Effects of radial conductivity variation on the Ponomarenko dynamo

We study the effects of conductivity variation on the Ponomarenko dynamo. Taking monotonically increasing, decreasing and sinusoidally varying radial profiles, we study the kinematic dynamo problem. The threshold of the dynamo, given by the critical magnetic Reynolds number Rm_{c}, is found to strongly depend on the form of conductivity variation near the discontinuity of the velocity field where the shear is dominant. For monotonically varying profiles in the limit of sharp variation, this threshold maps to the problem of the dynamo with a jump in conductivity, while for the sinusoidal profile with large variations or wave numbers, the threshold increases due to effective diffusivity arising from the rapid changes in conductivity. Far from the threshold, it is found that the growth rate depends only on the local gradient of the conductivity near the discontinuity, with the opposite sign of shear and conductivity gradient leading to a larger growth rate, thereby aiding the dynamo instability in regenerating B_{ϕ} from B_{r}. We calculate the expression for the growth rate in the limit of large Rm(≫1) and find that the conductivity variation leads to a modification proportional to the local gradient of conductivity times Rm^{-1/2}. This correction is found to depend only on the local gradient of the conductivity at the discontinuity. Similar dependencies are found for realistic, smooth velocity profiles and for different magnetic boundary conditions.