Fundamentals of Geophysical Hydrodynamics by Felix V. Dolzhansky

Author:Felix V. Dolzhansky
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

Felix V. Dolzhansky1

(1)http://www.springer.com

Abstract

And yet, in certain important cases, the method of normal modes turns out to be exhaustive. The reason is that the integration is performed along the real axis of variable z. So a singular point occurs for real values of c, i.e., for neutral oscillations. The continuous spectrum filling interval [U min,U max] also belongs to the real axis, and thus it does not give rise to unstable oscillations. Real eigenvalues c of the discrete spectrum can cause instability only if they are repeated. Then there are “secular” perturbations, linearly increasing in time, as a consequence of the non-self-adjoint property of the operator of linear stability. Let us state without proof the following theorem (Dikii 1976).

A two-dimensional plane-parallel flow of a homogeneous incompressible fluid with a monotonic velocity profile, whose boundary values U(a) and U(b) are not eigenvalues of the reduced operator of stability can be unstable only if the problem has either non-real eigenvalues in the discrete spectrum or repeated ones.

However, one should not get carried away by the method of normal modes. Returning to the original formulation of the problem, we can encounter solutions that are not covered by the reduced problem and that grow over time not exponentially but polynomially (the so-called algebraic instability). This may either modify or supplement the conclusions drawn on the basis of the reduction. Such an example is studied in the next chapter.

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