Mean‐field dynamo with cubic non‐linearity

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Abstract
The turbulent mean‐field dynamo of αω‐type with a mean helicity quadratically dependent on the magnetic field is investigated. A nonlinear system of ordinary differential equation is derived for the amplitudes of the magnetic field expansion over the eigenvectors of the linear problem.In a one‐mode approximation the non‐linear supercritical solution is stable when dγ/dD> 0, where γ is the growth rate of the linear solution andDis the dynamo number.Non‐linear interation between two modes of dipole and quadrupole symmetry is considered. The conditions are found for the synchronization and beats of these modes under the assumption that the quadrupole mode is weaker than the dipole one.

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