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On the linear stability study of zonal incompressible flows on a sphere
Author(s) -
Skiba Yuri N.,
Adem Julián
Publication year - 1998
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/(sici)1098-2426(199809)14:5<649::aid-num7>3.0.co;2-i
Subject(s) - mathematics , mathematical analysis , spherical harmonics , legendre polynomials , nonlinear system , invariant (physics) , mathematical physics , physics , quantum mechanics
The normal mode (linear) stability of zonal flows of a nondivergent fluid on a rotating sphere is considered. The spherical harmonics are used as the basic functions on the sphere. The stability matrix representing in this basis the vorticity equation operator linearized about a zonal flow is analyzed in detail using the recurrent formula derived for the nonlinear triad interaction coefficients. It is shown that the zonal flow having the form of a Legendre polynomial P n (μ) of degree n is stable to infinitesimal perturbations of every invariant set I m with | m | ≥ n . For each zonal number m , I m is here the span of all the spherical harmonics $Y^{m}_{k}(x)$, whose degree k is greater than or equal to m . It is also shown that such small‐scale perturbations are stable not only exponentially, but also algebraically. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 649–665, 1998