  At the heart of today's solar magnetic field evolution models lies the alpha
dynamo description. In this work, we investigate the fate of alpha-dynamos as
the magnetic Reynolds number $Rm$ is increased. Using Floquet theory, we are
able to precisely quantify mean field effects like the alpha and beta effect
(i) by rigorously distinguishing dynamo modes that involve large scale
components from the ones that only involve small scales, and by (ii) providing
a way to investigate arbitrary large scale separations with minimal
computational cost. We apply this framework to helical and non-helical flows as
well as to random flows with short correlation time. Our results determine that
the alpha-description is valid for $Rm$ smaller than a critical value $Rm_c$ at
which small scale dynamo instability starts. When $Rm$ is above $Rm_c$ the
dynamo ceases to follow the mean field description and the growth rate of the
large scale modes becomes independent of the scale separation while the energy
in the large scale modes is inversely proportional to the square of the scale
separation. The results in this second regime do not depend on the presence of
helicity. Thus alpha-type modeling for solar and stellar models needs to be
reevaluated and new directions for mean field modeling are proposed.
