Motivated by appearances of multiple equilibrium states in some models of atmospheric flows (Charney and DeVore (1979), Matsuda (1980)), we classify critical points (transition points) appearing in steady problems of fluid dynamics in connection with symmetry break-down in the velocity field.
For this purpose, we rewrite the Navier-Stokes equations in a spectral form. In the expansion of the velocity field in terms of normal modes, we divide the modes into two groups; the first group consists of modes which have the same symmetry as the external conditions imposed on the system, while the second group consists of other asymmetric modes. By postulating that there exists always a steady solution which has the same symmetry as the external conditions in the system, we restrict nonlinear coupling between the two groups of modes in the spectral equations.
Using the spectral equations obtained in this way, we examine structure of critical points. If the symmetry breaks down in the velocity field, critical points of two types can appear in a fluid system. The first type is the critical point at which a symmetric solution is destabilized and other solutions with lowered symmetry branch from it. The second type is the critical point at which a symmetric solution and an asymmetric solution exchange its stability. If the degree of symmetry is preserved, the critical point of another type can appear in a fluid system; this type of critical point is characterized by coalescence of a stable solution and an unstable one and its subsequent disappearence in the phase space as the relevant parameter is varied.
By constructing a local potential in the vicinity of each critical point, we examine the correspondence of critical points examined in this study to Thom's (1972) elementary catastrophes.
In the last section, in the light of our classification theory we re-examine multiplicity and stability of the equilibrium states obtained in Charney and DeVore's model (1979) of blocking phenomena.
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