## 1. Introduction

The Madden–Julian oscillation (MJO) is an eastward-propagating intraseasonal mode of variability, with its variance concentrated in the 30–90-day time-scale band. It is a physical phenomenon first characterized by Madden and Julian (1971), involving interactions between tropical deep convection focused within the Maritime Continent, moisture, and atmospheric dynamics (Zhang 2005). Since this initial study there has been a vast array of research into this physical phenomenon, with recent comprehensive reviews in Zhang (2005) and DeMott et al. (2015). In the current study we decompose atmospheric reanalysis into a series of scales using normal mode functions (NMF), and on the basis of these results propose a reduced-order theory of the MJO. The following literature review hence focuses on the spatiotemporal-scale properties of the MJO and the historical theories explaining its dynamics.

The life cycle of the MJO is typically characterized as being initiated by a strong large-scale deep convective event and associated precipitation over the Maritime Continent. This strong deep convection is linked to surrounding regions of weaker convection via the circulation in the vertical–longitudinal plane extending throughout the troposphere (Madden and Julian 1972; Rui and Wang 1990). In the lower troposphere, at an altitude of approximately 850 hPa, the zonal winds converge toward the convective center. In the upper troposphere, at around 200 hPa, the zonal winds diverge and travel away from the source of convection. This large-scale convective system propagates eastward and decays in the western Pacific. This first-order view of the MJO suggests a simple baroclinic structure, since the anomalous zonal velocity at the upper and lower levels are of opposite sign. In reality, however, this perceived large-scale eastward propagation is due to a hierarchy of smaller spatiotemporal-scale convective systems that on average are initiated eastward of the earlier systems (Nakazawa 1988; Hendon and Salby 1994; Chen et al. 1996). At present this finer vertical-scale structure is less well understood. One of the goals of the current study is to infer a hierarchy of important vertical scales from the NMFs. In doing so we aim to clarify the nonlinear processes that couple westward-propagating Rossby waves, fast eastward inertial gravity waves, and extratropical influences, which together give rise to the intraseasonal eastward propagation of the MJO.

The large-scale horizontal features of the MJO convection were characterized in Wheeler and Kiladis (1999) via the spectral properties of the outgoing longwave radiation (OLR). Here, the meridionally symmetric and antisymmetric components of OLR were spectrally decomposed into longitudinal wavenumber (*k*) and temporal-frequency (*f*) components. The MJO was identified as an eastward-propagating symmetric component of OLR. In the (*k*, *f*) plane it has significant variance for wavenumbers *k* = 1–4 for time scales longer than 30 days, with *k* = 1 having the largest contribution. Faster eastward-propagating Kelvin waves were also identified, which were aligned with solutions of the shallow-water equations of equivalent heights (or depths) between 12 and 50 m. The fast propagation speed (or equivalently short phase period) of Kelvin waves is commonly cited as a reason for why they are not representative of the MJO. Following on from this physical understanding of the MJO, Wheeler and Hendon (2004) devised a real-time multivariate MJO (RMM) index involving the singular value decomposition of OLR, and the longitudinal velocity at 200 and 850 hPa, all meridionally averaged within the region 15°S–15°N. The first two principal components (referred to as RMM1 and RMM2) are shown to have dominant variance within the intraseasonal time-scale band.

Theories of the MJO typically involve prescribed relationships between convection, moisture, and atmospheric dynamics. Initial models were developed on the basis of wave–conditional instability of the second kind (CISK) theory, in which Kelvin waves become unstable because of low-level convective heating (Lindzen 1974; Lau and Peng 1987; Chang and Lim 1988). These models, however, did not predict the appropriate longitudinal scale of the MJO. The wind-induced surface heat exchange (WISHE) theory of Emanuel (1987) proposed eastward-propagating tropical waves were perturbed via enhanced evaporation. The WISHE model produces modes of appropriate longitudinal scale, and propagation speed, but with inconsistent perturbation structure during the Northern Hemisphere winter. In the frictional wave-CISK theory equatorial Rossby waves are also included, the interaction of which produces MJO-like properties of appropriate length and time scales (Wang and Rui 1990; Wang and Li 1994).

Another prevailing theory of the MJO is that it is driven by a moisture mode instability. This type of instability depends on the evolution of the humidity field, with its growth governed by feedbacks that further moisten the atmosphere (Neelin and Yu 1994; Fuchs and Raymond 2005). Raymond and Fuchs (2009) identify that in general this moistening process occurs in the presence of negative gross moist stabilities (GMSs). GMS is a vertically integrated measure of the moist static stability in the atmosphere (Neelin and Held 1987). Sobel and Maloney (2012) also find in their idealized model of the MJO that eastward-propagating waves can only be growing if the effective GMS is less than zero.

Majda and Stechmann (2009) developed a minimal dynamical model of the MJO capturing the interactions between lower-tropospheric moisture and convectively coupled wave dynamics of prescribed meridional and baroclinic structure. Within this framework they identify a mechanism for fast Kelvin waves and westerly propagating equatorial Rossby waves to couple to each other and become dispersive. This model produced an MJO with a small group velocity, intraseasonal phase speed, and a horizontal quadrapole vortex structure. Chen et al. (2016) extended on this work to model both the onset and decay of the MJO via interactions with extratropical and barotropic modes.

In Frederiksen and Lin (2013) the MJO is characterized by a coupled tropical–extratropical mode energized by moist baroclinic–barotropic instability within a three-dimensional basic state. The linear primitive-equation instability model of Frederiksen (2002) is used to simulate eternal-January basic states. Within this framework the MJO is a single mode of dominant zonal wavenumber *k* = 1, first internal baroclinic structure in the tropics, equivalent barotropic structure in the extratropics, and a phase period of 34.4 days (Frederiksen and Lin 2013). Other intraseasonal modes have also been identified using this approach with phase periods between 28 and 60 days (Frederiksen and Frederiksen 1993; Frederiksen 2002; Frederiksen and Lin 2013).

There are multiple theories that arrive at length and time scales consistent with the MJO, each of which adopts different approaches and hence has an alternate explanation of the nature of the MJO. At present there is no consensus on the appropriate framework, nor as to which minimal set of scales are required to reproduce the MJO dynamics. Yano and Tribbia (2017) also make the point that the results of certain successful theories are sensitive to parameter selection. Here we divorce ourselves from any parameter selection issues by processing three-dimensional global reanalysis data to identify which atmospheric scales have the appropriate spatiotemporal MJO properties, and the observed relationships between them.

Žagar and Franzke (2015) undertook a scale decomposition of the MJO present in ERA-Interim on the basis of NMFs. NMFs are the eigenvectors of the linearized primitive equations on a sphere, simultaneously capturing the variability in the horizontal velocity field and geopotential height, with the eigenvalues being the temporal period of the mode. The NMFs are characterized by a zonal wavenumber *k*, meridional wavenumber *n*, and vertical mode number *m*, which are in turn decomposed into Rossby waves and westerly and easterly traveling inertial gravity waves. Žagar and Franzke (2015) calculated the MJO-associated spectrum of NMF variance using with the RMM index. They found the largest percentage of the global variance is in the *n* = 1 Rossby mode of baroclinic structure with vertical modes *m* > 3. Using a similar approach Castanheira and Marques (2015) isolated the MJO from ERA-Interim via a cross-spectral analysis between the NMF decomposed *k* = 1 and *n* = 1 Rossby wave, and the OLR field. However, in isolation (i.e., not interacting with any other modes nor itself), Rossby waves travel westward. In addition, the eigenvalues associated with this mode indicates that the phase periods are too short to be that of the MJO. While the dominant projection of the MJO is onto this particular mode, the dynamics cannot be described by it alone. However, we propose that the pertinent dynamics can be described via its interaction with other NMFs.

In the present paper we extend the previous NMF applications and show that there are smaller-scale vertical modes that do have the appropriate phase periods, and other physical properties representative of the MJO. Unlike the Žagar and Franzke (2015) and Castanheira and Marques (2015) studies, the approach presented here does not require the RMM index nor the OLR field to isolate the MJO. We do, however, compare our results to the RMM index and also generate composites of the OLR field as means of validation. The NMF decomposition presented within is not intended to be a real-time index of the MJO as in Wheeler and Hendon (2004). Instead the intention is to use the NMFs to further understand the three-dimensional nature of the MJO, and its multiscale interactions.

The manuscript is organized as follows. In section 2 we outline the NMF theory, with the NMFs themselves calculated in section 3. Candidate NMFs representative of the MJO are determined on the basis of their temporal and spatial properties. The NMFs are then projected onto the daily fields of the Japanese 55-year Reanalysis (JRA-55; Kobayashi et al. 2015) with the spectral decomposition of the variance presented in section 4. A cross-spectral analysis is undertaken in section 5 to infer the potential for interactions between the pertinent modes. In section 6, one of the MJO-like NMF modes is used to produce three-dimensional phase averages. This same mode is then used to composite large and persistent MJO events in section 7. This analysis demonstrates the propagation of the velocity potential, and oscillation of the OLR and precipitation fields with the appropriate intraseasonal period. The temporal evolution of a combination of NMF modes are shown to be highly correlated with the RMM index in section 8. Finally, concluding remarks are made in section 9.

## 2. Normal mode theory

*g*= 9.81 m s

^{−2}is gravity,

*t*is time,

*λ*is longitude,

*ϕ*is latitude, and the vertical coordinate

*p*is the pressure and

*p*

_{s}the surface pressure. These equations govern the linear evolution of the state vector

*z*is the standard geopotential height,

*σ*level, and

*R*= 287 J kg

^{−1}K

^{−1}is the gas constant in air. Note,

*z*is a function of both temperature and specific humidity. The static stability profile

*c*

_{p}= 1004.6 J kg

^{−1}K

^{−1}is the specific heat of air at constant pressure. Equations (1)–(3) are augmented with the surface boundary condition of

*m*th VSF (i.e., eigenvector), and

_{m}is given by

*k*, meridional index

*n*, vertical index

*m*, and mode type

*q*. The associated dimensionless frequency (i.e., eigenvalue) is

Since there are three components in the state vector, there are three eigensolutions (or mode types, *q*) per scale triplet *n*, *n*, the symmetry properties are swapped. The remaining two solutions are high-frequency inertial gravity (IG) waves, with symmetry properties swapped in comparison to the BAL modes. One solution is an eastward-propagating IG wave (EIG), and the other a westward-propagating IG wave (WIG). We use the index *q* of values (1, 2, 3), to refer to the mode types (EIG, WIG, BAL). Throughout the manuscript we use the subscript

## 3. Normal mode functions of the Japanese 55-year Reanalysis

The VSF and HSF modes derived in the previous section are calculated here using the MODES code developed and described in Žagar et al. (2015). Upon defining the vertical discretization on the terrain-following coordinate levels *σ*, the only input data required for the calculation of the VSF modes are the static stability profiles *σ* grid is defined such that for a nominal surface pressure of 1013 hPa, there are 7 levels below 800 hPa, 6 levels between 800 and 500 hPa, 13 levels from 500 to 100 hPa, 9 levels between 100 and 10 hPa, and 8 levels from 10 to 0.5 hPa, for a total of 43 levels.

Daily temperature, wind and moisture fields of JRA-55 from 1 January 1958 to 31 December 2016 are interpolated to the above-defined *σ* levels. JRA-55 temperature fields are first formed using 37 isobaric levels defined from 1 to 1000 hPa, and augmented by the surface temperature. These fields are then interpolated onto the above-defined *σ* grid, and averaged across each *σ* level to produce

The eigenvalues of the VSF EVP in (8) are the equivalent heights *m*. The *m*th VSF has *D*_{1} = 10 km. The *D*_{2} = 7 km. These modes are illustrated in Fig. 1c, along with the *D*_{8} = 324 m, *D*_{15} = 92 m, and *D*_{28} = 7 m. As the equivalent height decreases (or equivalently *m* increases) the structures of the VSFs are also concentrated closer to the surface.

The HSF EVP requires *m* has a unique set of HSFs and associated eigenvalues *D*_{m} > 1 m. The periods *k*. Both the WIG (red dots) and BAL (black dots) modes have a negative *k* and hence propagate westerly. The bold dots represent the *m* increases (Žagar et al. 2015). As *n* increases, the time scale of the IG modes decrease, while that of the BAL modes increase. Two other subclasses are also explicitly illustrated. The mixed Rossby–gravity (MRG) modes are by definition the BAL modes with *k* and *m*). Likewise the Kelvin wave are the EIG modes with

Horizontal structure function eigensolutions: (a) time scales *n*), mixed Rossby–gravity (MRG; equivalent to BAL:

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Horizontal structure function eigensolutions: (a) time scales *n*), mixed Rossby–gravity (MRG; equivalent to BAL:

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Horizontal structure function eigensolutions: (a) time scales *n*), mixed Rossby–gravity (MRG; equivalent to BAL:

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

At this stage we look to identify a set of modes representative of the MJO. To restate, the MJO is an eastward-propagating tropics centric intraseasonal oscillation with horizontal velocity components of dominant zonal wavenumber

The abovementioned NMFs have a meridional structure appropriate for the MJO as indicated by the zonal velocity component of the HSFs. Four HSFs are illustrated in Fig. 2c: the Kelvin wave with a 56-day phase period, *m* increases as well as with increasing *k*.

## 4. Scale and mode type decomposition

*M*, and

*K*is the zonal truncation wavenumber,

*N*is the meridional truncation wavenumber, and

*q*takes on the values (1, 2, 3), referring to mode types (EIG, WIG, BAL). Note, this notation is different from Žagar et al. (2015) with

*N*one-third of the size, because of the introduction of index

*q*to explicitly represent the BAL, EIG, and WIG motions in the summation. The scalar complex expansion coefficients

*q*for a given scale triplet is

*q*, as a function of a particular index (e.g.,

*k*), is given by the sum of

*n*and

*m*). The total energy is the sum of energy from all mode classes.

The expansion coefficients are calculated for the daily JRA-55 fields from 1 January 1958 to 31 December 2016. All of the analysis throughout the paper is undertaken over this period. The time-averaged zonal wavenumber spectrum of each mode type and the total energy is illustrated in Fig. 3a. The total energy exhibits a

Time-averaged spectral contribution from each of the NMFs: (a) zonal wavenumber spectra of total energy (i.e., potential and kinetic) summed over *n* and *m* decomposed into EIG, WIG, and BAL components, with the dashed black line representing the *k* and *n* as a function of equivalent height; (c) as in (b), but for only the

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Time-averaged spectral contribution from each of the NMFs: (a) zonal wavenumber spectra of total energy (i.e., potential and kinetic) summed over *n* and *m* decomposed into EIG, WIG, and BAL components, with the dashed black line representing the *k* and *n* as a function of equivalent height; (c) as in (b), but for only the

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Time-averaged spectral contribution from each of the NMFs: (a) zonal wavenumber spectra of total energy (i.e., potential and kinetic) summed over *n* and *m* decomposed into EIG, WIG, and BAL components, with the dashed black line representing the *k* and *n* as a function of equivalent height; (c) as in (b), but for only the

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

The wave energy (i.e., excluding the *D*_{m} = *D*_{1} = 10 km) is the most energetic. The BAL modes are again dominant when considering only the *D*_{m} = *D*_{2} = 7 km) has the greatest energy contribution for *D*_{m} = 324 m. The latter is the *m*, as illustrated in Fig. 3d. The most energy occurs at *D*_{m} = 324 m, which is associated with the

## 5. Cross-spectral analysis

From the analysis in section 3, *D*_{m} = 324 m is the most energetic, but has an eigenvalue-derived phase period too short to be associated with the MJO. The spectral structure of energy contained within these scales of EIG modes is illustrated via the variance distribution in the *f* is the temporal frequency. Fourier transforms are calculated for the expansion coefficient *D*_{m} = 324 m, and *D*_{m} = 7 m is illustrated in Fig. 4b, and indicates that the slow component is dominant.

Spectral coherence properties of the NMFs: (a) power spectral density in the zonal wavenumber (*k*)–frequency (*f*) plane of

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Spectral coherence properties of the NMFs: (a) power spectral density in the zonal wavenumber (*k*)–frequency (*f*) plane of

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Spectral coherence properties of the NMFs: (a) power spectral density in the zonal wavenumber (*k*)–frequency (*f*) plane of

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

The covariability and potential for interaction between

The coherence output with respect to

The associated coherence output for

In summary four pertinent mode clusters are identified as interacting with each other, and having properties compatible with the MJO. The cluster centered on the

We hypothesize that the nonlinear interaction between each of these mode four groups ensures that a component of the energetic modes propagate eastward with an intraseasonal phase period. A minimal model of the MJO would require the interacting triads formed by at least one mode central to each of the abovementioned clusters. The reduced-order model equations would be a set of ordinary differential equations (ODEs), derived from the projection of these orthogonal modes onto the nonlinear equations of motion, akin to what is done in the reduced-order modeling of canonical fluid flows (Noack et al. 2003; Kitsios 2010). These ODEs would be solved numerically to return the time-varying contribution of each of these modes to the MJO dynamics. To refine the representation of the MJO, the projection could be systematically expanded to include additional equivalent heights radiating out from the aforementioned four modes, and also finer-scale longitudinal waves. This framework incorporates the interactions between multiple vertical scales, and also between the tropics and extratropics via the varying meridional extents of the HSFs. Since

## 6. Phase averages

To provide further details on the three-dimensional spatial structure of the MJO, phase averages are constructed using *j* = 1–8, with phase angle ranges defined between

The phase average anomalies are illustrated by three-dimensional isosurfaces of the velocity potential overlaid with contours of OLR in Fig. 5. To account for the change in magnitude of the velocity potential with level, the fields are normalized by the standard deviations on each level. The dark and light gray structures represent isosurfaces two standard deviations above and below the mean, respectively. From phase 1 to phase 8, this figure demonstrates a consistent eastward propagation of the velocity potential structures, and a dipole type behavior of the OLR field dominant over the Maritime Continent. The phases are also in line with those presented in Wheeler and Hendon (2004). In phase 1 convection from the previous MJO event (positive convection anomaly) is centered over the east of the Maritime Continent. Subsequent phases see convection ramping up in magnitude initially over the east of Africa. There is a strong coupling between the OLR and velocity potential fields in phase 5 with the interface between positive and negative anomalies in both fields occurring at approximately 90°E. The imprint of the upper-level velocity potential is also evident outside of the Maritime Continent in the OLR, with anomalies of like sign aligned.

Anomalies of phase-averaged OLR and velocity potential fields, calculated using the EIG ^{−2} (yellow). Velocity potential isosurfaces are shaded for two standard deviations below the mean (light gray) and two standard deviations above the mean (dark gray). The vertical axis, *p*_{0} = 1013 hPa.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Anomalies of phase-averaged OLR and velocity potential fields, calculated using the EIG ^{−2} (yellow). Velocity potential isosurfaces are shaded for two standard deviations below the mean (light gray) and two standard deviations above the mean (dark gray). The vertical axis, *p*_{0} = 1013 hPa.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Anomalies of phase-averaged OLR and velocity potential fields, calculated using the EIG ^{−2} (yellow). Velocity potential isosurfaces are shaded for two standard deviations below the mean (light gray) and two standard deviations above the mean (dark gray). The vertical axis, *p*_{0} = 1013 hPa.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

The velocity potential also changes sign in the vertical, which is representative of convection being associated with upper-level divergence (positive velocity potential) and lower-level convergence (negative velocity potential). This is again consistent with the results of Wheeler and Hendon (2004), in which the first two EOFs have their 200-hPa zonal wind component of opposite sign to the 850-hPa zonal wind.

## 7. Composite fields

By construction phase averages, such as those presented in the previous section, contain only spatial structure and no temporal information. A more stringent test is to determine if the spatial and temporal properties of fields composited on the basis of

The magnitude of the selected events is illustrated as the faint gray lines in Fig. 6a. The ensemble mean and standard deviation is calculated for each day offset from day 0. The solid bold line in this figure is the ensemble-averaged magnitude, with the dashed lines showing the ensemble average plus and minus one ensemble standard deviation. There is a local maximum in magnitude in the vicinity of day 0, which indicates that these events are stronger than the background variability of this mode. The ensemble-averaged phase angles are illustrated in Fig. 6b, and indicate a persistent eastward propagation.

Composite-averaged normal mode function complex expansion coefficients

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Composite-averaged normal mode function complex expansion coefficients

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Composite-averaged normal mode function complex expansion coefficients

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

To confirm the spatial structure, ensemble-averaged composites of velocity potential, OLR, and precipitation are calculated for the days leading up to and following day 0. First, anomalous daily fields are calculated to have the yearly cycle removed and also ENSO time scales filtered out. The fields of dates associated with day 0 are averaged together to produce an ensemble-averaged day 0. Likewise, the same process is applied to produce ensemble-averaged composites for the nonzero day shifts. The ensemble-averaged velocity potential at a pressure level of 200 hPa is illustrated in Fig. 7 for a selection of days from 20 days before to 20 days after. There is a clear propagation of the structures over this 41-day period. The ensemble-averaged OLR is illustrated in Fig. 8 over the same relative day range. The OLR fields demonstrate a dipole type oscillation over the Maritime Continent. Composites of precipitation are illustrated in Fig. 9, which demonstrates that this field has structures very similar to those observed in the OLR fields, but of opposite sign. The observed strong anticorrelation between these two fields is consistent with the view that anomalously low (high) OLR leads to high (low) temperatures and increased (decreased) evaporation and water vapor concentration, which finally produces anomalously high (low) rainfall. By design the structure at day 0, of approximate phase angle 0°, is similar to that of phase 5 illustrated in Fig. 5e. Consistent with Fig. 6a, the magnitude of the velocity potential, OLR, and precipitation fields all decrease the further in time they are away from day 0.

Composite-averaged velocity potential field at 200 hPa based upon the EIG ^{6} (blue) to 2.5 × 10^{6} m^{2} s^{−1} (yellow). Minimum (min) and maximum values (max) over the illustrated domain for each plot are listed in their respective titles.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Composite-averaged velocity potential field at 200 hPa based upon the EIG ^{6} (blue) to 2.5 × 10^{6} m^{2} s^{−1} (yellow). Minimum (min) and maximum values (max) over the illustrated domain for each plot are listed in their respective titles.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Composite-averaged velocity potential field at 200 hPa based upon the EIG ^{6} (blue) to 2.5 × 10^{6} m^{2} s^{−1} (yellow). Minimum (min) and maximum values (max) over the illustrated domain for each plot are listed in their respective titles.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Composite-averaged outgoing longwave radiation field based upon the EIG ^{−2} (yellow). Min and max values over the illustrated domain for each plot are listed in their respective titles.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Composite-averaged outgoing longwave radiation field based upon the EIG ^{−2} (yellow). Min and max values over the illustrated domain for each plot are listed in their respective titles.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Composite-averaged outgoing longwave radiation field based upon the EIG ^{−2} (yellow). Min and max values over the illustrated domain for each plot are listed in their respective titles.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Composite-averaged precipitation field based upon the EIG ^{−1} (yellow). Min and max values over the illustrated domain for each plot are listed in their respective titles.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Composite-averaged precipitation field based upon the EIG ^{−1} (yellow). Min and max values over the illustrated domain for each plot are listed in their respective titles.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Composite-averaged precipitation field based upon the EIG ^{−1} (yellow). Min and max values over the illustrated domain for each plot are listed in their respective titles.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

## 8. Characterization of individual MJO events

We next contrast the NMF MJO characterization with the RMM index of Wheeler and Hendon (2004). The evolution of the NMF energy

Akin to what is done in Wheeler and Hendon (2004), 91-day centered moving averages are calculated. The square root of these moving averages are illustrated in Fig. 10a for the period from 2000 to 2016. The

MJO time series: (a) square root of the 91-day centered moving averages ^{2} + RMM2^{2}), the NMF energy

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

MJO time series: (a) square root of the 91-day centered moving averages ^{2} + RMM2^{2}), the NMF energy

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

MJO time series: (a) square root of the 91-day centered moving averages ^{2} + RMM2^{2}), the NMF energy

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

From the perspective of the vertical structure, the high energy modes,

We have identified four MJO events defined as local maxima of the RMM index, which in both order of increasing magnitude and chronological order are in November 2002, January 2006, February 2008, and March 2012. Note these dates also correspond to local maxima in our

## 9. Concluding remarks

The MJO is described here as a combination of interacting balanced (BAL) modes (i.e., Rossby waves) and eastward-propagating inertial gravity (EIG) waves of varying vertical scale and meridional extent. The MJO was isolated from JRA-55 via a normal mode function (NMF) scale and mode class decomposition. The only NMFs that have the appropriate MJO-like horizontal structure are the Rossby wave BAL modes of zonal wavenumber

A cross-spectral analysis indicated that the above mode groups are coherent and presumably interact with one another. The only way that these orthogonal NMFs can interact is through nonlinear triads. We, therefore, hypothesize that the nonlinear interaction between the above mode groups conspire such that on average the energetic modes propagate eastward with an intraseasonal time scale, as observed. In this representation of the MJO a minimal set of interacting triads is formed by the Kelvin waves

On the basis of the

These findings may in some way explain the influence of vertical resolution on the representation of the MJO in general circulation models (GCMs). Crueger et al. (2013) undertook a meta-analysis of 42 experiments of varying configuration and resolution, including both atmospheric and coupled GCMs. One of their key findings was that eastward propagation was enhanced when vertical and horizontal resolution were increased together, while changing only vertical resolution had a lesser influence. However, as stated in Liess and Bengtsson (2004), the relative lesser importance of the vertical grid could be due to having an inappropriate ratio of vertical to horizontal resolution. Regardless of the quantum of the influence of the vertical grid, we speculate that associated improvements in the representation of the MJO is related to the importance of the

## Acknowledgments

The authors were supported by the Australian Commonwealth Scientific and Industrial Research Organisation Decadal Forecasting Project (research.csiro.au/dfp). The authors would also like to acknowledge NCI for providing the computational resources.

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