  Upper bounds are derived on the amount of magnetic energy that can be
generated by dynamo action in collisional and collisionless plasmas with and
without external forcing. A hierarchy of mathematical descriptions is
considered for the plasma dynamics: ideal MHD, visco-resistive MHD, the
double-adiabatic theory of Chew, Goldberger and Low (CGL), kinetic MHD, and
other kinetic models. It is found that dynamo action is greatly constrained in
models where the magnetic moment of any particle species is conserved. In the
absence of external forcing, the magnetic energy then remains small at all
times if it is small in the initial state. In other words, a small "seed"
magnetic field cannot be amplified significantly, regardless of the nature of
flow, as long as the collision frequency and gyroradius are small enough to be
negligible. A similar conclusion also holds if the system is subject to
external forcing as long as this forcing conserves the magnetic moment of at
least one plasma species and does not greatly increase the total energy of the
plasma (i.e., in practice, is subsonic). Dynamo action therefore always
requires collisions or some small-scale kinetic mechanism for breaking the
adiabatic invariance of the magnetic moment.
