1. Introduction
Madden–Julian oscillation (MJO; Madden and Julian 1971, 1972) is the dominant mode of tropical intraseasonal variability (ISV, 20–100 days) with worldwide impacts (Zhang 2005, 2013) and serves as the most important predictability source of subseasonal prediction as well known (e.g., Waliser et al. 2003; Vitart et al. 2017). Among numerous indices for monitoring and predicting MJO, the real-time multivariate MJO (RMM) index (Wheeler and Hendon 2004, hereafter WH04) is generally appreciated as the most popular one and has been widely applied in MJO-related studies (e.g., Zhang et al. 2009; Jia et al. 2011; Zhang 2013; Straub 2013; Kiladis et al. 2014; Wei et al. 2019, 2020a,b; Dasgupta et al. 2021; Zhang et al. 2021; Wang et al. 2022) and operational subseasonal predictions (e.g., Lin et al. 2008; Gottschalck et al. 2010; Kim et al. 2014; Vitart et al. 2017; Pegion et al. 2019; Mariotti et al. 2020; Xiang et al. 2022).
The RMM index is originally designed to characterize salient features of MJO as efficiently as possible, such as intraseasonal/planetary-scale selection, slow eastward propagation, the tight coupling of convection and circulation, and baroclinicity (Jiang et al. 2020; Zhang et al. 2020). However, as noted by previous works (e.g., Roundy 2008, 2012a,b), certain aspects of MJO features are also shared by the high-frequency convectively coupled equatorial waves (Wheeler and Kiladis 1999; Kiladis et al. 2009). For example, the slow, moist equatorial Kelvin waves bear a strong resemblance to the MJO in dynamical structures over the Indo-Pacific warm pool with shallow equivalent depths (Roundy 2012c). In addition, the symmetric feature of equatorial Rossby waves about the equator also facilitates them to be easily projected onto the two RMM eigenmodes, thereby contributing to the day-to-day noise embedded in the RMM index (Roundy et al. 2009).
Besides high-frequency (e.g., period < 20 days) equatorial waves, the convection and/or circulation signals of low-frequency (e.g., period > 100 days) variability (LFV) may also bear high resemblances to the two RMM eigenmodes and distort the major component of RMM index arising from the MJO. For example, El Niño–Southern Oscillation (ENSO) as the major interannual variability of tropical ocean–atmosphere coupled system has convection structures like intraseasonal modes when MJO propagates through the Maritime Continent (MC) to the western Pacific (WP) (Lau and Chan 1986; Lo and Hendon 2000). A strong similarity between the leading modes of intraseasonal and interannual variabilities of the Indian Ocean (IO) convection and circulation was also reported (Hoell et al. 2012). The RMM phase was recently found to be able to persist extremely long (∼58 days), explaining the record-breaking mei-yu rainfall over the Yangtze during the boreal summer of 2020 (Zhang et al. 2021). This exceptional persistence of RMM phases 1–2 (thus intraseasonal signals over the IO) during June–July 2020 was further suggested to be partially induced by the LFV over the IO [i.e., Indian Ocean dipole (IOD)] (Wang et al. 2022). With these results, we put forward the key scientific questions in this study: how and to what extent the LFV can contribute to the RMM index?
The above reviews motivate us to revisit the calculation procedures of the RMM index (WH04). One essential aspect noted here for deriving the RMM index is the removal of the most recent 120-day mean. This step is originally designed to further remove any LFV after subtracting the ENSO-associated variability. However, we note in this study that such a real-time strategy may not necessarily eliminate all the LFV signals. This is especially true when the low-frequency background state is changing rapidly with time. On the other hand, if the low-frequency background state is in a fixed condition for several months and ongoing, the LFV can be well represented by the most recent 120-day mean. Moreover, this study unravels that the removal of the prior 120-day mean will inevitably introduce some “artificial” LFVs (hereafter LFVartificial) that affect the RMM index. The mathematical definition of LFVartificial will be given afterward, which facilitates us to quantitatively evaluate its contribution to the total RMM index.
There have been several studies that mentioned the deficiency of RMM calculation procedures in removing the LFV. For example, Lyu et al. (2019) noted that the long-term trend of MJO days in RMM phases 4–6 in recent decades might be exaggerated by the removal of the previous 120-day mean. Arcodia et al. (2020) found that removing the prior 120-day mean of a target day might cause some unwanted interannual variabilities and slight phase shifts of the LFV signals. Thus, we hypothesize that the LFV retained in the equatorial circulation and convection anomalies may sometimes project onto patterns of the two MJO eigenmodes and distort the RMM index. Such a case is represented in Figs. 1a and 1b, which show the RMM index evolutions from days −20 to 20 with day 0 occurring on November 19 in 2006. Although the total RMM amplitude maximizes in phases 2–3, the “pure” MJO defined by the ISV component peaks in phase 1. Besides, the ISV component only reflects a moderate MJO event, whose strength only explains one-half of the total RMM amplitude (Fig. 1a). Upon further consideration of the LFV component (green line in Fig. 1b), the reconstructed MJO index (i.e., ISV + LFV) well fit the total RMM, as evidenced by the closer phase points of day −20 and also day 0 between reconstructed (red) and original (blue) RMM indices (Fig. 1b). The LFV component in this same period is very strong and shows an anticlockwise rotation like the MJO, with amplitude increasing toward phase 3. Therefore, the observed large-amplitude RMM episode, especially in phases 2–3, for this case is largely rooted in LFV.
A case of RMM phase diagram from day −20 (solid dots) to day 20, in which day 0 (hollow dots) denotes the date of 19 Nov in 2006. (a),(b) Original RMM index (blue), where the red line in (a) denotes the ISV component while that in (b) denotes the sum of ISV and LFV components (green). (c),(d) As in (a) and (b), but for the NRMM index. See the methods (section 2b) for the details.
Citation: Journal of Climate 36, 7; 10.1175/JCLI-D-22-0368.1
As compared to the high-frequency equatorial waves that add day-to-day noises to the RMM index (Roundy et al. 2009), the LFV in some circumstances of strong MJO episodes may largely affect both the RMM amplitude and the RMM phase. In this regard, we should be more careful when monitoring strong RMM cases since they possibly do not represent adequately the MJO but are partially obscured by the remnant LFV that may be real or artificial. Although bandpass filtering (e.g., 20–100 days) as a standard technique can well eliminate the LFV, here we put forward a more convenient method that can largely remove the remnant LFV from the RMM index. Instead of the prior 120-day mean, we choose to remove the 120-day running mean centered on the target day. More mathematical details of the method are given in following section 2, which introduces the data and methods. The main results of this study are represented in section 3. Summary and discussions are given in section 4.
2. Data and methods
a. Data
The data used in this study include the daily Advanced Very High-Resolution Radiometer (AVHRR) interpolated outgoing longwave radiation (OLR) from the National Oceanic and Atmospheric Administration (NOAA) (Liebmann and Smith 1996). The daily zonal winds at 850 hPa (U850) and 200 hPa (U200) are from the reanalysis products of the National Centers for Environmental Prediction (NCEP) and the National Center for Atmospheric Research (NCAR) (Kalnay et al. 1996). We choose to use the NCEP–NCAR reanalysis, instead of others such as the ECMWF reanalysis products, mainly to keep consistency with the official RMM index online (http://www.bom.gov.au/climate/mjo). All gridded data have a spatial resolution of 2.5° × 2.5° and cover the period from 1979 to 2015. OLR is used as a proxy of the equatorial deep convection, and U850 and U200 are used to represent the baroclinic circulation, which together could capture the baroclinic, convectively driven circulation in the equatorial plane of MJO (WH04). Two sea surface temperature (SST) products of NOAA are also used in this study, including the monthly extended reconstructed SST, version 5 (ERSSTv5), on a 2° grid (Huang et al. 2017) and the daily Optimum Interpolation SST, version 2 (OISSTv2), on a 1/4° grid (Reynolds et al. 2007).
b. Methods
Figure 2 elaborates on the schematic diagram illustrating the main methods used in this study. Starting from the original OLR, U850, and U200, we first derive the equatorial (15°S–15°N) averaged “anomaly” by removing the mean and the first three harmonics of the annual cycle during 1979–2001 as well as the ENSO-associated variability, which is calculated as the reconstructed fields based on the seasonal regression relationships between ENSO and raw data with climatology removed only. The ENSO mode is extracted by the rotated empirical orthogonal function (EOF) analysis of Indo-Pacific (55°S–60°N, 30°E–70°W) SST anomalies from ERSSTv5 during 1949–91 (WH04). Second, the most recent 120-day mean is further subtracted from the anomaly. We label the outputs on the second step as “Anomaly1.” Finally, a multivariate EOF analysis is performed on the combined “Anomaly1” fields to derive the first two eigenmodes and principal components (PCs) associated with the RMM index. Before the EOF analysis, all “Anomaly1” fields are normalized by their global standard deviations (1.81 m s−1 for U850, 4.81 m s−1 for U200, and 15.1 W m−2 for OLR), which ensures that each of them contributes equally to the EOF modes.
Flowchart to calculate the RMM and NRMM indices. See methods (section 2b) for the details.
Citation: Journal of Climate 36, 7; 10.1175/JCLI-D-22-0368.1
After these preparations, we derive the contribution of LFV to the RMM index. The Anomaly1 fields are subjected to a 100-day low-pass filtering using the Lanczos filter with 201 weights (Duchon 1979). Then, the filtered Anomaly1 fields of OLR, U850, and U200 are combined and projected onto the two standard RMM eigenmodes of WH04. We name the derived PCs the “LFV RMM.” To facilitate comparison, we also calculate the “ISV RMM” using similar computing procedures as LFV RMM, except for the 20–100-day bandpass filtering (Fig. 2). Thus, we have defined the intraseasonal anomaly using the bandpass filtering in this study and the so-derived ISV RMM reflects the pure MJO characteristics, since both high-frequency noises and LFV have been filtered out. Previous studies also used bandpass filtering before the multivariate EOF analysis to derive an MJO index (e.g., Yoo and Son 2016) as the ISV RMM defined here. The LFV RMM, ISV RMM, and the total RMM indices can be directly compared in Fig. 1b. Compared to the total RMM with day-to-day noise, the LFV and ISV RMMs are smoother due to the usage of filtering. As has been put, we are going to quantify the extent to which the LFV contributes to the total RMM in the following section.
We call the index derived from Anomaly2, in which both ENSO and
3. Results
Because one of the chief goals of our study is to understand how the removal of prior 120-day mean will influence the RMM index, in the first two subsections we will examine the reconstructed LFVresidual RMM and also the LFV RMM derived using the 100-day low-pass filtering. In the last subsections, only the LFV RMM will be investigated, although the differences between the reconstructed LFVresidual and filtered LFV RMMs are negligibly small (as will be shown later).
a. Real versus artificial LFVs
We first compare the power spectrum of “anomaly” with that of “Anomaly1” and “Anomaly2.” The data used here for the spectral analysis are the normalized OLR, U850, and U200 in the latitude band of 15°S–15°N. For the raw anomaly with only the climatology and ENSO-associated variability removed, both the ISV and LFV signals can be observed clearly. The ISV signals mainly prevail over the Eastern Hemisphere, and the second peak centered at 100°W reflects the eastern Pacific ISV that is mainly active in the boreal summer (Maloney and Esbensen 2003). For the LFV, the power variance is strong for the period > 300 days but is weak in the period range of 100–300 days.
How will these characteristics change upon the removal of the prior 120-day mean? We then analyze the spectrum of Anomaly1 in Fig. 3b. To make a quantitative comparison, the differences of globally averaged spectra between Figs. 3b and 3a are also attached along the right axis of Fig. 3b. For the Anomaly1 that has been originally designed to contain LFV as little as possible, however, there remains strong LFV power. Moreover, the LFV power of 120–300 days even becomes stronger than that in the raw anomaly, although the longer period (e.g., >500 days) LFV has been largely reduced. This suggests that removing the prior 120-day mean in the calculation procedure of the RMM index is inefficient to filter out the LFV thoroughly, and even artificially bring new LFV into the RMM index, which likely projects substantially onto the RMM eigenmodes and distort the real MJO behaviors. We have performed the EOF analysis on the 100-day low-pass-filtered Anomaly1, and the leading modes indeed bear a strong resemblance to the RMM eigenmodes (not shown). Although the ISV power variance is also slightly weakened or strengthened depending on the period range, the variation magnitude is much smaller than the LFV.
Mean power spectrum of normalized OLR, U850, and U200 in the latitude band of 15°S–15°N as a function of longitude. (a) Raw anomaly, (b) Anomaly1, and (c) Anomaly2. (b),(right) The differences in the globally averaged spectrum between (b) and (a). (c),(right) The difference between (c) and (a). The horizontal dashed line denotes the 100-day period that partitions the ISV and LFV.
Citation: Journal of Climate 36, 7; 10.1175/JCLI-D-22-0368.1
What is the case for the spectrum of Anomaly2, in which the mean from lead 60 days to lag 60 days of the target day is removed? As seen in Fig. 3c, the LFV with a period longer than 120 days has all been substantially eliminated in Anomaly2, benefiting from the simple but effective procedure proposed here (Fig. 2), and more importantly, the strengthened 120–300-day LFV introduced by the removal of the prior 120-day mean has now been largely weakened. Additionally, some nonnegligible changes are also evident for the ISV component, especially for those in the low-frequency tails of the ISV, which are intensified as compared with those in the raw anomaly (Fig. 3c). These noisy features of ISV in the Anomaly2 as well as those in the Anomaly1, as compared with that of the raw anomaly, may arise from the ringing effect of the running-mean filtering. In a word, removing the centered 120-day mean will likely weaken the LFV on one hand and also strengthen the ISV on the other hand in the derived multivariate MJO index (i.e., the NRMM index). This conjecture will be confirmed robustly in the following.
Enlightened from the above analysis, both the LFVreal and LFVartificial exist in the Anomaly1. What are their relative contributions to the LFVresidual? Furthermore, how will these two obscure the total RMM index? Based on Eq. (1), we have derived the LFVreal and LFVartificial using the NCEP reanalysis data during 1980–2015. Using the same formatting as Fig. 3, we show the power spectrum of LFVresidual as well as its real and artificial components in Fig. 4. We find that the LFVresidual (Fig. 4a) reconstructed using the LFVreal and LFVartificial shows a power spectrum highly similar to that in Fig. 3b with a period > 100 days. Therefore, despite the bias [i.e., the ε in Eq. (1d)] of the reconstructed LFVresidual, it can adequately represent the remnant LFV signals in the Anomaly1. Furthermore, the spectrum of the LFVresidual can be well fitted by the sum of the LFVreal and LFVartificial. For the LFVreal, the power variance is mainly seen for the period < 200 days, while the signals with the period > 200 days have been strongly eliminated (Fig. 4b), benefiting from the removal of the prior 120-day mean. For the LFVartificial, however, the signals are mainly seen for the period > 150 days, and the signals in the period range of 100–150 days are nearly zero.
As in Fig. 3, but for the power spectra of (a) LFVresidual, (b) LFVreal, and (c) LFVartificial derived using Eq. (1).
Citation: Journal of Climate 36, 7; 10.1175/JCLI-D-22-0368.1
The efficiency of the reconstructed LFV in representing the 100-day low-pass-filtered Anomaly1 is further diagnosed in Figs. 5a and 5b, but focusing on the so-derived RMM indices using the methodology illustrated in Fig. 2. We find that the statistical characterizations of the filtered LFV RMM can be reasonably represented by the reconstructed LFVresidual RMM. For example, the bivariate correlation (Gottschalck et al. 2010) between these two kinds of RMM indices exceeds 0.96. Therefore, we can directly diagnose the RMM components from LFVreal and LFVartificial [see Eq. (1)] to understand how the LFVresidual contributes to the RMM index. Figure 5c compares the histograms of the amplitude of the LFVreal and LFVartificial RMM indices. Clearly, it shows that the LFVartificial RMM amplitude is often larger than the LFVreal RMM and the histogram of the former is more like the reconstructed LFVresidual RMM, suggesting that the removal of the prior 120-day mean causes LFVartificial that might play a major role in explaining the LFV component of the original RMM index. Thus, removing this LFVartificial through subtracting the centered 120-day mean can likely eliminate the LFV from the RMM index to a large extent.
(a) Scatter diagram of the filtered LFV RMM vs the reconstructed LFV RMM indices. Red (blue) color is for the RMM2 (RMM1). (b) Histogram analysis of filtered (dashed) and reconstructed (solid) LFV RMM amplitude. (c) Histogram analysis of LFVresidual RMM amplitude (black;
Citation: Journal of Climate 36, 7; 10.1175/JCLI-D-22-0368.1
Seasonality of the LFV RMM strength. (a) The standard deviation of LFVresidual (black), LFVreal, (red) and LFVartificial (blue) RMM amplitude in 12 calendar months during 1980–2015. (b) The fractional contribution (i.e., rm) of LFVreal (red) and LFVartificial (blue) RMM to the LFVresidual RMM.
Citation: Journal of Climate 36, 7; 10.1175/JCLI-D-22-0368.1
b. Possible low-frequency background control on the LFVartificial RMM
From section 3a, we see that the LFVresidual RMM is mainly contributed by the LFVartificial RMM component. Why are there LFVartificial signals in the RMM index? Keep in mind that the LFVartificial is the difference between
Figure 7 compares the LFVartificial RMM indices with the daily measure of ENSO, that is, the Niño-3 index derived from the OISSTv2 product. To facilitate the readers, the LFVartificial RMM1 and RMM2 indices have been multiplied by a factor of 2.5. As has been deduced, the LFVartificial RMM indices are indeed strong when the Niño-3 index grows sharply or decays quickly. For example, during the rapid warming stage of 1982/83 El Niño, the LFVartificial RMM2 component evolves from positive to strong negative. Similar coevolutions of the LFVartificial RMM2 and Niño-3 indices are also clear in other warming stages of El Niño years, such as 1986/87, 1991/92, 1994/95, 1997/98, 2002/03, 2004/05, 2006/07, 2009/10, and 2014/15. In these El Niño cases, however, the RMM1 component does not show consistent variations. The weak RMM1 amplitude is mostly seen, such as 1982/83, 1986/87, 2002/03, and 2009/10. The strong positive to the negative transition of LFVartificial RMM1 is seen in 1991/92 and 2014/15, while in other El Niño years, they support strong negative to positive LFVartificial RMM1 transition. When the Niño-3 index decays quickly in the rapid cooling stage of La Niña, the LFVartificial RMM mostly manifests a negative to positive transition, such as 1984/85, 1995/96, 1999/00, 2010/11, and 2011/12.
(a)–(c) Time series of the daily mean SST anomaly averaged over the Niño-3 region (gray area) and the LFVartificial RMM1 (red) and RMM2 (blue) indices during 1982–2015. The vertical line denotes the start of each year.
Citation: Journal of Climate 36, 7; 10.1175/JCLI-D-22-0368.1
To more clearly observe the possible controlling effect of ENSO on the LFVartificial RMM, we perform a simple composite analysis here. We identify those dates (hereafter, lag 0) meeting the following criterion: the Niño-3 index reaching a local maximum exceeding 0.5°C and lasting for more than 5 months. The Niño-3 index and the LFVartificial RMM1 and RMM2 indices are averaged each day from lag −200 to lead 200 days. As seen in Fig. 8a, the composite Niño-3 index shows an El Niño evolution, featuring a rapid warming stage before lag 0 and the mature and decaying stages after lag 0. The LFVartificial RMM2 index displays a robust transition from weak positive to strong negative, while the evolution of the RMM1 component is weak, although a weak negative to positive transition is marginally detectable. Similarly, we also composite the La Niña scenarios in Fig. 8b. Accompanied by the rapid cooling of the Niño-3 index, both the RMM1 and RMM2 components of LFVartificial RMM show a negative to positive transition. These composite results are consistent with the case studies in Fig. 7.
(a) Lead–lag composite of LFVartificial RMM1 (red) and RMM2 (blue) indices from lag −120 to lead 120 days when the daily mean SST anomaly averaged over the Niño-3 region is growing rapidly. The shading represents the uncertainty range of one standard deviation. (b) As in (a), but for the composite when the daily mean SST anomaly averaged over the Niño-3 region is decaying rapidly.
Citation: Journal of Climate 36, 7; 10.1175/JCLI-D-22-0368.1
We also examined the SST anomalies in the Niño-3 region after the ENSO-associated variability was removed, and most of the warming (e.g., 1982/83, 1991/92, 1994/95, 1997/98, 2002/03, 2006/07, 2009/10) and cooling (e.g., 1984/85, 1995/96) stages shown in Fig. 7 remained, albeit with reduced amplitude (figure not shown). This implies that the developing El Niño or La Niña might support rapid changes of low-frequency background states. However, these rapid changes of low-frequency background convection and circulation cannot be fully eliminated simply with the removal of ENSO-associated variability (e.g., the LFVreal in Fig. 4b), which thus play a role in influencing the LFVartificial RMM.
c. Quantitative evaluation of LFV RMM’s contribution to total RMM
In this section, we explore to what extent the LFV RMM can contribute to the total RMM. Following WH04, we show the 91-day running-mean variance (i.e., RMM12 + RMM22) of the Total, ISV, and LFV RMMs from 1980 to 2015 in Figs. 9a–c. The correlation of the Total RMM with the ISV RMM reaches 0.8 but less than 0.3 with the LFV RMM. The ISV RMM indeed can well explain the Total RMM in many cases, while there are also plenty of episodes in which the former is much smaller than the latter and even no more than one-half of the latter. For example, such cases can be seen in 1981, 1983/84, 1994/95, 1997/98, 1999/2000, 2002/03, 2007, and 2014/15. Moreover, in these periods, the LFV RMM is also strong and thus plays a crucial role in explaining the Total RMM.
(a)–(c) The 91-day running-mean variance (i.e., RMM12 + RMM22) of the total RMM index (black) as well as the ISV (blue) and LFV (red) components from 1980 to 2015. (d)–(f) As in (a)–(c), but for the NRMM index.
Citation: Journal of Climate 36, 7; 10.1175/JCLI-D-22-0368.1
As a comparison, we further show the running-mean variance of the NRMM index derived using Anomaly2 in Figs. 9d–f. In the periods aforementioned, the strength of LFV NRMM has been largely weakened, since the LFVartificial has been eliminated [Eq. (1)]. As expected from the weakened LFV NRMM, the Total NRMM variance of these cases also becomes smaller, and thus the NRMM contribution by the ISV turns dominant. The Total NRMM is more highly correlated (>0.9) with its ISV component. We also compared the power spectra of RMM and NRMM indices, and the LFV with a period > 100 days has been largely weakened in the latter (not shown). The same case of Figs. 1a and 1b has been reexamined here but using the NRMM indices. As seen in Fig. 1c, the ISV NRMM is closer to the total NRMM index as compared to Fig. 1a. The strong RMM signals in phases 2–3 in Fig. 1a are reduced due mainly to the weakening of the contribution of LFVresidual (Fig. 1d). As a result, the maximum strength of NRMM index is rectified to phase 1, in consistence with the ISV component.
To robustly confirm the case study results, we also conduct a composite analysis. From the original RMM amplitude time series, we identify all the “day 0” denoting the date when the amplitude of the LFV component exceeds 0.8 and reaches a local maximum. In this way, there are a total of 33 cases selected. Then, the lag composite of the Total RMM amplitude and its LFV and ISV components can be derived and shown in Fig. 10a. Using the same set of “day 0,” the composites for the NRMM index are also conducted and shown in Fig. 10b. We can see that the difference between the Total and ISV components of the original RMM index becomes much smaller in the NRMM index. This is mainly attributed to two aspects of reasons: 1) the overall intensity of LFV NRMM index has been strongly reduced by about 0.5 while 2) that of ISV NRMM index increases by about 0.1, as compared to the original RMM index (Fig. 10a).
(a) Composites of the amplitude evolution for the original RMM amplitude (black), and the LFV (red) and ISV (blue) components from day −30 to day 30, where day 0 indicates the date when the amplitude of the LFV component reaches a local maximum exceeding 0.8 standard deviations. The shading denotes the uncertainty range of one standard deviation. (b) As in (a), but for the NRMM amplitude.
Citation: Journal of Climate 36, 7; 10.1175/JCLI-D-22-0368.1
To further understand the effects of centered 120-day running-mean filtering, we compare the mean power spectrum of RMM1 and RMM2 with that of NRMM1 and NRMM2. The results are shown in Fig. 11 in such a way so that the area below the black curve represents the magnitude of the power spectrum (WH04). The most evident contrast is that the LFV with period > 120 days has been largely weakened in the NRMM index (Fig. 11b). The NRMM index shows enhanced ISV variability of 60–90 days, as compared with the RMM index (Fig. 11a). In other frequency bands of ISV and also high-frequency synoptic-scale variability, the power spectrum is less affected after subtracting the centered 120-day mean. These results are similar to that in Fig. 3.
(a) The mean power spectrum of RMM1 and RMM2 during 1980–2015. The red dotted line denotes the spectrum of the background red noise. The y axis indicates the multiplication of power and frequency [cycles per day (CPD)] so that the area below the black curve represents the magnitude of the power spectrum. The two dashed blue lines outline the MJO frequency band of 20–100 days. (b) As in (a), but for the mean power spectrum of NRMM1 and NRMM2.
Citation: Journal of Climate 36, 7; 10.1175/JCLI-D-22-0368.1
To quantitatively compare the RMM and NRMM indices, we diagnose the histograms of the amplitude time series of Total multivariate MJO indices as well as their ISV and LFV components. The amplitude of Total RMM features a Gaussian-like distribution peaking at 1.1 (Fig. 12a), which is smaller than its mean value (∼1.3, Fig. 12d) due to a weakly positive skewness toward infinity. The distribution of ISV RMM amplitude is similar to that of Total RMM, but with a smaller peak (∼0.7) and mean value (∼1.0), due to the reduction of large amplitude (>1.0) episodes and the increase of small amplitude (<1.0) episodes (Fig. 12b). For LFV RMM amplitude, although its peak (∼0.4) and mean (∼0.5) values become much smaller, there still exist some episodes with amplitude exceeding 1.0 and even approaching 2.0 (Fig. 8c). More specifically, the cases with the LFV RMM amplitude larger than 0.8 occupies a fraction of 10% of total cases (Table 1).
(a)–(c) Histograms of the RMM amplitude and (d) its mean and standard deviation (std). (a) Total RMM amplitude, (b) RMM amplitude contributed by the ISV, and (c) RMM amplitude contributed by the LFVresidual. The black color represents the original RMM index, while the red color is for the NRMM index. (e)–(h) As in (a)–(d), but for the amplitude ratio between the total RMM index and its two ISV and LFV components.
Citation: Journal of Climate 36, 7; 10.1175/JCLI-D-22-0368.1
Quantitative comparison of original RMM and NRMM indices in representing their LFV and ISV components. The values indicate the percentage of days occupying all days from 1980 to 2015, on which the comparison standard is met.
For the NRMM index, the peak value of total amplitude decreases to ∼1.0 (Fig. 12a); the ISV NRMM amplitude skews more toward infinity (Fig. 12b), leading to an enlarged mean value (Fig. 12d); the most significant change occurs in the LFV component, in which the cases of small LFV NRMM amplitude (∼0.25) increase sharply and the histogram curve becomes narrower (Fig. 12c), suggesting that the LFV component of NRMM index is almost weakened to a constant near zero (see Fig. 10b). For instance, the aforementioned percentage (i.e., 10%) has been reduced to 1.0% in the NRMM index (Table 1).
We also diagnose the histogram of the amplitude ratio of each two among Total, ISV, and LFV multivariate MJO indices. For the original RMM index, a Gaussian-like distribution is also observed for the ISV/Total ratio, with peak and mean values of 0.7 and 0.9 (Figs. 12g,h). The cases with the ISV/Total ratio exceeding the threshold of 0.8 occupy 49.4% (Table 1). The importance of the LFVresidual is further revealed by referring to the amplitude ratio of LFV/Total and LFV/ISV (Figs. 12e,f). For these two ratio curves, although their peak and mean values are relatively small, the cases with the LFV/Total ratio >0.5 and the LFV/ISV ratio > 1.0 occupy percentages of 33.4% and 19.3%, respectively (Table 1). For the NRMM index (red color in Figs. 12e,h), because the ISV component is strengthened and meantime the LFV component is substantially weakened, the above percentage of 49.4% increases to 60.0% while the other two ones (i.e., 33.4% and 19.3%) decrease largely to 17.6% and 8.1%, respectively.
In summary, the RMM calculation procedure is subjected to the LFV remaining in the “Anomaly1” fields, thereby making a nonnegligible contribution to the amplitude and evolution of the RMM index. This suggests that besides the high-frequency equatorial waves that explain the burr-like disturbances of RMM (Roundy et al. 2009), the possible obscuring effect of LFVresidual should be also taken into account when one monitors or predicts an MJO since the LFVresidual RMM might substantially influence the amplitude of the total RMM index in some circumstances.
d. How does the LFVresidual affect the RMM-defined MJO?
In this section, we explore the influences of LFVresidual on the eastward-propagating MJO events during 1980–2015. Following previous studies (e.g., Ling et al. 2013; Guan and Waliser 2015; Kerns and Chen 2016; Wei and Ren 2019; Hsu et al. 2021; Wei et al. 2022; Wei and Pu 2021; Huang et al. 2022; and many others), we select the MJO events initiated from the IO by two criteria: 1) RMM2 reaches a local minimum (the corresponding date is flagged as “day 0”) smaller than one negative standard deviation; 2) from day 0 to day 7, the RMM amplitude exceeds 1.0 and anticlockwise rotation reflecting eastward propagation is observed from the RMM phase diagram. The same set of “day 0” is also used to make composites based on the NRMM index. As shown in Fig. 13a, the amplitude of the total RMM is slightly larger than that of the total NRMM, although all MJO cases have been considered in the composite. This small amplitude difference mainly results from the LFV component (Fig. 13b), since the ISV components of the RMM and NRMM are nearly identical (Fig. 13c).
(a) Lead–lag composite of the phase diagram of the total RMM (blue) and NRMM (red) indices for all MJO cases, where “day 0” (hollow markers) denotes the date when the RMM2 reaches a local minimum smaller than −1 standard deviation. (b),(c) As in (a), but for the LFV and ISV components, respectively. (d)–(f) As in (a)–(c), but for the composite of MJO cases with the LFV RMM amplitude exceeding 0.8.
Citation: Journal of Climate 36, 7; 10.1175/JCLI-D-22-0368.1
Because the major object of this study is the LFVresidual, thus we further add another criterion to select those cases in which the LFV RMM amplitude exceeds 0.8. The composite results using the new set of “day 0” are shown in Figs. 13d–f. As expected, the difference between the total RMM and NRMM indices is enlarged (Fig. 12d), due to the strengthened contribution of LFV RMM (Fig. 13e). Besides, the LFV RMM also causes an eastward phase shift between the RMM and NRMM indices, similar to the case result of Fig. 1. Without discerning the potential influences of LFVresidual, one would treat this case as a strong MJO episode, while the ISV component reflecting the “pure” MJO signal only explains ∼50% of the total RMM index (Fig. 13f). In contrast, for the NRMM index, the LFV component has been largely reduced, and the amplitude of the total NRMM index thus becomes comparable to its ISV component.
The other way to appreciate MJO is to diagnose the Hovmöller diagram of equatorially averaged (e.g., 15°S–15°N) convection. The composite OLR “Anomaly1” and “Anomaly2” along with their LFV and ISV components for those MJO cases with the LFV RMM amplitude exceeding 0.8 are shown in Fig. 14. An organized eastward propagation of deep convection represented by the Anomaly1 appears from day −10, with strongly suppressed convection over the western–central Pacific (Fig. 14a). These behaviors are similar to the MJO mode composed using the Anomaly1 subjected to the 20–100-day filtering (Fig. 14c). However, some differences are also clear: the “MJO” in Fig. 14a is stronger, slower, and bigger (i.e., a larger zonal scale); the moist or dry phase of “MJO” persists longer. These differences are well explained by the composite of the LFV component in the Anomaly1 (Fig. 14b): from day −10, strong LFV also appears over the IO, slowly migrates eastward, and gradually vanishes when approaching 120°E; meanwhile, a locally suppressed LFV convection stands over the central-western Pacific. In contrast, the composite MJO using the “Anomaly2” (Fig. 14d) bears a high resemblance with the ISV component (Fig. 14f) in both the strength and propagation of convection, primarily benefiting from the largely weakened LFV convection (Fig. 14e).
(a) Lead–lag composite of equatorially (15°S–15°N) averaged OLR “Anomaly1” (W m−2) for the MJO cases with the LFV RMM amplitude exceeding 0.8. The magenta line shows the reference speed of 5 m s−1. (b),(c) As in (a), but for the 100-day low-pass-filtered and 20–100-day bandpass filtered “Anomaly1,” respectively. (d)–(f) As in (a)–(c), but for the “Anomaly2.”
Citation: Journal of Climate 36, 7; 10.1175/JCLI-D-22-0368.1
4. Summary and discussion
The calculation procedures of the RMM index are revisited in this study, with the primary focus on the real-time extraction of major ISV by removing the most recent 120-day mean after the removal of ENSO-associated variability (WH04). For the first time, our results indicate that this strategy is subjected to nonnegligible remnants of LFV, which primarily consist of two parts. The first part reflects the remnant LFV with periods of 100–200 days and mainly results from the insufficient elimination of the LFVreal existing already before removing the prior 120-day average. The second part is newly detected and reflects the remnant LFV with periods of longer than 150 days, which is “artificially” produced after subtracting the prior 120-day mean. The LFVreal and LFVartificial are formulated mathematically, and their summation (i.e., LFVresidual) can well fit the LFV remaining in the Anomaly1 fields. The LFVartificial plays a major role in explaining the LFVresidual, especially in the boreal summer when 70% of the LFVresidual is rooted in the LFVartificial. The developing stages of El Niño or La Niña might support rapid changes of low-frequency background convection and circulation, which cannot be fully eliminated simply with the removal of ENSO-associated variability and thus influences the LFVartificial RMM.
We propose an alternative method to recalculate the input data of the multivariate EOF analysis, that is, subtracting the centered 120-day mean on the target day. The so-derived index is named the NRMM index. The LFVartificial has been removed in Anomaly2, as expected from the definition of LFVartificial [Eq. (1)], although the LFVreal that weakly influences the NRMM index remains. The removal of the LFVartificial also slightly strengthens the ISV component of the NRMM index, possibly due to the ringing effect of running mean filtering (e.g., https://sites.google.com/oregonstate.edu/jenney/research/running-mean-filtering). We acknowledge that the best way to remove the LFVresidual is the bandpass filtering (such as the ISV RMM defined in this study) since both the high-frequency noises and LFV can be filtered out. However, the chief goal of this study is to understand the LFVresidual and how it will influence the RMM index. Thus, based on the definition of LFVartificial, we have chosen to remove the centered 120-day mean, which is more convenient than the bandpass filtering and can strongly eliminate the LFVresidual (see Fig. 9), especially in the boreal summer season.
To advance our understanding of the RMM index, we have quantified the contribution of the LFVresidual RMM to the total RMM and compared it with the NRMM. The results suggest that when one chooses the NRMM index instead of the RMM index: 1) the cases of large-amplitude (e.g., >0.8) LFV signals will change from a percentage of 10% to nearly zero; 2) the cases in which the LFV component contributes to more than one-half of the total index amplitude will reduce by 16%; 3) the cases in which the residual LFV explains an even larger fraction than the ISV will reduce by 11%. Again, we stress here that we are not to create a new index to replace the original RMM index of WH04; however, we want to highlight that one should be more careful when preparing to relate the monitored or predicted RMM behaviors to the “pure” MJO features since the non-MJO LFV signals might be working to exert an obscuring effect, especially when the low-frequency background state (such as ENSO) is rapidly changing with time.
We further carry out a composite analysis for the RMM-defined MJO events selected when both the total RMM amplitude and its LFV component are strong. In the composite “MJO” based on the RMM index, approximately 50% of the amplitude is sourced from the LFVresidual, and thus the “true” MJO composed using the ISV RMM index cannot be well represented by the total RMM index. However, when deriving the composite results using the NRMM index, the represented “MJO” is closer to that using the ISV NRMM index, because the LFV NRMM component has been largely weakened. The LFVresidual obscures the RMM-defined MJO events in such a way so that the composite “MJO” is stronger, slower, and persists longer, as compared to the MJO subjected to the 20–100-day bandpass filtering without the interference of LFV. These issues, however, do not exist by composing the Anomaly2 data with the centered 120-day mean removed.
Some caveats in this work should be also noted. For example, to keep consistency with WH04, we have defined the climatology by referring to the period of 1979–2001, although the major findings of this study are insensitive to the definition of the climatology. The reason of why LFVartificial RMM amplitude peaks during the boreal summer is not answered, but it may because of that the low-frequency background states usually adjust upward or downward in the summer season and thus the prior 120-day mean cannot efficiently represent the real LFV signals. A thorough analysis of this question is needed in the future. There are also some strong LFVartificial RMM episodes during the mature phase of ENSO and even neutral-ENSO years. Thus, besides ENSO, the rapid change of low-frequency convection and circulation might be also expected in other LFV modes (such as IOD and the stratosphere quasi-biennial oscillation).
Previous studies have used the RMM index to explore the MJO interannual (e.g., Pohl and Matthews 2007; Lin et al. 2015; Liu et al. 2016; Dasgupta et al. 2021) and decadal variations (e.g., Lafleur et al. 2015; Lyu et al. 2019; Roxy et al. 2019; Dasgupta et al. 2020; Hsu et al. 2021). Since the interannual–decadal variation of original RMM includes the nonnegligible contribution of LFVresidual, whether and how the conclusions drawn from these previous studies will vary after the strongly weakening of the LFVresidual (e.g., the NRMM index) is not clear and thus deserves further studies. MJO, as stochastic forcing, may play some roles in triggering extreme climate events (e.g., Lau and Chan 1986; McPhaden 1999; Zhang et al. 2001; Hendon et al. 2007; Rao et al. 2009), since the enhanced RMM amplitude can be sometimes observed during the developing phases of such as ENSO or IOD (e.g., Hu et al. 2022; Huang et al. 2022). However, our study suggests that the observed large-amplitude RMM evolution may not adequately reflect the MJO but is largely rooted in the LFVartificial created by the RMM calculation procedures.
Acknowledgments.
The authors thank the three anonymous reviewers for their comments and suggestion of a personal blog for the interesting parallel research information, which helped to improve the manuscript. This work was jointly supported by the Natural Science Foundation of China (U2242206, 41975094, and 41775066) and the Basic Research and Operational Special Project of CAMS (2021Z007).
Data availability statement.
The reanalysis data are from the National Centers for Environmental Prediction/National Weather Service/NOAA 1994, updated monthly; NCEP–NCAR Global Reanalysis Products, 1948–present; Research Data Archive at NOAA/PSL:/data/gridded/data.ncep.reanalysis.html. The interpolated OLR data were provided by NOAA/OAR/ESRL PSL, Boulder, Colorado, from their website at https://psl.noaa.gov/thredds/catalog/Datasets/interp_OLR/catalog.html. The OISSTv2 data were retrieved from https://climatedataguide.ucar.edu/climate-data/sst-data-noaa-high-resolution-025x025-blended-analysis-daily-sst-and-ice-oisstv2. The ERSSTv5 data were retrieved from https://climatedataguide.ucar.edu/climate-data/sst-data-noaa-extended-reconstruction-ssts-version-5-ersstv5.
REFERENCES
Arcodia, M. C., B. P. Kirtman, and L. S. Siqueira, 2020: How MJO teleconnections and ENSO interference impacts U.S. precipitation. J. Climate, 33, 4621–4640, https://doi.org/10.1175/JCLI-D-19-0448.1.
Dasgupta, P., A. Metya, C. V. Naidu, M. Singh, and M. K. Roxy, 2020: Exploring the long-term changes in the Madden Julian oscillation using machine learning. Sci. Rep., 10, 18567, https://doi.org/10.1038/s41598-020-75508-5.
Dasgupta, P., M. K. Roxy, R. Chattopadhyay, C. V. Naidu, and A. Metya, 2021: Interannual variability of the frequency of MJO phases and its association with two types of ENSO. Sci. Rep., 11, 11541, https://doi.org/10.1038/s41598-021-91060-2.
Duchon, C. E., 1979: Lanczos filtering in one and two dimensions. J. Appl. Meteor., 18, 1016–1022, https://doi.org/10.1175/1520-0450(1979)018<1016:LFIOAT>2.0.CO;2.
Gottschalck, J., and Coauthors, 2010: A framework for assessing operational Madden–Julian oscillation forecasts: A CLIVAR MJO Working Group project. Bull. Amer. Meteor. Soc., 91, 1247–1258, https://doi.org/10.1175/2010BAMS2816.1.
Guan, B., and D. E. Waliser, 2015: Detection of atmospheric rivers: Evaluation and application of an algorithm for global studies. J. Geophys. Res. Atmos., 120, 12 514–12 535, https://doi.org/10.1002/2015JD024257.
Hendon, H. H., M. C. Wheeler, and C. Zhang, 2007: Seasonal dependence of the MJO–ENSO relationship. J. Climate, 20, 531–543, https://doi.org/10.1175/JCLI4003.1.
Hoell, A. H., M. Barlow, and R. Saini, 2012: The leading pattern of intraseasonal and interannual Indian Ocean precipitation variability and its relationship with Asian circulation during the boreal cold season. J. Climate, 25, 7509–7526, https://doi.org/10.1175/JCLI-D-11-00572.1.
Hsu, P.-C., Z. Fu, H. Murakami, J.-Y. Lee, C. Yoo, N. C. Johnson, C.-H. Chang, and Y. Liu, 2021: East Antarctic cooling induced by decadal changes in Madden-Julian oscillation during austral summer. Sci. Adv., 7, eabf9903, https://doi.org/10.1126/sciadv.abf9903.
Hu, H., R. Wang, F. Liu, W. Perrie, J. Fang, and H. Bai, 2022: Cumulative positive contributions of propagating ISO to the quick low-level atmospheric response during El Niño developing years. Climate Dyn., 58, 569–590, https://doi.org/10.1007/s00382-021-05924-4.
Huang, B., and Coauthors, 2017: Extended Reconstructed Sea Surface Temperature, version 5 (ERSSTv5): Upgrades, validations, and intercomparisons. J. Climate, 30, 8179–8205, https://doi.org/10.1175/JCLI-D-16-0836.1.
Huang, Z., W. Zhang, C. Liu, and M. F. Stuecker, 2022: Extreme Indian Ocean dipole events associated with El Niño and Madden–Julian oscillation. Climate Dyn., 59, 1953–1968, https://doi.org/10.1007/s00382-022-06190-8.
Jia, X., L. Chen, F. Ren, and C. Li, 2011: Impacts of the MJO on winter rainfall and circulation in China. Adv. Atmos. Sci., 28, 521–533, https://doi.org/10.1007/s00376-010-9118-z.
Jiang, X., and Coauthors, 2020: Fifty years of research on the Madden-Julian oscillation: Recent progress, challenges, and perspectives. J. Geophys. Res. Atmos., 125, e2019JD030911, https://doi.org/10.1029/2019JD030911.
Kalnay, E., and Coauthors, 1996: The NCEP/NCAR 40-Year Reanalysis Project. Bull. Amer. Meteor. Soc., 77, 437–472, https://doi.org/10.1175/1520-0477(1996)077<0437:TNYRP>2.0.CO;2.
Kerns, B. W., and S. S. Chen, 2016: Large-scale precipitation tracking and the MJO over the Maritime Continent and Indo-Pacific warm pool. J. Geophys. Res. Atmos., 121, 8755–8776, https://doi.org/10.1002/2015JD024661.
Kiladis, G. N., M. C. Wheeler, P. T. Haertel, K. H. Straub, and P. E. Roundy, 2009: Convectively coupled equatorial waves. Rev. Geophys., 47, RG2003, https://doi.org/10.1029/2008RG000266.
Kiladis, G. N., and Coauthors, 2014: A comparison of OLR and circulation-based indices for tracking the MJO. Mon. Wea. Rev., 142, 1697–1715, https://doi.org/10.1175/MWR-D-13-00301.1.
Kim, H. M., P. J. Webster, V. E. Toma, and D. Kim, 2014: Predictability and prediction skill of the MJO in two operational forecasting systems. J. Climate, 27, 5364–5378, https://doi.org/10.1175/JCLI-D-13-00480.1.
Lafleur, D. M., B. S. Barrett, and G. R. Henderson, 2015: Some climatological aspects of the Madden–Julian oscillation (MJO). J. Climate, 28, 6039–6053, https://doi.org/10.1175/JCLI-D-14-00744.1.
Lau, W. K. M., and P. H. Chan, 1986: The 40–50 day oscillation and the El Niño/Southern Oscillation: A new perspective. Bull. Amer. Meteor. Soc., 67, 533–534, https://doi.org/10.1175/1520-0477(1986)067<0533:TDOATE>2.0.CO;2.
Liebmann, B., and C. Smith, 1996: Description of a complete (interpolated) outgoing longwave radiation dataset. Bull. Amer. Meteor. Soc., 77, 1275–1277, https://doi.org/10.1175/1520-0477-77.6.1274.
Lin, H., G. Brunet, and J. Derome, 2008: Forecast skill of the Madden–Julian oscillation in two Canadian atmospheric models. Mon. Wea. Rev., 136, 4130–4149, https://doi.org/10.1175/2008MWR2459.1.
Lin, H., G. Brunet, and B. Yu, 2015: Interannual variability of the Madden-Julian oscillation and its impact on the North Atlantic Oscillation in the boreal winter. Geophys. Res. Lett., 42, 5571–5576, https://doi.org/10.1002/2015GL064547.
Ling, J., C. Zhang, and P. Bechtold, 2013: Large-scale distinctions between MJO and non-MJO convective initiation over the tropical Indian Ocean. J. Atmos. Sci., 70, 2696–2712, https://doi.org/10.1175/JAS-D-13-029.1.
Liu, F., L. Zhou, J. Ling, X. Fu, and G. Huang, 2016: Relationship between SST anomalies and the intensity of intraseasonal variability. Theor. Appl. Climatol., 124, 847–854, https://doi.org/10.1007/s00704-015-1458-2.
Lo, F., and H. H. Hendon, 2000: Empirical extended-range prediction of the Madden–Julian oscillation. Mon. Wea. Rev., 128, 2528–2543, https://doi.org/10.1175/1520-0493(2000)128<2528:EERPOT>2.0.CO;2.
Lyu, M., X. Jiang, and Z. Wu, 2019: A cautionary note on the long‐term trend in activity of the Madden-Julian oscillation during the past decades. Geophys. Res. Lett., 46, 14 063–14 071, https://doi.org/10.1029/2019GL086133.
Madden, R. A., and P. R. Julian, 1971: Detection of a 40–50 day oscillation in the zonal wind in the tropical Pacific. J. Atmos. Sci., 28, 702–708, https://doi.org/10.1175/1520-0469(1971)028<0702:DOADOI>2.0.CO;2.
Madden, R. A., and P. R. Julian, 1972: Description of global-scale circulation cells in the tropics with a 40–50 day period. J. Atmos. Sci., 29, 1109–1123, https://doi.org/10.1175/1520-0469(1972)029<1109:DOGSCC>2.0.CO;2.
Maloney, E. D., and S. K. Esbensen, 2003: The amplification of east Pacific Madden–Julian oscillation convection and wind anomalies during June–November. J. Climate, 16, 3482–3497, https://doi.org/10.1175/1520-0442(2003)016<3482:TAOEPM>2.0.CO;2.
Mariotti, A., and Coauthors, 2020: Windows of opportunity for skillful forecasts subseasonal to seasonal and beyond. Bull. Amer. Meteor. Soc., 101, E608–E625, https://doi.org/10.1175/BAMS-D-18-0326.1.
McPhaden, M. J., 1999: Genesis and evolution of the 1997–98 El Niño. Science, 283, 950–954, https://doi.org/10.1126/science.283.5404.950.
Pegion, K., and Coauthors, 2019: The Subseasonal Experiment (SubX): A multimodel subseasonal prediction experiment. Bull. Amer. Meteor. Soc., 100, 2043–2060, https://doi.org/10.1175/BAMS-D-18-0270.1.
Pohl, B., and A. J. Matthews, 2007: Observed changes in the lifetime and amplitude of the Madden–Julian oscillation associated with interannual ENSO sea surface temperature anomalies. J. Climate, 20, 2659–2674, https://doi.org/10.1175/JCLI4230.1.
Rao, S. A., J. J. Luo, S. K. Behera, and T. Yamagata, 2009: Generation and termination of Indian Ocean dipole events in 2003, 2006 and 2007. Climate Dyn., 33, 751–767, https://doi.org/10.1007/s00382-008-0498-z.
Reynolds, R. W., T. M. Smith, C. Liu, D. B. Chelton, K. S. Casey, and M. G. Schlax, 2007: Daily high-resolution blended analyses for sea surface temperature. J. Climate, 20, 5473–5496, https://doi.org/10.1175/2007JCLI1824.1.
Roundy, P. E., 2008: Analysis of convectively coupled Kelvin waves in the Indian Ocean MJO. J. Atmos. Sci., 65, 1342–1359, https://doi.org/10.1175/2007JAS2345.1.
Roundy, P. E., 2012a: Tracking and prediction of large-scale organized tropical convection by spectrally focused two‐step space–time EOF analysis. Quart. J. Roy. Meteor. Soc., 138, 919–931, https://doi.org/10.1002/qj.962.
Roundy, P. E., 2012b: The spectrum of convectively coupled Kelvin waves and the Madden–Julian oscillation in regions of low-level easterly and westerly background flow. J. Atmos. Sci., 69, 2107–2111, https://doi.org/10.1175/JAS-D-12-060.1.
Roundy, P. E., 2012c: Observed structure of convectively coupled waves as a function of equivalent depth: Kelvin waves and the Madden–Julian oscillation. J. Atmos. Sci., 69, 2097–2106, https://doi.org/10.1175/JAS-D-12-03.1.
Roundy, P. E., C. J. Schreck III, and M. A. Janiga, 2009: Contributions of convectively coupled equatorial Rossby waves and Kelvin waves to the real-time multivariate MJO indices. Mon. Wea. Rev., 137, 469–478, https://doi.org/10.1175/2008MWR2595.1.
Roxy, M. K., P. Dasgupta, M. J. McPhaden, T. Suematsu, C. Zhang, and D. Kim, 2019: Twofold expansion of the Indo-Pacific warm pool warps the MJO life cycle. Nature, 575, 647–651, https://doi.org/10.1038/s41586-019-1764-4.
Straub, K. H., 2013: MJO initiation in the real-time multivariate MJO index. J. Climate, 26, 1130–1151, https://doi.org/10.1175/JCLI-D-12-00074.1.
Vitart, F., and Coauthors, 2017: The subseasonal to seasonal (S2S) prediction project database. Bull. Amer. Meteor. Soc., 98, 163–173, https://doi.org/10.1175/BAMS-D-16-0017.1.
Waliser, D. E., K. M. Lau, W. Stern, and C. Jones, 2003: Potential predictability of the Madden–Julian oscillation. Bull. Amer. Meteor. Soc., 84, 33–50, https://doi.org/10.1175/BAMS-84-1-33.
Waliser, D. E., and Coauthors, 2009: MJO simulation diagnostics. J. Climate, 22, 3006–3030, https://doi.org/10.1175/2008JCLI2731.1.
Wang, Y., H.-L. Ren, Y. Wei, F.-F. Jin, P. Ren, L. Gao, and J. Wu, 2022: MJO phase swings modulate the recurring latitudinal shifts of the 2020 extreme summer-monsoon rainfall around Yangtse. J. Geophys. Res. Atmos., 127, e2021JD036011, https://doi.org/10.1029/2021JD036011.
Wei, Y., and H.-L. Ren, 2019: Modulation of ENSO on fast and slow MJO modes during boreal winter. J. Climate, 32, 7483–7506, https://doi.org/10.1175/JCLI-D-19-0013.1.
Wei, Y., and Z. Pu, 2021: Moisture variation with cloud effects during a BSISO over the eastern Maritime Continent in a cloud-permitting-scale simulation. J. Atmos. Sci., 78, 1869–1888, https://doi.org/10.1175/JAS-D-20-0210.1.
Wei, Y., M. Mu, H.-L. Ren, and J. X. Fu, 2019: Conditional nonlinear optimal perturbations of moisture triggering primary MJO initiation. Geophys. Res. Lett., 46, 3492–3501, https://doi.org/10.1029/2018GL081755.
Wei, Y., H.-L. Ren, M. Mu, and J. X. Fu, 2020a: Nonlinear optimal moisture perturbations as excitation of primary MJO events in a hybrid coupled climate model. Climate Dyn., 54, 675–699, https://doi.org/10.1007/s00382-019-05021-7.
Wei, Y., Z. Pu, and C. Zhang, 2020b: Diurnal cycle of precipitation over the Maritime Continent under modulation of MJO: Perspectives from cloud‐permitting scale simulations. J. Geophys. Res. Atmos., 125, e2020JD032529, https://doi.org/10.1029/2020JD032529.
Wei, Y., F. Liu, H.-L. Ren, G. Chen, C. Feng, and B. Chen, 2022: Western Pacific premoistening for eastward-propagating BSISO and its ENSO modulation. J. Climate, 35, 4979–4996, https://doi.org/10.1175/JCLI-D-21-0923.1.
Wheeler, M., and G. N. Kiladis, 1999: Convectively coupled equatorial waves: Analysis of clouds and temperature in the wavenumber–frequency domain. J. Atmos. Sci., 56, 374–399, https://doi.org/10.1175/1520-0469(1999)056<0374:CCEWAO>2.0.CO;2.
Wheeler, M., and H. Hendon, 2004: An all-season real-time multivariate MJO index: Development of an index for monitoring and prediction. Mon. Wea. Rev., 132, 1917–1932, https://doi.org/10.1175/1520-0493(2004)132<1917:AARMMI>2.0.CO;2.
Xiang, B., and Coauthors, 2022: S2S prediction in GFDL SPEAR: MJO diversity and teleconnections. Bull. Amer. Meteor. Soc., 103, E463–E484, https://doi.org/10.1175/BAMS-D-21-0124.1.
Yoo, C., and S.-W. Son, 2016: Modulation of the boreal wintertime Madden-Julian oscillation by the stratospheric quasi-biennial oscillation. Geophys. Res. Lett., 43, 1392–1398, https://doi.org/10.1002/2016GL067762.
Zhang, C., 2005: Madden–Julian oscillation. Rev. Geophys., 43, RG2003, https://doi.org/10.1029/2004RG000158.
Zhang, C., 2013: Madden–Julian oscillation: Bridging weather and climate. Bull. Amer. Meteor. Soc., 94, 1849–1870, https://doi.org/10.1175/BAMS-D-12-00026.1.
Zhang, C., H. H. Hendon, W. S. Kessler, and A. J. Rosati, 2001: A workshop on the MJO and ENSO. Bull. Amer. Meteor. Soc., 82, 971–976, https://doi.org/10.1175/1520-0477(2001)082<0971:MSAWOT>2.3.CO;2.
Zhang, C., Adames, Á. F., B. Khouider, B. Wang, and D. Yang, 2020: Four theories of the Madden-Julian oscillation. Rev. Geophys., 58, e2019RG000685, https://doi.org/10.1029/2019RG000685.
Zhang, L., B. Wang, and Q. Zeng, 2009: Impact of the Madden–Julian oscillation on summer rainfall in southeast China. J. Climate, 22, 201–216, https://doi.org/10.1175/2008JCLI1959.1.
Zhang, W., Z. Huang, F. Jiang, M. F. Stuecker, G. Chen, and F.-F. Jin, 2021: Exceptionally persistent Madden–Julian oscillation activity contributes to the extreme 2020 East Asian summer monsoon rainfall. Geophys. Res. Lett., 48, e2020GL091588, https://doi.org/10.1029/2020GL091588.
