Long-Wave Stability And Instability Of Periodic Shear Flows For The 2D Navier-Stokes Equations On The β-Plane
Abstract
We study the spectral stability of periodic shear flows for the two-dimensional Navier--Stokes equations on the β-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 β≥ 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 β and the rescaled longitudinal wavenumber ε 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 β/ε. These results provide a rigorous analysis of how viscosity, shear, and planetary rotation interact to determine long-wave stability on the β-plane.
Related
- Time periodic problem of the Navier-Stokes equations in an exterior domain with periodically moving boundary
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- Stability threshold of the Poiseuille flow in 2D Navier-Stokes Boussinesq equations
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- Stability Analysis of Bounded Solutions to 2-D Incompressible Navier-Stokes Equations in L^2_uloc Spaces
BibTeX
@misc{chen2026longwave,
title = {{Long-Wave Stability And Instability Of Periodic Shear Flows For The 2D Navier-Stokes Equations On The $β$-Plane}},
author = {Robin Ming Chen and Tian Dai and Dehua Wang and Weiqiang Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.06899},
eprint = {2608.06899},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.06899},
}
RIS
TY - UNPB
TI - Long-Wave Stability And Instability Of Periodic Shear Flows For The 2D Navier-Stokes Equations On The β-Plane
AU - Robin Ming Chen
AU - Tian Dai
AU - Dehua Wang
AU - Weiqiang Wang
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2608.06899
UR - https://arxiv.org/abs/2608.06899
AB - We study the spectral stability of periodic shear flows for the two-dimensional Navier--Stokes equations on the β-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 β≥ 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 β and the rescaled longitudinal wavenumber ε 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 β/ε. These results provide a rigorous analysis of how viscosity, shear, and planetary rotation interact to determine long-wave stability on the β-plane.
ER -