Dark-soliton asymptotics for a repulsive nonlinear system in a baroclinic flow

In geophysical hydrodynamics, baroclinic instability denotes the process in which the perturbations draw the energy from the mean flow potential power. Researchers focus their attention on the baroclinic instability in the Earth's atmosphere and oceans for the meteorological diagnosis and prediction. Under investigation in this paper is a repulsive nonlinear system modeling the marginally unstable baroclinic wave packets in a baroclinic flow. With respect to the amplitude of the baroclinic wave packet and correction to the mean flow resulting from the self-rectification of the baroclinic wave, we present a Lax pair with the changeable parameters and then derive the N-dark-dark soliton solutions, where N is a positive integer. Asymptotic analysis on the N-dark-dark solitons is processed to obtain the algebraic expressions of the N-dark-dark soliton components. We find that the obtained phase shift of each dark-dark soliton component is relevant with the N − 1 spectral parameters. Furthermore, we take N = 3 as an example and graphically illustrate the 3-dark-dark solitons, which are consistent with our asymptotic-analysis results. Our analysis may provide the explanations of the complex and variable natural mechanisms of the baroclinic instability.