  Variational turbulence is among the few approaches providing rigorous results
in turbulence. In addition, it addresses a question of direct practical
interest, namely the rate of energy dissipation. Unfortunately, only an upper
bound is obtained as a larger functional space than the space of solutions to
the Navier-Stokes equations is searched. Yet, in general, this upper bound is
in good agreement with experimental results in terms of order of magnitude and
power law of the imposed Reynolds number. In this paper, the variational
approach to turbulence is extended to the case of dynamo action and an upper
bound is obtained for the global dissipation rate (viscous and Ohmic). A simple
plane Couette flow is investigated. For low magnetic Prandtl number $P_m$
fluids, the upper bound of energy dissipation is that of classical turbulence
(i.e. proportional to the cubic power of the shear velocity) for magnetic
Reynolds numbers below $P_m^{-1}$ and follows a steeper evolution for magnetic
Reynolds numbers above $P_m^{-1}$ (i.e. proportional to the shear velocity to
the power four) in the case of electrically insulating walls. However, the
effect of wall conductance is crucial : for a given value of wall conductance,
there is a value for the magnetic Reynolds number above which energy
dissipation cannot be bounded. This limiting magnetic Reynolds number is
inversely proportional to the square root of the conductance of the wall.
Implications in terms of energy dissipation in experimental and natural dynamos
are discussed.
