trends in solar activity, and consequently to derive the related mechanisms of the solar influence on the Earth's climate e.g.. Of the semi-regular solar cycles, the most prominent is the approximate 11-year periodicity which manifests in the solar magnetic field or through fluctuations of sunspot number, but also in the total solar irradiance (TSI) or solar wind properties. For the dynamics of the middle atmosphere, where most of the ozone production and destruction occur, the changes in the spectral solar irradiance (SSI) are the most influential, since the TSI as the integral over all wavelengths exhibits variations of orders lower than the ultraviolet part of the spectrum . This fact was supported by original studies e.g. that suggested the solar cycle (SC) influence on the variability of the stratosphere. have shown, with the fixed dynamical heating model, that the response of temperature in the photochemically controlled region of the upper tropical stratosphere is due to both direct solar heating and an indirect effect caused by the ozone changes.

Numerous studies have identified temperature and ozone changes linked to the 11-year cycle by multiple linear regression. The use of ERA-40 reanalysis pointed to a manifestation of annually averaged solar signal in temperature, exhibited predominantly around the Equator with amplitudes up to 2 K around the stratopause and with a secondary amplitude maximum of up to 1 K in the lower stratosphere. , and have used satellite ozone data sets to characterise statistically significant responses in the upper and lower stratosphere. The observed double-peaked ozone response in the vertical profile around the Equator was reproduced in some chemistry climate models, although concerns about the physical mechanism of the lower stratospheric response were expressed .

The ozone and temperature perturbations associated with the SC have an impact on the middle atmospheric circulation. They produce a zonal wind anomaly around the stratopause (faster subtropical jet) during solar maxima through the enhanced meridional temperature gradient. Since planetary wave propagation is affected by the zonal mean flow , we can suppose that a stronger subtropical jet can deflect planetary waves propagating from higher latitudes. Reduced wave forcing can lead to decreasing/increasing or upwelling/downwelling motions in the equatorial or higher latitudes respectively . The Brewer–Dobson circulation (BDC) is weaker during solar maxima , although this appears to be sensitive to the state of the polar winter. Observational studies, together with model experiments e.g., suggest a so-called “top-down” mechanism where the solar signal is transferred from the upper to lower stratosphere, and even to tropospheric altitudes.

Statistical studies e.g. have also focused on the lower stratospheric solar signal in the polar regions and have revealed modulation by the Quasi-Biennial Oscillation (QBO), or the well known Holton–Tan relationship modulated by the SC. Proposed mechanisms by suggested that the solar signal induced during early winter in the upper equatorial stratosphere propagates poleward and downward when the stratosphere transits from a radiatively controlled state to a dynamically controlled state involving planetary wave propagation . The mechanism of the SC and QBO interaction, which stems from reinforcing each other or canceling each other out , has been verified by WACCM3.1 model simulations . These proved the independence of the solar response in the tropical upper stratosphere from the response dependent on the presence of the QBO at lower altitudes. However, fully coupled WACCM-4 model simulations by raised the possibility of occurrence by chance of the observed solar-QBO response in the polar region. The internally generated QBO was not fully realistic though. In particular, the simulated internal QBO descended down to only about 50 hPa.

It has been shown that difficulties in the state-of-the-art climate models arise when reproducing the solar signal influence on winter polar circulation, especially in less active sun periods . The hypothesis is that solar UV forcing is too weak in the models. Satellite measurements indicate that variations in the solar UV irradiance may be larger than previously thought . However, the measurements by from the SORCE satellite may have been affected by instrument degradation with time and so may be overestimating the UV variability see the review by. The latter authors have also concluded that the SORCE measurements probably represent an upper limit on the magnitude of the SSI variation. Consequent results of general circulation models, forced with the SSI from the SORCE measurements, have shown a larger stratospheric response than for the NRL SSI data set. Thus, coordinated work is needed to have reliable SSI input data for GCM and CCM simulations , and also to propose robust conclusions concerning SC influence on climate .

At the Earth's surface, the detection of the SC influence is problematic since there are other significant forcing factors, e.g. greenhouse gases, volcanoes and aerosol changes e.g., as well as substantial variability attributable to internal climate dynamics. However, several studies detected the solar signal in sea level pressure and sea surface temperature, which supports the hypothesis of a troposphere–ocean response to the SC. Some studies e.g. suggest a so-called “bottom-up” solar forcing mechanism that contributes to the lower stratospheric ozone and temperature anomaly in connection with the lower stratosphere deceleration of the BDC.

The observed double-peaked ozone anomaly in the vertical profile around the Equator was supported by the simulations of coupled chemistry climate models . However, the results presented by suggest the contribution of SC variability could be smaller since two major volcanic eruptions are aligned with solar maximum periods and also given the shortness of the analysed time series (in our case, 35 years). These concerns related to the lower stratospheric response of ozone and temperature derived from observations have already been raised e.g.. However, another issue is whether or not the lower stratospheric response could depend on the model employed in the simulations .

Several past studies e.g. used multiple linear regression to extract the solar signal and separate other climate phenomena like the QBO, the effect of aerosols, North Atlantic Oscillation (NAO), El Niño–Southern Oscillation (ENSO) or trend variability. Apart from this conventional method, it is possible to use alternative approaches to isolate and examine particular signal components, such as wavelet analysis or empirical mode decomposition . The nonlinear character of the climate system also suggests potential benefits from the application of fully nonlinear attribution techniques to study the properties and interactions in the atmosphere. However, such nonlinear methods have been used rather sporadically in the atmospheric sciences e.g., mainly due to their several disadvantages such as the lack of explanatory power .

To examine middle atmospheric conditions, it is necessary to study reliable and sufficiently vertically resolved data. Systematic and global observations of the middle atmosphere only began during the International Geophysical Year (1957–1958) and were later expanded through the development of satellite measurements . Supplementary data come from balloon and rocket soundings, though these are limited by their vertical range (only the lower stratosphere in the case of radiosondes) and the fact that the in situ observations measure local profiles only. By assimilation of these irregularly distributed data and discontinuous measurements of particular satellite missions into an atmospheric/climatic model, we have modern basic data sets available for climate research, so-called reanalyses. These types of data are relatively long, globally gridded with a vertical range extending to the upper stratosphere or the lower mesosphere and thus suitable for 11-year SC research. In spite of their known limitations such as discontinuities in ERA reanalysis –, they are considered an extremely valuable research tool .

Coordinated intercomparison has been initiated by the SPARC (Stratospheric Processes and their Role in Climate) community to understand them, and to contribute to future reanalysis improvements . Under this framework, have examined nine reanalysis data sets in terms of 11-year SC, volcanic, ENSO and QBO variability. Complementing their study, we provide here a comparison with nonlinear regression techniques, assessing robustness of the results obtained by multiple linear regression (MLR). Furthermore, EP flux diagnostics are used to examine solar-induced response during the winter season in both hemispheres, and solar-related variations of assimilated ozone are investigated.

The paper is arranged as follows. In Sect.  the used data sets are described. In Sect.  the analysis methods are presented along with regressor terms employed in the regression model. Section  is dedicated to the description of the annual response results. In Sect.  solar response in MERRA reanalysis is presented. Next, in Sect.  other reanalyses are compared in terms of SC. Comparison of linear and nonlinear approaches is presented in Sect. . Section  describes monthly evolution of SC response in the state variables. Section  is aimed at dynamical consequences of the SC analysed using the EP flux diagnostics.

Our analysis was applied to the most recent generation of three reanalysed data sets: MERRA (Modern Era Reanalysis for Research and Applications, developed by NASA) , ERA-Interim (ECMWF Interim Reanalysis) and JRA-55 (Japanese 55-year Reanalysis) . We have studied the series for the period 1979–2013. All of the data sets were analysed on a monthly basis. The Eliassen–Palm (EP) flux diagnostics (described below) was computed on a 3-hourly basis from MERRA reanalysis and subsequently monthly means were produced. A similar approach has already been used by and . The former study proposed that even 6-hourly data are sufficient to diagnose tropical upwelling in the lower stratosphere. The vertical range extends to the lower mesosphere (0.1 hPa) for MERRA, and to 1 hPa for the remaining reanalyses. The horizontal resolution of the gridded data sets was 1.25∘ × 1.25∘ for MERRA and JRA-55 and 1.5∘ × 1.5∘ for ERA-Interim respectively.

In comparison with previous generations of reanalyses, it is possible to observe a better representation of stratospheric conditions. This improvement is considered to be connected with increasing the height of the upper boundary of the model domain . For example, the Brewer–Dobson circulation was markedly overestimated by ERA-40; an improvement was achieved in ERA-Interim, but the upward transport remains faster than observations indicate . Interim results of JRA-55 suggest a less biased reanalysed temperature in the lower stratosphere relative to JRA-25 .

In addition to the standard variables provided in reanalysis, i.e. air temperature, ozone mixing ratio and circulation characteristics – zonal, meridional or omega velocity – we have also analysed other dynamical variables. Of particular interest were the EP flux diagnostics – a theoretical framework to study interactions between planetary waves and the zonal mean flow . Furthermore, this framework allows the study of the wave propagation characteristics in the zonal wind and the induced (large-scale) meridional circulation as well. For this purpose the quasi-geostrophic approximation of transformed Eulerian mean (TEM) equations were used in the form employed by , i.e. using their formula (3.1) for EP flux vectors, (3.2) for EP flux divergence and (3.4) for residual circulation. These variables were then interpolated to a regular vertical grid. For the visualisation purposes, the EP flux arrows were scaled by the inverse of the pressure. The script was publicly released .

To detect variability and changes due to climate-forming factors, such as the 11-year SC, we have applied an attribution analysis based on multiple linear regression (MLR) and two nonlinear techniques. The regression model separates the effects of climate phenomena that are supposed to have an impact on middle atmospheric conditions. Our regression model of a particular variable X as a function of time t, pressure level p, latitude φ and longitude λ is described by the following equation: X(t,z,φ,λ)=α(t;z,φ,λ)+β(z,φ,λ)TREND(t)+γ(z,φ,λ)SOLAR(t)+δ1(z,φ,λ)QBO1(t)+δ2(z,φ,λ)QBO2(t)+δ3(z,φ,λ)QBO3(t)+ε(z,φ,λ)ENSO(t)+ζ(z,φ,λ)SAOD(t)+η(z,φ,λ)NAO(t)+e(t,z,φ,λ).

After deseasonalising, which can be represented by the α index for every month in a year, the individual terms represent a trend regressor TREND(t) either in linear form or including the equivalent effective stratospheric chlorine (EESC) index (this should be employed due to the ozone turnover trend around the middle of the 90s), a SOLAR(t) represented by the 10.7 cm radio flux as a proxy for solar ultraviolet variations at wavelengths 200–300 nm that are important for ozone production and radiative heating in the stratosphere, and which correlates well with sunspot number variation (the data were acquired from Dominion Radio Astrophysical Observatory (DRAO) in Penticton, Canada).

We have also included the quasi-biennial proxies QBO1,2,3(t) as another stratosphere-related predictor. Similar studies have represented the QBO in multiple regression methods in several ways. Our approach involves three separate QBO indices extracted from each reanalysis. These three indices are the first three principal components of the residuals of our linear regression model () excluding QBO predictors applied to the equatorial zonal wind. The approach follows the paper by or the study by to avoid contamination of the QBO regressors by the solar signal or other regressors. The three principal components explain 49, 47 and 3 % of the total variance for the MERRA; 60, 38 and 2 % for the JRA-55; and 59, 37 and 3 % for the ERA-Interim. The extraction of the first two components reveals a 28-month periodicity and an out-of phase relationship between the upper and lower stratospheres. The out-of phase relationship or orthogonality manifests approximately in a quarter period shift of these components. The deviation from the QBO quasi-regular period represented by the first two dominant components is contained in the residual variance. Linear regression analysis of the zonal wind with the inclusion of the first two principal components reveals a statistically significant linkage between the third principal component and the residuals of this analysis. Furthermore, the regression coefficient of this QBO proxy was statistically significant for all variables p value <0.05 (see below for details about significance testing techniques). Wavelet analysis for the MERRA demonstrates three statistically significant but non-stationary periods exceeding the level of the white noise wavelet spectrum (not shown): an approximate annual cycle (a peak period of 1 year and 2 months), a cycle with a peak period of 3 years and 3 months and a long-period cycle (a peak period between 10 and 15 years). Those interferences can be attributed to the possible nonlinear interactions between the QBO itself and other signals like the annual cycle or long-period cycle such as the 11-year SC at the equatorial stratosphere.

The El Niño–Southern Oscillation is represented by the multivariate ENSO index ENSO(t) which is computed as the first principal component of the six main observed variables over the Pacific Ocean: sea level pressure, zonal and meridional wind, sea surface temperature, surface air temperature and total cloudiness fraction of the sky . The effect of volcanic eruptions is represented by the stratospheric aerosol optical depth SAOD(t). The time series was derived from the optical extinction data . We have used globally averaged time series in our regression model. The North Atlantic Oscillation has also been included through its index NAO(t) derived by rotated principal component analysis applied to the monthly standardised 500 hPa height anomalies obtained from the Climate Data Assimilation System (CDAS) in the Atlantic region between 20 and 90∘ N .

The robustness of the solar regression coefficient has been tested in terms of including or excluding particular regressors in the regression model; e.g. the NAO term was removed from the model and the resulting solar regression coefficient was compared with the solar regression coefficient from the original regression set-up. The solar regression coefficient seems to be highly robust since neither the amplitude nor the statistical significance field was changed significantly when NAO or QBO3 or both of them were removed. However, cross-correlation analysis reveals that the correlation between NAO and TREND, SOLAR and SAOD regressors is statistically significant, but small (not shown).

The multiple regression model of Eq. () has been used for the attribution analysis, and supplemented by two nonlinear techniques. The MLR coefficients were estimated by the least squares method. To avoid the effect of autocorrelation of residuals and to obtain the best linear unbiased estimate (BLUE) according to the Gauss–Markov theorem , we have used an iterative algorithm to model the residuals as a second-order autoregressive process. A Durbin–Watson test confirmed that the regression model was sufficient to account for most of the residual autocorrelations in the data.

As a result of the uncorrelated residuals, we can suppose the standard deviations of the estimated regression coefficients not to be diminished . The statistical significance of the regression coefficients was computed with a t test.

The nonlinear approach, in our case, consisted of a multi-layer perceptron (MLP) and the relatively novel epsilon support vector regression (ε–SVR) technique with the threshold parameter ε=0.1. The MLP as a technique inspired by the human brain is capable of capturing nonlinear interactions between inputs (regressors) and output (modelled data) e.g.. The nonlinear approach is achieved by transferring the input signals through a sigmoid function in a particular neuron and within a hidden layer propagating to the output (a so-called feed–forward propagation). The standard error back–propagation iterative algorithm to minimise the global error has been used.

The support vector regression technique belongs to the category of kernel methods. Input variables were nonlinearly transformed to a high-dimensional space by a radial basis (Gaussian) kernel, where a linear classification (regression) can be constructed . However, cross-validation must be used to establish a kernel parameter and cost function searched in the logarithmic grid from 10-5 to 101 and from 10-2 to 105 respectively. We have used 5-fold cross-validation to optimise the SVR model selection for every point in the data set as a trade-off between the recommended number of folds and computational time. The MLP model was validated by the holdout cross-validation method since this method is more expensive in order of magnitude in terms of computational time. The data sets were separated into a training set (75 % of the whole data set) and a testing set (25 % of the whole data set). The neural network model was restricted to only one hidden layer with the maximum number of neurons set up to 20.

The earlier mentioned lack of explanatory power of the nonlinear techniques in terms of complicated interpretation of statistical models mainly comes from nonlinear interactions during signal propagation and the impossibility to directly monitor the influence of the input variables. In contrast to the linear regression approach, the understanding of relationships between variables is quite problematic. For this reason, the responses of our variables have been modelled by a technique originating from sensitivity analysis studies and also used by e.g. . The relative impact RI of each variable was computed as RI=Ik∑Ik, where Ik=σ(y^-yk^). σ(y^-yk^) is the variance of the difference between the original model output y^ and the model output yk^ when the k-input variable was held at its constant level. There are many possibilities with regard to which constant level to choose. It is possible to choose several levels and then to observe the sensitivity of model outputs varying for example on minimum, median and maximum levels. Our sensitivity measure (relative impact) was based on the median level. The primary reason comes from purely practical considerations – to compute our results fast enough as another weakness of the nonlinear techniques lies in the larger requirement of computational capacity. In general, this approach was chosen because of their relative simplicity for comparing all techniques to each other and to be able to interpret them too. The contribution of variables in neural network models has already been studied and produced a review and comparison of these methods.

In this section, we discuss the dynamical impact of the SC and its influence on middle atmospheric winter conditions. Linear regression was applied to the EP diagnostics. suggested that the solar signal produced in the upper stratosphere region is transmitted to the lower stratosphere through the modulation of the internal mode of variation in the polar-night jet and through a change in the Brewer–Dobson circulation (prominent in the equatorial region in the lower stratosphere). In our analysis, we discussed the evolution of the winter circulation with an emphasis on the vortex itself rather than the behaviour of the jets. Furthermore, we try to describe the possible processes leading to the observed differences in the quantities of state between the solar maximum and minimum period. Because the superposition principle only holds for linear processes, it is impossible to deduce the dynamics merely from the fields of differences. As noted by Kodera and Kuroda (2002), the dynamical response of the winter stratosphere includes highly nonlinear processes, e.g. wave–mean flow interactions. Thus, both the anomaly and the total fields, including climatology, must be taken into account.

We start the analysis of solar maximum dynamics with the period of the northern hemispheric winter circulation formation. The anomalies of the ozone, temperature, geopotential in the lower stratosphere only and Eliassen–Palm flux divergence mostly in the upper stratosphere support the hypothesis of weaker BDC during the solar maximum due to the less intensive wave pumping. This is possible through the “downward control” principle when modification of wave–mean flow interaction in the upper levels governs changes in residual circulation below . The finding about weaker BDC during the solar maximum is consistent with previous studies . The causality is unclear, but the effect is visible in both branches of BDC as is illustrated by Fig.  and summarised schematically in Fig. .

During the early Northern Hemisphere (NH) winter (including November) when westerlies develop in the stratosphere, we can observe a deeper polar vortex and consequent stronger westerly winds both inside and outside the vortex. However, only the westerly anomaly outside the polar region and around 30∘N from 10 hPa to the lower mesosphere is statistically significant (see the evolution of zonal wind anomalies in Fig. e–h). The slightly different wind field has a direct influence on the vertical propagation of planetary waves. From the Eliassen–Palm flux anomalies and climatology we can see that the waves propagate vertically with increasing poleward instead of equatorward meridional direction with height. This is then reflected in the EP flux divergence field, where the region of maximal convergence is shifted poleward and the anomalous convergence region emerges inside the vortex above approximately 50 hPa (Fig. m–p).

The poleward shift of the maximum convergence area further contributes to the reduced BDC. This is again confirmed by the temperature and ozone anomalies. The anomalous convergence inside the vortex induces anomalous residual circulation, the manifestation of which is clearly seen in the quadrupole-like temperature structure (positive and negative anomalies are depicted schematically in Fig.  using red and blue boxes respectively). This pattern emerges in November and even more clearly in December. In December, the induced residual circulation leads to an intrusion of the ozone-rich air into the vortex at about the 1 hPa level (Fig. r). The inhomogeneity in the vertical structure of the vortex is then also pronounced in the geopotential height differences. This corresponds to the temperature analysis in the sense that above and in the region of the colder anomaly there is a negative geopotential anomaly and vice versa. The geopotential height difference has a direct influence on the zonal wind field (via the thermal wind balance). The result is a deceleration of the upper vortex parts and consequent broadening of the upper parts (due to the conservation of angular momentum).

Considering the zonal wind field, the vortex enters January approximately with its average climatological extent. The wind speeds in its upper parts are slightly higher. This is because of the smaller geopotential values corresponding to the negative temperature anomalies above approximately 1 hPa. This probably results from the absence of adiabatic heating due to the suppressed BDC, although the differences in the quantities of state (temperature and geopotential height) are small and insignificant (see the temperature anomalies in Fig. c). It is important to note that these differences change sign around an altitude of 40 km inside the vortex further accentuating the vertical inhomogeneity of the vortex. This might start balancing processes inside the vortex, which is confirmed by analysis of the dynamical quantities, i.e. EP flux and its divergence (Fig. o).

Significant anomalies of the EP flux indicate anomalous vertical wave propagation resulting in the strong anomalous EP flux convergence being significantly pronounced in a horizontally broad region and confined to upper levels (convergence (negative values) drawn by green or blue shades in Fig. m–p). This leads to the induction of an anomalous residual circulation starting to gain intensity in January. The situation then results in the disruption of the polar vortex visible in significant anomalies in the quantities of state in February – in contrast to January. Further strong mixing of air is suggested by the ozone fields. The quadrupole-like structure of the temperature is visible across the whole NH middle atmosphere in February (indicated in the lower diagram of Fig. ), especially in the higher latitudes. This is very significant and well pronounced by the stratospheric warming and mesospheric cooling.

The hemispheric asymmetry of the SC influence can be especially documented in winter conditions, as was already suggested in Sect. . Since the positive zonal wind anomaly halts at approximately 60∘ S and intensifies over 10 m s-1, one would expect the poleward deflection of the planetary wave propagation to be according to NH winter mechanisms discussed above. This is actually observed from June to August when the highest negative anomalies of the latitudinal component of EP flux are located in the upper stratosphere and in the lower mesosphere (Fig. m–p). The anomalous divergence of EP flux develops around the stratopause between 30 and 60∘ S. Like the hypothetical mechanism of weaker BDC described above, we can observe less wave pumping in the stratosphere and consequently less upwelling in the equatorial region. In line with that, we can see in the lower stratosphere of equatorial region (Figs. b and b) a more pronounced temperature response in August (above 1 K) than in December (around 0.5 K) as already mentioned in previous observational or reanalysis studies. Although this can point to a weaker BDC, the residual circulation (Fig. q–t) as a proxy for BDC does not reveal this signature. Hypothetically, this could be due to a higher role of unresolved wave processes in reanalysis (small-scale gravity waves) or due to the worse performance of residual circulation as a proxy for the large-scale transport in SH (e.g. larger departure from steady waves approximation comparing to NH), or because of the other processes than BDC leading to the temperature anomaly, e.g. aliasing with volcanic signal.

Overall, the lower stratospheric temperature anomaly is more coherent for the SH winter than for the NH winter, where the solar signal is not so apparent or statistically significant in particular months and reanalysis data sets.

We have analysed the changes in air temperature, ozone and circulation characteristics driven by the variability of the 11-year solar cycle's influence on the stratosphere and lower mesosphere. Attribution analysis was performed on the three reanalysed data sets, MERRA, ERA-Interim and JRA-55, and aimed to compare how these types of data sets resolve the solar variability throughout the levels where the “top-down” mechanism is assumed. Furthermore, the results that originated in linear attribution using MLR were compared with other relevant attribution studies and supported by nonlinear attribution analysis using SVR and MLP techniques.

The nonlinear approach to attribution analysis, represented by the application of the SVR and MLP, largely confirmed the solar response computed by linear regression. Consequently, these results can be considered quite robust regarding the statistical modelling of the solar variability in the middle atmosphere. This finding indicates that linear regression is a sufficient technique to resolve the basic shape of the solar signal through the middle atmosphere. However, some uncertainties could partially stem from the fact that the SVR and MLP techniques are highly dependent on an optimal model setting that requires a rigorous cross-validation process (which places a high demand on computing time). As a benefit, nonlinear techniques show an ability to simulate the middle atmosphere variability with higher accuracy than linear regression.

The solar signal extracted from the temperature field from MERRA and ERA-Interim reanalysis using linear regression has the amplitudes around 1 and 0.5 K, in the upper stratospheric and in the lower stratospheric equatorial region respectively. However, the peak amplitudes of the temperature response in the equatorial upper stratosphere occur at different levels (about 4 and 2 hPa respectively). These signals, statistically significant at a p value < 0.01, are in qualitative agreement with previous attribution studies e.g.. A statistically significant signal was only observed in the lower part of the stratosphere in the JRA-55 reanalysis, however with similar amplitudes as the other data sets.

Similar to the temperature response, the double-peaked solar response in ozone was detected in satellite measurements e.g., although concerns were expressed about the physical mechanism of the lower stratospheric response e.g.. However, the exact position and amplitude of both ozone anomalies remain a point of disagreement between models and observations. The results of our attribution analysis point to large differences in the upper stratospheric ozone response to the SC in comparison with the studies mentioned above and even between reanalyses themselves. The upper stratospheric ozone anomaly reaches 2 % in the SBUV(/2) satellite measurements e.g.Fig. 5 which were assimilated as the only source of ozone profiles in MERRA reanalysis. This fact is remarkable since the same signal was not detected in the upper stratosphere in the MERRA results. However, the solar signal in the ozone field seems to be shifted above the stratopause where similar and statistically significant solar variability was attributed. Concerning the solar signal in the ERA-Interim, there is a negative ozone response via a regression coefficient in the upper stratosphere, although the solar variability expressed as relative impact appears to be in agreement with satellite measurements. The negative ozone response in the tropical upper stratosphere is not consistent with physical expectations for a nominal positive change in solar UV irradiance e.g..

Furthermore, the lower stratospheric solar response in the ERA-Interim's ozone around the Equator is reduced in this data set and shifted to higher latitudes. Another difference was detected in the monthly response of the zonal wind in October and November in the equatorial region of the lower mesosphere between the results for the MERRA series and ERA-40 data studied by . While in the MERRA reanalysis we have detected an easterly anomaly, a westerly anomaly was identified in the ERA-40 series.

A similar problem with the correct resolving of the double-peaked ozone anomaly was registered in the study of which investigated the solar response in the tropical stratospheric ozone using a 3-D chemical transport model. The upper stratospheric solar signal observed in SBUV/SAGE and SAGE-based data could only be reproduced in model runs with unrealistic dynamics, i.e. with no inter-annual meteorological changes.

The reanalyses have proven to be extremely valuable scientific tools . On the other hand, they have to be used with caution, for example, due to the existence of large discontinuities occurring in 1979, 1985 and 1998 that translated into errors in the derived solar coefficients. Our revised analysis with the adjustments from resulted in an 0.2 K/(Smax-Smin) reduction in the temperature solar regression coefficients in tropical latitudes of the upper stratosphere.

In the dynamical effects discussion, we described the dynamical impact of the SC on middle atmospheric winter conditions. The relevant dynamical effects are summarised in schematic diagrams (Fig. ). Both diagrams depict average conditions and anomalies induced by the SC. The first one summarises how equatorward wave propagation is influenced by the westerly anomaly around the subtropical stratopause. The quadrupole-like temperature structure is explained by anomalous residual circulation in the higher latitudes together with the anomalous branch heading towards the equatorial region already hypothesised by . The second diagram concludes the transition time to vortex disruption during February. Again, a very apparent quadrupole-like temperature structure is even more pronounced, especially in the polar region, and seems to be more extended to lower latitudes.

Fields of residual circulation and EP flux divergence in February are opposite to what would be expected from the suppressed BDC in the SC max. There is an enhanced downwelling in the polar and an enhanced upwelling in the equatorial region below 1 hPa. This suggests a need to diagnose the influence of SC on transport at least on a monthly scale because the changes in the underlying dynamics (compare the upper and lower diagrams in Fig. ) would make the transport pathways more complicated. The negative zonal wind response in late northern winter may be caused by an increased likelihood of major stratospheric warmings later in the winter under solar maximum conditions when the polar vortex in early winter is stronger, on average, and therefore less susceptible to disruption e.g.. Since GCMs have not yet successfully simulated this pattern e.g. and due to the short (35-year) time series, it is possible that this pattern is not really solar in origin but is instead a consequence of internal climate variability or aliasing from effects of the two major volcanic eruptions aligned to solar maximum periods.

However, we can strongly assume that the dynamical effects are not zonally uniform, as is shown here using two-dimensional (2-D) EP diagnostics and TEM equations. Hence, it would be interesting to extend the discussion of dynamical effects for other relevant characteristics, for example, for the analysis of wave propagation and wave–mean flow interaction using the 3-D formulation .

This paper is fully focused on the SC influence, i.e. on decadal changes in the stratosphere and lower mesosphere, although a huge number of results concerning other forcings was generated by attribution analysis. The QBO phenomenon in particular could be one of the points of future interest since the solar–QBO interaction and the modulation of the Holton–Tan relationship by the SC are regarded as highly challenging, especially in global climate simulations .