Nonlinear stability of quasi-Keplerian Taylor–Couette flow for large axisymmetric disturbances at arbitrary Reynolds numbers: Rigorous mathematical proof with novel physical insights

Abstract The quasi-Keplerian (q-K) regime of Taylor–Couette (T–C) flow is of substantial relevance to accretion disc rotations, while it is an exceptionally surprising flow regime in which extensive experiments and simulations have not captured any sustained hydrodynamic turbulence up to date. It led to a long-standing open problem on nonlinear hydrodynamic stability of q-K flows at high Reynolds numbers, which is fundamentally important in both astrophysics and fluid dynamics but challenging to solve. The present study establishes a novel viscous energy-Casimir identity for nonlinear disturbances that analytically implies, for the first time, nonlinear hydrodynamic stability in the entire q-K regime for infinitely large axisymmetric disturbances at arbitrarily high Reynolds numbers. In addition, the novel viscous identity theoretically pinpoints the exact stability mechanisms that perfectly match numerical simulations. The rigorous analysis sets the first-principle theoretical cornerstone for nonlinear hydrodynamic stability of q-K flows.