Dynamically relevant recurrent flows obtained via a nonlinear recurrence function from two-dimensional turbulence

Abstract

This paper demonstrates the efficient extraction of unstable recurrent flows from two-dimensional turbulence by using nonlinear triads to diagnose recurrence in direct numerical simulations. Nearly recurrent episodes are identified from simulations and then converged using a standard Newton- GMRES-hookstep method, however with much greater diversity than previous studies which performed this 'recurrent flow analysis'. Unstable periodic and relative periodic orbits are able to be identified which span larger values of dissipation rate, i.e. corresponding to extreme bursting events. The triad variables are found to provide a more natural way to weight the greater variety of spatial modes active in such orbits than a standard Euclidian norm of complex Fourier amplitudes. Moreover the triad variables build in a reduction of the continuous symmetry of the system which avoids the need to search over translations when obtaining relative periodic orbits. Armed with these orbits we investigate optimal weightings when reconstructing the statistics of turbulence and suggest that, in fact, a simple heuristic weighting based on the solution instability provides a very good prediction, provided enough dynamically relevant orbits are included in the expansion.