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# Communications in Mathematical Sciences

## Volume 8 (2010)

### Number 2

### Special Issue on the Occasion of Andrew Majda’s Sixtieth Birthday: Part II

### Intraseasonal multi-scale moist dynamics of the tropical atmosphere

Pages: 519 – 540

DOI: http://dx.doi.org/10.4310/CMS.2010.v8.n2.a11

#### Authors

#### Abstract

We derive a multi-scale model of moist tropical dynamics which is valid on horizontal synoptic scales, zonal planetary scales, and synoptic and intraseasonal time scales. The Intraseasonal Multi-Scale Moist Dynamics (IMMD) framework builds on the IPESD framework of A.J. Majda and R. Klein, *J. Atmos. Sci.*, 60, 393-408, 2003. It generalizes the latter by allowing for strong zonal winds (the Trade Winds) and the pressure and stratification variations that they generate. The framework consists of three pieces. The first, called TH, are planetary scale climatology modulation equations which govern the Trade Winds and Hadley Circulation. Self-consistency of the asymptotic theory requires that the meridional component of the Hadley Circulation is an order of magnitude weaker than the zonal component. The second piece, S, is a linear system of equations which govern synoptic scale velocity, temperature, and pressure fluctuations forced by synoptic scale heating fluctuations. Unlike the IPESD theory, these fluctuations are advected by part of the planetary scale climatology from TH. Since the meridional component of TH is an order of magnitude weaker than the zonal component, the synoptic scale fluctuations are only advected by the latter. The third, P, govern the planetary scale anomalies which, like IPESD, are driven both by planetary scale mean heating and by upscale fluxes from the synoptic scales. These planetary scale anomalies are advected both by the zonal component of the Trade Winds and by the meridional component of the Hadley Circulation and, furthermore, respond to an in-scale flux from the mean climatology. We also present an asymptotic analysis of the equations of bulk cloud thermodynamics in order to lay out a self-contained path for incorporating synoptic scale cloud models into the IMMD framework. This framework has potentially important implications for the development of models describing the Madden-Julian Oscillation (MJO) since the MJO manifests itself as planetary scale anomalies from a mean climatology which it modulates on intraseasonal time scales.

#### Keywords

Multi-scale asymptotics; tropical atmospheric dynamics; Madden-Julian oscillation

#### 2010 Mathematics Subject Classification

34E13, 76B60, 86A10