Towards a poloidal magnetic field theorem

The paper presents evidence that a kinematic dynamo with a purely poloidal magnetic field does not exist. In particular, it is shown that the poloidal scalar S, if it satisfies a certain regularity condition, decays monotonically with respect to the norm maxz (max x, y S—min x, y S) in the case of a plane fluid layer and with respect to the norm maxr[r(max|r|=r S—min|r|=r S)] in the case of a spherical fluid volume. The result is valid for nonsteady, radially varying conductivity, possibly moving boundaries and a nonsteady, compressible flow field constrained in such a way that no toroidal magnetic field is generated. Similar decay results apply to the nonradial derivatives of S. Since the above norm is equivalent to the uniform max-norm, we have also (not necessarily monotonic) decay of S in the max-norm. For steady or time-periodic flow fields these results imply exponential decay of S and its derivatives to zero. The steady case, finally, is ruled out by application of standard maximum principles for elliptic partial differential equations.