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    Green's function of the screened Poisson's equation on the sphere

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    Type
    Preprint
    Authors
    Tanios, Ramy
    Mohtar, Samah El
    Knio, Omar
    Lakkis, Issam
    KAUST Department
    King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia
    Applied Mathematics and Computational Science Program
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2019-11-21
    Permanent link to this record
    http://hdl.handle.net/10754/660746
    
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    Abstract
    In geophysical fluid dynamics, the screened Poisson equation appears in the shallow-water, quasi geostrophic equations. Recently, many attempts have been made to solve those equations on the sphere using different numerical methods. These include vortex methods, which solve a Poisson equation to compute the stream-function from the (relative) vorticity. Alternatively, the stream-function can be computed directly from potential vorticity (PV), which would offer the possibility of constructing more attractive vortex methods because PV is conserved along material trajectories in the inviscid case. On the spherical shell, however, the screened Poisson equation does not admit a known Green's function, which limits the extension of such approaches to the case of a sphere. In this paper, we derive an expression of Green's function for the screened Poisson equation on the spherical shell in series form and in integral form. A proof of convergence of the series representation is then given. As the series is slowly convergent, a robust and efficient approximation is obtained using a split form which isolates the singular behavior. The solutions are illustrated and analyzed for different values of the screening constant.
    Sponsors
    This work is supported by the University Research Board of the American University of Beirut. The authors would like to acknowledge Professor Leila Issa of the Lebanese American University-Beirut for her insightful feedback on the mathematical derivation of the convergence of the Green’s Function.
    Publisher
    arXiv
    arXiv
    1911.10944
    Additional Links
    https://arxiv.org/pdf/1911.10944
    Collections
    Preprints; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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