Homogeneous Planar and Two-Dimensional Mean-Field Antidynamo Theorems with Zero Mean Flow

In an electrically conducting fluid two types of turbulence with a preferred direction are distinguished: planar turbulence, in which every velocity of the turbulent ensemble of flows has no component in the given direction; and two-dimensional turbulence, in which every velocity in the turbulent ensemble is invariant under translation in the preferred direction. Under the additional assumptions of two-scale and homogeneous turbulence with zero mean flow, the associated alphaand beta-effects are derived in the second-order smoothing approximation when the electrically conducting fluid occupies all space. Two antidynamo theorems, which establish necessary conditions for dynamo action, are shown to follow from the special structures of these alpha and beta effects. The theorems are analogues of the laminar planar velocity and two-dimensional antidynamo theorems. The mean magnetic field is general in the planar theorem but only two-dimensional in the two-dimensional theorem. The laminar theorems imply decay of the total magnetic field for any velocity of the associated turbulent ensemble. However, the mean-field theorems are not fully consistent with this, because further conditions beyond those arising from the turbulence must be imposed on the beta-effect to establish decay of the mean magnetic field. The two mean-field theorems relax the previous restriction to turbulence which is both two-dimensional and planar.