Pseudospectra of the Orr-Sommerfeld Operator

This paper investigates the pseudospectra and the numerical range of the OrrSommerfeld operator for plane Poiseuille flow. A number z E C is in the E-pseudospectrum of a matrix or operator A if Il(zI A)-'II 2 E-l, or, equivalently, if z is in the spectrum of A + E for some perturbation E satisfying IlEll 5 E. The numerical range of A is the set of numbers of the form (Au, u), where (., .) is the inner product and u is a vector or function with llull = 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 that also has highly sensitive eigenvalues. It is shown that the sensitivity of the eigenvalues can be related qualitatively to solutions of the Airy equation that satisfy boundary conditions to within an exponentially small factor. As an application, the growth of initial perturbations is considered. The near-linear dependence of the eigenfunctions implies that there is potential for energy growth, even when all the eigenmodes decay exponentially. Necessary and sufficient conditions for no energy growth can be stated in terms of both the pseudospectra and the numerical range. Bounds on the growth can be obtained using the pseudospectra.