Abstract

Diagnostics obtained as an extension of empirical orthogonal function (EOF) analysis are shown to address many disadvantages of using EOF-based indices to assess the state of the Madden–Julian oscillation (MJO). The real-time multivariate MJO (RMM) index and the filtered MJO OLR (FMO) index are used to demonstrate these diagnostics. General characteristics of the indices, such as the geographical regions that most heavily influence each index, are assessed using the diagnostics. The diagnostics also identify how a given field, at various geographical locations, influences the index value at a given time. Termination (as defined by the RMM index) of the October 2011 MJO event that occurred during the Cooperative Indian Ocean Experiment on Intraseasonal Variability in the Year 2011 (CINDY) Dynamics of the MJO (DYNAMO) field campaign is shown to have resulted from changes in zonal wind anomalies at 200 hPa over the eastern Pacific Ocean, despite the onset of enhanced convection in the Indian Ocean and the persistence of favorable lower- and upper-level zonal wind anomalies near this region. The diagnostics objectively identify, for each specific geographical location, the index phase where the largest MJO-related anomalies in a given field are likely to be observed. This allows for the geographical variability of anomalous conditions associated with the MJO to be easily assessed throughout its life cycle. In Part II of this study, unique physical insight into the moist static energy and moisture budgets of the MJO is obtained from the application of diagnostics introduced here.

1. Introduction

Despite four decades of research, no consensus currently exists as to what method is “best” for objectively assessing the state, either past or present, of the Madden–Julian oscillation (MJO) (Straub 2013). This lack of consensus results from the diverse needs of the various user groups, differing views of the pertinent features that describe the MJO, and the advantages and disadvantages of the numerous techniques used to assess the MJO. Most MJO indices result from the application of one of (or a combination of) two general methods:

  • wavenumber–frequency filtering of a field, such as OLR, to spatial and temporal time scales associated with the MJO (Kiladis et al. 2005), and

  • projection of data onto a pair of EOFs or combined EOFs (CEOFs), as in Wheeler and Hendon (2004).

Previous studies (Straub 2013; Kiladis et al. 2014) have highlighted notable advantages and disadvantages of each method, which inevitably determine their suitability for a range of research applications.

A distinct advantage of wavenumber–frequency filtering methods is that disparate geographical locations can be compared in a relatively straightforward manner. Local indices based on the maximum or minimum of a filtered field can be used to define a “day 0” at various locations, which can then be used to compare and contrast the evolution of events at these different locations. An example of such an index is used in Kiladis et al. (2005). Unfortunately, it is often the periods of time that precede or follow this day 0 that are of interest. When considering multiple MJO events, variability in the propagation speed and duration of the events can substantially reduce the signal to noise ratio and sample size as the period of interest moves further away from day 0. One example of this is the “smearing” of composites at longer lags. Another problem is that local indices based on a single field may not be representative of the presence of the large-scale multivariate structure of the MJO. If the circulation anomalies associated with the MJO are considered to be a pertinent feature, does a cloudiness-based index provide an accurate assessment of the state of the MJO? Other problems exist that are more directly related to the application of wavenumber–frequency filtering. Spectral ringing associated with large amplitude events can introduce substantial spurious anomalies in filtered fields that are not present in their unfiltered counterparts (Kiladis et al. 2014). Robust filtered anomalies in the MJO band that compare well with anomalies in the unfiltered data are not necessarily associated with the MJO. Wavenumber–frequency filtering also complicates use of such indices in real-time applications (Wheeler and Weickmann 2001), which is a potential disadvantage for certain user groups (e.g., forecasters).

Indices constructed using an EOF-/PC-based methodology avoid many disadvantages of wavenumber–frequency filtered indices. Such indices can be constructed for use in real-time and forecasting applications (Roundy 2012), can assess the multivariate structure of the MJO (through use of CEOFs), and can be targeted at a specific sequential spatial–temporal evolution through use of extended EOFs (Kikuchi et al. 2012). Additionally, the signal-to-noise ratio in composite analyses that use such indices is not reduced by the inclusion of events with varying propagation speed and duration. The real-time multivariate MJO (RMM) index (Wheeler and Hendon 2004), which is based on the CEOF analysis of latitudinally averaged OLR and zonal wind at 850 and 200 hPa (herein U850 and U200, respectively), and the velocity potential MJO (VPM) index (Ventrice et al. 2013), which is based on the CEOF analysis of latitudinally averaged U850, U200, and velocity potential at 200 hPa, are examples of such indices.

While indices constructed using an EOF-/PC-based methodology have many advantages over those constructed using wavenumber–frequency filtering, their application does have some substantial disadvantages. Many of these disadvantages stem from the lack of a direct ability to discern the relationship between physical observations and the index. For example, while OLR is a component of the RMM index, Straub (2013) used bivariate correlation analysis to demonstrate that the index is dominated by the U850 and U200 components. Such characteristics of an index are not immediately apparent to a user who is unfamiliar with the index. Similarly, the user is left to speculate about which fields and, more specifically, what geographical locations are driving the index on a particular day. For example, it is possible for westerly U200 anomalies in the central Pacific Ocean to produce an RMM magnitude and phase identical to that produced by a region of suppressed OLR over the Indian Ocean, yet these two situations cannot be distinguished from each other with the information currently provided by the RMM index. Similarly, a low index magnitude may reflect non-MJO conditions or a noncanonical MJO event, with no way for the user to distinguish the two conditions. In addition, variability not associated with the MJO can project onto real-time indices (Roundy et al. 2009), yet the indices provide no direct indication of the extent to which this may be occurring. Perhaps the largest limitation of EOF-/PC-based indices is that they only give a vague location of the regions where anomalous conditions may exist (e.g., enhanced convection over the Western Hemisphere and Africa in the RMM index). This limits the user’s ability to compare data from different geographical locations in context of the life cycle of the MJO.

Some indices, such the filtered MJO OLR (FMO) index (Kiladis et al. 2014), combine both wavenumber–frequency filtering and EOF-/PC-based methodologies. The FMO index results from the projection of latitudinally averaged 20–96-day filtered OLR onto the EOF structures of OLR derived from similarly filtered data. Filtering is implemented because the EOF structures of OLR insufficiently isolate variability associated with the MJO in all seasons. The univariate structure of the FMO allows the convective signal of the MJO to be appraised in isolation from the dynamical signal, providing a targeted assessment that is more physically consistent from event to event. As the convective signal of the MJO is largely confined to the Eastern Hemisphere, the FMO also provides a more regionally restricted assessment than global indices such as the RMM. Yet the FMO suffers many of the drawbacks of both wavenumber–frequency filtering and EOF-/PC-based methodologies, including an inability to directly relate anomalous conditions at specific geographical locations with the value of the index at a particular time.

This study introduces new objective diagnostics that are obtained as an extension of the current EOF-/PC-based methodology and that address many of the method’s disadvantages in assessing the state of the MJO. This study does not seek to put forward a new “best” index but instead introduces techniques and diagnostics that can be applied to EOF-based indices currently in use, addressing some of their shortcomings and utilizing information previously overlooked. These diagnostics provide valuable insight to indices that are already widely used (e.g., the RMM index) and allow for new applications of the indices in MJO research. The datasets and methods used to demonstrate these diagnostics and techniques are introduced in section 2. The diagnostics are formally developed in section 3, where brief examples of their potential application are also provided. Discussion is presented in section 4. A companion paper (Wolding and Maloney 2015, manuscript submitted to J. Climate, hereinafter Part II) applies these new diagnostics to the moist static energy (MSE) and moisture budgets of the MJO. It will be shown in Part II that the diagnostics and techniques introduced in this study provide unique and useful physical insight into the life cycle of the MJO.

2. Data and methodology

a. MJO indices

Both the RMM index and the FMO index will be used to demonstrate and assess the diagnostics introduced in this study. The RMM index, the CEOFs used in its calculation, and the fractional contribution of each field can be obtained from http://cawcr.gov.au/staff/mwheeler/maproom/RMM/. In this study, an RMM-like index was calculated following the method of Wheeler and Hendon (2004), with the following exceptions: only the removal of the previous 120-day mean from each field at each grid point was used in the removal of inter annual variability [consistent with Lin et al. (2008) and Gottschalck et al. (2010)], the CEOFs were calculated for the time period of 1979–2012, the principal components (PCs) were normalized according to Kiladis et al. (2014), and National Centers for Environmental Prediction (NCEP)–U.S. Department of Energy (DOE) Reanalysis 2 dynamical fields were used instead of NCEP–NCAR Reanalysis 1 dynamical fields. Removal of the previous 120-day mean was chosen for consistency with previous studies, though errors associated with rapidly changing inter annual patterns may still be present. The chosen normalization, whereby the second PC (PC2) is normalized using the same scaling as the first PC (PC1), allows PC2 to retain its relative weighting with respect to PC1 (Kiladis et al. 2014), prevents errors in the assessment index phase, and allows the calculations presented in Fig. 1a to remain strictly valid and reversible (i.e., to be able to reconstruct the fields from the PCs and the CEOFs). Daily 2.5° × 2.5° NCEP–DOE Reanalysis 2 dynamical fields and interpolated OLR (Liebmann and Smith 1996), obtainable at http://www.esrl.noaa.gov/psd/, were used in the calculation of the first two CEOF structures, which are presented in Fig. 2. These structures are consistent with those of Wheeler and Hendon (2004), and the index derived from these will be referred to simply as the RMM index. The FMO index and the EOFs used in its calculation can be obtained from http://www.esrl.noaa.gov/psd/mjo/mjoindex/. To match the phase convention of the RMM index, EOF1 and PC1 were multiplied by −1, and then the order of the first two EOFs and PCs were reversed. The structures of the FMO EOFs (not shown) closely match the OLR component of the RMM CEOF structures.

Fig. 1.

(a) An illustration of how the PCs used in various diagnostics introduced in this study are calculated. In this example, a CEOF composed of OLR (blue), U850 (red), and U200 (green) is used. The top of (a) is the traditional calculation of a PC, the middle of (a) is the calculation of an individual field PC, and the bottom of (a) is the calculation of the PC for a single grid point of a single field illustrated here. (b) An illustration of how the field vectors, calculated using field PCs, can be projected on the index vector in PC1–PC2 phase space. An equivalent method can be used with gridpoint vectors. (c) An illustration of the relative phase for a grid point with a gridpoint phase equal to 90°. It is assumed that the eastward propagation of the MJO is represented by counterclockwise rotation of the index vector. This would imply that negative relative phases would likely correspond with a transition from negative anomalies (blue shading) to positive anomalies (red shading) at the grid point. Similarly, positive relative phases would likely correspond with a transition from positive to negative anomalies at the grid point. This is illustrated by the circular thick arrows in the figure.

Fig. 1.

(a) An illustration of how the PCs used in various diagnostics introduced in this study are calculated. In this example, a CEOF composed of OLR (blue), U850 (red), and U200 (green) is used. The top of (a) is the traditional calculation of a PC, the middle of (a) is the calculation of an individual field PC, and the bottom of (a) is the calculation of the PC for a single grid point of a single field illustrated here. (b) An illustration of how the field vectors, calculated using field PCs, can be projected on the index vector in PC1–PC2 phase space. An equivalent method can be used with gridpoint vectors. (c) An illustration of the relative phase for a grid point with a gridpoint phase equal to 90°. It is assumed that the eastward propagation of the MJO is represented by counterclockwise rotation of the index vector. This would imply that negative relative phases would likely correspond with a transition from negative anomalies (blue shading) to positive anomalies (red shading) at the grid point. Similarly, positive relative phases would likely correspond with a transition from positive to negative anomalies at the grid point. This is illustrated by the circular thick arrows in the figure.

Fig. 2.

(top) First and (bottom) second CEOF structure of OLR, U850, and U200 latitudinally averaged from 15°N to 15°S from 1979 to 2012. The percent of the total variance explained by each CEOF structure is presented in the top-right corner of each panel. The structures of the first two EOFs of the FMO index (not shown) closely match the OLR components of the CEOFs presented here.

Fig. 2.

(top) First and (bottom) second CEOF structure of OLR, U850, and U200 latitudinally averaged from 15°N to 15°S from 1979 to 2012. The percent of the total variance explained by each CEOF structure is presented in the top-right corner of each panel. The structures of the first two EOFs of the FMO index (not shown) closely match the OLR components of the CEOFs presented here.

b. Additional datasets

Composites of precipitation, U850, and OLR are presented in subsequent sections. TRMM 3B42 daily precipitation, at 2.5° × 2.5° resolution, was obtained for the years 1998–2012. ERA-Interim U850, provided by the European Centre for Medium-Range Weather Forecasts, was obtained at 1.5° × 1.5° resolution and at 6-hourly temporal resolution then averaged to daily for the years 1979–2012. As previously mentioned, NOAA interpolated OLR was obtained for the years 1979–2012. Unless otherwise stated, the first three harmonics of the seasonal cycle have been removed from all fields before compositing. Only extended boreal winter months (1 October–30 April) are considered in this study.

3. Diagnostics

While the following diagnostics are derived in the context of CEOF-based indices (e.g., RMM index), they are similarly applicable to univariate EOF indices (e.g., FMO index). Furthermore, they are applicable to both one-dimensional (i.e., latitudinally averaged) and two-dimensional EOF/CEOF spatial structures.

CEOF1 and CEOF2 are, by construction, orthogonal. Therefore PC1 and PC2 can be used to define a two-dimensional phase space, and an index vector can be defined within that phase space such that its magnitude and phase are given by

 
formula
 
formula

Traditionally, the value of a PC at a given time is calculated by multiplying the data at a given time by the CEOF spatial structure. It is just as valid to multiply subsets of the spatial dimension of the data by corresponding subsets of the CEOFs in order to calculate spatial dimension subset PCs. These spatial dimension subset PCs can be summed to obtain the same PC value as obtained using the traditional calculation (see Fig. 1a). The ability to deconstruct PCs into spatial dimension subset PCs can be leveraged in PC1–PC2 phase space using vector calculus, providing insight to various properties of the index.

Calculations of the following diagnostics are generalized to be applicable to any index calculated from a pair of CEOFs/PCs containing J fields (e.g., OLR, U850, and U200), each having K grid points, such that

 
formula

Here, fPC1j and fPC2j correspond to the spatial dimension subset PC values of the jth field in the CEOF structure and gpPC1j,k and gpPC2j,k correspond to the spatial dimension subset PC values of the kth grid point in the jth field in the CEOF structure. The spatial dimension subset PCs associated with an individual field are preceded with a letter f, and those associated with an individual grid point of an individual field are preceded by the letters gp. This notation has been introduced to help distinguish diagnostics calculated from an individual field from those calculated from an individual grid point of an individual field in subsequent sections. Vectors in PC1–PC2 phase space can be defined for each field and grid point of each field such that

 
formula
 
formula

These vectors will be referred to as field vectors and gridpoint vectors, respectively.

Each of the diagnostics will now be introduced and their potential applications demonstrated. Illustrations to aid visualization of the following diagnostics are presented in Fig. 1. Remember, for arbitrary vectors A and B,

 
formula

where is the norm (i.e., the magnitude of A) and the angled brackets represent the inner product of two vectors,

 
formula

where θ is the angle between the two vectors and

 
formula

is the scalar projection of A onto B.

a. Field and gridpoint contribution

The contribution of the jth field to the overall magnitude of the index vector is simply the projection of that field vector onto the index vector (see Figs. 1a,b). Using the previously introduced notation, this is given in PC units by

 
formula

or, dividing by the index magnitude,

 
formula

results in units of percent contribution to total index magnitude. This diagnostic has been in use for several years (see Matt Wheeler’s website at http://cawcr.gov.au/staff/mwheeler/maproom/RMM/) and is included here as a conceptual building block for subsequent diagnostics. Figure 1b illustrates how the contribution of various field vectors to the overall magnitude of the index is assessed. The contributions of the OLR and U850 field vectors (blue and red arrows, respectively) to the overall magnitude of the index vector (black arrow) are given by their projections (dashed color lines) onto the index vector. In this example, both the OLR and U850 field vectors contribute positively to the overall index magnitude, while the U200 field vector (green arrow) is perpendicular to the index vector and therefore makes zero contribution to the overall index magnitude.

The relative influence of each field in determining the magnitude and phase of an index can be assessed by examining the scaled field contributions [see Eq. (4)] averaged over time. The RMM index is disproportionately influenced by U200 and U850, whose average scaled field contributions are ~43% and ~40%, respectively. In contrast, the average scaled field contribution of OLR is only ~17%. This is consistent with the findings of Straub (2013) regarding the RMM index. The average scaled field contributions are fairly insensitive to the period over which averaging is applied, remaining approximately constant when the mean is taken only during periods where the index magnitude exceeds a value of 1, is less than 1, or over the whole record.

Just as the contribution of a single field to the overall magnitude of the index is simply the projection of that field vector onto the index vector, the contribution of a single grid point of a single field to the overall magnitude of the index is simply the projection of the gridpoint vector onto the index vector. In PC units, this is

 
formula

or, dividing by the index magnitude,

 
formula

results in units of percent contribution to total index magnitude. It is worth reiterating that the sum of the gridpoint contributions on a given day is equal to the magnitude of the index on that day.

The geographical regions where each field preferentially contributes to the magnitude and phase of an index can be assessed using the time mean scaled gridpoint contribution [see Eq. (6)], which is shown in Fig. 3 for both the RMM and FMO indices. Note that the integral of the OLR curve is 100% for the FMO index, while the sum of the integrals of the three curves is 100% for the RMM index. In both the RMM and FMO indices, the influence of OLR begins east of 60°E and ceases near the date line, with large maxima evident near 80°E. As the magnitude and phase of the FMO index are determined solely by OLR, Fig. 3 shows that, despite spanning all longitudes, the FMO really is a regional index. In contrast, the influence of U200 on the RMM index spans the globe, with maxima over the western Indian Ocean and eastern Pacific Ocean. The influence of U850 on the RMM index spans from the western Indian Ocean to the west coast of South America, with its strongest contributions coming from the region of the Maritime Continent. The mean gridpoint contribution of each field closely coincides with the square root of the sum of the squares of the first two EOFs/CEOFs (not shown).

Fig. 3.

Scaled gridpoint contribution for each variable used in the RMM and FMO indices, averaged from 1980 to 2012.

Fig. 3.

Scaled gridpoint contribution for each variable used in the RMM and FMO indices, averaged from 1980 to 2012.

The field and gridpoint contribution can also provide valuable insight into what an index is “seeing” on a given day. Kiladis et al. (2014) highlighted major differences in how the RMM index and OLR-based indices portrayed the November 2011 MJO event that occurred during the Cooperative Indian Ocean Experiment on Intraseasonal Variability in the Year 2011 (CINDY) Dynamics of the MJO (DYNAMO) field campaign (Gottschalck et al. 2013; Johnson and Ciesielski 2013; Yoneyama et al. 2013). As various indices are often implemented by researchers to designate the initiation, evolution, and cessation of specific events, understanding such differences between indices is crucial. During this event, the RMM index maintained large magnitude throughout early and mid-October, despite little corresponding signal in OLR-based indices such as the FMO (Fig. 4). The RMM index then dramatically decreased between 18 and 25 October, just as convection built up over the Indian Ocean and the OLR-based indices quickly increased in magnitude (Kiladis et al. 2014). Kiladis et al. (2014) and Gottschalck et al. (2013) note that the large magnitude of the RMM in mid-October was due almost entirely to the circulation components of the index. Figure 5 shows OLR, U850, and U200 anomalies, as well as their respective gridpoint contributions to the RMM and FMO indices, over this time period. From 15 to 20 October, when the RMM magnitude was large, easterly U850 and westerly U200 anomalies over the Maritime Continent, as well as easterly U200 anomalies over the eastern Pacific, are the primary drivers of the RMM index. U200 contributed over 50% of the amplitude to the RMM index throughout this period. The RMM amplitude dramatically decreased after 20 October, despite the persistence of substantial easterly U850 and westerly U200 anomalies over the Maritime Continent as well as the onset of enhanced convection over the Indian Ocean (Fig. 5), all of which contributed positively to the index magnitude. This dramatic decrease in RMM amplitude coincided with a change from easterly to westerly U200 anomalies over the eastern Pacific Ocean, which contributed negatively to the index magnitude, so much so that the net contribution of U200 to the index magnitude became negative on 27 October. While the westerly U200 anomalies that developed over the eastern Pacific Ocean are not part of the canonical MJO structure, they may still have played an important role in the development of subsequent Indian Ocean basin MJO activity. Yet, given that the DYNAMO field campaign documented a strong October MJO event in the Indian Ocean (Gottschalck et al. 2013; Johnson and Ciesielski 2013; Yoneyama et al. 2013), the FMO index appears to provide a better representation of the October MJO event in this region than the RMM index. Figure 5 supports this assertion and helps explain the differences between these indices during this period.

Fig. 4.

Phase plots for 1 Oct–15 Dec 2011 of the (a) FMO and (b) RMM indices. Dates divisible by 5 are labeled, with the red, blue, and green lines corresponding with October, November, and December, respectively.

Fig. 4.

Phase plots for 1 Oct–15 Dec 2011 of the (a) FMO and (b) RMM indices. Dates divisible by 5 are labeled, with the red, blue, and green lines corresponding with October, November, and December, respectively.

Fig. 5.

OLR, U850, and U200 anomalies (contours) averaged from 15°N to 15°S and corresponding scaled gridpoint contribution (shading) for the RMM and FMO indices from 15 to 31 Oct. Positive (negative) anomalies are given by solid (dashed) contours. In the top panel, OLR has been bandpass filtered to 20–96 days. Contours for the filtered OLR, unfiltered OLR, U850, and U200 anomalies are, respectively, as follows: every 5 W m−2 beginning at 10 W m−2, every 10 W m−2 beginning at 20 W m−2, every 2 m s−1 beginning at 2 m s−1, and every 5 m s−1 beginning at 5 m s−1.

Fig. 5.

OLR, U850, and U200 anomalies (contours) averaged from 15°N to 15°S and corresponding scaled gridpoint contribution (shading) for the RMM and FMO indices from 15 to 31 Oct. Positive (negative) anomalies are given by solid (dashed) contours. In the top panel, OLR has been bandpass filtered to 20–96 days. Contours for the filtered OLR, unfiltered OLR, U850, and U200 anomalies are, respectively, as follows: every 5 W m−2 beginning at 10 W m−2, every 10 W m−2 beginning at 20 W m−2, every 2 m s−1 beginning at 2 m s−1, and every 5 m s−1 beginning at 5 m s−1.

It has been shown here that the field and gridpoint contribution can be used to assess general characteristics of an index, such as which fields and geographical locations play particularly influential roles in determining the index magnitude and phase. Understanding such characteristics can help the user determine if an index is appropriate for its intended use (Straub 2013). The field and gridpoint contribution can also provide insight into the evolution of individual MJO “events,” as perceived by an index. Information about the “what and where” of anomalies projecting onto an index can be useful to forecasters and researchers in helping determine the physical nature and accurate representation of events identified by an index.

b. Gridpoint phase

The index phase where the largest MJO-related anomalies in a given field occur depends on the geographical location being considered. In this section, a diagnostic that can be used to identify this phase for a given geographical location is introduced. In the next section, this diagnostic is used to develop a compositing technique that allows different geographical locations to be easily compared at the same stage of the MJO life cycle.

At every spatial location (i.e., grid point), the ratio of CEOF2 to CEOF1 is fixed. The ratio of gpPC2j,k to gpPC1j,k, which determines the phase of gpPCj,k, results from the projection of data onto these fixed CEOF structures. In other words, each grid point of each field can only produce gridpoint vectors that project along a fixed phase in PC1–PC2 phase space. This phase is given by

 
formula

where CEOF1j,k and CEOF2j,k are the values of the first two CEOF structures at the kth grid point of the jth field. Note that the gridpoint phase has been defined such that 0° is the positive axis of PC1 in PC1–PC2 phase space, corresponding to the Maritime Continent phase of the RMM index [see Fig. 7 in Wheeler and Hendon (2004)]. While the ratio of CEOF2j,k to CEOF1j,k is fixed, the data projected onto the CEOF structures may be either positive or negative and vary in magnitude, thereby determining the magnitude of gpPCj,k and whether it projects positively or negatively along the gridpoint phase.

Remember that the magnitude of the index is simply the sum of the individual gridpoint contributions, and Eqs. (5) and (6) showed that a gridpoint vector will contribute maximally to the index vector when they have the same phase. The result is that, to the extent that the index is capturing a coherent structure associated with the MJO, the index is most likely to reflect anomalous conditions occurring in locations whose gridpoint vectors have a phase similar to that of the index. These locations can be objectively identified using the gridpoint phase. In other words, the real usefulness of the gridpoint phase is that it provides an objective estimate, for each geographical location, of the index phase where the largest MJO-related anomalies in a given field are likely to occur. When the index phase equals the gridpoint phase at a given location, the largest positive MJO-related anomalies that occur at that location are likely to be observed. When the index phase is 180° of phase away from the gridpoint phase at a given location, the largest negative MJO-related anomalies that occur at that location are likely to be observed.

Figure 6 shows phase composites of each field (shading) included in the RMM and FMO indices as well as the corresponding gridpoint phase (black markers). Remember, the gridpoint phase does not identify the longitude where the maximum anomalies in a field occur at a given index phase but instead identifies the index phase where the maximum anomalies in a field occur at a given location. In regions where the gridpoint contribution of a field is high (Fig. 3), the gridpoint phase corresponds fairly well with the index phase where the largest positive anomalies are observed at each longitude. The largest negative anomalies at each longitude tend to occur when the index phase is approximately 180° of phase away from the gridpoint phase. This will be shown in more detail in the next section. The gridpoint phase tends to perform poorly in locations where the gridpoint contribution of a field is low (e.g., OLR in the Western Hemisphere), reflecting the low amount of variance in a field explained by the MJO at those locations. This should be kept in mind when applying compositing techniques introduced in the next section, as they are based on the gridpoint phase. There is a tendency for the gridpoint phase of any field to increase eastward, consistent with the eastward propagation of the MJO being represented by counterclockwise rotation in PC1–PC2 phase space. As the MJO propagates eastward, associated anomalies project onto grid points with increasing gridpoint phase, driving the counterclockwise rotation of the index in PC1–PC2 phase space.

Fig. 6.

Phase composites of each field (shading) included in the RMM and FMO indices as well as the corresponding gridpoint phase (black markers). Days when the magnitude of the respective index did not exceed a value of 1 were excluded. Latitudinally averaged anomalies of each field were binned by the phase of the respective index, with bins spanning 30° calculated every 15°.

Fig. 6.

Phase composites of each field (shading) included in the RMM and FMO indices as well as the corresponding gridpoint phase (black markers). Days when the magnitude of the respective index did not exceed a value of 1 were excluded. Latitudinally averaged anomalies of each field were binned by the phase of the respective index, with bins spanning 30° calculated every 15°.

As a final example of how the gridpoint phase works, consider the gridpoint phase of OLR at 80°E in the RMM index. Figure 2 shows that at 80°E, the magnitude of the OLR component of CEOF1 is zero, while that of CEOF2 is positive. Positive OLR anomalies projected onto these CEOF structures will result in PC1 having a value of zero and PC2 having a positive value, producing a gridpoint vector with a phase of 90°, the gridpoint phase. Note that this corresponds to the transition between phases 6 and 7 of the RMM index, where the most enhanced OLR at 80°E is observed [see Fig. 8 in Wheeler and Hendon (2004)]. Negative OLR anomalies projected onto these CEOF structures will result in PC1 having a value of zero and PC2 having a negative value, producing a gridpoint vector with a phase of 270°. Note that this corresponds to the transition between phases 2 and 3 of the RMM index, where the most suppressed OLR at 80°E is observed [see Fig. 8 in Wheeler and Hendon (2004)].

It has been shown here that the gridpoint phase can objectively identify the index phase where the largest MJO-related anomalies in a given field are likely to be observed at a given location. Application of the gridpoint phase appears to be limited to locations where the MJO explains a sufficient amount of variance in a given field. In addition, it is worth noting that the structure of noncanonical MJO events may not be well represented by the EOFs/CEOFs used in a given index (Kessler 2001; Roundy 2014), and therefore the applicability of the gridpoint phase to noncanonical MJO events may be limited. As will be shown in the next section, the gridpoint phase can be leveraged to assess the geographical variability of anomalous conditions associated with the MJO throughout its life cycle.

c. Relative phase

Transition of the index through 360° of phase in PC1–PC2 phase space represents a full life cycle of the MJO. The gridpoint phase objectively identifies, for each geographical location, the index phase where the largest MJO-related anomalies in a given field are likely to be observed. The separation in phase space between the gridpoint phase and the index phase indicates how far away, in phase space, the likely occurrence of such anomalous conditions is for a given location. This distance in phase space can be viewed as a portion of a full 360° MJO life cycle, with 10° of phase corresponding to about 1 day in an MJO life cycle of typical duration (~40 days). The separation between the index phase and gridpoint phase, which indicates the stage of the MJO life cycle that is most likely to be occurring at each location, is given by

 
formula
 
formula

Figure 1c illustrates the relative phase for a location with a gridpoint phase of 90° (e.g., OLR at 70°E in the RMM index). As shown in the previous section, the largest positive MJO-related anomalies that occur at that location are likely to be observed when this index phase equals the gridpoint phase. This corresponds to a relative phase of 0°. The largest negative MJO-related anomalies that occur at that location are likely to be observed when this index phase is 180° of phase away from the gridpoint phase. This corresponds to a relative phase of ±180°. Given the counterclockwise rotation of the index vector when the MJO is present, negative relative phases represent a transition from negative anomalies to positive anomalies at a given location. Similarly, positive relative phases represent a transition from positive anomalies to negative anomalies at a given location.

The relative phase diagnostic is a particularly useful tool, as it allows geographically disparate locations to be objectively compared in the context of an MJO life cycle. The same stage in the MJO life cycle of a given field can be viewed at different geographical locations simply by comparing them at the same relative phase. Compositing techniques based on this diagnostic will be briefly demonstrated here and are applied to the MSE and moisture budgets of the MJO in Part II. In the following figures, fields are composited as a function of relative phase for both the RMM and FMO indices. The relative phase was computed for each longitude at each time. Fields were binned by relative phase, with bins spanning 60° of relative phase calculated every 30° of relative phase. Days when the magnitude of the respective index did not exceed a value of 1 were excluded. As the EOF structures of the RMM and FMO indices are calculated using 2.5° data, the relative phase for TRMM and ERA-Interim data was calculated using a linearly interpolated gridpoint phase.

Figure 7 illustrates the relationship between traditional phase composites and relative phase composites. Composite latitudinally averaged (15°N–15°S) NOAA OLR anomalies (shading) are shown as a function of phase of the FMO index (Fig. 7a) and relative phase of OLR for the FMO index (Fig. 7b). The boldface number adjacent to the brackets in Fig. 7a indicates the corresponding RMM phase as defined by Wheeler and Hendon (2004). The block arrows in Fig. 7a indicate how the data are “shifted” in the transition from a traditional phase composite to a relative phase composite. The gridpoint phase (solid black line) corresponds fairly well with the index phase of maximum positive OLR anomalies at each longitude. The physical meaning of composites made using the relative phase, such as Fig. 7b, will now be discussed.

Fig. 7.

Composite of latitudinally averaged (15°N–15°S) NOAA OLR anomalies (shading) as a function of (a) phase of the FMO index and (b) relative phase of OLR in the FMO index. Days when the magnitude of the FMO index did not exceed a value of 1 were excluded. Latitudinally averaged OLR anomalies were binned by (a) phase and (b) relative phase, with bins spanning 60° calculated every 30°. The solid black line is the gridpoint phase of OLR in the FMO index. The boldface number adjacent to the brackets in (a) indicates the corresponding RMM phase as defined by Wheeler and Hendon (2004). The block arrows indicate the data are “shifted” in the transition from (a) to (b).

Fig. 7.

Composite of latitudinally averaged (15°N–15°S) NOAA OLR anomalies (shading) as a function of (a) phase of the FMO index and (b) relative phase of OLR in the FMO index. Days when the magnitude of the FMO index did not exceed a value of 1 were excluded. Latitudinally averaged OLR anomalies were binned by (a) phase and (b) relative phase, with bins spanning 60° calculated every 30°. The solid black line is the gridpoint phase of OLR in the FMO index. The boldface number adjacent to the brackets in (a) indicates the corresponding RMM phase as defined by Wheeler and Hendon (2004). The block arrows indicate the data are “shifted” in the transition from (a) to (b).

Figure 8 shows composite latitudinally averaged (15°N–15°S) NOAA OLR anomalies and ERA-Interim U850 anomalies as a function of longitude and relative phase of OLR for the FMO index. Note that in order to present negative OLR anomalies in the center of Fig. 8, the vertical axis has been shifted relative to Fig. 7. As expected, the largest positive anomalies occur near a relative phase of 0°, the largest negative anomalies occur near ±180°, and near-zero anomalies occur around 90° and −90°, demonstrating the effectiveness of the compositing technique. Moving downward from the top of the figure corresponds to a transition from suppressed convection to enhanced convection and back to suppressed convection, showing a full composite life cycle of the MJO from the perspective of OLR. The magnitudes of OLR anomalies are largest over the Indian Ocean and Maritime Continent, then decrease moving eastward into the Pacific Ocean. This is similar to where previous studies (Wheeler and Hendon 2004; Kiladis et al. 2005) have noted high-amplitude MJO-related convection. A major benefit of this compositing technique is that it allows for a clear assessment of geographical changes in the relationship between various processes throughout the MJO life cycle. Figure 8 shows that, over the Indian Ocean and Maritime Continent, U850 anomalies transition from easterly to westerly about 30° of phase prior to the largest negative OLR anomalies. This corresponds to about 3 days given the typical MJO duration and is consistent with observations by Kiladis et al. (2005). U850 anomalies are approximately out of phase with OLR anomalies east of the date line, which is consistent with previous studies (Wheeler and Hendon 2004; Zhang 2005), but consideration should be given to the low amount of OLR variance explained by the MJO east of the date line when considering these results. It is important to remember that Fig. 8 has not been filtered to intraseasonal time scales, demonstrating the effectiveness of this compositing technique in isolating variability associated with the MJO. This is a major advantage when trying to analyze temporally discontinuous data, to which bandpass filtering could not be applied, in context of the MJO.

Fig. 8.

Composite of latitudinally averaged (15°N–15°S) NOAA OLR anomalies (shading) and ERA-Interim U850 anomalies (arrows) as a function of relative phase of OLR in the FMO index. A reference arrow is provided in the bottom-right corner. The relative phase was computed for each longitude at each time. Days when the magnitude of the FMO index did not exceed a value of 1 were excluded. Latitudinally averaged OLR and U850 were binned by relative phase, with bins spanning 60° calculated every 30°. The vertical dashed lines indicate the approximate locations of Africa and the Maritime Continent.

Fig. 8.

Composite of latitudinally averaged (15°N–15°S) NOAA OLR anomalies (shading) and ERA-Interim U850 anomalies (arrows) as a function of relative phase of OLR in the FMO index. A reference arrow is provided in the bottom-right corner. The relative phase was computed for each longitude at each time. Days when the magnitude of the FMO index did not exceed a value of 1 were excluded. Latitudinally averaged OLR and U850 were binned by relative phase, with bins spanning 60° calculated every 30°. The vertical dashed lines indicate the approximate locations of Africa and the Maritime Continent.

A unique advantage of the relative phase is that it allows processes occurring at different geographical locations to be easily compared at the same stage of the MJO life cycle (e.g., mature phase). Figure 9 shows composite TRMM 3B42 dataset precipitation anomalies at each location at a relative phase of OLR of ±180° in the FMO index. In other words, this composite shows the precipitation anomalies that occur at each location when the largest negative OLR anomalies (latitudinally averaged) occur at their respective longitudes. This corresponds to halfway down the vertical axis of Fig. 8. Figure 9 succinctly illustrates the geographical distribution of precipitation enhancement during the mature phase of the MJO. Precipitation anomalies are largest equatorward of 10° in the eastern Indian Ocean and the region surrounding the Maritime Continent, as well as the South Pacific convergence zone (SPCZ). Keep in mind that peak precipitation anomalies at a specific geographical location may not occur at the same stage of the MJO life cycle as the largest latitudinally averaged OLR anomalies. For example, precipitation over the Maritime Continent, which shows little enhancement during this period, peaks earlier during the transition from suppressed to enhanced convection (not shown), though the magnitude of these anomalies is much less than those of surrounding oceanic areas. The distribution of precipitation anomalies, and their lower magnitude over the Maritime Continent, are consistent with variance maps of TRMM precipitation presented in Sobel et al. (2010). Interestingly, precipitation over the ITCZ is reduced at the same time that precipitation in the SPCZ in enhanced.

Fig. 9.

Composite TRMM 3B42 precipitation at ±180° relative phase of OLR for the FMO index. This corresponds to the period of largest negative OLR anomalies (latitudinally averaged from 15°N to 15°S) at each location. The relative phase was computed for each longitude at each time. Days when the magnitude of the FMO index did not exceed a value of 1 were excluded. Precipitation was binned by relative phase, with the bin spanning 60°.

Fig. 9.

Composite TRMM 3B42 precipitation at ±180° relative phase of OLR for the FMO index. This corresponds to the period of largest negative OLR anomalies (latitudinally averaged from 15°N to 15°S) at each location. The relative phase was computed for each longitude at each time. Days when the magnitude of the FMO index did not exceed a value of 1 were excluded. Precipitation was binned by relative phase, with the bin spanning 60°.

Another unique advantage of this compositing technique is that the MJO life cycle can be viewed from the perspective of any variable included in the CEOF structure. Figure 10 shows composite latitudinally averaged (15°N–15°S) U850 total fields (shading) and anomalies from climatology (contours) as a function of longitude and relative phase of U850 for the RMM index. At a relative phase of 0°, which corresponds with the strongest westerly U850 anomalies, total U850 is weak and westerly over the Indian Ocean and western Maritime Continent, while weak easterlies prevail east of the Maritime Continent. At a relative phase of ±180°, which corresponds with the strongest easterly U850 anomalies, total U850 is weak and easterly over the Indian Ocean and western Maritime Continent, while strong easterlies prevail east of the Maritime Continent. While maximum westerly wind anomalies are relatively uniform from the central Indian Ocean to near the date line, their superposition with mean state winds results in total winds that vary from about 2 m s−1 westerly over the central Indian Ocean to about 3 m s−1 easterly near the date line. As will be shown in Part II, this superposition of intraseasonal wind anomalies and mean state winds plays an important role in determining how processes such as surface latent heat flux interact with moisture anomalies to help destabilize the MJO.

Fig. 10.

Composite of latitudinally averaged (15°N–15°S) ERA-Interim U850 anomalies (contours) and unfiltered U850 (shading) as a function of relative phase of U850 for the RMM index, from 1979 to 2012. Solid (dashed) contours correspond to positive (negative) anomalies every 0.6 m s−1 beginning at 1.2 m s−1. The relative phase was computed for each longitude at each time. Days when the magnitude of the RMM index did not exceed a value of 1 were excluded. Latitudinally averaged U850 was binned by relative phase, with bins spanning 60° calculated every 30°. The vertical dashed lines indicate the approximate locations of Africa and the Maritime Continent.

Fig. 10.

Composite of latitudinally averaged (15°N–15°S) ERA-Interim U850 anomalies (contours) and unfiltered U850 (shading) as a function of relative phase of U850 for the RMM index, from 1979 to 2012. Solid (dashed) contours correspond to positive (negative) anomalies every 0.6 m s−1 beginning at 1.2 m s−1. The relative phase was computed for each longitude at each time. Days when the magnitude of the RMM index did not exceed a value of 1 were excluded. Latitudinally averaged U850 was binned by relative phase, with bins spanning 60° calculated every 30°. The vertical dashed lines indicate the approximate locations of Africa and the Maritime Continent.

As demonstrated above, the use of relative phase in composite analysis has many unique benefits. It allows the life cycle of the MJO, and its relationship to various processes, to be viewed across a large geographical range within single figure. As the relative phase is applicable at any stage of the MJO life cycle, compositing is no longer required to be based on periods where some variable (e.g., OLR) has a maximum or minimum value. This allows a high signal-to-noise ratio, as well as a large sample size, to be maintained throughout the MJO life cycle. For example, approximately the same number of days is included in each stage of the life cycle presented in Figs. 8 and 10, all of which meet the criteria of the FMO index exceeding a value of 1. Use of this technique with indices such as the RMM or FMO effectively isolates variability associated with the MJO, even in unfiltered data. This is a large advantage when analyzing data that are temporally or spatially discontinuous and to which traditional filtering cannot be applied. Worth noting is that information regarding the propagation of the MJO has been lost—a deficiency of this compositing technique.

4. Discussion

When assessing the state of the MJO, EOF-/CEOF-based indices (e.g., the RMM index) offer many advantages over indices derived from wavenumber–frequency filtered fields: they are often available in real time, can be designed to assess the large-scale multivariate structure of the MJO, and their use in composite analysis mitigates the negative effects that varying propagation speeds and lifetimes introduce when comparing multiple events. Unfortunately, an EOF-/CEOF-based MJO index can appear to be a “black box” to the unfamiliar user. The index magnitude and phase (i.e., direction the index vector points in PC1–PC2 phase space) provide estimates of the strength and location of MJO activity respectively, but those estimates can sometimes appear to be at odds with observations and other MJO indices. While such indices are powerful tools in MJO research, a black-box approach limits the physical insight that can be gained from their application. The utility of EOF-/CEOF-based indices has been restricted by a lack of understanding of their basic characteristics, difficulty in relating their value to observations, and their failure to provide geographically specific information. This study has introduced novel objective diagnostics that specifically address each of these issues.

It has been shown that the field and gridpoint contribution can be used to assess basic characteristics of an index, such as which variables and geographical locations influence the index most readily. It has also been shown that the field and gridpoint contribution can directly relate observations to the value of the index on a given day, providing users with insight to what is driving the index, pertinent features the index may be missing, and why an index may provide an assessment of the MJO that may differ from other indices. The gridpoint phase, which is calculated from the ratio of the first two EOFs/CEOFs, objectively identifies, for each geographical location, the index phase where the largest MJO-related anomalies in a given field are likely to be observed. Composite analysis that implements the gridpoint phase allows the life cycle of the MJO to be viewed from the perspective of any variable included in the CEOF and allows various processes to be assessed at the same stage of the MJO life cycle across a large geographical region. The effectiveness of this compositing technique in isolating variability associated with the MJO has been demonstrated and its applicability to temporally and spatially discontinuous datasets highlighted. Specific examples, using both the RMM and FMO indices, have been provided for each diagnostic introduced in this study.

No single index can address the needs of the full range of user groups involved in MJO research. The diagnostics introduced in this study are applicable to any EOF-/CEOF-based index, allowing different user groups to fully utilize information provided by the index that best serves their needs. These diagnostics contribute to a deeper understanding of commonly used indices and provide new utility for these indices that make them more powerful tools for MJO research. As with any research tool, an honest assessment of the limitations and weaknesses of the techniques introduced here must be combined with prudent application if true insight is to be gained from their use. Remember that EOFs/CEOFs capture a canonical MJO structure and are limited in their ability to represent variations in zonal structure and spatial scale that may occur in MJO events (Kessler 2001; Roundy 2014). The diagnostics introduced here are only useful if the EOF/CEOF to which they are applied successfully isolates and represents the variability of interest. These diagnostics are most appropriately applied to EOFs/CEOFs where the fields of interest play an influential role in determining the index magnitude and phase. For example, if the convective signal of the MJO is of primary interest, use of an index where OLR plays a dominant role, such as the FMO, may provide better results than an index such as the RMM, where the wind components dominate. The fidelity of the gridpoint phase in representing a real physical relation to the MJO should be questioned in regions where the EOF/CEOF structures have little magnitude as well as during periods when the magnitude of the index is small. Compositing methods that incorporate the gridpoint phase should not be applied in these regions or during these periods.

Finally, this study has shown that the RMM and FMO indices, combined with the diagnostics introduced here, can be used successfully to assess the geographical variability of processes related to the MJO. These indices and diagnostics will be used in Part II to investigate the geographical variability of the MSE and moisture budgets of the MJO.

Acknowledgments

Special thanks to Paul Roundy, Carl Schreck, George Kiladis, Juliana Dias, and an anonymous reviewer, whose insightful comments helped improve this manuscript. Interpolated OLR and NCEP–DOE Reanalysis 2 data were provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, from their website at http://www.esrl.noaa.gov/psd/. The ERA-Interim data were obtained from the ECMWF data server. This work was supported by the Climate and Large-Scale Dynamics Program of the National Science Foundation under Grant AGS-1062161 and by the NOAA ESS Program under Contract NA13OAR4310163. The statements, findings, conclusions, and recommendations do not necessarily reflect the views of NOAA or NSF.

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