This study focusses on the variability of temperature, ozone and circulation characteristics in the stratosphere and lower mesosphere with regard to the influence of the 11-year solar cycle. It is based on attribution analysis using multiple nonlinear techniques (support vector regression, neural networks) besides the multiple linear regression approach. The analysis was applied to several current reanalysis data sets for the 1979–2013 period, including MERRA, ERA-Interim and JRA-55, with the aim to compare how these types of data resolve especially the double-peaked solar response in temperature and ozone variables and the consequent changes induced by these anomalies. Equatorial temperature signals in the tropical stratosphere were found to be in qualitative agreement with previous attribution studies, although the agreement with observational results was incomplete, especially for JRA-55. The analysis also pointed to the solar signal in the ozone data sets (i.e. MERRA and ERA-Interim) not being consistent with the observed double-peaked ozone anomaly extracted from satellite measurements. The results obtained by linear regression were confirmed by the nonlinear approach through all data sets, suggesting that linear regression is a relevant tool to sufficiently resolve the solar signal in the middle atmosphere. The seasonal evolution of the solar response was also discussed in terms of dynamical causalities in the winter hemispheres. The hypothetical mechanism of a weaker Brewer–Dobson circulation at solar maxima was reviewed together with a discussion of polar vortex behaviour.

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

Numerous studies have identified temperature and ozone changes linked to the
11-year cycle by multiple linear regression. The use of ERA-40 reanalysis

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

Statistical studies

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

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

The observed double-peaked ozone anomaly in the vertical profile around the
Equator was supported by the simulations of coupled chemistry climate models

Several past studies

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

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

The paper is arranged as follows. In Sect.

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)

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

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

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

After deseasonalising, which can be represented by the

We have also included the quasi-biennial proxies

The El Niño–Southern Oscillation is represented by the multivariate ENSO
index

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

The multiple regression model of Eq. (

As a result of the uncorrelated residuals, we can suppose the standard
deviations of the estimated regression coefficients not to be
diminished

The nonlinear approach, in our case, consisted of a multi-layer perceptron
(MLP) and the relatively novel epsilon support vector regression
(

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

The earlier mentioned lack of explanatory power of the nonlinear techniques
in terms of complicated interpretation of statistical
models

Figure

From a relative impact point of view (in Fig.

The annually averaged response of the solar signal in the MERRA,
ERA-Interim and JRA-55 zonal-mean temperature

The annually averaged solar signal in the zonal mean of zonal wind
(Figs.

The pattern of the solar response in geopotential height
(Figs.

The annually averaged response of the solar signal in the MERRA
zonal-mean temperature

Figure

Comparison of the results for the MERRA, ERA-Interim and JRA-55 temperature,
zonal wind and geopotential height shows that the annual responses to the
solar signal are in qualitative agreement (compare individual plots in
Fig.

However, upper stratospheric temperature response could be less than accurate
due to the existence of discontinuities in 1979, 1985 and 1998

The annually averaged response of the solar signal in the
ERA-Interim zonal-mean temperature

The variability of the solar signal in the MERRA stratospheric ozone series
was compared with the ERA-Interim results. The analysis points to large
differences in the ozone response to the SC between the reanalyses and in
comparison with satellite measurements by

The lower stratospheric ozone response in the ERA-interim is not limited to
the equatorial belt

The annually averaged response of the solar signal in the JRA-55
zonal-mean temperature

In this paper, we have applied and compared one linear (MLR) and two
nonlinear attribution (SVR and MLP) techniques. The response of the studied
variables to the solar signal and other forcings was studied using the
sensitivity analysis approach in terms of averaged response deviation from
the equilibrium represented by the original model output

In conclusion, the comparison of various statistical approaches (MLR, SVR and MLP) should actually contribute to the robustness of the attribution analysis including the statistically assessed uncertainties. These uncertainties could partially stem from the fact that the SVR and neural network techniques are dependent on an optimal model setting which is based on a rigorous cross-validation process, which places a high demand on computing time.

The major differences between the techniques can be seen in how much of the temporal variability of the original time series is explained, i.e. in the coefficient of determination. For instance, the differences of the explained variance reach up to 10 % between linear and nonlinear techniques, although the zonal structure of the coefficient of determination is almost the same. To conclude, nonlinear techniques show an ability to simulate the middle atmosphere variability with a higher accuracy than cross-validated linear regression.

The monthly averaged response of the solar signal in the MERRA
zonal-mean temperature

As was pointed out by

The monthly averaged response of the solar signal in the MERRA
zonal-mean temperature

Statistically significant upper stratospheric equatorial anomalies in the
temperature series (winter months in Figs.

The above described monthly anomalies of temperature correspond to the zonal
wind anomalies throughout the year (Figs.

In the Southern Hemisphere, this poleward motion of the positive zonal wind
anomaly halts approximately at 60

When comparing the results from the MERRA and ERA-40 series studied
by

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.

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

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

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.

Solar cycle modulation of the winter circulation: schema of the related mechanisms. The upper and lower figure show early and later winter respectively. The heating and cooling anomalies are drawn with red and blue boxes. The EP flux divergence and convergence are drawn with green and yellow boxes. The wave propagation anomaly is expressed as a wavy red arrow in contrast to the climatological average drawn by a wavy grey arrow. The induced residual circulation according to the quasi-geostrophic approximation is highlighted by the bold black lines.

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.

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.

The hemispheric asymmetry of the SC influence can be especially documented in
winter conditions, as was already suggested in Sect.

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

Similar to the temperature response, the double-peaked solar response in
ozone was detected in satellite measurements

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

A similar problem with the correct resolving of the double-peaked ozone
anomaly was registered in the study of

The reanalyses have proven to be extremely valuable scientific tools

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.

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.

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

The authors would like to thank the relevant working teams for the reanalysis
data sets: MERRA (obtained from NASA,