Complex patterns in rotating Rayleigh-B\'{e}nard convection.

Flows induced by thermal buoyancy in rotating systems play an important role in many industrial processes as well as in numerous problems in geophysical and astrophysical fluid dynamics. Thermal convection in a horizontal fluid layer heated from below and rotating about a vertical axis has also become a prime example in theories of pattern formation and of the transition to spatio-temporal chaos. The K\"{u}ppers-Lortz instability occurs in a rotating Rayleigh-B\'{e}nard convection and is a paradigmatic example of spatiotemporal chaos [G. K\"{u}ppers and D. Lortz, J. Fluid Mech. 35, 609 (1969)]. Surprisingly and contrary to this established scenario, Bajaj et al. 1998 [K. Bajaj, et al., Phys. Rev. Lett. 81 (1998)] observed experimentally in a cylinder square patterns in the range of parameters where K\"{u}ppers-Lortz instability was expected. In this work we study numerically square patterns properties by taking into account realistic boundary conditions. The Navier-Stokes and heat transport equations have been solved in the Oberbeck-Boussinesq approximation using an efficient pseudo-spectral technique. All the characteristics of the pattern show that it appears when the flow is laterally confined.