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 that only exhibit a linear dependence of the dependent variables in their infinitesimals, the associated differential condition to generate invariant solutions poses a linear eigenvalue problem. If the combined operator of these appendant symmetries is further 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 respectively. By first studying the one-dimensional (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 particular or general solutions for linear dynamical 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 a common set of symmetries that either induces a purely temporal or a purely spatial decomposition, and thus, besides these common ones, 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, which act as complementary coordinates to the ones in physical space, each single mode can also acquire non-system parameters, so-called regularization parameters, which can be picked up during the construction process depending on the choice of its underlying symmetry. As will be demonstrated, these symmetry-induced parameters are all irrelevant in physical space, 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.