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We investigate the Lagrangian mechanism of the kinematic ``fluctuation''magnetic dynamo in turbulent plasma flow at small magnetic Prandtl numbers. Thecombined effect of turbulent advection and plasma resistivity is to carryinfinitely many field lines to each space point, with the resultant magneticfield at that point given by the average over all the individual line vectors.As a consequence of the roughness of the advecting velocity, this remains trueeven in the limit of zero resistivity. We show that the presence of dynamoeffect requires sufficient angular correlation of the passive line-vectors thatarrive simultaneously at the SAME space point. We demonstrate this in detailfor the Kazantsev-Kraichnan model of kinematic dynamo with a Gaussian advectingvelocity that is spatially rough and white-noise in time. In the regime wheredynamo action fails, we also obtain the precise rate of decay of the magneticenergy. These exact results for the model are obtained by a generalization ofthe ``slow-mode expansion'' of Bernard, Gawedzki and Kupiainen to non-Hermitianevolution. Much of our analysis applies also to magnetohydrodynamic turbulence.

Small-Scale Kinematic Dynamo and Non-Dynamo in Inertial-Range Turbulence