Atmosphere-Ocean Dynamics Academic Press New York

Author(s): Gill A

Abstract

In traditional mechanics courses we learn to solve ideal problems, ones without friction-like effects, by transforming to normal coordinates. Such problems are typically Hamiltonian and the transformation theory of Hamiltonian systems is exploited. If the system under consideration describes linear motion about a stable equilibrium point, then in terms of normal coordinates the Hamiltonian has the following normal form: H(p, q) = k νk 2 q2 k + p2 k  = k νkJk. (1.1) Here (q, p) denotes a set of canonical variables and the νk’s are the natural frequencies of oscillation of the system. In (1.1) the Hamiltonian is written in two ways: as a sum over independent simple harmonic oscillators and as a sum over the action variables, Jk, with their conjugate variables being absent from the Hamiltonian, i.e. these variables are ignorable. It is well-known that the equations that describe ideal fluid flow are Hamiltonian (see e.g. Morrison 1998, 1982 and references therein) and thus one would expect that a similar transformation to normal coordinates should exist for inviscid fluid problems. Indeed this is the case and we demonstrate this here for the problem of shear flow in a channel. This shear flow problem is complicated by the fact that it possesses a continuous spectrum associated with the presence of critical levels. Continuous spectra cannot occur in finite degree-of-freedom systems and so new tools are required. We will describe these new tools and use them to show that the transformation to normal coordinates can be constructed using the singular eigenfunctions associated with the continuous spectrum. In particular, we show that this normal coordinate transformation is a linear integral transform that is a generalization of the Hilbert transform. For convenience we consider flow profiles for which our shear flow problem is stable and possesses only a continuous spectrum. That is, we consider profiles for which there is no discrete component to the spectrum. According to Rayleigh’s criterion this is guaranteed to be the case if the profile contains no inflection points. A more general criterion for assuring this type of spectrum is given in Balmforth & Morrison (1999), which treats necessary and sufficient conditions for stability

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