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.

Since its discovery in the early 1970s, the crucial role of the Madden‐Julian Oscillation (MJO) in the global hydrological cycle and its tremendous influence on high‐impact climate and weather extremes have been well recognized. The MJO also serves as a primary source of predictability for global Earth system variability on subseasonal time scales. The MJO remains poorly represented in our state‐of‐the‐art climate and weather forecasting models, however. Moreover, despite the advances made in recent decades, theories for the MJO still disagree at a fundamental level. The problems of understanding and modeling the MJO have attracted significant interest from the research community. As a part of the AGU's Centennial collection, this article provides a review of recent progress, particularly over the last decade, in observational, modeling, and theoretical study of the MJO. A brief outlook for near‐future MJO research directions is also provided.

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The two leading principal components of the daily 850-minus 150-hPa global velocity potential in the Interim European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-Interim) data are used as time-varying Madden-Julian oscillation (MJO) indices. Regression maps and meridional cross sections based on these indices are used to document the structure and evolution of the zonal wind (u) and geopotential height (Z) anomalies in the MJO cycle. The data are daily, and they are not separated by season. At upper-tropospheric levels the MJO signature is dominated by eastward-propagating planetary wave packets consisting of equatorial Kelvin waves flanked by Rossby waves centered along 288N/S, for which the westerly jet streams serve as waveguides. At lower-tropospheric levels the pattern more closely resembles the response to a pulsating heat source over the Maritime Continent, where the Andes block the eastwardpropagating Kelvin wave pulse. The contrasting upper-and lower-tropospheric patterns are made up of the same building blocks: a deep, baroclinic modal structure with a node at the 400-hPa level, which dominates the tropical signature, and a barotropic residual field consisting mainly of extratropical wave trains oriented along great circles. The extratropical wave trains emanate from the flanking Rossby waves in the baroclinic modal structure. The strongest of them, which resembles the Pacific-North America (PNA) pattern, extracts kinetic energy from the climatological-mean flow in the jet exit region. At other longitudes the jet stream seems to act as a barrier to the poleward propagation of MJO-related wave activity.

The large-scale circulation features that determine the structure and evolution of MJO-related moisture and precipitation fields are examined using a linear analysis protocol based on daily 850- minus 150-hPa global velocity potential data. The analysis is augmented by a compositing procedure that emphasizes the structural features over the Indo-Pacific warm pool sector (60°E–180°) that give rise to the eastward propagation of the enhanced moisture and precipitation. It is found that boundary layer (BL) convergence in the low-level easterlies to the east of the region of maximum ascent produces a deep but narrow plume of equatorial ascent that moistens the midtroposphere, while weakly diffluent flow above the BL spreads moisture away from the equator. Vertical advection of moisture from this plume of ascent accounts for the eastward propagation of the positive moisture anomalies across the Maritime Continent into the western Pacific. When the convection is first developing over the Indian Ocean, horizontal moisture advection contributes to both the eastward propagation and the amplification of the positive moisture anomalies along the equator to the east of the region of enhanced convection. Neither horizontal advection nor the net moistening from vertical advection and the apparent moisture sink exhibit significant westward tilt with height in the equatorial plane, but when they are superposed they explain the westward tilt of the moisture field. The strong spatial correlation between relative humidity and vertical velocity underscores the important role of equatorial wave dynamics in shaping the structure and evolution of the MJO.