An approximate analytic solution of a set of nonlinear model \(\alpha\) \(\omega\)-dynamo equations for marginally unstable systems (Q1115306)

We obtain an approximate analytic solution of a set of nonlinear model \(\alpha\) \(\omega\)-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 \(\omega\)-effect, the effect of the Lorentz force on the \(\alpha\)-effect is neglected in this study. The solution represents a nonlinear oscillation with the amplitude and period determined by the dynamo number N. The amplitude is proportional to N-1, while the period is almost exactly the same as the dissipation time of the unstable mode [proportional to N; note the linear oscillation period is proportional to N/(N-1) which is quite different for the solar situation where \(N\sim 1]\).