Nonlinear evolution of three-dimensional unstable disturbances in a sharply stratified shear flow with an inflection-free velocity profile

The horizontal plane-parallel flow with an inflection-free velocity profile is considered in ideal, incompressible fluid which is stably stratified in a thin layer. Such a flow is linearly unstable for an arbitrary bulk Richardson number, and it is three-dimensional disturbances that are most unstable within a wide range of parameters. In the paper, the weakly-nonlinear temporal development of an unstable disturbance in the form of a pair of oblique waves is studied. For this purpose, the evolution equation is derived which has the form of a nonlinear integral equation and is valid for both thin and thick critical layers, including the case where the critical layer width exceeds the stratification layer thickness. Solutions of this equation are studied asymptotically and numerically, and it is shown that during the nonlinear stage of development the disturbance grows, as a rule, explosively.