The adjoint Rayleigh and Orr-Sommerfeld equations: Green function and eigenmodes

The Rayleigh and Orr-Sommerfeld equations are ODEs which arise from the linearized Euler and Navier-Stokes equation around a shear flow. In this paper, we consider the adjoints of the Rayleigh and Orr-Sommerfeld equations on $[0,\mathrm{\infty })$ with respect to the complex $L^2$ product. In the viscous case, we consider a family of viscosity-dependent Navier boundary conditions, which in the limit corresponds to the no-slip condition. We rigorously establish existence and asymptotic properties of their eigenvalues, eigenmodes and Green functions away from critical layers. The adjoint operators prove useful because they also allow us to deduce properties about the kernels and images of the original operators.