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

## TL;DR

This paper links orthogonal polynomial roots to shear instability in quasi-geostrophic shallow water equations, proving unique unstable modes for each wave number and estimating their growth rates.

## Contribution

It establishes a novel connection between orthogonal polynomials and fluid instability analysis, providing rigorous proofs of unstable modes in QG equations.

## Key findings

- Existence of a unique unstable mode for each wave number 0<k<1
- Mathematically rigorous estimates of growth rates
- Connection between polynomial roots and shear instability

## Abstract

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.