Bending instabilities of m=1 mode in disc galaxies: interplay between dark matter halo and vertical pressure

A self-gravitating, differentially rotating galactic disc under vertical hydrostatic equilibrium is supported by the vertical pressure gradient force against the gravitational collapse. Such discs are known to support various bending modes e.g., warps, corrugation, or scalloping (typically, higher order bending modes) of which m=1 bending modes (warps) are the most prevalent ones in galactic discs. Here, we present a detailed theoretical analysis of the bending instability in realistic models of disc galaxies in which an exponential stellar disc is under vertical equilibrium and residing in a cold rigid dark matter halo. A quadratic eigenvalue equation describing the bending modes is formulated and solved for the complete eigen spectrum for a set of model disc galaxies by varying their physical properties such as disc scale-height, and dark matter halo mass. It is shown that the vertical pressure gradient force can excite unstable bending modes in such a disc as well as large scale discrete modes. Further, it is shown that the unstable eigen-modes in a thinner disc grow faster than those in a thicker disc. The bending instabilities are found to be suppressed in discs dominated by massive dark matter halo. We estimate the growth timescales and corresponding wavelength of the m=1 unstable bending modes in Milky Way like galaxies and discuss its implication.