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arXiv · 2512.18443

A Toy Model of the Madden-Julian Oscillation

Ian Folkins
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Abstract

We discuss a simple three layer model of the tropical atmosphere. The rainfall variance of the model is dominated by a rainfall mode moving parallel to the equator having the approximate size and propagation speed of the Madden-Julian Oscillation (MJO). The origin of the convective aggregation in the model is the imposition of distinct length scales for the deep updraft and stratiform downdraft circulations. Subsidence induced by the deep updraft circulation suppresses convective instability on a scale of $\sim$ 1000 km, while ascent induced by the downdraft circulation promotes convective instability on a scale of $\sim$ 500 km. Within the MJO envelope, high rainfall rates are maintained both by increased column relative humidity, and increased variance in lower tropospheric vertical motion. Each of the three model layers has a prescribed target pressure thickness. Convective mass fluxes introduce a mass excess into grid cells where there is net detrainment, and a mass deficit into grid cells from which there is net entrainment. Horizontal transport in the model is based on export of mass from grid cells where there is an excess, and import of mass toward grid cells where there is a deficit. The resulting patterns of horizontal convergence and divergence generate vertical motions between model levels. The simulated MJO events propagate eastward when there is a slight preference for mass deficits in the boundary layer to be compensated by inward flow from the west. The forward propagation of the MJO is limited by the rate at which the downdraft circulation within the MJO is able to generate net upward motion and promote new convective activity in advance of the leading edge. We also offer some guidance on how convective parameterizations that are implemented in models with more realistic dynamical schemes might be designed to exhibit stronger MJO variance.

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Ian Folkins. 2025-12-20. A Toy Model of the Madden-Julian Oscillation. https://arxiv.org/abs/2512.18443

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