On Vainshtein's dynamo conjecture

Vainshtein's conjecture in dynamo theory states that all fluid motions that are not precluded from dynamo action by known anti-dynamo theorems act as dynamos at least in some part of the parameter space. We disprove this conj ecture by analysing in detail two dynamo models with very simple flow field s. In both models: fluid motion is represented by a rigidly rotating cylind er of infinite length. In the first model, this cylinder is surrounded by a n infinite expanse of fluid at rest with a different conductivity than the moving fluid. In the second model, the cylinder is enclosed in a cylindrica l gap of identical fluid at rest, which itself is surrounded by a vacuum re gion extending to infinity. The models are sufficiently complicated so that none of the known anti-dynamo theorems excludes dynamo action. In fact, th e latter model has been claimed to be a working dynamo. It is shown in this paper, by a combination of analytical and numerical methods, that neither of these models operates as a dynamo. This result is valid in the entire pa rameter space, in particular for arbitrarily large Reynolds numbers.