Long-Wave Stability And Instability Of Periodic Shear Flows For The 2D Navier-Stokes Equations On The $\beta$-Plane

Abstract

We study the spectral stability of periodic shear flows for the two-dimensional Navier--Stokes equations on the $\beta$-plane in the long-wave regime. While it is known that non-rotating periodic shear flows are generically unstable to sufficiently long-wave perturbations, our results show that planetary rotation suppresses this instability mechanism. Using a perturbative analysis based on Kato's reduction, we derive an asymptotic expansion for the principal eigenvalue of the linearized operator that is uniform in the Rossby parameter $\beta \ge 0$, and establish an explicit stability criterion in terms of the shear profile, the viscosity, and the Rossby parameter. Under the critical scaling where the Rossby parameter $\beta$ and the rescaled longitudinal wavenumber $\epsilon$ are of comparable size, this criterion extends Yudovich's classical long-wave instability threshold to rotating flows and reveals a sharp transition between stability and instability governed by the range of the ratio $\beta /\epsilon$. These results provide a rigorous analysis of how viscosity, shear, and planetary rotation interact to determine long-wave stability on the $\beta$-plane.

28 pages