Toward detailed prominence seismology II. Charting the continuous magnetohydrodynamic spectrum

Context. Starting from accurate magnetohydrodynamic flux rope equilibria containing prominence condensations, we initiate a systematic survey of their linear eigenoscillations. This paves the way for more detailed prominence seismology, which thus far has made dramatic simplifications about the prevailing magnetic field topologies. Aims. To quantify the full spectrum of linear MHD eigenmodes, we require knowledge of all flux-surface localized modes, charting out the continuous parts of the MHD spectrum. We combine analytical and numerical findings for the continuous spectrum for realistic prominence configurations, where a cool prominence is embedded in a hotter cavity, or where the flux rope contains multiple condensations supported against gravity. Methods. The equations governing all eigenmodes for translationally symmetric, gravitating equilibria containing an axial shear flow, are analyzed, along with their flux-surface localized limit. The analysis is valid for general 2.5D equilibria, where either density, entropy, or temperature vary from one flux surface to another. We analyze the intricate mode couplings caused by the poloidal variation in the flux rope equilibria, by performing a small gravity parameter expansion. We contrast the analytical results with continuous spectra obtained numerically. Results. For equilibria where the density is a flux function, we show that continuum modes can be overstable, and we present the stability criterion for these convective continuum instabilities. Furthermore, for all equilibria, a four-mode coupling scheme between an Alfvenic mode of poloidal mode number m and three neighboring (m - 1, m, m + 1) slow modes is identified, occurring in the vicinity of rational flux surfaces. For realistically structured prominence equilibria, this coupling is shown to play an important role, from weak to stronger gravity parameter g values. The analytic predictions for small g are compared with numerical spectra, and progressive deviations for larger g are identified. Conclusions. The unstable continuum modes could be relevant for short-lived prominence configurations. The gaps created by poloidal mode coupling in the continuous spectrum need further analysis, as they form preferred frequency ranges for global eigenoscillations.