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Systematic Multiscale Models for the Tropics

Andrew J. MajdaCourant Institute of Mathematical Sciences, and Center for Atmosphere/Ocean Science, New York University, New York, New York

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Rupert KleinFB Mathematik and Informatik, Freie Universität Berlin, Berlin, and Potsdam Institut für Klimafolgenforschung, Potsdam, Germany

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Abstract

Systematic multiscale perturbation theory is utilized to develop self-consistent simplified model equations for the interaction across multiple spatial and/or temporal scales in the Tropics. One of these models involves simplified equations for intraseasonal planetary equatorial synoptic-scale dynamics (IPESD). This model includes the self-consistent quasi-linear interaction of synoptic-scale generalized steady Matsuno–Webster–Gill models with planetary-scale dynamics of equatorial long waves. These new models have the potential for providing self-consistent prognostic and diagnostic models for the intraseasonal tropical oscillation. Other applications of the systematic approach reveal three different balanced weak temperature gradient (WTG) approximations for the Tropics with different regimes of validity in space and time: a synoptic equatorial-scale WTG (SEWTG); a mesoscale equatorial WTG (MEWTG), which reduces to the classical models treated by others; and a new seasonal planetary equatorial WTG (SPEWTG). Both the SPEWTG and MEWTG model equations have solutions with general vertical structure, yet have the linearized dispersion relation of barotropic Rossby waves; thus, these models can play an important role in theories for midlatitude connections with the Tropics. The models are derived both from the equatorial shallow water equations in a simplified context and also as distinguished limits from the compressible primitive equations in general.

Corresponding author address: Prof. Andrew J. Majda, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012. Email: jonjon@cims.nyu.edu

Abstract

Systematic multiscale perturbation theory is utilized to develop self-consistent simplified model equations for the interaction across multiple spatial and/or temporal scales in the Tropics. One of these models involves simplified equations for intraseasonal planetary equatorial synoptic-scale dynamics (IPESD). This model includes the self-consistent quasi-linear interaction of synoptic-scale generalized steady Matsuno–Webster–Gill models with planetary-scale dynamics of equatorial long waves. These new models have the potential for providing self-consistent prognostic and diagnostic models for the intraseasonal tropical oscillation. Other applications of the systematic approach reveal three different balanced weak temperature gradient (WTG) approximations for the Tropics with different regimes of validity in space and time: a synoptic equatorial-scale WTG (SEWTG); a mesoscale equatorial WTG (MEWTG), which reduces to the classical models treated by others; and a new seasonal planetary equatorial WTG (SPEWTG). Both the SPEWTG and MEWTG model equations have solutions with general vertical structure, yet have the linearized dispersion relation of barotropic Rossby waves; thus, these models can play an important role in theories for midlatitude connections with the Tropics. The models are derived both from the equatorial shallow water equations in a simplified context and also as distinguished limits from the compressible primitive equations in general.

Corresponding author address: Prof. Andrew J. Majda, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012. Email: jonjon@cims.nyu.edu

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