On physically redundant and irrelevant features when applying Lie-group symmetry analysis to hydrodynamic stability analysis

Abstract

Every linear system of partial differential equations (PDEs) admits a scaling symmetry in its dependent variables. In conjunction with other admitted symmetries of linear type, the associated invariant solution condition poses a linear eigenvalue problem. If this problem is structured such that the spectral theorem applies, then the general solution of the considered linear PDE system is obtained by summing or integrating the invariant eigenfunctions (modes) over all eigenvalues, depending on whether the spectrum of the operator is discrete or continuous. By first studying the 1-D diffusion equation as a demonstrating example, this method is then applied to a relevant 2-D problem from hydrodynamic stability analysis. The aim of this study is to draw attention to the following two independent facts that need to be addressed in future studies when constructing solutions for linear PDEs with the method of Lie-symmetries: (i) Although each new symmetry leads to a mathematically different spectral decomposition, they may all be physically redundant to standard ones and do not reveal a new physical mechanism behind the overall considered dynamical process, as incorrectly asserted, for example, in the recent studies by the group of Oberlack et al. Hence, with regard to linear stability analysis, no physically "new" or more "general" modes are generated by this method than the ones already established. (ii) Next to the eigenvalue parameters, each single mode can also acquire non-system parameters, depending on the choice of its underlying symmetry. These symmetry-induced parameters, however, are all physically irrelevant, since their effect on a single mode will cancel when considering all modes collectively. In particular, the collective action of all single modes is identical for all symmetry-based decompositions and thus indistinguishable when considering the full physical fields.