Instability and nonlinear evolution of a density-driven coastal current with a surface front in a two-layer ocean

Abstract

Abstract Coupled equations describing the nonlinear evolution of a disturbance in a two-layer coastal current with a surface front are derived by assuming that the alongshore scale of the disturbance is much longer than the current width, the lower layer is much thicker than the upper layer and the potential vorticity in the upper layer is zero. Linear stability and the characteristics of the cnoidal wave and soliton solutions governed by these equations are examined analytically, and the nonlinear evolution of disturbance is discussed using numerical methods. The main results are as follows. (1) The current is linearly stable when the current speed is greater than a certain critical value. (2) Even in such a situation, there is no stable nonlinear solution whose amplitude lies between certain critical values, a c and a′c, which are functions of the wavelength and the basic flow speed. (3) A solitary disturbance whose amplitude lies between a c and a′c grows with an eddy-pair-like structure in the lower layer. After its amplitude exceeds a′c, a soliton with a large amplitude separates from the eddy-pair-like structure and propagates by itself. (4) In the linearly unstable case, a wave-like perturbation with small amplitude grows. However, this unstable mode does not grow indefinitely but reaches a maximum amplitude, having exceeded the critical amplitude, a′c. Its amplitude then oscillates with time. (5) When several unstable waves have been given as the initial condition, the unstable waves interact nonlinearly with each other and solitary disturbances are produced. These solitary disturbances also repeat the growth and decay, and a turbulence-like phenomenon occurs.