Stability of Quasi-two-dimensional Shear Flows with Arbitrary Velocity Profiles

  • Felix V. Dolzhansky
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 103)


Thus one can conclude the following: the linear stability theory of the strictly two-dimensional Kolmogorov flow, i.e., constructed without taking into account external friction, is structurally unstable with respect to the inclusion of the latter and, conversely, the linear theory of the quasi-two-dimensional flow, i.e., constructed by taking into account external friction, qualitatively is not sensitive to the inclusion or exclusion of internal viscosity. Moreover, it is easy to see that for \(\lambda_{0}=\widehat{\lambda}/\widehat{\nu}=R_{\nu}/R_{\lambda} \gg 1\), the results in the quasi-two-dimensional linear theory are almost self-similar in R ν . What is most important is that this conclusion holds for the nonlinear stability theory, as we shall see below.

Is the situation described above an exclusive feature of the Kolmogorov flow, or is it typical for shear flows with arbitrary profiles? To answer this seemingly very difficult question, we will give a new interpretation of the results in the preceding chapter, which will allow us to draw certain conclusions regarding the stability of quasi-two-dimensional shear flows, using the well-developed stability theory of strictly two-dimensional flows.


Shear Flow Dispersion Curve Potential Vorticity Neutral Stability Critical Curve 
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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Felix V. Dolzhansky
    • 1
  1. 1.

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