Título del libro: Computational Mathematics: Theory, Methods And Applications
Título del capítulo: Analytical and numerical methods in the linear stability study of ideal flows on a sphere
IOURI SKIBA SKIBA;
Autores externos:
Idioma:
Inglés
Año de publicación:
2011
Palabras clave:
Ideal incompressible fluid; Linear instability; Vorticity equation solutions
Resumen:
Analytical and numerical spectral methods are developed for the linear (normalmode) instability study of steady solutions to the nonlinear barotropic vorticity equation (BVE) governing the motion of a two-dimensional ideal incompressible fluid on a rotating sphere. The four types of BVE solutions known up to now are considered, namely, the zonal (one-dimensional) flows in the form of a Legendre-polynomial (LP) of degree n, and such non-zonal (two-dimensional) flows as Rossby-Haurwitz (RH) waves, Wu-Verkley (WV) waves and modons. A unified approach to the normal-mode instability study of these solutions is suggested. A conservation law for disturbances of each steady solution is derived and then used to obtain a necessary condition for its normal-mode (exponential) instability. According to these conditions, Fjörtoft's [1] average spectral number of the amplitude of any unstable mode must be equal to a special value. In the case of the LP flows or RH waves, this value is related only with the basic flow degree. Unlike these results, the above-mentioned value for the WV waves and modons depends both on the basic flow degree and on the spectral distribution of the mode energy in the inner and outer regions of the basic flow. Peculiarities of the instability conditions for different types of modons are also discussed. The new instability conditions specify the spectral structure of growing normal-mode disturbances localizing them in the phase space. Note that for the LP flows, the new condition complements the well-known Rayleigh-Kuo and Fjörtoft conditions related to the zonal flow profile. As to more complicated two-dimensional steady flows (RH waves, WV waves and modons), the instability conditions obtained here have no analogues. The maximum growth rate of unstable modes is estimated too, and the orthogonality of any unstable, decaying and non-stationary mode to the basic flow is shown in the energy inner product. The numerical spectral method for the normal mode instability study consists of the three part: the calculation of the elements of stability matrix, the solution of the complete eigenvalue problem, and the construction of unstable normal modes on the sphere. The spectral method uses the spherical harmonics as the basic functions for representing solutions and disturbances by their Fourier series on the sphere. Some analytical and numerical examples are given too. It should be stressed that the analytical instability results obtained here can also be applied for testing the accuracy of computational programs and algorithms used for the numerical stability study. Also note that Fjörtoft's spectral number appearing both in the instability conditions and in the maximum growth rate estimates is considered as the key parameter in the linear instability problem of ideal flows on a sphere. © 2010 by Nova Science Publishers, Inc. All rights reserved.
Entidades citadas de la UNAM:
ISBN:
9781608762712
Editorial:
Nova Science Publishers, Inc. Nombre de la publicación:
Analytical and numerical methods in the linear stability study of ideal flows on a sphere
Página inicial:
1
Página final:
34
Tipo de producto:
Capítulo de un Libro
