We analyze time series stemming from experiments and direct numerical simulations of hydrodynamic and magnetohydrodynamic turbulence. Simulations are done in periodic boxes, but with a volumetric forcing chosen to mimic the geometry of the flow in th
e experiments, the von Karman swirling flow between two counter-rotating impellers. Parameters in the simulations are chosen to (within computational limitations) allow comparisons between the experiments and the numerical results. Conducting fluids are considered in all cases. Two different configurations are considered: a case with a weak externally imposed magnetic field, and a case with self-sustained magnetic fields. Evidence of long-term memory and $1/f$ noise is observed in experiments and simulations, in the case with weak magnetic field associated with the hydrodynamic behavior of the shear layer in the von Karman flow, and in the dynamo case associated with slow magnetohydrodynamic behavior of the large scale magnetic field.
