A Connection Between Orthogonal Polynomials and Shear Instabilities in the Quasi-geostrophic Shallow Water Equations

In this paper we demonstrate a connection between the roots of a certain sequence of orthogonal polynomials on the real line and the linear instability of a $x$-directionally homogeneous background velocity profile $u^b(x,y) = \cos(y)$ in the quasi-geostrophic shallow water (QG) equation in a domain with periodic boundaries in the $y$-direction. Using the relationship we establish, we then prove that there exists a unique unstable mode for each horizontal wave number $0<k<1$ and provide mathematically rigorous estimates of the associated growth rate.