PreprintHomogeneous planar and twodimensional meanfield antidynamo theorems with zero mean flowD.J. Ivers and C.G. Phillips,AbstractIn 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 twodimensional turbulence, in which every velocity in the turbulent ensemble is invariant under translation in the preferred direction. Under the additional assumptions of twoscale and homogeneous turbulence with zero mean flow, the associated alpha and betaeffects are derived in the secondorder 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 twodimensional antidynamo theorems. The mean magnetic field is general in the planar theorem but only twodimensional in the twodimensional theorem. The laminar theorems imply decay of the total magnetic field for any velocity of the associated turbulent ensemble. However, the meanfield theorems are not fully consistent with this, because further conditions beyond those arising from the turbulence must be imposed on the betaeffect to establish decay of the mean magnetic field. The two meanfield theorems relax the previous restriction to turbulence which is both twodimensional and planar. Keywords: magnetohydrodynamics, dynamo theory, meanfield electrodynamics, alphaeffect, betaeffect, antidynamo theorem.AMS Subject Classification: Primary 76W05.
This paper is available as a pdf (328kB) file.
