An Investigation of Shear Instability in a Shallow Water

Part II. Numerical experiment

Abstract

Using a channel model in a shallow water, we numerically consider characteristics of finite-amplitude gravity waves which were found to be unstable waves in a constant shear flow by the linear analysis. Numerical integration is performed at Froude number equals to five for both inviscid and viscous case. In the inviscid run, it is shown that the first appeared mode has the growth rate and the structure which are the same as those expected from the linear analysis. Energy budget shows that disturbances extract their energy from the additional part of the mean kinetic energy as in the linear analysis.
In the viscous run (Re=3000), it is shown that disturbance energy reaches a quasisteady state. Energy budget shows that energy is supplied to the mean kinetic energy, converted to the eddy kinetic energy, and then dissipated by the viscosity acting on the disturbances. Momentum budget indicates that these gravity waves can mix the averaged momentum permanently.
At a later stage of time integration, the disturbance energy oscillates. A linear stability analysis for the quasi-steady state is examined, and it shows that the oscillation is produced by unstable sub-harmonics.
Shape of disturbance depth near the boundaries and change of mean depth are also discussed.