Pseudospectra of the Orr–Sommerfeld Operator

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Abstract
This paper investigates the pseudospectra,and,the numerical,range of the Orr- Sommerfeld,operator for plane Poiseuille flow. A number,, 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 that also has highly sensitive eigenvalues. It is shown,that the sensitivity of

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