Barotropic instability of shear flows. Issue 3 (17th February 2020)

• Record Type: Journal Article
• Authors: Lin, Zhiwu; Yang, Jincheng; Zhu, Hao
• Is Part Of: Studies in applied mathematics. Volume 144:Issue 3(2020)
• Page Start: 289
• Page End: 326
• Publication Date: 2020-02-17
• Subjects: barotropic instability -- fluid dynamics -- Hamiltonian structure -- shear flow; Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathématique; Mathématique appliquée; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519
• DOI: 10.1111/sapm.12297
• Languages: English
• ISSNs: 0022-2526

Abstract

Abstract: We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more accurate numerical computation. The addition of the Coriolis force is found to bring fundamental changes to the stability of shear flows. Moreover, we study dynamical behaviors near the shear flows, including the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian partial differential equations. The last one is to study the singular and nonresonant neutral modes using Sturm‐Liouville theory and hypergeometric functions.