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Nonlinear steady waves in planetary vortex dynamics. (English) Zbl 0663.76014

In this paper we introduce a class of nonlinear, nonlocal evolution equations in two-space dimensions, and we study steady translational waves governed by these equations. The class that we consider constitutes a mathematical generalization of a standard equation arising in geophysical fluid dynamics, and it is this special case that motivates our general theory. In the terminology of the geophysical applications, we are concerned with planetary vortex dynamics and, in particular, with nonlinear planetary (or Rossby) waves in a nonuniform (sheared) zonal current. The governing equation is given by the so-called \(\beta\)-plane model which provides an approximation to the motion of a layer of ideal fluid (an ocean or atmosphere) on a rotating sphere (a planet) at appropriate length and velocity scales (a quasigeostrophic flow). This equation is an evolution equation for the relative vorticity, and, in essence, it specifies that the sum of the relative vorticity and the ambient vorticity of rotation (the Coriolis parameter) be advected by the flow; it is commonly referred to as the barotropic, quasi-geostrophic (potential) vorticity equation.

MSC:

76B65 Rossby waves (MSC2010)
86A10 Meteorology and atmospheric physics
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