LIMIT SPECTRAL CURVES FOR THE ORR-SOMMERFELD OPERATOR AND QUASICLASSICAL EIGENVALUE DISTRIBUTION ALONG THESE CURVES

A A SHKALIKOV

ashkalikov@yahoo.com

Department of Mechanics and Mathematics
Moscow Lomonosov State University
Moscow, 198 899, Russia

We study the well-known in hydrodynamics Orr-Sommerfeld problem on the finite interval $[-1,1]$. We are interested in the question: how do the $\lambda$-eigenvalues of this problem behave as the Reynolds number $R\to\infty$? To answer this question we consider an auxillary model problem

$$i\varepsilon ^2 y''+q(x)y=\lambda y,$$

$$y(-1)=y(1)=0,$$

where $\varepsilon =(\alpha R)^{-1/2}$. The limit behaviour of the eigenvalues of this problem as $\varepsilon \to0$ depends on analytic properties of $q(z)$, namely, on the topology the Stokes lines, which are defined in the complex plane by the equation

$$\mathop{\mathrm{Re}} S(z)=0,\qquad S(z)=\int\limits_{\xi_{\lambda }}^1 \sqrt{i\left(q(\xi)-\lambda \right)}\,d\xi,$$

where $\xi_\lambda$ is the root of $q(\xi)-\lambda =0$.

We investigate the model problem in two cases: 1) an analytic monotone profile $q(z)$, 2) the so-called Couette-Poiseuille profile $q(z)=az^2+bz+c$, $a,b,c\in{\mathbb{R}}$. We prove that the spectrum of the model problem is concentrated along some critical curves in the complex plane, defined by the Stokes lines, which for a monotone profile form a ``spectral tie''. We find the formulas for the eigenvalue distribution along the limit curves which are uniform with respect to $\varepsilon$. Then we prove that the limit spectral curves for the original Orr-Sommerfeld problem coincide with those for the model problem, as well as the main terms in the formulas for the eigenvalue distribution along the curves.