126a Lermontov St., Irkutsk, 664033, Russia
Institute of Solar-Terrestrial Physics
Churilov S. M.
Resonant three–wave interaction of waves having a common critical layer
2011, Vol. 7, No. 2, pp. 257-282
Within the context of the weakly nonlinear approach, the leading nonlinear contribution to the development of unstable disturbances in shear flows should be made by resonant three-wave interaction, i.e., the interaction of triplets of such waves that have a common critical layer (CL), and their wave vectors form a triangle. Surprisingly, the subharmonic resonance proves to be the only such interaction that has been studied so far. The reason for this is that in many cases, the requirement of having a common CL produces too rigid selection of waves which can participate in the interaction. We show that in a broad spectral range, Holmboe waves in sharply stratified shear flows can have a common CL, and examine the evolution of small ensembles consisting of several interrelated triads of those waves. To do this, the evolution equations are derived which describe the three-wave interaction and have the form of nonlinear integral equations. Analytical and numerical methods are both used to find their solutions in different cases, and it is shown that at the nonlinear stage disturbances increase, as a rule, explosively.
Churilov S. M.
Nonlinear evolution of three-dimensional unstable disturbances in a sharply stratified shear flow with an inflection-free velocity profile
2009, Vol. 5, No. 2, pp. 159-182
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.