An approximate analytic solution of a set of nonlinear model αω-dynamo equations for marginally unstable systems

Abstract

We obtain an approximate analytic solution of a set of nonlinear model αω-dynamo equations. The reaction of the Lorentz force on the velocity shear which stretches and, hence, amplifies the magnetic field, is incorporated into the model. To single out the effect of the Lorentz force on the ω-effect, the effect of the Lorentz force on the α-effect is neglected in this study. The solution represents a nonlinear oscillation with the amplitude and period determined by the dynamo numberN. The amplitude is proportional toN−1, while the period is almost exactly the same as the dissipation time of the unstable mode [proportional toN; note the linear oscillation period is proportional toN/(N−1) which is quite different for the solar situation whereN∼1].