Linear stability of a simple shear layer between two parallel streams in a shallow water flow

The instability of shear layers in fluid flows is a crucial factor in forming vortices, leading to the development of turbulence. Analyzing shear layer instabilities of atmospheric and oceanic flows contributes to accurately forecasting weather and predicting tsunamis. Our research focuses on the stability of a shear layer sandwiched between two semi-infinite layers. The velocity profile of the shear layer is assumed to be linearly dependent on the vertical coordinate, while the velocity of the other layers remains uniform with different strengths. We employ the shallow water equations to analyze the interface stability of fluid layers. The dispersion relation between wave frequency and other wave characteristics is obtained involving the Whittaker functions and their first derivatives. The stability is analyzed by using the appropriate limits of the Whittaker functions. Our study provides a deeper understanding of the stability of shear layers and their implications in our lives.