发表机构
University of Pittsburgh（匹兹堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对β平面二维纳维-斯托克斯方程的周期剪切流，通过摄动分析推导线性化算子主特征值的渐近展开，明确了黏度、剪切与旋转对长波稳定性的竞争作用，给出了旋转流的显式稳定性准则。
AI 中文摘要
我们研究β平面上二维纳维-斯托克斯方程周期剪切流在长波 regime 下的谱稳定性。已知非旋转周期剪切流通常对足够长波的扰动不稳定。我们证明行星旋转可抑制这种不稳定性：利用基于Kato约化的摄动分析，我们推导线性化算子主特征值的渐近展开，并得到关于剪切剖面、黏度和科里奥利参数的显式稳定性准则。在科里奥利效应与长波扰动量级相当的临界标度下，该准则将Yudovich的经典长波不稳定性阈值推广至旋转流，并揭示由比值K决定的稳定性与不稳定性之间的尖锐转变。这些结果严谨阐释了黏度、剪切和旋转如何相互作用以确定β平面上的长波稳定性。
英文摘要
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 $β\geq 0$, 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 $\varepsilon$ 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 $β/\varepsilon$. These results provide a rigorous analysis of how viscosity, shear, and planetary rotation interact to determine long-wave stability on the $β$-plane.
Comments28 pages
