Vortex-wave interactions: long-wavelength streaks and spatial localization in natural convection

The unidirectional shear flow driven by buoyancy effects in a vertical channel when a temperature difference is maintained between the walls of the channel is considered. The flow is unstable to waves which interact to reinforce the original flow and make it 'streaky'. Such 'vortex-wave' interactions have been the subject of much recent research but little is yet known about what happens when the wavelength of the roll/streak flow becomes large. An asymptotic structure for long-wavelength interactions is derived and the tendency of the fluid to resist this state and the flow to become localized is revealed. Here the high-Grashof-number limit is considered and it is shown how a self-sustained process can occur with vortices interacting with a wave system in a manner similar to that discussed by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641-666) and Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178-205). The work is closely related to numerical simulations of self-sustained processes in for example Couette flow but the fact that the basic flow here is generated by buoyancy effects enables us to make analytical progress. It is shown that the wave part of the interaction process has a flat critical layer and its wavelength is twice that of the streaky flow which supports it. Such subharmonic vortex-wave/self-sustained process interactions have not been previously identified.