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 normal-mode 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 westward-propagating 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.

The Madden–Julian oscillation (MJO) and the quasi-biennial oscillation (QBO)
are two of the main elements of atmospheric low-frequency variability in
the tropics. The MJO acts on intraseasonal timescales on the troposphere and
impacts tropical monsoons, with global impacts

The stratosphere can act as a mediator between solar forcing and the climate
variability of the troposphere. It is conjectured that stratospheric influence
on the troposphere exists via the so-called top-down mechanism

Stratospheric control of tropospheric phenomena in middle to high latitudes was
addressed in several papers. For instance,

The study of QBO effects on the MJO has gained a lot of interest in the last few
years, since new evidence pointed out this connection

Recent studies have given a normal-mode description of the MJO

In this article, we study the interactions between the stratosphere and the
tropical troposphere, with particular emphasis on the MJO. A time series
analysis causality method, partial directed coherence (PDC)

To resolve the spectrum of the different timescales, timescale separation was applied to the data. We split the data into a fast timescale (periods shorter than 1 year) and a slow timescale (periods greater than 1 year). This was done by performing a resampling procedure on the data with a 10 d rate for the fast timescale. A 6-month window was applied for the slow timescale.

The causality between the QBO, tropical stratospheric ozone, and the MJO was
studied using the PDC method. PDC roughly corresponds to a frequency domain
counterpart of the Granger causality test

We search for normal modes that might contribute to the interactions between
stratospheric and tropospheric phenomena by performing a linear regression
with the MJO indices and stratospheric zonal winds. We then perform the PDC
analysis with the time series for the energies associated with each of the
Hough modes responsible for the MJO dynamics (as in

The concept of causality is a central question in science. One possible
definition of causality related to the predictability of two or more distinct
processes was introduced in

Consider a vector-valued signal

It is enough to say that

In practice, given a trivariate time series

Therefore, we can say that the

Partial directed coherence (PDC) is an extension of the concept of Granger
causality to the frequency domain as a measure of information flow. Thus,
PDC incorporates advantages of the Granger causality and of the classical
coherence methods with the additional advantage that it can be generalized to
more than two time series, enabling us to explicitly pinpoint the directed
information flow from mere indirect interactions

Again, consider a trivariate time series

Note that there is a duality between the Granger causality and PDC, as
demonstrated in

The partial directed coherence and Granger causality quantities
are linear measures, and a natural question is whether these methods are able
to capture the interaction between signals that arise from nonlinear
problems. There are several publications addressing this question such as
the possible nonlinear extension of this technique

The main advantage of PDC and Granger causality is that they are theoretically
related to the mutual information rate (MIR) between signals

On the right are the time series resampled at a 10

Finally, although PDC is a stochastic linear method, it correctly reconstructs
the topology of networks of nonlinear oscillators; see

The PDC is a function of the coefficients of a vector autoregressive
model. Given that the coefficients are asymptotically jointly normally
distributed, we can use the delta method

Based on the methodology of

PDC between tropical stratospheric ozone and stratospheric zonal wind (SZW) as well as the RMM MJO index at the fast (

From the time series of the amplitudes of the normal-mode functions we compute
the energy within a group of modes, consisting of the sum of the squares of
their amplitudes weighted by their equivalent depths

Time series of the stratospheric zonal wind at 30 Mb, the equatorial ozone
concentration in the stratosphere, and the RMM index are presented in
Fig.

PDC analysis between the MJO and stratospheric zonal wind (SZW) at the slow (

In order to investigate the interaction between the stratospheric variables
and the MJO index we performed a

Several studies point to the role of the interaction of waves with
different vertical structures in the dynamics of the MJO. For instance,

We initially perform a linear regression analysis between the time series
associated with the MJO indices and the stratospheric zonal wind
representative of the QBO, aiming to find which normal modes best represent
such oscillations. This analysis was introduced by

We search for interactions between the MJO and QBO normal modes. In order to do so, we calculate the time series of the energy associated with each of the modes (i.e., a weighted sum of the square of the absolute value of each of the modes). We begin by describing the interaction between modes associated with the MJO and the QBO as well as tropical stratospheric ozone forcing on sub-annual timescales. Due to the large number of variables we split the analysis into three sets, each containing all the “stratospheric variables” against one of the variables associated with the MJO. Since the most important interactions between QBO modes and MJO modes are through the QBO-related WIG waves, we restrict the analysis to these modes.

PDC analysis of the interaction of Kelvin, asymmetric Rossby, and westward gravity modes with ozone at the fast timescale (periods given in days). Significant interactions (red curve) between the MJO and ozone as well as QBO-related modes are found on intraseasonal, semi-annual, and annual timescales. Panels in the main diagonal show the power spectral density of each time series. Off-diagonal panels indicate the PDC values between the time series; PDC direction is from the time series indicated in the column to the one indicated in the row. For each panel, the

PDC analysis of the interaction of symmetric Rossby

In Fig.

Finally, we perform a PDC analysis of the interaction between symmetric
Rossby waves (the dominant mode in the MJO decomposition), asymmetric Rossby
waves, WIG waves, and stratospheric ozone on the fast timescale. The
corresponding PDC plot is presented in Fig.

PDC analysis of the interaction of ozone modes and Kelvin waves (KWs) at the slow timescale (periods given in years). The results show that KWs influence ozone on the annual timescale, while ozone influences KWs on decadal timescales. Figure conventions are the same as in Fig.

PDC analysis of the interaction of Kelvin modes (KWs) and westward gravity modes (WIGs) at the slow timescale (periods given in years). The results show a strong influence of the WIG mode on KWs on biennial and decadal timescales. Figure conventions are the same as in Fig.

PDC analysis of the interaction of symmetric Rossby modes (meridional index 1, denoted by RWSY1) and westward gravity modes (WIG1) at the slow timescale (periods given in years). Important interactions are found on annual to interannual timescales. Figure conventions are the same as in Fig.

We proceed by analyzing the PDC between the modes associated with
stratospheric zonal wind and stratospheric ozone vs. MJO-related modes on slow
timescales (annual–decadal timescales). Most importantly, we search for
stratospheric influences on the MJO on decadal and biennial timescales. The
analysis of the interaction between Kelvin waves, associated with the MJO and
tropical stratospheric ozone is presented in Fig.

PDC analysis of the interaction of westward gravity modes and ozone at the slow timescale (periods given in years). Important interactions are found on annual–biennial timescales and on the decadal timescale. Figure conventions are the same as in Fig.

Previous studies point to different MJO behavior depending on the phase
of the QBO (east or west)

Reconstruction at 200 Mb of the velocity and geopotential height fields associated with ROT modes with (positive stratospheric zonal wind) SZW30

Reconstruction at 200 Mb of the velocity and geopotential height fields associated with ROT modes with (negative stratospheric zonal wind) SZW30

Difference between the velocity and geopotential height fields associated with ROT modes with SZW30

Figures

Reconstruction at 200 Mb of the velocity and geopotential height fields associated with Kelvin modes with SZW30

Reconstruction at 200 Mb of the velocity and geopotential height fields associated with Kelvin modes with SZW30

Difference between the velocity and geopotential height fields associated with Kelvin modes with SZW30

Figures

The PDC results show strong coupling between tropical ozone, stratospheric
zonal wind, and the MJO. Most notable are the effects of tropical stratospheric
winds and ozone influencing the MJO on both intra-annual and interannual
timescales. The PDC analysis shows that tropical stratospheric ozone
influences the MJO in periods of 30–60

By the definition of Granger causality, one signal causes a second signal if
the information from the first helps to predict the future of the other after
taking into account the past of the second signal. In this sense, we confirm
the results of the recent studies cited above. We also show that tropical
stratospheric ozone improves MJO predictability on interannual and
decadal timescales. The periods of interaction suggest that the QBO might be
an important process in troposphere–stratosphere coupling through the MJO. This
conclusion agrees with numerical studies such as that of

It was also found that the MJO can affect stratospheric ozone, a possible
mechanism for this being the impact of deep convection on the tropopause
height

As for physical mechanisms that could link stratospheric heating driven by
solar UV forcing and tropical convection, tropopause changes
caused by ozone absorption are possible candidates.

We performed a linear regression analysis of the MJO index and stratospheric
zonal winds against the time series of the amplitudes of the Hough modes. We
confirm that the MJO is explained mainly by the first symmetric Rossby mode
(meridional index

A composite analysis of the velocity and geopotential height of the Kelvin and
Rossby modes associated with the MJO reveals the differences in the
characteristics of these modes during MJO events when the winds are positive
at 30 Mb and when they are negative. For the Rossby modes, differences
(Fig.

The time series for the amplitudes of the selected normal modes studied here is provided under the following DOI:

The supplement related to this article is available online at:

BR proposed the study, wrote the paper, and did the statistical analysis. DYT and LM worked on the PDC analysis. AST performed the normal-mode decomposition analysis, and PLdSD helped with the discussion and the interpretation of the analysis.

The authors declare that they have no conflict of interest.

This research was supported by the FAPESP-PACMEDY project (grant nos. 2015/50686-1 and 2017/23417-5), CAPES IAG/USP PROEX (grant no. 0531/2017), and the Meteorology Graduate Program at IAG-USP (CAPES finance code 001).

This paper was edited by Ben Kravitz and reviewed by Christian Franzke and two anonymous referees.