The impossibility of complete decay of a symmetric toroidal field in a compressible dynamo (Q1086106)

In this work we address the question of self-sustained dynamo action in an electrically conducting fluid. Namely, can the toroidal field actually vanish in a finite time? Under mild assumptions on the velocity field v(x,t) the answer is no. To establish this result we show that if the toroidal field vanishes everywhere at some time instant then it must be zero everywhere for all times before and after that instant. Such an approach has been employed in other fluid dynamics problems. While it might be argued decay of the toroidal part is sufficient for practical purposes to establish that dynamo action cannot be sustained, the fact that it cannot decay to zero is undoubtedly of interest.