ON THE NONLINEAR STABILITY OF THE 1/1/1 ABC FLOW

ABC flows which can be considered as prototypes for the study of the onset of three-dimensional spatio-temporal turbulence are known both analytically and numerically to be linearly unstable. We analyze the nonlinear evolution of the ABC flow A1 with A = B = C = 1 and with characteristic wavenumber k0 = 1 in the interval of Reynolds number 13 less-than-or-equal-to R less-than-or-equal-to 50. We solve numerically the forced Navier-Stokes equations with periodic boundary conditions for up to 9.9 X 10(4) eddy turnover times. Bifurcations towards progressively more complex flows obtain, with a relaminarization window, loss of symmetries, and chaotic oscillations probably revealing an underlying heteroclinic structure. In the chaotic regime, only three steady solutions emerge besides A1; they consist of a perturbed ABC flow A2 with A = B not-equal C plus cyclic permutations. At 23 less-than-or-equal-to R less-than-or-equal-to 50 an unstructured temporal chaos is observed with the flow still dominated by the largest scales.