The Miles Theorem and New Particular Solutions to the Taylor--Goldstein Equation

Abstract

The linear stability problem of steady-state plane-parallel shear flows of a continuously stratified inviscid incompressible fluid in the gravity field between two immovable impermeable solid planes is studied in and without the Boussinesq approximation. Using the Lyapunov direct method, it is proved that these flows are absolutely unstable in the theoretical sense with respect to small plane perturbations. The applicability domain boundaries of the known necessary condition of the linear instability of steady-state plane-parallel shear flows of a continuously stratified inviscid incompressible fluid in the gravity field is strictly determined in the Boussinesq approximation and without it (Miles theorem). It is found that this theorem is, by its character, both sufficient and necessary statement with respect to some uncompleted unclosed subclasses of studied perturbations. The analytical examples are constructed with the view of illustrations of the mentioned stationary flows and small plane perturbations imposed on these flows. These perturbations are not under the Miles theorem and they increase with time irrespective of the validity of the theoretical linear stability criterion in and without the Boussinesq approximation. Therefore, the results derived earlier by other authors with the help of the method of integral relations for the linear stability problems of steady-state plane-parallel shear flows of a continuously stratified inviscid incompressible fluid demand strict description for the studied partial classes of small plane perturbations as otherwise they can be mistaken.

Keywords

• ideal stratified fluid
• Boussinesq approximation
• stationary plane-parallel shear flows
• stability
• Lyapunov direct method
• instability
• small plane perturbations
• a priori estimate
• Miles theorem
• analytical solutions
• Bessel functions
• Whittaker functions