Periodic orbits in high symmetric flow

Recently several attempts have been made to identify and analyse periodic orbits in realistic fluid dynamical models. Analysis of such periodic orbits and surfaces connecting them helps to understand phenomena like bursting in shear flows [1] and transition from regular behaviour to spatio-temporal chaos in thermal convection [2]. In the current work we try to identify periodic orbits embedded in turbulence, i.e. periodic orbits which have certain statistical properties in common with fully developed turbulence. In particular we measure the time average energy dissipation rate as a function of the viscosity. The main problem one faces when computing periodic orbits in fluid dynamical problems is the large number of degrees of freedom. In order to perform Newton-Raphson iterations and continuation in one or more parameters we need to integrate the variational equations and therefore the computational effort and memory size required are quadratic in the number of modes. This makes the study of periodic orbits in models of fluid flow with a realistic resolution a hard problem indeed. One way to deal with this problem is to impose spatial symmetries on the solutions. On a triply periodic domain the maximal reduction, still allowing for turbulent flow, is obtained by imposing three finite rotations and three reflection