Symmetries of the linearized Navier-Stokes equation and their impact on the eigenvalue spectrum of stability theory

It has long been known that the symmetries of the linearized Navier-Stokes equations (LNSE) form the basis for their linear stability theory. In particular, the translation in time and space plus the scaling symmetry of the dependent variables implies the Fourier, i.e. normal-mode approach. Here we re-consider symmetries of the LNSE on its significance for the dispersion relation D(α,ω;Re)=0. If these symmetries are transferred to the description in Fourier space, important physical restrictions arise. For example, it follows from the Galilei invariance that the dispersion relation must be locally analytic or holomorphic at the complex eigenvalues ω. Singular eigenvalues are thus physically excluded, similarly to the constraints on the contour of the integration path in the complex $\omega$-plane induced by the causality condition, avoiding the crossing of branch cuts and discontinuities982. The above result is also independent of whether the instabilities are purely temporal or spatio-temporal, as used in Briggs theory, for example. In fact, however, it turns out that purely spatially developing modes cannot be used in a meaningful way.