Steady-state bifurcation analysis of a fully nonlinear quasi-geostrophic vorticity equation

Chen, Zhi-Min (2015) Steady-state bifurcation analysis of a fully nonlinear quasi-geostrophic vorticity equation. Journal of Mathematical Analysis and Applications, 1-24. (doi:10.1016/j.jmaa.2015.05.035).

Record type: Article

Abstract

The quasi-geostrophic vorticity equation studied in the present paper is a simplified form of the atmospheric circulation model introduced by Charney and DeVore [J. Atmos. Sci. 36(1979), 1205–1216] on the existence of multiple steady states to the understanding of the persistence of atmospheric blocking. The fluid motion defined by the equation is driven by a zonal thermal forcing and an Ekman friction forcing measured by ?. It is proved that the steady-state solution is globally unique for large ? values while multiple steady-state solutions branch off the basic steady-state solution for ?<?crit where the critical value ?crit is less than one. Without involvement of viscosity, the equation has fully non-linear property as its non-linear part contains the highest order derivative term. Steady-state bifurcation analysis is essentially based on the compactness, which can be simply obtained for semilinear equations such as the Navier–Stokes equations but is not available for the fully nonlinear quasi-geostrophic vorticity equation in the Euler formulation. Therefore the Lagrangian formulation of the equation is employed to gain the required compactness.

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