Stability of shallow flows: A weakly nonlinear approach

Linear and weakly nonlinear analysis of shallow mixing layers with free surface is performed in the present paper. It is assumed that the resistance force is not constant in the transverse direction of the flow. Linear stability problem is solved numerically for different values of the parameters of the problem. Assuming that the bed friction number is slightly smaller than the critical value we derive an amplitude evolution equation for the most unstable mode. It is shown that the amplitude equation is the complex Schrodinger equation.