An optimal transient growth of small perturbations in thin gaseous discs

A thin gaseous disc with an almost keplerian angular velocity profile, bounded by a free surface and rotating around point-mass gravitating object is nearly spectrally stable. Despite that the substantial transient growth of linear perturbations measured by the evolution of their acoustic energy is possible. This fact is demonstrated for the simple model of a non-viscous polytropic thin disc of a finite radial size where the small adiabatic perturbations are considered as a linear combination of neutral modes with a corotational radius located beyond the outer boundary of the flow.

💡 Research Summary

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The paper investigates the possibility of large transient amplification of small adiabatic perturbations in a thin, non‑viscous, polytropic accretion disc whose angular velocity profile is almost Keplerian. The disc is bounded radially by a free surface where the pressure vanishes, and its half‑thickness is much smaller than its radial extent (δ ≪ 1). The authors adopt a power‑law rotation law Ω ∝ r⁻ᵠ with 𝑞 = 3/2 + ε²/2 (ε ≪ 1), so that the flow is only slightly steeper than Keplerian. The equilibrium structure (enthalpy, density, surface height) is derived analytically, leading to explicit expressions for the inner (x₁) and outer (x₂) radial boundaries of the disc.

The linear perturbation problem is formulated for axisymmetric disturbances of the form exp(−i ω t + i m φ). After vertical integration, the governing equation reduces to a one‑dimensional Sturm‑Liouville type equation (8) for the pressure perturbation amplitude W(x). The key parameters are the shifted frequency (\bar ω = ω - mΩ) and the epicyclic frequency κ, combined into D = κ² − (\bar ω²). The free‑surface boundary condition (7) requires the Lagrangian pressure perturbation to vanish at x = x₁, x₂.

Instead of seeking the full complex eigen‑spectrum (which would contain exponentially growing or decaying modes), the authors deliberately restrict attention to neutral modes whose corotation radius x_c satisfies x_c > x₂, i.e., the corotation point lies outside the disc. In this regime D < 0 everywhere, the WKBJ approximation is valid, and the solutions are oscillatory with a short radial wavelength comparable to the disc thickness (λ ∼ H). The WKBJ solution (11) provides the local phase integral S₀ and amplitude factor S₁, while the constants C₀ and φ₀ are fixed by the regularity condition at the boundaries.

The central idea is that, although each neutral mode conserves its own energy, the set of modes is non‑orthogonal in the natural energy inner product. Consequently, a suitably chosen linear combination of a finite number of such modes can experience a substantial transient increase in the total acoustic energy. This is the classic non‑modal growth mechanism associated with the non‑normality of the linear operator governing the dynamics.

To quantify the optimal growth, the authors construct the evolution operator U(T) that maps an initial perturbation vector a = {a_j} (coefficients of the selected modes) to its state at time T. The maximal possible amplification over a fixed interval