Evolution of a magnetic blob in a helical flow

Abstract

An intermediate stage of magnetic field evolution in kinematic dynamo systems, during which the growing field does not coincide with the leading eigensolution, can be physically important in those cases, if the difference of the growth rates of individual dynamo eigenfunctions is small in comparison with the growth rates themselves. The eigenvalue problem is also insufficient for complete analysis of evolution of an initial distribution of magnetic field which is initially concentrated within a small portion of the dynamo volume. In this case there arise magnetic fronts which spread over the whole volume. Here we solve the Cauchy problem, which describes such situations, for the case of helical axisymmetric flow, the screw dynamo. We consider evolution of an initially small magnetized region (called here a “blob” or “magnetic packet”) embedded in the axisymmetric helical flow. A prototype of such flow is a helical flow in a tube or a swirling jet. As shown below, the group velocity of such magnetic packet is equal to the fluid velocity at the radius where the eigenfunction concentrates which has the maximal growth rate. The field within the packet grows at this rate. On the contrary, the growth rate of the field outside the packet (within the jet) grows at another rate given by the growth rate of the eigenfunction which has the minimal oscillation frequency v, i.e. dv/dk = 0, where k is the longitudinal wave number. If the latter growth rate is negative, the field within the jet decays and the growing magnetic packet moves through the jet leaving it unmagnetized.