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Abstract

This work identifies instabilities and computes the growth rate of a linear time-varying system using the Lyapunov method. The linear system describes 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 formulation is 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. We also use the Lyapunov method to analyze the instantaneous principal direction of instabilities and compare the computational resources required by the Lyapunov method, numerical simulations, and the Floquet theory.