Arbitrarily Slow Damping of Internal Waves in a Stably Stratified Flow in Two- and Three -dimensional Bounded Domains1

We study the damping of internal gravity waves in a stably stratified fluid with constant viscosity in twoand three-dimensional bounded domains. For the linearized Navier-Stokes equations for incompressible flow with no-slip boundary conditions that model this fluid, we prove there are non-oscillatory normal modes with arbitrarily small exponential decay rates. The proof in a twoand three-dimensional bounded domain is very different from that for a horizontal periodic layer [4] and depends on a structure theorem for compact operators which are self-adjoint with respect to an indefinite scalar product in a Hilbert space. The proof uses the method of Greenlee [1] and is closely related to results of Pontryagin [5].