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A non-oscillatory balanced scheme for an idealized tropical climate model

Part I: Algorithm and validation

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Abstract

We propose a non-oscillatory balanced numerical scheme for a simplified tropical climate model with a crude vertical resolution, reduced to the barotropic and the first baroclinic modes. The two modes exchange energy through highly nonlinear interaction terms. We consider a periodic channel domain, oriented zonally and centered around the equator and adopt a fractional stepping–splitting strategy, for the governing system of equations, dividing it into three natural pieces which independently preserve energy. We obtain a scheme which preserves geostrophic steady states with minimal ad hoc dissipation by using state of the art numerical methods for each piece: The f-wave algorithm for conservation laws with varying flux functions and source terms of Bale et al. (2002) for the advected baroclinic waves and the Riemann solver-free non-oscillatory central scheme of Levy and Tadmor (1997) for the barotropic-dispersive waves. Unlike the traditional use of a time splitting procedure for conservation laws with source terms (here, the Coriolis forces), the class of balanced schemes to which the f-wave algorithm belongs are able to preserve exactly, to the machine precision, hydrostatic (geostrophic) numerical-steady states. The interaction terms are gathered into a single second order accurate predictor-corrector scheme to minimize energy leakage. Validation tests utilizing known exact solutions consisting of baroclinic Kelvin, Yanai, and equatorial Rossby waves and barotropic Rossby wave packets are given.

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Authors and Affiliations

  1. Mathematics and Statistics, University of Victoria, PO BOX 3045 STN CSC, Victoria, B.C., V8W 3P4, Canada

    Boualem Khouider

  2. Department of Mathematics and Center for Atmosphere/Ocean Sciences, Courant Institute, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Andrew J. Majda

Authors
  1. Boualem Khouider
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  2. Andrew J. Majda
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Correspondence to Boualem Khouider.

Additional information

Communicated by R. Klein

PACS 02.70.Bf; 02.30.Jr; 92.60.Bh; 92.60.Dj; 92.60.Jq

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Khouider, B., Majda, A.J. A non-oscillatory balanced scheme for an idealized tropical climate model. Theor. Comput. Fluid Dyn. 19, 331–354 (2005). https://doi.org/10.1007/s00162-005-0170-8

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  • DOI: https://doi.org/10.1007/s00162-005-0170-8

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