Global transient dynamics of three-dimensional hydrodynamical disturbances in a thin viscous accretion disk

Thin viscous Keplerian accretion disks are considered asymptotically stable, even though they can show significant dynamic activity on short timescales. In this paper the dynamics of non-axisymmetric hydrodynamical disturbances of disks are investigated analytically building upon the steady state three-dimensional structure and evolution of axisymmetric perturbations explored in previous work. Assuming a polytropic equation of state solutions are found by means of an asymptotic expansion in the small parameter measuring the ratio of the disk thickness to characteristic radius. In-depth analysis shows that every perturbation that disturbs the radial velocity induces significant transient growth in the (acoustic) energy of the evolving disturbance. This effect is most evident in the density and vertical velocity. The transient growth observed is tied to the non-separable nature of the solutions where, in particular, pattern evolution is controlled by a similarity variable composed of the radial coordinate and time. This leads to growing winding perturbations that display successive radial peaks and troughs. We argue that these transient non-axisymmetric structures may precipitate secondary instabilities which, consequently, may be a critical element for a new alternative picture of turbulence arousal in non-magnetized astrophysical disks.

💡 Research Summary

The paper investigates the linear dynamics of three‑dimensional, non‑axisymmetric hydrodynamic disturbances in a thin, viscous Keplerian accretion disk. While such disks are asymptotically stable in the classical sense, the authors demonstrate that certain perturbations can undergo substantial transient amplification on dynamical timescales, potentially seeding secondary instabilities that lead to turbulence in non‑magnetized disks.

The analysis builds on a previously derived steady‑state three‑dimensional structure and on the evolution of axisymmetric perturbations. Assuming a polytropic equation of state (P=K\rho^{1+1/n}), the authors perform an asymptotic expansion in the small parameter (\epsilon = H/R), where (H) is the disk semi‑thickness and (R) a characteristic radius. The expansion separates the governing equations into orders of (\epsilon); the zeroth‑order recovers the axisymmetric equilibrium, while the first‑order captures the non‑axisymmetric response.

A crucial feature of the solution is its non‑separability: the perturbation fields depend on a similarity variable (\xi = r - q\Omega t) (with (q) the shear parameter and (\Omega) the Keplerian angular velocity). This variable describes a winding pattern that drifts radially as time progresses, producing a series of alternating peaks and troughs in the radial direction. Because the governing equations cannot be factorised into pure radial and temporal parts, the disturbance energy evolves in a way that is not captured by simple exponential growth or decay.

When a radial velocity perturbation (\delta v_r) is introduced, the analysis shows that the associated acoustic energy—comprising kinetic and compressional contributions—grows algebraically with time (approximately as (t) or (t^2) depending on the mode). The density perturbation (\delta\rho) and the vertical velocity perturbation (\delta v_z) exhibit the most pronounced transient amplification. Viscous damping eventually limits the growth, but for realistic disk parameters (e.g., polytropic index (n=3/2) and an (\alpha) viscosity of order (10^{-3})) the amplification can reach an order of magnitude increase over a timescale of several to ten orbital periods.

The authors argue that this transient growth provides a reservoir of energy that can trigger secondary, non‑linear instabilities such as Kelvin‑Helmholtz, Rossby‑wave, or other shear‑driven mechanisms. In the absence of magnetic fields, these secondary processes could supply the missing pathway to sustained turbulence in astrophysical disks. Thus, the work proposes an alternative route to turbulence arousal that does not rely on the magnetorotational instability (MRI) but instead on the intrinsic hydrodynamic dynamics of thin, viscous disks.

In summary, the paper offers a rigorous analytical framework for understanding how non‑axisymmetric disturbances in thin Keplerian disks can experience significant transient growth, highlights the role of a similarity‑variable‑driven winding pattern, and suggests that such growth may be a crucial precursor to turbulence in non‑magnetized accretion environments.