  The instability of flows via two-dimensional perturbations is analyzed
theoretically and numerically in a nonmodal framework. The analysis is based on
results obtained in [Verschaeve et al. (2018)] showing the inviscid character
of the growth mechanism of these waves. In particular, it is shown that the
formulation of this growth mechanism naturally reduces to the eigenvalue
problem for the energy bound formulated by [Davis and von Kerczek (1973)]. This
eigenvalue equation thus allows for a broader interpretation. It provides the
discrete growth rates for the base flow in question. In addition to this
eigenvalue problem, a corresponding eigenvalue problem for the phase speed of
the perturbations can be extracted from the equations found in [Verschaeve et
al. (2018)]. These two eigenvalue equations relate to the Hermitian and
skew-Hermitian part, respectively, of the nonmodal equations, cf. [Schmid
(2007)]. In contrast to traditional Orr-Sommerfeld modal analysis, the above
eigenvalue equations define an orthogonal set of eigenfunctions allowing to
decompose the perturbation into base perturbations. As a result of this
decomposition, it can be shown that the evolution of two-dimensional
perturbations is governed by two mechanisms: A first one, responsible for
extracting and returning energy from and to the base flow, in addition to
viscous dissipation and, a second one, responsible for dispersing energy among
the different base perturbations constituting the perturbation. As a general
result, we show that the stability of a flow is not only determined by the
growth rates of the base perturbations, but it is also closely related to its
ability to disperse energy away from the base perturbations with positive
growth. We illustrate the above results by means of three shear flows.
