Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 75 / 3 / 3 / Solution

In Gaussian units, the ensemble-averaged magnetic energy density isLet $M_q={\int }_0^{\infty }{B}^{q}FdB$, and assume the relevant moment exists and endpoint terms vanish. Twice applying integration by parts to the conservative magnitude equation givesIn particular, the magnetic stretching moment growth for $q=2$ isThis is a kinematic magnetic dynamo result with prescribed velocity statistics and no Ohmic loss. Once the Lorentz force appreciably changes the flow, this indefinitely exponential kinematic approximation is no longer self-consistent.