Scale analysis for the Madden$-$Julian oscillation

Abstract

This study performs a scale analysis for the dynamics of equatorial flow systems like the Madden$-$Julian oscillation (MJO). This analysis makes three assumptions about the geometry of the flow and focuses on the leading order terms of the mass continuity and momentum equation system. First, the flow is meridionally narrow. Second, the zonal structure is planetary wavenumber 1. Last, the flow is vertically shallow. When constrained by these geometric assumptions, the system shows a time scale much longer than a day depending on the meridional narrowness. Specifically, for equatorial Kelvin waves, the oscillation period ranges from 36 to 64 days given a ratio of the planetary radius to the meridional width scale from 6 to 8 (i.e., an e-folding latitude from 9.59$^\circ$ to 7.18$^\circ$ either side of the equator). Given the long, narrow, and shallow geometry, the dynamics can sustain Kelvin waves only at low frequencies. This result suggests that the intraseasonal time scale of the MJO arises from the Kelvin wave frequency limit due to the planetary wave geometry. This planetary intraseasonal scale relation is independent from thermodynamics, and an open question remains on what physical processes coordinate the thermodynamics in the same length or time scale.