A critical assessment of the parabolized stability equations

The parabolized stability equations (PSE) are a ubiquitous tool for studying the stability and evolution of disturbances in weakly nonparallel, convectively unstable flows. The PSE method was introduced as an alternative to asymptotic approaches to these problems. More recently, PSE has been applied with mixed results to a more diverse set of problems, often involving flows with multiple relevant instability modes. This paper investigates the limits of validity of PSE via a spectral analysis of the PSE operator. We show that PSE is capable of accurately capturing only disturbances with a single wavelength at each frequency and that other disturbances are not necessarily damped away or properly evolved, as often assumed. This limitation is the result of regularization techniques that are required to suppress instabilities arising from the ill-posedness of treating a boundary value problem as an initial value problem. These findings are valid for both incompressible and compressible formulations of PSE and are particularly relevant for applications involving multiple modes with different wavelengths and growth rates, such as problems involving multiple instability mechanisms, transient growth, and acoustics. Our theoretical results are illustrated using a generic problem from acoustics and a dual-stream jet, and the PSE solutions are compared to both global solutions of the linearized Navier-Stokes equations and a recently developed alternative parabolization.