Inertial instability of Couette flow with Coriolis force
Yanlong Fan, Daozhi Han, Quan Wang
TL;DR
The paper addresses inertial instability of 3D Couette flow under Coriolis forcing by showing nonlinear Hadamard instability in a precise range of Coriolis strength. It develops a linear theory where the operator $\mathscr{L}$ has only continuous spectrum, yet unstable dynamics arise from $\varepsilon$-pseudo-eigenfunctions localized near fixed frequencies, enabling a bootstrap argument. The main contribution is proving nonlinear velocity instability for $f\in\left(\frac{2}{17}(5-2\sqrt{2}),\frac{2}{17}(5+2\sqrt{2})\right)$ with $\nu>0$, i.e., a Hadamard-unstable Couette flow despite the absence of genuine eigenmodes; this reveals a significant non-normal growth mechanism driven by the Coriolis force. The results emphasize that linear pseudo-modes can dominate nonlinear dynamics in rotating shear flows and highlight the limitations of inviscid damping and enhanced dissipation as stabilizing effects in this regime, with potential implications for geophysical and astrophysical rotating systems.
Abstract
We analyze the nonlinear inertial instability of Couette flow under Coriolis forcing in \(\mathbb{R}^{3}\). For the Coriolis coefficient \(f \in (0,1)\), we show that the non-normal operator associated with the linearized system admits only continuous spectrum. Hence, there are no exponentially growing eigenfunctions for the linearized system. Instead, we construct unstable solutions in the form of pseudo-eigenfunctions that exhibit non-ideal spectral properties. Then through a bootstrap argument and resolving the challenges posed by the non-ideal spectral behavior of pseudo-eigenfunctions, we establish the velocity instability of Couette flow in the Hadamard sense for $ f \in \Big(\frac{2}{17} \left(5-2 \sqrt{2}\right), \frac{2}{17} \left(5 + 2 \sqrt{2}\right) \Big)$.