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Recent Advances Concerning Certain Class of Geophysical Flows

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Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Abstract

This chapter is devoted to reviewing several recent developments concerning certain class of geophysical models, including the primitive equations (PEs) of atmospheric and oceanic dynamics and a tropical atmosphere model. The PEs for large-scale oceanic and atmospheric dynamics are derived from the Navier–Stokes equations coupled to the heat convection by adopting the Boussinesq and hydrostatic approximations, while the tropical atmosphere model considered here is a nonlinear interaction system between the barotropic mode and the first baroclinic mode of the tropical atmosphere with moisture.It is mainly concerned with the global well-posedness of strong solutions to these systems, with full or partial viscosity, as well as certain singular perturbation small-parameter limits related to these systems, including the small aspect ratio limit from the Navier–Stokes equations to the PEs, and a small relaxation parameter in the tropical atmosphere model. These limits provide a rigorous justification to the hydrostatic balance in the PEs and to the relaxation limit of the tropical atmosphere model, respectively. Some conditional uniqueness of weak solutions, and the global well-posedness of weak solutions with certain class of discontinuous initial data, to the PEs are also presented.

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Acknowledgements

J.L. is thankful to the kind hospitality of Texas A&M University where part of this work was completed. This work was supported in part by the ONR grant N00014-15-1-2333 and the NSF grants DMS-1109640 and DMS-1109645.

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Authors and Affiliations

  1. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel

    Jinkai Li & Edriss S. Titi

  2. Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, China

    Jinkai Li

  3. Department of Mathematics, Texas A&M University, 3368 TAMU, 77843-3368, College Station, TX, USA

    Edriss S. Titi

  4. Department of Mathematics, University of California, Mathematics, Mechanical and Aerospace Engineering, 92697-3875, Irvine, CA, USA

    Edriss S. Titi

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  1. Jinkai Li
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Correspondence to Jinkai Li .

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Editors and Affiliations

  1. Tokyo, Tokyo, Japan

    Yoshikazu Giga

  2. Toulon, Var, France

    Antonin Novotny

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Li, J., Titi, E.S. (2016). Recent Advances Concerning Certain Class of Geophysical Flows. In: Giga, Y., Novotny, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-10151-4_22-1

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  • DOI: https://doi.org/10.1007/978-3-319-10151-4_22-1

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  • Online ISBN: 978-3-319-10151-4

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