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Article Type:
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The MJO as a Dispersive, Convectively Coupled Moisture Wave: Theory and Observations

Ángel F. Adames Department of Atmospheric Sciences, University of Washington, Seattle, Washington

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Daehyun Kim Department of Atmospheric Sciences, University of Washington, Seattle, Washington

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Print Publication:
01 Mar 2016
DOI:
https://doi.org/10.1175/JAS-D-15-0170.1
Page(s):
913–941
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Abstract

A linear wave theory for the Madden–Julian oscillation (MJO), previously developed by Sobel and Maloney, is extended upon in this study. In this treatment, column moisture is the only prognostic variable and the horizontal wind is diagnosed as the forced Kelvin and Rossby wave responses to an equatorial heat source/sink. Unlike the original framework, the meridional and vertical structure of the basic equations is treated explicitly, and values of several key model parameters are adjusted, based on observations. A dispersion relation is derived that adequately describes the MJO’s signal in the wavenumber–frequency spectrum and defines the MJO as a dispersive equatorial moist wave with a westward group velocity. On the basis of linear regression analysis of satellite and reanalysis data, it is estimated that the MJO’s group velocity is ~40% as large as its phase speed. This dispersion is the result of the anomalous winds in the wave modulating the mean distribution of moisture such that the moisture anomaly propagates eastward while wave energy propagates westward. The moist wave grows through feedbacks involving moisture, clouds, and radiation and is damped by the advection of moisture associated with the Rossby wave. Additionally, a zonal wavenumber dependence is found in cloud–radiation feedbacks that cause growth to be strongest at planetary scales. These results suggest that this wavenumber dependence arises from the nonlocal nature of cloud–radiation feedbacks; that is, anomalous convection spreads upper-level clouds and reduces radiative cooling over an extensive area surrounding the anomalous precipitation.

Corresponding author address: Ángel F. Adames, Department of Atmospheric Sciences, University of Washington, 408 ATG Building Box 351640, Seattle, WA 98195-1640. E-mail: angelf88@atmos.washington.edu

Abstract

A linear wave theory for the Madden–Julian oscillation (MJO), previously developed by Sobel and Maloney, is extended upon in this study. In this treatment, column moisture is the only prognostic variable and the horizontal wind is diagnosed as the forced Kelvin and Rossby wave responses to an equatorial heat source/sink. Unlike the original framework, the meridional and vertical structure of the basic equations is treated explicitly, and values of several key model parameters are adjusted, based on observations. A dispersion relation is derived that adequately describes the MJO’s signal in the wavenumber–frequency spectrum and defines the MJO as a dispersive equatorial moist wave with a westward group velocity. On the basis of linear regression analysis of satellite and reanalysis data, it is estimated that the MJO’s group velocity is ~40% as large as its phase speed. This dispersion is the result of the anomalous winds in the wave modulating the mean distribution of moisture such that the moisture anomaly propagates eastward while wave energy propagates westward. The moist wave grows through feedbacks involving moisture, clouds, and radiation and is damped by the advection of moisture associated with the Rossby wave. Additionally, a zonal wavenumber dependence is found in cloud–radiation feedbacks that cause growth to be strongest at planetary scales. These results suggest that this wavenumber dependence arises from the nonlocal nature of cloud–radiation feedbacks; that is, anomalous convection spreads upper-level clouds and reduces radiative cooling over an extensive area surrounding the anomalous precipitation.

Corresponding author address: Ángel F. Adames, Department of Atmospheric Sciences, University of Washington, 408 ATG Building Box 351640, Seattle, WA 98195-1640. E-mail: angelf88@atmos.washington.edu
Keywords: Circulation/ Dynamics; Convection; Dispersion; Madden-Julian oscillation; Physical Meteorology and Climatology; Moisture/moisture budget

1. Introduction

The Madden–Julian oscillation [MJO, after Madden and Julian (1971, 1972)] is the dominant mode of tropical intraseasonal variability. It is a distinct type of convectively coupled tropical wave that is often characterized by (i) a zonal scale of several thousand kilometers, (ii) a phase speed of about 5 m s−1, and (iii) eastward propagation. Although its influence on the global weather–climate system is well documented (Zhang 2005; Lau and Waliser 2011; Zhang 2013) our understanding of the MJO remains incomplete (Wang 2011), and general circulation models (GCMs) simulate the MJO with an unsatisfactory skill (Lin et al. 2006; Kim et al. 2009; Hung et al. 2013). Recent predictability studies have shown that there is a gap of 20–25 days between the potential and practical predictability of the MJO (H.-M. Kim et al. 2014; Neena et al. 2014), suggesting that improving MJO simulation would result in a more accurate forecast of it as well as other phenomena that are affected by it. However, in order to improve MJO simulation, a better understanding of the fundamental processes driving its dynamics is required.

There is a growing body of theoretical work in which the MJO is considered to be a “moisture mode” (Neelin and Yu 1994; Raymond 2000b, 2001; Sobel et al. 2001; Sobel and Gildor 2003; Fuchs and Raymond 2002, 2005, 2007; Sugiyama 2009a,b; Majda and Stechmann 2009; Sobel and Maloney 2012, 2013, hereafter SM refers to the two papers collectively; Sukhatme 2014). In the moisture-mode paradigm, humidity anomalies are of first-order importance, convection is assumed to be tightly coupled to column-integrated water vapor (Bretherton et al. 2004), and temporal and horizontal variations in temperature are often neglected, which is referred to as the weak temperature gradient (WTG) approximation (Charney 1963; Sobel et al. 2001). Under these approximations, the processes that determine the growth and propagation of the large-scale moisture envelope also explain those of the MJO.

The moisture-mode theoretical framework has been actively used to study the dynamics of the MJO in reanalysis (Benedict and Randall 2007; Kiranmayi and Maloney 2011; Hannah and Maloney 2014; among others) and GCM simulations (Benedict and Randall 2009; Maloney et al. 2010; Hsu and Li 2012; Hsu et al. 2014; Pritchard and Bretherton 2014; among others). Results of these studies have suggested the following:

  1. Modulation of longwave cooling by cloud and moisture anomalies (Hu and Randall 1994, 1995) plays an important role in destabilizing the MJO (Andersen and Kuang 2012; Arnold et al. 2013; Sobel et al. 2014; Crueger and Stevens 2015). As pointed out by Raymond (2001), the suppression of outgoing longwave radiation by increased water vapor and cloud cover results in a heating anomaly that enhances vertical motion, which in turn strengthens the anomalous vertical advection of moisture.

  2. Eastward propagation of the MJO is driven by horizontal advection associated with anomalous lower-tropospheric winds and vertical advection associated with frictionally driven boundary layer convergence (Maloney 2009; Zhao et al. 2013; Andersen and Kuang 2012; Wu and Deng 2013; D. Kim et al. 2014; Chikira 2014; Zhu and Hendon 2015; Adames and Wallace 2015).

Based on the results of the modeling and observational studies, SM developed a 1D moisture-mode model to represent the MJO. The model uses zonally varying column-integrated water vapor as the single prognostic variable to represent the processes that modulate the large-scale, low-frequency moisture variability. Unlike other moisture-mode theories, SM diagnose the horizontal wind field instantaneously from the precipitation field by assuming that the low-frequency circulation anomaly is in quasi equilibrium with anomalous heating, which further simplifies the system.

The importance of the work of SM lies in their derivation of a linear wave equation for the MJO and their representation of complex dynamic and thermodynamic processes by means of simple relationships involving column-integrated water vapor. Furthermore, the linearity of the theory makes it possible to analytically estimate the contribution of these processes to the growth and propagation of the MJO. However, with their representation of the moist processes that lead to eastward propagation, they obtained the result that the role of the anomalous flow is negligibly small compared to the zonal advection of the moisture anomaly by the background wind field, which is inconsistent with observations (D. Kim et al. 2014; Tseng et al. 2015; Adames and Wallace 2015).

In this study we will build upon the model of SM and make several critical revisions based on observations, which will enable us to obtain a more robust theoretical representation of the MJO. The modified framework presented in this study explains the most salient characteristics of the MJO, including

  1. a dispersion relationship for the wave and

  2. a potential mechanism for horizontal scale selection.

The paper is structured as follows. Section 2 discusses the data and methods of analysis employed in this study. In section 3, a dispersion relation is derived that can account for the propagation characteristics and zonal scale selection of the MJO. Section 4 shows observational evidence that the MJO is a dispersive wave. Section 5 relates the derived linear moist wave to the observed MJO, seeking to explain and interpret its dispersion relation. A mechanism that can lead to growth in the MJO is discussed in section 6. The MJO’s signature in the wavenumber–frequency spectrum is discussed in section 7. The implications of the results presented here for the definition of the MJO and the interpretation of its life cycle are discussed in section 8. Concluding remarks are offered in section 9.

2. Data and methods

Four datasets are extensively used to carry out the observational analysis in this study. The first is the 1.5° longitude × 1.5° latitude horizontal resolution, daily pressure level fields from the European Centre for Medium-Range Weather Forecasts (ECMWF) interim reanalysis (ERA-Interim; Dee et al. 2011). Vertical velocity ω and the horizontal wind components u and υ are used as field variables. Outgoing longwave radiation (OLR) data (2.5° × 2.5° horizontal resolution) from NOAA’s polar-orbiting satellites (Liebmann and Smith 1996) are used for the same period as the ERA-Interim data. Daily, 1.0° resolution precipitation rate P from the Global Precipitation Climatology Project (GPCP; Huffman et al. 2001) for the period 1996–2011 is used as an indicator of the region where the strongest convection is occurring. Finally, column-integrated water vapor is obtained from the daily, 0.25° horizontal resolution, combined precipitable water (PW) product from the Special Sensor Microwave Imager (SSM/I) and the Tropical Rainfall Measurement Mission Microwave Imager (TMI) (obtained from Remote Sensing Systems; http://www.remss.com/) for the time period 1998–2011. The temporal tendency in , is calculated by taking the 2-day centered differences in . For the MJO filtering procedure discussed below, the SSM/I–TMI PW anomalies have been interpolated in space in order to remove missing data over the landmasses.

Anomaly fields for the variables mentioned above are calculated by removing the mean and first three harmonics of the annual cycle, based on the 1979–2011 reference period for OLR and ERA-Interim data, 1996–2011 for GPCP rainfall, and 1998–2011 for SSM/I–TMI PW. Additionally, a 101-point Lanczos filter (Duchon 1979) is used to extract the anomalies on the 20–100-day time scale. For anomalies described as “MJO filtered,” the fields are calculated the same way but only the eastward-propagating, zonal wavenumber-1–5 signal is retained, based on the protocol of Hayashi (1981). Following the methods described in Adames and Wallace (2014a), regression maps for each anomaly field are obtained through the equation
eq1
where is the regression pattern, in dimensional units, for a two-dimensional matrix that represents a variable field S, is a standardized time series, and N is the sample size in days. Many of the patterns shown in this study are obtained through a lag-regression analysis upon box-averaged OLR time series based on data for all 12 calendar months. The contour intervals in all the plots are scaled to the approximate value of the 95% confidence interval based on a two-sided Student’s t test. We have verified that the results presented here are consistent and reproducible using various MJO indices such as those discussed in the appendix of Adames and Wallace (2014a).

In section 4 we estimate the phase speeds and group velocities of different MJO-related anomaly fields occurring within 25 days of the reference time (lag 0). These calculations are performed only for MJO-filtered anomalies in order to prevent smaller-scale features such as interactions with the Maritime Continent from affecting these calculations. Choosing extrema that occur within 25 days of the reference time ensures that only the most representative anomalies are used in the calculation. Phase speeds are calculated for each time–longitude section by averaging the MJO-filtered anomaly fields across the longitude intervals 80°–100°, 100°–120°, 120°–140°, and 140°–160°E. The time when a statistically significant extremum occurs is calculated within each interval for each field. Phase speed is then calculated by linear least squares fit for each propagating extremum. If multiple propagating envelopes are found, the phase speed is estimated by averaging all of them.

For the group velocity calculation, we calculate the zonal and temporal position of each local extremum. A local extremum is defined here as a local maximum/minimum occurring within 25 days of the reference time that is statistically significant at the 95% confidence interval. For a local maximum/minimum to be considered, it must be the largest anomaly in space and time within an interval of 10 days. Limiting the calculation to this time interval prevents more than one extremum to be obtained for a single propagating envelope. After all the local extrema are identified, the group velocity is calculated through a linear least squares fit.

For the spectral analysis shown in section 7, we calculate the space–time power spectrum of NOAA OLR through the use of fast Fourier transforms (FFTs), as in Wheeler and Kiladis (1999) and Hendon and Wheeler (2008). We divide the time series of the anomaly fields into 180-day segments that overlap by 90 days, as in Waliser et al. (2009) and Kim et al. (2009). Then, the space–time mean and linear trends are removed by least squares fits, and the ends of the series are tapered to zero through the use of a Hanning window. After tapering, the complex FFTs are computed in longitude and then in time. Finally, the power spectrum is averaged over all segments and over the 15°N–15°S belt. The number of degrees of freedom is calculated to be 133 [2 (amplitude and phase) × 33 (years) × 365 (days)/180 (segment length)]. The signal strength is calculated as , where is the red spectrum, calculated analytically using Eq. (1) of Masunaga (2007), and values above 0.4 are considered to be statistically significant in this study. We will show only the equatorially symmetric component (Yanai and Murakami 1970) in order to facilitate comparison with other studies, though we have verified that the results presented here are insensitive to separation of the anomaly fields into symmetric and antisymmetric components.

3. A linear moist wave theory for the MJO

a. Formulation of the governing equations

In this section we will build upon the model developed by SM in order to construct a dispersion relation that can describe the characteristics of the MJO. We will begin by developing a set of basic equations that can describe convectively coupled equatorial perturbation. A thorough derivation and scaling of the governing equations is shown in the appendix, and the most important variables and definitions used in this section are summarized in Tables 1 and 2. In deriving these equations we have linearized the field variables into time-mean (denoted by an overbar) and perturbation (denoted by a prime) components. The field variables are also truncated meridionally using parabolic cylinder functions (Majda and Shefter 2001; Majda and Khouider 2001; Majda and Stechmann 2009) and vertically using basis functions (Neelin and Zeng 2000; Haertel et al. 2008):
eq2
where and are the mean and anomalous latent energy (specific humidity scaled by the latent heat of vaporization ), respectively; b is the vertical structure function associated with moisture; is the equatorial Rossby radius of deformation, defined as in Gill (1980); and c is the phase speed of dry Kelvin waves. To obtain the simplest wave solution that can resemble the MJO, we will assume that the amplitude of the mean and anomalous q fields decay exponentially with the square of the distance from the equator, and thus their meridional variations can simply be expressed in terms of the zeroth-order parabolic cylinder function :
eq3
Furthermore, the governing equations are vertically integrated in order to simplify them and explicitly represent diabatic heating in terms of observable surface and top of the atmosphere (TOA) variables, as in Kiranmayi and Maloney (2011), Andersen and Kuang (2012), and Sobel et al. (2014), among many others. Variables with angle brackets will denote a mass-weighted vertical integral from 100 to 1000 hPa:
eq4
where hPa and hPa. From here on the vertically integrated basis functions will be shown explicitly and variables with primes and overbars will implicitly incorporate the parabolic cylinder functions .
Table 1.

Basic variables and definitions used for sections 3 and 5.

Table 1.
Table 2.

Summary of variables and definitions used for sections 3 and 5.

Table 2.
The linearized, truncated, vertically integrated equations for momentum, mass continuity, and latent heat (moist static energy) on an equatorial beta (β) plane take the form
e1a
e1b
e1c
e1d
where ε is a Rayleigh damping coefficient, is the vertical structure function associated with a first internal baroclinic mode (a function exhibiting opposite polarities in the upper and lower troposphere; see appendix), is the anomalous geopotential, is the net moistening associated with ascent forced by frictionally driven boundary layer convergence, and is the net moistening associated with the horizontal moisture advection by MJO-related high-frequency eddy activity. The terms , , and (W m−2) are the anomalous precipitation, longwave radiation, and surface latent heat flux, respectively. The normalized gross moist stability is denoted by [Neelin and Held (1987); see also Raymond et al. (2009) for a review on gross moist stability], defined as
e2a
where and are the mean gross dry stability and the gross moisture stratification, respectively, defined as
e2b
e2c
where is the mean dry static energy, assumed constant, and is a structure function associated with deep convective ascent. In this framework, c is expressed as in Sugiyama (2009a):
e3
where is the dry gas constant, is the specific heat of air at constant pressure, and a is the basis function for temperature. For J m−2 and kg m−2, values similar to those used by Neelin and Zeng (2000) and Sugiyama (2009a) (scaled by , = 900 hPa), m s−1, a value similar to those obtained by Bantzer and Wallace (1996) and Milliff and Madden (1996), which yields an equivalent depth value of m.

It is worth noting that Eqs. (1a)–(1c) are the dimensional, modal forms of Eqs. (2.6)–(2.8) in Gill (1980), which correspond to the steady-state wave response of , , and to an equatorial heat source . In this case, however, Eqs. (1a)–(1c) are coupled to a prognostic moisture equation [Eq. (1d)] that determines the evolution of the heat source. Therefore, as in SM, the momentum and mass fields are in quasi equilibrium with the heating at all instances, making Eqs. (1) a two-dimensional version of SM’s governing equation. In the absence of a heat source, Eqs. (1a)–(1c) decouple from Eq. (1d). If ε is also neglected and the temporal tendencies in , , and are included, then Eqs. (1) would yield the free wave solutions of Matsuno (1966).

b. Derivation of a single moisture wave equation

To obtain a simple linear moisture equation that can describe the MJO, some of the terms in Eq. (1d) need to be parameterized in a simple manner. The term in the budget that corresponds to moistening by frictionally induced moisture convergence was shown by Adames and Wallace (2014b) to be associated with the second EOF in the vertical velocity profile, which indicates whether the profile of vertical velocity is shallow or elevated. This second baroclinic mode is in turn associated with the difference in divergence between the boundary layer and the free troposphere, which is coupled to through wave-induced shallow meridional circulations [Figs. 3, 7, and 12 of Adames and Wallace (2014b)], where regions of equatorial easterlies to the east of the strongest convection induces shallow ascent and moistening, while frictional divergence induces low-level descent and drying (see also Benedict and Randall 2007; Liu and Wang 2012; Hsu and Li 2012). Hence, the equation for can be approximated in terms of using the following relation:
e4a
where
e4b
where is the convective adjustment time scale (Manabe and Strickler 1964), assumed to be constant in this study.

In defining , we introduce an effective vertical moisture gradient corresponding to the net moistening due to frictional convergence, recognizing the fact that much of the moisture that is advected upward condenses into cloud liquid water, so that only a fraction of the vertically advected moisture contributes to the moisture tendency. We define as a function of the mean moisture and , a parameter corresponding to the amount would be reduced by wave-induced boundary layer convergence; is dimensionally expressed as the change in per meter per second change in . The term is identical to the frictional convergence term in Eq. (12) of SM except the terms have been expressed differently. Condensation from frictionally driven ascent also modulates the horizontal and vertical structure of diabatic heating, but for the sake of keeping the analysis linear it will be assumed that its contribution is negligible compared to .

As discussed in SM, there is evidence that the low-level zonal wind anomalies in the MJO modulate synoptic-scale high-frequency eddy activity (Maloney and Dickinson 2003; Maloney 2009; Andersen and Kuang 2012), which results in anomalous moistening in regions of easterlies and drying in regions of westerlies. We can approximate this moistening process through the following relation:
e5
where is a parameter that indicates the amount of eddy moistening per meter per second change in and can be thought as an effective zonal moisture gradient imparted by the modulated eddy activity.
The term involving surface latent heat fluxes is expressed as in SM, in terms of the by assuming that fluctuations in the wind field dominate the surface latent heat flux anomalies
e6
where is a bulk formula for latent heat fluxes, whose sign depends on the nature of the mean flow (negative for , positive for ).
The longwave radiative cooling term is expressed in terms of through the following relation:
e7
where r is the greenhouse enhancement parameter [Kim et al. (2015); also known as the cloud–radiative feedback parameter], which describes the amount of suppressed OLR per unit of precipitation arising from the presence of anomalous water vapor and upper-level cloudiness.
With the aforementioned approximations, we can write the latent heat equation as follows:
e8
where is the effective gross moist stability, , and
e9
is the sum of all the moistening processes associated with the zonal wind anomalies. Note that is analogous to the term Eq. (17) of Sobel and Maloney (2013).
We will now proceed to write the terms of the right-hand side of Eq. (8) in terms of and seek a solution in the form of a wave:
e10
where is a constant defining the amplitude of the initial moisture perturbation, k is the zonal wavenumber, and is the angular frequency (a tilde is used to distinguish it from the pressure velocity ω). The anomalous precipitation is expressed as in SM:
e11
In observations, P is an exponential function of (Bretherton et al. 2004; Raymond et al. 2007; Holloway and Neelin 2009) but is approximated as linear here, which is justifiable provided that the q perturbations are small.
Because the heating fields and decay exponentially with distance squared from the equator (arising from ), we can solve Eqs. (1a)–(1c) to obtain expressions for and in the form of Kelvin and n = 1 equatorial Rossby wave responses to , which can be expressed as follows:
e12a
e12b
where is a complex function that describes the horizontal structure of the Matsuno–Gill response to an equatorially trapped heat source (expressed in terms of ) for each field, and the subscripts K and R denote the Kelvin and Rossby wave components, respectively:
e13a
e13b
e13c
where is a dimensional constant, and the length scale L is the distance that free Kelvin waves travel in the presence of Rayleigh damping in the free troposphere . Further details on how these structures are obtained, as well as the solution for can be found in the appendix.
Substituting the aforementioned terms, we can write the latent energy anomaly equation [Eq. (8)] in the following form:
e14
We will complete the meridional truncation of the moisture equation by projecting it onto the zeroth-order parabolic cylinder function , following the notation of Majda and Shefter (2001):
e15
where we will express and in order to make their meridional dependence explicit.

c. The linear moist wave’s dispersion relation

By assuming that has a solution in the form of a zonally propagating wave [Eq. (10)], we obtain the following dispersion relation for the moisture equation projected onto using Eq. (15):
e16
where
e17
is the projected effective gross moist stability. As noted by SM, this dispersion relation is complex and only the terms involving , , and , which arise from the horizontal wind field through Eq. (12) can be real and determine the propagation of the moist wave, while the other terms determine the growth and dissipation. Separating the dispersion relation into its real and imaginary parts yields the following:
e18a
e18b
The real component of the dispersion relation is solely due to the Kelvin and Rossby waves inducing moistening and drying by horizontal and vertical moisture advection and modulating the mean surface latent heat fluxes and high-frequency eddy activity. We define this induced moistening/drying as
e19a
e19b
which have units of per meter and can be thought of as the Kelvin () and Rossby wave () “moisture advection parameters”: the rate at which the anomalous winds per unit of heating increase or decrease the amount of column-integrated moisture. The terms and are weighting functions of value 1 that account for the damping effect of free-tropospheric dissipation on the Kelvin and Rossby wave-related wind fields:
e20a
e20b
The difference between the two weights results from Rossby waves being damped at 3 times the rate in which Kelvin waves are damped, hence the factor of 9 in the denominator of . We also define the phase angle α of the zonal winds in the Kelvin and Rossby waves with respect to :
e21a
e21b
Finally, we have defined as the projected total effective gross moist stability, since advection of mean moisture by the “Sverdrup” component of the anomalous meridional winds acts to destabilize the moist region:
e22
We can obtain the real phase speed and group velocity of the moist wave from Eq. (18a):
e23a
e23b
where and are weighting functions similar to and , which account for the damping effects of free-tropospheric dissipation on the propagation of wave energy in the Kelvin and Rossby waves, respectively:
e24a
e24b

d. The moist wave in the limit

The derived equations for the moist wave’s dispersion in Eqs. (18)–(24) contain many terms that depend on the dissipation length scale L, which shapes the horizontal wind field. The impact of L in the moist wave’s dispersion relation can be further understood after analyzing the wave’s propagation characteristics in the absence of any free-tropospheric dissipation. In this limit the dispersion relation simplifies to
e25a
e25b
where is the combined contribution from Kelvin and Rossby waves to propagation. The horizontal structure of the moist wave and the individual Kelvin and Rossby wave components is shown in Fig. 1a, and the structure of the terms that lead to growth and eastward propagation is shown in Fig. 2. The relative contribution of the shaded terms in Fig. 2 are chosen so that they are qualitatively consistent with the results of Andersen and Kuang (2012), D. Kim et al. (2014), and Adames and Wallace (2015), among others. In the absence of any free-tropospheric damping, the linear moist wave will exhibit the following properties:

  1. The phase speed of the disturbance depends upon its wavelength, the moisture advection parameter , and the convective time scale . Larger values of will result in faster eastward propagation while larger values of k (shorter wavelengths) and result in a slower phase speed.

  2. Only a negative total effective gross moist stability can lead to growth of the wave disturbance. If , then moisture is conserved following the wave since the meridional and vertical advection of moisture balances precipitation.

  3. Because changes polarity poleward of 1, it is the meridional advection of moisture by that reduces poleward of 1, increasing moisture in this region. Thus, meridional moisture advection can be thought of as widening the anomalous moist region, as suggested by Adames and Wallace (2015).

  4. The wave is dispersive. In the limit , assuming , the wave exhibits a group velocity that is equal and opposite to its phase speed, .

  5. Moistening by the anomalous winds is dominated by the Rossby wave contribution, as seen in Figs. 2a, 2e, and 2f. This dominance is due to the in the Rossby waves being 3 times stronger than in the Kelvin waves and being completely due to Rossby waves.

Fig. 1.
Fig. 1.

(left) Horizontal structure of a zonal wavenumber-2 linear moist wave (as derived in section 3), (center) its Rossby wave contribution, and (right) Kelvin wave contribution for values of the dissipation length scale L of (a) , (b) 13 200, and (c) 5000 km. Here is shown as the shaded field, is contoured, and the horizontal wind field is shown as arrows. Contour interval 0.75 m s−1. The largest arrows correspond to wind anomalies of ~1.6 m s−1. Magnitudes correspond to an initial moisture perturbation of J m−2 and . The red circle depicts the region where is a maximum. The wind and ϕ anomalies are scaled to a surface value of .

Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0170.1

Fig. 2.
Fig. 2.

Propagation of the linear zonal wavenumber-2 moist wave solution for . (a) Moisture anomalies after using the projection operator [Eq. (15)] (solid line), contribution from to the moisture tendency (dashed line), and the Kelvin (dotted–dashed line) and Rossby (dotted line) wave contributions to the moisture tendency. Horizontal maps of the contribution of (b) moistening by the zonal flow, where , , and J m−3, (c) meridional moisture advection, (d) effective gross moist stability, (e) Kelvin wave, and (f) Rossby wave to the moisture tendency. The term is contoured in all panels. Contour interval 0.25 J m−2. The horizontal wind field is shown as arrows. The largest arrows correspond to wind anomalies of ~1.6 m s−1. The wind and ϕ anomalies are scaled to a surface value of .

Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0170.1

These properties account for some aspects of the observed and simulated MJO. Studies in which the MJO’s background state is modulated (Arnold et al. 2013; Kang et al. 2013; Arnold et al. 2015) have suggested that increasing by increasing the horizontal and vertical moisture gradients results in faster MJO propagation, consistent with bullet (i). Many studies have shown the importance of a negative in destabilizing the MJO (Hu and Randall 1994; Sobel and Gildor 2003; Kim et al. 2011; Benedict et al. 2014), supporting bullet (ii). However, little evidence has been presented showing that the MJO is a dispersive wave with a westward group velocity. The dispersive properties of the MJO are documented in the next section and the relevance of points (iv) and (v) to the observed MJO are discussed in sections 5 and 6.

4. Evidence that the MJO is a dispersive wave

The MJO’s westward group velocity is apparent in time–longitude sections of individual events, without the need for compositing. Figure 3 shows a selection of sections for the 20–100-day-filtered and MJO-filtered [using the Hayashi (1981) method] OLR anomalies. The sequences shown in each column correspond to the three strongest events that occurred in three separate geographical regions, defined as the Indian Ocean (15°N–15°S, 60°–100°E), the Maritime Continent (15°N–15°S, 100°–140°E), and the western Pacific (15°N–15°S, 140°E–180°). In many of these sequences, the waves tend to be arranged in westward-migrating packets. The first wave in the packet is first discernible near or to the east of the Maritime Continent, between 120°E and the date line, and successive waves develop farther and farther toward the west. The behavior is analogous to the eastward dispersion of midlatitude Rossby waves (Chang 1993) and the behavior of dispersive equatorially trapped waves such as the mixed Rossby–gravity and inertio-gravity modes [see Wheeler et al. (2000), their Figs. 12, 16, and 20].

Fig. 3.
Fig. 3.

Time–longitude diagrams of MJO events for 20–100-day-filtered OLR (shaded) and MJO-filtered OLR (contoured). (left) The three strongest events for OLR anomalies averaged over the Indian Ocean sector (15°N–15°S, 60°–100°E), (center) the three strongest events over the Maritime Continent sector (15°N–15°S, 100°–140°E) that do not overlap with Indian Ocean events, and (right) the three strongest events over the western Pacific sector (15°N–15°S, 140°E–180°) that do not overlap with the other two sectors. The reference time (lag day 0) corresponds to the date in the diagram’s title. Contour interval 7.5 W m−2.

Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0170.1

To further elucidate the westward migration of the MJO-related anomalies, lag-regression analysis based on the time series of OLR anomalies for the three aforementioned sectors was performed on OLR, GPCP precipitation , SSM/I–TMI precipitable water , and ERA-Interim 300-hPa vertical velocity . Time–longitude diagrams of the regressed fields are shown in Fig. 4. Westward-migrating wave packets are clearly evident as in Fig. 3. The group velocity is within the range from to m s−1, and all fields exhibit phase speeds within 5 or 6 m s−1. The group velocity to phase speed ratio for these four fields ranges between −0.3 and −0.5. The time–longitude diagrams of Knutson and Weickmann (1987), Hendon and Salby (1994), Wheeler and Kiladis (1999), Roundy and Frank (2004), Kiladis et al. (2005), Straub (2013), and D. Kim et al. (2014) also show a westward migration of the extrema in the MJO-related anomaly fields, similar to that in these 12 panels.

Fig. 4.
Fig. 4.

Time–longitude diagrams of filtered (shaded) and MJO-filtered (contoured) (a) OLR, (b) GPCP precipitation, (c) SSM/I–TMI precipitable water, and (d) ERA-Interim 300-hPa vertical motion. The columns correspond to OLR anomalies that peak in the (left) Indian Ocean sector, (center) Maritime Continent, and (right) west Pacific. The reference time (day 0) corresponds to the time when the MJO-filtered OLR anomalies are a minimum. The gray dashed lines are linear least squares fit estimates of the phase speed and group velocity for each field, and the circles represent a local extremum in the MJO-filtered field. The values of the calculated phase speed and group velocities, along with their uncertainties, are shown in the top-left corner of each panel.

Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0170.1

5. The linear moist wave in relation to MJO structure and propagation

In section 3 we derived a linear wave solution for a linear moist wave whose propagation is determined by the anomalous wind field in the Matsuno–Gill wave response to an equatorial heat source. In the previous section we showed observational evidence that the MJO is characterized by a dispersion relation in which the phase speed is eastward and the group velocity westward, consistent with the linear moist wave solution derived in section 3. In this section, we will compare this model with the observed MJO. To make this comparison, however, a value for the dissipation length scale L and the convective adjustment time scale needs to be estimated.

a. Estimation of the dissipation length scale L in the tropical free troposphere and the convective adjustment time scale

For free-tropospheric motions, L depends on the phase speed of free Kelvin waves and on the dissipation time scale (). The variable is not one that can easily be measured with direct observations. The large uncertainty is reflected in the values ranging from 1 to 20 days that have been used in previous studies (Neelin et al. 1987; Seager 1991; Lee et al. 2009). We can estimate a value of L from Eq. (23), by approximating the relative contribution of Kelvin () and Rossby waves () to propagation, and using the approximated group velocity to phase speed ratio implied by Fig. 4, which is
e26
for an MJO with an effective horizontal wavenumber . To estimate , we perform a spectral analysis in longitude for each field in Fig. 4, within 10°N–10°S and for all days within 25 days of the reference time. The power spectrum is then averaged for all the latitudes and days included and then normalized using the formula . The approximate wavenumber is obtained by summing the zonal wavenumbers, weighting each one by its normalized power , which yields a value of 1.81.
The relative contributions of and to propagation can be inferred from previous observational studies (Kiranmayi and Maloney 2011; Adames and Wallace 2015; among others). These studies have shown that moistening processes related to the zonal wind anomalies are comparable to those from meridional moisture advection. As such, it can be inferred from Eqs. (19a) and (19b) that Rossby waves contribute about 4 times more to the propagation of the moist wave than the Kelvin wave does. Substituting the values of onto Eqs. (20) and (24), and solving Eq. (26) for L, noting that , we obtain
e27
Values of L from to m can be obtained for values of ranging from 2.5 to 6, and our results are insensitive to values of L within this range. Our estimated value is nearly an order of magnitude larger than that assumed by SM, which is more appropriate for the boundary layer rather than for the free troposphere. Our estimated value of L is also approximately equivalent to the distance that the Kelvin wave response travels in Fig. 5 of Matthews (2000), where it propagates ~130° of longitude from the Maritime Continent to the coast of South America. For a free Kelvin wave phase speed of ~50 m s−1, we obtain a dissipation time scale of ~3 days. This time scale is longer than the 1-day time scale obtained by SM, smaller than the 5-day time scale used by Sugiyama (2009b), and within the range of the convective damping time scale for large-scale circulations suggested by Romps (2014).
Another important parameter that needs to be estimated is the convective adjustment time scale , which in turn determines the magnitude of the phase speed, group velocity, and wind-driven damping. The term was estimated to be on the order of 12–16 h by Bretherton et al. (2004), while SM estimated to be on the order of 2.4 days using the following relationship:
e28
where J m−2 is the saturation-column-integrated latent energy, and and mm day−1. We can estimate using the same values as in SM but estimating by averaging the annual mean SSM/I–TMI PW over the warm pool region E and 10°N–10°S, which yields a value of of 13.7 h for a warm pool value of of 51.5 J m−2.

b. An approximate dispersion relation for the observed MJO

We can now obtain an approximate dispersion relation for the moist wave by using the ratio. Once again defining a total moisture advection parameter , and noting that , we can obtain the following:
e29a
e29b
where an approximate weight function for the dissipation of the total anomalous wind field takes the following form:
e29c
and the phase angle of the combined Kelvin and Rossby wave-related zonal winds
e29d
The simplified phase speed and group velocities take the form
e30a
e30b
where is a weighting function similar to of the form
e30c

c. The phase relationship between wind and moisture

To compare whether our moist wave solution, scaled by our estimated value of L, is consistent with the observed MJO, we calculated the zonal phase angle α between and using Eq. (29d). The importance of α in the maintenance and propagation of the linear moist wave is discussed in SM, but it is worth discussing here in relation to the observed MJO. In an atmosphere with no dissipation , Eq. (29d) would tend to infinity and thus , which corresponds to and being in spatial quadrature at the equator, as discussed in section 3d and shown in the top panel of Fig. 1.

As damping is increased, α decreases and the westerly wind anomalies are shifted eastward toward the heat source, as seen in Figs. 1b and 1c and shown in Fig. 5 for the first 10 zonal wavenumbers. This displacement can be interpreted as follows: In the presence of free-tropospheric dissipation, both Rossby and Kelvin wave components influence near the heat source. However, because the Rossby wave is characterized by a stronger zonal wind response [see Eq. (A14a)] and exhibits a stronger shift toward the heat source than the Kelvin wave response does, then it follows that it will dominate near the heat source, as seen in Figs. 1b and 1c.

Fig. 5.
Fig. 5.

Phase angle between and , as obtained from Eq. (29d) for values of L of , 25 000, 13 200, and 5000 km. The black triangle corresponds to α as inferred from SSM/I–TMI precipitable water and ERA-Interim 850 hPa (see text for further details). The 95% confidence interval based on a two-tailed t test is depicted as an error bar.

Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0170.1

An estimate of α for the observed MJO , indicated as a black triangle in Fig. 5, is obtained by calculating the spectral phase angle between ERA-Interim 850 hPa and SSM/I–TMI PW , for regressions based on the three sectors presented in Fig. 4. The phase angle is obtained by calculating the cross power spectrum in longitude between and within 10°N–10°S and for all days within 25 days of the reference time (Fig. 4). The cross spectrum is then averaged for all latitudes and days, and the phase angle is calculated as . An effective value of for the MJO can be estimated by summing the values of θ for each wavenumber, weighting each one by its normalized power , which yields a value of of ~66°. A close correspondence is observed between the value of α from ERA-Interim and SSM/I–TMI and the linear wave solution for L = 13 200 km. Based on the above discussion, this result indicates that westerlies from the Rossby wave response dominate the region of enhanced moisture and convection in the MJO, as was noted by Rui and Wang (1990), Hendon and Salby (1994), Kiladis et al. (2005), Adames and Wallace (2014b), among many others.

d. Propagation characteristics of the moist wave

Dissipation in the free troposphere acts to slow down the moist wave by weakening its horizontal wind field, as seen in Fig. 1. But because of the east/west asymmetry in the wind field arising from the dissipation scale L, the phase speed is reduced much less than the group velocity. This relationship is illustrated graphically in Fig. 6 for a moist wave disturbance with a mean moisture advection parameter m−1 and a convective time scale of 13.7 h, as calculated above. The value of is obtained through Eq. (30), using the inferred values of and L, the mean phase speed estimated from Fig. 4, and an effective zonal wavenumber .

Fig. 6.
Fig. 6.

Plots of (a) , (b) (the sign has been reversed in order to facilitate comparison with the phase speed), and (c) wave damping obtained from Eqs. (29) for a linear moist wave with a moisture advection parameter of 34.7 mm km−2, a convective time scale of 13.7 h, and a dissipation length scale L of values , 25 000, 13 200, and 5000 km. The black triangles correspond to the mean phase speed and group velocity inferred from the linear least squares fit in Fig. 4, and the error bars correspond to the range of uncertainty in the and values.

Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0170.1

For a free troposphere with no dissipation, a moist wave with will be characterized by a phase speed and group velocity of ~8.7 m s−1. Including dissipation with the value of L obtained in section 5a decreases the phase speed to ~5.8 m s−1 while the group velocity is reduced to −2.3 m s−1 (indicated in Fig. 6 by a black triangle). However, for the inferred value of L, the phase speed decreases more quickly with increasing k than the group velocity as a result of α increasing with wavenumber. Thus, the ratio between phase speed and group velocity is largest at the largest scales. It is worth noticing that, for the inferred value of L, the moist wave’s group velocity is ~0 for zonal wavenumber 1, the MJO group velocity inferred by Raymond (2001), Majda and Stechmann (2009), among others, but it is westward for all other zonal wavenumbers.

As discussed by SM, a value of α of less than 90° will lead to damping of the moist wave. This wave-driven damping corresponds to the first term on the right-hand side of Eq. (29b). Its contribution to the total growth rate is shown in the right panel of Fig. 6, from which it is evident that the moist wave would be damped by these wind anomalies at a rate of ~0.07 day−1, or roughly half the rate in which the anomalous wind field induces eastward propagation (0.14 day−1).

To further illustrate the propagation mechanism of the moist wave, Fig. 7 shows the contribution of Kelvin and Rossby waves to propagation. Compared to the case where , the drying anomalies in the Rossby waves (Fig. 7c) are shifted eastward toward the heat source. In comparison, the Kelvin wave response is only weakly affected by the increased free-tropospheric damping. Thus, it is the Rossby waves that dominate the increase in damping and the changes in phase speed and group velocity that we see for the inferred value of L. Observational evidence by Benedict and Randall (2007), Johnson and Ciesielski (2013), and Adames and Wallace (2015) is qualitatively consistent the idea of Rossby waves producing damping within the moist region in the MJO.

Fig. 7.
Fig. 7.

As in Figs. 2a, 2e, and 2f, but for L = 13 200 km. Contour interval 0.25 J m−2. The horizontal wind field is shown as arrows. The largest arrows correspond to wind anomalies of ~1.6 m s−1.

Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0170.1

6. Greenhouse enhancement feedbacks and MJO selection of horizontal scale

Because wind-driven processes can only damp an eastward-propagating moist wave, then destabilization can only occur when is negative. Additionally, wave-driven damping is strongest at the largest scales and, thus, cannot account for the observed scale of the MJO. Thus, both growth and scale selection may arise from a scale dependence in . In previous studies it has generally been assumed that and r and, hence, are scale independent and therefore have no bearing on the MJO’s preferred scale. In this section we will reconsider the validity of this assumption.

The greenhouse enhancement parameter r is obtained from Eq. (7). Using the 20–100-day-filtered OLR and GPCP rainfall anomaly fields at all points in time within 15°N–15°S, 60°E–180° yields a value of r of ~0.17 (top-left panel of Fig. 8), consistent with the values obtained by Peters and Bretherton (2005) and Bretherton et al. (2005). To determine whether r could be zonal wavenumber dependent, we further decompose the 20–100-day-filtered OLR and P anomalies into contributions from each zonal wavenumber, without separating them into eastward- and westward-propagating components. A value of r is then calculated for each individual wavenumber through linear least squares fit. Scatterplots for the density distribution of points for and are shown in the top panel of Fig. 8 for reference. The value of r for all zonal wavenumbers from 1 to 20 is shown in the bottom panel of Fig. 8. It is clear that r decreases as k increases. We can obtain an empirical relation for the k dependence on r through least squares fit, which yields the following relation:
e31
where is the value of the greenhouse enhancement parameter for zonal wavenumber zero and km is a length scale that characterizes the changes of r with k. The above fit has the advantage that r asymptotically approaches zero for large k and has a real value for . Similar results were obtained for daily anomalies without filtering. A similar analysis was performed to seek a wavenumber dependence on with ERA-Interim data, but no clear relationship emerged.
Fig. 8.
Fig. 8.

(a) Scatterplot of 20–100-day-filtered precipitation (W m−2) against OLR anomalies. The shaded field in the scatterplot corresponds to the base-10 logarithm of the amount of points located within 2 W m−2 × 2 W m−2 bins. The best-fit linear regression is depicted as a dashed line, where the slope corresponds to the cloud–radiation feedback parameter r. (b),(c) As in (a), but for 100–20 filtered anomalies decomposed into zonal wavenumbers 1 and 4, respectively. (d) Cloud–radiation feedback parameter r as a function of zonal wavenumber k. The best-fit regression is depicted as a dashed line. Error bars correspond to the 95% confidence interval. For all plots the best-fit regression equation and the correlation coefficient are shown in the bottom-left corner.

Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0170.1

A spatial relationship for and can be obtained by using the inverse Fourier transform of Eq. (31), which has the following form:
e32
The accuracy of the OLR (R) anomalies estimated through Eqs. (31) and (32) was verified by comparing them to NOAA OLR anomalies. A robust fit between the observed OLR and estimated from Eqs. (31) and (32) was seen, with a slope that is close to 1 and a correlation coefficient of ~0.9 (not shown).

To elucidate the physical mechanism that leads to the observed wavenumber dependency in r, Fig. 9a shows an example of the OLR response to a point region of precipitation, as obtained from Eq. (32). The OLR response has a shape similar to a Gaussian, with the anomaly decaying to half its original magnitude at a distance from the region of precipitation. This shape is likely the result of longwave heating by upper-level anvil clouds spreading horizontally away from the precipitating region, possibly enhanced by the environmental vertical wind shear (Ackerman et al. 1988; Lin and Mapes 2004). For several evenly spaced sources of precipitation (Fig. 9b), the OLR response is larger in zonal extent but also more than twice the amplitude. This is likely a result of cirrus clouds covering a greater fraction of the sky and forming thicker anvils in regions of aggregated, organized convection, leading to an enhanced greenhouse effect [see Fig. 13 of Ackerman et al. (1988)].

Fig. 9.
Fig. 9.

(a) OLR response (dashed line) to a point source of precipitation (solid line) at 180°, as estimated from Eq. (32). (b) As in (a), but for three point sources of precipitation located at 178°, 180°, and 182°. (c) OLR response to a Gaussian distribution of anomalous precipitation. The dotted–dashed line in this panel corresponds to . The OLR field has been multiplied by a factor of 100 in (a) and (b), and a factor of 7 in (c) in order to facilitate comparison.

Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0170.1

Figure 9c shows the OLR response for a Gaussian distribution in anomalous precipitation. Because the magnitude of precipitation decays more quickly with distance from the center of convection than the OLR anomalies diagnosed through Eq. (32) do, then regions of weaker precipitation within a convective complex exhibit a larger r, consistent with results from Kim et al. (2015). Their study also found, using 29 climate model simulations, that the greenhouse enhancement parameter in the weak precipitation regime has a robust statistical relationship with the MJO simulation capability of the models.

The profile of with values of r as determined by Eq. (31) is shown in Fig. 10 together with an for a constant value of r of 0.17. With the profile obtained from a wavenumber-dependent r, zonal wavenumber 1 grows at nearly twice the rate of zonal wavenumber 6 and nearly 5 times the rate of zonal wavenumber 10. Thus, a wavenumber-dependent profile of favors the growth of the largest-scale moist waves. It is worth noting that a wavenumber-dependent value of r causes to vary with k, but these variations are small.

Fig. 10.
Fig. 10.

Total gross moist stability , as obtained from Eq. (22), as a function of zonal wavenumber k for r determined from Eq. (31) (solid line) and (dashed). Other values used in the calculation of are shown in Tables 1 and 2.

Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0170.1

Finally, the combined contribution of Rossby wave damping and is shown in Fig. 11 as a function of k and . For the lowest values of considered, wind-driven damping is weak and the growth rate favors the development of moist waves with the largest zonal scales. As increases, so does the contribution from wind-driven damping to the growth rate and wavenumber 1 becomes unfavorable. For larger values of (upper-right corner of Fig. 11), roughly corresponding to periods of 20–30 days, wind-driven damping favors the development of smaller-scale moist waves of zonal wavenumber 5 and larger, which might correspond to the “zonally narrow MJO” described by Roundy (2014).

Fig. 11.
Fig. 11.

Growth rate, as obtained from Eq. (29b), as a function of k and for a linear moist wave where L = 13 200 km, h, and r as determined by Eq. (31). Other values used in the calculation of the growth rate are shown in Tables 1 and 2.

Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0170.1

7. The MJO’s dispersion relation in the wavenumber–frequency spectrum

Making use of the values of L and estimated in section 5, we can use the dispersion relation in Eq. (29a), prescribing different values of , and compare it with the MJO’s spectral signature in observations. The signal strength of OLR overlain by the dispersion relation derived in Eq. (29a) is shown in Fig. 12. Black triangles in Fig. 12a depict the growth rate from while the inverted triangles in Fig. 12b correspond to wave-induced damping. The sum of Figs. 12a and 12b is shown in Fig. 12c. A robust fit between the observed signal and the derived dispersion relationship is observed. Most of the signal is concentrated between zonal wavenumbers 1 and 3 and at periods between 40 and 60 days, which roughly correspond to a value of between 20 and 40 m−1. An MJO with an effective zonal wavenumber and m−1, which best matches the phase speed and group velocity inferred from Fig. 4, is located near the centroid of the MJO’s signal strength in all three diagrams (indicated by an asterisk).

Fig. 12.
Fig. 12.

Signal strength of symmetric OLR averaged over the 15°N–15°S latitude belt. The curves correspond to frequencies obtained from Eq. (29a) for L = 13 200 km, = 13.7 h, and of 25, 40, 60, and 80 m−1 (appearing from bottom to top in the diagrams). (a) Open black triangles depict the growth rate using Eq. (31) and are sized according to their magnitudes, (b) inverted triangles correspond to wave damping [first term in Eq. (29b)], and (c) the sum of the two terms. The asterisk corresponds to an MJO with 44 days and = 1.81. Shading interval is 0.05 with the first contour beginning at 0.65.

Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0170.1

An interesting feature of the MJO’s signal in Fig. 12 is the extension of the signal strength to higher frequencies for zonal wavenumber 1, resulting in a triangular shape in the signal strength, a feature that has not been previously documented. The triangular shape of the wavenumber–frequency spectrum indicates that the MJO is a dispersive wave with an eastward phase speed and a westward group velocity, as documented in section 4. The strongest signal is located over zonal wavenumbers 1 and 2, consistent with the growth rate from (black triangles in Fig. 12a). It is also notable that the dispersion curves corresponding to larger values of (top curves) are associated with a weaker signal in zonal wavenumber 1 relative to the signal in wavenumbers 3 and 4, consistent with wave-driven damping being stronger at the largest scales, as shown by the larger size of the inverted triangles over those locations in Fig. 12b. However, when both processes are considered together (Fig. 12c), wind-driven damping exceeds the growth from at the largest scales. The largest growth occurs instead in wavenumbers 4–6, whereas in the region where the spectral signature is strongest, wind-driven damping and growth from nearly cancel one another. This discrepancy between the linear moist wave and the observed MJO may be due to a time delay between the growth from and the onset of wind-induced wave damping in observations (Zhao et al. 2013; Ling et al. 2013), which is not taken into account in our linear analysis. It is possible that the MJO’s zonal scale is selected by the scale-dependent r [Eq. (31)] during initiation, well before the wind anomalies in the Rossby waves develop and begin to damp the moisture anomalies.

The MJO-derived dispersion curve alongside the dispersion curves for Kelvin, equatorial Rossby (ER), and tropical depression (TD) waves is shown in Fig. 13 in the format suggested by Wheeler and Kiladis (1999). In this diagram, plays a role analogous to the equivalent depth characteristic of equatorial Kelvin and Rossby waves [see Kiladis et al. (2009)]. That the dispersion curve of the MJO so closely fits its observed spectrum, which is clearly separated from that of Kelvin waves, provides further evidence that it is driven by dynamics distinct from that of convectively coupled Rossby and Kelvin modes.

Fig. 13.
Fig. 13.

Signal strength of symmetric OLR anomalies averaged over the 15°N–15°S latitude belt. The MJO-related dispersion curves (solid lines) correspond to frequencies obtained from Eq. (29a) for L = 13 200 km, = 13.7 h, and of 25, 40, 60, and 80 m−1. Dispersion curves are also plotted for Kelvin and ER (dashed lines) for of 12, 25, 50, and 90 m. Dotted lines indicate constant phase speeds of 7.0, 9.0, and 11.0 m s−1, which are representative of westward-propagating TD and easterly waves [see also Yasunaga and Mapes (2012)]. Contour interval is every 0.05 signal strength fraction beginning at 0.55.

Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0170.1

8. Synthesis

In this study, we expanded upon the theoretical work developed by Sobel and Maloney (2012, 2013) and made several important modifications in order to explain many of the observed characteristics of the MJO. These modifications are summarized as follows:

  1. Variations in the meridional and vertical direction have been included in the form of parabolic cylinder functions and vertical basis functions, respectively. By doing this, the horizontal structure of the Matsuno–Gill response can be treated explicitly and processes such as meridional moisture advection can be included.

  2. A mean state that only varies meridionally and vertically is assumed, and the basic equations are scaled accordingly (see appendix).

  3. Values for the free-tropospheric dissipation length scale L and the convective adjustment time scale have been determined directly from observational and reanalysis data.

As in Sobel and Maloney (2012, 2013), many of the processes included here are parameterized and meridionally truncated in order to obtain simpler representations of the physical processes associated with the MJO, which is a caveat of this study. In spite of this recourse to parameterization, a dispersion relation for the MJO is obtained that is largely in agreement with observations. This correspondence between theory and observations provides a strong case that the MJO is a moist wave whose horizontal structure resembles the wave response to an equatorial heat source, as described in Matsuno (1966) and Gill (1980). Through the use of time–longitude diagrams, we have shown that the MJO is characterized by a westward group velocity that is about ⅖ as large as its eastward phase speed—a feature of the MJO that has not been previously documented. This result is at odds with many previous studies (Wang and Rui 1990; Raymond 2001; Majda and Stechmann 2009), who have suggested that the MJO is either nondispersive or has zero group velocity. That the dispersion can be observed even without the use of composites, as seen in Fig. 3, attests to its significance.

The theoretical framework in section 3 shows that the observed group velocity is a consequence of the anomalous wind field in the Matsuno–Gill response. In a free troposphere characterized by weak dissipation, the Rossby wave–related wind response will cause the westerlies to shift eastward toward the moist region (as seen in Fig. 1), resulting in wind-induced drying and a dispersion relation in which the westward group velocity is a fraction of its eastward phase speed. Furthermore, Rossby wave–induced damping is strongest at the largest scales and when the moisture advection parameter is larger. This wave damping at the largest scales is offset, to some extent, by the total gross moist stability, which favors the growth of the largest scales through feedbacks between clouds, moisture, and longwave radiation.

The mechanism by which the group velocity enhances eastward propagation of the MJO is illustrated in Fig. 14. At lag −15 days, the dipole configuration of enhanced and suppressed convection results in strong wind anomalies from the interaction between the Kelvin wave response to the enhanced Indian Ocean convection and the anticyclonic Rossby wave response to the suppressed convection. This superposition of the eastward phase propagation and the westward energy dispersion favors the amplification of the moist anomalies as they propagate across the Indian Ocean into the Maritime Continent, as seen at lag day −5 (roughly a quarter cycle later). The phase speed and group velocity at the time when the MJO-related enhanced convection is centered over the Maritime Continent (lag day −5) favor the development of a suppressed phase in the Indian Ocean as the region of enhanced convection propagates into the west Pacific, which leads to the development of a dipole structure of reversed polarity, as seen in the bottom panel of Fig. 14.

Fig. 14.