On the modulation instability of the shallow wake flows via a higher-order extended Ginzburg–Landau model

This paper focuses on the modulation instability and nonlinear evolution of shallow wake flows. Under the f-plane approximation, starting from the two-dimensional viscous shallow water equations, a higher-order extended Ginzburg–Landau equation is derived using the multi-scale expansion method. This equation retains the effects of weak viscosity, bottom friction, and higher-order nonlinear coupling on the evolution of the wake flow envelope, thereby providing a theoretical basis for revealing the formation mechanism and evolution process of the fluid structure behind obstacles in real oceans or lakes. Subsequently, the equation is solved using the finite difference method combined with a semi-implicit time stepping. Further, based on the continuous wave background, the dispersion relation and gain spectrum of the modulation instability are obtained. The results show that the system exhibits a typical symmetric double-peak gain structure, where the stability parameter S serves as a key role for the gain peak and unstable bandwidth. Additionally, the evolution of the stream function is mainly governed by the stability parameter S and the velocity ratio R. As the stability parameter increases, the flow field structure gradually evolves from weak stripe-like wave trains to localized wave packets near the wake center, thereby promoting the formation of vortex pairs.