Flow Generation via Catastrophic Loss of Equilibrium in Weakly-Rotating Self-Gravitating Fluids: A Minimal Idealized Model

TL;DR

The paper develops a minimal idealized gravito-magnetofluid model for catastrophic flow generation in slowly rotating, self-gravitating neutral fluids around massive objects, drawing a strong analogy to gravito-electromagnetism and Beltrami-Bernoulli equilibria. It derives a Double Beltrami (DB) equilibrium that decomposes into macro- and micro-scale components, and shows that slow parameter evolution can trigger a catastrophic loss of equilibrium when one component vanishes, converting gravito-magnetic energy into kinetic flow energy. The critical condition is obtained from a quadratic in the macro/micro-scale eigenvalues, yielding a critical parameter $\lambda_{crit}$ that marks the DB-to-SB transition and the associated energy transfer, which may drive macro-scale flows or micro-scale turbulence. The findings highlight self-gravity as a viable energy source for astrophysical flows and jets, motivate time-dependent dynamical analyses, and point to links with dynamo-like processes, with relevance for accretion disks, galactic flows, and rotating stellar atmospheres.

Abstract

This paper explores the catastrophic energy transformations, in particular the ones leading to the generation of a flow in a weakly rotating self-gravitating fluid/gas found, for instance, in the vicinity of a massive compact object. Because of the similarity in the governing equations, the system dynamics is worked out exactly in parallel to the methods developed for investigating catastrophic relaxation in stellar plasmas [1-3]. In the latter a more ``complex" equilibrium state, on slow changes in the environment, can lose its equilibrium (catastrophe), and transform to a less complex state with a very different energy mix from the original. It is shown that a similar transformation in the weakly rotating self-gravitating fluid/gas will convert much of its gravitation energy into kinetic energy in the flow. Since flows are a perennial ingredient of high-energy astrophysical systems, the energy transformation processes revealed in present study, can advance our understanding of a variety of them. Some particularly relevant examples are: macro-scale flows / structures in galaxies, accretion discs, and the dynamics and stability of a rotating star / its atmosphere.

Figures (6)

• Figure 1: Plots for micro-scale $\mu$ and $C_{\lambda,\mu}$ vs macro-scale $\lambda$ for catastrophic rearrangement of the original state; $E = -3 > E_{crit}, \ h_{min} = - 0.5 , \ \kappa = 0.04 , \ b = 0.5$. The critical point $\lambda_{crit} = -0.298$ at which the transition happens can be observed on the plot of $C_{\mu}$; the macro-scale constituent is fully vanished.
• Figure 2: Plots for flow and gravito-magnetic energies vs $\lambda$ for catastrophe-prone parameters (see Fig1) - the micro-scale/turbulent flow generation is clearly seen.
• Figure 3: Plots for micro-scale $\mu$ and $C_{\lambda,\mu}$ vs macro-scale $\lambda$ for catastrophic rearrangement of the original state; $E = -13 , \ h_{min} = - 0.5 , \ \kappa = 0.2 , \ b = 5$. The point $\lambda_{crit} = -3.14$ at which the transition to turbulent state happens can be observed on the plot for $C_{\lambda }$.
• Figure 4: Plots for flow and gravito-magnetic energies vs $\lambda$ for the scenario leading to final turbulent state (see parameters in Fig.3) - flow energy is totally micro-scale.
• Figure 5: Plots for micro-scale $\mu$ and $C_{\lambda,\mu}$ vs macro-scale $\lambda$ for catastrophic rearrangement of the original state; $E = -13 , \ h_{min} = - 0.5 , \ \kappa = 0.2 , \ b = 5$. The point $\lambda_{crit} = -1.86$ at which the transition to macro-scale structure dominant state happens can be observed on the plot for $C_{\lambda }$ (when we follow the control parameter $\lambda$ variation from right to left - following the increase of inverse-length scale $|\lambda |$).