A Toy Model of the Madden-Julian Oscillation

TL;DR

This work presents a toy three-layer model of the tropical troposphere designed to reproduce Madden-Julian Oscillation (MJO)–like rainfall variance. The core idea is convective aggregation arising from two distinct circulations—deep updrafts and stratiform downdrafts—with different horizontal length scales, radiative subsidence, and humidity/CAPE feedbacks that together generate an eastward-propagating mode near MJO scales. The model reproduces MJO features such as multiscale structure, coastal-like surface pressure patterns, and propagation speeds around $6\ \mathrm{m\,s^{-1}}$, while providing diagnostics that align with TRMM and radiosonde observations; it also offers guidance for improving convective parameterizations in more realistic dynamical models. Limitations include hardwired, latitude-dependent length scales and instantaneous horizontal transport, suggesting future work to allow scale evolution and dynamical coupling to reduce reliance on fiat assumptions.

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.

Figures (17)

• Figure 1: Overview of the two main convective circulations in the tropics. Deep convective updrafts drive a large scale overturning circulation in which subsidence warming and drying inhibit convective development on the scale of several thousand km. Some of the precipitation falling from stratiform anvil clouds evaporates in the cloud free air below cloud base, generating downdrafts and induced ascent at intermediate spatial scales, and favoring the development of congestus clouds.
• Figure 2: The trimodal temperature response to tropical deep convection: upper tropospheric warming (450 hPa - 150 hPa), lower tropospheric cooling (800 hPa - 500 hPa), and boundary layer cooling (below 900 hPa). Deep convection also cools the Tropical Tropopause Layer (150 hPa - 80 hPa). This temperature anomaly response pattern has motivated the choice of an upper tropospheric horizontal length scale $L_{UT} \sim 1100$ km, and lower tropospheric length scale $L_{LT} \sim 500$ km, near the equator.
• Figure 3: The solid black curve shows the prescribed latitudinal variation of the length scale for horizontal transport for the upper tropospheric and boundary layers of the model ($L_{UT}$ and $L_{BL}$). It is therefore the spatial scale of the deep circulation. The dashed black curve shows the length scale for horizontal transport of the lower troposphere $L_{LT}$, and defines the spatial scale of the downdraft congestus circulation. The blue curve shows shows the latitudinal variation of the sea surface temperature.
• Figure 4: The model uses sigmoidal functions to characterize the nonlinear dependence of several parameters on model variables. This plot shows a generic sigmoidal function $f(x)$ defined by four constants: the value $f_{min}$ of $f(x)$ at low values of x, the value $f_{min} + f_{add}$ of $f(x)$ at large values of $x$, the value of $x$$x_{half}$ where $f(x)$ changes most rapidly, and a parameter $x_{scale}$ prescribing the steepness of $f(x)$ about $x_{half}$.
• Figure 5: Three of the sigmoidal function used in the model. $f_{cg}(cape_{LT})$ expresses the dependence of the congestus mass flux on lower tropospheric CAPE. $f_{dp}(cape_{UT})$ expresses the dependence of the deep convective mass flux on upper tropospheric CAPE. $f_{sub}(\delta p)$ characterizes the dependence of the radiative subsidence mass flux from the upper to lower troposphere on $\delta p$. This function modulates the subsidence mass flux between these two layers in such a way that the pressure thicknesses of grid cells do not deviate too strongly from their target values.