Two-dimensional Theorem Research Articles

Division of $$n$$-Dimensional Euclidean Space into Circumscribed $$n$$-Cuboids

In 1970, Bohm formulated a three-dimensional version of his two-dimensional theorem that a division of a plane by lines into circumscribed quadrilaterals necessarily consists of tangent lines to a given conic. Bohm did not provide a proof of his three-dimensional statement. The aim of this paper is to give a proof of Bohm’s statement in three dimensions that a division of three-dimensional Euclidean space by planes into circumscribed cuboids consists of three families of planes such that all planes in the same family intersect along a line, and the three lines are coplanar. Our proof is based on the properties of centers of similitude. We also generalize Bohm’s statement to the four-dimensional and then $$n$$ -dimensional case and prove these generalizations.

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 in 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 magnetohydrodynamic alpha- and beta-effects are derived in the second-order correlation approximation (SOCA) when the electrically conducting fluid occupies all space. Limitations of the SOCA are well known, but alpha- and beta-effects of a turbulent flow are useful in interpreting the dynamo effects of the turbulence. 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, which are analogues of the laminar planar velocity and two-dimensional antidynamo theorems, apply to all turbulent ensembles with the prescribed alpha- and beta-effects, not just the planar and two-dimensional ensembles. The mean magnetic field is general in the planar theorem but only two-dimensional in the two-dimensional theorem. The two theorems relax the previous restriction to turbulence which is both two-dimensional and planar. 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 the laminar theorems because further conditions beyond those arising from the turbulence must be imposed on the beta-effect to establish decay of the mean magnetic field. In particular, negative turbulent magnetic diffusivities must be restricted. It is interesting that there is no inconsistency in the alpha-effects. The failure of the SOCA with the two-scale approximation to simply preserve the laminar antidynamo theorems at the beta-effect level is a further demonstration of the restricted validity of the theory and shows that negative diffusivity effects derived by approximation methods must be treated cautiously.

On geostrophic motion of a non-homogeneous fluid

A general theoretical relationship between the (three-dimensional) vcLacity field and density field is established for geostrophic flow of a non-homogeneous fluid. The practical value of the relationship – which reduces to the celebrated two-dimensional theorem due to Proudman and Taylor when the fluid is homogeneous – is illustrated by means of three examples of flows in rapidly rotating fluids, namely (i) baroclinic waves in laboratory systems and in the atmosphere, (ii) the gyroscopic ‘steering’ processes in Jupiter's atmosphere that are implied by the ‘Taylor column’ theory of the Great Red Spot, and (iii) certain striking properties of ocean currents revealed by recent observations in the Atlantic and Mediterranean.