Three-dimensional absolute and convective instabilities, and spatially amplifying waves in parallel shear flows
At least two roots inl of the systemD(k, l, ω)=0,D k (k, l, ω)=0, originating on opposite sides of the reall-axis, collide on thel-plane for the parameter valuesk0,l0,ω0, asω is brought down toω0. Every point on thek-plane, that corresponds to a point on the collision paths on thel-plane, is itself a coalescence point ofk-roots for a fixedl ofD(k, l, ω)=0, that originate on opposite sides of the realk-axis.
At least two roots ink of the systemD(k, l, ω)=0,D l ,(k, l, ω)=0, originating on opposite sides of the realk-axis, collide on thek-plane for the parameter valuesk0,l0,ω0, asω is brought down toω0. Every point on thel-plane, that corresponds to a point on the collision paths on thek-plane, is itself a coalescence point ofl-roots for a fixedk ofD(k, l, ω)=0, that originate on opposite sides of the reall-axis.
Consequently, the causality condition for spatially amplifying 3-D waves in absolutely stable, but convectively unstable flow is derived as follows. We denote by (α, β) a unit vector on the (x, y) plane. The contributions to amplification in the direction of this vector come from the end points of the trajectories that consist of the coalescence roots on thel1-plane, given byl1,=−βk+αl, of the systemD=0,−βD k +gaD1=0. Thek1-components of these trajectories have to pass from above to below the real axis on ak1-plane, given byk1=αk+βl, asω moves down toω0. Hereω0 is the real frequency of excitation. At each point of such trajectories the group velocity vector (D k ,D l ) is collinear with the direction vector (α, β). There exists a direction for which the spatial amplification rate reaches its maximum.
The formalism is illustrated with a simple model example. A procedure for computing theN-factor in thee N -method, which is based on the wave packet approach is developed.
KeywordsDispersion Relation Wave Packet Group Velocity Real Axis Direction Vector
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- R. J. Briggs,Electron-Stream Interactions in Plasmas. MIT Press 1964.Google Scholar
- L. Brevdo,A study of absolute and convective instabilities with an application to the Eady model. Geophys. Astrophys. Fluid Dyn.40, 1–92 (1988).Google Scholar
- M. Gaster,The development of three-dimensional wave packets in a boundary layer. J. Fluid Mech.32, 173–184 (1968).Google Scholar
- A. Bers,Theory of absolute and convective instabilities, Int. Congress on Waves and Instabilities in Plasmas (G. Auer and F. Cap, eds.), Innsbruck, Austria, B1–B52 (1973).Google Scholar
- H. B. Squire,On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proc. Roy. Soc.A 142, 621–628 (1933).Google Scholar
- M. A. Lavrentiev and B. V. Shabat,Methods of the Theory of Functions of a Complex Variable (in Russian). Nauka, Moscow 1973.Google Scholar
- A. Lang,Algebra. Addison-Wesley, Reading 1965.Google Scholar
- J. D. Murray,Asymptotic Analysis. Springer-Verlag, New York 1984.Google Scholar
- D. Gottlieb, M. Y. Hussaini and S. A. Orszag,Theory and applications of spectral methods. InSpectral Methods for Partial Differential Equations (Eds. R. G. Voigt, D. Gottlieb and M. Y. Hussaini), pp. 1–54. SIAM 1984.Google Scholar