### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 331-354 |

Number of pages | 24 |

Journal | Theoretical and Computational Fluid Dynamics |

Volume | 19 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2005 |

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### Keywords

- Barotropic-baroclinic nonlinear interactions
- Large scale equatorial waves
- Non-oscillatory balanced schemes
- Precipitation fronts

### ASJC Scopus subject areas

- Fluid Flow and Transfer Processes
- Oceanography
- Computational Mechanics
- Mechanics of Materials
- Applied Mathematics
- Physics and Astronomy(all)
- Physics and Astronomy (miscellaneous)
- Condensed Matter Physics

### Cite this

**A non-oscillatory balanced scheme for an idealized tropical climate model Part I : Algorithm and validation.** / Khouider, Boualem; Majda, Andrew J.

Research output: Contribution to journal › Article

*Theoretical and Computational Fluid Dynamics*, vol. 19, no. 5, pp. 331-354. https://doi.org/10.1007/s00162-005-0170-8

}

TY - JOUR

T1 - A non-oscillatory balanced scheme for an idealized tropical climate model Part I

T2 - Algorithm and validation

AU - Khouider, Boualem

AU - Majda, Andrew J.

PY - 2005/10

Y1 - 2005/10

N2 - 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.

AB - 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.

KW - Barotropic-baroclinic nonlinear interactions

KW - Large scale equatorial waves

KW - Non-oscillatory balanced schemes

KW - Precipitation fronts

UR - http://www.scopus.com/inward/record.url?scp=28344452560&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=28344452560&partnerID=8YFLogxK

U2 - 10.1007/s00162-005-0170-8

DO - 10.1007/s00162-005-0170-8

M3 - Article

AN - SCOPUS:28344452560

VL - 19

SP - 331

EP - 354

JO - Theoretical and Computational Fluid Dynamics

JF - Theoretical and Computational Fluid Dynamics

SN - 0935-4964

IS - 5

ER -