Stratospheric ozone and quasi-biennial oscillation (QBO) interaction with the tropical troposphere on intraseasonal and interannual timescales: a normal-mode perspective

The Madden–Julian oscillation (MJO) is the main controller of the weather in the tropics on intraseasonal timescales, and recent research provides evidence that the quasi-biennial oscillation (QBO) influences the MJO interannual variability. However, the physical mechanisms behind this interaction are not completely understood. Recent studies on the normal-mode structure of the MJO indicate the contribution of global-scale Kelvin and Rossby waves. In this study we test whether these MJO-related normal modes are affected by the QBO and stratospheric ozone. The partial directed coherence method was used and enabled us to probe the direction and frequency of the interactions. It was found that equatorial stratospheric ozone and stratospheric zonal winds are connected with the MJO at periods of 1–2 months and 1.5–2.5 years. We explore the role of normalmode interactions behind the stratosphere–troposphere coupling by performing a linear regression between the MJO–QBO indices and the amplitudes of the normal modes of the atmosphere obtained by projections on a normal-mode basis using ERA-Interim reanalysis data. The MJO is dominated by symmetric Rossby modes but is also influenced by Kelvin and asymmetric Rossby modes. The QBO is mostly explained by westwardpropagating inertio-gravity waves and asymmetric Rossby waves. We explore the previous results by identifying interactions between those modes and between the modes and the ozone concentration. In particular, westward inertio-gravity waves, associated with the QBO, influence the MJO on interannual timescales. MJO-related modes, such as Kelvin waves and Rossby waves with a symmetric wind structure with respect to the Equator, are shown to have significantly different dynamics during MJO events depending on the phase of the QBO.

Here we provide an example of PDC analysis for a synthetically generated time series.

PDC analysis of a synthetic time series
In order to exemplify the use of partial directional coherence (PDC) method we present an analysis of a times series generated by iterated functions in such a way that we know beforehand the direction of the causality. Consider a bivariate time series ( 1 ( ), ( )) ∈ ℝ 2 for all times , generated iteratively: 1 ( ) = 1.601 1 ( − 1) − 0.98 1 ( − 2) + 1 ( ) 2 ( ) = 0.85 2 ( − 1) − 0,05 1 ( − 1) + 2 ( ) (S1) Where 1 and 2 are uncorrelated Gaussian white noise with distribution (0,1). From equation (S1) we see that the present (time ) of the componet depend only on the past component of the component itself at time ( − 1) and ( − 2). For the component the present state depends on the past of at lag one ( 2 ( − 1)) and of the past of the component at lag one ( 1 ( − 1)). Therefore for this time series 1 causes 2 but 2 doesn't cause 1 .
The generated time series were simulated with 1024 points. A VAR(2) model (bivariate autoregressive model with two lags) was fitted to the data using the Hannan-Quinn criterion. The fitted model passed Portmanteau test with 5% confidence level.
In figure (S1) we present the PDC analysis of the time series generated by formula (S1). A detailed explanation on how to read a PDC plot is given in the caption of the figure. Figure S1: PDC analysis of a synthetically generated time series through equation (S1). The horizontal axis is given in frequency (non dimensional) and the vertical axis represents the iPDC coefficient. 11 ( ) represents the power spectral density of the variable 1 and is presented at the top to the left of the panel, respectively 22 ( ) represents the power spectral density of the variable 2 and is presented at the bottom to the right of the panel. Coefficient 12 ( ) represents the influence of the variable 2 on 1 presented at the top to the right of the graph. Coefficient 21 ( ) represents the influence of the variable 1 on 2 presented at the bottom to the left of the graph. The value of the PDC at a given frequency is given by the red(green) curve. If the curve is red at a given frequency it means that there is causality from one variable to the other at that frequency within the chosen confidence level, conversely if the curve is green at a given frequency it means there is no causality from one variable to the other at that frequency. The dashed curve represent the statistical threshold above which the causality becomes statistically significant, in other words, if the PDC curve is above the dashed curve at a given frequency the causality is significant and the PDC curve is red, respectively if the PDC curve is under the dashed curve causality is not statistically significant and the curve is green.
From the graph on figure (S1) we conclude that variable 1 influences the variable 2 at the whole spectrum since the PDC curve (at the bottom to the left) is red at all frequencies using 5% of confidence level. On the other hand 2 doesn't influence 1 because PDC curve (at the top to the right) is green at all frequencies. This agrees with what we already knew from the definition of the time series from equation (S1).