Oscillating dynamo models with a time-nonlocal $α$ effect
The traditional $α$ effect is extended by the first temporal derivative, so that in turbulent flows the magnetic field entering the induction process is effectively sampled in the past. As a consequence, dynamo excitation becomes easier and cycle times are prolonged relative to solutions obtained from the local standard formulation. The temporal shape of the magnetic cycles is also modified, with rise times becoming shorter than decline times, so that the profiles acquire a characteristic ``sawtooth'' appearance. In the nonlocal formulation, the induced magnetic-field amplitudes acquire an additional dependence on the correlation time. For fixed dynamo number, the field amplitude reaches a maximum at a certain correlation time. For correlation times larger than this peak value, the model can reproduce Waldmeier-type trends known from solar sunspot observations. If the actual correlation time fluctuates around this value, the long-term magnetic cycle can occasionally enter weak-cycle episodes reminiscent of grand minima such as the Maunder minimum. For smaller dynamo numbers close to the critical one, however, rather long correlation times are required for this behaviour to occur.