Conditional Nonlinear Optimal Perturbations of MoistureTriggering Primary MJO InitiationYuntao Wei1,2,3 , Mu Mu1,4, Hong‐Li Ren3 , and Joshua‐Xiouhua Fu3,4

1Key Laboratory of Ocean Circulation andWaves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao, China,2College of Earth and Planetary Sciences, University of Chinese Academy of Sciences, Beijing, China, 3Laboratory forClimate Studies and CMA‐NJU Joint Laboratory for Climate Prediction Studies, National Climate Center, ChinaMeteorological Administration, Beijing, China, 4Institute of Atmospheric Sciences and Department of Atmospheric andOceanic Sciences, Fudan University, Shanghai, China

Abstract TheMadden‐Julian Oscillation (MJO) is a dominant intraseasonal variability originating in thetropics and influencing global weather and climate. Despite this dominance, we still lack reliable precursorsignals triggering nonlinear initiation of primary MJO events. Here for the first time, we deal with thisquestion from the point of view of nonlinearity, with successful implementation of conditional nonlinearoptimal perturbation (CNOP). Our results show that relative to random perturbations, CNOPs of moisturehave the most potential to trigger the strongest primary MJO events more than 15–20 days in advance fromthe given non‐MJO reference states. The CNOPs manifest as a moist equator and aggregate in the lowertroposphere, with stronger signals over the western Indian Ocean than over the eastern one. These CNOPscan also capture the observational moist precursor conditions of primaryMJO events. This work also impliesthe necessity of utilizing nonlinear optimal moist initialization in subseasonal predictions.

Plain Language Summary As a dominant intraseasonal (20–90 days) variability prevailing in thetropics, the Madden‐Julian Oscillation (MJO) casts significant impacts on global weather and climate.However, we still lack reliable precursor signals for MJOs initiating over the western Indian Ocean,especially for primary events, that is, those without an immediately preceding MJO event. Here for the firsttime, we deal with this topic from the point of view of nonlinearity, with the successful implementation ofconditional nonlinear optimal perturbation (CNOP). In particular, we find that our CNOPs can identifythe optimal precursors of primary MJO initiation. We examined four non‐MJO reference states, selectedbased on both circulation‐ and convection‐based MJO indices. Although different reference states producedifferent CNOPs, all tend to manifest as a moist equator and aggregate in the lower troposphere, withstronger signals over the western Indian Ocean than over the eastern Indian Ocean. These CNOP‐type moistprecursor conditions are also found preceding the onset of observed primary events. We find that, relativeto random perturbations, CNOPs of moisture have the most potential to trigger the strongest MJO eventmore than 15–20 days in advance. This work implies the necessity of utilizing nonlinear optimal moistinitialization to improve subseasonal (15–60 days) predictions.

1. Introduction

A remarkable feature of tropical mesoscale cloud clusters is their tendency to be organized as slowly east-ward propagating (~5 m/s) convective envelops at a planetary scale (Nakazawa, 1988). This phenomenonis known as the Madden‐Julian Oscillation (MJO; Madden & Julian, 1971, 1972). Widely recognized as thedominant intraseasonal (20–90 days) variability prevailing over the tropics (Zhang, 2005), the MJO can castsignificant impacts not only on tropical systems but also on a variety of climate phenomena across differentspatial and temporal scales. These phenomena include (1) modulating the occurrence of extreme weather(Ren et al., 2018; Ren & Ren, 2017; Xavier et al., 2014), (2) affecting the onsets and breaks of global monsoonsystems (Wang, Liu, et al., 2017; Zhou & Chan, 2010), (3) interacting with the extratropics by exciting Rossbywave train and teleconnection (Henderson et al., 2017; Jones et al., 2004), and (4) contributing to the life cycleof El Niño events (McPhaden, 1999). The successful prediction of the MJO, which bridges weather and cli-mate (Zhang, 2013), is thus invaluable for saving and improving human lives around the world.

The MJO has intrigued scientific communities since it was first documented. Over the past four decades ofscientific pursuit, the MJO has become well understood as an equatorially trapped, unstably initiating,

©2019. American Geophysical Union.All Rights Reserved.

RESEARCH LETTER10.1029/2018GL081755

Special Section:Bridging Weather andClimate: Subseasonal‐to‐Seasonal (S2S) Prediction

Key Points:• We first investigate the initiation of

primary MJO events from the pointof view of nonlinearity

• CNOPs of moisture have the mostpotential to trigger the strongestprimary MJO initiation more than15–20 days in advance

• CNOPs of moisture can also capturethe observational precursorconditions of primary MJO events

Supporting Information:• Supporting Information S1

Correspondence to:M. Mu and H.‐L. Ren,mumu@fudan.edu.cn;renhl@cma.gov.cn

Citation:Wei, Y., Mu, M., Ren, H.‐L., & Fu, J.‐X.(2019). Conditional nonlinear optimalperturbations of moisture triggeringprimary MJO initiation. GeophysicalResearch Letters, 46, 3492–3501. https://doi.org/10.1029/2018GL081755

Received 20 DEC 2018Accepted 30 JAN 2019Accepted article online 5 FEB 2019Published online 22 MAR 2019

WEI ET AL. 3492

planetary‐scale circulation system (Li & Zhou, 2009; Wei et al., 2017), coupled with a multiscale convectivecomplex (Majda & Biello, 2004), moving eastward slowly with a rearward‐tilted (Kiladis et al., 2009), mixedKelvin‐Rossby wave structure (Chen &Wang, 2018a; Wang et al., 2018; Wang & Chen, 2016). In spite of sig-nificant progress in simulating and predicting the MJO, there are still numerous long‐standing, fundamen-tal, yet unresolved scientific questions about the MJO. One of the most challenging and least studied amongthese is the identification of the optimal precursors that trigger the initiation of large‐scale, organized,eastward propagating, deep convection of the MJO over the western Indian Ocean (Ling et al., 2017).Building on previous works, two major schools of thought have developed as follows. The first schoolconsiders the triggering of MJO initiation to arise exclusively from tropical dynamic and thermodynamicprocesses, including regeneration from upstream circumnavigating Kelvin waves (Lau & Peng, 1987;Sakaeda & Roundy, 2015, 2016; Zhang et al., 2017); the recharge of moist static energy through localizednonlinear interaction among radiation, convection, and evaporation (Bladé & Hartmann, 1993; Kemball‐Cook & Weare, 2001; Maloney & Wolding, 2015); air‐sea interactions (Fu & Wang, 2004; Li et al., 2008;Rydbeck & Jensen, 2017; Rydbeck et al., 2017; Webber et al., 2010, 2012); and downstream Rossby wavedynamic effects (Li et al., 2015; Wang et al., 2005; Zhao et al., 2013). The second school emphasizes the cri-tical role of tropical‐extratropical interaction (Hsu et al., 1990; Ray et al., 2009; Ray & Zhang, 2010).

A consensus on the essential mechanisms forMJO initiation has not yet been reached. One possible reason forthe lack of agreement is the intrinsic diversity and sporadic nature of the MJO. Each MJO event can be clas-sified as either primary, that is, with no immediately preceding MJO event, or successive, that is, immediatelyfollowing a preceding event (Matthews, 2008). Unlike the smooth variation of the real‐time multivariate MJO(RMM) index (Wheeler & Hendon, 2004) throughout both the initiation and development in successive MJOevents, a fast, localized growth of the composite RMM index always occurs prior to the eastward propagationof primary events (e.g., Figure 4 in Ling et al., 2013, and Figure 6 in Straub, 2013). A newMJO is then triggeredafter sufficient predestabilization, possibly by the advective moistening processes associated with equatorialwave dynamics (Zhao et al., 2013) and/or three‐way interaction among radiation, evaporation, and convection(Bladé & Hartmann, 1993). This kind of sudden initiation from non‐MJO states strongly suggests that somenonlinear processes play important roles. Thus, the linearized theoretical framework has previously experi-enced difficulty in simulating MJO initiation that occurs without a strong predecessor signal (e.g., Adames& Kim, 2016; Wang & Chen, 2016; Wang, Wei, et al., 2017; Wei et al., 2017). This may be a result of linearmodels' omission of key components of MJO initiation and amplification that have not yet been identified.However, nonlinear processes are perhaps critical to primary MJO initiation and linear models should notbe expected to performwell under these circumstances. Consequently, a new nonlinear frameworkmethodol-ogy is urgently needed to study the initiation of MJO events, especially that of primary MJO events.

This study aims to demonstrate that primary MJO events can be triggered in a process of nonlinear optimalmoist initialization. Roles of moist processes have been shown to be essential in initiating (e.g., Li et al., 2015;Zhao et al., 2013) and maintaining the MJO (e.g., Adames & Kim, 2016; Hsu & Li, 2012; Sobel & Maloney,2012, 2013) and also in improving subseasonal to seasonal prediction (e.g., Ham et al., 2012; Ren et al., 2016).Our efforts will be devoted to mathematically identifying moist optimal precursors that trigger primary MJOinitiation in a coupled global climate model (CGCM). Benefiting from this advanced CGCM, we have devel-oped a new nonlinear framework methodology to study the initiation of the MJO. In this work, the keyaccomplishment is the successful implementation and application of “conditional nonlinear optimal pertur-bation” (CNOP), a technique proposed by Mu et al. (2003).

In section 2, we introduce the data and model used in this paper. Section 3 describes a new nonlinearmethodological framework for studying primary MJO initiation. The results are provided in section 4, andthe final section gives a brief summary and discussion of this research.

2. Data and Model2.1. Data

Tomeasure convective activity, we use the daily mean outgoing longwave radiation (OLR) interpolated fromthe National Oceanic and Atmospheric Administration's Advanced Very High Resolution Radiometer satel-lite (Liebmann & Smith, 1996). The daily mean winds and specific humidity are from the reanalysis projectof the National Centers for Environmental Prediction/the National Center for Atmospheric Research

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(Kalnay et al., 1996). The time period analyzed herein is from 1979 to 2013. All data sets have been interpo-lated to horizontal resolution of 2.5° × 2.5° before analysis. For simplicity, we refer to them as “observations”throughout this paper.

2.2. The Hybrid CGCM

The model used in this study is a hybrid CGCM (Fu & Wang, 2004). The atmospheric component is thestandard T42 version of the ECHAM‐4 model (Roeckner et al., 1996), which resolves 19 vertical layersextending from the surface to 10 hPa. The oceanic component is a tropical upper ocean model of intermedi-ate complexity (Fu &Wang, 2001). The ECHAM‐4 model was coupled with the ocean model once per day byexchanging surface heat fluxes, surface wind stress, and sea surface temperature over the tropical globalocean without heat flux corrections. A 30‐year free run was carried out and is analyzed below. The CGCMshows substantial capability in simulating the first two combined empirical orthogonal function (EOF)modes (Wheeler & Hendon, 2004), the nonlinear features of the MJO (Majda & Stechmann, 2011), andthe primary MJO events (Matthews, 2008); for further details please see supporting information Text S1and Figures S1–S3.

3. A New Methodological Framework for Studying Primary MJO Initiation

After some unsuccessful attempts to derive suitable linear solutions, we have directly chosen the nonlinearmethod, that is, CNOP (Appendix A). Using a CNOP framework, we constructed MJO initiation as a unifiednonlinear optimization problem. We first identified eight non‐MJO reference states (NMRSs; Appendix B),half of which were randomly chosen for examination in this study. Each NMRS has a 42‐day duration, withvery weak (<1) index values for both the OLR‐based MJO index (OMI; Kiladis et al., 2014) and the RMMindex. Neither convection nor circulation displays an organized, large‐scale, eastward propagating MJOsignal. It is noteworthy that this kind of convective initiation from non‐MJO to MJO event, if any, mustbe primary (Ling et al., 2013; Straub, 2013).

To find optimal precursors mathematically, an indicator must be constructed in advance to (i) clearly dis-tinguish non‐MJO and MJO events and (ii) measure the strength of the triggered MJO events. Takingadvantage of the 42‐day power spectra of convection and circulation, an indicator can be formulatedas follows:

I ¼ ∑3

i¼1

ρiπi

∬ΩMJOPi ω; kð Þdωdk; (1)

where i(=1, 2, 3) denotes OLR, 850‐hPa zonal wind (U850) and coherence square between them, respec-tively. ρi ∈ [0, 1] is the weight coefficient. Here we take ρ1 = ρ2 = ρ3 = 1/3, which guarantees a more precisedefinition of MJO initiation that further considers the tight coupling between circulation and convection.Pi(ω, k) is power variance at frequency ω and wavenumber k. (ω, k) ∈ ΩMJO = [1/100, 1/20] × [0, 3], whichoutlines the spatiotemporal range of the eastward propagating MJO signal. To satisfy the additivity require-ment, prenormalization is necessary. πi is the standard deviation of the total power spectrum in ΩMJO.Remarkably, under the “perfect model” assumption, I is totally controlled by the initial conditions of eachindividual NMRS.

To identify the optimal moist precursors, we let q0 be the initial specific humidity of the prechosen reference

state andq′0 be a finite‐amplitude disturbance of q0, that is, q′0�� ��≤δ. We term “ctr” and “per” as the respective

numerical integrations initiated from q0 and fromq0 þ q′0. Contributions of initial moist perturbations to theMJO can be derived as follows:

J ¼ J q′0� � ¼ Iper q′0 þ q0

� �−Ictr q0ð Þ (2)

From equation (2), a stronger MJO event must correspond to a larger value of J. Thus, searching for the opti-

mal moist precursoreq′0 that triggers the strongest primary MJO initiation can be generalized as the followingunified nonlinear optimization problem:

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J eq′0� � ¼ maxq′0k k≤δ

J q′0� �

(3)

Equation (3) tells us that the moist optimal precursor should be the global optimal point in the feasibleregion q′0

�� q′0�� ��≤δ� �

, that is, CNOP.

Upon obtaining the CNOP of each given NMRS, we performed three reforecasts: The first run (“Controlrun”) began with unperturbed initial conditions, the second run (“CNOP run”) had CNOP‐perturbed initialconditions, and the third run (“RP run”) was initialized with random perturbations. To facilitate compari-son, we scale any random perturbations to have the same norm as the CNOPs. The role of CNOP in trigger-ing primary MJO initiation was then revealed by comparing the three runs.

4. Results4.1. CNOPs and Observational Evidence

In this section, we first discuss the three‐dimensional (3‐D) structures of the CNOPs in all four cases. Todemonstrate the ability of CNOPs to capture the “true” precursor conditions of primary MJO initiation,we then search for these signals in observational data.

In general, the 3‐D distributions of CNOPs (Figure 1) show that different NMRSs produced different CNOPs,which implies that different MJO events should have distinctive preceding optimal growing initial perturba-tions. Despite the different patterns overall, all CNOPs tend to be wet along the equatorial Indian Ocean butdry away from the equator. The strongest signals aggregate over the lower troposphere, with stronger pertur-bations in the equatorial western Indian Ocean (WIO) than in eastern Indian Ocean (EIO). To search forthese signals from observations, we calculated the Hovmöller diagrams of the equatorially averaged (10°S

Figure 1. Three‐dimensional structure of conditional nonlinear optimal perturbation (in grams per kilogram) in cases 1 to4, from left to right. The six subplots in each column represent different pressure levels, that is, 500, 600, 700, 850, 925,and 1,000 hPa, from top to bottom. The wet anomaly is represented as red shading. The dry anomaly is shown by bluedotted contours at intervals of 0.02 g/kg. The thick black contour denotes the zero value. The x axis is longitude (indegrees), and the y axis is latitude (in degrees).

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to 10°N), 20‐ to 90‐day bandpass‐filtered OLR and 850‐hPa specific humidity (q850) anomalies in eachobserved primary MJO event. We further examined any anomalous moisture preceding the onset of eachMJO event.

The results showed that 23 of 49 events are clearly preconditioned by a moisture anomaly. The composite ofthese 23 events (Figure 2) shows that preceding the onset of large‐scale, eastward propagating, deep convec-tion, a significant moisture signal originates from the EIO and moves toward the WIO. Since day −10, thismoisture signal tends to be reflected near eastern Africa. Then it continuously moves eastward and triggersprimary MJO initiation at day −5. In addition, there also exists a small, localized, narrow, and suppressedconvection over the EIO at day −15, which may also play a role in initiating the MJO through a midtropo-spheric, cooling‐induced, predestabilization effect (Matthews, 2008). The horizontal structure of q850 andthe vertical distribution of specific humidity have been also diagnosed (Figures S4 and S5). Together, theabove findings demonstrate the existence, well captured by the CNOPs, of significant moisture anomaliesaggregating in the lower troposphere over eastern Africa and the WIO.

4.2. CNOPs Triggering Primary MJO Initiation

Despite displaying distinctive 3‐D structures, all the CNOPs tend to trigger a similar spatiotemporal variancepattern, especially over ΩMJO. This pattern is illustrated by the composite zonal wavenumber‐frequencyspectra of both convection (OLR) and circulation (U850) in Figure 3. As expected, both the Control run

Figure 2. (a) Composite Hovmöller diagrams of equatorially averaged (10°S to 10°N), 20‐ to 90‐day bandpass‐filtered OLR(contour in watts per squaremeter) and 850‐hPa specific humidity (shading in grams per kilogram) anomalies using the 23observed primary MJO events. The solid (dashed) contour is from 5 (−5) W/m2 at intervals of 5 W/m2. The thin blackcontour denotes the zero values. The two magenta lines outline the zonal region for calculating the conditional nonlinearoptimal perturbations. The dotted area and contours represent results passing Student's t test at the 95% confidencelevel. The x axis is longitude (in degrees), while the y axis is lead time (in days). (b) Composite RMM2 index of observedprimary Madden‐Julian Oscillation events. The x axis is RMM2, while the y axis is lead time (in days). OLR = outgoinglongwave radiation; RMM = real‐time multivariate MJO; SHUM = specific humidity.

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and RP run show very weak power spectral variance, and the westward signal is comparable with theeastward one. However, when the nonlinear optimal moist initialization with the CNOPs is included, thevariances of convection and circulation associated with the eastward propagating MJO increase intensely,leaving a largely reduced westward signal. In a further difference from the Control and RP runs, theCNOP run's ΩMJO shows a strong variance of the coherence square between OLR and U850, whichimplies that strong coupling between convection and circulation has been also triggered by the CNOPs.

To demonstrate the triggering of primary MJO initiation, we also investigated the phase diagram of theRMM index (Figure S6). The results show that, in contrast to the RMM index's small growth in theControl and RP runs, the CNOP run's RMM index increased quickly and crossed the unit circle after 15–20 days of development, which is a typical initiation of primary MJO events (Ling et al., 2013; Straub, 2013).

The triggering of primary MJO initiation can be observed more clearly from the time‐longitude sections ofOLR and U850 (Figures 4a–4c). During the first 10 days, the perturbations in the three runs show very

Figure 3. Composite zonal wavenumber‐frequency spectrum of OLR (left column), U850 (middle column), and coher-ence square between them (right column) in the Control run (first row), the CNOP run (second row), and the RP run(third row). The x axis is zonal wavenumber (2π/40,000 km), and the y axis is frequency (cycles per day). The blue shadingdenotes ΩMJO, which outlines the wavenumber and frequency ranges of eastward propagating MJO signals. The redshading denotes the results passing the Student's t‐test at the 95% confidence level. to keep color legend consistent, theOLR spectrum has beenmultiplied by 0.02. Contour starts from 0.1 at intervals of 0.1. OLR= outgoing longwave radiation;CNOP = conditional nonlinear optimal perturbation; RP = random perturbation.

Figure 4. Hovmöller diagrams of equatorially averaged (10°S to 10°N) 20‐ to 90‐day bandpass‐filtered outgoing longwave radiation (shading) and U850 (contour)anomalies in the Control run (first row), the conditional nonlinear optimal perturbation run (second row), and the random perturbation run (third row). Solid(dashed) contours denote westerly (easterly) wind anomalies at intervals of 2 m/s. The thin black contour shows the zero values. Panels (a)–(c) are case 1, (d)–(f)case 2, (g)–(i) case 3, and (j)–(l) case 4. To fully represent the life cycle of triggered Madden‐Julian Oscillation events, the time shown has been extended to day 62.

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similar patterns. After days 10 to 15, the results start to diverge. The differences increase with lead time. Afterdays 25 to 30, an organized, large‐scale, slowly eastward propagating, convectively coupled, primary MJOevent is triggered only in the CNOP run. In contrast, for the Control run and RP run, as implicated by theirsmall index evolutions, there are only very weak and unorganized anomalies. This distinction with respect torandom perturbations demonstrates the power of nonlinear fast growth of the CNOPs in triggering the stron-gest primary MJO event from a given NMRS. A strong, new, primary MJO event is also triggered by theCNOPs in other NMRSs (Figures 4d–4l), although the initiation time, strength, duration, and phase speedare somewhat different, reflecting MJO diversity and dependence on basic flow (Lorenz, 1965).

5. Conclusions and Discussion

A newly developed nonlinear methodological framework was applied to identify the optimal precursors trig-gering primary MJO initiation. This framework mathematically identified optimal precursors from the fourprechosen NMRSs using a hybrid CGCM that can accurately simulate observed primary MJO events and anovel concept of CNOP. In this new framework, searching for the optimal precursors of primary MJO initia-tion can be generalized as solving for optimal solutions, that is, CNOPs, of a unified constraint nonlinearoptimization problem.

Our results show that although different CNOPs have been found in different NMRSs, they tended to man-ifest as a moist equator and aggregate over the lower troposphere, with stronger signals in the WIO than inthe EIO. Preceding the initiation of observed primary MJO events, we found significant lower‐troposphericmoisture signals moving toward the equatorial WIO, implying the key role of moisture‐convection feedbackwhile demonstrating the true‐to‐life quality of the CNOPs. The GCM reforecast experiments show thatCNOPs have the most potential to initiate the strongest primary MJO events from any given NMRSs, whilerandom perturbations only induce a non‐MJO or weak‐MJO initiation.

CNOPs trigger a new primary MJO event after a development of more than 15–20 days, which may implythat we can predict the onset of primary MJO initiation more than 15–20 days in advance, if the CNOPsare implemented in the initial conditions. Comparing the linear and nonlinear evolutions of CNOPs, wefound that nonlinear growth is always dominant on the 20‐ to 90‐day intraseasonal timescale (see Text S2,Figure S7, and Table S1). This findingmay suggest the necessity of utilizing nonlinear optimal moisture initi-alization in subseasonal (15‐ to 60‐day timescale) prediction.

Under the conditions of limited computational resources and as a preliminary application of this newmethodological framework, we examined only humidity field with limited vertical levels and four NMRSs.In addition, we eschewed the problem of identifying the dominant physical processes that control thenonlinear growth of CNOPs. These issues deserve in‐depth study, which will be part of our future work.

Appendix A

We write the evolution equations for the state vector U, which may represent wind, specific humidity,temperature, etc., as follows

∂U∂t

¼ F U; Pð ÞUjt¼0 ¼ U0

8<: ; (A1)

where U0 is the initial state, P is model parameter that is independent with time t, and F is a nonlineardifferential operator. Assuming that equation (A1) is well defined, the solution to equation (A1) for the statevector U at time τ is given by

U τð Þ ¼ Mτ Pð Þ U0ð Þ: (A2)

HereMτ(P) is the propagator of equation (A1) with the parameter P and “propagates” the initial value U0 tothe time τ in the future. Now we consider the situation in which there exist both initial perturbation u0 andparameter perturbation p′ in equation (A2). Then we have

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U τð Þ þ u u0; p′; τ

� � ¼ Mτ P þ p′� �

U0 þ u0ð Þ; (A3)

where u(u0, p′; τ) is the departure from the reference stateU(τ) caused by the combined error mode (u0, p

′). Anonlinear optimization problem is defined as follows.

J u0; p′

� � ¼ Mτ P þ p′� �

U0 þ u0ð Þ−Mτ Pð Þ U0ð Þ�� ��; (A4)

and

J u0δ; p′

σ

� � ¼ maxu0∈Cδ ;p′∈Cσ

J u0; p′

� �: (A5)

Here u0 ∈ Cδ and p′ ∈ Cσ are, respectively, the constraint conditions of the initial perturbation and para-meter perturbations, where Cδ and Cσ are closed and δ and σ distinguish the constraints of initial pertur-bation and parameter perturbation. The optimal combination mode of initial perturbation and parameter

perturbation, that is, u0δ; p′σ� �

, of the constrained maximization problem (A5) is called the CNOP (Mu

et al., 2010).

In this work, we consider the perfect model assumption, that is, p′= 0. In this case, the optimization problemis reduced to

Ju0 uI0δ� � ¼ max

u0∈CδMτ Pð Þ U0 þ u0ð Þ−Mτ Pð Þ U0ð Þk k: (A6)

The initial perturbation uI0δ satisfying equation (A6) is just the CNOP defined in Mu et al. (2003).

Appendix B

The NMRSs have been selected based on both the RMM and OMI indices. First, the RMM amplitude iscalculated. To facilitate selection, we use a 5‐day running mean to remove high‐frequency variability.Second, we identify day 0 when the amplitude reaches its minimum. Third, the amplitudes from day 0to day 31 should be smaller than a prescribed threshold of one standard deviation. In some scenarios,however, the convection may be still active, albeit with a weak RMM index (Straub, 2013). Thus, the5‐day running mean OMI is also examined to adjust the weak event selected by the RMM index. To bespecific, we have removed those events that continuously maintain a high‐amplitude (>1) OMI lastingfor more than 10 days. In the present article, we focus solely on the extended boreal winter from 15October to 15 May

Appendix C

To improve computational efficiency, an EOF‐based strategy is carried out to reduce the model dimen-sionality. We assume that the role of low‐frequency basic state (> 90 days) is minor in triggering thelarge‐scale, deep convection initiation on intraseasonal time scales (20 to 90 days). The variance of the90‐day highpass‐filtered specific humidity prefers the Indo‐Pacific region and aggregates over the loweratmosphere (500–1,000 hPa). Considering that MJO events are often initiated over the equatorialIndian Ocean (Matthews, 2008), we choose that precise region for the extraction of possible precursor sig-nals, in particular the lower‐level, equatorially symmetric Indian Ocean from 15°S to 15°N, from 40° to110°E, tiered vertically at 1,000, 925, 850, 700, 600, and 500 hPa. The EOF results demonstrate that thefirst 50 PCs can explain more than 80% of the total variance. Most of them can pass the North test(North et al., 1982) and thus are well separated. Any initial perturbation can be expanded orthogonallyby the first 50 EOF modes

q′0≈q′

0 λ′0� � ¼ ∑

50

i¼1λi0′ei; (C1)

where λi0′ is the initial time coefficient disturbance corresponding to the ith EOF mode ei.

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AcknowledgmentsThe authors gratefully acknowledgeFei‐Fei Jin, Fei Liu, and Wansuo Duanfor their kind discussions about thiswork. Constructive comments by threeanonymous reviewers led to substantialimprovements in the presentation. Thiswork was jointly supported by theNational Basic Research (973) Programof China under grant 2015CB453203,the China National Science Foundationunder grants 41775066, 41375062 and41875064, the NSFC Innovative Groupunder grant 41421005, and the NSFC‐Shandong Joint Fund for MarineScience Research Centers under grantU1406401. NCEP Reanalysis data wereprovided by the NOAA/OAR/ESRLPSD, Boulder, Colorado, USA, fromtheirWeb site at https://www.esrl.noaa.gov/psd/. Daily interpolated OLRs wereobtained from NOAA/OAR/ESRL PSD,Boulder, Colorado, USA, https://www.esrl.noaa.gov/psd/.

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