Stability Analysis of Thermohaline Convection With a Time-Varying Shear Flow Using the Lyapunov Method

Abstract

This work demonstrates that the Lyapunov method can effectively identify the growth rate of a linear time-periodic system describing cold fresh water on top of hot salty water with a periodically time-varying background shear flow. We employ a time-dependent weighting matrix to construct a Lyapunov function candidate, and the resulting linear matrix inequalities are discretized in time using the forward Euler method. As the number of temporal discretization points increases, the growth rate predicted from the Lyapunov method or the Floquet theory will converge to the same value as that obtained from numerical simulations. Additionally, the Lyapunov method is used to analyze the most dangerous disturbance, and we also compare computational resource usage for the Lyapunov method, numerical simulations, and the Floquet theory.