Uniform growth rates for the magnetic field in a kinematic dynamo

The growth rate of a magnetic field within a conducting fluid with a certai n fixed kinematic velocity is an indication of the efficiency of the dynamo effect. A possible way to measure this growth rate consists of considering the maximum size of the magnetic field for all possible initial conditions and for every instant of time. Both the maximum and the minimum in time of these values are completely characterized here in terms of the numerical r ange and the spectrum of the induction operator for all the usual norms. Am ong the consequences of these results it is found that the classical bounds obtained by energy inequalities are optimal, that the maximal uniform grow th rate for the energy norm essentially coincides for the magnetic field an d the perturbed velocity, and that the minimal uniform growth rate for the magnetic field is precisely the classical maximal growth rate.