Título del libro: Advances In Mathematics Research
Título del capítulo: Liapunov and normal-mode instability of the rossby-haurwitz wave
IOURI SKIBA SKIBA;
Autores externos:
Idioma:
Inglés
Año de publicación:
2010
Palabras clave:
Ideal incompressible flow; Liapunov and exponential instability; Rossby-Haurwits wave
Resumen:
The dynamics of perturbations to the Rossby-Haurwitz (RH) wave is analytically analyzed. These waves being of great meteorological importance are exact solutions to the nonlinear vorticity equation describing the motion of an ideal incompressible fluid on a rotating sphere. Each RH wave belongs to a space H1 ? Hn where Hn is the subspace of homogeneous spherical polynomials of degree n. It is shown that any perturbation of the RH wave evolves in such a way that its energy K(t) and enstrophy ?(t) decrease, remain constant or increase simultaneously. A geometric interpretation of variations in the perturbation energy is given. A conservation law for arbitrary perturbations is obtained and used to classify all the RH-wave perturbations in four invariant setsMn -,Mn +, Hn andMn 0 -Hn depending on the value of their mean spectral number ?(t) = ?(t)/K(t). The energy cascade of growing (or decaying) perturbations has opposite directions in the sets Mn - and Mn + due to a hyperbolic dependence between K(t) and ?(t). A quotient space with a quotient norm of the perturbations is introduced using the invariant subspace Hn of neutral perturbations as the zero quotient class. While the energy norm controls the perturbation part belonging to Hn, the quotient norm controls the perturbation part orthogonal to Hn. It is shown that in the set Mn- (?(t) < n(n + 1) ), any non-zonal RH wave of subspace H1 ? Hn (n = 2 ) is Liapunov unstable in the energy norm. This instability has nothing in common with the orbital (Poincaré) instability and is caused by asynchronous oscillations of two almost coinciding RH-wave solutions. It is also shown that the exponential instability is possible only in the invariant set Mn 0 - Hn. A necessary condition for this instability is given. The condition states that the spectral number ?(t) of the amplitude of each unstable mode must be equal to n(n+1) where n is the RH wave degree. The growth rate is estimated and the orthogonality of the unstable normal modes to the RH wave are shown in two Hilbert spaces. The instability in the invariant setMn + of small-scale perturbations (?(t) < n(n + 1)) is still open problem. © 2010 Nova Science Publishers, Inc.
Entidades citadas de la UNAM:
ISBN:
9781612092126
Editorial:
Nova Science Publishers, Inc. Nombre de la publicación:
Liapunov and normal-mode instability of the rossby-haurwitz wave
Página inicial:
1
Página final:
28
Tipo de producto:
Capítulo de un Libro
