Pseudospectra of the Orr-Sommerfeld Operator

This paper investigates the pseudospectra and the numerical range of the Orr–Sommerfeld operator for plane Poiseuille flow. A number $z \in {\bf C}$ is in the $\epsilon $-pseudospectrum of a matrix or operator A if $\| ( zI - A )^{ - 1} \| \geq \epsilon ^{ - 1} $, or, equivalently, if z is in the spectrum of $A + E$ for some perturbation E satisfying $\| E \| \leq \epsilon $. The numerical range of A is the set of numbers of the form $( Au,u )$, where $( \cdot , \cdot )$ is the inner product and u is a vector or function with huh $\| u \| = 1$.The spectrum of the Orr–Sommerfeld operator consists of three branches. It is shown that the eigenvalues at the intersection of the branches are highly sensitive to perturbations and that the sensitivity increases dramatically with the Reynolds number. The associated eigenfunctions are nearly linearly dependent, even though they form a complete set.To understand the high sensitivity of the eigenvalues, a model operator is considered, related to the Airy equation tha...